Patent Publication Number: US-2012042217-A1

Title: State feedback control apparatus, state feedback controller, and state feedback control method

Description:
BACKGROUND OF THE INVENTION 
     1. Technical Field 
     The present invention relates to a state feedback control apparatus, a state feedback controller, and a state feedback control method for state-feedback controlling a control object. The present invention is applied to a damping force control apparatus for suppressing and controlling a vibration of a suspension apparatus of a vehicle by controlling a damping force for example. 
     2. Related Art 
     A state feedback control apparatus for state-feedback controlling a control object is practically utilized. For example, state feedback control is often used for damping force control of a suspension apparatus of a vehicle. 
     Nonlinear H-infinity state feedback control is sometimes used for the damping force control of the suspension apparatus of the vehicle. For example, Japanese Patent Application Publication No. 2000-148208 discloses a damping force control apparatus for obtaining a variable damping coefficient representing a variable amount of a damping force based on a control input calculated by a state feedback controller designed by applying a nonlinear H-infinity control theory to a system represented by a state space model of a vibration system including a variable damping type suspension apparatus (the control object). 
     SUMMARY OF THE INVENTION 
     In the case where a control object is state-feedback controlled, a system is required to be controllable as a premise thereof. That is, a controllable matrix of a state space model (a state space representation) representing the system is required to have full rank. However, there may be the case where the system cannot be designed to be controllable. Particularly, in the case where the number of a motion equation serving as a basis in designing of the state space model of the control object is less than the number of a control input calculated by a state feedback controller, the system represented by the state space model becomes uncontrollable. 
     For example, considering a situation that a two wheel model of a vehicle is a control object, and a state space model of the control object is designed on a basis of a motion equation in the vertical (up and down) direction of an sprung member (above-spring member) obtained from the control object. In this case, the number of the motion equation serving as a basis in designing the model is one (only the vertical motion equation of the sprung member). Meanwhile, the number of the control input calculated by the state feedback controller is two (variable damping coefficients of dampers used in left and right suspension apparatuses). Since the number of the motion equation is less than the number of the control input, the system represented by the designed state space model becomes uncontrollable. 
     Further, considering another situation that a four wheel model of the vehicle is the control object, and a state space model of the control object is designed on a basis of motion equations relating to heave motion (vertical motion), pitch motion, and roll motion of the sprung member obtained from the control object. In this case, the number of the motion equation serving as a basis in designing the model is three (a heave motion equation, a pitch motion equation, and a roll motion equation). Meanwhile, the number of the control input is four (variable damping coefficients of dampers used in suspension apparatuses respectively attached to front left and right portions of the sprung member and rear left and right portions of the sprung member). In this case as well, since the number of the motion equation is less than the number of the control input, the system becomes uncontrollable. 
     When the system is uncontrollable, a state quantity cannot be controlled by the control input. Thus, the control object cannot be state-feedback controlled. In this case, conventionally, the state space model is reviewed and the model is redesigned such that the system becomes controllable. However, in the case where the model is redesigned, new parameters are required to be identified, and the redesigned model becomes complicated. Therefore, there is a problem that a lot of time is required for redesigning the model. Another method of obtaining controllability is that a pseudo error is set into the model. According to this method, the model can be designed within a relatively short time since it is only necessary to add the error into the model. However, the error is conventionally added into the input and output sides of the model (such as an input matrix or an output matrix). Thus, there is a problem that the error greatly influences an output. Further, according to the conventional method, the error is added into a plurality of points of the model. Since the error is added into a plurality of points of the model, a magnitude of error elements are larger due to buildup of the error, and deviation between the designed model and the model of the control object is increased. Therefore, highly precise state-feedback control of the control object cannot be performed. 
     The present invention has been accomplished in order to solve the above problems, and its object is to provide a state feedback control apparatus and a state feedback control method capable of highly precisely state-feedback controlling a control object by a simple model correction even when a system represented by a state space model is uncontrollable, and a state feedback controller used in such state feedback control apparatus and method. 
     An aspect of the present invention is a state feedback control apparatus for state-feedback controlling a control object, including a state feedback controller for calculating a control input of a system based on a state quantity of the system represented by a corrected state space model, the corrected state space model being formed so as to represent a controllable system by adding an error matrix Δ to a state matrix of a state space model of the control object representing an uncontrollable system, and control means for controlling the control object based on the control input calculated by the state feedback controller. 
     According to the above invention, the control object is controlled based on the control input calculated by the state feedback controller in the system represented by the corrected state space model. The corrected state space model is designed so as to represent the controllable system by adding the error matrix Δ to the state matrix of the state space model of the control object which represents the uncontrollable system. In this corrected state space model, the state matrix to be multiplied by the state quantity is finely corrected by an addition of the error matrix Δ. By this fine correction, rank deficiency of the controllable matrix of the corrected state space model is prevented. Thereby, the system represented by the corrected state space model becomes controllable. 
     According to the present invention, even in the case where the state space model of the control object is designed as the model representing the uncontrollable system, the control object can be state-feedback controlled based on the control input calculated by the state feedback controller in the system represented by the corrected state space model corrected such that the system becomes controllable by an introduction of the error matrix Δ. Further, a basic structure of the corrected state space model is the same as the state space model of the control object except that the error matrix Δ is only added into the state space model. Therefore, there is no need for time required for redesigning the model. Further, since the error matrix Δ is added to the state matrix which is less influential on the output of the model, the error matrix Δ does not greatly influence the output. In addition, since only one error matrix Δ is added into the state space model, the buildup of the error is not generated. Therefore, the deviation between the corrected state space model and the state space model of the actual control object is decreased, and thereby highly precise state-feedback control of the control object can be performed. Since an error examination point is one point, time required for examining the error can be shortened. Since the present invention has many advantages described above, even in the case where the state space model is uncontrollable, highly precise state-feedback control of the control object can be performed by the simple model correction. 
     In the present invention, as long as the system represented by the corrected state space model becomes controllable, the error matrix Δ may be a matrix having a positive error element or a negative error element. The error matrix Δ may be designed such that an element or elements influencing a calculation of a rank of the controllable matrix of the corrected state space model is/are changed. In this case, the error matrix Δ may be designed such that elements in a row of the controllable matrix are not the same with the elements in another row of the controllable matrix. According to the configuration described above, the rank deficiency due to the fact that the elements in the row of the controllable matrix are the same with the elements in another row of the controllable matrix is prevented. 
     The present invention can be applied to the case where the number of the motion equation of the control object is less than the number of the control input calculated by the state feedback controller. For example, the present invention can be applied to the case where the damping force of the right suspension apparatus of the vehicle and the damping force of the left suspension apparatus of the vehicle are controlled at the same time based on one motion equation when vibrations of the suspension apparatuses are controlled by controlling damping forces of the suspension apparatuses by state feedback. The present invention can also be applied to the case where the damping forces of the four suspension apparatuses attached to the front left and right portions of the sprung member and the rear left and right portions of the sprung member are controlled at the same time based on the heave motion equation, the pitch motion equation, and the roll motion equation. 
     In the present invention, the state feedback controller may calculate the control input by applying H-infinity state feedback control to a generalized plant designed based on the system represented by the corrected state space model. In this case, the H-infinity state feedback control may be linear H-infinity state feedback control or nonlinear H-infinity state feedback control. According the configuration described above, the control object is state-feedback controlled based on the control input calculated by the state feedback controller (H-infinity state feedback controller) designed such that H-infinity norm ∥G∥ ∞  of the generalized plant (L 2  gain from a disturbance w to an output z of the system in a case of the nonlinear H-infinity state feedback control) becomes less than a predetermined positive constant γ. Thus, disturbance suppression and robust stabilization are improved. 
     Elements in the error matrix Δ may include zero element. However, all the elements must not be the zero elements. In the case where a value of a non-zero element, which is an element other than the zero element (that is, an error element) is large, an influence of the error on the system is larger than that in the case where the value is small. Therefore, the value of the non-zero element in the error matrix Δ may be as a small value as possible. However, when the value of the non-zero element in the error matrix Δ is very small, the error matrix Δ is regarded as a zero matrix, and the system represented by the corrected state space model becomes substantially uncontrollable. Therefore, it is preferable that the value of the non-zero element in the error matrix Δ is appropriately small. In this case, a magnitude of the non-zero element in the error matrix Δ may be 1/10 to 1/100 of a magnitude of a non-zero element in the state matrix. According to this configuration, the system can obtain sufficient controllability and a influence rate of the error on the system is sufficiently reduced. In addition, the non-zero element in the error matrix Δ and the non-zero element in the state matrix are different from each other in terms of the number of digits. Thus, when the error matrix Δ is added to the state matrix, an addition element is prevented from being zero due to a setoff. The addition element is used for a calculation of the elements of the controllable matrix of the corrected state space model. Thus, since the addition element is not zero, the rank deficiency is not easily generated in the controllable matrix. 
     Elements in the error matrix Δ, an element which does not influence a calculation of a rank of a controllable matrix of the corrected state space model may be set to zero element. According to this configuration, the influence of the error matrix Δ on the system is more reduced by setting the value of the element unnecessary for a rank calculation of the controllable matrix of the corrected state space model to zero. Therefore, an amount of the deviation between the corrected state space model and the state space model of the actual control object is further decreased. 
     The control object may include a suspension apparatus provided with a damper and a spring interposed between an sprung member and an unsprung member (below-spring member) of a vehicle, and the control means may control a damping force for damping a vibration of the suspension apparatus. According to this configuration, the vibration of the suspension apparatus is suppressed by controlling the damping force of the suspension apparatus. Therefore, riding quality of the vehicle is improved. 
     One of other aspects of the present invention is a state feedback controller for calculating a control input of a system based on a state quantity of the system represented by a state space model, wherein the state feedback controller calculates the control input based on a state quantity of the system represented by a corrected state space model which is formed so as to represent a controllable system by adding an error matrix Δ to a state matrix of a state space model representing an uncontrollable system. In this case, the state feedback controller may calculate the control input by applying H-infinity state feedback control to a generalized plant designed based on the system represented by the corrected state space model. A magnitude of a non-zero element of the error matrix Δ may be 1/10 to 1/100 of a magnitude of a non-zero element of the state matrix. Elements in the error matrix Δ, an element which does not influence a calculation of a rank of a controllable matrix of the corrected state space model may be set to a zero element. According to the present invention of such a state feedback controller, the same operations and effects as the invention of the above state feedback control apparatus are also obtained. 
     One of other aspects of the present invention is a state feedback control method for state-feedback controlling a control object, including a control input calculating step for calculating a control input of a system based on a state quantity of the system represented by a corrected state space model, the corrected state space model being formed so as to represent a controllable system by adding an error matrix Δ to a state matrix of a state space model of the control object representing an uncontrollable system, and a control step for controlling the control object based on the control input calculated in the control input calculating step. In this case, the control input may be calculated by applying H-infinity state feedback control to a generalized plant designed based on the system represented by the corrected state space model in the control input calculating step. According to the present invention of such a method, the same operations and effects as the invention of the above state feedback control apparatus are also obtained. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a system represented by a state space model of a certain control object; 
         FIG. 2  is a block diagram of a system represented by a corrected state space model obtained by adding an error matrix Δ to the state space model of  FIG. 1 ; 
         FIG. 3  is a block diagram showing a state feedback loop of the system represented by the corrected state space model of  FIG. 2 ; 
         FIG. 4  is an entire schematic diagram of a suspension apparatus of a vehicle according to an embodiment of the present invention; 
         FIG. 5  is a flowchart showing a flow of a variable damping coefficient calculation processing executed by a nonlinear H-infinity controller of a micro computer; 
         FIG. 6  is a flowchart showing a flow of a requested damping force calculation processing executed by a requested damping force calculation section of the micro computer; 
         FIG. 7  is a flowchart showing a flow of a requested step number determination processing executed by a requested step number determination section of the micro computer; 
         FIG. 8  is a block diagram of a closed loop system S in which a state quantity of a generalized plant G is fed back; 
         FIG. 9  is a diagram showing motion of suspension apparatuses according to the present embodiment as a two wheel model of the vehicle; 
         FIG. 10  is a block diagram of the system represented by the state space model of the control object according to the present embodiment in the case where the two wheel model is the control object; 
         FIG. 11  is a block diagram of the system represented by the corrected state space model according to the present embodiment; 
         FIG. 12  is a block diagram of the closed loop system in which state feedback is performed in the state of the generalized plant designed based on the corrected state space model; and 
         FIG. 13  is a block diagram of a system represented by another corrected state space model according to the present embodiment. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Hereinafter, an embodiment of the present invention will be described. 
     A state space model (a state space representation) of a control object is described for example as in the following equation (eq. 1) with using a control input u, an output z, and a state quantity x. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             x 
                             . 
                           
                           = 
                           
                             Ax 
                             + 
                             Bu 
                           
                         
                       
                     
                     
                       
                         
                           z 
                           = 
                           
                             Cx 
                             + 
                             Du 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     wherein: {dot over (x)}=dx/dt 
     It should be noted that the equation (eq. 1) shows a model of a linear time-invariant system. 
     In the above equation (eq. 1), A, B, C, D denote system coefficient matrices of the state space model. The matrix A is called a state matrix (or a system matrix), the matrix B is called an input matrix, the matrix C is called an output matrix, and the matrix D is called a transfer matrix. 
       FIG. 1  is a block diagram of a system represented by the state space model shown as the equation (eq. 1). In the figure, a block represented as I/s indicates a time integral, and blocks represented by A, B, C, D indicate the system coefficient matrices. 
     A necessary and sufficient condition for determining that the system represented by the state space model is controllable is that a controllable matrix U c (n×nm) of the state space model has full rank (rankU c =n). The controllable matrix U c  of the state space model shown as the equation (eq. 1) is represented as in the following equation (eq. 2). 
         U   c   =[BAB . . . A   n-1   B ],( n×nm )  (eq. 2)
 
     The state matrix A and the input matrix B are for example represented as in the following equation (eq. 3). 
     
       
         
           
             
               
                 
                   
                     A 
                     = 
                     
                       [ 
                       
                         
                           
                             2 
                           
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               - 
                               1 
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                       
                   
                    
                   
                     B 
                     = 
                     
                       [ 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
     In this case, the controllable matrix U c  is represented as in the following equation (eq. 4). 
     
       
         
           
             
               
                 
                   
                     U 
                     c 
                   
                   = 
                   
                     
                       [ 
                       
                         B 
                          
                         
                             
                         
                          
                         AB 
                       
                       ] 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               - 
                               1 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
     A rank of the controllable matrix U c  represented by the equation (eq. 4) is 1(rank U c =1). Since full rank is 2(Full Rank=2), the controllable matrix U c  does not have full rank. Therefore, in the case where the state matrix A and the input matrix B of the state space model are represented by the above equation (eq. 3), the system represented by that state space model is uncontrollable. 
     The following equation (eq. 5) is a corrected state space model obtained by correcting the state space model by adding an error matrix Δ to the state matrix A of the state space model shown in the equation (eq. 1). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             x 
                             . 
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   A 
                                   + 
                                   Δ 
                                 
                                 ) 
                               
                                
                               x 
                             
                             + 
                             Bu 
                           
                         
                       
                     
                     
                       
                         
                           z 
                           = 
                           
                             Cx 
                             + 
                             Du 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
     As understood from the equation (eq. 5), the state matrix to be multiplied by the state quantity x in the state equation is corrected by the error matrix Δ. The corrected matrix A+Δ is called a corrected state matrix in the present specification.  FIG. 2  is a block diagram of a system represented by the corrected state space model. As shown in  FIG. 2 , the error matrix Δ is added into the corrected state space model as an additive error of the state matrix A. 
     The error matrix Δ has the same form as the state matrix A. In the case where the state matrix A is a 2-by-2 matrix, the error matrix Δ is for example represented as in the following equation (eq. 6). 
     
       
         
           
             
               
                 
                   Δ 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             Δ 
                             11 
                           
                         
                         
                           
                             Δ 
                             12 
                           
                         
                       
                       
                         
                           
                             Δ 
                             21 
                           
                         
                         
                           
                             Δ 
                             22 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     When the state matrix A and the input matrix B are represented as in the above equation (eq. 3), a controllable matrix U c * of the corrected state space model is represented as in the following equation (eq. 7) with using the corrected state matrix A+Δ and the input matrix B. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           U 
                           c 
                           * 
                         
                         = 
                           
                          
                         
                           [ 
                           
                             B 
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 A 
                                 + 
                                 Δ 
                               
                               ) 
                             
                              
                             B 
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           [ 
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       1 
                                     
                                   
                                 
                                 ] 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         2 
                                         + 
                                         
                                           Δ 
                                           11 
                                         
                                       
                                     
                                     
                                       
                                         Δ 
                                         12 
                                       
                                     
                                   
                                   
                                     
                                       
                                         1 
                                         + 
                                         
                                           Δ 
                                           21 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           1 
                                         
                                         + 
                                         
                                           Δ 
                                           22 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     1 
                                   
                                 
                               
                               ] 
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 
                                   Δ 
                                   12 
                                 
                               
                             
                             
                               
                                 1 
                               
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                   + 
                                   
                                     Δ 
                                     22 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     In the above equation (eq. 7), when Δ 12  is a non-zero element (an element which is not zero), a rank of the controllable matrix U c * is 2(rankU c *=2). That is, the controllable matrix U c * has full rank, and thereby the system represented by the corrected state space model becomes controllable. In such a way, controllability of the then-uncontrollable system is recovered by correcting the state matrix A by the error matrix Δ. 
       FIG. 3  is a block diagram showing a state feedback loop of the controllable system represented by the corrected state space model. As shown in this closed loop system, a state feedback controller K calculates the control input u of the system based on the state quantity x of the system represented by the corrected state space model. By the calculated control input u, the control object is state-feedback controlled. 
     However, even if the error matrix Δ is added to the state matrix, sometimes the controllable matrix U c * does not have full rank. For example, in the case where Δ 12  is zero in the above example, even when other elements are non-zero, first row elements of the controllable matrix U c * are all zero. Thus, the rank is 1(rankU c * is =1). In this case, the system becomes uncontrollable. Therefore, there is a need for setting the elements of the error matrix Δ such that the system represented by the corrected state space model becomes controllable. That is, there is a need for setting the elements of the error matrix Δ such that the controllable matrix U c * of the corrected state space model has full rank. 
     The non-zero elements in the elements of the error matrix Δ may have so small values so as to have the different number of digits from non-zero elements of the state matrix A. If absolute values of the elements of the error matrix Δ are in a similar range to absolute values of the elements of the state matrix A, there is a possibility that rank deficiency is generated in the controllable matrix U c * of the corrected state space model, thereby the controllable matrix U c * does not have full rank. For example, in the case where the state matrix A and the input matrix B are represented as in the following equation (eq. 8) and the error matrix Δ is represented as in the above equation (eq. 6), the controllable matrix U c * is represented as in the following equation (eq. 9). 
     
       
         
           
             
               
                 
                   
                     A 
                     = 
                     
                       [ 
                       
                         
                           
                             
                               a 
                               11 
                             
                           
                           
                             
                               a 
                               12 
                             
                           
                         
                         
                           
                             
                               a 
                               21 
                             
                           
                           
                             
                               a 
                               22 
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                       
                   
                    
                   
                     B 
                     = 
                     
                       [ 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
     wherein: a 11 , a 12 , a 21 , a 22 ≠0 
     
       
         
           
             
               
                 
                   
                     U 
                     c 
                     * 
                   
                   = 
                   
                     
                       [ 
                       
                         B 
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             A 
                             + 
                             Δ 
                           
                           ) 
                         
                          
                         B 
                       
                       ] 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             0 
                           
                           
                             
                               
                                 a 
                                 12 
                               
                               + 
                               
                                 Δ 
                                 12 
                               
                             
                           
                         
                         
                           
                             1 
                           
                           
                             
                               
                                 a 
                                 22 
                               
                               + 
                               
                                 Δ 
                                 22 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     When a value of Δ 12  is equal to “−a 12 ” in the equation (eq. 9), the first row elements are all zero, and the rank deficiency is generated in the controllable matrix U c *. Therefore, the controllable matrix U c * does not have full rank. 
     Meanwhile, when a magnitude of the elements of the state matrix A and a magnitude of the elements of the error matrix Δ are different from each other in terms of the number of digits, the additional elements in the controllable matrix U c * do not become zero due to a setoff by an addition. Therefore, the rank deficiency generated by including a lot of zero elements in the elements of the controllable matrix U c * is prevented. 
     When the non-zero elements of the error matrix Δ have too small values, the error matrix Δ approximates a zero matrix. Thus, substantial controllability cannot be given to the system. Therefore, it is preferable that the non-zero elements of the error matrix Δ have appropriately small values. In this case, when the non-zero elements of the error matrix Δ have a magnitude of about 1/10 to 1/100 of the non-zero elements of the state matrix A, the controllability of the system represented by the corrected state space model is not deteriorated, and an influence of the error due to an addition of the error matrix Δ is sufficiently suppressed. 
     Hereinafter, a mode in which the present invention is applied to damping force control of suspension apparatuses of a vehicle will be described. 
     [Configuration of Suspension Control Apparatus] 
       FIG. 4  is an entire schematic diagram of a suspension control apparatus of the vehicle. This suspension control apparatus  1  is provided with a right side suspension apparatus SP R , a left side suspension apparatus SP L , and an electric control apparatus EL. The right side suspension apparatus SP R  is attached on the side of a right wheel of the vehicle, and the left side suspension apparatus SP L  is attached on the side of a left wheel of the vehicle. Structures of the right side suspension apparatus SP R  and the left side suspension apparatus SP L  are the same. In the following description, terms indicating the left and right sides of the configurations will be omitted when configurations of both the suspension apparatuses are collectively described. 
     The suspension apparatuses SP R , SP L  are provided with suspension springs  10 R,  10 L, and dampers  20 R,  20 L. The suspension springs  10 R,  10 L and the dampers  20 R,  20 L are interposed between a sprung member HA and unsprung members LA R , LA L  of the vehicle, one ends (lower ends) thereof are connected to the unsprung members LA R , LA L , and the other ends (upper ends) thereof are connected to the sprung member HA. The suspension springs  10 R,  10 L absorb (buffer) relative vibrations between the unsprung members LA R , LA L  and the sprung member HA. The dampers  20 R,  20 L are arranged in parallel to the suspension springs  10 R,  10 L, and damp the vibration by generating resistance to a vibration of the sprung member HA relative to the unsprung members LA R , LA L . It should be noted that knuckles coupled to the wheels, lower arms with one ends coupled to the knuckles, and the like correspond to the unsprung members LA R , LA L . The sprung member HA is supported by the suspension springs  10 R,  10 L and the dampers  20 R,  20 L. A vehicle body is included in the sprung member HA. 
     The dampers  20 R,  20 L are provided with cylinders  21 R,  21 L, pistons  22 R,  22 L, and piston rods  23 R,  23 L. The cylinders  21 R,  21 L are hollow members in which a viscous fluid such as oil is filled. Lower ends of the cylinders  21 R,  21 L are connected to the lower arms serving as the unsprung members LA R , LA L . The pistons  22 R,  22 L are arranged in the cylinders  21 R,  21 L. The pistons  22 R,  22 L are movable in the axial direction inside the cylinders  21 R,  21 L. The piston rods  23 R,  23 L are bar shape members. The piston rods  23 R,  23 L are connected to the pistons  22 R,  22 L at one ends, and extend upward in the axial direction of the cylinders  21 R,  21 L to protrude outward from upper ends of the cylinders  21 R,  21 L. The piston rods  23 R,  23 L connect to the vehicle body serving as the sprung member HA at the other ends. 
     As shown in the figure, upper chambers R 1   R , R 1   L , and lower chambers R 2   R , R 2   L  are separately formed in the cylinders  21 R,  21 L by the pistons  22 R,  22 L arranged inside the cylinders  21 R,  21 L. Communication passages  24 R,  24 L are formed in the pistons  22 R,  22 L. The upper chambers R 1   R , R 1   L  communicate with the lower chambers R 2   R , R 2   L  via the communication passages  24 R,  24 L. 
     In the dampers  20 R,  20 L with the above structure, when the sprung member HA is vibrated in the vertical direction (up and down direction) relative to the unsprung members LA R , LA L  upon the vehicle traveling over an uneven portion of a road surface or the like, the pistons  22 R,  22 L connected to the sprung member HA via the piston rods  23 R,  23 L are relatively displaced in the axial direction in the cylinders  21 R,  21 L connected to the unsprung members LA R , LA L . In accordance with the relative displacement, the viscous fluid flow through the communication passages  24 R,  24 L. When the viscous fluid flow through the communication passages  24 R,  24 L, resistance forces generated. The resistance forces act as damping forces against the vibration in the vertical direction. Thereby, the vibration of the sprung member HA relative to the unsprung members LA R , LA L  is damped. It should be noted that a magnitude of the damping forces is increased more as vibration speeds of the pistons  22 R,  22 L relative to the cylinders  21 R,  21 L (these speeds are corresponding to sprung-unsprung relative speeds described later) are increased more. 
     Variable throttle mechanisms  30 R,  30 L are attached to the suspension apparatuses SP R , SP L . The variable throttle mechanisms  30 R,  30 L have valves  31 R,  31 L, and actuators  32 R,  32 L. The valves  31 R,  31 L are provided in the communication passages  24 R,  24 L. A path sectional area of the communication passages  24 R,  24 L, or the number of the communication passages  24 R,  24 L are changed by actuating the valves  31 R,  31 L. That is, an opening degree OP of the communication passages  24 R,  24 L is changed by actuating the valves  31 R,  31 L. The valves  31 R,  31 L are for example formed by rotary valves built into the communication passages  24 R,  24 L. By means of changing the rotational angle of the rotary valve, the path sectional area of the communication passages  24 R,  24 L or the number of the connection passages  24 R,  24 L can be changed. The actuators  32 R,  32 L are connected to the valves  31 R,  31 L. In accordance with the actuation of the actuators  32 R,  32 L, the valves  31 R,  31 L are actuated. In the case where the valves  31 R,  31 L are the rotary valves as described above, the actuators  32 R,  32 L may be a motors for rotating the rotary valves. 
     When the Opening degree OP is changed as a result of the valves  31 R,  31 L being operated by the actuators  32 R,  32 L, the magnitude of the resistance which acts on the viscous fluid flowing through the communication passages  24 R,  24 L changes. The resistance forces serves as the damping forces against the vibration as described above. Therefore, when the opening degree OP is changed, the damping force characteristics of the dampers  20 R,  20 L change. It should be noted that the damping force characteristics refers to a characteristic which determines change in the magnitude of the damping forces with speeds of the pistons  22 R,  22 L in relation to the cylinders  21 R,  21 L (that is, the sprung-unsprung relative speeds). In the case where the damping forces are proportional to the speeds, the damping force characteristics are represented by damping coefficients. 
     In the present embodiment, the opening degree OP is set stepwise. Therefore, changing of the opening degree OP results in a stepwise change in the damping force characteristics of the dampers  20 R,  20 L. The damping force characteristics are represented by the set step numbers of the set opening degree OP. That is, the damping force characteristics are expressed in the form of step numbers in accordance with the set step numbers of the opening degree OP such as first, second, . . . . In this case, each step number representing a damping force characteristics can be set such that the greater the numeral representing the step numbers, the greater the damping forces. The set step numbers representing the damping force characteristics is changed through operation of the variable throttle mechanisms  30 R,  30 L as described above. 
     Next, the electric control apparatus EL will be described. The electric control apparatus EL includes a sprung acceleration sensor  41 , a right side unsprung acceleration sensor  42 R, a left side unsprung acceleration sensor  42 L, a right side stroke sensor  43 R, a left side stroke sensor  43 L, and a micro computer  50 . 
     The sprung acceleration sensor  41  is attached to the vehicle body, detects a sprung member acceleration d 2 y/dt 2  serving as acceleration in the vertical direction of the sprung member HA in relation to an absolute space, and outputs a signal representing the detected sprung acceleration d 2 y/dt 2 . The right side unsprung acceleration sensor  42 R is attached to the right side unsprung member LA R , detects a right side unsprung acceleration d 2 r R /dt 2  serving as an acceleration in the vertical direction of the right side unsprung member LA R  in relation to the absolute space, and outputs a signal representing the detected right side unsprung acceleration d 2 r R /dt 2 . The left side unsprung acceleration sensor  42 L is attached to the left side unsprung member LA L , detects a left side unsprung acceleration d 2 r L /dt 2  serving as an acceleration in the vertical direction of the left side unsprung member LA L  in relation to the absolute space, and outputs a signal representing the detected left side unsprung acceleration d 2 r L /dt 2 . 
     The right side stroke sensor  43 R is attached between the sprung member HA and the right side unsprug member LA R , detects a sprung-right side unsprung relative displacement r R −y, and outputs a signal representing the detected sprung-right side unsprung relative displacement r R −y. The sprung-right side unsprung relative displacement r R −y is a difference between a sprung member displacement y serving as a displacement in the vertical direction of the sprung member HA from a reference position and a right side unsprung member displacement r R  serving as a displacement in the vertical direction of the right side unsprung member LA R  from a reference position. It should be noted the displacement r R −y is equal to a displacement of the right side piston  22 R relative to the right side cylinder  21 R in the right side damper  20 R (right side stroke amount). The left side stroke sensor  43 L is attached between the sprung member HA and the left side unsprung member LA L , detects a sprung-left side unsprung relative displacement r L −y, and outputs a signal representing the detected sprung-left side unsprung relative displacement r L −y. The sprung-left side unsprung relative displacement r L −y is a difference between the sprung displacement y and a left side unsprung displacement r L  serving as a displacement in the vertical direction of the left side unsprung member LA L  from a reference position. It should be noted that the displacement r L −y is equal to a displacement of the left side piston  22 L relative to the left side cylinder  21 L in the left side damper  20 L (left side stroke amount). 
     Each of the sprung acceleration sensor  41  and the unsprung acceleration sensors  42 R,  42 L detects upward acceleration as positive acceleration, and downward acceleration as negative acceleration. Each of the stroke sensors  43 R,  43 L detects relative displacement, for the case where upward displacement of the sprung member HA from the reference position is detected as positive displacement, downward displacement of the sprung member HA from the reference position is detected as negative displacement, upward displacement of each of the unsprung members LA R , LA L  from the reference position is detected as positive displacement, and downward displacement of each of the unsprung members LA R , LA L  is detected as negative displacement. 
     The micro computer  50  is electrically connected to the sprung acceleration sensor  41 , the unsprung acceleration sensors  42 R,  42 L, and the stroke sensors  43 R,  43 L. The micro computer  50  determines a right side requested step number D reqR  representing a target step number corresponding to a target damping force characteristic of the right side damper  20 R, and a left side requested step number D reqL  representing a target step number of a target damping force characteristic of the left side damper  20 L on the basis of the signals output from the sensors. The micro computer  50  respectively output a command signal corresponding to the determined right side requested step number D reqR  to the right side actuator  32 R, and a command signal in corresponding to the determined left side requested step number D reqL  to the left side actuator  32 L. Both the actuators  32 R,  32 L are actuated based on the above command signals. As a result, the right side valve  31 R and the left side valve  31 L are actuated. In such a way, the micro computer  50  variously controls the damping force characteristics of the right side damper  20 R and the left side damper  20 L by controlling the right side variable throttle mechanism  30 R and the left side variable throttle mechanism  30 L to control the damping forces of the right side suspension apparatus SP R  and the left side suspension apparatus SP L  at the same time. 
     As can be understood from  FIG. 4 , the micro computer  50  includes a nonlinear H-infinity controller  51 , a requested damping force calculation section  52 , and a requested step number determination section  53 . The nonlinear H-infinity controller  51  acquires the signals from the sensors  41 ,  42 R,  42 L,  43 R,  43 L, and calculates a right side variable damping coefficient C vR  and a left side variable damping coefficient C vL  as the control input u on the basis of the nonlinear H-infinity control theory. The right side variable damping coefficient C vR  corresponds to a coefficient of a variable damping force (a right side variable damping force) relative to a vibration speed (a sprung-right side unsprung relative speed described later) which is varied by controlling. The right side variable damping force represents a variable force portion of the entire right side damping force to be generated in the right side suspension apparatus SP R  The left side variable damping coefficient C vL  corresponds to a coefficient of a variable damping force (a left side variable damping force) relative to a vibration speed (a sprung-left side unsprung relative speed described later) which is varied by the controlling. The left side variable damping force represents a variable force portion of the entire left side damping force to be generated in the left side suspension apparatus SP L . The requested damping force calculation section  52  inputs the variable damping coefficients C vR , C vL , and calculates a right side requested damping force F reqR  serving as a target damping force to be generated in the right side suspension apparatus SP R , and a left side requested damping force F reqL  serving as a target damping force to be generated in the left side suspension apparatus SP L  based on the input variable damping coefficients C vR , C VL . The requested damping force calculation section  52  outputs both the calculated requested damping forces F reqR , F reqL . The requested step number determination section  53  inputs the requested damping forces F reqR , F reqL , and determines the right side requested step number D reqR  and the left side requested step number D reqL  both serving as the control target step numbers of the damping force characteristics based on the input requested damping forces F reqR , F reqL . The requested step number determination section  53  outputs signals corresponding to the determined requested step numbers D reqR , D reqL  to the right side actuator  32 R and the left side actuator  32 L as instruction signals. 
     [Damping Force Control of Suspension Apparatuses] 
     In the suspension control apparatus  1  formed as described above, when a detected value of the sprung acceleration sensor  41  exceeds a predetermined threshold value (that is, when there is a need for vibration suppression control of the suspension apparatuses SP R , SP L ), the nonlinear H-infinity controller  51  of the micro computer  50  executes a variable damping coefficient calculation processing, the requested damping force calculation section  52  executes a requested damping force calculation processing, and the requested step number determination section  53  executes a requested step number determination processing respectively repeatedly every predetermined short time. 
     The nonlinear H-infinity controller  51  calculates the variable damping coefficients C vR , C vL  as the control input u by executing the variable damping coefficient calculation processing shown in a flowchart of  FIG. 5 . This processing will be described based on  FIG. 5 . The nonlinear H-infinity controller  51  starts the processing in Step  100  (hereinafter, a step number is abbreviated as S) of  FIG. 5 . In the next S 102 , the nonlinear H-infinity controller  51  acquires the sprung acceleration d 2 y/dt 2  from the sprung acceleration sensor  41 , the right side unsprung acceleration d 2 r R /dt 2  from the right side unsprung acceleration sensor  42 R, the left side unsprung acceleration d 2 r L /dt 2  from the left side unsprung acceleration sensor  42 L, the sprung-right side unsprung relative displacement r R −y from the right side stroke sensor  43 R, and the sprung-left side unsprung relative displacement r L −y from the left side stroke sensor  43 L. Next, in S 104 , the nonlinear H-infinity controller  51  respectively time-integrates the sprung acceleration d 2 y/dt 2  and the unsprung accelerations d 2 r R /dt 2 , d 2 r L /dt 2  to thereby obtain a sprung speed dy/dt serving as a vertical speed of the sprung member HA, a right side unsprung speed dr R /dt serving as a vertical speed of the right side unsprung member LA R , and a left side unsprung speed dr L /dt serving as a vertical speed of the left side unsprung member LA L . Further, the nonlinear H-infinity controller  51  time-differentiates the sprung-right side unsprung relative displacement r R −y to obtain a sprung-right side unsprung relative speed dr R /dt−dy/dt serving as a difference between the sprung speed dy/dt and the right side unsprung speed dr R /dt, and time-differentiates the sprung-left side unsprung relative displacement r L −y to obtain a sprung-left side unsprung relative speed dr L /dt−dy/dt serving as a difference between the sprung speed dy/dt and the left side unsprung speed dr L /dt. Each of the sprung speed dy/dt and the unsprung speeds dr R /dt, dr L /dt is calculated as positive speed when it is the speed in upward direction, and calculated as negative speed when it is the speed in downward direction. Each of the sprung-unsprung relative speeds dr R /dt−dy/dt, dr L /dt−dy/dt is calculated as positive speed when it is the relative speed in the direction in which a gap between the sprung member HA and the unsprung members LA R , LA S  is reduced, that is, speed toward the side where the dampers  20 R,  20 L are compressed, and calculated as negative speed when it is the relative speed in the direction in which the gap is extended, that is, speed toward the side where the dampers  20 R,  20 L are expanded. It should be noted that the sprung-unsprung relative speeds dr R /dt−dy/dt, dr L /dt−dy/dt represent vibration speeds of the suspension apparatuses SP R , SP L  due to external inputs. The speeds are equal to the speeds of the pistons  22 R,  22 L relative to the cylinders  21 R,  21 L described above. 
     Next, in S 106 , the nonlinear H-infinity controller  51  calculates the right side variable damping coefficient C vR  and the left side variable damping coefficient C vL  based on the nonlinear H-infinity control theory. The variable damping coefficients C vR , C vL  represent the variable amount of the damping coefficient which is varied by controlling. In this case, although detailed description will be given later, the nonlinear H-infinity controller  51  calculates the control input u that is the variable damping coefficients C vR , C vL , such that L 2  gain (L 2  gain from a disturbance w to an evaluation output z) of a system (a generalized plant) represented by the corrected state space model in which the control input u is represented by the variable damping coefficients C vR , C VL  becomes less than a positive constant γ. After calculating the variable damping coefficients C vR , C VL  in S 106 , the nonlinear H-infinity controller  51  outputs the variable damping coefficients C vR , C VL  in S 108 . After that, the nonlinear H-infinity controller  51  advances to S 110  and finishes this processing. The nonlinear H-infinity controller  51  has functions corresponding to the state feedback controller of the present invention. A step of executing the variable damping coefficient calculation processing shown in  FIG. 5  corresponds to a control input calculating step of the present invention. 
       FIG. 6  is a flowchart showing a flow of the requested damping force calculation processing executed by the requested damping force calculation section  52 . The requested damping force calculation section  52  starts this processing in S 200  of  FIG. 6 , and in the next S 202 , the requested damping force calculation section  52  inputs the variable damping coefficients C vR , C vL . Next, in S 204 , the requested damping force calculation section  52  calculates a right side requested damping coefficient C reqR  and a left side requested damping coefficient C reqL . The right side requested damping coefficient C reqR  is calculated by adding a preliminarily set right side linear damping coefficient C sR  to the right side variable damping coefficient C vR . The left side requested damping coefficient C reqL  is calculated by adding a preliminarily set left side linear damping coefficient C sL  to the left side variable damping coefficient C VL . The linear damping coefficients C sR , C sL  represent fixed amount (linear amount) of damping coefficients not varied by the control. Next, the requested damping force calculation section  52  calculates the right side requested damping force F reqR  and the left side requested damping force F reqL  in S 206 . The right side requested damping force F reqR  is calculated by multiplying the right side requested damping coefficient C reqR  by the sprung-right side unsprung relative speed dr R /dt−dy/dt. The left side requested damping force F reqL  is calculated by multiplying the left side requested damping coefficient C reqL  by the sprung-left side unsprung relative speed dr L /dt−dy/dt. Then, the requested damping force calculation section  52  goes on to S 208  and outputs the requested damping forces F reqR , F reqL . After that, the requested damping force calculation section  52  advances to S 210  and finishes this processing. 
       FIG. 7  is a flowchart showing a flow of the requested step number determination processing executed by the requested step number determination section  53 . The requested step number determination section  53  starts this processing in S 300  of  FIG. 7 , and in the next S 302 , the requested step number determination section  53  inputs the requested damping forces F reqR , F reqL . Next, the required step number determination section  53  determines the right side requested step number D reqR  and the left side requested step number D reqL  in S 304 . It should be noted that the micro computer  50  has a right side damping force characteristic table and a left side damping force characteristic table. The right side characteristic table stores a characteristic profile of the magnitude of damping forces generated in the right side damper  20 R in relation to the sprung-right side unsprung relative speeds dr R /dt−dy/dt for each of the step numbers representing the damping force characteristics of the right side damper  20 R. The left side damping force characteristic table stores a characteristic profile of the magnitude of the damping forces generated in the left side damper  20 L in relation to the sprung-left side unsprung relative speeds dr L /dt−dy/dt for each of the step numbers representing the damping force characteristics of the left side damper  20 L. In S 304 , the requested step number determination section  53  refers to the right side damping force characteristic table so as to determine the right side requested step number D reqR  and refers to the left side damping force characteristic table so as to determine the left side requested step number D reqL . Specifically, in S 304 , the requested step number determination section  53  selects the damping forces corresponding to the sprung-right side unsprung relative speeds dr R /dt−dy/dt for each of the step numbers with reference to the right side damping force characteristic table. Then, the closest damping force to the right side requested damping force F reqR  is picked out from the selected damping forces. The step number corresponding to the damping force picked out is determined as the right side requested step number D reqR . Further, the requested step number determination section  53  selects the damping forces corresponding to the sprung-left side unsprung relative speeds dr L /dt−dy/dt for each of the step numbers with reference to the left side damping force characteristic table. Then, the closest damping force to the left side requested damping force F reqL  is picked out from the selected damping forces. The step number corresponding to the damping force picked out is determined as the left side requested step number D reqL . 
     After determining the requested step numbers D reqR , D reqL  in S 304 , the requested step number determination section  53  advances to S 306  and outputs command signals corresponding to the requested step numbers D reqR , D reqL  to the actuators  32 R,  32 L. After that, the requested step number determination section  53  advances to S 308  and finishes this processing. Upon receiving the command signals, the actuators  32 R,  32 L act based on the command signals. As a result, the valves  31 R,  31 L are actuated, and the variable throttle mechanisms  30 R,  30 L are controlled such that the step numbers representing the damping force characteristics of the dampers  20 R,  20 L become the requested step numbers D reqR , D reqL . In such a way, the damping forces of the suspension apparatuses SP R , SP L  are controlled at the same time. 
     As understood from the above description, the requested damping force calculation section  52  and the requested step number determination section  53  control the damping forces of the suspension apparatuses SP R , SP L  based on the variable damping coefficients C vR , C vL  calculated by the nonlinear H-infinity controller  51  serving as the state feedback controller. By the above described damping force control, the vibrations of the suspension apparatuses SP R  and SP L  are controlled. The requested damping force calculation section  52  and the requested step number determination section  53  correspond to control means of the present invention. A step of executing the requested damping force calculation processing shown in  FIG. 6  and a step of executing the requested step number determination processing shown in  FIG. 7  correspond to a control step of the present invention. The micro computer  50  provided with the nonlinear H-infinity controller  51 , the requested damping force calculation section  52 , and the requested step number determination section  53  corresponds to a state feedback control apparatus of the present invention. 
     [Control Theory of Variable Damping Coefficients C vR , C vL ] 
     The variable damping coefficients C vR , C vL  are calculated by the nonlinear H-infinity controller  51 . Whether a riding quality of the vehicle is good or bad is determined by a manner in which an ideal variable damping coefficients C vR , C vL  are calculated in accordance with the traveling state of the vehicle and the damping forces are controlled on the basis of the calculated variable damping coefficients. In the present embodiment, the variable damping coefficients C vR , C vL  are calculated as the control input u on the basis of the nonlinear H-infinity state feedback control to the system. A calculation method of the variable damping coefficients C vR , C vL  by using the nonlinear H-infinity state feedback control in the present embodiment will be briefly described below. 
     1. Nonlinear H-infinity State Feedback Control Theory 
     Firstly, a nonlinear H-infinity state feedback control theory will be described. 
     1-1. Bilinear System 
       FIG. 8  is a block diagram of a closed loop system S in which the state quantity x of a generalized plant G is fed back. In this closed loop system S, w denotes the disturbance, z denotes the evaluation output, u denotes the control input, and x denotes the state quantity. A state space model (a state space representation) of the generalized plant G can be represented as in the following equation (eq. 10) with using the disturbance w, the evaluation output z, the control input u, and the state quantity x. 
     
       
         
           
             
               
                 
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     In a special case where the state space model is represented by a form shown in the following equation (eq. 11), the state space model is called a bilinear system. 
     
       
         
           
             
               
                 
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     1-2. Nonlinear H-infinity State Feedback Control Problem 
     A nonlinear H-infinity state feedback control problem, that is, a control target in the nonlinear H-infinity state feedback control, is to design the state feedback controller K of the system such an influence of the disturbance w of the closed loop system S is prevented from appearing in the evaluation output z to a possible extent. This problem is equal to designing the state feedback controller K (=u=K(x)) such that the L 2  gain (∥S∥ L2 ) from the disturbance w to the evaluation output z of the closed loop system S becomes less than a given positive constant γ, that is, the following equation (eq. 12) is satisfied. 
     
       
         
           
             
               
                 
                   
                     
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     1-3. Solution of the Nonlinear H-infinity State Feedback Control Problem 
     A necessary and sufficient condition to solve the nonlinear H-infinity state feedback control problem is that a positive definite function V(x) and a positive constant c satisfying a Hamilton-Jacobi partial differential inequality shown in an equation (eq. 13) exist. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                        
                       
                         
                           ∂ 
                           V 
                         
                         
                           ∂ 
                           
                             x 
                             T 
                           
                         
                       
                        
                       
                         g 
                         1 
                       
                        
                       
                         g 
                         1 
                         T 
                       
                        
                       
                         
                           ∂ 
                           V 
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                     - 
                     
                       
                         1 
                         4 
                       
                        
                       
                         
                           ∂ 
                           V 
                         
                         
                           ∂ 
                           
                             x 
                             T 
                           
                         
                       
                        
                       
                         g 
                         2 
                       
                        
                       
                         g 
                         2 
                         T 
                       
                        
                       
                         
                           ∂ 
                           V 
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                     + 
                     
                       
                         h 
                         1 
                         T 
                       
                        
                       
                         h 
                         1 
                       
                     
                     + 
                     
                       ɛ 
                        
                       
                           
                       
                        
                       
                         x 
                         T 
                       
                        
                       x 
                     
                   
                   ≤ 
                   0 
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     13 
                   
                   ) 
                 
               
             
           
         
       
     
     In this case, one of the state feedback controller K (=u=K(x)) is given by the following equation (eq. 14). 
     
       
         
           
             
               
                 
                   u 
                   = 
                   
                     
                       - 
                       
                         1 
                         2 
                       
                     
                      
                     
                       
                         g 
                         x 
                         T 
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                      
                     
                       
                         ∂ 
                         V 
                       
                       
                         ∂ 
                         x 
                       
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     14 
                   
                   ) 
                 
               
             
           
         
       
     
     It is said that solving the Hamilton-Jacobi partial differential inequality is almost impossible. Therefore, the state feedback controller K cannot be solved analytically. However, in the case where the state space model is the bilinear system, if a positive definite symmetric matrix P satisfying a Riccati inequality shown in the following equation (eq. 15) is existing, it is known that the nonlinear H-infinity state feedback control problem can be approximately solved. This Riccati inequality can be solved analytically. 
     
       
         
           
             
               
                 
                   
                     PA 
                     + 
                     
                       
                         A 
                         T 
                       
                        
                       P 
                     
                     + 
                     
                       
                         1 
                         
                           γ 
                           2 
                         
                       
                        
                       
                         PB 
                         1 
                       
                        
                       
                         B 
                         1 
                         T 
                       
                        
                       P 
                     
                     + 
                     
                       
                         C 
                         11 
                         T 
                       
                        
                       
                         C 
                         11 
                       
                     
                     + 
                     
                       
                         C 
                         12 
                         T 
                       
                        
                       
                         C 
                         12 
                       
                     
                   
                   &lt; 
                   0 
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     15 
                   
                   ) 
                 
               
             
           
         
       
     
     In this case, one of the state feedback controller K (=u=K(x)) is given by the following equation (eq. 16). 
         u=−D   122   −1 {(1+ m ( x ) x   T   C   11   T   C   11   x ) D   122   −T   B   2   T ( x ) P+C   12   }x   (eq. 16)
 
     In the equation (eq. 15) and the equation (eq. 16), C 11  is a matrix to be multiplied by the state quantity x in an output equation representing an output obtained by a frequency weight W s  acting on the evaluation output, and C 12  is a matrix to be multiplied by the state quantity x in an output equation representing an output obtained by a frequency weight W u  acting on the control input. D 122  is a matrix to be multiplied by the control input u in the output equation representing the output obtained by the frequency weight W u  acting on the control input. In addition, m(x) is an arbitrary positive definite scalar function influencing a constrained condition of a nonlinear weight to be multiplied by the frequency weights W s , W u . In the case where the nonlinear weight does not act as a weight, m(x) can be set to 0. 
     Therefore, in the case where the state space model is the bilinear system, the state feedback controller K can be designed by solving the Riccati inequality. Thus, the control object can be state-feedback by the control input u calculated by the designed state feedback controller K. 
     2. Designing of State Space Model 
     2-1. Derivation of Motion Equation of Suspension Apparatuses 
       FIG. 9  is a diagram in which the suspension apparatuses SP R , SP L  shown in  FIG. 4  are represented as a two wheel model of the vehicle. The two wheel model shows a vibration system serving as the control object in the present example. In the figure, M denotes a mass of the sprung member HA, K R  denotes a spring constant of the right side suspension spring  10 R, K L  denotes a spring constant of the left side suspension spring  10 L, C sR  denotes the linear damping coefficient of the right side damper  20 R, C sL  denotes the linear damping coefficient of the left side damper  20 L, C vR  denotes the variable damping coefficient of the right side damper  20 R, C VL  denotes the variable damping coefficient of the left side damper  20 L, y denotes the vertical displacement of the sprung member HA (the sprung vertical displacement), r R  denotes the vertical displacement of the right side unsprung member LA R  (the right side unsprung displacement), and r L  denotes the vertical displacement of the left side unsprung member LA L  (the left side unsprung displacement). 
     In the two wheel model shown in  FIG. 9 , a motion equation of the sprung member HA is represented by the following equation (eq. 17). 
         Mÿ=K   R ( r   R   −y )+ K   L ( r   L   −y )+ C   sR ( {dot over (r)}   R   −{dot over (y)} )+ C   sL ( {dot over (r)}   L   −{dot over (y)} )+ C   vR ( {dot over (r)}   R   −{dot over (y)} )+ C   vL ( {dot over (r)}   L   −{dot over (y)} )  (eq. 17)
 
     wherein:
 
ÿ=d 2 y/dt 2 , {dot over (y)}=dy/dt, {dot over (r)} R =dr R /dt, {dot over (r)}=dr L /dt
 
     2-2. Designing of State Space Model 
     Based on the equation (eq. 17), a state space model of the two wheel model is designed as shown in  FIG. 9 . In this case, a state quantity x p  is represented by the sprung-right side unsprung relative displacement r R −y, the sprung-left side unsprung relative displacement r L −y, and the sprung speed dy/dt. The disturbance w is represented by the right side unsprung speed dr R /dt, and the left side unsprung speed dr L /dt. The control input u is represented by the right side variable damping coefficient C vR , and the left side variable damping coefficient C vL . A state equation is described as in the following equation (eq. 18). 
         {dot over (x)}   p   =A   p   x   p   +B   p1   w+B   p2 ( x   p ) u   (eq. 18)
 
     wherein: 
     
       
         
           
             
               
                 x 
                 p 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           r 
                           R 
                         
                         - 
                         y 
                       
                     
                   
                   
                     
                       
                         
                           r 
                           L 
                         
                         - 
                         y 
                       
                     
                   
                   
                     
                       
                         y 
                         . 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
                 
             
              
             
               w 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           r 
                           . 
                         
                         R 
                       
                     
                   
                   
                     
                       
                         
                           r 
                           . 
                         
                         L 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
                 
             
              
             
               u 
               = 
               
                 [ 
                 
                   
                     
                       
                         C 
                         vR 
                       
                     
                   
                   
                     
                       
                         C 
                         vL 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               
                 A 
                 p 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                   
                     
                       
                         
                           K 
                           R 
                         
                         M 
                       
                     
                     
                       
                         
                           K 
                           L 
                         
                         M 
                       
                     
                     
                       
                         - 
                         
                           
                             
                               C 
                               sR 
                             
                             + 
                             
                               C 
                               sL 
                             
                           
                           M 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
                 
             
              
             
               
                 B 
                 
                   p 
                    
                   
                       
                   
                    
                   1 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                   
                   
                     
                       
                         
                           C 
                           sR 
                         
                         M 
                       
                     
                     
                       
                         
                           C 
                           sL 
                         
                         M 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
              
             
               
                 
                   B 
                   
                     p 
                      
                     
                         
                     
                      
                     2 
                   
                 
                  
                 
                   ( 
                   
                     x 
                     p 
                   
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         
                           
                             
                               r 
                               . 
                             
                             R 
                           
                           - 
                           
                             y 
                             . 
                           
                         
                         M 
                       
                     
                     
                       
                         
                           
                             
                               r 
                               . 
                             
                             L 
                           
                           - 
                           
                             y 
                             . 
                           
                         
                         M 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               
                 x 
                 . 
               
               p 
             
             = 
             
               
                  
                 
                   x 
                   p 
                 
               
               / 
               
                  
                 t 
               
             
           
         
       
     
     wherein: x p  denotes state quantity, w denotes disturbance, u denotes control input. 
     An output equation is described as in the following equation (eq. 19). 
         z   p   =C   p1   x   p   +D   p12   u   (eq. 19)
 
     In the case where an evaluation output z p  is set to the sprung-right side unsprung relative displacement r R −y and the sprung-left side unsprung relative displacement r L −y, z p , C p1 , and D p12  are represented as follows. 
     
       
         
           
             
               
                 z 
                 p 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           r 
                           R 
                         
                         - 
                         y 
                       
                     
                   
                   
                     
                       
                         
                           r 
                           L 
                         
                         - 
                         y 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
                 
             
              
             
               
                 C 
                 
                   p 
                    
                   
                       
                   
                    
                   1 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
                 
             
              
             
               
                 D 
                 
                   p 
                    
                   
                       
                   
                    
                   12 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Notably, the evaluation output z p  may be set to the sprung acceleration d 2 y/dt 2  or the sprung speed dy/dt. A term in relation to the disturbance w may be added to the output equation so that the output equation is rewritten as “z p =C p1 x p +D p11 w+D p12 u”. 
     With the equation (eq. 18) and the equation (eq. 19), the state space model of the control object shown in  FIG. 9  is described as in the following equation (eq. 20). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               x 
                               . 
                             
                             p 
                           
                           = 
                           
                             
                               
                                 A 
                                 p 
                               
                                
                               
                                 x 
                                 p 
                               
                             
                             + 
                             
                               
                                 B 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               w 
                             
                             + 
                             
                               
                                 
                                   B 
                                   
                                     p 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     p 
                                   
                                   ) 
                                 
                               
                                
                               u 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             z 
                             p 
                           
                           = 
                           
                             
                               
                                 C 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 x 
                                 p 
                               
                             
                             + 
                             
                               
                                 D 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   12 
                                 
                               
                                
                               u 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     20 
                   
                   ) 
                 
               
             
           
         
       
     
     The state space model shown in the equation (eq. 20) is the bilinear system.  FIG. 10  is a block diagram of the system represented by the equation (eq. 20). 
     3. Controllability of System Represented by State Space Model 
     A necessary and sufficient condition to obtain controllability of the system represented by the state space model of the equation (eq. 20) is that the controllable matrix U c  of this state space model has full rank. The controllable matrix U c  is represented as in the following equation (eq. 21). 
         U   c   =[B   p2 ( x   p ) A   p   B   p2 ( x   p ) A   p   2   B   p2 ( x   p )]  (eq. 21)
 
     In the case where a state matrix A p  and an input matrix B p2 (x p ) are represented by the above equation (eq. 18), the controllable matrix U c  is represented as in the following equation (eq. 22). 
     
       
         
           
             
                 
             
              
             
               ( 
               
                 eq 
                 . 
                 
                     
                 
                  
                 22 
               
               ) 
             
           
         
       
       
         
           
             
               U 
               c 
             
             = 
             
               [ 
               
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       
                         
                           
                             
                               r 
                               . 
                             
                             R 
                           
                           - 
                           
                             y 
                             . 
                           
                         
                         M 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           
                             
                               r 
                               . 
                             
                             L 
                           
                           - 
                           y 
                         
                         M 
                       
                     
                   
                   
                     
                       α 
                       R 
                     
                   
                   
                     
                       α 
                       L 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       
                         
                           
                             
                               r 
                               . 
                             
                             R 
                           
                           - 
                           
                             y 
                             . 
                           
                         
                         M 
                       
                     
                   
                   
                     
                       - 
                       
                         
                           
                             
                               r 
                               . 
                             
                             L 
                           
                           - 
                           y 
                         
                         M 
                       
                     
                   
                   
                     
                       α 
                       R 
                     
                   
                   
                     
                       α 
                       L 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             r 
                             . 
                           
                           R 
                         
                         - 
                         
                           y 
                           . 
                         
                       
                       M 
                     
                   
                   
                     
                       
                         
                           
                             r 
                             . 
                           
                           L 
                         
                         - 
                         y 
                       
                       M 
                     
                   
                   
                     
                       - 
                       
                         α 
                         R 
                       
                     
                   
                   
                     
                       - 
                       
                         α 
                         L 
                       
                     
                   
                   
                     
                       
                         - 
                         
                           
                             ( 
                             
                               
                                 
                                   r 
                                   . 
                                 
                                 R 
                               
                               - 
                               
                                 y 
                                 . 
                               
                             
                             ) 
                           
                           
                             M 
                             2 
                           
                         
                       
                        
                       β 
                     
                   
                   
                     
                       
                         - 
                         
                           
                             ( 
                             
                               
                                 
                                   r 
                                   . 
                                 
                                 L 
                               
                               - 
                               
                                 y 
                                 . 
                               
                             
                             ) 
                           
                           
                             M 
                             2 
                           
                         
                       
                        
                       β 
                     
                   
                 
               
                
               
                   
               
               ] 
             
           
         
       
     
     wherein: 
     
       
         
           
             
               
                 α 
                 R 
               
               = 
               
                 
                   
                     ( 
                     
                       
                         C 
                         sR 
                       
                       + 
                       
                         C 
                         sL 
                       
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           r 
                           . 
                         
                         R 
                       
                       - 
                       
                         y 
                         . 
                       
                     
                     ) 
                   
                 
                 
                   M 
                   2 
                 
               
             
             , 
             
                 
             
              
             
               
                 α 
                 L 
               
               = 
               
                 
                   
                     ( 
                     
                       
                         C 
                         sR 
                       
                       + 
                       
                         C 
                         sL 
                       
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           r 
                           . 
                         
                         L 
                       
                       - 
                       
                         y 
                         . 
                       
                     
                     ) 
                   
                 
                 
                   M 
                   2 
                 
               
             
             , 
             
               
 
             
              
             
               β 
               = 
               
                 
                   K 
                   R 
                 
                 + 
                 
                   K 
                   L 
                 
                 + 
                 
                   
                     
                       C 
                       sR 
                     
                     + 
                     
                       C 
                       sL 
                     
                   
                   M 
                 
               
             
           
         
       
     
     As understood from the equation (eq. 22), the controllable matrix U c  is represented as a 3-by-6 matrix. Therefore, full rank of the controllable matrix U c  is 3(Full rank=3). First row elements and second row elements in the controllable matrix U c  are all the same. Thus, the rank deficiency is generated, and the rank of the controllable matrix U c  becomes 2(rankU c =2). That is, the controllable matrix U c  does not have full rank. Therefore, the system represented by the state space model shown in the equation (eq. 20) is uncontrollable. 
     The reason for that the controllable matrix U c  does not have full rank is that the number of motion equation serving as a basis in designing of the model is one, nevertheless the number of the control input u is two (the right side variable damping coefficient C vR  and the left side variable damping coefficient C VL ). That is, the number of the motion equation is less than the number of the control input u. 
     4. Designing of Corrected State Space Model 
     In the present embodiment, a corrected state space model obtained by correcting the state space model of the control object shown in the equation (eq. 20) is proposed. This corrected state space model is described as in the following equation (eq. 23). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               x 
                               . 
                             
                             p 
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   
                                     A 
                                     p 
                                   
                                   + 
                                   Δ 
                                 
                                 ) 
                               
                                
                               
                                 x 
                                 p 
                               
                             
                             + 
                             
                               
                                 B 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               w 
                             
                             + 
                             
                               
                                 
                                   B 
                                   
                                     p 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     p 
                                   
                                   ) 
                                 
                               
                                
                               u 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             z 
                             p 
                           
                           = 
                           
                             
                               
                                 C 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 x 
                                 p 
                               
                             
                             + 
                             
                               
                                 D 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   12 
                                 
                               
                                
                               u 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     23 
                   
                   ) 
                 
               
             
           
         
       
     
     As understood from the equation (eq. 23), the state quantity x p  of the state equation is multiplied by a corrected state matrix (A p +Δ) obtained by adding the error matrix Δ to the state matrix A p  of the state space model of the equation (eq. 20). The error matrix Δ is a preliminarily designed matrix, and gives an error (perturbation) to the state matrix A p . That is, the corrected state space model shown in the equation (eq. 23) is a model obtained by correcting the state space model by adding the error matrix Δ to the state matrix A p  of the state space model representing the uncontrollable system shown in the equation (eq. 20). 
       FIG. 11  is a block diagram of a system represented by this corrected state space model. As shown in  FIG. 11 , the error matrix Δ is added into the corrected state space model as an additive error of the state matrix A p . The error matrix Δ is added to the state matrix A p  at an adding point Q 1 . The error matrix Δ has the same form as the state matrix A p  (3-by-3). 
     A necessary and sufficient condition for obtaining controllability of the system represented by the corrected state space model is that the controllable matrix U c * of the corrected state space model has full rank. The controllable matrix U c * of the corrected state space model is represented as in the following equation (eq. 24). 
         U   c   *=[B   p2 ( x   p )( A   p +Δ) B   p2 ( x   p )( A   p +Δ) 2   B   p2 ( x   p )]  (eq. 24)
 
     In order to avoid a complicated calculation, the state matrix A p  and the input matrix B p2 (x p ) represented by the equation (eq. 18) are respectively described as in the following equations (eq. 25) and (eq. 26). 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       p 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               - 
                               1 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               - 
                               1 
                             
                           
                         
                         
                           
                             
                               a 
                               31 
                             
                           
                           
                             
                               a 
                               32 
                             
                           
                           
                             
                               a 
                               33 
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     25 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       B 
                       
                         p 
                          
                         
                             
                         
                          
                         2 
                       
                     
                      
                     
                       ( 
                       
                         x 
                         p 
                       
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     26 
                   
                   ) 
                 
               
             
           
         
       
     
     wherein: 
     
       
         
           
             
               
                 a 
                 31 
               
               = 
               
                 
                   K 
                   R 
                 
                 
                   M 
                    
                   
                       
                   
                 
               
             
             , 
             
                 
             
              
             
               
                 a 
                 32 
               
               = 
               
                 
                   K 
                   L 
                 
                 M 
               
             
             , 
             
                 
             
              
             
               
                 a 
                 33 
               
               = 
               
                 - 
                 
                   
                     
                       C 
                       sR 
                     
                     + 
                     
                       C 
                       sL 
                     
                   
                   M 
                 
               
             
             , 
             
                 
             
              
             
               
                 
                   b 
                   1 
                 
                  
                 
                   ( 
                   
                     x 
                     p 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       r 
                       . 
                     
                     R 
                   
                   - 
                   
                     y 
                     . 
                   
                 
                 M 
               
             
             , 
             
               
 
             
              
             
               
                 
                   b 
                   2 
                 
                  
                 
                   ( 
                   
                     x 
                     p 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       r 
                       . 
                     
                     L 
                   
                   - 
                   
                     y 
                     . 
                   
                 
                 M 
               
             
           
         
       
     
     The error matrix Δ is for example represented as in the following equation (eq. 27). 
     
       
         
           
             
               
                 
                   Δ 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             0.1 
                              
                             
                               a 
                               33 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     27 
                   
                   ) 
                 
               
             
           
         
       
     
     As understood from the equation (eq. 27), a non-zero element 0.1a 33  of the error matrix Δ has a magnitude of 1/10 of a non-zero element a 33  of the state matrix A p . In the case where the state matrix A p , the input matrix B p2 (x p ), and the error matrix Δ are respectively represented as in the equation (eq. 25), the equation (eq. 26), and the equation (eq. 27), the following equations (eq. 28) and (eq. 29) are established. The controllable matrix U c * is represented as in the following equation (eq. 30). 
     
       
         
           
             
                 
             
              
             
               ( 
               
                 eq 
                 . 
                 
                     
                 
                  
                 28 
               
               ) 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   A 
                   p 
                 
                 + 
                 Δ 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         0.1 
                          
                         
                           a 
                           33 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                   
                     
                       
                         a 
                         31 
                       
                     
                     
                       
                         a 
                         32 
                       
                     
                     
                       
                         a 
                         33 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               ( 
               
                 eq 
                 . 
                 
                     
                 
                  
                 29 
               
               ) 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   ( 
                   
                     
                       A 
                       p 
                     
                     + 
                     Δ 
                   
                   ) 
                 
                 2 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           0.01 
                            
                           
                             a 
                             33 
                             2 
                           
                         
                         - 
                         
                           a 
                           31 
                         
                       
                     
                     
                       
                         - 
                         
                           a 
                           32 
                         
                       
                     
                     
                       
                         
                           - 
                           1.1 
                         
                          
                         
                           a 
                           33 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         
                           a 
                           31 
                         
                       
                     
                     
                       
                         - 
                         
                           a 
                           32 
                         
                       
                     
                     
                       
                         - 
                         
                           a 
                           33 
                         
                       
                     
                   
                   
                     
                       
                         1.1 
                          
                         
                           a 
                           31 
                         
                          
                         
                           a 
                           33 
                         
                       
                     
                     
                       
                         
                           a 
                           32 
                         
                          
                         
                           a 
                           33 
                         
                       
                     
                     
                       
                         
                           - 
                           
                             a 
                             31 
                           
                         
                         - 
                         
                           a 
                           32 
                         
                         + 
                         
                           a 
                           33 
                           2 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               ( 
               
                 eq 
                 . 
                 
                     
                 
                  
                 30 
               
               ) 
             
           
         
       
       
         
           
             
               U 
               c 
               * 
             
             = 
             
                 
               
                   
               
                
               
                   
                 
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           - 
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           - 
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             1.1 
                           
                            
                           
                             a 
                             33 
                           
                            
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             1.1 
                           
                            
                           
                             a 
                             33 
                           
                            
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           - 
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           - 
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               a 
                               33 
                             
                           
                            
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               a 
                               33 
                             
                           
                            
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             b 
                             1 
                           
                            
                           
                             ( 
                             
                               x 
                               p 
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             b 
                             2 
                           
                            
                           
                             ( 
                             
                               x 
                               p 
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             a 
                             33 
                           
                            
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             a 
                             33 
                           
                            
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           γ 
                            
                           
                               
                           
                            
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           γ 
                            
                           
                               
                           
                            
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                    
                   
                       
                   
                   ] 
                 
               
             
           
         
       
     
     wherein: γ=−a 31 −a 32 +a 33   2    
     As understood from the equation (eq. 30), elements in fifth and sixth columns in a first row of the controllable matrix U c * are different from elements in fifth and sixth columns in a second row. Therefore, the rank deficiency due to the fact that the first row elements and the second row elements are all the same elements is prevented, and the rank of the controllable matrix U c * becomes 3(rankU c *=3). That is, the controllable matrix U c * has full rank, and the system represented by the corrected state space model becomes controllable. Therefore, a state feedback control system of the corrected state space model can be designed. 
     5. Design Example of Error Matrix Δ 
     The error matrix Δ is designed such that the controllable matrix U c * of the corrected state space model has full rank as in the above example. A design example of such an error matrix Δ will be considered. For example, the corrected state matrix A p +Δ is represented by the following equation (eq. 31) and the input matrix B p2 (x p ) is represented by the following equation (eq. 32). 
     
       
         
           
             
               
                 
                   
                     
                       
                         A 
                         p 
                       
                       + 
                       Δ 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               a 
                               11 
                             
                           
                           
                             
                               a 
                               12 
                             
                           
                           
                             
                               a 
                               13 
                             
                           
                         
                         
                           
                             
                               a 
                               21 
                             
                           
                           
                             
                               a 
                               22 
                             
                           
                           
                             
                               a 
                               23 
                             
                           
                         
                         
                           
                             
                               a 
                               31 
                             
                           
                           
                             
                               a 
                               32 
                             
                           
                           
                             
                               a 
                               33 
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     31 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       B 
                       
                         p 
                          
                         
                             
                         
                          
                         2 
                       
                     
                      
                     
                       ( 
                       
                         x 
                         p 
                       
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             b 
                             1 
                           
                         
                         
                           
                             b 
                             2 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     32 
                   
                   ) 
                 
               
             
           
         
       
     
     The following equations (eq. 33) and (eq. 34) are established. 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         ( 
                         
                           
                             A 
                             p 
                           
                           + 
                           Δ 
                         
                         ) 
                       
                        
                       
                         
                           B 
                           
                             p 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             p 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 a 
                                 13 
                               
                                
                               
                                 b 
                                 1 
                               
                             
                           
                           
                             
                               
                                 a 
                                 13 
                               
                                
                               
                                 b 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               
                                 a 
                                 23 
                               
                                
                               
                                 b 
                                 1 
                               
                             
                           
                           
                             
                               
                                 a 
                                 23 
                               
                                
                               
                                 b 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               
                                 a 
                                 33 
                               
                                
                               
                                 b 
                                 1 
                               
                             
                           
                           
                             
                               
                                 a 
                                 33 
                               
                                
                               
                                 b 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     33 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             A 
                             p 
                           
                           + 
                           Δ 
                         
                         ) 
                       
                       2 
                     
                      
                     
                       
                         B 
                         
                           p 
                            
                           
                               
                           
                            
                           2 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           p 
                         
                         ) 
                       
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               
                                 b 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       a 
                                       11 
                                     
                                      
                                     
                                       a 
                                       13 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       12 
                                     
                                      
                                     
                                       a 
                                       23 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       13 
                                     
                                      
                                     
                                       a 
                                       33 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 b 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       a 
                                       11 
                                     
                                      
                                     
                                       a 
                                       13 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       12 
                                     
                                      
                                     
                                       a 
                                       23 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       13 
                                     
                                      
                                     
                                       a 
                                       33 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 b 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       a 
                                       21 
                                     
                                      
                                     
                                       a 
                                       13 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       22 
                                     
                                      
                                     
                                       a 
                                       23 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       23 
                                     
                                      
                                     
                                       a 
                                       33 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 b 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       a 
                                       21 
                                     
                                      
                                     
                                       a 
                                       13 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       22 
                                     
                                      
                                     
                                       a 
                                       23 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       23 
                                     
                                      
                                     
                                       a 
                                       33 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 b 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       a 
                                       31 
                                     
                                      
                                     
                                       a 
                                       13 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       32 
                                     
                                      
                                     
                                       a 
                                       23 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       33 
                                     
                                      
                                     
                                       a 
                                       33 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 b 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       a 
                                       31 
                                     
                                      
                                     
                                       a 
                                       13 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       32 
                                     
                                      
                                     
                                       a 
                                       23 
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       33 
                                     
                                      
                                     
                                       a 
                                       33 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     34 
                   
                   ) 
                 
               
             
           
         
       
     
     The controllable matrix U c * is represented by the following equation (eq. 35). 
     
       
         
           
             
               
                 
                   
                     U 
                     c 
                     * 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             
                               a 
                               13 
                             
                              
                             
                               b 
                               1 
                             
                           
                         
                         
                           
                             
                               a 
                               13 
                             
                              
                             
                               b 
                               2 
                             
                           
                         
                         
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     a 
                                     11 
                                   
                                    
                                   
                                     a 
                                     13 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     12 
                                   
                                    
                                   
                                     a 
                                     23 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     13 
                                   
                                    
                                   
                                     a 
                                     33 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     a 
                                     11 
                                   
                                    
                                   
                                     a 
                                     13 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     12 
                                   
                                    
                                   
                                     a 
                                     23 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     13 
                                   
                                    
                                   
                                     a 
                                     33 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             
                               a 
                               23 
                             
                              
                             
                               b 
                               1 
                             
                           
                         
                         
                           
                             
                               a 
                               23 
                             
                              
                             
                               b 
                               2 
                             
                           
                         
                         
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     a 
                                     21 
                                   
                                    
                                   
                                     a 
                                     13 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     22 
                                   
                                    
                                   
                                     a 
                                     23 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     23 
                                   
                                    
                                   
                                     a 
                                     33 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     a 
                                     21 
                                   
                                    
                                   
                                     a 
                                     13 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     22 
                                   
                                    
                                   
                                     a 
                                     23 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     23 
                                   
                                    
                                   
                                     a 
                                     33 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             b 
                             1 
                           
                         
                         
                           
                             b 
                             2 
                           
                         
                         
                           
                             
                               a 
                               33 
                             
                              
                             
                               b 
                               1 
                             
                           
                         
                         
                           
                             
                               a 
                               33 
                             
                              
                             
                               b 
                               2 
                             
                           
                         
                         
                           
                             
                               b 
                               1 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     a 
                                     31 
                                   
                                    
                                   
                                     a 
                                     13 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     32 
                                   
                                    
                                   
                                     a 
                                     23 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     33 
                                   
                                    
                                   
                                     a 
                                     33 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               b 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     a 
                                     31 
                                   
                                    
                                   
                                     a 
                                     13 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     32 
                                   
                                    
                                   
                                     a 
                                     23 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     33 
                                   
                                    
                                   
                                     a 
                                     33 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     35 
                   
                   ) 
                 
               
             
           
         
       
     
     In this case, when the following equality (eq. 36) is established, the first row elements and the second row elements of the controllable matrix U c * shown in the equation (eq. 35) are the same. Therefore, the rank deficiency is generated, and the controllable matrix U c * does not have full rank. 
     
       
         
           
             
               
                 
                   
                     
                       a 
                       13 
                     
                     
                       a 
                       23 
                     
                   
                   = 
                   
                     
                       
                         
                           a 
                           11 
                         
                          
                         
                           a 
                           13 
                         
                       
                       + 
                       
                         
                           a 
                           12 
                         
                          
                         
                           a 
                           23 
                         
                       
                       + 
                       
                         
                           a 
                           13 
                         
                          
                         
                           a 
                           33 
                         
                       
                     
                     
                       
                         
                           a 
                           21 
                         
                          
                         
                           a 
                           13 
                         
                       
                       + 
                       
                         
                           a 
                           22 
                         
                          
                         
                           a 
                           23 
                         
                       
                       + 
                       
                         
                           a 
                           23 
                         
                          
                         
                           a 
                           33 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     36 
                   
                   ) 
                 
               
             
           
         
       
     
     The above equation (eq. 36) can be represented as in the following equation (eq. 37). 
         a   11   a   13   a   23   +a   12   a   23   2   =a   13   2   a   21   +a   13   a   22   a   23   (eq. 37)
 
     The error matrix Δ can be designed such that the rank deficiency is not generated in the controllable matrix U c * by determining the corrected state matrix (Δ p +Δ) so that the above equation (eq. 37) is not established and by subtracting the state matrix A p  from the determined corrected state matrix (Δ p +Δ). For example, in the corrected state matrix (Δ p +Δ) represented by the above equation (eq. 28), elements relating to the equation (eq. 36) which influence the rank of the controllable matrix U c *(a 11 , a 12 , a 13 , a 21 , a 22 , a 23 ) are set as shown in the following equation (eq. 38). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             a 
                             11 
                           
                           = 
                           
                             0.1 
                              
                             
                               a 
                               33 
                             
                           
                         
                       
                       
                         
                           
                             a 
                             12 
                           
                           = 
                           0 
                         
                       
                       
                         
                           
                             a 
                             13 
                           
                           = 
                           
                             - 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             a 
                             21 
                           
                           = 
                           0 
                         
                       
                       
                         
                           
                             a 
                             22 
                           
                           = 
                           0 
                         
                       
                       
                         
                           
                             a 
                             23 
                           
                           = 
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     38 
                   
                   ) 
                 
               
             
           
         
       
     
     In the case where the elements are set as in the above equation (eq. 38), a left side value of the equation (eq. 37) becomes −0.1a 33 , and a right side value becomes zero. Therefore, the equation (eq. 37) is not established. Thus, the rank deficiency is not generated but the controllable matrix U c * has full rank. 
     The above design example is one example of designing the error matrix Δ in the case where the input matrix B p2 (x p ) is represented as in the equation (eq. 32). There is sometimes the case where the input matrix B p2 (x p ) is represented by a form other than the above equation (eq. 32). In that case, the error matrix Δ is individually designed such that the rank deficiency is not generated in the controllable matrix U c *. 
     6. Designing of State Feedback Control System 
       FIG. 12  is a block diagram of the closed loop system S (the state feedback control system) in which state feedback is performed in the state of the generalized plant G designed based on the system represented by the corrected state space model. A portion shown by M* of  FIG. 12  is the system represented by the corrected state space model. The corrected state space model is represented by the following equation (eq. 39). This equation is the same as the above equation (eq. 23). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               x 
                               . 
                             
                             p 
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   
                                     A 
                                     p 
                                   
                                   + 
                                   Δ 
                                 
                                 ) 
                               
                                
                               
                                 x 
                                 p 
                               
                             
                             + 
                             
                               B 
                               
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                             + 
                             
                               
                                 B 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               w 
                             
                             + 
                             
                               
                                 
                                   B 
                                   
                                     p 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     p 
                                   
                                   ) 
                                 
                               
                                
                               u 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             z 
                             p 
                           
                           = 
                           
                             
                               
                                 C 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 x 
                                 p 
                               
                             
                             + 
                             
                               
                                 D 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   12 
                                 
                               
                                
                               u 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     39 
                   
                   ) 
                 
               
             
           
         
       
     
     As understood from  FIG. 12 , the frequency weight W s  which is a weight varied by a frequency acts on the evaluation output z p . A state space model of the frequency weight W s  is expressed as in the following equation (eq. 40) with using a state quantity x w , an output z w , and constant matrices A w , B w , C w , D w . 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               x 
                               . 
                             
                             w 
                           
                           = 
                           
                             
                               
                                 A 
                                 w 
                               
                                
                               
                                 x 
                                 w 
                               
                             
                             + 
                             
                               
                                 B 
                                 w 
                               
                                
                               
                                 z 
                                 p 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             z 
                             w 
                           
                           = 
                           
                             
                               
                                 C 
                                 w 
                               
                                
                               
                                 x 
                                 w 
                               
                             
                             + 
                             
                               
                                 D 
                                 w 
                               
                                
                               
                                 z 
                                 p 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     40 
                   
                   ) 
                 
               
             
           
         
       
     
     wherein:
 
{dot over (x)} w =dx w /dt
 
     The equation (eq. 40) can be modified as in the following equation (eq. 41). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               x 
                               . 
                             
                             w 
                           
                           = 
                           
                             
                               
                                 A 
                                 w 
                               
                                
                               
                                 x 
                                 w 
                               
                             
                             + 
                             
                               
                                 B 
                                 w 
                               
                                
                               
                                 C 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 x 
                                 p 
                               
                             
                             + 
                             
                               
                                 B 
                                 w 
                               
                                
                               
                                 D 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   12 
                                 
                               
                                
                               u 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             z 
                             w 
                           
                           = 
                           
                             
                               
                                 C 
                                 w 
                               
                                
                               
                                 x 
                                 w 
                               
                             
                             + 
                             
                               
                                 D 
                                 w 
                               
                                
                               
                                 C 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 x 
                                 p 
                               
                             
                             + 
                             
                               
                                 D 
                                 w 
                               
                                
                               
                                 D 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   12 
                                 
                               
                                
                               u 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     41 
                   
                   ) 
                 
               
             
           
         
       
     
     The frequency weight W u  varied by the frequency acts on the control input u. A state space model of the frequency weight W u  is represented as in the following equation (eq. 42) with using a state quantity x u , an output z u , and constant matrices A u , B u , C u , D u . 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               x 
                               . 
                             
                             u 
                           
                           = 
                           
                             
                               
                                 A 
                                 u 
                               
                                
                               
                                 x 
                                 u 
                               
                             
                             + 
                             
                               
                                 B 
                                 u 
                               
                                
                               u 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             z 
                             u 
                           
                           = 
                           
                             
                               
                                 C 
                                 u 
                               
                                
                               
                                 x 
                                 u 
                               
                             
                             + 
                             
                               
                                 D 
                                 u 
                               
                                
                               u 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     42 
                   
                   ) 
                 
               
             
           
         
       
     
     wherein:
 
{dot over (x)} u =dx u /dt
 
     From the equations (eq. 39) to (eq. 42), the state space model representing the generalized plant is described as in the following equation (eq. 43). This state space model includes is corrected model corrected by the error matrix Δ. Therefore, the generalized plant is controllable. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             x 
                             . 
                           
                           = 
                           
                             Ax 
                             + 
                             
                               
                                 B 
                                 1 
                               
                                
                               w 
                             
                             + 
                             
                               
                                 
                                   B 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                                
                               u 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             z 
                             w 
                           
                           = 
                           
                             
                               
                                 C 
                                 11 
                               
                                
                               x 
                             
                             + 
                             
                               
                                 D 
                                 121 
                               
                                
                               u 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             z 
                             u 
                           
                           = 
                           
                             
                               
                                 C 
                                 12 
                               
                                
                               x 
                             
                             + 
                             
                               
                                 D 
                                 122 
                               
                                
                               u 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     43 
                   
                   ) 
                 
               
             
           
         
       
     
     wherein: 
     
       
         
           
             
               x 
               = 
               
                 [ 
                 
                   
                     
                       
                         x 
                         p 
                       
                     
                   
                   
                     
                       
                         x 
                         w 
                       
                     
                   
                   
                     
                       
                         x 
                         u 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               A 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           A 
                           p 
                         
                         + 
                         Δ 
                       
                     
                     
                       o 
                     
                     
                       o 
                     
                   
                   
                     
                       
                         
                           B 
                           w 
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                     
                       
                         A 
                         w 
                       
                     
                     
                       o 
                     
                   
                   
                     
                       o 
                     
                     
                       o 
                     
                     
                       
                         A 
                         u 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
              
             
               
                 B 
                 1 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         B 
                         
                           p 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                   
                   
                     
                       o 
                     
                   
                   
                     
                       o 
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 
                   B 
                   2 
                 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           B 
                           
                             p 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             p 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           w 
                         
                          
                         
                           D 
                           
                             p 
                              
                             
                                 
                             
                              
                             12 
                           
                         
                       
                     
                   
                   
                     
                       
                         B 
                         u 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               
                 C 
                 11 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           D 
                           w 
                         
                          
                         
                           C 
                           
                             p 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                     
                       
                         C 
                         w 
                       
                     
                     
                       o 
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 D 
                 121 
               
               = 
               
                 [ 
                 
                   
                     D 
                     w 
                   
                    
                   
                     D 
                     
                       p 
                        
                       
                           
                       
                        
                       12 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
              
             
               
                 C 
                 12 
               
               = 
               
                 [ 
                 
                   
                     
                       o 
                     
                     
                       o 
                     
                     
                       
                         C 
                         u 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 D 
                 122 
               
               = 
               
                 D 
                 u 
               
             
           
         
       
     
     7. Designing of State Feedback Controller 
     The state space model represented as in the above equation (eq. 43) is the bilinear system. Therefore, when a positive definite symmetric matrix P satisfying the Riccati inequality shown in the following equation (eq. 44) exists in relation to the preliminarily set positive constant γ, the closed loop system S of  FIG. 12  is internally stabilized and the L 2  gain μSμ L2  of the closed loop system S representing robustness against the disturbance can be made less than γ. 
     
       
         
           
             
               
                 
                   
                     PA 
                     + 
                     
                       
                         A 
                         T 
                       
                        
                       P 
                     
                     + 
                     
                       
                         1 
                         
                           γ 
                           2 
                         
                       
                        
                       
                         PB 
                         1 
                       
                        
                       
                         B 
                         1 
                         T 
                       
                        
                       P 
                     
                     + 
                     
                       
                         C 
                         11 
                         T 
                       
                        
                       
                         C 
                         11 
                       
                     
                     + 
                     
                       
                         C 
                         12 
                         T 
                       
                        
                       
                         C 
                         12 
                       
                     
                   
                   &lt; 
                   0 
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     44 
                   
                   ) 
                 
               
             
           
         
       
     
     At this time, one of the state feedback controller K (=K(x)) is represented as shown in the following equation (eq. 45). 
         K ( x )= u=−D   122   −T ( D   122   −T   B   2   T ( x ) P+C   12 ) x   (eq. 45)
 
     The equation (eq. 45) is described as in an equation (eq. 47) under a condition represented by an equation (eq. 46). 
         C   12   =o, D   122   =I   (eq. 46)
 
         K ( x )= u=−B   2   T ( x ) Px   (eq. 47)
 
     The control input u is calculated by the state feedback controller K (=K(x)) designed as in the above equation (eq. 47) as one example, that is, the state feedback controller K (=K(x)) designed such that the L 2  gain of the closed loop system S becomes less than the positive constant γ. By the calculated control input u, the right side variable damping coefficient C vR  and the left side variable damping coefficient C vL  are obtained. In the present embodiment, the damping force characteristic of the right side damper  20 R and the damping force characteristic of the left side damper  20 L are controlled on the basis of the right side variable damping coefficient C vR  and the left side variable damping coefficient C vL  obtained as described above. By controlling the damping force as described in the present embodiment, the vibrations of the right suspension apparatus SP R  and the left suspension apparatus SP L  are controlled. 
     According to the above present embodiment, the micro computer  50  as the state feedback control apparatus is provided with the state feedback controller K (the nonlinear H-infinity controller  51 ) for calculating the control input of the system based on the state quantity of the system represented by the corrected state space model, and the control means (the requested damping force calculation section  52 , the requested step number determination section  53 ) for controlling the vibrations of the suspension apparatuses SP R , SP L  by controlling the damping forces of the suspension apparatuses SP R , SP L  (the dampers  20 R,  20 L) based on the control input calculated by the state feedback controller K. 
     The above corrected state space model is obtained by correcting the state space model by adding the error matrix Δ to the state matrix of the state space model of the suspension apparatuses SP R , SP L  represented as the uncontrollable system. The error matrix Δ is designed such that the controllable matrix of the corrected state space model has full rank by adding the error matrix Δ to the state matrix. Therefore, the system represented by the corrected state space model (or the generalized plant) becomes controllable, and the control object can be state-feedback controlled. 
     A basic structure of the corrected state space model is the same as the original state space model of the control object except that the error matrix Δ is only added. Therefore, there is no need for redesigning time of the model. Further, since the error matrix Δ is added to the state matrix which is less influential on the output of the model, the error matrix Δ does not greatly influence the output. In addition, since only one error matrix Δ is added into the corrected state space model, buildup of the error is not generated. Therefore, deviation between the corrected state space model and the state space model of the actual control object is small, to thereby highly precisely state-feedback control is achieved. Since an error examination point is one point, time required for examining the error can be shortened. That is, according to the present embodiment, the control object can be highly precisely state-feedback controlled by a simple model correction. 
     The error matrix Δ is set such that the elements of the state matrix influencing the rank of the controllable matrix U c * of the corrected state space model are changed. Therefore, the error is added to the elements serving as a cause of the rank deficiency. By such an element correction, the corrected state space model can be made controllable. 
     The nonlinear H-infinity controller  51  calculates the control input by applying the nonlinear H-infinity state feedback control to the generalized plant G designed based on the system represented by the corrected state space model. Thereby, the suspension apparatuses SP R , SP L  can be state-feedback controlled such that disturbance suppression and robust stabilization are improved. 
     As understood from the equation (eq. 27), the magnitude of the non-zero element of the error matrix Δ is 1/10 of the magnitude of the non-zero element of the state matrix A p . Therefore, the system represented by the corrected state space model can sufficiently obtain the controllability, and an influence rate of the error on the system is sufficiently reduced. In addition, the non-zero element of the error matrix Δ and the non-zero element of the state matrix are different from each other in terms of the number of digits. Thus, when the error is added to the state matrix A p , an addition element is prevented from being zero due to the setoff. This addition element is used for an element calculation of the controllable matrix U c *. Thus, since the addition element is not zero, the rank deficiency is not easily generated in the controllable matrix U c *. 
     Further, according to the present embodiment, the elements in the error matrix Δ not influencing the rank of the controllable matrix U c * of the corrected state space model are set to zero. By setting the elements not relating to the rank deficiency of the controllable matrix U c * to zero in such a way, the influence of the error matrix Δ on the system can be reduced, and the deviation between the corrected state space model and the original state space model of the control object can be more decreased. 
     In the present embodiment, the control object is consisted of the vibration system including the sprung member of the vehicle, the unsprung members, and the suspension apparatuses SP R , SP L  having the dampers and the springs interposed between the sprung member and the unsprung members. The above vibration system is controlled by controlling the damping forces of the suspension apparatuses SP R , SP L  by the micro computer  50 . Thereby, the riding quality of the vehicle is improved. 
     From the above embodiment, the following inventions can be proposed. 
     (1) A corrected state space model formed so as to represent a controllable system by adding an error matrix Δ to a state matrix of a state space model of a control object representing an uncontrollable system.
 
(2) A designing method of a state feedback controller for calculating a control input based on a state quantity of the system represented by a state space model, wherein the state feedback controller is designed by applying H-infinity control to a generalized plant designed based on a system represented by a corrected state space model formed so as to represent a controllable system by adding an error matrix Δ to a state matrix of a state space model of a control object representing an uncontrollable system.
 
(3) In the invention (1) or (2), a magnitude of a non-zero element of the error matrix Δ is 1/10 to 1/100 of a magnitude of a non-zero element of the state matrix.
 
     The present invention is not limited to the above embodiment. For example, the two wheel model of the vehicle is taken as an example in the above embodiment, and the state feedback control apparatus capable of obtaining two control inputs from one motion equation is disclosed. The present invention can be applied to control other than such state feedback control. For example, with using three motion equations relating to heave motion, pitch motion, and roll motion of an sprung member of the vehicle derived from a four wheel model of a vehicle, a corrected state space model can be formed so as to represent a controllable system by correcting a state space model representing an uncontrollable system. In this case, control inputs are set to variable damping coefficients of dampers respectively provided in four suspension apparatuses attached to front left and right portions and rear left and right portions of the sprung member. An error matrix Δ is added to a state matrix of the uncontrollable state space model which represents the four wheel model to design the corrected state space model which is controllable. Four control inputs are calculated from a state feedback controller obtained by applying the H-infinity control or the like to a generalized plant designed based on a system represented by the corrected state space model, and damping forces of the four suspension apparatuses can also be controlled based on the calculated inputs. In this case, the three motion equations serving as bases in designing of the state space model are for example represented by the following equation (eq. 48), and a control input u is represented by the following equation (eq. 49). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             Heave 
                              
                             
                               : 
                             
                              
                             
                                 
                             
                              
                             M 
                              
                             
                                 
                             
                              
                             
                               x 
                               ¨ 
                             
                           
                           = 
                           
                             
                               F 
                               
                                 f 
                                 r 
                               
                             
                             + 
                             
                               F 
                               
                                 f 
                                 l 
                               
                             
                             + 
                             
                               F 
                               rr 
                             
                             + 
                             
                               F 
                               rl 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Roll 
                              
                             
                               : 
                             
                              
                             
                                 
                             
                              
                             
                               I 
                               r 
                             
                              
                             
                               
                                 θ 
                                 ¨ 
                               
                               r 
                             
                           
                           = 
                           
                             
                               
                                 I 
                                 2 
                               
                                
                               
                                 
                                   T 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       F 
                                       fr 
                                     
                                     - 
                                     
                                       F 
                                       fl 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                 
                                   T 
                                   r 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       F 
                                       rr 
                                     
                                     - 
                                     
                                       F 
                                       rl 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Pitch 
                              
                             
                               : 
                             
                              
                             
                                 
                             
                              
                             
                               I 
                               p 
                             
                              
                             
                               
                                 θ 
                                 ¨ 
                               
                               p 
                             
                           
                           = 
                           
                             
                               I 
                               2 
                             
                              
                             
                               L 
                                
                               
                                 ( 
                                 
                                   
                                     F 
                                     fr 
                                   
                                   + 
                                   
                                     F 
                                     fl 
                                   
                                   - 
                                   
                                     F 
                                     rr 
                                   
                                   - 
                                   
                                     F 
                                     rl 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     48 
                   
                   ) 
                 
               
             
           
         
       
     
     wherein:
 
M: a mass of the sprung member;
 
x: a vertical displacement of the sprung member;
 
F fr : a vertical force acting on the right front side of the sprung member;
 
F fl : a vertical force acting on the left front side of the sprung member;
 
F rr : a vertical force acting on the right rear side of the sprung member;
 
F rl : a vertical force acting on the left rear side of the sprung member;
 
I r : roll inertia moment;
 
I p : pitch inertia moment;
 
L: a wheelbase;
 
θ r : a roll angle;
 
θ p : a pitch angle;
 
T f : a tread (front side); and
 
T r : a tread (rear side).
 
     
       
         
           
             
               
                 
                   u 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             C 
                             vfr 
                           
                         
                       
                       
                         
                           
                             C 
                             vfl 
                           
                         
                       
                       
                         
                           
                             C 
                             vrr 
                           
                         
                       
                       
                         
                           
                             C 
                             vrl 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     eq 
                     . 
                     
                         
                     
                      
                     49 
                   
                   ) 
                 
               
             
           
         
       
     
     wherein:
 
C vfr : a variable damping coefficient of a right side front damper;
 
C vfl : a variable damping coefficient of a left side front damper;
 
C vrr : a variable damping coefficient of a right side rear damper; and
 
     C vrl : a variable damping coefficient of a left side rear damper. 
     In the above embodiment, the error matrix Δ is added to the state matrix A p  as the additive error as shown in  FIG. 11 . However, the error matrix Δ may be added to the state matrix A p  as a multiplicative error as shown in  FIG. 13 . In this case, the corrected state matrix is represented as in the following equation (eq. 50). 
       CSM= A   p   +A   p Δ  (eq. 50)
 
     wherein: CSM denotes the corrected state matrix. 
     In this case, the error matrix is represented by A p Δ. 
     Although the present invention is described taking the damping force control of the suspension apparatuses of the vehicle as an example in the above embodiment, the present invention can be applied to other state feedback control. Although the present invention is described taking the nonlinear H-infinity state feedback control as an example in the present embodiment, the present invention may be applied to linear H-infinity state feedback control. Further, the present invention is applied to the control which is not H-infinity control. The present invention can be modified as long as the invention does not depart from the scope of the invention.