Abstract:
Method, article of manufacture and system using minimum data to determine whether a sliding-mode control should be applied in a plant. First measure the plant in an open-loop control fashion, and using the measured data, describe a state equation of the plant by system identification and order determination methods. Then design a switching hyperplane for sliding-mode control. Next, calculate higher order statistics on the difference between an output of a linear model on the hyperplane and an output of the sliding-mode control model in the measured data; When any of the higher order moments is larger than a predetermined threshold, configure a controller as a sum of the linear control input term and the nonlinear control input term. If both higher-order moments are smaller than the threshold, then configure the controller using only a linear control input term.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority under 35 U.S.C. §119 from Japanese Patent Application No. 2010-267280 filed Nov. 30, 2010, the entire contents of which are incorporated herein by reference. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a sliding-mode control scheme, and more particularly to a method, article of manufacture, and system for configuring a controller in the sliding-mode control scheme, including a technique to automatically determine whether or not a sliding-mode control should be applied from least minimum measured experimental data. 
     2. Description of Related Art 
     Conventionally, it is often the case that a controller is required to be sufficiently robust against non-linear characteristics and parameter variations/load variations included in a system. As one solution to cope with such requirement, a sliding-mode control (SMC) has been widely used. The SMC causes the state of a controlled object to reach a preset hyperplane and to be constrained thereto while switching the control input, and forces the state to slide to an equilibrium point thereby stabilizing the controlled object against uncertainty, nonlinearity, noises of parameters. 
     For example, in a position/attitude control of a satellite, a satellite has 6 degrees of freedom of movement of translations and rotations in a 3-dimensional space. Because of the many degrees of freedom, the control needs to be able to cope with variation uncertainties of a nonlinear equation for matching the attitude of the satellite to the attitude of an observation target and for controlling the position of the satellite so that it circles around an observation target while maintaining a constant distance therebetween. 
     Furthermore, along with the progress of exhaust emission regulation of engines, in order to accurately control an exhaust gas recirculation (EGR) mechanism and the like, a design by a plant model based on a feedback control of multi-input and multi-output system becomes necessary in place of a design based on open-loop control, so that a variable structure system control by a piecewise switching of control input is required. 
     However, it is often the case that the decision of the introduction of SMC for the targeted control relies on the model designer, and it is problematic that in order for such decision, the existence of nonlinearity needs to be verified based on a great deal of experimental results. Further, in SMC, while the nonlinearity is utilized in the control law for reaching a hyperplane, in a design which assumes an ideal, instantaneous switching of control-input switching, chattering will have occurred in the control input. To avoid that, although various smoothing has been used, it is also problematic that nonlinearity that defines space is not used. 
     While physical law description often results in nonlinearity, if, based on that, it is assumed that control strategy is also nonlinear to advance the design of the controller, extra cost will be spent on that. Moreover, the need for collecting a large number of experimental data for verifying the necessity of SMC has been a factor of increasing the cost. Further, it is problematic that in a reaching control law in which nonlinearity is not exploited, a sufficient control effect may not be obtained. 
     Japanese Patent Laid-Open No. 05-80805 discloses a technique for enabling the introduction of sliding-mode control and adaptive control by utilizing a conventional linear control technology, in which the phase surface of a sliding mode is represented by a form in which a torque command determined by conventional linear control is divided by an integration gain K 2 , and a Liapunov function is adapted to take into consideration estimated values of an inertia, a coefficient of dynamic friction and a gravity term of an object to be controlled, so that respective linear control gains Kp, K 1  and K 2  are determined so as to always make the Liapunov function negative and an auxiliary input is determined by changing the above described respective estimated values. 
     Japanese Patent Laid-Open No. 2003-15703 discloses that to provide a control apparatus of a plant capable of identifying a model parameter by modeling a plant to be controlled, and further stabilizing control at the time of executing sliding-mode control by using the identified model parameter, a model parameter identifier calculates a model parameter vector θ in a format in which an updated vector dθ is added to a reference vector θbase of the model parameter, corrects the updated vector dθ by multiplying the past value of at least one element of the updated vector dθ by a predetermined value which is larger than 0 and smaller than 1, and calculates the model parameter vector θ by adding the updated vector dθ after correction to the reference vector θbase. 
     Further, in the section 3.1 of the first edition of Kenzo Nonami and Hong-Qi Tian, “Sliding-mode Control (in Japanese)”, CORONA PUBLISHING CO LTD., Oct. 20, 1994, a design method of switching hyperplane is described, and in the section 3.2, a design method of sliding-mode controller is described. 
     However, although the above described references disclose a design technique for sliding-mode control, they do not suggest a technique for easily determining whether or not the sliding-mode control should be applied. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a technique to automatically determine whether or not a sliding-mode control should be applied in a plant from least minimum measured experimental data. 
     In one aspect, a method according to the present invention first measures the plant in an open-loop control fashion. Then, using the measured data, the method describes a state equation of the plant by an existent system identification method and an order determination method. Further, the method designs a switching hyperplane for sliding-mode control based on a state equation of the plant which has undergone system identification. 
     Next, the method calculates a higher order statistic on the difference between the output of a nominal model when constrained to the hyperplane (that is, the output when only a control input of a linear model is used), and the output when a control input of sliding-mode is used, where, particularly, a third-order moment and a fourth-order moment are calculated. Then, when the value of the third-order moment is larger than a predetermined threshold, or the value of the fourth-moment is larger than a predetermined threshold, the method determines that sliding-mode control is valid, and calculates nonlinear control input term that is updated at each sampling time by using the value of the fourth moment. Then, the method configures a controller as a sum of the linear control input term and the nonlinear control input term. 
     On one hand, when having determined that both of the value of the third-order moment and the value of the fourth-order moment are smaller than a predetermined threshold, the method configures a controller by using only a linear control input term. 
     In another aspect, an article of manufacture tangibly embodying computer readable non-transitory instructions first measures the plant in an open-loop control fashion. Then, using the measured data, the article describes a state equation of the plant by an existent system identification method and an order determination method. Further, the article of manufacture having the instructions designs a switching hyperplane for sliding-mode control based on a state equation of the plant which has undergone system identification. 
     Next, the article of manufacture allows calculation of a higher order statistic on the difference between the output of a nominal model when constrained to the hyperplane (that is, the output when only a control input of a linear model is used), and the output when a control input of sliding-mode is used, where, particularly, a third-order moment and a fourth-order moment are calculated. Then, when the value of the third-order moment is larger than a predetermined threshold, or the value of the fourth-moment is larger than a predetermined threshold, the article determines that sliding-mode control is valid, and calculates nonlinear control input term that is updated at each sampling time by using the value of the fourth moment. Then, the article configures a controller as a sum of the linear control input term and the nonlinear control input term. 
     On one hand, when having determined that both of the value of the third-order moment and the value of the fourth-order moment are smaller than a predetermined threshold, the article of manufacture configures a controller by using only a linear control input term. 
     In a further aspect, a system according to the present invention first measures the plant in an open-loop control fashion. Then, using the measured data, the system describes a state equation of the plant by an existent system identification method and an order determination method. Further, the system designs a switching hyperplane for sliding-mode control based on a state equation of the plant which has undergone system identification. 
     Next, the system calculates a higher order statistic on the difference between the output of a nominal model when constrained to the hyperplane (that is, the output when only a control input of a linear model is used), and the output when a control input of sliding-mode is used, where, particularly, a third-order moment and a fourth-order moment are calculated. Then, when the value of the third-order moment is larger than a predetermined threshold, or the value of the fourth-moment is larger than a predetermined threshold, the system determines that sliding-mode control is valid, and calculates nonlinear control input term that is updated at each sampling time by using the value of the fourth moment. Then, the system configures a controller as a sum of the linear control input term and the nonlinear control input term. 
     On one hand, when having determined that both of the value of the third-order moment and the value of the fourth-order moment are smaller than a predetermined threshold, the system configures a controller by using only a linear control input term. 
     In an advantage according to the present invention, because the validity of the introduction of sliding-mode control can be determined from least minimum measured data, the need for repeating data collection experiments is obviated. Moreover, it becomes possible to design a controller more appropriate than conventional one based on a statistic which is utilized when the above described determination is made. 
     A further advantage of the present invention is it becomes possible to avoid chattering and perform stable control for an accidental disturbance with an inexpensive controller. The advantage holds even for a system such as a satellite, in which it is difficult to install a high-speed switching function, etc. . . . 
     Other characteristics and advantages of the invention will become obvious in combination with the description of accompanying drawings, wherein the same number represents the same or similar parts in all figures. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  illustrates a schematic block diagram of an embodiment of the computer to be used in the present invention; 
         FIG. 2  illustrates a functional block diagram of a program etc. for executing the processing to determine a controller according to an embodiment of the present invention; 
         FIG. 3  shows an exemplary flowchart for executing the processing to determine a controller according to an embodiment of the present invention; and 
         FIG. 4  shows an exemplary relationship between a plant and a controller according to an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Hereafter, according to the drawings, embodiments of the present invention will be described. It is to be understood that these embodiments are for the sake of explaining preferable modes of the present invention, and the scope of the present invention is not intended to be limited to what is shown here. Moreover, throughout the drawings shown below, unless otherwise stated, like reference symbols refer to the like objects. 
     The basic principle of the invention is to provide a technique to automatically determine whether or not a sliding-mode control should be applied in a plant by using least amount of measured experimental data. Detailed description of the invention is made in combination with the following embodiments. 
     Referring to  FIG. 1 , a block diagram of hardware of a control computer for controlling a controller  118  according to an embodiment of the present invention is shown. In  FIG. 1 , a CPU  104 , a main memory (RAM)  106 , a hard disk drive (HDD)  108 , a keyboard  110 , a mouse  112 , and a display  114  are connected to a system bus  102 . The CPU  104 , which is preferably based on a 32-bit or 64-bit architecture, can utilize, for example, PENTIUM™ 4 of Intel Corporation, Core™ 2 DUO of Intel Corporation, Athlon™ of Advance Micro Devices, and the like. The main memory  106  preferably has a capacity of 2 GB or more, and more preferably a capacity of 4 GB or more. 
     In the hard disk drive  108 , although not shown individually, operating system and processing programs, etc. relating to the present invention are prestored. The operating system can be any one that conforms to the CPU  104 , such as Linux™; Windows™ 7, Windows XP™, and Windows™ 2000 of Microsoft Corporation; and Mac OS™ of Apple Inc., etc. 
     The keyboard  110  and the mouse  112  are used to manipulate graphic objects such as icons, a task bar, a window, etc. which are displayed on the display  114  according to a graphic user interface supported by the operating system. The keyboard  110  and the mouse  112  are also used to operate a data recording program to be described below. 
     The display  114  is preferably, though not limited to, an LCD monitor having a resolution of 1024×768 or more with 32 bit true color. The display  114  is used for displaying, for example, a waveform of the operation of the plant  120  through the control by the controller  118 . 
     Further, the hard disk drive  108  prestores a measurement module  204 , a system identification module  206 , a higher-order statistic calculation module  208 , a nonlinear control input term calculation module  210 , a controller configuration module  212 , and a main program  202  (i.e. computer readable instructions) that integrates the whole processing, which are to be described below. These modules can be created by using any existing programming language such as C, C++, C#, Java®, etc. These modules are appropriately loaded onto the main memory  106  and executed by the working of the operating system. Although not shown, the main program  202  can display a window that is to be operated by the operator on the display  114 , so that the operator can start or stop the processing by using the keyboard  110  and the mouse  112 , etc. 
     The controller  118  and the plant  120  are connected to the bus  102  via an appropriate interface board  116 . The controller  118  has a register in which a control input obtained from the result of calculation can be set through the interface board  116 . 
     The measurement module  204  has a function of supplying a series of signals to the plant  120  via the bus  102  and the interface board  116 , and measuring responses thereto. 
     The system identification module  206  uses, although not limited to, for example, a system identification method which uses a state equation in its parametric model. In the case of a parametric model, model order determination processing is also performed collectively as a structure determination step. Next, processing such as estimating parameters by a prediction error method, and estimating a state space model by a subspace method, etc. . . are performed. As the system identification module  206 , although not limited to, the System Identification Toolbox of MATLAB® can be used. The system identification module  206  generates a state space equation as a result of system identification. For details of system identification processing, refer to, for example, Shuichi Adachi, “Foundation of system identification (in Japanese)”, TOKYO DENKI UNIVERSITY, Sep. 10, 2009. 
     Further, in this embodiment, the system identification module  206  designs a formula of a switching hyperplane for sliding-mode control from the generated state space equation. It is noted that the design of a switching hyperplane for sliding-mode control can be referred to Kenzo Nonami and Hong-Qi Tian, “Sliding-mode Control (in Japanese)”, section 3.1, CORONA PUBLISHING CO LTD., Oct. 20, 1994 and so on. 
     The higher-order statistic calculation module  208  has a function of calculating higher order statistics, particularly a third-order moment and a fourth-order moment of the output signal of the plant  120 . 
     The nonlinear control input term calculation module  210  has a function of calculating a nonlinear control input term peculiar to sliding-mode control at each sampling time. During this calculation, the value of fourth-order moment calculated at the higher-order statistic calculation module  208  is used. 
     The controller configuration module  212  has a function of configuring the controller  118  by using only the control input term based on a nominal model resulted from system identification, that is, a linear model when constrained to a hyperplane, or both of the control input term based on the linear model when constrained to the hyperplane and the nonlinear control input term. 
     The plant  120  can be a mechanical apparatus to be controlled, such as an engine, brake and an air conditioner of automobile, a satellite, and the like. 
     The plant  120  is typically a real machine, but may be a software-type model created by a simulation modeling tool such as Simulink®, Modelica, and the like. When the plant  120  is a software-type model, it will be a module stored in the hard disk drive  108 , and accordingly interfacing via the interface board  116  is unnecessary since the controller  118  will be a module stored in the hard disk drive  108  as well. 
     Next, with reference to the flowchart in  FIG. 3 , details of the processing executed by a functional logic configuration of  FIG. 2  will be described. 
     In step  302 , the main program  202  calls the measurement module  204 , and performs an open-loop measurement for the plant  120  by supplying measured signals at a predetermined sampling period. In the case of a closed loop measurement, since it is difficult to grasp a cause and effect relationship for a signal input, an open-loop measurement is preferable from the stand point of simply grasping a response output corresponding to a measured signal. 
     When response outputs are accumulated in this way, the main program  202  calls the system identification module  206  to identify a state space equation of the plant  120  preferably by using a parametric model and then an order-determination technique. The system identification module  206  designs a hyperplane for sliding-mode control from the thus obtained state space equation of the plant  120 . Thus, a model described by the following formulas is obtained:
 
 {dot over (X)}=AX+BU  
 
Y=CX
 
σ=SX
 
     Where, X in input, Y is output, U is control input, and A, B, and C determined by system identification are a coefficient matrix. Further, σ is a hyperplane, and S is a matrix to define the switching hyperplane determined based on A, B, and C. 
     Here, the main program  202  calls the higher-order statistic calculation module  208  to calculate a data observation sequence {S}=y tempSMC −y 1inear  when letting the output of the nominal mode, that is, a linear model when constrained to the hyperplane be y 1inear , and the output of a temporary sliding-mode control model be y tempSMC . 
     Further, the main program  202  calculates an average value of the data observation sequence {S}, letting it be 
     
       s 
     
     and calculates a data sequence {x} whose average value is made zero as shown below.
 
 x ( j )= s ( j )−   s ,j= 1 , . . . , n  
 
     Then, the higher-order statistic calculation module  208  calculates a third-order moment:
 
 c   3 ( d   1   ,d   2 )= E[x ( n ) x ( n+d   1 ) x ( n+d   2 )]
 
and a fourth-order moment:
 
 c   4 ( d   1   ,d   2   ,d   3 )= E[x ( n ) x ( n+d   1 ) x ( n+d   2 ) x ( n+d   3   ]−E[x ( n ) x ( n+d   1 )] E[x ( n+d   2 ) x ( n+d   3   ]−E[x ( n ) x ( n+d   2 )] E[x ( n+d   1 ) x ( n+d   3   ]−E[x ( n ) x ( n+d   3 )] E[x ( n+d   1 ) x ( n+d   2 ]
 
by using the data sequence {x}. Where, E[ ] is an expected value, and d 1 , d 2 , and d 3  are lags or delays. The values of d 1 , d 2 , and d 3  are appropriately determined in advance, and for example, they can be such that d 1 =d 2 =d 3 =0.
 
     The main program  202  determines a validity of the sliding-mode control model using thus calculated values of the third-order moment and the fourth-order moment, in step  308 . 
     That is, the main program  202  predetermines a threshold Th 3  for the value c 3  of the third-order moment, and a threshold Th 4  for the value c 4  of the fourth-order moment, and when determining that |c 3 |&gt;Th 3  or |C 4 |&gt;Th 4 , the main program  202  determines that using the sliding-mode control model is valid. On one hand, when |c 3 |&lt;=Th 3  and |c 4 |&lt;=Th 4 , the main program  202  determines that using the sliding-mode control model is not valid. 
     When the main program  202  determines that using the sliding-mode control model is not valid in step  310 , the process moves to step  316 , and the main program  202  calls the controller configuration module  212  to configure the controller  118  with a control input of the linear model when constrained to the hyperplane, as shown in the following formula.
 
 u   linear =−( SB ) −1 ( SAX )
 
     When the main program  202  determines that using the sliding-mode control model is valid in step  310 , the process moves to step  312 , and the main program  202  calls the nonlinear control input term calculation module  210  to calculate a coefficient k t  of the control input whose coefficient k is updated at each sampling time t according to the following formula, by using the fourth-order moment calculated in step  306 .
 
 k   t   =k   t-1 −μ{1+(−sign(κ 4 )tan  h (∥ X ∥)−∥ X ∥)∥ X∥}k   t-1  
 
     Where, μ is a step constant, for example, 0.17. The sign( ) is a function to return a sign. Moreover, κ 4  is the fourth-order moment calculated in step  306 , and in one example, κ 4 =c 4 (0, 0, 0). This formula is a recurrence formula, and k t  is established at the end of sampling time so that the value is used as k for nonlinear control input term. 
     It is noted that the coefficient k t  may not be calculated at each sampling time t, and the number of sampling times t can be decimated to calculate the coefficient as follows:
 
 k   m   =k   m-1 −μ{1+(−sign(κ 4 )tan  h (∥ X ∥)−∥ X ∥)∥ X∥}k   m-1   m←m+ 1 if mod( t, 3)=1
 
     Such a formula is decided under the consideration that since the characteristic of space leading to the hyperplane has a bias in terms of higher-order moment, controlling the approach to the hyperplane via a gradient that takes the bias into consideration will make chattering less likely to occur. This also has a meaning that it is an updating formula to approximate a partial differential of negentropy. To add further description, this is a formula based on the result of consideration that deviation of the sliding-mode control model from a nominal model corresponds to bringing the difference between the entropy of Gaussian distribution and the entropy that the norm of state quantity X closer to zero. As a result of this, the nonlinear input term is determined as follows: 
     
       
         
           
             
               u 
               nonlinear 
             
             = 
             
               
                 - 
                 
                   
                     
                       k 
                       t 
                     
                     ⁡ 
                     
                       ( 
                       SB 
                       ) 
                     
                   
                   
                     - 
                     1 
                   
                 
               
               ⁢ 
               
                 σ 
                 
                    
                   σ 
                    
                 
               
             
           
         
       
     
     The main program  202 , in step  314 , calls the controller configuration module  212 , and configures the controller  118  with the sum of the control input term of the linear model when constrained to the hyperplane, and the nonlinear control input term calculated in step  312 , as shown in the following formula: 
     
       
         
           
             u 
             = 
             
               
                 
                   u 
                   linear 
                 
                 + 
                 
                   u 
                   nonlinear 
                 
               
               = 
               
                 
                   
                     - 
                     
                       
                         ( 
                         SB 
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     SAX 
                     ) 
                   
                 
                 - 
                 
                   
                     
                       
                         k 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         SB 
                         ) 
                       
                     
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     σ 
                     
                        
                       σ 
                        
                     
                   
                 
               
             
           
         
       
     
       FIG. 4  is a schematic diagram to show the situation where a plant is controlled by the controller  118  which is configured as described above. The control input u of the controller  118  is fed back to the plant  120 . 
     While the present invention has been described with reference to what are presently considered to be the preferred embodiments, it is to be understood that the invention is not limited to the disclosed embodiments. On the contrary, the invention is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions. In particular, the present invention will not be limited to a specific platform of hardware and software etc., and can be implemented on any platform.