Patent Publication Number: US-6658338-B2

Title: Enhanced ratio control in a toroidal drive

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a method and a system for enhanced ratio control in a toroidal drive. 
     2. Description of the Background Art 
     Continuously variable transmissions (CVT&#39;s) are transmissions that change a speed ratio continuously, not in discrete intervals. This continuous nature of CVT&#39;s gives them an infinite number of speed ratios, making them very attractive for automotive use. 
     Various types of CVT are known. One such example is a CVT with pulley/V-belt power transfer. Another example is a CVT with disc/roller power transfer. The CVT of this type is often referred to as a toroidal-type CVT (T-CVT) because it transmits torque from one rotating semi-toroidal disc to another semi-toroidal disc by traction rollers through a traction force. The two semi-toroidal discs form a toroidal cavity. In each toroidal cavity, it is preferred to have two traction rollers in equiangularly spaced relationship engaging the discs for transmission of motion therebetween. While three or four traction rollers may be disposed in spaced relationship in each toroidal cavity and will provide increased life for contact surfaces as the total surface area is increased, two traction rollers are preferred for simplicity. 
     Each traction roller is rotatably supported by a pivot trunnion, respectively. The pivot trunnions, in turn, are supported to pivot about their respective pivot axis. In order to controllably pivot the pivot trunnions for a ratio change, a hydraulic control means is provided. The hydraulic control means is included in a hydraulic cylinder at each pivot trunnion and includes a control volume defined in the hydraulic cylinder between a piston and an axial end of the hydraulic cylinder The pistons within the hydraulic cylinders are connected to the pivot trunnions along their pivot axis by rods. The piston and its associated rod are thereby rotatable about the pivot axis with the associated pivot trunnion. Variation of the control volume causes the piston to move relative to the hydraulic cylinder, and applies a control force to displace the pivot trunnions. Control forces applied displace the pivot trunnions in the opposite directions along their pivot axis. As a result, the pivot trunnions are caused to pivot about their respective pivot axis, due to the forces present in the rotating toroidal discs, for initiating ratio change. 
     For terminating the ratio change when a desired ratio has been obtained, a feedback structure is provided. The feedback structure preferably includes a source of hydraulic pressure, and a ratio control valve for controlling the flow of hydraulic fluid for initiating ratio change. The feedback structure further includes a mechanism associated with at least one pivot trunnion to adjust the ratio control valve upon pivotal movement of the pivot trunnion to a desired ratio. The mechanism is preferably a cam connected to a pivot trunnion. The cam may be linked mechanically and/or electronically to operate the ratio control valve upon reaching a desired rotation. 
     Various ratio control strategies have been proposed. One such example is proposed by the assignee of the present invention in U.S. Pat. No. 5,669,845 (=JP-A 8-270772) issued Sep. 23, 1997 to Muramoto et al. According to this known control strategy, a feedback structure includes a source of hydraulic pressure, a ratio control valve, a bell crank, and a cam. The ratio control valve has a valve sleeve connected to a stepper motor. The ratio control valve further has a valve spool disposed within the valve sleeve. The valve spool has a rod projecting out of the valve sleeve for engagement with the bell crank. The bell crank is connected to the rod at one end. At the other end, the bell crank engages the cam connected to a pivot trunnion. At a middle point between the two ends, the bell crank is supported to pivot about the middle point. 
     The valve sleeve is positionable in response to an actuator command from a T-CVT controller to establish various speed ratios between input and output shafts of the T-CVT. The actuator command is indicative of motor steps of the stepper motor. The axial displacement of the valve sleeve has one-to-one and onto any selected number of motor steps. 
     To compute the number of motor steps, the T-CVT controller determines a desired engine or input shaft speed against vehicle speed and throttle position using a look-up table map. The desired input shaft speed is used in cooperation with actual output shaft speed to determine a desired ratio. Using a predetermined relationship, the T-CVT controller determines a desired trunnion angular position. Using the desired trunnion angular position, the T-CVT controller computes a feedforward term and a feedback term by carrying out proportional and integral control actions. Besides, the T-CVT controller computes a damping term using an estimated value of trunnion axial displacement given by a state observer. Combining the feedforward, feedback and damping terms gives the motor steps. 
     This known ratio control is satisfactory to some extent. As far as the inventors are aware of, huge amount of computer simulation and field test would be needed in designing such a T-CVT controller to ensure quick reduction of error in estimation, if occurred, by state observer, requiring increased cost and time in developing a desired control system. 
     Accordingly, a need remains for enhanced ratio control in a toroidal drive of a T-CVT, which does not require increased cost and time in developing a desired control system. 
     An object of the present invention is to provide a method and a system for enhanced ratio control in a toroidal drive of a T-CVT to meet the above-mentioned need. 
     SUMMARY OF THE INVENTION 
     According to one aspect of the present invention, a method for enhanced ratio control in a toroidal drive of a toroidal-type continuously variable transmission (T-CVT) is provided. The T-CVT includes a ratio control element positionable in response to an actuator command to establish various ratios between input and output shaft speeds of the T-CVT. The toroidal drive has toroidal discs defining a toroidal cavity, and traction roller assemblies having pivot trunnions rotatably supporting traction rollers disposed in the toroidal cavity and engaged between the toroidal discs. The method comprises: 
     computing a factor of proportionality by which a first physical quantity and a second physical quantity are related, 
     the first physical quantity being a trunnion axial displacement of a predetermined one of the pivot trunnions, the second physical quantity being indicative of a ratio rate of the ratio between the input and output shaft speeds of the T-CVT; 
     establishing a filter in the form of a characteristic equation that includes a third physical quantity and a fourth physical quantity, as inputs, a quasi-state quantity, as a state quantity, and coefficients including a transition coefficient for the quasi-state quantity, the transition coefficient including an observer gain, 
     the third physical quantity being indicative of the ratio between the input and output shaft speeds of the T-CVT, the fourth physical quantity being indicated by the actuator command; 
     computing the quasi-state quantity using the filter; 
     computing an estimated quantity of a system state quantity of the T-CVT using the quasi-state quantity, the observer gain, and a fifth physical quantity indicative of a trunnion angular position of the predetermined pivot trunnion, the system state quantity including at least the first physical quantity; and 
     correcting the observer gain in response to the factor of proportionality to keep the transition coefficient unaltered. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Further objects and advantages of the invention will be apparent from reading of the following description in conjunction with the accompanying drawings. 
     FIG. 1 is a schematic top view of a traction drive system, in the form of a T-CVT, including a dual cavity toroidal drive, a planetary drive, and a hydraulic drive. 
     FIG. 2 is a schematic side view of a pair of traction roller assemblies disposed in the rearward cavity of the toroidal drive of FIG. 1 and a schematic representation of a pressure control for a traction drive. 
     FIG. 3 is a block diagram showing the relationship between a T-CVT controller and an actuator. 
     FIG. 4 is a control diagram of a T-CVT controller according to the present invention. 
     FIG. 5 is a T-CVT map used to determine a desired engine speed command (ω e *) appropriate for a selected set of operating variables including vehicle speed (VSP) and accelerator pedal position (APS). 
     FIG. 6 is a graphical representation of a ratio vs., trunnion angular position characteristic, which may be used, as a conversion map, to relate a desired ratio command (ic*) and a desired trunnion angular position command (φ*). 
     FIG. 7 is a graphical representation of a trunnion angular position vs., motor steps characteristic, which may be used, as a conversion map, to relate a desired trunnion angular position command (φ*) and a motor steps command (u FF ) that is indicated by the actuator command. 
     FIG. 8A is a graphical representation of a proportional gain (K P ) versus factor of proportionality (f) characteristic. 
     FIG. 8B is a graphical representation of an integral gain (K I ) versus factor of proportionality (f) characteristic. 
     FIG. 8C is a graphical representation of a D gain versus factor of proportionality (f) characteristic. 
     FIG. 9 is a control diagram of a state observer according to the present invention. 
     FIGS. 10 to  15  are flow diagrams implementing the present invention. 
     FIG. 16 is a control diagram of another T-CVT controller according to the present invention. 
     FIG. 17 is a control diagram of another state observer according to the present invention. 
     FIGS. 18A is a simplified view of a portion of FIG. 16 illustrating the situation where there occurs a continuous difference (Δu), in stepper motor, between commanded motor steps (u) and actual motor steps (u P ). 
     FIG. 18B depicts timing diagrams of estimated values (y m  &amp; u m ) upon and after the occurrence of such a continuous difference (φ*) during a time period T1-T2. 
     FIGS. 19 to  24  are flow diagrams implementing the present invention. 
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     Referring to FIGS. 1 and 2, a T-CVT  10  includes a dual cavity toroidal drive  12  coaxially connected to a forward positioned input gear section  14  and connected also to a rearward positioned output gear section  16 . For purpose of clarification, the terms front or forward refer to the right side, and rear or rearward refer to the left side of the view shown in FIG.  1 . All three elements  12 ,  14  and  16  are enclosed in a housing  18  and driven by an input or turbine shaft  20  that is powered by an engine (not shown) through a torque converter  22  or a lock-up clutch  24 . Housing  18  has three chambers, one for each element  12 ,  14  and  16 , separated by walls  26  and  28 . Torque converter  22  is a conventional torque converter including a pump impeller  30  connected to the engine, a turbine runner  32  connected to input shaft  20 , and a stator  34  grounded via a one-way brake  36 . Lock-up clutch  24  is a conventional lock-up clutch including a clutch element  38  connected to input shaft  20 . 
     Dual cavity toroidal drive  12  includes first and second outboard traction discs  40  and  42  mounted on a torque sleeve  44  via ball splines  46  and  48  to rotate in unison. Toroidal drive  12  further includes two inboard traction discs  50  and  52 , which are positioned back-to-back and rotatably mounted on sleeve  44 , and both coupled to an output gear  54  to rotate in unison. Two inboard traction discs  50  and  52  may be formed as one integral element formed with output gear  54 . In this case, the integral element is a dual-faced single disc element rotatably supported by sleeve  44 . One example of a dual cavity toroidal drive having dual inboard discs positioned back-to-back is disclosed in U.S. Pat. No. 5,902,208 issued May 11, 1999 to Nakano, which is incorporated in its entirety herein by reference. Another example of a dual cavity toroidal drive having a dual-faced single disc element formed with an output gear is disclosed in co-pending U.S. patent application Ser. No. 09/940,875 commonly assigned herewith, which is incorporated in its entirety herein by reference. A toroidal cavity is defined between each outboard discs  40  and  42  and one of the inboard discs  50  and  52 . A pair of motion transmitting traction rollers  56 ,  58 ,  60  and  62  is disposed in each toroidal cavity, with one roller being disposed transversely on either side of each cavity (see FIGS.  1  and  2 ). Each pair of traction rollers  56 ,  58 ,  60  and  62  are mirror images of the other pair; therefore, only the one pair of rollers  60  and  62  are illustrated in FIG.  2 . Each pair of traction rollers  56 ,  58 ,  60  and  62  are engaged between each outboard discs  40  and  42  and one of the inboard discs  50  and  52  in circles of varying diameters depending on the transmission ratio. Traction rollers  56 ,  58 ,  60  and  62  are so supportive that they can be moved to initiate a change in the ratio. That is, each roller  56 ,  58 ,  60  and  62  can be actuated to vary its diameter and provide a substantial normal force at their points of contact with the corresponding discs to sufficiently support the traction forces needed to effect the change in ratio. With outboard discs  40  and  42  being rotated continuously by the engine, outboard discs  40  and  42  impinge on traction rollers  56 ,  58 ,  60  and  62 , causing the traction rollers to rotate. As they rotate, the traction rollers impinge on and rotate inboard discs  50  and  52  in a direction opposite to that of rotating outboard discs  40  and  42 . The structure and operation of the other elements of the toroidal drive  12  will be discussed later on in the specification, 
     With continuing reference to FIG. 1, toroidal drive  12  employs a cam loading system to control normal force between toroidal discs ( 50 ,  52 ,  60 ,  62 ) and traction rollers ( 56 ,  58 ,  60 ,  62 ). The cam loading system operates on outboard discs  40  and  42  to apply an axial force that is a linear function of the input torque. Describing, in detail, the cam loading system, torque sleeve  44  extends beyond the backs of outboard discs  40  and  42  and has flanges (not shown) at its front and rear ends to carry thrust bearings  64  and  66 . The cam loading system includes a disc spring (Belleville spring)  68 , which is supported on torque sleeve  44  between thrust bearing  66  and the back of toroidal disc  42  to operate on the disc. The cam loading system also includes a drive plate  70  rotatably supported by torque sleeve  44  via thrust bearing  64 . The cam loading system further includes cam rollers  72 , which are disposed between drive plate  70  and toroidal disc  40 . An example of a cam loading system having cam rollers between a drive plate and one of outboard toroidal discs is disclosed in U.S. Pat. No. 5,027,668 issued Jul. 2, 1991 to Nakano, which is incorporated in its entirety herein by reference. 
     Drive plate  70  of the cam loading system is drivingly connected to input shaft  20  through input gear section  14 . Input gear section  14  includes a dual-pinion planetary gear system (DPGS)  74 , a forward clutch  76 , and a reverse brake  78 . DPGS  74  includes, in a conventional manner, a sun gear  80 , a carrier  82 , a ring gear  84 , and a plurality of pairs of intermeshed planet pinions  86  and  88  rotatably supported by pins of carrier  82 . Pinions  86  and  88  are disposed between sun and ring gears  80  and  84 , with inner pinions  86  in engagement with sun gear  80  and outer pinions  88  in engagement with ring gear  84 . Sun gear  80  is coupled with input shaft  20  to rotate in unison. Carrier  82  is connected to drive plate  70  of the cam loading system for rotation in unison. Carrier  82  is connectable to input shaft  20  through forward clutch  76 . Ring gear  84  is connectable to housing  18  through reverse brake  78 . 
     Input gear section  14  including DPGS  74  functions to establish torque transmission in forward drive mode or reverse drive mode. In the forward drive mode, forward clutch  76  is engaged with reverse brake  78  released. In the reverse drive mode, reverse brake  78  is applied with forward clutch  76  disengaged. In this manner, input torque is applied to drive plate  70  to continuously rotate outboard toroidal discs  40  and  42  in the same direction as that of input shaft  20  in the forward drive mode, but in a direction opposite to that of input shaft  20  in the reverse drive mode. The input torque is transmitted from outboard discs  40  and  42  to inboard discs  50  and  52  to rotate output gear  54 . 
     Output gear section  16  including an input gear  90  of a counter shaft  92  functions to provide torque transmission from output gear  54 . Output gear  54  is in engagement with input gear  90  of counter shaft  92 , which has an output gear  94 . 
     Output gear section  16  also includes a gear  96  of an output shaft  98 . Output gear section  16  may include an idler gear (not shown) between output gear  94  and gear  96 . Rotation of inboard toroidal discs  50  and  52  is transmitted via output gear  54 , gear  90 , counter shaft  92 , gear  94  and gear  96  to output shaft  98 . 
     Referring to FIG. 2, toroidal drive  12  in this embodiment includes two traction rollers  60 ,  62  in each toroidal cavity. Each of the rollers  60 ,  62  is rotatably supported by a pivot trunnion  100 ,  102 , respectively. Pivot trunnions  100 ,  102 , in turn, are supported to pivot about their respective pivot axis  104 ,  106 . Each of traction rollers  60 ,  62  and the corresponding pivot trunnion  100 ,  102  are components of traction roller assemblies  108 ,  110 . 
     As is well known to those skilled in the art, the surfaces of toroidal discs  40 ,  42 ,  50 ,  52  defining cavities have a radius of curvature, the origin of which coincides with the pivot axis  104 ,  106 . This geometry permits the pivot trunnions and traction rollers to pivot and maintain contact with the surfaces of the toroidal discs. 
     Traction roller assemblies  108 ,  110  each also include a hydraulic piston assembly  112 ,  114  in addition to the pivot trunnion  100 ,  102 . Pivot trunnions  100 ,  102  each have a backing plate  116 ,  118  that supports traction roller  60 ,  62  rotatably. Bearings  120 ,  122 , positioned between plate  116 ,  118  and traction roller  60 ,  62 , permit relative rotation between backing plate  116 ,  118  and traction roller  60 ,  62 . Backing plates  116 ,  118  each have an extension  124 ,  126  that supports traction roller  60 ,  62 , on a bearing not shown, for rotation. 
     Hydraulic piston assembly  112 ,  114  includes a housing  128 ,  130  enclosing a cylinder  132 ,  134  in which is slidably disposed a piston and rod  136 ,  138 . Piston and rod  136 ,  138  divides cylinder  132 ,  134  into equal area chambers including a first chamber  140 ,  142  and a second chamber  144 ,  146 . Piston and rod  136 ,  138  is disposed so that its centerline  148 ,  150  is disposed substantially along pivot axis  104 ,  106 , respectively. So positioned, piston and rod  136 ,  138  is able to pivot about pivot axis  104 ,  106  with pivot trunnion  100 ,  102 , respectively. 
     The pressure in first and second chambers  140 ,  142 ;  144 ,  146  is established by a hydraulic control system  152 . Hydraulic control system  152  includes a pump, not shown, an electro-hydraulic control  154 , a ratio control valve  156 , and a feedback structure  158 . The pump is a conventional pump that draws hydraulic fluid from a reservoir  160  and delivers the fluid to electro-hydraulic control  154  from which the fluid is delivered to ratio control valve  156 . 
     Control  154  delivers system (or line) pressure to a passage  162  that is connected to an inlet port  164  of ratio control valve  156 . Ratio control valve  156  has a spool  166  slidably disposed in a valve bore  168 . Valve bore  168  is in fluid communication with passage  162  via inlet port  164 . Valve bore  168  is also in fluid communication with a first control passage  170  via a first control port  172 , and with a second control passage  174  via a second control port  176 . Valve bore  168  is further in fluid communication with a first drain passage  178  via a first drain port  180 , and with a second drain passage  182  via a second drain port  184 . 
     Spool  166  is connected to a feedback lever  186 , which is a component of feedback structure  158 . A ratio actuator  188 , in the form of a stepper motor, for example, receives a control signal. The control signal is an actuator command indicative of motor steps if a stepper motor is used as the actuator. In response to the control signal, actuator  188  moves feedback lever  186 , connected to an actuator shaft  190 , to initiate the ratio change in toroidal drive  12 . Feedback lever  186  is connected to actuator shaft  190  at one end and to a bell crank  192  at the other end. At a point between the two ends, the feedback lever  186  is pivotally connected to a spool rod  194 , which is connected to spool  166  to move in unison. Bell crank  192  has one end  196  pivotally connected to the other end of feedback lever  186  and the other end  198 . The other end  198  of bell crank  192  is controlled by the angular position about pivot axis  104  of traction roller assembly  108  through contact with a cam  200  formed on piston and rod  136 . As actuator  188  moves feedback lever  186 , valve  156 , in response to movement of valve rod  194 , alters the hydraulic pressure in lines  170  and  174 . Hydraulic pressure is provided to the valve  156  through line  162 , which is supplied with system or line pressure. As the pressure in lines  170  and  174  is altered, traction roller assemblies  108  and  110  move along pivot axis  104  and  106  in the opposite directions and then pivot about pivot axis  104  and  106 , changing the ratio in toroidal drive  12 . As traction roller assembly  108  pivots, lever  186  moves, due to rotation of cam  200  and movement of bell crank  192 , repositioning valve rod  194 , providing means for valve  156  to reinstate the pressure in lines  170  and  174  to stop traction roller assemblies  108  and  110  from pivoting. 
     Actuator  188  controls displacement of actuator shaft  190 , which, in turn, controls the ratio in toroidal drive  12 . If actuator  188  is in the form of a stepper motor, as is in exemplary embodiments of the present invention, controlling angular displacement of stepper motor in terms of motor steps controls the ratio in toroidal drive  12 . 
     As shown in FIG. 1, the speed of the input shaft  20  is detected by an input speed sensor  202  and fed, as an input shaft speed signal ω i , to a T-CVT controller  210  (see FIG.  3 ). The speed of the output shaft  98  is detected by an output speed sensor  204  and fed, as an output shaft speed signal ω o , to the controller  210 . An accelerator pedal position signal APS from an accelerator or gas pedal (not shown) is also fed to the controller  210 . 
     With reference now to FIG. 3, in an exemplary embodiment of the present invention, the T-CVT controller  210  comprises a microprocessor-based controller with an associated microprocessor, represented by a microprocessor  212 . The microprocessor  212  communicates with associated computer-readable storage media  214 . As will be appreciated by one of ordinary skilled in the art, the computer-readable storage media  214  may include various devices for storing data representing instructions executable by the microprocessor to control the T-CVT  10 . For example, the computer-readable storage media  214  may include a random access memory (RAM)  216 , a read-only memory (ROM)  218 , and/or a keep-alive memory (KAM)  220 . These functions may be carried out through any of a number of known physical devices including EPROM, EEPROM, flash memory, and the like. The present invention is not limited to a particular type of computer-readable storage medium, examples of which are provided for convenience of description only. 
     Controller  210  may also include appropriate electronic circuitry, integrated circuits, and the like to carry out control of the T-CVT  10 . As such, controller  210  is used to carry out control logic implemented in terms of software (instructions) and/or hardware components, depending upon the particular application. Additional details of control logic implemented by controller  210  are provided with reference to FIGS. 4 to  10 . 
     Controller  210  receives various signals to monitor driver power or torque demand, and operating conditions of T-CVT  10 . The various signals include accelerator pedal position signal APS on line  222 , output shaft speed signal ω o  on line  224 , and input shaft speed signal ω i  on line  226 . Controller  210  may communicate directly with various sensors including input shaft speed sensor  202  and output shaft speed sensor  204 . 
     Controller  210  processes the signals to determine motor steps u and generates an actuator command indicative of the determined motor steps u. Controller  210  applies the actuator command to actuator  188  via line  228 . 
     An exemplary embodiment of the present invention can be understood with reference to control diagram shown in FIG.  4 . In FIG. 4, motor steps u indicated by actuator command on line  228  is used as a system input to the toroidal drive  12  illustrated by a block diagram. An angular position φ of the trunnion  100  of traction roller assembly  108  about its pivot axis  104  is used as a system output. 
     From the preceding description, it is to be noted that the T-CVT  10  includes a ratio control element, in the form of the actuator shaft  190  (see FIG.  2 ). The ratio control element is positionable in response to an actuator command applied to the actuator  188  to establish various speed ratios between input and output shafts  20  and  98  of the T-CVT  10 . The axial displacement of the actuator shaft  190  may be expressed as 
     
       
           x=−a   1   φ−a   2 ( y+Δy )+ bu   (1) 
       
     
     where: 
     x is the axial displacement of actuator shaft  190 ; 
     y is the time integral of x and thus the axial displacement, along the pivot axis  104 , of trunnion  100  of the traction roller assembly  108 ; 
     Δy is the external disturbance; 
     u is the motor steps indicated by the actuator command applied to actuator  188 ; 
     φ is the angular position, about the pivot axis  104 , of the trunnion  100  of the traction roller assembly  108 ; 
     a 1  and a 2  are feedback gains, which are determined by the feedback structure  158  including the cam  200 ; and 
     b is a gain. 
     The axial displacement y of trunnion  100  will be hereinafter referred to as the trunnion axial displacement, and the axial displacement x of actuator shaft  190  will be hereinafter referred to as the actuator displacement. Trunnion axial displacement y is the time integral of the actuator displacement x. This relationship may be expressed as 
     
       
         dy=x  (2). 
       
     
     Similarly, the angular position φ of trunnion  100  will be hereinafter referred to as the trunnion angular position. The trunnion angular position φ is one example of various physical quantities indicative of a speed ratio between the input and output shafts  20  and  98 . Another example of such physical quantities is a ratio, which may be hereinafter referred to by the reference character G (=ic), computed from the outputs of input and output speed sensors  202  and  204 . 
     The term “a ratio rate” will be hereinafter used to mean the velocity at which the ratio changes. Mathematically, the ratio rate means the first time derivative of the ratio because the ratio is variable with respect to time. As the trunnion angular position φ and the ratio G are indicative of the speed ratio between the input and output shafts  20  and  98 , their first time derivatives dφ and dG are physical quantities indicative of ratio rate. 
     The physical quantity dφ indicative of ratio rate and another physical quantity that is trunnion axial displacement y are related by a factor of proportionality f as 
      dφ=fy  (3). 
     The factor of proportionality f is computed using a physical quantity indicative of trunnion angular position φ and another physical quantity indicative of the speed ω o  of the output shaft  98 . The output shaft speed ω o  is proportional to the speed of the inboard discs  50 ,  52  of toroidal drive  12 . The factor of proportionality f may be computed using an equation expressed as              f   =           cos        (       c   g1     -   φ     )            {       c   g0     -     cos        (       2        c   g1       -   φ     )         }         c   f            ω   o               (   4   )                         
     where: 
     c g1 , c g0 , and c f  are coefficients, which specify the toroidal drive  12 . 
     Accounting for the above equations (1), (2) and (3), the mathematical model of the system including the toroidal drive  12  may be expressed as                    d                   x   A       =         A   A          x   A       +       B   A        u              
            x   A     =     [         φ           y         ]            
          φ   =       C   0        x       ,       C   o     =     [         1       0         ]              
            A   A     =         [         0       f             -     a   i             -     a   2             ]          
          B   A       =     [         0           b         ]                 (   5   )                         
     where: 
     u is the input, x A  is the system state quantity and includes, as its matrix elements, the trunnion angular position φ and the trunnion axial displacement y, x is the displacement of the ratio control element in the form of actuator shaft  190 , C o  is a matrix by which φ and x are related, and A A , B A  are matrices that specify the system including toroidal drive  12 . 
     With continuing reference to FIG. 4, controller  210  is now described. Accelerator pedal position signal APS on line  222  is used as one of two inputs to a speed command generator  230 . Output shaft speed signal ω o  on line  224  is used as the other input to speed command generator  230 . The speed command generator  230  outputs a desired engine speed command signal ω e * on line  232 . In the speed command generator  230 , the vehicle speed VSP is determined by calculating the following equation. 
     
       
           VSP=k   V ω o   
       
     
     WHERE: 
     k V  is the constant determined accounting for the overall gear ratio from the output shaft  98  to the tire and the diameter of the tire. 
     The speed command generator  230  may include a look-up map in computer-readable storage media  214  (see FIG.  3 ). One example of such a look-up map is a T-CVT map depicted in FIG.  5 . In FIG. 5, a plurality of contour lines are illustrated for different accelerator pedal positions APS, such as  0 / 8 ,  1 / 8 , . . .  8 / 8 , each contour line representing a designed varying of desired engine speed commands ω e * with different vehicle speeds VSP. Using software technique available in the art to implement the relationships illustrated in FIG. 5, a desired engine speed command ω e * may be obtained against a given set of accelerator pedal position signal APS and vehicle speed VSP. 
     Desired engine speed command ω e * on line  232  is used as one of the two inputs to a trunnion angle or angular position command generator  234 . The output shaft speed signal ω o  on line  224  is used as the other input to the trunnion angle command generator  234 . The trunnion angle command generator  234  outputs a desired trunnion angular position command φ* on line  236  and a feedforward motor steps command u FF  on line  238 . In the trunnion angle command generator  234 , a desired ratio ic* is determined. The desired ratio may be expressed as 
     
       
           ic*=ω   e */ω o   (7). 
       
     
     The trunnion angle command generator  234  may include a look-up map in computer-readable storage media  214  for use in determining desired trunnion angular position command φ*. One example of such a look-up map is a conversion map depicted in FIG.  6 . In FIG. 6, a single contour line is illustrated, which contour line represents a designed varying of desired trunnion angular position command φ* with different desired ratio commands ic*. Using software technique available in the art to implement the relationship illustrated in FIG. 6, a desired trunnion angular position command φ* may be obtained against a given desired ratio command ic*. The trunnion angle command generator  234  may also include another look-up map in computer-readable storage media  214  for use in determining feedforward motor steps command u FF . One example of such a look-up map is a conversion map depicted in FIG.  7 . In FIG. 7, a single contour line is illustrated, which contour line represents a designed varying of feedforward motor steps command u FF  with different desired trunnion angular position commands φ*. Using software technique available in the art to implement the relationship illustrated in FIG. 7, a feedforward motor steps command UFF may be obtained against a desired trunnion angular position command φ*. 
     Desired trunnion angular position command φ* on line  236  is used as one of two inputs to a PI controller  240 . An actual trunnion angular position signal φ from an encoder (not shown) arranged around trunnion  100  (see FIG. 2) is applied to line  242 . Actual trunnion angular position φ may be obtained indirectly if the provision of such encoder is discouraged due to some reasons. The actual trunnion angular position φ may be obtained by calculating an actual ratio ic and by using the conversion map depicted in FIG.  6 . The actual ratio ic may be expressed as 
     
       
           ic=ω   i /ω o   (8). 
       
     
     Actual trunnion angular position signal φ on line  242  is used as the other input to PI controller  240 . PI controller  240  determines an error e by calculating the equation as follows 
     
       
           e=φ*−φ   (9). 
       
     
     PI controller  240  carries out proportional plus integral control action and outputs a PI motor steps command u PI  on line  244 . The proportional plus integral action may be expressed as                  u   PI     =         K   p        e     +         K   I     s        e              
            where   :     
            K   p                   is                 the                 proportional                 gain       ;          
              K   I                   is                                the                 integral                 gain     ;   and          
          s                 is                 the                 complex                   variable   .               (   10   )                         
     In the embodiment, as shown in FIG. 8A, proportional gain K P  is greater than 0 (zero) and proportional to factor of proportionality f, and, as shown in FIG. 8B, integral gain K I  is greater than 0 (zero) and proportional to factor of proportionality f. 
     In addition to the PI controller  240 , a D controller  246  is provided. An estimated value y m  of trunnion axial displacement y on line  248  is used as an input to D controller  246 . D controller  246  processes the trunnion axial displacement estimated value y m  and outputs a D motor steps command U D  on line  250 . Rather than measuring the trunnion axial displacement y, a state observer  252  makes the estimation by processing actual trunnion angular position signal φ on line  242 , output shaft speed ω o  on line  224 , and actuator command indicative of motor steps u on line  228 . 
     In one embodiment of the present invention, D controller  246  carries out the control action, which may be expressed as 
     
       
           u   D   =K   D   y   m   (11) 
       
     
     where: 
     u D  is the damping motor steps, K D  is the damping (D) gain, and y m  is the estimated value of trunnion axial displacement y. 
     In the embodiment, as shown in FIG. 8C, D gain K D  is less than 0 (zero) and inversely proportional to the factor of proportionality f. 
     Assuming now that the estimated value y m  is accurate enough to approximate the actual trunnion axial displacement y, the equation (3) may be rewritten as 
     
       
           dφ   m   =fy   m   (12) 
       
     
     where: 
     dφ m  is the estimated value of the first time derivative dφ that is the physical quantity indicative of the ratio rate. 
     Consider now a derivative control action in which the output is proportional to the first time derivative of the input. Let us assume that the damping motor steps u D  is the output and the trunnion angular position φ is the input. Then, the derivative control action may be expressed as 
       u   D   =cdφ   (13) 
     where: 
     c is the coefficient (c&lt;0). 
     As the first time derivative dφ of trunnion angular position is not measurable, it is approximated by the estimated value dφ m . Thus, we obtain 
     
       
           u   D   =cdφ   m   (14). 
       
     
     Using the equations (12) and (14), we obtain 
     
       
           u   D   =cfy   m   (15). 
       
     
     Comparing the equation (15) to the equation (11), we obtain 
     
       
           K   D   =cf   (16). 
       
     
     From the preceding description, it is noted that the D gain K D  is the differential gain. With reference to FIG. 8C, it is noted that the coefficient c is the gradient of the illustrated D gain vs., factor of proportionality characteristic. 
     With continuing reference to FIG. 4, at a summation point  254 , feedforward motor steps command u FF  on line  238 , PI motor steps command u PI  on line  244 , and D motor steps command u D  are processed to give commanded motor steps u indicated by actuator command on line  228 . The action at the summation point  254  may be expressed as 
     
       
           u=u   FF   +u   PI   +u   D   (17). 
       
     
     This equation (17) clearly states that D motor steps command u D , which is less than 0 (zero), acts as a damping component of the commanded motor steps u. As will be appreciated by one ordinary skill in the art, the provision of such damping component permits controller designers to choose sufficiently large proportional and integral gains K P  and K I  as desired. 
     An exemplary implementation of the present invention can be understood with reference to the control diagram of FIG.  9 . State observer  252  receives actuator command indicative of commanded motor steps u, trunnion angular position signal φ, and output shaft speed signal ω o , and processes them to give an estimated value y m  of trunnion axial displacement y. 
     Before making further description on the manner of giving the estimated value y m  of trunnion axial displacement y, we have to remember that the characteristic equation (5) expresses the mathematical model the system including the toroidal drive  12 . The system state quantity x A  of the characteristic equation (5) is a matrix including, as matrix elements, the trunnion axial displacement y and the trunnion angular position φ. To estimate the system state quantity x A , we now consider the mathematical model of a state observer. The mathematical model of state observer may be expressed as 
     
       
           dx   Ae   =A   A   x   Ae   +B   A   u+H   A ( dφ−dφ   m )  (18) 
       
     
     where: 
     x Ae  is the estimated quantity of the system state quantity x A  and the state quantity of the characteristic equation (18); u is the commanded motor steps and the input; φ is the trunnion angular position and the other input; A A , B A  and H A  are coefficients in the form of matrices that specify the system; dφ is the first time derivative of the trunnion angular position φ; dφ m  is the estimated value of the first time derivative dφ; and H A  is the coefficient called observer gain,          x   Ae     =     [           φ   m               y   m           ]                 d                 φ     =       C   A          x   A         ,       d                   φ   m       =       C   A          x   Ae                   C   A     =     [     0                 f     ]                             H   A     =       [           h     1      A                 h     2      A             ]     .                     
     In the above equation (18), the first time derivative dφ of trunnion angular position φ is not directly measurable, a state change is made as explained below 
     
       
           x   Ae   =q   A   +H   A φ  (19) 
       
     
     where: 
     φ is the input, x Ae  is the output, q A  is the quasi-state quantity, and H A  is the observer gain,          q   A     =       [           q     1      A                 q     2      A             ]     .                     
     Using the equation (18) and the equation (19) as modified by differentiating both sides of the latter, we obtain                      d                   q   A       =       d                   x     A                 e         -     d                   H   A        φ     -       H   A        d                 φ                   =         A   A          x     A                 e         +       B   A        u     +       H   A          (       d                 φ     -     d                   φ   m         )       -     d                   H   A        φ     -       H   A        d                 φ                   =         A   A          q   A       +       A   A          H   A        φ     +       B   A        u     +       H   A        d                 φ     -       H   A          C   A          x     A                 e         -     d                   H   A        φ     -       H   A        d                   φ              .                     =         A   A          q   A       +       A   A          H   A        φ     +       B   A        u     -       H   A          C   A          q   A       -       H   A          C   A          H   A        φ     -     d                   H   A        φ                   =         (       A   A     -       H   A          C   A         )          q   A       +       B   A        u     +       (         A   A          H   A       -       H   A          C   A          H   A       -     d                   H   A         )        φ                     (   20   )                         
     From the equation (20), we establish a filter, which is a characteristic function of the quasi-state quantity q A . The filter is expressed as 
     
       
           dq   A   =A   obA   q   A   +D   A   φ+B   A   u   (21) 
       
     
     where: 
     q A  is the state quantity, u and φ are the inputs, A obA  is the transition coefficient in the form of a transition matrix, and D A  is the coefficient in the form of a matrix,          A     o                 b                 A       =         A   A     -       H   A          C   A         =     [         0         f        (     1   -     h     1      A         )                 -     a   1               -     a   2       -       h     2      A          f             ]                       
     
       
           D   A =A A H A −H A C A H A −dH A . 
       
     
     The filter as expressed by equation (21) is used to compute or estimate quasi-state quantity q A . From the preceding description, it is to be noted that the mathematical model expressed by equation (18) has been manipulated into the form as expressed by equations (21) and (19). 
     In order to compute estimated quantity x Ae  that is the state quantity of equation (18), the state observer  252  uses the above-mentioned filter (21) to give the first time derivative dq A  of quasi-state quantity q A . The first time derivative dq A  is integrated to give quasi-state quantity q A , which is then put into the equation (19) to give the estimated quantity x Ae  of system state quantity x A . As it is one of two matrix elements of the estimated quantity x Ae , estimated value y m  of trunnion axial displacement y is given after calculation of equations (21) and (19). 
     With reference now to FIG. 9, state observer  252  according to one exemplary embodiment of the present invention will be described. Output shaft speed signal ω o  on line  224  and trunnion angular position signal φ on line  242  are used as inputs to a factor of proportionality f generator  260 . Factor of proportionality generator  260  calculates equation (4) and outputs factor of proportionality f on line  262 . Factor of proportionality f on line  262  is used as an input to a correction coefficient generator  264 , which outputs correction coefficients h 1A * and h 2A * on line  266 . As will be understood as discussion proceeds, correction coefficients h 1A * and h 2A * will work to keep transition coefficient A obA  of filter (21) unaltered. To determine such correction coefficients h 1A * and h 2A *, the generator  264  calculates the equations as follows                h     1      A     *     =     1   -       k     1      A       f               (22-1)                 h     2      A     *     =     -         k     2      A       +     a   2       f               (22-2)                         
     where: 
     k 1A  and k 2A  are coefficients that determine speed at which state observer  252  makes the estimation. Correction coefficients h 1A * and h 2A * will be further described later. 
     Factor of proportionality f on line  262  is used as one input to a quasi-state quantity generator  268 . Actuator command on line  228  indicative of motor steps u is used as another input to generator  268 . Trunnion angular position signal φ on line  242  is used as still another input to generator  268 . Observer gain matrix H A , which includes observer gains h 1A  and h 2A , on line  270  and observer gain first time derivative matrix dH A , which includes observer gain first time derivatives dh 1A  and dh 2A , on line  272  are used as the other inputs to generator  268 . Using them, generator  268  calculates equation (21) to determine quasi-state quantity q A . Generator  268  outputs quasi-state quantity q A  on line  274 . 
     Quasi-state quantity q A  on line  274  is used as one input to a trunnion axial displacement estimator  276 . Trunnion angular position signal φ on line  242  is used as another input to estimator  276 . Observer gain H A  on line  270  is used as the other input to estimator  276 . Estimator  276  calculates equation (19) to give estimated quantity x Ae  of system state quantity x A . Estimated quantity x Ae  includes, as one of its matrix elements, estimated value y m  of trunnion angular displacement y. Estimator  276  outputs the estimated value y m  on line  248 . 
     The before-mentioned correction coefficients h 1A * and h 2A * will now be described in relation to transition matrix A obA . Correction coefficients h 1A * and h 2A * on line  266  are used as inputs to a gain and gain time derivative generator  278 . Generator  278  outputs observer gains h 1A  and h 2A  on line  270 , and observer gain first time derivatives dh 1A  and dh 2A  on line  272 . 
     In one embodiment of the present invention, generator  278  uses correction coefficients h 1A * and h 2A * as observer gains h 1A  and h 2A , respectively. Instead of differential operation on observer gains h 1A  and h 2A , generator  278  uses a pseudo-differentiator to give the first time derivatives dh 1A  and dh 2A  thereof. Using correction coefficients h 1A * and h 2A * as observer gains h 1A  and h 2A , respectively, the transition matrix A obA  is kept unaltered as shown below                A   obA     =       [         0         k     1      A                 -     a   1             k     2      A             ]     .             (   23   )                         
     Using eigenvalue ω ob  of state observer  252 , the matrix elements k 1A  and k 2A  of transition coefficient A obA  are expressed as                k     1      A       =       ω   ob   2       a   1               (24-1)                 k     2      A       =       -   2            ω   ob     .               (24-2)                         
     Rewriting the matrix elements of equation (23) using equations (24-1) and (24-2), we obtain −ω ob , −ω ob  as eigenvalue of transition matrix A obA  of equation (21). 
     Consider now an error e obA  between the systeml state quantity x A  of equation (5) and the estimated quantity x Ae  given by the state observer  252 . The error e obA  is given as 
     
       
           e   obA   =x   A   −x   Ae   (25). 
       
     
     Subtracting the equation (18) from equation (5) gives the relationship as 
       dx   A   −dx   Ae   =A   A ( x   A   −x   Ae )− H   A   C   A ( x   A   −x   Ae )  (26-1) 
     Using the relationship expressed by equation (25), equation (26-1) may be written as 
     
       
           de   obA =( A   A   −H   A   C   A ) e   obA   (26-2). 
       
     
     As transition matrix A obA  is (A A −H A C A ), equation (26-2) may be written as 
     
       
           de   obA   =A   obA   e   obA   (26-3), 
       
     
     As the eigenvalue of transition matrix A obA  is −ω ob , −ω ob , the equation (26-3) clearly states that the error e obA  will converge to zero with linear response exhibiting a certain time constant provided by the eigenvalue −ω ob , −ω ob . 
     With reference again to the gain and gain time derivative generator  278  shown in FIG. 9, in the embodiment, correction coefficients h 1A * and h 2A * on line  266  are used as observer gains h 1A  and h 2A . 
     In another exemplary embodiment of the present invention, corrected coefficients h 1A * and h 2A * are not equal to observer gains h 1A  and h 2A , respectively. Instead, they are used as inputs to low pass filters, respectively, of a gain and gain time derivative generator  278 . The low pass filters are expressed as 
     
       
           dh   1A   =−a   01   h   1A   +a   01   h   1A *  (27-1) 
       
     
     
       
           dh   2A   =a   02   h   2A   +a   02   h   2A *  (27-2) 
       
     
     where: 
     a 01  and a 02  are the filter coefficients, respectively. 
     In this embodiment, integrating low pass filters (27-1) and (27-2) outputs observer gains h 1A  and h 2A , respectively, on line  270 . Gain and gain time derivative generator  278  use outputs of filters (27-1) and (27-2), which may be regarded as mid values for calculation of observer gains, as the first time derivatives h 1A , dh 2A  of observer gains h 1A , h 2A . 
     In the embodiment employing low pass filters, observer gains h 1A  and h 2A  are not completely equal to correction coefficients h 1A * and h 2A *, respectively, because each of the observer gains is given by multiplying the corresponding one of the correction coefficients with the associated low pass filter. This indicates that transition coefficient A obA  is not completely unaltered. However, in the embodiment, transition coefficient A obA  is kept substantially unaltered by selecting filter coefficients a 01  and a 02  to sufficiently large enough to reduce a deviation of each of observer gains h 1A  and h 2A  from the associated one of correction coefficients h 1A * and h 2A * toward zero. 
     Employing low pass filters (27-1) and (27-2) is advantageous in suppressing error between y m  and y caused due to noise because differential operation is no longer needed. As is well known by one of ordinary skill in the art, differential calculation is considered to amplify noise if it is included in the output of a sensor. 
     In the embodiment, factor of proportionality f and observer gain H A  may include noise of the above kind. Factor of proportionality f includes trunnion angular position signal φ and output shaft speed signal ω o , which are outputted by the encoder and speed sensor, respectively. The factor of proportionality f is used to correct observer gain H A . Thus, inclusion of noise into observer gain H A  inevitably takes place. In the case where observer gain H A  is subject to differential operation, the amplification of such noise is unavoidable, making it difficult to tell the time derivative component out of the noise component. 
     An example of how a controller, such as the T-CVT controller  210  (see FIG.  3 ), would implement the present invention can be understood with reference to FIGS. 10-15. The flow diagrams in FIGS. 10-15 illustrate a main routine and the associated sub routines of one exemplary implementation of the present invention. As will be appreciated by one of ordinary skill in the art, the steps illustrated with respect to FIGS. 10-15 are preferably repeated at predetermined intervals, such as 10 milliseconds, for example. Further, the steps illustrated with respect to FIGS. 10-15 are preferably stored in computer readable storage media  214  (see FIG.  3 ). Computer readable storage media  214  have data stored thereon representing instructions executable by T-CVT controller  210  to control toroidal drive  12  in a manner illustrated with respect to FIGS. 10-15. 
     In FIG. 10, the main routine is generally designated by the reference numeral  300 . 
     In step  302 , the controller executes a sub routine  320  in FIG. 11 to monitor operating variables indicative of output shaft speed ω o , vehicle speed VSP, accelerator pedal position APS, input shaft speed ω i , actual speed ratio i c  and actual trunnion angular position φ. 
     In step  304 , the controller executes a sub routine  340  in FIG. 12 to determine desired trunnion angular position command φ*. 
     In step  306 , the controller computes factor of proportionality f that is expressed by equation (4). 
     In step  308 , the controller determines feedforward motor steps command u FF  by referring to the illustrated relationship in FIG. 7 against the desired trunnion angular position φ*, which has been determined in step  304 . 
     In step  310 , the controller executes a sub routine  360  in FIG. 13 to carry out PI (proportional and integral) control action to determine PI (or feedback) motor steps command up,. 
     In step  312 , the controller executes a sub routine  380  in FIG. 14 to estimate trunnion axial displacement y, giving an estimated value y m  thereof. 
     In step  314 , the controller executes a sub routine  400  in FIG. 15 to carry out D control action to determine D motor steps command u D . 
     In step  316 , the controller combines in appropriate manner the feedforward motor steps command u FF , PI or feedback motor steps command u PI  and D motor steps command u D  to determine actuator command, in the form of motor steps, u by, for example, using equation (17). 
     In step  318 , the controller computes the first time derivative dq A  of quasi-state quantity q A , which is expressed as                dq   A     =       [           dq     1      A                 dq     2      A             ]     =         [         0         f        (     1   -     h     1      A         )                 -     a   1               -     a   2       -       h     2      A          f             ]          [           q     1      A                 q     2      A             ]       +     
            [             fh     2      A       -       h     1      A            fh     2      A         -     dh     1      A                       -     a   1            h     1      A         -       a   2          h     2      A         -     fh     2      A     2     -     dh     2      A               ]                   φ     +       [         0           b         ]                     u   .                   (   28   )                         
     In equation (28), it is noted that the matrix          [         0         f        (     1   -     h     1      A         )                 -     a   1               -     a   2       -       h     2      A          f             ]                        
     is transition coefficient A obA  used in equation (21). Transition coefficient A obA  is kept unaltered if the correction coefficients h 1A * and h 2A *, see equations (22-1) and (22-2), are used as observer gains h 1A  and h 2A . In this matrix, using the correction coefficients h 1A * and h 2A *, for example, as observer gains h 1A  and h 2A , equation (28) may be written as                [                      d                   q     1      A                   d                   q     2      A               ]     =         [                    0           ω     o                 b     2       a   1                 -     a   1               -   2          ω     o                 b               ]                [                      q     1      A                 q     2      A             ]     +       [                        f                   h     2      A         -       h     1      A          f                   h     2      A         -     d                   h     1      A                         -     a   1            h     1      A         -       a   2          h     2      A         -     f                   h     2      A     2       -     d                   h     2      A                 ]        φ     +       [         0           b         ]          u   .                 (   29   )                         
     In the embodiment, for simplicity of mathematical operation, the controller uses equation (29) instead of equation (28) to give the first time derivatives dq 1A  and dq 2A . 
     Referring to FIG. 11, the sub routine  320  to monitor operating variables is now described. As mentioned above, this sub routine is executed in step  302  in FIG.  10 . 
     In step  322 , the controller inputs information of output shaft speed by receiving output shaft speed signal ω o  from sensor  204  (see FIG.  1 ). 
     In step  324 , the controller computes vehicle speed VSP using the output shaft speed by calculating equation (6). 
     In step  326 , the controller inputs information of accelerator pedal position by receiving accelerator pedal position signal APS from encoder associated with the vehicle accelerator pedal. 
     In step  328 , the controller inputs information of input shaft speed by receiving input shaft speed signal ω i  from sensor  202  (see FIG.  1 ). 
     In step  330 , the controller computers actual speed ratio ic that is expressed by equation (8). 
     In step  332 , the controller inputs information of actual trunnion angular position by receiving actual trunnion angular position signal φ from the encoder arranged around trunnion  100  (see FIG.  2 ). If the provision of such encoder is discouraged, actual trunnion angular position φ may be obtained by using the actual ratio ic determined in step  330  in retrieving the illustrated relationship in FIG.  6 . 
     Referring to FIG. 12, the sub routine  340  to determine desired trunnion angular position command φ* is now described. This sub routine is executed in step  304  in FIG.  10 . 
     In step  342 , the controller determines desired engine speed command ω o  by looking into the illustrated data in FIG. 5, each being indexed by vehicle speed VSP and accelerator pedal position APS. 
     In step  344 , the controller computes desired speed ratio ic*, which is expressed by equation (7). 
     In step  346 , the controller determines desired trunnion angular position command φ* by looking into the illustrated data in FIG. 6, each being indexed by desired ratio command ic*. 
     Referring to FIG. 13, the sub routine  360  to carry out PI control action is now described. This sub routine is executed in step  310  in FIG.  10 . 
     In step  362 , the controller computes the time integral eI of an error e (e=ic*−ic) by calculating the following formula. 
     
       
           eI←eI+Te   (30) 
       
     
     where: 
     T represents a period of time of each of the predetermined intervals at which the steps illustrated in FIGS. 10-15 are executed and is equal to 0.01, indicative of 10 milliseconds. 
     As will be noted by one of ordinary skill in the art, the error e that was computed during the previous execution by the controller is used in calculating equation (30) in step  362 . 
     In step  364 , the controller computes error e between the desired ratio ic* (determined in step  344  during the current execution) and the actual ratio ic (determined in step  330  during the current execution). In the embodiment, error e is expressed as 
     
       
           e=ic*−ic   (31). 
       
     
     In step  366 , the controller determines a proportional gain K P  against factor of proportionality f computed in step  306  using the illustrated relationship in FIG.  8 A. 
     In step  368 , the controller determines an integral gain K I  against factor of proportionality f computed in step  306  using the illustrated relationship in FIG.  8 B. 
     In step  370 , the controller computes PI motor steps command u PI  by calculating the equation as follows 
     
       
           u   PI   =K   p   e+K   I   eI   (32). 
       
     
     Referring to FIG. 14, the sub routine  380  to estimate trunnion axial displacement is now described. This sub routine is executed in step  312  in FIG.  10 . 
     In step  382 , the controller computes correction coefficients h 1A * and h 2A *. Using equations (24-1), (24-2), (22-1) and (22-2), the correction coefficients h 1A * and h 2A * are given as                h     1        A   *         =     1   -       ω   ob   2         a   1        f                 (     33-1     )                 h     2        A   *         =     -           2        ω   ob       -     a   2       f     .               (     33-2     )                         
     In step  382 , the controller calculates the equations (33-1) and (33-2) using factor of proportionality f to give correction coefficients h 1A * and h 2A *. 
     In step  384 , the controller computes observer gains h 1A  and h 2A  by integrating the time derivatives of observer gains dh 1A  and dh 2A  that were obtained in step  318  in FIG. 10 during the last execution by the controller. To give observer gains h 1A  and h 2A , using the previously obtained values of h 1A , h 2A , dh 1A  and dh 2A , the controller calculates the following formulas 
     
       
           h   1A   ←h   1A   +Tdh   1A   (34-1) 
       
     
     
       
           h   2A   ←h   2A   +Tdh   2A   (34-2). 
       
     
     In step  386 , the controller computes the first time derivatives dh 1A  and dh 2A  of observer gains by calculating low pass filters expressed by equations (27-1) and (27-2) using corrected coefficients h 1A * and h 2A * that have been given in step  382  and observer gains h 1A  and h 2A  that have been given in step  384 . 
     In step  388 , the controller computes quasi-state quantity q 1A , q 2A  by integrating the first time derivatives dq 1A  and dq 2A  that were given in step  318  during the last cycle of execution. To give quasi-state quantities q 1A  and q 2A , using the previously obtained values q 1A , q 2A , dq 1A  and dq 2A , the controller calculates the following formulas 
     
       
           q   1A   ←q   1A   +Tdq   1A   (35-1) 
       
     
     
       
           q   2A   ←q   2A   +Tdq   2A   (35-2). 
       
     
     In step  390 , the controller computes estimated value y m  of trunnion axial displacement y. To give estimated value y m , using quasi-state quantity q 2A , actual trunnion angular position φ, and observer gain h 2A , the controller calculates the equation as follows 
     
       
           y   m   =q   2A   +h   2A φ  (36). 
       
     
     Referring to FIG. 15, the sub routine  400  to carry out D control action is now described. This sub routine is executed in step  314  in FIG.  10 . 
     In step  402 , the controller determines a D gain K D  against factor of proportionality f computed in step  306  using the illustrated relationship in FIG.  8 C. 
     In step  404 , the controller computes D motor steps command u D  by calculating equation (11). 
     With reference again to FIG. 9, generator  268  of state observer  252  calculates the filter expressed by equation (21) to give the first time derivative dq A  of quasi-state quantity q A . The first time derivative dq A  is integrated to give quasi-state quantity q A , which is then put into equation (19) to give estimated quantity x Ae  of system state quantity x A . Estimator  276  calculates equation (19). State quantity x A  contains, as its matrix elements, trunnion angular position φ as well as trunnion axial displacement y, making it possible for state observer  252  to give an estimated value φ m  of trunnion angular position φ as well. 
     In an exemplary embodiment where trunnion angular position φ is measurable, a low order state observer may replace such a high order state observer as expressed by equations (21) and (19). The mathematical model of a lower order state observer is manipulated into the form expressed as 
     
       
           dq   r =(− a   2   −h   r   f ) q   r +(− a   1   −a   2   h   r   −fh   r   2   −dh   r )φ+ bu   (37) 
       
     
     
       
           y   m   =q   r   +h   r φ  (38) 
       
     
     where: 
     q r  is the quasi-state quantity; 
     h r  is the observer gain; 
     (−a 2 −h r f) is the transition coefficient. 
     In one embodiment of the present invention, equations (37) and (38) have replaced equations (21) and (19), respectively. This has brought about a drop, in the rank of state observer  252 , from the second order to the first order. 
     In this embodiment, state observer  252  has an eigenvalue of ω ob . Thus, a correction coefficient h r * is given as                h   r   *     =           ω   ob     -     a   2       f     .             (   39   )                         
     Using correction coefficient h r * as the observer gain h r , equation (37) may be written as 
       dq   r =−ω ob   q   r +(− a   1   −a   2   h   r   *−fh   r * 2   −dh   r *)φ+ bu   (40) 
     As the eigenvalue is −ω ob , an error e y  between estimated value y m  of trunnion axial displacement and actual value y thereof will diverge toward zero, exhibiting the dynamic characteristic as expressed as 
     
       
           de   y +ω ob   e   y =0  (41). 
       
     
     An example of how T-CVT controller  210 , incorporating the low order state observer mentioned above, would implement the present invention can be understood with reference to FIGS. 10-15 only by listing alterations needed. 
     In FIG. 10, at step  318 , the controller computes the first time derivative dq r  of quasi-state quantity q r  by calculating equation (40) instead of calculating equation (29). 
     In FIG. 14, at step  382 , the controller gives correction coefficient h r * by calculating equation (39) instead of calculating equations (33-1) and (33-2). 
     In FIG. 14, at step  384 , instead of calculating formulas (34-1) and (34-2), the controller gives observer gain hr by calculating the following formula 
     
       
           h   r   ←h   r   +Tdh   r   (42). 
       
     
     In FIG. 14, at step  386 , instead of calculating equations (27-1) and (27-2), the controller gives the first time derivative dh r  of observer gain by calculating the equation as follows 
     
       
           dh   r   =−ah   r   +ah   r *  (43). 
       
     
     In FIG. 14, at step  388 , instead of calculating formulas (35-1) and (35-2), the controller gives quasi-state quantity q r  by calculating the following formula 
     
       
           q   r   ←q   r   +Tdq   r   (44). 
       
     
     In FIG. 14, at step  390 , instead of calculating equation (36), the controller gives estimated value y m  of trunnion axial displacement y by calculating equation (38). 
     Another exemplary embodiment of the present invention can be understood with reference to control diagram shown in FIG.  16 . In FIG. 16, motor steps rate or speed vindicated by actuator command on line  228  is used as a system input to toroidal drive  12 . A trunnion angular position φ of traction roller assembly  108  about its pivot axis  104  is used as a system output. 
     In the embodiment, actuator  188  is in the form of a stepper motor. The stepper motor  188  can perform integral operation of motor steps rate v to provide motor steps u and move actuator shaft  190  in its axial direction in proportional relationship to the motor steps u. 
     Motor steps rate v and motor steps u have the following relationship 
     
       
         du=v  (45) 
       
     
     where: 
     du is the first time derivative of motor steps u. 
     Similarly to equation (5), the dynamic characteristic of the system including the toroidal drive  12  may be expressed as 
     
       
           dx   B   =A   B   x   B   +B   B   v   (46) 
       
     
     where: 
     v is the input, x B  is the system state quantity and includes, as its matrix elements, trunnion angular position φ, trunnion axial displacement y and motor steps u, x is the displacement of the ratio control element, C B  is a matrix by which φ and x are related, and A B  and B B  are matrices that specify the system including the toroidal drive  12           φ   =       C   B        x       ,       C   B     =           [   1         0         0   ]                       x   B     =         [         φ           y           u         ]          
          A   B       =         [         0       f       0             -     a   1             -     a   2           b           0       0       0         ]          
          B   B       =       [         0           0           1         ]     .                         
     With continuing reference to FIG. 16, a T-CVT controller  210 A receives various signals, processes them to determine motor steps rate v, and generates an actuator command in the form of motor steps rate v on line  228 . Controller  210 A is similar to controller  210  in FIG. 4, so that like reference numerals are used to denote like component parts in FIGS. 4 and 16. 
     Controllers  210 A and  210  have substantially the same speed command generators  230 , each of which outputs a desired engine speed command ω e * on line  232 . Besides, they have like trunnion angle command generators  234  and  234 A. 
     In FIG. 16, desired engine speed command ω e * on line  232  is used as one of two inputs to a trunnion angle or angular position command generator  234 A. Output shaft speed signal ω o  on line  224  is used as the other input to trunnion angle command generator  234 A. Trunnion angle command generator  234 A is substantially the same as trunnion angle command generator  234  in FIG.  4  and outputs a desired trunnion angular position command φ* on line  236 . But, trunnion angle command generator  234 A is different from trunnion angle command generator  234  in that the former does not output a feedforward motor steps command u FF . 
     However, controllers  210  and  210 A are different from each other in the following respect. Controller  210 A includes a state observer  420 , a diffeomorphic transform  424  and a sliding mode controller  428 , in the place of the component parts, such as state observer  232 , D controller  246 , PI controller and summation point  254 , of controller  210 . 
     In the embodiment illustrated in FIG. 16, motor steps rate v is used as actuator command applied to stepper motor  188  (see FIG.  2 ). In the embodiment, as actual trunnion angular position φ is directly measurable, state observer  420  is in the form of a low order state observer is used. The low order state observer gives an estimated value y m  of trunnion axial displacement y and motor steps u. 
     In this case, a system state quantity w is considered, which includes, as its elements, trunnion axial displacement y and motor steps u. Using motor steps rate v and trunnion angular displacement φ as inputs, the mathematical model expressed by equation (46) may be simplified as 
     
       
           dw=A   22   w+B   2   v+A   21 φ  (47) 
       
     
     where: 
     w is the state quantity, v and φ are the inputs, and A 22 , B 2  and A 21  are the coefficients that specify the system        w   =     [         y           u         ]                 A   B     =     [           A   11           A   12               A   21           A   22           ]       ,       A   11     =   0     ,       A   12     =     [     f                 0     ]       ,     
            A   21     =     [           -     a   1               0         ]       ,       A   22     =     [           -     a   2           b           0       0         ]                   B   B     =     [           B   1               B   2           ]       ,       B   1     =   0     ,       B   2     =     [         0           1         ]                       
     To estimate system state quantity w, we now consider a state observer  420 . The mathematical model of state observer  420  may be expressed as 
     
       
           dW   e   =A   22   w   e   +B   2   v+A   21   φ+H   B ( dφ−dφ   m )  (48) 
       
     
     where: 
     w e  is the estimated quantity of system state quantity w, y m  is the estimated value of trunnion axial displacement y, u m  is the estimated value of motor steps, and H B  is the observer gain and has the matrix elements h 1B  and h 2B            w   e     =     [           y   m               u   m           ]               d                 φ     =       A   12        w               d                   φ   m       =       A   12          w   e                 H   B     =       [           h     1      B                 h     2      B             ]     .                     
     As trunnion angular position dφ is not directly measurable, a state change is made as explained below 
     
       
           w   e   =q   B   +H   B φ  (49) 
       
     
     where: 
     q B  is the quasi-state quantity          q   B     =       [           q     1      B                 q     2      B             ]     .                     
     Using the equation (48) and the equation (49) as modified by differentiating both sides of the latter, we obtain the equation as follows                      d                   q   B       =                  d                   w   e       -     d                   H   B        φ     -       H   B        d                 φ                   =                    A   22          w   e       +       B   2        v     +       A   21        φ     +       H   B          (       d                 φ     -     d                   φ   m         )       -     d                   H   B        φ     -       H   B        d                 φ                                =                    A   22          q   B       +       A   22          H   B        φ     +       B   2        v     +       A   21        φ     +       H   B        d                 φ     -                                    H   B          A   12          w   e       -     d                   H   B        φ     -       H   B        d                 φ                   =                    A   21          q   B       +       A   22          H   B        φ     +       B   2        v     +       A   21        φ     -       H   B          A   12          q   B       -                                  H   B          A   12          H   B        φ     -     d                   H   B        φ                   =                    (       A   22     -       H   B          A   12         )          q   B       +       B   2        v     +       (       A   21     +       A   22          H   B       -       H   B          A   12          H   B       -     d                   H   B         )        φ                     (   50   )                         
     From the equation (50), we obtain a filter, i.e., a characteristic equation for quasi-state quantity q B , expressed as 
     
       
           dq   B   =A   obB   q   B   +D   B   φ+B   2   v   (51) 
       
     
     where: 
     v and φ are the inputs, q B  is the state quantity, A obB  is the transition coefficient, and D B  and B 2  are the coefficients          A     o                 d                 B       =         A   22     -       H   B          A   12         =     [             -     a   2       -       h     1      B          f           b               -     h     2      B            f         0         ]                       
     
       
           D   B =A 21 +A 22 H B −H B A 12 H B −dH B .  
       
     
     In order to compute estimated quantity we, the state observer  420  calculates the filter (51) to give the first time derivative dq B  of quasi-state quantity q B . The first time derivative dq B  is integrated to give quasi-state quantity q B , which is then put into the equation (49) to give the estimated quantity w e  of system state quantity w. As they are two matrix elements of the estimated quantity w e , estimated value y m  of trunnion axial displacement y and estimated value u m  of motor steps u are given after calculation of equations (51) and (49). State observer  420  outputs estimated values y m  and u m  on line  422 . 
     The estimated values y m  and u m  on line  422 , and trunnion angular displacement φ on line  242  are used as inputs into diffeomorphic transform  424 . Diffeomorphic transform  424  outputs an estimated value dφ m  (={dot over (φ)} m ) of the first time derivative (speed) of trunnion angular position φ, and an estimated value of the second time derivative αφ m  (={umlaut over (φ)} m ) of trunnion angular position φ by calculating the equations as follows 
     
       
           dφ   m ={dot over (φ)} m   =f·y   m   (52-1) 
       
     
     
       
         αφ m ={umlaut over (φ)} m   =df·y   m   +f ( bu   m   −a   1   φ−a   2   y   m )  (52-2) 
       
     
     In equation (52-2), df is the first time derivative of factor of proportionality f To give the first time derivative df, a pseudo-differentiator may be used. But, the first time derivative df may be given by calculating the equation as follows              df   =         [       sin                   (       c   g1     -   φ     )          {       c   g0     -     cos                   (       2        c   g1       -   φ     )         }       -     sin                   (       2        c   g1       -   φ     )                     cos        (       c   g1     -   φ     )           ]                       fy   ω0       c   f         +         cos                     (       c   g1     -   φ     )          [       c   g0     -     cos                   (       2        c   g1       -   φ     )         ]           c   f              d   ω0     .                 (   53   )                         
     In equation (53), the first time derivative dω o  may be obtained by a pseudo-differentiator. However, the variation of output shaft speed signal ω o  is negligibly small due to inertia of the vehicle because output shaft  98  (see FIG. 1) is directly connected to the vehicle output shaft. Thus, dω o  may be set equal to approximately 0. Setting dω o =0, equation (53) may be simplified as              df   =       [       sin                   (       c   g1     -   φ     )          {       c   g0     -     cos                   (       2        c   g1       -   φ     )         }       -     sin                   (       2        c   g1       -   φ     )                   cos                   (       c   g1     -   φ     )         ]                         fy   ω0       c   f       .               (   54   )                         
     In the embodiment, diffeomorphic transform  424  calculates equation (54) to give the first time derivative df. Diffeomorphic transform  424  outputs the estimated values dφ m  (={dot over (φ)} m ) and αφ m  (={umlaut over (φ)} m ) of the first and second time derivatives on line  426 . 
     The first and second time derivatives dφ m  (={dot over (φ)} m ) and αφ m  (={umlaut over (φ)} m ) on line  426 , trunnion angular position φ on line  242 , and desired trunnion angular position φ* on line  236  are used as inputs into a sliding mode controller  428 . Sliding mode controller  428  outputs actuator command in the form of motor steps rate (or speed) v on line  228 . To determine motor steps rate (or speed) v, the sliding mode controller  428  computes a control error σ and then motor steps rate v by calculating equations as follows.              σ   =       α                   φ   m       +     2      ζ                   ω   n          dφ   m       +       ω   n   2          (     φ   -     φ   *       )                 (55-1)               v   =       -   K                     σ        σ                    (55-2)                         
     where: 
     ζ is the damping coefficient; 
     ω n  is the natural frequency; 
     K is the switching gain. 
     If, in equation (55-2), switching gain K is increased sufficiently, σ converges to zero. From equation (55-1), we obtain the equation as follows 
     
       
         αφ m =−2ζω n   dφ   m −ω n   2 φ+ω n   2 *  (56). 
       
     
     Equation (56) clearly states that trunnion angular position φ responds against desired trunnion angular position φ* with the second order delay of damping coefficient ζ and natural frequency ω n . 
     With reference now to FIG. 17, state observer  420  according to an embodiment of the present invention will be described. State observer  420  illustrated in FIG. 17 is substantially the same as state observer  252  illustrated in FIG.  9 . Accordingly, the like reference numerals are used to denote like component parts or portions throughout FIGS. 9 and 17. 
     As different from state observer  252 , instead of motor steps u, motor steps rate v on line  228  is used as an input to a quasi-state quantity generator  268 . Besides, in FIG. 17, a trunnion axial displacement estimator  276  outputs estimated value u m  of motor steps u in addition to estimated value y m  of trunnion axial displacement, while, in FIG. 9, trunnion axial displacement estimator  276  outputs estimated value y m  of trunnion axial displacement y only. Other differences may become apparent as discussion proceeds. 
     As described before, in order to compute estimated quantity w e , the state observer  420  calculates the filter (51) to give the first time derivative dq B  of quasi-state quantity q B . The first time derivative dq B  is integrated to give quasi-state quantity q B , which is then put into the equation (49) to give the estimated quantity w e  of system state quantity w. The calculation of equation (51) and the subsequent integral operation are carried out by quasi-state quantity generator  268  to give quasi-state quantity q B . The calculation of equation (49) is carried out by trunnion axial displacement estimator  276  to give estimated quantity w e . Estimator  276  outputs estimated value y m  of trunnion axial displacement y and estimated value u m  of motor steps u because they are matrix components of the estimated quantity w e  of the system state quantity 
     With continuing reference to FIG. 17, output shaft speed signal ω o  on line  224  and trunnion angular position signal φ on line  242  are used as inputs to a factor of proportionality f generator  260 . Factor of proportionality generator  260  calculates equation (4) and outputs factor of proportionality f on line  262 . Factor of proportionality f on line  262  is used as an input to a correction coefficient generator  264 , which outputs correction coefficients h 1B * and h 2B * on line  266 . As will be understood as discussion proceeds, correction coefficients h 1B * and h 2B * will work to keep transition matrix A obB  of equation (51) unaltered. To determine such correction coefficients h 1B * and h 2B *, the generator  264  calculates the equations as follows                h     1      B     *     =         -     a   2       -     k     1      B         f             (57-1)                 h     2      B     *     =     -         k     2      B       f     .               (57-2)                         
     In the above equations (57-1) and (57-2), k 1B  and k 2B  are the coefficients that determine speed at which state observer  420  makes the estimation. Correction coefficients h 1B * and h 2B * will be described later. 
     Factor of proportionality f on line  262  is used as one input to generator  268 . Actuator command on line  228  indicative of motor steps rate v is used as another input to generator  268 . Trunnion angular position signal φ on line  242  is used as still another input to generator  268 . Observer gain matrix H B , which includes observer gains h 1B  and h 2B , on line  270  and observer gain first time derivative matrix dH B , which includes observer gain first time derivatives dh 1B  and dh 2B , on line  272  are used as the other inputs to generator  268 . Using them, generator  268  calculates equation (51) and integrates the result to determine quasi-state quantity q B . Generator  268  outputs quasi-state quantity q B  on line  274 . 
     Quasi-state quantity q B  on line  274  is used as one input to estimator  276 . Trunnion angular position signal φ on line  242  is used as another input to estimator  276 . Observer gain H B  on line  270  is used as the other input to estimator  276 . Estimator  276  calculates equation (49) to determine estimated quantity w e  of system state quantity w. Estimated quantity w e  includes, as its matrix elements, estimated value y m  of trunnion angular displacement y and estimated value u m  of motor steps u. Estimator  276  outputs the estimated values y m  and u m  on line  422 . 
     The before-mentioned correction coefficients h 1B * and h 2B * will now be described in relation to transition coefficient A obB . Correction coefficients h 1B * and h 2B * on line  266  are used as inputs to a gain and gain time derivative generator  278 . Generator  278  outputs observer gains h 1B  and h 2B  on line  270 , and observer gain first time derivatives dh 1B  and dh 2B  on line  272 . 
     In one embodiment of the present invention, generator  278  uses correction coefficients h 1B * and h 2B * as observer gains h 1B  and h 2B , respectively. Instead of differential operation, generator  278  uses a pseudo-differentiator to give the first time derivatives dh 1B  and dh 2B . Using correction coefficients h 1B * and h 2B * as observer gains h 1B  and h 2B , respectively, the transition coefficient A obB  is kept unaltered as shown below                A   obB     =       [           k     1      B           b             k     2      B           0         ]     .             (   58   )                         
     Using eigenvalue ω ob  of state observer  420 , the matrix elements k 1B  and k 2B  of transition coefficient A obB  are expressed as                k     1      B       =       -   2          ω   ob               (59-1)                 k     2      B       =     -         ω   ob   2     b     .               (59-2)                         
     Rewriting the matrix elements of equation (58) using equations (59-1) and (59-2), we obtain −ω ob , −ω ob  as eigenvalue of transition matrix A obB  of equation (51). 
     Consider now an error e obB  between the system state quantity w of equation (47) and the estimated quantity w e  given by the state observer  420 . The error e obB  is given as 
     
       
           e   obB   =w−w   e   (60). 
       
     
     Subtracting the equation (48) from equation (47) gives the following equation. 
     
       
           dw−dw   e   =A   22 ( w−w   e )− H   B   A   12 ( w−w   e )  (61-1) 
       
     
     Using the relationship expressed by equation (60), equation (61-1) may be written as 
     
       
           de   obB =( A   22   −H   B   A   12 ) e   obB   (61-2). 
       
     
     As transition coefficient A obB  is (A 22 −H B A 12 ), equation (61-2) may be written as 
     
       
           de   obB   =A   obB   e   obB   (61-3) 
       
     
     As the eigenvalue of transition matrix A obB  is −ω ob , −ω ob , the equation (61-3) clearly states that the error e obB  will converge to zero with linear response exhibiting a certain time constant provided by the eigenvalue −ω ob , −ω ob . 
     With reference again to the gain and gain time derivative generator  278  shown in FIG. 17, in the embodiment, correction coefficients h 1B * and h 2B * on line  266  are used as observer gains h 1B  and h 2B . 
     In another exemplary embodiment of the present invention, corrected coefficients h 1B * and h 2B * are not equal to observer gains h 1B  and h 2B , respectively. Instead, they are used as inputs to low pass filters, respectively, of a gain and gain time derivative generator  278 . The low pass filters are expressed as 
     
       
           dh   1B   =−a   01   h   1B   +a   01   h   1B *  (62-1) 
       
     
     
       
           dh   2B   =−a   02   h   2B   +a   02   h   2B *  (62-2) 
       
     
     where: 
     a 01  and a 02  are the filter coefficients, respectively. 
     In this embodiment, integrating low pass filters (62-1) and (62-2) outputs observer gains h 1B  and h 2B , respectively, on line  270 . Gain and gain time derivative generator  278  use outputs of filters (62-1) and (62-2), which may be regarded as mid values for calculation of observer gains, as the first time derivatives dh 1B , dh 2B  of observer gains h 1B , h 2B . 
     In the embodiment employing low pass filters, observer gains h 1B  and h 2B  are not completely equal to correction coefficients h 1B * and h 2B *, respectively, because each of the observer gains is given by multiplying the corresponding one of the correction coefficients with the associated low pass filter. This indicates that transition coefficient A obB  is not completely unaltered. However, in the embodiment, transition coefficient A obB  is kept substantially unaltered by selecting filter coefficients a 01  and a 02  to sufficiently large enough to reduce a deviation of each of observer gains h 1B  and h 2B  from the associated one of correction coefficients h 1B * and h 2B * toward zero. 
     The preceding description on FIG. 17 in comparison with the description on FIG. 9 clearly reveals various other differences between state observers  420  and  252 . According to one of such differences, in state observer  420 , quasi-state quantity generator  268  calculates equation (51) and integrates the result to give quasi-state quantity q B , while, in state observer  252 , quasi-state quantity generator  268  calculates equation (21) and integrates the result to give quasi-state quantity q A . According to another difference, in state observer  420 , trunnion axial displacement estimator  276 , calculates equation (49) to give estimated quantity w e , while, in state observer  252 , trunnion axial displacement estimator  276  calculates equation (19) to give estimated quantity x Ae . According to still another difference, in state observer  420 , correction coefficient generator  264  uses equations (57-1) and (57-2) to give correction coefficients h 1B * and h 2B *, while, in state observer  252 , correction coefficient generator  264  calculates equations (22-1) and (22-2) to give correction coefficients h 1A * and h 2A *. According to further difference, in state observer  420 , gain and gain time derivative generator  278  uses equations (59-1) and (59-2) in rewriting matrix elements k 1B  and k 2B  of transition matrix A obB  expressed by equation (58), while, in state observer  252 , gain and gain time derivative generator  278  uses equations (24-1) and (24-2) in rewriting matrix elements k 1A  and k 2A  of transition matrix A obA  expressed by equation (23). According to the other difference, in state observer  420 , generator  278  may use low pass filters as expressed by equations (62-1) and (62-2), while, in state observer  252 , generator  278  may low pass filters as expressed by equation (27-1) and (27-2). 
     From the preceding description, it will now be appreciated that observer gains h 1A  and h 2A  (or h 1B  and h 2B ) and their first time derivatives are corrected with correction coefficients h 1A * and h 2A * (or h 1B * and h 2B *) in a manner as previously described to keep transition matrix A obA  (or A obB ) unaltered. As transition matrix A obA  (or A obB ) is kept unaltered, error e obA =x A −x Ae  (or e obB =w−w e ) converges to zero. This means that, with very high accuracy, state observers  420  and  252  can estimate trunnion axial displacement y and, if needed, motor steps u, too. 
     With reference again to control diagram shown in FIG. 16, motor steps rate v is used as system input to toroidal drive  12 , and this motor steps rate v is controlled using state observer  420 . Motor steps rate v is used also as an input to state observer  420  that may be expressed by equation (48). State observer  420  estimates motor steps u accounting for dynamic characteristic of stepper motor  188  and provides estimated value u m  of motor steps u. Referring to FIG. 18A, we will describe on the degree of accuracy with which state observer  420  may estimate actual motor steps u P . FIG. 18A is a portion of FIG. 16, illustrating the situation where there occurs a continuous difference Δu, in stepper motor  188 , between commanded motor steps u, which is given after integrating motor steps rate v indicated by actuator command on line  228 , and actual motor steps u P . Normally, the difference Δu is zero and the commanded motor steps u is equal to the actual motor steps u P . Generally, the actual motor step u P  may be expressed as 
     
       
           u   P   =u+Δu   (63). 
       
     
     As equation (47) expresses the model of the system including toroidal drive  12 , it may be modified to express a change in the dynamic characteristic due to the difference in motor steps Δu. Equation (47) may be modified as 
     
       
           dw=A   22   w+B   2   v+A   21   φ+B   u   Δu   (64) 
       
     
     where:          B   u     =       [         b           0         ]     .                     
     State observer  420  is expressed by equation (48). Error e obB  is given by equation (60). Subtracting equation (48) from equation (64) and using equation (60) gives the relationship as 
     
       
           de   obB   =A   obB   e   obB   +B   u   Δu   (65). 
       
     
     In equation (65), it is assumed that de obB =0 because the error e obB  under consideration remains unaltered. Then, we obtain the equation as follows 
     
       
           A   obB   e   obB   +B   u   Δu= 0 
       
     
     
       
         
           
             
               
                 
                   
                     
                       
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     Using the equation (60), the above equation (66) may be simplified as                e   obB     =       w   -     w   e       =       [           y   -     y   m                 u   -     u   m             ]     =       [         0               -   Δ                   u           ]     .                 (   67   )                         
     Equation (67) clearly states the relationship that u−u m =−Δu. Using this relationship and equation (63), we obtain an error between actual motor steps u P  and estimated value u m  as 
     
       
           u   P   −u   m =( u+Δu )− u   m   =−u+Δu= 0  (68). 
       
     
     This equation (68) clearly states that there occurs no error so that the estimated value u m  corresponds exactly to the actual motor steps up under the presence of continuous difference Δu of motor steps in stepper motor  188  that remains unaltered. 
     FIG. 18B illustrates the situation where, during a period of time from T1 to T2, a change in trunnion axial displacement y to initiate a ratio change in toroidal drive  12  has caused a continuous difference Δu of motor steps in stepper motor  188 . This difference Δu remains unaltered after the moment T2. In FIG. 18B, the fully drawn line illustrates the variation of actual motor steps u P , the dashed line illustrates the variation of commanded motor steps u, and the one-dot chain line illustrates the variation of the estimated value u m  given by state observer  420 . As illustrated in FIG. 18B, the estimated value u m  corresponds exactly the actual motor steps u P  under the presence of continuous difference Δu of motor steps in stepper motor  188  that remains unaltered after the moment T2. 
     An example of how a controller, such as the T-CVT controller  210 A (see FIG.  16 ), would implement the present invention can be understood with reference to FIGS. 19-24. The flow diagrams in FIGS. 19-24 illustrate a main routine and the associated sub routines of one exemplary implementation of the present invention. As will be appreciated by one of ordinary skill in the art, the steps illustrated with respect to FIGS. 19-24 are preferably repeated at predetermined intervals, such as 10 milliseconds, for example. Further, the steps illustrated with respect to FIGS. 19-24 are preferably stored in computer readable storage media  214  (see FIG.  3 ). Computer readable storage media  214  have data stored thereon representing instructions executable by T-CVT controller  210 A to control toroidal drive  12  in a manner illustrated with respect to FIGS. 19-24. 
     In FIG. 19, the main routine is generally designated by the reference numeral  440 . 
     In step  442 , the controller executes a sub routine  460  in FIG. 20 to monitor operating variables indicative of output shaft speed ω o , vehicle speed VSP, accelerator pedal position APS, input shaft speed ω i , actual speed ratio i c  and actual trunnion angular position φ. 
     In step  444 , the controller executes a sub routine  480  in FIG. 21 to determine desired trunnion angular position command φ*. 
     In step  446 , the controller computes factor of proportionality f that is expressed by equation (4). 
     In step  448 , the controller executes a sub routine  500  in FIG. 22 to estimate trunnion axial displacement y, giving an estimated value y m  thereof, and motor steps u, giving an estimated value u m  thereof. 
     In step  450 , the controller executes a sub routine  520  in FIG. 23 to computer diffeomorphic transform to give the first and second time derivatives dφ m  (={dot over (φ)} m ) and αφ m  (={umlaut over (φ)} m ) of trunnion angular position φ. 
     In step  452 , the controller executes a sub routine  540  in FIG. 24 to carry out sliding mode control to give actuator command in the form of motor steps rate v. 
     In step  454 , the controller computes the first time derivative dq B  of quasi-state quantity q B , which is expressed as                      dq   A     =                [           dq     1      B                 dq     2      B             ]                 =                    [             -     a   2       -       h     1      B          f           b               -     h     2      B            f         0         ]                [           q     1      B                 q     2      B             ]     +                                  [             -     a   1       -       a   2          h     1      B         +     bh     2      B       -     fh     1      B     2     -     dh     1      B                       -     fh     1      B              h     2      B         -     dh     2      B               ]                   φ     +       [         0           1         ]                   v                     (   69   )                         
     In equation (68), it is noted that the matrix        [             -     a   2       -       h     1      B          f           b               -     h     2      B            f         0         ]                   
     is transition matrix A obB  used in equation (51). Transition matrix A obB  becomes unaltered if the correction coefficients h 1B * and h 2B * see equations (57-1) and (57-2), are used as observer gains h 1B  and h 2B . In this matrix, using the correction coefficients h 1B * and h 2B *, for example, as observer gains h 1B  and h 2B , equation (69) may be written as                      [           dq     1      B                 dq     2      B             ]     =                    [             -   2          ω   ob           b             -       ω   ob   2     b           0         ]                [           q     1      B                 q     2      B             ]     +                                  [             -     a   1       -       a   2          h     1      B         +     bh     2      B       -     fh     1      B     2     -     dh     1      B                       -     fh     1      B              h     2      B         -     dh     2      B               ]                   φ     +           .                                [         0           1         ]                   v                   (   70   )                         
     In this case, for simplicity of computation, the controller may use equation (70) instead of equation (69) to give the first time derivatives dq 1B  and dq 2B . 
     Referring to FIG. 20, the sub routine  460  to monitor operating variables is now described. As mentioned above, this sub routine is executed in step  442  in FIG.  19 . 
     In step  462 , the controller inputs information of output shaft speed by receiving output shaft speed signal ω o  from sensor  204  (see FIG.  1 ). 
     In step  464 , the controller computes vehicle speed VSP using the output shaft speed by calculating equation (6). 
     In step  466 , the controller inputs information of accelerator pedal position by receiving accelerator pedal position signal APS from encoder associated with the vehicle accelerator pedal. 
     In step  468 , the controller inputs information of input shaft speed by receiving input shaft speed ω i  from sensor  202  (see FIG.  1 ). 
     In step  470 , the controller computers actual speed ratio ic that is expressed by equation (8). 
     In step  472 , the controller inputs information of actual trunnion angular position by receiving actual trunnion angular position signal φ from the encoder arranged around trunnion  100  (see FIG.  2 ). If the provision of such encoder is discouraged, actual trunnion angular position φ may be obtained by using the actual ratio ic determined in step  470  in retrieving the illustrated relationship in FIG.  6 . 
     Referring to FIG. 21, the sub routine  480  to determine desired trunnion angular position command φ* is now described. This sub routine is executed in step  444  in FIG.  19 . 
     In step  482 , the controller determines desired engine speed command ω o  by looking into the illustrated data in FIG. 5, each being indexed by vehicle speed VSP and accelerator pedal position APS. 
     In step  484 , the controller computes desired speed ratio ic*, which is expressed by equation (7). 
     In step  486 , the controller determines desired trunnion angular position command φ* by looking into the illustrated data in FIG. 6, each being indexed by desired ratio command ic*. 
     Referring to FIG. 22, the sub routine  500  to estimate trunnion axial displacement and motor steps is now described. This sub routine is executed in step  448  in FIG.  19 . 
     In step  502 , the controller computes correction coefficients h 1B * and h 2B *. Using equations (59-1), (59-2), (57-1) and (57-2), the correction coefficients h 1B * and h 2B * are given as                h     1      B     *     =         -     a   2       +     2        ω   ob         f             (71-1)                 h     2      B     *     =         ω   ob   2     bf     .             (71-2)                         
     In step  502 , the controller calculates the equations (71-1) and (71-2) using factor of proportionality f to give correction coefficients h 1B * and h 2B *. 
     In step  504 , the controller computes observer gains h 1B  and h 2B  by integrating the time derivatives of observer gains dh 1B  and dh 2B  that were obtained in step  454  in FIG. 19 during the last execution by the controller. To give observer gains h 1B  and h 2B , using the previously obtained values h 1B , h 2B , dh 1B  and dh 2B , the controller calculates the formulas as follows 
     
       
           h   1B   ←h   1B   +Tdh   1B   (72-1) 
       
     
     
       
           h   2B   ←h   2B   +Tdh   2B   (72-2) 
       
     
     where: 
     T is the period of time of each of the predetermined intervals at which the steps illustrated in FIGS. 19-24 are executed and is equal to 0.01, indicative of 10 milliseconds. 
     In step  506 , the controller computes the first time derivatives dh 1B  and dh 2B  of observer gains by calculating low pass filters expressed by equations (61-1) and (61-2) using corrected coefficients h 1B * and h 2B * that have been given in step  502  and observer gains h 1B  and h 2B  that have been given in step  504 . 
     In step  508 , the controller computes quasi-state quantity q 1B , q 2B  by integrating the first time derivatives dq 1B  and dq 2B  that were given in step  454  during the last cycle of execution. To give quasi-state quantities q 1B  and q 2B , using the previously obtained values q 1B , q 2B , dq 1B  and dq 2B , the controller calculates the formulas as follows 
     
       
           q   1B   ←q   1B   +Tdq   1B   (73-1) 
       
     
     
       
           q   2B   ←q   2B   +Tdq   2B   (73-2). 
       
     
     In step  510 , the controller computes estimated value y m  of trunnion axial displacement y and estimated value u m  of motor steps u. To give estimated values y m  and u m , using quasi-state quantities q 1B  and q 2B , actual trunnion angular position φ, and observer gains h 1B  and h 2B , the controller calculates the equation as follows 
     
       
           y   m   =q   1B   +h   1B φ  (74-1) 
       
     
     
       
           u   m   =q   2B   +h   2B φ  (74-2). 
       
     
     Referring to FIG. 23, the sub routine  520  to compute diffeomorphic transform is now described. This sub routine is executed in step  450  in FIG.  19 . 
     In step  522 , the controller computes the first time derivative df (={dot over (f)}) of factor of proportionality f by using a pseudo-differentiator or calculating equation (54). 
     In step  524 , the controller computes an estimated value dφ m  (={dot over (φ)} m ) of the first time derivative (speed) of trunnion angular position φ by calculating equation (52-1). 
     In step  526 , the controller computes an estimated value of the second time derivative αφ m  (={umlaut over (φ)} m ) of trunnion angular position φ by calculating equation (52-2). 
     Referring to FIG. 24, the sub routine to carry out sliding mode control  540  is now described. This sub routine is executed in step  452  in FIG.  19 . 
     In step  542 , the controller computes a control error σ using equation (55-1). 
     In step  544 , the controller computes actuator command in the form of motor steps rate v using equation (55-2). 
     In the preceding embodiments of the present invention, trunnion angular position φ has been used as a physical quantity indicative of ratio established in toroidal drive  12 . The present invention is not limited to the use of trunnion angular position and may be implemented by using any one of other various physical quantities indicative of ratio in toroidal drive  12 . One such example is a ratio ic (=ω i /ω o ) that is expressed by equation (8). This ratio ic is hereinafter indicated by the reference character G. 
     Referring to FIGS. 4 and 9, in another exemplary embodiment of the present invention, the ratio G is used instead of trunnion angular position signal φ. The ratio G may be expressed as a predetermined function of trunnion angular position φ as follows.              G   =       h                   (   φ   )       =         c   g0     -     cos        (       2        c   g1       -   φ     )             c   g0     -     cos                 φ                   (   75   )                         
     where: 
     c g1  and c g0  are the coefficients, which specify the toroidal drive  12 . 
     The physical quantity dG indicative of ratio rate and trunnion axial displacement y are related by a factor of proportionality f′ as 
     
       
           dG=f′y   (76) 
       
     
     The factor of proportionality f′ may be expressed as                      f   ′     =                    ∂   h       ∂   φ          f                 =                    ∂   h       ∂   φ                cos        (       c   g1     -   φ     )            {       c   g0     -     cos        (       2        c   g1       -   φ     )         }         c   f            ω   0                     (   77   )                         
     where: 
     c f  is the coefficient, which specifies the toroidal drive  12 . 
     In a similar manner to obtain the mathematical model of state observer as expressed by equation (18), let us now consider the mathematical model of a state observer that may give an estimated quantity x Ae  of system state quantity x A . The mathematical model of state observer may be expressed as 
     
       
           dx   Ae   =A   A   x   Ae   +B   A   u+H   A ′( dG−dG   m )  (78) 
       
     
     where: 
     G m  is the estimated value of ratio G, and H A ′ is the observer gain          dG   =       C   A   ′          x   A   ′         ,       dG   m     =       C   A   ′          x   Ae   ′                   C   A   ′     =     [     0                   f   ′       ]               H   A   ′     =       [           h     1      A     ′               h     2      A     ′           ]     .                     
     In the equation (78), as the first time derivative dG is not directly measurable, a state change is made as explained below 
     
       
           x   Ae   =q   A   +H   A   ′G   (79) 
       
     
     where: 
     H A ′ is the observer gain. 
     Using the equation (78) and the equation (78) as modified by a differentiating both sides of the latter, we obtain a filter as 
     
       
           dq   A   =A   obA   ′q   A   +D   A   ′G+B   A   u   (80) 
       
     
     where: 
     A obA ′ is the transition coefficient 
     
       
           A   obA   ′=A   A   −H   A   ′C   A ′ 
       
     
     
       
           D   A   ′=A   A   H   A   ′−H   A   ′C   A   ′H   A   ′−dH   A . 
       
     
     With continuing reference to FIGS. 4 and 9, in order to compute estimated quantity x Ae , a state observer  252  uses the filter (80) to give the first time derivative dq A  of quasi-state quantity q A . The first time derivative dq A  is integrated to give quasi-state quantity q A , which is then put into the equation (79) to give the estimated quantity x Ae  of system state quantity x A . The calculation of filter (80) and the subsequent integral operation are carried out by an quasi-state quantity generator  268  to give quasi-state quantity q A . The calculation of equation (79) is carried out by a trunnion axial displacement estimator  276  to give estimated quantity x Ae . Estimator  276  outputs estimated value y m  of trunnion axial displacement y because it is one of matrix elements of the estimated quantity x Ae . 
     In order to keep transition coefficient A obA ′ of filter (80) unaltered, factor of proportionality f′ is used at a correction coefficient generator  264  in determining correction coefficients h 1A ′* and h 2A ′*. To determine such correction coefficients h 1A ′* and h 2A ′*, the generator  264  calculates the equations as follows                h     1      A       ′   *       =     1   -       k     1      A         f   ′                 (81-1)                 h     2      A       ′   *       =     -         k     2      A       +     a   2         f   ′                 (81-2)                         
     The above equations (81-1) and (81-2) are substantially the same as the before-mentioned equations (22-1) and (22-2), respectively, except the provision of factor of proportionality f′ instead of factor of proportionality f. 
     Correction coefficients h 1A ′* and h 2A ′* are used as inputs into a gain and gain time derivative generator  278 . In the embodiment of the present invention, generator  278  uses correction coefficients h 1A ′* and h 2A ′* as observer gains h 1A ′ and h 2A ′, respectively. The observer gain matrix H A ′ in equation (78) may be written as                H   A   ′     =       [           h     1      A       ′   *                 h     2      A       ′   *             ]     =     [           1   -       k     1      A         f   ′                   -         k     2      A       +     a   2         f   ′               ]               (   82   )                         
     Instead of differential operation, generator  278  uses a pseudo-differentiator to give the first time derivatives dh 1A ′ and dh 2A ′. Using the correction coefficients h 1A ′* and h 2A ′* as observer gains h 1A ′ and h 2A ′, respectively, the transition matrix A obA ′ is kept unaltered as shown below                A   obA   ′     =     [         0         k     1      A                 -     a   1             k     2      A             ]             (   83   )                         
     where: 
     the matrix elements k 1A  and k 2A  are the coefficients that determine speed at which state observer  252  makes the estimation. 
     Using eigenvalue ω ob  of state observer  252 , the matrix elements k 1A  and k 2A  of transition coefficient A obA ′ of equation (80) are expressed by the before mentioned equations (24-1) and (24-2), respectively. 
     In another exemplary embodiment of the present invention, a low order state observer outputs an estimated value y m  of trunnion axial displacement y using motor steps u and ratio G as inputs. The mathematical model of such a lower order state observer may be expressed as 
     
       
           dy   m   =−a   2   y   m   +bu−a   1   h   −1 ( G )+ h   r ′( dG−dG   m )  (84) 
       
     
     where: 
     h −1 (G) is an inverse function of the function expressed by equation (74); 
     h r ′ represents an observer gain; 
     dG=f′y 
     dG m =f′y m . 
     The mathematical model expressed by the equation (84) is manipulated into the form as expressed as 
     
       
           dq   r ′=(− a   2   −h   r   ′f ′) q   r   ′+bu +(− a   1   h   −1 ( G )− a   2   h   r   ′G+f′h   r   ′G−dh   r   ′G )  (85) 
       
     
     
       
           y   m   =q   r   ′+h   r   ′G   (86) 
       
     
     where: 
     q r ′ is the quasi-state quantity; 
     (−a 2 −h r ′f′) is the transition coefficient. 
     This state observer has an eigenvalue of ω ob . Thus, a correction coefficient h r ′* may be expressed as                h   r     ′   *       =           ω   ob     -     a   2         f   ′       .             (   87   )                         
     Using the correction coefficient h r ′* as observer gain h r , the transition matrix (−a 2 −h r ′f′) becomes (−ω ob ) and is kept unaltered. 
     Referring to FIGS. 16 and 17, in an exemplary embodiment of the present invention, a state observer  420  outputs an estimated value y m  of trunnion axial displacement y and an estimated value u m  of motor steps u, using motor steps rate (or speed) v and ratio G as inputs. The mathematical model of state observer may be expressed as 
     
       
         dw e   =A   22   w   e   +B   2   v+A   21   h   −1 ( G )+ H   B ′( dG−dG   m )  (88) 
       
     
     where: 
     dG=A 12 ′w 
     dG m =A 12 ′w e                 A   12   ′     =     [           f   ′         0         ]                   H   B   ′     =     [           h     1      B     ′               h     2      B     ′           ]                           
     As the first time derivative dG is not directly measurable, a state change is made as explained 
     
       
           w   e   =q   B   +H   B   ′G   (89). 
       
     
     Using the equation (88) and the equation (89) as modified by differentiating both sides of the latter, we obtain a filter expressed as 
     
       
           dq   B   =A   obB   ′q   B   +D   B   ′G+B   2   v   (90). 
       
     
     where: 
     A obB ′ is the transition coefficient          A   obB   ′     =         A   22     -       H   B   ′          A   12   ′         =     [             -     a   2       -       h     1      B     ′          f   ′             b               -     h     2      B     ′            f   ′           0         ]                       
     
       
           D   B ′=A 22 H B ′G+A 12 h −1 (G)−H B ′A 12   ′H   B ′G−dH B ′G.  
       
     
     With continuing reference to FIGS. 16 and 1 7 , in order to compute estimated quantity we, state observer  420  uses filter (90) to give the first time derivative dq B  of quasi-state quantity q B . The first time derivative dq B  is integrated to give quasi-state quantity q B , which is then put into the equation (89) to give the estimated quantity w e  of state quantity w. The calculation of filter (90) and the subsequent integral operation are carried out by an quasi-state quantity generator  268  in FIG. 17 to give quasi-state quantity q B ′. The calculation of equation (89) is carried out by a trunnion axial displacement estimator  276  in FIG. 17 to give estimated quantity w e . Estimator  276  outputs estimated value y m  of trunnion axial displacement y and estimated value u m  of motor steps u because they are matrix elements of the estimated quantity w e . 
     In order to keep transition matrix A obB ′ of equation (89) unaltered, factor of proportionality f′ is used at a correction coefficient generator  264  in FIG. 17 in determining correction coefficients h 1B ′* and h 2B ′*. To determine such correction coefficients h 1B ′* and h 2B ′*, the generator  264  calculates the equations as follows                h     1      B       ′   *       =         -     a   2       -     k     1      B           f   ′               (91-1)                 h     2      B       ′   *       =     -         k     2      B         f   ′       .               (91-2)                         
     The above equations (91-1) and (91-2) are substantially the same as the before-mentioned equations (57-1) and (57-2), respectively, except the provision of factor of proportionality f′ instead of factor of proportionality f. 
     Correction coefficients h 1B ′* and h 2B ′* are used as inputs into a gain and gain time derivative generator  278  in FIG.  17 . In the embodiment of the present invention, generator  278  uses correction coefficients h 1B ′* and h 2B ′* as observer gains h 1B ′ and h 2B , respectively. The observer gain matrix H B ′ in equation (88) may be written as                H   B   ′     =       [           h     1      B       ′   *                 h     2      B       ′   *             ]     =       [               -     a   2       -     k     1      B           f   ′                 -       k     2      B         f   ′               ]     .               (   92   )                         
     Instead of differential operation, generator  278  uses a pseudo-differentiator to give the first time derivatives dh 1B ′ and dh 2B ′. Using the correction coefficients h 1B ′* and h 2B ′* as observer gains h 1B ′ and h 2B ′, respectively, the transition matrix A obB ′ is kept unaltered. 
     While the present invention has been particularly described, in conjunction with exemplary embodiments, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art in light of the foregoing description. It is therefore contemplated that the appended claims will embrace any such alternatives, modifications and variations as falling within the true scope and spirit of the present invention. 
     This application claims the priority of Japanese Patent Application No. P2001-029547, filed Feb. 6, 2001, the disclosure of which is hereby incorporated by reference in its entirety.