Patent Publication Number: US-7711453-B2

Title: Positioning control system and filter

Description:
BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to a positioning control system having a disturbance observer for estimating a disturbance from at least one observed value of a controlled object and feeding back an estimated disturbance. The present invention also relates to a filter that is used in such a positioning control system. 
   2. Description of the Related Art 
   Positioning control systems are widely used in production facility equipment. In clean room environments, particularly, pneumatic cylinders employed by positioning control systems are highly advantageous, since they are free of liquid leakage and can easily be serviced for maintenance. 
   However, pneumatic cylinders do not lend themselves to highly accurate and highly rigid positioning owing to their behavior, on account of air compressibility and limitations on the pneumatic pressure that can be used. In particular, a pneumatic cylinder having a seal mounted on a piston, for preventing leakage of air and entry of external foreign matter, is unable to perform highly accurate positioning due to a so-called stick slip phenomenon caused by frictional forces of the seal. 
   Japanese Laid-Open Patent Publication No. 2004-144196 discloses a positioning system comprising a pneumatic cylinder, which includes a piston incorporating a static pressure bearing in order to reduce frictional forces and provide increased positioning accuracy. 
   For increased control accuracy, some applications employ a disturbance observer for estimating a disturbance from at least one observed value of a controlled object and feeding back the estimated disturbance. For example, Japanese Laid-Open Patent Publication No. 2000-347738 reveals a positioning system employing a variable-gain disturbance observer for stably controlling an object irrespective of whether static friction or dynamic friction is involved in moving the object. 
   It is known in the art that it is effective to increase a control gain in order to increase positioning control accuracy. However, in order to protect the actuator, it is necessary to provide a suitable saturating element for preventing the command values from becoming too large. 
   The viscoelastic characteristics due to frictional forces of a seal in a pneumatic cylinder and the effect of such characteristics on control performance are described in “Viscoelasticity in Displacement of a Pneumatic Cylinder and its Effect on Control Performance,” by Osamu Oyama et al., a collection of articles of the Japan Fluid Power System Society, November 1998, Vol. 29, No. 7, pp. 19(155)-25(161). The article shows that the viscoelasticity of a seal is effective for stabilizing piston displacement more than when the piston slides, and it is possible, using this effect, to increase control sensitivity and thereby increase settling speed, so that displacement of the piston can be stabilized as a result of such viscoelasticity. 
   The positioning system disclosed in Japanese Laid-Open Patent Publication No. 2004-144196 is complex in structure and highly expensive, since it employs a static pressure bearing. Since the static pressure bearing causes too small an amount of friction, it has poor damping characteristics, tending to increase the convergence time for positioning. 
   The disturbance observer employed by the positioning system disclosed in Japanese Laid-Open Patent Publication No. 2000-347738 generally includes a low-pass filter, and exhibits integrator characteristics in a low frequency range. It is also known in the art that if a control system includes an integrator, then it suffers from a phenomenon known as integrator windup. 
   Integrator windup is a situation where, when a given positive deviation is steadily applied, the positive deviation is integrated excessively beyond a saturated value L of the manipulated variable applied to a controlled object, as shown in  FIG. 35  of the accompanying drawings. If a negative deviation is steadily applied after time T 1 , then it is desirable under normal circumstances for the manipulated variable to be reduced immediately from time T 1 , as indicated by the broken-line curve  1 . However, because of windup, the manipulated variable starts being reduced from an integral S, at time T 1 , as indicated by the solid-line curve  2 , and the controlled object actually changes its operation from time T 2 . Therefore, the manipulated variable remains saturated at the saturated value L from time T 1  to time T 2 . The controlled object thus suffers a delay in its operation, tending to result in a reduction in its control performance, e.g., increased overshooting. In order to protect the control system against integrator windup, the integrator may have a function to stop its integrating action beyond the saturated value L. 
   A simple integrator may relatively easily be designed to incorporate a means for stopping its integrating action beyond the saturated value L. However, no attempts have heretofore been made to add such a means to a disturbance observer, and to determine a location where such a means is provided with respect to a disturbance observer. Hence, it has not been possible to prevent integrator windup in disturbance observers. 
   SUMMARY OF THE INVENTION 
   It is an object of the present invention to provide a positioning control system incorporating a disturbance observer for increasing control accuracy and preventing integrator windup, as well as to provide a filter for use in such a positioning control system. 
   According to an aspect of the present invention, there is provided a positioning control system comprising a disturbance observer for estimating a disturbance from at least one observed value of a controlled object and feeding back an estimated disturbance, and a saturation element disposed in a feedback loop for feeding back the estimated disturbance from the disturbance observer, the feedback loop having a main loop based on an output value of the controlled object, and a minor loop for performing positive feedback based on a predetermined parameter, wherein the saturation element is disposed in the minor loop. 
   The positioning control system produces an integrator windup, which is restrained for an increased control capability, with the saturation element disposed in the positive-feedback minor loop of the disturbance observer. The disturbance observer operates to compensate for the disturbance in order to highly accurately and rigidly position the controlled object. 
   The minor loop may perform the positive feedback through two subtractors. 
   The saturation element may be saturated with positive and negative values whose absolute values are equal to each other. 
   If the positioning control system has a saturated value changer for changing a saturated value of the saturation element based on a control deviation, then the saturated value is appropriately set based on the control variation, thereby providing an increased control capability. 
   The controlled object may comprise a cylinder having a seal on a slidable component thereof, and the saturated value of the saturation element may have an absolute value that is greater when the control deviation falls outside of the viscoelastic displacement range of the seal than when the control deviation falls within a viscoelastic displacement range of the seal. If the control deviation is small, then a small saturated value is set for sufficiently restraining integrator windup. If the control deviation is large, then a large saturated value is set for enabling increased dynamic characteristics. 
   The cylinder may comprise a pneumatic cylinder. 
   If the seal comprises a piston seal and a cap seal, then a viscoelastic displacement range may be established based on a hypothetical combined seal, including the piston seal and the cap seal. 
   The cylinder may be bidirectionally actuatable by a proportional valve. 
   If the saturated value changer changes the saturated value stepwise based on the control deviation, then the saturated value can easily be calculated. 
   According to another aspect of the present invention, there is also provided a filter in a control system having a main feedback loop, comprising a minor loop for performing positive feedback within the main feedback loop, a low-pass element disposed in a feedback path of the minor loop, and a saturation element disposed in a forward path of the minor loop. 
   The filter operates in the same manner as an integrating process so as to restrain integrator windup, with the low-pass element being disposed in the feedback path of the minor loop and the saturation element being disposed in the forward path thereof. 
   As described above, in the positioning control system according to the present invention, integrator windup is restrained for enabling increased control capability, with the saturation element being disposed in the positive-feedback minor loop of the disturbance observer. The disturbance observer operates to compensate disturbances, so as to highly accurately and rigidly position the controlled object. 
   As described above, the filter according to the present invention operates in the same manner as an integrating process so as to restrain integrator windup, with the low-pass element being disposed in the feedback path of the minor loop and the saturation element being disposed in the forward path thereof. 
   The above and other objects, features, and advantages of the present invention shall become more apparent from the following description when taken in conjunction with the accompanying drawings in which preferred embodiments of the present invention are shown by way of illustrative example. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a block diagram, partly in cross section, of a positioning control system according to an embodiment of the present invention; 
       FIG. 2  is an enlarged fragmentary cross-sectional view of a piston; 
       FIG. 3  is a block diagram of a controller; 
       FIG. 4  is a block diagram of a positive-feedback filter; 
       FIG. 5  is a Bode diagram showing gain characteristics of the positive-feedback filter; 
       FIG. 6  is a Bode diagram showing phase characteristics of the positive-feedback filter; 
       FIG. 7  is a block diagram of a saturated positive-feedback filter; 
       FIG. 8  is a diagram showing gain characteristics of the saturated positive-feedback filter; 
       FIG. 9  is a diagram showing simulated outputs of the positive-feedback filter and the saturated positive-feedback filter at the time an input amplitude X is 0.31 at a low frequency; 
       FIG. 10  is a diagram showing simulated outputs of the positive-feedback filter and the saturated positive-feedback filter at the time an input amplitude X is 0.35 at a low frequency; 
       FIG. 11  is a diagram showing simulated outputs of the positive-feedback filter and the saturated positive-feedback filter at the time an input amplitude X is 0.5 at a low frequency; 
       FIG. 12  is a diagram showing simulated outputs of the positive-feedback filter and the saturated positive-feedback filter at the time an input amplitude X is 1.0 at a low frequency; 
       FIG. 13  is a diagram showing simulated outputs of the positive-feedback filter and the saturated positive-feedback filter at the time an input amplitude X is 2.35 at a low frequency; 
       FIG. 14  is a diagram showing simulated outputs of the positive-feedback filter and the saturated positive-feedback filter at the time an input amplitude X is 2.8 at a low frequency; 
       FIG. 15  is a diagram showing simulated outputs of the positive-feedback filter and the saturated positive-feedback filter at the time an input amplitude X is 3.5 at a low frequency; 
       FIG. 16  is a diagram showing simulated outputs of the positive-feedback filter and the saturated positive-feedback filter at the time an input amplitude X is 5.0 at a low frequency; 
       FIG. 17  is a block diagram of a positioning system employing a PID controller; 
       FIG. 18  is a block diagram of a positioning system similar to the control system shown in  FIG. 17 , except that a positive-feedback filter is used instead of the integrator that is used in the control system shown in  FIG. 17 ; 
       FIG. 19  is a block diagram of a positioning system similar to the control system shown in  FIG. 17 , except that a saturated positive-feedback filter is used instead of the integrator that is used in the control system shown in  FIG. 17 ; 
       FIG. 20  is a diagram showing simulated outputs at the time a saturation element is not saturated in the positioning systems shown in  FIGS. 17 and 18 ; 
       FIG. 21  is a diagram showing a simulated input signal of the saturation element, which produces the simulated outputs shown in  FIG. 20  at the time the saturation element is not saturated; 
       FIG. 22  is a diagram showing simulated outputs at the time a saturation element is saturated in the positioning systems shown in  FIGS. 17 and 18 ; 
       FIG. 23  is a diagram showing a simulated input signal of the saturation element, which produces the simulated outputs shown in  FIG. 20  at the time the saturation element is saturated; 
       FIG. 24  is a diagram showing simulated outputs at the time a saturation element is saturated in the positioning systems shown in  FIGS. 17 and 18 , wherein the saturated value of the saturation element is smaller than a disturbance; 
       FIG. 25  is a diagram showing a simulated input signal of the saturation element, which produces the simulated outputs shown in  FIG. 24  at the time the saturation element is saturated; 
       FIG. 26  is a block diagram of a system wherein a saturated positive-feedback filter is not incorporated in a disturbance observer; 
       FIG. 27  is a block diagram of a system wherein a saturated positive-feedback filter is incorporated in a disturbance observer; 
       FIG. 28  is a diagram showing simulated outputs at the time a saturation element is not saturated in the systems shown in  FIGS. 26 and 27 ; 
       FIG. 29  is a diagram showing a simulated estimated disturbance value at the time a saturation element is not saturated in the systems shown in  FIGS. 26 and 27 ; 
       FIG. 30  is a diagram showing simulated outputs at the time a saturation element is saturated in the systems shown in  FIGS. 26 and 27 ; 
       FIG. 31  is a diagram showing a simulated estimated disturbance value at the time a saturation element is saturated in the system shown in  FIG. 26 ; 
       FIG. 32  is a diagram showing a simulated estimated disturbance value at the time a saturation element is saturated in the system shown in  FIG. 27 ; 
       FIG. 33  is a graph showing saturated values set by a saturated value changer; 
       FIG. 34  is a graph showing modified saturated values set by the saturated value changer; and 
       FIG. 35  is a timing chart of integrals produced by an integrator, which are illustrative of integrator windup. 
   

   DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Positioning control systems and filters according to embodiments of the present invention shall be described below with reference to  FIGS. 1 through 34 . 
   As shown in  FIG. 1 , a positioning control system  10  according to an embodiment of the present invention generally comprises a pneumatic cylinder  12  and a proportional valve  40  as a controlled object P, and a controller  14  for controlling the pneumatic cylinder  12 . According to the present embodiment, the controller  14  incorporates a saturated positive-feedback filter  110  (see  FIG. 7 ). 
   The pneumatic cylinder  12  is used, for example, to position a workpiece  20  in a clean room of a semiconductor fabrication factory. The pneumatic cylinder  12  has a grip  22  for gripping the workpiece  20 , a rod  24  supporting the grip  22  on one end thereof, a piston  26  mounted on the other end of the rod  24 , a tube  28  in which the piston  26  slidably moves, a cap  30  mounted on an end of the tube  28  and supporting the rod  24 , a piston seal  32  disposed around the piston  26 , a cap seal  34  mounted on the cap  30 , and a sensor  36  for detecting a displaced distance Y of the rod  24 . The piston  26  is a single-sided rod type piston. The tube  28  is divided by the piston  26  into a rod chamber  28   a , in which the rod  24  is disposed, and a bottom chamber  28   b  opposite to the rod chamber  28   a  across the piston  26 . The rod chamber  28   a  and the bottom chamber  28   b  are held in fluid communication with respective ports  38 . The ports  38  are controlled by a proportional valve  40  for selective fluid control. The proportional valve  40  is controlled by the controller  14  to supply compressed air from an air pressure source  41  to one of the ports  38 , and to vent the other port  38  to atmosphere. The piston  26  is slidingly movable within the tube  28  to a maximum displaced distance Ymax. 
   As shown in  FIG. 2 , the piston seal  32  serves to prevent leakage of air between the rod chamber  28   a  and the bottom chamber  28   b . The piston seal  32  has an inner circumferential portion, which is fitted into an annular groove  26   a  defined in an outer circumferential surface of the piston  26 , and an outer circumferential portion that is slightly compressed and held against an inner circumferential surface of the tube  28 . The piston seal  32  may comprise any of various configurations, and should preferably have a substantially O-shaped cross section, or a substantially X-shaped cross section, depending on the design conditions. 
   When the rod  24  is still at rest, the outer circumferential portion of the piston seal  32  is held against the inner circumferential surface of the tube  28  and undergoes a static frictional force. Therefore, when the bottom chamber  28   b  is pressurized to move the piston  26  slightly toward the cap  30 , the outer circumferential surface of the piston seal  32  remains unmoved under the static frictional force, and is viscoelastically deformed as the rod  24  is displaced. When the rod  24  is further moved until the pressure applied to the piston  26  exceeds the static frictional force imposed thereon by the piston seal  32 , the piston seal  32  starts to move, and the tube  28  experiences a dynamic frictional force that is smaller than the static frictional force. The pneumatic cylinder  12  operates similarly when the rod chamber  28   a  is pressurized to move the piston  26  away from the cap  30 . 
   As described above, while the piston  26  and the rod  24  are moving in a viscoelastic displacement range Yo due to the viscoelasticity of the piston seal  32 , i.e., within a range in which the piston seal  32  is elastically deformed but is held at rest without sliding against the tube  28 , a resistive force is generated due to the viscoelasticity of the piston seal  32 , which provides appropriate damping characteristics. In addition to the piston seal  32  that has been described above, the cap seal  34  also provides similar damping characteristics, because it is viscoelastically held against the rod  24 . Therefore, the viscoelastic displacement range Yo is determined by the combined viscoelasticity of the piston seal  32  and the cap seal  34 . 
   As shown in  FIG. 3 , the controller  14  comprises a position feedback loop  42 , a speed feedback loop  44 , a disturbance observer  46 , a saturation element  48  (also referred to as a limiter), and a saturated value changer  50 . A first subtractor  52  subtracts the displaced distance Y from a command value R to determine a deviation (control deviation) ε. The displaced distance Y is supplied from the sensor  36 . The deviation ε is converted by an element C into a parameter C·ε, which is supplied to a second subtractor  54 . The element C amplifies the deviation ε and performs a predetermined compensating process on the deviation ε. 
   The speed feedback loop  44  includes a differentiating element B for differentiating the displaced distance Y in order to determine a speed v. A second subtractor  54  subtracts the speed v from the parameter C·ε to determine a parameter Uv, which is supplied to a third subtractor  56 . 
   The disturbance observer  46  comprises a known control means for estimating a parameter Do from the displaced distance Y as observed by the sensor  36  and a manipulated variable (a given parameter) Us for the controlled object P, and feeding back the estimated parameter Do. The disturbance observer  46  comprises an element  60  for processing the displaced distance Y, a fourth subtractor  62 , and an element Q. The element  60  performs a process represented by 1/Pn, i.e., the element  60  determines a parameter Y/Pn by dividing the displaced distance Y by Pn, and supplies the parameter Y/Pn to the fourth subtractor  62 . The parameter Pn represents a model that approximates the controlled object P with an nth-order function. If the model Pn has an appropriate level of accuracy, and there is no disturbance D (D=0), then the displaced distance Y is expressed as Y←Us·P. Therefore, the parameter Y/Pn is expressed as Y/Pn=Us (≈Us·P/Pn). 
   The fourth subtractor  62  subtracts the manipulated variable Us from the parameter Y/Pn in order to determine a parameter Di, and supplies the parameter Di to the element Q. The element Q is a low-pass element. The element Q performs a low-pass process on the parameter Di in order to determine a parameter Do (hereinafter referred to as an estimated disturbance value Do), and supplies the parameter Do to the third subtractor  56 . If the element Q is expressed by a transfer function, then its denominator degree is set to a value equal to or greater than the denominator degree of Pn. 
   The third subtractor  56  subtracts the parameter Do from the parameter Uv in order to determine a basic manipulated variable U, and supplies the basic manipulated variable U to the saturation element  48 . The saturation element  48 , the fourth subtractor  62 , the element Q, and the third subtractor  56  jointly make up a minor loop  70 . In the minor loop  70 , the manipulated variable Us acts as the subtrahend in the fourth subtractor  62 , and then goes through the element Q and acts again as the subtrahend in the third subtractor  56 . Therefore, the process in the minor loop  70  is equivalent to a process of multiplying “−1” twice and hence multiplying “1” once. Consequently, the minor loop  70  is a positive-feedback loop. The low-pass element Q is placed in a feedback path of the positive-feedback minor loop  70 , whereas the saturation element  48  is placed in a forward path of the positive-feedback minor loop  70 . 
   The saturation element  48  limits the basic manipulated variable U with the saturated value L if the basic manipulated variable U exceeds a positive value of the saturated value L, and further limits the basic manipulated variable U with a negative value −L of the saturated value L if the basic manipulated variable U is smaller than the negative value −L. Therefore, the limited manipulated variable Us falls within a range of −L&lt;Us&lt;L. If the basic manipulated variable U is not limited, then U=Us (the gain is 1). The positive and negative saturated values L and −L are changed by the saturated value changer  50  depending on the absolute value of the deviation ε. Operational details of the saturated value changer  50  shall be described later. 
   Since the manipulated variable Us is supplied to the proportional valve  40 , the piston  26  of the cylinder  12  is actuated such that the command value R equals the displaced distance Y (R=Y). 
   If there is no disturbance D, then Y/Pn=Us. Therefore, the parameters Di and Do are nil, and the feedback process of the disturbance observer  46  does not operate, with the result that the controlled object P is positionally controlled by the position feedback loop  42  and the speed feedback loop  44 , in the same manner as a classical control system. 
   Actually, however, the controlled object P suffers an error due to the disturbance D, wherein the disturbance D is compensated for by the disturbance observer  46 . Specifically, if the disturbance D is not nil (D≠0), then the parameter Di has a value depending on the disturbance D. The parameter Di is processed in accordance with a low-pass process by the element Q to result in the parameter Do, which is subtracted from the parameter Uv by the third subtractor  56 . Even if the parameter Uv is nil, the parameter Do, depending on the disturbance D, produces values Us=U=−Do (not limited with the threshold by the saturation element  48 ), Us=L (limited with the positive threshold by the saturation element  48 ), or Us=−L ((limited with the negative threshold by the saturation element  48 ), which control the controlled object P so as to cancel out the disturbance D. 
   Simulations for confirming operations of the positioning control system  10  thus constructed shall be described below with reference to  FIGS. 4 through 32 . The simulations have been carried out by the inventor of the present invention using a computer, for the purpose of confirming the effects of the saturation element  48  provided mainly within the minor loop  70 . 
   1) Positive-Feedback Filter: 
   First, a positive-feedback filter  100  (see  FIG. 4 ), which is obtained by removing the saturation element  48  from the minor loop  70  and adding a proportional element ka instead, will be analyzed. For generalizing the filter arrangement, a proportional element kq is inserted into a path leading to the input of the positive-feedback filter  100 . 
   As shown in  FIG. 4 , the positive-feedback filter  100  has a proportional element kq in the input stage, an adder  102 , a proportional element ka in the loop, and an element Q in the loop. The loop Q is a low-pass element. The proportional element ka is placed in a forward path of the loop, whereas the element Q is placed in a feedback path of the loop. An output from the element Q is added to an output from the proportional element kq by the adder  102 . The positive-feedback filter  100  has a transfer function G(s) expressed by the following equation (1): 
   
     
       
         
           
             
               
                 
                   G 
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   Assuming the element Q is typically given as Q=100/(s+10) 2 , ka=1, and kq=0.2, then the transfer function G(s) is given as follows: 
   
     
       
         
           
             
               
                 
                   
                     
                       
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   It can be seen from equation (2) that the positive-feedback filter  100  has a proper transfer function with a pole at the origin, and has characteristics equal to an integrator of 1/s in a low frequency range. If ka≠1, then the positive-feedback filter  100  has no origin pole, and has a stable and proper transfer function, exhibiting characteristics similar to those of an ordinary phase delay compensating circuit. Frequency characteristics of the positive-feedback filter  100  are shown in  FIGS. 5 and 6 . 
   If ka=1, then the positive-feedback filter  100  is equivalent to the form 1/(1−Q) of the minor loop  70  of the disturbance observer  46 . 
   2) Saturated Positive-Feedback Filter: 
   As shown in  FIG. 7 , a saturated positive-feedback filter  110  having a saturation element  48  with a neutral gain of 1, which is inserted in place of the proportional element ka in the forward path of the positive-feedback filter  100  shown in  FIG. 4 , shall be analyzed below. The saturated positive-feedback filter  110  is equivalent in configuration to the minor loop  70 . 
   When the magnitude of an input of the saturation element  48  is in an unsaturated range near a neutral point, the saturated positive-feedback filter  110  has frequency characteristics equal to those of the positive-feedback filter  100  shown in  FIG. 4  in which ka=1, and has a pole at the origin. 
   If the input of the saturation element  48  is represented by x, and the output thereof by y, then y=x in an unsaturated range of |x|&lt;x 0 , y=x 0  in a saturated range of x&gt;x 0 , and y=−x 0  in a saturated range of x&lt;−x 0 . If an output y produced when an input x=Xsin(ωt) is applied is represented by y=Ysin(ωt+θ), then a describing function of the saturation element  48  is expressed as follows: 
   If α=X/x 0 ≦1, then the gain of the saturation element  48  is Y/X=1 and the phase shift θ thereof is θ=0. 
   If α=X/x 0 &gt;1, then the gain of the saturation element  48  is Y/X=k(α) and the phase shift θ thereof is θ=0 where 
   
     
       
         
           
             
               
                 
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   As shown in  FIG. 8 , as the input amplitude increases, a increases. If α≦1, then Y/X=1. As a increases further, k(α) decreases in inverse proportion. If the input amplitude of the saturation element  48  becomes greater than the saturated value (x 0 ), then the equivalent gain of the saturation element  48  becomes smaller than 1. In this case, the frequency characteristics of the saturated positive-feedback filter  110  (see  FIG. 7 ) are presumed to be similar to those obtained when ka in the positive-feedback filter  100  (see  FIG. 4 ) is smaller than 1. In other words, integral characteristics are restrained. 
     FIGS. 9 through 16  show simulated cosine inputs, in the event that the output of the saturated positive-feedback filter  110  is represented by y 1  and the output of the positive-feedback filter  110  is represented by y 2 . In these simulations, the neutral point gain of the saturation element  48  of the saturated positive-feedback filter  110  is 1, and the saturated value (x 0 ) is 0.5.  FIGS. 9 through 12  show responses to a low frequency (0.2×π(rad/s)), whereas  FIGS. 13 through 16  show responses to a high frequency (2×π(rad/s)). The maximum input amplitude X is X=0.31 in  FIG. 9 , X=0.35 in  FIG. 10 , X=0.5 in  FIG. 11 , X=1.0 in  FIG. 12 , X=2.35 in  FIG. 13 , X=2.8 in  FIG. 14 , X=3.5 in  FIG. 15 , and X=5.0 in  FIG. 16 . In  FIGS. 9 through 16 , the left vertical axis represents graduations for the outputs y 1 , y 2 . For the sake of brevity, graduations for the input X have been omitted from illustration. 
   At the frequency ω=0.1 Hz (0.2×π(rad/s)) ( FIGS. 9 and 10 ), the cosine input amplitude X is equal to or smaller than about 0.1π and the input of the saturation element  48  is equal to or smaller than a saturated value of 0.5, with the output y 1  and the output y 2  being equal to each other. 
   At the frequency ω=0.1 Hz (0.2×π(rad/s)), the maximum amplitude Ya of the output y 1  of the positive-feedback filter  100  for the cosine input X=0.1π is determined as Ya=0.5015 in accordance with the above equation (2). 
   As can be seen from  FIGS. 9 and 13 , if no saturation occurs, the output y 1  of the saturated positive-feedback filter  110  and the output y 2  of the positive-feedback filter  100  are equal to each other. If saturation occurs, the response of the output y 1  for a low-frequency input is ahead in phase of the output y 2  having integral characteristics, and the amplitude is limited by the saturated value of 0.5. 
   For a high-frequency input, as can be seen from the frequency characteristics shown in  FIGS. 5 and 6 , the output y 1  and the output y 2  have similar forms, except that the amplitude of the output y 1  is limited by the saturated value. 
   From the above results, it has been confirmed that if the input amplitude is equal to or smaller than the saturated value, then the low-frequency gain of the saturated positive-feedback filter  110  has characteristics similar to the integration element (1/s). Further, in a range in which the input amplitude exceeds the saturated value, the output amplitude is limited and integral characteristics are restrained, resulting in a leading phase. 
   3) Integrator Windup Suppression by the Saturated Positive-Feedback Filter  110 : 
   As described above, if saturation occurs with respect to the integrator, integration windup is caused. If the saturated positive-feedback filter  110  is used instead of an ordinary integrator (1/s), then it functions as an integrator within a small-deviation range, and can restrain integral characteristics for large deviations, which would otherwise cause integrator windup, thereby suppressing integrator windup. An application of the saturated positive-feedback filter  110  to servo system PID control, for example, shall be described below. 
     FIG. 17  shows a first system model  200 , which is a positioning system employing a PID controller  202 .  FIG. 18  shows a second system model  210 , which is a positioning system employing the positive-feedback filter  100  in place of the integrator  204  of the first system model  200 .  FIG. 19  shows a third system model  220 , which is a positioning system employing the saturated positive-feedback filter  110  in place of the integrator  204  of the first system model  200 . It is assumed that a control valve  150 , which corresponds to the proportional valve  40 , has a saturation of ±0.2. In  FIGS. 17 through 19 , the reference numeral  152  represents a model of the cylinder  12 . 
   It is assumed that the command value R is a step input having a magnitude of 1, and the disturbance D is a step input having a magnitude of 0.5, which is generated at a time td later than the command value R. The PID controller  202  has a proportional element  206  of “3”, a differential element  208  of “0.06 s/(0.006 s+1)”, and a proportional element Kp of “0.6”. The cylinder  12  has a transfer function of 52000/(s 3 +125 s 2 +2500 s), and the saturation element  48  has a saturated value of ±0.6. For the sake of brevity, the speed feedback loop  44 , the element  60 , and the saturated value changer  50  in the positioning control system  10  (see  FIG. 3 ) have been omitted from illustration in  FIGS. 17 through 19 . 
   Simulated outputs S 1 , S 2 , S 3  of the respective first, second, and third system models  200 ,  210 ,  220  when kq=0.2 are shown in  FIG. 20 . As shown in  FIG. 21 , the input i of the saturation element  48  of the third system model  220  does not exceed the saturated range of ±0.6. Therefore, the outputs S 2  and S 3  are equal to each other. 
   Simulated outputs S 1 , S 2 , S 3  of the respective first, second, and third system models  200 ,  210 ,  220  when kq=2.0 are shown in  FIG. 22 . It can be observed from  FIG. 22  that integrator windup for the step input is large in the output S 2 , and is restrained in the output S 3 . Responses to the disturbance D after time td are not different from each other. Reasons for such responses are as follows: As shown in  FIG. 23 , the input i of the saturation element  48  of the third system model  220  has reached the saturated range (equal or higher than 0.6) of the saturation element  48  upon a response to the step input, so that the third system model  220  functions to restrain integrator windup. Upon a response to the disturbance D after time td, since the input i of the saturation element  48  has not reached the saturated range of 0.6, the outputs S 2  and S 3  are equal to each other. Naturally, the responses are different from each other depending on the saturated value and the magnitude of the disturbance. 
   Simulated outputs S 1 , S 2 , S 3  of the respective first, second, and third system models  200 ,  210 ,  220 , when kq=0.2 in the second system model  210 , kq=2.0 in the third system model  220 , and the saturated value of the saturation element is ±0.45, are shown in  FIG. 24 . Responses to the step input are the same as those in  FIGS. 20 and 22 , but responses to the disturbance D are different from each other. 
   The outputs S 1  and S 2  have a steady-state deviation of 0 with respect to the disturbance D due to the integral characteristics in a low-frequency range. However, the output S 3  of the third system model  220  suffers a steady-state deviation of Ea. This is because the input i of the saturated positive-feedback filter  110  exceeds the saturated value of “0.6” upon response to the disturbance D, as shown in  FIG. 15 , and is caused by the restrained integral characteristics in the low-frequency range of the saturated positive-feedback filter  110 . 
   Generally, if a plant with integral characteristics has a constant output value in a steady state, then the input of the plant has to be nil. Since the output of a differentiator  222  of the third system model  220  is nil, the output of the saturated positive-feedback filter  110  is 0.45, and the step quantity of the disturbance D is −0.5, the output of the proportional element  206  must be 0.05 (see  FIG. 19 ). 
   Therefore, a steady-state deviation with respect to the disturbance D remains with an Error=0.05/0.6=0.08333. Based on these simulated results, a trade-off to be described below is taken into consideration when setting a threshold value for the saturated positive-feedback filter  110 . To give priority to suppression of integrator windup due to saturation of the plant input and the integral element, the saturated value of the saturation element  48  in the saturated positive-feedback filter  110  is set to a low level, so as to cause the saturation element  48  to be saturated. In order to make the steady-state deviation with respect to the disturbance D nil or as small as possible, the saturated value is set to a sufficiently large level. Consequently, the saturated value may be set taking into consideration a balance between the suppression of integrator windup and minimization of the steady-state deviation, depending on the design conditions. 
   4) The Effect of the Saturated Positive-Feedback Filter  110  Applied to the Disturbance Observer: 
   The effect of the saturated positive-feedback filter  110  applied to the disturbance observer  46  shall be described below, based on simulations using two systems shown in  FIGS. 26 and 27 .  FIG. 26  shows a system model  300  as a comparative example, and  FIG. 27  shows a system model  310 , which incorporates the saturated positive-feedback filter  110  in place of the positive-feedback filter  100  used in the system model  300 . The system model  310  represents a model of the positioning control system  10  referred to above. In the system model  310 , the saturation element  48  has a saturated value of ±0.6, the proportional valve  40  has a saturated value of ±0.2, the element Q has a value of Q=100 3 /(s+100) 3 , the cylinder  12  has a transfer function of 52000/(s 3 +125 s 2 +2500 s), and the element C has a gain of 1. For the sake of brevity, the speed feedback loop  44  and the compensating circuit for the saturation characteristics, and response improvement of the model of the proportional valve  40  in the nominal model (element  60 ), have been omitted from illustration. The output of the system model  300  is represented by z 1 , and the output of the system model  310  is represented by z 2 . It is assumed that the command value R is a step input having a magnitude of 1, whereas the disturbance D is a step input having a magnitude of 0.5, which is generated at a time td later than the command value R. In the system model  310 , the element Q is shown as being divided into a portion corresponding to a pre-subtractor path  61  and the element  60  in  FIG. 3 . These divided components make up a circuit that is equivalent to the element Q. 
   As shown in  FIG. 28 , when the plant is not saturated, the outputs z 1 , z 2  are substantially equal to each other. As shown in  FIG. 29 , estimated disturbance values Do, Doc of the system models  300 ,  310  and the disturbance D are substantially equal to each other. Therefore, it is confirmed that the disturbance D is properly estimated. 
   As shown in  FIG. 30 , when the plant is saturated to a saturated value of ±0.2, the outputs z 1  and z 2  are clearly different from each other. Specifically, the output of the system model  300  suffers from integrator windup due to the plant saturation. As shown in  FIG. 31 , the estimated disturbance value Doc has an inaccurate waveform compared with the disturbance D. 
   On the other hand, as shown in  FIG. 30 , the output z 2  of the system model  310  indicates a suppressed integrator windup. It is thus confirmed that the saturated positive-feedback filter  110  is effective. As shown in  FIG. 32 , the estimated disturbance value Do converges to a proper estimated value within a short period of time. 
   As described above, the system model  310 , which incorporates the saturated positive-feedback filter  110  in the disturbance observer  46 , has integrator windup restrained as with the third system model  220  under PID control. Since the disturbance observer  46  is employed, the disturbance D is compensated for. If the step quantity of the command value R is reduced, then the integrator windup is further restrained. 
   Operation and advantages of the saturated value changer  50  in the positioning control system  10  shall be described below. The saturated value changer  50  serves to change the saturated value L (and −L) of the saturation element  48  based on an absolute value of the deviation ε. 
   Specifically, as shown in  FIG. 33 , if the absolute value |ε| of the deviation ε is |ε|&lt;ε 1 , then the saturated value L is set to L←L 1  or the saturated value −L is set to −L←−L 1 . The set value L 1  is a sufficiently small value capable of sufficiently restraining integrator windup. If the absolute value |ε| is in the range of ε 1 ≦|ε|&lt;ε 2 , then the saturated value L is proportionally increased (or the saturated value −L is proportionally reduced) dependent on the absolute value |ε| from the set value L 1  as a reference for the positive saturated value (or from the set value −L 1  as a reference for the negative saturated value). When |ε|=ε 2 , the saturated value L is set to L=L 2  or the saturated value −L is set to −L=−L 2 . When the absolute value |ε| is in the range of |ε|&gt;e 2 , the saturated value L is set to L←L 2  or the saturated value −L is set to −L←−L 2 . 
   The reference value ε 1  is of a numerical value for defining a range wherein the deviation ε is relatively small, and may be set to the viscoelastic displacement range Yo (see  FIG. 2 ) due to the piston seal  32  (or a hypothetical combined seal including the piston seal  32  and the cap seal  34 ), or to a value that is slightly smaller than the viscoelastic displacement range Yo. When the piston  26  is displaced slightly within the viscoelastic displacement range Yo, integrator windup is sufficiently restrained, and the piston  26  is stably moved due to the viscoelastic characteristics of the piston seal  32  (or the hypothetical combined seal). Consequently, steady-state deviation is minimized for highly accurately and rigidly positioning the controlled object or, stated otherwise, thereby increasing the static characteristics. 
   The reference value E 2  is of a numerical value for defining a range wherein the deviation ε is sufficiently large, and the maximum displacement is set to be equal to Ymax (see  FIG. 1 ). As the absolute value |ε| increases, the saturated value L is also set to a larger value, making it possible to increase the manipulated variable Us for the proportional valve  40 . In other words, when the absolute value |ε| of the deviation ε is large, the proportional valve  40  is actuated with a large stroke to move the piston  26  at a high speed, thereby enabling the displaced distance Y to quickly approach the command value R or, stated otherwise, thereby increasing the static characteristics. 
   The saturated values L and −L set by the saturated value changer  50  may be changed stepwise between the reference value ε 1  and the reference value ε 2 , as shown in  FIG. 34 . Specifically, as the absolute value |ε| increases, the saturated value L is increased stepwise and the saturated value −L is reduced stepwise. The saturated value L may be changed stepwise by a simple comparison process, without requiring a proportional calculation process. The process of setting the saturated values L and −L stepwise, as shown in  FIG. 34 , provides the same advantages as those obtained when they are set as shown in  FIG. 33 . The same advantages are also achieved if the absolute value of the saturated value L of the saturation element  48  is greater when the deviation ε exceeds the viscoelastic displacement range Yo than when the deviation ε falls within the viscoelastic displacement range Yo. 
   As described above, the positioning control system  10  according to the present invention restrains integrator windup for enabling an increased control capability, with the saturation element  48  being disposed in the positive-feedback minor loop  70  of the disturbance observer  46 . The disturbance observer  46  operates to compensate for the disturbance D in order to highly accurately and rigidly position the controlled object. 
   The saturated positive-feedback filter  110  according to the present invention operates in the same manner as an integrating process so as to restrain integrator windup, with the low-pass element Q being disposed in the feedback path of the positive-feedback minor loop, and the saturation element  48  being disposed in the forward path thereof. 
   Although certain preferred embodiments of the present invention have been shown and described in detail, it should be understood that various changes and modifications may be made to the embodiments without departing from the scope of the invention as set forth in the appended claims.