Abstract:
A mechanism for empirically deriving the values of the damping ratio and frequency of the mechanism driven by a servo-controlled control system is disclosed. In accordance with the illustrative embodiment, the values of the damping ratio and frequency are continually re-generated based on empirical data derived from sensor feedback of the maximum-amplitude switch and the linear second-order servo. Because the values of the damping ratio and frequency are generated from empirical data, it is not necessary that they be known, and because the values of the damping ratio and frequency are continually re-generated, variances in their values are continually noticed and compensated for.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    The application claims the benefit of U.S. Provisional Patent Application 60/938,064, filed May 15, 2007, entitled “Robust Timed Switching Control with State Measurement,” (Attorney Docket: 711-120us), which is also incorporated by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The present invention relates to control systems in general, and, more particularly, to servo controllers. 
       BACKGROUND OF THE INVENTION 
       [0003]      FIG. 1  depicts a block diagram of control system  100  in the prior art as taught, for example, by J. H. Fu, U.S. Pat. No. 6,611,119, which is incorporated by reference. Control system  100  comprises: three-degree-of-freedom mechanism  101  and degree-of-freedom controllers  102 - 1 ,  102 - 2 , and  102 - 3 , interrelated as shown. 
         [0004]    Three-degree-of-freedom mechanism is a mechanical device (e.g., robotic manipulator, gun turret, antenna dish, hard-disk drive, etc.) that comprises three independent degrees of freedom, x 1 (t), x 2 (t), and x 3 (t). Each of the degrees of freedom is driven by one of degree-of-freedom controller  102 -i, wherein i ε {1, 2, 3}. Degree-of-freedom controller  102 -i takes as input a time-varying signal A i (t) and drives the corresponding degree of freedom x i (t) of mechanism  101  to that value. 
         [0005]      FIG. 2  depicts a block diagram of degree-of-freedom controller  102 -i, as depicted in  FIG. 1 . Degree-of-freedom controller  102 -i is an open-loop controller that comprises: maximum-amplitude switch  201 -i and linear second-order servo  202 -i, which drives one degree of freedom of three-degree-of-freedom mechanism  101  as shown. 
         [0006]    Maximum-amplitude switch  201 -i takes as input: 
         [0007]    i. a signal A i (t), which is a time-varying desired setting degree of freedom x i (t) of mechanism  101 , 
         [0008]    ii. a signal p i , which is the damping ratio of degree of freedom i of mechanism  101 , and 
         [0009]    iii. a signal ω i , which is the frequency of degree of freedom i of mechanism  101 . Maximum-amplitude switch  201 -i takes these three values and generates a time-varying output signal a i (t), which is the input to linear-second-order servo  202 -i. It will be clear to those skilled in the art how to generate a i (t) given A i (t), p i , and ω i . See, for example, J. H. Fu, U.S. Pat. No. 6,611,119. 
         [0010]    Linear second-order servo  102  receives a i (t) and generates a time-varying output x i (t), drives one degree of freedom of three-degree-of-freedom mechanism  101 , in well-known fashion. 
         [0011]    In the prior art, the values of p i  and ω i  are assumed to be known parameters of mechanism  101  and to be constant. In many situations, however, the values of p i  and ω i  are not known, and in some situations, the values of p i  and ω i  vary, even if somewhat slightly. For these reasons, the need exists for a solution when either the values of p i  and ω i  are not known or the values vary or both. 
       SUMMARY OF THE INVENTION 
       [0012]    The present invention provides a solution to the problem when the values of p i  and ω i  are not known or the values vary or both. In accordance with the illustrative embodiment, the values of p i  and ω i  are continually re-generated based on empirical data derived from sensor feedback of the maximum-amplitude switch and the linear second-order servo. Because the values of p i  and ω i  are generated from empirical data, it is not necessary that they be known, and because the values of p i  and ω i  are continually re-generated, variances in their values are continually noticed and compensated for. 
         [0013]    It will be clear to those skilled in the art, after reading this disclosure, how to make and use embodiments of the present invention that apply to rotary mechanisms that can be modeled as second-order linear control systems. Furthermore, it will be clear to those skilled in the art, after reading this disclosure, how to make and use embodiments of the present invention to nonlinear mechanisms that, via standard coordinate transformation techniques (e.g., feedback linearization techniques in non-linear control theory, etc.) can be modeled as linear or non-linear control systems of higher dimensions. 
         [0014]    The illustrative embodiment comprises: a linear second-order servo that drives one degree of freedom of a mechanism; a sensor for ascertaining the velocity and position of the degree of freedom of the mechanism at instant s 0 , wherein the velocity at instant s 0  is represented by y 0  and wherein the position at instant s 0  is represented by x 0 ; a real-time system parameter identifier for generating the damping ratio p i  and frequency ω 1  of the one degree of freedom of the mechanism based on x 0  and y 0 ;and a maximum-amplitude switch for controlling the linear second-order servo based on the damping ratio p i  and frequency ω i  of the one degree of freedom of the mechanism. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]      FIG. 1  depicts a block diagram of control system  100  in the prior art as taught, for example, by J. H. Fu, U.S. Pat. No. 6,611,119, which is incorporated by reference. 
           [0016]      FIG. 2  depicts a block diagram of degree-of-freedom controller  102 -i, as depicted in  FIG. 1 . 
           [0017]      FIG. 3  depicts a block diagram of control system  300  in accordance with the illustrative embodiment of the present invention. 
           [0018]      FIG. 4  depicts a block diagram of degree-of-freedom controller  302 -i, as depicted in  FIG. 3 . 
       
    
    
     DETAILED DESCRIPTION 
       [0019]      FIG. 3  depicts a block diagram of control system  300  in accordance with the illustrative embodiment of the present invention. Control system  300  comprises: three-degree-of-freedom mechanism  301  and degree-of-freedom controllers  302 - 1 ,  302 - 2 , and  302 - 3 , interrelated as shown. 
         [0020]    Three-degree-of-freedom mechanism is a mechanical device (e.g., robotic manipulator, gun turret, hard-disk drive, etc.) that comprises three independent degrees of freedom, x 1 (t), x 2 (t), and x 3 (t). Each of the degrees of freedom is driven by one of degree-of-freedom controller  302 -i, wherein i ε {1, 2, 3}. It will be clear to those skilled in the art, after reading this specification, how to make and use alternative embodiments of the present invention in which mechanical device  301  comprises any number of degrees of freedom (e.g., 1 degree of freedom, 2 degrees of freedom, 4 degrees of freedom, 5 degrees of freedom, 6 degrees of freedom, etc.). Degree-of-freedom controller  302 -i takes as input a time-varying signal A i (t) and the corresponding degree of freedom x i (t) as feedback and drives the corresponding degree of freedom x i (t) of mechanism  301  to A i (t). 
         [0021]      FIG. 4  depicts a block diagram of degree-of-freedom controller  302 -i, as depicted in  FIG. 3 . Degree-of-freedom controller  302 -i is an closed-loop control system that comprises: maximum-amplitude switch  401 -i, linear second-order servo  402 -i, sensor  403 -i, and real-time system parameter identifier  404 -i, interrelated as shown. 
         [0022]    Maximum-amplitude switch  401 -i is identical to maximum-amplitude switch  201 -i in the prior art, and takes as input: 
         [0023]    i. the signal A i (t), 
         [0024]    ii. a signal p i , which is the damping ratio of degree of freedom i of mechanism  101 , and 
         [0025]    iii. a signal ω i , which is the frequency of degree of freedom i of mechanism  101 . Maximum-amplitude switch  401 -i takes these three values and generates a time-varying output signal a i (t), which is the input to linear-second-order servo  402 -i. It will be clear to those skilled in the art how to generate a i (t) given A i (t), p i , and ω 1 . See, for example, J. H. Fu, U.S. Pat. No. 6,611,119. In accordance with the illustrative embodiment, maximum-amplitude switch  201  feeds the signal a i (t) to real-time system parameter identifier  205  for periodic instants s n−1 , s n−2 , s n , s n+1 , s n+2 , etc., which are designed by a n−1 , a n−2 , a n , a n+1 , a n+2 , etc., wherein n is an integer. 
         [0026]    Linear second-order servo  202  is identical to linear second-order servo  102  in the prior art, and also receives a i (t) and generates output a time-varying output x i (t), which drives one degree of freedom of mechanism  203 , in well-known fashion. It will be clear to those skilled in the art how to make and use linear second-order servo  202 . 
         [0027]    Mechanism  203  is a mechanism with one degree of freedom, as in the prior art. It will be clear to those skilled in the art, however, after reading this disclosure, how to make and use alternative embodiments of the present invention that have any number of degrees of freedom. In those cases, there is a set of maximum-amplitude switch, linear second-order servo, sensor, and real-time system parameter identifier for each degree of freedom. 
         [0028]    Sensor  204  continually samples x i (t) and continually provides real-time system parameter identifier  205  with real-time periodic estimates of the position and velocity of x i (t). It will be clear to those skilled in the art how to make and use sensor  204 . In accordance with the illustrative embodiment of the present invention, the sampling rate—designated f—is much higher than the switching rate of maximum-amplitude switch  201 . 
         [0029]    In accordance with the illustrative embodiment, the consecutive estimates of the position and velocity of x i (t) are provided for instants s n−1 , s n−2 , s n , s n+1 , s n+2 , etc., and the corresponding estimates of the position and velocity are designed by x n−1 , x n−2 , x n , x n+1 , x n+2 , etc. and y n−1 , y n−2 , y n , y n+1 , y n+2 , etc., respectively. Table 1 depicts the time correlation of five consecutive instants and the corresponding values for the signal a i (t), the velocity of x i (t), x′ i (t),and the position of x i (t). 
         [0000]    
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Time Correlation of s(t), a i (t), x′ i (t), and x i (t) 
               
             
          
           
               
                   
                 s i (t) 
                 a i (t) 
                 x′ i (t) 
                 x i (t) 
               
               
                   
                   
               
               
                   
                 s 0   
                 a 0   
                 y 0   
                 x 0   
               
               
                   
                 s 1   
                 a 1   
                 y 1   
                 x 1   
               
               
                   
                 s 2   
                 a 2   
                 y 2   
                 x 2   
               
               
                   
                 s 3   
                 a 3   
                 y 3   
                 x 3   
               
               
                   
                 s 4   
                 a 4   
                 y 4   
                 x 4   
               
               
                   
                   
               
             
          
         
       
     
         [0030]    Real-time system parameter identifier  205  takes as input: 
         [0031]    i. the signal a i (t) output from maximum-amplitude switch  201 , 
         [0032]    ii. the estimates of position x n−1 , x n−2 , x n , x n+1 , x n+2 , etc., provided by sensor  204 , and 
         [0033]    iii. the estimates of velocity y n−1 , y n−2 , y n , y n+1 , y n+2 , etc., provided by sensor  204   
         [0034]    and continually regenerates estimates for the values of p i  and ω i . When A 1 (t) is constant, the values of p i  and ω i  are generated in accordance with Equations (1) and (2): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ω 
                       i 
                     
                     = 
                     
                       
                         
                           
                             y 
                             1 
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                           - 
                           
                             
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         [0000]    where |A max | is the maximum output of the maximum-amplitude switch, and f=1/sampling rate (i.e., f=1/(s 1 −s 0 )). 
         [0035]    To suppress the transients in x i (t), maximum-amplitude switch  401 -i outputs one value (+A max ) for one interval from t B  to t S  and outputs a second value (−A max ) for a second interval from t S  to t E . In solving for t S  and t E , maximum-amplitude switch  401 -i solves two simultaneous algebraic equations for the two unknown time intervals (t S -t B ) and (t E -t S ): 
         [0000]        x ( t   E )= x   p ( t   E )  (Eq. 3) 
         [0000]        x ′( t   E )= x′   p ( t   E )  (Eq. 4) 
         [0036]    Equations (3) and (4) generalize the prior art in J. H. Fu, U.S. Pat. No. 6,611,119 in that Equations (3) and (4) teach the specification of this application that at the time t E  the position and the velocity states of the mechanism must be equal to the intended or desired position and velocity states. 
         [0037]    As in J. H. Fu, U.S. Pat. No. 6,611,119, Equations (3) and (4) can be derived as explicit algebraic equations by virtue of linearity of the control system model. Specifically, the algebraic expressions for the position and the velocity of the mechanism to be controlled can be given explicitly, given the initial (time t B ) position and velocity, at time t S  as well as at time t E . 
         [0038]    This application recognizes and generalizes J. H. Fu, U.S. Pat. No. 6,611,119 as a special case that teaches driving the controlled mechanism to a fixed value servo command in minimum time without incurring the undesirable overshoots and undershoots by open-loop strictly time-based switching. In addition, this application teaches as an embodiment how to utilize the state data available from sensor(s). 
         [0039]    In accordance with the illustrative embodiment, the values of p i  and ω i  are regenerated once for each sample and within s n -s n−1  seconds of instant s n . 
         [0040]    It is to be understood that the disclosure teaches just one example of the illustrative embodiment and that many variations of the invention can easily be devised by those skilled in the art after reading this disclosure and that the scope of the present invention is to be determined by the following claims.