2-D Analysis for Iterative Learning Controller for Discrete-Time Systems With Variable Initial Conditions Yong FANG 1, and Tommy W. S.

Transcription

1 -D Analysis for Iterative Learning Controller for Discrete-ime Systems With Variable Initial Conditions Yong FANG, and ommy W. S. Chow Abstract In this aer, an iterative learning controller alying to linear discrete-time multivariable systems with variable initial conditions is investigated based on two-dimensional (-D) system theory. he aer first introduces a -D tracing error system, and shows the effect of tracing errors against variable initial conditions. he sufficient conditions for the convergence of the learning control rules are derived and discussed. Based on the roosed ILC rule, we have shown that the convergence of the learning rule is guaranteed with less restriction. An imroved iterative learning control (ILC) rule is roosed. As a resul the convergence is robust with resect to small erturbations of the system arameters. wo numerical simulation examles are used to validate the effectiveness of the roosed methodologies. Key wards: Discrete-time systems; variable initial conditions; iterative learning control; -D system theory Introduction Iterative learning control (ILC) has received considerable attention in recent years in the alications of robotic maniulators and dis drive systems [-4]. Because of its aealing ability, various schemes of ILC have been roosed [-3]. he obective of ILC is to use the reetitive nature of the rocess to rogressively imrove the tracing erformance. he control inuts are udated iteratively after each oeration using the error measurements in the revious cycle. hese controllers are able to deal with dynamic systems with imerfect nowledge of dynamics structures and/or arameters oerating reetitively over a fixed time interval. he technical difficulty of ILC lies in the two-dimensionality (in the mathematical sense) of the overall system [7]. Amid the iterative learning rocess, the interaction between the system dynamics and the iterative learning rocess oses an imortant issue for ILC. wo-dimensional (-D) iterative learning models rovide a clear descrition on the dynamics of the control system and the behavior of the learning rocess. Effective learning rules were roosed based on the -D system theory [-4, ]. For linear Fang Yong was with City University of Hong Kong, and is now with the Shanghai University at PR China.

2 discrete-time systems, the -D Roesser s discrete model can be used as learning error system [-4]. While, for linear continuous-time systems, the -D continuous-discrete system model are able to describe clearly the iterative learning rocess []. As a resul we are now able to consider the control system dynamics together with the learning rocess when we are on the rocess of designing a learning control algorithm. Although ILC theory has well been develoed for both linear multivariable systems and certain classes of nonlinear systems, most research on ILC mainly focus on the dynamic systems under the same initial iterative conditions. It is not ractical to assume that each learning iteration starts at the same oin because it is imossible to reeat the same initial condition in ractical engineering alication. he initial condition always deviates from the initial condition of last iterative erformance. When the initial conditions vary, the learning dynamics becomes much more comlex. Hitherto, the effect on the interaction between the learning dynamics and the variable initial conditions remains an oen and imortant issue for ILC research. Lee and Bien [9] have identified an undesirable henomenon due to the mismatch on the initial conditions in learning control schemes. hey reorted that the control system can become unstable if the initial condition at the beginning of each iteration is different from the revious initial condition. hey identified that the matching conditions are necessary for erfect tracing. he stability of learning control with disturbances and uncertain initial condition was studied []. Initial shift roblem and its ILC solution for nonlinear systems with higher relative degree was discussed in []. In [8], a method called ILC with multi-modal inut was roosed to tacle the ILC roblem with variable initial conditions [8]. Desite all these romising results, these wor have been restricted to continuous-time systems. It is the maor contribution of this aer that an iterative learning controller for discrete-time multivariable systems is investigated when the initial conditions are variable. It is well now that when the initial state value is fixed but not same to the desired initial state value and the ILC algorithm is alied to the system, the converged outut traectory is aart from the desired outut traectory with constant error value. It is obvious that the learning dynamics are more comlex when the initial iterative state is different at each iteration. Our question is which conditions need to be satisfied when the iterative learning controller is used for the initial conditions with random distribution in a interval? In this aer, a -D analysis aroach is derived from the -D notion. he effect of initial conditions for control errors is studied based on the -D learning model and the -D system theory. he conditions of convergence of the learning control rule are derived. We also resent a modified ILC rule for effective tracing erformance. As a resul a new iterative learning algorithm for asymtotic tracing of a given reference traectory is roosed. wo numerical examles are resented to validate the effectiveness of the

3 roosed learning control algorithm. Iterative Learning Control and Its -D Reresentation Consider the linear time invariant multivariable system described by x ( t + ) = Ax( t) + Bu( t), () y ( t) = Cx( t), () n m where x( t) R, u ( t) R and y( t) R denotes the state, inut and outut vector resectively, A, B and C are real matrices with aroriate dimensions and, without loss of generality, it is assumed that matrices B and C are of full ran. he obective of ILC is to find an aroriate control inut { u( t), t =,, L, } such that the system outut y(t) follows the reference traectory y ( t) R, t =,, L,. Since only estimated values of the system matrices A, B and C are available, the system is required to execute the reetitive tas of tracing over the finite time interval t. In the th oeration/iteration, the control inu (t) u, may be udated iteratively in a certain way by using the error measurements in the revious oeration. his may lead to the fact that the system outut y (t) aroaches gradually to the given reference traectory y r (t), i.e., lim y (t) = (t). Many researchers have roosed various iterative y r learning algorithms for this roblem and much success have been achieved, but all these roosed methods are strictly restricted to the case that the initial state condition is the same at each iteration. In ractice, the initial condition may vary when the iteration is reeated, that is x ) = x, =,,,L. ( Obviously, the learning dynamics will be of much comlex because of the varying of initial conditions. his aer exlores the ILC algorithm for the cases of variable initial conditions from a -D ersective. According to the rincile of ILC, there are two indeendent dynamic rocesses: system time and learning iteration during learning rocess. Every variable can be exressed as a -D function, such as x (, which reresents x(t) in the th learning iteration. hus, the system (-) can be resented in the following -D form r x ( t +, = Ax( + Bu(, (3) y ( = Cx(, (4) where t =,,, L,, =,,, L. he general ILC rule can also be given 3

4 u ( + ) = u( + u(. (5) he -D system (3-5) describes a general -D model of the ILC rocess. Our obective is to introduce a learning rule u( such that the system tracs a given reference outut traectory. Before the iterative learning begins, the initial inut sequence u() may be arbitrarily chosen, i.e., u ( t,) = u ( t ), t =,,, L,. (6) While every control cycle starts at the different initial system state, i.e., x, = x, =,,, L. (7) ( (6) and (7) consist of the boundary conditions for the -D system (3-5). It is difficult to find a learning rule (5) so that lim y( = y ( t), for each time t because the initial condition is a variable. It is, r however, ossible to find a learning rule such that the system tracs asymtotically the reference traectory. he control error is as small as it is tolerable as t >>. In this aer, a learning rule (5) is n said to be convergent if for any initial condition x R and any initial control sequence { u()}, it generates a sequence { u( } for system (3-4) such that lim y( = y ( t), for << t. We assume that the system () is stable and the initial shifts are of random shifts, which is different from the consecutive shift in many motion control roblems exerienced in servo and robotics systems. It imlies that the initial condition x (, = x at every iteration is always in the neighbourhood D of, where D = { x x x Λ / }. (8) With this assumtion, we are able to derive the effective learning rule in the following section. r x 3 -D Iterative Learning Algorithm We first resent the following error equations. for e( = yr ( t) y(, (9) η ( = x( t, + ) x( t,, () t >. Using (3-5), and after a few straightforward maniulations, we obtain η ( t +, = Aη( + B u( t,, () and e ( + ) e( = CAη( CB u( t,. () 4

5 If we use the following control rule [] u ( = Ke( t +,, for t > and, (3) the -D control error system can be resented in the form of -D Roesser s tye model [4], η( t +, A = e( + ) CA I BK η(, (4) CBK e( where I denotes the identity matrix, which boundary conditions are given by η (, = x(, + ) x(,, for, (5) t + i t and e( ) = yr ( t) CA x t i CA Bu( i,), for t =,, L. (6) = Hence, the initial outut error e () is bounded and η (, = x(, + ) x(, Λ in accordance with the assumtion given in section. Based on the -D system theory [4], the -D state transition matrix, i, for the -D system (4) can be defined as follows: I n+ i = = i, = i, + i, i, ( i + ). (7) i < / < A BK, =,, =. (8) CA + RC I CBK For the state transition matrix, i,, it is rather straightforward to rove the following relationshi [4], l =. (9) i, l= i, l,, Using the state resonse formula of -D Roesser s model [5] and (9), we are able to obtain the solution of equation (4) in the following form, η( η(, ) t = t, + t i,, where t >. () e( = i= e( i,) For a matrix Q = [ q i ], let Q = [ qi ] and ρ(q) denote the sectral radius of Q. We can obtain the following theorem: heorem : If ρ ( ), ( ) ρ < and ρ[ ( I ) ], then the tracing error e (, <,, n+, < asymtotically converges to zero, i.e., lim ( =, for << t. e Proof: If ρ( ), <, then the series (, ) converges = 5

6 = (, ) = ( I n+, ) i h i,,,, ] = = h=. Hence, = ( ) [ ( ) also converges. his imlies lim for any < t. = According to the assumtion that the initial outut error e () is bounded, we have lim t i= (,) e i t i, =. () η(, ) x(, + ) x(, ) Let S( = t, = t,, we will rove lim S( =. Let us = = t consider the series t= S( = = ( I n t= = η(, ) = = t= η(, ) x(, + ) x(, ) + +, ) [, ( I n, ) ]. = Here, we use the formula = ( I ) [ ( I ) because of ρ( ) <. Hence, we have t= t= S n+,, n+, ] n+ = n+, ( ( I ) ( I )., n+, <,, Since ρ[ ( I ) ], the series S( is absolute convergent for all. his imlies lim S t ( =. According to the assumtion that ρ( ) < and equation (), the tracing error, e(, asymtotically converges to zero. he roof of the theorem is comleted. It needs to oint out that the initial erturbations u () are assumed to be of l tye. From equations (6) and (), it is obvious that a better learning convergence can be achieved if the initial erturbations are assumed to be of l tye. t=,, From the roof of heorem, if the initial state value is fixed but not same to the desired initial state and the iterative learning controller is alied to the system, η, thus for all t and. In this (, S( = case, lim e( = for all t > if ρ( ) <. In other word, the D-tye algorithm is convergent even, though there is constant initial error value, which convergence condition is same as the case that the initial error is always zero. 6

7 We also note that ρ( ) <, ρ( ) < if and only if ρ( A) < and ρ( CBK) <, and, ( I n, I n A BK +, ) = CA I CBK I CA( I n A) = CA I A ( n ), n+, <, BK + I, CBK thus, ρ [ ( I ) ] if and only if ρ [ CA( I A) BK + I CBK ] <. We obtain theorem. heorem : For the learning rule (3), the tracing error of system (3-4) asymtotically converges to zero if ρ( A) <, ρ( CBK) < and ρ [ CA( I A) BK + I CBK ] <. I n heorem shows the convergent conditions of the learning rule (3) for the learning matrix K when the initial iterative condition of the system (3-4) is varying. Also, in order to assure that the conditions of theorem are satisfied, a roer selection on the learning matrix K is essential. As it is nown that there exists a matrix K, which stabilizes the matrix I CBK, if and only if matrix CB has a full-row ran. Since n I ρ ( I CBK) I CBK CB K ( CB) [( CB)( CB) ] then ρ( CBK) < I if K ( CB) < [( CB)( CB) ] CB. () herefore, if all K in () satisfies ρ [ CA( I A) BK + I CBK ] <, then the learning matrix is n able to yield convergent tracing. However, in ractical alication, the system arameters are unnown. If the estimated values of system arameters of A, B, and C are available, Also, in order to assure that the conditions of theorem are satisfied, a roer selection on the learning matrix K is essential. It is, however, that the learning matrix K may be failure to the last condition in heorem. In order to imrove the leaning control erformance, a modified ILC rule is given. For the case of zero initial error [], we roosed an iterative learning control rule which assured that the outut error met the required tolerance for the whole reference traectory within a few of learning iterations. In this aer, the wor is extended to ractical cases that the initial conditions are variable [], u = K e( t +, + K [ x( x( )], (3) ( where x ( is the state vector of the following closed-loo system 7

8 with u * * x ( t +, = ( A BK ) x( + Bu ( ), (4) ( t, = u( + Ke( t +, + K x(, and the initial state: x (, = x(, + ). For the system defined in (3), we can show that x ( = x( + ). (5) In fac from (3), (,3), we have x( t +, + ) = Ax( + ) + B[ u ( K x( ] * = Ax( + ) + Ax( BK x( + Bu ( Ax( = ( A BK ) x( + Bu ( + A[ x( + ) x( ] = x( t +, + A[ x( + ) x( ]. * hus, x( t +, + ) x( t +, = A[ x( + ) x( ]. (6) Since x (, = x(, + ), namely, x (, + ) x(,, then x ( = x( + ). With the new learning control rule (3), one has y( + ) = Cx( + ) = = y( + C[ x( + ) x( ] y( + CA[ x( t, + ) x( t, ] + CB[ u( t, + ) u( t, ] = y ( + CA[ x( t, + ) x( t, ] + CB[ Ke( + K x( t, K x( t, ] = y( + CA[ x( t, + ) x( t, ] CBK [ x( t, + ) x( t, ] Namely, + CBKe( = y ( + ( CA CBK ) η( + CBKe(. e( + ) = ( CA CBK ) η ( + ( I CBK) e(. (7) On the other hand, it follows from () and (6) that η ( t +, = x( + ) x( = x ( x( ) = ( A BK )[ x( t, x( t, )] + Bu ( t, B u ( t, ) = ( A BK ) η( + BKe(. (8) 8

9 Combining (7) with (8) leads to the following -D Roesser model: η( t +, A BK = e( + ) CA + CBK I BK CBK η( e(. (9) From heorem, we can obtain the following convergence theorem. heorem 3: For the learning rule (3), the tracing error of system (3-4) asymtotically converges to zero if ρ( A BK ) <, ρ( CBK ) < and I ρ ( CA + CBK )( I A + BK ) BK + I CBK ]. (3) According to [ n < ρ [ ( CA + CBK )( I n A + BK ) BK + I CBK ] CA + CBK I n A + BK ) ( BK + I CBK, if we chose learning matrices K and K such that CA + CBK </ and I CBK </, I + ) n A BK BK <, A BK < ( then the conditions of theorem 3 can be satisfied. Similar to K K (), we need to chose and such that K ( CB) [( CB)( CB) ] < CB, (3) K ( CB) [( CB)( CB) ] CA < CB. (3) When the estimated system arameters are available, the better selections of K and ~~ ~~ ~~ ( CB) [ CB( CB)] and K C ~ A ~ resectively, if matrix C ~ B ~ has a full-row ran. From the above discussion, it is clear that the convergence of the learning control rule () is robust with resect to small erturbations of the system arameters. Based on the learning control rule (3), we roose the following algorithm. K are Algorithm : ~. Given the system (-) and the estimated system matrices A, B ~ and C ~ ; the reference outut traectory (t), < t ; and the traectory tolerance ε >. y r. Let =, u(t), t < are selected randomly, i.e., between and. Select the learning matrices and K such that CA + CBK </ and I CBK </, K 9

10 I + ) n A BK BK <, A BK < ~~ ~~ ~~. Usually, we can calculate K = ( CB) [ CB( CB)] ( ~~ and K K CA. = 3. x( ) = x, and aly the control u (t) to system (-). Measure x(t), y(t), < t. * 4. If su ( t) y( t) > ε, then calculate u t) = u( t) + K [ yr ( t + ) y( t + )] + K x( t) and aly y r t < t * u ( t) to the closed-loo system u (t) = u ( t ) K x (t), else go to ste = +, returns to ste End. ( * x (t + ) = ( A BK ) x( t) + Bu ( ) and measure x(t), let t 4 Simulation Examles In order to demonstrate the erformance of the roosed control rules, two simulation examles are used to validate our roosed methods. he reference traectory is defined as / 5 6t y r (t) =.5e t sin( ), for < t. (6) 5 All control inuts are started with zero. x x = ( µ µ ) ) A =.3 he variable initial conditions are given as ( =, where µ and µ are two random numbers assumed to be uniformly distributed in [ ]. his initial inut sequences are selected randomly between and. Examle : Consider the linear control system (-) with which are estimated as 8, = and C =. 5 B (.4 ) ~.35 ~ 7. 8 ~ A =, B = and C = (.44 ) ~~ ~~ ~~ We selected the learning matrix K = ( CB) [ CB( CB)] =.94, then ρ( A) =.5,.3, ρ( I CBK) =. 676 and ρ [ CA( I n A) BK + I CBK ] =. 33. Using the ILC rule described in (3), the tracing error converged to a very small level. After 5 trails, the average error of the tracing erformance was calculated. Figure shows the averages absolute tracing error after second, fifth and eighth iteration. Obviously, the tracing error is negligible after a time-ste of, when

11 the iterative control oerates for eight times. able shows the average absolute values of e () via the number of iterations. After twentieth iterations, the average absolute tracing error at t = is his examle illustrates that the learning rule (3) rovides an outstanding tracing erformance, even when the initial condition varies. he following examle, however, shows that the learning control rule described in (3) cannot always meet the requirement..6 Examle : Consider the system in Examle with the system matrix A =, which is estimated as A =. We used the iterative learning rule described in (3) to erform the.8.5 tracing tas and the learning matrix K was selected.94 as in Examle. he tracing error did not converge in this case. his resul however, has not come as a surrise because the learning control rule does not meet the convergence requirement of, ρ [ CA( A) BK + I CBK ] =.33 >. I n Using the learning control rule described in (), we selected K =.94, and ~~ K = K CA (.744 ). hus, ρ( A BK ) =. 5,.5, ρ( CBK ).676, and = ρ[ ( CA + CBK )( I n A + BK ) BK + I CBK ] =.68. We also designed an aroriate state observer for system (4) for the alication of iterative learning controller (3). he average absolute tracing errors after the second, the third and the fourth iteration for 5 trails are shown in Figure. he average absolute values of are also shown in able. After twentieth iterations, the average absolute tracing error at t = converged to a very tiny value of I = via the number of iterations. It is worth noting that the rate of convergence of Algorithm is faster than that of the learning rule described in (3). It taes only few iterations to trac the reference outut but it requires more comutational effort. () e 5 Conclusion his aer investigates an iterative learning controller alying to linear discrete-time multivariable systems with variable initial conditions from a -D notion ersective. he tracing errors are modelled as -D Roesser s tye models. he models not only describe the system dynamics and the learning rocess, they also describe the efforts of the error for the variable initial conditions. Based on the -D system theory, new sufficient conditions for the convergence of the learning control rules are determined. For the second learning rule described in (), we have illustrated that there always exist learning

12 matrices such that the learning control rule is convergen when the given matrix CB is a full-row ran. Acnowledgement: his wor is fully suorted by the City University of Hong Kong SRG grant of roect No References [] K. L. Moore, Iterative learning control for Deterministic systems, Sringer-Verlag, 993. [] S. Arimoto, S. Kawamura, and F. Miyazai, Bettering oeration of robots by learning, J. Robot. Syst., vol.,. 3-4,984. [3] P. Bondi, G. Casalino, and L. Gambardella, On the iterative learning control theory for robotic maniulators, IEEE J. Robot. Automat., vol. 4,. 4-, 988. [4] J. J. Craig, Adative control of maniulators through reeated trials, in Proc. Amer. Cont. Cong., San Diego, CA, 984, [5] D. Wang, Y. C. Soh and C. C. Cheah, Robust motion and force control of constrained maniulators by learning, Automatica, vol. 3, no.,. 57-6, 995. [6] N. Amann, D. H. Owens and E. Rogers, Iterative learning control for discrete-time systems with exonential rate of convergence, IEE Proc.-Control heory Al., vol. 43, no., 996. [7] D. H. Hwang, Z. Bien and S. R. Oh, Iterative learning control method for discrete-time dynamic systems, IEE Proc.-D, vol. 38, no., , 99. [8] H. S. Lee and Z. Bien, Study on robustness of iterative learning control with non-zero initial error, International Journal of Control, vol. 64, no. 3, , 996. [9] H. S. Lee and Z. Bien, Initial condition roblem of learning control, Proc. of the Institution of Electrical Engineers, Pt D, vol. 38, , 99. [] G. Heinzinger, D. Fenwic, B. Paden and F. Miyazai, Stability of learning control with disturbances and uncertain initial conditions, IEEE rans. Automatic Control, vol. 37, no.,. -4, 99. []. W. S. Chow and Y. Fang, An iterative learning control method for continuous-time systems based on -D system theory, IEEE rans. on Circuits & Systems, Part I, vol. 45, no. 6, , 998. [] Y. Fang and. W. S. Chow, Iterative learning control of linear discrete-time multivariable systems, Automatica, vol. 34, no., , 998. [3] Z. Geng, R., Carroll, and J. Xies, wo-dimensional model and algorithm analysis for a class of iterative learning control systems, Int. J. Control, vol. 5, , 99.

13 [4] J. E. Kure and M. B. Zaremba, Iterative learning control synthesis based on -D system theory, IEEE rans. On Automatic Control, vol. 38, no.,. -5,993. [5]. Kaczore, wo-dimensional Linear Systems. New Yor: Sringer-Verlag, 985. [6]. Hinamoto, -D Lyaunov equation and filter design based on the Fornasini-Marchesini second model, IEEE rans. Circuits and Systems, Part I, vol. 4 no., (Feb). -, 993. [7]. Bose and D. A. rautman, wo s comlement quantization in two-dimensional state-sace digital filters, IEEE rans. on Signal Processing, vol. 4 no., October, , 99. [8]. Hinamoto, Stability of -D discrete systems described by the Fornasini-Marchesini second model IEEE rans. on Circuits and Systems, Part I, vol. 44, no. 3, March, , 997. [9] R. P. Roesser, A discrete state-sace model for linear image rocessing, IEEE rans. Automatic Control, vol. AC-,. -, Feb [] Sun M, Wang D, Xu G. Initial Shift Problem and Its ILC Solution for Nonlinear Systems with Higher Relative Degree[C]. Proceedings of the American Control Conferences, Chicago, lllinois, June, [].W.S.Chow, Yong Fang, A recurrent Neural Networ Based Real-ime Learning Control Strategy Alying to Nonlinear Systems with Unnown Dynamics, IEEE rans On Industrial Electronics, Vol. 45, No.,.5-6, Feb,

14 able : Average absolute errors at time ste for examle and examle No of Iteration Examle Examle

15 =. Average absolute error time ste (a) =5.3 Average absolute error time ste (b) 5

16 =8.3 Average absolute error time ste (c) Figure : Averages absolute tracing error of Examle after the number of iterations, =, 5, and 8. =.4 Average absolute error time ste (a) 6

17 =3.3 Average absolute error time ste (b) =4.3 Average absolute error time ste (c) Figure : Averages absolute tracing error of Examle after the number of iteration, =, 3 and 4. 7