Robust_Iterative_Learning_Control_with_application.pdf

See discussions, stats, and author profiles for this publication at: Robust Iterative Learning Control with applications to injection molding process Article in Chemical Engineering Science December 2001 DOI: 10.1016/S0009-2509(01)00339-6 CITATIONS 96 READS 173 3 authors, including: Furong Gao The Hong Kong University of Science and Tec 323 PUBLICATIONS 4,874 CITATIONS SEE PROFILE Cheng Shao Dalian University of Technology 154 PUBLICATIONS 644 CITATIONS SEE PROFILE All content following this page was uploaded by Furong Gao on 20 January 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. Chemical Engineering Science 56 (2001) 70257034 Robust iterative learning control with applications to injection molding process Furong Gaoa; , Yi Yanga, Cheng Shaob aDepartment of Chemical Engineering, The Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong bResearch Center of Information and Control, Dalian University of Technology, Dalian 116024, Liaoning, Peoples Republic of China Abstract Convergence is an important issue in the design and application of iterative learning control (ILC) to batch processes. This paper presents a design of a robust iterative learning controller. Su.cient and necessary condition to ensure BIBO (bounded-input bounded-output) stability is derived for the optimal ILC when tracking arbitrary bounded output reference. A practical scheme of the weighting matrices selection is also proposed for the process with uncertain initial resetting and disturbances, to ensure system performance improvement from batch to batch. Finally, an application to the injection molding control is given to demonstrate the e5ectiveness of the proposed algorithm.?2001 Elsevier Science Ltd. All rights reserved. Keywords: Iterative learning control; Batch process; Injection molding 1. Introduction Iterative learning control (ILC) is motivated to mimic human learning process. It is originally developed for the manipulation of industrial robots, in which it is required to repeat a given task with high precision. By using the repetitive nature of the processes, ILC progressively and iteratively improves the control accuracy along two time dimensions for control input, that is the trial (or batch) index k from trial to trial, and the elapsed time index t during a trial from step to step. This two-dimensional learning results in advantages over the conventional feed- back control techniques where only time dimensional input actions are made along the time axis. The key for a learning control design is to 9nd an algorithm to ensure that the control input is generated for next trial in such a way that the performance improves for each succes- sive trial. The concept of iterative learning for generating such an input was 9rst introduced by Uchiyama (1978) and was later mathematically formulated by Arimoto, Kawamura, and Miyazaki (1984). Since then, consid- erable e5orts have been made on the development and Corresponding author. Tel.: +852-2358-7146; fax: +852-2358- 0054. E-mail address: kefgaoust.hk (F. Gao). analysis of iterative learning control. Recently, ILC has been applied to many repetitive processes, such as batch reactor, batch distillation, and injection molding (Lee, Bang, Yi, Son, Havlicsek this raises the expectation of wider applications to industrial set- ting. In reality, there always exist uncertainties in the process disturbances, and also, it is very likely that pro- cess initialization may not be exactly repeatable. These practical issues are important to many batch processes, for example, injection molding. These issues have not been addressed in Amanns original paper. 0009-2509/01/$-see front matter?2001 Elsevier Science Ltd. All rights reserved. PII: S0009-2509(01)00339-6 7026F. Gao et al./Chemical Engineering Science 56 (2001) 70257034 This paper is to extend the optimal iterative learning control algorithm of Amann, Owens, and Rogers (1996) for applications to general batch processes with uncer- tain disturbances where exact initial resetting is not avail- able. A su.cient and necessary condition to ensure the ILC to have the bounded-inputbounded-output (BIBO) stability is established. Analysis on the selection of the weightingmatricesforthecostfunctionisperformedcon- sequently. Finally, simulation and experimental applica- tion to control injection molding 9lling velocity, using the introduced optimal iterative learning control, are given to demonstrate the e5ectiveness of the proposed algorithm. 2. Optimal learning control background The background of the Amanns optimal learning al- gorithm is introduced as below. 2.1. Problem formulation The plant of interest is assumed to be described by the following sampledtime linear system with disturbances xk(t + 1)=Axk(t) + Buk(t) + ?k(t); 06t6N;k =0;1;2;:; yk(t)=Cxk(t) + !k(t);xkRn; ukRm; ykRp; (1) where the subscript k denotes the iteration number of operation corresponding to the trial index, for exam- ple, yk(t) is the value of the system output at time t; 06t6N; at the kth operation. ?k(t) and !k(t) denote the bounded state and external disturbance. Note that the exact state initialization of Eq. (1) for each iteration is not required in this work. The robustness to initial state variation and external disturbances will be discussed in this paper. The state-space matrices A; B; and C are as- sumed to be time-invariant for simplicity. It is possible, without any technical di.culties, to extend all results of this paper to time-varying systems. Based on the linear system theory, the following solution to Eq. (1) can be deduced: yk(t)= t1 ? l=0 CAt1lBuk(l) + ?k(t);(2) ?k(t)=CAtxk(0) + t1 ? l=0 CAt1l?k(l) + !k(t):(3) It can be observed in the above that initial actions of each trial and disturbances appear in the plant, extending the work of Amann, Owens, and Rogers (1996) to a more general case. In each trial, 9nite time intervals are in- volved. Eq. (2) can be made in a vector form by building supervectors yk; ukand ?kfrom yk(t); uk(t) and ?k(t) as below: yk=Guk+ ?k;(4) where yk= yk(1) yk(2) . . . yk(N) ;uk= uk(0) uk(1) . . . uk(N1) ; ?k= ?k(1) ?k(2) . . . ?k(N) ;G= CB0:0 CABCB:0 . . . . . . . . CAN1B CAN2B : CB : The supervectors are presented with the omission of the argument time t. During implementation of the iterative learning control, ykand ukof previous trials need to be memorized for the computation of uk+1(t) of the current trial. Matrix G, a lower-triangular block matrix known as a Toeplitz matrix, can be determined from its 9rst column. In this paper, it is postulated that kerGT=0, de9ned as regularity condition by Amann, Owens, and Rogers (1996). If plant, Eq. (1), has relative degree of one, i.e. CB ? =0, then G is invertible in SISO case. Oth- erwise, if CB=0, then a regularizing procedure can be performed as detailed in the works of Amann, Owen, and Rogers (1996) and Silverman (1969). This regular- ity condition ensures that GTG (or GGT) has at least one positive eigenvalue. Based on this postulation a conver- gence proof di5erent from Amann, Owen, and Rogers (1996) will be given in Section 3. The objective considered here, for a tracking control problem,isthatforanygivenreferencetrajectorydenoted by r(t); 16t6N, an iterative learning controller is de- rivedsuchthatwhenappliedtosystem(1)theclosed-loop tracking error is reduced iteratively from trial to trial, even with the existence of initial errors and uncertain dis- turbances. De?nition 2.1. (Amann, Owen, t and from previous trials. 2.2. Optimal iterative learning controller Considerthefollowingnominalsystemcom- posed of the coe.cient matrices A; B and Cof F. Gao et al./Chemical Engineering Science 56 (2001) 702570347027 Eq. (1): xk(t + 1)=A xk(t) + Buk(t); 06t6N;k =0;1;2;:; yk(t)=C xk(t); xkRn; ukRm; ykRp;(5) where the variables with superscript : denote the nom- inal system outputs, and they are initialized by zeros. They represent the system outputs of Eq. (1) in the ab- sence of any disturbances and initial errors. For the ref- erence trajectory (or desired system output) r(t), given over 16t6N, in the (k+1)th trial, the nominal optimal iterative learning control law is obtained by minimizing the following quadratic performance index with respect to uk+1(t): Jk+1= N ? t=1 r(t) yk+1(t)TQ(t)r(t) yk+1(t) + N1 ? t=0 Puk+1(t)TR(t)Puk+1(t);(6) where Puk+1(t)=uk+1(t)uk(t), and the weighting matrices Q(t) and R(t) are arbitrary symmetric positive de9nitive for all t. The index function Eq. (6) can be rewritten in matrix form as Jk+1=r yk+1TQr yk+1 + PuT k+1RPuk+1 (7) where Q=diagQ(1);Q(2);:;Q(N); R=diagR(0); R(1);:;R(N1), and yk= yk(1) yk(2) . . . yk(N) ;r = r(1) r(2) . . . r(N) :(8) By 9nding the partial derivative of Eq. (7) with respect to uk+1, one obtains the nominal optimal control input uk+1= uk+ R1GTQr yk+1:(9) However, it can be observed that the algorithm of Eq. (9) is not causal for the computation of uk+1, because by this control law, uk+1(t) would depend on values of yk+1(t?) for t6t?6N. Following Amann, Owens, and Rogers (1996) an equivalent form of Eq. (9) can be given below. S(t)=ATS(t + 1)IBBTS(t + 1)B + R(t + 1)1 BTS(t + 1)A + CTQ(t + 1)C; t =0;1;:;N1; S(N)=0;(10) ?k+1(t)=I + S(t)BR1(t)BT1AT?k+1(t + 1) +CTQ(t + 1) ek(t + 1); t =0;1;:;N1; ?k+1(N)=0(11) where ek(t+1)=r(t+1) yk(t+1). The nominal input update law thus becomes uk+1(t)= uk(t)BTS(t)B + R(t)1BT S(t)A xk+1(t) xk(t) + R1(t)BT?k+1(t): (12) This indicates that a causal nominal control input could be obtained iteratively if the nominal states xkand outputs ykwere introduced by the nominal system, Eq. (5). This iterative learning control algorithm is also optimal when applied to Eq. (4) where ?k=0, i.e. the disturbance-free case. This paper aims to develop an ILC algorithm in the presence of uncertain initials and disturbances. This can be achieved by calculating ukby replacing the nominal xkand ykin Eqs. (10)(12) with the measurements of xkand ykof the system, Eq. (1). Therefore, a causal iterative learning control algorithm of the interest can be summarized as uk+1=uk+ R1GTQek+1;(13) ?k+1(t)=I + S(t)BR1(t)BT1AT?k+1(t + 1) +CTQ(t + 1)ek(t + 1); t =0;1;:;N1; ?k+1(N)=0;(14) uk+1(t)=uk(t)BTS(t)B + R(t)1BT S(t)Axk+1(t)xk(t) + R1(t)BT?k+1(t); (15) where S(t) is obtained by Eq. (10). It can be seen that the control algorithm consisting of Eqs. (10), and (13)(15) iscausal.InEq.(15),uk+1(t)isobtainedbyimprovingthe last trial input uk(t) by incorporating a feedback action of the current trial (2nd term on the right-hand side of Eq. (15) and a feed-forward action 3rd term of Eq. (15), which represents the information from previous trials. In work of Amann, Owen, and Rogers (1996), there lacks also a guideline on the selection of the weighting matrices Q and R for the system convergence. This prac- tical consideration is important in the design and applica- tion of the ILC to batch processes. The convergence and robustness analysis of the above method is conducted be- low with uncertainties of initializations and disturbances, with a demonstrative application to injection velocity control. 3. Robust and convergence analysis The robust bounded-inputbounded-output stability as de9ned below will be investigated for the proposed algorithm. De?nition 3.1. An iterative learning control system is said to be robust BIBO (bounded-inputbounded-output) 7028F. Gao et al./Chemical Engineering Science 56 (2001) 70257034 stable if, and only if, for a system with any uncertain initialization and bounded disturbance, the outputs are bounded for each trial for any bounded control reference. The above de9nition considers the disturbances to the system and the initialization uncertainty along trial axis. The robust BIBO stability together with convergence and robustness of the iterative learning control algorithm is discussed. Theorem 3.1 (Robust BIBO Stability). Theapplica- tion of the iterative learning control algorithm of Eq. (10), and (13)(15) to plant (1) is robust BIBO stable if; and only if; I + GR1GTQ and I + R1GTQG have their all eigenvalues outside unit disc; or ?I + R1GTQG?1;(16) ?I + GR1GTQ?1:(17) Proof. Premultiplying Eq. (13) by G and in view of Eq. (4) and ek=rykone obtains ek+1=ekGR1GTQek+1P?k+1;(18) where P?k+1=?k+1?k. An iterative relationship for ek along the trial index k is then followed as ek+1=(I + GR1GTQ)1ek(I + GR1GTQ)1P?k+1 (19) Again substituting ek+1=ryk+1into Eq. (13) and using Eq. (4), it follows that uk+1=(I + R1GTQG)1uk +(I + R1GTQG)1R1GTQ(r?k+1):(20) The result follows by applying standard discrete time sys- tem theory. Theorem 3.2 (Convergence). Apply the iterative learn- ing control algorithm of Eqs. (10); and (13)(15) to plant Eq. (1); in which R and Q are chosen to satisfy Eqs. (16) and (17). If all trials are repeated in the sense that all xk(0); external disturbances ?k(t) and !k(t) are the same along the trial index k; then the following con- vergence results hold: lim k uk+1=(GTQG)1GTQ(r?);(21) lim k ek+1=0;(22) where ?is some constant vector. Proof. If all trials are repeated, it follows from Eq. (3) that there exists a constant vector ?such that ?k=?for all trial index k. Iteratively using Eqs. (20) and (19), one obtains uk+1=(I + R1GTQG)ku0 + k ? l=1 (I + R1GTQG)lR1GTQ(r?k+2l); (23) ek+1=(I + GR1GTQ)ke0 k ? l=1 (I + GR1GTQ)lP?k+2l;(24) respectively. Therefore, under the robust BIBO stable condition of Theorem 3.1 one obtains lim kuk+1 =(I +R1GTQG)1I(I +R1GTQG)11 R1GTQ(r?) =(R1GTQG)1R1GTQ(r?) =(GTQG)1GTQ(r?); which completes Eq. (21). Due to P?k+1=0 and in view of Eq. (24), the limit of Eq. (22) follows readily. 4. Selection of the weighting matrices The above stability derivation is based on the assump- tion that the initial error and disturbance are bounded. To achieveareasonabletransientperformance,theweighting matrices Q and R must be selected carefully. Let R=?I, Q= I where ? and are positive design constants, and let != =?. Notice that the performance of the optimal ILC is a5ected by the ratio of Q and R rather than their real values, as shown in Eqs. (16) and (17). A neces- sary condition has to be satis9ed by ? and is to ensure robust bounded-inputbounded-output stability. This is straightforward from Eqs. (16) and (17) if ? and are both positive, and GTG or GGThas at least one posi- tive eigenvalue. The following is to determine constants ? and such that the resulted control system cannot only reject uncertain disturbances but also track the desired reference with rapid convergence. It is derived from Eqs. (13)(15) that uk+1=(I + !GTG)ku0 + k ? l=1 !(I + !GTG)lGT(r?k+2l);(25) ek+1=(I + !GGT)ke0 k ? l=1 (I + !GGT)lP?k+2l: (26) F. Gao et al./Chemical Engineering Science 56 (2001) 702570347029 Therefore, with 9xed GGT, a large value of ! (equiva- lently, large ) will be helpful in reducing error of the 9rst trial e0, i.e. a quick convergence can be achieved trial by trial. However, from Eqs. (14) and (15) one obtains S(t)=ATS(t + 1)IBBTS(t + 1)B + ?I)1 BTS(t + 1)A + CTC; t =0;1;:;N1; S(N)=0;(27) ?k+1(t)=?I + S(t)BBT1 AT?k+1(t + 1) + CTek(t + 1); t =0;1;:;N1; ?k+1(N)=0;(28) uk+1(t)=uk(t)BTK(t)B + ?I1BT S(t)Axk+1(t)xk(t) + ?1BT?k+1(t): (29) It can be seen that a large ! (or large ) leads to a stronger feed-forward action on uk+1(t) through ?k+1(t), making the control system less sensitive to the variation of the output reference. A strong feed-forward action tends to accumulate stochastic errors resulted from uncertainties and external disturbances, resulting in strong Quctuations in control input. On the other hand, it can be seen from Eqs. (3), (19) and (20) that when A has eigenvalues outside unit disc, the initialization uncertainties and ex- ternal disturbances could cause a slow convergence, or even oscillatory control. A varying weighting scheme is thus suggested here to take these practical considerations into account. Let !k= k=?kbe a sequence approaching to zero with the increase of cycle number k, i.e., !k0 (or k0) when k . Then Eqs. (10) and (28) become Sk(t)=ATSk(t + 1)IBBTSk(t + 1)B + ?kI)1 BTSk(t + 1)A + kCTC; t =0;1;:;N1; Sk(N)=0;(30) ?k+1(t)=?k?kI + Sk(t)BBT1 AT?k+1(t + 1) + kCTek(t + 1); t =0;1;:;N1; ?k+1(N)=0:(31) It is obvious that Sk(t)0 and ?k(t)0 when k , which indicates that a rapid convergence of uk(t) and ek(t) can be ensured through Theorem 3.1 and Eq. (15). The suggested scheme of selecting the weight- ing matrices Q and R is veri9ed experimentally in the following sections. 5. Simulation and experimental application to injection velocity control 5.1. Injection molding process Injection molding is an important polymer processing technique. It transforms polymer granules into various shapes and types of products, ranging from simple cups to precision lens and compact discs. As a cyclic pro- cess, the injection molding comprises three stages: 9lling (injection), packing-holding and cooling. During 9lling, injection screw moves forward and pushes the polymer melt into the mold cavity. Once the mold is completely 9lled, the process switches to the packing-holding stage, during which additional polymer is added under a certain pressure to the mold to compensate for the shrinkage as- sociated with the material cooling and solidi9cation. The packing-holding stage continues until the gate, which is a narrow entrance to the mold cavity, free