ISA Transactions 51 (2012) 81–94
Contents lists available at SciVerse ScienceDirect
ISA Transactions
journal homepage: www...
82 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 1. The decentralized observer-based digital-redesign tracke...
J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 83
Fig. 2. The traditional OKID-based modeling for unknown stochast...
84 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 3. Observer-based linear quadratic analog tracker.
possible...
J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 85
Fig. 4. OKID-based digital tracker and observer for the determin...
86 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 7. Errors between: (a) outputs Yid11(kTs) and Yo11(kTs), (b...
J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 87
Fig. 8. Comparison between the actual output and its observer-ba...
88 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 9. Errors between: (a) outputs Yid11(kTs) and Yo11(kTs), (b...
J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 89
Fig. 10. Input responses via the proposed method at the 10th gen...
90 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 11. Comparison between the proposed method and the traditio...
J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 91
Fig. 12. Comparison between the proposed method and the traditio...
92 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 13. Errors of the traditional ILC and the well-initialized ...
J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 93
Fig. 14. Comparison between the proposed method and the traditio...
94 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 15. Outputs via the proposal method without the fault-toler...
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Efficient decentralized iterative learning tracker for unknown sampled data interconnected large-scale state-delay system with closed-loop decoupling property

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In this paper, an efficient decentralized iterative learning tracker is proposed to improve the dynamic performance of the unknown controllable and observable sampled-data interconnected large-scale state-delay system, which consists of NN multi-input multi-output (MIMO) subsystems, with the closed-loop decoupling property. The off-line observer/Kalman filter identification (OKID) method is used to obtain the decentralized linear models for subsystems in the interconnected large-scale system. In order to get over the effect of modeling error on the identified linear model of each subsystem, an improved observer with the high-gain property based on the digital redesign approach is developed to replace the observer identified by OKID. Then, the iterative learning control (ILC) scheme is integrated with the high-gain tracker design for the decentralized models. To significantly reduce the iterative learning epochs, a digital-redesign linear quadratic digital tracker with the high-gain property is proposed as the initial control input of ILC. The high-gain property controllers can suppress uncertain errors such as modeling errors, nonlinear perturbations, and external disturbances (Guo et al., 2000) [18]. Thus, the system output can quickly and accurately track the desired reference in one short time interval after all drastically-changing points of the specified reference input with the closed-loop decoupling property.

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Efficient decentralized iterative learning tracker for unknown sampled data interconnected large-scale state-delay system with closed-loop decoupling property

  1. 1. ISA Transactions 51 (2012) 81–94 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Efficient decentralized iterative learning tracker for unknown sampled-data interconnected large-scale state-delay system with closed-loop decoupling property Jason Sheng-Hong Tsaia,∗ , Fu-Ming Chena , Tze-Yu Yua , Shu-Mei Guob,∗∗ , Leang-San Shiehc a Control System Laboratory, Department of Electrical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC b Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC c Department of Electrical Engineering, University of Houston, University Park, Houston, TX 77204-4005, USA a r t i c l e i n f o Article history: Received 18 May 2011 Received in revised form 8 July 2011 Accepted 6 August 2011 Available online 27 August 2011 Keywords: Decentralized control Digital redesign Linear quadratic tracker Iterative learning control a b s t r a c t In this paper, an efficient decentralized iterative learning tracker is proposed to improve the dynamic performance of the unknown controllable and observable sampled-data interconnected large-scale state- delay system, which consists of N multi-input multi-output (MIMO) subsystems, with the closed-loop decoupling property. The off-line observer/Kalman filter identification (OKID) method is used to obtain the decentralized linear models for subsystems in the interconnected large-scale system. In order to get over the effect of modeling error on the identified linear model of each subsystem, an improved observer with the high-gain property based on the digital redesign approach is developed to replace the observer identified by OKID. Then, the iterative learning control (ILC) scheme is integrated with the high-gain tracker design for the decentralized models. To significantly reduce the iterative learning epochs, a digital- redesign linear quadratic digital tracker with the high-gain property is proposed as the initial control input of ILC. The high-gain property controllers can suppress uncertain errors such as modeling errors, nonlinear perturbations, and external disturbances (Guo et al., 2000) [18]. Thus, the system output can quickly and accurately track the desired reference in one short time interval after all drastically-changing points of the specified reference input with the closed-loop decoupling property. © 2011 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction In the recent years, the decentralized control of interconnected large-scale systems has been one of the popular research topics in control theory. Large-scale systems have very complex dynamic models due to the uncertain environment, the interconnected structure of the system, and the varying system parameters. All these issues make the stabilization of such large-scale systems a difficult control problem. Examples of practical systems including power systems, chemical processes, and communications systems are shown in [1,2]. Many researches focusing on the above issues have appeared in [3], and various methods have been proposed to deal with those problems. Decentralized adaptive control work started long before 1996 but Ioannou and Sun [4] is a good literature review up to that date. Ref. [4] also shows that even weak interconnections can make a decentralized adaptive controller ∗ Corresponding author. Tel.: +886 6 2757575x62360; fax: +886 6 2345482. ∗∗ Corresponding author. Tel.: +886 6 2757575x62525; fax: +886 6 2747076. E-mail addresses: shtsai@mail.ncku.edu.tw (J.S.-H. Tsai), guosm@mail.ncku.edu.tw (S.-M. Guo). unstable. In the main, the decentralized adaptive control methods need to utilize real system information to modify the model so as to improve the model-base optimum [5]. So, when the system information could not be obtained or measured at some time, the previous methods could not be used. Iterative learning control (ILC) was firstly proposed by Arimoto et al. [6,7]. The main advantage of the iterative learning control strategy requires less prior knowledge about the system dynamics and less computational effort than many other types of control strategies. Furthermore, the controller can exhibit the perfect control performance according to less prior knowledge of the system. Due to its advantages which present higher efficiency and more convenience, many researchers discuss ardently the subjects of ILC and usually apply the controller in several physical systems such as industrial robots [8], robotic manipulators [9], computer numerical control (CNC) machines [10], and antilock braking systems (ABS) [11], etc. An optimization paradigm of the iterative learning control that concentrated on linear systems and the potential of the use of optimization methods to achieve effective control has been proposed by Owens’s research group at Sheffield University [12]. In recent years, decentralized iterative learning controls [13,14] have been developed maturely and 0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2011.08.001
  2. 2. 82 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 Fig. 1. The decentralized observer-based digital-redesign trackers for the unknown sampled-data interconnected large-scale system (for N = 2) with the state-delay terms. embedded to many industrial processes, such as petrochemical processes and metallurgical processes [14,15]. The aim to embed decentralized ILC into the above procedures is to improve the dynamic performance of the transient response which limits to stable systems only. However, in general, the decentralized iterative learning control law is high-order and more complex for a class of large-scale interconnected dynamical systems. In this paper, an efficient decentralized iterative learning tracker is proposed to improve the dynamic performance of the un- known sampled-data interconnected large-scale state-delay sys- tem, which consists of N MIMO subsystems, with the closed-loop decoupling property. Yi et al. [16] derived the criteria for point- wise controllability and observability and developed a method to use the matrix Lambert W function-based solution form for the lin- ear time delay system with one state-delay. According to controlla- bility and observability of time delay system with the multi-state delays and interconnected state delays, the literature is not pre- sented so far. The purpose of this paper is to propose an efficient decen- tralized iterative learning tracker for the unknown sampled-data interconnected large-scale state-delay system with closed-loop decoupling property. First, the appropriate (low order) decen- tralized linear observers are determined by the off-line OKID method [17] for a class of (unknown) controllable and ob- servable sampled-data interconnected large-scale system with state-delay. The OKID method is a time-domain technique that identifies a discrete input–output mapping in the general coordi- nate form by using known input–output sampled data, through an extension of the eigensystem realization algorithm (ERA), so that the order-determination problem existing in the system identifica- tion problem can be solved. In order to get over the effect of mod- eling error on the identified linear model of each subsystem, an improved observer with high-gain property based on the digital redesign approach will be developed to replace the identified ob- server based on OKID. Then, the iterative learning control scheme is integrated with the high-gain tracker design for the decentralized models. To significantly reduce the iterative learning epochs, we propose the improved observer-based digital redesign tracker with the high-gain property [18] to generate the initial control input of ILC. Furthermore, it can suppress the uncertain errors such as non- linear perturbations and external disturbances as well as make the system output quickly and accurately track the desired reference in one short time interval after all drastically-changing points of the specified reference input. Finally, an example is given to demon- strate the high-performance trajectory tracking with the closed- loop decoupling property by the proposed methodology. 2. Problem description Consider the unknown controllable and observable system consisting of N interconnected MIMO subsystems with state delay shown as follows Si : ˙xdi(t) = αi− j=0 Aijxdi(t − τij) + N− k=1,k̸=i βk− j=0 εikj ¯Aikjxdk(t − ¯τikj) + Biudi(t), (1a) ydi(t) = Cixdi(t). (1b) Notation Si in (1a) denotes the ith subsystem of the interconnected large-scale system, where i = 1, 2, . . . , N. The first term presented by summation from j = 0 to αi in the right-hand side of (1a) denotes various linear combinations of the internal state delays of Subsystem Si. The second term presented by double summations in the right-hand side of (1a) denotes various linear combinations of the external state delays from Subsystem Sk to Subsystem Si. Notation τij is the jth the internal delay of Subsystem Si, ¯τikj is the jth external delay from Subsystem Sk to Subsystem Si, εikj is the jth weighting external disturbance gain from Subsystem Sk to Subsystem Si, and Aij and ¯Aikj are system matrices and external delay system matrices, respectively. Notation xdi(t) ∈ ℜni is the state vector of the Subsystem Si, udi(t) ∈ ℜmi is the input, and ydi(t) ∈ ℜpi is the output. The design procedure of this paper is then briefly described as follows. First, applying the off-line OKID method, the appropriate (low order) decentralized linear observers for the interconnected sampled-data large-scale system can be determined. Then, in order to overcome the effect of modeling error on the identified linear model of each subsystem, an improved observer with the high- gain property based on the digital redesign approach will be developed to take the place of the observer determined by the OKID. Subsequently, the decentralized digital redesign trackers with the high-gain property shown in Fig. 1 will be proposed, so that the closed-loop system has a good tracking performance and the decoupling property. To improve the dynamic performance of the transient response, the ILC scheme is embedded in the decentralized models. The digital redesign linear quadratic tracker with the high-gain property is then applied to generate the first input of ILC to significantly reducing the iterative learning epochs. All the detailed design procedures will be presented in the following sections, respectively.
  3. 3. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 83 Fig. 2. The traditional OKID-based modeling for unknown stochastic sampled-data systems. 3. Traditional OKID-based modeling for unknown sampled- data systems The traditional decentralized iterative learning control law is high-order and complicated for a class of large-scale intercon- nected dynamical systems with state delay. The appropriate (low order) decentralized linear observer for the sampled-data inter- connected large-scale linear system is then desired to be deter- mined by the off-line OKID [17] method. The discrete-time state-space model of a multi-variable linear system can be represented in the following general form x(k + 1) = Gx(k) + Hu(k), (2a) y(k) = Cx(k), (2b) where x(k) ∈ ℜn , u(k) ∈ ℜm and y(k) ∈ ℜp are state, output, and control input vectors, respectively, and G ∈ ℜn×n , H ∈ ℜn×m and C ∈ ℜp×n are system, input, and output matrices, respectively. The following result is summarized from the approach in [17,19]. Note that the Hankel matrix obtained from the combined Markov parameters is associated with the system and the observer as follows ¯H(k − 1) =     Υk Υk+1 · · · Υk+β−1 Υk+1 Υk+2 · · · Υk+β ... ... ... ... Υk+α−1 Υk+α · · · Υk+α+β−2     , (3) where α and β are two sufficiently large but otherwise arbitrary in- tegers, and Υk = [CGk−1 H CGk−1 F], and F is the observer gain to be determined based on input and output measurements. When the combined Markov parameters are determined, the eigensystem re- alization algorithm (ERA) is used to obtain the desired appropri- ate (low order) n∗ and the discrete system and observer realization [ˆG, ˆH, ˆC, F] through the singular value decomposition (SVD) of the Hankel matrix. The ERA processes a factorization of the block data matrix in (3), starting from k = 1, using the singular value decomposition formula ¯H(0) = V ∑ ST , where the columns of matrices V and S are orthonormal and ∑ is a rectangular matrix of the form − = − ˜n 0 0 0  , (4) where ∑ ˜n = diag[σ1, σ2, . . . , σn∗ , σn∗+1, · · · , σ˜n] contains mono- tonically non-increasing entries σ1 ≥ σ2 ≥ · · · ≥ σn∗ > σn∗+1 ≥ · · · ≥ σ˜n > 0. Here, some singular values σn∗+1, . . . , σ˜n are rel- atively small (σn∗+1 ≪ σn∗ ). In order to construct the low order observer of the system, let us define ∑ n∗ = diag [σ1, σ2, . . . , σn∗ ]. The realizations of the system parameters and observer parame- ters by the ERA are given as ˆG = −1/2− n∗ VT n∗ ¯H(1)Sn∗ −1/2− n∗ , (5a)  ˆH F  = First (m + p) columns of 1/2− n∗ ST n∗ , (5b) ˆC = First p rows of Vn∗ 1/2− n∗ . (5c) For system identification, SVD is very useful in determining the sys- tem order. In practice, the primary purpose of applying the OKID method is that the constructed observer satisfies the least-squares solution or acts the input–output map same as a Kalman filter. If the data length is sufficiently long and the order of the observer is sufficiently large, the truncation error is negligible. Now, we show the relationship between the identified observer and the Kalman filter. Let (2a) and (2b) be extended to include process and measurement noises described as x(k + 1) = ˆGx(k) + ˆHu(k) + w(k), (6a) y(k) = ˆCx(k) + v(k), (6b) where the process noise w(k)is a Gaussian, zero-mean white signal with covariance matrix Q, and the measurement noise satisfies the same assumption as w(k) with a different covariance matrix R. The sequences w(k) and v(k) are independent of each other. Then, a typical Kalman filter can be written as ˆx(k + 1) = ˆGˆx(k) + ˆHu(k) + Kεr (k), (7a) ˆy(k) = ˆC ˆx(k), (7b) where ˆx(k) is the estimated state and εr (k) is defined as the difference between the real measurement y(k) and the estimated measurement ˆy(k). Therefore, when the residual εr (k) is a white sequence of the Kalman filter residual, the observer gain F converges to the steady-state Kalman filter gain for F = −K, where K is the Kalman filter gain. The traditional OKID-based modeling for the unknown stochastic sampled-data system is shown in Fig. 2. 4. Prediction-based digital redesign observer for unknown deterministic sampled-data systems 4.1. Digital redesign of the observer-based linear quadratic analog tracker The conventional observer has the property that ˆxd (kTs) is reconstructed from yd (kTs − Ts) , yd (kTs − 2Ts) , . . . . It is also
  4. 4. 84 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 Fig. 3. Observer-based linear quadratic analog tracker. possible to derive an observer, which also uses yd (kTs) to estimate ˆxd (kTs), as follows. The digital estimator gain Ld will be indirectly designed in this section, rather than being directly estimated from the identified discrete-time parameters of the OKID model. The following result is summarized from [18]. First, consider the linear continuous-time deterministic system described as ˙xc (t) = Axc (t) + Buc (t), xc (0) = x0 (8a) yc (t) = Cxc (t), (8b) which is assumed to be both controllable and observable, where xc (t) ∈ ℜn , uc (t) ∈ ℜm , and yc (t) ∈ ℜp . One can take advantage of the observer to estimate the unmeasured system state. Let the linear observer with respect to the continuous-time system (8a) and (8b) be presented by ˙ˆxc (t) = Aˆxc (t) + Buc (t) + Lc [yc (t) − C ˆxc (t)], (9) where ˆxc (t) is the estimate of xc (t) and Lc ∈ ℜn×p is the observer gain [20], where Lc = PobCT R−1 ob , (10) in which Pob is the symmetric and positive definite solution of the following Riccati equation APob + PobAT − PobCT R−1 ob CPob + Qob = 0, (11) where Qob ≥ 0 and Rob > 0 with appropriate dimensions. The observer-based linear quadratic analog tracker is shown in Fig. 3. The digital redesign-based observer for the deterministic sampled-data system is then given by ˙ˆxd(kTs + Ts) = Gd ˆxd(kTs) + Hdud(kTs) + Ldyd(kTs + Ts) (12a) and ˆyd(kTs + Ts) = C ˆxd(kTs + Ts), (12b) where Ld = (G − In)A−1 Lc (Im + C(G − In)A−1 Lc )−1 , (13) Gd = G − LdCG, (14) Hd = (In − LdC)H, (15) G = eATs , (16) H = (G − In)A−1 B. (17) In view of practical implementation, the alternative discrete observer utilizes the current output yd(kTs) and the previously estimated state ˆxd(kTs − Ts) to estimate the current state ˆxd(kTs) as follows ˙ˆxd(kTs) = Gd ˆxd(kTs − Ts) + Hdud(kTs − Ts) + Ldyd(kTs), (18a) ˆyd(kTs) = C ˆxd(kTs). (18b) The observer-based digital tracker for the deterministic sampled- data system is shown in Fig. 4. 4.2. Linear quadratic analog tracker design Consider the linear analog system given in (8). The optimal state-feedback control law for the linear quadratic tracker is to minimize the following performance index J = ∫ ∞ 0  [Cxc (t) − r(t)]T Q [Cxc (t) − r(t)] + uT c (t)Ruc (t)  dt, (19) with Q ≥ 0 and R > 0. This optimal control [18] is given by uc (t) = −Kc xc (t) + Ec r(t), (20) where the analog feedback gain Kc ∈ ℜm×n and the forward gain Ec ∈ ℜm×m for m = p are Kc = R−1 BT P, (21) Ec = −R−1 BT [(A − BKc )−1 ]T CT Q . (22) Then, the resulting closed-loop system becomes ˙xc (t) = (A − BKc )xc (t) + BEc r(t). (23) Here, r(t) is a reference input or desired trajectory, and P is the positive definite and symmetric solution of the following Riccati equation as AT P + PA − PBR−1 BT P + CT QC = 0. (24) The closed-loop system (23) is asymptotically due to the property of LQR design (20). 4.3. Digital redesign of the linear quadratic analog tracker Let the continuous-time state-feedback controller be uc (t) = −Kc xc (t) + Ec r(t), (25) where Kc ∈ ℜm×n and Ec ∈ ℜn×m have been designed to satisfy some specified goals, and r(t) ∈ ℜm is a desired reference input vector. Thus, the analogously controlled system is ˙xc (t) = Ac xc (t) + BEc r(t), xc (0) = xc0 = x0, (26) where Ac = A − BKc . Let the state equation of a corresponding discrete-time equivalent model be ˙xd(t) = Axd(t) + Bud(t), xd(0) = xd0 = x0, (27) where ud(t) ∈ Rm is a piecewise-constant input vector, satisfying ud(t) = ud(kTs), for kTs ≤ t < (k + 1)Ts, (28) and Ts > 0 is the sampling period. Then, the discrete-time state- feedback controller is given by [18] as ud(kTs) = −Kdxd(kTs) + Edr∗ (kTs), (29) where
  5. 5. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 85 Fig. 4. OKID-based digital tracker and observer for the deterministic sampled-data system. Fig. 5. A memory based ILC. Fig. 6. Comparison between the actual output and estimated output obtained by OKID: (a) outputs Yid11(kTs) and Yo11(kTs), (b) outputs Yid12(kTs) and Yo12(kTs), (c) outputs Yid21(kTs) and Yo21(kTs), (d) outputs Yid22(kTs) and Yo22(kTs).
  6. 6. 86 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 Fig. 7. Errors between: (a) outputs Yid11(kTs) and Yo11(kTs), (b) outputs Yid12(kTs) and Yo12(kTs), (c) outputs Yid21(kTs) and Yo21(kTs), (d) outputs Yid22(kTs) and Yo22(kTs). Kd = (Im + Kc H)−1 Kc G, (30a) Ed = (Im + Kc H)−1 Ec , (30b) r∗ (kTs) = r(kTs + Ts), (30c) G = eATs , (30d) H = (G − In)A−1 B for nonsingular A, (30e) H = [ Ts + A (Ts)2 2! + A2 (Ts)3 3! + · · · ] B for singular A, (30f) Kd ∈ ℜm×n is a digital state-feedback gain, Ed ∈ ℜm×m is a digital feed-forward gain, and r∗ (kTs) ∈ ℜm is a piecewise-constant reference input vector determined in terms of r(kTs) for tracking purposes. It is well known that the high-gain (analog) controller/observer induces a high quality performance on trajectory tracking design/state estimation, and it can also suppress system uncer- tainties such as nonlinear perturbations, parameter variations, modeling errors and external disturbances. For these reasons, the sub-optimal analog controller and observer with a high-gain prop- erty is adopted in our approach. The high-gain property controller can be obtained by choosing a sufficiently high ratio of Q to R in (19) so that the system output can closely track a pre-specified trajectory. However, the high-gain property of the analog tracker usually yields large control signals, which might cause the system actuator to saturate and give an unsatisfactory system response. To overcome this difficulty, the tracker is redesigned based on the ad- vanced digital redesign technique equipped with a suitably large sampling period and zero hold, which yields an equivalent dig- ital controller but with a low-gain, without possibly losing the high quality performance. However, a large sampling period usu- ally induces a degradation of the tracking performance. Therefore, in general, a suitable compromise between the pre-specified per- formance and the selections of the sampling time Ts, weighting ma- trices (Qo, Ro) in (11) and (Q , R) in (19) should be considered. For simplicity in discussion, we neglect the actuator saturation prob- lem in this paper. 5. Iterative learning control In recent years, there have been many subjects and approaches in the field of ILC. The basic idea of ILC is that a system performs the same task repeatedly, and the control performance of the system can be improved by learning from previous iterations. Many existing control methods are not able to fulfill such a task, because they only warrant an asymptotic convergence, and being more essential, they are unable to learn from previous control trails, whether those succeeded or failed. Without learning, a control system can only produce the same performance without improvement, even if the task repeats consecutively. On the other hand, an initial state error occurs when the initial state of the system is different from the initial state that is implicitly given by the reference trajectory. It is shown that under this sufficient condition [21–23], the iterative learning control can ensure the system output converge to desired trajectories with bounded tracking errors. Besides, the proposed iterative learning controller [21,22] works with a reduced sampling rate that ensures
  7. 7. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 87 Fig. 8. Comparison between the actual output and its observer-based output by OKID: (a) outputs Yid11(kTs) and Yo11(kTs), (b) outputs Yid12(kTs) and Yo12(kTs), (c) outputs Yid21(kTs) and Yo21(kTs), (d) outputs Yid22(kTs) and Yo22(kTs). the reduction of an appropriate norm of the error trajectory from cycle to cycle. Notice that the convergence of ILC is directly influenced by the initial control input. To accelerate the convergence of ILC, the digital-redesign linear quadratic digital tracker with the high-gain property is proposed as the initial control input of ILC. 5.1. The design of the ILC controller ILC has earned a lot of interest in the current years. The basic concept of the ILC is to use the control information of the previous iterations to improve the control performance of the present iteration. This is realized through the memory based learning scheme which is shown in Fig. 5. Consider a discrete time system x(k + 1) = Gx(k) + Hu(k), x(0) = x0, (31a) y(k) = Cx(k), (31b) where G = eAc Ts and H =  T 0 eAc Ts Bdt. The above equations are the zero order holder equivalent of continuous time systems described by x(t) ∈ ℜn , Ac ∈ ℜn×n , Bc ∈ ℜn×m , and C ∈ ℜp×n . For the iterative learning cycle l with l = 0, 1, . . . , Nl, the system can be described by [21,22]. yl(0) = Cx0, (32) yl = L0x0 + Lul, (33) yl = [yl(1)yl(2)yl(3) · · · yl(Nl)]T , (34) ul = [ul(0)ul(1)ul(2) · · · ul(Nl − 1)]T , (35) with L0 =       CG CG2 CG3 ... CGNl       , (36) and L =       CH 0 · · · 0 CGH CH ... ... ... ... ... ... CGNl−1 H · · · CGH CH       (37) where CH is assumed to be non-singular. 5.2. Problem formulation and discrete ILC updating law According to the literature [21,22], the update learning law is presented as ul+1(k) = ul(k) + Γ el (k + 1) , (38) el(k) = yd(k) − yl(k), (39)
  8. 8. 88 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 Fig. 9. Errors between: (a) outputs Yid11(kTs) and Yo11(kTs), (b) outputs Yid12(kTs) and Yo12(kTs), (c) outputs Yid21(kTs) and Yo21(kTs), (d) outputs Yid22(kTs) and Yo22(kTs). where el is the error term, yd is the tracking trajectory, and Γ is the learning control gain. Then, substituting (38) into (33) yields the following error function El+1 = LeEl, (40) where El = [eT l (1)eT l (2)eT l (3) · · · eT l (N)]T , and the matrix Le is presented as Le =       I − CHΓ 0 · · · 0 −CGHΓ I − CHΓ ... ... ... ... ... 0 −CGNl−1 HΓ · · · −CGHΓ I − CHΓ       . (41) While we assign an appropriate learning gain Γ to induce the 1-norm of the matrix Le to be less than 1, i.e. ‖Le‖1 < 1, the tracking error el converges to zero as l approaches infinity. To assure the condition ‖Le‖1 < 1 can be satisfied, the literature [21,22] suggests reducing the matrix norm ‖G‖ via choosing a large sample time Ts. Moreover, ILC takes a lot of iterative learning epochs to update the input of the system owing to the input of the initial iterative learning epoch being set to zero, which is commonly used in literature. To significantly reduce the iterative learning epochs and greatly promote the tracking performance, we use the observer- based digital redesign linear quadratic analog tracker shown in Fig. 4, which has been shown [18] to be a high performance approach, to set up the primary cycle input of the control system. 6. Design procedure The design procedure is summarized as follows: Step 1: Perform the off-line observer/Kalman filter identification method to determine the appropriate (low order) decen- tralized linear system/observer models from the unknown sampled-data interconnected large-scale linear system with state delay. Step 2: Transform the obtained discrete-time linear system/obser- ver models to continuous-time linear system/observer models with the pre-specified sampling times. Step 3: Select appropriate weighting matrices (Q , R)and (Qob, Rob) for tracker and observer designs by choosing sufficiently high ratios of Q to R and Qob to Rob, so that the tracker and observer have the high-gain property. Step 4: Embed the iterative learning control scheme in the decen- tralized models. Extract the control input determined in Step 3 as the first generation control input of ILC. Step 5: Repeatedly apply the ILC algorithm ul(k), for l = 1, 2, 3, . . . , until it reaches the good performance tracking object.
  9. 9. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 89 Fig. 10. Input responses via the proposed method at the 10th generation: (a) Input responses of Subsystem S1: input ud11(t). (b) Input responses of Subsystem S1: input ud12(t). (c) Input responses of Subsystem S2: input ud21(t). (d) Input responses of Subsystem S2: input ud22(t). 7. An illustrative example To show the effectiveness of the newly proposed method, an ex- ample is given to explain the high performance trajectory tracking with the closed-loop decoupling property as following. From the academic study point of view, consider an unknown controllable and observable deterministic large-scale linear system which con- tains two interconnected 2-input–2-output subsystems with state delays as follows Si : ˙xd1(t) = A10xd1(t − τ10) + A11xd1(t − τ11) + A12xd1(t − τ12) + ε120 ¯A120xd2(t − ¯τ120) + ε121 ¯A121xd2(t − ¯τ121) + ε122 ¯A122xd2(t − ¯τ122) + B1ud1(t), S2 : ˙xd2(t) = A20xd2(t − τ20) + A21xd2(t − τ21) + A22xd1(t − τ22) + ε210 ¯A210xd1(t − ¯τ210) + ε211 ¯A211xd1(t − ¯τ211) + ε212 ¯A212xd2(t − ¯τ212) + B2ud2(t), where ud1(t) =  ud1,1(t) ud1,2(t)  , ud2(t) =  ud2,1(t) ud2,2(t)  , xd1(t) =  xd1,1(t) xd1,2(t)  , xd2(t) =  xd2,1(t) xd2,2(t)  , A10 =  −2.1172 0.0377 0.6958 −1.8828  , A11 =  −0.1672 −0.0184 −0.1198 −0.1328  , A12 =  −0.0331 −0.0074 −0.0479 −0.0469  , ¯A120 =  −0.0072 0.0068 −0.0220 −0.0403  , ¯A121 =  −0.0200 −0.0019 0.0063 −0.0104  , ¯A122 =  −0.0215 −0.0032 0.0103 −0.0060  , A20 =  −1.8994 0.2636 0.3433 −1.1006  , A21 =  −0.2264 −0.1994 −0.0097 −0.2736  , A22 =  −0.0153 −0.0399 −0.0019 −0.0247  , ¯A210 =  −0.0430 −0.0625 0.0015 −0.0196  , ¯A211 =  −0.0343 −0.0500 0.0013 −0.0157  , ¯A212 =  −0.0332 −0.0600 0.0015 −0.0108  ,
  10. 10. 90 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 Fig. 11. Comparison between the proposed method and the traditional ILC at the 10th generation for output responses: (a) Output responses of Subsystem S1: outputs Yd11(t) and reference r11(t). (b) Output responses of Subsystem S1: outputs Yd12(t) and reference r12(t). (c) Output responses of Subsystem S2: outputs Yd21(t) and reference r21(t). (d) Output responses of Subsystem S2: outputs Yd22(t) and reference r22(t). B1 =  −0.15 0.3 0.2 0.4  , B2 =  −0.2 0.2 0.21 0.15  , C1 =  0.1 0.1 −0.4 0.1  , C2 =  0.1 0.2 −0.2 0.1  , and the initial condition xd1(0) =  0.1 0.5 T , xd2(0) =  0.5 0.1 T . The time delays of the nonlinear interconnected terms are τ10 = ¯τ210 = 0, τ11 = ¯τ211 = 0.6Ts1, τ12 = ¯τ212 = 1.2Ts1, τ20 = ¯τ120 = 0, τ21 = ¯τ121 = 1.6Ts2, and τ22 = ¯τ122 = 2Ts2, where Ts1 = 0.05 s, Ts2 = 0.05 s and the simulation time for the off-line OKID is set as 3 s. Notations ε120 = ε121 = ε122 = 0.5, ε210 = ε211 = ε212 = 0.4. The eigenval- ues of A10 (denoted as σ(A10)) are {−2.1999, −1.8001} , σ(A11) = {−0.2, −0.1} , σ(A12) = {−0.0199, −0.0601} , σ(A20) = {−2, −1} , σ(A21) = {−0.2001, −0.2999} , σ(A22) = {−0.0101, −0.0299}. Since the given system model is unknown, it is desired to construct the Hankel matrix based on the off-line OKID to de- termine the appropriate order as 4, where ∑ 1 = diag[8.6588, 3.5468, 0.4780, 0.0107, 0.0000, . . .], ∑ 2 = diag[8.4041, 5.7287, 0.6776, 0.0592, 0.0000, . . .], and the system is excited by the white noise signal udi(t) =  udi1(t) udi2(t) T for i = 1, 2 with zero mean and covariance diag  cov (udi1(t)) cov (udi2(t))  = diag  0.2 0.2  , for Subsystem S1 and S2, respectively. The iden- tified system matrices and observer gains for Subsystem S1 and Subsystem S2 are respectively given as ˆG1 =    0.9522 −0.0133 0.0827 0 0.0236 0.9056 −0.100 −0.0254 0.2293 −0.0289 0.5991 −0.0133 0.0038 0.0425 0.0157 0.1155    , ˆH1 =    0.0025 −0.0032 −0.0015 −0.0025 0.0021 −0.0022 0 0    , ˆC1 = [ −0.2755 −0.8358 −0.1531 −0.0884 1.0030 −0.1978 0.4942 −0.0218 ] , F1 =    0.3047 −1.7287 1.1671 0.2419 −0.0677 0.5082 −0.0789 −0.0107    , ˆG2 =    0.9431 −0.0106 −0.0927 −0.0105 0.0860 0.9639 0.0645 −0.0325 0.2802 −0.0031 0.6257 0.0879 −0.0098 0.0799 −0.1007 0.2615    , ˆH2 =    0.0022 0 −0.0064 −0.0014 0.0018 0 0 −0.0002    ,
  11. 11. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 91 Fig. 12. Comparison between the proposed method and the traditional ILC at the 30th generation for output responses: (a) Output responses of Subsystem S1: outputs Yd11(t) and reference r11(t). (b) Output responses of Subsystem S1: outputs Yd12(t) and reference r12(t). (c) Output responses of Subsystem S2: outputs Yd21(t) and reference r21(t). (d) Output responses of Subsystem S2: outputs Yd22(t) and reference r22(t). ˆC2 = [ −0.9912 −0.4815 −0.5800 −0.0290 −0.4390 −0.7261 0.0605 −0.2081 ] , F2 =    1.402 −1.0201 −0.9319 1.0871 −0.4670 −0.3851 0.1014 −0.1590    . The OKID-based output compared with the actual system output for Subsystem S1 and Subsystem S2 are shown in Figs. 6 and 7, respectively. To overcome the effect of modeling error, an improved observer with the high-gain property based on the digital redesign approach has been used, where Gd1 =  In − Ld1 ˆC1  ˆG1 =    0.1516 −0.0027 −0.2440 0.0053 0.0058 0.0231 −0.0113 −0.0124 −0.2674 0.0041 0.4990 −0.0106 −0.0055 −0.0136 0.0140 0.1164    , Hd1 = (In − Ld1 ˆC1) ˆH1 =    −0.0001 0 0 −0.00005 −0.0004 −0.0003 0.0001 0    , Ld1 =    −0.1790 0.6993 −1.0877 −0.3079 −0.0786 −0.4435 −0.0711 −0.0125    , with Qob1 = 106 × I4, Rob1 = I2, Gd2 =  In − Ld2 ˆC2  ˆG2 =    0.1500 −0.0334 −0.2094 −0.0469 −0.1057 0.0735 0.1883 −0.0411 −0.3134 0.0998 0.5178 0.0606 0.0079 −0.0768 −0.0711 0.2623    , Hd2 = (In − Ld2 ˆC2) ˆH2 =    0.0002 0.00002 −0.0004 −0.0001 −0.0005 −0.0001 0.0003 0.0002    , Ld2 =    −0.5708 −0.4158 0.2789 −1.0629 −0.4837 −0.1807 0.0868 −0.1616    , for Qob2 = 106 × I4, Rob2 = I2. Then, the comparisons between the actual outputs and their observer-based outputs by digital redesign for two subsystems are shown in Figs. 8 and 9, respectively. The simulation result shows the proposed observer by digital redesign significantly improves the performance of the OKID- based observer. Here, we would like to point out that in this paper, for simplicity in design, an experience approach is utilized for selecting weight- ing matrices Qob and Rob. A more complicated and sophisticated approach for selecting the weighting matrices Qob and Rob can be found in [24–27].
  12. 12. 92 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 Fig. 13. Errors of the traditional ILC and the well-initialized ILC at each iteration: (a) errors ∑Nf k=1[|r11(k) − Yd11(k)|] at each iteration, (b) errors ∑Nf k=1[|r12(k) − Yd12(k)|] at each iteration, (c) errors ∑Nf k=1[|r21(k) − Yd21(k)|] at each iteration, (d) errors ∑Nf k=1[|r22(k) − Yd22(k)|] at each iteration. To demonstrate that the proposed ILC-based high-gain tracker is superior to the traditional ILC tracker, a digital-redesign linear quadratic digital tracker with the high-gain property as the initial control input of the modified ILC is proposed in this paper. The reference input is given by r(t) =  r11(t) r12(t) r21(t) r22(t) T , where r11(t) =  0.5 cos(πt) 0 ≤ t < 1 0.5 sin(πt) 1 ≤ t < 2 −0.5 − 0.5 sin(πt) 2 ≤ t < 4, r12(t) =  0.5 sin(πt) 0 ≤ t < 1.5 0.5 sin(πt), 1 ≤ t < 2.5 −0.5 − 0.5 sin(πt) 2.5 ≤ t < 4, r21(t) =  0.5 sin(πt) 0 ≤ t < 0.8 0.5 sin(πt) 0.8 ≤ t < 2.3 1 + 0.5 sin(πt) 2.3 ≤ t < 4 r22(t) =  0.5 cos(πt) 0 ≤ t < 0.5 0.5 sin(πt) 0.5 ≤ t < 1.8 −1 − 0.5 sin(πt) 1.8 ≤ t < 4. In the following, we apply the digital-redesign method to design an observer-based linear quadratic tracker for setting up the initial control input of the iterative learning control system. The feedback gain Kd and feed-forward gain Ed of the observer-based digital tracker for Subsystems S1 and S2 are respectively given as Kd1 = [ 161.9073 −227.8525 31.0387 3.1829 −105.3106 −182.3413 −19.9290 3.9588 ] , Ed1 = [ 244.1982 255.6082 251.7290 −23.2788 ] , Kd2 = [ 117.6028 −88.0957 30.6766 5.1789 −622.3522 −225.8409 −36.0860 −19.1845 ] , Ed2 = [ −127.6567 37.7943 347.3539 554.8802 ] , with Q1 = 107 × I2, R1 = I2, Q2 = 107 × I2, R2 = I2. Fig. 10 shows the control input with well-initialized ILC at the 10th generation. The simulation results of the novel iterative learning tracker and the traditional ILC at the 10th generation are shown in Fig. 11. The simulation results of the novel iterative learning tracker and the traditional ILC at the 30th generation are shown in Fig. 12. The comparison learning errors of the every iteration between the traditional ILC and the novel iterative learning tracker are shown in Fig. 13. From simulations, it shows the system outputs quickly and accurately track the desired reference in one short time interval after all drastically-changing points of the specified reference input via the proposed method. To further show that the newly proposed ILC-based high-gain tracker can improve the transient response and decrease the Q /R ratio of the controlled system under the traditional digital tracker, we consider the same system given in this example. When the
  13. 13. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 93 Fig. 14. Comparison between the proposed method and the traditional digital redesign tracker for output responses: (a) Output responses of Subsystem S1: outputs Yd11(t) and reference r11(t). (b) Output responses of Subsystem S1: outputs Yd12(t) and reference r12(t). (c) Output responses of Subsystem S2: outputs Yd21(t) and referencer21(t). (d) Output responses of Subsystem S2: outputs Yd22(t) and reference r22(t). Q /R ratio is sufficiently high, the newly proposed iterative learning tracker and the traditional digital redesign tracker have a good tracking performance in both the transient response and steady state response. However, when the Q /R ratio is not sufficiently high, the traditional digital redesign tracker has a poor tracking performance in both the transient response and steady state response. To overcome the above problem, the ILC-based high-gain tracker is newly proposed to improve the transient and steady state response. The simulation results (Q1 = 104 × I2, R1 = I2, Q2 = 104 × I2 and R2 = I2) of the traditional digital redesign tracker and the novel iterative learning tracker are shown in Fig. 14. To show the robustness of the proposed method, let the tracker have a good performance in the beginning, but the first subsystem input is artificially reduced to 30% of the determined input by external factor in 2.5–3.0 s without the fault-tolerant control. Fig. 15 shows that the decentralized controller induces a good robustness on the decoupling of the closed-loop controlled system. When the inputs of parts of the system are broken, the others are not influenced entirely, so the other digitally controlled systems still follow the reference inputs with quite a satisfactory performance. 8. Conclusions The efficient decentralized iterative learning tracker is pro- posed to improve the dynamic performance of the unknown sampled-data interconnected large-scale state-delay system, which consists of N multi-input multi-output subsystems, with the closed-loop decoupling property in this paper, which is regarded as an open problem in literature. The appropriate (low order) de- centralized linear observer for the sampled-data linear system is determined by the off-line OKID method. In order to get over the effect of modeling error on the identified linear models of each sub- system, an improved observer with high-gain property based on the digital redesign approach is developed to replace the identified observer based on OKID. Then, each subsystem of the large-scale decentralized systems is identified as a linear model; applying the PD-type ILC method trains the dynamics of decentralized models to trace the desired trajectory as fast as possible. In order to pursue a faster learning performance, the digital redesign linear quadratic tracker with the high-gain property is constructed to generate the initial control input for ILC. Indeed, the proposed technique greatly promotes the tracking performance and decreases the it- erative epochs thanks to its well-selected initial iterative learning epoch. Acknowledgment This work was supported by the National Science Council of Republic of China under contract NSC99-2221-E-006-206-MY3 and NSC98-2221-E-006-159-MY3.
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