AADECA 2006 XXº Congreso Argentino de Control Automático ITERATIVE LEARNING CONTROL APPLIED TO NONLINEAR BATCH REACTOR E.J.

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1 ITERATIVE LEARNING CONTROL APPLIED TO NONLINEAR BATCH REACTOR E.J. ADAM (1) Institute of Technological Development for the Chemical Industry (INTEC), CONICET Universidad Nacional del Litoral (UNL). Güemes 3450, (3000) Santa Fe Argentina. Tel.: (0342) /77 Fax: (0342) Abstract: This work presents an application example of iterative learning control (ILC) applied to nonlinear batch reactor. Two examples are studied; one of them shows that plants with important nonlinearities can lead to a non monotonic convergence of the error e k (t), and still worse, a unstable equilibrium signal e (t) can be obtained. However, with an adequate study, it is possible to obtain a better performance in the controlled variable with ILC than with the traditional feedback. Keywords: ILC, CSTR, Batch Reactor, Linear Control. to ILC. 1. INTRODUCTION In a batch process, the control problem is usually given as a tracking problem for a time varying reference trajectories defined over a finite interval. Usually, the engineers talk about that a batch process has three operative stages clearly different, start up, batch run and, shutdown. In addiction, one aspect of batch operation unexplored is, how the control engineer can use repetitive nature of the operation to reach a better performance in the controlled variable. And, this is exactly the central point in which ILC theoretical framework is supported. ILC associates three interesting concepts. Iterative refers to a process that executes the same setpoint trajectory over and over again. Learning refers to the idea that by repeating the same thing, the system should be able to improve the performance. Finally, control emphasizes that the result of the learning is used to control the plant. Thus, ILC constitutes the adequate theoretical framework to renew efforts in order to study new alternatives for the batch process control. The organization of this work is as follows. Section 2 included a theoretical framework presentation related 1 To whom all correspondence should be addressed. Then, Section 3 presents by means of numeric simulations the behavior of the batch reactor in closed loop when the designer pretends to apply the ILC linear theory to nonlinear system. Finally, in Section 4 the conclusions are summarized. 2. THEORETICAL FRAMEWORK 2.1. Basic Idea The ILC scheme was initially developed as a feedforward action applied directly to the open loop system (Arimoto, 1984; Kurek and Zaremba, 1993; among others). However, if the system is integrator or unstable to open loop, or well, it has wrong initial condition, the ILC scheme to open loop can be inappropriate. Thus, the feedback based ILC has been suggested in the literature as a more adequate structure (De Roover, 1996; Moon et al. 1998; Doh et al., 1999; Tayebi and Zaremba, 2003). The basic idea is shown in Fig. 1.

2 R E k C V k Memory Device Figure 1. Feedback based ILC diagram. Here, continuous lines denote the signals used during the k th trail and dashed lines denote signals will be used in the next iteration. This scheme operates as follows. Consider a plant which is operated iteratively with the same setpoint trajectory over and over again, as a robot or a industrial batch process. During the k th trail an inputsignal u k (t) is applied to the plant, producing the output signal y k (t). Both signals are stored in the memory devise. Thus, two vectors with length T f are available for the next iteration. If the system of Fig. 1 operates to open loop, using u k (t) in the k+1 th trail it is possible to obtain the same output again. But, if the k+1 iteration includes u k (t) and e k (t) information then, new u k+1 (t) and y k+1 (t) can be obtained. The importance of the input signal modification is to reduce the tracking error as the iterations are progressively increased. That is, e k 1 t e k t k 0. The purpose of an ILC algorithm is to find a unique equilibrium input signal u (t) which minimizes the tracking error. The ILC formulation uses an iterative updating formula for the control signal, given by u k+1 = f(r, y k,..., y k j, u k, u k 1,..., u k l ), j, l 0. (1) This formula can be categorized according to how the information from previous iteration is used (Norrlöf, 2000). The most common algorithm suggested by several authors (Arimoto, 1984; Horowitz, 1993; Bien and Xu; 1998; Tabeyi and Zaremba, 2003; among others), is that whose structure is given by Q U k V k+1 V k+1 = Q(V k, + CE k ), (2) where V 1 = 0, C denotes the controller transfer function and Q is a linear filter 2. Six postulates were originally formulated by different authors (Chen and Wen, 1999; Norrlöf, 2000; Scholten, 2000, among others). 1. Every iteration ends in a fixed time of duration T f. 2 In this paper variables in time domain are denoted with small letters and variables in s domain are denoted with capital letters. P Y k 2. Plant dynamics are invariant throughout the iterations. 3. The set point, r(t) with t [0, T f ] is given a priori. 4. For each trail the initial states are the same. That means that x k (0) = x 0 (0), k Z The plant output y(t) is observable. 6. There exists a unique input, u (t), that yields the desired output, r(t), with a minimum tracking error e (t). A major issue in ILC is the convergence, and each type of ILC has its own convergence criterion. Next, some definitions and remarks are introduced to present the tracking problem The Tracking ILC Formulation The tracking error e k (t) is defined as e k (t) := r(t) y k (t), (3) where the subscript k denotes the run number and e k represents (finite length) output error trajectory for k th trail. A tracking error e (t) is an equilibrium signal of the system. If the system has reached the signal e (t), it will have this error signal for all future trails. Definition 1. The equilibrium signal e (t) is said to be stable if B 0, b 0, e 0 t e t b => k 0, e k t e t B (4) where e 0 (t) is the initial tracking error. Definition 2. An equilibrium signal e (t) is said to be asymptotically stable if it is stable and b 0, e 0 t e t b => lim k e k t e t = A Simple Iterative Updating Formula. (5) Now, consider the iterative updating formula (2) and according to Fig. 1, U k = V k + CE k. Then, V k+1 = QU k (6) is a simple update formula in Laplace domain. Thus, U k+1 = QU k + CE k+1. (7)

3 Being E k+1 = R Y k+1 then, E k+1 = R Y k+1 = R PU k+1 = R P(QU k + Ce k+1 ), where P denotes the plant transfer function and (8) E = S 1 Q 1 SQ R. Similarly and according to Eqs. (13), (16) E k+1 (1+PC) = R PQU k = R PQU k + QE k QE k = R PQU k + QE k Q(R Y k ) = (1 Q)R PQU k + QE k + PQU k. (9) Being S := 1/(1+PC) the sensitivity function, the last equation can be written as, E k+1 = S(1 Q)R + SQE k. (10) According to last equation, it is possible to write (R Y k+1 ) = S(1 Q)R + SQ(R Y k ), (11) and being Y k+1 = PU k+1, the last equation can be rewritten as PU k+1 = R S(1 Q)R SQR + SQPU k = (1 S)R + SQPU k = TR + SQPU k, (12) where T := 1 S is denoted as complementary sensitivity function. In consequence, U k 1 = T P R SQU k. (13) Adam (2005) proofs the following remark for LTI plant without model uncertainty: Remark 1. Consider a feedback based ILC scheme in Fig. 1 with the updating formula (6) and the plant is a LTI system without model uncertainty. If there exists C(s) such that the nominal stability is satisfied, then by adopting Q such that SQ 1 the tracking error is reduced as k is increased and it is bounded for all k + and converges uniformly to e t =lim e k t = L 1 k when k in the sense of the L 2 norm. According to Eq. (10), for k, S 1 Q 1 SQ R, (14) E = S(1 Q)R + SQE. (15) or well or well U = T P R SQU. T U = P 1 SQ R. (17) (18) Based on Eqs. (16) and (18) the following remark can be enunciated (Adam, 2005): Remark 2. Consider the feedback based ILC scheme in Fig. 1 with the updating formula (6) and the plant is a LTI system without model uncertainty. If there exists C(s) such that the nominal stability is satisfied, then by adopting Q = 1 the perfect control can be reached as k. Based on the last remarks the following design procedure is enunciated: Design Procedure (Nominal Case): Step 1: Adopt a controller such that nominal stability and performance are satisfied. Step 2: Set Q = 1 or well Q(s) to be low pass filter such that, Q( ) 1 [0, c ],and Q( ) 0 c with c a cut off frequency. Step 3: Use the ILC updating formula (2) or (6). 3. ILC APPLIED TO BATCH REACTOR In this section, a non linear case or more specifically, a batch reactor with strong parametric uncertainty is studied under theoretical framework of linear ILC presented before Non Linear Batch Reactor Consider a batch reactor with a strong nonlinear dynamic where an exothermic and irreversible second order chemical reaction A B takes place. It is assumed that the reactor has a cooling jacket whose temperature can be directly manipulated. The goal is to control the reactor temperature by means of inlet coolant temperature.

4 32 q ci, T ci q i, c Ai, T i V, c A, T A B q co, T co cooling jacket and reactor temperature reactor temperature response cooling jacket temperature test q o, c Ao, T o Time (min.) Figure 2. A batch reactor scheme. The reader understand that the inlet and outlet process flow rate operates as batch process. The reactor dynamic is modeled by the following equations: d T dt dc A dt = H Mcp k 0e ER/T c A ² = k 0 e ER/T c A ², UA Mcp (T T c). (19) (20) Furthermore, it must be noted that the reaction rate kinetic is r A = kc A 2 with k = k 0 e E/RT. The nominal values of model parameters and variables are presented in the Table 1. They are based on data given by Lee and Lee (1997). Table 1. Nominal batch reactor Values. parameter nomenclature value feed concentration C Ae 0.9 mol m 3 feed temperature T e K inlet coolant T c K temperature UA/Mcp l min 1 reaction rate constant k l mol 1 s 1 activation energy E/R K 1 H/Mcp 5.79 K l mol Identification Experiment A simple test was applied for determining of the linear transfer function parameters. This test consists of introducing a step change in cooling jacket temperature (manipulated variable) and the reactor temperature time response is registered. This numerical experiment is showed in Fig. 3. The nonlinearity of the batch reactor is clearly evidenced. Figure 3. Batch reactor identification tests for different step changes in the cooling jacket temperature. In Fig. 3, the reader can notice that the transfer function structure of the batch reactor changes according to operation point of the reactor. For 28ºC T c < 31ºC a good linear approximation is a first order plus zero while, for 31ºC T c < 32ºC a better approximation is a simple first order. Thus, by simplicity and taking into a account that the batch reactor operates to 30ºC, a firth order transfer function was accepted. The transfer function parameters were computed using a Matlab optimization toolbox based on a multiparametric optimization algorithm. Thus, they result K = 1, T = Numerical Simulations The batch reactor has operation sequence divided in three stage; start up, run and shutdown. The controlled temperature inside the reactor is monitored during these three stage. The controlled temperature performance obtained by a traditional feedback is compared with the one obtained by means of feedback based ILC implementation Example 1 Consider the batch reactor presented in the Section 3.1 where the manipulated variable (cooling jacket temperature) can change between minimum and maximum limits without saturation. Figure 4 shows the ratio between integral square error (ISE) and the maximum ISE obtained when the traditional feedback is implemented alone (k = 0). Clearly, the ISE is reduced as k is incremented, and in consequence, the convergence is monotonic. Notice that, for the tenth iteration the ISE is 13% of the maximum ISE.

5 Unbounded problem 1.4 Bounded problem ISE / ISE max ISE / ISE Iteration Number Iteration Number Figure 4. ISE/ISE max vs. k trail. Figure 5 compares the performance obtained with the traditional feedback (k = 0) and tenth iteration for the feedback based ILC. Notice that, when the feedback based ILC is implemented during the start up and the shutdown, the ramp tracking error is very small. In addiction, during the 10 th batch run, the error is considerably reduced. Setpoint and Controlled Temperature (ºC) Start up k = 0 k =1 0 Run Setpoint Shutdown Time (min.) Figure 5. Setpoint and controlled temperature for iteration 0 (traditional feedback) and 10 (Feedback based ILC) without limit in the manipulated variable Example 2 Now consider the same batch reactor where the manipulated variable has minimum and maximum constrains. That is, T cmin T c T cmax where T cmin = 10 and T cmax = 20 (where T c is written in deviation variable). Figure 5 shows the ratio between integral square error (ISE) and the ISE obtained with the tradional feedfack (here denoted ISE 0 ). Notice that, there is not a monotonic decrease of the ISE (as in the Fig. 4) in the sense that errors do increase in early repetitions. In consequence, the equilibrium signal e (t) is not stable. Figure 6. ISE/ISE 0 vs. k trail. A similar result was reported by Owens and Hätönen (2005). The authors show with a simple example that the L 2 norm can grow to a very large value on some iteration before terminal convergence. This is, in most practical applications of ILC, not acceptable. Amann (1996) and Owens and Hätönen (2005) studied this problem and they propose a optimization problem for iteratively solving. At the present time, there is not much information in the literature about this problem and its possible solution. Figure 5 compares the performance obtained with the traditional feedback (k = 0) and tenth iteration for the feedback based ILC when there are constrains in the manipulated variable. For this case, and according to Fig. 6, the 10 th batch run reduces the ISE approximately a 19%. Notice that, the equilibrium signal is not stable for this case but, this fact does not imply that the control system is unstable. Setpoint and Controlled Temperature (ºC) Start up k = 0 Setpoint Run k =1 0 Shutdown Time (min.) Figure 7. Setpoint and controlled temperature for iteration 0 (tradicional feedback) and 10 (Feedback based ILC) with limit in the manipulated variable.

6 Although, the control system is strongly limited with the constrains in the manipulated variable, a correct formulation would have to lead to an stable equilibrium signal e (t) 0 with a monotonic convergence. The fact that the equilibrium signal e (t) reaches a value different from zero, it is due to the limits that the control system has in itself. 4. CONCLUSIONS Based on the results from different numeric simulations, and considering that the plant has not a linear behavior, it is possible to conclude that: (i), an stable equilibrium signal e (t) is difficult to be reached however, (ii) if it is possible to reach a stable equilibrium signal, a monotonic terminal convergence is little provable. Clearly, when the control engineer want to implement ILC in strongly nonlinear system, the convergence criteria must be reformulated in order to reach a monotonic decrease of the error e k (t) allowing the convergence of the feedback based ILC. In this sense, the proposals of Amann (1996) or Owens and Hätönen (2005) look to be the most appropriate forms to study the problem. This topic is being at the moment studying. ACKNOWLEDGMENT The author would like to thank the Universidad Nacional del Litoral for financial support received (CAI+D 2005 # 04/0016). REFERENCES Adam, E. J., Iterative Learning Control Applied to Linear Batch Processes, XI Reunión de Trabajo en Procesamiento de la Información y Control, Amann, N. Optimal Algorithms for Iterative Learning Control. Ph.D. Thesis, University of Exter (1996). Arimoto S., S. Kawamura and F. Miyazaki, Bettering Operation of Robots by Learning, Journal of Robotic System, 1, (2), (1984). Bien Z. and J X Xu. Iterative Learning Control: Analysis, Design, Iteration and Application. Kluwer Academic Publishers. (1998) Chen Y. and C. Wen, Iterative Learning Control. Springer Verlag (1999). De Roover D., Synthesis of a Robust Iterative Learning Control Using an H Approach, Proc. 35 th Conf. Decision Control, Kobe, Japan, (1996). Doh T. Y., J. H. moon, K. B. Jin and M. J. Chung, Robust ILC with Current Feedback for Uncertain Linear System, Int. J. Syst. Sci, 30, (1), (1999). Horowitz R., Learning Control of Robot Manipulators. Journal of Dynamic Systems, Measurement and Control, 115, (1993). Kurek J. E. and M. B. Zaremba, Iterative Learning Control Synthesis Based on 2 D System Theory, IEEE Trans. Automat. Contr., 38, (1), (1993). Lee K. S. and J. H. Lee, Model Predictive Control for Nonlinear Batch Processes with Asymptotically Perfect Tracking, Computers Chem. Engng., 21, Suppl.,S873 S879 (1997). Morari M. and E. Zafiriou, Robust Process Control. Prentice Hall, Inc. (1989). Moon J. H., T. Y. Doh and M. J. Chung, A Robust Approach to Iterative Learning Control Design for Uncertain System, Automatica, 34, (8), (1992). Norrlöf M. Iterative Learning Control. Analysis, design, and experiments. Ph. D. Thesis, Linköpings Universtet, Sweden (2000). Owens D. H. and Hätönen J., Iterative Learning Control An Optimization Paradigm, Annual Reviews in Control 29, Issue 1, (2005). Scholten P., Iterative Learning Control: Analysis. A design for a linear motor motion system. M. Sc. Thesis University of Twente (2000). Sánchez Peña R. S. and M. Sznaier, Robust System. Theory and application. John Wiley & Sons, Inc. (1998). Tayebi A. and M. B. Zaremba, Robust Iterative Learning Control Design is Straightforward for Uncertain LTI System Satisfying the Robust Performance Condition, IEEE Trans. Automat. Contr., 48, (1), (2003). Uchiyama M. Formulation of high speed motion pattern of a mechanical arm by trail, Trans. SICE, 14, (6), (1978). IN JAPANESE. Zhou K. with J. C. Doyle, Essentials of Robust Control. Prentice Hall, Inc. (1998).