모델 예측 반복 학습 제어

진인식, 이광순, 최진훈

서강대학교 화학공학과

Model Predictive Iterative Learning Control

In Shik Chin, Kwang Soon Lee, Jinhoon Choi

Dept. of Chemical Engineering, Sogang University

Introduction

Since the Arimoto's pioneering work[1] iterative learning control(ILC) has drawn continuous attention as a novel (open-loop) control technique for processes with repetitive operation[4]. ILC can be considered as a technique to improve the input trajectory for the upcoming run utilizing the measurements in the previous run. This concept makes it possible for a

repetitive process to overcome various uncertainties such as model error and

repetitive disturbances and to attain perfect tracking as the run number increases[1,5,8]. Despite the attractive features mentioned above, however, ILC has not been widely practiced owing to the several decisive drawbacks such as extreme high frequency sensitivity, excessive input movement, and relatively restrictive allowable model error bound for convergence, etc. Recently, as a way to cope with these drawbacks, Lee et. al.[5] and  Amann et. al.[1] proposed independently novel ILC algorithms based on a quadratic criterion which includes an input change (run-to-run) penalty term. For the purpose of application to nonlinear repetitive processes, Lee et. al.[6] developed several versions of algorithms for time-varying linear repetitive processes under the collective name, Q-ILC(quadratic criterion-based iterative learning control). They showed that the high frequency sensitivity and the input movement can be moderated in Q-ILC without sacrificing the ultimate tracking performance.

The real-time control counterpart of Q-ILC is the one-step ahead model predictive control(MPC). Driven by the fact that MPC with a longer prediction horizon exhibits better performance, Amann et. al.[2] proposed model predictive iterative learning control(MPILC) and established that a longer prediction horizon in MPILC enhances the performance. It is also known in MPC[7] that  longer prediction horizon also leads to better robustness. However, in this paper, we show that MPILC with a longer prediction horizon does not always exhibit better robustness. These results are illucidated by an example.

Belief Review of Q-ILC

Now, because the control strategy of Q-ILC is model based control, we should get the model of given batch process. In general, the relationship between the manipulated variables and process variables can be written as following:

[eqn: rm y_k =bar{y}_k + w_k = NU ``(u_k , d_k , it x_{0,k} rm ) +v_k]

()

where[eqn: rm y^T &=&[it y^T (1), y^T (2), cdots ``y^T (N`) ]# rm u^T &=&[it u^T (1), u^T (2), cdots ``u^T (N`) ]]

[eqn: rm {bar{y}}^T , d^T , w^T] are defined similarly.

We assume the nonlinear function [eqn: NU] is analytic and by linear approximation,

[eqn: rm {bary}_{k+1} = rm {bary}_{k} +G_k DELTA`u_k+1 +D_k DELTA `d_{k+1} +rm higher ~order ~term]

(2)

where[eqn: rm Delta {u}_{k+1} &=& {u}_{k+1}-{u}_k # rm Delta {d}_{k+1} &=& {d}_{k+1}-{d}_k]

If we let[eqn: rm {y}_d] denote the desired output sequence,

[eqn: rm{bar{e}}_k &=& {y}_d - {bar{y}}_k # rm {e}_k &=& {y}_d -{y}_k # rm {w}_{k+1} &=& {D}_k Delta {d}_{k+1} + {higher~ order~ terms}]

(3)

We can obtain the state-space model of errors which consist in those of successive batches.

[eqn: rm {bar{e}}_{k+1} &=& {bar{e}}_k - {G}_k Delta {u}_{k+1} + {w}_{k}# rm {e}_k &=& {bar{e}}_k + {v}_k]

(4)

In order to calculate sequence of manipulate variables,[eqn: rm Delta `u_k+1], we solve the least square problem

[eqn: rm {min} atop {Delta {u}_k+1 } LEFT [ J = {1} over {2} \{`` || {e}_{k+1} ||_{Q} ^2 + `||Delta{u}_{k+1} ||_{R} ^2 ``\} ` RIGHT ]]

(5)

subject to constraints.

Model Predictive Iterative Learning Control(MPILC)

After the operation of [eqn: k_th] batch, the predictor that predicts future batch's error sequence is formulated by

[eqn: rm e_{k+i|k}=e_{k|k}-SUM_{j=1}^{i}`G ` Delta `u_{k+i}.]

(6)

Employing the quadratic criterion, the control input can be calculated according to

[eqn: {min}atop{rm Delta` u_k+i} LEFT[ rm J=SUM_{i=1}^{m} rm e_{k+i|k}^T Q `e_{k+i|k} + SUM_{i=1}^{m} rm Delta u_{k+i}^T R `Delta u_{k+i|k} RIGHT ].]

For the unconstrained case, the solution to the above minimization is easily obtained and only [eqn: rm Delta` u_k+1] from [eqn: rmLEFT [ { Delta ` u_k+1^T}& { cdots}& {Delta ` u_k+p^T} RIGHT ]^T] is applied to the [eqn: k+1_th] batch of process.

[eqn: rm Delta ` u_k+1 = LEFT ( G^T {bar{Q}}_m G +R RIGHT ) ^-1 ``G^T {bar{Q}}_m]

(7)

From Riccati equation, recursion relationship of[eqn: rm {bar{Q}}_m] according to prediction horizon [eqn: m] can be obtained.

[eqn: rm {bar{Q}}_m = {bar{Q}}_m-1 -{bar{Q}}_m-1 G LEFT ( G^T {bar{Q}}_m-1 G +R RIGHT ) G^T {bar{Q}}_m-1 + Q]

(8)

Properties of MPILC and Real-time MPILC

Assumptions;

A.1 :[eqn: rm G] has a full row rank.

A.2 : Input constraints and [eqn: rm u_0] are such that the zeroing of the error is possible with an input in the feasible set.

A.3 : Output constraints are satisfied when [eqn: rm e=0].

A.4 :[eqn: rm R] is block-diagonal such that

[eqn: rm Delta`u_k^T (t)R Delta` u_k (t) = SUM from { {i}=t} to N-1 Delta itu_{rmk}^{rmT}(i)itR Delta itu_{rmk}(i).]

Theorem 1 :[eqn: rm {bar{Q}}_m] is positive definite and nondecreasing as the prediction horizon, [eqn: m], is increased.

Theorem 2 : Under the assumption A.1 and [eqn: rm w_k =v_k =0], system (4) converges to the origin with MPILC, i.e.[eqn: rm e_k ``->`` 0] and [eqn: rm Delta `u_k ``-> ``0] as [eqn: k `` -> `` inf].

Theorem 3 : Let [eqn: rm J_k+1] be the cost function with the input, [eqn: rm LEFT [ Delta ` u_k+2^* ~cdots ~Delta ` u_k+m^* ~ 0RIGHT]] such that

[eqn: rm J_k+1 =J_k^* - { 1} over {2 } e_{k|k}^*T Q e_{k|k}^*- { 1} over {2 } Delta ` u_{k+1}^*T R Delta `u_{k+1}^* + { 1} over {2 } e_{k+m|k}^*T Q e_{k+m|k}^* .]

(9)

The learning system is robustly convergent if

[eqn: rm { 1} over {2 } e_{k|k}^*T Q e_{k|k}^* - { 1} over {2 } e_{k+m|k}^*T Q e_{k+m|k}^* &>& alpha over 2 Delta `u_k+1^*T Delta_a^T LEFT[ R-G^T Q G RIGHT] Delta_a Delta `u_k+1^* #rm &+& { 1} over {2 } Delta `u_k+1^*T Delta_a^T R Delta_a Delta `u_k+1^* + e_{k+m|k}^*T Q(-G) Delta_a `Delta`u_{k+1}^*]

(10)

where [eqn: alpha `>=` 0] is a real number such that

[eqn: rm G^T { PARTIAL ^2 J_k+1 } over { PARTIAL e_{k+1|k}^*2} <=` alpha LEFT[ R-G^T QG RIGHT].]

Theorem 4 : Under the assumptions A.1-A.3 and [eqn: rm w_k =v_k =0], system (4) converges to the origin uder the constraed MPILC, i.e., [eqn: rm e_k ``->`` 0] and [eqn: rm Delta `u_k ``-> ``0] as [eqn: k `` -> `` inf].

Theorem 5 : Under the assumption A.1-A.4 and[eqn: rm w_k =v_k =0], system (4) converges to the origin with Real-time MPILC as [eqn: k `` -> `` inf].

Theorem 6 : The learning system is robustly convergent if

[eqn: rm 1 over 2 Delta itu_{rmk+1}^{*rmT} (rmt) rmalpha itu_{rmk+1}^{*} (rmt) - rm-e_k+1^*T (N|t)QG delta_a (t) Delta itu_{rmk+1}^* rm(t) ` <`0]

(11)

where [eqn: rm alpha ={itR}-delta_a^{T}(t)G^T QG delta_a (t).]

Theorem 7 : Under the assumption A.1-A.3 and [eqn: rm w_k =v_k =0], system (4) converges to the origin with Real-time MPILC, i.e., [eqn: rm e_k ``->`` 0] and [eqn: rm Delta `u_k ``-> ``0] as [eqn: k `` -> `` inf].

Acknowledgement

This work was supported by the Korea Science and Engineering Foundation(KOSEF) through the Automation Research Center at POSTECH.

Reference

1. N. Amann, D. H. Owens, and E. Rogers, ``Iterative Learning Control for Discrete-time System with Exponential Rate of Convergence,'' Proc. IEE Contr. Theory and Appl., Pt. D, 143, pp. 217-224, 1996.

2. N. Amann, D. H. Owens, and E. Rogers, ``Predictive Optimal Iterative Learning Control'' Int. J. Contr., 69, 2, 1998.

3. S. Arimoto, S. Kawamura, and F. Miyazaki, ``Bettering Operation of Robots by Learning,'' J. Robotic Syst., 1, pp.123-140, 1984.

4. Y. Chen, ``ILC bibliography List,'' http://shuya.ml.org:888/~yqchen/ILC/

ilcref.html, 1998.

5. K. S. Lee, W. C. Kim, and J. H. Lee, ``Model-based Iterative Learning Control with Quadratic Criterion for Linear Batch Processes,'' J. Contr. Autom. and Sys. Engng, 2, pp. 148-157, 1996.

6. K. S. Lee, J. H. Lee, and W. C. Kim, ``Model-based Iterative Learning Control with a Quadratic Criterion for Time-varying Linear Systems,'' submitted to Automatica, 1997. Manuscript is available at  http://procd.sogang.ac.kr.

7. G. D. Nicolao, L. Magni, and R. Scattolini, ``On the Robustness of Receding-Horizon Control with Terminal Constraints,'' IEEE Transactions on Automatic Control, 41, pp. 451-453, 1996.

8. E. Zafiriou, H. W. Chiou, and R. A. Adomaitis ``Nonlinear Model Based Run-to-run Control for Rapid Thermal Processing with Unmeasured Variable Estimation'', Proc. Symp. on Process Contr., Diagnostcs, and Modeling of Semiconductor Manufacturing, San Diego, California, pp. 18-31, 1995.