이차성능지수에 근거한 반복학습제어기법의 계산량 감소를 위한 Blocking 기법

진인식, 이광순

서강대학교 화학공학과

Blocking Technique for Quadratic criterion-based Iterative Learning Control for Computation Reduction

In Shik Chin, Kwang Soon Lee

Dept. of Chemical Engineering, Sogang University

Introduction

The batch process is very often characterized by its repeated operation over a finite time interval. During an operation, the process usually pursues a time-varying reference trajectory and swings over a wide operating regime. This again causes the process to expose its nonlinearity significantly.  Because the above mentioned characteristics, the control techniques developed for continuous processes  are not successful in many batch processes.

As a way to overcome the difficulties in batch process control, Lee  et.al.(1997) utilized the concept of iterative learning, which refines the input signal by feeding back the operation result in the previous batch run, and proposed various versions of the Q-ILC (quadratic criterion-based iterative learning control). Q-ILC has many advantages over the other existing iterative learning algorithms in the aspects of noise sensitivity, zero tracking offset, better robustness, easy incorporation of constraints, etc. However, it shares the disadvantage common to the existing iterative leaning algorithms in that it suffers heavy computational burden. The longer the batch horizon is, the heavier the computation load becomes.  Moreover, when the constraints are involved, a QP(quadratic programming) of a very large dimensionality should be solved after each batch run.

In this paper, we have studied the so-called blocking technique for Q-ILC to diminish the required computations. The blocking is a sort of approximation technique to reduce the dimension of a signal using a small number of appropriate basis vectors. To do this, we first investigate  how blocking modifies the original Q-ILC algorithm. Through this investigation, we first propose how the basis vectors should be chosen for optimal blocking. As a suboptimal but practical method, we next propose to use the KLT(Karhunen-Loeve transformation) or equivalently, the principal component analysis for blocking. The proposed blocking technique was evaluated through series of numerical examples. Computation reduction as well as performance degradation of blocked Q-ILC were compared with those of original Q-ILC. We also studied the blocking using DCT(discrete cosine  transformation) and compared its performance with others.

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 Delta{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} + Delta {w}_{k+1}# rm {e}_k &=& {bar{e}}_k + {v}_k]

(4)

In order to calculate sequence of manipulate variables,[eqn: rm Delta `u_k+1], we formulate objective function similarly in MPC.[2]

[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

Blocked Q-ILC using blocking technique

In the eq(4), we approximate the manipulate variables, [eqn: rm Delta `u_{k+1}] with [eqn: rm Delta`u_k+1 ^*].[4]

[eqn: rmDelta {u}_{k+1} &approx & B^T Delta `u_{k+1}^* # rmDelta `u_k+1^* &=& B Delta {u}_{k+1}]

(6)

where matrix [eqn: rm B_{(n times N)}] consists of nth basis vectors of transform matrix.

By blocking through [eqn: rm B_{(n times N)}], the size of[eqn: rm Delta`u_k+1 ^*] is decreased to [eqn: rm n times 1] from the size of[eqn: rm Delta `u_{k+1}], [eqn: rm N times 1]. It is most important to design the matrix B when blocking technique is applied to Q-ILC.

Given the state-space model in (5) can be rewritten approximately by blocking as follows:

[eqn: rm {{overlinee}^*}_{k+1} &=& {{overlinee}^*}_{k} - G_k ``B^T Delta `u_{k+1}^* + Delta{w}_{k+1} # rm e_{k}^* &=& {{overlinee}^*}_{k}+{v}_k]

(7)

The objective function for the blocked constrained Q-ILC is as follows:

[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 ]]

(8)

subject to constraints

Numerical Illustration

Example 1. Multi-Input Multi-Output(MIMO) process

It is assumed that [eqn: rm{G}^true] for the true process and nominal model [eqn: rm G^nom] are as follows:

[eqn: rm{G}^true (s) &=& BMATRIX { { { 2.5} over {300s^2 +35s +1 } }& { { 1.7} over {110s^2 +21s +1 } }# {{ 1.7} over {225s^2 +30s +1 } }& {{ 2.8} over {117s^2 +22s +1 } } } ,## rm{G}^nom (s) &=& BMATRIX { { { 1.9} over {240s^2 +31s +1 } }& { { 1.4} over {130s^2 +23s +1 } }# {{ 1.2} over {180s^2 +28s +1 } }& {{ 2.3} over {96s^2 +20s +1 } } } #]

(9)

The batch horizon was chosen [0,200] and sampling interval is 1.

The transform matrix is obtained by KLT. We choose 40 basis vectors, which means the size of QP problem to solve is decreased from [eqn: 400 times 400] to [eqn: 40 times 40].The reason of choosing 40 basis vectors is that the original signal can be reconstructed 90 percents.

In Figs. 1, 2 and 3, we show the original Q-ILC and the results of the Blocked Q-ILC using KLT and DCT with input constraints, respectively. We impose the constraints on the input magnitude and input changes. The tuning parameters are chosen as [eqn: rmQ_1 =I,~Q_2 =I] and [eqn: rmR_1 =0.1timesI,~R_2 =0.1timesI]

We measured the CPU-running time in each case and the result is as follows:

1. Original Q-ILC : 14229 seconds for 11 batches (in Pemtium-150)

2.. Blocked Q-ILC(KLT) : 447.42 seconds for 11 batches (in Pentium-150)

3.. Blocked Q-ILC(DCT) : 457.17 seconds for 11 batches (in Pentium-150)

We can see that there are no differences between the Blocked Q-ILC and the original Q-ILC in performances in this case. But, the CPU-running time is decreased to below the 5 percents.

Figure 1. Original Q-ILC             Figure 2. Blocked Q-ILC(KLT)

Conclusion

In this paper, we applied the blocking technique to quadratic criterion-based iterative learning control(named as Q-ILC) by KLT and DCT, and we also compare the case of the blocking technique applied and the case of original formulation. We can relieve efficiently the computational load in QP problem without sacrifice of any tracking performance using blocking technique. Moreover, we show that KLT is more efficient than DCT in the aspect of input regulation.

Figure 3. Blocked Q-ILC(DCT)

Reference

1. Lee, K. S.Jay H. Lee, ``Model-base refinement of input trajectories for batch and other transient processes'', Proc. of 1997 AIChE,paper 149i, 1997,

2. Zafiriou, E and Chiou, H. W.,``Output constraint softening for siso model

predictive control'', Proc. of 1993 ACC,372-276, 1993,

3. Amann, N., Owens, D. H. and Rogers, E..``Iterative learning control for

discrete-time system with exponential rate of convergence'',  Proc. of 1996 IEE, 217-224, 1996.

4. Ali N. Akansu and Reichard A. Haddad "Multiresolution signal Decomposition", pp 9-60, Academic Press, 1992.