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1 Multivariable Receding-Horizon Predictive Control for Adaptive Applications Tae-Woong Yoon and C M Chow y Department of Electrical Engineering, Korea University 1, -a, Anam-dong, Sungbu-u, Seoul 1-1, Korea twy@uccnxoreaacr y Department of Engineering Science, Oxford University Pars Road, Oxford OX1 PJ, UK billychow@engoxacu Abstract This paper presents a multivariable receding-horizon predictive control strategy It does not require nowledge of the interactor matrix of a process and therefore is suited to adaptive applications The proposed scheme also allows equality constraints to be imposed on each channel, thereby guaranteeing asymptotic stability 1 Introduction Industrial processes often consist of many interacting variables and are subject to changes in dynamics Thus designs using SISO (single-input single-output) xed controllers inevitably lead to limited performance, suggesting the necessity of MIMO (multi-input multioutput) adaptive control strategies As is usually the case, adaptive or self-tuning methods were developed for SISO plants, and then their MIMO versions were later sought This type of extension, however, is not so straightforward as one might expect When extending the celebrated minimum variance self-tuner of Astrom and Wittenmar (19), the delay matrix of a MIMO process needed to be considered as a generalization of the time delay (dead time) of a SISO system Since the structure of a delay matrix is not unique, MIMO minimum variance self-tuners were developed for a particular lower triangular structure (Borisson, 199; Goodwin et al,

2 1 Introduction 198; Dugard et al, 198; Dugard et al, 198) A delay matrix in such a lower triangular form is called an interactor matrix, and almost complete nowledge of the interactor is required in those self-tuners based on Astrom and Wittenmar's wor This is the main obstacle to the successful application of MIMO minimum variance adaptive controllers Another problem with minimum variance control, which is actually more fundamental, is the fact it cancels process zeros as in MRAC (model reference adaptive control) and therefore is not appropriate for non-minimum phase systems It is also nown to lac robustness, especially to process dead time As solutions to this problem, pole placement design and receding-horizon predictive control were suggested for adaptive applications The former prescribes the closed-loop poles, but computation of the left to right fraction description of the plant is required in addition to resolution of multivariable Diophantine identities More importantly, although arbitrary pole assignment yields stable control, it maes no attempt to reduce interactions; the appropriate choice for the closed-loop poles to give a reasonable amount of decoupling is in general nontrivial (Mohtadi, 198) On the other hand, receding-horizon predictive control, represented by GPC (generalized predictive control) (Clare et al, 198), minimizes a quadratic cost involving a set of predictions and their desired values The resulting control law is simple to realize, and is nown to be robust to process dead time as it is not based only on a single prediction Successful applications have also been reported in Clare (1988); see the references therein Despite these popular aspects of GPC, there is no clear theory guaranteeing closed-loop stability in terms of the GPC tuning nobs For this problem, solutions were later proposed independently by Clare and Scattolini (1991) and Mosca and Zhang (199) These methods, which are actually identical, add terminal constraints to the usual GPC cost, and are thus referred to here as CRHPC (constrained receding-horizon predictive control) The stability of CRHPC was proved on the basis of the pioneering wor on state-space receding-horizon control by Kwon and Pearson (19; 198) The eectiveness of adaptive CRHPC was demonstrated through a recent benchmar tas (Yoon and Clare, 199) Further renements of stability conditions for CRHPC were also obtained by Yoon and Clare (199a) together with a more general framewor For the robustness properties of receding-horizon predictive control, see Yoon and Clare (199b) The applicability of adaptive CRHPC seems promising However its MIMO extension is not as straightforward as that of GPC, which is due to the fact that it is not clear how to impose equality constraints on each channel so as to guarantee stability The aim of this

3 Plant description and prediction paper is thus to derive a general MIMO receding-horizon predictive control scheme which can lead to both GPC and CRHPC; note that such a formulation of predictive control has recently been given by Yoon and Clare (199a) for SISO processes Dierent sets of equality constraints are to be provided for dierent outputs so that the receding-horizon cost is monotonically non-increasing, thereby ensuring asymptotic stability Plant description and prediction Consider a MIMO discrete-time process described by: y(t) = M(q?1 )u(t? 1) + N(q?1 ) (t) ; (1) where u(t) and y(t) are the input and output vectors, (t) is a vector of uncorrelated random noise sequences, M(q?1 ) and N(q?1 ) are transfer function matrices, and is the dierence operator (ie 1? q?1 ) For simplicity, it is supposed here that the vectors y, u and have the same dimension n (ie y; u; R n ) The i; j-th element of M, which is the transfer function between the i-th output y i and the j-th input u j, is written in the form: M ij (q?1 ) = Bij (q?1 ) A ij (q?1 ) = B ij (q?1 ) A i (q?1 ) ; () where the polynomial A i is the least common multiple of A i1 ; A i ; ; A in It is also assumed that the noise model N is diagonal and its i; i-th element is given by: N ii (q?1 ) = T i (q?1 ) A i (q?1 ) : () From equations () and (), we now have the following CARIMA model for the i-th output: A i (q?1 )y i (t) = j=1 B ij (q?1 )u j (t? 1) + T i (q?1 ) i (t)=; () for i [1; n] Note that superscript i denotes the i-th element of a vector The optimal prediction ^y i (t + ) for y i (t + ) is then obtained by: ^y i (t + ) = j=1 G ij u j (t +? 1) + f i (t + );

4 Predictive control f i (t + ) = F i T i yi (t) + j=1 H ij T i uj (t? 1); () where the polynomials F i, G ij and H ij satisfy the Diophantine identities: Note that G ij T i = A i E i + q? F i ; E i Bij = G ij T i + q? H ij : () is a polynomial of order? 1 whose coecients are equivalent to the rst step responses of the system B ij =A i = B ij =A ij, ie: B ij 1X A i = l=1 g ij l q?l+1 ; G ij = X l=1 g ij l q?l+1 : On the basis of the predictions given in eqn(), the MIMO receding-horizon control law is derived below Predictive control As is standard in predictive control, we tae account of a quadratic cost function of the form: J = + X i=1 Ny?1 =N 1 i () [w i (t + )? ^y i (t + )] + X N u?1 = N y+mi?1 X i (N y ) =N y [w i (t + N y )? ^y i (t + )] 1 A j ()u j (t + ) : () This is a MIMO extension of the performance index considered in Yoon and Clare (199a): i () and j () are positive weighting sequences for the i-th tracing errors and j-th control increments, N 1 and N y are the lower and upper prediction horizons, N u is the control horizon, and is a non-negative number ( 1) introduced to place heavier weighting on the errors further ahead than N y Note that if =, then we have the following equality constraints: w i (t + N y ) = ^y i (t + ) for [N y ; N y +m i?1]; (8) and m i is the number of constraints for the output y i It is also assumed that the control u(t) does not move after the interval N u (ie u(t + ) = if N u ), and that the rst

5 Predictive control term of the cost () is ignored when N 1 equals N y The weighting sequences i () and j () are normally set to be constant; however, time-varying weighting can be specied to enhance performance: as in Yoon and Clare (199), it is suggested that: i () =? ; j () =? j : (9) This exponential weighting places the closed-loop poles within a circle of radius (if the control law is stabilizing) The cost function () is general in that it can lead to a wide range of predictive methods including GPC ( = 1) and CRHPC ( = ) The use of a nonzero number ( < 1) for is expected to have eects similar to those observed in receding-horizon control with nite end-point weighting as discussed in Kwon and Byun (1989) and Demircioglu and Clare (199) In order to rewrite the cost () in a simple vector form, we dene: U = [u(t) T u(t + 1) T u(t + N u? 1) T ] T ; w i 1 = [w i (t + N 1 ) w i (t + N 1 + 1) w i (t + N y? 1)] T ; ^y i 1 = [ ^y i (t + N 1 ) ^y i (t + N 1 + 1) ^y i (t + N y? 1)] T ; f i 1 = [ f i (t + N 1 ) f i (t + N 1 + 1) f i (t + N y? 1)] T ; w i = [w i (t + N y ) w i (t + N y ) w i (t + N y )] T ; ^y i = [ ^y i (t + N y ) ^y i (t + N y + 1) ^y i (t + N y + m i? 1)] T ; f i = [ f i (t + N y ) f i (t + N y + 1) f i (t + N y + m i? 1)] T : (1) Using these vectors, we re-express eqn() as: J = i=1 [^y i 1? w i 1] T Q [^y i 1? w i 1] +?Ny [^y i? w i ] T [^y i? w i ] + U T U ; (11) with the weighting matrices Q and being:! Q = diag [?N 1 ;?(N1+1) ; ;?(Ny?1) ]; = diag [ o ;? o ; ;?(Nu?1) o ]; (1) o = diag [ 1 ; ; ; n ]: This cost function is subject to the predictions: ^y i 1 = G i 1U + f i 1 ;

6 Predictive control ^y i = G i U + f i ; (1) where the matrices G i 1 and G i are given by: G i 1 = g i g i N 1 N 1?1 g i g i N 1 +1 N 1 g i g i gi Ny?1 Ny? Ny?Nu and g i is a row vector obtained as: ; G i = g i Ny g i Ny +1 g i Ny +m i?1 g i Ny?1 g i Ny?Nu+1 g i Ny g i Ny?Nu+ g i Ny +m i? gi Ny?Nu+m i ; (1) g i = [g i1 g i g in ]: (1) Minimizing the cost (11) results in the optimal U: U = + i=1 i=1 [G it [G it 1 Q Gi 1 +?N y 1 Q (wi 1? f i 1) +?N y G it Gi ]!?1 G it (w i? f i )]! : (1) The control law (1) may, however, cause numerical trouble as approaches zero, and it is obvious that cannot be made zero To deal with such a situation, introduce p i : p i =?Ny [G i U? (w i? f i )]: (1) From equations (1) and (1), it follows that U can also be found by forming the augmented linear equation: + i=1 G it Q 1 Gi 1 G 1T G nt G 1? Ny I m1 G n? Ny I mn U p 1 p n = i=1 G it 1 Q (wi 1? f i 1) w 1? f 1 w n? f n ; (18) where I mi is the identity matrix of dimension m i This description of the control law now allows to be zero, and thus is numerically superior to eqn(1) for 1 In order

7 Stability guarantees further to simplify eqn(18), we dene: G = G 1 G G n ; p = p 1 p p n ; w = w 1 w w n ; f = f 1 f f n : (19) Then the control law is rewritten as: + P n i=1 GiT 1 Gi 1 G T G? Ny I U p = P n i=1 GiT 1 Q (wi 1? f i 1) w? f ; () which is in exactly the same form as in Yoon and Clare (199a) When =, the vector p (p i ) turns out to be the Lagrange multiplier for the optimization problem subject to G U = w? f (G i U = w i? f i ) Having computed U using eqn(18) or (), only u(t) (the rst n elements of U) is actually applied at time t, and the whole procedure is repeated at the next sample instant: hence the name `receding-horizon' The resulting control law is simple to implement, and is independent of the input-output ordering in the system description as in MIMO GPC Stability guarantees As discussed in the introduction, imposing terminal constraints plays a ey role in guaranteeing stability of SISO predictive control systems The following is a MIMO extension of the stability theorem given in Yoon and Clare (199a) Theorem 1 For the MIMO process described by (1)-() and the receding-horizon predictive control law (18) or (), the closed-loop is guaranteed to be stable if: = ; m i = deg(a i ) + 1; N u max 1in m i; (1) N y = N u + max 1i;jn hdeg(bij )? deg(a i )i; where hi equals what is inside if that is positive and is zero otherwise hdeg(b ij )? deg(a i )i is the number of poles of B ij =A i at q =

8 Illustrative examples 8 Proof: This theorem can be proved by showing the monotonicity of the cost as in the proof of theorem of Yoon and Clare (199a), so is omitted here for lac of space A detailed proof is to be given in a later version rrr Remar 1 As = in theorem 1, it is not clear if these stability requirements lead to the nonsingularity of the matrix in eqn(18) or (), which is equivalent to G in () having full row ran However, it can be shown that the matrix equation in question actually has a solution under the conditions (1) Hence, we can always nd U satisfying (18) or () by using the pseudo inverse of the matrix Remar The overall MIMO process is regarded in this paper as a set of multi-input single output systems As a consequence, it has become straightforward to nd the number of constraints for each channel so as to result in a monotonic cost Flexible design is also possible: for instance, one may place constraints only on problematic outputs Remar As is well-nown, the observer polynomial T i (q?1 ) plays a crucial role in robust design We believe that its selection guideline for SISO predictive control given in Yoon and Clare (199b) still applies here; it is thus suggested that T i include A i assuming its stability Illustrative examples Although the proposed algorithm is developed for adaptive applications, simulations in this section highlight only the stabilizing eect of We thus assume no modelling error and set all the observer polynomials to 1 Suppose that the plant has unstable poles and zeros close to each other: A 11 (q?1 ) = (1? :8q?1 )(1? 1:q?1 ); A (q?1 ) = (1? :8q?1 )(1? 1:q?1 ); B 11 (q?1 ) = :q?1 (1? 1:q?1 ); B 1 (q?1 ) = :q?1 (1? 1:19q?1 ); B 1 (q?1 ) = :q?1 (1? 1:q?1 ); B (q?1 ) = :q?1 (1? 1:19q?1 ): () This creates diculties for GPC or equivalently the control law with = 1, which fails to stabilize the closed-loop as presented in Fig1 Stability is regained when = 1? : see Fig Finally the system response is much improved if is reduced to zero, as shown

9 Conclusions 9 in Fig The controller parameters (N 1, N y ( = N u ), 1,,, m 1, m ) are assumed to be (1, 1, 1, 1, 1,, ) Note that m 1 and m are both set to following theorem 1 and that exponential weighting is not employed here Output 1 (y1) Samples Output (y) Samples Figure 1: Simulation results when = 1 Conclusions This paper has presented a general framewor of MIMO predictive control within which a MIMO extension of CRHPC as well as MIMO GPC is obtained The proposed scheme allows equality constraints to be imposed on each channel, and stability conditions have been derived in terms of these constraints The proposed predictive strategy is particularly suited to adaptive applications as it requires no nowledge of the Delay matrix It does not only possess the positive features of MIMO GPC but also is stabilizing

10 Conclusions 1 1 Output 1 (y1) Samples Output (y) Samples Figure : Simulation results when = 1? 1 Output 1 (y1) Samples 1 Output (y) Samples Figure : Simulation results when =

11 References 11 Acnowledgment The rst author is grateful to nancial support by ACRC (Automatic Control Research Center), Seoul National University References Astrom, K J and Wittenmar, B (19) On self-tuning regulators Automatica, 9(), 18{199 Borisson, U (199) Self-tuning regulators for a class of multivariable systems Automatica, 1, 9{1 Clare, D W and Scattolini, R (1991) Constrained receding-horizon predictive control IEE Proc Part D, 18(), { Clare, D W, Mohtadi, C, and Tus, P S (198) Generalised predictive control part I: the basic algorithm and part II: extensions and interpretations Automatica, (), 1{1 Clare, D W (1988) Application of generalised predictive control to industrial processes IEEE Control Systems Magazine, 8(), 9{ Demircioglu, H and Clare, D W (199) Generalised predictive control with end-poing weighting IEE Proc Part D, 1(), {8 Dugard, L, Goodwin, G C, and De Souza, C E (198) Prior nowledge in model reference adaptive control of multi-input multi-output systems In Proceedings of nd IEEE CDC, San Antonio, USA Dugard, L, Goodwin, G C, and Xianya, X (198) The role of the interactor matrix in multivariable stochastic adaptive control Automatica, (), 1{9 Goodwin, G C, Ramadge, P J, and Caines, P E (198) Discrete-time multivariable adaptive control IEEE Trans on Auto Control, AC-(), 9{ Kwon, W H and Byun, D G (1989) Receding horizon tracing control as a predictive control and its stability properties Int J Control, (), 18{18

12 References 1 Kwon, W H and Pearson, A E (19) A modied quatratic cost problem and feedbac stabilization of a linear system IEEE Trans on Auto Control, AC-(), 88{8 Kwon, W H and Pearson, A E (198) On feedbac stabilization of time-varying discrete linear systems IEEE Trans on Auto Control, AC-(), 9{81 Mohtadi, C (198) Advanced self-tuning algorithms DPhil thesis, Department of Engineering Science, Oxford University Mosca, E and Zhang, J (199) Stable redesign of predictive control Automatica, 8(), 19{1 Yoon, T-W and Clare, D W (199) Receding-horizon predictive control with exponential weighting Int J Syst Sci, (9), 1{1 Yoon, T-W and Clare, D W (199) Adaptive predictive control of the benchmar plant Automatica, (), 1{8 Yoon, T-W and Clare, D W (199a) A reformulation of receding-horizon predictive control Int J Syst Sci, (), 18{1 Yoon, T-W and Clare, D W (199b) Observer design in receding-horizon predictive control Int J Control, 1(1), 11{191

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