On the link between generalized predictive control and deadbeat control for linear multivariable systems

Systems & Control Letters 16 (1991) 267-272 North-Holland

267

On the link between generalized predictive control and deadbeat control for linear multivariable systems Y o u b l n Peng * and Michel K i n n a e r t

* *

Laboratolre d'Automattque, C P 165, Umverslt~ Ltbre de Bruxelles, 50 Avenue F D Roosevelt, 1050-Brussels, Belgzum Received 2 June 1990 Revised 3 September 1990

Abstract Tlus paper studies a special case of multlvanable generahzed predictive control (GPC) m which a deadbeat controller is obtained It is shown that the multivanble GPC design can be made to be eqmvalent to the deadbeat control design by certain choices of the design parameters. The proposed procedure needs the introduction of different control horizons for different inputs. A tumng strategy ss used to aclueve deadbeat control for reai-ume tmplementauon_ Keywords Long-range predictive control, deadbeat control; multlvanable systems; adaptive control

1. Introduction

Recently, generalized preditive control (GPC) developed by Clark et al [1] was extended by several authors [6,7,8,11] in order to control linear multlvariable systems. In particular, the method proposed by Kinnaert [6,7] is described for a controlled autoregressive integrated moving average ( C A R I M A ) model, and for the equivalent state-space representation in its innovation form The resulting closed-loop system equations can be derived and analyzed by means of the state-space approach. The GPC design offers several possible tuning knobs (the design parameters of the criterion) such as costing horizons, control horizon and weighting factor In a special case where the weighting factor is zero, certain choices of the other design parameters can lead to a deadbeat controller. The conditions for the deadbeat settings of monovanable GPC can be found in [1,2,8,10]. Here, the results are extended to linear multlvarlable systems. It is shown that, to this end, different control horizons for different inputs must generally be introduced, which differs from the monovarlable case. Then, the general conditions for the deadbeat settings of multivariable GPC can be derived by means of the state-space approach. Moreover, a tuning strategy based on C A R I M A model is used to achieve deadbeat control for real-time implementation, the state-space calculation can thus be avoided In addition, the proposed procedure does not require the a priori knowledge of the system interactor matrix. An overview of the classical deadbeat controllers was given in [9]. The designs presented there can be divided into two classes, one is based on system controllability, the other is based on eigenvalue assignment However, the design based on system controllability requires the assumption that the system transition matrix A is nonslngular. This assumption is not necessarily true for some discrete time systems such as large time-delay systems. The design based on elgenvalue assignment does not requires the above assumption, but it requires the computation of a Slimlarity transformation that brings the system into a * On leave from the Department of Automatic Control and Computer Engmeenng, Huazhong Umvers]ty of Science and Technology, Wuhan, Cluna ** Senior Research Assistant of the Belgian National Fund for Scientific Research_ 0167-6911/91/$03_50 © 1991 - Elsevier Science Pubhshers B V (North-Holland)

Y Peng, M Kmnaert / Generahzed predlctwe control and deadbeat control

268

controllable canomcal form. In adaptive control context, such a design does not look simpler than the proposed deadbeat controller via the GPC approach. For the deadbeat controller, the system output has generally a large over-shoot to reference and &sturbance changes. In order to reduce this over-shoot, the desired characteristic polynomial can be assigned independently by filtering the input and output signals and using the filtered signals in the cost function [4]. Combining the proposed deadbeat controller via the GPC approach with the filtering procedure should lead to an usual pole placement control design. This is another motive of the proposed method. This paper is organized as follows. In Section 2 some preliminaries are described. In Section 3 the general conditions for the deadbeat settings of multlvariable GPC are derived, and conclusions are given in Section 4.

2. Preliminaries Consider a multiple-input-multiple-output (MIMO) discrete linear tlme-invariant process described by the following C A R I M A model

A(q-l)A(q-1)y(t)

= B(q-l)A(q

l ) u ( t ) + C(q-')g;(t)

(2.1)

where ( y ( t ) } and (u(t)} are the rn × 1 output and input sequences respectively. (~(t)} denotes the m x 1 renovation sequence of random variables. The sequence (~(t)} is assumed to satisfy E{~(t)/F,_l} = O, E( ~(t)~(t)V/F,_ l } = Q, where F, is the sigma algebra generated by the data up to and including time t, and Q is a poslnve definite matrix In (2 1), A(q-~), B(q -~) and C(q -a) are m × rn polynomial matrices in the unit delay operator q - l , and det C(q -1) has all its roots strictly reside the unit orcle (in q-plane) A(q 1) is the differencing operator l - q 1. The multivariable GPC minimizes a multi-stage cost function of the f o r m

J=E

Ily(t+j)-w(t+j)[12~ J=

+ ~ IlAu(t+j--1)[12/F,

I

(2.2)

J=l

where ]]X[]R=XTRX, I , , i s t h e m X r n i d e n t i t y m a t r l x , h = d i a g ( h , } ( t = l , .,rn) l s a n m × m posmve diagonal matrix, N 1 is the minimum costing horizon, N 2 is the maximum costing horizon, and (w(t)} is the m × 1 sequence of reference signals. To derive the GPC, a control horizon iV, is usually introduced. That is, Au(t +j - 1) is assumed to be zero for j > N,. However, in order to derive the con&tlons for deadbeat settings of the multlvaraable GPC, m different control horizons, N,~ . . . . N,m, should be introduced for m inputs. That is, Au,(t +j - 1) is assumed to be zero for j > N,, (t = 1..... m). Taking this hypothesis into account, the rmnlrmzation of (2.2) with respect to the future incremental control sequence yields

f'(/t))

OT= lab,a, +

(2.3)

where

O=

au,(t

W=[w(t+N,) T y(/t)=[f(t+N,/t)

+ No, -

..

1)

---

w ( t + N , ) T ] T, T

T T

fi(t+Nz/t) ] ,

fi(t+j/t)

(j=N,,..

are predictions of the output assuming there are no future changes in Au, A=blockd,ag{A,}

( t = l . . . . m),

A , = ~ , I N . ,,

,N:),

Y Peng, M Kmnaert / Generahzedpredtctwe control and deadbeat control

and G 1 l s a n ( N 2 - N l + l ) m × ( N . l + [ g N _1(1)

GI

t

"'"

269

.- +N.,~)matrix

gN_N.~(1)

"-"

gN_a(m)

gN,-N~( m ) (2 4)

gN2_l(1)

with g, = [g,(1) . . . .

gN _N~(1)

--

"'

gN _l(rn)

g,(m)], g, (1 = 0, .., N 2 - 1), are the first N 2 elements of the impulse response of

[A(q-a)A(q-1)]-lqB(q-~). Notice that we take g, = 0/,1 if i < 0. This convention is also v a h d wherever a negative index in g, appears. F o r the GPC, the control law (2.3) is implemented in the receding horizon sense. At each sampling time, a new vector U is calculated using (2.3), but only the m elements contained in Au(t) are used. Hence, the G P C is finally

Au(t) = L(W-

Y(/t))

(2.5)

where L is the matrix m a d e of the m appropriate hnes of [G~G1 + AI]-1G T, i.e hnes 1, 1 + N , 1 , . . . , 1 + N,, + "" +Nu~m_,~

3. General conditions for the deadbeat settings In order to get an exphcit form for the charactenstic polynoimal of the closed-loop system, we can use a state-space model in the innovation form Such a model is proved to be equivalent to the C A R I M A model (2.1) under classical detectability and stablhzabihty conditions [3]. Let us n o w consider the state-space model in the innovation form

~(t + l/t) =A~(t/t-

1) + B A u ( t ) + K ~ ( t ) ,

y( t ) = C~( t / t - 1) + K~( t ),

(3.1a) (3.1b)

where the triplet { A, B, C } is a m i m m a l reahzatlon of the transfer function [ A ( q - 1)A ( q - l ) ] - aB ( q - 1). A, B and C are n × n, n x m and m × n matrices. K denotes the K a l m a n filter gain. Hence, J ( t + 1 / t ) ts actually the optimal estimate of x ( t + 1) given data up to time t. Using the same procedure of closed-loop system analysis as that described in [6,7], the characteristics polynomial of the closed-loop system obtained b y G P C is d e t ( q I - A + BLEA)det(qI - A + K C ) with E=[(ANI-1)TcT

'''

T T (A N2-11 C T]

(3.2)

C o m p a n n g (2.1) and (3.1) yields

[A(q-')A(q-')]-1C(q-1)

= I + C(ql- A)-'K

(3.3)

As we assume det C(q -1) has all its roots stnctly inside the u m t circle m q-plane, so does d e t ( q I - A + KC). However, d e t ( q I A + BLEA) is not always stable and more analysis must be undertaken W e note that d e t ( q I A + BLEA) depends on the choice of N1, N 2, Nu, (i = 1 . . . . . m) and h. In fact, it is not a simple task to adjust ~ in such a way that d e t ( q I A + BLEA) is stable. Here, we only discuss a special case where X = 0 I m. In tlus case, as the calculation of L involves the reverse of G~GI, G 1 should be of full rank Let us first show when tbas condition is satisfied. We can assume without loss of generahty that A is of the J o r d a n c a n o m c a l form, since it is always possible to find such a form with sirmlanty transformation. Hence, A can be represented as A = block diag( Ad, Aq }, where A d is the nad × nad nonsingular matrix which includes the J o r d a n blocks corresponding to the nonzero eigenvalues of A, and Aq is the naq X naq singular matrix which includes the J o r d a n -

-

-

270

Y Peng, M Kmnaert / Generahzed predtctwe control and deadbeat control

0]

blocks c o r r e s p o n d i n g to the zero etgenvalues of A If the geometric m u l t i p l i c i t y of the zero eigenvalues is equal to 1, Aq includes one J o r d a n b l o c k only, l e.

Aq =

]

...

(3 4)

.

0

If the geometric m u l t i p l i c i t y of the zero elgenvalues ~s larger than 1, Aq includes several J o r d a n blocks which have m o r e zero entries than (3.4) In b o t h cases, AqO~= OI, oq is verified. W e define also B = [B ff B:] T and C = [Ca Cq], w h e r e Bd, Bq, C d a n d Cq are had × m , naq X m , m × nod and m × naq matrices respectively N o t i c e that, if ( A , B, C } is minimal, so are ( A d, B a, C a ) a n d { Aq, Bq, Cq). Since g, (t = 0 . . . . . N 2 - 1) are the first N 2 e l e m e n t s of the i m p u l s e r e s p o n s e of [A(q-l)A(q-1)] l q B ( q 1), it can also be represented in terms of ( A , B, C } as follows: (3.5)

g, = C A ' B = CdA dB d + C q m t q n q

F o r t > n aq~ we have (3 6)

g, = CdA'dBd.

If we take N 1 > max{ N,, } + n ,q, G1 can be rewritten in terms of ( A a, B a, C a } as follows: Cd A N '

lbdm

N~, b dm ~ d ~dN1-'d

(3 7)

GI= Cd AN2

lbdl

.

"

['~ zIN2--Nul~ "~d"d ~dl

•.

CdA~2 'bdm

CdAU:-U"~ba,,,

where B a = [ba~.. barn], or In a concise form G 1 = GllG12 with

1)Tc:

G,1 : lIAr'

N 2 -- 1

(A d

G12=[bd, --Ab-N.'ba,--.

T

) Ca ]

bd, ' . - .

(3.8a)

, A~d

N'bdm]

(3.88)

T h e following l e m m a gives a sufficient c o n d i t i o n for G 1 to be full rank. L e m m a 1. I f the triplet ( Ad, B a, C a } ts both controllable a n d observable, and ~f N 1 > g + n aq" N2 >~ NI + P - 1 and N~, =/L, (l = 1 . . . . m), where { g , ) are the controllabthty mdtces o f t h e p a t r { A d 1, Bd}, g Is the controllabthty index o f the p a w ( A f t ~, Ba} (t.e. /~ = max{/~, }) and v lS the observabdtty index o f the p a w { Ad, C a }, then G 1 ts o f f u l l rank. Proof. F r o m the d e f i n m o n s of the c o n t r o l l a b i l i t y a n d o b s e r v a b l h t y indices, we have rank[ball r a n k [ C aT

.-Ala-~,bal _.

bdm

(A~d 1 ) T c T ] = n ~ a

"'"

Ala-"~bdm] = n a g ,

(3.9a) (3.9b)

Hence, for the values of N 1, N 2 a n d N,, (l = 1 . . . . m ) m e n t i o n e d in the t h e o r e m s t a t e m e n t , rank(G12) = n~d a n d r a n k ( G l l ) = n , a , i.e. rank(G1) = n~d, which concludes the p r o o f [] R e m a r k 1. If N 1 > max{Nu, } +naq, N2>N 1 + v - 1 a n d Nu, xs free, we have always that rank(G11) = n a a a n d rank(G1) < n a j . Hence, had is the u p p e r b o u n d on N,1 + . . . + N , , , such that G1 is of full rank. A l t h o u g h there are different choices of {N,, } for this u p p e r b o u n d , we p r o p o s e to c h o o s e N~, = #, (t = 1 . . . . . m ) which u n i f o r m l y treats the different system inputs. This result gives a w a y of tuning N~, w i t h o u t violating the full r a n k p r o p e r t y of G~. Now, we show when d e t ( q I - A + B L E A ) has all its roots at the origin

Y Peng, M

Kmnaert

/ Generahzedpredtctwe

271

control a n d d e a d b e a t control

T h e o r e m 1. I f the trtplet{ Ad, nd, C d } lS both controllable and observable, and tf N 1 > It + n aq, N2 ~-~N1 + lP -- 1, N,, = I~, ( t = 1, ., m ) and h =OIm, the closed-loop charactertstm polynomial resulting from the G P C approach reduces to q"det( qI - A + K C ) , t.e det(ql-

A + B L E A ) = q".

(3 10)

!1

Proof. F r o m L e m m a 1, GV~G~ is invertible Hence, we have

LGHG12 =

I! ool ..

o

0

0

...

0

o ..

1

.. (3.11) J

This can be d e c o m p o s e d into the following had column vector equations" LG,lbd~

=

[1

LG,abd, . = [0

O]T,

. .

--

L G ] I . . da-~,~+i-Odl =

..,

11T . . . . .

[0

. .

LGHAd~'-+'bd,,, = [0

-

(3.12)

O]T, O]T.

These lead to ( A a . B a L .G , , A d .) A d ' b d. ,

0] T ,

[0

( A a - B a L G H A a ) A f f ' b a m = [0

---

...,

0] T . . . . .

(Ad

--

BdLGH A d,-~t~ A a ~ " b d , = [ 0

( A a - BdLG11Ad)~"Aff~"bam = [0

0] T'

--

0] T(3 13)

C o m b i n i n g the above n~a equations yields (A d-

BdLGllAd)

" A -dI

Glz

=

0I, o .

(3.14)

As AdlGa2 is of full rank, A d - BdLGlaA a is a nilpotent matrix, i.e. A a - B d L G H A a has n~d zero elgenvalues Finally, we have from (3.2) and (3.8a) that E A = [G1aAd 0] and

A -- B L E A = [

B La.A

- BqLGIlAd

0]

(315)

Aq '

or

d e t ( q I - A + B L E A ) = det(qI.o~ - A d + B d L G a , A a ) det(qI.oq - A q ) which concludes the p r o o f

=q',

(3.16)

[]

If C ( q -1) is taken as identity matnx, which is often the case in adaptive control, the design parameters described in T h e o r e m 1 will lead to a deadbeat control law So T h e o r e m 1 also defines the conditions for the deadbeat settings of GPC. R e m a r k 2. T h e o r e m 1 imphes that it is sufficient to introduce only had degrees of freedom to A u ( t ) in order to set the rlad nonzero elgenvalues of A zero, while keeping the other naq z e r o eigenvalues of A. For real-time implementation, we do not need to calculate the minimal state-space realizatmn and state transformatmn into the J o r d a n canonical form. We k n o w n from R e m a r k 1 and T h e o r e m 1 that, if N 1 and

272

Y Peng, M Kmnaert / Generahzed predwtwe control and deadbeat control

N 2 are selected s u f f i c i e n t l y large, t h e r e exists an u p p e r b o u n d o n Nul q+ N , , , at w l u c h d e a d b e a t c o n t r o l is a c h i e v e d In o r d e r to find such an u p p e r b o u n d , we p r o p o s e the f o l l o w i n g t u n i n g s t r a t e g y 1. C a l c u l a t e G1 b y m e a n s o f the a l g o r i t h m s d e s c r i b e d in [6,7]. 2 Set N ~ > n , N 2 > N l + n - l a n d ) ~ = 0 1 m. 3. F o r t = 1 . . . . m, let N~, = 0_ 4 For t=l, . , m , let N u , = N ~ , + I If for t = j ( l < j < m ) , G1 loses its full r a n k p r o p e r t y , t h e n let #, = N , j - 1. 5. R e p e a t 4 b u t skip for t = j , w h e r e #/ are a l r e a d y f o u n d p r e v i o u s l y . If /~1. . . . bt,, are all f o u n d , terimnate. 6. F o r l = l , . , m set N ~ , = # ,

Remark 3. It is p o s s i b l e to d e r i v e an a l g o r i t h m for i n v e r t i n g GX~Gt r e c u r s t v e l y w~th r e s p e c t to N., (1 = 1 . . . .

rn) w h i c h lS similar to that d e s c r i b e d m [10] for m o n o v a r l a b l e GPC_

4. Conclusions T h e g e n e r a l c o n d i t t o n s for the d e a d b e a t settings o f m u l t l v a r i a b l e G P C are d e r i v e d . It ts s h o w n that if N 1 a n d N 2 are a p p r o p r i a t e l y selected, t h e r e exasts an u p p e r b o u n d o n N,1 + -- + N , , ~ at w h i c h d e a d b e a t c o n t r o l ts a c l u e v e d F o r r e a l - t i m e i m p l e m e n t a t i o n , a t u n i n g s t r a t e g y ts p r o p o s e d for s e a r c h i n g s u c h an u p p e r b o u n d T h e p r o p o s e d p r o c e d u r e d o e s n o t r e q u i r e the a p r i o r i k n o w l e d g e of the s y s t e m m t e r a c t o r matrix I n this p a p e r , we h a v e o n l y d i s c u s s e d the case of the d e a d b e a t c o n t r o l l e r via the G P C a p p r o a c h . H o w e v e r , c o m b ~ m n g the p r o p o s e d m e t h o d w i t h a s u i t a b l e filtering p r o c e d u r e s h o u l d l e a d to a n u s u a l p o l e p l a c e m e n t c o n t r o l design.

References [1] D W Clarke, C Mohtadt and P S Tuffs, generalized predictive control - I The basic algorithm, lI Extensions and mterpretaaons, Automatwa 23 (1987) 137-160 [2] D_W Clarke and C_ Mohtadl, Properties of generahzed predictive control, Automanca 25 (1989) 859-875 [3] G C Goodwln and K S Sin, Adaptwe Faltering Prediction and Control (Prentice-Hall, Englewood Chffs, N J, 1984) [4] E_ Irving, C M Fahnower and C Fonte, Adaptwe generalized predictive control with multiple reference model, 2rid IFAC Workshop on Adaptwe Systems in Control and Signals Processing, Lund, Sweden (1986) [5] T Kadath, Linear Systems (Prentlce-HaU, Englewood Cliffs, NJ, 1980) [6] M_ Kannaert, Adaptwe control of multiple-input/multiple-output linear systems, Ph D Thesis (in French), Universlt6 Llbre de Bruxelles, Brussels. Belgmm (1987) [7] M Kannaert, Adaptxve generahzed predLctive controller for MIMO systems, lnternat J Control 50 (1989) 161-172_ [8] C Mohtadi, Advanced self-tumng algorithms, Ph D. Thesis, Report No OUEL 1689/87, Umversity of Oxford, Oxford, UK (1987) [9] J O'Rellly, The discrete linear time lnvanant time-optimal control problem - an overview, Automattca 17 (1981) 363-370 [10] Y Peng and R Hanus, Pole placement via generalized predlctxve control, Proceedings of the 9th International Conference on Analysis and Opttmtzation of Systems, Lecture Notes in Control and Information Sciences No 144 (Spnnger-Verlag, Berlin-New York, 1990) 664-673 [11] S L Shah, C_ Mohtadi and D_W Clarke, Multivanable adaptive control without a pnon knowledge of the delay mamx, Systems Control Lett 9 (1987) 295-306_