automatica
automatica PERGAMON
Autornatica 37 (2001)297-302
Technical Communique
Fixed poles of simultaneous disturbance rejection and decoupling: a geometric approach* U~LLI&\
J.-F. Camarta, M. Malabre"?", J.-C. Martinez-Garciab?' " l ~ r . ~ r i t de ~ r / Rechcrchc en Corr1rt1~17icati,1n.c. e/ Cvhernitique de N ~ n t e s CNRS , UMR 6597, 1, r u e de lo ,Voi. B.P. 92/01, F44321 hluntes C ' n l u 0.3, I , r r ~ ~ i c r bUel~ur/omcwtode Control Autonl6rico, CINVESTAV-IPN,Au. ln.ctilt~loPoliticnico Nacional No. 2508, 07300 M h i c o U . K , M r x i t r ,
Kccc~vcd4
January 2000; rcvrscd 6 Junc 2000; rccc~vcdIn final form 30 Junc 2000
Abstract
When the simultaneous disturbance rejection and regular block decoupling problem by static state feedback is solv:~ble,and ~ ~ n d c r some unrestrictive minimality assumption. all the poles of the closed-loop system can be placed, except the so-called fiscd polcs (which are present for any solution). We present here a geometric characterization for this set of fixed poles a s well as ;i constructive proccrlurc (based on geometric tools) for designing a feedback solution of the problem which assigns the remaining polcs to a~-bi(rary prcspccificd locations. (G 2000 Elsevier Science Ltd. All rights reserved. h'~:a.~~~oril.s: Linear multivariable syslems; Disturbance rejeclion; Decoupling; Fixed
poles; S~ability;Pole assign men^; Ckoinelric approach: SI I - L I C ~ I I ~ : I ~
approach
T h e disturbancc rejection problem a n d the decoupling problem have both been of great interest in the control thcory literature a n d still remain challenging. F o r each >cp,irate problem, it has been shown (see Wonham & Morse (1 970), Grizzle & Isidori (1989), Koussiouris (1 980) arid Icart. Lafay & Malabre (1990) for the decoup11ng problem a n d Malabre, Martinez-Garcia & DelMuro-Cuellar (1997) for the disturbance rejection probIcm) t h a t there exist polcs, callcd the prohbnz f i e d poles, that a p p c a r in any closed-loop system after applying a n y lcedback s o l u t i o ~ iof the problem. In many practical situations, dislurbance rejection I S not the only control objeclive. O n e possible use of the remaining degrees of [reedom is orten to a d d other
l h i s papcr was not prcscntcd at any IFAC mccting. Tl~ispapcr was rccommcndcd for publication in reviscd form by Associate Editor 4.A. Stoorvogcl under thc dircction of Editor Paul Van dcn I-lof. *'Corresponding author. 'l'cl.: $- 33-2-40376912; fax: -t 33-2-
10376930. C-1nni1 oddr-m.r: micl~el.malal?re@ircey~~.ec-na111es.fr (M. Malabre). S u p p m e d by CONACYT Reseal-ch Projec~No. 21 1085-5-26450A. Vl~xico.
'
requirements such as decoupling, model malchiog. particular pole placement strategies,. .. I n this conlcst wc consider here the simultaneous disturbance ~rjectiorla n d regular block decoupling problem by static state tccdback for which a necessary a n d sufficient condition of solvability has been derived by C h a n g and Rhodcs (1975) and appeared t o b e equivalent t o the solvabiliiy of each separate problem. Algebraic a n d structural charactcrizations of the fixed poles of this simultaneous problem have been given in Koussiouris a n d Tzierakis (1996) within a frequency-domain approach in the cast whcrc the outputs are scalar-partitioned. T h e aim of this papcr is to provide a n alternative characterization of Lhese fiscd poles based o n the geometric approach as well :IS Lo provide another conslructive procedure to solve the problem while assigning the remaining free poles. This contribution is also a n extension of the previous c11ar:~terizations (including the result of Malabrc Sr Martincz Garcia (1995)) from t h c scalar case t o the block case. T h e problem is rormulated in Section 2 a n d results concerrlirig the fixed poles for the decoupling problem a n d for the disturbance rejection problem a r e rzc;tlled in Section 3. T h e main results a r e presented in Section 4 a ~ l d are illustrated in Section 5 through o n c cx:lmplc. Wc conclude i n Section 6 with some c o m p l e m c n t a ~ ~remarks y a n d with possible futurc extensions.
~~(~05-IOL~X!OI;%-scc front rnattcr f' 2000 Elscvicr Scicncc Ltd. All rights rcscrvcd. t'll: S 0 0 0 5 - IOOS(00)00143-6
2. Problem formulation Let us consider a h e a r time-invariant disturbed systern (A, B, C, 1:') described by
( klr) = Ax(t) + Bu(t) + Ed([),
.I v ( t ) = Cu(t),
n
O(A r E ~(SDRDP) (1)
where: X E 2' = R" denotes the state; u EJ& = Wm denotes the control input; y € : ? Y = WL denotes the output; r l ~ i i ;= RY denotes the disturbance. The output y(r) is partitioned in p block outputs y,(t), each of size li, with 1= l i . Wc can associate to the disturbed system (.3, B, C, E ) the undisturbed system (A, B, C) (i.e. d(.) = 0) and the combined .system (A,[B E],C) (i.e. d(.) is considered as a control input). The block suhsystons (A, B, C i ) o l (A, B, C) are also introduced where C iis the ith block o l C in connection with the splitting of y = [yT .. . y;lT. Script letters are used to denote the image of maps ( M : =lm B; 6':= In1 E), while 'ker' stands for the kernel. The combined system (A,[B E],C) is assumed to bc controllable, i.c. the controllable space (A I lm [B El):= Im[B E l +.AIin[B E l ... + A " ' I ~ [ BE l equals which is a nalural and unrestrictive assumption. We shall use the geometric approach, as described in Wonham (1 985) and Basile and Marro (1992). Let us clcnote: $"*, the suprcmal (A,:&)-invariant subspace included in kcr C. &. the supremal (A,#)-controllability subspace includod in ker C , .#:, the supremal (A.Im [B El)-controllability subspace included in kcr C and when needed, Y ' f ,:#? and the Samc subspaces with Ci in place of C. Let Y" be an (A,.#)-invariant subspace, .F(A,B, Y') dcnotcs the sct of maps P such that (A + B F ) Y c $ '. Y(A,B, C) denoles the set of finite invariant zeros for :I system (.4, B. C). Let us recall that a geometric definiA, B, C):= G(A + BFIY'*/:#*), tion for thcsc zcros is Y'( whcrc o stands for the spectrum and A + BI.'IY'*/W* is thc map induced by A + BP on the quotient space ""/:#*, wherc 1' is any map in .$(A, B;Y'*). The siinullaneous disturbance rejection and regular clecoupling problem (SDRDP) may be stated as follows:
x"=,
+
:,
Denoting @(SDRDP)the set of feedback salu~lon\of the SDRDP, we can formally define the SDRDP l n d by
Definition 1. Let (A,B,C,E) be given. The SDRDP amounts to tinding, if possible, a control law of the type ic(t) = I;x(t) + G,c.,(t)with G := [G, ... G,] rcgular such that
,
+ BFI.
A structural characterization (LC.in term4 of ~ nnrmnt \ zeros) of the S D R D P fixed polcs has becn proposcd by Koussiouris and 'Tzierakis (1996) using a polynoniial matrix approach, in the case where the outp~ttsJ . ~ ( / !arc scalars (Ii = 1). Our main result is a geometric wuiiterpart with extension to the "general block" casc. In Koussiouris and Tzierakis (1996), n diil'ertnt system description was adopted which explicitly splits coutrol inputs into two distinct sets (those which arc dircctly contaminated by disturbances (A) and thc remaining ones (B,)). That separation is not necessary (and tilcir description may be expressed in form (1) by t l c h i n g B:= [B, A] and E:= [D,A]) as will be secn in thc treatment of the example.
3. Background
Let us first recall the dislurbance rejection problem (DRP) definition:
Definition 2. G ~ v e na disturbed system (,4,13, ( , L ) , the tllsturbance rejection problem (by static s l a k I t t ~ l h : ~ c h ) amounts to f nding, if possible. a control law 01 ~ h 1) c pe ~ ( t=) Fx(t) such that for the closed-loop system, the transfer function matrix from d(.) to y( ) 14 ~dcntic:llly zero (i.e. C(sl - (A + BF))-'E = 0). The geometric solvability condition for thc cxistcucc ol solutions to the D R P is very well known (scc. Sol. instance, Wonham. 1985; Basile & Marro, 190'). Inclccd. this problem is solvable if and only if 8' c 3
Autornatica 37 (2001)297-302
Technical Communique
Fixed poles of simultaneous disturbance rejection and decoupling: a geometric approach* U~LLI&\
J.-F. Camarta, M. Malabre"?", J.-C. Martinez-Garciab?' " l ~ r . ~ r i t de ~ r / Rechcrchc en Corr1rt1~17icati,1n.c. e/ Cvhernitique de N ~ n t e s CNRS , UMR 6597, 1, r u e de lo ,Voi. B.P. 92/01, F44321 hluntes C ' n l u 0.3, I , r r ~ ~ i c r bUel~ur/omcwtode Control Autonl6rico, CINVESTAV-IPN,Au. ln.ctilt~loPoliticnico Nacional No. 2508, 07300 M h i c o U . K , M r x i t r ,
Kccc~vcd4
January 2000; rcvrscd 6 Junc 2000; rccc~vcdIn final form 30 Junc 2000
Abstract
When the simultaneous disturbance rejection and regular block decoupling problem by static state feedback is solv:~ble,and ~ ~ n d c r some unrestrictive minimality assumption. all the poles of the closed-loop system can be placed, except the so-called fiscd polcs (which are present for any solution). We present here a geometric characterization for this set of fixed poles a s well as ;i constructive proccrlurc (based on geometric tools) for designing a feedback solution of the problem which assigns the remaining polcs to a~-bi(rary prcspccificd locations. (G 2000 Elsevier Science Ltd. All rights reserved. h'~:a.~~~oril.s: Linear multivariable syslems; Disturbance rejeclion; Decoupling; Fixed
poles; S~ability;Pole assign men^; Ckoinelric approach: SI I - L I C ~ I I ~ : I ~
approach
T h e disturbancc rejection problem a n d the decoupling problem have both been of great interest in the control thcory literature a n d still remain challenging. F o r each >cp,irate problem, it has been shown (see Wonham & Morse (1 970), Grizzle & Isidori (1989), Koussiouris (1 980) arid Icart. Lafay & Malabre (1990) for the decoup11ng problem a n d Malabre, Martinez-Garcia & DelMuro-Cuellar (1997) for the disturbance rejection probIcm) t h a t there exist polcs, callcd the prohbnz f i e d poles, that a p p c a r in any closed-loop system after applying a n y lcedback s o l u t i o ~ iof the problem. In many practical situations, dislurbance rejection I S not the only control objeclive. O n e possible use of the remaining degrees of [reedom is orten to a d d other
l h i s papcr was not prcscntcd at any IFAC mccting. Tl~ispapcr was rccommcndcd for publication in reviscd form by Associate Editor 4.A. Stoorvogcl under thc dircction of Editor Paul Van dcn I-lof. *'Corresponding author. 'l'cl.: $- 33-2-40376912; fax: -t 33-2-
10376930. C-1nni1 oddr-m.r: micl~el.malal?re@ircey~~.ec-na111es.fr (M. Malabre). S u p p m e d by CONACYT Reseal-ch Projec~No. 21 1085-5-26450A. Vl~xico.
'
requirements such as decoupling, model malchiog. particular pole placement strategies,. .. I n this conlcst wc consider here the simultaneous disturbance ~rjectiorla n d regular block decoupling problem by static state tccdback for which a necessary a n d sufficient condition of solvability has been derived by C h a n g and Rhodcs (1975) and appeared t o b e equivalent t o the solvabiliiy of each separate problem. Algebraic a n d structural charactcrizations of the fixed poles of this simultaneous problem have been given in Koussiouris a n d Tzierakis (1996) within a frequency-domain approach in the cast whcrc the outputs are scalar-partitioned. T h e aim of this papcr is to provide a n alternative characterization of Lhese fiscd poles based o n the geometric approach as well :IS Lo provide another conslructive procedure to solve the problem while assigning the remaining free poles. This contribution is also a n extension of the previous c11ar:~terizations (including the result of Malabrc Sr Martincz Garcia (1995)) from t h c scalar case t o the block case. T h e problem is rormulated in Section 2 a n d results concerrlirig the fixed poles for the decoupling problem a n d for the disturbance rejection problem a r e rzc;tlled in Section 3. T h e main results a r e presented in Section 4 a ~ l d are illustrated in Section 5 through o n c cx:lmplc. Wc conclude i n Section 6 with some c o m p l e m c n t a ~ ~remarks y a n d with possible futurc extensions.
~~(~05-IOL~X!OI;%-scc front rnattcr f' 2000 Elscvicr Scicncc Ltd. All rights rcscrvcd. t'll: S 0 0 0 5 - IOOS(00)00143-6
2. Problem formulation Let us consider a h e a r time-invariant disturbed systern (A, B, C, 1:') described by
( klr) = Ax(t) + Bu(t) + Ed([),
.I v ( t ) = Cu(t),
n
O(A r E ~(SDRDP) (1)
where: X E 2' = R" denotes the state; u EJ& = Wm denotes the control input; y € : ? Y = WL denotes the output; r l ~ i i ;= RY denotes the disturbance. The output y(r) is partitioned in p block outputs y,(t), each of size li, with 1= l i . Wc can associate to the disturbed system (.3, B, C, E ) the undisturbed system (A, B, C) (i.e. d(.) = 0) and the combined .system (A,[B E],C) (i.e. d(.) is considered as a control input). The block suhsystons (A, B, C i ) o l (A, B, C) are also introduced where C iis the ith block o l C in connection with the splitting of y = [yT .. . y;lT. Script letters are used to denote the image of maps ( M : =lm B; 6':= In1 E), while 'ker' stands for the kernel. The combined system (A,[B E],C) is assumed to bc controllable, i.c. the controllable space (A I lm [B El):= Im[B E l +.AIin[B E l ... + A " ' I ~ [ BE l equals which is a nalural and unrestrictive assumption. We shall use the geometric approach, as described in Wonham (1 985) and Basile and Marro (1992). Let us clcnote: $"*, the suprcmal (A,:&)-invariant subspace included in kcr C. &. the supremal (A,#)-controllability subspace includod in ker C , .#:, the supremal (A.Im [B El)-controllability subspace included in kcr C and when needed, Y ' f ,:#? and the Samc subspaces with Ci in place of C. Let Y" be an (A,.#)-invariant subspace, .F(A,B, Y') dcnotcs the sct of maps P such that (A + B F ) Y c $ '. Y(A,B, C) denoles the set of finite invariant zeros for :I system (.4, B. C). Let us recall that a geometric definiA, B, C):= G(A + BFIY'*/:#*), tion for thcsc zcros is Y'( whcrc o stands for the spectrum and A + BI.'IY'*/W* is thc map induced by A + BP on the quotient space ""/:#*, wherc 1' is any map in .$(A, B;Y'*). The siinullaneous disturbance rejection and regular clecoupling problem (SDRDP) may be stated as follows:
x"=,
+
:,
Denoting @(SDRDP)the set of feedback salu~lon\of the SDRDP, we can formally define the SDRDP l n d by
Definition 1. Let (A,B,C,E) be given. The SDRDP amounts to tinding, if possible, a control law of the type ic(t) = I;x(t) + G,c.,(t)with G := [G, ... G,] rcgular such that
,
+ BFI.
A structural characterization (LC.in term4 of ~ nnrmnt \ zeros) of the S D R D P fixed polcs has becn proposcd by Koussiouris and 'Tzierakis (1996) using a polynoniial matrix approach, in the case where the outp~ttsJ . ~ ( / !arc scalars (Ii = 1). Our main result is a geometric wuiiterpart with extension to the "general block" casc. In Koussiouris and Tzierakis (1996), n diil'ertnt system description was adopted which explicitly splits coutrol inputs into two distinct sets (those which arc dircctly contaminated by disturbances (A) and thc remaining ones (B,)). That separation is not necessary (and tilcir description may be expressed in form (1) by t l c h i n g B:= [B, A] and E:= [D,A]) as will be secn in thc treatment of the example.
3. Background
Let us first recall the dislurbance rejection problem (DRP) definition:
Definition 2. G ~ v e na disturbed system (,4,13, ( , L ) , the tllsturbance rejection problem (by static s l a k I t t ~ l h : ~ c h ) amounts to f nding, if possible. a control law 01 ~ h 1) c pe ~ ( t=) Fx(t) such that for the closed-loop system, the transfer function matrix from d(.) to y( ) 14 ~dcntic:llly zero (i.e. C(sl - (A + BF))-'E = 0). The geometric solvability condition for thc cxistcucc ol solutions to the D R P is very well known (scc. Sol. instance, Wonham. 1985; Basile & Marro, 190'). Inclccd. this problem is solvable if and only if 8' c 3
