- Email: [email protected]
On the Decoupling of Square Systems
Copyright © IFAC System Structure and Control, Bucharest, Romania, 1997
ON THE DE COUPLING OF SQUARE SYSTEMS* J. Ruiz-Leon
Centro de Investigacion y de Estudios A vanzados, Unidad Guadala"jam. Apartado postal 31-438, Plaza la Luna. 44550 Guadalajara, Jalisco. Mexico. E-mail: [email protected]
Abstract. A new simple proof of the well known result that. a square invertible linear multivariable system is decouplable by stat.ic state feedback if and only if the sum of its infinite zero orders equals t.he sum of it.s row-infinit.e zero orders is presented. In t.his work, we take advantage of the properties of t.he ring of proper rational functions IRp(s) and IRp(s)unimodular matrices. Copyright © 199BlFAC Keywords. interactor.
Linear systems, decoupling, static state feedback , infinite structure, system
l. INTRODUCTION
2. PRELIMINARIES
The problem of oecoupliug a square liuear system (a system with the same uumber of iuputs allo outputs) by static state feeoback was solveo by Falb allo Wolovich (1967), the solutioll beillg stateo ill terms of the iuvertibility of a special matrix coustructeo from the matrices of the system.
We cousioer iu t his work square liuear multivariable systems oescribeo by (C, A, n) { ;i:(t)
1/(t)
= A.r(t) + nu(t) = C.r(t.)
where :1' E JR", 11 E !RP auo y E ]RP are, respectively, the state, iuput allo output vectors of the system.
Later on, iu the framework of the so-calleo structural approach, Descusse auo Diou (1982) preseuteo a uice iuterpretatiou of Falb allo Wolovich's result ill terms of the ilifiliite stl'llctllfe of the system. They showeo that this famous problem is solvable if ano only if the sum of the infinite Llero oroers of the system equals the sum of its rowiufinite zero oroers.
The system (C, A, n) is saio to be oecouplable by static state feeoback (or simply, oecouplable) if there exists (F, C):
1/.(1)
= Fx(t) + ClI(t)
where F alld Care coustaut matrices, slIch that the tralisfer function TFC(S) of the c1oseo-loop system (C, A + nF, nc) is of the form
By giving a new characteriLlatioll of the rowinfinite zero orders of the system amI using the properties of the ring of proper ratioual functions, we preseut in this lIote a simpler proof of the result given by Descusse ano Dion, where it is uot evell necessary to consioer output permutatious as they 00.
for a proper rational matrix
'Part of this work was realized during a st.ay at the Institute of Information Theory and Automation, Czt'ch Academy of Sciences.
Proper rational functions The set of propel' rational functions JRp(s) is
271
Lemma 1 The matrix U ( .~) appearillg iu (1) state feedback realizable.
kuowu to be a Euclideau rillg, the degree of the proper fuuctiou 1/1(.~) = E Rp(s), hereafter deuoted Dcyp 1/1(8), takeu to be the uumber of iufinite ~eros, i.e.,
*+
P1'Ooj. From (1) we have that
The result follows from Propositiou 1, sillce is biproper, and
where a(8) alld b(8) are coprime polYllomials.
The interactor of the system Let T (s) be the trausfer fuuctioll of the systeul (C, A, n) . Theu there exist a IRp(s)-uuilllodular (biproper) matrix U(.~) aud a 111lique lower triallgular matrix <1>-1 (s), the Hermite form of T(s ) over Rp(s), such that 1
T(S)U(8)
=
-I(.~)
= [
Thus, we may thiuk of the process of applyillg state feedback to the system (C, A, n) iu order to realize the actiou of the biproper matrix U (.~) as to that of reducillg the systcm to its" simplest expressiou·' . What we are doiug actually is usiug state feedback to reuder the system maximally Iloll-observable by producillg pole-~ero callcellatiolls ami shiftiug the remaiuiug poles. to the poillt 8 = O. From this poiut of view, the matrix <1>-\ (.~ ) call be cOllsidered as the trallsfer fUllctioll of this " maximally uou-observaGle system" .
-;JP
(1) where 'P ij (.~) are proper ratioual fuuctiolls with a uuique pole at s = 0, satis(yillg, for i > .i,
= 0,
or
Dcyp
<
Dr' yp
= fi
Infinite zero orders The iufiuite zero or'uers of the system cau be uetilled as follows. There exist IRp (s)-uuimodular matrices UJ(8) alld U2('~)' allu a ullique uiagoual matrix AI (.~), the Smith form of T(.~) over IRp (s) (also kllOWll as the Slllith-McMillau form at illfiuity of T(.~)), such that
(2)
The matrix <1>(8), the iuverse of -I(s ), is a polylIomial matrix kllowll as the illteractor of the systeru (Wolovich alld Falb, 1976). This matrix plays all importallt role ill the solutioll of several problems ill cOlltrol theory, sillce it is a complete illvariallt of the system llI1der state feedback. 1 Actually, the matrix <1> - 1(s ) call be regarded as the trausfer fUllctioll of the llIaximally 1l01lobservable system obtailled from (C , A, n) by state feedback . This meaus that the matrix U(s) which reveals the system iuteractor ill (1) is state feedback reali~able, i.e., that there exists a state feedback (F, G) snch that [J(.~) = [I - F(sl - A)-Inl-IG . To show this we ueed the followiug result.
n;
where ~ TI;+\, i = 1, . . . ,p - 1, are the illtillite ~ero oruers of the system.
Row-infinite zero orders The row-illfiuite zero oruers of the systelll , uelIoteu in;} , are the orders of the illtiuite ~eros of the subsystellls (C; , A , n), where Ci is the ith row of the matrix C . These illtegers are usually defilled ill the literature (Varuulakis, 1980) as
Proposition 1 (Hautus aud HeYlllaull , 1978) Let the matrices N 1 (.~) aud D (.~) be a right coprime matrix fractiou descriptiou of the systeIll (In,A,n), i.e., T(s) = CN1(S)D-l( .~), aud let C (.~) be a Ilollsingnlar com peusator. Thell C (.~) is state fcedback realizable ou (C, A , n) if and ouly if •
C(.~)
U(.~)
•
is a polyuomial matrix.
(~I 1
;7l
IS
ni
= lIIinU
: C i Aj-l
n i= 0,
j
= I, . . . , }
i=l, .. . ,p.
For our purposes , the followillg characterizatioll will be lIIore useful (Ruiz-Leou, 1996, Lelllllla 2.3)
is biproper, aud ni
• C-l(.~)D(s) is a polyuomiallllatrix.
= 'IIIin{Dr'yp
t. ;j(.~),
.i
= 1, ... ,p} , i =
1, . .. , p
where t;j(8) is the ijth elemeut of T(.~) .
•
Sillce the Rp(S)-UllillJodular lIIatrix U (8) does IlOt modify the iufoflllatioll at iufiuity of T( .~), frolll (1) we call see that
1 To
be more precise, the syst.em interact.or is a complet.e invariant under regular st.at.e feed hack (t.he st.at.e feedback (F,G) wit.h t.he matrix G being invertible), which is the kind of feedback we are considering in this work; (s) is no longer invariant for nonregular st.ate feedback.
ni = min{D('flp 'Pii(.~)'
272
j = 1, .. . , i},
(4)
i = 1, ... ,po
Suppose that <{>-I (8) is uot diagoual. Theu there exists at least oue elemeut 'Pij (8) # 0, i, j =
Notice also, because of property (2), that
1, ... ,p,i>j.
i = 1, . . . ,p
From (2) aud (4) we have that
(5)
11,;
< f; for some
i = 1, .. . ,p, implying also that
The followiug result will be ueeded iu the sequel.
p
p
Ln < Lfi i
Lemma 2 Let <{>-1 (8) aud A!(.~) be, respectively, the Henuite form aud the Smith form of T( 8) over JRp(s) as giveu by (1) aud (3). Theu
;=1
;=1
Followiug Lemllla 2. this lueaus that p
p
;=1
,=1
2:11,; < Ln; ;=1
;= 1
coutradictiug our assnlllptiou. Pmof. The result cau be readily seen from (l) aud
•
(3) , siuce 4. REFERENCES Descnsse, .1., aud .1.M. Diou (H)82). Ou the Structure at Iufiuity of Lillear Square Decon pled Systellls. IEEE Tmn8. A 1ltornot. Contr·., Vo\. AC-27, No. 4, pp. 971-974.
due to the fact that the degree of the deterllliuaut of a IRp(s)-nllimodular matrix (takeu as a proper rational flluctioll) is zero. •
Falb. P.L., aud W .A. Wolovich (1967). Deconpliug iu the Desigu aud Syuthesis of Mnltivariable Coutrol Systellls. IEEE Tmns. Automat. Contr. , Vo!. AC-12, No. 6, pp. 651659 .
3. MAIN RESULT
Theorem 1 The system (C, A, B) is deconplable if aud ouly if p
p
2:11,; = 2:11,; ;=1
Hantlls, M.L ..J., aud M. HeYlllauu (1978). Liuear Feedback: Au Algebraic Approach . SIA M .I. Contr·. Optirniz., Vo!. 16, No. 1, pp. 83-105.
(6)
;.=1
Rni:.-;-Leou . .1. (1996) . Decollpliug of Liuear SystelllS. PhD dissertatiou, C:.-;ech Tecllllical U uiversity, C:.-;ech Republic .
Pr·()of. Necessity. Suppose that (C, A, B) is decouplable. Then there exists a state feedback (F , G) such that
TFC(S)
= C(.~I = T(s)[I
A - BF)-IBe - F(sI - A)-IBl- I e
Vardlllakis, A.I.G . (1980) Ou Iufiuite Zeros. Int . .!. Contr. , Vo\. 32, pp. 849-866.
= W(s)
(7) where [I - F(sI - A)-I Bl-Ie is easily seell to be lRp(s)-ullimodnlar aud W(8) = diu.Y{W;(8)}f=1 is a proper ratiollal matrix.
Wolovich , W .A .. aud P.L. Falb (1976) . Iuvariauts aud Cauouical FOrluS Uuder Dyualllic COllipeusatiou . SIA M .!. Contr·. Optimiz., Vo!. 14, No. 6, pp . 996-1008.
Theu, from (1) and (7) it follows that <{>-1(.~) is diagollal, implyillg also that i = 1, ... ,p
Thus, from Lemma 2 p
2: i=1
p
ni
= 2: 11; ;.=1
Sllfficieucy. From Lemma 1, it cau be seell that the system (C,A, B) is decollplable if the system illteractor is diagoual. Theu, it will be sllfficieut to prove that <{>-I(8) is diagoual givell that (6) holds.
273
ON THE DE COUPLING OF SQUARE SYSTEMS* J. Ruiz-Leon
Centro de Investigacion y de Estudios A vanzados, Unidad Guadala"jam. Apartado postal 31-438, Plaza la Luna. 44550 Guadalajara, Jalisco. Mexico. E-mail: [email protected]
Abstract. A new simple proof of the well known result that. a square invertible linear multivariable system is decouplable by stat.ic state feedback if and only if the sum of its infinite zero orders equals t.he sum of it.s row-infinit.e zero orders is presented. In t.his work, we take advantage of the properties of t.he ring of proper rational functions IRp(s) and IRp(s)unimodular matrices. Copyright © 199BlFAC Keywords. interactor.
Linear systems, decoupling, static state feedback , infinite structure, system
l. INTRODUCTION
2. PRELIMINARIES
The problem of oecoupliug a square liuear system (a system with the same uumber of iuputs allo outputs) by static state feeoback was solveo by Falb allo Wolovich (1967), the solutioll beillg stateo ill terms of the iuvertibility of a special matrix coustructeo from the matrices of the system.
We cousioer iu t his work square liuear multivariable systems oescribeo by (C, A, n) { ;i:(t)
1/(t)
= A.r(t) + nu(t) = C.r(t.)
where :1' E JR", 11 E !RP auo y E ]RP are, respectively, the state, iuput allo output vectors of the system.
Later on, iu the framework of the so-calleo structural approach, Descusse auo Diou (1982) preseuteo a uice iuterpretatiou of Falb allo Wolovich's result ill terms of the ilifiliite stl'llctllfe of the system. They showeo that this famous problem is solvable if ano only if the sum of the infinite Llero oroers of the system equals the sum of its rowiufinite zero oroers.
The system (C, A, n) is saio to be oecouplable by static state feeoback (or simply, oecouplable) if there exists (F, C):
1/.(1)
= Fx(t) + ClI(t)
where F alld Care coustaut matrices, slIch that the tralisfer function TFC(S) of the c1oseo-loop system (C, A + nF, nc) is of the form
By giving a new characteriLlatioll of the rowinfinite zero orders of the system amI using the properties of the ring of proper ratioual functions, we preseut in this lIote a simpler proof of the result given by Descusse ano Dion, where it is uot evell necessary to consioer output permutatious as they 00.
for a proper rational matrix
'Part of this work was realized during a st.ay at the Institute of Information Theory and Automation, Czt'ch Academy of Sciences.
Proper rational functions The set of propel' rational functions JRp(s) is
271
Lemma 1 The matrix U ( .~) appearillg iu (1) state feedback realizable.
kuowu to be a Euclideau rillg, the degree of the proper fuuctiou 1/1(.~) = E Rp(s), hereafter deuoted Dcyp 1/1(8), takeu to be the uumber of iufinite ~eros, i.e.,
*+
P1'Ooj. From (1) we have that
The result follows from Propositiou 1, sillce is biproper, and
where a(8) alld b(8) are coprime polYllomials.
The interactor of the system Let T (s) be the trausfer fuuctioll of the systeul (C, A, n) . Theu there exist a IRp(s)-uuilllodular (biproper) matrix U(.~) aud a 111lique lower triallgular matrix <1>-1 (s), the Hermite form of T(s ) over Rp(s), such that 1
T(S)U(8)
=
-I(.~)
= [
Thus, we may thiuk of the process of applyillg state feedback to the system (C, A, n) iu order to realize the actiou of the biproper matrix U (.~) as to that of reducillg the systcm to its" simplest expressiou·' . What we are doiug actually is usiug state feedback to reuder the system maximally Iloll-observable by producillg pole-~ero callcellatiolls ami shiftiug the remaiuiug poles. to the poillt 8 = O. From this poiut of view, the matrix <1>-\ (.~ ) call be cOllsidered as the trallsfer fUllctioll of this " maximally uou-observaGle system" .
-;JP
(1) where 'P ij (.~) are proper ratioual fuuctiolls with a uuique pole at s = 0, satis(yillg, for i > .i,
= 0,
or
Dcyp
<
Dr' yp
= fi
Infinite zero orders The iufiuite zero or'uers of the system cau be uetilled as follows. There exist IRp (s)-uuimodular matrices UJ(8) alld U2('~)' allu a ullique uiagoual matrix AI (.~), the Smith form of T(.~) over IRp (s) (also kllOWll as the Slllith-McMillau form at illfiuity of T(.~)), such that
(2)
The matrix <1>(8), the iuverse of -I(s ), is a polylIomial matrix kllowll as the illteractor of the systeru (Wolovich alld Falb, 1976). This matrix plays all importallt role ill the solutioll of several problems ill cOlltrol theory, sillce it is a complete illvariallt of the system llI1der state feedback. 1 Actually, the matrix <1> - 1(s ) call be regarded as the trausfer fUllctioll of the llIaximally 1l01lobservable system obtailled from (C , A, n) by state feedback . This meaus that the matrix U(s) which reveals the system iuteractor ill (1) is state feedback reali~able, i.e., that there exists a state feedback (F, G) snch that [J(.~) = [I - F(sl - A)-Inl-IG . To show this we ueed the followiug result.
n;
where ~ TI;+\, i = 1, . . . ,p - 1, are the illtillite ~ero oruers of the system.
Row-infinite zero orders The row-illfiuite zero oruers of the systelll , uelIoteu in;} , are the orders of the illtiuite ~eros of the subsystellls (C; , A , n), where Ci is the ith row of the matrix C . These illtegers are usually defilled ill the literature (Varuulakis, 1980) as
Proposition 1 (Hautus aud HeYlllaull , 1978) Let the matrices N 1 (.~) aud D (.~) be a right coprime matrix fractiou descriptiou of the systeIll (In,A,n), i.e., T(s) = CN1(S)D-l( .~), aud let C (.~) be a Ilollsingnlar com peusator. Thell C (.~) is state fcedback realizable ou (C, A , n) if and ouly if •
C(.~)
U(.~)
•
is a polyuomial matrix.
(~I 1
;7l
IS
ni
= lIIinU
: C i Aj-l
n i= 0,
j
= I, . . . , }
i=l, .. . ,p.
For our purposes , the followillg characterizatioll will be lIIore useful (Ruiz-Leou, 1996, Lelllllla 2.3)
is biproper, aud ni
• C-l(.~)D(s) is a polyuomiallllatrix.
= 'IIIin{Dr'yp
t. ;j(.~),
.i
= 1, ... ,p} , i =
1, . .. , p
where t;j(8) is the ijth elemeut of T(.~) .
•
Sillce the Rp(S)-UllillJodular lIIatrix U (8) does IlOt modify the iufoflllatioll at iufiuity of T( .~), frolll (1) we call see that
1 To
be more precise, the syst.em interact.or is a complet.e invariant under regular st.at.e feed hack (t.he st.at.e feedback (F,G) wit.h t.he matrix G being invertible), which is the kind of feedback we are considering in this work; (s) is no longer invariant for nonregular st.ate feedback.
ni = min{D('flp 'Pii(.~)'
272
j = 1, .. . , i},
(4)
i = 1, ... ,po
Suppose that <{>-I (8) is uot diagoual. Theu there exists at least oue elemeut 'Pij (8) # 0, i, j =
Notice also, because of property (2), that
1, ... ,p,i>j.
i = 1, . . . ,p
From (2) aud (4) we have that
(5)
11,;
< f; for some
i = 1, .. . ,p, implying also that
The followiug result will be ueeded iu the sequel.
p
p
Ln < Lfi i
Lemma 2 Let <{>-1 (8) aud A!(.~) be, respectively, the Henuite form aud the Smith form of T( 8) over JRp(s) as giveu by (1) aud (3). Theu
;=1
;=1
Followiug Lemllla 2. this lueaus that p
p
;=1
,=1
2:11,; < Ln; ;=1
;= 1
coutradictiug our assnlllptiou. Pmof. The result cau be readily seen from (l) aud
•
(3) , siuce 4. REFERENCES Descnsse, .1., aud .1.M. Diou (H)82). Ou the Structure at Iufiuity of Lillear Square Decon pled Systellls. IEEE Tmn8. A 1ltornot. Contr·., Vo\. AC-27, No. 4, pp. 971-974.
due to the fact that the degree of the deterllliuaut of a IRp(s)-nllimodular matrix (takeu as a proper rational flluctioll) is zero. •
Falb. P.L., aud W .A. Wolovich (1967). Deconpliug iu the Desigu aud Syuthesis of Mnltivariable Coutrol Systellls. IEEE Tmns. Automat. Contr. , Vo!. AC-12, No. 6, pp. 651659 .
3. MAIN RESULT
Theorem 1 The system (C, A, B) is deconplable if aud ouly if p
p
2:11,; = 2:11,; ;=1
Hantlls, M.L ..J., aud M. HeYlllauu (1978). Liuear Feedback: Au Algebraic Approach . SIA M .I. Contr·. Optirniz., Vo!. 16, No. 1, pp. 83-105.
(6)
;.=1
Rni:.-;-Leou . .1. (1996) . Decollpliug of Liuear SystelllS. PhD dissertatiou, C:.-;ech Tecllllical U uiversity, C:.-;ech Republic .
Pr·()of. Necessity. Suppose that (C, A, B) is decouplable. Then there exists a state feedback (F , G) such that
TFC(S)
= C(.~I = T(s)[I
A - BF)-IBe - F(sI - A)-IBl- I e
Vardlllakis, A.I.G . (1980) Ou Iufiuite Zeros. Int . .!. Contr. , Vo\. 32, pp. 849-866.
= W(s)
(7) where [I - F(sI - A)-I Bl-Ie is easily seell to be lRp(s)-ullimodnlar aud W(8) = diu.Y{W;(8)}f=1 is a proper ratiollal matrix.
Wolovich , W .A .. aud P.L. Falb (1976) . Iuvariauts aud Cauouical FOrluS Uuder Dyualllic COllipeusatiou . SIA M .!. Contr·. Optimiz., Vo!. 14, No. 6, pp . 996-1008.
Theu, from (1) and (7) it follows that <{>-1(.~) is diagollal, implyillg also that i = 1, ... ,p
Thus, from Lemma 2 p
2: i=1
p
ni
= 2: 11; ;.=1
Sllfficieucy. From Lemma 1, it cau be seell that the system (C,A, B) is decollplable if the system illteractor is diagoual. Theu, it will be sllfficieut to prove that <{>-I(8) is diagoual givell that (6) holds.
273