We consider quadruples of matrices (E; A;B;C),
representing singular linear time invariant systems in
the form
Ex_ (t) = Ax + Bu
y = Cx
¾
(1)
with E;A 2 Mp£n(C), B 2 Mp£m(C) and C 2
Mq£n(C) under proportional and derivative feedback
and proportional and derivative output injection.
In this paper we present a canonical reduced form preserving
the structure of the system and provides a decomposition
of the system into two independent systems,
one being a maximal regular system and the second
one a minimal completely singular one.Postprint (published version

Díaz, Adolfo

García Planas, María Isabel

English

UPCommons. Portal del coneixement obert de la UPC

PHYSCON 2009, Catania, Italy, September, 1–September, 4 2009A CANONICAL REDUCED FORM FOR SINGULAR TIMEINVARIANT LINEAR SYSTEMSM. Isabel Garcı´a-PlanasDepartament Matema`tica Aplicada IUniversitat Polite`cnicaSpainmaria.isabel.garcia@upc.eduAdolfo Dı´azDepartament Matema`tica Aplicada IUniversitat Polite`cnica de CatalunyaSpainadiaz pr@yahoo.comAbstractWe consider quadruples of matrices (E,A,B,C),representing singular linear time invariant systems inthe formEx˙(t) = Ax+Buy = Cx}(1)with E,A ∈ Mp×n(C), B ∈ Mp×m(C) and C ∈Mq×n(C) under proportional and derivative feedbackand proportional and derivative output injection.In this paper we present a canonical reduced form pre-serving the structure of the system and provides a de-composition of the system into two independent sys-tems, one being a maximal regular system and the sec-ond one a minimal completely singular one.Key wordsSingular Systems, equivalence relation, canonicalform.1 IntroductionWe denote by Mp×q(C) the space of complex matri-ces having p rows and q columns, and in the case whichp = q we write Mn(C). We consider the set M ofquadruples of matrices (E,A,B,C) representing fam-ilies of singular linear time invariant systems in theformEx˙ = Ax+Bu, y = CxwithE,A ∈Mp×n(C),B ∈ Mp×m(C) and C ∈ Mq×n(C), and we define anequivalence relation that permit us to decompose thesystem into two independent systems one being a max-imal regular system and the second a minimal com-pletely singular one. Each one of the subsystems de-compose into independent systems in the form.{x˙1 = N2x1 +B1uy1 = C1x1(2){x˙2 = N3x2 +B2u (3){x˙3 = N4x3y3 = C2x3(4){x˙4 = Jx4 (5){N1x˙5 = x5 (6){L1x˙6 = R1x6 (7){Lt2x˙7 = Rt2x7 (8){B3u3 = 0 or {C3x8 = 0. (9)System (2) is a maximal controllable and observablesubsystem, system (3) is the maximal controllable noobservable subsystem, system (4) is the maximal ob-servable no controllable subsystem , system (5) is astandard system with no modifiable finite zeros, system(6) is a system containing all no transferable infinitezeros, and finally systems (7) and (8) are completelysingular systems.The equivalence relation considered is the one that ac-cept one or more, of the following transformations: ba-sis change in the state space, input space, output space,feedback, derivative feedback, output injection, deriva-tive output injection and premultiplication by an invert-ible matrix.In the sequel we will use the following notations.- In denotes the n-order identity matrix,- N denotes a nilpotent matrix in its reduced form N =diag (N1, . . . , N`), Ni =(0 Ini−10 0)∈Mni(C),- J denotes the Jordan matrix J = diag (J1, . . . , Jt),Ji = diag(Ji1 , . . . , Jis), Jij = λiIij +N ,-L denotes the diagonal matrixL = diag (L1, . . . , Lq),where Lj =(Inj 0) ∈Mnj×(nj+1)(C),- R denotes the diagonal matrix R =diag (R1, . . . , Rp), where Rj =(0 Inj) ∈Mnj×(nj+1)(C),- t, rc y ro determine the quantity of controllable andobservable, controllable no observable and observableno controllable blocks that appear in an standard tripleof matrices.e2 Equivalence of singular systemsWe consider singular linear as in (1), many interestingand useful equivalence relations between singular sys-tems have been defined. As pointed in the introduction,we deal with the equivalence relation accepting one ormore, of the following transformations: basis change inthe state space, input space, output space, operations ofstate and derivative feedback, state and derivative out-put injection and to premultiply the first equation in (1),by an invertible matrix. That is to say.Definition 1. Two quadruples (Ei, Ai, Bi, Ci) ∈M, i = 1, 2, are equivalent if and only if there ex-ist matrices P ∈ Gl(n;C), Q ∈ Gl(p;C), R ∈Gl(m;C), S ∈ Gl(q;C), FBE , FBA ∈ Mm×n(C),FCE , FCA ∈Mp×q(C) such thatE2 = QE1P +QB1FBE + FCE C1P,A2 = QA1P +QB1FBA + FCAC1P,B2 = QB1R,C2 = SC1P.Given a quadruple of matrices (E,A,B,C) ∈ M, wecan associate the following matrix pencilH(λ) =λE +A λB BλC 0 0C 0 0 ,and we haveProposition 1. Two quadruples are equivalent underequivalent relation considered if and only if the asso-ciates matrix pencils are strictly equivalent.So, we can apply kroneckers theory of singular pencils(see [5]).Corollary 1. Let H(λ) be a matrix pencil associatedto the quadruple (E,A,B,C) ∈ M. Then H(λ) it isequivalent to the pencil λF +G withF =LLtI1N , G =RRtJI2Then, we can reduce any quadruple to a simpler form.This reduced form does not preserve the structure ofthe system, so, we propose the following reduced formas starting point to obtain the desired reduction.Theorem 1. Let (E,A,B,C) ∈ M be a quadrupleof matrices. Then it is equivalent under equivalencerelation considered, to a quadruple (Eω, Aω, Iω, Iω)in the following form:((E0),(A0),(0Ib),(0Ic))where (E,A) is a pair in its Kronecker reduced form.A collection of structural invariants to the quadruple(Eω, Aω, Iω, Iω) characterizing equivalent quadruplesis the following collection of numbersi) ω1 ≥ ω2 ≥ . . . ≥ ωz ≥ 1 nilpotency indicesii) k1(λ) ≥ k2(λ) ≥ . . . ≥ kjλ(λ) ≥ 1 Segre charac-teristic corresponding to eigenvalue λiii) ²1 ≥ . . . ≥ ²r² > ²r²+1 = . . . = ²rk = 0 columnminimal indicesiv) η1 ≥ . . . ≥ ηlη > ηlη+1 = . . . = ηlk = 0 rowminimal indices3 New canonical reduced formTheorem 2. Let (E,A,B,C) ∈ M be a quadrupleof matrices. Then it can be reduced under equiv-alence relation considered, to the following reducedform (Ec, Ac, Bc, Cc):((I1 00 Ek),(AeAk),(Be 00 0),(Ce 00 0))where (Ae, Be, Ce) and (Ek, Ak) are in its Kroneckerreduced form (see [6], [3] respectively).Proof. Let (E,A,B,C) be a quadruple and(Eω, Aω, Iω, Iω) its reduced form given in The-orem 1. So, the quadruple (Eω, Aω, Iω, Iω) ispartitioned in the following mannerEr Es0,Ar As0,0 00 0Ib 0,(0 0 Ic0 0 0)with(Er, Ar) =I1 I2N ,J J0I3and(Es, As) =((LLt),(RRt))We proceed by performing the following steps.Step 1: We consider the subquadruple(E′, A′, B′, C ′) :L Lt0 ,R Rt0 ,00Ib , (0 0 Ic)and we take the subtriple:((L0),(R0),(0Ib))and we distinguish two cases depending on relation be-tween rk and b.a) rk ≤ bIn this case the triple is writtenL00 ,R00 , 0 0Irc 00 Itwith rc = rk, t = b− rc ≥ 0. Then, it is controllable.Detailing the subtriple((L0),(R0),(0Irc)), wehave(L0)=L1 00 0L2 00 0. . ....Lr² 00 00(0Irc)=0 01 00 01 0. . ....0 01 00 0 0 Irc−r²(R0)=R1 00 0R2 00 0. . ....Rr² 00 00It is easy to verify that the controllability indices ofeach subsystem((Li0),(Ri0),(0Bi))=((I²i 00 0),(0 I²i0 0),(01))i = 1, . . . , rc are kcoi = ²i + 1.Fixed t, we have that the maximal quantity of ob-servable no controllable blocks in the pair (Lt, Rt) isro = c− t. Then if lk > ro, we have that the subtriple((Lt 0), (Rt 0), (0 Ic)) is completely no observable.We write this triple in the form((Lt1 0Lt2 0),(Rt1 0Rt2 0),(0 Ir0 0 00 0 0 It))where the pair (Lt1, Rt1) contains the first ro-blocks andthe pair (Lt2, Rt2) contains the rest of l = lk−ro-blocks.The subtriple ((Lt1 0), (Rt1 0), (0 Iro)), written in theform((Lt1 0)(0 Iro))=Lt11 0Lt12 0. . .Lt1rη 000 10 1. . .0 10 0 0 0 . . . 0 0 Iro−lη((Rt1 0)(0 Iro))=Rt11 0Rt12 0. . .Rt1rη 000 10 1. . .0 10 0 0 0 . . . 0 0 Iro−lη,is observable and observability indices of each subsys-tem((Lt1i 0), (Rt1i 0), (0 C1i)) =((Iηi 00 0),(0 0Iηi 0),(0 1))i = 1, . . . , ro are kcoi = ηi + 1.We observe that, if rk ≤ b, then the pencil(λEk +Ak IbIc 0)has column full rank. It have row fullrank if and only if lk = ro, that is to say, lk ≤ c.b) rk > bIn this case, if lk > c, the subquadruple(E′, A′, B′, C ′) is no controllable and no observableand it can be decomposed into two independent sub-triples: a triple no controllableL10L2 ,R10R2 ,0Ib0 ,where the pair (L1, R1) contains the first rc − b blocksand the pair (L2, R2) contains the rest r = rk−b blocksand the other no observable triple((Lt1 0Lt2),(Rt1 0Rt2),(0 Ic 0)),where the pair (Lt1, Rt1) contains the firsts ro−c blocksand (Lt2, Rt2) contains the rest of the l = lk − c blocks.Obviously, t = 0.Now, we take the subtriple((Lt 0), (Rt 0), (0 Ic))whose study is analogous to the previous one. If lk ≤ c,then the pencil(λEk +Ak IbIc 0)has row full rank. Itwill have column full rank if and only if rk = b − t,that is to say, rk ≤ b, where t = c− lk.Step 2: Now we consider the quadruple(E′, A′, B′, C ′) obtained in step 1. If t 6= 0, weseparate the subquadruple (E1, A1, B1, C1) :((0N),(0I3),(It0),(It 0))the nilpotent matrix N has z blocks, N =diag (N1, . . . , Nz).a) z ≤ tMaking elementary transformations, the subquadruple(E1, A1, B1, C1) is reduced toE1 =0N1. . .0Nz0A1 =0I31. . .0I3z0B1 =10. . .10It−sandC1 =10. . .10It−sIt is easy to verify that each subsystem(E1i, A1i, B1i, C1i), i = 1, . . . , z in the formE1i =(0N), A1i =(0I),B1i =(10), C1i =(1 0),is controllable and observable and the controllable-observable indices are kcoi = ωi + 1.b) z > tIn this case, the subquadruple (E1, A1, B1, C1) can bedecomposed in the form0 N1N2 ,0 I31I32 ,It00 , (It 0 0)where (N1, I31) contains the first t blocks of nilpotencyand (N2, I32) the rest s = z − t-blocks.4 ConclusionIn this paper we present a canonical reduced formpreserving the structure of the system providing a de-composition of the system into independent subsys-tems where the structural properties of the system ascontrollability, observability for example, can be easilydescribed.ReferencesS. L. Campbell. “Singular Systems of DifferentialEquations”. Pitman, San Francisco, (1980).L. Dai “Singular Control Systems”. Springer Verlag.New York (1989).A. Dı´az, M. I. Garcı´a-Planas, An alternative completesystem of invariants for matrix pencils under strictequivalence. To appear.A. Dı´az, M. I. Garcı´a-Planas, A canonical reducedform for singular time invariant linear systems. PartI. To appear.F. R. Gantmacher, The Theory of Matrices, Vol. 1, 2,Chelsea, New York, (1959).Ma I. Garcı´a-Planas, M.D. Magret, An alternativeSystem of Structural Invariants for Quadruples of Ma-trices, Linear Algebra and its Applications. 291, (1-3),pp. 83-102, (1999).A.S. Morse, Structural invariants of linear multivari-able systems, SIAM J. Contr. 11, pp. 446-465, (1973).Misra, Van Dooren, Varga Computation of StructuralInvariants of Generalized State Space Systems, Auto-matica, vol. 30, pp. 1921-1936, (1994).Varga On Stabilization Methods of Descriptor Sys-tems, Systems and Control Letters, vol 24. pp. 133-138, (1995).

A canonical reduced form for singular time invariant linear systems

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A canonical reduced form for singular time invariant linear systems

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