Quasi feedback forms for differential-algebraic systems
aa r X i v : . [ m a t h . O C ] F e b IMA Journal of Mathematical Control and Information
Page 1 of 27doi:10.1093/imamci/dnaaxxx
Quasi feedback forms for differential-algebraic systems T HOMAS B ERGER
Institut f¨ur Mathematik, Universit¨at Paderborn, Warburger Str. 100, 33098 Paderborn,Germany, [email protected] A CHIM I LCHMANN
Institut f¨ur Mathematik, Technische Universit¨at Ilmenau, Weimarer Straße 25, 98693Ilmenau, Germany, [email protected] S TEPHAN T RENN
Systems, Control and Applied Analysis – Bernoulli Institute, University of Groningen,Nijenborgh 9, 9747 AG Groningen, The Netherlands, [email protected]
We investigate feedback forms for linear time-invariant systems described by differential-algebraic equa-tions. Feedback forms are representatives of certain equivalence classes. For example state space trans-formations, invertible transformations from the left, and proportional state feedback constitute an equiva-lence relation. The representative of such an equivalence class, which we call proportional feedback formfor the above example, allows to read off relevant system theoretic properties. Our main contribution is toderive a quasi proportional feedback form. This form is advantageous since it provides some geometricinsight and is simple to compute, but still allows to read off the relevant structural properties of the controlsystem. We also derive a quasi proportional and derivative feedback form. Similar advantages hold.
Keywords :Differential-algebraic systems, descriptor systems, feedback forms, Wong sequences
Dedication
We dedicate the present note to the memory of Nicos Karcanias – a friend and colleague. Nicos has hada fundamental impact in diverse areas of systems and control theory , in particular in matrix pencil theoryof linear systems . He has had a special interest in system structure , leading to invariants and canonicalforms. See for example the very early paper [13] and many more. Our present note is in this spirit.Some of our results are closely related to his seminal work [16]. We will refer to this in due place.
1. Introduction
We study structured matrix pencils of the form s [ E , ] − [ A , B ] with E , A ∈ R ℓ × n and B ∈ R ℓ × m for whichwe write [ E , A , B ] ∈ Σ ℓ, n , m . Such pencils are typically associated with differential-algebraic controlsystems of the form E ˙ x ( t ) = Ax ( t ) + Bu ( t ) . (1.1)An essential difference between differential-algebraic systems (1.1) and ordinary differential sys-tems (by this we mean (1.1) with square and invertible E ∈ R n × n ) is their solution behaviour (see [18])and the different controllability concepts (see [3]). To address various control problems of systems (1.1),the well known (quasi) Kronecker form (cf. [5, 11, 14]) of the augmented pencil s [ E , ] − [ A , B ] is not“good enough”. A finer structure, which takes into account the input matrix B , is required. This isachieved by “(quasi) canonical” forms. The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. of 27
T. B
ERGER ET AL . A very early contribution in this spirit is by Nicos Karcanias and coworkers [16]; we will explaintheir achievements in due course. Our approach to “(quasi) canonical” forms is via augmented Wongsequences; this tool is fundamental and introduced in Section 2.In Section 3, we allow for state space transformations of (1.1), invertible transformations fromthe left, and proportional state feedback, where the latter means to add the algebraic relation u ( t ) = F P x ( t ) + v ( t ) to the system (1.1) for F P ∈ R m × n and v as new input. These transformations constitutean equivalence relation and the representatives of the equivalence classes are called P-feedback forms.We derive a quasi P-feedback form which – when compared to the previous form – has the advantagesthat it provides some geometric insight and is simpler to compute, but still allows to read off the mostrelevant structural properties of the control system.In Section 4, we extend the class of allowed transformations by also considering derivative feedback,i.e. u ( t ) = F P x ( t ) + F D ˙ x ( t ) + v ( t ) . Again, we derive a quasi form for the corresponding equivalencerelation.
2. Augmented Wong sequences
The Wong sequences have been introduced as a fundamental geometric tool for the analysis of ma-trix pencils and the derivation of quasi canonical forms – see [2, 5, 6]. This approach has been ex-tended to control systems [ E , A , B ] ∈ Σ ℓ, n , m to derive a Kalman controllability decomposition – see [3,7]. Compared to matrix pencils sE − A ∈ R [ s ] ℓ × n , the augmented pencil [ sE − A , − B ] with B ∈ R ℓ × m contains additional independent variables, which are typically associated with the input of the controlsystem (1.1) . We like to emphasize that the augmented Wong sequences are projections of the Wongsequences corresponding to the augmented matrix pencil s [ E , ] − [ A , B ] as shown in Proposition 2.2.The augmented Wong limits are related to the concepts of reachable and controllable spaces forthe DAE control system [ E , A , B ] ∈ Σ ℓ, n , m . These spaces are some of the most important notionsfor (DAE) control systems and have been considered in [15] for regular systems. Further usage ofthese concepts can be found in the following: in [17] generalized reachable and controllable subspacesof regular systems are considered; Eliopoulou and Karcanias [9] consider reachable and almost reach-able subspaces of general DAE systems; Frankowska [10] considers the reachable subspace in terms ofdifferential inclusions. However, to the best of our knowledge, the interplay between the (augmented)Wong-sequences and (quasi) canonical forms has not been investigated so far and the present contribu-tion aims to close this gap. Definition 2.1 (Augmented Wong sequences) . Let [ E , A , B ] ∈ Σ ℓ, n , m . The sequences ( V i [ E , A , B ] ) i ∈ N and ( W i [ E , A , B ] ) i ∈ N defined as V [ E , A , B ] : = R n , V i + [ E , A , B ] : = A − ( E V i [ E , A , B ] + im B ) ⊆ R n , W [ E , A , B ] : = { } , W i + [ E , A , B ] : = E − ( A W i [ E , A , B ] + im B ) ⊆ R n , are called augmented Wong sequences and V ∗ [ E , A , B ] : = \ i ∈ N V i [ E , A , B ] , W ∗ [ E , A , B ] : = [ i ∈ N W i [ E , A , B ] , are called the augmented Wong limits . ⋄ Of course, input constraints may be present, but we ignore this for purpose of motivation.
UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS
Proposition 2.2 (Properties of augmented Wong sequences) . Consider [ E , A , B ] ∈ Σ ℓ, n , m and the se-quences ( V i [ E , A , B ] ) i ∈ N , ( W i [ E , A , B ] ) i ∈ N and ( X i [ E , A , B ] ) i ∈ N . Then we have:(a) The sequences are nested and terminate, i.e., there exist i ∗ , j ∗ n such that, for all i , j ∈ N , V [ E , A , B ] ) V [ E , A , B ] ) · · · ) V i ∗ [ E , A , B ] = V i ∗ + i [ E , A , B ] = V ∗ [ E , A , B ] = A − ( E V ∗ [ E , A , B ] + im B ) , (2.1a) W [ E , A , B ] ( W [ E , A , B ] ( · · · ( W j ∗ [ E , A , B ] = W j ∗ + j [ E , A , B ] = W ∗ [ E , A , B ] = E − ( A W ∗ [ E , A , B ] + im B ) , (2.1b)and hence their limits are well defined.(b) The augmented Wong limits V ∗ [ E , A , B ] , W ∗ [ E , A , B ] ⊆ R n are linear subspaces and satisfy E W ∗ [ E , A , B ] ⊆ A W ∗ [ E , A , B ] + im B , A V ∗ [ E , A , B ] ⊆ E V ∗ [ E , A , B ] + im B , (2.2) E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) = E V ∗ [ E , A , B ] ∩ ( A W ∗ [ E , A , B ] + im B ) , A ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) = ( E V ∗ [ E , A , B ] + im B ) ∩ A W ∗ [ E , A , B ] , (2.3)and E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) + im B = ( E V ∗ [ E , A , B ] + im B ) ∩ ( A W ∗ [ E , A , B ] + im B )= A ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) + im B . (2.4)(c) The augmented Wong limits V ∗ [ E , A , B ] , W ∗ [ E , A , B ] ⊆ R n are related to the Wong limits V ∗ [[ E , ] , [ A , B ] , ] , W ∗ [[ E , ] , [ A , B ] , ] ⊆ R n + m of the augmented pencil s [ E , ] − [ A , B ] as follows: V ∗ [ E , A , B ] = [ I n , ] · V ∗ [[ E , ] , [ A , B ] , ] and W ∗ [ E , A , B ] = [ I n , ] · W ∗ [[ E , ] , [ A , B ] , ] . (2.5) Proof. (a) The proof of this statement is straightforward and hence omitted.(b) By (2.1) the two relations in (2.2) follow, which in turn immediately yield the subset inclusion “ ⊆ ”of the equations in (2.3). To show the converse inclusions, let z ∈ E V ∗ [ E , A , B ] ∩ ( A W ∗ [ E , A , B ] + im B ) .Then there exist v ∈ V ∗ [ E , A , B ] , w ∈ W ∗ [ E , A , B ] , and u ∈ R m such that Ev = z = Aw + Bu . By (2.1) we have E W ∗ [ E , A , B ] = ( A W ∗ [ E , A , B ] + im B ) ∩ im E and since Aw + Bu ∈ ( A W ∗ [ E , A , B ] + im B ) ∩ im E there exists w ∈ W ∗ [ E , A , B ] such that Ew = Aw + Bu and hence z = Ev = Ew . Therefore, v − w ∈ ker E ⊆ W ∗ [ E , A , B ] , which gives v ∈ V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] , thus z = Ev ∈ E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) .This shows E V ∗ [ E , A , B ] ∩ ( A W ∗ [ E , A , B ] + im B ) ⊆ E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) . The inclusion ( E V ∗ [ E , A , B ] + im B ) ∩ A W ∗ [ E , A , B ] ⊆ A ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) can be shown similarly and its proof is omitted.We show (2.4): For the first equality, observe that “ ⊆ ” follows from (2.2). For “ ⊇ ” let x ∈ of 27 T. B
ERGER ET AL . ( E V ∗ [ E , A , B ] + im B ) ∩ ( A W ∗ [ E , A , B ] + im B ) , i.e., x = Ev + b = Aw + b for some v ∈ V ∗ [ E , A , B ] , w ∈ W ∗ [ E , A , B ] , b , b ∈ im B . Then v ∈ E − { Aw + b − b } ⊆ E − ( A W ∗ [ E , A , B ] + im B ) = W ∗ [ E , A , B ] and hence x ∈ E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) + im B . The second equality in (2.4) can be proved similarly;we omit the proof.(c) The proof of this statement can be easily inferred from the proof of [1, Lem. 2.1]. (cid:3)
3. P-feedback forms
In this section, we recall the concept of P-feedback which allows a decoupling of the DAE (1.1). Thishas been successfully used for various purposes, cf. the survey [3]. After that, we present the P-feedbackform from [16], which is a canonical form. As a new contribution, we derive a quasi P-feedback formusing the augmented Wong sequences. Relevant system theoretic information can be read off this form.Apart from allowing a calculation via the simple subspace sequences, this also provides some geometricinsight in the decoupling.3.1
P-feedback equivalence
We recall the notion of P-feedback equivalence for systems [ E , A , B ] ∈ Σ ℓ, n , m , see e.g. [3]. Here and inthe following, GL p ( R ) denotes the set of all invertible matrices in R p × p , p ∈ N . Definition 3.1 (P-feedback equivalence) . Two systems [ E , A , B ] , [ E , A , B ] ∈ Σ ℓ, n , m are called P-feedback equivalent , if ∃ S ∈ GL ℓ ( R ) , T ∈ GL n ( R ) , V ∈ GL m ( R ) , F P ∈ R m × n : (cid:2) sE − A , − B (cid:3) = S (cid:2) sE − A , − B (cid:3) (cid:20) T F P V (cid:21) ; (3.1)we write [ E , A , B ] ∼ = P [ E , A , B ] or, if necessary, [ E , A , B ] S , T , V , F P ∼ = P [ E , A , B ] . ⋄ Remark 3.2.
P-feedback equivalence is an equivalence relation on Σ ℓ, n , m :• Reflexivity: Clear with S = I , T = I , V = I , F P = [ E , A , B ] S , T , V , F P ∼ = P [ E , A , B ] we have that [ E , A , B ] S − , T − , V − , − V − F P T − ∼ = P [ E , A , B ] , which can be verified by observing that (cid:20) T F P V (cid:21) − = (cid:20) T − − V − F P T − V − (cid:21) . UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS [ E , A , B ] S , T , V , F ∼ = P [ E , A , B ] S , T , V , F ∼ = P [ E , A , B ] we have [ E , A , B ] S S , T T , V V , e F ∼ = P [ E , A , B ] where e F = F T + V F . ⋄ The augmented Wong sequences change under P-feedback as shown in the following result.
Lemma 3.3 (Augmented Wong sequences under P-feedback) . If the systems [ E , A , B ] , [ E , A , B ] ∈ Σ ℓ, n , m are P-feedback equivalent [ E , A , B ] S , T , V , F P ∼ = P [ E , A , B ] , then ∀ i ∈ N : V i [ E , A , B ] = T − V i [ E , A , B ] and W i [ E , A , B ] = T − W i [ E , A , B ] . Proof.
We prove the first statement by induction. It is clear that V [ E , A , B ] = T − V [ E , A , B ] . Assumethat V i [ E , A , B ] = T − V i [ E , A , B ] for some i >
0. Then (3.1) yields V i + [ E , A , B ] = A − ( E V i [ E , A , B ] + im B )= ( x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∃ y ∈ V i [ E , A , B ] ∃ u ∈ R m : ( SA T + SB F P ) x = SE Ty + SB Vu ) = n x ∈ R n (cid:12)(cid:12)(cid:12) ∃ z ∈ V i [ E , A , B ] ∃ v ∈ R m : A T x = E z + B v o = T − (cid:16) A − ( E V i [ E , A , B ] + im B ) (cid:17) = T − V i + [ E , A , B ] . The proof of the second statement is similar and omitted.3.2
P-feedback form (PFF)
For the definition of the P-feedback form we need to introduce some further notation. For k ∈ N ,consider the matrices N k : = (cid:20)
01 1 0 (cid:21) ∈ R k × k , K k : = (cid:20) (cid:21) , L k : = (cid:20) (cid:21) ∈ R ( k − ) × k , where K k = L k = × for k =
1. We set, for some multi-index α = ( α , . . . , α k ) ∈ N k , | α | = α + . . . + α k and introduce the notation N α : = diag ( N α , . . . , N α k ) ∈ R | α |×| α | , K α : = diag ( K α , . . . , K α k ) ∈ R ( | α |− k ) ×| α | , L α : = diag ( L α , . . . , L α k ) ∈ R ( | α |− k ) ×| α | . By e [ n ] i we denote the i -th unit vector in R n and define E α : = diag ( e [ α ] α , . . . , e [ α k ] α k ) ∈ R | α |× k , for α = ( α , . . . , α k ) ∈ N k . of 27 T. B
ERGER ET AL . Definition 3.4 (P-feedback form) . The system [ E , A , B ] ∈ Σ ℓ, n , m is said to be in P-feedback form (PFF),if [ E , A , B ] = I | α | L β K ⊤ γ L ⊤ δ N κ
00 0 0 0 0 I n c , N ⊤ α K β L ⊤ γ K ⊤ δ I | κ |
00 0 0 0 0 A c , E α E γ
00 0 00 0 00 0 0 , (3.2)where α ∈ N n α , β ∈ N n β , γ ∈ N n γ , δ ∈ N n δ , κ ∈ N n κ are multi-indices and A c ∈ R n c × n c . ⋄ We like to note that the PFF of a system [ E , A , B ] can be viewed as a Kronecker canonical form (KCF)of the augmented pencil s [ E , ] − [ A , B ] with some additional structure, as shown in [3, Rem. 3.10]. Thisis remarkable because P-feedback equivalence induces an equivalence relation on R ℓ × ( n + m ) [ s ] which isa subrelation of the system equivalence used to obtain the Kronecker canonical form, and hence it isnot clear whether the Kronecker canonical form of s [ E , ] − [ A , B ] is contained in each of the smallerequivalence classes.We use the connection between the KCF and the PFF to show that two P-feedback equivalent sys-tems have the same PFF up to permutation of the entries of α , β , γ , δ , κ and similarity of A c . Proposition 3.5 (Uniqueness of indices for PFF) . Let [ E i , A i , B i ] ∈ Σ ℓ, n , m , i = ,
2, be in PFF (3.2) withcorresponding multi-indices α i ∈ N n α i , β i ∈ N n β i , γ i ∈ N n γ i , δ i ∈ N n δ i , κ i ∈ N n κ i , and A c , i ∈ R n c , i × n c , i . If [ E , A , B ] ∼ = P [ E , A , B ] , then α = P α α , β = P β β , γ = P γ γ , δ = P δ δ , κ = P κ κ and n c , = n c , , A c , = H − A c , H , for permutation matrices P α , P β , P γ , P δ , P κ of appropriate sizes and H ∈ GL n c , ( R ) . Proof.
This result is a consequence of Lemma 3.3 and [4, Thm. 2.2 & Prop. 2.3].We are now in the position to show that any [ E , A , B ] ∈ Σ ℓ, n , m is P-feedback equivalent to a systemin PFF. Theorem 3.6 (PFF) . For any system [ E , A , B ] ∈ Σ ℓ, n , m there exist S ∈ GL ℓ ( R ) , T ∈ GL n ( R ) , V ∈ GL m ( R ) , F P ∈ R m × n such that [ SET , SAT + SBF P , SBV ] is in PFF (3.2).The proof of Theorem 3.6 is omitted. It relies on subtle transformations and is proved in [16,Thm. 3.1] – a paper coauthored by Nicos Karcanias. Example 3.7 (PFF, Example 3.7 revisited) . For an illustration of Theorem 3.6 we consider the system [ E , A , B ] ∈ Σ , , with E = , A = − − − − − − , B = . UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS F P = h − − − − − i ∈ R × which con-verts A into A + BF P = − . (3.3)Then we observe that E = diag ( I , L , K ⊤ , K ⊤ , N , I ) , A + BF P = diag ( N ⊤ , K , L ⊤ , L ⊤ , I , A c ) , B = diag ( e [ ] , e [ ] , e [ ] , × ) , where A c = [ − ] , and therefore [ E , A , B ] I , I , I , F P ∼ = P [ E , A + BF P , B ] , where [ E , A + BF P , B ] is in the form (3.2) with α = ( ) , β = ( ) , γ = ( , ) , n δ = κ = ( ) and n c = ⋄ Quasi P-feedback form (QPFF)
We will now weaken P-feedback forms to quasi
P-feedback forms. Roughly speaking, the latter is “lesscanonical” than the former, it contains less zeros and ones. However – and this is the important mes-sage – the relevant system theoretic properties can be read off the quasi P-feedback form and, moreover,the form provides a geometric insight (as it is obtained via the augmented Wong sequences) and can beeasily computed.
Definition 3.8.
The system [ E , A , B ] ∈ Σ ℓ, n , m is said to be in quasi P-feedback form (QPFF) , if [ E , A , B ] = E E E E E E , A A A A A A , B B B , (3.4)where(i) [ E , A , B ] ∈ Σ ℓ , n , m with ℓ < n + m and rk E = rk C [ λ E − A , B ] = ℓ for all λ ∈ C ,(ii) E , A ∈ R ℓ × n with ℓ = n and E ∈ GL n ( R ) ,(iii) [ E , A , B ] ∈ Σ ℓ , n , m satisfies rk C [ λ E − A , B ] = n + m for all λ ∈ C and the remaining matrices have suitable sizes.Furthermore, a QPFF (3.4) with zero off-diagonal blocks (i.e. E = A = E = A = B = E = A =
0) is called decoupled QPFF . ⋄ Remark 3.9.
The three conditions in Definition 3.8 describe control theoretic properties as follows (seethe survey [3] for the different notions of controllability):(i) The system [ E , A , B ] in the QPFF (3.4) is completely controllable and the input is not con-strained. of 27 T. B
ERGER ET AL . (ii) The ODE system [ E , A , ] is uncontrollable.(iii) The system [ E , A , B ] has only the trivial solution; in particular, the system is trivially behav-iorally controllable but the input corresponding to B is maximally constrained (because it has tobe zero). ⋄ The following result will be helpful in due course.
Proposition 3.10.
Any system [ E , A , B ] ∈ Σ ℓ, n , m in QPFF (3.8) is P-feedback equivalent to a system in decoupled QPFF with identical diagonal blocks.
Proof.
The proof is structurally similar to the proof of [5, Thm. 2.6], however, due to the presence ofthe input some technical adjustments are necessary. Two technical results required for the proof arecollected in the Appendix A.To show that any QPFF can be decoupled, we prove existence of matrices G S , H S , F S and G xT , G uT , H xT , H uT , F xT of appropriate sizes such that sE − A sE − A sE − A sE − A sE − A sE − A I G xT H xT I F xT I + B B B (cid:20) G uT H uT (cid:21) = I − G S − H S I − F S I sE − A sE − A
00 0 sE − A and B B B = I − G S − H S I − F S I B
00 00 B . This holds if, and only if, the following matrix equations have solutions:0 = A + [ A , − B ] h G xT G uT i + G S A , = E + [ E , ] h G xT G uT i + G S E ; (3.5a)0 = [ A , ] + A [ F xT , ] + F S [ A , − B ] , = [ E , ] + E [ F xT , ] + F S [ E , ] ; (3.5b)0 = A F xT + A + [ A , − B ] h H xT H uT i + H S A , = E F xT + E + [ E , ] h H xT H uT i + H S E , = − B − H S B . (3.5c)In the following we show that each of the sets of equations above admits a solution, where we useLemmas A.1 and A.3. We show that (3.5a) has a solution.
Clearly, (3.5a) has the form (A.1) with A = [ A , − B ] , D = A , C = [ E , ] , B = E . Since E UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS λ ∈ R such that λ E − A is also invertible and hence the assumption ofLemma A.1 is satisfied. Therefore, it suffices to show solvability of the associated generalized Sylvesterequation (A.2). By assumption, rank [ λ E − A , B ] = ℓ for all λ ∈ C ∪ { ∞ } , i.e. no rank drop occursat all, so all assumptions of Lemma A.3 are satisfied and existence of a solution h G xT G uT i and G S of (3.5a)is shown. We show that (3.5b) has a solution.
We consider first a relaxed version of (3.5b) by replacing [ F xT , ] by [ F xT , F uT ] in both equations of (3.5b).With A = A , D = [ A , − B ] , C = E , B = [ E , ] we again see that all assumptions of Lemmas A.1and Lemma A.3 are satisfied ensuring solvability of the relaxed version of (3.5b). From the secondrelaxed equation of (3.5b) we see that, in particular,0 = + E F uT + , which, due to the invertibility of E , immediately yields that F uT =
0. This shows solvability of theoriginal equations (3.5b).
We show that (3.5c) has a solution.
Since by assumption [ A , − B ] has full column rank, there exists an invertible row operation R ∈ R ℓ × ℓ such that [ A , − B ] = R (cid:20) A x I m (cid:21) , A x ∈ R ( ℓ − m ) × n . Define [ H xS , H uS ] : = H S R , then the last equation of (3.5c) simplifies to0 = − B + H uS . This is solvable with H uS = B . Let h E x E u i : = R − E with E x ∈ R ( ℓ − m ) × n and E u ∈ R m × n , thenthe first two equations of (3.5c) have the form0 = e A + [ A , − B ] h H xT H uT i + H xS A x , = e E + [ E , ] h H xT H uT i + H xS E x , (3.6)where e A = A F xT + A and e E = E F xT + E + H uS E u . Since A x has full column rank, the pencil sE x − A x has full polynomial column rank and by assumption rank ( λ [ E , ] − [ A , − B ]) = ℓ for all λ ∈ C ∪ { ∞ } . So Lemma A.3 together with Remark A.2 is applicable to (3.6) and guarantees existenceof H xT , H uT , H xS satisfying (3.6). Now set H S = R − [ H xS , H uS ] , and the proof is complete.A notable observation from the proof of Proposition 3.10 is that no input transformation is needed(i.e., V = I in (3.1)) to arrive at a decoupled QPFF.We stress some important properties of the augmented Wong limits for systems in decoupled QPFF. Lemma 3.11.
For any [ E , A , B ] ∈ Σ ℓ, n , m in decoupled QPFF, the augmented Wong sequences satisfy: V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] = R n × { } n + n , V ∗ [ E , A , B ] = R n + n × { } n and E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) = R ℓ × { } ℓ + ℓ , E V ∗ [ E , A , B ] = R ℓ + ℓ × { } ℓ . T. B
ERGER ET AL . Furthermore, we have that dim (cid:16) im B ∩ ( { } ℓ + ℓ × R ℓ ) (cid:17) = m . (3.7)In particular (in view of Lemma 3.3 and Proposition 3.10), two QPFFs which are P-feedback equivalenthave the same block sizes in E , A and B . Proof. Step 1 : We show V ∗ [ E , A , B ] = R n + n × { } n .Since [ E , A , B ] is in decoupled QPFF, V ∗ [ E , A , B ] = V ∗ [ E , A , B ] × V ∗ [ E , A , ] × V ∗ [ E , A , B ] . Since both E and E have full row rank, they are surjective. It follows inductively that V i [ E , A , B ] = R n and V i [ E , A , ] = R n for all i >
0. It remains to show that V ∗ [ E , A , B ] = { } n . Since λ [ E , ] − [ A , B ] has full column rank for all λ ∈ C , its quasi-Kronecker form [6] has only a nilpotent and a overdeter-mined part, in particular, V ∗ [[ E , ] , [ A , B ] , ] = { } n + m . Now the claim follows from (2.5). Step 2 : We show R n × { } n × { } n ⊆ W ∗ [ E , A , B ] ⊆ R n × { } n × R n .Again, since [ E , A , B ] is in decoupled QPFF we have W ∗ [ E , A , B ] = W ∗ [ E , A , B ] × W ∗ [ E , A , ] × W ∗ [ E , A , B ] . Thus, it suffices to show that W ∗ [ E , A , B ] = R n and W ∗ [ E , A , ] = { } . The latter is asimple consequence of invertibility of E and that W [ E , A , ] = { } ; for the former we observe thatthe augmented matrix pencil s [ E , ] − [ A , B ] is an underdetermined DAE in the sense of [5] andhence W ∗ [[ E , ] , [ A , B ] , ] = R n + m . Invoking again (2.5) we can conclude the claim. Step 3 : We conclude the claimed properties of the augmented Wong-sequences in the statement of thelemma. The first two equations follow from Steps 1 and 2. The third and fourth equation follow fromthe block structure of E and full row rank of E and E . Step 4 : We show (3.7). It is easy to see thatim B ∩ ( { } ℓ + ℓ × R ℓ ) = { } ℓ + ℓ × im B , and the full column rank of B yields dim im B = m . This proves (3.7). (cid:3) In the following we derive the QPFF by choosing basis matrices according to the augmented Wongsequences. This has the advantage that the transformation provides some geometric insight. We are nowin the position to show that any system [ E , A , B ] is equivalent to a system in QPFF. Theorem 3.12 (Quasi P-feedback form) . Consider [ E , A , B ] ∈ Σ ℓ, n , m with corresponding augmentedWong limits V ∗ [ E , A , B ] and W ∗ [ E , A , B ] . Choose full column rank matrices U T ∈ R n × n , R T ∈ R n × n , O T ∈ R n × n , U S ∈ R ℓ × ℓ , R S ∈ R ℓ × ℓ , O S ∈ R ℓ × ℓ such thatim U T = V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] , im R T ⊕ im U T = V ∗ [ E , A , B ] , im O T ⊕ im R T ⊕ im U T = R n , im U S = E V ∗ [ E , A , B ] ∩ ( A W ∗ [ E , A , B ] + im B ) , im R S ⊕ im U S = E V ∗ [ E , A , B ] , im O S ⊕ im R S ⊕ im U S = R ℓ UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS
11 of 27and, additionally, im B ⊆ im [ U S , O S ] . (3.8)Let T : = [ U T , R T , O T ] , S : = [ U S , R S , O S ] − and further choose (not necessarily full rank) matrices F ∈ R m × n , F ∈ R m × n such that [ , I ℓ + ℓ ] S ( AU T + BF ) = [ , , I ℓ ] S ( AR T + BF ) = , (3.9)and let F P = [ F , F , ] . Finally, let V = [ V , V ] , where V , V are full column rank matrices such thatim V = ker [ , , I ℓ ] SB and im V ⊕ im V = R m . Then [ SET , S ( AT + BF P ) , SBV ] is in QPFF (3.4). Proof. Step 1 : We show that the block structure of the QPFF (3.4) is achieved. The subspace inclu-sions (2.2) imply thatim EU T ⊆ im U S , im AU T ⊆ im U S + im B , im ER T ⊆ im [ U S , R S ] , im AR T ⊆ im [ U S , R S ] + im B , im EO T ⊆ im [ U S , R S , O S ] = R n , im AO T ⊆ im [ U S , R S , O S ] = R ℓ , hence there exists matrices E , E , E , E , E , E , A , A , A , A , A , A , F , F such that EU T = U S E , AU T = U S A − BF , ER T = U S E + R S E , AR T = U S A + R S A − BF , EO T = U S E + R S E + O S E , AO T = U S A + R S A + O S A . (3.10)Note that, in particular, F and F satisfy the equations (3.9), which hence have solutions. Con-versely, for any solution F and F of (3.9), the matrices A : = [ I ℓ , , ] S ( AU T + BF ) and h A A i : =[ I ℓ + ℓ , ] S ( AR T + BF ) are suitable choices for satisfying (3.10).Observe that (3.10) implies that SET and S ( AT + BF ) have the desired block structure ofa QPFF (3.4). Furthermore, sinceim U S ∩ im B = E V ∗ [ E , A , B ] ∩ im B = im [ U S , R S ] ∩ im B , one may always choose some O S such that (3.8) holds. Therefore, we may choose matrices e B , e B suchthat B = U S e B + O S e B holds, or, equivalently, SB = e B e B . Finally, by construction we have [ , , I ℓ ] SBV =
0, hence
SBV = B B B , T. B
ERGER ET AL . which concludes Step 1. For later use we note that 0 = B v = [ , , I ℓ ] SBV v implies that V v ∈ ker [ , , I ℓ ] SB ∩ im V = im V ∩ im V = { } , thus B has full column rank. Step 2 : We show that [ E , A , B ] satisfies Definition 3.8 (i). Step 2a : We show that rk E = ℓ . Observe that U S E = EU T yieldsim U S E = im EU T = E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) Prop . . = E V ∗ [ E , A , B ] ∩ ( A W ∗ [ E , A , B ] + im B ) = im U S . As a consequence, the full column rank of U S gives that E has full row rank. Step 2b : We show that V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] = R n .Set [ b E , b A , b B ] : = [ SET , S ( AT + BF P ) , SBV ] . Invoking Lemma 3.3 we can conclude that V ∗ [ b E , b A , b B ] ∩ W ∗ [ b E , b A , b B ] = T − ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) = T − ( im U T ) = R n × { } n + n . From (2.4) we may further infer that b A ( V ∗ [ b E , b A , b B ] ∩ W ∗ [ b E , b A , b B ] ) ⊆ b E ( V ∗ [ b E , b A , b B ] ∩ W ∗ [ b E , b A , b B ] ) + im b B . Hence for all x ∈ R n , there exist y , z ∈ R n and U T , v ∈ R m , U S , v ∈ R m such that b A x = b E y + b B (cid:18) U T U S (cid:19) and b A y = b E z + b B (cid:18) v v (cid:19) . From the block diagonal structure, we see that the former relation is equivalent to the two equations A x = E y + B U T + B U S , = B U S . Since ker B = { } , we find that U S =
0, thus we may conclude that x ∈ A − ( E { y } + im B ) ⊆ V [ E , A , B ] . With the same reasoning, one may show that y ∈ A − ( E { z } + im B ) ⊆ V [ E , A , B ] and hence x ∈ V [ E , A , B ] . Continuing this reasoning, it finally follows that x ∈ V k [ E , A , B ] forall k ∈ N , and, since x is arbitrary, we have shown that V ∗ [ E , A , B ] = R n .Now, let j ∗ , r ∗ ∈ N be such that W ∗ [ b E , b A , b B ] = W j ∗ [ b E , b A , b B ] and W ∗ [ E , A , B ] = W r ∗ [ E , A , B ] and set q ∗ : = max { j ∗ , r ∗ } . Again, take arbitrary x = ( x ⊤ , , ) ⊤ ∈ V ∗ [ b E , b A , b B ] ∩ W ∗ [ b E , b A , b B ] , then x ∈ W q ∗ [ b E , b A , b B ] and Defini-tion 2.1 give that there exist y k ∈ W k [ b E , b A , b B ] and u k ∈ R m , k = , . . . , q ∗ −
1, such that b Ex = b Ay q ∗ − + b Bu q ∗ − and b Ey k = b Ay k − + b Bu k − for all k = , . . . , q ∗ −
1. Now we find that b Ay q ∗ − = b Ex − b Bu q ∗ − ∈ b E (cid:0) V ∗ [ b E , b A , b B ] ∩ W ∗ [ b E , b A , b B ] (cid:1) + im b B (2.4) = b A (cid:0) V ∗ [ b E , b A , b B ] ∩ W ∗ [ b E , b A , b B ] (cid:1) + im b B , UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS
13 of 27whence y q ∗ − ∈ b A − (cid:16) b A (cid:0) V ∗ [ b E , b A , b B ] ∩ W ∗ [ b E , b A , b B ] (cid:1) + im b B (cid:17) = V ∗ [ b E , b A , b B ] ∩ W ∗ [ b E , b A , b B ] + b A − ( im b B ) ⊆ V ∗ [ b E , b A , b B ] . Therefore, y q ∗ − ∈ V ∗ [ b E , b A , b B ] ∩ W q ∗ − [ b E , b A , b B ] . With the same reasoning, we may now conclude inductively y k ∈ V ∗ [ b E , b A , b B ] ∩ W k [ b E , b A , b B ] , k = q ∗ − , . . . , . This implies that y k = ( y ⊤ k , , , ) ⊤ and u k = ( u ⊤ k , , u ⊤ k , ) ⊤ for some y k , ∈ R n , u k , ∈ R m , u k , ∈ R m , k = , . . . , q ∗ −
1, and hence E y , = B u , + B u , , E y , = A y , + B u , + B u , , . . . , E y q ∗ − , = A y q ∗ − , + B u q ∗ − , + B u q ∗ − , , E x = A y q ∗ − , + B u q ∗ − , + B u q ∗ − , , and 0 = B u , , = B u , , . . . , = B u q ∗ − , , = B u q ∗ − , . Therefore, we obtain u k , = k = , . . . , q ∗ − x ∈ W ∗ [ E , A , B ] . Since x was arbitrary we have proved that V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] = R n . Step 2c : We show that V ∗ [[ E , ] , [ A , B ] , ] ∩ W ∗ [[ E , ] , [ A , B ] , ] = R n + m .It follows from Proposition 2.2 and Step 2b that [ I n , ] V ∗ [[ E , ] , [ A , B ] , ] ∩ [ I n , ] W ∗ [[ E , ] , [ A , B ] , ] = R n . As shown in the proof of Proposition 2.2 we have that W ∗ [[ E , ] , [ A , B ] , ] = [ I n , ] W ∗ [[ E , ] , [ A , B ] , ] × R m = R n × R m . Furthermore, V ∗ [[ E , ] , [ A , B ] , ] = [ A , B ] − (cid:0) [ E , ] V ∗ [[ E , ] , [ A , B ] , ] (cid:1) = [ A , B ] − (cid:0) im E (cid:1) Step 2a = [ A , B ] − (cid:0) R ℓ (cid:1) = R n + m , which proves the claim. Step 2d: Conclusion of Step 2.
The result of Step 2c implies that the augmented matrix pencil s [ E , ] − [ A , B ] is in quasi-Kronecker form, see [5], and consists only of an underdeterminedblock. In particular, ℓ < n + m and rk ( λ [ E , ] − [ A , B ]) = ℓ for all λ ∈ C . Step 3 : We show that E is square and invertible. Step 3a : We show that im U S ⊕ im ER T = E V ∗ [ E , A , B ] .Since im R T ⊆ V ∗ [ E , A , B ] and im U S ⊆ E V ∗ [ E , A , B ] it follows that im U S + im ER T ⊆ E V ∗ [ E , A , B ] . Furthermore, E V ∗ [ E , A , B ] = E (cid:0) ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) ⊕ im R T (cid:1) ⊆ E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) + im ER T (2.3) = im U S + im ER T , T. B
ERGER ET AL . hence it remains to be shown that the intersection of im U S and im ER T is trivial. Towards this goal,let x ∈ im U S ∩ im ER T , then there exists y ∈ im R T with x = Ey and, in view of (2.3), there exists z ∈ V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] such that x = Ez . Hence z − y ∈ ker E ⊆ W ∗ [ E , A , B ] . From z ∈ W ∗ [ E , A , B ] it then followsthat y ∈ W ∗ [ E , A , B ] and, therefore, y ∈ W ∗ [ E , A , B ] ∩ im R T = { } . This implies x = Step 3b : We show ℓ = n .From Step 3a we have ℓ = rk ER T n and hence it suffices to show that ER T has full column rank.Let v ∈ R n be such that ER T v =
0, then R T v ∈ im R T ∩ ker E ⊆ im R T ∩ W ∗ [ E , A , B ] = { } and due to fullcolumn rank of R T the claim follows. Step 3c : We show full column rank of E .Let v ∈ R n be such that E v =
0. Then by (3.10) we have ER T v = U S E v and hence, invoking Step3a, ER T v ∈ im ER T ∩ im U S = { } . As already shown in Step 3b, ER T v = v = E is shown. Step 4 : We show that [ E , A , B ] satisfies Definition 3.8 (iii).Assume there exist λ ∈ C , x ∈ C n , u ∈ C m such that ( λ E − A ) x + B u =
0. Then we have,according to (3.10), that ( λ E − A ) O T x = U S ( λ E − A ) x + R S ( λ E − A ) x − O S B u . Writing the complex variables in terms of their real and imaginary parts, i.e., λ = µ + i ν , x = x + i b x and u = u + i b u , we can conclude that ( µ E − A ) O T x − ν EO T b x + O S B u ∈ im [ U S , R S ] , ( µ E − A ) O T b x + ν EO T x + O S B b u ∈ im [ U S , R S ] . Furthermore, im ( O S B + U S B ) = im [ U S , R S , O S ] (cid:20) B B (cid:21) = im BV ⊆ im B , and hence ( µ E − A ) O T x − ν EO T b x ∈ im [ U S , R S ] + im B = E V ∗ [ E , A , B ] + im B , ( µ E − A ) O T b x + ν EO T x ∈ im [ U S , R S ] + im B = E V ∗ [ E , A , B ] + im B . Assume now inductively that O T x , O T b x ∈ V k [ E , A , B ] (which is trivially satisfied for k = AO T x = µ EOx − ν EO b x + Ev + Bu , AO T b x = µ EO b x + ν EOx + E b v + B b u , for some v , b v ∈ V ∗ [ E , A , B ] ⊆ V k [ E , A , B ] and u , b u ∈ R m . Consequently, O T x ∈ A − ( E V k [ E , A , B ] + im B ) = V k + [ E , A , B ] and O T b x ∈ A − ( E V k [ E , A , B ] + im B ) = V k + [ E , A , B ] . Altogether, we can conclude that O T x , O T b x ∈ im O T ∩ V ∗ [ E , A , B ] = { } , which in view of full columnrank of O T implies that x =
0. Therefore, also B u = B , impliesthat u =
0. This completes the proof. (cid:3)
UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS
15 of 27We stress that condition (3.8) in Theorem 3.12 cannot be omitted in general, as the following exam-ple shows.
Example 3.13.
Consider the system [ E , A , B ] = (cid:20)(cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21)(cid:21) and calculate that V ∗ [ E , A , B ] = R and W ∗ [ E , A , B ] = { } . Then we may choose T = R T = [ ] and S = [ R S , O S ] − = (cid:20) α (cid:21) − = (cid:20) − α
11 0 (cid:21) , α ∈ R . Furthermore, we may choose V = [ ] and F P = F = [ α ] so that (3.9) is satisfied. Then [ SET , S ( AT + BF P ) , SBV ] = (cid:20)(cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) − α (cid:21)(cid:21) , which is not in QPFF (3.4), if α =
0. However, we obtain α = ⋄ Finally, we stress that the QPFF (3.4) is unique in the following sense.
Proposition 3.14.
Consider the system [ E , A , B ] ∈ Σ ℓ, n , m and assume there are S , S ∈ GL ℓ ( R ) , T , T ∈ GL n ( R ) , V , V ∈ GL m ( R ) , F P , F P ∈ R m × n such that for i = , [ E , A , B ] S i , T i , V i , F iP ∼ = P E i E i E i E i E i E i , A i A i A i A i A i A i , B i B i B i . Then the corresponding diagonal blocks (which have the same corresponding sizes as established al-ready in Lemma 3.11) are P-feedback equivalent, i.e. [ E kk , A kk , B kk ] ∼ = P [ E kk , A kk , B kk ] for k = , , B i ∈ R ℓ × ). Proof.
Consider any system [ E , A , B ] and the following P-feedback equivalent systems [ E QPFF , A QPFF , B QPFF ] S , T , V , F ∼ = P [ E , A , B ] and [ E WQPFF , A WQPFF , B WQPFF ] S W , T W , V W , F W ∼ = P [ E , A , B ] where [ E QPFF , A QPFF , B QPFF ] is any decoupled QPFF (not necessarily obtained via the Wong-sequenceapproach, but probably utilizing Proposition 3.10) which is P-feedback equivalent to [ E , A , B ] , andthe QPFF [ E WQPFF , A WQPFF , B WQPFF ] is obtained from [ E , A , B ] via the Wong-sequence approach (The-orem 3.12). We will now show that the diagonal blocks of [ E QPFF , A QPFF , B QPFF ] are P-feedbackequivalent to the corresponding diagonal blocks of [ E WQPFF , A WQPFF , B WQPFF ] , from which the claim ofProposition 3.14 follows.First observe that [ E QPFF , A QPFF , B QPFF ] S , T , V , F ∼ = P [ E WQPFF , A WQPFF , B WQPFF ] with S = S ( S W ) − , T = ( T W ) − T , V = ( V W ) − V , F = ( V W ) − (cid:0) F − F W T (cid:1) . Denote by V ∗ and W ∗ the Wong sequences of the original system [ E , A , B ] . Then, by construction (cf.Theorem 4.12), im U WT = V ∗ ∩ W ∗ , im [ U WT , R WT ] = V ∗ , im U WS = E ( V ∗ ∩ W ∗ ) , im [ U WS , R WS ] = E V ∗ . T. B
ERGER ET AL . Lemma 3.11 in conjunction with Lemma 3.3 yields that the decoupled QPFF [ E QPFF , A QPFF , B QPFF ] satisfies T − ( V ∗ ∩ W ∗ ) = im h I i , T − V ∗ = im h I I i , E QPFF T − ( V ∗ ∩ W ∗ ) = im h I i , E QPFF T − V ∗ = im h I I i . This gives, for some invertible M TU , M TR , M TO , M SU , M SR and M SO , T = [ U WT , R WT , O WT ] " M TU ∗ ∗ M TR ∗ M TO , S − = [ U WS , R WS , O WS ] " ( M SU ) − ∗ ∗ ( M SR ) − ∗ ( M SO ) − . Therefore, S = " M SU ∗ ∗ M SR ∗ M SO , T = " M TU ∗ ∗ M TR ∗ M TO and it follows from E QPFF = SE WQPFF T that E = M SU E W M TU , E = M SR E W M TR , E = M SO E W M TO , where E ii , E Wii , i = , ,
3, are the corresponding diagonal blocks of the block diagonal matrices E QPFF and E WQPFF . This shows the desired P-feedback equivalence for the entries of the E -matrix.Writing V = h V V V V i and multiplying the equation B QPFF = SB WQPFF V from the left by [ , , I ] andfrom the right by (cid:2) I (cid:3) gives M SO B W V =
0. Invertibility of M SO and full column rank of B W yields V =
0, i.e. V = h V V V i with V and V invertible.Furthermore, from B QPFF = SB WQPFF V we see that B = M SU B W V and B = M SO B W V as desired.Writing F = h F U , F R , F O , F U , F R , F O , i , we have SB WQPFF F = " M SU ∗ ∗ M SR ∗ M SO B W F U , B W F R , B W F O , B W F U , B W F R , B W F O , . (cid:21) . We will show that F U , and F R , are both zero, since this yields the desired form A QPFF = S ( A WQPFF T + B WQPFF F ) = " M SU ( A W M TU + B W F U , ) ∗ ∗ M SU A W M TU ∗ M SO ( A W M TO + B W F O , ) . To show that F U , =
0, we first observe that in view of (3.9) we have [ , , I ] S W ( AT W + BF W ) h I i = . (3.11)Let F W = (cid:20) F WU , F WR , F WU , F WR , (cid:21) . Then due to the special block structure of S W ( AT W + BF W ) it followsfrom (3.11) that F W =
0. Furthermore, since S W AT W = S − S ( AT + BF ) T − − S − SBFT − it follows UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS
17 of 27from (3.11) that 0 = [ , , ( M SO ) − ]( A QPFF − SBF + SBF W T ) (cid:20) ( M TU ) − (cid:21) = [ , , ( M SO ) − ] SBVV − ( V W ) − (cid:18) F W h I i − F (cid:20) ( M TU ) − (cid:21)(cid:19) , where we have used that [ , , ( M SO ) − ] A QPFF (cid:20) ( M TU ) − (cid:21) = [ , , ( M SO ) − ] (cid:20) A ( M TU ) − (cid:21) = . Let F = h F U , F R , F O , F U , F R , F O , i . Due to the special block structures of SBV , V and V W , we may further concludethat 0 = F WU , − F U , ( M TU ) − = − F U , ( M TU ) − , i.e. F U , =
0. Therefore, F U , = [ , I ] F h I i = [ , I ] (cid:20) ( V W ) − ∗ ( V W ) − (cid:21) (cid:18)h F U , F R , F O , F R , F O , i − (cid:20) F WU , ∗ ∗ F WR , ∗ (cid:21)(cid:19) h I i = . It also follows from (3.9) that [ , , I ] S W ( AT W + BF W ) h I i = . (3.12)The block structure of S W ( AT W + BF W ) gives F WR , =
0. Similar as above, we can conclude from (3.12)that 0 = [ , , ( M SO ) − ] SBVV − ( V W ) − (cid:18) F W h I i − F (cid:20) ∗ ( M TR ) − (cid:21)(cid:19) and, taking the block structure of SBV , V , V W , and F WU , = F U , = F R , = F R , = [ , I ] (cid:20) ( V W ) − ∗ ( V W ) − (cid:21) (cid:16)h F U , F R , F O , F O , i − h F WU , ∗ ∗ ∗ i(cid:17) h I i = . This completes the proof.
4. PD-feedback forms
In this section, we investigate PD-feedback which allows for a simpler “(quasi) canonical” form com-pared to the (quasi) P-feedback form since the set of allowed transformations is larger. However, unlikethe latter, the quasi PD-feedback form can be derived directly from the Kalman controllability decom-position presented in [7] by decomposing the first block row.4.1
PD-feedback equivalence
The following concept of PD-feedback equivalence enlarges the transformation class associated withP-feedback equivalence as in Definition 3.1.8 of 27
T. B
ERGER ET AL . Definition 4.1 (PD-feedback equivalence) . Two systems [ E , A , B ] , [ E , A , B ] ∈ Σ ℓ, n , m are called PD-feedback equivalent , if ∃ S ∈ GL ℓ ( R ) , T ∈ GL n ( R ) , V ∈ GL m ( R ) , F P , F D ∈ R m × n : (cid:2) sE − A , B (cid:3) = S (cid:2) sE − A , B (cid:3) (cid:20) T sF D − F P V (cid:21) ; (4.1)we write [ E , A , B ] ∼ = PD [ E , A , B ] or, if necessary, [ E , A , B ] S , T , V , F P , F D ∼ = PD [ E , A , B ] . ⋄ Remark 4.2.
Similarly as for P-feedback equivalence, it can easily be shown that PD-feedback equiva-lence is reflexive, symmetric and transitive, so it is indeed an equivalence relation. ⋄ For later use we show how the augmented Wong sequences change under PD-feedback.
Lemma 4.3 (Augmented Wong sequences under PD-feedback) . If the systems [ E , A , B ] , [ E , A , B ] ∈ Σ ℓ, n , m are PD-feedback equivalent [ E , A , B ] S , T , V , F P , F D ∼ = PD [ E , A , B ] , then ∀ i ∈ N : V i [ E , A , B ] = T − V i [ E , A , B ] and W i [ E , A , B ] = T − W i [ E , A , B ] . Proof.
We prove the statement by induction. It is clear that V [ E , A , B ] = T − V [ E , A , B ] . Assume that V i [ E , A , B ] = T − V i [ E , A , B ] for some i >
0. Then (4.1) yields V i + [ E , A , B ] = A − ( E V i [ E , A , B ] + im B )= ( x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∃ y ∈ V i [ E , A , B ] ∃ u ∈ R m : ( SA T + SB F P ) x = ( SE T + SB F D ) y + SB Vu ) = n x ∈ R n (cid:12)(cid:12)(cid:12) ∃ z ∈ V i [ E , A , B ] ∃ v ∈ R m : A T x = E z + B v o = T − (cid:16) A − ( E V i [ E , A , B ] + im B ) (cid:17) = T − V i + [ E , A , B ] . The proof of the statements about W i [ E , A , B ] and W i [ E , A , B ] is similar and omitted.4.2 PD-feedback form (PDFF)
Definition 4.4 (PD-feedback form) . The system [ E , A , B ] ∈ Σ ℓ, n , m is said to be in PD-feedback form (PDFF), if [ E , A , B ] = L α L ⊤ β N γ
00 0 0 I n c , K α K ⊤ β I | γ |
00 0 0 A c , I r , (4.2)where r = rk B , α ∈ N n α , β ∈ N n β , γ ∈ N n γ denote multi-indices, and A c ∈ R n c × n c . ⋄ UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS
19 of 27Two PD-feedback equivalent systems have the same PDFF up to permutation of the entries of α , β , γ and similarity of A c . Proposition 4.5 (Uniqueness of indices for PDFF) . Let [ E i , A i , B i ] ∈ Σ ℓ, n , m , i = ,
2, be in PDFF (4.2)with corresponding r i = rk B i , multi-indices α i ∈ N n α i , β i ∈ N n β i , γ i ∈ N n γ i , and A c , i ∈ R n c , i × n c , i . If [ E , A , B ] ∼ = PD [ E , A , B ] , then α = P α α , β = P β β , γ = P γ γ , and r = r , n c , = n c , , A c , = H − A c , H , for permutation matrices P α , P β , P γ of appropriate sizes and H ∈ GL n c , ( R ) . Proof.
It follows from (4.1) that B = SB V and hence r = r . Furthermore, sE − A = S ( sE − A ) T + SB ( sF D − F P ) , which gives sL α − K α sL ⊤ β − K ⊤ β sN γ − I | γ |
00 0 0 sI nc , − A c , = S sL α − K α sL ⊤ β − K ⊤ β sN γ − I | γ |
00 0 0 sI nc , − A c , sF − G sF − G sF − G sF − G T for some matrices F i and G i of appropriate sizes. Set r : = r = r and write S = (cid:20) S S S S (cid:21) , S ∈ R r × r , and S , S , S of appropriate size. Then B = SB V yields (cid:20) I r (cid:21) = (cid:20) S S S S (cid:21) (cid:20) I r (cid:21) V = (cid:20) S S (cid:21) V , and hence S ∈ GL r ( R ) and S =
0. Then again S ∈ GL ℓ − r ( R ) and we have sL α − K α sL ⊤ β − K ⊤ β sN γ − I | γ |
00 0 0 sI nc , − A c , = S sL α − K α sL ⊤ β − K ⊤ β sN γ − I | γ |
00 0 0 sI nc , − A c , T , which in view of [6, Rem. 2.8] implies the assertion. (cid:3) We are now in the position to show that any [ E , A , B ] ∈ Σ ℓ, n , m is PD-feedback equivalent to a systemin PDFF. This result has already been observed in the seminal work [16] co-authored by Nicos Karca-nias; it simply consists of a left transformation S together with an input space transformation V whichputs the matrix B into the form SBV = (cid:2) I r (cid:3) and an additional transformation which puts the pencil sNE − NA , where N = [ I l − r , ] W , into Kronecker canonical form. Theorem 4.6 (PDFF [16]) . For any system [ E , A , B ] ∈ Σ ℓ, n , m there exist S ∈ GL ℓ ( R ) , T ∈ GL n ( R ) , V ∈ GL m ( R ) , F P , F D ∈ R m × n such that [ SET + SBF D , SAT + SBF P , SBV ] is in PDFF (4.2).0 of 27 T. B
ERGER ET AL . Example 4.7 (PDFF) . We revisit Example 3.7 to illustrate Theorem 4.6 and consider again the system [ E , A , B ] ∈ Σ , , . As already performed in Example 4.7 we may convert the matrix A into A + BF P in (3.3) by the proportional feedback matrix F P . Then we may apply the derivative feedback given by F D = h − − i ∈ R × which converts E into E + BF D = E = . (4.3)We may observe that E + BF D = diag ( L , L , L ⊤ , N , L ⊤ , L ⊤ , N , I ) , A + BF P = diag ( K , K , K ⊤ , I , K ⊤ , K ⊤ , I , A c ) , where A c = [ − ] , and therefore [ E , A , B ] I , I , I , F P , F D ∼ = PD [ E + BF D , A + BF P , B ] , where [ E + BF D , A + BF P , B ] is, after a suitable permutation of blocks, in the form (4.2) with r = α = ( , ) , n β = γ = ( , ) and n c = ⋄ Remark 4.8 (PDFF from PFF) . We may directly derive the PDFF (4.2) from the PFF (3.2). To this end,observe that the system [ I α i , N ⊤ α i , e [ α i ] α i ] can be written as dd t K α i x c [ i ] ( t ) = L α i x c [ i ] ( t ) , dd t x c [ i ] , α i ( t ) = u c [ i ] ( t ) , and hence it is PD-feedback equivalent to the system (cid:20)(cid:20) K α i (cid:21) , (cid:20) L α i (cid:21) , (cid:20) (cid:21)(cid:21) . On the other hand, the system [ L ⊤ γ i , K ⊤ γ i , e [ γ i ] γ i ] can be written as dd t N γ i − x ob [ i ] ( t ) = x ob [ i ] ( t ) , dd t x ob [ i ] , γ i − ( t ) = u ob [ i ] ( t ) , and hence it is PD-feedback equivalent to the system (cid:20)(cid:20) N γ i − (cid:21) , (cid:20) I γ i − (cid:21) , (cid:20) (cid:21)(cid:21) . It is now easy to see that we have obtained a system in the form (4.2). ⋄ Quasi PD-feedback form (QPDFF)
We will now weaken the PD-feedback form to a quasi
PD-feedback form, again exploiting the aug-mented Wong sequences as the crucial tool to achieve the necessary geometric insight.
UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS
21 of 27
Definition 4.9.
A system [ E , A , B ] ∈ Σ ℓ, n , m is said to be in quasi PD-feedback form (QPDFF) , if [ E , A , B ] = E E E E E E , A A A A A A , B , (4.4)where(i) E , A ∈ R ℓ × n with ℓ < n and rk E = rk C [ λ E − A ] = ℓ for all λ ∈ C ,(ii) E , A ∈ R ℓ × n with ℓ = n and E ∈ GL n ( R ) ,(iii) E , A ∈ R ℓ × n satisfy rk C ( λ E − A ) = n for all λ ∈ C ,(iv) ˆ B ∈ GL m ( R ) for m = rk B and the remaining matrices have suitable sizes. Furthermore, we call a QPDFF decoupled , if all off-diagonal blocks are zero. ⋄ The control theoretic interpretation of the QPDFF is that the system can be decomposed in fourparts. The first three parts contain only state variables, they form a homogeneous DAE. The first partconsists of an underdetermined DAE which is completely controllable, the second part is actually anuncontrollable ODE and the third part is a DAE which has only the trivial solution (and is thereforetrivially behaviorally controllable). The fourth part contains only the input u = ( u , u ) , where thefirst component of the input is completely free (but does not influence the state) and the second inputcomponent is maximally constrained ( = u ) . Remark 4.10.
Utilizing Sylvester equations in a very similar way as in Proposition 3.10 it can be shownthat any QPDFF is PD-feedback equivalent to a decoupled QPDFF with identical diagonal blocks. Sincethe input and state are comletely decoupled in the QPDFF this decoupling is actually much easier toachieve than the decoupling in the QPFF. ⋄ For later use we derive some properties of the augmented Wong limits for a system [ E , A , B ] whichis in QPDFF (4.4). Lemma 4.11.
Assume [ E , A , B ] ∈ Σ ℓ, n , m is in decoupled QPDFF (4.4), then V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] = R n × { } n + n , V ∗ [ E , A , B ] = R n + n × { } n (4.5)and im B = { } ℓ + ℓ + ℓ × R m , E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) + im B = R ℓ × { } ℓ + ℓ × R m , E V ∗ [ E , A , B ] + im B = R ℓ + ℓ × { } ℓ × R m . In particular (in view of Lemma 4.3), two QPDFFs which are PD-feedback equivalent have the sameblock sizes in E , A and B .The proof utilizes the observation that V ∗ [ E , A , B ] = V ∗ [ E , A , ] and W ∗ [ E , A , B ] = W ∗ [ E , A , ] and is very similar to the proof of Lemma 3.11. It is therefore omitted.We will now show that any system [ E , A , B ] is PD-feedback equivalent to a system in QPDFF andthat the transformation matrices can be obtained from the augmented Wong sequences; this providessome geometric insight in the decoupling.2 of 27 T. B
ERGER ET AL . Theorem 4.12 (Quasi PD-feedback form) . Consider [ E , A , B ] ∈ Σ ℓ, n , m with corresponding augmentedWong limits V ∗ [ E , A , B ] and W ∗ [ E , A , B ] . Choose full column rank matrices U T ∈ R n × n , R T ∈ R n × n , O T ∈ R n × n , Q S ∈ R ℓ × m , U S ∈ R ℓ × ℓ , R S ∈ R ℓ × ℓ , O S ∈ R ℓ × ℓ such thatim U T = V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] , im R T ⊕ im U T = V ∗ [ E , A , B ] , im O T ⊕ im R T ⊕ im U T = R n , im Q S = im B , im U S ⊕ im Q S = E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) + im B , im R S ⊕ im U S ⊕ im Q S = E V ∗ [ E , A , B ] + im B , im O S ⊕ im R S ⊕ im U S ⊕ im Q S = R ℓ . Let T : = [ U T , R T , O T ] , S : = [ U S , R S , O S , Q S ] − and choose (not necessarily full rank) matrices F P , F D ∈ R m × n such that 0 = [ , I m ] S ( ET + BF D ) , = [ , I m ] S ( AT + BF P ) , and choose full column rank matrices V ∈ R m × m , V ∈ R m × m such thatim V = ker B , im V ⊕ im V = R m . Then [ S ( ET + BF D ) , S ( AT + BF P ) , SBV ] is in QPDFF (4.4). Proof. Step 1 : We show that the block structure of the QPDFF (4.4) is achieved. The subspace inclu-sions (2.2) and the equalities (2.4) imply thatim EU T ⊆ im [ U S , Q S ] , im AU T ⊆ im [ U S , Q S ] , im ER T ⊆ im [ U S , R S , Q S ] , im AR T ⊆ im [ U S , R S , Q S ] , im EO T ⊆ im [ U S , R S , O S , Q S ] = R n , im AO T ⊆ im [ U S , R S , O S , Q S ] = R ℓ , hence there exists matrices E , E , E , E , E , E , A , A , A , A , A , A , F E , F E , F E , F A , F A , F A such that EU T = U S E + Q S F E , AU T = U S A + Q S F A , ER T = U S E + R S E + Q S F E , AR T = U S A + R S A + Q S F A , EO T = U S E + R S E + O S E + Q S F E , AO T = U S A + R S A + O S A + Q S F A . Furthermore, B = Q S ¯ B for some ¯ B ∈ R m × m and ˆ B : = ¯ BV ∈ R m × m since rk V = m − dim ker B = rk B = m . Then 0 = ˆ Bx = ¯ BV x yields V x ∈ ker ¯ B ∩ im V = ker B ∩ im V = { } , thus ˆ B ∈ GL m ( R ) .Therefore, SBV = S [ , BV ] = S [ U S , R S , O S , Q S ] BV = B UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS
23 of 27has the block structure as in (4.4). Set F E : = [ F E , F E , F E ] and F A : = [ F A , F A , F A ] , then im [ , I m ] SB = im [ , ˆ B ] = R m and hence the equations F E + [ , I m ] SBF D = , F A + [ , I m ] SBF P = F D , F P ∈ R m × n which satisfy [ , I m ] S ( ET + BF D ) = F E + [ , ˆ B ] F D = , [ , I m ] S ( AT + BF P ) = F A + [ , ˆ B ] F P = . This proves that S ( ET + BF D ) and S ( AT + BF P ) have the block structure as in (4.4). Step 2 : We show that E , A satisfy Definition 4.9 (i).Denote by V i [ E , A , ] , W i [ E , A , ] , V ∗ [ E , A , ] , W ∗ [ E , A , ] the Wong sequences and Wong limits corre-sponding to the matrix pencil sE − A . By choice of U T we have V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] = im U T = T ( R n × { } n + n ) . It follows from Lemma 4.3 that for [ ˜ E , ˜ A , ˜ B ] : = [ S ( ET + BF D ) , S ( AT + BF P ) , SBV ] we have V ∗ [ E , A , B ] = T V ∗ [ ˜ E , ˜ A , ˜ B ] , W ∗ [ E , A , B ] = T W ∗ [ ˜ E , ˜ A , ˜ B ] , hence V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] = R n × { } n + n . Now let x ∈ V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] . Then x = ( x ⊤ , , ) ⊤ for some x ∈ R n and˜ Ax ∈ ˜ A (cid:0) V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] (cid:1) + im ˜ B (2.4) = ˜ E (cid:0) V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] (cid:1) + im ˜ B , and thus there exist y = ( y ⊤ , , ) ⊤ ∈ V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] and u ∈ R m such that A x = E y and0 = ˆ Bu , thus u =
0. This implies x ∈ A − ( E { y } ) ⊆ V [ E , A , ] . A similar reasoning yields y ∈ V [ E , A , ] , and therefore x ∈ A − ( E { y } ) ⊆ A − ( E V [ E , A , ] ) ⊆ V [ E , A , ] . Again, we conclude similarly that y ∈ V [ E , A , ] and thus x ∈ V [ E , A , ] . Proceeding in this way weobtain x ∈ V ∗ [ E , A , ] .Now let j ∗ , r ∗ ∈ N be such that W ∗ [ ˜ E , ˜ A , ˜ B ] = W j ∗ [ ˜ E , ˜ A , ˜ B ] and W ∗ [ E , A , ] = W r ∗ [ E , A , ] and set q ∗ : = max { j ∗ , r ∗ } . Since x ∈ W ∗ [ ˜ E , ˜ A , ˜ B ] = W q ∗ [ ˜ E , ˜ A , ˜ B ] , it follows from Definition 2.1 that there exist y k ∈ W k [ ˜ E , ˜ A , ˜ B ] and u k ∈ R m , k = , . . . , q ∗ −
1, such that ˜ Ex = ˜ Ay q ∗ − + ˜ Bu q ∗ − and ˜ Ey k = ˜ Ay k − + ˜ Bu k − for all k = , . . . , q ∗ −
1. We find that˜ Ay q ∗ − = ˜ Ex − ˜ Bu q ∗ − ∈ ˜ A (cid:0) V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] (cid:1) + im ˜ B (2.4) = ˜ A (cid:0) V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] (cid:1) + im ˜ B , thus y q ∗ − ∈ ˜ A − (cid:16) ˜ A (cid:0) V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] (cid:1) + im ˜ B (cid:17) = V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] + ˜ A − ( im ˜ B ) ⊆ V ∗ [ ˜ E , ˜ A , ˜ B ] . T. B
ERGER ET AL . Therefore, y q ∗ − ∈ V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W q ∗ − [ ˜ E , ˜ A , ˜ B ] . Analogously, we may show that y k ∈ V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W k [ ˜ E , ˜ A , ˜ B ] , k = , . . . , q ∗ − . This implies that y k = ( y ⊤ k , , , ) ⊤ for some y k , ∈ R n , k = , . . . , q ∗ −
1, and hence, in particular, E y , = , E y , = A y , , . . . , E y q ∗ − , = A y q ∗ − , , E x = A y q ∗ − , . Therefore, we obtain x ∈ W ∗ [ E , A , ] . This proves (cid:0) V ∗ [ E , A , ] ∩ W ∗ [ E , A , ] (cid:1) × { } n + n = V ∗ [ ˜ E , ˜ A , ˜ B ] ∩ W ∗ [ ˜ E , ˜ A , ˜ B ] = R n × { } n + n . To conclude Step 2, it remains to show that rk E = ℓ , then the assertion follows from [5, Thm. 2.3].To this end, observe that U S E + Q S F E = EU T and Q S ¯ B = B imply thatim [ U S , Q S ] (cid:20) E F E ¯ B (cid:21) = im EU T + im B = E ( V ∗ [ E , A , B ] ∩ W ∗ [ E , A , B ] ) + im B = im [ U S , Q S ] . As a consequence, full column rank of [ U S , Q S ] gives that h E F E ¯ B i has full row rank, by which rk E = ℓ . Step 3 : The proof of ℓ = n , invertibility of E , and the property rk C λ E − A = n for all λ ∈ C issimilar to the proof of [7, Thm. 3.3] and omitted. (cid:3) Proposition 4.13 (Uniqueness of QPDFF) . Let [ E , A , B ] ∈ Σ ℓ, n , m and S , S ∈ GL ℓ ( R ) , T , T ∈ GL n ( R ) , V , V ∈ GL m ( R ) , F P , F P , F D , F D ∈ R m × n be such that, for i = , [ E , A , B ] S i , T i , V i , F iP , F iD ∼ = PD [ E i , A i , B i ] = E , i E , i E , i E , i E , i E , i , A , i A , i A , i A , i A , i A , i , B i , where [ E i , A i , B i ] is in QPDFF (4.4). Then the corresponding diagonal blocks (which have matching sizesaccording to Lemma 4.11) are equivalent in the sense that there exist invertible matrices S ii , i = , , , T ii , i = , ,
3, and V such that E ii , = S ii E ii , T ii , i = , , , and b B = S b B V . Proof.
Without loss of generality we assume that S = I ℓ , T = I n , V = I m and F P = F D = Step 1 : By Lemma 4.3 we have V ∗ [ E , A , B ] = T V ∗ [ E , A , B ] , W ∗ [ E , A , B ] = T W ∗ [ E , A , B ] , and from Lemma 4.11 we obtain V ∗ [ E i , A i , B i ] ∩ W ∗ [ E i , A i , B i ] = R n , i × { } n , i + n , i for i = , UASI FEEDBACK - FORMS FOR DIFFERENTIAL - ALGEBRAIC SYSTEMS
25 of 27This implies n , = n , and T = T T T T T T T for T ∈ GL n , ( R ) , T ∈ R n , × n , , T ∈ R n , × n , and T , T , T , T of appropriate sizes. Furthermore, Lemma 4.11 gives R n , + n , × { } n , = V ∗ [ E , A , B ] = T V ∗ [ E , A , B ] = T ( R n , + n , × { } n , ) , which, together with n , = n , , yields n , = n , , n , = n , and T = , T ∈ GL n , ( R ) , T ∈ GL n , ( R ) . Step 2 : Partitioning S = S S S S S S S S S S S S S S S S for S ∈ R ℓ , × ℓ , , S ∈ R ℓ , × ℓ , , S ∈ R ℓ , × ℓ , , S ∈ R m , × m , and V = (cid:20) V V V V (cid:21) for V ∈ R m , × m , , V ∈ R m , × m , and off-diagonal block matrices of appropriate sizes, the equation S B V = B in conjunction withˆ S : = [ S ⊤ , S ⊤ , S ⊤ ] ⊤ givesˆ S ˆ B V = , ˆ S ˆ B V = , S ˆ B V = , S ˆ B V = ˆ B . Since ˆ B [ V , V ] has full row rank, it follows that S = , S = S =
0. By Definition 4.9we have m , = rk B = m , and hence also m , = m , . Since S is invertible, this implies that S ∈ GL m , ( R ) , thus V = V ∈ GL m , ( R ) and, since V is invertible, V ∈ GL m , ( R ) .The equation S ( E T + B F P ) = E yields h S S i E , T = E , im-plies S = S =
0. Since S is invertible, it follows that ℓ , ℓ , . Reversing the rolesof [ E , A , B ] and [ E , A , B ] gives ℓ , > ℓ , , whence ℓ , = ℓ , . We further have the equation S E , T = T and E , gives that S =
0. This finally implies ℓ , = ℓ , = n , = n , , ℓ , = ℓ , , S ∈ GL ℓ , ( R ) , S ∈ GL ℓ , ( R ) , and hence the proof of the proposition is complete.
5. Conclusion
We have presented the novel concepts of quasi P-feedback and quasi PD-feedback forms for DAE con-trol systems which reveal the key structural properties of the control system under P(D)-feedback trans-formations. Furthermore, the forms are easily obtained via the augmented Wong-sequences, whichadditionally provides a geometric insight.6 of 27
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Lemma A.1 ([5, Lem. 4.14], c.f. [8]) . Consider the the two matrix equations0 = E + A Y + Z D0 = F + C Y + Z B (A.1)for some A , C ∈ R m × n , B , D ∈ R p × q , E , F ∈ R m × q . Assume that the pencil s B − D has full polynomialcolumn rank, i.e. there exists λ ∈ R such that ( λ B − D ) has a left inverse ( λ B − D ) † . Then (A.1) issolvable, if the generalized Sylvester equationA X B − C X D = − E + ( λ E − F )( λ B − D ) † Dis solvable.
Remark A.2.
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