J factorizations of a general discrete-time system

Automatica 49 (2013) 2221–2228

Contents lists available at SciVerse ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

J factorizations of a general discrete-time system✩ Cristian Oară 1 , Raluca Marinică Department of Automatic Control and Systems Engineering, Faculty of Automatic Control and Computers, University ‘‘Politehnica’’ Bucharest, Splaiul Independenţei 313, Sector 6, RO 060042, Bucharest, Romania

article

info

Article history: Received 4 November 2011 Received in revised form 16 January 2013 Accepted 24 March 2013 Available online 28 April 2013

abstract The problems of extending the J-lossless conjugation, the (J , J ′ )-spectral factorization, and the (J , J ′ )lossless factorization to a completely general discrete-time system are considered. Existence conditions together with a numerically-sound prototype algorithm for computing the factors are provided for a system having any type of singularity, including arbitrary normal rank, poles and zeros at infinity, at zero, or on the unit circle. © 2013 Elsevier Ltd. All rights reserved.

Keywords: Discrete-time systems Singular systems Factorization methods Matrix Riccati equation Spectral factorization

1. Introduction The (J , J ′ )-spectral and (J , J ′ )-lossless factorizations (and the related Riccati equation techniques) are the main technical tools in the classical state-space approaches of Kimura (1997) and DGKF (Doyle, Glover, Khargonekar, & Francis, 1989) to the H ∞ control problem—see Ionescu and Weiss (1993) and Kongprawechnon and Kimura (1998) for their discrete-time variants. However, these classical approaches are restricted by the so-called regularity assumptions, which require for the transfer function from controls to controlled outputs to be left invertible, for the transfer function from disturbances to measured outputs to be right invertible, and for both transfer functions to have no zeros on the boundary of the stability domain (be it either the extended imaginary axis in the continuous-time case or the unit circle in the discrete-time case). The regularity assumptions, although quite unnatural from an engineer’s viewpoint are essentially dictated by the technical limitations: the underlying factorizations can be explicitly solved in state-space only if an associated algebraic Riccati equation

✩ This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PN-II-ID-PCE-2011-3-0235. The material in this paper was partially presented at the 51st IEEE Conference on Decision and Control (CDC), December 10–13, 2012, Maui, Hawaii, USA. This paper was recommended for publication in revised form by Associate Editor Mario Sznaier under the direction of Editor Roberto Tempo. E-mail addresses: [email protected] (C. Oară), [email protected] (R. Marinică). 1 Tel.: +40 212674458; fax: +40 212674458.

0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.03.023

can be written down and solved for its stabilizing solution. Since these assumptions hinder the application of the H ∞ -control theory in real-life applications, an important research effort has been invested towards their removal (Gahinet & Apkarian, 1994; Scherer, 1992; Stoorvogel, 1991). However, the only numerically feasible solution for the general case is (Gahinet & Apkarian, 1994; Iwasaki & Skelton, 1994) and relies on three Linear Matrix Inequalities (LMIs) and their solutions. Nevertheless, although extremely elegant, the LMIs solution suffers from a severe drawback: its complexity may be up to O (n6 )—compare with the O (n3 ) complexity of the classical solutions. Apart from this central role played in H ∞ -control by these factorizations, the (J , J ′ )-spectral factorization and its state-space computation are an essential technical tool in a variety of results in signals and systems theory: filtering, game theoretic situations, control and estimation problems formulated in Krein spaces with indefinite metric, system identification, network and circuit theory, to mention just a few (see for example Basar and Olsder (1995), Hassibi, Sayed, and Kailath (1999), Ionescu, Oară, and Weiss (1999), Kimura (1997) and Zhou, Doyle, and Glover (1996)). However, precisely as for H ∞ -control, the effective computation of the spectral factorization in state-space can be done only for a full rank system, having no zeros on the boundary of the stability domain, restricting therefore its application. In order to extend the scope of these factorization techniques to cover the full range of possible real-life applications, these restrictive assumptions need to be removed. Removing the regularity assumptions leads to a variety of problems, like the need to solve Riccati equations with indefinite sign quadratic terms for their semi-stabilizing solutions – a notoriously ill-conditioned

2222

C. Oară, R. Marinică / Automatica 49 (2013) 2221–2228

problem – or, even worse, to the impossibility of even writing down the Riccati equations due to the singularity of the quadratic term. Due to their importance, these factorizations have been investigated in tens of papers under various technical hypotheses, both in the continuous or discrete-time settings. Among the continuous-time approaches closer to the present development we mention (Aliev & Larin, 1997; Chu & Ho, 2005; Chu & Tan, 2008; Green, Glover, Limebeer, & Doyle, 1990; Kawamoto & Katayama, 2003; Kwakernaak & Sebek, 1994; Xin & Kimura, 1994a,b), with the most general constructive solution given in Oară and Andrei (2011a). Until now, the discrete-time variant has been solved either in different particular instances or for a particular underlying system: in Oară (2005) the unitary case (J = J ′ = I) has been approached for a completely general system, in Kongprawechnon and Kimura (1996) the (J , J ′ ) case is solved for a system fulfilling several regularity assumptions (left invertible, without poles at infinity or at zero and without zeros on the unit circle or at infinity), while some of these assumptions are relaxed to include zeros at 0 in Hung and Chu (1998a), and zeros on the unit circle in Hung and Chu (1998b). The main purpose of this work is to extend (Oară & Andrei, 2011a) to obtain the genuine discrete-time construction of the factors, under the same level of generality. This implies the avoidance of the well-known brute force approach of using bilinear transformations, which has the drawbacks of complicating the resulting formulas, unnecessarily inverting possibly ill-conditioned matrices, and leaving apart some points in which the transformations are not defined. The needed extension is far from trivial since, in contrast to the continuous-time case, the point at infinity is in the antistable region and therefore both poles and zeros at infinity need to be moved in 0 (the symmetric point with respect to the unit circle). Apart from this additional feature that allows for inclusion of polynomial or improper systems, possibly with zeros at infinity, our solution encompasses for the first time systems that might have arbitrary normal rank (are not necessarily left invertible), and poles and zeros on the unit circle. Although the technical machinery and the resulting formulas for the factors are more involved than their continuous-time counterparts, the numerical algorithm has the same nice features, being based on stable unitary transformations only. In particular, our results pave the way to a numerically traceable O (n3 ) statespace solution to the H ∞ -control problem formulated for a general discrete-time system, and contain in particular the solution to the important problem of (J , J ′ )-spectral factorization of a general matrix polynomial. A conference version of this work (without proofs) appeared in Oară and Marinică (2012). The paper is organized as follows. In Section 2 we state the factorization problems and review a couple of notions related to generalized state-space realizations of rational matrices. In Section 3 we put ground for the main results by giving a spectral decomposition of a generalized state-space realization of the system. Section 4 contains the main results (the technical proofs are deferred to the Appendix). In Section 5 we illustrate our approach on a numerical example for the (J , J ′ )-lossless factorization. We draw some conclusions in Section 6. 2. Preliminaries We start with some basic definitions. A square invertible matrix J having complex elements is called a signature matrix provided J = J −1 = J ∗ , where ∗ denotes the conjugate transpose. A rational matrix function (rmf) with complex coefficients Θ (z ) is called (J , J ′ )-unitary if Θ (z )∗ J Θ (z ) = J ′ , at every point on the unit circle at which Θ is analytic, where J and J ′ are two signature matrices, not necessarily of the same dimension. By analytic continuation, Θ (z )# J Θ (z ) = J ′ , ∀z ∈ C, where G# (z ) := G∗ ( 1z ). If, in addition,

Θ (z )∗ J Θ (z ) ≤ J ′ for every point of analyticity of Θ in the exterior of the closed unit disk, then Θ is called (J , J ′ )-lossless. If J = J ′ , Θ (z ) is called J-unitary and J-lossless, respectively. The normal rank of a rmf G(z ) is its rank for almost all z ∈ C. We denote by RL∞ the set of rational matrices having no poles on the unit circle. By D we denote the open unit disk and Dc = C \ D stands for the exterior of the closed unit disk containing infinity. Here ‘‘overbar’’ denotes closure. By ⋆ we denote irrelevant matrix entries. We investigate the following factorizations problems formulated for a completely general discrete-time system G(z ) (which may be strictly proper/improper/polynomial, and may have arbitrary normal rank, poles and zeros): J-lossless conjugation. Find a minimal McMillan degree J-lossless rmf Θ (z ) such that G(z ) = Θ (z )Π (z ),

(1)

where Π (z ) has only marginally stable (located in the closed unit disk) poles; (J , J ′ )-spectral factorization. Find a rmf Π (z ) which has full row normal rank and only marginally stable zeros such that G# (z )JG(z ) = Π # (z )J ′ Π (z ),

(2)

where G(z )Π (+) (z ) has no poles on the unit circle, Π (+) (z ) stands for the Moore–Penrose pseudoinverse of Π (z ); (J , J ′ )-lossless factorization. Find a (J , J ′ )-lossless rmf Θ (z ) (without poles on the unit circle) and a rmf Π (z ) which has full row normal rank and only marginally stable poles and zeros such that G(z ) = Θ (z )Π (z ).

(3)

′

In (1)–(3) J and J are two signature matrices. For any p × m rmf G(z ) with coefficients in C (even improper or polynomial) one can write down a generalized state-space realization of the form (see Verghese, Lévy, and Kailath (1981) and Verghese, Van Dooren, and Kailath (1979)) G(z ) = D + C (zE − A)−1 B =:





A − zE B , C D

(4)

where A, E ∈ Cn×n , B ∈ Cn×m , C ∈ Cp×n , D ∈ Cp×m , and A − zE is a regular pencil, i.e., det(A − zE ) ̸≡ 0. With any realization (4) we associate two matrix pencils: the   pole pencil P (z ) = A − zE and the system pencil S (z ) =

A − zE C

B D

. The Weierstrass and Kronecker

canonical forms (Gantmacher, 1960) of the pole and system pencil play an important role in the sequel as they are in correspondence with the Smith–McMillan form and the minimal indices to the left and right of the transfer function matrix. For an exhaustive overview of these notions and the relations among them we refer the interested reader to Verghese et al. (1981, 1979). The union of finite and infinite generalized eigenvalues (Gantmacher, 1960) of a general pencil A − zE is denoted by Λ(A − zE ). The generalized state-space realization (4) of G(z ) is called minimal if its dimension is as small as possible and it is called irreducible if it satisfies the following conditions (Verghese et al.,    1981): (i) rank A − zE B = n, ∀z ∈ C; (ii) rank E B = n;

T

T

(iii) rank (A − zE )T C T = n, ∀z ∈ C; (iv) E T C T = n. These conditions are usually known as finite and infinite controllability, and finite and infinite observability, respectively. We end this section by a technical result which will be used in the sequel to characterize (J , J ′ )-unitary and (J , J ′ )-lossless rational matrices given by a special type of generalized realizations (see for example Gohberg, Kaashoek, and Ran (1992) and Oară and Varga (1999)).





Lemma 2.1. Let a rmf without poles at 1 given by the minimal realization G(z ) = C (zE − A)−1 B(1 − z ) + D. Then G(z ) is (J , J ′ )unitary ((J , J ′ )-lossless) iff D∗ JD = J ′

(5)

C. Oară, R. Marinică / Automatica 49 (2013) 2221–2228

and there is an invertible (positive definite) Hermitian matrix X such that A∗ XA − E ∗ XE + C ∗ JC = 0,

(6)

D∗ JC + B∗ X (A − E ) = 0.

3. Unitary spectral decomposition

Ab − zEb 0  G(z ) = 0 Cb

Abg − zEbg Ag − zEg 0 Cg

Abn − zEbn Agn − zEgn An Cn

Bb Bg  Bn , D



(7)

Eb 0

Ebg Eg

Ebn Egn

= rank E . Clearly, Ab and Ab − Eb are both

invertible. Theorem 3.1. Let G(z ) be a p × m rmf given by an irreducible realization (7). Then there exist two constant unitary matrices U and Z such that





Ab − zEb 0 U0  0 I  0 Cb

Abg − zEbg Ag − zEg 0 Cg

Abn − zEbn Agn − zEgn An Cn

Arg − zErg 0  = 0 0

B1 − zF1 Abℓ − zEbℓ 0 Cbℓ

B2 − zF2 Bbℓ (1 − z ) 0 Dbℓ







Bb Bg  Z Bn D B3 − zF3 B4 − zE4  B5 B6

 (8)

where: (I) The pencil Arg − zErg has full row rank for z ∈ Dc (contains the right singular structure and the good zeros in D) and Erg has full row rank; (II) Abℓ − Ebℓ and B5 are invertible, Dbℓ has full column rank, the pencil



Abℓ − zEbℓ Cb ℓ

Bbℓ (1 − z ) Dbℓ

Theorem 4.1 (J-Lossless Conjugation). Let G(z ) be a general rmf given by an irreducible realization (7). The J-lossless conjugation (1) has a solution if and only if the Stein equation (10)

has a unique invertible solution Y > 0. In this case, we have in (1)

 Θ (z ) :=

K (1 − z ) I 0

Ab − zEb 0 Cb

0 I I

 ,

(11)

Π (z ) in (12) in Box I, where K := Y −1 (Ab − Eb )−∗ Cb∗ J ,

(13)

Aˆ b := Ab + KCb and Eˆ b := Eb + KCb .

where Ab − zEb contains the bad poles (in Dc ), Ag − zEg contains the good poles (in D), An contains the nondynamic modes, and  rank

exists) can be used to write down state-space formulas for the factors Π (z ) and Θ (z ) solving the three factorization problems under investigation.

A∗b YAb − Eb∗ YEb + Cb∗ JCb = 0

We introduce a spectral decomposition of the generalized state-space realization of G(z ), which can be achieved by unitary transformations only and will play a key role in expressing our main results. Assume G(z ) is given by an irreducible realization (4) which, by a mere unitary transformation (Oară & Varga, 1999), can always be brought to



2223



Since Λ(Ab − zEb ) ⊂ Dc , Ab is invertible. Therefore, (10) can be converted into a standard Stein equation which can be solved, for its unique solution, by any numerically-sound algorithm available in the literature (see for example Penzl (1998) for a version that avoids any matrix inversion). Theorem 4.1 is actually an extension to the case of an arbitrary rmf of the discrete-time J-conjugation method given in Kongprawechnon and Kimura (1996). However, even for the standard proper case presented in Kongprawechnon and Kimura (1996), our method is more efficient since we solve a Stein equation – rather than a Riccati one –, whose dimension is smaller whenever G(z ) has marginally stable poles. Remark 4.2. Compared with its continuous-time counterpart, the expression (12) is more complicated. This should come as no surprise, since in contrast to the continuous-time in which the poles at infinity are on the boundary of the stability domain and need not to be moved, in the discrete-time case the poles at ∞ are in the antistable region and should be symmetrized in 0. An additional obstacle is brought by the fact that the number of eigenvalues at ∞ of the pole pencil A − zE of an improper system (4) is always greater than the number of poles at ∞, the excess being equal to the number of nondynamic modes that have been separated in An (see (7)).

(9)

has full column rank for z ∈ D (contains  the left singular  structure and the bad zeros in C − D), and the pair Abℓ − zEbℓ Bbℓ is controllable. The existence of the matrices U and Z and a numerically stable algorithm to compute them can be obtained from Theorem 7 in Oară and Sabău (2009) (for α = β = 1). Remark 3.2. Apart from the different properties, the decomposition (8) has a different form with respect to its continuous-time counterpart (see Oară and Andrei (2011a, Eq. (4) and Remark 3.1)). Actually, the projected system (9) for which we solve in fact the factorization problem has the pencil Bbℓ (1 − z ) as its ‘‘B’’ matrix, and Lemma 2.1 will be used for enforcing the (J , J ′ )-unitary property. Nevertheless, both continuous and discrete-time decompositions may be obtained in a quite similar fashion using solely unitary transformations (Oară & Sabău, 2009).

Theorem 4.3 ((J , J ′ )-Spectral Factorization). Let G(z ) be a general rmf given by an irreducible realization (7), and let U and Z be two constant unitary matrices such that (8) holds. The (J , J ′ )-spectral factorization problem (2) has a solution if and only if there is a constant invertible matrix V such that D∗bℓ JDbℓ = V ∗ J ′ V

(14)

and the matrix Riccati equation A∗bℓ XAbℓ − Eb∗ℓ XEbℓ − ((Abℓ − Ebℓ )∗ XBbℓ

+ Cb∗ℓ JDbℓ )(D∗bℓ JDbℓ )−1 (B∗bℓ X (Abℓ − Ebℓ ) + D∗bℓ JCbℓ ) + Cb∗ℓ JCbℓ = 0

(15)

has a unique invertible stabilizing solution X , i.e., Λ(Abℓ − zEbℓ + Bbℓ F (1 − z )) ⊂ D, where F := −(D∗bℓ JDbℓ )−1 (B∗bℓ X (Abℓ − Ebℓ ) + D∗bℓ JCbℓ ). In this case, the spectral factor in (2) is given by





A − zE B , CΠ DΠ

4. Main results

Π (z ) :=

Starting from the decomposition (8), we get the coefficients of two equations, a Stein and a Riccati, whose solutions (when they

and the partition of the matrix 0 the block column partition of (8).



CΠ DΠ





  := V 0 −F I 0 Z ∗ ,

−F

I



(16)

0 is conformably with

2224

C. Oară, R. Marinică / Automatica 49 (2013) 2221–2228



Abg + KCg − z (Ebg + KCg ) Ag − zEg Cg

Aˆ b − z Eˆ b 0 Π (z ) :=  Cb



∗ Bb + Y −1 A−∗ b Cb D  Bg ∗ [I + Cb (Ab − Eb )−1 Y −1 A−∗ C ] D b b

(12)

Box I.

Remark 4.4. The theorem can be equivalently stated in the following form: The (J , J )′ -spectral factorization for G(z ) has a solution if and only if the (J , J )′ -spectral factorization for the system (9) has one. Apart from using a discrete version of the Riccati equation, the expression of the spectral factor in (16) is identical to its continuous-time counterpart (see Theorem 4.2 in Oară and Andrei (2011a)). However, (15) is not a completely standard discrete-time algebraic Riccati equation (see chapter 5 in Ionescu et al. (1999)). Nevertheless, its stabilizing solution may be computed by employing unitary transformations on symplectic matrix pencils in a fashion very similar to the standard case (Oară & Andrei, 2012). Special care should be taken if some antistable eigenvalues of Abℓ − zEbℓ are close to the unit circle, as this can lead numerically to a loss of dichotomy. Notice that the above result may be applied in particular to a polynomial matrix, and provides a numerically sound statespace construction of the (J , J ′ )-spectral factor (Aliev & Larin, 1997; Kwakernaak & Sebek, 1994). A slightly different attempt to solve the general (J , J ′ )-spectral factorization was made in Oară and Andrei (2011b). Theorem 4.5 ((J , J ′ )-Lossless Factorization). Let G(z ) be a general rmf given by an irreducible realization (7), and let U and Z be two constant unitary matrices such that (8) holds. The (J , J ′ )-lossless factorization problem (3) has a solution if and only if: (1) The Stein equation (10) has a unique invertible solution Y > 0. (2) There is a constant invertible matrix V such that (14) holds. (3) The matrix Riccati equation Aˆ ∗bℓ X Aˆ bℓ − Eˆ b∗ℓ X Eˆ bℓ − ((Abℓ − Ebℓ )∗ X Bˆ bℓ

+ Cb∗ℓ JDbℓ )(D∗bℓ JDbℓ )−1 (Bˆ ∗bℓ X (Abℓ

has a unique invertible stabilizing solution X > 0, i.e., Λ(Aˆ bℓ − z Eˆ bℓ + Bˆ bℓ F (1 − z )) ⊂ D, where Aˆ bℓ := Abℓ + K2 Cbℓ , Eˆ bℓ := Ebℓ + K2 Cbℓ , Bˆ bℓ := Bbℓ + K2 Dbℓ ,

K1T



K2T

T

 := U K T

T

0

K := Y −1 (Ab − Eb )−∗ Cb∗ J .

which has a realization (7) given by   −1 0 0 0 0 0 0 0 −1/3z 1 0 −5 0 0 0 0 0 0 0 0 0   0 −3 −1 − 2/3z   z −2 2/3z 0 0 0 0 0 0 0 0 0     0 0 3 − z −1 0 1 0 0 0 0 0   0   0 0 −1 −z 0 0 1 0 0 0 0   0   0 0 16 −6 −3 − z 0 0 0 1 0 0   0 ,  0 0 −6 2 1 −z 0 0 0 1 0   0   0 0 5 −2 −1 0 −z 0 0 0 1   0    0 0 0 0 0 0 0 0 −1 0 1 4     0 0 0 0 −1 0 0 0 0 0 0 0    4 −1 0 1 0 0 0 0 0 0 0 0 0 −6 −2 0 0 0 0 0 0 0 0 0 where the dotted line delineates the corresponding matrix blocks in (7). The structural elements of G(z ) are: a pole at ∞ with multiplicity 2, a pole at −1 with multiplicity 2, a pole at 1 with multiplicity 2, a pole at 0, a pole at − 32 , a zero at −1, a zero at ∞, a left minimal index equal to 5, a right minimal index equal to 1, and normal rank r = 2. Eq. (10) has the unique invertible positive definite solution 36/5 16/5 −24/5

Y = (17)

F := −(D∗bℓ JDbℓ )−1 (Bˆ ∗bℓ X (Abℓ − Ebℓ ) + D∗bℓ JCbℓ ),

We exemplify our results on the (J, J ′ )-lossless factorization of a system having all types of singularities: zeros and poles at infinity (improper), zeros and poles on the unit circle, and row and column deficient normal rank. For illustrative simplicity we will use nonunitary transformations as well. Throughout this section we take J ′ = diag(1, −1), J = diag(1, J ′ ). Let   1/z (z + 1)2 1/z (z + 1)2 −(z + 2)/z (z + 1)2 2 2 2  G(z ) =  1/ (z − 1) (z + 2)/ (z − 1) −1/(z − 1) −3z 2 /z + 3/2 (z 3 − 2z 2 )/z + 3/2 (4z 3 + 7z 2 )/z + 3/2

16/5 36/5 −24/5



− Eb ℓ )

+ D∗bℓ JCbℓ ) + Cb∗ℓ JCbℓ = 0

5. A numerical example

0 0 0

K =

,

 −1/2 −1/2 , −5/6

0 0 0

0 0 3/2

0 0 −1/2

 K2 = (19)

In this case, the factors in (3) are given by (20) and (21) in Box II. Notice that in contrast to the (J , J ′ )-spectral factorization, in the (J , J ′ )-lossless case the sign of the matrix Riccati equation should be positive definite. Remark 4.6. The more complicated expressions of the discretetime factors (20) and (21) in comparison with their continuoustime counterparts (see Theorem 4.1 (Oară & Andrei, 2011a)) is partially due to the reasons explained in Remark 4.2 to which we add now the necessity to symmetrize the zeros at infinity into 0.

(22)

With expressions given in Box III, we get the decomposition (8) in the form (23) in Box IV, where the dotted line delineates the corresponding matrix blocks in the right-hand side of (8). Using (22) in (19) we get

 (18)

 −24/5 −24/5 . 36/5

0 0 −5/6

0 0 3/2

0 0 0

T

0 0 0

The matrix Riccati equation (17) has a stabilizing positive definite solution 78.272  0.059   4.226 Xs =  −74.443   −5.408 28.386



 With V =



0.059 0.281 −0.161 −0.056 0.053 −0.031

4.226 −0.161 0.506 −3.961 −0.168 1.753

11/4

(8/5)2 − (1/4)2

1

−74.443 −0.056 −3.961 70.853 5.320 −26.861

−5.408 0.053 −0.168 5.320 1.056 −1.584

28.386 −0.031   1.753  . −26.861  −1.584  11.729





0

in (14), we get from (20) and (21)

the expressions in (24) in Box IV.

C. Oară, R. Marinică / Automatica 49 (2013) 2221–2228

K (Cbℓ + Dbℓ F )(1 − z ) KDbℓ V −1 (1 − z ) Abℓ − zEbℓ + (Bbℓ + K2 (Cbℓ + Dbℓ F ))(1 − z ) (Bbℓ + K2 Dbℓ )V −1 (1 − z ) 0 I Cbℓ + Dbℓ F 0



Ab − zEb  0 Θ (z ) :=  0 −C b Aˆ b − z Eˆ b 0  Π (z ) :=  0  0 CΠ 1



Abg − zEbg + KCg (1 − z ) Abn − zEbn + KCn (1 − z ) Bb + KD(1 − z ) Ag − zEg Agn − zEgn Bg 0 An Bn 0 0 I C C 0 Π 2 Π 3     CΠ 1 CΠ 2 CΠ 3 DΠ := V 0 −F I 0 Z ∗

2225



0  0 −I  Dbℓ V −1

(20)



0 0  0   −I DΠ

(21)

Box II.

0 1 −3  0 U = 0  −3  0 0



0 0 0 1 0 0 0 0

0

−2

0 0  0  0  0 Z = 0 0  0  1  0 0

0 0 0 0 1 0 0 3 3 −1 1

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 3 0 14 −12 3 0 0 16 −4

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 −4 1

1 −1 3 0 0 3 0 0

0 0 0 0 0 0 1 0

0 1/2 −9/10 0 0 0 0 0 0 0 0 0

0 5  −14  0   0   −15  4 0



−4/3 −2/3 6/5 0 0 0 0 0 0 0 −16 4

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 −1/2 0 −2 1/2 0 0 8 −2

0 0 0 0 0 0 1 0 0 0 0 0

−4/3 −2/3 6/5 0 1/4 0 −5/2 1/4 0 0 8 −2

0 0 0 1 0 5 −5 1 0 0 8 −2

0 0 0  0  0  0  0 0  0  0  1 0



Box III.

0 0 0 0 3 0 −3/2 1−z 4/3 1/3 − 5/4z 0 0 0 2 − 5/2z 0 −1 1 0 0 0 −7 + 7z 0 −7/2(1 − z ) 1−z 4/5(1 − z ) 0 −3/5(1 − z ) 4/5(1 − z ) −1 − 32 z 0 0 0 4/3(1 − z ) 0 −1 4/3(1 − z ) 0 0 0 0 0 0 −4 0 −15/2(1 − z ) 1 −15/4(1 − z ) 1−z 0 0 0 0 3 −z −3/2(1 − z ) −(1 − z ) 1/2(1 + z ) 1/4(1 − z ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/2 0 0 0 0 0 −1/4 0 0 0 0 −3/2 1 −11/4 −1 0 0 8/5 8/5 0 0 −6/5 −2 0 0 0

 1 −z 6 − 14z  0 0 −1 − z 0 0 0 0 0 0  0 0 0  0 0 0 0 0 0  0 0 0  0 0 0  0 0 0 0 0

0 0

2 (2z +3)(z 2 +0.771z +0.289) Θ (z ) = 1/z  (2z + 3)(−0.221z 2 + 0.378z − 0.157) (2z + 3)(−0.394z 2 + 0.113z − 0.045) ×  z (2z + 3)(−0.367z 2 − 1.065z − 0.628) z (2z + 3)(0.228z 2 − 0.047z − 0.180)  z 4 (1.040z + 0.203)(z − 1)(z + 2/3) z 4 (3.613z 2 + 1.983z + 0.661)(z + 2/3)  −9.558z 4 − 2.614z 3 − 5.450z 2 − 4.457z + 2.080 2 2 1 Π (z ) = /(z −1) (z +1) (3z +2) (−2.229z 3 − 6.310z 2 + 5.286z + 7.227)(z − 1) 5 −6.052z − 25.969z 4 − 16.227z 3 − 8.346z 2 − 4.791z + 1.386 (3.402z 4 + 5.568z 3 − 3.329z 2 + 1.461z + 4.818)(z − 1)  3.506z 5 + 5.319z 4 − 2.933z 3 + 12.463z 2 + 6.498z − 4.854 . (5.632z 4 + 18.568z 3 + 10.314z 2 − 21.624z − 16.863)(z − 1)

0 0 0 0 0 0 1 0 1 0 0 0

              

(23)

(24)

Box IV.

A direct check shows that indeed Θ (z ) is (J , J ′ )-lossless and belongs to RL∞ , and Π (z ) has full row rank and marginally stable

poles and zeros (a pole at 1 with multiplicity 2, a pole at −1 with multiplicity 2, a pole at − 23 , a zero at 0, a zero at −1, a

2226

C. Oară, R. Marinică / Automatica 49 (2013) 2221–2228

pair of two complex conjugated zeros at −0.38577786896230 ± 0.37441778265865i, and a right minimal index equal to 1). 6. Conclusions

and (29) shown in Box V. To prove that (28) and (29) are the same transfer matrix we notice first the identity (30) in Box V. Multiplying to the right with Z ∗ and by using (8), (21), (19), and the notation in Theorem 4.5 leads to

 We have given state-space formulas for three essential factorization problems formulated for a completely general discretetime system: J-lossless conjugation, (J , J ′ )-spectral factorization and (J , J ′ )-lossless factorization. Precisely as in the standard case (under regularity assumptions), the numerically-sound construction of the factors employs solutions to Stein and Riccati equations, with the only difference now that the equations are written in terms of a unitary projection of a generalized state-space realization of the system. The computational burden lies in obtaining the initial unitary projection which is a particular type of Kroneckerlike form and may be achieved by successive rank decisions. Due to the inherently ill-conditioned nature of these computations, very accurate rank revealing algorithms (Chan, 1987) should be used at intermediate steps to guarantee the relevance of the final results. The resulting numerical complexity of the overall algorithm is O (n3 ) (see Oară and Andrei (2011a) for more details). Appendix

 Θ1 (z ) :=

Ab − zEb 0 Cb

K (1 − z ) I 0

0 I I

 ,

(25)





A˜ bℓ − z E˜ bℓ Bˆ bℓ V −1 (1 − z ) 0 0 I −I , Θ2 (z ) :=  Cbℓ + Dbℓ F 0 Dbℓ V −1

(26)

where A˜ bℓ := Aˆ bℓ + Bˆ bℓ F and E˜ bℓ := Eˆ bℓ + Bˆ bℓ F . Instead of (3), we will show equivalently that

Θ1−1 (z )G(z ) = Θ2 (z )Π (z )

(27)

by relating the associated system pencils in the left and right terms by a transformation matrix  that leaves invariant the underlying rmf. Let Aˆ bg − z Eˆ bg := Abg − zEbg Abn − zEbn , Aˆ g − z Eˆ g := Ag − zEg 0



CΠ 2

Agn − zEgn An

CΠ 3



, Bˆ g :=

  Bg Bn

, Cˆ g :=





Cg

Cn , CΠ 23

:=

, Bˆ b := Bb + KD. We have



Θ1 (z )G(z )  ˆ Ab − z Eˆ b  0 (7),(25)  =   0

Aˆ b − z Eˆ b × 0 V −1 C Π 1

Aˆ bg − z Eˆ bg + K Cˆ g (1 − z ) Aˆ g − z Eˆ g V −1 CΠ 23

 = (A˜ bℓ − z E˜ bℓ ) X21 





X11 X12 := 0 I X13



X21

X22



 U0



0 := 0

0 I I



0 ,



X22



Bˆ b − KDz  Bˆ g V −1 D Π



(31)

,

0

 (Eˆ bℓ + Bˆ bℓ F )−1 Fˆ4 Z ∗ .

(32)

We use now (31) to define an equivalence transformation on (29) and get after removing the uncontrollable part (33) and (34) in Box VI. The system pencils associated with the realizations (28) of Θ1−1 (z )G(z ) and (33) of Θ2 (z )Π (z ) are related through   0 0 U13 Im diag(U , I ) − (35) in Box VI, where U := D − Dbℓ V −1 DΠ , U13

:= [−B6 + (Cbℓ + Dbℓ F )(Eˆ bℓ +   1 ˆBbℓ F )−1 Fˆ4 ]B− 0 0 U13 in the definition 5 , and the partition 

0

of U is conformably to (8). Indeed, (35) follows straightforwardly by it to the left and right with diag(U , I , Im , −Ip ) and  multiplying  Z

−Zm

0 I

, respectively, where Zm denotes the last m lines of the

matrix Z , and by using (8), (19), (32), and the definitions of Ce1 and Ce2 in (34). However, the first matrix in the left-hand side of (35) is a Rosenbrock transformation on the system pencil of (28) that leaves unchanged the associated transfer matrix function (see Theorem 3.1 in Rosenbrock (1970)), from where we conclude that (27) holds. To end the sufficiency part, we still need to show that Θ (z ) and Π (z ) fulfill the required properties of the (J , J ′ )-lossless factorization. We show first that Θ (z ) = Θ1 (z )Θ2 (z ) is (J , J ′ )lossless by proving that Θ1 (z ) is J-lossless and Θ2 (z ) is (J , J ′ )lossless. Using the explicit form Θ1 (z ) = I − Cb (zEb − Ab )−1 K (1 − z ) and the positive definite solution Y to the Stein equation (10), one can check directly that (5) and (6) of Lemma 2.1 are fulfilled for G(z ) := Θ1 (z ), J = J ′ , and X := Y . Therefore, Θ1 (z ) is J-lossless. Similarly, using the explicit form Θ2 (z ) = Dbℓ V −1 + (Cbℓ + Dbℓ F )(z E˜ bℓ − A˜ bℓ )−1 Bˆ bℓ V −1 (1 − z ), and the positive definite solution X to the Riccati equation (17), we verify that (5) and (6) of Lemma 2.1 are fulfilled for G(z ) := Θ2 (z ). Indeed, (14)

(Dbℓ V −1 )∗ J (Dbℓ V −1 ) = J ′ , A˜ ∗bℓ X A˜ bℓ − E˜ b∗ℓ X E˜ bℓ + (Cbℓ + Dbℓ F )∗ J (Cbℓ + Dbℓ F )

= Aˆ ∗bℓ X Aˆ bℓ − Eˆ b∗ℓ X Eˆ bℓ + Cb∗ℓ JCbℓ + ((Abℓ − Ebℓ )∗ X Bˆ bℓ + Cb∗ℓ JDbℓ )F + F ∗ (Bˆ ∗bℓ X (Abℓ − Ebℓ ) + D∗bℓ JCbℓ ) + F ∗ D∗bℓ JDbℓ F (17), (18)

=

(18)

F ∗ (Bˆ ∗bℓ X (Abℓ − Ebℓ ) + D∗bℓ JCbℓ ) + F ∗ D∗bℓ JDbℓ F = 0,

(Dbℓ V −1 )∗ J (Cbℓ + Dbℓ F ) + (Bˆ bℓ V −1 )∗ X (Aˆ bℓ − Eˆ bℓ )

−1

Cb

 −Bˆ bℓ (1 − z )

X12





We provide here the proof of the three main theorems in Section 4. However, we need to prove explicitly only Theorem 4.5, since Theorems 4.1 and 4.3 are particular cases of it, involving either only the poles, or the zeros and the rank compression of the original system G(z ), respectively. As the one and only exception we mention the signature of the solutions to the matrix Riccati equations (15) and (17), which in Theorem 4.3 may be arbitrary, while in Theorem 4.5 should be positive. Anyway, this small difference plays no role in the technical machinery of the proof and is due to the (J , J ′ )-unitary and (J , J ′ )-lossless transformations, respectively, used in the two different cases. Sufficiency. We start to prove that the conditions in Theorem 4.5 are sufficient, and (20) together with (21) form a solution to the (J , J ′ )-lossless factorization (3). A direct check shows that Θ (z ) = Θ1 (z )Θ2 (z ), where



X11

(18)

Aˆ bg − z Eˆ bg + K Cˆ g (1 − z )

Bˆ b − KDz

Aˆ g − z Eˆ g

Bˆ g

0

I

, (28) −I 

Cˆ g

0

D

0



0  

= V −∗ Bˆ ∗bℓ X (−(Abℓ − Ebℓ ) + (Aˆ bℓ − Eˆ bℓ )) = 0.

Hence Θ2 (z ) is (J , J ′ )-lossless. From the expressions (25) and (26) it follows in addition that Θ (z ) is in RL∞ . Further, the poles of Π (z ) belong to Λ(Ab − zEb + KCb (1 − z )) ∪ Λ(Ag − zEg ) ⊂ D (the

C. Oară, R. Marinică / Automatica 49 (2013) 2221–2228

 Θ2 (z )Π (z )

(21),(26)

=

   

A˜ bℓ − z E˜ bℓ 0 0 0 Cbℓ + Dbℓ F

Bˆ bℓ V −1 CΠ 1 (1 − z ) Bˆ bℓ V −1 CΠ 2 (1 − z ) Bˆ bℓ V −1 DΠ (1 − z ) ˆAb − z Eˆ b ˆAbg − z Eˆ bg + K Cˆ g (1 − z ) Bˆ b − KDz ˆ ˆ 0 Ag − z Eg Bˆ g 0 0 I D b ℓ V −1 C Π 1 Dbℓ V −1 CΠ 23 0

Arg − zErg   0 −Bˆ bℓ (1 − z ) ×  0 0





0

I

X13

 = (A˜ bℓ − z E˜ bℓ ) 0

I

⋆ Aˆ bℓ − z Eˆ bℓ

Bˆ bℓ (1 − z ) 0 I

0 −F

(Eˆ bℓ + Bˆ bℓ F )−1 Fˆ4

0

⋆

2227

0 0 0 −I Dbℓ V −1 DΠ

    

(29)

 ⋆ Bˆ 4 − z Fˆ4   B5 0



(30)

1 where Bˆ 4 := B4 + K2 B6 , Fˆ4 := F4 + K2 B6 , X13 := [−Bˆ 4 + (Aˆ bℓ + Bˆ bℓ F )(Eˆ bℓ + Bˆ bℓ F )−1 Fˆ4 ]B− 5 .

Box V.

Aˆ bg − z Eˆ bg + K Cˆ g (1 − z ) Aˆ g − z Eˆ g 0 Ce2

Aˆ b − z Eˆ b  0 Θ2 (z )Π (z ) =  0 Ce1



Bˆ b − KDz Bˆ g I 0



0  0  I −1 Dbℓ V

(33)

where Ce1 Ce2 := Dbℓ V −1 CΠ 1 CΠ 23 + (Cbℓ + Dbℓ F ) X21 X22 .













  Aˆ b − z Eˆ b Aˆ bg − z Eˆ bg + K Cˆ g (1 − z ) Bˆ b − KDz  I 0  0 Aˆ g − z Eˆ g Bˆ g  0 0 I U¯ I 0 Cb Cˆ g  Aˆ b − z Eˆ b Aˆ bg − z Eˆ bg + K Cˆ g (1 − z ) Bˆ b − KDz  0 Aˆ g − z Eˆ g Bˆ g = 0 Ce1

0 Ce2

I 0

(34)



0 0  −I  D  0  0 . −I −1 Dbℓ V DΠ

(35)

Box VI.

infinite generalized eigenvalues of the diagonal blocks An and I in (21) correspond all to nondynamic modes, which cancel out and do not appear as poles of Π (z ) when forming its explicit expression). Finally, we have for the  system pencil of (21) the identity (36).

ˆ (z ) := Θ+−1 (z )G(z ) = Θ− (z )Π (z ). G

(37)

Θ+−1 (z )

rank and their zeros are located in D (see the properties of the decomposition given in Theorem 3.1 and the stabilizing property of the Riccati feedback F ), (36) shows that Π (z ) has full row normal rank and all zeros located in D. This ends the sufficiency part.

Since the right-most side of (37) is marginally stable and −1 is (−J )-lossless, the pair (Θ+ (z ), Θ− (z )Π (z )) is a solution to the minimal degree left coprime factorization with (−J )-lossless denominator for G(z ) (see Theorem 6.2 in Oară and Varga (1999)). Therefore, the Stein equation (10) has a unique invertible solution Y > 0, which is precisely condition (1) in the statement (the −1 unicity follows from Λ(Ab − zEb ) ⊂ Dc ). Further, Θ+ (z ) has the form



Θ+−1 (z ) = WCb (z Eˆ b − Aˆ b )−1 K (1 − z ) + W

Since Arg − zErg and

Aˆ bℓ − z Eˆ bℓ −F

Bˆ bℓ (1 − z ) I

have full row normal



U0

0 0 0 0  Im 0  0 0 V −1 DΠ −V −1

0 I 0 0

Aˆ b − z Eˆ b Aˆ bg − z Eˆ bg + K Cˆ g (1 − z ) Bˆ b  0 Aˆ g − z Eˆ g ×  0 0 CΠ 1 CΠ 23



 (8),(19)

=

   

Arg − zErg 0 0 0 0

⋆ Aˆ bℓ − z Eˆ bℓ 0 0 −F

⋆

(see (6.4) in Oară and Varga (1999)), with K from (19), and W a constant J-unitary matrix. The right hand-side of (37) is a (J , J ′ )ˆ (z ) whose lossless factorization of the marginally stable system G system pencil is given by

 − KDz 0   Bˆ g 0   Z 0 I −I  −Zm I 0

Bˆ bℓ (1 − z ) 0 0 I

Aˆ b − z Eˆ b  0 WCb



DΠ

⋆ Bˆ 4 − z Fˆ4 B5 0 0

0 0 0 −Im 0

(38)

   . 

(36)

Necessity. Assume now G(z ) has a (J , J ′ )-lossless factorization (3), where Θ (z ) is (J , J ′ )-lossless and belongs to RL∞ , Π (z ) has marginally stable poles and zeros and full row normal rank. Since Θ (z ) is (J , J ′ )-lossless and in RL∞ , it can be written as Θ (z ) := Θ+ (z )Θ− (z ), where Θ+ (z ) is J-lossless with poles in Dc and Θ− is (J , J ′ )-lossless with poles in D (see Theorem 4.1 in Gohberg et al. (1992) and Lemma 4.9 in Kimura (1997)). We have

Aˆ bg − z Eˆ bg + K Cˆ g (1 − z ) Aˆ g − z Eˆ g W Cˆ g

Bˆ b − KDz , Bˆ g WD



(39)

where we have used (7) and (38). Let U and Z be two unitary matrices that provide the decomposition (8). Then the corˆ (z ) in (39) responding decomposition of the system pencil of G ˆ (z ) has a (J , J ′ )-lossless factorization, is given in (40). Since G there is an invertible (−J )-lossless factor R(z ) which cancels ˆ (z ) all left minimal indices and the unin the product R(z )G ˆ stable zeros of G(z ). Using Theorem 26 in Oară and Sabău (2009), such an invertible factor exists if and only if (14) holds and the matrix Riccati equation (17) has an invertible unique stabilizing solution X > 0. This ends the proof of the necessity part.

2228



C. Oară, R. Marinică / Automatica 49 (2013) 2221–2228

U 0 0



 0 I 0  0 0 Ip



Aˆ b − z Eˆ b Aˆ bg − z Eˆ bg + K Cˆ g (1 − z ) Bˆ b − KDz Z 0 Aˆ g − z Eˆ g Bˆ g



WCb

W Cˆ g

WD

Ab − zEb Aˆ bg − z Eˆ bg 0 I 0 K (1 − z )  0 0  0 I 0 Aˆ g − z Eˆ g 0 0 I 0 0 W Cb Cˆ g

U 0





= 0 I   = 

Arg − zErg ⋆ ⋆ ⋆ 0 Aˆ bℓ − z Eˆ bℓ Bˆ bℓ (1 − z ) Bˆ 4 − z Fˆ4  .  0 0 0 B5 0 WCbℓ WDbℓ WB6



Bb Bˆ g Z D



(40)

References Aliev, F., & Larin, V. (1997). Algorithm of J-spectral factorization of polynomial matrices. Automatica, 33, 2179–2182. Basar, T., & Olsder, G. (1995). Dynamic noncooperative game theory. London: Academic Press. Chan, T. (1987). Rank revealing QR factorizations. Linear Algebra and its Applications, 88/89, 67–82. Chu, D., & Ho, W. (2005). Computation of the (J , J ′ )-lossless factorization for general rational matrices. SIAM Journal on Control and Optimization, 44(1), 240–267. Chu, D., & Tan, R. (2008). The extended J-spectral factorization for descriptor systems. Automatica, 44, 585–597. Doyle, J., Glover, K., Khargonekar, P., & Francis, B. (1989). State-space solutions to standard H 2 and H ∞ control problems. IEEE Transactions on Automatic Control, 34(8), 831–847. Gahinet, P., & Apkarian, P. (1994). A linear matrix inequality approach to H ∞ control. International Journal of Robust and Nonlinear Control, 4, 421–448. Gantmacher, F. R. (1960). The theory of matrices. New York: Chelsea. Gohberg, I., Kaashoek, M. A., & Ran, A. C. M. (1992). Factorizations of and extensions to J-unitary rational matrix functions on the unit circle. Integral Equations Operator Theory, 15(2), 262–300. Green, M., Glover, K., Limebeer, D., & Doyle, J. C. (1990). A J-spectral factorization approach to H ∞ -control. SIAM Journal on Control and Optimization, 28, 1350–1371. Hassibi, B., Sayed, A., & Kailath, T. (1999). Indefinite-quadratic estimation and control: a unified approach to H 2 and H ∞ Theories. Philadelphia: SIAM. Hung, Y. S., & Chu, D. (1998a). (J , J ′ )-lossless factorization for discrete-time systems. International Journal of Control, 71(3), 517–533. Hung, Y. S., & Chu, D. (1998b). On extended (J , J ′ )-lossless factorization. Linear Algebra and its Applications, 271, 117–138. Ionescu, V., Oară, C., & Weiss, M. (1999). Generalized Riccati theory and robust control: a Popov function approach. New York: John Wiley & Sons. Ionescu, V., & Weiss, M. (1993). Two Riccati formulae for the discrete-time H ∞ control problem. International Journal of Control, 57(1), 141–195. Iwasaki, T., & Skelton, R. E. (1994). All controllers for the general H ∞ control problem: LMI existence conditions and state space formulas. Automatica, 30, 1307–1317. Kawamoto, A., & Katayama, T. (2003). A non-standard J-spectral factorization of rational matrices via zero compensator approach. International Journal of Robust and Nonlinear Control, 13, 1–33. Kimura, H. (1997). Chain-scattering approach to H ∞ -control. Boston: Birkhauser. Kongprawechnon, W., & Kimura, H. (1996). J-lossless conjugation and factorization for discrete-time systems. International Journal of Control, 65, 867–884. Kongprawechnon, W., & Kimura, H. (1998). J-lossless factorization and H ∞ -control for discrete-time systems. International Journal of Control, 70(3), 423–446. Kwakernaak, H., & Sebek, M. (1994). Polynomial J-spectral factorization. IEEE Transactions on Automatic Control, 39, 315–328. Oară, C. (2005). Constructive solutions to spectral and inner–outer factorizations with respect to the disk. Automatica, 41(11), 1855–1866.

Oară, C., & Andrei, R. (2011a). Computation of the general (J , J ′ )-lossless factorization. IEEE Transactions on Automatic Control, 56(3), 710–717. Oară, C., & Andrei, R. (2011b). The (J , J ′ )-spectral factorization of a general discretetime system. In Proceedings of the ACC. San Francisco, California. Oară, C., & Andrei, R. (2012). The descriptor discrete-time Riccati equation: numerical solution and applications. In Proceedings of the MTNS. Melbourne, Australia. Oară, C., & Marinică, R. (2012). The general (J , J ′ )-lossless factorization for descriptor discrete-time systems. In Proceedings of the CDC. Maui, Hawaii. Oară, C., & Sabău, S. (2009). Minimal indices cancellation and rank revealing factorizations for rational matrix functions. Linear Algebra and its Applications, 431(10), 1785–1814. Oară, C., & Varga, A. (1999). Minimal degree coprime factorization of rational matrices. SIAM Journal on Matrix Analysis and Applications A, 21(1), 245–278. Penzl, T. (1998). Numerical solution of generalized Lyapunov equations. Advances in Computational Mathematics, 8(1–2), 33–48. Rosenbrock, H. H. (1970). State-space and multivariable theory. New York: Wiley. Scherer, C. (1992). H ∞ -optimization without assumptions on finite or infinite zeros. SIAM Journal on Control and Optimization, 30, 143–166. Stoorvogel, A. A. (1991). The singular H ∞ control problem with dynamic measurement feedback. SIAM Journal on Control and Optimization, 29, 160–184. Verghese, G., Lévy, B., & Kailath, T. (1981). A generalized state-space for singular systems. IEEE Transactions on Automatic Control, 26, 811–831. Verghese, G., Van Dooren, P., & Kailath, T. (1979). Properties of the system matrix of a generalized state-space system. International Journal of Control, 30, 235–243. Xin, X., & Kimura, H. (1994a). Singular (J , J ′ )-lossless factorization for strictly proper functions. International Journal of Control, 59, 1383–1400. Xin, X., & Kimura, H. (1994b). (J , J ′ )-lossless factorization for descriptor systems. Linear Algebra and its Applications, 205/206, 1289–1318. Zhou, K., Doyle, J. C., & Glover, K. (1996). Robust and optimal control. New Jersey: Prentice Hall.

Cristian Oară was born in Bucharest, Romania, in 1968. He received the Diploma Engineer Degree and Ph.D. Degree from the University ‘‘Politehnica’’ Bucharest, Romania, in 1992 and 1995, respectively. From September 1995 to April 1997 he was a postdoctoral Fellow and a NATO Research Fellow at Universite Catholique de Louvain, Louvain-la-Neuve, Belgium. From May 1997 to June 1999 he was an Alexander von Humboldt Fellow at Institute for Robotics and System Dynamics, DLR, Germany. Since 1992 he is a staff member of the Faculty of Automatic Control and Computers, Politehnica University of Bucharest, where he is now Professor of Systems Theory and Head of the Department of Automatic Control and Systems Engineering. He has served on the editorial boards of IEEE Transactions on Automatic Control (1999–2003) and International Journal of Control (2000–2007). He has also served in the IPC of various international conferences. Dr. Oarã is the recipient of ‘‘In Hoc Signo Vinces’’ (2000) for young researchers. His research interest span the broad areas of Systems Theory, Linear Algebra, and Numerical Algorithms.

Raluca Marinică is a postdoctoral researcher at the Delft Center for Systems and Control, Delft University of Technology. She obtained a diploma in Automatic Control and Computer Science from University ‘‘Politehnica’’ Bucharest in 2005, an M.Sc. in Intelligent Systems and a Ph.D. in Control Systems from the Department of Automatic Control and Systems Engineering, University ‘‘Politehnica’’ Bucharest in 2010. Her research interests include control, numerical linear algebra, descriptor systems and adaptive optics.