h2 optimization problem for general discrete-time systems

Politehnica University of BucharestFaculty of Automatic Control and Computer Science

Department of Automatic Control and Systems Engineering

DIPLOMA DISSERTATION

H2 Optimization Problem for GeneralDiscrete-Time Systems

StudentFlorin Sebastian Tudor

AdviserProfessor Cristian Oara

Bucharest, 2013

Contents

Glossary iii

1 Introduction 1

2 Preliminaries 32.1 Matrix pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 The regular pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 The singular pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Rational Matrix Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Zeros, poles and structural indices . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Realization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Constructing centered realizations . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Structural elements in terms of realizations . . . . . . . . . . . . . . . . . 13

2.3 Discrete-Time Descriptor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 An evaluation of the H2−norm . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Optimality and Descriptor Riccati Equations 203.1 Popov Quadruples and associated objects . . . . . . . . . . . . . . . . . . . . . . 203.2 The stabilizing solution of the Descriptor Riccati Equation . . . . . . . . . . . . 223.3 Descriptor Linear Quadratic Problem . . . . . . . . . . . . . . . . . . . . . . . . 24

4 H2 Optimization Problem 284.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Special Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.1 The Full Information problem . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.2 The Two-Block problem and its dual . . . . . . . . . . . . . . . . . . . . . 334.3.3 The case D11 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4 A second proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Numerical Examples 43

Bibliography 48

ii

Glossary

DDTARE(Σ) Descriptor Discrete-Time Algebraic Riccati Equation associated with the PopovQuadruple Σ. ii, 21–23, 43

DDTARS(Σ) Descriptor Discrete-Time Algebraic Riccati System associated with the PopovQuadruple Σ. ii, 21

DLQP(Σ) Descriptor Linear Quadratic Problem. ii, 24, 25

DSP(Σ) Descriptor Pymplectic Pencil associated with the Popov Quadruple Σ. ii, 22, 23, 43

FI Full Information. ii, 31, 33–35, 37

KSPYS(Σ, J) Kalman-Szego-Popov-Yackubovich system associated with the Popov QuadrupleΣ and the sign matrix J . ii, 21, 22

LFT Linear Fractional Transformation. ii, 18, 19

LLFT Lower Linear Fractional Transformation. ii, 19

LMIDP Left Minimal Indices Displacement Problem. ii

RMF Rational Matrix Function. ii, 3, 6, 7, 9–14, 17–19, 25, 30, 40, 43, 44

TFM Transfer Function Matrix. ii, 22

iii

1 Introduction

Abstract. In this thesis we solve the H2 optimization problem for discrete-time descriptorsystems. A state-space characterization is provided for the optimal controller in terms of thesolutions of two generalized descriptor algebraic Riccati equations. The main result is expressedusing centered state-space realizations, that exhibit the same attractive features and allows forformulas that bear the same elegant simplicity of the standard (proper) case. It turns out thatthe optimal controller is proper and has the well-known observer-based structure. Therefore,our results are natural extensions of the H2 optimization problem for general rational matrixfunctions, even improper or polynomial. Three numerical examples of growing complexity arealso included in order to show the applicability of our results.

A descriptor system provides a great tool for modeling general physical systems, since theycan include non-dynamic constraints, impulsive elements, and algebraic dynamical systems. Theassociated transfer matrix is an arbitrary rational matrix function, even improper or polynomial.Descriptor systems have received large attention since the middle of the 1970s, see, e.g., Brenan,Campbell, and Petzold 1989; Dai 1989; Kunkel and Mehrmann 2006 and references therein. Inthe literature they appear under various different names such as singular systems, generalizedstate-space systems, differential-algebraic equations, semi-state systems, constrained systems,degenerate systems, etc. The range of applications of descriptor systems varies from engineeringincluding power systems, electrical networks, aerospace engineering, mechanical systems, chem-ical processes (Dai 1989; Kunkel and Mehrmann 2006; Gunther and Feldmann 1999; Rabier andRheinboldt 2000), to economics (Luenberger 1977).

As a consequence, the control system design of the descriptor systems has received muchattention in the past few decades. In fact, the linear quadratic (LQ) linear regulator problem hasbeen extensively studied in the literature (Bender and Laub 1987; Mehrmann 1991; Katayamaand Minamino 1992; Geerts 1994). There are also papers on the output feedback control problem.Kucera was the first to consider the LQG (Kucera 1986) and H2 (Kucera 1992) control problemsfor a singular (descriptor) system, using the Diophantine equation approach. The main drawbackhere is that there has been given no clear relationship between the Diophantine equation andthe state-space descriptor realization of the plant.

The model matching technique has been extensively used in the literature as an alternativefor solving H2 optimization problem, in both state-space and transfer function setting (Doyle1984; Francis 1982; Vidyasagar 1985). An explicit solution to the H2 optimization problem fora multivariable continuous-time descriptor system (based on the model matching technique) isavailable in the article (Takaba and Katayama 1998).

In this thesis, we provide a solution to the H2 optimization problem for a multivariablediscrete-time descriptor system using a novel approach, based on Popov’s positivity theory(Popov 1973) and the original results in (Ionescu, Oara, and Weiss 1999). The generalizedeigenvalue problem also plays a key role in this thesis. The recent results, e.g. numerically soundalgorithms, realization theory, centered realizations, factorization of rational matrices, developedin the book (Oara, Van Dooren, and Varga 2000), have been extensively used in this thesis. Themain tool for the problem at hand is a special type of algebraic Riccati equation, namely thedescriptor discrete-time algebraic Riccati equation, investigated in the article (Oara and Andrei2013). A slightly more general form of the descriptor discrete-time algebraic Riccati equation

1

Chapter 1. Introduction

also emerged in (Oara and Sabau 2009). Furthermore, we shall use as main technical tools inour proofs several mathematical objects (typically, associated with the optimality requirement)well-known in the literature, e.g. Ionescu, Oara, and Weiss 1999; Zhou, Doyle, and Glover 1996,namely the Kalman-Szego-Popov-Yackubovich system and the Hamiltonian system.

Throughout the thesis, we use the standard notations. By C, D, and ∂D we denote thecomplex plane, the open unit disk, and the unit circle, respectively, and let C := C∪∞ be theclosed complex plane. λ is a complex variable, and z ∈ C will be used to explicitly specify thediscrete-time case. For a constant matrix A with elements in C we denote by A∗ its conjugatetranspose. If A is invertible A−∗ is its conjugate transpose inverse. A hermitian matrix satisfiesA = A∗, and we denote by A > 0 (A < 0) if it is in addition positive definite (negative definite).A matrix has full column (row) rank if its rank equals the number of columns (rows). Theidentity matrix is denoted by I. The set of p × m rational matrices with coefficients in C isdenoted by Cp×m(λ).

Acknowledgements

First of all, I would like to express my gratitude to Professor Cristian Oara, my adviser, forhis guidance, productive comments and strong inspiration. His interesting ideas, good advicesand careful checking of the proofs made this dissertation possible. I would also like to thankProfessor Dumitru Popescu and Professor Dan Stefanoiu, for the opportunity of working onmany interesting research topics and novel ideas in the field of renewable energetical systems.

2

2 Preliminaries

In this chapter we shall present briefly some results on the generalized eigenvalue and eigen-tructure problem, respectively. Further, we introduce Rational Matrix Functions, on which

our theory relies. We shall define and describe the set of invariants induced by the generalizedSmith-MicMillan form, their structural indices, and left (right) minimal indices. RealizationTheory of RMF also plays a central role in our developments. We will introduce a new type ofrealizations, namely centered realizations, having many useful properties, as it will be discussedin the sequel. Finally, we will analyze in the time-domain setting the discrete-time descriptorsystems.

2.1 Matrix pencils

In this section we study first order polynomial matrices in the complex variable λ,

A− λE,

called matrix pencils, where A,E ∈ Cm×n are two rectangular matrices. For m × n matrixpencils we present canonical forms and corresponding canonical invariants (Gantmacher 1960)induced by the following relation of equivalence. Two pencils A − λE and A − λE are strictlyequivalent if there exists two invertible matrices Q ∈ Cm×m, Z ∈ Cn×n, called transformationmatrices, such that

Q(A− λE)Z = A− λE. (2.1)

We discuss successively two problems of growing complexity:

(i) the regular pencil case, with A−λE square and det(A−λE) 6≡ 0, leading to the Weierstrasscanonical form and to generalized eigenvalue problems;

(ii) the singular pencil case, with A − λE arbitrary, leading to the Kronecker canonical formand to generalized eigenstructure problems.

We pay special attention to the particular case of unitary equivalence, where transfor-mation matrices Q, Z in (2.1) are restricted to be unitary, i.e., Q∗Q = I, Z∗Z = I. Unitarytransformations are primarily important for numerical computations as they may yield numericalstability of many computational algorithms.

With the help of unitary transformations, we can deduce quasi-canonical forms, e.g. theWeierstrass-like and Kronecker-like forms, that exhibit the complete set of canonical invariantsinduced by general strict equivalence. With invertible transformations performed on quasi-canonical forms, one can obtain the corresponding canonical forms. Although the canonicalforms are important, their accurate numerical computation is usually difficult because the trans-formation matrices are often ill-conditioned.

For more details on matrix pencils and their canonical forms, see the excellent books(Gantmacher 1960; Horn and Johnson 1991) and the paper (Van Doreen 1981b).

3

Chapter 2. Preliminaries

2.1.1 The regular pencil

Here, A and E are square matrices of dimension n and and the pencil A−λE has non-vanishingdeterminant. The problem of solving for λ ∈ C the polynomial equation

χ(λ) := det(A− λE) = 0 (2.2)

is called the generalized eigenvalue problem and χ(λ) is called the characteristic polynomial.Since E may be singular, χ(λ) has generally degree nf ≤ n. The nf roots of χ(λ) are the finiteeigenvalues of A− λE. We say that λ =∞ is a generalized eigenvalue of A− λE if λ = 0 is aneigenvalue of the reciprocal pencil E−λA, or equivalently, if E is singular. The multiplicity n∞of the infinite generalized eigenvalue of A− λE is by definition the multiplicity of λ = 0 as theeigenvalue of E − λA. Let χ(λ) = a0 + a1λ+ · · ·+ anfλ

nf , with anf 6= 0. Thus, we can write

det(E − λA) = (−1)nλn−nf (a0 + a1λ+ · · ·+ anfλnf ).

It follows that n∞ = n − nf . Therefore, an n × n matrix pencil has always n generalizedeigenvalues (finite and infinite) which form the spectrum of A− λE, denoted Λ(A− λE).

For a generalized eigenvalue λ0 of A − λE there always exists a nonzero vector x ∈ Cn,called the generalized eigenvector, such that

Ax = λ0Ex, if λ0 is finite,Ex = 0, if λ0 is infinite. (2.3)

In the regular case, the equivalence relation (2.1) with Q,Z invertible induces the Weier-strass canonical form

AW − λEW :=[In∞ − λE∞ 0

0 Af − λInf

], (2.4)

where Af and E∞ are in the Jordan canonical form, with E∞ nilpotent. Recall that a matrixAf is in the Jordan canonical form if

Af =

A11(λ1). . .

Akk(λk)

, with Aii(λi) :=

Js

(i)1

(λi). . .

Js

(i)hi

(λi)

.Here, λ1, . . . , λk is the set of k distinct eigenvalues of the matrix Af and Js(λi) is an s × smatrix of the form

Js(λi) :=

λi 1

λi. . .. . . 1

λi

,

called an elementary Jordan block. For an eigenvalue λi, the sizes s(i)j , j = 1, . . . hi are called

the partial multiplicities of λi, the positive integer hi is the geometric multiplicity and the sumni := s

(i)1 + s

(i)2 + . . . s

(i)hi

is called the algebraic multiplicity. Note that nf =∑ki=1 ni. E∞ is a

nilpotent matrix in Jordan canonical form

E∞ =

Js∞

1(0)

. . .Js∞

h∞(0)

,

4


and Js(0) is an s × s elementary nilpotent Jordan block. We define the partial, geometricaland algebraic multiplicities of the infinite generalized eigenvalue of A−λE as the correspondingmultiplicity of the zero eigenvalue of E∞. Thus, for λ =∞ the partial multiplicities are s∞i , i =1, . . . , h∞, the geometric multiplicity is h∞ and the algebraic multiplicity is n∞ :=

∑h∞i=1 s

∞i .

If we restrict ourselves to unitary equivalence transformations, we can derive a quasi-canonical form, called the complex Weierstrass-like form, which reveals the generalized eigenval-ues of a regular pencil and its partial multiplicities. Therefore, the complete set of Weierstrasscanonical invariants can be computed with a numerically reliable algorithm, namely the QZalgorithm. For a complete discussion, see (Oara, Van Dooren, and Varga 2000).

The following concepts will be useful in the sequel. We give first the definition of a deflatingsubspace.

Definition 2.1. The linear space V ⊂ Cn is called a deflating subspace of the regular n × nmatrix pencil zM −N if dim(MV +NV) = dim(V).

For a deflating subspace V of zM −N denote by (zM −N)|V and by Λ(zM −N)|V themap and the spectrum of the pencil restricted to V, respectively. The next result, taken from(Oara and Andrei 2013), gives a useful characterization of deflating subspaces in terms of basismatrices.

Lemma 2.2. Let zM −N be a regular n× n matrix pencil.

1. If V = Im(V ) is an `−dimensional deflating subspace of zM − N , where V is a basismatrix for V, then there exists a regular `× ` pencil zT − S which is strictly equivalent to(zM −N)|V such that MV S = NV T.

2. Conversely, if MV S = NV T holds for a certain n× ` basis matrix V and a regular `× `pencil zT − S, then V = Im(V ) is a deflating subspace of zM −N and zT − S is strictlyequivalent to (zM −N)|V .

2.1.2 The singular pencil

We discuss briefly the case of an arbitrary m× n singular pencils. A singular pencil A− λE iseither non-square, i.e. m 6= n or has a vanishing determinant det(A− λE) ≡ 0.

The normal rank r of the pencil, denoted rank n(A−λE), is the rank of A−λE for almostall λ ∈ C (but a finite number of points). If νl := m − r > 0 then we say that the pencil hasa (nontrivial) left singular structure. If νd := n− r > 0 then the pencil has a (nontrivial) rightsingular structure.

Strict equivalence leads, for an appropriate choice of the invertible transformation matricesQ and Z in (2.1), to the Kronecker canonical form of an arbitrary pencil,

Q(A− λE)Z = AKR − λEKR,

AKR − λEKR :=

Lε1. . .

LενrIn∞ − λE∞

Af − λInfLTη1

. . .LTηνl

. (2.5)

5


Here Lk, k ≥ 0 denotes the bidiagonal k × (k + 1) pencil

Lk :=

λ −1. . . . . .

λ −1

,Af and E∞ are in the Jordan canonical form, with E∞ nilpotent. The regular part of A − λEis defined by the regular pencil diag (In∞ , Af )−λ diag (E∞, Inf ). The partial multiplicities, thegeometric multiplicity and the algebraic multiplicity of the finite and infinite eigenvalues havethe same definitions as for the regular pencil. Note that the finite generalized eigenvalues of thesingular pencil A− λE are the eigenvalues of the matrix Af .

The singular part of the pencil is defined by the right and left singular Kronecker structureas follows. The εi × (εi + 1) blocks Lεi , i = 1, . . . , νr are the right elementary Kronecker blocks,and εi ≥ 0 are called the right Kronecker indices. The (ηj + 1)× ηj blocks LTηj , j = 1, . . . , νl arethe left elementary Kronecker blocks, and ηj ≥ 0 are called the left Kronecker indices. Noticethat εi and ηj can be zero.

From the Kronecker canonical form (2.5), the normal rank of the pencil A − λE can beexpressed as

r = nr + n∞ + nf + nl, (2.6)

where nr :=νr∑i=1

εi and nl :=νl∑j=1

ηj . If the pencil A − λE is regular, there are no Kronecker

indices, i.e., the pencils Lεi and LTηj are void and the Kronecker canonical form reduces to theWeierstrass canonical form.

Starting from a singular pencil, we can always achieve the Kronecker-like form, that canbe obtained using only unitary transformations gaining therefore benefits in terms of the numer-ical reliability. The Kronecker-like form displays the same information as the canonical form.Precisely, any matrix pencil A− λE with A,E ∈ Cm×n can be reduced by unitary transforma-tions Q ∈ Cm×m, Z ∈ Cn×n to the block upper triangular form (Van Doreen 1979; Oara andVan Doreen 1997)

AKR − λEKR :=

Aε − λEε ∗ ∗ ∗

0 A∞ − λE∞ ∗ ∗0 0 Af − λEf ∗0 0 0 Aη − λEη

, (2.7)

where the regular part is determined by the regular pencil A∞ − λE∞ and Af − λEf (finiteand infinite generalized eigenvalues). The singular part is determined by the pencil Aε − λEεcontaining the right Kronecker indices and having full row rank ∀λ ∈ C, and by Aη−λEη, whichcontains the left Kronecker indices and has full column rank ∀λ ∈ C.

A more detailed discussion on this topic (staircase condensed form, computation of theKronecker-like form) can be found in (Oara, Van Dooren, and Varga 2000). We do not elaboratefurther in this direction.

2.2 Rational Matrix Functions

In this section we study rational matrix functions RMF in the variable λ ∈ C, i.e. matricesT(λ) ∈ Cp×m(λ) whose elements are rational functions, i.e.,

T(λ) :=[pij(λ)qij(λ)

]i=1,p j=1,m

,

6


where pij(λ) and qij(λ) are scalar polynomials with coefficients in C, with qij(λ) 6≡ 0, ∀i, j.

The values for which T(λ) is undefined or losses rank play a central role in the theoryof RMF, and are associated with the poles and zeros of T(λ). We also need to define theirstructural indices and minimal indices for the left and right null spaces.

We shall also discuss the realization theory, which is a topic of interest for this thesis. Theminimality requirement will be further analyzed.

2.2.1 Zeros, poles and structural indices

For the definition of poles and zeros and the associated indices of a rational matrix we needfirst Smith-McMillan’s theory. The key idea of Smith-McMillan theory is to perform certainequivalence transformations defined next in order to reduce a rational matrix T(λ) to a diagonalform, called the Smith-McMillan form, from which poles and zeros can be further defined. Westart with several definitions.

A square rational matrix T(λ) is called regular at λ0 ∈ C if the constant matrix T(λ0) iswell defined (with finite entries) and invertible, and is called regular at ∞ if the limit T(∞) :=limλ→∞ T (λ) exists and is invertible. A rational matrix which is regular ∀λ ∈ C (but infinity)is called unimodular (or ∞−unimodular). Extending the definition of unimodular matrices, wesay that a rational matrix is λ0−unimodular if it is regular ∀λ ∈ C but λ0. It is easy to see thata square rational matrix is λ0−unimodular if and only if it is square, each of its elements hasno poles in C \λ0, and has an inverse with the same property.

Definition 2.3. Let T1(λ) and T2(λ) be p × m rational matrices. Then T1(λ) and T2(λ)are called equivalent under λ0−unimodular transformations if there exist two λ0−unimodularmatrices U(λ) and V (λ) such that

U(λ)T1(λ)V (λ) = T2(λ). (2.8)

Furthermore, two RMF are strongly equivalent (denoted ~) if they are equivalent under λ0−unimodulartransformations, ∀λ ∈ C (including infinity).

The central result on equivalence under unimodular transformations is the generalizedSmith-McMillan form, which describes the simplest rational matrix in each equivalence class, asfollows, see (Oara, Van Dooren, and Varga 2000).

Theorem 2.4. Generalized Smith-McMillan Form. Let T(λ) be a p×m rational matrixwith coefficients in C and having normal rank r, and let λ0 ∈ C be fixed. Let two constants α, βbe as follows:

α = 1, β = 0, for λ0 =∞,α = λ0, β = 1, for λ0 ∈ C.

(2.9)

Then there exist two λ0−unimodular matrices U(λ) and V (λ), both with coefficients in C, thatbring T(λ) to the Smith-McMillan form (with respect to λ0)

S(λ) = U(λ)T(λ)V (λ), (2.10)

S(λ) :=[

D(λ) 00 0

]

=

ε1(λ)η1(λ)(α− λβ)k1

. . . 0r×(m−r)εr(λ)ηr(λ)(α− λβ)kr

0(p−r)×r 0(p−r)×(m−r)

.

7


The polynomials εi(λ), ηi(λ) with coefficients in C are monic, are pairwise coprime for i =1, . . . , r, have no root at λ0, satisfy the divisibility properties

εi(λ) | εi+1(λ)ηi+1(λ) | ηi(λ) , i = 1, . . . , r − 1,

and the indices ki, i = 1, . . . , r are such that

∂ηi − ∂εi − ki , i = 1, . . . , r

form a nondecreasing sequence. Moreover, the rational factors εi(λ)ηi(λ) , called the invariant factors

of T(λ) (with respect to λ0), and the indices ki are uniquely defined by T(λ) and λ0 and togetherthey form a set of invariants under λ0−unimodular transformations, called the Smith-McMillaninvariants.

From the theorem, it follows that two rational matrices are equivalent in the sense of Def-inition 2.3 if and only if they have the same sets of Smith-McMillan invariants (or, equivalently,the same Smith-McMillan form with respect to λ0) at all λ ∈ C.

The following theorem taken from (Oara, Van Dooren, and Varga 2000) gives an importantresult, namely that one does not have to compute the Smith-McMillan form at each point λ0 ∈ Cin order to conclude that two rational matrices are equivalent.

Theorem 2.5. Let T(λ) ∈ Cp×m(λ) of normal rank r, and let λ0 ∈ C be fixed. Then theSmith-McMillan canonical invariants are independent of the particular choice of λ0 satisfyingthe conditions

T(λ0) is well-defined and rank nT(λ) = rank T(λ0). (2.11)

We are now ready to define another set of invariants for a rational matrix: poles and zeros,and their structural indices. Note that not every λ0 ∈ C is suitable, and thus we shall use theSmith-McMillan form with respect to λ0 given in (2.9), satisfying the condition (2.11).

Definition 2.6. Zeros. The point λ ∈ C is called a (Smith) zero of T(λ) if it is a zero of aninvariant factor εi

ηi. We say that λ =∞ is a (Smith) zero of T(λ) if ∂ηi− ∂εi− ki > 0 for some

index i = 1, . . . , r. The set of all zeros of T(λ) is denoted by Z (T).

Definition 2.7. Poles. The point λ ∈ C is called a pole of T(λ) if it is a pole of an invariantfactor εi

ηi. We say that λ =∞ is a pole of T(λ) if ∂ηi− ∂εi− ki < 0 for some index i = 1, . . . , r.

The set of all poles of T(λ) is denoted by P (T).

With these definition we can equivalently express condition (2.11) as λ0 is neither a polenor a zero of T(λ). Notice also that if T(λ) is regular at λ0 ∈ C then it has neither poles norzeros at λ0. This follows immediately from the generalized Smith-McMillan form (2.10). Inprinciple, a pole is a point in C where an entry of the rational matrix becomes infinite, while azero is a point where the rank of the matrix drops below the normal rank.

Further, we define the structural indices, the degree and the order of a certain zero/poleof T(λ). Let λj , j = 1, . . . , N be N distinct finite poles/zeros of T(λ) and consider the Smith-McMillan form (2.10) where λ0 is neither a pole nor a zero of T(λ). Then D(λ) an be uniquelyfactorized as

D(λ) = Dλ0

N∏j=1

Dλj , (2.12)

where Dλj (λ) := diag((λ− λj)α1(λj), (λ− λj)α2(λj), . . . , (λ− λj)αr(λj)

)∈ Cr×r(λ), with j =

1, . . . , N and Dλ0(λ) := diag((α− λβ)k1 , (α− λβ)k2 , . . . , (α− λβ)kr

)∈ Cr×r(λ) and α, β are

8


chosen according to (2.9). We can now associate with the RMF T(λ) the index set defined∀λ ∈ C as

I(T, λ) :=

α1(λj), . . . , αr(λj), for λj ∈ Z(T) ∪ P(T), j = 1, . . . , N ;α1(∞), . . . , αr(∞), for λ =∞;0, 0, . . . , 0︸︷︷︸

r

, for λ ∈ C−Z(T) ∪ P(T).(2.13)

where αi(∞) := ∂ηi − ∂εi − ki, called the zero-pole excess.

We call the elements of the set I(T, λ) the structural indices of T(λ) at λ. The set ofall structural indices (considered for all λ ∈ C) completely determine and are determined bythe Smith-McMillan invariants and thus they form a complete set for the equivalence relationintroduces by Defintion 2.3. We now state an interesting result, taken from (Oara, Van Dooren,and Varga 2000).

Theorem 2.8. Two rational matrices T1(λ) and T2(λ) with coefficients in C are stronglyequivalent if and only if they have the same dimensions, P(T1) = P(T2), Z(T1) = Z(T2) andI(T1, λ) = I(T2, λ), ∀λ ∈ C.

We introduce further the notions of order and degree of a zero and pole, and the McMillandegree of a rational matrix.

The order ωz(T, λk) of the finite zero λk is the largest positive power of λ− λk occurringin Dλk(λ) - see the factorization (2.12) -, i.e. ωz(T, λk) := αr(λk). The degree is simply thesum δz(T, λk) :=

∑i,αi>0 αi(λ). Note that if αr(λk) ≤ 0, then both the order and the degree

of the zero λk are null. Analogously, the order ωp(T, λk) of the finite pole λk is the largestnegative power of λ − λk occurring in Dλk(λ), i.e. ωp(T, λk) := −α1(λk) and the degree isδp(T, λk) := −

∑i,αi<0 αi(λ). If α1(λk) ≥ 0, then both the order and the degree of the pole λk

are null.

The order of the zero (pole) at infinity is thus the largest positive (negative) zero-poleexcess considered for all scalar rationals εi(λ)

ηi(λ)(α− λβ)ki , i = 1, . . . , r.

By definition, the McMillan (polar) degree of T is the sum of the degrees of all its poles(finite and infinite), i.e.,

δ(T) :=∑λ∈C

δp(T, λ).

Thus the McMillan degree of a rational matrix equals its number of poles (counting degrees andincluding infinity). Finally, notice that when applying the above definitions to a scalar rationalfunction, the classical notion of order (in this case degree equals also order) of poles and zeros,and McMillan degree are retrieved. The present definitions thus give a generalization of theseconcepts to the matrix case.

We define further the right and left minimal indices of a RMF, see for details (Forney1975). Let C(λ) be the field of rational functions in λ with coefficients in C and Cn(λ) be thevectorspace of n−tuples over C(λ). For any rational vectorspace one can always find polynomialbases that feature some properties as explained further. The degree of a polynomial vector isthe largest power of λ occurring in its components. The order of a polynomial basis is the sumof the degrees of its elements. The next result is taken from (Oara and Sabau 2009).

Theorem 2.9. Each vectorspace over Cn(λ) has a minimal polynomial basis, i.e., a basis whoseorder is minimal. The degrees (arranged in a non decreasing order) of two minimal polyno-mial bases of the same vectorspace χ are equal and are called the minimal indices of χ . Thecorresponding order is called the minimal order of χ.

9


Further, let Nr (Nl) be the right (left) null space of T(λ), i.e., the rational vectorspaceof all column vectors v(λ) ∈ Cm(λ) (v(λ) ∈ Cp(λ)) satisfying T(λ)v(λ) = 0 (vT (λ)T(λ) = 0).The dimension of Nr (Nl) is m − r (p − r), where r = rank nT(λ). The minimal indices of Nr(Nl) are called the right (left) minimal indices of the rational matrix function T(λ). Denote bynr(T) (nl(T)) the sum of the right (left) minimal indices of T(λ) and by γ(T) the sum of thetotal multiplicities of all zeros (finite and infinite) of T(λ).

Clearly, the minimal McMillan degree of a rational basis matrix (see Oara, Van Dooren,and Varga 2000) for the right (left) null space of T equals nr(T) (nl(T)). Then, for a RMFT(λ) the following relation among its structural elements holds true, see e.g. (Theorem 3 inVerghese, Van Dooren, and Kailath 1979):

δ(T) = γ(T) + nr(T) + nl(T). (2.14)

2.2.2 Realization Theory

In this subsection we develop the theory of realizations for rational matrices. We introducefirst the notion of centered realization and further we discuss briefly realizations for polynomialmatrices. We also indicate how to construct a centered realization for a given RMF and how toconvert between realizations. Throughout the section T(λ) denotes an arbitrary p×m rationalmatrix function.

Any p ×m RMF T(λ) (even improper or polynomial) has a descriptor realization of theform (see for example Rosenbrock 1970):

T(λ) = D + C(λE −A)−1B =:[A− λE B

C D

], (2.15)

where A − λE is a regular pencil, A,E ∈ Cn×n, B ∈ Cn×m, C ∈ Cp×n,and D ∈ Cp×m. Wecall the positive integer n the order (or the dimension) of the realization (2.15). Note that theright-hand side of (2.15) is a rational matrix, not to be confused with a block matrix.

Although (2.15) can represent any rational matrix it has a couple of drawbacks for theproblems under investigation. For example, if ∞ is a pole of T(λ) than the minimum order ofa realization is strictly greater than the McMillan degree of T(λ) while D does not representthe value of T(λ) at any particular point. To circumvent this, we will work with a slightly moregeneral type of realizations called centered.

Centered Realizations

In order to introduce the notion of centered realization we have to fix a certain point λ0 ∈ C inwhich we actually "center" the realization. In this thesis we shall discuss the theory of realizationscentered at λ0 with the assumption that T(λ) has no poles at λ0, which we call the proper case.

Let λ0 ∈ C be fixed and choose two constants α, β as follows:α = 1, β = 0, for λ0 =∞,α = λ0, β = 1, for λ0 ∈ C.

(2.16)

A realization centered at λ0 of the RMF T ∈ Cp×m(λ) is a representation of the form

T(λ) = D + C(λE −A)−1B(α− βλ) =:[A− λE B

C D

]λ0

, (2.17)

10


where the matrix pencil A−λE is regular, and the parameter matrices have appropriate dimen-sion as in (2.15). Whenever we use realizations centered at λ0 we assume the implicit choice ofα and β according to (2.16). The positive integer n is called the order (or the dimension) of therealization (2.17). A realization is called minimal if its order is as small as possible.

Two realizations of the same RMF

T(λ) =[A− λE B

C D

]λ0

=[A− λE B

C D

]λ0

are called equivalent if they have the same order n and there are two invertible matrices Q ∈Cn×n, Z ∈ Cn×n, such that

A− λE = Q(A− λE)Z, B = QB, C = CZ, D = D. (2.18)

We say that the realization (2.17) is proper if the constant matrix αE − βA is invertible.Notice that T(λ) has a proper realization centered at λ0 only if it has no poles at λ0. We callthe realization (2.17) normalized if we have αE − βA = I. Clearly, a normalized realization isautomatically proper.

If the realization (2.17) is proper then D = limλ→λ0 T(λ). Furthermore, for a properrealization we have that

rankD = rank nT(λ).

In particular, for proper realizations T(λ) is invertible if and only if D is invertible, and therealization for the inverse is proper as well. Observe that these properties of proper realizationsare key for solving several control problems.

A realization centered at λ0 (proper or not) can be converted into a realization centeredat a different λ0 . However, these conversions are highly unreliable from a numerical viewpointand should be avoided (Oara, Van Dooren, and Varga 2000).

Polynomial Matrices

Consider a polynomial matrix function P(λ) ∈ Cp×m(λ) having the form

P(λ) := P0 + P1λ+ P2λ2 + · · ·+ Pdλ

d.

Then an (improper) realization centered at infinity of P(λ) is given by

P(λ) =[A∞ − λE∞ B∞

C∞ 0

]=

I Pd

−λI . . . Pd−1. . . . . . ...

−λI I P00 · · · 0 I 0

. (2.19)

Further, notice that a polynomial matrix has all poles at infinity. The change of variableλ← 1

λ is appropriate. Thus, another way to obtain a realization centered at infinity for P(λ) isto consider instead the rational matrix

R(λ) := − 1λ

P( 1λ

), (2.20)

which satisfies R(∞) = 0, i.e. is strictly proper, and has all poles at 0. Thus R(λ) has a

11


realization of the formR(λ) = C(λI −N)−1B =

[N B

C 0

], (2.21)

with N nilpotent. Note that a minimal realization for R(λ) can be obtained (i.e., the matricesN,B,C) in a standard manner. Further, from (2.20) and (2.21), P(λ) can be written as

P(λ) = − 1λR

(1λ

)= − 1

λC(

1λI −N

)−1B = −C(I − λN)−1B

= C(λN − I)−1B.(2.22)

In conclusion, P(λ) has a realization centered at infinity which can be obtained with standardalgorithms, given by

P(λ) =[I − λN B

C 0

]. (2.23)

Remark 2.10. Let h be the nilpotent index of N , i.e. Nh = 0. Expanding the transfer matrixof P(λ) in a Taylor series we get that

P(λ) = C(λN − I)−1B = −CB − CNBλ− · · · − CNh−1Bλh−1.

Therefore, the transfer matrix of the system P(λ) is a polynomial matrix with the degree h− 1.

Constructing centered realizations

We present here a method to obtain from a given RMF a proper realization centered at λ0 6∈Λ(A − λE). Let T ∈ Cp×m(λ) be a known rational matrix having McMillan degree n. Thenthere always exist a decomposition (see, e.g. Theorem 2-6.2 in Dai 1989)

T(λ) = G(λ) + P(λ), (2.24)

where G(∞) = 0 and P(λ) is a matrix polynomial. Thus, the problem of constructing arealization for T(λ) may be decomposed in two problems of constructing realizations for G(λ),which has no poles at infinity and is strictly proper, and P(λ), which is a polynomial and hasall poles at infinity. The realizations are given by

G(λ) =[A1 − λI B1C1 0

], P(λ) =

[I − λN B2C2 D2

]. (2.25)

Let λ0 ∈ C (finite) not a pole of G(λ). Therefore, the matrices λ0I − A1 and I − λ0N areinvertible. Note that a minimal realization for G(λ) can be obtained in a standard manner,while a simply procedure for obtaining realization for P(λ) was already detailed, see (2.23). Wehave successively

G(λ) = C1(λI −A1)−1B1 = C1(λI −A1)−1(λ0I −A1)(λ0I −A1)−1B1= C1(λI −A1)−1(λ0I −A1 + λI − λI)(λ0I −A1)−1B1= C1(λ0I −A1)−1B1 + C1(λI −A1)−1(λ0I −A1)−1B1(λ0 − λ).

. (2.26)

In a similar manner, we obtain that

P(λ) = λ0C2(I − λ0N)−1B2 + C2(λN − I)−1(I − λ0N)−1B2(λ0 − λ) +D2. (2.27)

12


Note that the new realizations for G(λ) and P(λ) are centered at λ0, i.e.,

G(λ) =[A1 − λI (λ0I −A1)−1B1C1 C1(λ0I −A1)−1B1

]λ0

,

P(λ) =[I − λN (I − λ0N)−1B2C2 λ0C2(I − λ0N)−1B2 +D2

]λ0

.

(2.28)

Furthermore,

T(λ) =

A1 − λI 0 (λ0I −A1)−1B10 I − λN (I − λ0N)−1B2C1 C2 C1(λ0I −A1)−1B1 + λ0C2(I − λ0N)−1B2 +D2

λ0

. (2.29)

If the realizations (2.25) to start with are minimal, then the realization (2.29) is minimal as well.Moreover, the dimension of the minimal realization is exactly n, the McMillan degree of T(λ).

Further, we will normalize the centered minimal realization. Consider that T(λ) is givenby (2.29) and denote

T(λ) =[A− λE B

C D

]λ0

= D + C(λE −A)−1B(λ0 − λ).

Note that if λ0 is not a pole of G(λ), then λ0E −A is invertible. Thus we can write

T(λ) = D + C(λE −A)−1(λ0E −A)−1(λ0E −A)B(λ0 − λ):= D + C(λE − A)−1B(λ0 − λ),

where E and A are such that λ0E − A = I. With this, we obtain

T(λ) = D + C(

λλ0−λE −

1λ0−λA

)−1B

= D + C[

λλ0−λE −

1λ0−λ(λ0E − I)

]−1B

= D + C(

1λ0−λI − E

)−1B.

Notice that this expression exhibits the same features as the proper case, having a single polematrix A := E, i.e.,

T(λ) = D + C

( 1λ0 − λ

I − A)−1

B.

and furthermorelimλ→λ0

C

( 1λ0 − λ

I − A)−1

B = 0, T(λ0) = D.

2.2.3 Structural elements in terms of realizations

In this subsection we define the structural elements of a RMF, i.e., poles, zeros and their struc-tural indices, as well as left and right minimal indices in terms of a particular realization. Itturns out that if the realization exhibits certain properties, like minimality or irreducibility, thenall the structural elements of T(λ) can be read from the singular structure of two polynomialmatrices associated with any realization. We shall discuss this topic for centered realizations.

We will also characterize the structural properties of a linear descriptor system, namelycontrollability, observability, minimality, irreducibility, etc.

13


Let T(λ) be an arbitrary RMF and let

T(λ) =[A− λE B

C D

]λ0

(2.30)

be a realization centered at λ0 ∈ C. Assume that the realization is proper, i.e. λ0 is not a poleof T. For this realization we define the pole pencil P(λ) := A− λE, and the system pencil

S(λ) :=[A− λE B(α− λβ)C D

]=[A αBC D

]− λ

[E βB0 0

], (2.31)

where α and β are chosen according to (2.16). We give now an important theorem whichcharacterizes all the structural indices of a rational matrix function.

Theorem 2.11. Let T(λ) be given by a proper minimal realization of the form (2.30) andassociate with it the pole pencil A − λE and the system pencil S(λ) given in (2.31). Then wehave:

(i) Poles. The poles of T(λ) are the generalized eigenvalues of the pole pencil. Moreover, themodulus of the strictly negative structural indices of T(λ) at a pole µ ∈ C are pairwiseequal to the multiplicities of µ as a generalized eigenvalue of the pole pencil.

(ii) Finite zeros. The finite zeros of T(λ) are the finite generalized eigenvalues of the systempencil. Moreover, the strictly positive structural indices of T(λ) at a zero µ ∈ C arepairwise equal to the partial multiplicities of µ as a generalized eigenvalue of the systempencil.

(iii) Infinite zeros. T(λ) has infinite zeros if and only if the system pencil has an infinite gen-eralized eigenvalue with a partial multiplicity strictly greater than 1. Moreover, the strictlypositive structural indices of T(λ) at ∞ are pairwise equal to the partial multiplicities ofthe infinite generalized eigenvalues of the system pencil minus 1.

(iv) Minimal indices. The minimal indices to the left (right) of T(λ) are pairwise equal tothe elementary left (right) Kronecker indices of the system pencil.

The previous theorem shows that, provided we have a minimal realization for T(λ), theproblem of computing the Smith-McMillan invariants and the left and right minimal indices ofT(λ) is reducible to the simpler problem of computing the eigenstructure of two matrix pencils,for which there are available numerically sound algorithms, see for example Van Doreen 1979;Oara and Van Doreen 1997; Oara, Van Dooren, and Varga 2000.

Controllability and observability (and the less restrictive properties stabilizability and de-tectability) are important structural properties in systems theory. In the case of linear systemsthey admit purely algebraic characterizations in terms of the system matrices. The characteri-zations of controllability and observability given in the following definition are an extension ofthe Popov-Belevitch-Hautus tests.

Definition 2.12. Let λ0 ∈ C and Ω ⊂ C. Consider that the order of the realization (2.30) tostart with is n.

Controllability & Stabilizability. The pair (A− λE,B) is called Ω−controllable if

rank[A− λE B

]= n, ∀λ ∈ C− Ω,

rank[E B

]= n, λ =∞,

(2.32)

and simply controllable if Ω = ∅. Furthermore, if Ω = D the pair is called stabilizable.

14


Observability & Detectability. The pair (C,A− λE) is called Ω−observable if

rank[A− λEC

]= n, ∀λ ∈ C− Ω,

rank[EC

]= n, λ =∞,

(2.33)

and simply observable if Ω = ∅. Furthermore, if Ω = D the pair is called detectable.

Controllability and observability are dual properties in the following sense. The pair(A − λE,B) is controllable if and only if the pair (B∗, A∗ − λE∗) is observable, and the pair(C,A− λE) is observable if and only if the pair (A∗ − λE∗, C∗) is controllable.

A realization that is both controllable and observable is called irreducible. The realizationis called minimal if its order is as small as possible. Notice that an irreducible proper realizationis automatically minimal.

However, the most important property of proper realizations (2.30) is that their minimalorder coincides with the McMillan degree of T(λ) as shown in Theorem 4 in (Oara and Sabau2009) which is a straightforward extension of the Kalman canonical decomposition in the stan-dard case (in which the realization is centered at λ0 =∞, see for example Verghese, Van Dooren,and Kailath 1979).

2.3 Discrete-Time Descriptor Systems

In this section we will discuss discrete-time descriptor systems (λ = z) from the dynamicalpoint of view. A linear, time-invariant, discrete-time descriptor system may be described by thefollowing equations:

Exk+1 + βBuk+1 = Axk + αBukyk = Cxk +Duk

, k = 0, 1, . . . (2.34)

where xk ≡ x(tk) ∈ Cn is the state vector, uk ∈ Cm is the control input, and yk ∈ Cp is themeasured output. Here, A,E ∈ Cn×n, B ∈ Cn×m and C ∈ Cp×n are constant matrices. Withoutloss of generality, assume that the matrix E is singular having rankE < n.

Apply the Z Transform to the system (2.34) to obtain

(zE −A)x(z) = B(α− βz)u(z) ⇒ x(z) = (zE −A)−1B(α− βz)u(z)y(z) = Cx(z) +Du(z) ⇒ y(z) =

(D + C(zE −A)−1B(α− βz)

)u(z) ,

where z ∈ C is a complex variable and by abuse of notation x(z) is the unilateral Z Transform ofx. Further, notice that the transfer function for the input u to the output y, i.e., y(z) = Tu(z),is given by the expression

T(z) = D + C(zE −A)−1B(α− βz).

But this is exactly a centered realization of a general rational matrix function, also given in(2.17). Here, α and β are chosen such that λ0 := α

β is not an generalized eigenvalue of the polepencil A− zE.

As pointed out in the previous section, there are two invertible matrices Q and Z that

15


bring the pole pencil to the Weierstrass canonical form

Q(A− zE)Z =[A1 − zIn1 0

0 In2 − zN

], (2.35)

with N nilpotent having the nilpotent index h, A1 ∈ Cn1×n1 , N ∈ Cn2×n2 , n1 + n2 = n. Withthe change of variables

x = Z

[x1x2

],

with x1 ∈ Cn1 , x2 ∈ Cn2 , the system (2.34) can be written asEZ

[σx1σx2

]+ βBσu = AZ

[x1x2

]+ αBu

y = CZ

[x1x2

]+Du

, (2.36)

where by σ we denote the shfit-operator, defined on the space of vector valued sequences w =(wk)k∈Z as (σw)k := wk+1, with k ∈ Z. Pre-multiply by Q the first equation of (2.36) to obtain[

I 00 N

] [σx1σx2

]+ βQBσu =

[A1 00 I

] [x1x2

]+ αQBu. (2.37)

Finally, the system (2.34) can be written equivalently asσx1 + βB1σu = A1x1 + αB1u

Nσx2 + βB2σu = x2 + αB2uy = C1x1 + C2x2 +Du

, (2.38)

with QB =:[B1B2

], CZ =:

[C1 C2

]. Note that for an improper realization (centered at

λ0 =∞) we have that α = 1, β = 0 and the equations can be written asσx1 = A1x1 +B1uy1 = C1x1 +Du

,

Nσx2 = x2 +B2u

y2 = C2x2. (2.39)

Further, observe that the form (2.39) exhibits two subsystems, a standard one, which we willcall the forward subsystem and a subsystem with all poles at ∞, the backward subsystem.

The forward subsystem is a discrete-time standard system whose state is

x1(k) = Ak1x1(0) +k−1∑i=0

Ak−i−11 B1u(i), (2.40)

in which there exists a causal relationship between state and inputs. The backward systemis a backward recurrence of state. By repeatedly left multiplying the first equation withI,N,N2, . . . , Nh−1, and sum all up (see Dai 1989) we have

x2(k) = −h−1∑i=0

N iB2u(k + i), (2.41)

for which we see that to determine x2(k) future inputs are needed, and thus the system isanti-causal. Combining (2.40) and (2.41) we can easily obtain the general state for the system.

An excellent discussion on various topics of interest for this thesis, e.g. stability in Lya-

16


punov’s sense, stabilizability, (R-, impulse) controllability, detectability, (R-, Y-) observability,pole-placement etc. can be found in the book of Dai 1989. We will further state a result ofinterest for this study.

Theorem 2.13. The discrete-time system (2.34) is stable iff its pole set Λ(A − zE) is insidethe open unit disk on the complex plane, which is denoted D. The closure of this set, i.e., theunit circle, will be denoted ∂D.

Remark 2.14. According to the theorem, a stable system has the matrix E invertible, sincethere are no poles at infinity (the realization is assumed minimal). Thus the descriptor systemis a standard system, being causal as well.

Let u = Fx + v, where F ∈ Cm×n is an arbitrary matrix. With this change of variable,the descriptor system (2.34) becomes

(E + βBF )σx+ βBσv = (A+ αBF )x+ αBvy = (C +DF )x+Dv

. (2.42)

The pole pencil for this system is (A+αBF )− z(E + βBF ) ≡ A− zE +BF (α− βz). But thisis the generalized eigenvalue assignment problem. The solution is available in the literature, seefor example (Dai 1989; Oara and Varga 1999). We state now a result of interest.

Lemma 2.15. Let (A− zE,B) be a controllable pair, and let Γ ⊂ C be a set of n elements (notnecessarily distinct, and possibly containing infinity), and let α, β ∈ C, not both zero, such thatαβ 6∈ Λ(A− zE) and α

β 6∈ Γ. Then there exists a matrix F such that

Λ (A− zE +BF (α− βz)) = Γ.

The next result is given in (Theorem 2, Oara and Sabau 2011) and it provides existenceconditions in order to stabilize a given matrix pencil.

Lemma 2.16. Let (A− zE,B) be a stabilizable pair and λ0 := αβ ∈ C−Λ(A− zE). Then there

exists a matrix F such that

Λ(A− zE +BF (α− βz)

)⊂ D. (2.43)

2.3.1 An evaluation of the H2−norm

We shall assume that the control energy is finite, i.e., u ∈ `+2 . Here, `2 is an infinite dimensionalHilbert space, consisting in the set of all complex square summable sequences u ≡ ukk∈Z, i.e.,

∞∑k=−∞

|uk|2 <∞, 〈u,w〉 :=∞∑

k=−∞uk wk.

Naturally, `+2 is the closed subspace of `2 of sequences with positive support k ≥ 0. Let L2(∂D)be the Hilbert space of matrix-valued complex functions T(z) for which the L2(∂D)−norm,

‖T‖22 := 12π

∫ 2π

0Trace

[T∗(ejθ)T(ejθ)

]dθ (2.44)

is finite. If T(z) is a RMF, then its L2(∂D)−norm is finite if and only if it has no poles on theunit circle. The H2−norm is defined for stable rational matrices and has the same expression(2.44).

We give now a useful evaluation of the H2−norm of a stable system in terms of theoutput yi in the case in which its input is a unit impulse, ui = δei i = 1, . . . ,m, where δ is thediscrete-time unit impulse and ei, i = 1, . . . ,m is the canonical basis for Rm.

17


Lemma 2.17. LetT(z) =

[A− zE B

C 0

]λ0

(2.45)

be a minimal realization of an arbitrary discrete-time descriptor system with no poles on theunit circle, A− zE stable, and λ0 ∈ ∂D. Then

‖T‖22 =m∑i=1‖yi‖22. (2.46)

Proof. Since the system has no poles on the unit circle and A− zE is stable, the H2−norm iswell-defined and finite. Note that the matrix E is invertible and the system T(z) is causal, seeRemark 2.14. Let Tk, k = 0, 1, . . . be the causal impulse matrix

yik = Tkei, i = 1, . . .m, k = 0, 1, . . .

We obtain successivelym∑i=1‖yi‖22 =

m∑i=1

∑k∈Z

(yik)T yik =∑k∈Z

m∑i=1

eTi T Tk Tkei =∑k∈Z

Trace(T Tk Tk

). (2.47)

Using the discrete-time version of the Parseval’s formula

∑k∈Z

Trace(T Tk Tk

)= 1

2π

∫ 2π

0Trace

[T∗(ejθ)T(ejθ)

]dθ

we obtain (2.46). This completes the proof.

2.3.2 Connections

We will discuss here some useful connections in the sequel, namely the Linear Fractional Trans-formation LFT and the Redheffer product.

The LFT can be viewed as an extension to RMF of the well-known conformal mappingcalled Moebius transformation, namely

f(z) = a+ bz

c+ dz, a, b, c, d ∈ C.

Consider two discrete-time descriptor systems given byEσx+ βB1σw + βB2σu = Ax+ αB1w + αB2u

z = C1x+D11w +D12uy = C2x+D21w

, (2.48)

and EKσxK + βBKσuK = AKxK + αBKuK

yK = CKxK +DKuK(2.49)

having the RMF

T(z) =[

T11 T12T21 T22

]=

A− zE B1 B2C1 D11 D12C2 D21 0

λ0

(2.50)

18


andK(z) =

[AK BKCK DK

]λ0

, (2.51)

respectively. A partitioned system of type (2.50) will be called a generalized system. If T22(z)has the number of inputs and outputs equal to the numbers of outputs and inputs of K(z),respectively, then we define the lower linear fractional transformation (LLFT) of T(z) withK(z) (in this order) as the system with input w and output z obtained by setting u = yK anduK = y. Note that the connection is automatically well-posed, since the matrix[

I 0DK I

]

is invertible. Thus the RMF of the resulting system is defined as

TR(z) = LLFT(T,K) := T11 + T12K(I −T22K)−1T21 (2.52)

and has a realization given by

TR(z) =

A− zE +B2DKC2(α− βz) B2CK(α− βz) B1 +B2DKD21BKC2(α− βz) AK − zEK BKD21C1 +D12DKC2 D12CK D11 +D12DKD21

λ0

. (2.53)

We shall define now the Redheffer Product. This connection is an extension of the LFT.Consider two generalized systems (2.48) and

EKσxK + βBK1σwK + βBK2σuK = AKxK + αBK1w + αBK2uKzK = CK1xK +DK11wK +DK12uKyK = CK2xK +DK21wK

, (2.54)

having the transfer function matrices given by (2.50) and

K(z) =[

K11 K12K21 K22

]=

AK − zEK BK1 BK2CK1 DK11 DK12CK2 DK21 0

λ0

, (2.55)

respectively. If the number of inputs and outputs of T22(z) is equal to the number of outputsand inputs of K11(z), then we define the Redheffer product of T(z) with K(z) (in this order)

as the system with input[wuK

]and output

[zyK

]obtained by setting wK = y and u = zK .

The connection is automatically well-posed, since the matrix[I 0

DK11 I

]

is invertible. Thus the RMF of the resulting system is defined as TR(z) = T ⊗K and has arealization given by

TR(z) =

A− zE +B2DK11C2(α− βz) B2CK1(α− βz) B1 +B2DK11D21 B2DK12

BK1C2(α− βz) AK − zEK BK1D21 BK2C1 +D12DK11C2 D12CK1 D11 +D12DK11D21 D12DK12

DK21C2 CK2 DK21D21 0

λ0

.

(2.56)Note that the formulas are a generalization of the standard case with E = I, α = 1, β = 0, see(Ionescu, Oara, and Weiss 1999; Zhou, Doyle, and Glover 1996).

19

3 Optimality and Descriptor Riccati Equations

We introduce in this chapter the notion of a Popov Quadruple and various associated ob-jects, which provides a general and elegant framework for the theoretical developments

in this thesis. Further, we will present a special type of Riccati equation, namely the descriptordiscrete-time Riccati equation (DDTARE). We will also give computable formulas and solv-ability conditions for the DDTARE. The linear quadratic problem for descriptor discrete-timesystems will be stated and solved. We shall denote throughout the thesis z := λ ∈ C the complexvariable for discrete-time systems.

3.1 Popov Quadruples and associated objects

Definition 3.1. Popov Quadruples. A quadruple of matrices

Σ := (A− zE,B;P ), P :=[Q LL∗ R

]= P ∗, (3.1)

where A,E ∈ Cn×n, B ∈ Cn×m, Q = Q∗ ∈ Cn×n, L ∈ Cn×m, R = R∗ ∈ Cm×m and P ∈C(n+m)×(n+m) is called a Popov quadruple. The more detailed form, i.e., Σ = (A−zE,B;Q,L,R)will be also frequently used.

A Popov quadruple can be seen as a synthetic representation of a discrete-time descriptorsystem and an associated quadratic performance (cost) index:

Eσx+ βBσu = Ax+ αBu, x0 =: ξ, (3.2)

JΣ(ξ, u) :=∞∑k=0

[x∗k u∗k

]P

[xkuk

], (3.3)

where λ0 := αβ is not a generalized eigenvalue of the pole pencil A− zE. Further, we associate

with Σ = (A−zE,B;Q,L,R) and the criterion (3.3) several mathematical objects. The discrete-time descriptor system

Eσx + βBσu = Ax + αBuβQσx − A∗σλ + βLσu = αQx − E∗λ + αLu

v = L∗x − B∗λ + Ru, (3.4)

is called the descriptor discrete-time Hamiltonian system associated with Σ, denoted DDTHS(Σ).We shall be interested in those triplets (x, λ, u) that annihilate the output v of the Hamiltoniansystem.

Consider a hermitian matrix X = X∗ ∈ Cn×n. The dissipation matrix associated with Σis defined as

DΣ(X) :=[E∗XE −A∗XA+Q (αE − βA)∗XB + LB∗X(αE − βA) + L∗ R

]. (3.5)

20

Chapter 3. Optimality and Descriptor Riccati Equations

We call the system of equations

DΣ(X)[IF

]= 0 (3.6)

in the unknowns F ∈ Cn×m, X ∈ Cn×n, the descriptor discrete-time algebraic Riccati systemassociated with Σ, and we denote it as DDTARS(Σ). A pair (X,F ) with X hermitian is calleda stabilizing solution if it satisfies (3.6), and the matrix pencil z(E + βBF ) − (A + αBF ) isstable, i.e., Λ (z(E + βBF )− (A+ αBF )) ⊂ D. It is well-known that the stabilizing solution isunique. The proof of this claim is very similar to the standard case, see e.g. (Ionescu, Oara,and Weiss 1999), and therefore is omitted.

Consider now that the matrix R is nonsingular, i.e., invertible. Writing down the twoequations from (3.6) we get that

F := −R−1(B∗X(αE − βA) + L∗), (3.7)

E∗XE −A∗XA+Q+ ((αE − βA)∗XB + L)F = 0. (3.8)

Substituting (3.7) in (3.8) we obtain

E∗XE −A∗XA+Q− ((αE − βA)∗XB + L)R−1 (B∗X(αE − βA) + L∗) = 0, (3.9)

known as the descriptor discrete-time algebraic Riccati equation associated with Σ, and denotedDDTARE(Σ). The hermitian matrix X = X∗ is called the stabilizing solution to DDTARE(Σ)if the matrix pencil z(E + βBF )− (A+αBF ) is stable. It is proved in (Oara and Andrei 2013,Theorem 3) that the stabilizing solution is unique. Moreover, the equation (3.9) is associatedwith a state-space system of the form

T(z) = D + C(zE −A)−1B(α− βz).

and the quadratic criterion (3.3). We will further analyze this remark in the sequel.

A slightly more general form of the Riccati objects can be found in (Oara and Sabau2009), namely DDTARS and DDTARE in J−form, respectively, where J is a sign matrix,

J =[−Im1

Im2

], m = m1 +m2.

For any sign matrix J introduce the Kalman-Szego-Popov-Yakubovich system in J-form KSPYS(Σ, J):

R = V ∗JV(αE − βA)∗XB + L = W ∗JVE∗XE −A∗XA+Q = W ∗JW

, (3.10)

where R ∈ Cm×m is nonsingular. Here X = X∗ ∈ Cn×n, V ∈ Cm×m and W ∈ Cm×n. SinceR is assumed invertible, it follows from the first equation of (3.10) that a necessary solvabilitycondition for the KSPYS(Σ, J) is the invertibility of J . Moreover, R should satisfy sgn (R) = J .A triplet of matrices (X = X∗, V,W ) satisfying (3.10) such that z(E + βBF ) − (A + αBF ) isstable, for

F := −V −1W (3.11)

is called a stabilizing solution to the KSPYS(Σ, J) and F is called the stabilizing feedback.

The following result is a slightly modified version of Lemma 2.1 in (Ionescu 1999) andgives a relation between DΣ(X) and the KSPYS(Σ, J).

Lemma 3.2. The following statements are equivalent.

21


(i) There exists X = X∗ for which DΣ(X) admits a J-factorization

DΣ(X) =[W ∗

V ∗

]J[W V

].

(ii) The KSPYS(Σ, J) has a solution (X = X∗, V,W ).

The proof follows mutatis mutandis from the standard case. We are now ready to give animportant definition, on which Popov’s positivity theory is based, see (Popov 1973).Definition 3.3. For J = Im, the KSPYS (3.10) is known as the KSPYS(Σ, I) in positivityform and it reads

R = V ∗V,(αE − βA)∗XB + L = W ∗V,E∗XE −A∗XA+Q = W ∗W.

(3.12)

Moreover, the Popov Quadruple Σ = (A− zE,B;Q,L,R) will be called positive if there exists atriplet of matrices (X = X∗, V,W ) such that (3.12) holds for J = Im.

The following particular matrix pencil will be used to characterize and compute the so-lutions to the DDTARE(Σ). It is the corresponding extension, for the problem at hand, of thesymplectic pencils introduced in (Van Doreen 1981a).Definition 3.4. The (2n+m)× (2n+m) matrix pencil zMΣ −NΣ, with

MΣ =

E 0 βBβQ −A∗ βL0 0 0

, NΣ =

A 0 αBαQ −E∗ αLL∗ −B∗ R

(3.13)

is called the descriptor symplectic pencil associated with Σ, denoted DSP(Σ).

Notice that the DSP(Σ) can be regarded as the transmission pencil of the Hamiltoniansystem (3.4). If we take the output v ≡ 0 and denote

p :=

xλu

,we can write the DDTHS(Σ) in (3.4) in the descriptor form MΣσp = NΣp. Apply the Ztransform to obtain the DSP(Σ) zMΣ −NΣ. The transfer function matrix (TFM) from u to vof the Hamiltonian system (3.4) is called the Popov function associated with Σ (Popov 1973),and it is given by

ΠΣ(z) :=

zE −A 0 B−Q(α− βz) zA∗ − E∗ L

L∗ B∗ R

λ0

, (3.14)

where λ0 := αβ /∈ Λ(zE −A) ∪ Λ(zA∗ − E∗). It can be easily checked that

ΠΣ(z) =[B∗(E∗ − zA∗)−1(α− βz)∗ I

] [ Q LL∗ R

] [(zE −A)−1B(α− βz)

I

]. (3.15)

3.2 The stabilizing solution of the Descriptor Riccati Equation

In this section we investigate the numerical solution of the DDTARE(Σ) in (3.9). We will givenumerically checkable conditions and computable formulas for the unique stabilizing solution.The discussion is based on (Oara and Andrei 2013).

22


We claim that if λ0 := αβ is not among the generalized eigenvalues of the matrix pencil

zE −A, with α, β chosen according to (2.16), then the matrix (αE − βA) is invertible. Indeed,if λ0 /∈ Λ(zE −A) then det(λ0E −A) 6= 0. But this is equivalent to det(αE − βA) 6= 0 and theclaim is proved.

A useful characterization of the spectrum of the DSP(Σ) is given in (Proposition 7 in Oaraand Andrei 2013). Also, Lemma 2.2 and Proposition 7 from the same reference implies that theDSP(Σ) has an n− dimensional deflating subspace with stable spectrum (included in D) if andonly if there the DSP(Σ) has no finite generalized eigenvalues with modulus equal to 1. We givenow an important result for this exposition.

Theorem 3.5. Assume that αE − βA is invertible. The following statements are equivalent.

(i) R is invertible and DDTARE(Σ) (3.9) has a stabilizing solution.

(ii) The DSP(Σ) is regular and has a maximal n−dimensional stable deflating subspace havinga basis matrix

V =

V1V2V3

nnm

, (3.16)

with invertible V1. Moreover, the stabilizing solution can be computed from

X = V2V−1

1 (αE − βA)−1 (3.17)

and the stabilizing Riccati feedback can be computed from

F = V3V−1

1 . (3.18)

Proof. (i)⇒ (ii) Since αE− βA and R are both invertible, (Proposition 7 in Oara and Andrei2013) shows that the DSP(Σ) is regular. Let X be the stabilizing solution of the DDTARE andF the stabilizing Riccati feedback. Then, from Lemma 2.2 there is a regular pencil zT −S withΛ(zT − S) ⊂ D such that

(E + βBF )S = (A+ αBF )T (3.19)

Further, the DDTARE (3.9) can be written as E∗XE−A∗XA+Q+((αE − βA)∗XB + L)F = 0.Post-multiply this expression with βS − αT , pre-multiply (3.19) with (βA − αE)∗X and sumthe two expressions to obtain

(A∗X(αE − βA) + βLF + βQ)S = (E∗X(αE − βA) + αLF + αQ)T. (3.20)

From the DDTARE we can also write B∗X(αE − βA) + L∗ + RF = 0 which, together with(3.19) and (3.20), leads to E 0 βB

βQ −A∗ βL0 0 0

IX(βA− αE)

F

S =

A 0 αBαQ −E∗ αLL∗ −B∗ R

IX(βA− αE)

F

T. (3.21)

But this is equivalent to MΣV S = NΣV T , where zMΣ −NΣ is the DSP(Σ), and Im(V ) =: V isits stable n−dimensional deflating subspace. This ends the first part of the proof.

The second part of the proof, i.e., (ii)⇒ (i) follows the same line as the proof of Theorem10 in Oara and Andrei 2013.

The following lemma gives a useful relation between the solutions of the DDTARE(Σ) andthe solutions of the KSPYS(Σ, I) in positivity form.

23


Lemma 3.6. The DDTARE (3.9) has a hermitian stabilizing solution X if and only if theKSPYS (3.12) has a stabilizing solution (X = X∗, V,W ).

Proof. First, consider that (X = X∗, V,W ) satisfies the KSPYS (3.12). From the first and thelast equation in (3.12) we get

V = V ∗ = R1/2,

W = W ∗ = (E∗XE −A∗XA+Q)1/2.(3.22)

Subtitute this in the unused equation of (3.12) to obtain

L+ (αE − βA)∗XB = (E∗XE −A∗XA+Q)1/2R1/2,

(L+ (αE − βA)∗XB)R−1/2 = (E∗XE −A∗XA+Q)1/2.(3.23)

Multiply the last expression with R−1/2 (L∗ +B∗X(αE − βA)) to arrive at the DDTARE (3.9).This completes the first part of the proof.

Conversely, if X is a hermitian solution to (3.12), then we can choose a V such that thefirst equation of (3.12) is satisfied. Further, define W such that it satisfies the second equationand it follows that it satisfies also the third equation in (3.12). Also, it is easy to see that thestabilizing Riccati feedbacks are equal:

F = −R−1 (L∗ +B∗X(αE − βA))= −V −1W

. (3.24)

This completes the proof.

3.3 Descriptor Linear Quadratic Problem

In this section we deal with the optimal control of discrete time descriptor systems with aquadratic performance criterion. Consider the system given in (3.2) with the initial conditionarbitrary fixed x(0) := ξ. The objective is to find a control sequence uk defined on Z+ such thatthe state xk is driven to the origin at k → ∞ while the performance index (3.3) is minimized,that is

minu

JΣ(ξ, u) ≡ minu

∞∑k=0

[x∗k u∗k

] [ Q LL∗ R

] [xkuk

], (3.25)

for some Q = Q∗, R = R∗ > 0. This is a generalization of the standard Linear QuadraticProblem originated in (Kalman 1964), which we call the Descriptor Linear Quadratic Problemassociated with the Popov quadruple Σ, DLQP(Σ). Here we have assumed that R > 0 toemphasis that the control energy has to be finite, i.e., u ∈ `+2 . So this is the space over whichthe sum is minimized. It is also generally assumed that P ≥ 0, and therefore can be factored as

P :=[Q LL∗ R

]=[C∗1D∗12

] [C1 D12

]=[C∗1C1 C∗1D12D∗12C1 D∗12D12

]. (3.26)

for some C1 ∈ Cp×n, D12 ∈ Cp×m having appropriate dimensions. Note that with the factoriza-tion (3.26) the Popov Quadruple can be written as Σ12 := (A − zE,B;C∗1C1, C

∗1D12, D

∗12D12)

and the KSPYS(Σ12, I) (3.12) in the positivity form becomes

D∗12D12 = V ∗V(αE − βA)∗XB + C∗1D12 = W ∗VE∗XE −A∗XA+ C∗1C1 = W ∗W

(3.27)

24


As we will see in the sequel, it is useful to consider realizations centered on the unit circle∂D, i.e. λ0 ∈ ∂D. For this purpose, consider α = ejθ0 , θ0 ∈ R and let β := α = e−jθ0 . Thus

λ0 = α

α= α2

|α|= α2 = e2jθ0 ∈ ∂D,

where λ0 = α2 is not among the generalized eigenvalues of the matrix pencil A− zE. It impliesthat α2 is not a pole of the rational matrix function T12(z) := D12 + C1(zE − A)−1B(α − αz)and thus the centered realization on the unit circle of T12(z) is proper.

We are now ready to state the DLQP(Σ). Consider the discrete time descriptor systemEσx+ αBσu = Ax+ αBu , x(0) =: ξ,

z = C1x+D12u ,(3.28)

where α ∈ ∂D (the unit circle) and assume that the parameter matrices satisfy the followinghypotheses:

(H1) The pair (A− zE,B) is stabilizable;

(H2) rank[A− ejθE B(α− αejθ)

C1 D12

]= n+m, ∀θ ∈ [0, 2π).

Find an optimal control sequence u ∈ `+2 such that the feedback system is internally stable andthe performance criterion JΣ(ξ, u) ≡ ‖z‖22 is minimized, that is,

minu∈`+2

‖C1x+D12u‖22.

Remark 3.7. The hypothesis (H1) is clearly necessary for the existence of a stabilizing controlsequence u.

Remark 3.8. If the hypothesis (H2) holds true, then for θ = 2θ0 where θ0 is such that α = ejθ0

we get

rank[A− e2jθ0E 0

C1 D12

]= n+m ⇒ rankD12 = m, (3.29)

or D12 ∈ Cp×m has full column rank. Thus R ≡ D∗12D12 is invertible.

Remark 3.9. The hypothesis (H2) ensure that the RMF T12(z) has no zeros on the unit circleand therefore the associated DSP(Σ12)

z

E 0 αBαC∗1C1 −A∗ αC∗1D12

0 0 0

− A 0 αBαC∗1C1 −E∗ αC∗1D12D∗12C1 −B∗ D∗12D12

(3.30)

has no generalized eigenvalues on the unit circle. With Lemma 2.2 and (Proposition 7 in Oaraand Andrei 2013), it follows that the DSP(Σ12) has an n− dimensional stable deflating subspace,with a basis matrix partitioned as in (3.16). It was shown in (Oara and Andrei 2013) that V1in (3.16) is invertible, and thus the stabilizing hermitian solution X exists.

Therefore, provided that the hypotheses (H1) and (H2) hold true, andD12 has full columnrank, we can conclude that the DDTARE(Σ12)

E∗XE−A∗XA−((αE − αA)∗XB + C∗1D12) (D∗12D12)−1 (D∗12C1 +B∗X(αE − αA))+C∗1C1 = 0

has a stabilizing hermitian solution X = X∗. We are now ready to state the main result of thissection.

25


Theorem 3.10. There exists a unique optimal control sequence for the DLQP(Σ12) problem,namely uk = Fxk, with

F = − (D∗12D12)−1 (D∗12C1 +B∗X(αE − αA)) . (3.31)

Moreover,minu∈`+2

‖z‖22 = −ξ∗Xξ . (3.32)

Proof. Consider the KSPYS(Σ12, I) in (3.27) associated with the DDTARE(Σ12). Then with(H1), (H2), Remark 3.9, and Lemma 3.6 it follows that the KSPYS(Σ12, I) has a stabilizingsolution (X = X∗, V,W ), with the stabilizing Riccati feedback F = V −1W . Note that thematrix pencil z(E + αBF )− (A+ αBF ) is stable, i.e.,

Λ (z(E + αBF )− (A+ αBF )) ⊂ D. (3.33)

With the change of variable vk := uk − Fxk, the system (3.28) can be written as(E + αBF )xk+1 + αBvk = (A+ αBF )xk + αBvk, x(0) = ξ

zk = (C1 +D12F )xk.(3.34)

Denote AF := A + αBF , EF := E + αBF and CF := C1 + D12F . The closed-loop system isthus stable. Further, insert the initial condition in the closed-loop dynamics to get a descriptorsystem with x(0) = 0:

EFxk+1 + αBvk + αξδk+1 = AFxk + αBvk + αξδk ,zk = CFxk ,

k = 0, 1, . . . (3.35)

where δkk∈Z is the unit discrete impulse. It is obvious that the optimality of the 2-normis reached, as in the standard case, for vk = 0, ∀k ∈ Z, see (Chapter 14 in Zhou, Doyle,and Glover 1996). Furthermore, note that in order to recover the DSP(Σ12) from the DescriptorHamiltonian System DDTHS (3.4), we are interested in the triplets (x, λ, u) for which the outputv ≡ 0. Moreover, it is easy to see that elimination of x and λ in DDTHS (3.4) leads to u = Fx.

We will further evaluate the 2-norm for the sequence zkk∈Z+ and obtain successivelyusing the KSPYS (3.27):

‖zk‖2 = ‖C1xk +D12uk‖2 = (C1xk +D12uk)∗(C1xk +D12uk)= x∗kC

∗1C1xk + x∗kC

∗1D12uk + u∗kD

∗12C1xk + u∗kD

∗12D12uk

(3.27)= ‖Wxk + V uk‖2 + x∗kA∗XAxk − x∗kE∗XExk − 2 Re [x∗k(αE − αA)∗XBuk] .

Let uk = Fxk = −V −1Wxk for which the minimum is attained. The above expression triviallybecomes

min ‖zk‖2 = x∗kA∗XAxk − x∗kE∗XExk − 2 Re [x∗k(αE − αA)∗XBFxk]

= [x∗kA∗XAxk + 2 Re (αx∗kA∗XBFxk)]− [x∗kE∗XExk + 2 Re (αx∗kE∗XBFxk)] .

Note that

x∗k(A+ αBF )∗X(A+ αBF )xk = x∗kA∗XAxk + 2 Re (αx∗kA∗XBFxk) + x∗kF

∗B∗XBFxk,

x∗k(E + αBF )∗X(E + αBF )xk = x∗kE∗XExk + 2 Re (αx∗kE∗XBFxk) + x∗kF

∗B∗XBFxk.

With this, obtain that min ‖zk‖2 = x∗kA∗FXAFxk−x∗kE∗FXEFxk. But from the dynamics of the

system (3.35) with vk = 0,∀k we know that

EFxk+1 = AFxk − ξ∆k , x(0) = 0, ∆k := αδk+1 − αδk .

26


Thus AFxk = EFxk+1 + ξ∆k. We evaluate the expression further to get

min ‖zk‖2 = (EFxk+1 + ξ∆k)∗X(EFxk+1 + ξ∆k)− x∗kE∗FXEFxk= x∗k+1E

∗FXEFxk+1 − x∗kE∗FXEFxk + 2 Re(∆∗kξ∗XEFxk+1) + ξ∗Xξ|∆k|2.

(3.36)Sum the both sides of the equality (3.36) from k = 0 to ∞, i.e., in the `+2 space:

min ‖z‖22 = −2 Re(αξ∗XEFx1) + |α|2ξ∗Xξ. (3.37)

Here, we have considered that∞∑k=0

(x∗k+1E

∗FXEFxk+1 − x∗kE∗FXEFxk

)= 0, since the system

(3.35) is stable and therefore limk→∞ xk = 0. Further, observe that for k = 0 we get from (3.35)

EFx1 = αξ ⇒ x1 = αE−1F ξ.

With the expression of x1, (3.37) and |α|2 = 1 (because α = ejθ0 ∈ ∂D) we obtain the mainresult min ‖z‖22 = −ξ∗Xξ. This ends the whole proof.

27

4 H2 Optimization Problem

In this chapter we consider the H2 Optimization Problem which consists in finding a controller,for a given descriptor system, that stabilizes the resulting closed-loop system and minimizes

its input-output H2−norm. First of all, we formulate the problem. The main result is given inSection 4.2 and its proof is based on some special problems, each solved in the following section.We will also give a second proof of our main result, which is based on a successive reduction tosimpler problems.

4.1 Problem formulation

Consider a discrete-time descriptor system with input[wu

]and output

[zy

]having the

rational matrix function (RMF) T(z) ∈ Cp×m(z) partitioned as[zy

]= T

[wu

]=[

T11 T12T21 T22

] [wu

], (4.1)

where Tij ∈ Cpi×mj (z) with i, j ∈ 1, 2 and m := m1 +m2, p := p1 + p2. Associate with T(z)a centered minimal realization given by

T(z) =

A− zE B1 B2C1 0 D12C2 D21 0

λ0

=[

0 D12D21 0

]+[C1C2

](zE −A)−1

[B1 B2

](α− αz),

(4.2)

where A,E ∈ Cn×n, Bi ∈ Cn×mi , Cj ∈ Cpj×n and Dij ∈ Cpi×mj , with i, j ∈ 1, 2. The minimalrealization is centered at λ0 := α2 ∈ ∂D, having |α|2 = 1. We will assume in the sequel that thematrix pencil A− zE is regular, i.e., det(zE−A) 6= 0, and that λ0 is not among the generalizedeigenvalues of A− zE.

In the time domain setting, the generalized plant is given byEσx+ αB1σw + αB2σu = Ax+ αB1w + αB2u

z = C1x+D12uy = C2x+D21w

, (4.3)

where by σ is the shfit-operator. Further, define a controller for the plant (4.3) as the lineardiscrete-time descriptor system

EKσxK + αBKσuK = AKxK + αBKuKyK = CKx+DKuK

, (4.4)

28

Chapter 4. H2 Optimization Problem

having the RMF

K(z) =[AK − zEK BK

CK DK

]λ0=α2

. (4.5)

The closed-loop system is obtained by connecting the controller (4.4) to the plant (4.3) suchthat u ≡ yK and uK ≡ y. After some algebra we find that the closed-loop system with input wand output z, i.e. z = TCLw is given by

TCL = LLFT(T,K) = T11 + T12K(I −T22K)−1T21, (4.6)

TCL(z) =

A− zE +B2DKC2(α− αz) B2CK(α− αz) B1 +B2DKD21BKC2(α− αz) AK − zEK BKD21C1 +D12DKC2 D12CK D12DKD21

λ0=α2

. (4.7)

We can now formulate the optimal H2 Control Problem: given a plant T(z), find anappropriate controller K(z) such that the closed-loop system is internally stable, i.e.

Λ(ACL − zECL) ⊂ D

and has minimum H2 norm, i.e. ‖TCL‖2 attains its minimum over the class of stabilizingcontrollers. In this formulation we have implicitly assumed that T11(α2) = D11 = 0 andT22(α2) = D22 = 0. The assumptions are made for simplifying the formulas, with no loss ofgenerality. Indeed, if K is a solution to the problem with D22 = 0, then K(I + D22K)−1 is asolution for the original problem. The case D11 6= 0 will be discussed in the sequel.

4.2 Main result

We give directly the main result and divide its proof into several steps.

Theorem 4.1. Consider the system T(z) be given by (4.2) and let λ0 = α2 ∈ ∂D not a pole ofA− zE. Assume the following hypotheses hold:

(H1) The pair (A− zE,B2) is stabilizable and

rank[A− ejθE B2(α− αejθ)

C1 D12

]= n+m2, ∀θ ∈ [0, 2π) . (4.8)

(H2) The pair (C2, A− zE) is detectable and


C2 D21

]= n+ p2, ∀θ ∈ [0, 2π) . (4.9)

Then the DDTARE (Σ12) and the DDTARE (Σ21)

E∗XE−A∗XA−((αE − αA)∗XB2 + C∗1D12) (D∗12D12)−1 (D∗12C1 +B∗2X(αE − αA))+C∗1C1 = 0(4.10)

EY E∗−AY A∗−((αE − αA)Y C∗2 +B1D∗21) (D21D

∗21)−1 (D21B

∗1 + C2Y (αE − αA)∗)+B1B

∗1 = 0(4.11)

have stabilizing hermitian solutions X = X∗ and Y = Y ∗, i.e.,

Λ ((A+ αB2F )− z(E + αB2F )) ∪ Λ ((A+ αK∗C2)− z(E + αK∗C2)) ⊂ D,

29


where

F := − (D∗12D12)−1 (D∗12C1 +B∗2X(αE − αA)) , (4.12)K := − (D21D

∗21)−1 (D21B

∗1 + C2Y (αE − αA)∗) (4.13)

are the stabilizing Riccati feedbacks. Moreover, there exists a proper controller given by

K(z) =[A− zE + (B2F +K∗C2)(α− αz) −K∗

F 0

]λ0=α2

(4.14)

that solves the optimal H2 control problem. Moreover, the optimal value of the H2−norm is

minK stabilizing

‖TCL‖22 = 2 Trace(Re(α2WE−1

K AKYW∗)−WYW ∗

)+ η, (4.15)

where (X = X∗, V,W ) is the stabilizing solution of KSPYS(Σ12), and η ∈ C is given by

η = 2 Trace[Re(α2B∗1X(AE−1

F + I)B2FE−1K (B1 +K∗D21) +AE−1

F B1)−B∗1XB1

], (4.16)

with AF = A+ αB2F , EF = E + αB2F , AK = A+ αK∗C2, and EK = E + αK∗C2.Remark 4.2. The hypotheses (A− zE,B2) stabilizable and (C2, A− zE) detectable are neces-sary conditions for the existence of a stabilizing controller. Indeed, it is well-known that K(z)internally stabilizes T(z) if and only if it internally stabilizes

T22(z) =[A− zE B2C2 0

]λ0=α2

.

The proof of this claim is similar with the proof in (Chapter 6, Zhou, Doyle, and Glover 1996).Remark 4.3. If the hypothesis (4.8) holds true, then for θ = 2θ0 where θ0 is such that α = ejθ0

we get

rank[A− e2jθ0E 0

C1 D12

]= n+m2 ⇒ rankD12 = m2, (4.17)

or D12 has full column rank. Thus D∗12D12 is invertible. Further, if (4.9) holds true we get thatD21 has full row rank, thus D21D

∗21 is invertible.

Remark 4.4. The positiveness Popov quadruples associated with T12(z) and T21(z) are

Σ12 := (A− zE,B2;C∗1C1, C∗1D12, D

∗12D12),

Σ21 := (A− zE,C2;B1B∗1 , B1D

∗21, D21D

∗21)

Remark 4.5. If the pencil A − zE has no poles in 1 or −1 we can chose α ∈ −1, 1, thusα = α. In this case, the constant matrices to start with can be chosen with real elements.Remark 4.6. The hypotheses (H1) and (H2) ensure that the RMFs T12(z) and T21(z) hasno zeros on the unit circle. Therefore, the DSP(Σ12) and the DSP(Σ21) have no generalizedeigenvalues on the unit circle. Thus, the DDTARE(Σ12) and DDTARE(Σ21), i.e., (4.10) and(4.11), have stabilizing hermitian solutions X = X∗ and Y = Y ∗, see also Remark 3.9.Remark 4.7. Since F and K are the stabilizing Riccati feedbacks, the matrix pencil A− zE+(B2F +K∗C2)(α−αz) is stable, and thus the controller is proper, i.e., K(∞) is finite. Therefore,the controller can be easily implemented on a hardware device.Remark 4.8. The well-known separation property of the H2 optimal solution is reflected inour main result. Indeed, the controller (4.14) dynamics can be written in the observer form

Eσx+ αB2σu+ αK∗(C2σx− σy) = Ax+ αB2u+ αK∗(C2x− y)u = Fx

, (4.18)

30


where x is the optimal estimate of the state x. It remains to investigate if there are no guaranteedstability margins for the H2 optimization problem, as in the standard case.

We proceed now with the proof which is based on the solution of some special problems,each stated as a separate proposition. First of all, we solve a simpler problem, namely the FullInformation (FI) problem. Further, we solve two particular H2 optimization problems, namelythe two-block problem and its dual. Finally, we reduce the general H2 optimization problemto the dual two-block problem, using the KSPYS as the main technical tool. The optimizationproblems are developed under the simplifying assumption D11 = 0. The case when D11 6= 0 willbe discussed in detail.

4.3 Special Problems

4.3.1 The Full Information problem

The corresponding plant is given by

TFI(z) =

A− zE B1 B2C1 0 D12[I0

] [0I

] [00

]λ0=α2

. (4.19)

We shall assume that T12(z) has no zeros on the unit circle ∂D and that α2 is not a generalizedeigenvalue of the pencil A − zE. Note that for the FI problem the state and the disturbancesare available:

y =[I0

]x+

[0I

]w =

[xw

].

Proposition 4.9. Consider the descriptor system TFI(z) given in (4.19) and let λ0 = α2 ∈ ∂Dnot a pole of T. Assume that the pair (A− zE,B2) is stabilizable and


C1 D12

]= n+m2, ∀θ ∈ [0, 2π). (4.20)

Then the DDTARE (4.10) has a stabilizing hermitian solution X = X∗. Moreover, there existsa static controller given by

KFI =[F 0

](4.21)

that solves the FI H2 optimization problem, where F := − (D∗12D12)−1 (D∗12C1 +B∗2X(αE − αA))is the corresponding stabilizing Riccati feedback. Furthermore, the optimal value of the H2−normis

minK stabilizing

‖TCL‖22 = 2 Trace(Re(α2B∗1E

−∗F A∗FXB1)−B∗1XB1

), (4.22)

where AF = A+ αB2F, EF = E + αB2F .

Proof. Notice that Σ12 := (A − zE,B2;C∗1C1, C∗1D12, D

∗12D12) is the positiveness Popov

Quadruple associated with T12(z). Then with the assumptions (A − zE,B2) stabilizable and(4.20) it follows that the DDTARE(Σ12) has a stabilizing hermitian solution X, see Remark 3.9.

The control signal u is simply

u =[F 0

] [ xw

]= Fx.

31


Therefore, the closed-loop dynamics is given by(E + αB2F )σx+ αB1σw = (A+ αB2F )x+ αB1w

z = (C1 +D12F )x . (4.23)

Denote AF := A+ αB2F , EF := E + αB2F and CF := C1 +D12F . Thus

TCL(z) =[A− zE +B2F (α− αz) B1

C1 +D12F 0

]α2

=[AF − zEF B1

CF 0

]α2

. (4.24)

But F is the Riccati stabilizing feedback having

Λ ((A+ αB2F )− z(E + αB2F )) ≡ Λ(AF − zEF ) ⊂ D,

and therefore the closed-loop system is internally stable.

It remains to prove that the optimality requirement is satisfied as well. Note from Theorem3.10 that the minimality of the H2−norm is attained for u = Fx. Furthermore, the HamiltonianSystem associated with Σ12

Eσx + αB2σu = Ax + αB2uαC∗1C1σx − A∗σλ + αC∗1D12σu = αC∗1C1x − E∗λ + αC∗1D12u

v = D∗12C1x − B∗2λ + D∗12D12u(4.25)

has the output v ≡ 0 for u = Fx. The optimality is therefore reached. Recall that the DDTARE(4.10) has a stabilizing solution iff the KSPYS in the positivity form,

D∗12D12 = V ∗V,(αE − αA)∗XB2 + C∗1D12 = W ∗V,E∗XE −A∗XA+ C∗1C1 = W ∗W,

(4.26)

has a solution (X = X∗, V,W ), see Lemma 3.6. The Riccati stabilizing feedback is givenequivalently by F = V −1W . We evaluate successively the optimal value for the H2−norm.

‖zk‖2 = ‖C1xk +D12uk‖2 = (C1xk +D12uk)∗(C1xk +D12uk)= x∗kC

∗1C1xk + x∗kC

∗1D12uk + u∗kD

∗12C1xk + u∗kD

∗12D12uk

(4.26)= ‖Wxk + V uk‖2 + x∗kA∗XAxk − x∗kE∗XExk − 2 Re [x∗k(αE − αA)∗XB2uk] .

(4.27)Let zk := Wxk + V uk and note that

x∗kA∗XAxk + 2 Re (αu∗kB∗2XAxk) + u∗kB

∗2XB2uk = (Axk + αB2uk)∗X(Axk + αB2uk),

−x∗kE∗XExk − 2 Re (αu∗kB∗2XExk)− u∗kB∗2XB2uk = −(Exk + αB2uk)∗X(Exk + αB2uk).

Add the two equations given above to obtain with (4.27):

‖zk‖2 = ‖zk‖2 − (Exk + αB2uk)∗X(Exk + αB2uk) + (Axk + αB2uk)∗X(Axk + αB2uk). (4.28)

Let now uk = Fxk = −V −1Wxk for which the optimality is reached. With this, zk = 0,and (4.28) becomes

min ‖zk‖2 = x∗kA∗FXAFxk − x∗kE∗FXEFxk. (4.29)

From the dynamics of the closed-loop system (4.23) we have

EFxk = AFxk−1 +B1(αwk−1 − αwk) (4.30)

Consider now the impulsive input wk = wik := δkei, where δ is the discrete unit impulse, with

32


δ0 = 1 and δk = 0,∀k 6= 0, and ei, i = 1, . . .m is the canonical basis for Rm. Define

∆k := αδk−1 − αδk

and note that ∆0 = −α, ∆1 = α, and 0, otherwise. Thus, the dynamics of the system (4.30)becomes

EFxk = AFxk−1 +B1ei∆k, i = 1, . . . ,m, k = 0, 1, . . . (4.31)

With this, (4.29) can be rewritten as

min ‖zik‖2 = x∗kA∗FXAFxk − (AFxk−1 +B1ei∆k)∗X(AFxk−1 +B1ei∆k)

= x∗kA∗FXAFxk − x∗k−1A

∗FXAFxk−1 − 2 Re(x∗k−1A

∗FXB1ei∆k)−

−e∗iB∗1XB1e1|∆k|2,(4.32)

where zik is the response to the impulsive input wik. Sum both sides of (4.32) from k = 0 to ∞to obtain

min ‖zi‖22 = −2 Re(αx∗0A∗FXB1ei)− 2e∗iB∗1XB1ei. (4.33)

Here, we have considered that∞∑k=0

(x∗kA

∗FXAFxk − x∗k−1A

∗FXAFxk−1

)= 0, since the system is

stable and therefore limk→∞ xk = 0. Further, notice that for k = 0, (4.31) becomes

EFx0 = −αB1ei ⇒ x0 = −αE−1F B1ei.

Recall that a stable system has all poles inside the unit circle (thus no poles at infinity) andtherefore the system is standard, having EF invertible. We further obtain that

min ‖zi‖22 = 2e∗i(Re(α2B∗1E


)ei. (4.34)

Summ both sides of the equality (4.34) from i = 1 to m and recall Lemma 2.17 to obtain

min ‖TCL‖22 = 2 Trace(Re(α2B∗1E


).

This ends the proof.

4.3.2 The Two-Block problem and its dual

This specific problem deals with the case when D21 is square, i.e., p2 = m1, and invertible.Also, it is assumed that T12(z) has no zeros on the unit circle ∂D and that the matrix pencil(A−αB1D

−121 C2)−z(E−αB1D

−121 C2) is stable. In order to prove the main result of this section,

we need the following lemma.

Lemma 4.10. Consider the FI system TFI(z) given in (4.19) and the system T(z) (4.2), withD21 ∈ Cm1×m1 invertible, and (A− αB1D

−121 C2)− z(E − αB1D

−121 C2) stable. We have that

TFI(z) = T(z)⊗G(z), (4.35)

where ⊗ is the Redheffer product and

G(z) :=

A− zE −B1D

−121 C2(α− αz) B1D

−121 B2

0 0 I[I

−D−121 C2

] [I

−D−121

] [00

]α2

.

33


Proof. Let x and x denote the state of T(z) and G(z), respectively. Define e := x− x and let

x :=[ex

]be the state of the interconnected system. After a bit of algebra, the realization of

the interconnected system is

T(z)⊗G(z) =

A− zE −B1D

−121 C2(α− αz) 0 0 0

B1D−121 C2(α− αz) A− zE B1 B2

C1 C1 0 D12[0

−D−121 C2

] [I0

] [0I

] [00

]α2

. (4.36)

Removing the stable uncontrollable states we obtain TFI(z) given in (4.19). This ends the proof.

We can now easily obtain the controller for the two-block problem, knowing the solutionof the FI problem. We will now state the main result of this subsection.Proposition 4.11. Consider the descriptor system T(z) given in (4.2) and let λ0 = α2 ∈ ∂Dnot a pole of T(z). Assume the following hypotheses hold:

(H1−2b) The pair (A− zE,B2) is stabilizable and


C1 D12

]= n+m2, ∀θ ∈ [0, 2π) . (4.37)

(H2−2b) D21 ∈ Cm1×m1 is square and invertible, and the matrix pencil (A − αB1D−121 C2) −

z(E − αB1D−121 C2) is stable.

Then the DDTARE (4.10) has a stabilizing hermitian solution X = X∗. Moreover, there existsa proper controller

K(z) =[A− zE + (B2F −B1D

−121 C2)(α− αz) B1D

−121

F 0

]α2

(4.38)

that solves the H2 optimization problem, where F is the corresponding stabilizing Riccati feedback.Moreover, the optimal value of the H2−norm is

minK stabilizing

‖TCL‖22 = 2 Trace(Re(α2B∗1E


), (4.39)

where AF = A+ αB2F, EF = E + αB2F .

Proof. With the hypothesis (H1−2b) it follows that the DDTARE(Σ12)

E∗XE−A∗XA−((αE − αA)∗XB2 + C∗1D12) (D∗12D12)−1 (D∗12C1 +B∗2X(αE − αA))+C∗1C1 = 0

has a stabilizing hermitian solution X, see Remarks 3.9 and 4.6.

From Lemma 4.10 we know that TFI(z) = T(z) ⊗G(z). With this expression, we canobtain the controller for the H2 two-block optimization problem. The closed-loop system is

TCL(z) = LLFT (T(z)⊗G(z),KFI)= LLFT (T(z),LLFT (G(z),KFI)) .

(4.40)

with KFI =[F 0

]. Notice that the controller for the H2 two-block optimization problem is

simplyK(z) = LLFT

(G(z),

[F 0

]).

34


After a bit of algebra, we arrive exactly at (4.38). Furthermore, the closed-loop system isidentical for the two mentioned problems (FI and H2 two-block), see equation (4.40), and thusthe H2−norm of the closed-loop system has the same optimal value, see Proposition 4.9.

It remains to show that the closed-loop system is indeed stable. With (4.7) and (4.38),the realization is

TCL(z) =

A− zE B2F (α− αz) B1B1D

−121 C2(α− αz) A− zE + (B2F −B1D

−121 C2)(α− αz) B1

C1 D12F 0

α2

. (4.41)

Perform an equivalence transformation with Q =[I 0I −I

]= Z. Then the closed-loop system

becomes

TCL(z) =

A− zE +B2F (α− αz) −B2F (α− αz) B10 A− zE −B1D

−121 C2(α− αz) 0

C1 +D12F −D12F 0

α2

=[A− zE +B2F (α− αz) B1

C1 +D12F 0

]α2

,

(4.42)

where the last equality follows by removing the stable uncontrollable part. It is now obviousthat the closed-loop system is internally stable, since F is the stabilizing Riccati feedback, i.e.,

Λ (A− zE +B2F (α− αz)) ⊂ D.

This ends the whole proof.

We now consider the dual two-block problem, having p1 = m2, i.e. D12 is square.

Proposition 4.12. Consider the descriptor system T(z) given in (4.2) and let λ0 = α2 ∈ ∂Dnot a pole of T(z). Assume the following hypotheses hold:

(H1−d2b) D12 ∈ Rm2×m2 is invertible and (A− αB2D−112 C1)− z(E − αB2D

−112 C1) is stable.

(H2−d2b) The pair (C2, A− zE) is detectable and


C2 D21

]= n+ p2, ∀θ ∈ [0, 2π) . (4.43)

Then the DDTARE (4.11) has a stabilizing hermitian solution Y = Y ∗. Furthermore, thereexists a proper controller

K(z) =[A− zE + (−B2D

−112 C1 +K∗C2)(α− αz) −K∗−D−1

12 C1 0

]α2

(4.44)

that solves the optimal H2 control problem, where K = − (D21D∗21)−1 (D21B

∗1 + C2Ys(αE − αA)∗)

is the corresponding Riccati stabilizing feedback. Moreover, the optimal value of the H2−normis

minK stabilizing

‖TCL‖22 = 2 Trace(Re(α2C1E

−1K AKY C

∗1 )− C1Y C

∗1

), (4.45)

where AK := A+ αK∗C2 and EK := E + αK∗C2.

35


Proof. The result follows by duality from Proposition 4.11. Indeed, if T(z) satisfies the hypothe-ses (H1−d2b) and (H2−d2b), then TD(z) satisfies the hypotheses (H1−2b) and (H2−2b),where the suprascript stands for "dual" and it is defined as

TD(z) =

A∗ − zE∗ C∗1 C∗2B∗1 0 D∗21B∗2 D∗12 0

α2

. (4.46)

Further, write down the controller (4.38) for TD(z) to obtain

KD(z) =[A∗ − zE∗ + (C∗2K − C∗1D−∗12 B

∗2)(α− αz) C∗1D

−∗12

K 0

]α2

K(z) =[A− zE + (−B2D

−112 C1 +K∗C2)(α− αz) K∗

D−112 C1 0

]λ0=α2

.

(4.47)

But this is exactly the controller (4.44), up to a sign reduction. The rest of the proof is a matterof straightforward substitutions.

Proof of Theorem 4.1. Notice that Σ12 and Σ21 are the positiveness Popov quadruplesassociated with T12(z) and T21(z), respectively. Then with (H1) and (H2) it follows thatthe DDTARE(Σ12) and the DDTARE(Σ21) have the stabilizing hermitian solutions X and Y ,respectively, see Remarks 3.9 and 4.6. Further, the DDTARE (4.10) has a stabilizing solutioniff the KSPYS(Σ12, I) given by

D∗12D12 = V ∗VC∗1D12 + (αE − αA)∗XB2 = W ∗VC∗1C1 + E∗XE −A∗XA = W ∗W

(4.48)

has a stabilizing solution (X = X∗, V,W ), with F = −V −1W the corresponding stabilizingfeedback. Further, replace the output z = C1x+D12u in (4.3) with the expression

z = Wx+ V u (4.49)

to obtain the new system

T(z) =[T11(z) T12(z)T21(z) T22(z)

]=:

A− zE B1 B2W 0 VC2 D21 0

α2

. (4.50)

Since V is invertible and the matrix pencil A−zE−B2V−1W (α−αz) ≡ A−zE−B2F (α−αz)

is stable, the system T(z) satisfies the hypotheses (H1−d2b) and (H2−d2b) of Proposition4.12. Write the controller (4.44) for the system T(z) in (4.50) to obtain

K(z) =[A− zE + (−B2V

−1W +K∗C2)(α− αz) −K∗−V −1W 0

]α2

=[A− zE + (B2F +K∗C2)(α− αz) −K∗

F 0

]α2

(4.51)

36


But this is exactly our controller, given in (4.14). The closed-loop system is calculated to be

TCL(z) =

A− zE +B2F (α− αz) −B2F (α− αz) B10 A− zE +K∗C2(α− αz) B1 +K∗D21

C1 +D12F −D12F 0

λ0=α2

. (4.52)

Recall that F and K are the stabilizing Riccati feedbacks, i.e.,

Λ(A− zE +B2F (α− αz)

)∪ Λ

(A− zE +K∗C2(α− αz)

)⊂ D,

and therefore the resulting closed loop system is internally stable.

It only remains to show that K(z) is indeed optimal and to get the evaluation (4.15).Recall from the FI problem that (4.28) holds, i.e.,

‖zk‖2 = ‖zk‖2 − (Exk + αB2uk)∗X(Exk + αB2uk) + (Axk + αB2uk)∗X(Axk + αB2uk). (4.53)

It is easy to see that, in this case, the evaluation (4.28) also hold. From the dynamics (4.3) ofthe general descriptor system we can write

Exk + αB1wk + αB2uk = Axk−1 + αB1wk−1 + αB2uk−1, k = 0, 1, . . .

Let wk = wik := δkei, for i = 1, . . . ,m. We obtain that

Exk + αB2uk = Axk−1 +B1ei∆k + αB2uk−1, (4.54)

with ∆k := αδk−1 − αδk. Substitute (4.54) in (4.53) to get

‖zik‖2 = ‖zik‖2 − (Axk−1 + αB2uk−1)∗X(Axk−1 + αB2uk−1) + (Axk + αB2uk)∗X(Axk + αB2uk)−e∗iB∗1XB1ei|∆k|2 − 2 Re

[e∗iB

∗1X(Axk−1 + αB2uk−1)∆k

].

(4.55)Notice that ∆0 = −α, ∆1 = α, and ∆k = 0, for k 6∈ 0, 1. Sum the above expression fromk = 0 to ∞ to get

‖zi‖22 = ‖zi‖22 − 2e∗iB∗1XB1ei − 2 Re[αe∗iB

∗1X(Ax0 + αB2u0)

]. (4.56)

We have considered once more that∞∑k=0

[(Axk + αB2uk)∗X(Axk + αB2uk)− (Axk−1 + αB2uk−1)∗X(Axk−1 + αB2uk−1)

]= 0,

since the closed-loop system is stable. Further, we need to evaluate x0 and u0, respectively. Forthis purpose, consider the closed-loop dynamics[

EF −αB2F0 EK

] [x(k)xK(k)

]+[

αB1eiα(B1 +K∗D21)ei

]δk =

[AF −αB2F0 AK

] [x(k − 1)xK(k − 1)

]+

+[


]δk−1

at k = 0 to obtain [EF −αB2F0 EK

] [x(0)xK(0)

]+[


]= 0. (4.57)

Here, we have denoted by xK the internal state of the controller K. From (4.57) we get the

37


evaluationsxK(0) = −αE−1

K (B1 +K∗D21)ei,

x(0) = αE−1F (αB2FxK(0)−B1ei)

= −αE−1F

[αB2FE

−1K (B1 +K∗D21) +B1

]ei.

(4.58)

Further, from the dynamics of the controller (4.51) we know that u(0) = FxK(0). We can nowfurther evaluate the closed-loop H2−norm. Equation (4.56) becomes

‖zi‖22 = ‖zi‖22 − 2e∗iB∗1XB1ei+

+2e∗i Re[α2B∗1X

(AE−1

F (B2FE−1K (B1 +K∗D21) +B1) +B2FE

−1K (B1 +K∗D21)

)]ei.

(4.59)Sum the above equation from i = 1 to m and recall Lemma 2.17. We get that

‖TCL‖22 = ‖TCL‖22+2 Trace

[Re(α2B∗1X(AE−1

F + I)B2FE−1K (B1 +K∗D21) +AE−1

F B1)−B∗1XB1

],

(4.60)where TCL(z) is the system (4.50). Since TCL(z) satisfies the hypotheses of a two-block H2optimization problem, min ‖TCL‖22 can be computed using the result in Proposition 4.12. Thisends the whole proof of the result stated in Theorem 4.1.

4.3.3 The case D11 6= 0

Consider the general system

T(z) =

A− zE B1 B2C1 D11 D12C2 D21 0

α2

. (4.61)

In the time domain setting, the system is described byEσx+ αB1σw + αB2σu = Ax+ αB1w + αB2u

z = C1x+D11w +D12uy = C2x+D21w

, (4.62)

Substitute the control input u with the new input

ε := u−Ry,

where R ∈ Cp×m is an arbitrary constant matrix. In other words, connect a constant feedbackmatrix R to the system T(z) to obtain a new system T(z) having the inputs z and ε and thesame outputs as the original system. Substituting in (4.62) the expressions

u = ε+Ryu = ε+RC2x+RD21wσu = ε+RC2σx+RD21σw

(4.63)

we obtain the new system as

T(z) =

(A+ αB2RC2)− z(E + αB2RC2) B1 +B2RD21 B2C1 +D12RC2 D11 +D12RD21 D12

C2 D21 0

λ0=α2

. (4.64)

38


Requiring that D11 := D11 + D12RD21 ≡ 0 and assuming that D12 has full column rank andD21 has full row rank, we get

R = −(D∗12D12)−1(D∗12D11D∗21)(D21D

∗21)−1. (4.65)

Further, we can write the controller in (4.14), i.e., the main result, for T(z) because it hasD11 = 0. After some algebra, we obtain the controller

K(z) =[A− zE + (B2F +K∗C2 −B2RC2)(α− αz) B2R−K∗

F −RC2 0

]α2

. (4.66)

Connect now the controller K(z) with the constant feedback matrix −R to obtain a new con-troller denoted with K, i.e., K = K−R. We have that K = K +R, i.e.,

K(z) =[A− zE + (B2F +K∗C2 −B2RC2)(α− αz) B2R−K∗

F −RC2 R

]α2

. (4.67)

In other words, the controller (4.67) solves the H2 optimization problem for the general systemin (4.61) (the constant feedbacks cancel each other, R−R = 0).

4.4 A second proof

We provide in this section an alternative proof for our main result given in Theorem 4.1, whichis based on a successive reduction to simpler problems. The main tool for this reduction is theKalman-Szego-Popov-Yackubovich System.

We begin with internal stability. The next result is a slightly modified version of (Theorem4, Oara and Sabau 2011) and it provides the class of all stabilizing controllers for a generalizedplant in the form (4.61).

Proposition 4.13. 1. Let (A− zE,B) be a stabilizable pair and λ0 = α2 in ∂D−Λ(A− zE) .Then there exists a matrix F such that

Λ(A− zE +BF (α− αz)

)⊂ D. (4.68)

2. Let T(z) be a general p ×m system given by (4.61) and assume that λ0 = α2 ∈ ∂D is nota pole of T(z). Then the class of all controllers K(z) achieving an internally stable feedbacksystem is

K(z) =[A− zE + (B2F +KC2 −B2RC2)(α− αz) B2R−K

F −RC2 R

]α2

, (4.69)

where (4.61) is a proper stabilizable and detectable realization of T(z), F and K are two matricessuch that

Λ(A− zE +B2F (α− αz)

)∪ Λ

(A− zE +KC2(α− αz)

)⊂ D, (4.70)

and R is an arbitrary p×m complex matrix.

Remark 4.14. Notice that the controller that achieves internal stability has the same form asthe controller we obtained in the case D11 6= 0, see (4.67). Moreover, notice that the optimalcontroller has an extra condition on R ∈ Cp×m, given by (4.65) (obviously, in order to fulfill theoptimality requirement).

39


Remark 4.15. The closed-loop system is

TCL(z) =

A− zE +B2F (α− αz) B2(RC2 − F )(α− αz) B1 +B2RD210 A− zE +KC2(α− αz) B1 +KD21

C1 +D12F D12(RC2 − F ) D11 +D12RD21

α2

(4.71)

Note from (4.70) that the closed-loop system is stable, i.e., has all its poles inside the open unitdisk D.

With this result, we can further solve the one-block H2 optimization problem. The one-block problem is a particular case with p1 = m2 and p2 = m1, i.e. D12 and D21 are square.Therefore, the RMF is also square, having T(z) ∈ Cm×m(z), with m := m1 +m2.

Proposition 4.16. Consider the system T(z) given in (4.61) and assume that the followinghypotheses hold:

(H1−1b) D12 ∈ Cm2×m2 is invertible and (A− αB2D−112 C1)− z(E − αB2D


(H2−2b) D21 ∈ Cm1×m1 is invertible and (A− αB1D−121 C2)− z(E − αB1D


Then the controllerK(z) =

[AK − zEK BK

CK DK

]α2

(4.72)

withAK = A− αB2D

−112 C1 − αB1D

−121 C2 + αB2D

−112 D11D

−121 C2,

EK = E − αB2D−112 C1 − αB1D

−121 C2 + αB2D

−112 D11D

−121 C2,

BK = (B1 −B2D−112 D11)D−1

21 ,

CK = D−112 (D11D

−121 C2 − C1),

DK = −D−112 D11D

−121

(4.73)

is a solution to the H2 optimization problem. Moreover, the optimal value of the H2−norm is

minK stabilizing

‖TCL‖2 = 0.

Proof. Consider the stabilizing controller (4.69). The closed-loop system is given by (4.71) andthe closed-loop poles are Λ

(A− zE +B2F (α−αz)

)∪Λ

(A− zE +KC2(α−αz)

). According to

hypotheses (H1−1b), (H2−1b), D12 and D21 are invertible. Thus we can take

F = −D−112 C1, K = −B1D

−121 . (4.74)

With (4.74), the closed-loop poles are

Λ(A− zE −B2D

−112 C1(α− αz)

)∪ Λ

(A− zE −B1D

−121 C2(α− αz)

)⊂ D.

The inclusion follows from the stability of the two matrix pencils in hypotheses (H1−1b),(H2−1b). Thus the closed loop system is internally stable. Further, substitute (4.74) in (4.71)

40


to get

TCL(z) =

A− zE −B2D−112 C1(α− αz) B2(RC2 +D−1

12 C1)(α− αz) B1 +B2RD210 A− zE −B1D

−121 C2(α− αz) B1 −B1D

−121 D21

C1 −D12D−112 C1 D12(RC2 +D−1

12 C1) D11 +D12RD21

α2

=

A− zE −B2D−112 C1(α− αz) B2(RC2 +D−1

12 C1)(α− αz) B1 +B2RD210 A− zE −B1D

−121 C2(α− αz) 0

0 D12(RC2 +D−112 C1) D11 +D12RD21

α2

= D11 +D12RD21,(4.75)

where, to obtain the last equality, we remove the stable uncontrollable and unobservable parts.Notice that TCL(z) is a constant matrix and thus its H2−norm is simply

‖TCL‖2 = ‖D11 +D12RD21‖2.

Requiring ‖TCL‖2 = 0 (the achievable minimum) we get

R = −D−112 D11D

−121 . (4.76)

Substituting (4.74) and (4.76) in (4.69), we get the expressions (4.73). Moreover, ‖TCL‖2 = 0.This completes the first part of the proof.

Conversely, assume that K(z) is given by (4.73). After some algebra, the closed-loopsystem (4.7) becomes (4.75), where R := DK = −D−1

12 D11D−121 . It follows that the closed loop

system is internally stable and ‖TCL‖2 = 0. This ends the whole proof.

Remark 4.17. Assume that T12(z) ∈ Cm2×m2(z) has an invertible D12 and that it has nounstable zeros, i.e., ZT12 ⊂ D. This implies that T−1

12 (z) has no unstable poles. We get that

T−112 (z) =

[A− αB2D

−112 C1 − z(E − αB2D

−112 C1) B2D

−112

−D−112 C1 D−1

12

]α2

(4.77)

is stable, and thus the pole pencil A − αB2D−112 C1 − z(E − αB2D

−112 C1) has all generalized

eigenvalues inside the open unit disk D. But this is exactly the hypothesis (H1−1b). Thehypothesis (H2−1b) can be derived in the same manner.

We will now give an alternative proof to the two-block H2 optimization problem,stated in Proposition 4.11. Recall that the existence of a stabilizing hermitian solution for theDDTARE(Σ12) is equivalent to the existence of a stabilizing solution (X = X∗, V, F ) to theKSPYS(Σ12, I) with F := −V −1W the corresponding stabilizing feedback. Replace the outputz = C1x+D12u in (4.62) (consider that D11 = 0) with the expression

z = Wx+ V u (4.78)

to obtain the new system

T(z) =[T11(z) T12(z)T21(z) T22(z)

]=:

A− zE B1 B2W 0 VC2 D21 0

α2

. (4.79)

Since V is invertible and the matrix pencils

A− zE +B1D−121 C2(α− αz),

A− zE −B2V−1W (α− αz) ≡ A− zE −B2F (α− αz)

41


are stable, the system T(z) satisfies the hypotheses (H1−1b), (H2−1b) of Proposition 4.16.Write the controller (4.73) with D11 = 0 for the system T(z) in (4.79) to obtain the controller(4.38) given by Proposition 4.11. Notice that TCL(z) and TCL(z) share the same pole penciland that the closed-loop system is internally stable.

The only point that needs a different argument is the proof of optimality for the two-blockproblem. We illustrate this aspect only. First of all, notice that the evaluation (4.56) holds fora two-block problem, since no dynamics was necessary to derive it. The evaluation is

‖zi‖22 = ‖zi‖22 − 2e∗iB∗1XB1ei − 2 Re[αe∗iB

∗1X(Ax0 + αB2FxK(0)

)]. (4.80)

Since the pencil A − zE + B1D−121 C2(α − αz) is stable, xK(0) = x0. Further, consider the

closed-loop dynamics for wk = wik := δkei, i = 1, . . .m, i.e.,

EFxk +B1eiδk = AFxk−1 +B1eiδk−1,

where AF = A+αB2F and EF = E+αB2F . For k = 0 we get that x0 = −E−1F B1ei. Introduce

this in (4.80) and sum both sides from i = 1, . . . ,m (recall Lemma 2.17) to arrive at

‖TCL‖22 = ‖TCL‖22 + 2 Trace(Re(α2B∗1E


). (4.81)

Further, it is easy to see from Proposition 4.16 that min ‖TCL‖2 = 0 (u = Fx = −V −1Wx).This is yet another argument for the fact that optimality is reached when u = Fx. Therefore,

min ‖TCL‖22 = 2 Trace(Re(α2B∗1E


), (4.82)

which is exactly the evaluation (4.39). This ends the proof.

The dual two-block H2 optimization problem follows by duality. The main result can beproved in the same manner as before, and therefore is omitted.

42

5 Numerical Examples

We illustrate in this chapter the proposed approach by three numerical examples of growingcomplexity. In the first example we present a controller that solves a H2 two-block

optimization problem, while the second example provides a controller for a four-block problem.Finally, we treat the general case with D11 6= 0. Moreover, we show that the controllers areproper in each case. We begin with a brief discussion on the numerical solution of DDTARE(Σ).

The numerical solution of the DDTARE(Σ) is based on the n−dimensional stable deflatingsubspace of the DSP(Σ), zMΣ−NΣ. Perform an ordered QZ complex factorization with respectto the unit disk of the DSP(Σ) to obtain an upper triangular form

Q(zMΣ −NΣ)Z = zMΣ − NΣ.

If the matrices are restricted to be real, the QZ algorithm produces upper quasi-triangularmatrices MΣ, NΣ. Further, note that a basis matrix for the n−dimensional stable deflatingsubspace of the DSP(Σ) is V := Z(:, 1 : n) and thus

V1 = Z(1 : n, 1 : n),V2 = Z(n+ 1 : 2n, 1 : n),V3 = Z(2n+ 1 : 2n+m, 1 : n).

Moreover, if V1 is invertible, we can compute from Theorem 3.5 the stabilizing hermitian solu-tion X and the corresponding stabilizing feedback F .

Example 1. Consider the Rational Matrix Function

T(z) =

20z2 + 2z − 22z + 1

−4z2 − z + 1z + 1

10z2 − z − 9 −2z2 + 3

20z3 + 8z2 − 13z − 5z + 1

−4z3 − 2z2 + 5z + 1z + 1

∈ R3×2(z), (5.1)

with the partitions given by p1 = 2 and m1 = 1 (thus, m2 = 1 control input and p2 = 1measured output). The RMF has a proper minimal realization centered at λ0 = 1 (note thatT(1) is finite), i.e, with α = α = 1 ∈ ∂D, given by

T(z) =

−1− z 0 1 −13 10 −z 1 −1 00 −1 0 −10 21 1 0 0 −20 1 1 0 11 0 2 5 0

λ0=1

. (5.2)

The discrete-time system T(z) is improper having one pole at −1 with multiplicity 1 and onepoles at ∞ with multiplicity 2. Clearly, the system is unstable. Note that D21 = 5, thus it is

43

Chapter 5. Numerical Examples

square and invertible, and that the matrix pencil A− zE −B1D−121 C2(1− z) is stable:

Λ(A− zE −B1D

−121 C2(1− z)

)= 0.8010,−0.8208,−0.3802 ⊂ D.

Moreover, it can be easily checked that the pair (A − zE,B2) is stabilizable, see PBH tests inDefinition 2.12. Therefore, we deal with the H2 two-block optimization problem. The stablizingsolution of the DDTARE (4.10) is

X =

−1.4574 −1.3623 1.7877−1.3623 −2.7686 3.2746

1.7877 3.2746 −5.3911

,with the corresponding stablizing Riccati feedback

F =[−0.4472 0.9615 1.2610

].

The generalized eigenvalues for the pencil AF − zEF are

Λ (A− zE +B2F (1− z)) = 0.6633,−0.5851,−0.4861 ⊂ D. (5.3)

We construct now the optimal H2 controller using (4.38) in Proposition 4.11. With F givenabove, we obtain the following proper controller:

K(z) = 0.1994z3 + 0.06138z2 − 0.2135z − 0.04723z3 − 0.575z2 − 0.2573z − 0.009621 .

The closed-loop system is given by

TCL(z) =

−z3 − 3.828z2 + 1.862z + 2.966z3 + 0.4079z2 − 0.4262z − 0.1887

12.04z3 + 1.856z2 − 12.04z − 1.856z3 + 0.4079z2 − 0.4262z − 0.1887

=

[−(z + 4.106)(z + 0.7224)(z − 1)

12.0398(z − 1)(z + 1)(z + 0.1541)

](z − 0.6633)(z + 0.5851)(z + 0.4861) .

Thus the closed-loop system is internally stable and proper, having the closed-loop poles given in(5.3) and the optimal value of the H2−norm min ‖TCL‖2 = 15.0623. Moreover, the evaluationfor the H2−norm holds true:

minK stabilizing

‖TCL‖2 =[2 Trace

(B∗1E

−∗F A∗FXB1 −B∗1XB1

)] 12 = 15.0623.


T(z) =

4z5−5z4−14z3−17z2+34z−22z3−4z2−6z

2z3−5z2−4z+7z−3

z4−z3−10z2+15z−1z2−2z−3

−z4+5.5z3−8.5z2+3z+1z2−3z

−4z5−7z4+35z3−23z2−7z+62z3−4z2−6z

−2z3+4z2−2zz−3

−z4+z3−2z2+11z−9z2−2z−3

2z4−7z3+7z−62z2−6z

8z5+6z4−23z3−8z2+11z−22z3−4z2−6z

4z3−7z2+4z−5z−3

2z4−z3−4z2+z+2z2−2z−3

−4z4+16z3−7z2−7z+22z2−6z

4z4−14z3+29z2−26z+72z3−4z2−6z

−3z2+15z−14z−3

−2z3+12z2−18z+8z2−2z−3

−2z2+9z−72z2−6z

.

The RMF has the dimension 4 × 4, with partitions given by p1 = 2 and m1 = 2, and thusm2 = 2, p2 = 2. The system has a proper minimal realization centered at λ0 = 1 (note that

44


T(1) is finite) given by

T(z) =

1 z 0 −3 0 0 0 1 1 00 1 0 0 z − 1 0 1 0 0 10 0 1 2− 3z 0 0 1 −1 2 −10 0 0 z + 1 0 0 1 0 −1 00 0 0 0 z − 3 −1 3 2 1 00 0 0 0 0 −2z 1 0 0 −11 1 0 0 −3 0 0 0 −1 0−1 0 −1 1 0 −2 0 0 0 1

2 0 1 −1 1 1 1 2 0 00 1 1 0 3 −1 0 1 0 0

λ0=1

. (5.4)

The discrete-time system T(z) is improper having 3 poles at∞, one with multiplicity 2, and theother with multiplicity 1. It also has a pole outside the unit disc in 3, a pole on the unit circlein −1, and a pole in 0. Note that D11 = D22 = O2×2. Using the PBH tests in Definition 2.12,one can easily check that the pair (A − zE,B2) is stabilizable, and that the pair (C2, A − zE)is detectable. This is the full H2, i.e., the four-block, optimization problem.

The stabilizing symmetrical solutions of the DDTAREs (4.10) and (4.11) are given by

X =

−2.2098 4.3416 3.8799 7.2063 −0.1378 −0.43014.3416 −21.5281 −24.9643 −57.4610 −14.3250 7.45663.8799 −24.9643 −30.0833 −70.5164 −18.6616 9.50157.2063 −57.4610 −70.5164 −168.9155 −48.6195 23.0002−0.1378 −14.3250 −18.6616 −48.6195 −19.4334 7.4257−0.4301 7.4566 9.5015 23.0002 7.4257 −3.6191

,

Y =

−9.4304 2.5613 17.8654 −0.4177 −8.3482 −0.03592.5613 −4.7028 −2.5846 −0.9806 3.9250 0.8079

17.8654 −2.5846 −37.1363 1.0878 14.7981 −0.1537−0.4177 −0.9806 1.0878 −0.8280 0.1672 0.2831−8.3482 3.9250 14.7981 0.1672 −8.0996 −0.3592−0.0359 0.8079 −0.1537 0.2831 −0.3592 −0.1874

,

with the corresponding Riccati stabilizing feedbacks

F =[−0.7941 −3.2451 −4.4319 −17.2468 13.5498 2.2783

1.8919 −3.1285 −3.3825 −19.1828 6.1782 1.9407

], (5.5)

K =[−1.5440 −0.1081 0.6180 0.7587 −2.2340 0.1666

0.2089 3.8960 4.2653 −0.8660 13.1367 0.1387

]. (5.6)

With (5.5) and (5.6), the generalized eigenvalues for the closed-loop system are given by

Λ ((A+B2F )− z(E +B2F )) = −0.4690, 0.5453± 0.6801i, 0.1881, 0.3852, 0.6319 ⊂ D,Λ((A+KTC2)− z(E +KTC2)

)= −0.5176± 0.5580i, 0.7510± 0.0778i, 0.4011± 0.1114i ⊂ D.

(5.7)The controller that solves the H2 optimization problem is given in Theorem 4.1 and for thisexample is computed to be

K(z) =[

K1(z) K2(z)],

45


where

K1(z) :=

−0.1928z6+0.4834z5−0.4621z4+0.2311z3+0.03315z2−0.1701z+0.07735

z6−3.805z5+5.158z4−2.422z3−0.6477z2+0.9484z−0.2323

0.0498z6−0.3187z5+0.2607z4+0.5309z3−0.8115z2+0.2809z+0.007801z6−3.805z5+5.158z4−2.422z3−0.6477z2+0.9484z−0.2323

,

K2(z) :=

0.2843z6−0.5647z5−0.05319z4+0.5993z3−0.1842z2−0.1276z+0.04612

z6−3.805z5+5.158z4−2.422z3−0.6477z2+0.9484z−0.2323

0.043z6+0.04807z5−0.3647z4+0.2996z3+0.06979z2−0.0957zz6−3.805z5+5.158z4−2.422z3−0.6477z2+0.9484z−0.2323

.Note that the controller is proper, having K(∞) finite, and therefore it can be easily

implemented on a hardware device. The closed-loop system is

TCL(z) =

[T1(z) T2(z)

](z − 0.6319)(z − 0.1881)(z2 − 1.502z + 0.57)(z2 + 1.035z + 0.5792)(z2 − 1.091z + 0.7599) ,

where

T1(z) =[

−4.9483(z − 4.958)(z − 1.117)(z − 1.056)(z − 1)(z2 + 1.406z + 0.9365)(z2 − 1.278z + 1.142)−22.0291(z − 1)(z − 0.9881)(z − 1.423)(z − 0.6303)(z2 + 1.52z + 1.037)(z2 − 1.578z + 1.404)

],

T2(z) =[

3.8703(z + 3.301)(z − 1.27)(z − 1)(z − 0.4241)(z2 − 1.309z + 1.058)(z2 − 0.107z + 0.8058)−2.0858(z − 5.423)(z − 1)(z − 0.6304)(z − 0.4073)(z2 − 1.991z + 1.831)(z2 − 0.5327z + 1.095)

].

Thus the closed-loop system is internally stable and proper, having the closed-loop poles insidethe unit circle and the optimal value of the H2−norm

min ‖TCL‖2 = 106.4345.


T(z) =

−2z4 + 14z3 − 11z2 − z + 6z3 − z2 + 2

−4z5 + 6z4 − 3z3 − 3.75z2 + 5.75z + 4z4 − 0.75z3 − 0.25z2 + 2z + 0.5

−z4 + 3z3 + 2z2 − 13z + 11z2 − 2z + 2

−2z5 + 3.5z4 − 3z3 + 7.25z2 − 13z + 12.25z3 − 1.75z2 + 1.5z + 0.5

−z5 + 2z4 + 6z3 − 8z2 + 10z + 1z3 − z2 + 2

−2z6 + 1.5z5 + 3.5z4 − 11.5z3 + 3.5z2 + 6.25z − 1.25z4 − 0.75z3 − 0.25z2 + 2z + 0.5

,

with p = 3 outputs and m = 2 inputs.The partitions are obtained by setting p1 = 2 and m1 = 1,i.e., we have m2 = 1 control input and p2 = 1 measured output. The proper minimal realizationcentered at λ0 = 1 is

T(z) =

−z − 1 0 1 0 0 0 0 0 10 −z 1 0 0 0 0 1 00 −1 0 0 0 0 0 1 20 0 0 1− z 1 0 0 1 00 0 0 −1 1− z 0 0 −2 10 0 0 0 0 −z − 1

4 1 −1 30 0 0 0 0 0 −z − 1

4 0 11 1 0 1 3 0 1 3 20 1 1 −2 1 0 5 2 41 0 1 2 1 0 −1 5 0

λ0=1

.

46


The system is improper having one pole at ∞ with multiplicity 2, one pole on the unit circle in−1, one complex conjugate pair of poles in 1 ± i, and one stable pole in −1

4 with multiplicity

2. This is a slightly more general case, having D11 =[

32

]6= 0, and D22 = 0. Further, we

compute the stabilizing symmetrical solutions of the DDTAREs (4.10) and (4.11):

X =

−4.7399 −3.3377 11.2379 1.6074 −14.2981 0 −2.2016−3.3377 −52.8326 97.4204 −24.2231 −155.8582 0 3.243211.2379 97.4204 −235.1286 125.7092 351.8098 0 1.60961.6074 −24.2231 125.7092 −296.3267 −134.5666 0 −1.1783

−14.2981 −155.8582 351.8098 −134.5666 −606.8626 0 6.33340 0 0 0 0 0 0

−2.2016 3.2432 1.6096 −1.1783 6.3334 0 −27.2810

,

Y =

−175.0319 −139.0325 −17.0295 44.0751 44.9552 −3.6573 0−139.0325 −115.4501 −36.1056 36.6614 42.8593 −0.7925 0−17.0295 −36.1056 −125.0332 9.9053 52.0302 8.1370 0

44.0751 36.6614 9.9053 −17.6005 −10.7246 0.6549 044.9552 42.8593 52.0302 −10.7246 −33.2278 −1.4922 0−3.6573 −0.7925 8.1370 0.6549 −1.4922 −1.0608 0

0 0 0 0 0 0 0

.

The stabilizing Riccati feedbacks are

F =[−0.2236 3.0356 1.8062 −4.1396 5.3640 0 0.1456

],

K =[

1.9076 −0.6506 2.1582 1.4923 0.0778 −0.0149 0],

and

Λ ((A+B2F )− z(E +B2F )) = −0.2500, 0.6642, 0.4912± 0.5726i,−0.8397,−0.3069± 0.4489i,Λ((A+KTC2)− z(E +KTC2)

)= −0.2500, 0.4508± 0.7459i,−0.4089,−0.0927, 0.2984,−0.2500,

inside the unit circle. The controller that solves the H2 optimization problem with D11 6= 0 hasa state-space realization given in (4.67). With F and K given above, and R = 0.14 from (4.65),K(z) was computed to be

K(z) = 0.08149z6 + 0.008825z5 − 0.3495z4 + 0.1826z3 + 0.3201z2 − 0.1881z − 0.06283z6 − 0.9192z5 − 1.893z4 + 2.392z3 + 0.2586z2 − 0.9428z + 0.1571

The closed-loop system

TCL =

8.5(z−0.9731)(z+1.388)(z2+1.223z+0.4148)(z2−2.046z+1.195)(z2−1.128z+1.245)(z2−1.79z+4.692)

(z+0.8397)(z+0.4089)(z−0.6642)(z−0.2984)(z2+0.6138z+0.2957)(z2−0.9824z+0.5691)(z2−0.9017z+0.7596)

−5.636(z+2.106)(z+0.9751)(z−1.011)(z−1.768)(z2−2.098z+1.351)(z2−1.218z+1.28)(z2+1.997z+2.848)(z+0.8397)(z+0.4089)(z−0.6642)(z−0.2984)(z2+0.6138z+0.2957)(z2−0.9824z+0.5691)(z2−0.9017z+0.7596)

is internally stable and proper. The optimal value of the H2−norm was computed to be

min ‖TCL‖2 = 216.1535.

47

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h2 optimization problem for general discrete-time systems

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