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H∞ CONTROL PROBLEM FOR DISCRETE–TIMEALGEBRAIC DYNAMICAL SYSTEMS∗
SEBASTIAN F. TUDOR† AND CRISTIAN OARA‡
Abstract. This paper considers the suboptimal H∞ control problem for linear discrete–timealgebraic dynamical systems. Such systems can be formally described as LTI (linear and time–invariant) discrete–time systems, whose transfer function matrix is allowed to be improper or poly-nomial. The parametrization of output feedback controllers is given in a realization–based settinginvolving two generalized algebraic Riccati equations and features the same elegant simplicity of thestandard (proper) case. Two relevant numerical examples prove the effectiveness of our approach.
Key words. Robust control, H∞ control, algebraic dynamical systems, discrete–time systems.
AMS subject classifications.
1. Introduction. Ever since it emerged in the 1980’s in the seminal paper ofZames [38], the H∞ control problem (also known as the disturbance attenuationproblem) has drawn much attention, mainly due to the wide range of applications ittriggered. The design problem is concerned with finding the class of controllers, fora given system, that stabilizes the closed–loop system and makes its input–outputH∞–norm bounded by a prescribed threshold. Various solutions based on differentmathematical formalisms have been given along a period covering about three decades:analytic functions or operator theory based [30, 1, 7, 6], state–space representation[5, 32, 31], chain scattering approach [15], generalized Popov theory [14, 12], to namejust a few.
The importance of models described by algebraic dynamical systems is well–known [16]. In the literature, algebraic dynamical systems are often called descriptor(or singular) systems [4], differential algebraic systems [3], improper and/or polyno-mial systems. An algebraic dynamical system covers a wide class of physical systems,e.g., mechanical systems featuring non–dynamic algebraic constraints [2], impulsivebehavior in circuits with inconsistent initial conditions [34], or hysteresis, to namejust a few. Cyber–physical systems under attack, mass/gas transportation networks,power systems and advanced communication systems can also be modeled as an alge-braic dynamical system, see [26] and the references therein. The range of applicationsspans wide topics, from engineering, e.g., aerospace industry, robots, path prescribedcontrol, mechanical multi–body systems, network theory [11, 27, 17], to economics[18].
H∞ controllers for continuous–time algebraic dynamical systems were obtainedin a couple of papers: an extended model matching technique was employed in [33], asolution expressed in terms of two generalized algebraic Riccati equations was given in[37], while a matrix inequality approach was considered in [19]. Having the motivationthat a discrete–time controller is more suitable for real–time applications (since most
∗This work was supported by the Romanian Authority for Scientific Research, CNCS - UEFISCDI,project number PN-II-ID-PCE-2011-3-0235. The first author’s work was funded by the SectoralOperational Program Human Resources Development 2007-2013 of the Ministry of European Fundsthrough the Financial Agreement POSDRU/159/1.5/S/132397.
†Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken,NJ 07030, USA. The first author is on leave from University ”Politehnica” of Bucharest, Romania.E–mail: [email protected].
‡Department of Automatic Control and Systems Engineering, University ”Politehnica” ofBucharest, Romania. E–mail: [email protected].
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modern control algorithms are implemented on digital computers) we consider in thispaper the H∞ control problem for discrete–time algebraic dynamical systems. Ourapproach is to generalize the indefinite sign Popov theory in [12] to cover this classof discrete–time systems. The solution we will provide is realization–based, involvesa novel type of descriptor algebraic Riccati equation (recently investigated in [20]),and exhibits a numerical easiness similar with the celebrated DGKF solution [5] inthe proper case.
The paper is organized as follows. In Section 2 we give preliminary definitionsand results, while in Section 3 we state the suboptimal H∞ output feedback controlproblem. The main result giving realization–based formulae for the class of all stabi-lizing and γ−contracting controllers for a general discrete–time system is provided inSection 4. Two relevant numerical examples showing the applicability of our resultsare presented in Section 5. We sketch the technical proofs in a separate Section. Thepaper ends with several conclusions.
2. Preliminaries. Let C, D, and ∂D denote the complex plane, the open unitdisc, and the unit circle, respectively. Let z ∈ C denote the complex variable. Wedenote with In the n × n identity matrix. When the dimension is clear from thecontext, we will drop the index.
For a complex matrix A ∈ Cp×m, A∗ stands for the conjugate transpose; fora square matrix (p = m), A−1 denotes the inverse of A, and A1/2 is such thatA1/2A1/2 = A. We say that A is Hermitian if A = A∗. If A is Hermitian thenall its eigenvalues are real, and for an invertible A we introduce the sign matrix assgn (A) = diag (−In−
, In+), where n+ and n− are the number of positive and nega-
tive eigenvalues, respectively. ρ(A) denotes the spectral radius of a square matrix Aand it is defined as the absolute of the largest eigenvalue. The union of generalizedeigenvalues (finite and infinite, multiplicities counting) of the matrix pencil A − zEis denoted with Λ(A− zE). The matrix pencil A− zE is called regular if it is squareand det(A− zE) 6≡ 0.
We say that a matrix function T : C → Cp×m is rational if its elements arerational functions, i.e.,
(2.1) Tij(z) =aij(z)
bij(z), i = 1, . . . , p, j = 1, . . . ,m,
where aij(z) and bij(z) are scalar polynomials with coefficients in C, having arbitrarybut fixed degrees. The class of all complex p×m rational matrix functions is denotedwith Cp×m(z). By definition, the complex conjugate is given by T#(z) := T∗(1/z).Let RLp×m
∞ (∂D) be the Banach space of complex p ×m RMF bounded on the unitcircle, having the H∞ norm defined as
(2.2) ‖T‖∞ := supθ∈[0,2π)
σmax
(T(ejθ)
),
where σmax(·) is the largest singular value of a constant matrix.In the control theory framework, a multi-input multi-output (MIMO) dynamical
system is formally described as a transfer function matrix (TFM) from some m inputsu ∈ Cm to some p outputs y ∈ Cp, i.e., y = Tu. In the standard (proper) case, theTFM is proper rational, having deg aij ≤ deg bij , for all i = 1, . . . , p, j = 1, . . . ,m,where deg(·) is the degree of a scalar polynomial function. In this paper, we considerdynamical systems having general TMFs, possibly improper (deg aij ≥ deg bij) and/or
USING SIAM’S LATEX MACROS 3
polynomial (bij(z) ≡ 1 for some i, j). A system described by a general TMF hastwo parts: a dynamical part and an algebraic part, which contains the non–dynamicconstraints (see, e.g., [4] and the decomposition in [21], Section 3).
To represent a polynomial or improper discrete–time system T ∈ Cp×m(z), wewill use in this paper a general type of realization called centered:
(2.3) T(z) = D + C(zE −A)−1B(α− βz) =:
[A− zE B
C D
]
z0=αβ
,
where z0 = α/β ∈ C is fixed, A,E ∈ Cn×n, rankE ≤ n, B ∈ Cn×m, C ∈ Cp×n,D ∈ Cp×m, n is called the order (or the dimension) of the realization, and the matrixpencil A−zE is regular. For an improper or polynomial TFM, the matrix E is alwayssingular. Note that any TFM T ∈ Cp×m(z) has a realization of the form (2.3), see[29] for a more general representation. It is easy to notice that for α = 1 and β = 0we recover the well–known descriptor realization [4] which actually is a realizationcentered at z0 = α/β = ∞. For a fixed z0, we call the realization (2.3) minimal if itsorder is as small as possible among all realizations of this type.
Centered realizations have some nice properties, due to the flexibility in choosingz0 always disjoint from the set of poles of T and, if needed, also disjoint from theset of zeroes. For example, the order of such a centered minimal realization alwaysequals the McMillan degree of T (see [39], Definition 3.12), denoted δ(T), while T(z0)always equals the matrix D in (2.3). Realizations of type (2.3) have been widely usedin the literature for various problems related to generalized systems [9, 8, 28, 24].
Switching back and forth between realizations centered at infinity (standard de-scriptor realizations) and realizations centered at a finite z0 ∈ C can be done bystraightforward manipulations, see Section 5 in [22]. Alternatively, a direct methodto obtain a centered minimal realization starting from the rational transfer matrixrepresentation is presented in [36], Section 2.
Throughout the paper we consider realizations centered at z0 ∈ ∂D (thus featuringspecific symmetries with respect to the unit disk), where z0 is not a pole of theunderlying TFM. Accordingly, let α ∈ ∂D, β := α, and thus z0 = α/α = α2 ∈ ∂D.
Example. To highlight better the advantages of centered realizations versusdescriptor ones for this general class of systems, consider the following discrete–timepolynomial system:
(2.4) T(z) =
−z2 + z + 1 −z + 2z3 − z + 4 z2 + 1z3 − z − 2 z2 + 3
∈ C
3×2(z).
The system has two poles at ∞, one with multiplicity 3 and one with multiplicity2, and thus δ(T) = 5. The number of partial multiplicities is np = 2. A minimal
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descriptor realization (i.e., having the smallest order) is given by
(2.5)
T(z) = D + C(zE − A)−1B =
[A− zE B
C D
]
=
1 0 0 0 0 0 −z 0 1−z 1 0 0 0 0 0 −1 00 −z −1 0 0 0 0 0 20 0 0 1 0 0 0 1 00 0 0 −z −1 0 0 0 10 0 0 0 −z 1 0 0 10 0 0 0 0 0 1 1 01 1 0 0 2 0 0 0 10 1 1 −1 0 1 −1 1 01 0 1 0 1 0 1 −1 1
.
Note that the dimension of the realization (2.5) is n = 7 > δ(T), since n = δ(T)+np.
Moreover, the pencil A− zE has 3 eigenvalues at ∞ (two have multiplicity 3 and the
remaining one is simple), while the matrix D has no particular significance. A minimalcentered realization as in (2.3) can be computed from (2.5) using the formulae in [22],Section 5. With z0 = 1 we obtain:
(2.6) T(z) = D + C(zE −A)−1B(1− z) =
−1 −z 0 0 0 0 −10 1 −z 0 0 −1 −10 0 1 0 0 −1 00 0 0 1 −z 1 10 0 0 0 −1 −1 00 1 1 0 2 1 11 1 0 1 0 4 21 0 1 0 1 −2 4
1
.
In this case, n = 5 = δ(T). Moreover, it is easy to check that D = T(1), and thatthe pencil A − zE has 2 eigenvalues at ∞ (one with multiplicity 3 and one withmultiplicity 2), which are exactly the poles of T. Thus, using centered realizationswe obtain a standard–like characterization of the TFM.
The problem of computing the complete structure of a TFM T(z) (the set of polesand zeros, their partial multiplicities and the set of left and right minimal indices)is equivalent to the simpler problem of computing the eigenstructure of two matrixpencils, namely the pole pencil and the system pencil (assuming that the realizationto start with is minimal):
(2.7) PT(z) := A− zE, ST(z) :=
[A− zE B(α − αz)
C D
],
respectively. For a comprehensive overview we refer to Section 2 in [23] and thereferences therein.
We say that a system T(z) is stable provided all its poles are inside the unit diskor, if T(z) is given by a realization (2.3), if Λ(A − zE) ⊂ D. Note that any stablesystem has the corresponding TFM analytic in C\D ∪ ∞. We will denote the setof all stable TFMs with RH∞. Furthermore, the set of all TFMs that are stable and
USING SIAM’S LATEX MACROS 5
bounded in H∞–norm by a threshold γ > 0 is denoted by
BH(γ)∞ :=
T ∈ RH∞ : ‖T‖∞ < γ
.
We say that the system given by (2.3) is stabilizable if the following two conditionsare met: (i) rank
[A− zE B
]= n, for all z ∈ C\D, and (ii) rank
[E B
]= n,
for z = ∞. We call the pair (C,A − zE) detectable if the pair (A∗ − zE∗, C∗) isstabilizable.
Consider now the collection of matrices Σ := (A− zE,B;Q,L,R), where A,E ∈Cn×n, B,L ∈ Cn×m, Q = Q∗ ∈ Cn×n, Q ≥ 0, R = R∗ ∈ Cm×m. Σ can be seen as anabbreviated representation of a dynamical system (2.3) with C = I and D = 0, i.e.,having the states available for measurement, and a quadratic performance index
(2.8)∑
k≥0
[xk
uk
]∗ [Q LL∗ R
] [xk
uk
],
where x is the state variable and u is the controlled input, see for example [12, 35].The collection Σ may be seen as an extension of the so–called Popov triplets, see [12].We associate with Σ two mathematical objects. The matrix equation
(2.9) A∗XA− E∗XE +Q−[(αA− αE)∗XB + L
]R−1
[L∗ +B∗X(αA− αE)
]= 0
is called the descriptor discrete–time algebraic Riccati equation, denoted DDTARE(Σ, α).We say that the Hermitian square matrix X = X∗ ∈ C
n×n is the unique stabilizingsolution to DDTARE(Σ, α) if Λ
(A− zE +BF (α− αz)
)⊂ D, where
(2.10) F := −R−1(B∗X(αA− αE) + L∗)
is called the stabilizing feedback. Necessary and sufficient existence conditions to-gether with computable formulae for the stabilizing solution are given in [20], Theorem10 and Theorem 11.
We shall define now a parahermitian TFM ΠΣ ∈ Cm×m(z) associated with thecollection of matrices Σ, known as the Popov function [13, 12]:
(2.11) ΠΣ(z) =
A− zE 0 BQ(α− αz) E∗ − zA∗ L
L∗ B∗ R
z0
.
It can be easily checked that ΠΣ is exactly the TFM of the Hamiltonian systemassociated with T, see [35]. Moreover, the descriptor symplectic pencil, see Definition6 in [20], is exactly the system pencil SΠΣ
(see equation (2.7)) associated with therealization (2.11) of ΠΣ.
We say that Σ = (A− zE,B; Q,L,R) and Σ = (A− zE, B; Q, L, R) are feedback
equivalent if there is a matrix F such that A− zE = A− zE +BF (α− αz), B = B,
Q = Q + LF + F ∗L∗ + F ∗RF , L = L + F ∗R, R = R. Note that this is indeedan equivalence relation, satisfying the corresponding axioms. Also notice that thistransformation can be obtained with the change of variables u = Fx+ v, and that itleaves unchanged the value of the quadratic functional (2.8).
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3. Problem formulation. Let T ∈ Cp×m(z) be a general LTI discrete–timesystem, possibly improper or polynomial, with input u and output y, written inpartitioned form as:
(3.1)
[y1y2
]= T
[u1
u2
]=
[T11 T12
T21 T22
] [u1
u2
],
where uj ∈ Cmj , yi ∈ Cpi , and Tij ∈ Cpi×mj (z), with i, j ∈ 1, 2, m := m1 + m2,p := p1 + p2. As usual, u2 is the control input and y2 is the measured output.
The suboptimal H∞ control problem consists in finding the class of all outputfeedback controllers K ∈ Cm2×p2(z), u2 = Ky2, for which the closed–loop system
(3.2) TCL = LFT(T,K) := T11 +T12K(I −T22K)−1T21
is well–posed (that is, all the closed–loop matrices are well–defined), stable, and
‖TCL‖∞ < γ, i.e., TCL ∈ BH(γ)∞ .
We shall say that such controller K is stabilizing and γ−contracting. If in additionγ = 1, we simply say that K is contracting. Let
(3.3) T(z) =
[T11(z) T12(z)T21(z) T22(z)
]=
A− zE B1 B2
C1 D11 D12
C2 D21 0
z0
be a realization centered at z0 ∈ ∂D\Λ(A − zE), having the partitions conformablywith (3.1). We make a set of assumptions on T.(H1) The pair (A− zE,B2) is stabilizable and the pair (C2, A− zE) is detectable.(H2) For all θ ∈ [0, 2π), we have that
(3.4) rank
[A− ejθE B2(α− αejθ)
C1 D12
]= n+m2.
(H3) For all θ ∈ [0, 2π), we have that
(3.5) rank
[A− ejθE B1(α− αejθ)
C2 D21
]= n+ p2.
Remark 1. (H1) is not restrictive as it is a necessary condition for the existence ofstabilizing controllers, see [24]. Moreover, K stabilizes T if and only if it stabilizesT22. The proof of this claim for general systems follows mutatis mutandis from thestandard case, see Chapter 6 in [39]. We will assume throughout the paper that (H1)always holds.Remark 2. (H2) and (H3) are regularity assumptions, see [39, 12] for the standardcase. In particular, it follows from (H2) that T12 has no zeros on the unit circle,p1 ≥ m2, and that rankD12 = m2 (thus D∗
12D12 is invertible). Dual conclusionsfollow from (H3): T21 has no zeros on ∂D, m1 ≥ p2, rankD21 = p2, and D21D
∗21 is
invertible. (H2) and (H3) are reminiscent from the general H2 problem [36] and areby no means necessary conditions for the existence of a solution to the general H∞control problem. If either of these two assumptions does not hold, we get a singularH∞ control problem, which is beyond the scope of this paper.
USING SIAM’S LATEX MACROS 7
Remark 3. Without restricting generality, we have implicitly assumed T22(z0) =D22 = 0. Indeed, if K is a solution to the problem with D22 = 0, then K(I+D22K)−1
is a solution to the original problem. In particular, it also follows from this assumptionthat the closed–loop system is automatically well–posed, since the following matrix isalways invertible:
(3.6)
[I D22
K(z0) I
]=
[I 0
K(z0) I
].
4. Main result. We state in this section the main result of the paper giving nec-essary and sufficient conditions for the existence of a stabilizing and γ−contractingcontroller, together with analytical formulae for the parametrization of all such con-trollers.Theorem 4. Consider a general discrete–time system T ∈ Cp×m(z), having a real-ization (3.3), centered in z0 = α2 ∈ ∂D\Λ(A−zE). Assume that the hypotheses (H1),(H2), and (H3) from Section 3 hold. Define the following collections of matrices:
(4.1)
Σc :=(A− zE,
[γ− 1
2B1 γ12B2
]; Qc, Lc, Rc
),
Σo :=(A∗ − zE∗,
[γ− 1
2C∗1 γ
12C∗
2
]; Qo, Lo, Ro
),
where
Qc = γ−1C∗1C1, Qo = γ−1B1B
∗1 , Lc = γ− 1
2C∗1
[γ−1D11 D12
], Lo = γ− 1
2B1
[γ−1D∗
11 D∗21
],
Rc =
[γ−2D∗
11
D∗12
] [D11 D12
]−[
Im10
0 0
], Ro =
[γ−2D11
D21
] [D∗
11 D∗21
]−[
Ip10
0 0
].
Then we have:(I) There exists K ∈ Cm2×p2(z) that solves the suboptimal H∞ control problem if
and only if the following conditions simultaneously hold:(C1) The DDTARE(Σc, α) has a stabilizing semidefinite Hermitian solution
X = X∗ ≥ 0 and sgn (Rc) = diag (−Im1, Im2
).(C2) The DDTARE(Σo, α) has a stabilizing semidefinite Hermitian solution
Y = Y ∗ ≥ 0 and sgn (Ro) = diag (−Ip1, Ip2
).(C3) ρ
(X(αA− αE)2Y
)< γ2.
(II) Assume that the conditions (C1), (C2), and (C3) hold. Let Fc =:[F ∗1 F ∗
2
]∗be the stabilizing feedback corresponding to DDTARE(Σc, α), see (2.10), let
CF := γ12C2 +D21F1, and
(4.2) Z := Y(I − γ−2X(αA− αE)2Y
)−1
.
Then the class of all controllers K ∈ Cm2×p2(z) that solve the suboptimal H∞control problem can be expressed as K = LFT(G,Q), where Q ∈ BH(γ)
∞ is anarbitrary parameter and
(4.3) G(z) =
Ag − zEg Bg1 Bg2
Cg1 Dg11 Dg12
Cg2 Dg21 0
z0
,
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with(4.4)
Ag − zEg = A− zE +(γ− 1
2B1F1 + γ12B2F2 +Bg1CF
)(α− αz),
Bg1 = −B1V−11 D∗
21(D21V−11 D∗
21)−1 − γB2Dg11 − γ
12 (αA− αE)ZC∗
g2Dg21,
Bg2 = γ−1(−B1V
−11
(D∗
11D12(D∗12D12)
−1 +D∗21D
∗g11
)− γ
12 (αA− αE)ZC∗
g1
)×
×(D∗12D12)
12V
− 12
3 −B2Dg12,
Cg1 = −γ12F2 + γ
12Dg11CF ,
Cg2 = γ− 12 (D21V
−11 D∗
21)− 1
2CF ,Dg11 = −(D∗
12D12)−1D∗
12D11V−11 D∗
21(D21V−11 D∗
21)−1,
Dg12 = (D∗12D12)
−1/2V1/23 ,
Dg21 = (D21V−11 D∗
21)−1/2,
where
V1 := γ−2D∗11D12(D
∗12D12)
−1D∗12D11 − γ−2D∗
11D11 + Im1= V ∗
1 ,V2 := γ−1(D∗
12D12)−1/2D∗
12D11,V3 := V2V
−11 D∗
21(D21V−11 D∗
21)−1D21V
−11 V ∗
2 − V2V−11 V ∗
2 + Im2= V ∗
3 .
Corollary 5. Assume that the conditions (C1) to (C3) in Theorem 4 hold. Then thecentral controller (i.e., Q = 0) under the following normalizing conditions:
D11 = 0, D∗12
[C1 D12
]=
[0 I
],
[B1
D21
]D∗
21 =
[0I
],
is given by(4.5)
K0(z) =
[A− zE +
[
(γ−1B1B∗1 − γB2B
∗2 )X(αA− αE)− γ(αA− αE)ZC∗
2C2
]
(α− αz) −γ12 (αA− αE)ZC∗
2
γ12B∗
2X(αA− αE) 0
]
z0
.
Remark 6. Notice that for D11 = 0, the formulae are considerably simplified. Weget with this assumption that V1 = Im1
, V2 = 0, V3 = Im2, and Dg11 = 0. Moreover,
Rc and Ro are block diagonal matrices. In particular, D11 = 0 implies that sgn (Rc) =diag (−Im1
, Im2) and sgn (Ro) = diag (−Ip1
, Ip2). The complete solution with D11 = 0
and γ = 1 is given in Section 6, see the system G4(z) in equation (6.28).Remark 7. Condition (C1) holds true if and only if the system of matrix equations:
(4.6)Rc = V ∗
c JcVc,(αA− αE)∗XB + Lc = W ∗
c JcVc,A∗XA− E∗XE +Qc = W ∗
c JcWc,
has a stabilizing solution(X = X∗ ≥ 0, Vc,Wc), with Fc = −V −1
c Wc, and with Vc oflower–left block triangular form. We have denoted with Jc := diag (−Im1
, Im2) the
signature matrix. Note that the system (4.6) is an extension of the well-known KalmanYakubovich Popov system, see e.g. [12]. Further, since D∗
12D12 > 0, sgn (Rc) = Jc ifand only if the Schur complement [?] of the (2, 2) block of Rc is negative definite, i.e.,−D∗
11D12(D∗12D12)
−1D∗12D11 +D∗
11D11 − Im1< 0. Similar conclusions follow for the
condition (C2).Remark 8. Note that our controller is expressed in terms of two dual and decoupledDDTAREs. Therefore, the solution presented in Theorem 4 exhibits the well-knownseparation structure of the H∞ control, see [39] for the standard case. As expected,there is yet another necessary condition, namely (C3), involving a bound on a specificspectral radius.
USING SIAM’S LATEX MACROS 9
Remark 9. It is interesting to notice that if γ ≫ 1 the H∞ central controller inCorollary 5 becomes the H2 optimal controller. The claim can be easily checkedby a simple inspection of the formula (4.5): when γ ≫ 1, Z ≈ Y in (4.2) andγ−1B1B
∗1 ≈ 0. Using this approximations, we obtain the controller in [36], Theorem
1 (under normalizing assumptions).
5. Numerical examples. In order to show the applicability of our results, wepresent in this section two numerical examples, namely a system having both improperand polynomial elements and a system having only polynomial entries.
Example 1. Consider the discrete–time algebraic dynamical system T ∈ C3×3(z):
(5.1) T(z) =
2z2 − 2z z − 1 2z2 − 2z − 1
−z + 1 − z − 1
3z + 2−z + 2
8z3 − 8z2 + 2z − 112z3 + 2z2 − 3z − 1
3z + 28z3 − 8z2 + 2z − 2
,
with m1 = 2,m2 = 1, p1 = 2, and p2 = 1. Note that the system has both improperand polynomial entries. A minimal centered realization with z0 = 1 (thus α = α = 1)is given by:
(5.2) T(z) =
1 2z − 1 0 0 1 1 10 1 0 0 −1 0 −1
−2z 0 1 0 1 1 10 0 0 3z + 2 0 −1 01 0 0 0 0 0 −10 1 0 1 0 0 10 0 2 −1 1 2 0
z0=1
.
The system has three poles: two at ∞ (one simple and one with multiplicity 2) andone real pole at −2/3. Note that the McMillan degree equals the dimension of theminimal realization, i.e., δ(T) = n = 4. Moreover, it is easy to check that T(1) = D.
We shall design now a stabilizing and γ−contracting controller with γ = 4 for thesystem (5.1) using Theorem 4. It is easy to check that the hypotheses (H1), (H2), and(H3) from Section 3 hold. Moreover, the DDTARE(Σc, 1) and the DDTARE(Σo, 1)have stabilizing solutions X = X∗ ≥ 0 and Y = Y ∗ ≥ 0, respectively:
X =
0.9169 −0.3155 0.2885 0.0542−0.3155 1.6994 0.1003 0.10410.2885 0.1003 0.2196 −0.02640.0542 0.1041 −0.0264 0.0510
, Y =
0.1112 −0.0936 0.2519 0.0292−0.0936 0.0845 −0.2175 −0.03570.2519 −0.2175 0.5945 0.08820.0292 −0.0357 0.0882 0.0480
.
Furthermore, ρ(X(A − E)2Y
)= 0.0366 < γ2 = 16. We conclude that the necessary
conditions for the existence of a stabilizing and γ−contracting controller are fulfilled,see Part (I) in Theorem 4. We obtain the following central controller:
(5.3) K(z) =−0.1874z4 − 0.1153z3 + 0.0533z2 + 0.1627z + 0.08668
z4 − 0.1236z3 − 0.3647z2 − 0.1411z − 0.1599.
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Note that the controller is proper. The closed–loop system is given by
TCL(z) =1
(z + 0.6667)(z + 0.6606)(z2 − 1.583z + 0.6287)(z2 − 0.0916z + 0.09543)(z2 + 0.6293z + 0.2261)×
[(z + 1.578)(z + 0.6667)(z + 0.6601)(z + 0.4363)(z − 0.8699)(z − 1)(z2 − 0.1058z + 0.1462)−(z + 0.6667)(z + 0.6604)(z − 1)(z − 0.8725)(z2 − 0.1216z + 0.1513)(z2 + 0.9488z + 1.223)
−0.5(z − 2.128)(z − 1)(z − 0.529)(z + 0.6604)(z + 0.9861)(z + 0.143)(z2 + 0.8154z + 0.2537)0.16667(z − 2.813)(z − 1)(z − 0.5482)(z + 1.09)(z + 0.6618)(z + 0.31)(z2 + 1.414z + 0.6185)
].
We observe that TCL(z) is proper and stable, having all its poles inside the unit disk,i.e., 0.7916 ± 0.0447j, 0.0458± 0.3055j,−0.3147± 0.3565j,−0.6606,−0.6667 ⊂ D.Moreover, we have that ‖TCL‖∞ = 3.3065 < γ = 4. The singular value plot for Tand TCL is given in Figure 1.
Example 2. Consider the following polynomial system T ∈ C5×4(z):
T(z) =
−13z2 + 16z − 3 −4z + 4 −11z2 + 11z + 1 26z2 − 28z6z2 − 6z 2z − 2 6z2 − 6z + 2 −12z2 + 11z
z2 − 3z + 2 0 −z2 + z + 1 −2z2 + 4z − 57z2 − 9z + 3 2z − 1 5z2 − 5z −14z2 + 15z − 1−12z3 + 12z2 −4z2 + 3z + 2 −12z3 + 12z2 − 2z + 2 24z3 − 22z2 − 5z + 3
,
having m1 = m2 = 2, p1 = 3, and p2 = 2. A minimal realization centered at z0 = 1is given by:
T(z) =
1 0 0 0 0 1 0 1 −2−6z 1 0 0 0 0 2 0 −1
0 −2z 1 0 0 1 1 3 10 0 0 1 0 0 0 −1 0
−z 0 0 −2z 1 −2 0 0 21 −2 0 1 −1 0 0 1 −20 1 0 0 0 0 0 2 −10 0 0 0 1 0 0 1 −30 1 0 0 1 1 1 0 01 0 −1 0 0 0 1 0 0
z0=1
.
The system has two poles at ∞, one with multiplicity 2 and one with multiplicity 3.We notice that δ(T) = n = 5, and T(1) = D.
We will design now a stabilizing and γ-contracting controller for T(z), with γ = 6.The hypotheses (H1), (H2), and (H3) from Section 3 hold. The DDTARE(Σc, 1) andthe DDTARE(Σo, 1) have stabilizing semidefinite Hermitian solutions:
X =
7.3768 1.0840 0.2523 1.3247 0.41331.0840 0.1856 0.0659 0.1335 −0.00890.2523 0.0659 0.2100 0.3078 −0.15741.3247 0.1335 0.3078 2.1904 0.37570.4133 −0.0089 −0.1574 0.3757 0.4299
= X∗ ≥ 0,
USING SIAM’S LATEX MACROS 11
10−3
10−2
10−1
100
101
−80
−60
−40
−20
0
20
40
Example 1. Singular value plot for T and TCL
Frequency (rad/s)
Sin
gula
r V
alue
s (d
B)
TCL
T
10−3
10−2
10−1
100
101
−50
−40
−30
−20
−10
0
10
20
30
40
Example 2. Singular value plot for T and TCL
Frequency (rad/s)
Sin
gula
r V
alue
s (d
B)
TCL
T
Fig. 1. Singular value plots for Example 1 (left) and Example 2 (right)
Y =
0.0200 0.0258 −0.0461 −0.0057 0.03770.0258 0.1220 0.0748 −0.0997 −0.1202
−0.0461 0.0748 0.3108 −0.1269 −0.3439−0.0057 −0.0997 −0.1269 0.0978 0.16530.0377 −0.1202 −0.3439 0.1653 0.3940
= Y ∗ ≥ 0.
Moreover, ρ(X(A − E)2Y
)= 0.1620 < γ2 = 36. Thus, the necessary conditions in
Theorem 4 hold true. The central controller was computed to be:
K(z) =
[−0.242z5 + 0.238z4 + 0.388z3 − 0.120z2 − 0.440z + 0.175 −0.794z5 − 0.257z4 + 1.71z3 + 0.289z2 − z + 0.0600.397z5 − 2.192z4 + 3.072z3 − 1.347z2 + 0.192z − 0.123 0.061z5 − 0.920z4 + 1.313z3 − 0.173z2 − 0.264z − 0.015
]
z5 − 4.121z4 + 1.707z3 + 4.083z2 − 2.777z + 0.07476.
Note that K(z) is proper. Moreover, the closed–loop system is proper and stable,having the poles 0.8853,−0.0511± 0.1102j, 0.4239± 0.2854j, 0.2664± 0.2922j ⊂ D.The controller is also γ−contracting, since ‖TCL‖∞ = 5.2141 < γ = 6. The singularvalue plot is given in Figure 1.
6. Proof of the main result. We proceed now with the proof of our mainresult stated in Theorem 4. In order to simplify the formulae, we assume throughoutthis section that D11 = 0 and γ = 1, see Remark 6. The extension for the generalcase will be detailed in the sequel. We shall need some preliminary results.
6.1. Preliminary results.Lemma 10. Let (C,A−zE) be a detectable pair and assume that there exists a matrixX = X∗ such that the following descriptor Stein equation holds: A∗XA − E∗XE +C∗C = 0. Then X ≥ 0 if and only if Λ(A− zE) ⊂ D.Proof. Assume that Λ(A − zE) ⊂ D. Then the matrix E is nonsingular and the
descriptor Stein equation becomes a regular Stein equation A∗XA −X + C∗C = 0,where A = AE−1 and C = CE−1. The rest of the proof follows mutatis mutandisfrom linear matrix equations theory, see e.g. Theorem 1.5.5 in [12].
Proposition 11. Consider the collection of matrices Σ := (A − zE,B; Q,L,R).Assume that Λ(A− zE) ⊂ D. Let ΠΣ be the Popov function as in (2.11). Then the
12 TEX PRODUCTION
following are equivalent.(i) ΠΣ(e
jθ) < 0, for all θ ∈ [0, 2π).(ii) R < 0 and the DDTARE(Σ, α) (2.9) has a unique stabilizing solution X = X∗.Proof. (i) ⇒ (ii): Since z0 ∈ ∂D is implicitly assumed, we have that ΠΣ(z0) =R < 0. Further, if ΠΣ(e
jθ) < 0, ∀θ ∈ [0, 2π), then ΠΣ has no zeros on ∂D. ThusSΠΣ
(see (2.7)), which is the descriptor symplectic pencil (DSP) associated with theDDTARE(Σ, α) (see [20] Definition 6), has no generalized eigenvalues on ∂D. This inturn implies that the DSP has an n−dimensional stable deflating subspace, denotedby V , see [20], Propositions 5, 7, and Remark 8. Moreover, since ΠΣ(e
jθ) < 0, weget that V is disconjugate, i.e., its basis matrix partitioned conformably with theDSP has the n × n upper partition invertible, see the Appendix in [20]. Thus theDDTARE(Σ, α) has a unique stabilizing Hermitian solution, see [20], Theorem 10.
(ii) ⇒ (i): Let F be the stabilizing Riccati feedback as in (2.10). Consider thespectral factor given by:
S(z) :=
[A− zE B−F I
]
z0
.
It can be easily checked that the factorization ΠΣ(z) = S#(z)RS(z) holds. More-over, S ∈ RH∞ and S−1 ∈ RH∞, since Λ(A − zE) ⊂ D and Λ(A − zE + BF (α −αz)) ⊂ D. Thus S is a unity in RH∞ (i.e., unimodular). Since R < 0, ΠΣ(e
jθ) =S#(ejθ)RS(ejθ) < 0, for all θ ∈ [0, 2π).
Proposition 12. (Bounded–Real Lemma) Consider T ∈ Cp×m(z) having a real-ization as in (2.3) and let Σ = (A−zE,B; C∗C,C∗D,D∗D−Im). Then the followingstatements are equivalent.(i) T ∈ BH(1)
∞ , i.e., Λ(A− zE) ⊂ D and ‖T‖∞ < 1.(ii) D∗D − Im < 0 and DDTARE(Σ, α) has a stabilizing solution X = X∗ ≥ 0.Proof. (i) ⇒ (ii): Note that ‖T‖∞ < 1 ⇔ T#(ejθ)T(ejθ)− Im < 0, ∀θ ∈ [0, 2π). Bystraight-forward manipulations we get that T#(z)T(z)−Im = ΠΣ(z), where ΠΣ is as
in (2.11). ThusΠΣ(ejθ) < 0, ∀θ ∈ [0, 2π). Since Λ(A−zE) ⊂ D (because T ∈ BH(γ)
∞ ),it follows from Proposition 11 that R := D∗D − Im < 0 and DDTARE(Σ, α) has astabilizing solution X = X∗.
It remains to prove that X ≥ 0. It is easy to check that the DDTARE(Σ, α) hasa stabilizing solution X = X∗ and D∗D − Im < 0 if and only if the following systemof matrix equations
(6.1)D∗D − I = −V ∗V
(αA− αE)∗XB + C∗D = −W ∗VA∗XA− E∗XE + C∗C = −W ∗W
has a solution (X = X∗, V,W ), with the stabilizing feedback F = −V −1W. Further,note that the last equation in (6.1) can be written as
(6.2) A∗XA− E∗XE +
[CW
]∗ [CW
]= 0.
Also note that Λ(A−zE−V −1W (α−βz)
)⊂ D, since F = −V −1W is the stabilizing
feedback. Therefore, the pair (W,A− zE) is detectable and furthermore, the pair
(6.3)
([CW
], A− zE
)
USING SIAM’S LATEX MACROS 13
is also detectable. Using this conclusion and the fact that the Stein equation (6.2)holds, it follows from Lemma 10 that X ≥ 0, since Λ(A− zE) ⊂ D.
(ii) ⇒ (i): Following a similar reasoning as above, we have from (ii) that thepair in (6.3) is detectable. Since X ≥ 0 and the equality (6.2) holds, we get fromLemma 10 that Λ(A− zE) ⊂ D. Using the implication (ii) ⇒ (i) in Proposition 11,we have that ΠΣ(e
jθ) < 0, ∀θ ∈ [0, 2π). But this is equivalent with ‖T‖∞ < 1. Thus
T ∈ BH(1)∞ and the proof is complete.
We say that a system T ∈ Cp×m(z) is unitary on the unit circle if T#(z)T(z) = I,for all z ∈ ∂D\Λ(A− zE), where T#(z) := T∗(1/z). If in addition T ∈ RH∞, thenT is called inner. The following definition will be used in the sequel to characterizeinner systems given by centered realizations (see for example [10, 25, 12]).Definition 13. Assume that T ∈ Cp×m(z) is given by a centered realization (2.3),with z0 ∈ ∂D\Λ(A− zE). If D∗D = Im and there is an invertible (positive definite)matrix X = X∗ such that
(6.4)A∗XA− E∗XE + C∗C = 0,D∗C +B∗X(αA− αE) = 0,
then we say that the realization of T is unitary (inner). It can be easily checked thatif T has a unitary (inner) realization, then T is unitary (inner), that is, T#T = I(and T ∈ RH∞). The converse is not generally true. In the sequel, we shall makethe abuse to call a system unitary (inner) iff its realization is unitary (inner).
We state now a crucial result toward the solution, known in the literature asRedheffer’s theorem. It basically states that, under some assumptions, theH∞ controlproblem for inner systems is solved by any stable controller having its H∞ normbounded by the prescribed threshold γ > 0.Theorem 14. (Redheffer theorem) Consider a centered realization for T ∈ Cp×m(z)partitioned as in (3.3). Assume that T is unitary, D21 is square and nonsingular,Λ(A − zE − B1D
−121 C2(α − αz)) ⊂ D, and let K be a controller for T. Then the
following statements are equivalent.(i) TCL := LFT(T,K) ∈ BH(1)
∞ .
(ii) T is inner and K ∈ BH(1)∞ .
Proof. (ii) ⇒ (i) Let
K(z) =
[AK − zEK BK
CK DK
]
z0
be a realization for K ∈ Cm2×p2(z). Since K ∈ BH(1)∞ , we have from Proposition
12 that D∗KDK − I < 0 and DDTARE(ΣK , α) has a stabilizing Hermitian solution
XK = X∗K ≥ 0, where ΣK := (AK − zEK , BK ; C∗
KCK , C∗KDK , D∗
KDK − I). Further,since T is inner we have with Definition 13 that
D∗D =
[0 D∗
21
D∗12 0
] [0 D12
D21 0
]= I
and there is X = X∗ ≥ 0 such that (6.4) holds. Compute now a centered real-ization for TCL = LFT(T,K), see Section 2.3.2 in [35]. After lengthy but simplealgebraic manipulations we obtain that the realization of TCL satisfies condition (ii)in Proposition 12, with
(6.5) XCL :=
[X 00 XK
]= X∗
CL ≥ 0, RCL := D∗21(D
∗KDK − I)D21 < 0,
14 TEX PRODUCTION
and ΣCL := (ACL − zECL, BCL; C∗CLCCL, C
∗CLDCL, D
∗CLDCL − I). Therefore, it
follows from Proposition 12 that TCL ∈ BH(1)∞ .
(i) ⇒ (ii) : Since TCL ∈ BH(1)∞ , we have that T#
CL(z)TCL(z) − I < 0, for allz ∈ ∂D. Using the LFT definition in (3.2) and the fact that T21 is unimodular, i.e.,a unity in RH∞ (see the Theorem statement), we get after some manipulations thatK#(z)K(z) − I < 0, for all z ∈ ∂D, which is equivalent with ‖K‖∞ < 1. Thus K isbounded on ∂D, i.e., K ∈ RL∞. The stability of K is a direct consequence of the factthat (H1) is fulfilled, that TCL is stable, and that K ∈ RL∞. Therefore, we conclude
that K ∈ BH(1)∞ .
It remains to prove that T is inner. K ∈ BH(1)∞ implies with Proposition 12
that the DDTARE(ΣK , α) has a stabilizing solution XK = X∗K ≥ 0. Using the same
argument, the DDTARE(ΣCL, α) has a stabilizing solution with XCL = X∗CL ≥ 0 as
in (6.5). Thus X ≥ 0. Further, recall that (C2, A− zE) is detectable, see Remark 1.Then (C,A − zE) is also detectable. Since T is assumed to be unitary, i.e., satisfiesthe equations (6.4) with X ≥ 0, it follows with Lemma 10 that Λ(A − zE) ⊂ D.Therefore, T is inner and the proof is complete.
6.2. Proof of necessity. We proceed now with the proof of Part (I) of Theo-rem 4. In order to prove the necessity of (C1) and (C2), we show that the existenceof a controller which solves the H∞ problem is equivalent with a specific signaturecondition, which in turn implies that the DDTARE has a stabilizing solution. Thenecessity of the signature conditions on Rc and Ro will be proven in the next sub-section, together with the extension for D11 6= 0. The second part of the proof, i.e.,the necessity of (C3), follows a more intricate route: we find an auxiliary necessarycondition (C4), and then show that (C2) and (C3) are equivalent with (C4).
Necessity of (C1) and (C2). Assume that there exists a stabilizing and contractingcontroller K for the system T having a realization as in (3.3). Let Σc be as in (4.1)with D11 = 0, γ = 1, i.e.,
(6.6) Σc :=
(A− zE,
[B1 B2
]; C∗
1C1,[0 C∗
1D12
],
[−I 00 D∗
12D12
]).
Note that in this case sgn (Rc) = diag (−Im1, Im2
) automatically holds. It can bechecked that the Popov function associated with Σc can be written in partitionedform as:
(6.7)
ΠΣc(z) =:
[Π11 Π12
Π21 Π22
]=
[T#
11
T#12
] [T11 T12
]−[
I 00 0
]
=
A− zE 0 B1 B2
C∗1C1(α− αz) E∗ − zA∗ 0 C∗
1D12
0 B∗1 −I 0
D∗12C1 B∗
2 0 D∗12D12
z0
,
where T11 and T12 have realizations as in (3.3). Let M := K(I − T22K)−1T21 ∈RH∞, since the closed–loop system is stable. Then we have that
(6.8)[I M#
]ΠΣc
[IM
]= T#
CLTCL − I < 0, ∀z ∈ ∂D, since ‖TCL‖∞ < 1.
USING SIAM’S LATEX MACROS 15
At this point we need a condition for the existence of a stabilizing solution for thematrix equation DDTARE(Σc, α).Lemma 15. (The signature condition) Let ΠΣc
be given in partitioned form asin (6.7). Assume that Λ(A− zE) ⊂ D. If there exists an M ∈ RH∞ such that
(6.9)[I M#
]ΠΣc
[IM
]< 0 on ∂D,
then the DDTARE(Σc, α) has a stabilizing hermitian solution X = X∗ ≥ 0.Proof. Note that Π22(e
jθ) = T12(ejθ)#T12(e
jθ) > 0, ∀θ ∈ [0, 2π). Clearly, we have:
[Q L2
L∗2 R22
]:=
[C∗
1
D∗12
] [C1 D12
]≥ 0.
Since Λ(A− zE) ⊂ D, E is nonsingular. We conclude that ΠΣcsatisfies the hypothe-
ses of Theorem 4.12.9 in [12]. The rest of the proof follows mutatis mutandis fromthe standard case.
In general, Λ(A − zE) is not necessarily a subset of D. To cope with this, let
Σc be the feedback equivalent of Σc, where F =[0 F ∗
2
]∗is chosen such that the
matrix pencil
A− zE = A− zE +[B1 B2
] [ 0F2
](α− αz) = A− zE +B2F2(α− αz)
has all its eigenvalues inside the unit disc (such F always exists, provided that the pair(A− zE,B2) is stabilizable). Moreover, the following signature condition is verified:
(6.10)[I M#
]ΠΣc
[I
M
]=
[I M#
]ΠΣc
[IM
]< 0, ∀z ∈ ∂D,
where M is such that
(6.11)
[I
M
]=
A− zE B1 B2
0 I 0−F2 0 I
z0
[IM
], M ∈ RH∞.
Therefore, we can apply Lemma 15 for ΠΣcto conclude that DDTARE(Σc, α) has a
stabilizing solution X = X∗ ≥ 0. It is easy to show that X is a stabilizing solution tothe DDTARE(Σc, α) if and only if X is a stabilizing solution to the DDTARE(Σc, α).Therefore, (C1) in Theorem 4 is indeed a necessary condition for the existence of astabilizing and contracting controller.
Further, notice that the condition (C2) simply states that the dual system
(6.12) Tdual(z) =
A∗ − zE∗ C∗1 C∗
2
B∗1 0 D∗
21
B∗2 D∗
12 0
z∗
0
satisfies (C1), with
(6.13) Σo :=
(A∗ − zE∗,
[C∗
1 C∗2
]; B1B
∗1 ,[0 B1D
∗21
],
[−I 00 D21D
∗21
]).
16 TEX PRODUCTION
Note that sgn (Ro) = diag (−Ip1, Ip2
). The line of reasoning for proving the necessityof (C2) is identical, and thus omitted.
Necessity of (C3). We shall prove now the necessity of (C3), using an auxiliarynecessary condition. With the solution of DDTARE(Σc, α) and the correspondingstabilizing feedback (see equation (2.10)):
(6.14) Fc :=
[F1
F2
]=
[B∗
1X(αA− αE)−(D∗
12D12)−1
(D∗
12C1 +B∗2X(αA− αE)
)],
we introduce the following systems:
(6.15)
TI(z) =
A− zE +B2F2(α− αz) B1 B2(D∗12D12)
− 12
C1 +D12F2 0 D12(D∗12D12)
− 12
−F1 I 0
z0
,
TO(z) =
A− zE +B1F1(α− αz) B1 B2
−(D∗12D12)
12F2 0 (D∗
12D12)12
C2 +D21F1 D21 0
z0
.
After lengthy but simple manipulations, we can check that T = TI ⊗ TO, whereT,TI and TO are given by the minimal realizations (3.3), (6.15). We denoted by⊗ the Redheffer star product of TFMs, see [39, 12] for a standard definition andSection 2.3.2 in [35] for centered realization–based formulae. The following lemma isan interesting result by itself.Lemma 16. A controller K is stabilizing and contracting for T if and only if it isstabilizing and contracting for TO, i.e., LFT(T,K) ∈ BH(1)
∞ ⇔ LFT(TO,K) ∈ BH(1)∞ .
Proof. It is easy to show that the system TI having a minimal realization as in (6.15)is inner, in the sense that it satisfies the equations in Definition 13, with X = X∗ ≥ 0the solution of DDTARE(Σc, α). Moreover, it can be checked that TI satisfies thehypotheses of Theorem 14 (Redheffer).
Notice that LFT(T,K) = LFT(TI ⊗TO,K) = LFT(TI ,LFT(TO,K)) ∈ BH(1)∞ .
Further, since TI is inner and satisfies the hypotheses of Theorem 14, it follows thatLFT(TO,K) ∈ BH(1)
∞ . Conversely, let LFT(TO,K) ∈ BH(1)∞ be a controller for the in-
ner systemTI . Then, we have from Theorem 14 that LFT(TI ,LFT(TO,K)) ∈ BH(γ)∞ .
But this is equivalent with LFT(T,K) ∈ BH(1)∞ , since TI ⊗TO = T, q.e.d.
Note that Lemma 16 holds for any realization of T,TI and TO (not necessarilyminimal). It can be easily proved that K is a stabilizing and contracting controller forT iff it is stabilizing and contracting for TI ⊗TO. We give now additional necessaryconditions for the solvability of the H∞ problem.Proposition 17. Assume that the condition (C1) in Theorem 4 holds. Let Fc be thestabilizing feedback as in (6.14). Define(6.16)
Σ× =
(A∗ − zE∗ + F ∗
1B∗1(α− αz),
[−(D∗
12D12)1/2F2
C2 +D21F1
]∗;B1B
∗1 , [ 0 B1D
∗21 ] ,
[−I 00 D21D
∗21
]).
Then the suboptimal H∞ control problem has a solution if and only if (C1) holds and:(C4) The DDTARE(Σ×, α) has a stabilizing solution Z = Z∗ ≥ 0.
Proof. We have to prove the following statement: if the suboptimal H∞ controlproblem has a solution and (C1) holds, then (C4) also holds. According with Lemma
USING SIAM’S LATEX MACROS 17
16, it is enough to solve the problem for TO. One can easily check that TO satisfiesthe hypotheses (H1), (H2), and (H3) in Section 3. Further, note that TO given in(6.15) satisfies (C2) in Theorem 4, which is a necessary condition for the existence ofa stabilizing and contracting controller. Write Σo as in (6.13) for TO to get (6.16).Thus, the condition (C2) for TO becomes condition (C4), i.e., the DDTARE(Σ×, α)has a stabilizing semidefinite Hermitian solution. This completes the proof.
It remains to prove the necessity of (C3). We will show that if (C1) holds, then(C2) and (C3) are equivalent with (C4). The formal result is stated as follows.Proposition 18. Assume that (C1) in Theorem 4 holds. Then the following state-ments are equivalent.
(i) The DDTARE(Σo, α) has a stabilizing semidefinite Hermitian solution Y =Y ∗ ≥ 0 and ρ
(X(αA− αE)2Y
)< 1, where Σo is given in (6.13).
(ii) The DDTARE(Σ×, α) has a stabilizing semidefinite Hermitian solution Z =Z∗ ≥ 0, with Z satisfying (4.2), and Σ× given in (6.16).
Proof. The idea of the proof is to relate the system pencils SΠΣcand SΠΣ
×
associatedwith the Popov functions ΠΣc
and ΠΣ×, in order to find a relation between the
stabilizing solutions of the DDTARE(Σo, α) and the DDTARE(Σ×, α). According to(2.7), (2.11), (6.13), and (6.16) we have that(6.17)
SΠΣo=
A∗ 0 αC∗1 αC∗
2
αB1B∗1 E 0 αB1D
∗21
0 C1 −I 0D21B
∗1 C2 0 D21D
∗21
− z
E∗ 0 αC∗1 αC∗
2
αB1B∗1 A 0 αB1D
∗21
0 0 0 00 0 0 0
=: No − zMo ,
SΠΣ×
=
A∗ + αF ∗1B
∗1 0 −αF ∗
2 (D∗12D12)
12 α(C∗
2 + F ∗1D
∗21)
αB1B∗1 E + αB1F1 0 αB1D
∗21
0 −(D∗12D12)
12F2 −I 0
D21B∗1 C2 +D21F1 0 D21D
∗21
−
−z·
E∗ + αF ∗1B
∗1 0 −αF ∗
2 (D∗12D12)
12 α(C∗
2 + F ∗1D
∗21)
αB1B∗1 A+ αB1F1 0 αB1D
∗21
0 0 0 00 0 0 0
=: N× − zM×.
Since Mo, No ∈ C(2n+p)×(2n+p) and M×, N× ∈ C(2n+m2+p2)×(2n+m2+p2), the systempencils (6.17) are not of the same dimension. We shall work with their augmentedversions defined by
(6.18)Na
o := diag (No,−Im1,−Im2
), Mao = diag (Mo, 0m1
, 0m2);
Na× := diag (N×,−Im1
,−Ip1), Ma
× = diag (M×, 0m1, 0p1
).
Thus Nao ,M
ao , N
a×,M
a× are all (2n+m+p)× (2n+m+p) matrices. With the solution
18 TEX PRODUCTION
X ≥ 0 of the DDTARE(Σc, α), introduce the following nonsingular matrices:(6.19)
Ua :=
I X(αA− αE) −F ∗2 (D
∗12D12)
12 0 0 −C∗
1
0 I 0 0 0 00 0 0 0 0 I0 0 0 I 0 00 0 0 0 I 00 0 I 0 0 0
,Wa :=
I −X(αA− αE) 0 0 0 00 I 0 0 0 0
0 −(D∗12D12)
12F2 0 0 0 I
0 0 0 I 0 00 0 0 0 I 00 −C1 I 0 0 0
.
Using the DDTARE(Σc, α) it can be checked by rather lengthy manipulations that
(6.20) Ua(Na× − zMa
×)Wa = Nao − zMa
o .
Thus the two augmented matrix pencils are strictly equivalent. We shall proceed nowwith the proof.
(ii) ⇒ (i) : Since the DDTARE(Σ×, α) has a stabilizing solution Z ≥ 0, it followsfrom [20] (specifically, Proposition 5 and Theorem 10) that there exists T× and S×such that
(6.21) M×V×S× = N×V×T×(6.18)⇐==⇒ Ma
×Va×S× = Na
×Va×T×.
where
V× =
I(αA− αE)Z
⋆
, V a
× =
V×00
.
Using (6.20), (6.21) yieldsMaoW
−1a V a
×S× = NaoW
−1a V a
×T×. Further, compute W−1a V a
×to obtain
(6.22)
Mao
I +X(αA− αE)2Z(αA− αE)Z
⋆
S× = Na
o
I +X(αA− αE)2Z(αA− αE)Z
⋆
T×
⇒ Mo
I(αA− αE)Y
⋆
So = No
I(αA− αE)Y
⋆
To,
where we have denoted
(6.23) Y := Z(I +X(αA− αE)2Z
)−1 ≥ 0,
So :=(I + X(αA − αE)2Z
)S×, and To :=
(I + X(αA − αE)2Z
)T×. But (6.22)
shows by [20], Theorem 10, that Y is the stabilizing solution to the DDTARE(Σo, α).Further, with (6.23) we obtain
(6.24) I −X(αA− αE)2Y =(I +X(αA− αE)2Z
)−1.
Since X ≥ 0 and Z ≥ 0, all the eigenvalues of I +X(αA− αE)2Z are positive. Thiswith (6.24) implies that ρ
(X(αA − αE)2Y
)< 1. Moreover, from (6.23) and (6.24)
we obtain that
Z = Y(I −X(αA− αE)2Y
)−1,
which is exactly (4.2). Thus, the first implication is proved. The ”only if” part of theproof follows by reversing the above reasoning. The necessity of the condition (C3)was completely proven.
USING SIAM’S LATEX MACROS 19
6.3. Proof of sufficiency. We proceed now with the proof of Part (II) in The-orem 4, which is based on a successive reduction to simpler problems, called theone–block problem and the two–block problem, borrowing the terminology from themodel matching problem.
Consider the one–block problem, for which p1 = m2, p2 = m1, i.e., D12 and D21
are square, and T12 and T21 are invertible, having only stable zeros, i.e.,
(A1) D12 ∈ Cm2×m2 is invertible and Λ(A− zE −B2D
−112 C1(α− αz)
)⊂ D.
(A2) D21 ∈ Cm1×m1 is invertible and Λ(A− zE −B1D
−121 C2(α− αz)
)⊂ D.
Proposition 19. For the one–block problem the class of all controllers that solve theH∞ control problem is K = LFT(G1,Q), Q ∈ BH(1)
∞ is arbitrary and(6.25)
G1(z) =
A− zE − (B2D−112 C1 +B1D
−121 C2)(α− αz) B1D
−121 B2D
−112
−D−112 C1 0 D−1
12
−D−121 C2 D−1
21 0
z0
.
Proof. Let TR := T ⊗ G1. With G1 from (6.25), compute a centerd realizationfor TR, using the formulae in Section 2.3.2 in [35]. We further get, after a changeof coordinates in the state–space (equivalence transformation) and by removing thestable uncontrollable and undetectable parts, that
TR = T⊗G1 =
[0 II 0
].
Thus TCL = LFT(T,K) = LFT(T,LFT(G1,Q)) = LFT(TR,Q) = Q ∈ BH(1)∞ .
Conversely, let K be such that TCL ∈ BH(1)∞ . Take TCL ≡ Q ∈ BH(γ)
∞ be an arbitrarybut fixed parameter. Then LFT(G1,Q) = LFT(G1,TCL) = LFT(G1,LFT(T,K)) =LFT(G1 ⊗T,K). It can be checked that in this case G1 ⊗T = T⊗G1 = TR (thisis not generally true). Thus LFT(G1,Q) = K.
Consider now the two–block problem, for which p2 = m1, and the hypotheses (H2)in Section 3 and (A2) are fulfilled. Let Σc be as in (6.6).Proposition 20. Assume that the DDTARE(Σc, α) has a stabilizing solution X =X∗ ≥ 0. Then the two–block problem has a solution. Moreover, the class of allstabilizing and contracting controllers is K = LFT(G2,Q), with Q ∈ BH(1)
∞ and(6.26)
G2(z) =
A− zE + (B2F2 −B1D−121 C2)(α− αz) B1D
−121 B2(D
∗12D12)
− 12
F2 0 (D∗12D12)
− 12
−D−121 C2 −B∗
1X(αA− αE) D−121 0
z0
.
Proof. Let Fc be the stabilizing feedback corresponding to DDTARE(Σc, α), givenin (6.14). Consider the systems TI and TO in (6.15). Recall that T = TI ⊗TO andthat TI is inner, since its realization (6.15) satisfies the equations given in Definition13, where X = X∗ ≥ 0 is the unique stabilizing solution of the DDTARE(Σc, α).
Since TI is inner, we have from Lemma 16 that it is enough to find the classof controllers for TO. Further, it is easy to show that, for the two–block problem,TO in (6.15) satisfies the assumptions (A1) and (A2) from the one–block problem.Compute G1 in (6.25) for TO to get G2 in (6.26). This completes the proof.
20 TEX PRODUCTION
Proposition 21. Assume p1 = m2, (A1), (H3), and that DDTARE(Σo, α) has astabilizing solution Y = Y ∗ ≥ 0. Then the dual two–block problem has a solution.Moreover, the class of all controllers is K = LFT(G3,Q), where Q ∈ BH(γ)
∞ and(6.27)
G3(z) =
A− zE + (H2C2 −B2D−112 C1)(α − αz) H2 −B2D
−112 − (αA− αE)Y C∗
1
D−112 C1 0 D−1
12
(D21D∗21)
− 12C2 (D21D
∗21)
− 12 0
z0
,
where H2 = −(B1D∗21 + (αA− αE)Y C∗
2 )(D21D∗21)
−1.Proof. The result follows by duality from Proposition 20. Indeed, if T(z) satisfies(A1) and (H3), then Tdual(z) in (6.12) satisfies the assumptions in Proposition 20,with Σo given in (6.13). Further, write down the system G2 in (6.26) for Tdual toobtain the system G3 in (6.27).
Proof of Part (II), Theorem 4. Assume that (H1), (H2), and (H3) in Section3 hold. Suppose that (C1) in Theorem 4 holds. Consider now the systems TI andTO, given in (6.15). We have shown in Lemma 16 that it is enough to find thethe class of controllers for TO. It is easy to check that, in the general case, TO
satisfies (A2). Write now Σo in (6.13) and DDTARE(Σo, α) for TO to obtain Σ×and DDTARE(Σ×, α). Since there exists a stabilizing and contracting controller,then (C4) must hold true. Therefore, TO satisfies the assumptions in Proposition 21.Write G3(z) in (6.27) for TO to obtain
(6.28)
G4(z) =
A− zE + (BFc +BZCF )(α− αz) BZ −B2(D∗12D12)
− 12 + (αA− αE)ZF ∗
2 (D∗12D12)
12
−F2 0 (D∗12D12)
− 12
(D21D∗21)
− 12CF (D21D
∗21)
− 12 0
z0
,
where Fc =[F ∗1 F ∗
2
]∗is given in (6.14),
B :=[B1 B2
], CF := C2+D21F1, BZ := −
(B1D
∗21+(αA−αE)ZC∗
F
)(D21D
∗21)
−1.
We conclude that the parametrization of all controllers that solves the suboptimalH∞ control problem for D11 = 0 and γ = 1 is K = LFT(G4,Q), with Q ∈ BH(1)
∞ ,and G4 given in (6.28).
To obtain the formulae for γ > 0 arbitrary, one can use a scaling procedure, seeRemark 10.3.2 in [12]. Consider the initial problem ‖TCL‖∞ < γ, see Section 3. Thisis equivalent with ‖Tsc
CL‖∞ < 1, where TscCL := 1
γTCL. The latter expression can bewritten as:
TscCL =
1
γT11+T12
1
γK
(I − γT22
1
γK
)−1
T21 = Tsc11+T12K
sc (I −Tsc22K
sc)−1
T21,
with Ksc = 1γK. Thus, the conversion to a problem with γ > 0 arbitrary can be
achieved by the following scaling transformation on the initial problem:
Tsc(z) =
[Tsc
11(z) T12(z)T21(z) Tsc
22(z)
]=
[ 1γT11(z) T12(z)
T21(z) γT22(z)
]=
A− zE 1√γB1
√γB2
1√γC1
1γD11 D12√
γC2 D21 0
z0
.
USING SIAM’S LATEX MACROS 21
Further, note that, for the scaled systemTsc, the stabilizing and contracting controlleris Ksc = LFT(G4,Q
sc), with Qsc ∈ BH(1)∞ , and G4 as in (6.28). Thus:
K = γKsc = LFT
([γG4,11 G4,12
G4,211γG4,22
],Q
),
where Q = γQsc ∈ BH(γ)∞ . After straightforward manipulations, we can obtain with
(6.28) the stabilizing and γ−contracting controller for γ > 0 arbitrary.The case D11 6= 0. The formulae for the controller are obtained by employing
a transformation called loop-shifting, see Sections 17.1 and 17.2 in [39]. The loop-shifting procedure uses two steps. First, normalize D12 and D21. Note that this isalways possible, since D12 and D21 have full rank. Second, solve a unitary matrixdilatation problem, see Lemmas 17.2 and 17.3 in [39]. Notice that we can use the sameline of reasoning, since we work with centered realizations for which K(z0) = DK .
It remains to prove that sgn (Rc) = diag (−Im1, Im2
) and sgn (Ro) = diag (−Ip1, Ip2
)are indeed necessary conditions. It is enough to show that the statement in Remark7 holds true in Lemma 15 and Proposition 18.
In Lemma 15, notice that R22 := Rc22 = Π22(z0) = D∗12D12 > 0. Further, letting
DM := M(z0), we obtain with (6.9)
(6.29)[I D∗
M
] [ D∗11D11 − Im1
D∗11D12
D∗12D11 D∗
12D12
] [I
DM
]< 0.
With this inequality and the fact that R22 > 0 we get −D∗11D12(D
∗12D12)
−1D∗12D11+
D∗11D11 − Im1
< 0. But this is equivalent with sgn (Rc) = diag (−Im1, Im2
) =: Jc.Moreover, we can write Rc = V ∗
c JcVc, with Vc of lower–left block triangular form.We still need to prove that, in Proposition 18, we have (see Remark 7):
sgn (R×) = diag (−Im2, Ip2
) ⇔ sgn (Ro) = diag (−Ip1, Ip2
).
But this follows mutatis mutandis from the standard case, see Lemma 10.7.1 in [12].This completes the whole proof of our main result stated in Theorem 4.
7. Conclusions. Necessary and sufficient conditions for the existence of sub-optimal H∞ controllers for a discrete–time algebraic dynamical system have beenprovided. A realization–based characterization for the class of all stabilizing and con-tracting controllers was given. Our simple and numerically reliable formulae havebeen illustrated on two numerical examples. The easiness in obtaining centered real-isations and the similarity with the solution from the standard case, once the appro-priate equivalents are taken, recommends these realisations for virtually any problemin control dealing with general systems.
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