On the distributed stable full information H minimax problem

Staffans, Olof J.

doi:10.1002/(sici)1099-1239(19981230)8:15<1255::aid-rnc386>3.0.co;2-p

Cited by 8 publications

(6 citation statements)

References 3 publications

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“…Regular linear systems were introduced in [40], and most well-posed systems encountered in applications are regular. There is now a rich literature using this concept, ranging from partial differential equations to engineering applications, see for instance Avalos et al [2], Hinrichsen and Pritchard [8], Jacob and Zwart [10], Logemann et al [14], Logemann and Ryan [15], Logemann, Ryan and Townley [16,17], Logemann and Townley [18,19], Morris [20], Rebarber [25,26], Rebarber and Townley [27], Staffans [30,32,33,34], Weiss and Curtain [44], Weiss and Häfele [46] and Weiss 2 [45]. The concept of a well-posed linear system is closely related to that of a scattering semigroup, as studied in Lax and Phillips [12,13] -we shall say more about this connection later.…”

Section: Introductionmentioning

confidence: 99%

Transfer functions of regular linear systems Part II: The system operator and the Lax–Phillips semigroup

Staffans

2002

Trans. Amer. Math. Soc.

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Abstract. This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as "Part I". We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described byẋ = Ax + Bu, y = Cx + Du would be the s-dependent matrixIn the general case, S Σ (s) is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks A−sI and B, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of S Σ (s) where the right lower block is the feedthrough operator of the system. Using S Σ (0), we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the "initial time" is −∞. We also introduce the Lax-Phillips semigroup T induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ω ∈ R which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of A and also the points where S Σ (s) is not invertible, in terms of the spectrum of the generator of T (for various values of ω). The system Σ is dissipative if and only if T (with index zero) is a contraction semigroup.

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Section: Introductionmentioning

confidence: 99%

Transfer functions of regular linear systems Part II: The system operator and the Lax–Phillips semigroup

Staffans

2002

Trans. Amer. Math. Soc.

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“…The litterature for the continuous time case is considerably richer. Recent continuous time papers, somewhat parallelling our work, are [1], [6], [19], [21], [22], [24], [20], [29], [10]. The papers [1], [6] contain also short reviews of the history and development of the theories connecting the spectral factorization and feedback control; the latter for the discrete time systems, too.…”

Section: Introductionmentioning

confidence: 88%

Minimax control of distributed discrete time systems through spectral factorization

Malinen

1997

1997 European Control Conference (ECC)

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In this paper we introduce a Riccati equation theory for (a class of) well posed (I/O-stable) discrete time linear systems Φ as presented in [9].We tie together three different notions: The first notion is the general question under which conditions it is possible to solve a minimax control problem associated to Φ by static state feedback. The second notion concerns the existence of a certain spectral factorization of the I/O-map of Φ. The third notion is about a particular (stabilizing) solution of a Riccati equation system associated with Φ.We show that these three notions are in fact equivalent under fairly mild stability assumptions of Φ, namely input-output stability. Furthermore, this equivalence does not require any finite dimensional structure in any of the operators of the system.

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“…where D is a given bounded linear feedthrough operator. We use diagrams of the type drawn in Figure 1 to represent the relation between the state x(t), the observation y, the initial value x , and the control u defined in (4). Throughout in our diagrams we use the following conventions:…”

Section: Definition 21mentioning

confidence: 99%

“…Proof. If is J-coercive, then > D*JD > can be inverted, and we get (6) from (5). By substituting this value for…”

Section: Lemma 34mentioning

confidence: 99%

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Feedback representations of critical controls for well-posed linear systems

Staffans

1998

Int. J. Robust Nonlinear Control

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This is the first part in a three part study of the suboptimal full information H∞ problem for a well‐posed linear system with input space U, state space H, and output space Y. We define a cost function Q(x0,u)=∫ boldR +〈y(s),Jy(s)〉Yds, where y∈L2loc(R+; Y) is the output of the system with initial state x0∈H and control u∈L2loc(R+; U), and J is a self‐adjoint operator on Y. The cost function Qis quadratic in x0 and u, and we suppose (in the stable case) that the second derivative of Q(x0, u) with respect to u is non‐singular. This implies that, for each x0∈H, there is a unique critical control ucrit such that the derivative of Q(x0, u) with respect to u vanishes at u=ucrit. We show that ucrit can be written in feedback form whenever the input/output map of the system has a coprime factorization with a (J, S)‐inner numerator; here S is a particular self‐adjoint operator on U. A number of properties of this feedback representation are established, such as the equivalence of the (J, S)‐losslessness of the factorization and the positivity of the Riccati operator on the reachable subspace. © 1998 John Wiley & Sons, Ltd.

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On the distributed stable full information H minimax problem

Cited by 8 publications

References 3 publications

Transfer functions of regular linear systems Part II: The system operator and the Lax–Phillips semigroup

Transfer functions of regular linear systems Part II: The system operator and the Lax–Phillips semigroup

Minimax control of distributed discrete time systems through spectral factorization

Feedback representations of critical controls for well-posed linear systems

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