Time-Varying Discrete Linear Systems

Halanay, A.; Ionescu, Vlad

doi:10.1007/978-3-0348-8499-0

Cited by 112 publications

(43 citation statements)

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“…Their interpretation as representative for factorization problems of the external or inner-outer type have been well documented also in the recent literature, see [22], [25]. In the context of the Darlington theory they get a particularly nice interpretation, since, as we have argued, the Darlington embedding problem reduces to a factorization problem.…”

Section: Darlington Embedding Of a Contractive Time-varying Tranmentioning

confidence: 59%

Generalized Darlington synthesis

Dewilde

1999

IEEE Trans. Circuits Syst. I

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Abstract-The existence of a "Darlington embedding" has been the topic of vigorous debate since the time of Darlington's original attempts at synthesizing a lossy input impedance through a lossless cascade of sections terminated in a unit resistor. This paper gives a survey of present insights in that existential question. In the first part it considers the multiport, time invariant case, and it gives the necessary and sufficient conditions for the existence of the Darlington embedding, namely that the matrix transfer scattering function considered must satisfy a special property of analyticity known as "pseudomeromorphic continuability" (of course aside from the contractivity condition which ensures lossiness). As a result, it is reasonably easy to construct passive impedances or scattering functions which do not possess a Darlington embedding, but they will not be rational, i.e. they will have infinite dimensional state spaces. The situation changes dramatically when time-varying systems are concerned. In this case also Darlington synthesis is possible and attractive, but the anomalous case where no synthesis is possible already occurs for systems with finite dimensional state spaces. We give precise conditions for the existence of the Darlington synthesis for the time-varying case as well. It turns out that the main workhorse in modern Darlington theory is the geometry of the so called Hankel map of the scattering transfer function to be embedded. This fact makes Darlington theory of considerably larger importance for the understanding of systems and their properties than the original synthesis question would seem to infer. Although the paper is entirely devoted to the theoretical question of existence of the Darlington embedding and its system theoretic implications, it does introduce the main algorithm used for practical Darlington synthesis, namely the 'square root algorithm' for external or inner-outer factorization, and discusses some of its implications in the final section.

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Section: Darlington Embedding Of a Contractive Time-varying Tranmentioning

confidence: 59%

Generalized Darlington synthesis

Dewilde

1999

IEEE Trans. Circuits Syst. I

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“…Some of our results appear to be new. The previous standard results (e.g., Halanay and Ionescu [3]) assume a type of "exponential" stabilizability that is too strong for our purposes. In section 4 we review the relevant theory on Lyapunov-balanced realizations for discrete-time infinite-dimensional systems.…”

Section: Introductionmentioning

confidence: 79%

“…Discrete-time infinite-dimensional systems have been treated in a number of texts (e.g., [1], [3], [12]). However, the standard treatments of the linear quadratic theory assume the strong concept of power stabilizability, i.e., the existence of an F such that (A + BF ) n ≤ M λ n for some constants M > 0, 0 < λ < 1 and all positive integers n. Unfortunately, this concept is not suitable for a nice theory of LQG-balanced realizations.…”

Section: Discrete-time Infinite-dimensional Systemsmentioning

confidence: 99%

Linear Quadratic Gaussian Balancing for Discrete-Time Infinite-Dimensional Linear Systems

Opmeer¹,

Curtain²

2004

SIAM J. Control Optim.

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In this paper, we study the existence of linear quadratic Gaussian (LQG)-balanced realizations for discrete-time infinite-dimensional systems. LQG-balanced realizations are those for which the smallest nonnegative self-adjoint solutions of the control and filter Riccati equations are equal. We show that the control (filter) Riccati equation has a nonnegative self-adjoint solution if and only if the system is output (input) stabilizable. Our main result is that the transfer function of a discrete-time linear system has an approximately controllable and observable LQG-balanced realization iff it has an input and output stabilizable realization. The corresponding control and filter Riccati equations have unique nonnegative self-adjoint solutions. Moreover, approximately controllable and observable LQG-balanced realizations are unique up to a unitary state-space transformation. Finally, we show that the spectrum of the product of the smallest nonnegative self-adjoint solutions of the control and filter Riccati equations is independent of the particular realization.

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“…Thus, the theorem is simply a statement regarding the stability of linear time-varying systems and can be found in [14]. …”

Section: Appendixmentioning

confidence: 99%

Linear dynamically varying LQ control of nonlinear systems over compact sets

Bohacek

Jonckheere²

2001

IEEE Trans. Automat. Contr.

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Abstract-Linear-quadratic controllers for tracking natural and composite trajectories of nonlinear dynamical systems evoluting over compact sets are developed. Typically, such systems exhibit "complicated dynamics," i.e., have nontrivial recurrence. The controllers, which use small perturbations of the nominal dynamics as input actuators, are based on modeling the tracking error as a linear dynamically varying (LDV) system. Necessary and sufficient conditions for the existence of such a controller are linked to the existence of a bounded solution to a functional algebraic Riccati equation (FARE). It is shown that, despite the lack of continuity of the asymptotic trajectory relative to initial conditions, the cost to stabilize about the trajectory, as given by the solution to the FARE, is continuous. An ergodic theory method for solving the FARE is presented. Furthermore, it is shown that wrapping the LDV controller around the nonlinear system secures a stable tracking dynamics. Finally, an example of controlling the Hénon map to follow an aperiodic orbit is presented.

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Time-Varying Discrete Linear Systems

Cited by 112 publications

References 0 publications

Generalized Darlington synthesis

Generalized Darlington synthesis

Linear Quadratic Gaussian Balancing for Discrete-Time Infinite-Dimensional Linear Systems

Linear dynamically varying LQ control of nonlinear systems over compact sets

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