Marc Andre Peters

Minimum Entropy Control for Time-Varying Systems

From the Publisher: In this book we provide a time-domain theory of the entropy criterion. For linear time-invariant systems, this time-domain notion of entropy is equivalent to the usual frequency domain criterion. Moreover, this time-domain notion of entropy enables us to define a suitable entropy for other classes of systems, including the class of linear time-varying systems. Furthermore, by working with this time-domain definition of the entropy we are able to gain new interpretations of the advantages of minimum entropy control. In particular we consider the connections between the time-varying minimum entropy control problem and the time-varying analogues to the H[subscript 2], H[symbol not reproducible] and risk-sensitive control problem. This book is aimed at researchers in mathematical control theory. It will be particularly attractive to those working in both H[symbol not reproducible] control theory as well as linear time-varying systems. While the approach used has its basis in operator theory, it can be followed easily by graduate students with a basis in linear systems theory.

Minimum Entropy Control

In this chapter we provide results for the minimum entropy optimal control problem for linear discrete-time time-varying systems. Our approach breaks down to the analysis of a series of simpler control problems, along the line of [31] from which the general result follows. In fact, we exhibit the same separation principle in the entropy as was seen in [53].

A Spectral Test for Observability and Reachability of Time-Varying Systems

A spectral test for the observability and reachability of linear time-invariant systems---the Popov--Belevitch--Hautus test---is well known and serves as a powerful characterization of these properties. In this paper it is shown that similar tests exist for linear time-varying systems. The test presented here involves a check over a subset of the spectrum of the weighted block shift known as the set of almost eigenvalues.

A spectral test for observability and detectability of discrete-time linear time-varying systems

The PBH test is a well known and powerful spectral test for the observability of linear time-invariant systems. In this paper we show that a similar test exists for linear time-varying systems. The test presented here involves a check over part of the spectrum of the weighted block shift known as approximate eigenvalues.

The relationship between minimum entropy control and risk-sensitive control for time-varying systems

The connection between minimum entropy control and risk-sensitive control for linear time-varying systems is investigated. For time-invariant systems, the entropy functional and the linear exponential quadratic Gaussian cost are the same. In this paper, it is shown that this is not true for general time varying systems. It does hold, however, when the system admits a state-space representation.

On the induced norms of discrete-time and hybrid time-varying systems

Characterizations for the induced norms of two types of systems are considered. It is first shown that the induced `2 norm of a discrete-time, linear time-varying system may be characterized by the existence requirement on solutions to operator algebraic Riccati equations. A similar result is derived for systems that arise in sampled-data systems, involving a mixture of continuousand discrete-time signals.

An entropy formula for nonlinear systems

Abstract Minimum entropy control theory provides a powerful means of trading off the advantages of optimal H 2 and H ∞ controllers. In this paper, a framework for the nonlinear minimum entropy control problem is suggested, based on the nonlinear spectral factorization problem. It is also shown that, for a large class of H ∞ suboptimal controllers, the central controller minimizes the closed-loop entropy. The techniques used are similar to the time-domain notion of the entropy that was introduced by the authors for time-varying systems.

Continuous-time time-varying entropy

In this paper we present a definition for the entropy for time-varying continuous-time systems. This entropy differs significantly from the entropy for discrete-time systems, which is defined as the memoryless component of the spectral factor of an operator related to the system. Properties of this entropy are discussed, the relationships with ℋ2 and ℋ∞, as well as the relationship with the discrete-time entropy.

Connections between minimum entropy control and mixed H 2 / H ∞ control for time-varying systems

Abstract In this paper two properties of minimum entropy control for linear time-varying systems are investigated. An entropy formulation for anti-causal systems is first considered. Minimizing the norm of an anticausal system is straightforward because of the fact that the norm of an operator is equal to the norm of its adjoint. This is not true for the entropy measure. For this reason a separate formula for the entropy of anti-causal systems is needed in order to consider dual problems that arise in optimal control theory. Secondly, we investigate the relationship between the entropy measure for time-varying systems and the H 2 and H ∞ norms. For time-invariant systems, minimum entropy control has been shown to be a good mixed H 2/ H ∞ controller. In this paper we show that this is also true in the time-varying case.

The minimum entropy controller as a mixed H/sub 2//H/sub /spl infin// controller: the time-varying case

In this paper we have given an interpretation of the entropy, defined for discrete-time time-varying systems. The interpretation shows why minimum entropy controllers have an advantage over other H/sup /spl infin// controllers, in the sense that the difference in norms of the input and output signals will be relatively high for signals not equal to the worst-case one. This is in favor of the goal that H/sup /spl infin// control wants to achieve. Furthermore we showed that minimizing the entropy in general results in a relatively low quadratic cost, which indicates why minimum entropy controllers are in general good mixed H/sup 2//H/sup /spl infin// controllers.