Synthesis of dissipative systems using quadratic differential forms: Part I

Abstract

Abstract-The problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative. Our formulation and solution of this problem is completely representation-free. The system dynamics are specified by a differential behavior, and the performance is specified through a quadratic differential form. We view control as interconnection: a controller constrains a distinguished set of system variables, the control variables. The resulting behavior of the to-be-controlled variables is called the controlled behavior. The constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior, obtained by setting the control variables equal to zero, and the plant behavior, obtained by leaving the control variables unconstrained. The main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications. The essential requirement is a coupling condition, an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior.Index Terms-Behaviors, controller implementability, coupling condition, dissipative systems, hidden behavior, quadratic differential forms, storage functions.

On Lyapunov functions and data-driven dissipativity

“…Necessary and sufficient conditions for the existence of a controller C implementing K are given in Th. 1, p. 55 of Willems and Trentelman (2002).…”

On the regular implementability of nD systems

“…Necessary and sufficient conditions for implementability by partial interconnection for a given K ⊆ A q for the case n = 1 are given in [14]. In [12] it was shown that these conditions are also necessary and sufficient for more general classes of linear systems, including nD systems.…”

“…In the behavioral approach to control this problem has been formalized as the problem of characterizing all behaviors that are implementable with respect to the given plant behavior. Originally the problem was mainly studied for 1D systems, see for example [14,2,1,7], but also generalizations to more general classes of system behaviors, including nD systems, have been investigated, see [12,10,4,15].…”

“…Given a 1D full plant behavior, in [14] for the first time a characterization was given of all implementable behaviors. Later, in [12], this result was generalized to more general system classes, including nD system behaviors.…”

Dissipativity preserving model reduction by retention of trajectories of minimal dissipation

“…For an extensive treatment of dissipative systems in a behavioral context we refer to [21,27,30,31]. Here, we review the basic material.…”