Implementation of behavioral systems

In this chapter, we study control by interconnection of a given linear differential system (the plant behavior) with a suitable controller. The problem formulations and their solutions are completely representation free, and specified only in terms of the system dynamics. A controller is a system that constrains the plant behavior through a certain set of variables. In this context, there are two main situations to be considered: either all the system variables are available for control, i.e., are control variables (full control) or only some of the variables are control variables (partial control). For systems evolving over a time domain (1D) the problems of implementability by partial (regular) interconnection are well understood. In this chapter, we study why similar results are not valid in themultidimensional (nD) case. Finally, we study two important classes of controllers, namely, canonical controllers and regular controllers

Nyquist plots, finite gain/phase margins & dissipativity

The relation between the small gain theorem and 'infinite phase margin' is classical; in this paper we formulate a novel supply rate, called the 'not-out-of-phase' supply rate, to first prove that 'infinite gain margin' (i.e. non-intersection of the Nyquist plot of a transfer function and the negative half of the real axis) is equivalent to dissipativity with respect to this supply rate. Capturing non-intersection of half-line makes the supply rate system-dependent: a novel feature unobserved in the supply rates considered in the literature so far. We then show that the traditional finite and positive gain/phase margin conditions for stability are equivalent to dissipativity with respect to a frequency weighted convex combination of the not-out-of-phase supply rate and the small-gain supply rate; both frequency weightings and combining two supply-rates/performance-indices have been investigated in the literature in different contexts, but only as sufficient conditions. (C) 2013 Elsevier B.V. All rights reserved

Improper L-infinity Optimal/Suboptimal Controllers

We consider L-infinity-control of MIMO systems and address solvability of the problem over all finite dimensional LTI controllers: i.e., controllers whose transfer functions can be proper or improper. We show that improper controllers are easily dealt with using the behavioral approach, unlike the standard state-space/transfer-matrix methods, and argue that there are cases where an improper controller can outperform a proper controller. In this setting, we next formulate and prove necessary and sufficient conditions for suboptimal L-infinity-control problem solvability and relate this to existing results about system invariant zeros. Further, we infer that in our formulation, assuming suboptimal solvability conditions on the system, an optimal controller always exists, possibly with an improper transfer function. In other words, the infimum L-infinity-norm of the closed loop system is achievable when dealing with both proper and improper controller transfer functions. We illustrate these results through an example for which the optimal L-infinity-controller has an improper transfer function

DISSIPATIVITY OF UNCONTROLLABLE SYSTEMS, STORAGE FUNCTIONS, AND LYAPUNOV FUNCTIONS

Dissipative systems have played an important role in the analysis and synthesis of dynamical systems. The commonly used definition of dissipativity often requires an assumption on the controllability of the system. In this paper we use a definition of dissipativity that is slightly different ( and less often used in the literature) to study a linear, time-invariant, possibly uncontrollable dynamical system. We provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not "mixed"; i.e., no pair of uncontrollable poles is symmetric about the imaginary axis. This condition is known to be related to the solvability of a Lyapunov equation; we link Lyapunov functions for autonomous systems to storage functions of an uncontrollable system. The set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature. We show that for an uncontrollable system the set of storage functions is unbounded, and that the unboundedness arises precisely due to the set of Lyapunov functions for an autonomous linear system being unbounded. Further, we show that stabilizability of a system results in this unbounded set becoming bounded from below. Positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive. In this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions. In most of the results in this paper, we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis; we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity