On the Optimal Control of Relaxation Systems

Abstract

The relaxation systems are an important subclass of the passive systems that arise naturally in applications. We exploit the fact that they have highly structured state-space realisations to derive analytical solutions to some simple Hinfinity type optimal control problems. The resulting controllers are also relaxation systems, and often sparse. This makes them ideal candidates for applications in large-scale problems, which we demonstrate by designing simple, sparse, electrical circuits to optimally control large inductive networks and to solve linear regression problems.

Phase Analysis of MIMO LTI Systems

“…The concept of phase unifies the well-known notions of positive real systems [10], [12], [14], negative imaginary systems [15], [16], and relaxation systems [28], [29]. A system G ∈ RH m×m ∞ is said to be positive real or passive if G(jω)+G * (jω) ≥ 0 for all ω ∈ [0, ∞].…”

Passive and reciprocal networks: From simple models to simple optimal controllers

“…−1 B K + D K . We can thus find a lower bound on the achievable H ∞ performance (as suggested in [4]) by solving the static optimisation problem A least squares argument (see [31,Lemma 1]) shows that K = G (0)…”

Introduction to Jan C. Willems’, “Dissipative Dynamical Systems, Part II: Linear Systems With Quadratic Supply Rates”

“…In fact, the use of the terminology 'reciprocal' goes back to Maxwell (Maxwell's reciprocal rule) and Onsager (the Onsager reciprocal relations). Recently, there is renewed interest in the study of reciprocal systems, and relaxations systems in particular, motivated by neuro-computing [8], as well by obtaining simple control strategies for complex physical systems [13,14].…”