# Robust Stability of Interconnections

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## Abstract

mtns2006BFP.dvi Robust Stability of Interconnections Wenming Bian∗, Mark French† Harish Pillai‡ MTNS 2006 Extended Abstract 1 Introduction We begin by observing that the graph topology with its various metrizations plays a fundamental role in the theory of robust stability for classical LTI systems([1, 2, 6]. The contribution of this note is to develop the basic theory of robust stability involving the gap-distance directly from a behavioural perspective, observing that recent approaches to generalisations of the gap metric [2] have been purely trajectory based and hence are easily amenable to such an approach. There has been previous interest in developing behavioural notions of the gap metric, see e.g. [3] for an example. 1.1 The classical result Our concern is with the closed loop systems of equations as shown in Figure 1: [P,C] : y1 = Pu1, y0 = y1 + y2 u2 = Cy2, u0 = u1 + u2, where u0, u1, u2 ∈ H 2, y0, y1, y2 ∈ H 2 and P , C are transfer functions. For such a BIBO u0 u1 y1 P C y0 u2 y2 − + + − Figure 1: The closed-loop system [P,C]. system, the closed loop transfer function ΠP//C is of interest: w0 = ( u0 y0 ) ΠP//C 7→ ( u1 y1 ) = w1. ∗School of Electronics and Computer Science, University of Southampton, SO17 1BJ, UK, wb@ecs.soton.ac.uk †School of Electronics and Computer Science, University of Southampton, SO17 1BJ, UK, mcf@ecs.soton.ac.uk, ‡Department of Electrical Engineering, Indian Institute of Technology, Bombay, India hp@ee.iitb.ac.in. 1 The classical robust stability theorem of linear control is as follows: Theorem If [P,C] is BIBO stable, i.e. ‖ΠP//C‖H∞ <∞, [P1, C] is well posed, and ~δ(P, P1)‖ΠP//C‖ < 1, then [P1, C] is BIBO stable, i.e. ‖ΠP1//C‖H∞ <∞. Here ~δ(P1, P2) denotes the directedH 2 gap distance between P1 and P2. The gap measures the size of the smallest stable co-prime factor perturbation between normalised co-prime factor representations of P1 and P2. 1.2 A behavioural generalisation Within a b

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