Robust Stability of Interconnections

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Robust Stability of Interconnections



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, †School of Electronics and Computer Science, University of Southampton, SO17 1BJ, UK,, ‡Department of Electrical Engineering, Indian Institute of Technology, Bombay, India 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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