A bound on closed-loop performance based on finite-frequency response samples

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Abstract

A new, computationally tractable, bound is derived for the level of closed-loop performance achieved by a given finite-dimensional feedback compensator with a plant for which a finite number of frequency response samples are computable. The bound involves quantities reflecting the performance of the controller with a finite-dimensional, nominal model of the plant, quantities that can be determined from the finite number of frequency response samples of the true plant, and quantities related to the complexity (in the sense of Vinnicombe) of all systems involved. This bound can be used to ‘validate’ closed-loop performance in the case that the true plant frequency response samples are of a plant which is not completely known or, to measure the performance of a finite-dimensional controller with a computationally intractable (e.g. infinite-dimensional) model of the true plant.

Section snippets

Notation

Let D{zC:|z|<1} and D denote the boundary of D. The symbol L is used to denote the space of all (possibly matrix valued) functions F(z) that are essentially bounded on D and have finite norm FLesssupωσ¯(F(ejω)), where σ¯(·) represents the maximum singular value. The symbol H denotes the space of functions F(z) that are analytic in D and have finite norm FsupzD|f(z)|<. Given a system transfer function F(z), the transfer function of the adjoint system is denoted by F*(z)F(1/z)T,

Review of ν-gap metric robustness results

The ν-gap between two linear time-invariant plants P1 and P2 is defined asδν(P1,P2)=infQ,Q-1LG1-G2QifI(P1,P2)=0,1otherwise,where I(P1,P2)wnodet(GP2*GP1) and wno(g) denotes the winding number of g(z) evaluated on the standard Nyquist contour indented around any poles and zeros on D [6]. For a real rational transfer matrix X satisfying X,X-1L the winding number wnodet(X)=η(X-1)-η(X) where η(x) denotes the number of unstable poles of x. When the winding number condition is satisfied, δν(P1

A lower bound on achieved closed-loop performance

Below, a lower bound on ρ(Pt,C), at any intermediate frequency ω[ωi,ωi+1], is obtained using the definitions of system complexity defined above.

Proposition 1

Let C be a controller that stabilises both the true plant Pt and the nominal model Pm. Then for any ω[ωi,ωi+1], the following inequalities hold:ρ(Pt,C)(ejω)max{b(Pm,C),1-xi2}-yi,ρ(Pt,C)(ejω)max{b(Pm,C),1-Xi2}-Yi,whereximin{κ(Pm,-C*)(ejωi),κ(Pm,-C*)(ejωi+1)}+(vPmi+v-C*i)|ωi+1-ωi|,yimin{κ(Pm,Pt)(ejωi),κ(Pm,Pt)(ejωi+1)}+(vPti+vPmi)|ωi+1-ωi|Ximin

Computation of complexity and the lower bound

Although the results presented so far are established for discrete-time models, which is perhaps most suitable in a situation where Pt is not known completely and the frequency response samples are obtained from identification experiments, they are also applicable to continuous-time situations. This is because the results are all established in the frequency-domain and are essentially invariant under a conformal mapping of the jω axis to the unit circle, such as z=(1+sT/2)/(1-sT/2) for a

Conclusion

A new lower bound on the level of closed-loop performance achieved by finite-dimensional feedback controller C with plant Pt for which a finite number of frequency response samples can be computed, is presented. The lower bound involves quantities that can be computed from a model of the controller, a finite-dimensional nominal model of the plant, and a measure of the complexity (in the sense of Vinnicombe) of all systems involved. Computation of the lower bound is discussed for the case in

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Parts of this paper have been published in the proceedings of ECC ’03, Cambridge, UK.

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