A bound on closed-loop performance based on finite-frequency response samples☆
Section snippets
Notation
Let and denote the boundary of . The symbol is used to denote the space of all (possibly matrix valued) functions that are essentially bounded on and have finite norm , where represents the maximum singular value. The symbol denotes the space of functions that are analytic in and have finite norm . Given a system transfer function , the transfer function of the adjoint system is denoted by ,
Review of -gap metric robustness results
The -gap between two linear time-invariant plants and is defined aswhere and denotes the winding number of evaluated on the standard Nyquist contour indented around any poles and zeros on [6]. For a real rational transfer matrix X satisfying the winding number where denotes the number of unstable poles of x. When the winding number condition is satisfied,
A lower bound on achieved closed-loop performance
Below, a lower bound on , at any intermediate frequency , is obtained using the definitions of system complexity defined above. Proposition 1 Let C be a controller that stabilises both the true plant and the nominal model . Then for any , the following inequalities hold:where
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 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 axis to the unit circle, such as for a
Conclusion
A new lower bound on the level of closed-loop performance achieved by finite-dimensional feedback controller C with plant 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.