By Shankar P. Bhattacharyya
This publication is a learn monograph describing effects bought by means of the writer and his co-workers of the final years. It bargains with the suggestions stabilization of linear time invariant platforms containing a true parameter vector topic to perturbation. the consequences defined contain equipment for designing low order regulators for the nominal plant, the calculation of balance areas and balance margins within the parameter house, and systems to layout stabilizing controllers that tolerate prescribed levels of parameter perturbations. each one of those difficulties is handled within the move functionality and country area domain names. effects are offered on the theoretical point and on the algorithmic computational point. the aim of this monograph is to provide - at an straight forward point - a few promising initial effects and to attract the eye of researchers to this type of difficulties. The publication is meant for regulate theorists, utilized mathematicians and engineers taken with keep watch over structures study and development.
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Additional resources for Robust Stabilization Against Structured Perturbations
Example text
1 and the equation Xp=6 for the characteristic vector. e. X, fixed and the plant parameter p = p0 + Ap we have ~ = 6(x,p ° + Ap) and closed loop stability is equivalent to 6(x,p ° + Ap) 6 H , . 24) and let Sp(p °) denote the interior of the hypersphere of radius p(p0) centered at p0 in parameter space: Sp(P °) := {PIP 6 R A', p = pO + Ap, flApt[2 < p(p0)}. 12) Let S ~ ( p °) denote the boundary of this hypersphere: SB(p °) := {pip 6 R k, p -----pO + Ap, llApll2 _-- p(pO)}. 13) 44 The proof of the theorem now consists of showing that $(x,p) E H a V p E S .
In practice, this may not be the case, for example, when the perturbations are prescribed to lie within a rectangular polytope. This motivates us to consider weighted perturbations in the next two sections and to solve the problem of determining the largest stability ellipsoid. In Section 4, we use the stability margins calculated so far to describe a numerical procedure to design a controller for robust stability. These ideas are illustrated by some examples in section 5 which concludes with a brief discussion.
3) its perturbation, so t h a t a = a ° + Aa. 5) where A E R kxt and b E R k. W i t h o u t loss of generality, we can assume that A is of full column rank and therefore l _< k (otherwise the parameters a could be redefined). 6) which shows the arlene transformation m a p p i n g the p a r a m e t e r vector a into the characteristic vector/g. 16)) in the space of a as I/0, I / , and I](w). It is to be noted, however, that now some of these sets m a y be empty. 7) r,~ = co if 1-I. 8) r(w) = c~ if II(~) = 0 .