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By Francesco Amato

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"This publication presents a scientific remedy of the speculation of sturdy keep watch over of dynamical structures. … the writer has selected to think about simply the issues for which a definitive answer has been came upon, having in brain researchers or engineers operating in industries who are looking to observe those methodologies to useful regulate difficulties. … The presentation is self contained, all proofs are given in addition to genuine global examples. … The self contained presentation makes it compatible for use for academic post-graduate purposes." (Anna Maria Perdon, Zentralblatt MATH, Vol. 1142, 2008)

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Extra info for Robust Control of Linear Systems Subject to Uncertain Time-Varying Parameters

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This fact suggests to replace the nonlinear element described in Fig. 3 by the linear time-varying gain described in Fig. 4, where the function g(·) is allowed to be any member of the set g(·) ∈ C0 (R+ , R) : g(t) ∈ [gmin , 1] , t ∈ [0, +∞) . 24) u Fig. 4. The linear time-varying element This operation leads to the linear closed loop system in Fig. 5, which has the following state-space description x(t) ˙ = A0 − bcT g x(t) . 25) is quadratically stable; this means that there exists a positive definite matrix P such that A0 − bcT g T P + P A0 − bcT g < 0 , g ∈ [gmin , 1] .

Linear Time-Varying Systems An application of the Lyapunov theorem shows that when the system is sufficiently slowly varying in time the eigenvalues location in the left half of the complex plane is still sufficient to guarantee exponential stability of the system. A particular attention has been given to the definition of input-output gain of a linear time-varying system, as this concept will be exploited throughout the book to study the performance problem. In this context, the main result that has been stated is the time-varying version of the Bounded Real Lemma.

The procedure is composed of three steps. 49) by introducing the fictitious parameters δj ∈ Ij := [0, 1] if fj is not multi-affine , {0} if fj is multi-affine j = 1, . . 54) and substituting for fj (p), j = 1, . . , ν, the multi-affine function fjm (p, δj ) := (1 − δj )f j (p) + δj f j (p) if fj is not multi-affine fj (p) if fj is multi-affine . 55) Let D := I1 × I2 × · · · × Iν ; hence Ψ (p, δ) is defined over R × D. 49) there are at most ν products, the maximum exponent of each pi in Ψ (p, δ) cannot exceed ν.

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