By Alessandro Astolfi, Lorenzo Marconi

This e-book is a tribute to Prof. Alberto Isidori at the social gathering of his sixty fifth birthday. Prof. Isidori’s proli?c, pioneering and high-impact learn job has spanned over 35 years. all through his occupation, Prof. Isidori has built ground-breaking effects, has initiated researchdirections and has contributed towardsthe foundationofnonlinear controltheory.In addition,his commitment to provide an explanation for problematic concerns and di?cult innovations in an easy and rigorous method and to inspire younger researchers has been instrumental to the highbrow development of the nonlinear regulate neighborhood around the world. the amount collects 27 contributions written by means of a complete of fifty two researchers. The imperative writer of every contribution has been chosen one of the - searchers who've labored with Prof. Isidori, have in?uenced his examine task, or have had the privilege and honour of being his PhD scholars. The contributions deal with a signi?cant variety of regulate subject matters, together with th- retical matters, complicated purposes, rising keep watch over instructions and educational works. the range of the components lined, the variety of participants and their overseas status offer proof of the impression of Prof. Isidori within the regulate and structures conception groups. The publication has been divided into six elements: method research, Optimization equipment, suggestions layout, law, Geometric tools and Asymptotic research, re?ecting vital keep watch over parts which were strongly in- enced and, every so often, pioneered by means of Prof. Isidori.

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This means that it has two terminals, both of ‘electrical’ type, and that it is parameterized by a non-negative real number (the value of the resistor in Ohms). Since the terminals are electrical, there are two variables, a voltage and a current (counted positive when it runs into the device), on each terminal. This yields in total four real variables associated with a resistor: (V1 , I1 ) and (V2 , I2 ) . C. Willems Fig. 5. A resistor have an ohmic resistor, we know that the relation between these variables is V1 − V2 = RI1 , I1 + I2 = 0.

Cn such that S ⊆ i=1 Nci . It then follows that for any d ∈ S, there exists some 1 ≤ j ≤ n such that |Fd [v](tj )| ≥ τcj /2 , (15) where tj = tcj . Let μ = (μ0 , μ1 , . . , μk−1 ) ∈ Rmk be given. Let {ωj } be a sequence of analytic functions deﬁned on [0, T ] such that • • • (i) ωj (0) = μi for 0 ≤ i ≤ k − 1, j ≥ 1; ωj → v in the L1 norm (as functions deﬁned on [0, T ]); and for some M ≥ 1, ωj ∞ ≤ M for all j ≥ 1. ) Reducing the value of T if necessary, one may assume that (T, M ) is admissible for all d ∈ S.

For the feedback system, we obtain the module equations (u3 , y4 ) ∈ BG1 ; (u5 , y6 ) ∈ BG2 ; y2 = u 1 + u 6 ; y4 = u 2 + u 4 , and the interconnection equations u 3 = y3 ; u 4 = y4 ; u 5 = y6 ; u 6 = y6 . Here BG1 and BG2 denote the behavior of the input/output systems in respectively the forward loop and the feedback loop of the feedback system. 6 The Manifest Variable Assignment The ﬁnal step consists of the manifest variable assignment. This is a mapping that assigns the manifest variables as a function of the terminal variables.