By Alessandro Astolfi, Lorenzo Marconi

ISBN-10: 354074357X

ISBN-13: 9783540743576

ISBN-10: 3540743588

ISBN-13: 9783540743583

This booklet is a tribute to Prof. Alberto Isidori at the get together of his sixty fifth birthday. Prof. Isidori’s proli?c, pioneering and high-impact examine task has spanned over 35 years. all through his occupation, Prof. Isidori has constructed ground-breaking effects, has initiated researchdirections and has contributed towardsthe foundationofnonlinear controltheory.In addition,his commitment to provide an explanation for elaborate concerns and di?cult suggestions in an easy and rigorous approach and to encourage younger researchers has been instrumental to the highbrow development of the nonlinear regulate neighborhood around the globe. the amount collects 27 contributions written by way of a complete of fifty two researchers. The central writer of every contribution has been chosen one of the - searchers who've labored with Prof. Isidori, have in?uenced his study task, or have had the privilege and honour of being his PhD scholars. The contributions handle a signi?cant variety of regulate themes, together with th- retical concerns, complex purposes, rising keep an eye on instructions and educational works. the variety of the parts lined, the variety of participants and their overseas status offer facts of the effect of Prof. Isidori within the regulate and structures idea groups. The ebook has been divided into six elements: approach research, Optimization tools, suggestions layout, law, Geometric tools and Asymptotic research, re?ecting vital regulate parts which were strongly in- enced and, from time to time, pioneered by way of Prof. Isidori.

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**Extra resources for Analysis and Design of Nonlinear Control Systems: In Honor of Alberto Isidori**

**Example text**

Its parameters are (m, p, G), with m, p ∈ N and G a p × m matrix of proper real rational functions. This means that we have a system with m + p terminals, the ﬁrst m of ‘input’ type, the last p of ‘output’ type, and behavior described by the controllable input/state/output system d dt x = Ax + Bu, y = Cx + Du, with A, B, C, D such that G(ξ) = D + C(Iξ − A)−1 B. The behavior of this system consists of all (u, y) : R → Rm × IRp for which there exists x : R → Rn such that these equations are satisﬁed: the state serves as a latent variable.

Consider a set Θ = {X0 , X1 , . . , Xm }, whose elements will be thought as m + 1 non-commuting variables. We use Θ∗ to denote the free monoid generated by Θ, where the neutral element of Θ∗ is the empty word, and the product is concatenation. We deﬁne R[Θ] to be the R-algebra generated by Θ∗ , that is, the set of all polynomials in the Xi ’s. By a power series in the variables X0 , X1 , . . , Xm we mean a formal power series: c, w w , c = w∈Θ∗ where c, w ∈ R for each w ∈ Θ∗ . We use R[[Θ]] to denote the set of all power series in the Xi ’s.

Let S be an equi-convergent family. Then, for any r > 0, there exists some T1 > 0 such that for any T < T1 the map ψ : S × VT (r) → C[0, T ], (c, u) → Fc [u] is continuous. Proof. Let S be compact and equi-convergent. Then there exists ρ such that | d, w | ≤ M ρk k! ∀|w| = k , ∀d ∈ S . (11) 1 . Fix T ∈ [0, T1 ). Then Fd is deﬁned on VT (r) for each d ∈ S. Let T1 = rρ(m+1) For any c, d ∈ S, u, v ∈ VT (r), Fc [u] − Fd [v] ∞ ≤ Fc [u] − Fc [v] ∞ + Fc [v] − Fd [v] ∞ . Hence, by Lemma 2, it is enough to show that the map S → C[0, T ], c → Fc [v] (12) is equi-continuous for v ∈ VT (r), that is, for any c ∈ S, for any ε > 0, there exists a neighborhood N of c such that Fc [v] − Fd [v] ∞ <ε for all d ∈ N and all v ∈ VT (r).

### Analysis and Design of Nonlinear Control Systems: In Honor of Alberto Isidori by Alessandro Astolfi, Lorenzo Marconi

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