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By Bernard Brogliato

Dissipative platforms research and keep an eye on (second version) offers a completely revised and elevated remedy of dissipative platforms idea, constituting a self-contained, complicated creation for graduate scholars, researchers and training engineers. It examines linear and nonlinear structures with examples of either in every one bankruptcy; a few infinite-dimensional examples also are incorporated. all through, emphasis is put on using the dissipative houses of a method for the layout of solid suggestions regulate legislation. the idea is substantiated by means of experimental effects and via connection with its program in illustrative actual instances (Lagrangian and Hamiltonian structures and passivity-based and adaptive controllers are lined thoroughly).The moment variation is considerably reorganized either to house new fabric and to augment its pedagogical homes. a few of the alterations brought are: whole proofs of the most theorems and lemmas. The Kalman-Yakubovich-Popov Lemma for non-minimal realizations, singular platforms, and discrete-time platforms (linear and nonlinear). Passivity of nonsmooth structures (differential inclusions, variational inequalities, Lagrangian platforms with complementarity conditions). Sections on optimum keep watch over and H[Infin] concept. An enlarged bibliography with greater than 550 references, and an augmented index with greater than 500 entries. a stronger appendix with introductions to viscosity recommendations, Riccati equations and a few priceless matrix algebra.

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9. Note the difference between the condition Re[h(jω)] > 0 and the condition for input strict passivity in that there exists a δ > 0 so that Re[h(jω0 )] ≥ δ > 0 for all ω. 37) However there is no δ > 0 that ensures Re[h(jω0 )] ≥ δ > 0 for all ω ∈ [−∞, +∞]. 38) This implies that h1 (s) is not input strictly passive. 39) 1 + (ωT )2 which means that the system is output strictly passive with = 1. 10. Consider a system with the transfer function h2 (s) = s+c (s + a)(s + b) where a, b and c are positive constants.

As h(s) has no poles in Re [s] > 0 by assumption, it follows that the system is passive if and only if Re[h(jω)] ≥ 0 for all ω ∈ [−∞, +∞]. 17. 56) h2 (s) = 2 s + ω02 where all the poles are on the imaginary axis. 14 is satisfied. 58) ω0 − ω 2 so that condition 2 also holds in view of Re[h1 (jω)] = Re[h2 (jω)] = 0 for all ω so that jω is not a pole in h (s) . 14, the system with transfer function h2 (s) is passive, while h1 (s) is passive whenever a < ω0 . 18. 62) s The transfer function has no poles in Re [s] > 0, and Re[h(jω)] ≥ 0 for all ω = 0.

Assume that there is a δ so that Re[h(jω)] ≥ δ > 0 and a γ so that |h (jω)| ≤ γ for all ω ∈ [−∞, +∞]. 88) 4δ . Next assume that and the result follows with 0 < γ < min 1, (γ+1) 2 g(jω)|2 ≤ 1 − γ for all ω. 9. 85): the association of the new system with transfer function g(s) merely corresponds to writing t down uy = 14 (a + b)(a − b) = 14 (a2 − b2 ). Thus if 0 u(s)y(s)ds ≥ 0 one gets t 2 t a (s)ds ≥ 0 b2 (s)ds: the L2 -norm of the new output b(t) is bounded by 0 the L2 -norm of the new intput a(t).

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