By Bernard Brogliato
Dissipative platforms research and keep watch over (second variation) provides an absolutely revised and accelerated remedy of dissipative structures conception, constituting a self-contained, complicated advent for graduate scholars, researchers and training engineers. It examines linear and nonlinear structures with examples of either in each one bankruptcy; a few infinite-dimensional examples also are incorporated. all through, emphasis is put on using the dissipative homes of a process for the layout of solid suggestions keep an eye on legislation. the idea is substantiated through experimental effects and by way of connection with its software in illustrative actual instances (Lagrangian and Hamiltonian platforms and passivity-based and adaptive controllers are lined thoroughly).The moment variation is considerably reorganized either to house new fabric and to reinforce its pedagogical houses. the various adjustments brought are: entire proofs of the most theorems and lemmas. The Kalman-Yakubovich-Popov Lemma for non-minimal realizations, singular structures, and discrete-time structures (linear and nonlinear). Passivity of nonsmooth structures (differential inclusions, variational inequalities, Lagrangian structures with complementarity conditions). Sections on optimum keep watch over and H[Infin] conception. 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 helpful matrix algebra.
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Extra info for Dissipative Systems Analysis And Control
We assume that the plant h(s) and that the feedback controller hr (s) are strictly passive with ﬁnite gain. 6 we have |h0 (jω)| < 180◦ where h0 (s) := h(s)hr (s) is the loop transfer function, and the system is BIBO stable. A change of variables in now introduced to bring the system into a scattering formulation. The new variables are ∆ ∆ a = y + u and b = y − u for the plant and ∆ ∆ ar = ut + e and br = ut − e for the feedback controller. In addition input variables ∆ ∆ a0 = y0 + uf and b0 = y0 − uf are deﬁned.
Furthermore H(s − ) for = λ2 is also PR and thus H(s) is also SPR.
The contour C is closed with a semicircle into the right half plane. On the part of C that is on the imaginary axis Re[h(s)] ≥ 0 by assumption. 136) 46 2 Positive Real Systems Im C : Re r Fig. 15. Contour C of h(s) in the right half plane. As Re[s] ≥ 0 on the small semi-circles and Ress=jω 0 h(s) is real and positive according to condition 3, it follows that Re[h(s)] ≥ 0 on these semi-circles. On the large semi-circle into the right half plane with radius Ω we also have Re[h(s)] ≥ 0 and the value is a constant equal to limω→∞ Re[h(jω)], unless h(s) has a pole at inﬁnity at the jω axis, in which case h(s) ≈ sR∞ on the large semi-circle.
Dissipative Systems Analysis And Control by Bernard Brogliato