By Arjan van der Schaft
With recognize to the 1st variation as quantity 218 within the Lecture Notes in Con trol and data Sciences sequence the elemental suggestion of the second one variation has remained an identical: to supply a compact presentation of a few easy rules within the classical idea of input-output and closed-loop balance, including a call of contributions to the new concept of nonlinear powerful and 1foo regulate and passivity-based regulate. however, a few components of the ebook were completely revised and/or multiplied, which will have a extra balanced presen tation of the idea and to incorporate the various new advancements that have been taken position because the visual appeal of the 1st variation. I quickly learned, how ever, that it's not attainable to provide a extensive exposition of the prevailing literature during this quarter with out affecting the spirit of the e-book, that is accurately geared toward a compact presentation. in order a consequence the second one variation nonetheless displays a great deal my own style and learn pursuits. I belief that others will write books emphasizing varied elements. significant alterations with recognize to the 1st version are the next: • a brand new part has been extra in bankruptcy 2 referring to L2-gain and passivity through scattering, emphasizing a coordinate-free, geometric, therapy. • The part on balance in bankruptcy three has been completely improved, additionally incorporating a few fresh effects offered in [182J.
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Additional resources for L2 - Gain and Passivity Techniques in Nonlinear Control
Sample text
E. 49) CHAPTER 3. DISSIPATIVE SYSTEMS THEORY 44 which needs to be satisfied for all x EX. 49). Furthermore, it follows from the theory of dynamic programming that if Sa and Sr (assuming existence) are C l , they are actually solutions of the HamiltonJacobi (-Bellman) equality 1 1 Sx(x)f(x) + -2SAx)g(x)gT (x)S; (x) 2y 1 + -2hT (x)h(x) = O. 50) will be given in Chapter 8. 3 Stabilization of passive systems Consider a passive system (cf. 33) with storage function S, that is dS < uTy. 52) with v E IRm the new input.
81) has L2-gain ~ y from (u, u) to (y, ji). ). having largest singular value ~ ~. ). 80) that I; p has L2-gain ~ y. 0 This last corollary can be extended to dynamic perturbations lowing way. ). 81) has L2-gain ~ y from (u, u) to (y, ji». ). with state ~, inputs ji and outputs u, and having L2-gain ~ ~ with C I storage function S6 (~) ::: O. Then the closed-loop system has L2-gain ~ y from u to y, with storage function Sex) + y2 S6 (~). CHAPTER 3. y2 II it 112 -! :!. has L2-gain ::: ; with storage function StJ,.
E2 > O. Thus S is a candidate Lyapunov function for the closedloop system with el = e2 = O. In fact. 8. 7 Suppose 1:1 and 1:2 have L2-gain ::: YI and ::: Y2. with YI . Y2 < 1. 56) are C l and have strict local minima at xi = 0, xi = O. and 1:1 and 1:2 are zero-state detectable. Then x· = (xi, xi) is an asymptotically stable equilibrium of the closed-loop system 1:! ~ with el = e2 = 0, which is globally asymptotically stable if additionally "'I . "'2 SI. S2 have global minima at xi = 0, xi = 0 and are proper.