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 notion of the second one variation has remained a similar: to supply a compact presentation of a few uncomplicated principles within the classical idea of input-output and closed-loop balance, including a decision of contributions to the new idea of nonlinear powerful and 1foo keep watch over and passivity-based keep an eye on. however, a few components of the publication were completely revised and/or elevated, to be able to have a extra balanced presen tation of the idea and to incorporate many of the new advancements which were taken position because the visual appeal of the 1st variation. I quickly discovered, how ever, that it isn't attainable to offer a wide exposition of the present literature during this sector with out affecting the spirit of the booklet, that is accurately geared toward a compact presentation. in order a outcome the second one variation nonetheless displays greatly my own style and learn pursuits. I belief that others will write books emphasizing various elements. significant alterations with recognize to the 1st variation are the next: • a brand new part has been further in bankruptcy 2 touching on L2-gain and passivity through scattering, emphasizing a coordinate-free, geometric, therapy. • The part on balance in bankruptcy three has been completely elevated, additionally incorporating a few fresh effects offered in [182J.
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Additional resources for L2 - Gain and Passivity Techniques in Nonlinear Control
B. 14 Let G : L2e(U) G has finite L2-gain. -+ L2e(U) be strictly output passive. 23) P+ 1e IIUTII~ + ~ IIYTII~ whence ~IIYTII~ ~ 1e lIuTII~ + p, proving finite L2-gain. 14 enable us to state the following passivity theorems. CHAPTER 2. 15 Consider the feedback system I:Gf 1. 1, with G) : L2e(Ud --+ L2e(Ud, G2 : L2e(U2) --+ L2e(U2), and EI = E2 = UI = U2 =: U. (a) If GI and G2 are strictly output passive then the feedback system I:Gf 1. G2 has finite L2 -gain. (b) Assume that for any el E L2e(U) and e2 = 0 there are solutions UI, U2 E L2e (U).
G2 has finite L2 -gain. (b) Assume that for any el E L2e(U) and e2 = 0 there are solutions UI, U2 E L2e (U). If GI is passive and G2 is strictly input passive, or if GI is strictly output passive and G2 is passive, then I:~ 1. G2 with e2 = 0 and input el and output YI has finite L2-gain. 14, noting the definition of finite L2-gain of I:Gf 1. 6). 2. 14. 16 1. Note that part (a) can be alternatively formulated as: (a)' Assume that for any el, e2 E L2e(U) there exist solutions UI, U2 E L2e(U), If GI and G2 are strictly output passive then I:~ 1.
Closing the loop with a passive controller with storage function Sc, that is, dSc T -dt<-- Yc Uc results in a system which is passive with respect to (u e , Ye), since d T dt (S + Sc) ::: ue Yeo This can be generalized to arbitrary supply rates, and different types of interconnections; see also Chapter 4 for some developments in this direction. 55) have L2-gain ::: Yl, respectively::: Y2· Denote the storage functions of I:l, I:2 by Sl, S2, resulting in the dissipation inequalities 11'1 Sl (Xl (tI» - Sl (Xl (to» < 2 S2(X2(td) - S2(X2(tO» < ~ (I (Y;ll u2(t)1I 2 -IIY2(t)1I 2)dt.