Download L2 - Gain and Passivity Techniques in Nonlinear Control by Arjan van der Schaft PDF

By Arjan van der Schaft

With recognize to the 1st version as quantity 218 within the Lecture Notes in Con­ trol and data Sciences sequence the elemental suggestion of the second one variation has remained a similar: to supply a compact presentation of a few simple principles within the classical conception of input-output and closed-loop balance, including a decision of contributions to the new idea of nonlinear strong and 1foo keep an eye on and passivity-based keep an eye on. however, a few elements of the ebook were completely revised and/or accelerated, on the way to have a extra balanced presen­ tation of the speculation and to incorporate many of the new advancements which were taken position because the visual appeal of the 1st variation. I quickly learned, how­ ever, that it isn't attainable to provide a vast exposition of the present literature during this quarter with out affecting the spirit of the ebook, that is accurately aimed toward a compact presentation. in order a end result the second one variation nonetheless displays a great deal my own style and study pursuits. I belief that others will write books emphasizing assorted features. significant alterations with recognize to the 1st variation are the next: • a brand new part has been additional in bankruptcy 2 pertaining to L2-gain and passivity through scattering, emphasizing a coordinate-free, geometric, remedy. • The part on balance in bankruptcy three has been completely accelerated, additionally incorporating a few contemporary effects awarded in [182J.

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13 that the input-output maps Gxo for every initial condition x(O) = Xo have L2-gain ~ y (with bias 2S(xo». 11 . For simplicity of exposition we will throughout identify U and its norm IIl1u with R m together with its standard Euclidean norm 1111, and similarly Yand lilly with RP and the standard Euclidean norm 1111. y2l1ulf lIy112. The L2-gain of I: is defined as y(I:) = inf{y I I: has L2-gain ~ y}. I: is said to have L2-gain < y if there exists y < y such that I: has L2-gain ~ y. llul12 -lIyIl2.

DISSIPATIVE SYSTEMS THEORY 50 An example is a robotic mechanism interacting with its environment via generalized forces Ue and generalized velocities Ye, and controlled by collocated sensors (generalized velocities Yc) and actuators (generalized forces uc ). 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.

9, x* = 0 is a stable equilibrium. 43) then (by taking u = 0) SAx)J(x) ::::: -ellh(x)11 2 for a certain e > O. Hence, since S(x) ~ 0, h(x) = 0 whenever S(x) = 0, and stability follows as above. 3) the solutions of x = J(x) in a compact set B converge to the largest invariant set of x = f(x) contained in (x I h(x) = O} n B, which is {x* = OJ. 11 The property of x* = 0 being asymptotically stable conditionally to the largest positively invariant set contained in (x I h(x) = O} is very close to zero-state detectability.

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