NDLI: Optimal Adaptive Control—Contradiction in Terms or a Matter of Choosing the Right Cost Functional?

Content Provider	IEEE Xplore Digital Library
Author	Krstic, M.
Copyright Year	1963
Abstract	Approaching the problem of optimal adaptive control as ldquooptimal control made adaptive,rdquo namely, as a certainty equivalence combination of linear quadratic optimal control and standard parameter estimation, fails on two counts: numerical (as it requires a solution to a Riccati equation at each time step) and conceptual (as the combination actually does not possess any optimality property). In this note, we present a particular form of optimality achievable in Lyapunov-based adaptive control. State and control are subject to positive definite penalties, whereas the parameter estimation error is penalized through an exponential of its square, which means that no attempt is made to enforce the parameter convergence, but the estimation transients are penalized simultaneously with the state and control transients. The form of optimality we reveal here is different from our work in [Z. H. Li and M. Krstic, ldquoOptimal design of adaptive tracking controllers for nonlinear systems,rdquo Automatica, vol. 33, pp. 1459-1473, 1997] where only the terminal value of the parameter error was penalized. We present our optimality concept on a partial differential equation (PDE) example-boundary control of a particular parabolic PDE with an unknown reaction coefficient. Two technical ideas are central to the developments in the note: a nonquadratic Lyapunov function and a normalization in the Lyapunov-based update law. The optimal adaptive control problem is fundamentally nonlinear and we explore this aspect through several examples that highlight the interplay between the non-quadratic cost and value functions.
Sponsorship	IEEE Control Systems Society
Starting Page	1942
Ending Page	1947
Page Count	6
File Size	212540
File Format	PDF
ISSN	00189286
Volume Number	53
Issue Number	8
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2008-09-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Programmable control Adaptive control Optimal control Cost function Automatic control Parameter estimation Riccati equations Error correction Convergence State estimation distributed parameter systems backstepping boundary control
Content Type	Text
Resource Type	Article
Subject	Control and Systems Engineering Electrical and Electronic Engineering Computer Science Applications

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Optimal Adaptive Control—Contradiction in Terms or a Matter of Choosing the Right Cost Functional?

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