Page 941 - The Mechatronics Handbook

0066_frame_C31.fm Page 4 Wednesday, January 9, 2002 7:49 PM

Remark: Note that in order to show stability for nonautonomous systems, it is necessary to bound the
function V(x, t) by the class K functions that do not depend upon time t. A detailed treatment of all the
2
definitions and proof for this theorem can be found in Slotine and Li and Khalil. 3
Remark: In the recent years, several interesting converse Lyapunov results have been obtained. In par-
ticular, for every uniformly stable (or uniformly asymptotic stable) system, there exists a positive definite
4
Lyapunov function with a negative semidefinite time derivative (see Sastry and Bodson ). These results
are particularly useful from a closed-loop performance point of view because they allow us to explicitly
estimate the convergence rates in some cases of nonlinear adaptive control systems.
The application of Lyapunov’s stability theorem for nonautonomous systems arising out of adaptive
control often leads us to negative semidefinite time derivatives of the Lyapunov function. Therefore,
asymptotic stability analysis is a much harder problem and the following result, known as Barbalat’s
Lemma, is extremely useful in such situations.
Lemma: Barbalat.
Consider a uniformly continuous function f : R → R defined at all real values of t ≥ 0. If

lim ∫ t f s() ds
t→∞ 0

exists and is finite, then f(t) → 0 as t → ∞.
Remark: A consequence of this result is that if f ∈L 2 and ∈L ∞ , then f(t) → 0 as t → ∞ (see Slotinef
5
2
and Li and Tao for discussion and proof).

31.4 Adaptive Control Theory

In contrast to a fixed or ordinary controller, an adaptive controller is one with adjustable parameters and
an adjustment mechanism. The following are some basic concepts that are necessary for any discussion
of adaptive control theory.

Regulation and Tracking Problems
The desired objective for any control problem is to maintain the plant output either at its desired value
or within specified/acceptable bounds of the desired value. If these desired values are constant with respect
to time, we have a regulation problem, otherwise it is a tracking problem.

Certainty Equivalence Principle

This principle has been the bedrock of most adaptive control design methods and has received consid-
erable attention during the past two decades. 4, 6, 7 Adaptive controllers based on this approach are obtained
by independently designing a control law that meets the control objective assuming complete knowledge
of all the unknown plant parameters (deterministic case), along with a parameter update law, which is
usually a differential equation that generates online parameter estimates that are used to replace the
unknown parameters within the control law. Such a controller would have perfect output tracking
capability in the case when the plant parameters are exactly known. In the presence of parameter
uncertainty, the adaptation mechanism will adjust the controller parameters so that the tracking objective
is asymptotically achieved. The main issue, thus, in adaptive controller design is to synthesize the adaptation
mechanism (parameter update law) that will guarantee that the control system remains stable and the
output tracking error converges to zero as the parameter values are updated.

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