Adaptation and Parameter Estimation in Systems With Unstable Target Dynamics and Nonlinear Parametrization

Abstract

We propose a technique for the design and analysis of adaptation algorithms in dynamical systems. The technique applies both to systems with conventional Lyapunov-stable target dynamics and to ones of which the desired dynamics around the target set is nonequilibrium and in general unstable in the Lyapunov sense. Mathematical models of uncertainties are allowed to be nonlinearly parametrized, smooth, and monotonic functions of linear functionals of the parameters. We illustrate with applications how the proposed method leads to control algorithms. In particular we show that the mere existence of nonlinear operator gains for the desired dynamics guarantees that system solutions are bounded, reach a neighborhood of the target set, and mismatches between the modeled uncertainties and uncertainty compensator vanish with time. The proposed class of algorithms can also serve as parameter identification procedures. In particular, standard persistent excitation suffices to ensure exponential convergence of the estimated to the actual values of the parameters. When a weak, nonlinear version of the persistent excitation condition is satisfied, convergence is asymptotic. The approach extends to a broader class of parameterizations where the monotonicity restriction holds only locally. In this case excitation with oscillations of sufficiently high frequency ensure convergence.Results in adaptive control theory and systems identification are most frequently used in control engineering, but have potentially a much wider significance. In particular these theories are of great potential relevance for sciences such as physics and biology [55]. On the other hand, it is in these areas that the current limitations of control theory are most strongly felt. Whereas effective procedures are available in case the system is static [7],[22],[66],[28], adequate solutions for dynamical systems have been proposed under conditions that may not be adequate for most scientific applications. These conditions require that systems are linear in their parameters, the target dynamics is stable in the Lyapunov sense, and a Lyapunov function of the target dynamics can be given [54], [43], [30], [33],[16], [5], [40]. Each of these restrictions alone is limiting the role of control theory in the scientific arena; together they constitute the "standard" approach that confines control theory to a limited role, even within the realm of engineering. Whereas in artificial system design, nonlinear parametrizations could often be avoided, physical and biological models often require the inclusion of nonlinearly parametrized uncertainties [2],[49],[6],[13], [29].Proposed solutions to the nonlinear parametrization problem have cemented the standard approach, in that they eliminate any hopes of escaping from the stable target dynamics requirement. Nonlinearity is traditionally solved by invoking dominance of the nonlinear terms [32], [31]. Dominance inevitably overcompensates the nonlinearity inherent to the system. This is undesirable if the system's target moti...

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