Lipschitzian Parameterization-Based Approach for Adaptive Controls of Nonlinear Dynamic Systems with Nonlinearly Parameterized Uncertainties: A Theoretical Framework and Its Applications

The original and popular adaptive control theory usually deals with linear parameterizations (LP) of uncertainties, that is, it is assumed that uncertain quantities in dynamic systems are expressed linearly with respect to unknown parameters. Actually, most developed approaches such as gradient-based ones or recursive least squares [1, 2] rely heavily on this assumption and effective techniques have been proposed in this context [2]. However, LP is impossible in practical applications whose dynamic parameters are highly coupled with system states. Stribeck effect of frictional forces at joints of the manipulators [3] or nonlinear dynamics of space-robot in inertia space are typical examples [4]. Unfortunately, there were very few results in the literature addressing the adaptive control problem for NP in a general and direct manner. Recently, adaptation schemes for NP have been proposed [5, 6] with the assumption on the convexity/concavity and smoothness of the nonlinear functions in unknown parameters. In this approach, the controllers search a known compact set bounding the unknown parameters (i.e. the unknown parameter must belong to a prescribed closed and bounded set) for min-max parameter estimation. Also, the resulting controllers posse a complex structure and need delicate switching due to change of adaptation mechanism up to the convexity/concavity of the nonlinear functions. Such tasks may be hard to be implemented in a real-time manner. In this chapter, we propose novel adaptive control technique, which is applicable to any NP systems under Lipschitzian structure. Such structure is exploited to design linear-inparameter upper bounds for the nonlinear functions. This idea enables the design of adaptive controllers, which can compensate effectively for NP uncertainties in the sense that it can guarantee global boundedness of the closed-loop system signals and tracking error within any prescribed accuracies. The structures of the resulting controllers are simple since they are designed based on the nonlinear functions' upper bound, which depends only on