In parallel with developments in adaptive nonlinear control, there has been a tremendous amount of activity in neural control and adaptive fuzzy approaches. In these studies, neural networks or fuzzy approximators are used to approximate unknown nonlinearities. The input/output response of the approximator is modified by adjusting the values of certain parameters, usually referred to as weights. From a mathematical control perspective, neural networks and fuzzy approximators represent just two classes of function approximators. Polynomials, splines, radial basis functions, and wavelets are examples of other function approximators that can be used and have been used in a similar setting. We refer to such approximation models with adaptivity features as adaptive approximators, and control methodologies that are based on them as adaptive approximation based control. Adaptive approximation based control encompasses a variety of methods that appear in the literature: intelligent control, neural control, adaptive fuzzy control, memory-based control, knowledge-based control, adaptive nonlinear control, and adaptive linear control. |
During the last few years there have been significant developments in the control of highly uncertain, nonlinear dynamical systems. For systems with parametric uncertainty, adaptive nonlinear control has evolved as a powerful methodology leading to global stability and tracking results for a class of nonlinear systems. Advances in geometric nonlinear control theory, in conjunction with the development and refinement of new techniques, such as the backstepping procedure and tuning functions, have brought about the design of control systems with proven stability properties. In addition, there has been a lot of research activity on robust nonlinear control design methods, such as sliding mode control, Lyapunov redesign method, nonlinear damping, and adaptive bounding control. These techniques are based on the assumption that the uncertainty in the nonlinear functions is within some known, or partially known, bounding functions.
TABLE OF CONTENTS Adaptive Approximation Based Control - Appendix A Systems and Stablility Concepts
This appendix presents certain necessary concepts that are used in the main body of the book. This material is presented in the form of an appendix, as it may be familiar to many readers and will therefore not interrupt the main flow of the text. Proofs are not included. Proofs can be found in [119,134,169,249], which are the main references for this appendix. A.1 SYSTEMS CONCEPTS Many dynamic systems (all those of interest herein) can be conveniently represented by a finite number of coupled first-order ordinary differential equations:  where x ∈ In the special case where u(t) is a constant and ois not an explicit function of t, then eqn. (A.I) simplifies to  This equation, which is independent of time, is said to be autonomous. Solutions. The analyst is often interested in qualitative properties of the solutions of the system of equations defined by eqn. (A.I). For a given signal u(t), a solution to eqn. (A.1) over an interval t ∈ [t0 t1] is a continuous function x(t) : [t0, t1] Existence and Uniqueness of Solutions.The two questions of whether a differential equation has a solution and, if so, whether it is unique are fundamental to the study of differential equations. Discussion of the uniqueness of a solution requires introduction of the concept of the Lipschitz condition. |
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