Non-identifier-based high-gain adaptive control
Adaptive control without plant identification is sometimes referred to as universal adaptive control. In order to define what universal adaptive control is, let \({\mathfrak P}\) be a set of plants with \(m\) inputs \((u_ 1,\dots,u_ m)^ T= u\) and \(p\) outputs \((y_ 1,\dots,y_ p)^ T= y\). Usually, each element in \({\mathfrak P}\) is given by a controlled and observed ordinary differential equation. A single control law of the form \[ u(t)= f(t,k(t),y(t)),\quad\dot k(t)= g(t,k(t),y(t)),\quad k(t_ 0)\in\mathbb{R}^{\ell} \] is a \({\mathfrak P}\)-universal adaptive controller if it guarantees that every plant in \({\mathfrak P}\) exhibits some prescribed dynamic behaviour (for example, attractivity or stability of an equilibrium or asymptotic output tracking of some reference signal). The book gives a rather complete account of the state of the art of universal adaptive control for classes \(\mathfrak P\) of linear systems which are stabilizable by high-gain static output feedback. The assumption of high-gain stabilizability implies that the systems under consideration are necessarily minimum-phase and of relative degree \(\leq 2\). However, no assumptions on the McMillan degree of the system are imposed. The book contains 8 chapters, of which the first one is of an introductory nature-illustrating the problems under consideration by a simple example and outlining the contents of the book. Chapters 2 and 3 do not deal with the adaptive control problem, but contain a number of results on minimum-phase, high-gain stabilizable and almost strictly positive real systems which are needed in the rest of the book. In Chapter 4 the author studies the adaptive stabilization problem and presents a number of adaptive high-gain stabilization schemes for minimum-phase systems of relative degree 1. Moreover, stabilization results for systems of relative degree 1 or 2 and for systems with non- zero feedthrough are given. Chapter 5 is devoted to the adaptive tracking problem. Results on exact asymptotic tracking are obtained by applying the stabilization results of Chapter 4 to a precompensated plant, where the precompensator is minimum-phase and of relative-degree 0 and contains an internal model of the dynamic structure of the reference signals. Furthermore, several results on approximate asymptotic tracking are given. Chapter 6 deals with robustness issues. It is shown that the adaptive control laws derived in chapters 4 and 5 are robust with respect to various classes of nonlinear perturbations, such as for example sector bounded sensor and actuator nonlinearities. While chapters 4 to 6 are concerned with the asymptotic behaviour of the closed-loop system. Chapter 7 contains simulations and results addressing the closed-loop transient dynamics. In particular, it is shown that some of the control schemes discussed in chapters 4 to 6 can be modified to improve the transient behaviour considerably. Finally, in Chapter 8 it is shown that for a given minimum-phase single-input single-output system of relative degree 1 and a given non-zero initial value, there exists an open and dense set of switching sequences such that for any sequence in this set the piecewise continuous stabilizer introduced in Chapter 4 will lead to an exponentially stable terminal closed-loop system.
scientific article; zbMATH DE number 462639