Adaptive Stabilization of Nonlinear Systems

An overview of the various parametric approaches which can be adopted to solve the problem of adaptive stabilization of nonlinear systems is presented. The Lyapunov design and two estimation designs –equation error filtering and regressor filtering– are revisited. This allows us to unify and generalize most of the available results on the topic and to propose a classification depending on the required extra assumptions – matching conditions or growth conditions. 1 Problem Statement and Assumptions 1.1 Problem Statement We consider a dynamic system which admits a finite state-space representation and whose dynamics are described by an equation which involves uncertain constant parameters. We are concerned with the design of a dynamic state-feedback controller which ensures, in spite of that uncertainty, that solutions of the closed-loop system are bounded and their x-components converge to a desired set point. Example: (1) Consider the following one-dimensional system: . x = p x + u , (2) where p is a constant parameter. Would the value of p be known, we could use the following linearizing control law to globally stabilize the origin of the closed-loop system: u = −p x − x . (3) When only an approximate value p̂ of p is known and is used in the control law (3), we obtain the following closed-loop system: . x = −x + (p − p̂) x . (4) This system (4) has two equilibrium points: 1. x = 0, which is exponentially stable, 2. x = 1 p∗ − p̂ , which is exponentially unstable. 348 Praly, Bastin, Pomet, and Jiang Hence, as long as p̂ is not exactly equal to p, the global asymptotic stability of x = 0 is lost. We notice also that the simple linear control u = −x (5) gives exactly the same qualitative behavior. Assume now that we apply the following dynamic controller: . p̂ = x u = −p̂ x − x . (6) This is the linearizing controller where instead of the true value p of the parameter, we use an on-line updated estimate p̂. The corresponding closed-loop system we get is: . p̂ = x . x = −x + (p − p̂) x . (7) To study the stability, we consider the function: W (x, p̂) = 1 2 ( x + (p̂− p) ) . (8) Its time derivative along the solutions of (7) is: . W = −x . (9) It follows that any solution of the closed-loop system (7) is bounded and: lim t→∞ x(t) = 0 . (10) Therefore, the convergence of x to 0 is restored. ⊓⊔ In this example, the parameter p enters linearly in the dynamic equation (2). This assumption is fundamental throughout the paper. It is formalized as follows: Assumption Λ-LP (Λ-Linear Parameterization) (11) We can find a set of measured coordinates x in IR such that, given an integer k and a C function Λ : IR × IR+ → Mkn(IR) , there exist an integer l, two C functions: a : IR × IR → IR , A : IR × IR → Mnl(IR) , and an unknown parameter vector p in IR such that: 1. the functions Λ(x, t)a(x, u) and Λ(x, t)A(x, u) are known, i.e., can be evaluated, 2. the dynamics of the system to be controlled are described by: . x = a(x, u) + A(x, u) p , (12) where u is the input vector in IR. To deal with the case where a and A are affine in u, it is useful to introduce the following notation: a(x, u) = a0(x) + m ∑