Strong Lyapunov functions for two classical problems in adaptive control

Abstract

Strong Lyapunov functions for two classical problems in adaptive control and parameter identification are presented. These Lyapunov functions incorporate in their structure the classical persistency of excitation conditions, allowing to show global uniform asymptotic stability of the associated adaptive systems under sufficient and necessary conditions.

Introduction

Classical adaptive control deals with the identification of unknown and constant parameters. A good part of the classical problems can be studied through the following two Linear Time-Varying (LTV) systems (Narendra & Annaswamy, 1989 Sec. 2.8)

$$\dot{x}\left(t\right)=-\Gamma C^{\top }\left(t\right)C\left(t\right)x\left(t\right),$$

where $x\left(t\right)\in {\mathbb{R}}^n$ is the state, $C\left(t\right)\in {\mathbb{R}}^{m\times n}$ is the regressor and $\Gamma \in {\mathbb{R}}^{n\times n}$ is a design gain, and

$$\dot{z}\left(t\right)=\mathcal{A}\left(t\right)z\left(t\right),$$$$\mathcal{A}\left(t\right)=\left[\begin{array}{cc}\hfill A\left(t\right)\hfill & \hfill B\left(t\right)\hfill \\ \hfill -B^{\top }\left(t\right)P\left(t\right)\hfill & \hfill 0\hfill \end{array}\right],$$

where $z\left(t\right)\in {\mathbb{R}}^{n+m}$ is the state, $B\left(t\right)\in {\mathbb{R}}^{m\times n}$ is the regressor and $A\left(t\right),P\left(t\right)\in {\mathbb{R}}^{m\times m}$ are given known matrices. The regressors $C\left(\cdot \right)$ and $B\left(\cdot \right)$ are piecewise continuous functions. The first system, system (1), is the error dynamics of a parameter estimation process (Anderson, 1977, Morgan and Narendra, 1977b, Narendra and Annaswamy, 1989), whereas system (2) appears, for example, when a linear system is subjected to an adaptive control or as the error dynamics of an adaptive observer (Anderson, 1977, Morgan and Narendra, 1977a, Narendra and Annaswamy, 1989). In such situation, $A\left(t\right)$ and $P\left(t\right)$ represent the gains of the adaptive control/observer and are specified by the designer.

Necessary and sufficient conditions for the Global Uniform Asymptotic Stability (GUAS) of the zero equilibrium solution of (1), (2) have been obtained in Anderson (1977) and Morgan and Narendra, 1977a, Morgan and Narendra, 1977b. The proofs rely on weak Lyapunov functions (LFs), i.e., LFs having only negative semi-definite derivatives, and some geometric methods for Morgan and Narendra, 1977a, Morgan and Narendra, 1977b, and using the connections between uniform complete observability (UCO) and GUAS in Anderson (1977). Although for LTV systems GUAS (or equivalently Uniform Exponential Stability) implies the existence of a strong quadratic Lyapunov function (Khalil, 2002 Thm. 4.12) and (Anderson, 1977 Lem. 2), i.e., a LF having negative definite derivative, such functions have not been yet given explicitly for systems (1), (2) under the most general GUAS conditions for non-smooth regressors.

Recently, some explicit strong LFs for (1), (2) have been found (see e.g. Aranovskiy et al., 2019, Loría et al., 2019a, Loría et al., 2019b, Maghenem and Loría, 2017, Maghenem and Loría, 2017, Mazenc, de Queiroz et al., 2009), but under conditions much more restrictive than those required by the systems to be GUAS. The objective (and novelty) of this note is to exhibit, explicit, smooth and quadratic strong Lyapunov functions for systems (1), (2) under necessary and sufficient conditions for GUAS, i.e., the origin of these systems is GUAS iff the corresponding function is a strong LF. It is well-known that having strong LFs is advantageous for e.g. robust analysis, study of input/output properties as Input-to-State Stability (ISS), calculation of convergence velocity, etc.

The note structure is as follows: In Section 2, the necessary and sufficient conditions for GUAS of systems (1), (2) are recalled; this work contribution is also given in this section in item (iii) of Theorem 1, Theorem 3. In Section 3, the proposed LFs are discussed. In Section 4, the results available in the literature are reviewed and contrasted with the proposed ones; finally, in Section 5, the proof of the main result of this work is given.

Notation

Along the note, $\mathbb{R}$ denotes the set of real numbers, ${\mathbb{R}}^n$ the real $n$-dimensional Euclidean space and ${\mathbb{R}}^{n\times m}$ the set of real $n\times m$ matrices. ${\mathbf{I}}_n\in {\mathbb{R}}^{n\times n}$ denotes the identity matrix. For $A,B\in {\mathbb{R}}^{n\times n}$ symmetric, $A>B$ ($A\ge B$) means that $A-B$ is positive (semi) definite. For $v\in {\mathbb{R}}^n$, $\Vert v\Vert$ denotes $={\left(v^{\top }v\right)}^{1∕2}$ and for $B\in {\mathbb{R}}^{m\times n}$, $\Vert B\Vert$ denotes the induced norm of $B$, defined as ${sup}_{\Vert x\Vert =1}\Vert Bx\Vert$. For a $A=A^{\top }$, ${\lambda }_{min}\left(A\right)$ and ${\lambda }_{max}\left(A\right)$ denote the smallest and largest eigenvalues of $A$. The function space $PC\left(\left[0,\infty \right),{\mathbb{R}}^{n\times m}\right)$ is the set of all functions mapping non negative real values into ${\mathbb{R}}^{n\times m}$ which are piecewise continuous, i.e., they are continuous everywhere, except that they may have a finite number of discontinuity points on every bounded subinterval, where the one-sided limits are well defined and finite. Moreover, $R\left(t\right)\in P{C}^1\left(\left[0,\infty \right),{\mathbb{R}}^{n\times m}\right)$ if $\dot{R}\left(t\right)$ exists almost everywhere and $\dot{R}\left(t\right)\in PC\left(\left[0,\infty \right),{\mathbb{R}}^{n\times m}\right)$.