Necessary and sufficient conditions for convergence of DREM-based estimators with applications in adaptive control

Abstract

In this paper, a necessary and sufficient condition for finite-time identification of a regression model, obtained using the dynamic regressor extension and mixing (DREM) method, is established. Estimators designed to satisfy transient and robust specifications via a time-varying gain are then proposed to have this condition as necessary and sufficient for their convergence to the true values when continuous functions are involved. These estimators are then used as a part of an adaptive control scheme, following a modular approach, to solve a tracking control problem for a nonlinear system in the strict feedback form with parametric and non-parametric uncertainty. It is shown that the necessary and sufficient condition can be expressed without using closed-loop signals, which allows attaining finite-time identification, exponential convergence to the tracking aim, and local/global robustness to non-parametric perturbations with minimal excitation conditions on the tracked trajectory. An example is developed to illustrate the main results.

Introduction

Parameter estimation has remained a central and difficult problem in the theory of control and identification of systems, despite the variety of approaches to tackle it. In the identification setting, for instance, well-known methods such as gradient descent or least-squares (LS) can be applied to estimate the parameters under the common assumption that the unknown parameters of the underlying system are linearly related to measurable data. However, the resulting algorithms ensure the identification under the strong assumption of a persistent excitation (PE) of the measured data (Ioannou and Sun, 1996, Sastry and Bodson, 1994). Modifications relaxing this requirement to finite excitation (FE) (Ahmad and Guez, 1997, Cho et al., 2018, Parikh et al., 2019) are computationally demanding, and the transient and/or robust performance is not addressed but as a side consequence.

When applied to control problems, the estimation is required to have additional features. First, the estimation must avoid finite escape time for systems with non-Lipschitz terms (Teel et al., 1991). Second, the control signal built from the estimator must be free from singularities in every instant (controllability problem). And third, the control aim must be established with conditions based on external rather than closed-loop signals (Lin & Kanellakopoulos, 1998), entailing a hard problem since the estimator and the external signals are mediated by closed-loop signals (Adetola and Guay, 2008, Ahmad and Guez, 1997, Cho et al., 2018, Parikh et al., 2019).

A novel treatment of the regression equation, which is called dynamic regressor extension and mixing (DREM) (Ortega et al., 2020), reduces the estimation to a scalar problem. Estimators designed on this framework can improve the transient performance and provide a solution to the controllability problem by using that a monotonic decreasing of each component of the parametric error is ensured, while classic estimators ensure this only for its vector norm (Ortega et al., 2019).

The aim of this paper is to provide a solution to the third control problem described above within the DREM framework. In order to accomplish this task, the following technical contributions, together with their organization, are introduced. In Section 2, we establish a necessary and sufficient condition for the convergence of DREM-based estimators. Unlike other finite-time estimators (Adetola & Guay, 2008), the scalar nature, in addition to easing the finding of the convergence condition, avoids the numerical issues of matrices’ inversion and makes it possible to control the performance with a scalar time-varying gain, as stated in Section 3. In Section 4, we show how to apply the estimator in control of nonlinear systems, addressing the first and third control problems described above. In particular, by exploiting the necessary and sufficient condition, a minimal external excitation requirement for identification and exponential convergence of the closed-loop is attained. The finite estimation does not introduce chattering, and no bound or Lipschitz requirement on the parametric uncertainty is needed.

Notation: For $A\in {\mathbb{R}}^{n\times m}$, $A^{\prime }$ denotes its transpose. $I_n$ denotes the identity matrix in ${\mathbb{R}}^{n\times n}$. For a matrix $A\in {\mathbb{R}}^{n\times n}$, $adj\left(A\right)$ is its adjoint matrix and $det\left(A\right)$ its determinant. $\left|\cdot \right|$ stands for the Euclidean norm in ${\mathbb{R}}^n$, and ${\left|\cdot \right|}_F$ for the Frobenious norm in ${\mathbb{R}}^{n\times n}$. ${\mathcal{L}}_p$ denotes the Lebesgue space for $p\in [1,\infty ]$. A zero function is a constant function taking the value $0$ in all its domain. $\ast$ is the convolution operator. A function $f:[0,\infty )\to {\mathbb{R}}^n$ is PE if there exist $\delta ,\mu >0$ such that ${\int }_t^{t+\delta }f\left(\tau \right)f{\left(\tau \right)}^{\prime }d\tau \ge \mu I$ for any $t\ge 0$; $\mu$ is called the excitation degree of $f$.