Learning-based modular indirect adaptive control for a class of nonlinear systems

Abstract

We study in this paper the problem of adaptive trajectory tracking control for a class of nonlinear systems with parametric uncertainties. We propose to use a modular approach, where we first design a robust nonlinear state feedback which renders the closed loop input-to-state stable (ISS), where the input is considered to be the estimation error of the uncertain parameters, and the state is considered to be the closed-loop output tracking error. Next, we augment this robust ISS controller with a model-free learning algorithm to estimate the model uncertainties. We implement this method with two different learning approaches. The first one is a model-free multi-parametric extremum seeking (MES) method and the second is a Bayesian optimization-based method called Gaussian Process Upper Confidence Bound (GP-UCB). The combination of the ISS feedback and the learning algorithms gives a learning-based modular indirect adaptive controller. We show the efficiency of this approach on a two-link robot manipulator example. I. INTRODUCTIONClassical adaptive methods can be classified into two main approaches; 'direct approaches', where the controller is updated to adapt to the process, and 'indirect approaches', where the model is updated to better reflect the actual process. Many adaptive methods have been proposed over the years for linear and nonlinear systems, we could not possibly cite here all the design and analysis results that have been reported, instead we refer the reader to e.g., [1], [2] and the references therein for more details. Of particular interest to us is the indirect modular approach to adaptive nonlinear control, e.g., [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. In this approach, first the controller is designed by assuming that all the parameters are known and then an identifier is used to guarantee certain boundedness of the estimation error. The identifier is independent of the designed controller and thus the approach is called 'modular'. For example, a modular approach has been proposed in [3] for adaptive neural control of pure-feedback nonlinear systems, where the input-to-state stability (ISS) modularity of the controller-estimator is achieved and the closed-loop stability is guaranteed by the small-gain theorem (see also [13], [14]).

Learning‐based iterative modular adaptive control for nonlinear systems

“…We now consider the uncertain model (3), ie, when Δ f (t, x) ≠ 0. The corresponding exact linearized model is given by (6), where Δb(t, (t)) ≠ 0. The global asymptotic stability of the error dynamics (11) cannot be guaranteed anymore because of the additive uncertainty Δb(t, (t)).…”