Robust adaptive control for nonholonomic systems with nonlinear parameterization

doi:10.1016/j.nonrwa.2009.11.019

Nonlinear Analysis: Real World Applications

Volume 11, Issue 4, August 2010, Pages 3242-3250

https://doi.org/10.1016/j.nonrwa.2009.11.019 Get rights and content

Abstract

In this paper, an adaptive controller is designed for a class of nonholonomic systems in chained form with nonlinear parameterization. The robust adaptive control law is developed using parameter separation, state scaling and backstepping technique. Global asymptotic regulation of the closed-loop system states is achieved. The proposed control based switching strategy is proposed to overcome the uncontrollability problem associated with $x_{0} (t_{0}) = 0$ .

Introduction

In recent years, the control and stabilization of nonholonomic systems has been an active research area within the nonlinear control community. Due to Brockett’s theorem [1], it is well known that nonholonomic systems cannot be stabilized by stationary continuous state feedback, although it is controllable. As a consequence, the well-developed smooth nonlinear control theory and methodology cannot be directly used for such systems. The controller design for these systems is a challenging problem which has attracted an even increasing attention in the control community. A number of approaches have been proposed for the problem, see the survey paper [2] and the references therein for more details.

One commonly used approach for controller design of nonholonomic systems is to convert, with appropriate coordinates and state feedback transformation, the original systems into chained form systems for which controller design can be carried out easily [3]. Using the special algebra structures of the canonical forms, various feedback strategies have been proposed to stabilize nonholonomic systems without drifts or with weak nonlinear drifts in the literature [4], [5], [6], [7], [8], [9]. A class of nonholonomic systems perturbed by strong nonlinear uncertainties was studied in [10]. Discontinuous state and output feedback controllers were designed in it and achieved global exponential stability. However, the paper required that the $x_{0}$ -subsystem be Lipschitz. By using a novel switching scheme, these results have been extended and both Lyapunov stability and exponential convergence are achieved in [11], [12]. Recently, adaptive control strategies were proposed to stabilize the dynamic nonholonomic systems with modeling or parametric uncertainties, for instance, adaptive state feedback control was considered in [13], [14], [15], [16] and output feedback control in [17], [18]. However, it should be noticed that all these papers were concerned with systems with linear parameterization. There are very few reports in the literature for adaptive control of nonlinearly parameterized nonholonomic systems. However, many practical control systems such as biochemical processes [19] and machines with friction [20], often contain unknown parameters that enter the systems nonlinearly. Indeed, nonlinear parameterizations frequently arise and are inevitable in various realistic dynamic models of practical control problems, as illustrated in [21], [22], [23]. Since nonlinear parameterization is exceptionally difficult to estimate, from a theoretical viewpoint, adaptive control of nonlinearly parameterized nonholonomic systems is also interesting, because it represents a new challenge to the theory of nonlinear adaptive control.

This paper addresses the problem of stabilization of a class of nonholonomic systems in chained form with drift nonlinearity and unknown nonlinear parameters. The contribution of this paper are listed as follows: (i) adaptive state feedback stabilization using state scaling and backstepping is developed without imposing any restriction on the systems order and the growth of the drift nonlinearities; (ii) using parameter separation technique [24], the parameters nonlinearities is solved and the resulting adaptive regulator is minimum dimension (1D) independently of the parameter dimension; (iii) adaptive control based switching strategy is adopted to handle the technical problem of uncontrollability at $x_{0} (t_{0}) = 0$ , which prevents the finite escape of systems and guarantees that all the states converge to the origin and other signals are bounded.

The rest of this paper is organized as follows. In Section 2 preliminary knowledge and the problem formulation are given. Section 3 presents the state scaling technique and the backstepping design procedure, while Section 4 provides the switching control strategy and the main results. Finally, concluding remarks are proposed in Section 5.

Section snippets

Problem formulation

In this paper, we present an adaptive control design procedure for a class of uncertain nonholonomic systems with nonlinear parameterization ${\dot{x}}_{0} = d_{0} (t) u_{0} + f_{0} (x_{0}, θ) + ϕ_{0}^{d} (t, x_{0})$ ${\dot{x}}_{i} = d_{i} (t) x_{i + 1} u_{0} + f_{i} (x_{0}, x, θ) + ϕ_{i}^{d} (t, x_{0}, x, u_{0})$ ${\dot{x}}_{n} = d_{n} (t) u_{1} + f_{n} (x_{0}, x, θ) + ϕ_{n}^{d} (t, x_{0}, x, u_{0})$ where $x_{0}$ and $x = (x_{1}, \dots, x_{n}) \in R^{n}$ are system states, $u_{0}$ and $u_{1}$ are control inputs, $f_{0}$ and $f_{i}$ are continuous functions, of their arguments $θ \in R^{p}$ is an unknown constant vector. $d_{i} (t)$ are disturbed virtual control directions and $ϕ_{i}^{d}$ represent the unmodeled

Adaptive control design

In this section, we proceed to design an adaptive controller using backstepping. For clarity, the case that $x_{0} (t_{0}) \neq 0$ is considered first. Then the case where the initial $x_{0} (t_{0}) = 0$ is dealt later. The inherently triangular structure of system (1) suggests that we should design the control inputs $u_{0}$ and $u_{1}$ in two separate stages.

Consider the control $u_{0} (x_{0}, {\hat{Θ}}_{0}) = x_{0} g_{0} (x_{0}, {\hat{Θ}}_{0})$ $g_{0} (x_{0}, {\hat{Θ}}_{0}) = - \frac{1}{c_{01}} [{\bar{φ}}_{0} {\hat{Θ}}_{0} + \sqrt{k_{0}^{2} + {(k_{1} {\bar{φ}}_{0})}^{2} + {({\bar{φ}}_{0} {\hat{Θ}}_{0})}^{2}} + ϕ_{0}]$ where $k_{0}$ and $k_{1}$ is positive design parameter and $k_{1}$ defined later and ${\hat{Θ}}_{0}$ is

Switching controller and main results

In the preceding section, we have given controller design for $x_{0} (0) \neq 0$ . Now, we discuss how to select the control laws $u_{0}$ and $u_{1}$ when $x_{0} (0) = 0$ . Choose $u_{0}$ as $u_{0} = x_{0} g_{0} + u_{0}^{*}, u_{0}^{*} > 0$ where $g_{0}$ is given by (3), ${\hat{Θ}}_{0}$ is updated by (6).

Choosing the same Lyapunov function (4), its time derivative is given by ${\dot{V}}_{0} \leq - k_{0} x_{0}^{2} + u_{0}^{*} x_{0}$ which leads to the boundedness of $x_{0}$ , and consequently the boundedness of ${\hat{Θ}}_{0}$ as well.

Similar to the Lemma 3 in Section 2, the following lemma can be established.

Lemma 6

Using the control law(41),

Conclusions

In this paper, a constructive adaptive control strategy is presented for a class of nonholonomic systems in chained form with nonlinear parameterization. To deal with the nonlinear parameterization problem, a parameter separation technique is introduced to transform the nonlinear parameterized nonholonomic system into a linear-like parameterized nonholonomic system. We estimated only $Θ$ , the bound of the unknown parameters rather than the parameter vector $θ$ . This, in turn, resulted in a minimum

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^☆: This work is supported in part by National Nature Science Foundation of China under Grant (60674020), and in part by Nature Science Foundation of Henan Province under Grant (092300410145).

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Robust adaptive control for nonholonomic systems with nonlinear parameterization☆

Abstract

Introduction

Section snippets

Problem formulation

Adaptive control design

Switching controller and main results

Conclusions

Systems & Control Letters

System & Control Letters

Automatica

Journal of the Franklin Institute

Automatica

Asymptotic stability and feedback stabilization

Differentical Geometry Control Theory

Development in nonholonomic control problems

IEEE Control Systems Magazine

Nonholonomic motion planning: Steering using sinusoids

IEEE Transcations on Automatic Control

Feedback linearization and stabilization of sencond-order nonholonomic chained systems

International Journal of Control

Adaptive stabilization of uncertain nonholonomic mechanical systems

Robotica

A recursive techinque for tracking control of nonholonomic systems in chained form

IEEE Transcations on Automatic Control

A switching algorithm for global exponential stabilization of uncertain chained systems

IEEE Transcations on Automatic Control