Elsevier

Systems & Control Letters

Volume 56, Issues 11–12, November–December 2007, Pages 742-752
Systems & Control Letters

Nonsmooth feedback stabilizer for strict-feedback nonlinear systems that may not be linearizable at the origin

https://doi.org/10.1016/j.sysconle.2007.04.009Get rights and content

Abstract

We present a continuous feedback stabilizer for nonlinear systems in the strict-feedback form, whose chained integrator part has the power of positive odd rational numbers. Since the power is not restricted to be larger than or equal to one, the linearization of the system at the origin may fail. Nevertheless, we show that the closed loop system is globally asymptotically stable (GAS) with the proposed continuous (but, possibly not differentiable) feedback. We formulate a condition that enables our design by characterizing the powers of the given system. The condition also shows that our result is an extension of Qian and Lin [Non-lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization, Systems Control Lett. 42 (2001) 185–200] where the power of odd positive integers has been considered. New result on the global finite time stabilization problem is also presented.

Introduction

In practice, there exist systems that do not have the first approximation at the origin, e.g., a leaky bucket whose dynamics is given by h˙=-Ch [13, p. 41] or the hydraulic control systems [15]. Partly motivated by this fact, we construct a continuous (but possibly nondifferentiable) state feedback stabilizer which globally stabilizes a single-input nonlinear system in the strict-feedback form given byx˙1=x2r1+φ1(x1)x˙2=x3r2+φ2(x1,x2)x˙n=urn+φn(x1,,xn),where φi(x1,,xi), i=1,,n, are C1 functions vanishing at the origin and ri's are rational numbers whose numerators and denominators are all positive odd integers (we will call such ri a positive odd rational number). We stress that if ri<1, then the system is not linearizable at each point xRn with xi+1=0 as well as at the origin, hence the standard backstepping design, which requires smoothness of the vector field, does not work directly.

This is a sharp contrast to the previous works [2], [3], [4], [9], [11], [14] which have considered a system whose right-hand side is continuously differentiable in the state x, or all ri's are greater than or equal to 1 so that its linearization at the origin may be uncontrollable. In [9], [11], [14], they propose a state feedback controller for the system (1) in which all ri's are positive odd integers. Lin and Qian [9] explicitly construct, using a tool called adding a power integrator, a globally stabilizing smooth feedback control law for system (1) under the condition that the odd integer powers ri are in decreasing order (i.e., r1rn1), and under a growth condition that |φi(x1,,xi)|(|x1|ri++|xi|ri)γi(x1,,xi), i=1,,n, where each γi(·) is a smooth nonnegative function. The decreasing assumption and the growth condition have been removed in [4], [11] while a continuous (instead of smooth) feedback is obtained in [11] and a smooth but time-varying feedback is designed in [4]. More generally, a triangular systemx˙i=fi(x1,,xi+1),i=1,,n-1,x˙n=fn(x1,,xn)+uis studied in [2], [3], but it is assumed that all fi(·)'s are C so that its linearization at the origin does exist.

In this paper, we design a C0 state feedback control law as well as a C1 (positive definite and proper) Lyapunov function to make the origin globally asymptotically stable (GAS). The control law has an interesting feature: it contains some exponents (or powers) which are determined by a set of inequalities during the design procedure (thus, they are design parameters). These exponents are also closely related to the Lyapunov function used to prove the stability. The design procedure shows its efficiency when we discuss the finite time stabilization problem [1], [5] since a slight change of the inequalities involved in the asymptotic stabilization problem ensures the existence of finite time stabilizer. One drawback of the proposed existence condition (the inequalities) is that it is not easy to check in general (although it can be converted to linear matrix inequality (LMI)). Thus, in order to avoid further difficulties, we provide explicit design guidelines for some special cases of ri's.

The paper is organized as follows. In Section 2.1, we state our main theorem for global stabilization whose proof is given in Section 2.2. Global finite time stabilization problem is discussed in Section 2.3. In Section 2.4, the conditions proposed in the main theorems of Sections 2.1 and 2.3 are discussed in detail, where some relation to the previous work [11] is also pointed out. Finally, we conclude the paper in Section 3.

For convenience, let us define the set of all rational numbers whose numerators and denominators are all positive odd integers by Qodd. Note that the set Qodd is closed under multiplication, division and odd number of additions, but is not closed under even number of additions or subtraction, and that, therefore, xa+b or xc(a+b) for a, b, cQodd is a positive definite function of x.

Section snippets

Statement of main theorem

We now state our main theorem.

Theorem 1

Suppose that, for the system (1), riQodd, i=1,,n. If there exist μ0,μ1,,μnQodd such thatμ0,,μn1,r1μ11μ0,r2μ2min1μ0,1μ1,,rnμnmin1μ0,1μ1,,1μn-1,01μ0-r1μ11μ1-r2μ21μn-1-rnμn,then there exists a C0 feedback controller u=u(x) with u(0)=0 which renders the origin of the closed loop system GAS. In addition, if the assumption holds with all μi=1 (i=1,,n), then a smooth feedback controller u(x) exists.

Remark 1

Note that from (4), the condition μi=1 (i=0,,n) implies 1

Conclusion

In this paper, we have developed a C0 state feedback design method for the powers of the integrators perturbed by C1 lower triangular vector fields. The result extends the work [11] in the sense that the powers are not restricted to positive odd integers. Although the system considered here may not have the first order approximation around the origin, which has rarely been studied in the literature, the proposed controller guarantees GAS of the origin. The finite time stabilization problem is

Acknowledgments

The authors would like to thank the reviewers for their constructive and pertinent comments, in particular, for suggesting the finite time stabilization problem which has not been discussed in the original version.

References (16)

There are more references available in the full text version of this article.

Cited by (39)

  • Sampled-data stabilization of a class of lower-order nonlinear systems with input delays based on a multi-rate control scheme

    2019, Journal of the Franklin Institute
    Citation Excerpt :

    In [35–37], the authors considered a relatively simple case where each subsystem has a chained integrator with the same power. The case of different powers was investigated in [38,39], where although the authors found a way to construct feedback stabilizers, the proposed controllers are usually non-smooth and have very complex expressions. In this paper, we consider the low-order lower-triangular nonlinear systems where powers of chined integrators are different and delays of long length are considered in control input.

  • Robust control for a class of nonlinear systems with unknown measurement drifts

    2016, Automatica
    Citation Excerpt :

    Different from the backstepping method, the work (Lin & Qian, 2000; Qian & Lin, 2001) introduced the adding a power integrator technique, which relies on dominating, instead of cancelling the nonlinear terms. With this new tool, numerous stabilization results have been achieved for nonlinear systems with various structures and restrictions, for example, Back, Cheong, Shim, and Seo (2007), Fu, Ma, and Chai (2015), Zhai and Qian (2012) and the references therein. It is demonstrated in Rosier (1992) that homogeneous systems inherit some nice properties from linear systems and provide us a new viewpoint to deal with the nonlinear systems (Kawski, 1990).

  • Finite-time stabilization of a class of output-constrained nonlinear systems

    2015, Journal of the Franklin Institute
    Citation Excerpt :

    The finite-time stabilization is one of the most important problems of finite-time control and has drawn an increasing attention in recent years [1,12,13,22,31,36,41] because the finite-time stabilized systems usually practically demonstrate some desired features such as faster convergence rates, higher accuracies, and better disturbance rejection properties [4,5,25].

  • Switched adaptive control of switched nonlinearly parameterized systems with unstable subsystems

    2015, Automatica
    Citation Excerpt :

    One of the reasons for the rapid growth and continuing popularity of adaptive control is its clearly defined goal: to control systems with uncertainties by estimating unknown system’s parameters. For about two decades, there is a vast amount of literature on design and analysis of various adaptive control systems using rigorous proofs (Back, Cheong, Shim, & Seo, 2007; Haddad, Hayakawa, & Chellaboina, 2003; Hong, Wang, & Cheng, 2009; Spooner & Passino, 1999). In particular, adaptive control has proven its great capability in compensating for non-switched nonlinearly parameterized systems involving inherent nonlinearity on the basis of a parameter separation technique (Lin & Qian, 2002a,b).

View all citing articles on Scopus
View full text