Elsevier

Automatica

Volume 121, November 2020, 109198
Automatica

Technical communique
A necessary and sufficient condition for stability of a class of planar nonlinear systems

https://doi.org/10.1016/j.automatica.2020.109198Get rights and content

Abstract

In this paper, the stability problem of a class of planar nonlinear systems is investigated. Motivated by the Routh–Hurwitz stability criterion for planar linear systems, a necessary and sufficient condition for stability of a class of planar nonlinear systems is established. To prove the necessity, a new method called particular solution method is provided to justify instability. In addition, a sufficient condition for existence of oscillations is introduced. The necessary and sufficient condition can provide a simple way to design controllers by adjusting control parameters for some planar nonlinear systems.

Introduction

In this paper, we analyze stability problem of a class of planar nonlinear systems described by ẋ1=c1x1+c2x2p,ẋ2=c3x11p+c4x2,where x=(x1x2)R2 is state vector, p is a positive odd number, and c1,c2,c3,c4R are constants. As p=1, system (1) becomes a linear system ẋ1=c1x1+c2x2, ẋ2=c3x1+c4x2. It is well known that planar linear system ẋ=Ax is asymptotically stable if and only if its characteristic polynomial det|λIA| is Hurwitz, that is c1,c2,c3,c4 satisfy c1+c4<0 and c1c4>c2c3 (Routh–Hurwitz stability criterion, Hurwitz, 1964, Routh, 1877). For the planar nonlinear system (1) with p>1, a natural question is that if it has a result similar to the Routh–Hurwitz stability criterion.

Planar nonlinear systems are a kind of dynamical systems where the information propagation occurs in two independent directions (Dayawansa, Martin, & Knowles, 1990). Such systems are widely used to describe various practical and physical systems, like circuit analysis, image processing, seismographic data transmission, multidimensional digital filtering and thermal processes, etc. (Shtessel, Shkolnikov, & Levant, 2007). Stability problem is one of the most active topics in nonlinear control theory (Hahn, 1967). It is crucial as a step in achieving many other control problems such as output regulation, optimal control, disturbance decoupling and attenuation, etc. (Khalil & Grizzle, 2002).

Over the past few decades, a great number of interesting results have been obtained for stability of nonlinear systems (Li and Wu, 2016, Li et al., 2019, Ooba, 2012, Sontag and Sussmann, 1980, Wu et al., 2015), and many techniques, such as center manifold theory (Aeyels, 1985), the idea of zero dynamics (Byrnes & Isidori, 1991), homogeneous domination approach (Qian and Lin, 2006, Zhu and Qian, 2018), etc., have been proposed. Specially, by using sum of squares (SOS) techniques, Chesi (2013) provided an exact linear matrix inequality (LMI) condition for robust asymptotic stability of uncertain systems. In Aylward, Parrilo, and Slotine (2008), the robust stability properties of uncertain nonlinear systems with polynomial or rational dynamics were analyzed with convexity-based methods. In Lacerda and Seiler (2017), a systematic procedure was presented to investigate robust stability of uncertain systems in polytopic domains. However, due to lack of constructive applied approaches (Qian & Lin, 2001), there are few references to investigate necessary and sufficient conditions for stability of nonlinear systems.

For the following planar nonlinear system with input ẋ1=x1x23,ẋ2=u, Bacciotti (1992) showed that there is no continuously differentiable state feedback controller stabilizing system (2). Kawski (1989) designed a controller u=ax2+bx113 and proved that system (2) is asymptotically stable if b>a>1. It is easy to see that the system (2) with u=ax2+bx113 is a special case of (1). If there exists a necessary and sufficient condition for stability of system (1), then we can easily solve stabilization problem for system (2). Thus, it is meaningful to investigate necessary and sufficient conditions for stability of system (1).

Motivated by the Routh–Hurwitz stability criterion for planar linear control, we propose a necessary and sufficient condition for stability of system (1), that is system (1) is asymptotically stable if and only if c1, c2, c3 and c4 satisfy c1+c4<0 and c1c4p>c2c3p. We prove the necessary and sufficient condition with particular solution method and Lyapunov method. This paper is organized as follows: In Section 2, we give a number of basic concepts. The main results are presented in Section 3, where a necessary and sufficient condition for stability is given, the problem of oscillation detection is addressed, and a sufficient condition for oscillation existence is formulated. Additionally, the application of stability condition in state-feedback control design is analyzed, and three examples are given. Concluding remarks are given in Section 4.

Section snippets

Preliminaries

This section presents some fundamental theorems and some useful lemmas which will play important roles in obtaining the main results of this paper.

Theorem 1

Lyapunov Stability Theorem, Lyapunov (1992)

Given a nonlinear system ẋ=f(x),xRn,where f(x) is Lipschitz continuous and f(0)=0. System (3) is asymptotically stable, if there exists a continuously differentiable function V(x) such that V(x) is positive definite and V̇(x)V(x)xf(x)<0 for all x0.

Theorem 2

LaSalle’s Invariance Theorem, LaSalle (1960)

The nonlinear system (3) is asymptotically stable, if there exists a continuously differentiable

Main results

In this section, we shall give the main results of this paper. We first analyze a necessary and sufficient condition for stability of system (1).

Theorem 8

The nonlinear system (1) is asymptotically stable if and only if c1, c2, c3 and c4 satisfy c1+c4<0 and c1c4p>c2c3p.

Proof

(sufficiency) The sufficient proof is broken into two cases. Case 1: c1c40. Since c1+c4<0, then c10, c40 and (c1,c4)(0,0). Without loss of generality, assume c4<0. Choose the Lyapunov function V(x)=pp+1x1p+1p+bp+1x2p+1,b>0, which is

Conclusion

In this paper, we study stability problem for a class of planar nonlinear systems. A necessary and sufficient condition for stability is put forward. A new method called particular solution method is developed to justify instability. It is worth pointing out that the particular solution method proposed in this paper can also be exploited to justify instability of more general planar nonlinear systems. In addition, the analysis method for stability of the planar nonlinear system (1) can be

References (31)

  • ShtesselY.B. et al.

    Smooth second-order sliding modes: Missile guidance application

    Automatica

    (2007)
  • WuL. et al.

    Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings

    Automatica

    (2015)
  • ZhuJ. et al.

    A necessary and sufficient condition for local asymptotic stability of a class of nonlinear systems in the critical case

    Automatica

    (2018)
  • BacciottiA.

    Local stabilizability of nonlinear control systems

    (1992)
  • ChenT.Q. et al.

    Neural ordinary differential equations

  • This work is supported in part by the U.S. National Science Foundation under Grant No. 1826086 and by National Natural Science Foundation of China under Grant No. 61807030. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tingshu Hu under the direction of Editor André L. Tits.

    View full text