Stability analysis of nonlinear quadratic systems via polyhedral Lyapunov functions

Abstract

Quadratic systems play an important role in the modeling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems it is mandatory not only to determine whether the origin of the state space is locally asymptotically stable, but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the domain of attraction of the equilibrium. The proposed approach is based on polyhedral Lyapunov functions, rather than on the classical quadratic Lyapunov functions. An example shows that our methodology may return less conservative results than those obtainable with previous approaches.

Introduction

Nonlinear quadratic systems provide an appropriate tool for modeling phenomena in a wide range of applications both in engineering (electric power systems, chemical reactors (Rössler, 1976), and robots), and in other areas such as biology, ecology, economics and meteorology (Lorenz, 1963). In Biology, quadratic systems can capture the nonlinear behaviour of interacting (biochemical and biological) species, e.g. in enzymatic reaction kinetics (Murray, 2002), or in tumour progression models (Merola, Amato, & Cosentino, 2008).

In such applications, it is relevant to determine the set of initial conditions around an equilibrium point which steer the systems trajectories to the equilibrium itself. This requires to obtain an estimate of the domain of attraction (DA) of the equilibrium.

Concerning the classical approach to the DA estimation problem, Quadratic Lyapunov Functions (QLFs) are used to approximate the DA of nonlinear (quadratic, cubic and polynomial) systems, see e.g. (Chesi et al., 2005, Tesi et al., 1996, Tibken, 2000). QLFs may provide conservative results, especially in the case of a priori choice of the function itself. Some recent results enable to reduce the conservatism of the estimate of the DA. For instance, one of the first methods to recast the DA estimation problem in terms of LMIs and Sum of Squares (SOS) conditions was proposed in Chesi, Tesi, Vicino, and Genesio (1999). The same approach is used in Chesi (2007) where the DA estimate of polynomial systems is obtained through a family of polynomial Lyapunov functions, while non-polynomial systems are addressed in Chesi (2009).

Since the exact determination of the whole DA of the equilibrium point of a given quadratic system is a difficult or even impossible task (except for very simple cases), the recent papers Amato, Cosentino, and Merola (2006) and Amato, Cosentino, and Merola (2007) are aimed at solving the more practical problem of determining whether an assigned polytope $\mathcal{P}$ containing the origin of the state space belongs to the DA of the equilibrium. The approach proposed in Amato et al. (2006) and Amato et al. (2007) makes use of QLFs. Therefore, the given polytope $\mathcal{P}$ is shown to be contained into a level curve of the QLF which provides an ellipsoidal estimate of the DA. In extension to the results in Amato et al. (2006) and Amato et al. (2007), the idea exploited here is that of using polyhedral Lyapunov functions (Blanchini, 1995, Brayton and Tong, 1979, Brayton and Tong, 1980, Molchanov and Pyatnitskii, 1986) which are likely to improve the DA estimate. To this end, an optimization procedure is proposed here and its effectiveness is shown in comparison with existing results.

Other results in the literature on polyhedral functions concern the stabilization of discrete-time bilinear systems via linear state-feedback (Bitsoris & Athanasopoulos, 2008), where a stabilizing control law is obtained through the satisfaction of a condition on the positive invariance of polyhedral sets of nonlinear systems with second order polynomial nonlinearity.

The paper is organized as follows. In Section 2 the problem we deal with in the paper is precisely stated and some preliminary definitions and results about polytopes are provided. In Section 3 the main result of the paper is stated, namely a sufficient condition guaranteeing that a given polytope $\mathcal{P}$ belongs to the DA of a given quadratic system. In Section 4 a numerical example shows the goodness of the proposed technique over existing methods. Finally some concluding remarks are given in Section 5.