Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity

Pierluigi Di Franco, Giordano Scarciotti,, Alessandro Astolfi,

Abstract—The stability analysis for nonlinear differentialalgebraic systems is addressed using tools from classical control theory. Sufficient stability conditions relying on matrix inequalities are established via Lyapunov Direct Method. In addition, a novel interpretation of differential-algebraic systems as feedback interconnection of a purely differential system and an algebraic system allows reducing the stability analysis to a smallgain-like condition. The study of stability properties for constrained mechanical systems, for a class of Lipschitz differential-algebraic systems and for an academic example is used to illustrate the theory.

1. Introduction

DIFFERENTIAL-algebraic systems (also known as DAE systems, descriptor systems or singular systems) provide a generalization of the classical state-space framework which allows a simpler characterization of many physical phenomena,such as conservation of mass and flow, topological and environmental constraints and/or thermodynamical relations.The range of engineering applications which can be naturally described by DAE systems includes mechanical systems [1],robot manipulators with constrained end-effector [2],chemical processes [3], electrical networks with nonlinear elements [4], as well as models arising in social and economic sciences [5]. Recently, modern simulation tools based on object-oriented languages [6] have considerably spread the use of DAE systems for the modelling of physical systems to such an extent that the interest in studying the problems of numerical integration and control of dynamical systems in the DAE formulation has grown rapidly (see e.g., [7]–[10]).

Classical approaches to the stability analysis of DAE systems are based on index1or coordinates reduction techniques which, by means of multiple time differentiations and algebraic manipulations, reveal the underlying differential representation of the system to which classical results can be applied. One of the first systematic contributions in this area has been provided by [12], in which state-space equivalent forms for linear time-invariant DAE systems have been presented. In [13] a state space realization for index-3 nonlinear DAE systems has been derived and the feedback stabilization problem has been solved by means of linearization techniques. A similar approach has been adopted in [14], i.e., a state space realization and an output feedback stabilization methodology have been developed for nonlinear index-2 DAE systems. However, the multiple differentiation of the algebraic equation and the need for further algebraic manipulations required by these methods poorly suit the scale of many engineering problems. This is the case, for instance,in power system models and switching networks [15]. In addition, nonlinearities in the model equations and model uncertainties may prevent the applicability of coordinates reduction methods [16]. Therefore, an approach to the problems of stability analysis and control directly in the DAE formulation is needed.

Another approach to the stability analysis of DAE systems consists in extending tools from classical and modern control theory to this class of systems. Lyapunov stability theory has been extended to nonlinear DAE systems in [17], in which the robust control problem for DAE systems with uncertainties has also been discussed. Other examples of application of Lyapunov stability theory can be found in [18], in which DAE systems with delays are considered, in [19], in which estimations of the domain of attraction of equilibria of DAE systems are provided, in [10], in which the task of finding a Lyapunov function for a DAE system is transformed into an optimization problem subject to algebraic constraints, thus yielding sufficient stability conditions.

V. Conclusions

In this paper, exploiting the properties of the solution manifold, we have given sufficient stability conditions for classes of DAE systems. Using Lyapunov Direct Method we have shown that local asymptotic stability of the zero equilibrium can be inferred by the feasibility of a statedependant matrix inequality (Theorem 1), which assumes different forms depending on the selection of the design parameters. On the one hand, the proposed method allows recovering classical results based on the inversion of the algebraic equation (Proposition 1). On the other hand, such an inversion can be avoided by a suitable selection of the design parameters (Proposition 2). A numerical example motivated by the analysis of a nonlinear mechanical system has been used to validate the technique (Section II-A). We have also proposed a novel interpretation of DAE systems as feedback interconnection of a differential system and an algebraic system (Section III). In this framework, the algebraic variable assumes the role of an external disturbance and the stability analysis reduces to a small-gain-like condition (Theorem 2).The application of this method to the linear case yields classical results which can be reinterpreted in this framework as a particular case (Corollary 2). For a class of constrained mechanical systems we have shown that the problem of stability analysis can be formulated as a stabilization problem of a differential system (Lemma 4). By means of linearization techniques, we have also shown that such a problem can be solved when a detectability condition is satisfied (Proposition 3).Finally, using the feedback interpretation introduced in Section III, we have shown that the stability analysis of a class of Lipschitz DAE systems reduces to finding the feasibility of a linear matrix inequality (Proposition 4).

Motivated by the results of this paper [26] has shown that the stabilization problem for a general class of DAE systems can be addressed by means of the feedback decomposition introduced in Section III of the present study. In addition, the stabilization problem for a class of Lipschitz DAE systems has been studied in [30]. Numerical examples which exploit the results derived in Sections III and IV can be found in [25,Section 3.4], in [29, Section III.A], in [26, Section IV] and in[30, Section V]. Following the same line of research, future works will focus on the application to the observer design and to the output feedback stabilization problems for classes of DAE systems.