On Incremental Stability of Interconnected Switched Systems

In this letter, the incremental stability of interconnected systems is discussed. In particular, we consider an interconnection of switched nonlinear systems. The incremental stability of the switched interconnected system is a stronger property as compared to conventional stability. Guaranteeing such a notion of stability for an overall interconnected nonlinear system is challenging, even if individual subsystems are incrementally stable. Here, preserving incremental stability for the interconnection is ensured with a set of sufficient conditions. Contraction theory is used as a tool to achieve incremental convergence. For the feedback interconnection, the small gain characterisation is presented for the overall system's incremental stability. The results are derived for a special case of feedback, i.e., cascade interconnection. Matrix-measure-based conditions are presented, which are computationally tractable. Two numerical examples are demonstrated and supported with simulation results to verify the theoretical claims

Approximately bisimilar symbolic models for incrementally stable switched systems

Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. This paper describes one particular approach to address these different objectives based on the construction of approximately equivalent (bisimilar) symbolic models for switched systems. The main contribution of this paper consists in showing that under standard assumptions ensuring incremental stability of a switched system (i.e. existence of a common Lyapunov function, or multiple Lyapunov functions with dwell time), it is possible to construct a finite symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. To support the computational merits of the proposed approach, we use symbolic models to synthesize controllers for two examples of switched systems, including the boost DC-DC converter.Comment: 17 page

Polybius and the anger of the Romans

In this paper, incremental exponential asymptotic stability of a class of switched Carathéodory nonlinear systems is studied based on the novel concept of measure of switched matrices via multiple norms and the transaction coefficients between these norms. This model is rather general and includes the case of staircase switching signals as a special case. Sufficient conditions are derived for incremental stability allowing for the system to be incrementally exponentially asymptotically stable even if some of its modes are unstable in some time periods. Numerical examples on switched linear systems with periodic switching and on the synchronization of switched networks of nonlinear systems are used to illustrate the theoretical results

Contraction analysis of switched Filippov systems via regularization

We study incremental stability and convergence of switched (bimodal) Filippov systems via contraction analysis. In particular, by using results on regularization of switched dynamical systems, we derive sufficient conditions for convergence of any two trajectories of the Filippov system between each other within some region of interest. We then apply these conditions to the study of different classes of Filippov systems including piecewise smooth (PWS) systems, piecewise affine (PWA) systems and relay feedback systems. We show that contrary to previous approaches, our conditions allow the system to be studied in metrics other than the Euclidean norm. The theoretical results are illustrated by numerical simulations on a set of representative examples that confirm their effectiveness and ease of application.Comment: Preprint submitted to Automatic

Incremental stability of hybrid dynamical systems

International audienceThe analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. As we will show, defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump-and flow incremental asymptotic stability. To present Lyapunov conditions for these two notions, in both cases, we resort to an extended hybrid system and we prove that the stability of a well-defined set for this extended system implies incremental stability of the original system. We can then use available Lyapunov conditions to infer the set stability of the extended system. Various examples are provided throughout the paper, including an event-triggered control application and a bouncing ball system with Zeno behaviour, that illustrate incremental stability with respect to continuous time or discrete time, respectively

Observer design for piecewise smooth and switched systems via contraction theory

The aim of this paper is to present the application of an approach to study contraction theory recently developed for piecewise smooth and switched systems. The approach that can be used to analyze incremental stability properties of so-called Filippov systems (or variable structure systems) is based on the use of regularization, a procedure to make the vector field of interest differentiable before analyzing its properties. We show that by using this extension of contraction theory to nondifferentiable vector fields, it is possible to design observers for a large class of piecewise smooth systems using not only Euclidean norms, as also done in previous literature, but also non-Euclidean norms. This allows greater flexibility in the design and encompasses the case of both piecewise-linear and piecewise-smooth (nonlinear) systems. The theoretical methodology is illustrated via a set of representative examples.Comment: Preprint accepted to IFAC World Congress 201

Symbolic Models for Stochastic Switched Systems: A Discretization and a Discretization-Free Approach

Stochastic switched systems are a relevant class of stochastic hybrid systems with probabilistic evolution over a continuous domain and control-dependent discrete dynamics over a finite set of modes. In the past few years several different techniques have been developed to assist in the stability analysis of stochastic switched systems. However, more complex and challenging objectives related to the verification of and the controller synthesis for logic specifications have not been formally investigated for this class of systems as of yet. With logic specifications we mean properties expressed as formulae in linear temporal logic or as automata on infinite strings. This paper addresses these complex objectives by constructively deriving approximately equivalent (bisimilar) symbolic models of stochastic switched systems. More precisely, this paper provides two different symbolic abstraction techniques: one requires state space discretization, but the other one does not require any space discretization which can be potentially more efficient than the first one when dealing with higher dimensional stochastic switched systems. Both techniques provide finite symbolic models that are approximately bisimilar to stochastic switched systems under some stability assumptions on the concrete model. This allows formally synthesizing controllers (switching signals) that are valid for the concrete system over the finite symbolic model, by means of mature automata-theoretic techniques in the literature. The effectiveness of the results are illustrated by synthesizing switching signals enforcing logic specifications for two case studies including temperature control of a six-room building.Comment: 25 pages, 4 figures. arXiv admin note: text overlap with arXiv:1302.386