Input-to-state-K ℒ*-stability for a class of hybrid dynamical systems

Dramatic progress in computing capabilities has resulted in the synthesis and implementation of increasingly complex dynamical systems. Since such systems frequently exhibit simultaneously several kinds of dynamic behavior in different parts of the system, they are referred to as hybrid dynamical systems. Such systems frequently defy traditional modeling and analysis techniques since the different system components may evolve along different notions of "time", including real time, discrete time, and discrete events. Most investigations of such systems to date involve ad hoc models and tailor-made analysis results. In some recent work, however, a general model has been proposed which contains most of the different classes of hybrid dynamical systems considered in the literature as special cases. At the core of this general model of hybrid dynamical system, which is defined on an arbitrary metric space, is a notion of generalized time. For this class of general hybrid dynamical systems, a variety of Lyapunov and Lagrange stability results have been established and made public in a scattering of publications and workshop records. Our objective in this paper is to present a unified overview of the more important aspects of this work.

Stability of Dynamic Hybrid Systems With Descriptor Systems as Dynamic Model

In this paper, we develop an approach to inverse optimal control for a class of hybrid dynamical system with impacts. As it is usually posed, the problem of inverse optimal control is to find a cost function that is consistent with an observed sequence of decisions, under the assumption that these decisions are optimal. We assume instead that observed decisions are only approximately optimal and find a cost function that minimizes the extent to which these decisions violate first-order necessary conditions for optimality. For the hybrid dynamical system that we consider with a cost function that is a linear combination of known basis functions, this minimization is a convex program. In fact, it reduces to a simple least-squares computation that - unlike most other forms of inverse optimal control - can be solved very efficiently. We apply our approach to a dynamic bipedal climbing robot in simulation, showing that we can recover cost functions from observed trajectories that are consistent with two different modes of locomotion.

This work deals with nonlinear observer synthesis for a particular class of hybrid dynamic systems (HDS): autonomous switching systems with jumps. The jumps can result from the system’s dynamics or from the diffeomorphism, which makes it possible to lead the system to an observability canonical form. In this paper, our contribution relates to the design of a second order sliding mode based observer (“Super Twisting Algorithm”). It allows for estimating both continuous and discrete state related to the active dynamic. On the other hand, these observers ensure a finite time convergence of the estimation error.

Robust stability results for nominally linear hybrid systems are obtained from total stability theorems for purely continuous-time and discrete-time systems by using the powerful tool of fixed point theory. The class of hybrid systems dealt consists, in general, of coupled continuous-time and digital systems subject to state perturbations whose nominal (i.e., unperturbed) parts are linear and, in general, time-varying. The obtained sufficient conditions on robust stability under a wide class of harmless perturbations are dependent on the values of the parameters defining the over-bounding functions of those perturbations. The weakness of the coupling dynamics in terms of norm among the analog and digital substates of the whole dynamic system guarantees the total stability provided that the corresponding uncoupled nominal subsystems are both exponentially stable. Fixed point stability theory is used for the proofs of stability. A generalization of that result is given for the case that sampling is not uniform. The boundedness of the state-trajectory solution at sampling instants guarantees the global boundedness of the solutions for all time. The existence of a fixed point for the sampled state-trajectory solution at sampling instants guarantees the existence of a fixed point of an extended auxiliary discrete system and the existence of a global asymptotic attractor of the solutions which is either a fixed point or a limit "Equation missing" globally stable asymptotic oscillation.

Towards a general class of hybrid dynamical system with free terminal states a global optimal control problem is proposed in this paper. And we bring forward the mathematical method to solve the above problem?HDS? global maximum principle. Then we prove the theorem by means of general variational principle.

This paper proposes a hybrid controller with an event-based switching design for complex network systems, and analyzes the convergence result with network hybrid automaton models. Specifically, the main contributions of the paper are fourfold. First, a hybrid controller design is proposed for a wide spectrum of complex network systems. Second, a class of hybrid dynamical systems is described via a hybrid automaton model; and some invariance convergence results for this class of hybrid dynamical systems are proposed. Third, a network automaton model is presented to capture the details of the closed-loop system; and the convergence results are proved. Finally, the hybrid network controller is applied to the Kunder's two-area, four-machine power system, the implementation of the hybrid network controller for the power network in Simulink is presented, and the simulation is conducted to show the efficacy of the proposed hybrid controller design.

In this work, we study finite-time stability of hybrid systems with unstable modes. We present sufficient conditions in terms of multiple Lyapunov functions for the origin of a class of hybrid systems to be finite-time stable. More specifically, we show that even if the value of the Lyapunov function increases during continuous flow, i.e., if the unstable modes in the system are active for some time, finite-time stability can be guaranteed if the finite-time convergent mode is active for a sufficient amount of cumulative time. This is the first work on finite-time stability of hybrid systems using multiple Lyapunov functions. Prior work uses a common Lyapunov function approach, and requires the Lyapunov function to be decreasing during the continuous flows and non-increasing at the discrete jumps, thereby, restricting the hybrid system to only have stable modes, or to only evolve along the stable modes. In contrast, we allow Lyapunov functions to increasebothduring the continuous flows and the discrete jumps. As thus, the derived stability results are less conservative compared to the earlier results in the related literature, and in effect allow the hybrid system to have unstable modes.

Many flexible manufacturing systems have hybrid production structures, whereby discrete events and continuous evolutions are linked. In this paper, we present a model for a class of hybrid production systems composes of two interconnected sub-systems: one sub-system producing a discrete flow of parts and the other a continuous flow of materials. Regarding the nature of the interactions between the two sub-systems, converting machines or units transform a continuous flow or a discrete flow into a flow with a different nature. The analysis and the control of continuous systems and discrete events systems are based on different methodological and theoretical tools. We propose a modelling method for a class of hybrid dynamic systems, taking into account the interactions between two sub-systems having different characteristics. This method leads to a model used in simulation as a decision support system for supervision.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The 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 this paper, including an event-triggered control application and a bouncing ball system with Zeno behavior, that illustrate incremental stability with respect to continuous time or discrete time, respectively.

Finite time stability and stabilization of hybrid dynamic systems

This paper deals with stability analysis of hybrid systems. Such systems are characterized by a combination of continuous dynamics and logic based switching between discrete modes. Lyapunov theory is a well known methodology for the stability analysis of linear and nonlinear systems in control system literature. Construction of Lyapunov functions for hybrid systems is generally a difficult task, but once these functions are defined, stabilization of the system is straight-forward. The sum of squares (SOS) decomposition and semidefinite programming has also provided an efficient methodology for analysis of nonlinear systems. The computational method used in this paper relies on the SOS decomposition of multivariate polynomials. By using SOS, we construct a (some) Lyapunov function(s) for the hybrid system. The reduction techniques provide numerical solution of large-scale instances; otherwise they will be practically unsolvable. The introduced method can be used for hybrid systems with linear or nonlinear vector fields. Some examples are given to demonstrate the capabilities of the proposed approach.

In this note, we consider the asymptotical stability of hybrid dynamical systems (HDSs) in the sense of Lyapunov. Some necessary and sufficient conditions are derived which require only the Lyapunov function to be non-increasing along the sequences of time-interval of the active subsystems.

A numerical method for hybrid optimal control based on dynamic programming

This paper deals with qualitative analysis of the so-called switched fluid networks. Such networks are used to model communication, computer, and flexible manufacturing systems. We prove that for any deterministic network from a specific class, there exists a finite number of limit cycles attracting all the trajectories of the system. Furthermore, we determine this number.