Stability Analysis Of Hybrid Systems Research Articles

Path planning in formation and collision avoidance for multi-agent systems

This paper investigates, in a centralized manner, the motion planning problem for a team of unicycle-like mobile robots in a known environment. In particular, a multi-agent collision-free patrolling and formation control algorithm is presented, which combines outcomes of: (i) stability analysis of hybrid systems, (ii) algebraic geometry, and (iii) classical potential functions. The objective is achieved by designing a Lyapunov-based hybrid strategy that autonomously selects the navigation parameters. Tools borrowed from algebraic geometry are adopted to construct Lyapunov functions that guarantee the convergence to the desired formation and path, while classical potential functions are exploited to avoid collisions among agents and the fixed obstacles within the environment. The proposed navigation algorithm is tested in simulation and then validated by using the robots of a remote accessible robotic testbed.

Global robust attitude tracking with torque disturbance rejection via dynamic hybrid feedback

In this paper an approach to the design of robust global attitude tracking controllers for fully actuated rigid bodies is proposed. The challenge of simultaneously dealing with topological obstructions to global attitude tracking and with disturbances affecting the attitude dynamics is tackled by means of a hybrid hierarchical design that exploits the cascade structure of the underlying mathematical model. The proposed hierarchical strategy is based on an inner–outer loop paradigm comprising a dynamic control law for angular velocity tracking (inner loop) and a hybrid control law for attitude tracking (outer loop). By leveraging recent tools for the stability analysis of hybrid systems, we prove a robust global tracking property by assuming mild properties on the dynamics of the velocity feedback. We also discuss a few relevant examples satisfying these properties, encompassing harmonic disturbance compensators and conditional integrators, capable of rejecting unknown constant disturbances with an intrinsic anti-windup action.

Decomposable dissipativity and stability analysis of hybrid systems

This paper proposes definitions of decomposable dissipativity and decomposable passivity for a class of hybrid systems to investigate stability properties and dissipativity-based control of this kind of hybrid system. Decomposable dissipativity is established via employing multiple flow-storage functions, jump-storage functions, flow-supply rates and jump-supply rates, which allows each subsystem to have its own storage function and supply rate. The decomposable dissipativity reveals the energy changes of activated and inactivated subsystems. Based on the proposed definitions, sufficient conditions of guaranteeing the stability of hybrid systems are derived, which relax the constraint on non-positive supply rates. Then, dissipativity- and passivity-based controllers are designed. In order to demonstrate the usefulness and effectiveness of the obtained results, two examples are provided.

A Poincaré-Bendixson theorem for hybrid systems

The Poincaré-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincaré-Bendixson Theorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincaré return map, a useful object for the stability analysis of hybrid systems. We also prove a Poincaré-Bendixson Theorem for a class of one dimensional hybrid dynamical systems.

Stability of singularly perturbed nonlinear stochastic hybrid systems

ABSTRACTThe problem of exponential mean-square stability of nonlinear singularly perturbed, stochastic hybrid systems is studied in this article. Two groups of nonlinear systems are considered separately. To obtain the sufficient conditions of stability, two basic approaches of stability analysis for hybrid systems with a given Markovian switching rule and any Markovian switching rule and singularly perturbed non–hybrid systems were combined. The Lyapunov techniques were used in both approaches. The obtained results are illustrated by examples.

Local analysis of hybrid systems on polyhedral sets with state-dependent switching

Abstract This paper deals with stability analysis of hybrid systems. Various stability concepts related to hybrid systems are introduced. The paper advocates a local analysis. It involves the equivalence relation generated by reset maps of a hybrid system. To establish a tangible method for stability analysis, we introduce the notion of a chart, which locally reduces the complexity of the hybrid system. In a chart, a hybrid system is particularly simple and can be analyzed with the use of methods borrowed from the theory of differential inclusions. Thus, the main contribution of this paper is to show how stability of a hybrid system can be reduced to a specialization of the well established stability theory of differential inclusions. A number of examples illustrate the concepts introduced in the paper.

Stability of Hybrid Stochastic Systems with Time-Delay

This paper develops some criteria for a kind of hybrid stochastic systems with time-delay, which improve existing results on hybrid systems without considering noises. The improved results show that the presence of noise is quite involved in the stability analysis of hybrid systems. New results can be used to analyze the stability of a kind of stochastic hybrid impulsive and switching neural networks (SHISNN). Therefore, stability analysis of SHISNN can be turned into solving a linear matrix inequality (LMI).

Hybrid Systems with Memory: Modelling and Stability Analysis via Generalized Solutions

Hybrid systems with memory are dynamical systems exhibiting both hybrid and delay phenomena. We present a general modelling framework for such systems using hybrid functional inclusions, whose generalized solutions are defined on hybrid time domains and evolve in the phase space of hybrid memory arcs equipped with the graphical convergence topology. We prove a general existence result based on some basic conditions on the data of hybrid systems. We further establish sufficient conditions for the stability analysis of hybrid systems with memory using Lyapunov-Razumikhin (L-R) functions. We demonstrate the stability results on a general nonlinear system with switching dynamics and state jumps.

Input-to-state stability and interconnections of discontinuous dynamical systems

In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov’s solution concept, that is appropriate for ‘open’ systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.

Using Sum of Squares Decomposition for Stability of Hybrid 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.

Practical stability and stabilization of hybrid and switched systems

In this note, practical stability and stabilization problems for hybrid and switched systems are studied. The main results of this note include a direct method for the /spl epsi/-practical stability analysis of hybrid systems and sufficient conditions for the practical stabilizability of switched systems. We construct an /spl epsi/-practically stabilizing switching law in the proof of the practical stabilizability result and apply it to a tracking problem to show its effectiveness.

Polytopic Connective Stability

The purpose of this paper is to consider connective stability of large scale systems when the interconnection matrices are allowed to move within a convex polytope. Distinct sets of sufficient vertex conditions are derived for polytopic connective stability in terms of convex M-matrices. A brief overview of the literature on the connective stability concept is provided, because of its relevance in stability analysis of hybrid systems.

Stability of a spinning body containing elastic parts via Liapunov'sdirect method

A general and rigorous method for the stability analysis of a spinning body which is part rigid and part elastic is presented. The motion of the body is described by a system of equations, namely a system consisting of both ordinary and partial differential equations. The stability analysis is based on the Liapunov direct method and it works directly with the hybrid set of equations in contrast with the common practice of replacing the partial differential equations with ordinary differential equations by means of modal truncation or spatial discretization. This new approach permits a more rigorous analysis which not only produces sharper stability criteria but also avoids questions as to the effect of the truncation and discretization processes on the results. The general formulation can be used to test the stability of orbiting satellites with various types of elastic members and should also be applicable to the stability analysis of hybrid systems encountered in other areas.