Stability analysis of a general class of singularly perturbed linear hybrid systems

Motivated by a real problem in steel production, we introduce and analyze a general class of singularly perturbed linear hybrid systems with both switches and impulses, in which the slow or fast nature of the variables can be mode-dependent. This means that, at switching instants, some of the slow variables can become fast and vice-versa. Firstly, we show that using a mode-dependent variable reordering we can rewrite this class of systems in a form in which the variables preserve their nature over time. Secondly, we establish, through singular perturbation techniques, an upper bound on the minimum dwell-time ensuring the overall system's stability. Remarkably, this bound is the sum of two terms. The first term corresponds to an upper bound on the minimum dwell-time ensuring the stability of the reduced order linear hybrid system describing the slow dynamics. The order of magnitude of the second term is determined by that of the parameter defining the ratio between the two time-scales of the singularly perturbed system. We show that the proposed framework can also take into account the change of dimension of the state vector at switching instants. Numerical illustrations complete our study

Stability of Hybrid Singularly Perturbed Systems with Time Delay

Hybrid singularly perturbed systems (SPSs) with time delay are considered and exponential stability of these systems is investigated. This work mainly covers switched and impulsive switched delay SPSs . Multiple Lyapunov functions technique as a tool is applied to these systems. Dwell and average dwell time approaches are used to organize the switching between subsystems (modes) so that the hybrid system is stable. Systems with all stable modes are first discussed and, after developing lemmas to ensure existence of growth rates of unstable modes, these systems are then extended to include, in addition, unstable modes. Sufficient conditions showing that impulses contribute to yield stability properties of impulsive switched systems that consist of all unstable subsystems are also established. A number of illustrative examples are presented to help motivate the study of these systems

On Stability and Stabilization of Hybrid Systems

The thesis addresses the stability, input-to-state stability (ISS), and stabilization problems for deterministic and stochastic hybrid systems with and without time delay. The stabilization problem is achieved by reliable, state feedback controllers, i.e., controllers experience possible faulty in actuators and/or sensors. The contribution of this thesis is presented in three main parts. Firstly, a class of switched systems with time-varying norm-bounded parametric uncertainties in the system states and an external time-varying, bounded input is addressed. The problems of ISS and stabilization by a robust reliable $H_{\infty}$ control are established by using multiple Lyapunov function technique along with the average dwell-time approach. Then, these results are further extended to include time delay in the system states, and delay systems subject to impulsive effects. In the latter two results, Razumikhin technique in which Lyapunov function, but not functional, is used to investigate the qualitative properties. Secondly, the problem of designing a decentralized, robust reliable control for deterministic impulsive large-scale systems with admissible uncertainties in the system states to guarantee exponential stability is investigated. Then, reliable observers are also considered to estimate the states of the same system. Furthermore, a time-delayed large-scale impulsive system undergoing stochastic noise is addressed and the problems of stability and stabilization are investigated. The stabilization is achieved by two approaches, namely a set of decentralized reliable controllers, and impulses. Thirdly, a class of switched singularly perturbed systems (or systems with different time scales) is also considered. Due to the dominant behaviour of the slow subsystem, the stabilization of the full system is achieved through the slow subsystem. This approach results in reducing some unnecessary sufficient conditions on the fast subsystem. In fact, the singular system is viewed as a large-scale system that is decomposed into isolated, low order subsystems, slow and fast, and the rest is treated as interconnection. Multiple Lyapunov functions and average dwell-time switching signal approach are used to establish the stability and stabilization. Moreover, switched singularly perturbed systems with time-delay in the slow system are considered

Finite-time stochastic input-to-state stability and observer-based controller design for singular nonlinear systems

This paper investigated observer-based controller for a class of singular nonlinear systems with state and exogenous disturbance-dependent noise. A new sufficient condition for finite-time stochastic input-to-state stability (FTSISS) of stochastic nonlinear systems is developed. Based on the sufficient condition, a sufficient condition on impulse-free and FTSISS for corresponding closed-loop error systems is provided. A linear matrix inequality condition, which can calculate the gains of the observer and state-feedback controller, is developed. Finally, two simulation examples are employed to demonstrate the effectiveness of the proposed approaches

H ? filtering for stochastic singular fuzzy systems with time-varying delay

This paper considers the H? filtering problem for stochastic singular fuzzy systems with timevarying delay. We assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by Ito-type stochastic differential equations. Based on an auxiliary vector and an integral inequality, a set of delay-dependent sufficient conditions is established, which ensures that the filtering error system is e?t - weighted integral input-to-state stable in mean (iISSiM). A fuzzy filter is designed such that the filtering error system is impulse-free, e?t -weighted iISSiM and the H? attenuation level from disturbance to estimation error is belowa prescribed scalar.Aset of sufficient conditions for the solvability of the H? filtering problem is obtained in terms of a new type of Lyapunov function and a set of linear matrix inequalities. Simulation examples are provided to illustrate the effectiveness of the proposed filtering approach developed in this paper

Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Averaging in a Class of Stochastic Hybrid Dynamical Systems with Time-Varying Flow Maps

We present stability and recurrence results for a class of stochastic hybrid dynamical systems with oscillating flow maps. These results are developed by introducing averaging tools that parallel those already existing for ordinary differential equations and deterministic hybrid dynamical systems. Such tools can be used to examine the stability properties of the original dynamics based on the properties of a simpler dynamical system constructed from the average of the original oscillating vector field. In this work, we focus on a class of systems for which global stability and recurrence results are achievable under suitable smoothness assumptions on the dynamics. By studying the average stochastic hybrid dynamics using Lyapunov-Foster functions, we derive similar stability and recurrence results for the original stochastic hybrid system