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

Tikhonov theorem for differential equations with singular impulses

The paper considers impulsive systems with singularities. The main novelty of the present research is that impulses (impulsive functions) are singular. This is beside singularity of differential equations. The Lyapunov second method is applied to proof the main theorems. Illustrative examples with simulations are given to support the theoretical results

Tikhonov theorem for differential equations with singular impulses

The paper considers impulsive systems with singularities. The main novelty of the present research is that impulses (impulsive functions) are singular. This is beside singularity of differential equations. The Lyapunov second method is applied to proof the main theorems. Illustrative examples with simulations are given to support the theoretical results

A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE

Time scale separation is a natural property of many control systems that can be ex- ploited, theoretically and numerically. We present a numerical scheme to solve optimal control problems with considerable time scale separation that is based on a model reduction approach that does not need the system to be explicitly stated in singularly perturbed form. We present examples that highlight the advantages and disadvantages of the method

Feedback Nash Equilibria for Linear Quadratic Descriptor Differential Games

In this note we consider the non-cooperative linear feedback Nash quadratic differential game with an infinite planning horizon for descriptor systems of index one. The performance function is assumed to be indefinite. We derive both necessary and sufficient conditions under which this game has a Nash equilibrium.linear-quadratic games;linear feedback Nash equilibrium;affine systems;solvability conditions;Riccati equations

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