A general stability criterion for switched linear systems having stable and unstable subsystems

We report conditions on a switching signal that guarantee that solutions of a switched linear systems converge asymptotically to zero. These conditions are apply to continuous, discrete-time and hybrid switched linear systems, both those having stable subsystems and mixtures of stable and unstable subsystems

Minimally Constrained Stable Switched Systems and Application to Co-simulation

We propose an algorithm to restrict the switching signals of a constrained switched system in order to guarantee its stability, while at the same time attempting to keep the largest possible set of allowed switching signals. Our work is motivated by applications to (co-)simulation, where numerical stability is a hard constraint, but should be attained by restricting as little as possible the allowed behaviours of the simulators. We apply our results to certify the stability of an adaptive co-simulation orchestration algorithm, which selects the optimal switching signal at run-time, as a function of (varying) performance and accuracy requirements.Comment: Technical report complementing the following conference publication: Gomes, Cl\'audio, Beno\^it Legat, Rapha\"el Jungers, and Hans Vangheluwe. "Minimally Constrained Stable Switched Systems and Application to Co-Simulation." In IEEE Conference on Decision and Control. Miami Beach, FL, USA, 201

Contraction analysis of nonlinear systems and its application

The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents

Contraction analysis of nonlinear systems and its application

The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents

New advances in H∞ control and filtering for nonlinear systems

The main objective of this special issue is to summarise recent advances in H∞ control and filtering for nonlinear systems, including time-delay, hybrid and stochastic systems. The published papers provide new ideas and approaches, clearly indicating the advances made in problem statements, methodologies or applications with respect to the existing results. The special issue also includes papers focusing on advanced and non-traditional methods and presenting considerable novelties in theoretical background or experimental setup. Some papers present applications to newly emerging fields, such as network-based control and estimation

Stabilizability and optimal control of switched differential algebraic equations

In this thesis control of dynamical systems with switches is considered. Examples of such systems are electronic circuits and mechanical systems. The switches are induced by abrupt structural changes due to component failure or physical switches. In the case of constraints on the dynamics, the state of the system can only take certain values and not only differential equations are involved in modeling the system, but also algebraic equations. An important question in control problems is often how well a certain controller performs. Some controllers require little energy, but induce undesired behavior of the system, whereas others perform well in terms of the systems behavior but require a lot of energy. It turns out that in general an optimal controller does not exist. However, necessary and sufficient conditions for the existence of optimal controller given a quadratic cost functional are presented in this thesis. Besides quantitative properties also some qualitative properties are investigated. The systems considered exhibit discontinuous behavior and Dirac impulses, whereas especially Dirac impulses are practically undesirable. Dirac impulses occur in practice in the form of hydraulic shocks in fluid networks or sparks in electronic circuits. The possibility to avoid Dirac impulses is also studied and necessary and sufficient conditions are given

Real-time estimation for switched linear systems

International audienceWe extend previous works on real-time estimation, via algebraic techniques, to the recovering of the switching signal and of the state for switching linear systems. We characterize also singular inputs for which the switched systems become undistinguishable. Several convincing numerical experiments are illustrating our techniques which are easily implementable