Stability and Instability Matrices for Linear Evolution Variational Inequalities

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.

The Complementarity Class of Hybrid Dynamical Systems

Unlike linear dynamical systems the existence and uniqueness of solutions (wellposedness) for linear complementarity systems (LCS) is not trivial. It has been shown in literature that the consistent and jump space of an LCS (with zero input) plays an important role in establishing the wellposedness. In this paper we apply state and port feedback to an LCS to reorient these spaces. Sometimes it is desirable to increase the consistent space which means enlarging the set of states having continuous extension. At the same time it may be desirable to shrink the set of states which may have discontinuous extension, in other words, to decrease the jump space. Sufficient conditions in this direction are obtained in terms of feedback matrices.

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.

Linear complementarity problems and affine variational inequalities have been intensively investigated by different methods. Recently, some authors have shown that solution stability of these problems with respect to total perturbations can be effectively studied via a generalized linear constraint system. The present paper focuses on characterizing stability properties of the solution map of a linearly perturbed generalized linear constraint system. The obtained results lead to several stability conditions for parametric linear complementarity problems and affine variational inequalities in explicit forms.

This paper presents a dynamic variable structure system and its controller design conditions. The dynamic variable structure system, which is a class of hybrid dynamic systems, consists of plural subsystems which are switched by switching conditions. A key feature of the dynamic variable structure system is that each subsystem can have different degrees of freedom. In this paper, we employ switching fuzzy models to represent the nonlinear subsystem's dynamics, switching conditions and conservation laws with respect to states, momentum and/or energy of the system. We derive controller design conditions for dynamic variable structure systems.

This paper presents a novel switching control strategy with variable timing for multi-subsystem affine switching systems. The main challenge in such systems is to determine when and how transitions should occur to balance stability, efficiency, and performance. To address this, an adaptive algorithm is developed that selects switching instances based on the system state, thereby reducing unnecessary transitions and minimizing switching-induced losses. System stability is established within a Lyapunov-based framework and verified through Bilinear Matrix Inequalities (BMIs), ensuring rigorous theoretical guarantees. Unlike conventional methods that rely on fixed switching patterns or are limited to only a few modes, the proposed approach accommodates an arbitrary number of subsystems, offering greater flexibility and applicability across a wide class of hybrid dynamical systems. The strategy activates switches only when necessary, improving efficiency and reducing operational stress. Its effectiveness is validated through comprehensive simulations and experimental studies, demonstrating enhanced stability, lower transition frequency, and improved dynamic performance.

The differential equations with piecewise constant argument (DEPCAs, for short) is a class of hybrid dynamical systems (combining continuous and discrete). In this paper, under the assumption that the nonlinear term is partially unbounded, we study the bounded solution and global topological linearisation of a class of DEPCAs of general type. One of the purpose of this paper is to obtain a new criterion for the existence of a unique bounded solution, which improved the previous results. The other aim of this paper is to establish a generalized Grobman-Hartman-type theorem for the topological conjugacy between a nonlinear perturbation system and its linear system. The method is based on the new obtained criterion for bounded solution. The obtained results generalized and improved some previous papers. Some novel techniques are employed.

Stability of Dynamic Hybrid Systems With Descriptor Systems as Dynamic Model

Clocks are deeply integrated into practically every cyberphysical system either explicitly as provenance for timetriggered actions, or implicitly in cases where cyber components operate in lock step with physical dynamics. Recognizing the criticality of timing components, this paper investigates an analysis approach that allows a system designer to formally incorporate timing uncertainty as a factor when evaluating the uncertainty of the overall cyber-physical system. A set theoretic approach is considered in this paper that offers advantages in the form of computational scalability and in its ability to accommodate a general class of hybrid dynamic systems. A demonstration of the approach is provided via illustrative example using a charge pump phase locked loop and a second order dynamic system. We anticipate that the proposed approach is particularly applicable to systems where safety or reachability guarantees are required.

Linear complementarity problems provide a powerful framework to model nonsmooth phenomena in dynamical control systems. Mimicking the general strategy that led to the foundation of bifurcation theory in smooth maps, we introduce a novel notion of equivalence between linear complementarity problems that sets the basis for a theory of bifurcations in a large class of nonsmooth maps, including steady-state bifurcations in linear complementarity systems. Our definition leads to constructive algebraic conditions for identifying and classifying the nonsmooth singularities associated with nonsmooth bifurcations. We thoroughly illustrate our theory on an extended applied example and on the identification and classification of all possible equivalence classes in two-dimensional linear complementarity problems.

Considers the extension of the absolute stability problem and of the Lagrange-Dirichlet theorem, to the case when nonlinearities are multivalued monotone mappings. Some of the considered systems may also be recast in the framework of complementarity systems, which in turn may be seen as a special class of hybrid dynamical systems. Another, fruitful point of view, is that of differential inclusions, with or without measures. This class of systems encompasses mechanical systems with unilateral constraints and impacts.

We establish equivalences among five classes of hybrid systems, that we have encountered in previous research: mixed logical dynamical systems, linear complementarity systems, extended linear complementarity systems, piecewise affine systems, and max-min-plus-scaling systems. These results are of paramount importance for transferring properties and tools from one class to another.