Stabilization of switched neural networks with time-varying delay via bumpless transfer control

This paper investigates the stabilization of switched neural networks with time-varying delay. In order to overcome the drawback that the classical switching state feedback controller may generate the bumps at switching time, a new switching feedback controller which can smooth effectively the bumps is proposed. According to mode-dependent average dwell time, new exponential stabilization results are deduced for switched neural networks under the proposed feedback controller. Based on a simple corollary, the procedures which are used to calculate the feedback control gain matrices are also obtained. Two simple numerical examples are employed to demonstrate the effectiveness of the proposed results.Peer reviewe

Stochastic switching of delayed feedback suppresses oscillations in genetic regulatory systems

Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical, and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.Comment: updated: 13 pages, 5 figure

On stability of linear switched differential algebraic equations

This paper studies linear switched differential algebraic equations (DAEs), i.e., systems defined by a finite family of linear DAE subsystems and a switching signal that governs the switching between them. We show by examples that switching between stable subsystems may lead to instability, and that the presence of algebraic constraints leads to a larger variety of possible instability mechanisms compared to those observed in switched systems described by ordinary differential equations (ODEs). We prove two sufficient conditions for stability of switched DAEs based on the existence of suitable Lyapunov functions. The first result states that a common Lyapunov function guarantees stability under arbitrary switching when an additional condition involving consistency projectors holds (this extra condition is not needed when there are no jumps, as in the case of switched ODEs). The second result shows that stability is preserved under switching with sufficiently large dwell time

A Framework for Modeling Cyber-Physical Switching Attacks in Smart Grid

Security issues in cyber-physical systems are of paramount importance due to the often safety- critical nature of its associated applications. A rst step in understanding how to protect such systems requires an understanding of emergent weaknesses, in part, due to the cyber-physical coupling. In this paper, we present a framework that models a class of cyber-physical switching vulnerabilities in smart grid systems. Variable structure system theory is employed to effectively characterize the cyber-physical interaction of the smart grid and demonstrate how existence of the switching vulnerability is dependent on the local structure of the power grid. We identify and demonstrate how through successful cyber intrusion and local knowledge of the grid an opponent can compute and apply a coordinated switching sequence to a circuit breaker to disrupt operation within a short interval of time. We illustrate the utility of the attack approach empirically on the Western Electricity Coordinating Council three-machine, nine-bus system under both model error and partial state information.The open access fee for this work was funded through the Texas A&M University Open Access to Knowledge (OAK) Fund

Transparency in Port-Hamiltonian-Based Telemanipulation

After stability, transparency is the major issue in the design of a telemanipulation system. In this paper, we exploit the behavioral approach in order to provide an index for the evaluation of transparency in port-Hamiltonian-based teleoperators. Furthermore, we provide a transparency analysis of packet switching scattering-based communication channels

Qualitative Studies of Nonlinear Hybrid Systems

A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. Hybrid systems arise in a wide variety of important applications in diverse areas, ranging from biology to computer science to air traffic dynamics. The interaction of continuous- and discrete-time dynamics in a hybrid system often leads to very rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems. Investigating the dynamical behavior of hybrid systems is of great theoretical and practical importance. The objectives of this thesis are to develop the qualitative theory of nonlinear hybrid systems with impulses, time-delay, switching modes, and stochastic disturbances, to develop algorithms and perform analysis for hybrid systems with an emphasis on stability and control, and to apply the theory and methods to real-world application problems. Switched nonlinear systems are formulated as a family of nonlinear differential equations, called subsystems, together with a switching signal that selects the continuous dynamics among the subsystems. Uniform stability is studied emphasizing the situation where both stable and unstable subsystems are present. Uniformity of stability refers to both the initial time and a family of switching signals. Stabilization of nonlinear systems via state-dependent switching signal is investigated. Based on assumptions on a convex linear combination of the nonlinear vector fields, a generalized minimal rule is proposed to generate stabilizing switching signals that are well-defined and do not exhibit chattering or Zeno behavior. Impulsive switched systems are hybrid systems exhibiting both impulse and switching effects, and are mathematically formulated as a switched nonlinear system coupled with a sequence of nonlinear difference equations that act on the switched system at discrete times. Impulsive switching signals integrate both impulsive and switching laws that specify when and how impulses and switching occur. Invariance principles can be used to investigate asymptotic stability in the absence of a strict Lyapunov function. An invariance principle is established for impulsive switched systems under weak dwell-time signals. Applications of this invariance principle provide several asymptotic stability criteria. Input-to-state stability notions are formulated in terms of two different measures, which not only unify various stability notions under the stability theory in two measures, but also bridge this theory with the existent input/output theories for nonlinear systems. Input-to-state stability results are obtained for impulsive switched systems under generalized dwell-time signals. Hybrid time-delay systems are hybrid systems with dependence on the past states of the systems. Switched delay systems and impulsive switched systems are special classes of hybrid time-delay systems. Both invariance property and input-to-state stability are extended to cover hybrid time-delay systems. Stochastic hybrid systems are hybrid systems subject to random disturbances, and are formulated using stochastic differential equations. Focused on stochastic hybrid systems with time-delay, a fundamental theory regarding existence and uniqueness of solutions is established. Stabilization schemes for stochastic delay systems using state-dependent switching and stabilizing impulses are proposed, both emphasizing the situation where all the subsystems are unstable. Concerning general stochastic hybrid systems with time-delay, the Razumikhin technique and multiple Lyapunov functions are combined to obtain several Razumikhin-type theorems on both moment and almost sure stability of stochastic hybrid systems with time-delay. Consensus problems in networked multi-agent systems and global convergence of artificial neural networks are related to qualitative studies of hybrid systems in the sense that dynamic switching, impulsive effects, communication time-delays, and random disturbances are ubiquitous in networked systems. Consensus protocols are proposed for reaching consensus among networked agents despite switching network topologies, communication time-delays, and measurement noises. Focused on neural networks with discontinuous neuron activation functions and mixed time-delays, sufficient conditions for existence and uniqueness of equilibrium and global convergence and stability are derived using both linear matrix inequalities and M-matrix type conditions. Numerical examples and simulations are presented throughout this thesis to illustrate the theoretical results

Switched and hybrid systems with inputs: small-gain theorems, control with limited information, and topological entropy

In this thesis, we study stability and stabilization of switched and hybrid systems with inputs. We consider primarily two topics in this area: small gain theorems for interconnected switched and hybrid systems, and control of switched linear systems with limited information. First, we study input-to-state practical stability (ISpS) of interconnections of two switched nonlinear subsystems with independent switchings and possibly non-ISpS modes. Provided that for each subsystem, the switching is slow in the sense of an average dwell-time (ADT), and the total active time of non-ISpS modes is short in proportion, Lyapunov-based small-gain theorems are established via hybrid system techniques. By augmenting each subsystem with a hybrid auxiliary timer that models the constraints on switching, we enable a construction of hybrid ISpS-Lyapunov functions, and consequently, a convenient formulation of a small-gain condition for ISpS of the interconnection. Based on our small-gain theorem, we demonstrate the stabilization of interconnected switched control-affine systems using gain-assignment techniques. Second, we investigate input-to-state stability (ISS) of networks composed of n ≥ 2 hybrid subsystems with possibly non-ISS dynamics. Lyapunov-based small-gain theorems are established based on the notion of candidate ISS-Lyapunov functions, which unifies and extends several previous results for interconnected hybrid and impulsive systems. In order to apply our small-gain theorem to different combinations of non-ISS dynamics, we adopt the method of modifying candidate exponential ISS-Lyapunov functions using ADT and reverse ADT timers. The effect of such modifications on the Lyapunov feedback gains between two interconnected hybrid systems is discussed in detail through a case-by-case study. Third, we consider the problem of stabilizing a switched linear system with a completely unknown disturbance using sampled and quantized state feedback. The switching is assumed to be slow enough in the sense of combined dwell-time and average dwell-time, each individual mode is assumed to be stabilizable, and the data rate is assumed to be large enough but finite. By extending the approach of reachable-set approximation and propagation from an earlier result on the disturbance-free case, we develop a communication and control strategy that achieves a variant of input-to-state stability with exponential decay. An estimate of the disturbance bound is introduced to compensate for the unknown disturbance, and a novel algorithm is designed to adjust the estimate and recover the state when it escapes the range of quantization. Last, motivated by the connection between the minimum data rate needed to stabilize a linear time-invariant system and its topological entropy, we examine a notion of topological entropy for switched systems with a known switching signal. This notion is formulated in terms of the number of initial points such that the corresponding trajectories approximate all trajectories within a certain error, and can be equivalently defined using the number of initial points that are separable up to a certain precision. We first calculate the topological entropy of a switched scalar system based on the active rates of its modes. This approach is then generalized to nonscalar switched linear systems with certain Lie structures to establish entropy bounds in terms of the active rate and eigenvalues of each mode

Switched dynamical systems: Transition model, qualitative theory, and advanced control

Ph.DDOCTOR OF PHILOSOPH

Qualitative Properties of Hybrid Singular Systems

A singular system model is mathematically formulated as a set of coupled differential and algebraic equations. Singular systems, also referred to as descriptor or differential algebraic systems, have extensive applications in power, economic, and biological systems. The main purpose of this thesis is to address the problems of stability and stabilization for singular hybrid systems with or without time delay. First, some su cient conditions on the exponential stability property of both continuous and discrete impulsive switched singular systems with time delay (ISSSD) are proposed. We address this problem for the continuous system in three cases: all subsystems are stable, the system consists of both stable and unstable subsystems, and all subsystems are unstable. For the discrete system, we focus on when all subsystems are stable, and the system consists of both stable and unstable subsystems. The stability results for both the continuous and the discrete system are investigated by first using the multiple Lyapunov functions along with the average-dwell time (ADT) switching signal to organize the jumps among the system modes and then resorting the Halanay Lemma. Second, an optimal feedback control only for continuous ISSSD is designed to guarantee the exponential stability of the closed-loop system. Moreover, a Luenberger-type observer is designed to estimate the system states such that the corresponding closed-loop error system is exponentially stable. Similarly, we have used the multiple Lyapunov functions approach with the ADT switching signal and the Halanay Lemma. Third, the problem of designing a sliding mode control (SMC) for singular systems subject to impulsive effects is addressed in continuous and discrete contexts. The main objective is to design an SMC law such that the closed-loop system achieves stability. Designing a sliding surface, analyzing a reaching condition and designing an SMC law are investigated throughly. In addition, the discrete SMC law is slightly modi ed to eliminate chattering. Last, mean square admissibility for singular switched systems with stochastic noise in continuous and discrete cases is investigated. Sufficient conditions that guarantee mean square admissibility are developed by using linear matrix inequalities (LMIs)