Switched and Impulsive Systems

Modern complex large‐scale impulsive systems involve multiple modes of operation placing stringent demands on controller analysis of increasing complexity. In analyzing these large‐scale systems, it is often desirable to treat the overall impulsive system as a collection of interconnected impulsive subsystems. Solution properties of the large‐scale impulsive system are then deduced from the solution properties of the individual impulsive subsystems and the nature of the impulsive system interconnections. In this paper, we develop vector dissipativity theory for large‐scale impulsive dynamical systems. Specifically, using vector storage functions and vector hybrid supply rates, dissipativity properties of the composite large‐scale impulsive systems are shown to be determined from the dissipativity properties of the impulsive subsystems and their interconnections. Furthermore, extended Kalman‐Yakubovich‐Popov conditions, in terms of the impulsive subsystem dynamics and interconnection constraints, characterizing vector dissipativeness via vector system storage functions, are derived. Finally, these results are used to develop feedback interconnection stability results for large‐scale impulsive dynamical systems using vector Lyapunov functions.

In this paper we develop Lyapunov and invariant set stability theorems for non-linear impulsive dynamical systems. Furthermore, we generalize dissipativity theory to non-linear dynamical systems with impulsive effects. Specifically, the classical concepts of system storage functions and supply rates are extended to impulsive dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous-time system dynamics and dissipated energy over the resetting instants. Furthermore, extended Kalman‐Yakubovich‐Popov conditions in terms of the impulsive system dynamics characterizing dissipativeness via system storage functions are derived. Finally, the framework is specialized to passive and non-expansive impulsive systems to provide a generalization of the classical notions of passivity and non-expansivity for non-linear impulsive systems. These results are used in the second part of this paper to develop extensions of the small gain and positivity theorems for feedback impulsive systems as well as to develop optimal hybrid feedback controllers.

The stated objective of this book is to "develop a unified analysis and control design framework for impulsive and hybrid dynamical systems using a Lyapunov and dissipative systems approach." The book is organized into 13 chapters and includes an appendix section. Some of the topics covered include: stability theory for time-variant and time-varying impulsive systems; extending the notion of dissipative dynamical systems to impulsive dissipative dynamical systems; vector dissipativity for large-scale nonlinear impulsive dynamical systems; the stability of feedback interconnections of dissipative impulsive dynamical systems; qualitative analyses of special classes of impulsive hybrid feedback control systems; disturbance rejection control and robust control for nonlinear impulsive dynamical systems with bounded exogenous disturbances; and Poincare's theorem adapted to left-continuous dynamical systems. This book fills a void in the area of systems research and is a welcome addition to the literature on hybrid and impulsive systems. The book is well-organized, well written, and rigorous in the development of the subject on hand. It would be of great use to many researchers within the control systems community and could be used as the basis for a graduate course om control systems.

In this article, the exponential stability of impulsive control systems with time delay is studied. By using the average impulsive interval method, some sufficient Lyapunov-based conditions are established for the stability of impulsive time-delay systems, and the impacts of delay on the stability analysis method are further revealed. It is interesting to show that some unstable impulsive time-delay systems may be stabilized by increasing the time delay in continuous dynamics. More interestingly, it is proved that along with the increase of the delay within a certain range, the convergence rate of such impulsive time-delay systems also increases correspondingly. Further, a strict comparison principle for impulsive control systems with delay is established. Then by utilizing this comparison principle, it can be shown that for some stable impulsive systems with delay, under certain conditions, the stability is robust against any large but bounded delay. Compared with the previous results on delay-free impulsive systems, some potential impacts of delay on the stability are investigated. Particularly, the obtained results are extended to the case of impulsive control systems with hybrid impulses, which contain both stabilizing impulses and destabilizing impulses. Three illustrative examples are presented to reveal the potential impacts of delay on the stability of impulsive control systems.

In this paper, we discuss Lyapunov regularity and stability for linear non-instantaneous impulsive differential systems. In particular, we give sufficient conditions to guarantee any non-trivial solution has a finite Lyapunov exponent and we prove an impulsive system is stable using the Lyapunov exponent for the solution. A new version of Perron’s theorem is given by introducing the associated adjoint impulsive system and some criteria for the existence of non-uniform exponential behaviour are given. In addition, we present a stability result for a small perturbed nonlinear impulsive system when the linear impulsive system admits a non-uniform exponential contraction. Finally, we give a bound for the regularity coefficient.

In this paper, we present partial stability results; that is, stability with respect to part of the system's state, for nonlinear impulsive dynamical systems. Using these results, we provide unification between partial stability theory for (autonomous) state-dependent impulsive dynamical systems and stability theory for (non-autonomous) time-dependent impulsive dynamical systems. This unification allows for stability theory of time-dependent impulsive systems to be presented as a special case of partial stability theory for state-dependent impulsive systems.

This paper aims to review some uniform stability results for impulsive systems. For the review, we classify the models of impulsive systems into time-based impulsive systems and state-based ones, including continuous-time impulsive systems, discrete-time impulsive systems, stochastic impulsive systems, and impulsive hybrid systems. According to these models, we review, respectively, the related stability concepts and some representative results focused on uniform stability, including the results on uniform asymptotic stability, input-to-state stability (ISS), KLL -stability (uniform stability expressed by KLL -functions), event-stability, and event-ISS. And we formulate some questions for those not fully developed aspects on uniform stability at each subsection.

Some recent results of analysis and control for impulsive systems

An invariance principle for nonlinear hybrid and impulsive dynamical systems

RationaleCentral aspects of alcohol use disorder (AUD) are the irresistible desire for alcohol and impaired control over its intake. According to the triadic neurocognitive model of addiction, this arises from aberrant functioning of different neural and cognitive systems: an impulsive system, a reflective system, and the abnormal dynamics between both systems based on an insular-dependent system.ObjectivesIn this study, we examined the effects of a single dose of nalmefene on resting-state functional connectivity (rsFC) patterns within and between these addiction-related neural systems in AUD.MethodsNon-treatment seeking participants with AUD (N = 17; 19–66 years, 6 female) took part in a randomized, placebo-controlled, double-blind, crossover study and received either a single dose of 18 mg nalmefene or a placebo. Using seed-based correlation analyses on resting‐state functional magnetic resonance imaging data, we examined the effects of nalmefene on key nodes related to the (1) impulsive system; (2) reflective system; (3) salience network; and (4) default mode network.ResultsUnder nalmefene, participants showed reduced rsFC between components of the impulsive system (Nucleus accumbens–putamen/pallidum/insula). Reduced rsFC was found between elements of the reflective system and impulsive system (orbitofrontal cortex–insula/putamen/pallidum), salience network (orbitofrontal cortex–insula/inferior frontal gyrus), and default mode network (lateral prefrontal cortex–precuneus/cuneus). Components of the salience network showed both increased (anterior cingulate cortex) and decreased (insular cortex) rsFC to elements of the reflective system.ConclusionA single dose of nalmefene impacts rsFC and alters the interaction between key nodes of addiction-related neural systems in non-treatment seeking participants with AUD. Nalmefene may normalize rsFC patterns by weakening the impulsive system while strengthening the reflective system.Trial registration: clinicaltrials.gov: NCT02372318.

Impulsive systems are widely used in metallurgy, chemical, and oil and gas industries, heat power engineering, irrigation and other industries. Periodic nature of the discrete control devices operation is the reason for the appearance of pure delay in the control channels. Methods for studying linear impulsive time-delay systems in synchronous-in-phase modes of operation of discrete devices are developed with sufficient completeness. At the same time, the methods of modeling and research of multiple-time-scale, multivariable, nonlinear impulsive time-delay systems require their further development. In this paper we propose a method for modeling impulsive time-delay systems based on dynamic graphs. The result of simulation of pulse systems with delay based on dynamic graphs is completely the same as the results obtained by the modified Z-transformation. In the case of one-dimensional systems, calculations are much easier to perform on the basis of the proposed method of dynamic graphs. The application of the modified Z-transformation for analytical studies is particularly problematic in the case of multi-dimensional pulse systems with delay, as it is accompanied by difficulties of a fundamental nature due to the structural complexity and the problem of formalized mathematical description of the processes using the traditional approach.

Vector measure for explicit nonlinear impulsive system of glycerol bioconversion in fed-batch cultures and its parameter identification

General description of impulsive differential systems linear systems stability of solutions periodic and almost periodic impulsive systems integral sets of impulsive systems optimal control in impulsive systems asymptotic study of oscillations in impulsive systems a periodic and almost periodic impulsive system.

We provide discrete abstractions of impulsive systems on Banach spaces. Thereby we explicitly allow infinite-dimensional state and input spaces, which makes it possible to cover a crucially important class of dynamical systems modeled by partial differential equations with jumps, referred to as impulsive evolution systems. Using the notion of the so-called alternating simulation function, we prove, under an incremental stability assumption, that there exists an approximate alternating simulation relation between the impulsive system and its discrete abstraction. We also provide conditions for the existence of an approximate alternating bisimulation relation. A notable feature of our work is that we propose a time-varying alternating simulation function that allows the construction of discrete abstractions for a broad class of impulsive systems in which both the flow and jumps are possibly unstable. Our method also covers the classes of time-varying impulsive systems and impulsive systems with an output map.

We develop Lyapunov and invariant set stability theorems for nonlinear impulsive dynamical systems. Furthermore, we generalize the dissipativity theory to nonlinear dynamical systems with impulsive effects. Specifically, the classical concepts of system storage functions and supply rates are extended to impulsive dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous-time system dynamics, and dissipated energy over the resetting instances. Furthermore, extended Kalman-Yakubovich-Popov conditions in terms of the impulsive system dynamics characterizing dissipativeness in terms of system storage functions are derived. Finally, the framework is specialized to passive and nonexpansive impulsive systems to provide a generalization of the classical notions of passivity and nonexpansivity.