A new unified input‐to‐state stability criterion for impulsive stochastic delay systems with Markovian switching

In this paper, the problems of the input‐to‐state stability (ISS), the integral input‐to‐state stability (iISS), the stochastic input‐to‐state stability (SISS) and the e^λt(λ>0)‐weighted input‐to‐state stability (e^λt‐ISS) are investigated for nonlinear time‐varying impulsive stochastic delay systems with Markovian switching. We propose one unified criterion for the stabilizing impulse and the destabilizing impulse to guarantee the ISS, iISS, SISS and e^λt‐ISS for such systems. We verify that when the upper bound of the average impulsive interval is given, the stabilizing impulsive effect can stabilize the systems without ISS. We also show that the destabilizing impulsive signal with a given lower bound of the average impulsive interval can preserve the ISS of the systems. In addition, one criterion for guaranteeing the ISS of nonlinear time‐varying stochastic hybrid systems under no impulsive effect is derived. Two examples including one coupled dynamic systems model subject to external random perturbation of the continuous input and impulsive input disturbances are provided to illustrate the effectiveness of the theoretic results developed.     

Input‐to‐state‐stability‐type comparison principles and input‐to‐state stability for discrete‐time dynamical networks with time delays

This paper investigates the input‐to‐state stability (ISS) issue for discrete‐time dynamical networks (DDNs) with time delays. Firstly, a general comparison principle for solutions of DDNs is proposed. Then, based on this general comparison principle, three kinds of ISS‐type comparison principles for DDNs are established, including the comparison principle for input‐to‐state ‐stability, ISS, and exponential ISS. The ISS‐type comparison principles are then used to investigate stability properties related to ISS for three kinds (linear, affine, and nonlinear) of DDNs. It shows that the ISS property of a DDN can be derived by comparing it with a linear or lower‐dimension DDN with known ISS property. By using methods such as variation of parameters, uniform M‐matrix, and the ISS‐type comparison principle, conditions of global exponential ISS for time‐varying linear DDNs with time delays are derived. Moreover, the obtained ISS results for DDNs are extended to the hybrid DDNs with time delays. As one application, the synchronization within an error bound in the sense of ISS is achieved for DDNs with coupling time delays and external disturbances. Finally, two examples are given to illustrate the results. Copyright © 2011 John Wiley & Sons, Ltd.     

Input‐to‐state stability for discrete‐time nonlinear impulsive systems with delays

This paper aims to study the problem of input‐to‐state stability (ISS) for nonlinear discrete impulsive systems with time delays. Razumikhin‐type theorems, which guarantee ISS – asymptotically ISS and exponentially ISS – for the discrete impulsive ones with external disturbance inputs, are established. As applications, numerical examples are given to illustrate the effectiveness of the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

Event‐triggered control via impulses for exponential stabilization of discrete‐time delayed systems and networks

This paper investigates the stabilization issue via event‐triggered controls (ETCs) for discrete‐time delayed systems (DDSs) and networks. Based on the recently proposed ETC scheme for discrete‐time systems without time delays, improved ETC (I‐ETC) and event‐triggered impulsive control (ETIC) are proposed for DDS. The algorithms for ETC, I‐ETC, and ETIC are given respectively to derive criteria of exponential stabilization of DDS. Moreover, the exponential stabilization and stabilization to ISS for discrete‐time delayed networks is achieved by employing the algorithms of ETC and ETIC. The issue of stabilization via ETCs for dynamical networks where different subsystems have different sequences of event instants is solved by introducing the check‐period into ETCs and establishing general ISS estimate of discrete‐time delayed inequality. In order to assess the performances of the control schemes, discussions on nontriviality are given by proposing the concept of rate of control and the function of control cost. Finally, two examples with numerical simulations are presented to demonstrate the effectiveness of theoretical results. From the obtained results on stabilization and the simulations, the ETIC is shown to have clear advantages and well performances than the classical state feedback control, the ETC recently proposed, I‐ETC, and the time‐based impulsive control on aspects of nontriviality, lower rate of control, lower cost of control, and robustness w.r.t. external disturbances.     

Krasovskii and Razumikhin criteria on input‐to‐state stability of discrete‐time time‐varying switched delayed systems

In this article, we are concerned with the problem on input‐to‐state stability (ISS) for discrete‐time time‐varying switched delayed systems. Some Krasovskii and Razumikhin ISS criteria are provided by using the notions of uniformly asymptotically stable (UAS) function and mode‐dependent average dwell time (MDADT). With the help of the concept of UAS function, the advantage of our results in this article is that the coefficients of the first‐order difference inequalities for the mode‐dependent Krasovskii functionals and mode‐dependent Razumikhin functions are allowed to be time‐varying, mode‐dependent, and can even take both positive and negative values, and the whole switched system can be allowed to have both ISS subsystems and non‐ISS subsystems. With the aid of the notion of MDADT, each subsystem can have its own average dwell time. As an application, we also provide an ISS criterion for discrete‐time time‐varying switched delayed Hopfield neural networks with disturbance inputs. Numerical simulations verify the effectiveness of the established criteria.     

Input‐to‐state stability of nonlinear stochastic time‐varying systems with impulsive effects

This paper considers the input‐to‐state stability, integral‐ISS, and stochastic‐ISS for impulsive nonlinear stochastic systems. The Lyapunov function considered in this paper is indefinite, that is, the rate coefficient of the Lyapunov function is time‐varying, which can be positive or negative along time evolution. Lyapunov‐based sufficient conditions are established for ensuring ISS of impulsive nonlinear stochastic systems. Three examples involving one from networked control systems are provided to illustrate the effectiveness of theoretical results obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Input‐to‐state stability of impulsive switched nonlinear time‐delay systems with two asynchronous switching phenomena

This summary addresses the input‐to‐state stability (ISS) and integral ISS (iISS) problems of impulsive switched nonlinear time‐delay systems (ISNTDSs) under two asynchronous switching effects. In our investigated systems, impulsive instants and switching instants do not necessarily coincide with each other. Meanwhile, systems switching signals are not simultaneous with the corresponding controllers switching signals, which will induce instability seriously, and cause many difficulties and challenges. By utilizing methods of Lyapunov‐Krasovskii and Lyapunov‐Razumikhin, mode‐dependent average dwell time approach, and mode‐dependent average impulsive interval technique, some stability criteria are presented for ISNTDSs under two asynchronous switching effects. Our proposed results improve the related existing results on the same topic by removing some restrictive conditions and cover some existing results as special cases. Finally, some simulation examples are presented to illustrate the effectiveness and advantages of our results.     

Tracking control for sampled‐data systems with uncertain time‐varying sampling intervals and delays

In this paper, a solution to the approximate tracking problem of sampled‐data systems with uncertain, time‐varying sampling intervals and delays is presented. Such time‐varying sampling intervals and delays can typically occur in the field of networked control systems. The uncertain, time‐varying sampling and network delays cause inexact feedforward, which induces a perturbation on the tracking error dynamics, for which a model is presented in this paper. Sufficient conditions for the input‐to‐state stability (ISS) of the tracking error dynamics with respect to this perturbation are given. Hereto, two analysis approaches are developed: a discrete‐time approach and an approach in terms of delay impulsive differential equations. These ISS results provide bounds on the steady‐state tracking error as a function of the plant properties, the control design and the network properties. Moreover, it is shown that feedforward preview can significantly improve the tracking performance and an online extremum seeking (nonlinear programming) algorithm is proposed to online estimate the optimal preview time. The results are illustrated on a mechanical motion control example showing the effectiveness of the proposed strategy and providing insight into the differences and commonalities between the two analysis approaches. Copyright © 2009 John Wiley & Sons, Ltd.     

On the pth moment integral input‐to‐state stability and input‐to‐state stability criteria for impulsive stochastic functional differential equations

In this paper, several new Razumikhin‐type theorems for impulsive stochastic functional differential equations are studied by applying stochastic analysis techniques and Razumikhin stability approach. By developing a new comparison principle for stochastic version, some novel criteria of the pth moment integral input‐to‐state stability and input‐to‐state stability are derived for the related systems. The feature of the criteria shows that time‐derivatives of the Razumikhin functions are allowed to be indefinite, even unbounded, which can loosen the constraints of the existing results. Finally, some examples are given to illustrate the usefulness and significance of the theoretical results.     

Input/output‐to‐state stability of nonlinear impulsive delay systems based on a new impulsive inequality

In this paper, input/output‐to‐state stability (IOSS) and integral IOSS (iIOSS) are investigated for nonlinear impulsive systems with delay. Based on a new impulsive inequality, we propose some sufficient criteria for IOSS and iIOSS of impulsive delay systems. It is shown that the obtained results for IOSS and iIOSS are regardless of the length of the impulsive interval and the size of time delay if the impulsive gain satisfies a given condition. In addition, based on the average impulsive interval method, some more useful sufficient conditions are derived for IOSS and iIOSS of impulsive delay systems with persistent large‐scale destabilizing impulses. Furthermore, a relationship is established among the average impulsive interval, impulses, time delay, and the decay of the system without impulses such that the impulsive delay system is input/output‐to‐state stable and integral input/output‐to‐state stable, respectively. Two examples are given to show the validity of the obtained results.     

L2‐Gain Analysis for Synchronization of Stochastic Complex Network with Switched Topology and Time‐Varying Delay

We investigate the exponential stability and L₂‐gain analysis for the synchronization of stochastic complex networks under average dwell time switched topology with consideration of external disturbance, internal noise and fast time‐varying delay in the synchronized process. Based on the proposed stochastic network, a new L₂‐gain synchronization is proposed to solve the mean‐square exponential stable under switched topology with an H_∞ performance from the extrinsic disturbances to the synchronization error. The obtained results are applicable for the fast time‐varying case with larger‐than‐1 delay derivative. Finally, numerical simulations are performed to demonstrate the effectiveness of our strategies.     

Robust Exponential Stability of Discrete‐Time Delay Impulsive Systems with Parametric Uncertainties

This paper investigates robust exponential stability for discrete‐time delay impulsive systems with parametric uncertainties. The parametric uncertainties in the systems are assumed to be time varying and norm bounded. Using Lyapunov functionals, some robust exponential stability criteria are given. It is shown that the time interval between the nearest two impulses should be small enough, i.e., impulses must act frequently, when the impulses are employed to stabilize the original impulse‐free system that is not robustly stable. Conversely, when the original system without impulses is robustly stable, the time interval between the nearest two impulses should be large enough to let the system with impulsive perturbations retain its stability property. It should be noted that this is the first time that impulsive robust exponential stabilization results are given via Lyapunov functionals for discrete‐time uncertain delay impulsive systems. Some examples, including an example which cannot be studied by the existing results, are also presented to illustrate the effectiveness of the obtained results.     

Input‐to‐state stability with respect to inputs and their derivatives

A new notion of input‐to‐state stability involving infinity norms of input derivatives up to a finite order k is introduced and characterized. An example shows that this notion of stability is indeed weaker than the usual ISS . Applications to the study of global asymptotic stability of cascaded non‐linear systems are discussed. Copyright © 2003 John Wiley & Sons, Ltd.     

Delay‐range‐dependent control of nonlinear time‐delay systems under input saturation

This paper describes a delay‐range‐dependent local state feedback controller synthesis approach providing estimation of the region of stability for nonlinear time‐delay systems under input saturation. By employing a Lyapunov–Krasovskii functional, properties of nonlinear functions, local sector condition and Jensen's inequality, a sufficient condition is derived for stabilization of nonlinear systems with interval delays varying within a range. Novel solutions to the delay‐range‐dependent and delay‐dependent stabilization problems for linear and nonlinear time‐delay systems, respectively, subject to input saturation are derived as specific scenarios of the proposed control strategy. Also, a delay‐rate‐independent condition for control of nonlinear systems in the presence of input saturation with unknown delay‐derivative bound information is established. And further, a robust state feedback controller synthesis scheme ensuring L₂ gain reduction from disturbance to output is devised to address the problem of the stabilization of input‐constrained nonlinear time‐delay systems with varying interval lags. The proposed design conditions can be solved using linear matrix inequality tools in connection with conventional cone complementary linearization algorithms. Simulation results for an unstable nonlinear time‐delay network and a large‐scale chemical reactor under input saturation and varying interval time‐delays are analyzed to demonstrate the effectiveness of the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

Compensation of time‐varying input delay for discrete‐time nonlinear systems

We consider general discrete‐time nonlinear systems (of arbitrary nonlinear growth) with time‐varying input delays and design an explicit predictor feedback controller to compensate the input delay. Such results have been achieved in continuous time, but only under the restriction that the delay rate is bounded by unity, which ensures that the input signal flow does not get reversed, namely, that old inputs are not felt multiple times by the plant (because on such subsequent occasions, the control input acts as a disturbance). For discrete‐time systems, an analogous restriction would be that the input delay is non‐increasing. In this work, we do not impose such a restriction. We provide a design and a global stability analysis that allow the input delay to be arbitrary (containing intervals of increase, decrease, or stagnation) over an arbitrarily long finite period of time. Unlike in the continuous‐time case, the predictor feedback law in the discrete‐time case is explicit. We specialize the result to linear time‐invariant systems and provide an explicit estimate of the exponential decay rate. Carefully constructed examples are provided to illustrate the design and analytical challenges. Copyright © 2015 John Wiley & Sons, Ltd.     

Input‐to‐state stability of min‐max MPC scheme for nonlinear time‐varying delay systems

This paper studies the robustness problem of the min–max model predictive control (MPC) scheme for constrained nonlinear time‐varying delay systems subject to bounded disturbances. The notion of the input‐to‐state stability (ISS) of nonlinear time‐delay systems is introduced. Then by using the Lyapunov–Krasovskii method, a delay‐dependent sufficient condition is derived to guarantee input‐to‐state practical stability (ISpS) of the closed‐loop system by way of nonlinear matrix inequalities (NLMI). In order to lessen the online computational demand, the non‐convex min‐max optimization problem is then converted to a minimization problem with linear matrix inequality (LMI) constraints and a suboptimal MPC algorithm is provided. Finally, an example of a truck‐trailer is used to illustrate the effectiveness of the proposed results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Squared‐down passivity–based state synchronization of homogeneous continuous‐time multiagent systems via static protocol in the presence of time‐varying topology

This paper studies state synchronization of homogeneous multiagent systems (MAS) via a static protocol with partial‐state coupling in the presence of a time‐varying communication topology, which includes general time‐varying graphs as well as switching graphs. If the agents are squared‐down passive or squared‐down passifiable (via static output feedback or static input feedforward), then a static protocol can be designed for balanced, time‐varying graphs. Moreover, this static protocol works for arbitrary switching directed graphs if the agents are squared‐down minimum phase with relative degree one. The static protocol is designed for each agent such that state synchronization is achieved without requiring exact knowledge about the time‐varying network.     

On input‐to‐state stability of rigid‐body attitude control with quaternion representation

The concept of input‐to‐state stability (ISS) is important in robust control, as the state of an ISS system subject to disturbances can be stably regulated to a small region around the origin. In this study, the ISS property of the rigid‐body attitude system with quaternion representation is thoroughly investigated. It has been known that the closed loop with continuous controllers is not ISS with respect to arbitrarily small external disturbances. To deal with this problem, hybrid proportional‐derivative controllers with hysteresis are proposed to render the attitude system ISS. The controller is far from new, but it is investigated in a new aspect. To illustrate the applications of the results about ISS, 2 new robust hybrid controllers are designed. In the case of large bounded time‐varying disturbances, the hybrid proportional‐derivative controller is designed to incorporate a saturated high‐gain feedback term, and arbitrarily small ultimate bounds of the state can be obtained; in the case of constant disturbances, a hybrid adaptive controller is proposed, which is robust against small estimate error of inertia matrix. Finally, simulations are conducted to illustrate the effectiveness of the proposed control strategies.     

Exponential Consensus for Nonlinear Multi‐Agent Systems with Communication and Input Delays via Hybrid Control

This paper aims to investigate the exponential leader‐following consensus for nonlinear multi‐agent systems with time‐varying communication and input delays by using hybrid control. Based on the Lyapunov functional method, impulsive differential equation theory and matrix analysis, we show that all the followers can achieve leader‐following consensus with the virtual leader exponentially even if only a fraction of followers can obtain the leader's information. Two classes of exponential consensus criteria as well as the convergence rates for the controlled multi‐agent systems are presented under very relaxed interaction topology conditions, i.e., the directed interaction topology among the followers is only required to have p(p>1) disjoint strong components. Finally, two numerical examples are given to validate the proposed theoretical results.