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 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.     

Input/output‐to‐state stability for switched nonlinear systems with unstable subsystems

In this paper, a couple of sufficient conditions for input/output‐to‐state stability (IOSS) of switched nonlinear systems with non‐IOSS subsystems are derived by exploiting the multiple Lyapunov functions (MLFs) method. A state‐norm estimator–based small‐gain theorem is also established for switched interconnected nonlinear systems under some proper switching laws, where the small‐gain property of individual connected subsystems is not required in the whole state space instead only in some subregions of the state space. The state‐norm estimator for the switched system under study is explicitly designed via a constructive procedure by exploiting the MLFs method and the classical small‐gain technique. The presented results permit removal of a technical condition in existing literature, where all subsystems in switched systems are IOSS or some are IOSS. An illustrative example is also provided to illustrate the effectiveness of the theoretical results.     

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.     

Input‐to‐state stability of stochastic port‐Hamiltonian systems using stochastic generalized canonical transformations

As a practically important class of nonlinear stochastic systems, this paper considers stochastic port‐Hamiltonian systems (SPHSs) and investigates the stochastic input‐to‐state stability (SISS) property of a class of SPHSs. We clarify necessary conditions for the closed‐loop system of an SPHS to be SISS. Moreover, we provide a systematic construction of both the SISS controller and Lyapunov function so that the proposed necessary conditions hold. In the main results, the stochastic generalized canonical transformation plays a key role. The stochastic generalized canonical transformation technique enables to design both coordinate transformation and feedback controller with preserving the SPHS structure of the closed‐loop system. Consequently, the main theorem guarantees that the closed‐loop system obtained by the proposed method is SISS against both deterministic disturbance and stochastic noise. Copyright © 2017 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‐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 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     

Input‐to‐state exponents and related ISS for delayed discrete‐time systems with application to impulsive effects

This paper aims to investigate the input‐to‐state exponents (IS‐e) and the related input‐to‐state stability (ISS) for delayed discrete‐time systems (DDSs). By using the method of variation of parameters and introducing notions of uniform and weak uniform M‐matrix, the estimates for 3 kinds of IS‐e are derived for time‐varying DDSs. The exponential ISS conditions with parts suitable for infinite delays are thus established, by which the difference from the time‐invariant case is shown. The exponential stability of a time‐varying DDS with zero external input cannot guarantee its ISS. Moreover, based on the IS‐e estimates for DDSs, the exponential ISS under events criteria for DDSs with impulsive effects are obtained. The results are then applied in 1 example to test synchronization in the sense of ISS for a delayed discrete‐time network, where the impulsive control is designed to stabilize such an asynchronous network to the synchronization.     

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.     

Input/output‐to‐state stability of impulsive switched delay systems

This paper investigates the input/output‐to‐state stability (IOSS) and integral IOSS (iIOSS) of nonlinear impulsive switched delay systems where the switching moments and impulsive moments do not necessarily coincide with each other. Some Razumikhin‐type criteria are presented to guarantee the IOSS and iIOSS of the systems, where both destabilizing and stabilizing effects of switching behavior and impulses are considered simultaneously. The counterpart results for impulsive switched systems without delay can be naturally obtained. Several examples are provided to verify the effectiveness and superiority of the proposed results.     

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/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.     

Adaptive Integrated Guidance and Control Based on Backstepping and Input‐to‐State Stability

This paper presents an approach of integrated guidance and control (IGC) design for interception of maneuvering targets (evaders). An IGC model with uncertainties in the pitch plane is formulated, and, by adopting a backstepping scheme, an adaptive nonlinear IGC approach is developed. Theoretical analysis shows that the design approach makes the line‐of‐sight (LOS) rate be input‐to‐state stable (ISS) with respect to target maneuvers and missile model uncertainties, and the stability of the missile dynamics can be guaranteed as well. The numerical simulation confirms the effectiveness of the proposed design approach.     

Robustness analysis of systems' safety through a new notion of input‐to‐state safety

In this paper, we propose a new robustness notion that is applicable for certifying systems' safety with respect to external disturbance signals. The proposed input‐to‐state safety notion allows us to certify systems' safety in the presence of the disturbances, which is analogous to the notion of input‐to‐state stability for analyzing systems' stability.     

New noise‐to‐state stability and instability criteria for random nonlinear systems

This paper investigates the noise‐to‐state stability and instability criteria for random nonlinear affine systems. Firstly, some new noise‐to‐state stability theorems, which weaken the sufficient conditions in the existing stability criteria on random nonlinear systems, are given by means of the uniformly asymptotically stable function. Secondly, the noise‐to‐state instability definitions are introduced and the sufficient conditions of noise‐to‐state instability are provided based on a new established lemma and the uniformly asymptotically stable function. Finally, some examples show the feasibility of theoretical findings.     

On input‐to‐state stability‐based design for leader/follower formation control with measurement delays

This paper addresses the design of low‐level controllers for leader–follower formations of nonholonomic vehicles in the presence of bounded measurement delays. The concept of input‐to‐state stability is extended to encompass the effect of bounded delays and restrictions on the input. A method is proposed to integrate a Smith predictor in a backstepping design on the basis of nested saturations and nonlinear small‐gain assignment, which allows for time delays in the feedback loop. Robustness analysis under uncertain bounded time delays is provided, and design tradeoffs resulting from the use of bounded controls are discussed. Illustrative simulations are shown to validate the design and robustness analysis in the context of a simple leader–follower trailing control problem. Copyright © 2012 John Wiley & Sons, Ltd.