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.     

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.     

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.     

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.     

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.     

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 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.     

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.     

A complete parameterization of clf‐based input‐to‐state stabilizing control laws

Sontag's formula proves constructively that the existence of a control Lyapunov function implies asymptotic stabilizability. A similar result can be obtained for systems subject to unknown disturbances via input‐to‐state stabilizing control Lyapunov functions (ISS‐clfs) and the input‐to‐state analogue of Sontag's formula. The present paper provides a generalization of the ISS version of Sontag's formula by completely parameterizing all continuous ISS control laws that can be generated by a known ISS‐clf. When a simple inner‐product constraint is satisfied, this parameterization also conveniently describes a large family of ISS controls that solve the inverse‐optimal gain assignment problem, and it is proved that these controls possess Kalman‐type gain margins. Copyright © 2004 John Wiley & Sons, Ltd.     

An internal model approach to autonomous leader/follower trailing for non‐holonomic vehicles

The focus of this paper is on the design of a control architecture of decentralized type for controlling a leader/follower pair of autonomous non‐holonomic vehicles. A fundamental constraint in this trailing control requires that each agent employs local sensor information to process data on the relative position and velocity between its neighbouring vehicles, without relying on global communication with mission control. This constraint poses a challenge in the design of the control system because the reference trajectory to be tracked, which in the case considered in this paper is related to the motion of the leader, is not known a priori. It is shown in the paper that this specific control problem can be approached from the point of view of the internal model paradigm. In particular, once models of the autonomous dynamics of the leader are embedded in a decentralized dynamic controller, the design of the controller can be completed with a robust stabilizer, obtained by using ISS‐gain‐assignment techniques. It is shown that asymptotic convergence of the follower to an arbitrarily small neighbourhood of the desired steady‐state configuration is achieved, despite the presence of possibly large parameter uncertainties, while the motion of each agent remains confined into specified ‘sectors’ to avoid possible collision between neighbouring vehicles during transients. Simulation results are presented to illustrate the design methodology. Copyright © 2006 John Wiley & Sons, Ltd.     

Two‐stage stability control for strict‐feedback systems with input delays and input nonlinear uncertainties

In this paper, a two‐stage control procedure is proposed for stabilization of a class of strict‐feedback systems with unknown constant time delays and nonlinear uncertainties in the input. A nominal controller is first designed to compensate input time delays without considering input nonlinear uncertainties. Extended from backstepping algorithm, input delay compensation is realized by means of predicted states that are computed through integration of cascaded system dynamics, making the nominal closed‐loop system asymptotically stable. Based on the nominal controller presented for the input delay system, a multi‐timescale system is subsequently developed to estimate the unknown input nonlinearity and make the estimate approach the nominal control input as fast as possible. It is proved that the proposed control scheme can make states of the strict‐feedback systems converge to zero and all the signals of the closed‐loop systems are guaranteed to be bounded in the presence of input time delays and nonlinear uncertainties. Simulation verification is carried out to illuminate the effectiveness of the proposed control approach.     

Dynamic controllers for local input‐to‐state stabilization of discrete‐time linear parameter‐varying systems with delay and saturating actuators

We address the design of dynamic parameter‐dependent controllers with antiwindup action to locally stabilize in the input‐to‐state sense a class of discrete‐time linear parameter‐varying (LPV) systems. Such a class consists of systems with delayed state, saturating actuators, and subject to energy bounded disturbances. Moreover, the interval time‐varying delay can have a limited variation rate between two consecutive instants allowing to achieve less conservative design conditions. Differently from other conditions in the literature, the proposed convex synthesis methods allow to design dynamic controllers of different orders. Additionally, the user can choose to feed back only the current output of the system or its delayed ones. Thanks to the embedded (parameter dependent) antiwindup action, it is possible, for instance, to enlarge the region of admissible initial conditions or the maximum admissible disturbance energy. To illustrate the efficiency of our approach, we present numerical examples to compare with other methods from the literature.     

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.     

Dynamic self‐triggered output‐feedback control for nonlinear stochastic systems with time delays

In this paper, the dynamic self‐triggered output‐feedback control problem is investigated for a class of nonlinear stochastic systems with time delays. To reduce the network resource consumption, the dynamic event‐triggered mechanism is implemented in the sensor‐to‐controller channel. Criteria are first established for the closed‐loop system to be stochastically input‐to‐state stable under the event‐triggered mechanism. Furthermore, sufficient conditions are given under which the closed‐loop system with dynamic event‐triggered mechanism is almost surely stable, and the output‐feedback controller as well as the dynamic event‐triggered mechanism are co‐designed. Moreover, a dynamic self‐triggered mechanism is proposed such that the nonlinear stochastic system with the designed output‐feedback controller is stochastically input‐to‐state stable and the Zeno phenomenon is excluded. Finally, a numerical example is provided to illustrate the effectiveness of proposed dynamic self‐triggered output‐feedback control scheme.     

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.     

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/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.     

Observer‐predictor feedback for consensus of discrete‐time multiagent systems with both state and input delays

This article is concerned with the consensus problem for discrete‐time multiagent systems with both state and input delays. Single observer‐predictor‐based protocols and multiple observer‐predictors feedback protocols are simultaneously established to predict the future state such that the input delay that can be arbitrarily large yet bounded is completely compensated. It is shown that the consensus of the multiagent system can be achieved by the single/multiple observer‐predictors feedback protocol. Moreover, sufficient conditions guaranteeing the consensus of the multiagent system are provided in terms of the stability of some simple observer‐error systems, and the separation principle is discovered. Finally, a numerical example is worked out to illustrate the effectiveness of the proposed approaches.