Revisiting the IISS Small‐Gain Theorem through Transient Plus ISS Small‐Gain Regulation

Recently, the small‐gain theorem for input‐to‐state stable (ISS) systems has been extended to the class of integral input‐to‐state stable (iISS) systems. Feedback connections of two iISS systems are robustly stable with respect to disturbance if an extended small‐gain condition is satisfied. It has been proved that at least one of the two iISS subsystems needs to be ISS for guaranteeing globally asymptotic stability and iISS of the overall system. Making use of this necessary condition for the stability, this paper gives a new interpretation to the iISS small gain theorem as transient plus ISS small‐gain regulation. The observation provides useful information for designing and analyzing nonlinear control systems based on the iISS small‐gain theorem.     

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.     

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 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 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.     

Reduction of the small gain condition for large‐scale interconnections

The small gain condition is sufficient for input‐to‐state stability (ISS) of interconnected systems. However, verification of the small gain condition requires large amount of computations in the case of a large size of the system. To facilitate this procedure, we aggregate the subsystems and the gains between the subsystems that belong to certain interconnection patterns (motifs) by using three heuristic rules. These rules are based on three motifs: sequentially connected nodes, nodes connected in parallel, and almost disconnected subgraphs. Aggregation of these motifs keeps the structure of the mutual influences between the subsystems in the network. Furthermore, fulfillment of the reduced small gain condition implies ISS of the large network. Thus, such reduction allows to decrease the number of computations needed to verify the small gain condition. Finally, an ISS‐Lyapunov function for the large network can be constructed using the reduced small gain condition. Applications of these rules is illustrated on an example. Copyright © 2013 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.     

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.     

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.     

Fractional‐order–dependent global stability analysis and observer‐based synthesis for a class of nonlinear fractional‐order systems

This paper focuses on proposing novel conditions for stability analysis and stabilization of the class of nonlinear fractional‐order systems. First, by considering the class of nonlinear fractional‐order systems as a feedback interconnection system and applying small‐gain theorem, a condition is proposed for L₂‐norm boundedness of the solutions of these systems. Then, by using the Mittag‐Leffler function properties, we show that satisfaction of the proposed condition proves the global asymptotic stability of the class of nonlinear fractional‐order systems with fractional order lying in (0.5, 1) or (1.5, 2). Unlike the Lyapunov‐based methods for stability analysis of fractional‐order systems, the new condition depends on the fractional order of the system. Moreover, it is related to the H_∞‐norm of the linear part of the system and it can be transformed to linear matrix inequalities (LMIs) using fractional‐order bounded‐real lemma. Furthermore, the proposed stability analysis method is extended to the state‐feedback and observer‐based controller design for the class of nonlinear fractional‐order systems based on solving some LMIs. In the observer‐based stabilization problem, we prove that the separation principle holds using our method and one can find the observer gain and pseudostate‐feedback gain in two separate steps. Finally, three numerical examples are provided to demonstrate the advantage of the novel proposed conditions with the previous results.     

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‐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 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.     

ℒ︁2‐Stabilization of continuous‐time linear systems with saturating actuators

This paper addresses the problem of controlling a linear system subject to actuator saturations and to ??₂‐bounded disturbances. Linear matrix inequality (LMI) conditions are proposed to design a state feedback gain in order to satisfy the closed‐loop input‐to‐state stability (ISS) and the closed‐loop finite gain ??₂ stability. By considering a quadratic candidate Lyapunov function, two particular tools are used to derive the LMI conditions: a modified sector condition, which encompasses the classical sector‐nonlinearity condition considered in some previous works, and Finsler's Lemma, which allows to derive stabilization conditions which are adapted to treat multiple objective control optimization problems in a potentially less conservative framework. Copyright © 2006 John Wiley & Sons, Ltd.     

Separation principle for quasi‐one‐sided Lipschitz nonlinear systems with time delay

In this article, we address the problem of output stabilization for a class of nonlinear time‐delay systems. First, an observer is designed for estimating the state of nonlinear time‐delay systems by means of quasi‐one‐sided Lipschitz condition, which is less conservative than the one‐sided Lipschitz condition. Then, a state feedback controller is designed to stabilize the nonlinear systems in terms of weak quasi‐one‐sided Lipschitz condition. Furthermore, it is shown that the separation principle holds for stabilization of the systems based on the observer‐based controller. Under the quasi‐one‐sided Lipschitz condition, state observer and feedback controller can be designed separately even though the parameter (A,C) of nonlinear time‐delay systems is not detectable and parameter (A,B) is not stabilizable. Finally, a numerical example is provided to verify the efficiency of the main results.