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.     

Semi‐global stabilization via linear sampled‐data output feedback for a class of uncertain nonlinear systems

This paper develops a systematic design scheme to construct a linear sampled‐data output feedback controller that semi‐globally asymptotically stabilizes a class of uncertain systems with both higher‐order and linear growth nonlinearities. To deal with the uncertain coefficients in the systems, a robust state feedback stabilizer and a reduced‐order sampled‐data observer, both in the linear form, are constructed and then integrated together. The semi‐global attractivity and local stability are delicately proved by carefully selecting a scaling gain using the output feedback domination approach and a sampling period sufficiently small to restrain the state growth under a zero‐order‐holder input. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive output feedback control for a class of nonlinear uncertain systems with quantized input signal

In this paper, adaptive output feedback control for a class of nonlinear systems with quantized input is investigated. The nonlinearities of the nonlinear systems under consideration are assumed to satisfy linear growth condition on the unmeasured states multiplied by unknown growth rate and output polynomial function. By developing a dynamic high‐gain observer, a linear‐like output feedback controller is constructed, with which it is proved that the output of the quantized control system can be steered to within an arbitrarily small residual set while keeping all the other closed loop states bounded. In particular, if the growth rate is known, it is proved that all the states of the system can be steered to within an arbitrarily small neighborhood of the origin. Copyright © 2016 John Wiley & Sons, Ltd.     

Resilient decentralized stabilization of interconnected time‐delay systems with polytopic uncertainties

Decentralized delay‐dependent local stability and resilient feedback stabilization methods are developed for a class of linear interconnected continuous‐time systems. The subsystems are time‐delay plants which are subjected to convex‐bounded parametric uncertainties and additive feedback gain perturbations while allowing time‐varying delays to occur within the local subsystems and across the interconnections. The delay‐dependent local stability conditions are established at the subsystem level through the construction of appropriate Lyapunov–Krasovskii functional. We characterize decentralized linear matrix inequalities (LMIs)‐based delay‐dependent stability conditions by deploying an injection procedure such that every local subsystem is delay‐dependent robustly asymptotically stable with an γ‐level ??₂‐gain. Resilient decentralized state‐feedback stabilization schemes are designed, which takes into account additive gain perturbations such that the family of closed‐loop feedback subsystems enjoys the delay‐dependent asymptotic stability with a prescribed γ‐level ??₂‐gain for each subsystem. The decentralized feedback gains are determined by convex optimization over LMIs. All the developed results are tested on representative examples. Copyright © 2010 John Wiley & Sons, Ltd.     

A test for stability robustness of linear time‐varying systems utilizing the linear time‐invariant ν‐gap metric

A stability robustness test is developed for internally stable, nominal, linear time‐invariant (LTI) feedback systems subject to structured, linear time‐varying uncertainty. There exists (in the literature) a necessary and sufficient structured small gain condition that determines robust stability in such cases. In this paper, the structured small gain theorem is utilized to formulate a (sufficient) stability robustness condition in a scaled LTI ν‐gap metric framework. The scaled LTI ν‐gap metric stability condition is shown to be computable via linear matrix inequality techniques, similar to the structured small gain condition. Apart from a comparison with a generalized robust stability margin as the final part of the stability test, however, the solution algorithm implemented to test the scaled LTI ν‐gap metric stability robustness condition is shown to be independent of knowledge about the controller transfer function (as opposed to the LMI feasibility problem associated with the scaled small gain condition which is dependent on knowledge about the controller). Thus, given a nominal plant and a structured uncertainty set, the stability robustness condition presented in this paper provides a single constraint on a controller (in terms of a large enough generalized robust stability margin) that (sufficiently) guarantees to stabilize all plants in the uncertainty set. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Finite‐time stabilization of stochastic nonlinear systems with SiISS inverse dynamics

This paper investigates the finite‐time control problem for a class of stochastic nonlinear systems with stochastic integral input‐to‐state stablility (SiISS) inverse dynamics. Motivated by finite‐time stochastic input‐to‐state stability and the concept of SiISS using Lyapunov functions, a novel finite‐time SiISS using Lyapunov functions is introduced firstly. Then, by adopting this novel finite‐time SiISS small‐gain arguments, using the backstepping technique and stochastic finite‐time stability theory, a systematic design and analysis algorithm is proposed. Given the control laws that guarantee global stability in probability or asymptotic stability in probability, our design algorithm presents a state‐feedback controller that can ensure the solution of the closed‐loop system to be finite‐time stable in probability. Finally, a simulation example is given to demonstrate the effectiveness of the proposed control scheme. Copyright © 2017 John Wiley & Sons, Ltd.     

Peaking free output‐feedback exact tracking of uncertain nonlinear systems via dwell‐time and norm observers

The control of uncertain nonlinear systems by high‐gain observer based output feedback is addressed. Two tracking sliding mode controllers are designed for a broad class of uncertain nonlinear systems with arbitrary relative degree and unmatched polynomial nonlinearities in the unmeasured states. The proposed strategies are based either on dwell‐time for control activation or on simple norm state observers to remove the peaking phenomenon related with high‐gain observers, depending on the nonlinearity growth conditions. In contrast with previous works, exact tracking is also achieved by means of a switching strategy based on locally exact differentiators. Global or semi‐global stability is proved by using Lyapunov theory and on small‐gain analysis. Simulations show that the proposed methodologies provide better and uniform transient behavior, larger regions of attraction, performance recovery with significantly smaller observer gains and good robustness properties with respect to exogenous disturbances and measurement noise. Copyright © 2011 John Wiley & Sons, Ltd.     

Interconnected jumping time‐delay systems: Mode‐dependent decentralized stability and stabilization

In this paper, we deal with the problems of mode‐dependent decentralized stability and stabilization with ??_∞ performance for a class of continuous‐time interconnected jumping time‐delay systems. The jumping parameters are governed by a finite state Markov process and the delays are unknown time‐varying and mode‐dependent within interval. The interactions among subsystems satisfy quadratic bounding constraints. To characterize mode‐dependent local stability behavior, we employ an improved Lyapunov–Krasovskii functional at the subsystem level and express the stability conditions in terms of linear matrix inequalities (LMIs). A class of local decentralized state‐feedback controllers is developed to render the closed‐loop interconnected jumping system stochastically stable. Then, we extend the feedback strategy to dynamic observer‐based control and establish the stochastic stabilization via LMIs. It has been established that the developed results encompass several existing results as special cases which are illustrated by simulation of examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Direct Adaptive Fuzzy Backstepping Control for Stochastic Nonlinear SISO Systems with Unmodeled Dynamics

This paper discusses the input‐to‐state practical stability (ISpS) problem for a class of stochastic strict‐feedback systems which possess dynamic disturbances, unstructured uncertainties and unmodeled dynamics. The uncertain terms not only depend on the measurable output, but also are related with other unmeasurable states of the system. In the backstepping design, we use fuzzy logic systems directly to approach unknown control signals rather than unknown functions. A main advantage of the direct control method is that for an nth order strict‐feedback stochastic system, only four online parameters are needed. Moreover, it is proved that the closed‐loop system is ISpS in probability by using a stochastic small‐gain approach. Two simulation examples illustrate the effectiveness of the proposed scheme.     

A new methodology to compute stabilizing control laws for continuous‐time LTV systems

This paper is devoted to the problem of computing control laws for the stabilization of continuous‐time linear time‐varying systems. First, a necessary and sufficient condition to assess the stability of a linear time‐varying system based on the norm of the transition matrix computed over a sequence of successive finite‐time intervals is proposed. A link with a stability condition for an equivalent discrete‐time model is also established. Then, 3 approaches for the computation of stabilizing state‐feedback gains are proposed: a continuous‐time technique, ie, directly derived from the stability condition, not suitable for numerical implementation; a method based on the stabilization of the discrete‐time equivalent model along with a transformation to generate the desired continuous‐time gain; and the computation of stabilizing gains for a set of periodic discrete‐time systems. Finally, by adapting one of the existing methods for the stabilization of periodic discrete‐time systems, an algorithm for the computation of a stabilizing state‐feedback continuous‐time gain is proposed. A numerical example illustrates the validity of the technique.     

A Lyapunov‐based small‐gain approach on design of triggering conditions in event‐triggered control systems

This paper studies a Lyapunov‐based small‐gain approach on design of triggering conditions in event‐triggered control systems. The event‐triggered control closed‐loop system is formulated as a hybrid system model. Firstly, by viewing the event‐triggered control closed‐loop system as a feedback connection of two smaller hybrid subsystems, the Lyapunov‐based small‐gain theorems for hybrid systems are applied to design triggering conditions. Then, a new class of triggering condition, the safe, adjustable‐type triggering condition, is proposed to tune the parameters of triggering conditions by practical regulations. This is conducive to break the restriction of the conservation of theoretical results and improve the practicability of event‐triggered control strategy. Finally, a numerical example is given to illustrate the efficiency and the feasibility of the proposed results. Copyright © 2015 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.     

On the relation between absolute stability with respect to a set of linear time-varying elements and a set of nonlinear time-varying elements

Consider two feedback systems with identical plants. SystemLhas a time-varying linear gain and systemNhas a time-varying nonlinear gain; both gains satisfy the same sector condition. The absolute stability ofLimplies that ofNin the case of stability in the sense of Lagrange and quasi-asymptotic stability in the large; the implication does not hold in the case of stability in the sense of Lyapunov, as is shown by an example.     

Time‐varying feedback for stabilization in prescribed finite time

This paper provides a time‐varying feedback alternative to control of finite‐time systems, which is referred to as “prescribed‐time control,” exhibiting several superior features: (i) such time‐varying gain–based prescribed‐time control is built upon regular state feedback rather than fractional‐power state feedback, thus resulting in smooth (C^m) control action everywhere during the entire operation of the system; (ii) the prescribed‐time control is characterized with uniformly prespecifiable convergence time that can be preassigned as needed within the physically allowable range, making it literally different from not only the traditional finite‐time control (where the finite settling time is determined by a system initial condition and a number of design parameters) but also the fixed‐time control (where the settling time is subject to certain constraints and thus can only be specified within the corresponding range); and (iii) the prescribed‐time control relies only on regular Lyapunov differential inequality instead of fractional Lyapunov differential inequality for stability analysis and thus avoids the difficulty in controller design and stability analysis encountered in the traditional finite‐time control for high‐order systems.     

Synchronization of A Class of Uncertain Chaotic Systems with Lipschitz Nonlinearities Using State‐Feedback Control Design: A Matrix Inequality Approach

This paper proposes a new state‐feedback stabilization control technique for a class of uncertain chaotic systems with Lipschitz nonlinearity conditions. Based on Lyapunov stabilization theory and the linear matrix inequality (LMI) scheme, a new sufficient condition formulated in the form of LMIs is created for the chaos synchronization of chaotic systems with parametric uncertainties and external disturbances on the slave system. Using Barbalat's lemma, the suggested approach guarantees that the slave system synchronizes to the master system at an asymptotical convergence rate. Meanwhile, a criterion to find the proper feedback gain vector F is also provided. A new continuous‐bounded nonlinear function is introduced to cope with the disturbances and uncertainties and obtain a desired control performance, i.e. small steady‐state error and fast settling time. Several criteria are derived to guarantee the asymptotic and robust stability of the uncertain master–slave systems. Furthermore, the proposed controller is independent of the order of the system's model. Numerical simulation results are displayed with an expected satisfactory performance compared to the available methods.     

Analysis and control of switched linear systems via modified Lyapunov–Metzler inequalities

This paper addresses analysis and switching control problems of continuous/discrete‐time switched linear systems. A particular class of matrix inequalities, the so‐called Lyapunov–Metzler inequalities, will be modified to provide conditions for stability analysis and output feedback control synthesis under a relaxed min‐switching logic. The switching rule combined with switching output feedback controllers will be designed to stabilize the switched system and satisfy a prespecified gain performance. The proposed analysis and switching control approach could refrain frequent switches commonly observed in min‐switching based designs. The effectiveness of the proposed approach will be illustrated through numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

New results on stability analysis and stabilization of time‐delay continuous Markovian jump systems with partially known rates matrix

In this note, the problems of stability analysis and controller synthesis of Markovian jump systems with time‐varying delay and partially known transition rates are investigated via an input–output approach. First, the system under consideration is transformed into an interconnected system, and new results on stochastic scaled small‐gain condition for stochastic interconnected systems are established, which are crucial for the problems considered in this paper. Based on the system transformation and the stochastic scaled small‐gain theorem, stochastic stability of the original system is examined via the stochastic version of the bounded realness of the transformed forward system. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a precise approximation of the time‐varying delay and the new result on the stochastic scaled small‐gain theorem. The proposed stability condition is demonstrated to be much less conservative than most existing results. Moreover, the problem of stabilization is further solved with an admissible controller designed via convex optimizations, whose effectiveness is also illustrated via numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Semi‐global finite‐time observers for multi‐output nonlinear systems

Finite‐time stability is investigated for nonlinear systems, which satisfy uniqueness of solution. First, a new sufficient condition for local finite‐time stability is presented. Next, by using the high‐gain observers and carefully selecting the homogeneity powers and weights, the problem of semi‐global and finite‐time stable observers is studied for multi‐output nonlinear systems with uniform observability and a triangular structure. Then, a design procedure is worked out for such observers. Finally, two numerical examples further verify the validity of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.     

输出饱和线性系统的稳定性及L2增益性能

考虑输出饱和线性系统的稳定性以及L