Delay‐dependent Stability and Static Output Feedback Control of 2‐D Discrete Systems with Interval Time‐varying Delays

This paper is concerned with the problems of delay‐dependent stability and static output feedback (SOF) control of two‐dimensional (2‐D) discrete systems with interval time‐varying delays, which are described by the Fornasini‐Marchesini (FM) second model. The upper and lower bounds of delays are considered. Applying a new method of estimating the upper bound on the difference of Lyapunov function that does not ignore any terms, a new delay‐dependent stability criteria based on linear matrix inequalities (LMIs) is derived. Then, given the lower bounds of time‐varying delays, the maximum upper bounds in the above LMIs are obtained through computing a convex optimization problem. Based on the stability criteria, the SOF control problem is formulated in terms of a bilinear matrix inequality (BMI). With the use of the slack variable technique, a sufficient LMI condition is proposed for the BMI. Moreover, the SOF gain can be solved by LMIs. Numerical examples show the effectiveness and advantages of our results.     

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.     

Robust feedback stabilization using high‐gain observer via event triggering

In this article, the problem of robust output feedback stabilization of single‐input single‐output nonlinear systems is studied in the event‐triggering framework. In this work, an event‐triggered output feedback law based on a high‐gain observer is constructed, which guarantees the stability of closed‐loop system. First, the high‐gain observer with a triggering scheme is designed to estimate the plant state in the presence of external disturbances subject to any satisfactory accuracy of the estimation error. The observer‐based triggering mechanism decides the transmission of plant output to the observer by observing a certain event condition. Similarly, another triggering mechanism is designed using the estimated state of observer that triggers the control signal to be updated only when it is satisfied. Under this proposed event‐triggering framework, the stability of closed‐loop system is then analyzed. Here, we provide the simplified design technique, in which the high‐gain parameter and the triggering thresholds can be selected independently to achieve any desired bound for the plant trajectory. The results are finally demonstrated through simulation of a numerical example.     

A New Observer‐Type Consensus Protocol for Linear Multi‐Agent Dynamical Systems

The consensus problem is investigated in this paper for a class of multi‐agent systems with general linear node dynamics and directed communication topologies. A new distributed observer‐type consensus protocol is designed based only on the relative output measurements of neighboring agents. Compared with existing observer‐type protocols, the one presented here does not require information about the relative states of the observers. Tools from small gain theory and matrix analysis, some sufficient conditions are obtained for achieving consensus in such multi‐agent systems where the underlying network topology contains a directed spanning tree. Finally, some numerical examples including an application in low‐Earth‐orbit satellite formation flying are provided to illustrate the theoretical results.     

Gain scheduled control for discrete‐time systems depending on bounded rate parameters

In this paper we consider a linear, discrete‐time system depending multi‐affinely on uncertain, real time‐varying parameters. A new sufficient condition for the stability of this class of systems, in terms of a feasibility problem involving linear matrix inequalities (LMIs), is obtained under the hypothesis that a bound on the rate of variation of the parameters is known. This condition, obtained by the aid of parameter dependent Lyapunov functions, obviously turns out to be less restrictive than that one obtained via the classical quadratic stability (QS) approach, which guarantees stability in presence of arbitrary time‐varying parameters. An important point is that the methodology proposed in this paper may result in being less conservative than the classical QS approach even in the absence of an explicit bound on the parameters rate of variation. Concerning the synthesis context, the design of a gain scheduled compensator based on the above approach is also proposed. It is shown that, if a suitable LMI problem is feasible, the solution of such problem allows to design an output feedback gain scheduled dynamic compensator in a controller‐observer form stabilizing the class of systems which is dealt with. The stability conditions are then extended to take into account L₂ performance requirements. Some numerical examples are carried out to show the effectiveness and to investigate the computational burden required by the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Observer‐Based Consensus Tracking for Nonlinear Multi‐Agent Systems With Intermittent Communications

This paper addresses the observer‐based consensus tracking problem of multi‐agent systems with intermittent communications. The agent dynamics are modeled as general linear systems with Lipschitz nonlinearity. Under the assumption that each agent can intermittently share its relative output with neighbors, a class of an observer‐type protocol is proposed, and the consensus tracking problem can be converted further into the stability problem of the nonlinear switching systems. Using a combined tool from M matrix theory, switching theory and the averaging approach, a multi‐step algorithm is presented to construct the observer gains and protocol parameters, and the sufficient criteria established not only can ensure the state estimates convergence to the real values but also can guarantee the follower states synchronize to those of the leader. The obtained results reveal the relationships among the communication rate, the convergence rate, and the dwell time of switching topologies. Finally, the theoretical findings are validated by a numerical example.     

NON‐PDC Observer‐Based T‐S Fuzzy Tracking Controller Design and its Application in CHAOS Control

In many mechanical devices with chaotic behavior, stabilizing unstable periodic orbits (UPOs) of the system has positive effects in the lifetime and effectiveness of these devices. In this study, a new non‐parallel distributed compensation (non‐PDC) observer‐based tracking controller is presented for Takagi–Sugeno fuzzy systems to control the chaotic behavior of such systems. Asymptotic stability synthesis of the closed‐loop system is investigated using a fuzzy Lyapunov function to derive less conservative conditions than common quadratic Lyapunov function‐based approaches. To tackle the main drawback of the fuzzy Lyapunov‐based approaches, which assume some upper bounds on the derivatives of the fuzzy grade functions, we propose a new procedure by considering a constraint on the control signal. The new design conditions are given in the form of linear matrix inequalities (LMIs). The proposed control structure is applied to spinning disks in which chaos phenomena appear in lateral vibration. Simulation results are given to show the applicability of the proposed tracker to the UPO problem.     

Observer‐Based Reliable Control for Discrete‐Time Switched Linear Systems with Faulty Actuators

This paper deals with the issue of reliable control for discrete‐time switched linear systems with faulty actuators by utilizing a multiple Lyapunov functions method and estimate state‐dependent switching technique. A solvability condition for the reliable control problem is given in terms of matrix inequality with an extra matrix variable. This condition allows the reliable control problem for each individual subsystem to be unsolvable. For each subsystem of such a switched system, we design an observer and an observer‐based controller. A switching rule depending on the observer state is designed which, together with the controllers, can guarantee the stability of the closed‐loop switched system for all admissible actuator failures. The observers, controllers, and switching law are explicitly computed by solving linear matrix inequalities (LMIs). The proposed design method is illustrated by two numerical examples.     

Robust Observer and Observer‐Based Controller for Time‐Delay Singular Systems

In this paper we address the problems of observer and observer‐based controller design for a class of nonlinear time‐delay singular systems. The proposed methods use particular Lyapunov functions depending on the disturbances in order to avoid a specific obstacle in the stability analysis. Consequently, two linear matrix inequality (LMI) conditions ensuring the convergence of the estimation error and the closed loop system were presented. These LMIs were obtained by manipulating Young's inequality in order to linearize some bilinear terms.     

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.     

Design of Robust Discrete‐Time Observer‐Based Repetitive‐Control System

This paper concerns the design of a robust discrete‐time observer‐based repetitive‐control system for a class of linear plants with periodic uncertainties. A discrete two‐dimensional model is built that partially uncouples the control and learning actions of a repetitive‐control system, enabling their preferential adjustment. The combination of a singular‐value decomposition of the output matrix and Lyapunov stability theory is used to derive a linear‐matrix‐inequality‐based design algorithm that determines the control and state‐observer gains. A numerical example illustrates the main advantage of the method: easy, preferential adjustment of control and learning by means of two tuning parameters in an linear‐matrix‐inequality‐based condition.     

Output‐feedback tracking in causal nonminimum‐phase nonlinear systems using higher‐order sliding modes

Asymptotic output‐feedback tracking in a class of causal nonminimum phase uncertain nonlinear systems is addressed via sliding mode techniques. Sliding mode control is proposed for robust stabilization of the output tracking error in the presence of a bounded disturbance. The output reference profile and the unknown input/disturbance are supposed to be described by unknown linear exogenous systems of a given order. Local asymptotic stability of the output tracking error dynamics along with the boundedness of the internal states are proven. The unstable internal states are estimated asymptotically via the proposed multistage observer that is based on the method of extended system center. A higher‐order sliding mode observer/differentiator is used for the exact estimation of the input–output states in a finite time. The bounded disturbance is reconstructed asymptotically. A numerical example illustrates the efficiency of the proposed output‐feedback tracking approach developed for causal nonminimum phase nonlinear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

DISCRETE‐TIME OBSERVER‐BASED OUTPUT TRACKING CONTROLLER DESIGN FOR NONMINIMUM PHASE SYSTEMS WITH PERTURBATIONS

In this paper, an observer‐based output tracking controller for an SISO nonminimum phase discrete‐time system is proposed. When the disturbances between two consecutive sampling instances do not vary significantly, the observer algorithm can simultaneously estimate the system states and the unknown perturbation, and can render the estimation errors of system states and perturbation constrained in a small bounded region. The control law, including a feedforward term and a feedback input, can make the tracking error constrained in a small bounded region with guaranteed system stability. A numerical example is presented to demonstrate the applicability of the proposed control scheme.     

An improved multiple‐state observer of Boolean control networks

This paper explores the issue of state estimation for Boolean control networks (BCNs), and a kind of improved multiple‐state observer is proposed. The improved multiple‐state observer can be described by means of a specific BCN that overcomes the difficulty of the existing multiple state observers where it is difficult to find a general expression for the observer gain matrix. Next, based on the states that can possibly generate the output and those that are observed by the designed observer in current time step, an adaptive algorithm that completes the design of the multiple‐state observer is provided to update the observer states, and which makes the state estimation of Boolean control networks feasible. Finally, an example is presented to illustrate the effectiveness of the obtained results.     

Application of the dynamic high‐gain scaling methodology to servocompensator design

The error‐feedback servomechanism problem is addressed for a general class of strict‐feedback‐like systems. We provide two error‐feedback control designs based on our recent results on adaptive output‐feedback based on dynamic high‐gain scaling. One control design is of a dual high‐gain observer/controller structure, whereas the other control design utilizes a backstepping‐based controller in conjunction with a dynamic high‐gain scaling‐based observer. Owing to the particular robustness properties offered by a dynamic high‐gain scaling‐based controller and a backstepping‐based controller, the two designs require slightly different sets of assumptions on the system. Both design techniques allow the system to contain both unknown functions and uncertain appended input‐to‐state stable dynamics. Copyright © 2008 John Wiley & Sons, Ltd.     

Sliding mode observers for a class of linear parameter varying systems

In this paper, a new framework for the synthesis of a class of sliding mode observers for affine linear parameter varying (LPV) systems is proposed. The sliding mode observer is synthesized by selecting the design freedom via linear matrix inequalities ( LMIs ). Posing the problem from a small gain perspective allows existing numerical techniques from the literature to be used for the purpose of synthesizing the observer gains. In particular, the framework allows affine parameter‐dependent Lyapunov functions to be considered for analyzing the stability of the state estimation error dynamics, to help reduce design conservatism. Initially a variable structure observer formulation is proposed, but by imposing further constraints on the LMIs, a stable sliding mode is introduced, which can force and maintain the output estimation error to be zero in finite time. The efficacy of the scheme is demonstrated using an LPV model of the short period dynamics of an aircraft and demonstrates simultaneous asymptotic estimation of the states and disturbances.     

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.     

Observer‐based event‐triggered control for singularly perturbed systems with saturating actuator

This paper investigates an observer‐based event‐triggered control problem of singularly perturbed systems with saturating actuator. A strategy that consists of an observer‐based controller (OBC) and an event‐triggered mechanism (ETM) is considered. Firstly, sufficient conditions, which guarantee that the saturated SPSs are asymptotically stable excluding Zeno phenomenon, are derived via constructing an ε‐dependent Lyapunov‐Krasovskii functional. Then, the OBC and ETM are designed simultaneously based on the aforementioned criteria. Furthermore, an estimate of the basin of attraction and an ε‐bound are given by solving an optimization problem in the form of LMIs. Finally, an electric circuit system and a numerical example are presented to demonstrate the merits of the obtained method.