Time-varying State Delays Research Articles

Asynchronous repetitive control of switched systems via periodic event-based dynamic output feedback

Abstract This paper develops an asynchronous mode-dependent repetitive control strategy with periodic event-based dynamic output feedback for periodic trajectory tracking of continuous-time switched systems subject to time-varying switching delays between system modes and controllers and limited communication capacity in the feedback channel. By employing the input delay approach, the overall system is modelled as an augmented closed-loop switched system with both constant and time-varying state delays. A co-design framework is proposed for simultaneously designing the controller, event-triggering mechanism and mode switching signal. Under the co-design framework, using piecewise Lyapunov functional, free-weighting matrices and average dwell time technique, sufficient conditions are derived for ensuring the augmented closed-loop system to be exponentially stable with a prescribed $H_{\infty }$ attenuation level $\gamma $ for an exogenous disturbance input. The performance of the proposed controller design scheme is verified via numerical simulation results on a switched RLC series circuit system.

Robust $$ H_{\infty } $$ Fault-Tolerant Control for Discrete-Time Nonlinear System with Actuator Faults and Time-Varying Delays Using Nonlinear T–S Fuzzy Models

In this paper, the problem of fault estimation and fault-tolerant control for a class of nonlinear discrete-time system state time-varying delay and actuator fault is investigated. This class of systems is represented through the Takagi–Sugeno (T–S) fuzzy model with nonlinear functions satisfying some sector-bounded conditions. By adding these nonlinear functions in the local sub-models, the observer and controller can be designed with fewer rules and less computation burden. The method proceeds in two steps: first, a full-order fuzzy fault estimation observer (FFEO) design is proposed to estimate the actuator faults and the nonlinear functions in the T–S models. Second, based on the online fault estimation, a dynamic output feedback fault-tolerant controller (DOFTC) is then designed to compensate the effect of faults by stabilizing the closed-loop system. Furthermore, sufficient less conservative delay-dependent conditions for the existence of the desired FFEO and DOFTC are given in terms of linear matrix inequalities by employing the fuzzy Lyapunov–Krasovskii function and free-weighting approach. Finally, a practical example is given to show the effectiveness and advantages of the proposed approach.

Simultaneous stabilization of a collection of uncertain discrete-time systems with time-varying state-delay via discrete-time sliding mode control

This paper considers the simultaneous stabilization problem which concerns finding a single controller to simultaneously stabilize a finite collection of discrete-time nonlinear systems with time-varying state delay and norm-bounded uncertainties. For this purpose, a simultaneously stabilizing discrete-time sliding mode controller (DT-SMC) is synthesized. Based on the delay-dependent Lyapunov–Krasovskii functional method, sufficient conditions are derived which guarantee the existence of a common sliding surface in terms of a nonconvex feasibility problem. The cone complementarity linearization approach is exploited to transform the nonconvex conditions into a convex optimization problem. Finally, the robust simultaneously stabilizing DT-SMC is designed in order to draw the closed-loop trajectories of each system onto the common sliding surface in finite time and to maintain it thereafter. The effectiveness of the proposed method is investigated by simulation of a practical example.

Delay-Dependent Energy-to-Peak Stability of 2-D Time-Delay Roesser Systems With Multiplicative Stochastic Noises

This paper is concerned with the problem of energy-to-peak stochastic stability (EPSS) of two-dimensional (2-D) Roesser systems in the presence of state time-varying delays and multiplicative noises. First, a scheme that ensures a 2-D stochastic time-delay system is stochastically stable with an attenuation performance is proposed. The scheme presented in this paper can be regarded as an extension of the Lyapunov–Krasovskii functional method for 2-D stochastic time-delay systems, focusing on the EPSS problem. The proposed scheme is then utilized to derive delay-dependent EPSS conditions in terms of tractable linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the derived stability conditions.

Fuzzy Static Output Feedback Control for Discrete-time Nonlinear Time-varying Delay Systems in a Network Environment

This paper studies the problem of fuzzy output control of discrete-time nonlinear systems with immeasurable state variables, time-varying delay, and data packet dropout. A T–S fuzzy model is used to represent the original nonlinear systems with time-varying delay, and the Bernoulli distribution is employed to describe the dropout process of the random packet. The two uncertainties of time delay and data packet dropout are considered in the system design. By constructing a fuzzy Lyapunov–Krasovskii functional and introducing a summation inequality based on Cauchy–Schwartz form, the existence condition of fuzzy static output feedback controller is established by using linear matrix inequalities (LMIs). The simulation example verifies the effectiveness of the proposed method.

Internally positive representations and stability analysis of linear differential systems with multiple time‐varying delays

This chapter introduces the Internally Positive Representation of linear systems with multiple time-varying state delays. The technique, previously introduced for the undelayed case, aims at building a positive representation of systems whose dynamics is, in general, indefinite in sign. As a consequence, stability criteria for positive time-delay systems can be exploited to analyze the stability of the original system. As a result, an easy-to-check sufficient condition for the delay-independent stability is provided, that is compared with some well known conditions available in the literature.

Finite-time sensor fault diagnosis observer design for attitude control systems of satellite based on sampled-data descriptor systems

In this paper, a finite-time robust sensor fault diagnosis observer with non-singular structure is proposed for Attitude Control Systems of satellite based on Lipschitz non-linear sampled-data descriptor systems with state time-varying delay. Firstly, an Attitude Control Systems model based on sampled-data descriptor system with time-varying delay is proposed and transformed into a discrete-time non-singular one. Then, a sensor fault diagnosis observer is designed upon on the state estimation error and the measurement residual. The finite-time stability and robustness of the augmented error dynamic model is analyzed, the H infinity performance index in finite-time is satisfied. As the confining matrices of the observer parameters do not meet the Linear Matrix Inequality, a cone complementary linearization iteration algorithm is proposed to solve this problem. Residual evaluation function and the threshold are introduced for judging if the faults occur. At the end, simulation examples are given to illustrate the effectiveness of this method.

Distributed output-feedback controllers design for Markovian jump systems interconnected over an undirected graph with time-varying delay

ABSTRACTThis paper considers the problems of stochastic stability analysis and distributed output-feedback controllers design for Markovian jump systems interconnected over an undirected graph with state time-varying delay. Specifically, a sufficient condition for the well-posedness, stochastic stability and contractiveness of the resultant closed-loop system is first developed in the form of non-linear matrix inequalities. Then, to avoid the difficulty of solving non-linear matrix inequalities, the sufficient condition is relaxed to an equivalent one in terms of coupling linear matrix inequalities (LMIs), which provides an effective way to confirm the existence of a distributed output-feedback controller inheriting the structure of the given plant. Furthermore, a constructive algorithm is given for the design of such a distributed output-feedback controller to achieve the well-posedness, stochastic stability as well as the contractiveness of the corresponding closed-loop system. Finally, numerical simulations are given to illustrate the validity of the proposed method.

Guaranteed cost control for exponential stability of a nonlinear system with mixed time-varying delays in states and controls

This paper presents a new guaranteed cost control for exponential stability of a nonlinear system with mixed time-delays in state and feedback control. The considered mixed time-delays are both discrete and distributed time-varying delays, but not necessarily differentiable. The proposed conditions allow us to design the state feedback controllers which stabilize the closed-loop system. By constructing an appropriate Lyapunov–Krasovskii functional, new delay-dependent sufficient conditions for the existence of guaranteed cost control are given in terms of linear matrix inequalities (LMIs). Moreover, we design new quadratic cost functions and minimize their upper bound. Finally, numerical examples are given to illustrate the effectiveness and improvement over some existing results in the literature.

Stabilization of Nonlinear Systems With Time-Varying Input and State Delays by Approximate Predictor

In this paper, the stabilization problem of nonlinear systems with time-varying input and state delays is considered by an approximate predictor. Due to the introduction of state delays, the nonlinear dynamic without state delays is used to predict the plant state, and a new error term is introduced into the approximation error between the prediction value and the future state of the plant. By Lyapunov–Krasovskii functional and small-gain argument, the stability of the whole closed-loop system is guaranteed provided that the sampling period, the predictor accuracy, and the upper bound of state delays satisfy an inequality constraint. The effectiveness of the proposed results is illustrated by a numerical example.

Adaptive fuzzy decentralized control for a class of MIMO large-scale nonlinear state delay systems with unmodeled dynamics subject to unknown input saturation and infinite number of actuator failures

This paper addresses design of an adaptive fuzzy decentralized fault-tolerant controller for a class of uncertain multi-input multi-output (MIMO) large-scale nonlinear systems with unmodeled dynamics subject to unknown state time-varying delay, external disturbances, unknown input saturation and actuator faults. It is shown that the proposed fault-tolerant control (FTC) scheme can handle infinite number of actuator failures including partial and total loss of effectiveness. System uncertainties have been approximated by the fuzzy logic systems (FLSs). To cope with the unknown state time-varying delay, the Razumikhin lemma has been utilized and unmodeled dynamics has been tackled by introducing a dynamical signal. To avoid high computational burden, the dynamic surface control (DSC) technique has been used and the number of adaption laws has been reduced by updating the maximum norm of fuzzy weight vectors of each subsystem. It has been shown that all closed-loop signals are semi-globally uniformly ultimately bounded (SGUUB) and the output tracking errors converge to a small neighborhood of the origin by choosing the design parameters appropriately. Simulation results demonstrate the effectiveness of the proposed control scheme.

Robust Local Stabilization of Discrete-Time Systems with Time-Varying State Delay and Saturating Actuators

The robust local stabilization of uncertain discrete-time systems with time-varying state delayed and subject to saturating actuators is investigated in this work. A convex optimization method is proposed to compute robust state feedback control law such that the uncertain closed-loop is locally asymptotically stable if the initial condition belongs to an estimate of the region of attraction for the origin. The proposed procedure allows computing estimates of the region of attraction through the intersection of ellipsoidal sets in an augmented space, reducing the conservatism of the estimates found in the literature. Also, the conditions can handle the amount of delay variation between two consecutive samples, which is new in the literature for the discrete-time case. Although the given synthesis conditions are delay dependent, the proposed control law is delay independent, yielding to easier real time implementations. A convex procedure is proposed to maximize the size of the set of safe initial conditions. Numerical examples are provided to illustrate the effectiveness of our approach and also to compare it with other conditions in the literature.

Delay Dependent Local Stabilization Conditions for Time-delay Nonlinear Discrete-time Systems Using Takagi-Sugeno Models

We propose convex conditions for stabilization of nonlinear discrete-time systems with time-varying delay in states through a fuzzy Takagi-Sugeno (T-S) modeling. These conditions are developed from a fuzzy Lyapunov-Krasovskii function and they are formulated in terms of linear matrix inequalities (LMIs). The results can be applied to a class of nonlinear systems that can be exactly represented by T-S fuzzy models inside a specific region called the region of validity. As a consequence, we need to provide an estimate of the set of safe initial conditions called the region of attraction such that the closed-loop trajectories starting in this set are assured to remain in the region of validity and to converge asymptotically to the origin. The estimate of the region of attraction is done with the aid of two sets: one dealing with the current state, and the other concerning the delayed states. Then, we can obtain the feedback fuzzy control law depending on the current state, x k , and the maximum delayed state vector, xk−d. It is shown that such a control law can locally stabilize the nonlinear discrete-time system at the origin. We also develop convex optimization procedures for the computation of the fuzzy control gains that maximize the estimates of the region of attraction. We present two examples to demonstrate the efficiency of the developed approach and to compare it with other approaches in the literature.

Event-triggered state estimation for time-delayed complex networks with gain variations based on partial nodes

This paper designs the event-triggered non-fragile state estimator for a class of time-delayed complex networks with randomly occurring sensor saturations (ROSSs) and estimator gain variations on the basis of measurements from partial nodes. Both the time-varying state delays and the stochastic disturbances are considered in the network model. A Bernoulli-distributed white sequence is utilized to reflect the phenomenon of ROSSs. Two sequences of Gaussian distributed random variables combined with the multiplicative norm-bounded uncertainties are used to characterize the randomly occurring gain variations in the estimators. An event generator function is employed to regulate the transmission of data from the sensor to the estimator. The aim of this paper is to design an exponentially ultimately bounded state estimator in mean square through measurement outputs from a partial of network nodes under the event-triggered mechanism. With the help of Lyapunov–Krasovskii functional and stochastic analysis techniques, sufficient conditions are acquired for the existence of the desired state estimator which ensures that the estimation error dynamics is exponentially ultimately bounded in mean square, and then the estimator gain matrices can be computed via the software Matlab. A simulation example is provided to demonstrate the effectiveness of the proposed state estimation method.

Robust Stochastic Control for High-Speed Trains With Nonlinearity, Parametric Uncertainty, and Multiple Time-Varying Delays

In this paper, a novel delay-dependent robust $H_\infty $ control criterion for the velocity tracking control of high-speed trains is developed based on a multi-particle model and Lyapunov stability theory. The dynamic model of high-speed trains is proposed with consideration of the nonlinearity of aerodynamic drag, stochastic property of the measuring signals, uncertainties of train mass, resistance and stiffness of the couplers, multiple time-varying state delays, and input delays caused by the transmission process. Numerical simulations are carried out to verify the effectiveness and robustness of the proposed controllers, while the weights of delays are varying, and explore the influences of actuation power distribution on the velocity tracking performance and driving smoothness. Moreover, some useful results are obtained from the comparison between the proposed robust $H_{\infty }$ controller and an existing robust adaptive controller, which can also deal with parameter uncertainties and nonlinearity.

Gain-scheduled control of time-varying delay systems with input constraint

ABSTRACTThis paper investigates gain-scheduled control design for linear systems with time-varying state delays subject to actuator saturation and external disturbance. Assuming the disturbance is peak bounded, a sufficient delay-dependent condition is established to guarantee that a family of level sets, corresponding to a novel parameter-dependent Lyapunov–Krasovskii functional, are nested and invariant to the closed-loop system. The invariant sets are then used to obtain nested reachable sets (ellipsoids) to bound the closed-loop states. A family of continuous controllers are designed based on these nested ellipsoids. The controller with the best performance is selected, each time, based on the closed-loop state vector, while complying with the saturation bound, and the resulting closed-loop system is locally input-to-state stable. All conditions are represented in the form of linear matrix inequalities (LMIs) by the linear spline method. Finally, the benefit of the control method is illustrated by two examples.

Adaptive Fuzzy Tracking Control for Stochastic Nonlinear Systems with Time-Varying Input Delays Using the Quadratic Functions

In this paper, for the stochastic nonlinear systems the adaptive fuzzy tracking controllers are constructed by using the fuzzy logic systems (FLS) and the classical quadratic functions. Compared with the existing results for adaptive fuzzy control, the stochastic nonlinear systems investigated in this paper are much more complex since the systems not only have distributed state time-varying delays in the noise jamming intensity terms but also have the time-varying delays in the input signals. During the controller design procedure, through appropriate assumptions and a state transformation the system with time-varying input delay can be easily transformed into a system without input delay. The other main advantage is that quadratic functions are used as Lyapunov functions to analyze the stability of systems, other than the fourth moment approach proposed by H. Deng and M. Krstic, and the hyperbolic tangent functions are introduced to deal with the Hessian terms. The proposed adaptive fuzzy controller guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error can converge to a small residual set around the origin in the mean square sense.

ISS Stabilization of Discrete-time LPV Systems with Interval Time-varying State Delay and Saturating Actuators

Abstract We address the problem of input-to-state stabilization of linear parameter-varying (LPV) discrete-time systems with delayed state and subject to saturated actuators and to l2-limited disturbance. It is assumed that the time-varying delay belongs to a known interval and its maximum variation between two consecutive instants has limited rate. We propose convex delay-dependent conditions for the synthesis of LPV state feedback controllers that do not require the real-time knowledge of the delay and ensure the local input-to-state stability of the closed-loop system for a set of initial conditions and subject to energy bounded disturbance input signals. The approach is based on rewriting the delayed system as a switched uncertain augmented delay-free system with a dead-zone non-linearity and on the application of the generalized sector condition to deal with the saturating actuators. To illustrate the efficiency of our approach, we present a numerical example.

<inline-formula> <tex-math notation="LaTeX">$H_\infty$ </tex-math> </inline-formula>Synchronization for Uncertain Time-Delay Chaotic Systems With One-Sided Lipschitz Nonlinearity

This paper addresses $H_\infty $ synchronization for uncertain chaotic systems with one-sided Lipschitz nonlinearity under the output and intrinsic state delays. By utilizing the one-sided Lipschitz condition and quadratic inner boundedness, constructing an appropriate Lyapunov–Krasovskii (LKF), robust controller design conditions based on Lyapunov stability theory are derived for synchronization of chaotic systems under disturbances or perturbations bounded by $L_{2}$ norm. By introducing the delay-derivative limits and delay-interval bounds into LKF, the intrinsic state time-varying delay can be tackled by the delay-range-dependent strategy. Less conservative stability condition can be obtained by the further improved inequality of Jensen inequality and reciprocally convex approach, which can lead to the tighter upper bound for integral inequality. Numerical simulations are provided to verify the validity of the proposed methodology for synchronization of chaotic systems.

Observability of Boolean Control Networks with Time-Variant Delays in States

This paper gives an equivalent condition for the observability of Boolean control networks (BCNs) with time-variant delays in states under a mild assumption by using the graph-theoretic method under the framework of the semi-tensor product of matrices. First, the BCN under consideration is split into a finite number of subsystems with no time delays. Second, the observability of the BCN is verified by testing the observability of the so-called observability constructed path (a special subsystem without time delays) based on graph theory. These results extend the recent related results on the observability of BCNs. Examples are shown to illustrate the effectiveness of the results.