Delay-dependent Stability Conditions Research Articles

Stability and Stabilization of T-S Fuzzy Systems With Time-Varying Delays via Delay-Product-Type Functional Method.

This paper is concerned with the stability and stabilization problems of T-S fuzzy systems with time-varying delays. The purpose is to develop a new state-feedback controller design method with less conservatism. First, a novel Lyapunov-Krasovskii functional is constructed by combining delay-product-type functional method together with the state vector augmentation. By utilizing Wirtinger-based integral inequality and an extended reciprocally convex matrix inequality, a less conservative delay-dependent stability condition is developed. Then, the corresponding controller design method for the closed-loop delayed fuzzy system is derived based on parallel distributed compensation scheme. Finally, two classic numerical examples are given to show the effectiveness and merits of the proposed approaches.

Distributed robust output consensus for linear multi‐agent systems with input time‐varying delays and parameter uncertainties

This study addresses the leader-tracking problem for linear multi-agent systems in the presence of both parameter model uncertainties and time-varying communication delays. To solve the robust output consensus problem, a delayed distributed proportional–integral–derivative control is proposed and the overall closed-loop stability is proven by exploiting the Lyapunov–Krasovskii theory. Delay-dependent robust stability conditions are given via linear matrix inequalities which allow the proper tuning of robust control gains. The effectiveness of the theoretical derivation is confirmed through a numerical analysis in the practical application domain of cooperative driving for connected vehicles.

<i>H</i><SUB align="right">∞ control of asynchronous networked control systems with Markov time delays

In this paper, the solutions to the H∞ synthesis problems of stochastic asynchronous discrete-time (DT) networked control systems (NCSs) with random communication time delays are proposed according to the following steps. Firstly, a new Lyapunov-Krasovskii functional is explicitly constructed to analyse the stochastic stability of the systems. Both sensor-to-controller (S-C) and controller-to-actuator (C-A) random networked-induced delays, which are two Markov chains, are considered in the analysis. Moreover, the stochastic sampling of S-C and holding of C-A are also considered in the analysis. Secondly, based on a specified definition of the state with delay and a more rigorous inequality, sufficient delay-dependent stability conditions can be transformed into the form of linear matrix inequalities (LMIs), the LMIs can be conveniently solved by LMI tools. Finally, the proposed theory is validated by a simulation example.

Static anti-windup compensator design for nonlinear time-delay systems subjected to input saturation

In this paper, a novel technique for synthesizing static anti-windup compensator (AWC) is explored for dynamic nonlinear plants with state interval time-delays, exogenous input disturbance, and input saturation nonlinearity, by means of reformulated Lipschitz continuity property. A delay-range-dependent approach, based on Wirtinger-based inequality, is employed to derive a condition for finding the static AWC gain. By using the Lyapunov–Krasovskii functional, reformulated Lipschitz continuity property, Wirtinger-based inequality, sector conditions, bounds on delay, range of time-varying delay, and $$\mathcal {L}_2$$ gain reduction, several conditions are derived to guarantee the global and local stabilization of the overall closed-loop system. Further, when the lower time-delay bound is zero, the delay-dependent stabilization condition is derived for saturated nonlinear time-delay systems as a particular scenario of the suggested static AWC design approach. Furthermore, a static AWC design strategy is also provided when a delay-derivative bound is not known. An application to the nonlinear dynamical system is employed to demonstrate the usefulness of the proposed methodologies. A comparative numerical analysis with the existing literature is provided to show the superiority of the proposed AWC results.

A Predictor-Based Model Reference Adaptive Controller for Time-Delay Systems

This paper presents novel modifications to a predictor-based model reference adaptive controller (PMRAC) to compensate for input delays in uncertain nonlinear systems. A delay-dependent stability condition, derived using a Lyapunov–Krasovskii functional, indicates that there exists a time delay margin, below which the asymptotic stability of the tracking and estimation errors as well as the input-to-state stability of the closed-loop system are guaranteed. Numerical experiments are carried out with comparisons to the original PMRAC and to a finite spectrum assignment scheme to validate the theoretical claims. The results also illustrate the effectiveness of the proposed delay compensation formulation and the asymptotic behaviors of the tracking and estimation errors.

On Finite-Time Stability of Linear Positive Differential-Algebraic Delay Equations

This brief addresses finite-time stability of linear positive differential-algebraic equations with delay. Being different from the Lyapunov function method, delay-dependent sufficient conditions for positivity and finite-time stability of the system are established in terms of standard linear programming problems. A numerical example is given to illustrate the obtained results.

Further results on robust stability for uncertain neutral systems with distributed delay

This paper studies the problem of robust stability for uncertain neutral systems with distributed delay. By utilizing the incorporation of a new integral inequality technique and a novel Lyapunov–Krasovskii functional, some reduced conservative delay-dependent stability conditions for asymptotic stability are established. Then some special cases of neutral systems are discussed. Based on these delay-dependent stability conditions, the condition for robustness is obtained for uncertain linear delayed systems. All these stability conditions are given in terms of linear matrix inequalities (LMIs), which can easily be computed by the LMI toolbox of Matlab. Finally, several examples are discussed in detail to display the usefulness and superiority of the obtained results.

Robust functional observer for stabilising uncertain fuzzy systems with time-delay

This paper presents a new technique for stabilising a Takagi–Sugeno (T-S) fuzzy system with time-delay and uncertainty. A robust fuzzy functional observer is employed to design a controller when the system states are not measurable. The model uncertainty is norm bounded, and the time-delay is time-varying but bounded. The parallel distributed compensation method is applied for defining the fuzzy functional observer to design this controller. The proposed procedure reduces the observer order to the dimension of the control input. Improved stability conditions are provided for the observer compared with the existing results of functional observer-based stabilisation of T-S fuzzy models. Lyapunov–Krasovskii functionals are used to construct delay-dependent stability conditions as linear matrix inequalities. The solution of these inequalities is used for calculating the observer parameters. The sensitivity of the estimation error to the model uncertainty is reduced by minimising the $$L_2$$ gain. The new design method developed is illustrated and verified using two examples.

Delay-Dependent Stability in Uncalibrated Image-Based Dynamic Visual Servoing Robotic System

This paper addresses the stability problem of uncalibrated image-based visual servoing robotic systems. Both the visual feedback delay and the uncalibrated visual parameters can be the sources of instability for visual servoing robotic systems. To eliminate the negative effects caused by kinematic uncertainties and delays, we propose an adaptive controller including the delay-affected Jacobian matrix and design an adaptive law accordingly. Besides, the delay-dependent stability conditions are provided to show the relationship between the system stability and the delayed time in order to obtain less conservative results. A Lyapunov-Krasovskii functional is constructed, and a rigorously mathematic proof is given. Finally, the simulation results are presented to show the effectiveness of the proposed control scheme.

Delay-dependent supplementary damping controller of TCSC for interconnected power system with time-delays and actuator saturation

In this paper, a delay-dependent state feedback based supplementary damping controller of Thyristor Controlled Series Capacitor (TCSC) is designed to enhance the damping of inter-area oscillations of an interconnected power system. Although wide-area feedback signals are advantageous for damping enhancement of inter-area modes, but its usage as feedback signals in designing of a damping controller together with physical limitation of the actuator introduces time-delay and actuator saturation in the feedback loop. This results in degradation of the power system performance and even cause instability. The paper focuses on designing a damping controller considering the aforesaid time-delay and actuator saturation effects. Residue approach and Schur balanced truncation model reduction methods are used to obtain input-output control signals and reduced order power system model. The nonlinear effects of the actuator saturation are consider in the closed-loop system by using generalized sector condition. Based on Lyapunov–Krasovskii functional approach, a delay-dependent stability and stabilization conditions via linear matrix inequalities (LMIs) formulation are derived to guarantee the asymptotic stability of the closed-loop system. Then the problem of designing a state feedback control law which maximizes the region of attraction is formulated and solved as an optimization problem with LMI constraints. Simulation studies in the two-area four-machine power system demonstrate the effectiveness of the proposed controller for damping enhancement of the inter-area oscillations and compensating the effect of time-delays and actuator saturation.

Stability Analysis and Memory Control Design of Polynomial Fuzzy Systems with Time Delay via Polynomial Lyapunov-Krasovskii Functional

This paper investigates the problems of delay-dependent stability analysis and memory control design of polynomial fuzzy systems with time delay. Using polynomial Lyapunov-Krasovskii functional and slack polynomial matrix variables, delay dependent sufficient stability and stabilizability conditions are derived in terms of sum of squares (SOS) which can be numerically (partially symbolically) solved via the recently developed SOSTOOLS. The main advantage of the proposed design is the reduction of conservatism for three great reasons. The first one is that polynomial matrices are not only dependent on the system state vector but also on the state vector with time delay. The second one is that the design conditions are formulated in delay dependent SOS. It is well known that the delay-dependent conditions are less conservative than those independent of time delay. The third one is that only correlated terms are used in the design of SOS. The simulation and comparison are given to illustrate the lesser conservativeness of the proposed result.

Fuzzy sliding mode control design of Markovian jump systems with time-varying delay

This paper studies the robust stochastic stabilization problem for a class of fuzzy Markovian jump systems with time-varying delay and external disturbances via sliding mode control scheme. Based on the equivalent-input-disturbance (EID) approach, an online disturbance estimator is implemented to reject the unknown disturbance effect on the considered system. Specifically, to obtain exact EID estimation Luenberger fuzzy state observer and a low-pass filter incorporated to the closed-loop system. Moreover, novel fuzzy EID-based sliding mode control law is constructed to ensure the stability of the closed-loop system with satisfactory disturbance rejection performance. By employing Lyapunov stability theory and some integral inequalities, a new set of delay-dependent robust stability conditions is derived in terms of linear matrix inequalities (LMIs). The resulting LMI is used to find the gains of the state-feedback controller and the state observer a for the resulting closed-loop system. At last, numerical simulations based on the single-link arm robot model are provided to illustrate the proposed design technique.

Stochastic stability of mode-dependent Markovian jump inertial neural networks

The problem of stochastic stability analysis for Markovian jump inertial neural networks with mode-dependent time-varying delay is considered in this paper. The time-delay is assumed to be mode-dependent and time-varying. Using the suitable transformation technique, the second-order inertial neural network model is transformed into a system of first-order differential equations model. Delay-dependent stochastic stability condition is formulated in terms of linear matrix inequalities through the construction of Lyapunov–Krasovskii functional candidate involving mode-dependent time-varying delay. An integral inequality technique is used in the process to find the bounds of some integral terms. Finally, a numerical example is illustrated to show the effectiveness of the derived theoretical results.

Membership-Function-Dependent Stability and Stabilization Conditions for T–S Fuzzy Time-Delay Systems

ABSTRACTIn this paper, the stability analysis and stabilization of time-delay nonlinear systems represented by a Takagi–Sugeno fuzzy model are investigated. Combining with a recently developed integral inequality, relaxed delay-dependent stability conditions are derived by means of linear matrix inequalities. To obtain the further relaxed stability criteria, the membership-function-dependent method is applied via the piecewise membership functions (PMFs) and some slack variables. The PMFs are selected to approximate the membership functions (MFs), and then the characteristics of the MFs are taken into consideration adequately. Under the imperfect premise matching design scheme, this paper also proposes a flexible fuzzy controller design strategy that the fuzzy controllers do not employ the identical MFs and the number of rules as the fuzzy models. Finally, this proposed approach is tested successfully and its effectiveness is validated by the simulation examples.

Extended dissipative filter design for T-S fuzzy systems with multiple time delays

This paper deals with the fuzzy filtering problem for a class of nonlinear time-delay systems described by T-S fuzzy models. Different from the existing schemes in the literature, this paper aims to solve the fuzzy filtering problem by considering the H∞, L2 − L∞ and dissipative performance constraints in a unified way. To achieve this purpose, the recently proposed notion of extended dissipativity is applied, which provides an inequality covering the well-known H∞, L2 − L∞ and dissipative performances. Another purpose of this paper is to design filters involving communication delays. Such filters have a more general form than the delay-free filters that have been largely considered in the traditional studies. In order to design the fuzzy filters under consideration, a novel fuzzy Lyapunov-Krasovskii functional is employed, and delay-dependent conditions for stability and performance analysis of the filtering error system are obtained. Then, LMI-based conditions for the existence of the desired filters are presented. The filter parameters can be obtained by solving the presented LMIs. Finally, the effectiveness of the proposed method is substantiated with an illustrative example.

Network-based PI control for output tracking of continuous-time systems with time-varying sampling and network-induced delays

For a continuous-time linear system with constant reference input, the network-based proportional-integral (PI) control is developed to solve the output tracking control problem by taking time-varying sampling and network-induced delays into account. A traditional PI control system is introduced to obtain the equilibriums of state and control input. Using the equilibriums, a discrete-time PI tracking controller in a network environment is constructed. The resulting network-based PI control system is described by an augmented system with two input delays and the output tracking objective is transformed into ensuring asymptotic stability of the augmented system. A delay-dependent stability condition is established by a discontinuous augmented Lyapunov–Krasovskii functional approach. The PI controller design result of in-wheel motor as a case study is provided in terms of linear matrix inequalities. Matlab simulation and experimental results resorting to a test-bed for ZigBee-based control of in-wheel motor are given to validate the proposed method.

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 fuzzy control of nonlinear systems with its application to inverted pendulum systems

This paper is concerned with the event-triggered fuzzy control design and its application to an inverted pendulum system. First, all of the introduced time-varying delay information from the inverted pendulum system and event-triggered mechanism is considered in the stability analysis. The interval of the time-varying delay is then divided into l nonuniform subintervals by utilizing an improved delay-partitioning technique. Information on each subinterval is processed by employing a reciprocally convex method. The proposed delay-dependent stability condition of the inverted pendulum system is determined to be considerably less conservative than the existing results in the literature, and the reduction in the conservativeness becomes progressively noticeable as the delay partitioning number l becomes progressively thinner. Moreover, by employing the parallel distributed compensation law, conditions sufficient for the resulting fuzzy controller and the event-triggering fuzzy controller are presented for the nonlinear inverted pendulum system. Finally, the advantages and effectiveness of the proposed design schemes are demonstrated by three simulation examples.

New Criteria for Stability of Generalized Neural Networks Including Markov Jump Parameters and Additive Time Delays

This paper examines the problem of asymptotic stability criteria for Markovian jump generalized neural networks with successive time-varying delay components. Generalized neural networks consist of a finite number of modes, which may jump from one mode to another according to a Markovian chain with known transition probability. By constructing novel augmented Lyapunov–Krasovskii functionals (LKFs) with triple integral terms that contain more and more information on the state vectors of the NNs, the upper bound of the successive time-varying delays is formulated. By employing a new integral inequality technique, free-weighting matrix-based integral inequality approach, and Wirtinger double integral inequality technique and that is combined with the reciprocally convex combination approach to estimate the single and double integral terms in the time derivative of the LKFs, a new set of delay-dependent conditions for the asymptotic stability of the considered NNs are represented in the form of linear matrix inequalities. Finally, five numerical examples are given to verify the effectiveness of the proposed approach with a four-tank benchmark real-world problem.

Novel delay-dependent stability condition for mixed delayed stochastic neural networks with leakage delay signals

ABSTRACTIn this paper, the problem of stability condition for mixed delayed stochastic neural networks with neutral delay and leakage delay is investigated. A novel Lyapunov functional is constructed with double and triple integral terms. New sufficient conditions are derived to guarantee the global asymptotic stability of the concerned neural network. This paper is more general than the paper by Zhu et al. [Robust stability of Markovian jump stochastic neural networks with time delays in the leakage terms, Neural Process. Lett. 41 (2015), pp. 1–27]. In our paper, we considered both the neutral delay and leakage delay, but the paper by Zhu et al. is not considering the neutral delay. Also we employed triple integrals in the Lyapunov functional which is not used in the paper by Zhu et al. Finally, two numerical examples are provided to show the effectiveness of the theoretical results.