Delay-dependent Stability Criteria Research Articles

Novel Delay-Dependent Stabilization for Fuzzy Stochastic Systems with Multiplicative Noise Subject to Passivity Constraint

A novel delay-dependent stability criterion for Takagi-Sugeno (T-S) fuzzy systems with multiplicative noise is addressed in this paper subject to passivity performance. The general case of interval time-varying delay is considered for the practical control issue. For the criterion, an integral Lyapunov-Krasovskii function is proposed to derive some sufficient relaxed conditions and to avoid the derivative of the membership function. Moreover, a free-matrix inequality is adopted to deal with the delay terms such that the available derivative of time-varying delay is bigger than one. In order to employ a convex optimization algorithm to find the control gain, a projection lemma is applied to acquire the Linear Matrix Inequality (LMI) form of the sufficient conditions. With the obtained gains, a fuzzy controller is designed by the concept of Parallel Distributed Compensation (PDC) such that the delayed T-S fuzzy systems with multiplicative noise are asymptotically stable and passive in the mean square. Finally, a stabilization problem of the ship’s autopilot dynamic system and some comparisons are discussed during the simulation results.

Event-triggered consensus control and fault estimation for time-delayed multi-agent systems with Markov switching topologies

This paper focuses on the consensus control and fault estimation problems for a class of time-delayed multi-agent systems with Markov switching topologies. Two different event-triggered mechanisms are adopted with hope to reduce burden of shared network and improve energy efficiency. Under Markov process, by establishing the consensus control protocol and designing a novel adaptive fault estimation observer, the consensus control and fault estimation problems are transformed into two stochastic stability problems in different forms. Then, according to the switching Lyapunov function method and free-weighting matrix technique, two delay-dependent stability criteria on the consensus control and fault estimation are derived, respectively. However, the two criteria containing nonlinear coupling terms are not standard linear matrix inequalities (LMIs) and cannot be solved directly with the LMI toolbox. In order to eliminate the coupling terms, two improved path-following algorithms are presented. These algorithms depend on the initial conditions, so it is very crucial to choose the appropriate preset parameters. The computational complexity is increasing with the number of iterations, system size and matrix dimension, which is a fully new challenge for the study of consensus control and fault estimation of multi-agent systems. Based on the algorithms, the switching consensus controller gains and model gain matrices of fault estimation can be efficiently solved out. Finally, a simulation example of tailless fighter airplanes is given to illustrate the practicality and validity of the theoretical results.

New relaxed stability and stabilization conditions for T‐S fuzzy systems with time‐varying delays

This paper investigates the stability analysis and stabilization of T-S fuzzy systems with time-varying delays. First, a new augmented Lyapunov–Krasovskii functional is constructed, delay-dependent stability criteria in terms of linear matrix inequalities (LMIs) are obtained by combining them with the integral inequality technique and the reciprocally convex combination inequality. Based on the state space decomposition method, some piecewise membership functions are employed to approximate the membership functions. The piecewise membership functions can be locally represented in terms of the convex combinations of the supremum and infimum of some local basis functions. The boundary information of the membership functions is adequately taken into consideration in stability analysis, and then some relaxed membership-function-dependent stability results are obtained. Second, state feedback controllers for fuzzy systems with time-varying delays are presented under the imperfect premise-matching technique, whose membership functions and the number of fuzzy rules are allowed to be designed freely, consequently, the flexibility of controller design is improved. Finally, four numerical examples are given to demonstrate the effectiveness of the presented approaches.

Design of uncertainty and disturbance estimator based tracking control for fuzzy switched systems

In this paper, an uncertainty and disturbance estimator (UDE)-based control is employed to ensure the finite-time tracking and disturbance rejection performance for a class of Takagi–Sugeno fuzzy switched systems including additive time-varying delays, unknown time-varying uncertainties and disturbances. To be precise, the robust tracking performance is achieved by estimating unknown uncertainties and disturbances with the aid of low-pass filter by means of appropriate bandwidth choice of the desired filter. Moreover, a new delay-dependent finite-time stability criterion is derived by employing fuzzy-dependent Lyapunov–Krasovskii functional, extended Wirtinger's integral inequality and average dwell-time approach to enforce the addressed system to track the given reference system within a finite period of time. Finally, the developed theoretical results are ensured by presenting the numerical examples with simulation results.

Improved Results on Delay dependent Stability Criteria of Neural Networks with Interval Time Varying Delay

This paper examines the stability issue of continuous Neural Networks with a time varying delay. A Lyapunov Krasovskii functional consisting of some simple augmented terms and delay dependent terms is constructed. While calculating the derivative of Lyapunov functional, various integral inequalities such as Auxiliary Function Based Integral Inequality, Wirtinger-based integral inequality and an extended Jensen double integral inequality are jointly adopted and hence in terms of linear matrix inequality a new delay dependent stability criterion is obtained. Two numerical examples are taken to show that the derived result is less conservative than some existing ones.

A novel observer-based approach to delay-dependent LFC of power systems with actuator faults and uncertain communications conditions

In this paper, a novel observer-based approach is proposed for the delay-distribution-dependent (DDD) load frequency control (LFC) of multi-area power systems with actuator faults and probabilistic communication conditions. LFC is designed considering missing data which obey the Bernoulli distribution and actuator faults. After that, an observer gain matrix is constructed for realizing Lyapunov stability, where delay-dependent stability criteria are derived under the consideration of different factors such as packet dropout during data transmission and faults in control. The stability conditions are derived in terms of linear matrix inequalities to achieve less conservative results in the communication delay and H∞ performance. Finally, the efficiency of the proposed observer-based DDD LFC is demonstrated and compared through time-domain simulations.

Improved Results on Delay-Dependent Robust H ∞ Control of Uncertain Neutral Systems with Mixed Time-Varying Delays

In this paper, the problem of the delay-dependent robust H ∞ control for a class of uncertain neutral systems with mixed time-varying delays is studied. Firstly, a robust delay-dependent asymptotic stability criterion is shown by linear matrix inequalities (LMIs) after introducing a new Lyapunov–Krasovskii functional (LKF). The LKF including vital terms is expected to obtain results of less conservatism by employing the technique of various efficient convex optimization algorithms and free matrices. Then, based on the obtained criterion, analyses for uncertain systems and H ∞ controller design are presented. Moreover, on the analysis of the state-feedback controller, different from the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively, we can obtain the state-feedback gain directly by calculating the LMIs through the toolbox of MATLAB in this paper. Finally, the feasibility and validity of the method are illustrated by examples.

Further Robust Stability Analysis For Time-delayed Neutral-type Systems by a Modified Lyapunov-Krasovskii Functional

This paper further studies the stability of the uncertain delayed neutral-type system by constructing a novel Lyapunov-Krasovskii functional (LKF) with some delay-dependent and N-dependent items. Some new delay-dependent robust stability criteria are proposed based on linear matrix inequality (LMI). The criteria proposed in this paper are less conservative than some recent previous ones. Some numerical examples, including a linear neutral-type system, the partial element equivalent circuit and a general linear system, are presented to show the effectiveness of the proposed approach.

Composite slack-matrix-based integral inequality and its application to stability analysis of time-delay systems

This paper focuses on the delay-dependent stability problem of time-varying delay linear systems. A composite slack-matrix-based integral inequality (CSMBII) is presented in delay-product types. It overcomes the limitation of reciprocal convexity in Bessel–Legendre integral inequality such that CSMBII is more convenient to cope with time-varying delay systems. And it also overcomes the shortcoming that the slack matrices in previous works are independent of time-varying delay. Based on CSMBII, a new delay-dependent stability criterion is derived for time delay systems. A numerical example is given to illustrate the effectiveness of the stability criterion.

Robust Stabilization of Stochastic Markovian Jump Systems with Distributed Delays

This paper addresses the robust stabilization problem for a class of stochastic Markovian jump systems with distributed delays. The systems under consideration involve Brownian motion, Markov chains, distributed delays, and parameter uncertainties. By an appropriate Lyapunov–Krasovskii functional, the novel delay-dependent stabilization criterion for the stochastic Markovian jump systems is derived in terms of linear matrix inequalities. When given linear matrix inequalities are feasible, an explicit expression of the desired state feedback controller is given. The designed controller, based on the obtained criterion, ensures asymptotically stable in the mean square sense of the resulting closed-loop system. The convenience of the design is greatly enhanced due to the existence of an adjustable parameter in the controller. Finally, a numerical example is exploited to demonstrate the effectiveness of the developed theory.

Asymptotic stability of static neural networks with interval time-varying delay based on LMI

In this paper, the asymptotic stability of static neural networks with interval time-varying delay is studied. An improved integral inequality based on orthogonal polynomials is proposed, in which three free vectors can be selected independently. A novel Lyapunov–Krasovskii functional containing more delayed state information, instant state information and integrated state information is constructed. Based on the improved non-convex technique, two less conservative delay-dependent stability criteria in terms of linear matrix inequalities (LMIs) are derived. The validity and superiority of the theoretical results derived in this paper are verified by numerical simulation.

Reachable Set Estimation for Uncertain Markovian Jump Systems with Time-Varying Delay and Disturbances

In this paper, we are concerned with the problem of reachable set estimation for uncertain Markovian jump systems with time-varying delays and disturbances. The main consideration is to find a proper method to obtain the no-ellipsoidal bound of the reachable set of Markovian jump system as small as possible. Based on an augmented Lyapunov–Krasovskii functional, by dividing the time-varying delay into two nonuniform subintervals, more general delay-dependent stability criteria for the existence of a desired ellipsoid are derived. An optimized integral inequality which is based on distinguished Wirtinger integral inequality and reciprocally convex combination inequality is used to deal with the integral terms. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results.

New stability criterion for time-delay systems via an augmented Lyapunov–Krasovskii functional

This paper focuses on the stability of linear systems with time-delay. An augmented Lyapunov–Krasovskii functional (LKF) is first constructed, and then a generalized delay-dependent stability criterion is obtained by combining the newly constructed LKF and the generalized vectors-based multiple integral inequality (GVMII). By deriving one type of the concrete orthogonal function in the criterion, an easy-to-use criterion has been derived, which is less conservative than before. Finally, one numerical example is given to show the superiority of our method.

Stability Analysis for the Discrete-Time T-S Fuzzy System with Stochastic Disturbance and State Delay

In this paper, we study the stability for discrete-time Takagi and Sugeno (T-S) fuzzy systems with perturbation disturbance and time delay. Stochastic delay-dependent stability criteria are derived for stochastic T-S fuzzy systems with time-invariant and time-varying delays, respectively. For the time-varying delay case, a novel fuzzy Lyapunov–Krasovskii functional (LKF) without requiring all the involved symmetric matrices to be positive definite is constructed to reduce the conservatism. These stability conditions are then represented in terms of finite linear matrix inequalities (LMIs), which can be solved efficiently by using standard LMIS optimisation techniques. Two numerical examples are given to illustrate the feasibility of the proposed method.

Novel stability criterion for linear system with two additive time-varying delays using general integral inequalities

<abstract> The stability analysis strategy for continuous linear system with two additive time-varying delays is proposed in this paper. First, for the purpose of analysis, the novel Lyapunov-Krasovskii functional (LKF) consisting of integral terms based on the first-order derivative of the system state is constructed. Second, the derivative of LKF is estimated by utilizing the Wirtinger-based integral inequality and extended reciprocally convex matrix inequality. The delay-dependent stability criterions are established in terms of linear matrix inequalities (LMIs) framework. The results show that the system performances are improved based on both enlarging the maximum allowable upper bound of the time-delays and reducing the number of decision variables. Furthermore, the conservatism of obtained delay-dependent stability criterion is reduced. Finally, a numerical simulation is given to demonstrate the effectiveness of obtained theoretical results. </abstract>

Stability analysis of network controlled micro- grid systems with communication delays and nonlinear perturbations

In this paper, the problem of delay-dependent stability of micro-grid load frequency control systems under networked environment with time-invariant delays and bounded nonlinear perturbations has been addressed using the Lyapunov- Krasovskii functional approach. In the networked control environment, it is observed that transfer of feedback variable from the sensor to centralized controller, and the control effort from controller back to the actuator through communication links introduces time-delays in the feedback path. The time-delays adversely affect the overall performance of the closed-loop system paving way to system instability. In addition, in distributed generation scenario, the uncertainties in the time-delayed microgrid system brought about by the penetration of fluctuating power generators, viz., solar and wind power combined with perturbations in the system load also affect the performance of the overall system. To assess the impact of these time-varying uncertainties to the closed-loop stability of the micro-grid system, they are included in the mathematical model of the system as a norm-bounded nonlinear perturbation term. Subsequently, the stated problem is solved in a less conservative manner by employing the classical Lyapunov-Krasovskii functional approach combined with Wirtinger inequality. The analysis results in a delay-dependent stability criterion in linear matrix inequality (LMI) framework. In the sequel, the presented stability criterion is validated on a standard benchmark system and supported with extensive simulation results.

The Lyapunov-Razumikhin theorem for the conformable fractional system with delay

&lt;abstract&gt;&lt;p&gt;This paper explicates the Razumikhin-type uniform stability and a uniform asymptotic stability theorem for the conformable fractional system with delay. Based on a Razumikhin-Lyapunov functional and some inequalities, a delay-dependent asymptotic stability criterion is in the term of a linear matrix inequality (LMI) for the conformable fractional linear system with delay. Moreover, an application of our theorem is illustrated via a numerical example.&lt;/p&gt;&lt;/abstract&gt;

Stability and control of fuzzy time delay systems with unmatched preconditions

The stability and control of nonlinear time-delay systems of Takagi-Sugeno (T-S) fuzzy model are studied in this paper. The integral inequality of a free weight matrix is chosen to give a less conservative delay-dependent stability criterion in the form of linear matrix inequalities (LMIs). The premise mismatch strategy is applied, it is combined with Finsler lemma, a more flexible design method of fuzzy state feedback controller is proposed. This method does not require the controller and system to share the common premise membership function and the number of rules. The controller design strategy proposed in this paper can effectively solve the control problem of fuzzy systems when the number of state variables is not equal to the number of input variables (r≠c), or mi⁢(x⁢(t))≠hi⁢(x⁢(t)),i=1,2,…,r. Finally, two simulation examples are given to prove the advancement and effectiveness of the proposed theory.

Stability analysis of network-controlled generator excitation system with interval time-varying delays

This paper deals with the problem of stability analysis of generator excitation system (GES) with interval time-varying delays. Network-controlled GES involves transmitting measured data from the plant site to controller and control signals from the controller to the plant site. Open communication channels impart time-varying nature to the delays. These time-varying delays can vary between intervals which would affect the system stability. The model of a GES is developed with the proportional integral (PI) controller including delay effects. In this paper, a less conservative delay-dependent stability criterion is derived using Lyapunov-Krasvoskii functional (LKF) for GES with interval time-varying delays. The bounding technique for derivative of LKF is developed by using Wirtinger inequality and free-weighting matrices. The relationship between the delay margins of GES and gains of the PI controller are investigated. This delay margin is used to tune the PI controller. The adequacy of the proposed result is confirmed by using simulation studies.

Novel Robust Stability Criteria of Uncertain Systems with Interval Time-Varying Delay Based on Time-Delay Segmentation Method and Multiple Integrals Functional

Interval time-varying delay is common in control process, e.g., automatic robot control system, and its stability analysis is of great significance to ensure the reliable control of industrial processes. In order to improve the conservation of the existing robust stability analysis method, this paper considers a class of linear systems with norm-bounded uncertainty and interval time-varying delay as the research object. Less conservative robust stability criterion is put forward based on augmented Lyapunov-Krasovskii (L-K) functional method and reciprocally convex combination. Firstly, the delay interval is partitioned into multiple equidistant subintervals, and a new Lyapunov-Krasovskii functional comprising quadruple-integral term is introduced for each subinterval. Secondly, a novel delay-dependent stability criterion in terms of linear matrix inequalities (LMIs) is given by less conservative Wirtinger-based integral inequality approach. Three numerical comparative examples are given to verify the superiority of the proposed approach in reducing the conservation of conclusion. For the first example about closed-loop control systems with interval time-varying delays, the proposed robust stability criterion could get MADB (Maximum Allowable Delay Bound) about 0.3 more than the best results in the previous literature; and, for two other uncertain systems with interval time-varying delays, the MADB results obtained by the proposed method are better than those in the previous literature by about 0.045 and 0.054, respectively. All the example results obtained in this paper clearly show that our approach is better than other existing methods.