Conservative Robust Stability Research Articles

Improvement master–slave robustly synchronous criteria of uncertain chaotic Lur’e systems via an augmented Lyapunov–Krasovskii functional

The robust synchronization of uncertain master–slave Lur’e systems based on time-delayed feedback control is further investigated. A less conservative synchronization stability criterion and a less conservative robust synchronization stability criterion than some recent published references are proposed via Lyapunov stability theory. First, an augmented delay-dependent Lyapunov–Krasovskii functional (LKF) is constructed, where some single integral terms are augmented to the integrand component of a single-integral subfunction. This not only increases some coupling information on some system variables but also increases the coupling information on some necessary variables contained in the inequality lemmas, which can make full use of the inequality lemmas and reduce the conservatism of the synchronization stability criterion. Second, to overcome the nonlinear phenomena in the synchronization criterion, the novel negative definite inequality equivalent transformation lemma is used to transform the nonlinear inequalities to the linear matrix inequalities (LMIs) equivalently, which can be easily solved by the MATLAB LMI-Toolbox. Finally, some common numerical examples are presented to show the effectiveness of the proposed approach.

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.

Stability Analysis and Robust H∞ Controller Synthesis with Derivatives of Membership Functions for T-S Fuzzy Systems with Time-varying Delay: Input-output Stability Approach

This paper investigates the problem of delay dependent stability and H∞ control design with derivatives of membership functions of uncertain Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delay. A model transformation is employed by considering a three-term approximation of delayed state vector. Using Scaled Small Gain (SSG) theorem and fuzzy weighting-dependent Lyapunov functions with some useful slack variables, less conservative robust stability and stabilization criteria are formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by using standard numerical packages. Finally, numerical experiments are presented to illustrate the effectiveness of the proposed method.

Iterative Identification and Control Using Non-normalized Coprime Factors With Application in Wafer Stage Motion Control

Robustness against model uncertainty is essential in model-based controller design. It is well known that a relatively small uncertainty in lightly damped poles and zeros can result in a large distance measured in the $\nu $ -gap metric, leading to conservative robust stability and performance guarantees. This paper aims to develop an identification and control procedure that results in less conservative robust stability and performance conditions for linear systems with lightly damped poles and zeros. To achieve this, a connection is established between a distance measure based on a nonnormalized coprime factorization of the system and existing identification criteria in closed-loop system identification. A nominal model of the system is determined by minimizing this distance measure by means of a frequency-domain identification algorithm. Then, a controller synthesis method is proposed that addresses both nominal performance as robust stability. Improved robustness by using the proposed approach compared to existing approaches is confirmed in an experimental example for a system with lightly damped poles and zeros.

Stability analysis and design of nonlinear sampled‐data systems under aperiodic samplings

SummaryIn this paper, the problem of exponential stability analysis and the design of sampled‐data nonlinear systems have been studied using a polytopic linear parameter‐varying approach. By means of modeling a new double‐layer polytopic formulation for nonlinear sampled‐data systems, a modified form of piecewise continuous Lyapunov‐Krasovskii functional is proposed. This approach provides less conservative robust exponential stability conditions by using Wirtinger's inequality in terms of linear matrix inequalities. The distances between the real continuous parameters of the plant and the measured parameters of the controller are modeled by convex sets, and the analysis/design conditions are given at the vertices of some hyper‐rectangles. In order to get tractable linear matrix inequality conditions for the stabilization problem, we performed relaxation by introducing a slack variable matrix. Under the new stability criteria, an approach is introduced to synthesize a sampled‐data polytopic linear parameter‐varying controller considering some constraints on the location of the closed‐loop poles in the presence of uncertainties on the varying parameters. It is shown that the proposed controller guarantees the exponential stability of the closed‐loop system for aperiodic sampling periods smaller than a known value, ie, maximum allowable sampling period. Finally, the effectiveness of the proposed approach is verified and compared with some state‐of‐the‐art existing approaches through numerical simulations.

Robust controller design method for linear systems with parametric uncertainty using minimum eigenvalue approach

In this paper, a static state feedback robust controller design method for linear systems with parametric uncertainty is presented. The closed-loop stability analysis is carried out in the sense of Lyapunov and the controller design parameters with constraints are formulated in linear matrix inequality (LMI) framework. A less conservative robust stability result is obtained by introducing a new set of sufficient conditions developed which contains the information of minimum eigenvalue of the perturbation matrices. The control effort is also reduced by imposing additional constraints in the designed framework. The proposed design technique is illustrated by considering several numerical examples that show the superiority of the proposed technique over the existing methods.

Novel Stability Analysis for Uncertain Neutral-Type Lur’e Systems with Time-Varying Delays Using New Inequality

This paper considers the delay-dependent stability analysis of neutral-type Lur’e systems with time-varying delays and sector bounded nonlinearities. First of all, using constructed function methods, a new Jensen-like inequality is introduced to obtain less conservative results. Second, a new class of Lyapunov-Krasovskii functional (LKF) is constructed according to the characteristic of the considered systems. Third, combining with the new inequality and reciprocal convex approach and some other inequality techniques, the new less conservative robust stability criteria are shown in the form of linear matrix inequalities (LMIs). Finally, three examples demonstrate the feasibility and the superiority of our methods.

Less conservative robust absolute stability criteria for uncertain neutral-type Lur׳e systems with time-varying delays

This paper focuses on robust absolute stability analysis for uncertain neutral-type Lur׳e systems with time-varying delays. New delay-dependent and delay-rate-dependent stability criteria in terms of linear matrix inequalities (LMIs) are established by: (i) utilizing fully the information of delays to construct Lyapunov–Krasovskii functional (LKF) and augmented vectors; (ii) employing improved integral inequalities to estimate the derivative of LKF; and (iii) using the so-called reciprocally convex combination approach and convex combination approach simultaneously. The main contribution of the present result is to reduce conservativeness of previous results in the LMI framework. Several numerical examples are carried out to demonstrate the effectiveness and merits of the present result.

Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays

In this paper, we consider the problem of delay-interval-dependent robust stability and stabilization of a class of linear uncertain neutral-type systems with time-varying delay. By constructing a candidate Lyapunov–Krasovskii functional (LKF), that takes into account the delay-range information appropriately, less conservative robust stability criteria are proposed in terms of linear matrix inequalities (LMIs) to compute the maximum allowable upper bounds (MAUB) for the delay-interval within which the uncertain neutral-type system under consideration remains asymptotically stable. The verifiable stabilizability conditions and memoryless state feedback control design are stated. Finally, numerical examples are also designated to demonstrate the effectiveness and reduced conservatism of the developed results.

Stability and passivity analysis for uncertain discrete-time neural networks with time-varying delay

In this paper, the problem of stability and passivity analysis for uncertain discrete-time neural networks with time-varying delay is investigated. By constructing a new Lyapunov–Krasovskii functional and employing a novel summation inequality which is a discrete-time counterpart of the Wirtinger-based integral inequality, a less conservative robust stability criterion is derived in terms of linear matrix inequalities. Furthermore, a new sufficient condition is established to assure the considered neural networks to be passive. Several numerical examples are provided to demonstrate the effectiveness of the proposed method.

Improved robust exponential stability for Takagi–Sugeno fuzzy uncertain systems with time-varying delays

This paper concerns managing the robust exponential stability problem of uncertain Takagi–Sugeno fuzzy systems with time-varying delay by employing a further improved integral inequality matrix approach. Based on the linear matrix inequality (LMI) approach, delay-dependent robust exponential stability criteria have been developed. By taking the relationship among the time-varying delay, its upper bound and their difference into account, some less conservative LMI-based delay-dependent robust exponential stability criteria are obtained without ignoring any useful terms in the derivative of Lyapunov–Krasovskii functionals. Maximum allowable upper bound for time-varying delays is determined. Numerical examples are provided to show that the obtained results significantly improve the allowed upper bounds of delay size over some methods existing in the literature.

Improved stability conditions for uncertain neutral-type systems with time-varying delays

This paper investigates the robust stability problem for a class of uncertain neutral-type delayed systems. The systems under consideration contain parameter uncertainties and time-varying delays. We aim at designing less conservative robust stability criteria for such systems. A new second-order reciprocally convex inequality is first proposed in order to deal with double integral terms. Then, by constructing a new Lyapunov– Krasovskii functional and employing the improved Wirtinger-based integral inequality and the reciprocally convex combination approaches, novel stability criteria are obtained. Moreover, the stability conditions for standard time-delay system are obtained as by-product results. Comparisons in three numerical examples illustrate the effectiveness of our results.

Robust Delay-Dependent Stability Criteria for Time-Varying Delayed Lur’e Systems of Neutral Type

This paper deals with the problem of the robust delay-dependent stability of uncertain Lur'e systems with neutral-type time-varying delays. By constructing a set of Lyapunov---Krasovskii functional, less conservative robust stability criteria are derived in terms of linear matrix inequalities. The contribution in reduced conservation of the proposed stability criteria relies on the reciprocally convex method and Wirtinger inequality, which provides tighter upper bound than Jensen inequality. Three numerical examples are provided to show the effectiveness of the proposed method.

Robust stability analysis of a class of uncertain neutral T–S fuzzy systems with time delay

We consider the problem of the robust stability of a class of uncertain T–S fuzzy neutral systems with time delay under time-varying parametric uncertainties using the Lyapunov–Krasovskii approach, where the parametric uncertainty is assumed to be norm bounded. By choosing a new Lyapunov–Krasovskii function, we are able to propose less conservative robust stability criteria in terms of linear matrix inequalities (LMIs) such that the uncertain T–S fuzzy neutral system under consideration remains asymptotically stable. The reduced conservatism of the proposed stability criterion compared with recently reported results is attributed to our using the Jensen inequality. The obtained results can also reduce the computational complexity.

Less conservative robust stability criteria for uncertain discrete stochastic singular systems with time-varying delay

The problem of robust stabilisation for uncertain discrete singular systems with time-varying delay is discussed in this article. First, a new delay-distribution-dependent Lyapunov function is established. Then, sufficient conditions are proposed for guaranteeing that the uncertain discrete singular system with time-varying delay has the properties of regularity, causality and stability. Finally, a suitable robust state feedback controller is designed by linear matrix inequality. Numerical examples illustrate that the results derived from the proposed method are less conservative than the existing ones.

Improved delay-range-dependent robust stability criteria for a class of Lur’e systems with sector-bounded nonlinearity

In this paper, the problem of delay-dependent stability of a class of uncertain Lur’e systems of neutral type with interval time-varying state delay and sector-bounded nonlinearity has been considered based on Lyapunov–Krasovskii functional approach. By constructing a candidate Lyapunov–Krasovskii (LK) functional, less conservative robust stability criteria are proposed in terms of linear matrix inequalities (LMIs). The reduction in conservatism of the proposed stability criteria over recently reported results is attributed to the candidate LK functional used in the delay-dependent stability analysis, and to the tighter bounding of the time-derivative of the functional without neglecting any useful terms using minimal number of slack matrix variables. The proposed analysis, subsequently, yields a stability condition in convex LMI framework, and is solved non-conservatively at boundary conditions using standard numerical packages. The effectiveness of the proposed stability criterion is demonstrated through standard numerical examples.

Robust Stability Criteria for Uncertain Neutral Systems with Interval Time-Varying Delay

In this paper, we consider the problem of robust stability of a class of linear uncertain neutral systems with interval time-varying delay under (i) nonlinear perturbations in state, and (ii) time-varying parametric uncertainties using Lyapunov-Krasovskii approach. By constructing a candidate Lyapunov-Krasovskii (LK) functional, that takes into account the delay-range information appropriately, less conservative robust stability criteria are proposed in terms of linear matrix inequalities (LMI) to compute the maximum allowable bound for the delay-range within which the uncertain neutral system under consideration remains asymptotically stable. The reduction in conservatism of the proposed stability criterion over recently reported results is attributed to the fact that time-derivative of the LK functional is bounded tightly without neglecting any useful terms using a minimal number of slack matrix variables. The analysis, subsequently, yields a stability condition in convex LMI framework, that can be solved non-conservatively at boundary conditions using standard LMI solvers. The effectiveness of the proposed stability criterion is demonstrated through standard numerical examples.

Robust Delay-dependent Exponential Stability of Uncertain Stochastic System with Time-varying Delay

This paper discusses the problem of robust delay-dependent exponential stability for a class of uncertain stochastic systems with time-varying delay. The uncertainty is assumed to be of norm-bounded form. A less conservative robust exponential stability condition is derived by using a new Lyapunov–Krasovskii functional and a free-weighting matrix method in terms of linear matrix inequalities. Two numerical examples are given to illustrate the effectiveness of the proposed method.

Novel Delay-Dependent Robust Stability Analysis for Switched Neutral-Type Neural Networks With Time-Varying Delays via SC Technique

This paper studies a class of new neural networks referred to as switched neutral-type neural networks (SNTNNs) with time-varying delays, which combines switched systems with a class of neutral-type neural networks. The less conservative robust stability criteria for SNTNNs with time-varying delays are proposed by using a new Lyapunov-Krasovskii functional and a novel series compensation (SC) technique. Based on the new functional, SNTNNs with fast-varying neutral-type delay (the derivative of delay is more than one) is first considered. The benefit brought by employing the SC technique is that some useful negative definite elements can be included in stability criteria, which are generally ignored in the estimation of the upper bound of derivative of Lyapunov-Krasovskii functional in literature. Furthermore, the criteria proposed in this paper are also effective and less conservative in switched recurrent neural networks which can be considered as special cases of SNTNNs. The simulation results based on several numerical examples demonstrate the effectiveness of the proposed criteria.

A less conservative robust stability criteria for uncertain neutral systems with mixed delays

This paper is concerned with the problem of the delay-dependent robust stability of neutral systems with mixed delays and time-varying structured uncertainties. By considering the cross-terms with additional design parameters, a complete form of Lyapunov–Krasovskii functional is constructed. Then some free weighting matrices are introduced, and a sufficient stability condition is obtained in terms of linear matrix inequalities. The obtained criterion is dependent on the sizes of neutral delay and discrete delay and are less conservative than those produced by previous approaches. Numerical examples are given to illustrate both the improvement the proposed method provides over some existing ones and the relationship between the neutral- and discrete-delay.