A delay decomposition approach to delay‐dependent stability for linear systems with time‐varying delays

This paper is concerned with delay‐dependent stability for linear systems with time‐varying delays. By decomposing the delay interval into multiple equidistant subintervals, on which different Lyapunov functionals are chosen, and new Lyapunov‐Krasvskii functionals are then constructed. Employing these new Lyapunov‐Krasvskii functionals, some new delay‐dependent stability criteria are established. The numerical examples show that the obtained results are less conservative than some existing ones in the literature. Copyright © 2009 John Wiley & Sons, Ltd.     

Global tracking of nonlinear systems with simultaneous unknown time‐varying state and input delays

This paper presents the design of a robust control law for a class of nonlinear dynamical systems subjected to parametric uncertainty and simultaneous unknown, variable state and input delays. A novel controller is developed, which consists of a filtered tracking error and the integral of previous values of control input where the limits of integration are dependent on the known bound of the input delay. Lyapunov‐Krasovskii functionals–based stability analysis guarantees a global uniformly ultimately bounded tracking result where sufficient conditions on controller gains and maximum allowable delay are derived. The performance and robustness of the controller are evaluated by simulation on a two‐link robot manipulator for different combinations of time‐varying state and input delays.     

Less conservative stability criteria for linear systems with interval time‐varying delays

The problem of the stability of a linear system with an interval time‐varying delay is investigated. A new Lyapunov–Krasovskii functional that fully uses information about the lower bound of the time‐varying delay is constructed to derive new stability criteria. It is proved that the proposed Lyapunov–Krasovskii functional can lead to less conservative results than some existing ones. Based on the proposed Lyapunov–Krasovskii functional, two stability conditions are developed using two different methods to estimate Lyapunov–Krasovskii functional's derivative. Two numerical examples are given to illustrate that the two stability conditions are complementary and yield a larger maximum upper bound of the time‐varying delay than some existing results. Copyright © 2013 John Wiley & Sons, Ltd.     

Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

This paper studies mean square exponential stability of linear stochastic neutral‐type time‐delay systems with multiple point delays by using an augmented Lyapunov‐Krasovskii functional (LKF) approach. To build a suitable augmented LKF, a method is proposed to find an augmented state vector whose elements are linearly independent. With the help of the linearly independent augmented state vector, the constructed LKF, and properties of the stochastic integral, sufficient delay‐dependent stability conditions expressed by linear matrix inequalities are established to guarantee the mean square exponential stability of the system. Differently from previous results where the difference operator associated with the system needs to satisfy a condition in terms of matrix norms, in the current paper, the difference operator only needs to satisfy a less restrictive condition in terms of matrix spectral radius. The effectiveness of the proposed approach is illustrated by two numerical examples.     

Reduced‐order state estimation for linear time‐varying systems

We consider reduced‐order and subspace state estimators for linear discrete‐time systems with possibly time‐varying dynamics. The reduced‐order and subspace estimators are obtained using a finite‐horizon minimization approach, and thus do not require the solution of algebraic Lyapunov or Riccati equations. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A new result on the delay‐dependent stability of discrete systems with time‐varying delays

This paper proposes an improvement to the delay‐dependent stability of discrete systems with time‐varying delays. The approach is based on the observation that the positive definiteness of a chosen Lyapunov–Krasovskii functional does not necessarily require all the involved symmetric matrices to be positive definite, which has been overlooked in the literature. The derived delay‐dependent stability conditions are in terms of linear matrix inequalities. It is theoretically proved that our results are less conservative than the corresponding ones obtained by requiring the positive definiteness of all the symmetric matrices in a chosen Lyapunov–Krasovskii functional. The importance of the present approach is that a great number of delay‐dependent analysis and synthesis results obtained by the aforementioned requirement in the literature can be improved by the present approach without introducing any new decision variables. Copyright © 2013 John Wiley & Sons, Ltd.     

Stabilization of linear systems with time‐varying delays

This paper is concerned with the stability and stabilization problems for a class of time‐delayed systems, whose time‐varying delays are studied via Markovian approach. By separating the delay interval into several subintervals and by considering the inherent distribution of time‐varying delay, a new model is firstly developed. On the basis of the established model, a novel Lyapunov functional, which makes full use of each subinterval's delay bounds and the randomicity of time‐varying delay, is constructed to drive less conservative stability criteria. Especially sufficient conditions for the existence of stabilizing controllers are obtained as linear matrix inequalities, which are further used to deal with networked control systems. Finally, numerical examples are used to demonstrate the effectiveness of the proposed methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Polynomials‐based summation inequalities and their applications to discrete‐time systems with time‐varying delays

This paper proposes a novel summation inequality, say a polynomials‐based summation inequality, which contains well‐known summation inequalities as special cases. By specially choosing slack matrices, polynomial functions, and an arbitrary vector, it reduces to Moon's inequality, a discrete‐time counterpart of Wirtinger‐based integral inequality, auxiliary function‐based summation inequalities employing the same‐order orthogonal polynomial functions. Thus, the proposed summation inequality is more general than other summation inequalities. Additionally, this paper derives the polynomials‐based summation inequality employing first‐order and second‐order orthogonal polynomial functions, which contributes to obtaining improved stability criteria for discrete‐time systems with time‐varying delays. Copyright © 2017 John Wiley & Sons, Ltd.     

Fast adaptive fault estimation and accommodation for nonlinear time‐varying delay systems

This paper studies the problem of fault estimation and accommodation for a class of nonlinear time‐varying delay systems using adaptive fault diagnosis observer (AFDO). A novel fast adaptive fault estimation algorithm that does not need the derivative of the output vector is proposed to enhance the performance of fault estimation. Meanwhile, a delay‐dependent criteria is obtained based on free weighting matrix method with the purpose of reducing the conservatism of the AFDO design. On the basis of fault estimation, an observer‐based fault‐tolerant controller is designed to guarantee the stability of the closed‐loop system. In terms of matrix inequality, we derive sufficient conditions for the existence of the adaptive observer and fault‐tolerant controller. Simulation results are presented to illustrate the efficiency of the proposed method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Exponential H∞ filter design for stochastic time‐varying delay systems with Markovian jumping parameters

In this paper, the exponential H_∞ filter design problem is investigated for a general class of stochastic time‐varying delay system with Markovian jumping parameters. The stochastic uncertainties appear in both the dynamic and the measurement equations and the state delay is assumed to be time‐varying. Attention is focused on the design of mean‐square exponentially stable and Markovian jump filter such that the filtering error systems are mean‐square exponentially stable and the estimation error satisfies a given H_∞ performance. By introducing some slack matrix variables, delay‐dependent sufficient conditions for the solvability of the above problem are presented in terms of linear matrix inequalities (LMIs). In addition, the decay rate can be a given positive value without any other constraints. When the proposed LMIs are feasible, an explicit expression of the desired H_∞ filter can be given. A numerical example is provided to illustrate the effectiveness of the proposed design approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Consensus of multiagent systems with nonlinear dynamics and time‐varying communication delays

This paper deals with the local consensus of multiagent systems with nonlinear dynamics communication delays simultaneously. By introducing a weighted average state and applying the properties of the Laplacian matrix eigenvalues, the system is decoupled into several subsystems, firstly, to reduce complexity of theory analysis. Then, a new augmented vector containing single and double integral terms is constructed and the corresponding Lyapunov functional with triple integral terms is introduced. Meanwhile, in order to improve the estimating accuracy of the derivatives of constructed Lyapunov functional, single integral inequalities and double integral inequalities via auxiliary functions, an extended relaxed integral inequality and an reciprocally convex approach are used, as a result, stability criterion with less conservatism is derived, which guarantees the local consensus of the considered systems. Finally, numerical examples are provided to check the improvement of the proposed method over the existing works.     

Stability criteria for uncertain stochastic dynamic systems with time‐varying delays

In this paper, the problem of delay‐dependent stability for uncertain stochastic dynamic systems with time‐varying delay is considered. Based on the Lyapunov stability theory, improved delay‐dependent stability criteria for the system are established in terms of linear matrix inequalities. Three numerical examples are given to show the effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

New delay range–dependent stability criteria for interval time‐varying delay systems via Wirtinger‐based inequalities

Stability conditions for time‐delay systems using the Lyapunov‐based methodologies are generically expressed in terms of linear matrix inequalities. However, due to assuming restrictive conditions in deriving the linear matrix inequalities, the established stability conditions can be strictly conservative. This paper attempts to relax this problem for linear systems with interval time‐varying delays. A double‐integral inequality is derived inspired by Wirtinger‐based single‐integral inequality. Using the advanced integral inequalities, the reciprocally convex combination techniques and necessary slack variables, together with extracting a condition for the positive definiteness of the Lyapunov functional, novel stability criteria, have been established for the system. The effectiveness of the criteria is evaluated via 2 numerical examples. The results indicate that more complex stability criteria not only improve the stability region but also bring computational expenses.     

New insight into delay‐dependent stability of time‐delay systems

This paper presents a new insight into the delay‐dependent stability for time‐delay systems. Because of the key observation that the positive definiteness of a chosen Lyapunov–Krasovskii functional does not necessarily require all the involved symmetric matrices in the Lyapunov–Krasovskii functional to be positive definite, an improved delay‐dependent asymptotic stability condition is presented in terms of a set of LMIs. This fact has been overlooked in the development of previous stability results. The importance of the present method is that a vast number of existing delay‐dependent results on analysis and synthesis of time‐delay systems derived by the Lyapunov–Krasovskii stability theorem can be improved by using this observation without introducing additional variables. The reduction of conservatism of the proposed result is both theoretically and numerically demonstrated. It is believed that the proposed method provides a new direction to improve delay‐dependent results on time‐delay systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust stability of nonlinear time‐delay systems with interval time‐varying delay

This paper deals with the problem of obtaining delay‐dependent stability conditions and L₂‐gain analysis for a class of nonlinear time‐delay systems with norm‐bounded and possibly time‐varying uncertainties. No restrictions on the derivative of the time‐varying delay are imposed, though lower and upper bounds of the delay interval are assumed to be known. A Lyapunov–Krasovskii functional approach is proposed to derive novel delay‐dependent stability conditions which are expressed in terms of linear matrix inequalities (LMIs). To reduce conservatism, the work exploits the idea of splitting the delay interval in multiple regions, so that specific conditions can be imposed to a unique functional in the different regions. This improves the computed bounds for certain delay‐dependent integral terms, providing less conservative LMI conditions. Examples are provided to demonstrate the reduced conservatism with respect to the available results in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

Asymptotic stabilization of high‐order feedforward systems with delays in the input

This paper deals with the state feedback controller design for a class of high‐order feedforward (upper‐triangular) nonlinear systems with delayed inputs. The uncertainties in the systems are assumed to be dominated by higher‐order nonlinearities multiplying by a constant growth rate. The designed controller, which is a continuous but not smooth feedback, could achieve global asymptotical stability. Based on the appropriate state transformation of time‐delay systems, the problem of controller design can be converted into the problem of finding a parameter, which can be obtained by appraising the nonlinear terms of the systems. The nonlinear systems considered here are more general than conventional feedforward systems and they could be viewed as generalized feedforward systems. Two examples are given to show the effectiveness of the proposed design procedure. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive discrete neural observer design for nonlinear systems with unknown time‐delay

This paper focuses on the adaptive observer design for nonlinear discrete‐time MIMO systems with unknown time‐delay and nonlinear dynamics. The delayed states involved in the system are arguments of a nonlinear function and only the estimated delay is utilized. By constructing an appropriate Lyapunov–Krasovskii function, the delay estimation error is considered in the observer parameter design. The proposed method is then extended to the system with a nonlinear output measurement equation and the delayed dynamics. With the help of a high‐order neural network (HONN), the requirement for a precise system model, the linear‐in‐the‐parameters (LIP) assumption of the delayed states, the Lipschitz or norm‐boundedness assumption of unknown nonlinearities are removed. A novel converse Lyapunov technical lemma is also developed and used to prove the uniform ultimate boundedness of the proposed observer. The effectiveness of the proposed results is verified by simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

Stability of discrete‐time systems with time‐varying delay via a novel Lyapunov‐Krasovskii functional

This article is concerned with the problem of stability analysis of discrete‐time systems with time‐varying delay. Unlike previous articles concentrating on the improvement of summation inequalities, this article constructs a novel augmented Lyapunov‐Krasovskii (L‐K) functional by fully considering the coupling information among state‐related vectors. Based on the newly developed L‐K functional, a more relaxed stability criterion is derived by exploring the interest of existing inequalities. Three numerical examples are given to demonstrate the effectiveness of the proposed method.     

On robust stability of uncertain systems with multiple time‐delays

On the basis of an infinite to one mapping and the structure of the null space of a multivariate matrix polynomial (MMP), a novel sufficient condition is derived in this paper for the robust stability of a linear time‐invariant system with multiple uncertain time‐delays, parametric modelling errors and unmodelled dynamics. This condition depends on time‐delay bounds and is less conservative than the existing ones. An attractive property is that this condition becomes also necessary in some physically meaningful situations, such as the case that there is only one uncertain time‐delay and neither parametric perturbations nor unmodelling errors exist. Moreover, using ideas of representing a positive‐definite MMP through matrix sum of squares, an asymptotic necessary and sufficient condition is derived for the robust stability of this system. All the conditions can be converted to linear matrix inequalities. Copyright © 2009 John Wiley & Sons, Ltd.     

High‐order sliding‐mode observer for linear time‐varying systems with unknown inputs

In this article, an observer for linear time variant systems affected by unknown inputs is suggested. The proposed observer combines the deterministic least squares filter and the high‐order sliding‐mode differentiator to provide exact state reconstruction in spite of bounded unknown inputs and system instability. The cascade structure of the algorithm provides a correct state reconstruction for the class of linear time variant systems that satisfy the structural property of strong observability. Simulations illustrate the performance of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.