## No full-text available

To read the full-text of this research,

you can request a copy directly from the author.

The discretized Lyapunov functional method for the stability problem of time-delay systems is further refined using a combination of integral inequality and variable elimination technique. As a result, the computational requirement is further reduced for the same discretization mesh. For systems without uncertainty, the convergence to the analytical solution is greatly accelerated. For uncertain systems, the new method is much less conservative. Numerical examples are presented to illustrate the effectiveness of the method.

To read the full-text of this research,

you can request a copy directly from the author.

... More than a decade ago, some researchers paid much attention to the stability of coupled differential-difference system(CDDS) because of its specially mathematical characteristics. See, [3] [9][19] [20]. It is well known that many hyperbolic type of partial differential equations can be converted into the lossless propagation systems [1] [12] [18] ,the lossless propagation model can be represented the special case of coupled differential-difference equations(CDDEs) [21] [22] . ...

... It shows sufficiently that the difference operator D(φ) is Schur stable. , φ), the LKF V 0 is completely similar to the one in [3]. Then the positive definite characterization of S i , i = 1, 2, . . . ...

... Then the positive definite characterization of S i , i = 1, 2, . . . , N and LMI (18) corresponds to the equations (31) and (32) of Proposition 3 in [3]. The LKF condition (14) is established. ...

... Generally speaking, time delays are the sources of instability and poor control performance. Since stability is a basic requirement of control systems, the stability analysis of time-delay systems is the foundation for the analysis of time-delay systems and has drawn widely attention in the past few decades [1]- [9]. ...

... IV. STABILITY ANALYSIS OF TIME-DELAY SYSTEMS In this section, we construct a polynomial-type LKF and adopt the Jacobi-Bessel inequality to derive a stability criterion for system (1). ...

... Then, one may note that all derivatives of (1 − u) k u k up to the (k − 1)-st vanish at the endpoints of the interval [1,0]. Therefore, for any squareintegrable function f , after integration by parts k times, we have ...

To derive a less conservative stability criterion via Lyapunov-Krasovskii functional (LKF) method, in previous literature, multiple integral terms are usually introduced into the construction of LKFs. This paper generalizes the results of previous literature by proposing a polynomial-type LKF, which contains the LKFs with multiple integral terms as special cases. In addition, a Jacobi-Bessel inequality is presented to bound the derivative of such LKF. As a result, an improved stability criterion of time-delay systems is established. Finally, two numerical examples are given to illustrate the effectiveness and advantages of our method.

... 7. [62,66], [63, p. 321] Consider P 1 ∈ S n , P 2 , X ∈ S q , Q 1 ∈ R n×m , Q 2 ∈ R q×p , R 1 ∈ S m , and R 2 ∈ S p . The matrix inequalities given by ...

... 8. [62,66] Consider P ∈ S n , R ∈ S m , S ∈ S p , Q ∈ R n×m , X ∈ R n×p , V ∈ R m×p , and E ∈ R p×m . The matrix inequalities given by ...

... Proof. The proof is found in [66] and is very similar to the proof of Property 2. ...

Linear matrix inequalities (LMIs) commonly appear in systems, stability, and control applications. Many analysis and synthesis problems in these areas can be solved as feasibility or optimization problems subject to LMI constraints. Although most well-known LMI properties and manipulation tricks, such as the Schur complement and the congruence transformation, can be found in standard references, many useful LMI properties are scattered throughout the literature. The purpose of this document is to collect and organize properties, tricks, and applications related to LMIs from a number of references together in a single document. In this sense, the document can be thought of as an "LMI encyclopedia" or "LMI cookbook." Proofs of the properties presented in this document are not included when they can be found in the cited references in the interest of brevity. Illustrative examples are included whenever necessary to fully explain a certain property. Multiple equivalent forms of LMIs are often presented to give the reader a choice of which form may be best suited for a particular problem at hand. The equivalency of some of the LMIs in this document may be straightforward to more experienced readers, but the authors believe that some readers may benefit from the presentation of multiple equivalent LMIs.

... As is well known, the Lyapunov functional method has received more and more attention in recent years due to its effectiveness in many problems, such as the problem of control for linear neutral systems, the problem of delay-dependent stability (DDS) for lin-ear neutral systems (LNSS), etc. Based on this method, many interesting results on less conservative DDS conditions have been obtained (see, e.g., [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]). It is worth noting that [24][25][26][27][28][29][30]42] mainly considered the neutral and discrete delay. ...

... Based on this method, many interesting results on less conservative DDS conditions have been obtained (see, e.g., [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]). It is worth noting that [24][25][26][27][28][29][30]42] mainly considered the neutral and discrete delay. In fact, a lot of practical applications are modeled by systems with distributed delay. ...

... On the other hand, DDS conditions are often obtained by the Lyapunov functionals theorem accompanied by some important techniques. Many important methods include the bounding inequalities for the cross term [24], a descriptor model transformation [25], the free-weighting matrix technique [26], integral inequality (II) methods (see [27,28,42]), delay decomposition [29], and discretized LKFs [30]. Generally speaking, the II method has always played a very important role in acquiring a DDS condition. ...

Abstract 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.

... To make the system stable under bounded uncertainty, the form of free matrix is improved using Leibniz-Newton formula, and a convex optimization algorithm is proposed. 5, 6 Wu et al. 7 considered the influence of the Leibniz-Newton formula when replacing the delay term 8 and introduced the free weighting matrix (FWM) to eliminate the influence. The introduced FMW was determined by linear matrix inequality (LMI), and the criterion with less conservatism was obtained. ...

... The vehicle trajectory tracking pose error variable is derived along the trajectories equations (3) and (8). Get the tracking pose error equation as follows: ...

Trajectory tracking control of autonomous underwater vehicles in three-dimension always suffers disturbances such as input time delays and model uncertainties. Regarding this problem, an integral time-delay sliding mode control law is proposed in this article with dividing the vehicle’s input time delays model into cascade system consisting of a kinematics subsystem and a dynamics subsystem. Based on the established pose error equation and velocity error equation, a suitable Lyapunov–Krasovskii functional is given to analyze and guarantee the global stability of the whole system under reasonable assumptions. At last, comparative simulations are presented to demonstrate the effectiveness of the proposed method.

... The simulation study of [2] is used to analyze the influences of the model parameters on the stability of human-robot interaction. In [15,16] the authors analyze their system with the linear matrix inequality (LMI) approach of [24] and [25, p. 191] resulting in a numerical stability bound. The authors of [13] derive analytical stability bounds in the case of a neglected response time. ...

... For a graphical comparison, see Fig. 5. In addition to the conditions listed in TABLE II the figure also shows the stability bound of the numerical LMI approach [24] for time-delay systems with Proposition 5.22 from [25], the stability regions determined by simulations of a continuous system with a timedelay and the Frequency-sweeping test (FST) [33]. The figure [13] filter ...

In this paper, we present sufficient conditions for the stability analysis of a stationary point for a special type of nonlinear time-delay systems. These conditions are suitable for analyzing systems describing physical human-robot interaction (pHRI). For this stability analysis a new human model consisting of passive and active elements is introduced and validated. The stability conditions describe parametrization bounds for the human model and an impedance controller. The results of this paper are compared to stability conditions based on passivity, approximated time-delays and to numerical approaches. As a result of the comparison, it is shown that our conditions are more general than the Passivity condition of Colgate [1]. This includes the consideration of negative stiffness and nonlinear virtual environments. As an example, a pHRI including a nonlinear virtual environment with a polynomial structure is introduced and also successfully analyzed. These theoretical results could be used in the design of robust controllers and stability observers in pHRI.

... Therefore, stability analysis of delayed neural networks has become an important support for many research topics. Many methods have been proposed to optimize and improve this analysis,such as model transformation [7], free-weighting matrices technique [8], reciprocally convex optimization [9], and delaypartitioning approach [10,11]. ...

This article investigates the stability of time-delay neural network system by the free-weighting matrices based on variable-augmentation.In the process of accurately evaluating the stability of delayed neural network systems, the conservative simplification of stability criteria has received widespread attention. In this paper, a variable-augmented-based free-weighting matrix method is applied to time-varying delayed neural network systems, and unnecessary free weighting matrices are removed. Some results with fewer free-weighting matrices are obtained, while maintaining their stability and conservatism. Finally, a specific neural network numerical example was used to demonstrate the effectiveness of this method.

... The Jensen integral inequality was first used by Gu [15] to improve the stability conditions of time-delay systems. The Wirtinger-based integral inequality (WBII) [16] was introduced to obtain a tighter lower bound for the quadratic form in the integral term than that provided by Jensen's inequality [17]. Park [18] presented the Wirtinger-based double integral inequality for the quadratic double integral form in the stability of time-delay systems. ...

This work investigates the stability conditions for linear systems with time-varying delays via augmented Lyapunov-Krasovskii functional(LKF). Two types of augmented LKFs with cross terms in integral are suggested to improve the stability conditions for interval time-varying linear systems. Through this work, the compositions of the LKFs are considered to enhance the feasible region of stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality(WBII), zero equalities, reciprocally convex approach, and Finsler’s lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality(LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. And then simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in two examples. Two numerical examples show that the proposed Theorem 1 and 2 obtained less conservative results than existing outcomes. Also, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1’s in some specific systems.

... The Jensen integral inequality was first used by Gu [19] to improve the stability conditions of time-delay systems. The Wirtinger-based integral inequality (WBII) [20] was introduced to obtain a tighter lower bound for the quadratic form in the integral term than that provided by Jensen's inequality [21]. Park [22] presented the Wirtinger-based double integral inequality for the quadratic double integral form in the stability of time-delay systems. ...

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler’s lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems.

... More specifically, exponential stability is desirable for some applications [13,2,7,31]. To this end, various approaches have been developed to the subject, which include techniques of free weighting matrices [4], reciprocally convex optimization [23] and delay-partitioning [3,33]. Exponential stability analysis of time-delay neural networks aims at deriving an admissible delay upper bound (ADUB) such that the delayed neural networks are stable for all time-delays less than the obtained ADUB. ...

... It was also pointed out in [4] that systems with multiple commensurate delays may be rewritten as a coupled differential-difference equation with single delay. Indeed, several order of magnitude of saving of computational time have been reported in both the discretized Lyapunov-Krasovskii functional method [4] and sum-of-square method [15] due to reduced dimension of the delayed variable y as compared with the more traditional formulation (such as those given in [5] and [9]). ...

This article discusses the invariant subspaces that are restricted to be external direct sums. Some existence conditions are presented that facilitate finding such invariant subspaces. This problem is related to the decomposition of coupled differential-difference equations, leading to the possibility of lowering the dimensions of coupled differential-difference equations. As has been well documented, lowering the dimension of coupled differential-difference equations can drastically reduce the computational time needed in stability analysis when a complete quadratic Lyapunov-Krasovskii functional is used. Most known ad hoc methods of reducing the order are special cases of this formulation.

... The converse results of the theory of Krasovskii, which guarantee the existence of the functional when the system is stable, have been less popular, although the general form of the functional introduced by Repin (1965) and Datko (1972) has been a source of inspiration for the determination of sufficient stability conditions in Gu (2001) and Peet and Bliman (2011). ...

An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. The main results and proof are presented in details for the case of systems with multiple delays. The state of the art, ongoing research and potential extensions to other classes of delay systems are discussed.

... In recent years, complex programs have been developed with the aim of improving adaptability [8,9] and robustness [10,11]; consequently, the computational load increases, and the negative effects of the discretization emerge. As shown in the literature [12][13][14], discretization results in different dynamic behavior and is one of the reasons for the instability of digital control systems. Therefore, the task scheduling period should be selected carefully considering both the control performance and the limitations of the computational resources. ...

The clutch engagement process involves three phases known as open, slipping, and locked and takes a few seconds. The engagement control program runs in an embedded control unit, in which discretization may induce oscillation and even instability in the powertrain due to an improper scheduling period for the engagement control task. To properly select the scheduling period, a methodology for control–scheduling co-design during clutch engagement is proposed. Considering the transition of the friction state from slipping to being locked, the co-design framework consists of two steps. In the first step, a stability analysis is conducted for the slipping phase based on a linearized system model enveloping the driving and driven part of the clutch, feed-forward and feedback control loop together with a zero-order signal hold element. The critical period is determined according to pole locations, and factors influencing the critical period are investigated. In the second step, real-time hardware-in-the-loop experiments are carried out to inspect the dynamic response concerning the friction state transition. A sub-boundary within the stable region is found to guarantee the control performance to satisfy the engineering requirements. In general, the vehicle jerk and clutch frictional loss increase with the increase in the scheduling period. When the scheduling period is shorter than the critical period, the rate of increase is mild. However, once the scheduling period exceeds the critical period, the rate of increase becomes very high.

... Complete Lyapunov functional methods can provide necessary and sufficient conditions on the stability of linear systems with a constant time-delay [18][19][20][21]. Simple LKF methods only provide sufficient conditions on the stability of linear time-delay systems. ...

... Thus, much effort has been invested in reducing the conservatism of delay-dependent stability criteria. For example, in [7], [19], [20], [21], different sorts of LKF has been well considered to reduce the conservatism of developed stability conditions. Moreover, to develop a novel integral inequality for quadratic functions plays a significant role in the issue of stability of NNs. ...

This paper investigates the problem of delay-dependent stability analysis for Markovianjump static neural networks (MJSNNs) with mode-dependent time-varying delays.Thefundamental objective of this paper is to create novel stability criterion for the consideredMJSNNs with less conservatism.A suitable Lyapunov-Krasovskii functional (LKF) isconstructed with more system information.By employing integral inequality, a noveldelay-dependent sufficient condition is obtained to ensure the asymptotically stability of theequilibrium point. The obtained stability condition is derived and entrenched in terms of linearmatrix inequality (LMI) which can be clearly checked by MATLAB LMI control toolbox. Atlong last, two benchmark illustrative case are given to show the effectiveness of the theoreticalresult.

... · · · · · · · · · · · · * * · · · −(n!) 2 ...

This paper addresses the global stabilization of uncertain single input and single output (SISO) dynamical systems using sliding mode control with multiple delayed partial state feedback. It is shown that using only delayed partial state information, it is possible to stabilize a large class of SISO uncertain dynamical systems. Artificial stabilizing delay is introduced in the partial state feedback to make the reduced-order sliding mode dynamics
of a given system asymptotically stable. It is demonstrated that despite the order of dynamical systems, using the information of just three states together with their delayed terms, can asymptotically stabilize the uncertain SISO dynamical system. A numerical example of a ball and wheel system is provided to demonstrate the efficacy of the proposed method.

... • Les auteurs de [SG15] ont prouvé l'existence d'une hiérarchie des conditions de stabilité pour différentes valeurs de N , qui assure qu'augmenter le degré N des conditions de stabilité résultantes ne peut que réduire le pessimisme du résultat. • Enfin, les résultats obtenus dans [SG15] montrent un compromis des conditions de stabilité entre le nombres de variables à determiner et le pessimisme du résultat numérique par rapport aux méthodes existantes basées sur la discrétisation ou de partition de l'intervalle du retard [Gu97] ; [Gu01] ; [Zha+09] ou encore sur les sommes de carrées [PPL09] ; [Pee14]. ...

... For systems with a linear environment, the authors of Tsumugiwa et al. (2004) used a simulation study to analyse the influences of the model parameters on the stability. In Müller et al. (2015), the authors analyse their nonlinear system with the linear matrix inequality (LMI) approach of Gu (2001) and Gu, Kharitonov, and Chen (2003, p. 191) which results in a numerical stability bound for the parameters. This method is based on the Lyapunov-Krasovskii functional which is a time-domain approach. ...

Human-Robot-Man Interaction (HRH), understood as a physical human-robot interaction (pHRI) with two humans, can be applied when lifting heavy, bulky and large-sized objects with a robot. In combination with a virtual environment, this system can become non-linear. In this article we prove sufficient stability conditions for a stationary point of such a particular type of non-linear multiple time-delay systems. In addition, a new human model consisting of a passive and an active part will be introduced and validated on experimental data. The derived stability conditions are applied to a single-user pHRI system including this human model. The results indicate that these conditions are very conservative. Then four approaches for the analysis of a multi-user pHRI are introduced and compared with each other. Finally, a potential HRH application with a nonlinear environment in the form of a potential force field is presented.

... Specifically, there are complete Lyapunov functional methods and simple Lyapunov-Krasovskii functional methods for estimating the maximum admissible delay upper bound that the system can tolerate and still maintain stability. Complete Lyapunov functional methods can provide necessary and sufficient conditions on stability of linear systems with a constant time-delay [25][26][27][28]. Simple Lyapunov-Krasovskii functional methods only provide sufficient conditions on stability of linear time-delay systems. ...

This study provides an overview and in-depth analysis of recent advances in stability of linear systems with time-varying delays. First, recent developments of a delay convex analysis approach, a reciprocally convex approach and the construction of Lyapunov–Krasovskii functionals are reviewed insightfully. Second, in-depth analysis of the Bessel–Legendre inequality and some affine integral inequalities is made, and recent stability results are also summarised, including stability criteria for three cases of a time-varying delay, where information on the bounds of the time-varying delay and its derivative is totally known, partly known and completely unknown, respectively. Third, a number of stability criteria are developed for the above three cases of the time-varying delay by employing canonical Bessel–Legendre inequalities, together with augmented Lyapunov–Krasovskii functionals. It is shown through numerical examples that these stability criteria outperform some existing results. Finally, several challenging issues are pointed out to direct the near future research.

... The converse results of the theory of Krasovskii, that guarantee the existence of the functional when the system is stable has been less popular, although the general form of the functional introduced by Repin (1965) and Datko (1972), has been a source of inspiration for the determination of sufficient stability conditions in Gu (2001), Peet and Bliman (2011). ...

An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. The main results and proof strategies are outlined in the retarded type case. The state of the art and potential extensions to other classes of delay systems are discussed.

... In this paper, we study stability analysis problem of the delayed output feedback (DOF) with distributed terms implemented by numerical integration. Enlightened by the idea of [7], we will provide an effective stability analysis approach using LMIs. To this end, we first transform the closed-loop system implemented by numerical integration into a standard form that the feedback loop does not contain state delay terms. ...

... Remark 2: Compared with [7], [32], the integral inequalities (8) and (9) in Lemma 2 give much tighter lower bounds than the Jensen's inequalities do. ...

In this paper, the problem of stability analysis for linear continuous-time systems with constant discrete and distributed delays is investigated. First, an improved reciprocally convex lemma is presented, which is a generalization of the existing reciprocally convex inequalities and can be directly applied in the case that of the delay interval is divided into
$N\geq 2$
subintervals. Second, combining this with the auxiliary functions-based integral inequalities and the delay partition approach, a novel stability criterion of delay systems is given in terms of linear matrix inequalities. Finally, three numerical examples are given and their results are compared with the existing results. The comparison shows that the stability criterion proposed in this paper can provide larger upper bounds of delay than the other ones.

... lower bounds of integral term such as t t−h ˙ x (s ) R ˙ x (s ) ds, such as Jensens inequality [8] , Wirtinger inequality [9] , free-matrix-based inequality [10] , double integral inequality [11] , Bessel-Legendre inequality [12] and so on. Especially, relaxed Writinger inequality [13,14] which relaxes Wirtinger inequality [9] can derive lower bounds to estimate integral term. ...

This paper is concerned with the problem of the stability and stabilization for continuous-time Takagi–Sugeno(T–S) fuzzy systems with time delay. A novel Lyapunov–Krasovskii functional which includes fuzzy line-integral Lyapunov functional and membership-function-dependent Lyapunov functional is proposed to investigate stability and stabilization of T–S fuzzy systems with time delay. In addition, switching idea which can avoid time derivative of membership functions is introduced to deal with derivative term. Relaxed Wirtinger inequality is employed to estimate integral cross term. Sufficient stability and stabilization criteria are derived in the form of matrix inequalities which can be solved using the switching idea and LMI method. Several numerical examples are given to demonstrate the advantage and effectiveness of the proposed method by comparing with some recent works.

This article discusses invariant subspaces of a matrix with a given partition structure. The existence of a nontrivial structured invariant subspace is equivalent to the possibility of decomposing the associated system with multiple feedback blocks such that the feedback operators are subject to a given constraint. The formulation is especially useful in the stability analysis of time‐delay systems using the Lyapunov–Krasovskii functional approach where computational efficiency is essential in order to achieve accuracy for large scale systems. The set of all structured invariant subspaces are obtained (thus all possible decompositions are obtained as a result) for the coupled differential‐difference equations (DDE) associated with the DDE of retarded and neutral types, as well as systems with a time‐varying delay. It was shown that the known ad hoc methods of reducing the dimensions of delay channels can be considered as special cases of decomposition where one subsystem has trivial dynamics. The reduction of computational cost is demonstrated by a numerical example. For the general case, a recursive procedure is developed to obtain the set of all structured invariant subspaces. Based on this procedure, a method is presented to obtain a nontrivial structured invariant subspace that considers computational efficiency and increased possibility of terminating in a finite number of steps.

This paper proposes improved Lyapunov-Krasovskii functionals (LKFs) for asymptotic stability of generalized neural networks (GNNs) with time-varying delays. By utilizing generalized free-weighting matrix inequality (GFWMI) and some mathematical techniques, sufficient conditions which are dependent on the size of time delays are derived for guaranteeing the stability of GNNs. Additionally, the augmented zero equality approach (AZEA) is applied to enhance the results and eliminate the free variables. Three numerical examples show that the proposed method can be effective and provide less conservative results than previous researches.

The purpose of this paper is to study the coherent feedback control dynamics based on the network that an $N$-level atom is coupled with a cavity and the cavity is coupled with a single or multiple parallel waveguides through two semitransparent mirrors. When initially the atom is excited at the highest energy level, it can emit multiple photons into the cavity via the spontaneous emission, and the photons in the cavity can be transmitted into the waveguide and then re-interact with the cavity quantum electrodynamics (cavity-QED) system through the feedback channel. When the cavity is coupled with a single waveguide, the generation of multi-photon states in the waveguide can be characterized by the exponential stability of the linear control system with feedback delays determined by the feedback loop length. By tuning the feedback loop length, there can be zero or multiple photons in the waveguide. Besides, when the cavity-QED system is coupled with multiple parallel waveguides, the emitted photons oscillate among different waveguides and this process is influenced by the feedback loop length and coupling strengths among waveguides.

An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. The main results and proofs are presented in details for the case of systems with multiple delays. The state of the art, ongoing research and potential extensions to other classes of delay systems are discussed.

This paper proposes novel conditions based on linear matrix inequalities (LMI) for stability analysis of arbitrarily-fast time-varying delays systems. The time-varying delay interval is divided into smaller pieces in order to obtain an equivalent switched model with multiple time-varying delays of smaller interval, which differently from other existing approaches, the maximum switching frequency is not required for stability analysis. Thus, by the use of augmented Lyapunov-Krasovskii functionals and the Finsler’s lemma, together with some relationships among state variables intentionally defined, the inherent conservatism can be progressively reduced by refining more and more the delay partition. The superiority of the proposed method is illustrated through two benchmark examples.

This paper investigates the exponential stability of uncertain time delay systems using a novel descriptor redundancy approach based on delay partitioning. First, the original system is casted into an equivalent descriptor singular state–space representation by introducing redundant state variables so that the resulting delay is progressively reduced. From the equivalent model and applying Lyapunov Functional method, a sufficient condition based on Linear Matrix Inequalities (LMIs) for exponential stability with guaranteed decay rate performance is obtained. As a result, the inherent conservatism of Lyapunov–Krasovskii functional techniques can arbitrarily be reduced by increasing the number of delay partition intervals including decay rate performance and model uncertainties in polytopic form. Various benchmark examples are provided to validate the effectiveness of the proposed method, showing better trade-off between conservatism and performance in comparison to previous approaches.

We study exponential stability for a kind of neural networks having time-varying delay. By extending the auxiliary function-based integral inequality, a novel integral inequality is derived by using weighted orthogonal functions of which one is discontinuous. Then, the new inequality is applied to investigate the exponential stability of time-delay neural networks via Lyapunov-Krasovskii functional (LKF) method. Numerical examples are given to verify the advantages of the proposed criterion.

The paper investigates the formation control problem for high-order linear swarm systems with limited communications such as time-varying delays and switching interconnections. Firstly, the problem description is given including the dynamics of high-order swarm systems, formation protocol and the definitions of formation maintenance and tracking. Secondly, four conditions are obtained: the former three rely heavily on formation function, reference trajectory, auxiliary functions and eigenvalues configuration; the fourth one can be transformed into the time-delay systems stability. In order to get lower conservative criteria, the Free-weighting Matrices(FWM) approach is employed to analyze the stabilization problem. Finally, the allowance upper bound of delays is obtained through solving the feasible linear matrix inequalities (LMIs). Numerical examples and simulation results are given to demonstrate the effectiveness and benefit on reducing conservativeness of the proposed method.

This paper is concerned with the stability analysis of systems with additive time-varying delays. First, an extended reciprocally convex matrix inequality is presented, which is a generalization of the existing reciprocally convex matrix inequalities. Second, combining the proposed matrix inequality with the Wirtinger-based integral inequality, a new stability criterion of systems with additive time-varying delays is proposed. Meanwhile, an improved stability criterion of systems with a single time-varying in a range is also obtained. Finally, two numerical examples are employed to illustrate the advantage of the obtained theoretical results.

In this chapter, we show how the problem of controller synthesis can be posed as a form of convex optimization in an operator-theoretic framework. Furthermore, we show how to: (a) Parameterize the integral and multiplier operator-valued decision variables using finite-dimensional vectors; (b) Verify and enforce positivity and negativity of multiplier and integral operators using positive matrices; (c) Invert positive integral and multiplier operators through the use of a new formula based on algebraic manipulation. Finally, we show how these 3 parts can be combined into a computational procedure for finding a stabilizing state-feedback controller for systems defined by differential-difference equations—a class which includes differential systems with discrete delays. Finally, a numerical example is used to illustrate the form of the resulting stabilizing controller.

This paper studies the problem of stability and dissipativity analysis for uncertain Markovian jump systems (UMJSs) with random time-varying delays. Based on the auxiliary function-based integral inequality (AFBII) and with the help of some mathematical tools, a new double integral inequality (NDII) is developed. Then, to show the efficiency of the proposed inequality, a suitable Lyapunov-Krasovskii functional (LKF) is constructed with augmented delay-dependent terms. By employing integral inequalities, new delay-dependent sufficient conditions are derived in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are given to show the effectiveness and less conservatism of the results.

In this chapter we will briefly introduce some background knowledge related to Genetic Regulatory Networks (GRNs).

This paper investigates input-output finite-time reliable static output feedback (SOF) control of a time-varying system under the influence of both input time delay and actuator failures. An actuator fault model consisting of linear and nonlinear faults is considered during the time-varying control process. The objective is to design a reliable SOF controller that can ensure input-output finite-time stability (IO-FTS) of the resulting closed-loop system. An augmented time-varying Lyapunov functional is constructed, in which some Lyapunov matrices are variable function of time t. By dividing the time interval and delay interval into equal segments, the matrix-valued functions are expressed by a linear interpolation formula. Moreover, combining with the single and double Wirtinger-based integral inequalities, delay-dependent IO-FTS conditions are derived. It is shown that the SOF control issue is solved in forms of linear matrix inequalities. In the end, the effectiveness is demonstrated by simulations.

In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampleddata, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles – which are most closely related to the research area – are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic; Second, to suggest possible future research directions to be tackled by scientists and engineers in the field.

This paper considers the problem of stabilizing continuous-time
linear systems with time delays. Specifically, a fixed-order (i.e.,
full- and reduced-order) dynamic compensation problem is addressed for
systems with simultaneous state, input, and output delays. The principal
result involves sufficient conditions for characterizing fixed-order
dynamic controllers for delay systems via a system of modified coupled
Riccati equations. The controllers obtained are delay independent and
hence apply to systems with arbitrary unknown delay

This paper focuses on 'mixed' delay-independent/delay-dependent asymptotic stability problems of a class of linear systems described by delay-differential equations involving several constant but unknown delays. We give some sufficient conditions for characterizing unbounded stability regions in the delays parameter space. The proposed approach makes use of some appropriate Liapunov-Krasovskii functionals, and the results obtained are expressed in terms of matrix inequalities. We also discuss several ways to construct such analytic functionals. These results allow us to recover (or to improve) as limit cases previous delay-independent or/and delay-dependent conditions from the control literature.

This paper concerns H∞ control of systems under norm bounded uncertainties in all the system matrices. This is an extension of the work by Xie et al.(1992), where only the A matrix is allowed to be uncertain. It is found that the problem shares the same formulation with the H∞ control problem for systems without uncertainties. It can also be viewed as reducing the problem of dealing with systems with both structured uncertainties and unstructured uncertainties to one with unstructured uncertainties only

The stability problem of linear uncertain time-delay systems is considered using a quadratic Lyapunov functional. The resulting stability criterion is a constrained linear matrix inequality set. The condition is necessary and sufficient if it is applied to uncertainty-free systems. A discretization scheme is proposed to reduce the constrained LMI set to a regular LMI problem. Conservatism due to discretization can be made small through finer discretization. Comparison with a previous example shows significant improvements even under very coarse discretization.

The stability problem of linear uncertain time-delay systems is considered using a quadratic Lyapunov functional. The kernel of the functional, which is a function of two variables, is chosen as piecewise linear. As a result, the stability condition can be written as a linear matrix inequality, which improves the estimate of stability limit over the existing approaches. A number of constraints on the parameters can be introduced to reduce the computational effort needed with some compromised accuracy. For a particular choice of constraints, the existing discretization scheme can be recovered. Copyright © 1999 John Wiley & Sons, Ltd.

This paper examines the problems of robust stabilization and robust H∞ control for linear systems with a constant time delay in the state and subject to parameter uncertainty. Systems with norm-bounded parameter uncertainty will be considered and attention is focused on the design of linear memoryless state feedback controllers. We develop methods of robust stabilization and robust H∞ control which are dependent on the size of the delay and are based on the solution of linear matrix inequalities. The proposed methods can be easily extended to uncertain linear polytopic systems as well as to the case of multiple time delays.

This paper deals with the problem of robust stability analysis and robust stabilization for a class of uncertain linear systems with a time-varying state delay. The uncertainty is assumed to be norm-bounded and appears in all the matrices of the state-space model. We develop delay-dependent methods for robust stability analysis and robust stabilization via linear memoryless state feedback. The proposed methods are given in terms of linear matrix inequalities.

In this survey some recent contributions to stability and robust stability analysis of linear time delay systems with parameter uncertainty are discussed. The main aim of the paper is to discuss some basic techniques used for deriving tractable stability and robust stability conditions. The reference list presents rather a small portion of the large number of publications in this field.RésuméOn discute quelques contributions récentes a l'analyse de la estabilité et de la estabilité robuste des systemes linéares a retards a paramétres incertains. L'espirit de article est de présenter l'essential des techniques de base que sout utilisées pour obtenir des conditions vérifiables de stabilité et de stabilité robuste. Les références listées ne représenteut qu'une petite partie du gran nombre des publications dans ce domaine.

Some linear matrix inequalities in the stability problem of
time-delay systems can be partially solved by eliminating some
variables. A previous stability criterion is made less conservative by
allowing some parameters to depend on the uncertain system matrix. These
free parameters can be eliminated. As a result, the stability criterion
is simpler and less conservative

It is quite common in stability analysis of time-delay systems to
make special transformations of the system under investigation in order
to get stability conditions which depend on values of delays. We discuss
additional conditions for the asymptotic stability of the transformed
system which do not appear for that of the original system. These
conditions are results of the transformations and may be expressed in
the form of specific restrictions imposed on spectrum location of some
system matrices

Asymptotic stability of linear, time invariant regulators for plants with unknown but bounded parameters are investigated using minimax and extensions of Lyapanov's stability theory. Largest sets of parameters variations are determined such that a linear autonomous system retains stability. Several design procedures are proposed and applied, yielding regulators with satisfactory performance despite large variations in the plant parameters.

This paper considers the robust stability problem for a class of
time-delay systems with norm-bounded, and possibly time-varying
uncertainty. Based on the discretized Lyapunov functional approach, a
stability criterion is derived. The time-delay is assumed constant and
known. Numerical examples show that the results obtained by this new
criterion significantly improve the estimate of the stability limit over
some existing results in the literature

In studying the stability of time delay systems, many published
results use a transformation to transform a system with single time
delay to a system with distributed delay. In this article, the inherent
limitations of such approaches are studied. Specifically, it is shown
that such a transformation incurs additional dynamics that can be
characterized by appropriate additional eigenvalues. The critical delay
values when such additional eigenvalues cross the imaginary axis can be
explicitly calculated. If the smallest of such delays is less than the
stability delay limit of the original system, then any stability
criteria obtained using such transformation will be conservative. Some
examples are also included

A linear memoryless controller is developed for a class of
uncertain linear time-delay systems to track dynamic inputs of a
non-delay reference model. For matched uncertainties, the controller
designed guarantees uniform ultimate boundedness of the tracking error,
and the error bound can be made arbitrarily small. For mismatched
uncertainties a condition is derived under which the controller designed
ensures ultimate boundedness of the tracking error. However, in this
case there is no control over the tracking error bound. Finally,
examples are included to illustrate the results developed in this paper

This paper deals with the problem of robust stabilization for
uncertain systems with multiple state delays. The parameter
uncertainties are time-varying and unknown but are norm-bounded, and the
delays are time-varying. A new method for achieving robust stabilization
is presented for a class of uncertain time-delay systems via linear
memoryless state feedback control. The results depend on the size of the
delays and are given in terms of several linear matrix
inequalities

The design of stabilizing linear output feedback control of
uncertain systems is a computationally demanding optimization problem as
local minima may exist which are distinct from the global minimum.
However, in some special cases, quasiconvexity can be proven through a
simple redefinition of variables and/or a reformulation of the problem
in the dual form. Such special cases include state feedback control for
both discrete and continuous time systems