Delay-dependent Case Research Articles

Inspecting the stability of non-linear IS-LM model with dual time delay

Capital stock is a strong pillar in the macroeconomics paradigm and plays a significant role when dealing with macroeconomic problems. Also, implementation of the investment decisions and employment of capital into productive use are two ignored time delays in accumulation of capital. Hence, a mathematical model with dual time delays in capital accumulation is formulated in this work. Furthermore, by adhering to functions that do not follow regularity, i.e., considering non-linear functional forms will not only influence policy making but also broaden our viewpoint towards realistic scenarios. Therefore, non-linear functional forms for investment, liquidity preference and government expenditure are considered. First, the equilibrium point is established, which is unique in this case. For analysing stability around the equilibrium point, the work is divided into three delay-dependent cases. When the time delay reaches a critical value, the complex dynamics of the system, such as stability switches and Hopf bifurcation, are investigated. The system is sensitive to a few parameters, such as depreciation rate, propensity to invest in capital stock, and average tax rate. The behavioural change is observed by altering these parameters. This study establishes a time limit within which the available capital for production should be put to productive use, and investment decisions should be made, which will aid in effective policymaking.

Delay Dependent Robust Stability of A Discrete Time Recurrent Neural Network with Time Varying Delays

In this paper, the robust stability analysis of a problem is investigated for a class of discrete recurrent neural networks with distributed time varying delays for delay dependent case. The problem is to determine the robust stability by employing Lyapunov–Krasovskii stability theory. The class of neural network under some consideration is globally asymptotically stable if the quadratic matrix inequality involving several parameters is less than zero. Furthermore, a Linear Matrix Inequality (LMI) approach is provided to show the stability analysis. The numerical examples are given to show the usefulness of the proposed robust stability conditions. The numerical simulation is proved using MATLAB.

Finite-time stabilization of switched linear time-delay systems with saturating actuators

In this paper, finite-time stabilization of switched linear time-delay systems with saturating actuators is addressed. State feedback controllers are presented to make the closed-loop systems finite-time stable. Not only the gain matrix of the saturating actuator but also the switching signal is designed to guarantee the finite-time stability of the switched systems. Sufficient conditions for both delay independent and delay dependent cases are given. The results in this paper which are expressed by scalar inequalities show the transient performance of the controlled switched systems. The bound of the trajectory given by the proposed methods leads us easily to judge the finite-time stability of the closed-loop switched systems. An example is employed to verify the efficiency of the proposed method.

Basis-dependent and delay-dependent results on robust L1 filtering for networked control systems

Persistent and amplitude-bounded disturbances are nowadays considered for various engineering applications. Whereas the traditional H∞ and L2–L∞ filtering methods under the energy-bounded external disturbances assumption cannot receive good noise suppression effect. To effectively reflect the capability of disturbances suppression, this paper proposes a robust L1 filtering approach for a class of nonlinear networked control systems (NNCSs) described by Takagi–Sugeno (T–S) fuzzy model. According to the analysis of the typical time-delay issue induced by communication network and the parallel distributed compensation (PDC) technique, a fuzzy filtering error system is established. Attention is focused on designing the robust L1 filter that guarantees the fuzzy filtering error system to be asymptotically stable and to have a prescribed L1 noise attenuation level γ with respect to all persistent and amplitude-bounded disturbances. In particular, we concentrate on the delay-dependent case, using a basis-dependent Lyapunov–Krasovskii functional which refers to reducing the conservatism, the peak-to-peak performance criterion is first proposed. The integral inequality method and linear matrix inequality (LMI) technique are adopted during the process of robust L1 stability analysis. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed method.

Gain-scheduled L-one control for linear parameter-varying systems with parameter-dependent delays

This paper deals with the problem of gain-scheduled L-one control for linear parameter-varying (LPV) systems with parameter-dependent delays. The attention is focused on the design of a gain-scheduled L-one controller that guarantees being an asymptotically stable closed-loop system and satisfying peak-to-peak performance constraints for LPV systems with respect to all amplitude-bounded input signals. In particular, concentrating on the delay-dependent case, we utilize parameter-dependent Lyapunov functions (PDLF) to establish peak-to-peak performance criteria for the first time where there exists a coupling between a Lyapunov function matrix and system matrices. By introducing a slack matrix, the decoupling for the parameter-dependent time-delay LPV system is realized. In this way, the sufficient conditions for the existence of a gain-scheduled L-one controller are proposed in terms of the Lyapunov stability theory and the linear matrix inequality (LMI) method. Based on approximate basis function and the gridding technique, the corresponding controller design is cast into a feasible solution problem of the finite parameter linear matrix inequalities. A numerical example is given to show the effectiveness of the proposed approach.

Delay-Dependent L1 Filtering for LPV Systems With Parameter-Varying Delays

The paper investigates the problems of delay-dependent L1 filtering for linear parameter-varying (LPV) systems with parameter-varying delays, in which the state-space data and the time delays are dependent on parameters that are measurable in real-time and vary in a compact set with bounded variation rate. The attention is focused on the design of L1 filter that guarantees the filtering error system to be asymptotically stable and satisfies the worst-case peak-to-peak gain of the filtering error system. In particular, we concentrate on the delay-dependent case, using parameter-dependent Lyapunov function, the decoupled peak-to-peak performance criterion is first established for a class of LPV systems. Under this condition, the admissible filter can be found in terms of linear matrix inequality (LMI) technology. According to approximate basis function and the gridding technique, the filter design problem is transformed into feasible solution problem of the finite parameter LMIs. Finally, a numerical example is provided to illustrate the feasibility of the developed approach.

Advanced Clustering With Frequency Sweeping Methodology for the Stability Analysis of Multiple Time-Delay Systems

A new methodology, advanced clustering with frequency sweeping (ACFS), is introduced for the stability analysis of the most general class of linear time-invariant time-delay systems with multiple delays. ACFS is a cross fertilization of algebraic geometry, frequency sweeping technique and the root clustering paradigm, and it can directly extract the 2-D cross sections of the stability views in any two-delay domain. In this domain, ACFS can also test the delay-independent stability based on necessary and sufficient conditions, and can compute, for the delay-dependent case, the precise lower and upper bounds of the only parameter, the frequency, that ACFS sweeps. Case studies that are prohibitive to analyze with the existing analytical methods are presented to demonstrate the strengths of the method.

Convex Optimization Approaches to Robust ℒ1 Fixed-Order Filtering for Polytopic Systems with Multiple Delays

This paper investigates the robust ℒ1 fixed-order filtering problem for continuous polytopic systems with multiple state delays. Attention is focused on the design of robust full-order and reduced-order filters that guarantee the filtering error system to be asymptotically stable and satisfy the worst case peak-to-peak gain of the filtering error system for all admissible uncertainties and time delays. In particular, we concentrate on the delay-dependent case, and the peak-to-peak performance criterion is first established for polytopic systems with multiple state delays. Two different convex optimization approaches are proposed to solve this problem. One is the parameter-dependent Lyapunov approach in which the filter is not only dependent on the parameters (residing in a polytope), but also the Lyapunov matrices are different over the entire polytope domain. The other is the quadratic stability approach which obtains an admissible filter in the quadratic framework. Computational algorithms in terms of linear matrix inequalities (LMIs) are provided. It is shown that the parameter-dependent Lyapunov approach turns out to be less conservative than the quadratic stability approach, but the quadratic stability approach is computationally less demanding. Two numerical examples are presented to illustrate the proposed theory.

Dynamic output feedback H ∞ control problem for a class of linear neutral systems

This note presents the solution of dynamic output feedback H ∞ control problem for a class of linear neutral systems in delay dependent case. First, sufficient conditions for the existence of such controllers is obtained by Lyapunov method in terms of linear matrix inequalities (LMIs). A parameterized characterization of the controllers is given in terms of feasible solutions to the certain matrix inequalities by using Matlab LMI Toolbox. Finally, a numerical example is given to illustrate the proposed results.

Dynamic Output Feedback $H_{\infty}$ Control Problem for Linear Neutral Systems

This note presents the solvability conditions of the dynamic output feedback Hinfin control problem for linear neutral systems with time- varying state delays in the delay-dependent case, where the delay is assumed to be time-varying continuous or differentiable uniformly bounded function, but no restriction on the derivative of the delay is needed, which means that the delay may be the fast time-varying function. An improved delay-dependent bounded real lemma (BRL) for a closed-loop system is established. Based on the obtained BRL, the dynamic output feedback Hinfin controller is designed in terms of the linear matrix inequalities with inverse constraints. An iterative algorithm involving convex optimization methods is used to satisfy these constraints. The proposed results are illustrated in the examples.

Resilient L 2 – L ∞ filtering of polytopic systems with state delays

The problem of designing a resilient L2–L∞ filter for a class of linear uncertain state delay systems with uncertainties that belong to a convex-bounded polytopic domain is investigated. The objective is to derive tractable synthesis conditions for the resilient design of full-order and reduced-order filters such that a prescribed energy-to-peak disturbance-attenuation level is attained for all admissible uncertainties and gain perturbations. Both additive and multiplicative gain perturbations are considered. It is established that the filter design can be obtained from the solution of the convex optimisation problem over linear matrix inequalities. The design results are developed for both delay-independent and delay-dependent cases. In the latter case, two approaches are used: descriptor and extended Newton–Leibniz. Results on numerical simulations are presented to demonstrate the behaviour of the resilient filters.

DELAY-DEPENDENT STABILITY ANALYSIS OF LINEAR TIME DELAY SYSTEMS

AbstractStability of time delay systems is investigated considering the delay-dependent case. The system without delays is assumed stable and conservative conditions are derived for finding the maximal delay that preserves stability. The problem is treated in the quadratic separation framework and the resulting criteria are formulated as feasibility problems of Linear Matrix Inequalities. The construction of the results relies on a fractioning of the delay. As the fractioning becomes thinner, the results prove to be less and less conservative. An example show the effectiveness of the proposed technique.

Discussion on: “Parameter-Dependent Lyapunov Function Approach to Stability Analysis and Design for Uncertain Systems with Time-Varying Delay”

The paper by Cao and Xue considers the employment of the extended linear matrix inequality (LMI) approach of Oliveira et al. [1,2] to the stability analysis of linear systems with constant unknown parameters and time-varying delays. Both delay-independent and delay-dependent cases are considered. The extended LMI approach, incorporating additional instrumental matrix variables, opens the door to lower conservatism in the analysis of parameter-dependent systems as well as the synthesis of state-feedback controllers without requiring all plant polytope vertices to share the same Lyapunov matrix in the stability condition. The later point, of course, implies that quadratic stability is not required and thus the results obtained do not suffer from the conservatism that this assumption would induce for systems with constant parameters. Furthermore, at the price of introducing additional scalars ð, dÞ and losing convexity of the analysis problem, synthesis results for state/delayed-state feedback are obtained. As in the finite-dimensional case, the extension of the LMIs for time delay systems can be readily done by introducing new vector variables for signals previously defined from the state (e.g. the state derivative), imposing an algebraic constraint on the vector variables to maintain the system dynamics, and applying Finsler’s lemma [2] (although the paper of Cao and Xue does not precisely follow this formulation). In summary, the paper brings together in an interesting and cogent fashion a number of disparate, recent approaches to systems analysis and controller design so as to tackle a particularly difficult problem in delay-differential systems theory. Of particular interest to the reader may be the development of stability conditions for both the delaydependent and delay-independent cases without any requirement on the rate of change of the delay via Lyapunov–Razumikhin functions. It is interesting to note that for the constant delay case, both the Lyapunov–Krasovskii and Lyapunov–Razumikhin LMI results can be reinterpreted in terms of a scaled small gain condition [3]. For the time-varying delay case,

Robust Stability and Performance of Uncertain Lurie Systems with State Delays

In this paper, we consider the problem of persistent bounded disturbance rejection for uncertain Lurie systems with delay. We mainly focus on the problems of absolute stability (delay-independent stability and delay-dependent stability) and performance of bounded disturbance rejection using the Lyapunov-Krasovskii function method. In the two cases, we give sufficient conditions for guaranteeing simultaneously absolute stability and achieving certain performance of bounded disturbance rejection in terms of a linear matrix inequality (LMI). For the delay-dependent case, an estimate of the maximum admissible delay bound is established in terms of a generalized eigenvalue problem that can be solved numerically with the efficient LMI toolbox. Similarly, we study the corresponding problem for Lurie delay systems with uncertainties. Finally, two numerical examples are worked out to illustrate the effectiveness of the proposed results.

Passive control synthesis for uncertain systems with multiple-state delays

In this paper, we investigate the robust passivity synthesis problem for a class of uncertain systems with multiple state delays. Both the delay-independent and the delay-dependent cases are treated. In each case, a sufficient condition for which the uncertain multi-state delay system is robustly stable and strictly passive (SP) for all admissible uncertainties is provided in terms of a linear matrix inequality. Then, we propose a dynamic feedback design methodology based on either state measurements or output measurements. For both types of controllers, it is proven that the closed-loop uncertain time-delay system is asymptotically stable and SP for all admissible uncertainties. A detailed simulation example is given to illustrate the theoretical developments.

Passivity analysis and synthesis for uncertain time‐delay systems

In this paper, we investigate the robust passivity analysis and synthesis problems for a class of uncertain time‐delay systems. This class of systems arises in the modelling effort of studying water quality constituents in fresh stream. For the analysis problem, we derive a sufficient condition for which the uncertain time‐delay system is robustly stable and strictly passive for all admissible uncertainties. The condition is given in terms of a linear matrix inequality. Both the delay‐independent and delay‐dependent cases are considered. For the synthesis problem, we propose an observer‐based design method which guarantees that the closed‐loop uncertain time‐delay system is stable and strictly passive for all admissible uncertainties. Several examples are worked out to illustrate the developed theory.