Delay Systems Of Neutral Type Research Articles

Stability test for neutral type delay systems: a piecewise linear approximation scheme*

A necessary and sufficient test for the stability of neutral type linear time-delay systems is presented. Our approach combines the Lyapunov functional expressed in terms of the delay Lyapunov matrix for this class of systems with a piecewise linear approximation of the functional argument. As a result, a stability criterion is given in terms of the non-negative definiteness test of a matrix. An example is presented to validate the obtained result.

New Delay-Partitioning LK-Functional for Stability Analysis with Neutral Type Systems

This paper investigates the stability issues associated with neutral-type delay systems. Firstly, the delay-partitioning method is employed to construct a brand-new LK-functional candidate. The discrete delay and a neutral delay are divided into several piecewise points through a relaxable sequence of constant numbers, are increasing at a steady rate and are not larger than 1. Secondly, to fully use the interconnection information among the delayed state vectors, a new LK-functional is constructed. Thirdly, the recently published single/multiple integral inequalities are employed to bound the derivative of the newly developed LK function. Finally, a novel stability criterion for neutral systems is developed based on the above treatment. Furthermore, a new corollary is also proposed for the condition of τ=h. The benefits and productivities of our method are demonstrated by numerical examples.

LMI-Based Delayed Output Feedback Controller Design for a Class of Fractional-Order Neutral-Type Delay Systems Using Guaranteed Cost Control Approach

In this research work, we deal with the stabilization of uncertain fractional-order neutral systems with delayed input. To tackle this problem, the guaranteed cost control method is considered. The purpose is to design a proportional–differential output feedback controller to obtain a satisfactory performance. The stability of the overall system is described in terms of matrix inequalities, and the corresponding analysis is performed in the perspective of Lyapunov’s theory. Two application examples verify the analytic findings.

On exact controllability and complete stabilizability for linear systems

In this paper we consider linear systems with control described by the equation $\dot x = \mathcal A x +\mathcal B u$ where functions u and x take values in U and X respectively. For such object, a short review of results concerning relations between exact controllability and complete stabilizability (stabilizability with arbitrary decay rate) is given. The analysis is done in various situations: bounded or unbounded state and control operators $\mathcal A$ and $\mathcal B$, Banach or Hilbert spaces U and X. The well known equivalence between complete controllability and pole assignment in the situation of finite dimensional spaces is no longer true in general in infinite dimensional spaces. Exact controllability is not sufficient for complete stabilizability if U and X are Banach spaces. In Hilbert space setting this implication holds true. The converse also is not so simple: in some situations, complete stabilizability implies exact controllability (Banach space setting with bounded operators), in other situation, it is not true. The corresponding results are given with some ideas for the proofs. Complete technical development are indicated in the cited literature. Several examples are given. Special attention is paid to the case of infinite dimensional systems generated by delay systems of neutral type in some general form (distributed delays). The question of the relation between exact null controllability and complete stabilizability is more precisely investigated. In general there is no equivalence between the two notions. However for some classes of neutral type equations there is an equivalence. The question how the equivalence occurs for more general systems is still open. This is a short and non exhaustive review of some research on control theory for infinite dimensional spaces. Our works in this area were initiated by V. I. Korobov during the 70th of the past century in Kharkov State University.

Stability analysis of uncertain fractional-order neutral-type delay systems with actuator saturation

Stability analysis of uncertain fractional-order neutral-type delay systems with actuator saturation

Lyapunov–Krasovskii functionals for homogeneous time delay systems of neutral type

AbstractIn this article, we present a general construction of Lyapunov–Krasovskii functionals for a class of neutral‐type time delay systems with a linear difference part and homogeneous right‐hand sides of degree strictly greater than one. Under an assumption that the corresponding difference matrix equation as well as a system with zero delays are asymptotically stable, the functionals allow proving delay‐independent asymptotic stability. They are based on the Lyapunov functions of the corresponding delay free systems and constitute a generalization of constructions presented recently for homogeneous systems of retarded type. The functionals are applied to the robustness analysis with respect to uncertainties at the right‐hand sides and at the difference term as well.

New results on stability analysis of neutral-type delay systems

ABSTRACT This paper is concerned with stability analysis of linear neutral-type delay systems. The main novelty of this paper comes from proposing a new stability analysis method called a delay-mode-based functional (DMBF) method for nonliear neutral-type delay systems. This is achieved by weakening the condition that when the Lyapunov–Krasovskii functional (LKF) method is used, a functional is required for a neutral-type delay system with a delay varying in a delay interval. With the help of the DMBF method, new stability criteria of linear neutral-type delay systems are obtained. The effectiveness of the obtained criteria is illustrated by two examples from the literature.

Observer‐based control approach for fractional‐order delay systems of neutral type with saturating actuator

In practice the states of a system are not directly available that are difficult to measure by sensors. This article proposes an observer‐based controller for fractional‐order neutral‐type delay systems with actuator saturation. A feedback algorithm is constructed by applying the Lyapunov direct method. Moreover, based on the observer, the gains of controller and observer are determined by solving a set of nonlinear matrix inequalities. Therefore, an iterative algorithm including the convex optimization is also proposed to solve the inequalities. An example is given to illustrate the theoretical results.

Robust stability of uncertain fractional order systems of neutral type with distributed delays and control input saturation

Time delay occurs naturally due to the limited bandwidth of any real-world system. However, this problem can deteriorate the system performance and can even result in system instability. Input saturation is also an essential issue due to the energy constraint in real actuators that makes the control design procedure more difficult. This article concerns with the stability of uncertain fractional order (FO) delay systems of neutral type including structured uncertainties, distributed delays and actuator saturation. A Lyapunov–Krasovskii functional allows the formulation of the conditions to insure the asymptotic robust stability of such systems via the linear matrix inequalities (LMI) and to compute the gain of a state feedback controller. In addition, by using the cone complementarity linearization method, we obtain the controller gains that extend the domain of attraction. Several simulations validate the theoretical analysis.

An LMI based approach to stabilize a type of nonlinear uncertain neutral-type delay systems

This paper proposes a linear matrix inequality (LMI) based approach for stability analysis of a type of nonlinear uncertain neutral-type delay systems. The delays in states, state derivatives and inputs are known and constant. By constructing a Lyapunov–Krasovskii functional (LKF), a stability criterion in the form of LMI is derived. In this paper, we treat these nonlinear terms to be norm-bounded and construct a Lyapunov matrix inequality considering system states, delayed states, delayed state derivatives, and delayed inputs. The stability criterion of this nonlinear neutral-type delay system is then obtained. To this end, a numerical example is given to demonstrate the feasibility of the proposed approach. The developed method for stability analysis can potentially be applied to critical real-world applications such as the down-hole drilling system.

Stability of neutral type delay systems: A joint Lyapunov–Krasovskii and Razumikhin approach

In this paper, a new stability criterion for linear time-delay systems of neutral type is presented. By virtue of the well-known Krasovskii theorem, the system is exponentially stable, if and only if there exists a functional which (i) has a negative definite derivative along the solutions of the system, and (ii) admits a quadratic lower bound, of course under an assumption that the difference operator corresponding to the system is stable. We modify the second condition in the following manner: The functional is required to admit a quadratic lower bound only on the special set of initial functions in our stability result instead of the set of all appropriate functions in the Krasovskii theorem. This special set consists of the functions satisfying a Razumikhin-type inequality and a similar inequality on the derivative. Basing on such modification and staying within the framework of the functionals with a given derivative, we suggest a methodology for the stability analysis, which is described and tested on the examples in the case of a scalar equation. Unlike most of recent results, we use the functional with a derivative prescribed just as a negative definite quadratic form of the “current” state of a system, what does not impose restrictions on the approach.

Modal control of hybrid differential-difference systems and associated delay systems of neutral type in scales of differential-difference controllers

We study the statements and solvability of the modal control problem (the pole assignment problem) for linear time-invariant hybrid difference-differential systems in symmetric form and for the associated delay systems of neutral type. We obtain constructive necessary and sufficient parametric conditions for the modal controllability of the systems in question in various scales of difference-differential controllers. Methods for the construction of such controllers solving the corresponding modal controllability problem are indicated. The results are illustrated by examples and counterexamples.

New robustness bounds for neutral type delay systems via functionals with prescribed derivative

The paper is devoted to the robust stability analysis of linear neutral type time delay systems with a constant delay and norm-bounded uncertainties. The method is based on the Lyapunov–Krasovskii functional with a derivative prescribed as a negative definite quadratic form of the “current” system state, which is considered to be not suitable for the robustness analysis due to the fact that it does not admit a quadratic lower bound. Unlike existing results, our approach does not require the derivative of the functional along the solutions of the perturbed system to be negative definite. Instead, we need just an essential part of the integral of the derivative to be negative. The resulting stability condition is presented in the form of a simple inequality depending on the so-called Lyapunov matrix, under an assumption that the difference operator of the perturbed system is stable. The result is applicable to all exponentially stable systems.

Computation of the region of attraction for a class of nonlinear neutral type delay systems

An estimation of the region of attraction of the trivial solution of nonlinear neutral type delay systems is presented by using the so called Lyapunov-Krasovskii functionals of complete type. Results concerning bounds on the delay Lyapunov matrix allow simplifying computations in the estimation of this region. The result is illustrated by some examples.

Stability of neutral type fractional delay systems and its relation with stability of time‐delay and discrete systems

The bounded-input bounded-output stability of fractional delay systems of neutral type is investigated in this study. The proposed method calculates the number of unstable poles of the system for each delay value. All time-delay intervals in time-delay space where system is stable are precisely determined. The stability of systems with not only indefinitely large values of time-delay, but also with multiple poles on the imaginary axis has been studied. The proposed method is employed for investigating the stability of a fractional delay system when its fractional orders approach to the integer numbers or zero. It is proved that simple case of neutral types and time-delay systems have the same conditions for the independent stability. Two numerical examples are provided to illustrate the proficiencies of the proposed method.

Obtention of the functional of complete type for neutral type delay systems via a new Cauchy formula

In this paper, we present a new manner for computing the so-called new form of the functional of complete type for neutral type time-delay systems without assumptions of continuity and differentiability on the initial function. It is obtained by using a new Cauchy formula. We present briefly the necessary stability conditions depending on the delay Lyapunov matrix that this result makes possible to prove.

Robust absolute stability criteria for uncertain Lurie interval time-varying delay systems of neutral type

This study investigates the delay-dependent robust absolute stability analysis for uncertain Lurie systems with interval time-varying delays of neutral type. First, we divide the whole delay interval into two segmentations with an unequal width and checking the variation of the Lyapunov–Krasovskii functional (LKF) for each subinterval of delay, much less conservative delay-dependent absolute and robust stability criteria are derived. Second, a new delay-dependent robust stability condition for uncertain Lurie neutral systems with interval time-varying delays, which expressed in terms of quadratic forms of linear matrix inequalities (LMIs), and has been derived by constructing the LKF from the delayed decomposition approach (DDA) and integral inequality approach (IIA). Finally, three numerical examples are given to show the effectiveness of the proposed stability criteria.

BIBO stability of some classes of delay systems and fractional systems

A new test is presented for the BIBO stability of delay systems of neutral type with a single delay, specified in terms of their transfer functions, enabling us to decide on some cases that were previously open. Next, a class of fractional systems is considered, and a method is given for determining the stability intervals for such systems.

The Effect of Fractional Order on Oscillatory Behavior of Scalar Fractional Delay Systems of Neutral Type

The undamped oscillations of a scalar fractional neutral system are studied in this paper. For this purpose, it is proved that this kind of systems can have the oscillatory behavior. The necessary and sufficient condition is proposed to determine that there exists a delay value for which the system oscillates. The frequency, amplitude and required delay for the undamped oscillations depend on the fractional order. However, the behavior of dependencies is not specified. Indeed, when the system has oscillatory behavior, it is bounded-input bounded-output stable for delay less than the required delay. Moreover, when the order is approaching to an integer value, the necessary and sufficient condition is proposed to determine that the system can oscillate with the bounded value of delay. The obtained results are verified with respect to the classical theorems. Finally, two numerical examples demonstrate the proposed conditions.

Necessary and Sufficient Conditions for BIBO-Stability of Some Fractional Delay Systems of Neutral Type

In this note, bounded-input bounded-output (BIBO)-stability of a large class of neutral type fractional delay systems is investigated. Necessary and sufficient conditions of BIBO-stability are presented for the intended class of systems (the sufficient conditions have been provided for a more general case in the previous studies). Two lemmas are provided for checking a prerequisite imposed on the considered class of systems. Finally, two numerical examples are given to illustrate the obtained results.