Stability and estimate of solution to uncertain neutral delay systems

Abstract

The coefficients and delays in models describing various processes are usually obtained as a results of measurements and can be obtained only approximately. We deal with the question of how to estimate the influence of 'mistakes' in coefficients and delays on solutions' behavior of the delay differential neutral systemThis topic is known in the literature as uncertain systems or systems with interval defined coefficients. The goal of this paper is to obtain stability of uncertain systems and to estimate the difference between solutions of a 'real' system with uncertain coefficients and/or delays and corresponding 'model' system. We develop the so-called Azbelev W-transform, which is a sort of the right regularization allowing researchers to reduce analysis of boundary value problems to study of systems of functional equations in the space of measurable essentially bounded functions. In corresponding cases estimates of norms of auxiliary linear operators (obtained as a result of W-transform) lead researchers to conclusions about existence, uniqueness, positivity and stability of solutions of given boundary value problems. This method works efficiently in the case when a 'model' used in W-transform is 'close' to a given 'real' system. In this paper we choose, as the 'models' , systems for which we know estimates of the resolvent Cauchy operators. We demonstrate that systems with positive Cauchy matrices present a class of convenient 'models' . We use the W-transform and other methods of the general theory of functional differential equations. Positivity of the Cauchy operators is studied and then used in the analysis of stability and estimates of solutions.Results: We propose results about exponential stability of the given system and obtain estimates of difference between the solution of this uncertain system and theNew tests of stability and in the future of existence and uniqueness of boundary value problems for neutral delay systems can be obtained on the basis of this technique.

New results on reachable set bounding for linear time delay systems with polytopic uncertainties via novel inequalities

“…Some literature works researched the stability of second order delay differential equations; see, for example, references [ 29 – 33 ]. In the future, reachable set bounding for second order delay differential equations may be a hot issue, and methods similar to those in this paper may be used to estimate reachable set bounding for second order delay differential equations.…”

“…In the reference [ 29 ], Domoshnitsky discussed the stability of more complicated linear neutral systems with uncertain coefficients and uncertain delays. In further work, we will study reachable set bounding for this type of linear neutral systems.…”

Further results on robust stability for uncertain neutral systems with distributed delay

“…The stability analysis of neutral delay-differential systems has received considerable attention over the decades [ 1 – 18 ]. In the literature [ 1 – 18 ], the W-transform approach [ 1 ], the positivity-based approach [ 2 – 16 ], the characteristic equation method [ 17 ], the Lyapunov technique [ 18 ], and the state trajectory approach have been utilized to derive sufficient conditions for asymptotic stability and exponential stability of the systems. However, most of the criteria are expressed in terms of a matrix norm or matrix measure of the system matrices.…”

“…Unfortunately, the matrix norm operations usually make the criteria more conservative. Also the criteria in recent studies [ 16 – 18 ] require strong assumptions such as that the matrix measures of system matrices have to be negative. These assumptions often make it difficult to apply the criteria to various systems, such as neutral delay-differential systems with time-varying delay, uncertain neutral delay-differential systems with time-delay, and so on.…”

Robust NonfragileH∞Filtering for Uncertain T-S Fuzzy Systems with Interval Delay: A New Delay Partitioning Approach

“…Meanwhile, it is worthy of mentioning that the variable delay decomposition approach proposed in this paper can be applied to the further stability analysis along with a new model transformation [15], and the corresponding stability criteria with less conservatism and small computing burden may be derived. Moreover, the proposed results can be extended to reduced-order ∞ filtering for T-S fuzzy system based on the proposed method in [16], finite frequency ∞ filtering [17], and even the above analysis and filtering for 2-D systems [18], and neutral system [42], and the corresponding results will appear in the near future.…”