Discrete Interval Systems Research Articles

An appropriate discrete-transformation technique for order reduction methodology

Abstract Various transformation techniques are known for transforming continuous-time to discrete-time domains and vice-versa, among which only two techniques are used in application frequently. This paper attempts to access both the techniques presenting a suitable consequence of their sole importance. The assessment of these two techniques is made on the basis of computation of reduced order model of discrete-time interval systems. Approximate models obtained after the transformations lead to the same response, and the outcome in favor of one technique. As a conclusion, the appropriate transformation is mathematically preferred over another based on its simplicity and ease of computation. Examples illustrating the comparisons are accompanied. Eventually, the assessment demonstrated in this article would be helpful to the mathematicians, mathematics educators and researchers working in the area of model order reduction.

Model Order Diminution of Discrete Interval Systems Using Kharitonov Polynomials

In this research proposal, diminution of higher order (HO) discrete interval system (DIS) is accomplished by utilizing Kharitonov polynomials. The DIS is firstly, transformed into continuous interval system (CIS). The Markov-parameters (MPs) and time-moments (TMs) are exploited for determination of approximated models. The ascertainment of model order diminution (MOD) of DISs is done by Routh-Padé approximation. The Routh table is utilized to obtain the denominator of approximated model. The unknown numerator coefficients of desired approximated model are determined by matching MPs and TMs of DISs and desired model. This whole procedure of MOD is elucidated with the help of one test illustration in which third order system is reduced to first order model as well as second order model. To prove applicability of the proposed method, impulse, step and Bode responses are plotted for both system and model. For relative comparison, time-domain specifications of proposed model are tabulated for both upper and lower limits. Further, performance indices are specified for dissimilarities between responses of system and model. The obtained results depict the effectiveness and efficacy for the proposed method.

Model Order Reduction of Discrete Time Interval System Based on Time Moment Matching

Model Order Reduction of Discrete Time Interval System Based on Time Moment Matching

Stable Approximation of Discrete Interval Systems

The prime motive of this research article is to construct a new mathematical approach for approximation of higher-order discrete interval system into its corresponding lower-order discrete interval model (LODIM). In this approach, a stable approximation technique for discrete interval system is derived with the help of integrated derivative technique. The key feature of this approach is that it is eliminating both the reciprocal transformation and normalization process as carried out in existing methods. Additionally, it assures stability retention of LODIM and upholds the steady-state value of original system. In order to show the accuracy and effectiveness of the proposed approach, a standard sampled test system is taken from the literature. The proposed technique is compared with the recent and relevant techniques of model order reduction. Further, step response and Bode diagram are plotted to prove the efficacy of proposed approach. A comparative study is also undertaken to conclude the same.

Model order reduction of interval systems using an arithmetic operation

ABSTRACTThe paper presents an extension of the differentiation method for model order reduction of large-scale interval systems. This is an alternative approach to the existing differentiation method of interval systems. The proposed method has been applied for both continuous and discrete-time interval systems. The reduction of discrete-time interval systems is achieved by using simple linear transformation and bilinear transformation , where . The proposed method always generates stable reduced-order models, and also it retains the zeroth-order interval time moment. Four numerical examples exemplify the accuracy of the method and computational simplicity. Furthermore, the difficulties associated with the extension of Routh-based approximations to interval systems for obtaining stable reduced-order models are discussed. The stability of interval systems is verified by using Kharitonov's theorem.

Model Order Reduction of LTI Interval Systems Using Differentiation Method Based on Kharitonov’s Theorem

ABSTRACTA new reduction method is proposed to reduce the higher order interval system. The proposed reduced-order interval model is achieved based on the differentiation method using the Kharitonov’s theorem. This method is significant for both single-input single-output interval systems and multi-input multi-output interval systems. This method is simple in mathematical calculation and preserves the dominant characteristics of the original interval system in its reduced-order interval models. The suggested method provides stable reduced-order models if the original system is stable. To show the effectiveness of this technique, it is illustrated with benchmark problems. The results show the close approximation of the reduced-order model obtained by the proposed technique with the original system. Furthermore, the performance of the suggested technique validated qualitatively in terms of time response specifications and performance indices by comparing with other well-known reduction methods. These results exhibit the effectiveness and efficiency of the proposed technique. Moreover, this method also extended for discrete-time interval systems.

Model Order Reduction of Discrete-Time Interval Systems by Differentiation Calculus

The paper presents an algorithm for order reduction of a discrete-time interval system based on the conventional differential calculus approach. The procedure initiated for the continuous-time non-interval system advances to the discrete-time interval system in a novel manner in this paper. The proposed algorithm is computationally straightforward, simple and leads to an acceptable results as compared to the existing techniques. The examples play a significant role in establishment of the algorithm. Additionally, the limitation derived during the discourse of methodology is also accounted along with a possible future scope.

Reduction of discrete interval system using clustering of poles with Padé approximation: a computer-aided approach

This paper presents a method for the order reduction of discrete interval systems using pole-clustering method along with Padé approximation technique. For a given discrete high-order interval system (HOIS), the denominator of the reduced-order interval model (ROIM) is obtained by pole-clustering while the numerator is obtained by minimizing errors between first r time moments of the HOIS to those of its ROIM where r is the order of ROIM. Inverse distance measure (IDM) criterion is used for clustering of poles of HOIS whereas the time moments of HOIS (or ROIM) are obtained without inverting the denominator of transfer function and presented in generalized form. The error function which is weighted squared sum of errors between first r time moments of HOIS and those of its ROIM is carried out by the algorithm due to Luus-Jaakola. The distinctive feature of this method is that the numerator coefficients are treated free in the process of minimization. A numerical example is given to illustrate the procedure.Keywords: Discrete interval system; Inverse distance measure; Kharitonov polynomials; Model reduction; Padé approximation; Pole-clustering method

Analysis and Design of High Order Discrete Time Interval Systems using New Order Reduction Technique via Least Squares Method and Time Moments Technique

Analysis and Design of High Order Discrete Time Interval Systems using New Order Reduction Technique via Least Squares Method and Time Moments Technique

Model order reduction of discrete-time interval system based on Mikhailov stability criterion

Analysis of higher-order interval system is an appealing topic of research demanding an appropriate methodology to reduce them to their lower approximate. This paper attempts to fulfill the requirement by introducing two different algorithms interlaced with the property of Mikhailov stability criterion. Precisely, the stated approach derives the reduced order denominator polynomial, and two separate techniques compute the numerator polynomials. Overall the two proposed algorithms are justified for their novelty through their implication over the numerical examples followed by their rational acceptability to a real-time test system.

Sufficient Conditions for Robust Stability of Discrete Large-Scale Interval Systems with Multiple Time Delays

The robust stability analysis for discrete large-scale uncertain systems with multiple time delays is addressed in this paper. We establish a method for selecting properly a positive definite matrix Q to derive a very simple upper solution bound of the discrete algebraic Lyapunov equation (DALE). Then, using the Lyapunov equation approach method with this upper bound, several sufficient conditions are presented to guarantee the robust stability of the overall systems. Comparisons between the proposed results with a previous one are also given.

Revisiting Approximation Techniques to Reduce Order of Interval System

Abstract Large interconnected modules result in complex system of higher order and often of interval structure, making the overall study and analysis, time consuming and complicated. Accepting the challenge to state an approximate model of such system, both system analysts and control engineers, headed towards the model order reduction. Continuing the same, this paper revisits few noteworthy estimation techniques for simplification of discrete-time interval system. In particular, denominator is derived through reciprocal algorithm and numerator by two varied algorithms. The proposed algorithms are validated with examples from literatures and real-time test systems via assessment of error computation. Limitation encountered during the course is also taken into count.

시변 지연시간을 갖는 이산 구간 시스템의 안정조건

본 논문에서는 상태변수에 시변 지연시간이 있는 선형 이산 구간 시스템의 안정조건을 고려한다. 고려한 시스템은 시스템 행렬과 지연 상태변수에 대한 시스템 행렬이 구간 행렬로 표현되며, 지연시간도 구간에 대하여 시변인 특성을 갖는다. 제안된 안정조건은 리아프노프 안정 이론에 의하여 유도되며 매우 간단한 부등식의 형태로 표현된다. 기존의 시불변 구간행렬의 안정성 문제를 시변 지연 시간을 갖는 시스템으로 확장한 것이다. 더불어, 새로운 안정조건은 시불변 경우에 대하여 연구된 기존 결과를 포함할 수 있으며, 구간 시변 지연 시간과 시스템의 안정성과의 연관관계를 나타내는 것이다. 제안된 조건은 구간시스템에 대한 교란 변수의 크기를 구하는 문제에도 응용될 수 있다. 수치예제를 통하여 새로운 안정조건의 효용성을 확인할 수 있으며, 기존에 발표된 결과들과의 비교도 이루어진다. The stability condition of linear discrete interval systems with a time-varying delay time is considered. The considered system has interval system matrices for both non-delayed and delayed states with time-varying delay time within given interval values. The proposed condition is derived by using Lyapunov stability theory and expressed by very simple inequality. Compared to previous results, the stability issue on the interval systems is expanded to time-varying delay. Furthermore, the new condition can imply the existing results on the time-invariant case and show the relation between interval time-varying delay time and stability of the system. The proposed condition can be applied to find the stability bound of the discrete interval system. Some numerical examples are given to show the effectiveness of the new condition and comparisons with the previously reported results are also presented.

Robust Stability of Discrete Large-Scale Interval Time-Delay Systems

This paper addresses the robust stability analysis for discrete large-scale interval systems subjected to time delays. We establish a very simple upper solution bound of the discrete algebraic Lyapunov equation (DALE). Then, using the Lyapunov equation approach associated with this upper bound and some linear algebraic techniques, several simple criteria are presented to guarantee the robust stability of the overall systems. The featuresof these present results arethat they are independent of any Lyapunov equation although the Lyapunov equation approach is adopted.

RobustH∞Control for Discrete-Time Stochastic Interval System with Time Delay

The robustH∞control problem for discrete-time stochastic interval system (DTSIS) with time delay is investigated in this paper. The stochastic interval system is equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing the appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robustH∞controller for DTSIS are obtained in terms of linear matrix inequality (LMI) form, and the robustH∞controller is designed. Finally, a numerical example with simulation is given to show the effectiveness and correctness of the designed robustH∞controller.

Guardian map approach to robust stability of interval systems

A systematic approach for the robust Hurwitz and Schur stability of the dynamic interval systems is proposed. An interval matrix is expressed as a linear fractional transformation (LFT) of an interconnection matrix with structured real parametric uncertainties. Based on guardian map theory and µ-analysis, a new approach is provided to derive the necessary and sufficient conditions in terms of the structured singular value (μ) ensuring the stability robustness of interval systems. This approach is feasible for both continuous- and discrete-time interval systems by a unified LFT framework, and it is applicable directly to D-stability for performance requirements.

Model reduction of discrete interval system using clustering of poles

This paper presents a method for reduction of discrete interval systems using pole clustering and Pade approximation. For a given high-order system, denominator of the reduced-order model is obtained by pole clustering and the numerator is obtained by matching first r time moments of the system and its reduced-order model where r is the order of reduced-order model. Two numerical examples are included to illustrate the procedure.

Simultaneous Schur stability of interval matrices

Interval matrix structures are ubiquitous in nature and engineering. Ordinarily, in an uncertain system there is associated with a set of coupled interval matrices, a basic issue of exploring its asymptotic stability. Here we introduce the notion of simultaneous Schur stability by linking the concepts of the majorant and the joint spectral radius, and prove the asymptotic stability of a set of interval matrices governed by simultaneous Schur stability. The present result may lead to the stability analysis of discrete dynamical interval systems.

A Sufficient Condition for Asymptotic Stability of Discrete-Time Interval System with Delay

The asymptotic stability of discrete-time interval system with delay is discussed. A new sufficient condition for preserving the asymptotic stability of the system is presented by means of the inequality techniques. By mathematical analysis, the stability criterion is less conservative than that in previous result. Finally, one example is given to demonstrate the applicability of the present scheme.

Global asymptotic stability analysis of discrete-time Cohen–Grossberg neural networks based on interval systems

The global asymptotic stability of discrete-time Cohen–Grossberg neural networks (CGNNs) with or without time delays is studied in this paper. The CGNNs are transformed into discrete-time interval systems, and several sufficient conditions for asymptotic stability for these interval systems are derived by constructing some suitable Lyapunov functionals. The conditions obtained are given in the form of linear matrix inequalities that can be checked numerically and very efficiently by using the MATLAB LMI Control Toolbox. Finally, some illustrative numerical examples are provided to demonstrate the effectiveness of the results obtained.