Robust stability manifolds for multilinear interval systems

The stability of a class of multilinearly perturbed families of systems is considered. It is shown how the problem of checking the stability of the entire family can be reduced to that of checking certain subsets that are independent of the degrees of the polynomials involved. The extremal property of these subsets is established. The results point to the need for a complete study of the stability of manifolds of polynomials composed of products of simple surfaces     

Robust stability of polynomials with multilinearly dependentcoefficient perturbations

Analysis of the robust stability of a polynomial with multilinearly dependent coefficient perturbations is presented in this note. Some sufficient conditions for forming a convex polygon with the value set of the polynomials with multilinearly dependent coefficient perturbations are obtained. A zero-exclusion algorithm is then given to determine the D-stability of such polynomials. The well known Kharitonov's theorem and the edge theorem for stability analysis can be included as special eases of the authors' conclusions     

Robust Schur stability of interval polynomials

The investigation of Schur stability using a Kharitonov parameter box is discussed. The discrete counterpart of Kharitonov's theorem is obtained. The solution is based on the use of the Hollot-Bartlett-Huang theorem and the Hollott-Bartlett theorem. This made it possible to test for Schur stability only a subset of the edges. The Schur testing of the required edges of the cube is performed using three different methods, namely, the critical edge polynomial, edge stability as an eigenvalue problem, and edge stability using colinearity conditions. Comparison of these three methods is presented. It is believed that, with the Schur testing of the minimum number of edges and the use of the critical stability constraints, minimum computational effort can be achieved     

Robust stability with structured real parameter perturbations

This paper considers the problem of robust stabilization of a linear time-invariant system subject to variations of a real parameter vector. For a given controller the radius of the largest stability hypersphere in this parameter space is calculated. This radius is a measure of the stability margin of the closed-loop system. The results developed are applicable to all systems where the closed-loop characteristic polynomial coefficients are linear functions of the parameters of interest. In particular, this always occurs for single-input (multioutput) or single-output (multiinput) systems where the transfer function coefficients are linear or affine functions of the parameters. Many problems with transfer function coefficients which are nonlinear functions of physical parameters can be cast into this mathematical framework by suitable weighting and redefinition of functions of physical parameters as new parameters. The largest stability hyperellipsoid for the case of weighted perturbations and a stability polytope in parameter space are also determined. Based on these calculations a design procedure is proposed to robustify a given stabilizing controller. This algorithm iteratively enlarges the stability hypersphere or hyperellipsoid in parameter space and can be used to design a controller Io stabilize a plant subject to given ranges of parameter excursions. These results are illustrated by an example.     

Robust stability analysis of polynomials with linearly dependentcoefficient perturbations

A computational tractable procedure for robust pole location analysis of uncertain linear time-invariant dynamical systems, whose characteristic polynomial coefficients depend linearly on parameter perturbations, is proposed. It is shown that, in the case of linearly dependent coefficient perturbations, the stability test with respect to any unconnected domain of the complex plane can be carried out, and the largest stability domain in parameter space can be computed by using only a quick test on a particular set of polynomials named vertex polynomials. The procedure requires only one sweeping function and simple geometrical considerations at each sweeping step. This leads to a very short execution time, as is shown in an example. A unification with Kharitonov's theory and edge theorem is also provided     

Simple robust stability tests for polynomials with real parameter uncertainty

In this paper we consider several problems related to the robust stability of polynomials whose coefficients depend on real parameters. We first show that if these parameters appear linearly in the coefficients, the computation of the largest stability hypercube with a fixed centre does not require a frequency ‘sweep’. Its size can be determined by considering a finite set of frequencies. We also demonstrate that for some polynomials where the coefficients themselves are polynomials in the parameters, the determination of stability can be made in a manner that involves a single frequency sweep and where at each frequency a finite number of simple tests are carried out. The methods are based on the Nyquist theorem and ‘value set’ ideas, and the approach provides analytical solutions.     

Robust Schur stability of a polytope of polynomials

The main objective of the authors is to provide a necessary and sufficient condition for a polytope of polynomials to have all its zeros inside the unit circle. The criterion obtained serves as a discrete-time counterpart for results in S. Bialas (1985) and F. Fu and B.R. Barmish (1987) for the continuous case. Also, the results are reduced to operations on (n-1)×(n-1) matrices. It is concluded that, by the edge result of A.C. Bartlett et al. (1987), it suffices to check the exposed edges in order to determine whether a polytope of polynomials has all its zeros in a simply connected region D     

Robust stability of a family of disc polynomials

In his well-known theorem, V. L. Kharitonov established that Hurwitz stability of a set f1 of interval polynomials with complex coefficients (polynomials where each coefficient varies in an arbitrary but prescribed rectangle of the complex plane) is equivalent to the Hurwitz stability of only eight polynomials in this set. In this paper we consider an alternative but equally meaningful model of uncertainty by introducing a set fD of disc polynomials, characterized by the fact that each coefficient of a typical element P(s) in fD can be any complex number in an arbitrary but fixed disc of the complex plane. Our result shows that the entire set is Hurwitz stable if and only if the ‘center’ polynomial is stable, and the H _∞-norms of two specific stable rational functions are less than one. Our result can be readily extended to deal with the Schur stability problem and the resulting condition is equally simple.     

参数摄动混杂离散系统的鲁棒稳定性

采用线性矩阵不等式(LMI)方法研究离散事件状态转移条件为状态依赖的参数摄动线性混杂离散系统的鲁棒稳定性问题, 提出此类系统全局鲁棒渐近稳定性判定定理, 基于分段Lyapunov函数给出了一般混杂离散系统在Lyapunov意义下局部稳定的判定定理, 该定理可将线性混杂离散系统的稳定性问题转化为LMI问题, 在此基础上提出了参数摄动线性混杂离散系统在Lyapunov意义下局部鲁棒稳定的充分条件.     

Robust stability of parameter uncertain systems with limited data rates

This article investigates the robust stability problem for parameter uncertain systems with limited data rates. A time-varying recursive allocation (TVRA) algorithm is proposed to solve the modal decomposition and data-rate allocation problems that have to be faced for a multi-dimensional uncertain system. Sufficient conditions for robust stability of the feedback loop are derived. A positive critical lower bound of data rates is presented, below which there exists no quantisation, coding and control scheme to stabilise an unstable parameter uncertain plant. Simulation results show the validity of the proposed scheme.     

Robust stability of a class of polynomials with coefficientsdepending multilinearly on perturbations

Necessary and sufficient conditions are given for robust stability of a family of polynomials. Each polynomial is obtained by a multilinearity perturbation structure. Restrictions on the multilinearity are involved, but, in contrast to existing literature, these restrictions are derived from physical considerations stemming from analysis of a closed-loop interval feedback system. The main result indicates that all polynomials in the family of polynomials have their zeros in the strict left half-plane if and only if two requirements are satisfied at each frequency. The first requirement is the zero exclusion condition involving four Kharitonov rectangles. The second requirement is that a specially constructed &thetas;0-parameterized set of 16 intervals must cover the positive reals for each &thetas; ε[0,2π]     

Robust stability for linear time-delay systems with linear parameter perturbations

A new robust state feedback controller design in the time domain is developed for parametrically perturbed linear systems with time delay. The properties of the matrix measure and comparison theorem are employed to investigate the robust stability conditions of systems with linear structured or unstructured perturbations. A method is proposed to design the robust state feedback controller to satisfy the robustness requirement. Examples are given to illustrate the proposed method.     

Robust stability of Metzler operators under parameter perturbations

In this paper we study how the spectral bound of Metzler operators changes under parameter perturbations. Characterizations of the stability radii of Metzler operator with respect to this type of disturbances are established. The results generalize those obtained in (Vietnam J. Math. 2006; 34 :357–368; Vietnam J. Math. 1998; 26 :147–163). Copyright © 2009 John Wiley & Sons, Ltd.     

Necessary and sufficient conditions for the stability of polynomials with linear parameter dependencies

In this paper we show that Kharitonov's theorem will provide a necessary and sufficient condition for the stability of a particular family of polynomials where the coefficients are real and linearly dependent on a set of uncertain parameters. The main results are obtained by restricting the linear map from the parameter to the coefficient space to contain the Kharitonov polynomials. The results are applied to both continous-time and discrete-time polynomials and future extensions are discussed.     

参数不确定时滞系统的鲁棒稳定性

研究一类具有参数不确定时滞系统的鲁棒稳定性问题．假定其线性部分的参数不确定性由区间摄动模武描述，非线性部分的动态不确定性由积分二次约束(IQC)描述，给出了时滞互联系统鲁棒稳定的充分条件，并根据IQC乘子的特性，将无穷维稳定性检验问题转化为一维检验或有限检验问题．     

Robust stability analysis of singular linear system with delay and parameter uncertainty

This paper deals with th e problem of robust stability for continuous-time sin gular systems with state delay and parameter uncerta inty.The uncertain singular systems with delay consi dered in this paper are assumed to be regular and impulse free.By decomposing the systems into slow and fast subsystems,a robust delay-dependent asymptotic stability criteria based on linear matrix inequality is proposed,which is derived by using Lyapunov-Krasovskii functionals,neither model transformatio n nor bounding for cross terms is required in the derivation of our delay-dependent result.The r obust delay-dependent stability criterion proposed in th is paper is a sufficient condition.Finally,numerical examples and Matlab simulation are provided to illustrate the effectiveness of the proposed method.     

Robust stability analysis of singular linear system with delay and parameter uncertainty

1Introduction It is well known that the existence of a delay in adynamical system may induce instability or poorperformances in various systems suchas electric,pneumatic,and hydraulic networks,chemical processes,longtransmission lines,etc.For a survey of time_delay systems,the reader can refer to a recent overview paper[1].Control of singular systems has been extensively studied inthe past years due to the fact that singular systems betterdescribe physical systems than regular ones.Agreat numb…     

Robust stability and stabilization for singular systems with statedelay and parameter uncertainty

Considers the problems of robust stability and stabilization for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm bounded. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free, and stable for all admissible uncertainties, while the purpose of the robust stabilization is to design a state feedback control law such that the resulting closed-loop system is robustly stable. These problems are solved via the notions of generalized quadratic stability and generalized quadratic stabilization, respectively. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are derived. A strict linear matrix inequality (LMI) design approach is developed. An explicit expression for the desired robust state feedback control law is also given. Finally, a numerical example is provided to demonstrate the application of the proposed method