Real Stability Radius Research Articles

Stability Radii of Positive Linear Functional Differential Equations under Multi-Perturbations

We study stability radii of linear retarded systems described by general linear functional differential equations. A lower and an upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer function. Then, the class of positive linear retarded systems is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices.

Stability Radii of Linear Systems Under Multi-perturbations

In this paper, we study stability radii of linear systems under multi-perturbation of the coefficient matrices. Formulas for complex stability radius are derived. We then consider linear positive systems and prove that for this class of systems, the complex stability radius is equal to the real stability radius which can be computed via a simple formula. We illustrated the obtained results by two examples.

Robust measures for a large platoon of vehicles using non-identical controllers

This paper studies the robustness of a platoon of identical vehicles subject to decentralized control, where each control agent is assumed to only have knowledge of the distance between itself and its immediate forward neighbour. Two different types of controllers: (i) a decentralized PID controller and (ii) a decentralized robust servomechanism controller, are considered in the control of such a platoon of identical vehicles, where it is assumed that the models of the vehicles are subject to perturbation. In this case two types of perturbations are considered: (i) the perturbations applied to the vehicles are identical from vehicle to vehicle, (ii) the perturbations applied to the vehicles are non-identical. A study of two types of robustness measures is then considered: (i) the “real stability radius” robustness with respect to the perturbed closed loop system and (ii): the “string stability margin robustness” with respect to the perturbed closed loop system. Various numerical examples are presented to illustrate the type of results which may be obtained. In this case it is shown that the “string stability margin robustness” measures obtained are much more fragile than the corresponding “real stability robustness measures obtained.

Robust stability of a special class of polynomial matrices with control applications

The robust stability problem of uncertain continuous-time systems described by higher-order dynamic equations is considered in this paper. Previous results on robust stability of Metzlerian matrices are extended to matrix polynomials, with the coefficient matrices having exactly the same Metzlerian structure. After defining the structured uncertainty for this class of polynomial matrices, we provide an explicit expression for the real stability radius and derive simplified formulae for several special cases. We also report on alternative approaches for investigating robust Hurwitz stability and strong stability of polynomial matrices. Several illustrative examples throughout the paper support the theoretical development. Moreover, an application example is included to demonstrate uncertainty modeling and robust stability analysis used in control design.

Bounds on the real stability radius for affinely perturbed differential-algebraic systems

Bounds on the real stability radius for affinely perturbed differential-algebraic systems

On the Stability of a class of Polytopes of third order square matrices and stability radius

En este trabajo estudiamos las propiedads de estabilidad de un politopo simétrico y convexo de matrices de tercer orden invariantes en el tiempo, este politopo depende de un parámetro real positivo r. Los resultados que se obtienen son aplicados al cálculo de radio real de estabilidad, no dependiente del tiempo, de una matriz cuadrada de tercer orden sometida a cierta multiperturbación afin.

Real Structural Stability Radius of Infinite-Dimensional Linear Systems: Its Estimate

Lower estimate for the real structural stability radius of infinite-dimensional discrete- and continuous-time linear systems is determined. A metric similar to the Frobenius norm of matrices is introduced on the set of linear bounded operators.

A PARAMETER SPACE APPROACH FOR FIXED-ORDER ROBUST CONTROLLER SYNTHESIS BY SYMBOLIC COMPUTATION

In this paper we propose a new method of parameter space design for robust control synthesis, in particular in terms of real stability radius, using quantifier elimination (QE). We also aim at practicality by employing the scheme to combine the sign definition condition (SDC) and a special QE algorithm using Sturm-Habicht sequence. The validity of our approach is confirmed by some concrete examples.

A Bound on the Real Stability Radius of Continuous-Time Linear Infinite-Dimensional Systems

The article examines bounds on disturbance norms under which a linear infinite-dimensional dynamical system remains stable.

Real stability radii for linear neutral delay systems

The real stability radius for perturbed linear neutral delay systems with a single delay is investigated. We present the corresponding readily computable formulae with respect to an arbitrary stability region in the complex plane. The results are further extended to the case of linear neutral delay systems with multiple delays.

Robust output feedback pole assignment using genetic algorithm

In this paper, the problems of output feedback robust approximate pole assignment (OFRAPA) and robust output feedback stabilization (ROFS) are considered. The idea is to search for an optimal output feedback gain matrix such that objective functions defined via appropriate robustness measures and pole assignment constraints can be optimized using a genetic algorithm (GA). The nature of the GA renders it suitable for optimizing non-smooth objectives such as that involving computation of the spectral condition number or the real stability radius. Furthermore, constraints on the elements of the feedback gain matrix can be catered for naturally in the GA parameter encoding procedure.

Robust stability of linear systems

Computationally efficient estimates for the real stability radius of a system matrix are developed for continuous and discrete time systems. The Frobenius norm is used instead of the operator norm and the estimates involve only the stability reserve of the system matrix and the eigenvalues of the Gram matrix of real and imaginary parts of the eigenvectors of the system matrix. A version of the estimate for a class of infinite dimensional systems is presented.

An improved method for determining the stability of interval matrices

In this paper, the problem of the stability of interval matrices has been tackled using the properties of real stability radius. Based on these, a necessary and sufficient condition has been developed for the Hurwitz (Schur) stability of an interval matrix. An algorithm has been suggested on the basis of the above results to determine the stability of such a system. This work provides an alternative tool to that proposed by Wang et al. and it has been claimed, on the basis of comparative results, that it is a more efficient method in terms of both the computation time and the number of matrices to be checked.

Robust output feedback stabilization using genetic algorithm

The Robust Output Feedback Stabilization (ROFS) problem is tackled using Genetic Algorithm (GA) by finding an output feedback gain matrix such that a fitness function is maximized. The fitness function incorporates the real stability radius of the closed-loop system matrix and performance indices characterizing the assignability of the set of closed-loop poles within certain stable regions in the complex plane. The nature of GA renders it suitable for optimizing nonsmooth objectives such as that involving computation of the real stability radius. Constraints on the elements of the feedback gain matrix can also be catered naturally in the GA parameter encoding procedure.

The probabilistic real stability radius

In this paper we study the probabilistic real stability radius , which gives a description of the degradation of the probability of stability beyond the classical real stability radius. The main result of the paper is to determine the probability density function of the singular values of the perturbation matrix ▵. Subsequently, we propose an algorithm for uniform generation of matrix samples, so that the probability of stability can be effectively estimated via a randomized algorithm.

On the estimation of the real stability radius 1

Computationally efficient estimates for the real stability radius of a system matrix are developed for continuous and discrete time systems. The Frobenius norm is used instead of the operator norm and the estimates involve only the stability reserve of the system matrix and the eigenvalues of the Gram matrix of real and imaginary parts of the eigenvectors of the system matrix. A version of the estimate for a class of infinite dimensional systems is presented.

An easily computable estimate for the real unstructured F-stability radius

Computationally efficient estimates for the real stability radius of a system matrix are developed for continuous and discrete-time systems. The Frobenius norm is used instead of the operator norm and the estimates involve only the stability reserve of the matrix and the eigenvalues of the Gram matrix of real and imaginary parts of the eigenvectors of the system matrix.

Analysis of the local robustness of stability for flows

In this paper the problem of measuring the robustness of stability for a perturbed continuous-time nonlinear system at a singular fixed point is studied. Various stability radii are introduced and their values for the nonlinear system and its linearization are compared. It is shown that they generically coincide. This result may also be used to show generic continuity of linear real stability radii. Some examples are presented showing that it is sometimes necessary to consider the nonlinear system directly, and not simply to rely on the information provided by the linearization.

A Robust Stability Problem for Discrete-Time Systems Subject to an Uncertain Parameter

In this paper a linear, discrete-time uncertain system depending in norm-bounded form on an uncertain, real, time varying parameter, whose magnitude is between the quadratic stability radius and the real stability radius, is considered. In this case, stability of the system depends on the time rate of variation of the parameter. By utilizing parameter-dependent Lyapunov functions, one can take the parameter rate of change into account and obtain sufficient conditions for stability which are less restrictive than those imposed by parameter-independent Lyapunov functions.

Robust gamma stability of highly perturbed systems

The authors investigate the robust gamma stability problem of linear time invariant systems with state space representations. The perturbations of concern are assumed to be time invariant with some given structures. Based on real stability radii, they derive various stability robustness criteria such that all the eigenvalues of the perturbed systems are kept in a specified region, a wedge or a disc in the complex plane. Both the cases of continuous-time and discrete-time systems are discussed. The authors also propose a convergent algorithm to improve stability bounds iteratively. Examples illustrate that less conservative bounds can be obtained. Compared with the existing results, they improve the stability bounds by 20 to 48% after only a few iterations.