Computation of the structured stability radius via matrix sign function

Robustness of perturbed state space models of the form is considered, where B, C are given matrices, A is an asymptotically stable matrix and D is the unknown perturbation matrix. An efficient algorithm to compute the complex structured stability radius, which is based on the properties of the matrix sign function, is presented. A comparison with previous algorithms shows the efficiency of the new algorithm     

Robustness of network flow control against disturbances and time-delay

This paper studies robustness of Kelly's source and link control laws in (J. Oper. Res. Soc. 49 (1998) 237) with respect to disturbances and time-delays. This problem is of practical importance because of unmodelled flows, and propagation and queueing delays, which are ubiquitous in networks. We first show L_p-stability, for p[1,∞], with respect to additive disturbances. We pursue L_∞-stability within the input-to-state stability (ISS) framework of Sontag (IEEE Trans. Automat. Control 34 (1989) 435), which makes explicit the vanishing effect of initial conditions. Next, using this ISS property and a loop transformation, we prove that global asymptotic stability is preserved for sufficiently small time-delays in forward and return channels. For larger delays, we achieve global asymptotic stability by scaling down the control gains as in Paganini et al. (Proceedings of 2001 Conference on Decision and Control, Orlando, FL, December 2001, pp. 185–190)     

D-stability of polynomial matrices

Necessary and sufficient conditions are formulated for the zeros of an arbitrary polynomial matrix to belong to a given region D of the complex plane. The conditions stem from a general optimization methodology mixing quadratic and semidefinite programming, LFRs and rank-one LMIs. They are expressed as an LMI feasibility problem that can be tackled with widespread powerful interior-point methods. Most importantly, the D-stability conditions can be combined with other LMI conditions arising in robust stability analysis.     

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.     

Sufficient condition for asymptotic stability of discrete interval systems

A counter-example is given to a recent result on the stability analysis of interval matrices by Jiang (1988). A sufficient condition is then presented for the asymptotic stability of the discrete-time interval system X(k + 1) = A _I X(k), by testing the norms of extreme matrices of the interval matrix A _I such that they are all less than one.     

Stability radii of linear discrete-time systems and symplectic pencils

In this paper, we introduce and analyse robustness measures for the stability of discrete-time system x(t + 1) = Ax(t) under parameter perturbations of the form A→A + BDC where B,C are given matrices. In particular we characterize the stability radius of the uncertain system x(t + 1) = (A + BDC)x(t), D an unknown complex perturbation matrix, via an associated symplectic pencil and present an algorithm for the computation of that radius.     

A summability factor theorem involving an almost increasing sequence for generalized absolute summability

In this paper we prove a general theorem on |A;δ|_k-summability factors of infinite series under suitable conditions by using an almost increasing sequence, where A is a lower triangular matrix with non-negative entries. Also, we deduce a similar result for the weighted mean method.     

Real structured singular value conditions for the strong D-stability

Recently, Chen et al. (Systems Control Lett. 24 (1995) 19) proposed conditions for D-stability and strong D-stability in terms of structured singular values. In this paper, simpler conditions for the strong D-stability are derived.     

Adapted BDF Algorithms: Higher-order Methods and Their Stability

We present BDF type formulas of high-order (4, 5 and 6), capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential with parameter A and ordinary polynomials. For A = 0, the new formulas reduce to the classical BDF formulas. Theorems of the local truncation error reveal the good behavior of the new methods with stiff problems. Plots of their 0-stability regions in terms of the eigenvalues of the parameter A h are provided. Plots of their absolute stability regions that include the whole of the negative real axis are provided. The weights of the method usually require the evaluation of a matrix exponential. However, if the dimension of the matrix is large, we shall not perform this calculus and shall only approximate those coefficients once. Numerical examples underscore the efficiency of the proposed codes, especially when one is integrating stiff oscillatory problems.      

Optimal Finite Characterization of Linear Problems with Inexact Data

Abstract. For many linear problems, in order to check whether a certain property is true for all matrices A from an interval matrix A, it is sufficient to check this property for finitely many “vertex” matrices A ∈ A. J. Rohn has discovered that we do not need to use all 2ⁿ² vertex matrices, it is sufficient to only check these properties for 2²ⁿ⁻¹ ≪ 2ⁿ² vertex matrices of a special type A_yz. In this paper, we show that a further reduction is impossible: without checking all 2²ⁿ⁻¹ matrices A_yz, we cannot guarantee that the desired property holds for all A ϵ A. Thus, these special vertex matrices provide an optimal finite characterization of linear problems with inexact data.     

Stability by order stars for non-linear theta-methods based on means

The linear stability analysis of non-linear one-step methods based on means is studied by means of the concept of stability regions and order stars. Concretely, non-linear θ-methods based on harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are considered. Their stability diagrams and order stars show their A-stability for θ≥1/2, and L-stability in some cases. Order stars in the Riemann surface are a requirement for non-linear one-step methods. The advantages and disadvantages of this technique are presented.     

Two-DOF lifted LMI conditions for robust D-stability of polynomial matrix polytopes

This technical note investigates the problem of checking robust D-stability of polytopes of polynomial matrices. Lifted linear matrix inequality (LMI) conditions with two-DOF (two degree of freedom) positive integers (τ, κ) are derived to possess more flexible tradeoff between the conservatism and computational complexity. In the process of formulating the LMIs, the relevant region D is represented by a quadratic constraint in the complex plane. The matrix, composing the quadratic form with the vector of a variable, is called the region matrix. Then a variable substitution approach is put forward for the lifted LMI version by extending the dimensions of the region matrix and the Lyapunov matrix. The effectiveness and advantages of the proposed method have been illustrated by numerical examples.     

A Contribution to the Feasibility of the Interval Gaussian Algorithm

We apply the interval Gaussian algorithm to an n × n interval matrix [A] whose comparison matrix 〈[A]〉 is generalized diagonally dominant. For such matrices we prove conditions for the feasibility of this method, among them a necessary and sufficient one. Moreover, we prove an equivalence between a well-known sufficient criterion for the algorithm on H matrices and a necessary and sufficient one for interval matrices whose midpoint is the identity matrix. For the more general class of interval matrices which also contain the identity matrix, but not necessarily as midpoint, we derive a criterion of infeasibility. For general matrices [A] we show how the feasibility of reducible interval matrices is connected with that of irreducible ones. Dedicated to Professor Dr. H. J. Stetter, Wien, on the occasion of his 75th birthday     

Origin-shifted algorithm for matrix eigenvalues

In this paper an origin-shifted algorithm for matrix eigenvalues based on Frobenius-like form of matrix and the quasi-Routh array for polynomial stability is given. First, using Householder's transformations, a general matrix A is reduced to upper Hessenberg form. Secondly, with scaling strategy, the origin-shifted Hessenberg matrices are reduced to the Frobenius-like forms. Thirdly, using quasi-Routh array, the Frobenius-like matrices are determined whether they are stable. Finally, we get the approximate eigenvalues of A with the largest real-part. All the eigenvalues of A are obtained with matrix deflation. The algorithm is numerically stable. In the algorithm, we describe the errors of eigenvalues using two quantities, shifted-accuracy and satisfactory-threshold. The results of numerical tests compared with QR algorithm show that the origin-shifted algorithm is fiducial and efficient for all the eigenvalues of general matrix or for all the roots of polynomial.     

Matrix WA + AtW and its applications

For matrix A, with off–diagonal elements not necessarily of the same sign, conditions are obtained for the existence of a positive definite diagonal matrix W, such that matrix WA + A^tW is positive definite; and the applications of the latter to the determination of the stability of interval matrices are considered.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

Pointwise frequency responses framework for stability analysis in periodically time-varying systems

This paper explicates a pointwise frequency-domain approach for stability analysis in periodically time-varying continuous systems, by employing piecewise linear time-invariant (PLTI) models defined via piecewise-constant approximation and their frequency responses. The PLTI models are piecewise LTI state-space expressions, which provide theoretical and numerical conveniences in the frequency-domain analysis and synthesis. More precisely, stability, controllability and positive realness of periodically time-varying continuous systems are examined by means of PLTI models; then their pointwise frequency responses (PFR) are connected to stability analysis. Finally, Nyquist-like and circle-like criteria are claimed in terms of PFR's for asymptotic stability, finite-gain L_p-stability and uniformly boundedness, respectively, in linear feedbacks and nonlinear Luré connections. The suggested stability conditions have explicit and direct matrix expressions, where neither Floquet factorisations of transition matrices nor open-loop unstable poles are involved, and their implementation can be graphical and numerical. Illustrative studies are sketched to show applications of the main results.     

High-order stiffly stable methods for analysis of non-linear circuits and systems

Two easy-to-check conditions are given which, together, are sufficient conditions for A₀ stability, A(0)-stability and stiff stability of linear multistep integration formulae. As an example, these conditions are applied to high-order stiffly stable operators developed by Dill and Gear, and by Jain and Srivastava.     

Guaranteed stability with subspace methods

We show how stability of models can be guaranteed when using the class of identification algorithms which have become known as ‘subspace methods’. In many of these methods the ‘A’ matrix is obtained (or can be obtained) as the product of a shifted matrix with a pseudo-inverse. We show that whenever the shifted matrix is formed by introducing one block of zeros in the appropriate position, then a stable model results. The cost of this is some (possibly large) distortion of the results, but in some applications that is outweighed by the advantage of guaranteed stability.     

A simple graph theoretic characterization of reachability for positive linear systems

In this paper we consider discrete-time linear positive systems, that is systems defined by a pair (A,B) of non-negative matrices. We study the reachability of such systems which in this case amounts to the freedom of steering the state in the positive orthant by using non-negative control sequences. This problem was solved recently [Canonical forms for positive discrete-time linear control systems, Linear Algebra Appl., 310 (2000) 49]. However we derive here necessary and sufficient conditions for reachability in a simpler and more compact form. These conditions are expressed in terms of particular paths in the graph which is naturally associated with the system.