Real Stability Radius Research Articles

Subspace method for the estimation of large-scale structured real stability radius

We consider the autonomous dynamical system \(x^{\prime } = Ax,\) with \(A\in {\mathbb {R}}^{n\times n}.\) This linear dynamical system is asymptotically stable if all of the eigenvalues of A lie in the open left-half of the complex plane. In this case, the matrix A is said to be Hurwitz stable or shortly a stable matrix. In practice, the stability of a system can be violated because of perturbations such as modeling errors. In such cases, one deals with the robust stability of the system rather than its stability. The system above is said to be robustly stable if the system, as well as all of its perturbations from a certain perturbation class, are stable. To measure the robustness of the system subject to perturbations, a quantity of interest is the stability radius or in other words the distance to instability. In this paper, we focus on the estimation of the structured real stability radius for large-scale systems. We propose a subspace framework to estimate the structured real stability radius and prove that our new method converges at a quadratic rate in theory. Our method benefits from a one-sided interpolatory model order reduction technique, in the sense that the left and the right subspaces are the same. The quadratic convergence of the method is due to the certain Hermite interpolation properties between the full and reduced problems. The proposed framework estimates the structured real stability radius for large-scale systems efficiently. The efficiency of the method is demonstrated on several numerical experiments.

On the real stability radius of sparse systems

In this paper, we study robust stability of sparse LTI systems using the stability radius (SR) as a robustness measure. We consider real perturbations with an arbitrary and pre-specified sparsity pattern of the system matrix and measure their size using the Frobenius norm. We formulate the SR problem as an equality-constrained minimization problem. Using the Lagrangian method for optimization, we characterize the optimality conditions of the SR problem, thereby revealing the relation between an optimal perturbation and the eigenvectors of an optimally perturbed system. Further, we use the Sylvester equation based parametrization to develop a penalty based gradient/Newton descent algorithm which converges to the local minima of the optimization problem. Finally, we illustrate how our framework provides structural insights into the robust stability of sparse networks.

A globally convergent method to compute the real stability radius for time-delay systems

This paper presents a novel algorithm to compute the real stability radius for a linear delay system of retarded type with multiple delays. The real stability radius is the distance to instability, measured as the minimal real-valued perturbation that renders the system unstable. Our method is based on characterizing this distance to instability as the inverse of the global maximum of a real structured singular value function. We develop a criss-cross type algorithm that globally converges to this maximum, and whose convergence rate seems to be superlinear and sometimes quadratic in numerical experiments. The algorithm exploits that the intersections of these singular value functions with constant functions can be written as purely imaginary eigenvalues of certain delay eigenvalue problems (DEP) with positive and negative delays. This is an extension of the well-known linear case (without delays) where this results in algebraic eigenvalue problems. In addition, a novel numerical solver to compute all the imaginary eigenvalues of this DEP is also presented. It combines an approximation using spectral discretizations and an automatic procedure to determine the required number of discretization points. Finally, due to the presence of multiple eigenvalues at the maximum, these approximations are corrected with a block-Newton algorithm for nonlinear eigenvalue problems.

Stability analysis on flow parameters in coolant temperature control system of lead-cooled fast reactor

Lead-cooled Fast Reactor (LFR), one of the promising nuclear energy technologies for future advanced system, is well foreseen in meeting the goals of Generation IV fission reactor. In LFR, Coolant Temperature Control System (CTCS) is one of the critical parts, and the influence of coolant flow parameters on the stability analysis is worth investigating and studying. In this paper, we use the CLEAR-S, an integrated multifunctional non-nuclear test facility, as reference and the stability analysis of CTCS, concerned to coolant flow rate has been completed based on simplified coolant thermal circulation model. Location of eigenvalues and calculation results of real stability radius, both qualitatively and quantitatively reflect the influence of coolant flow rate on stability respectively. The analysis results show that the feed-water flow rate in steady state is a critical parameter for CLEAR-S, because larger feed-water flow rate can improve stability apparently. However, LBE coolant flow rate has little impact on system stability. The results of this work will provide an effective reference for design and operation of LFR.

Robust stability of differential-algebraic equations with an arbitrary unsolvability index

Consideration was given to the linear stationary systems of differential-algebraic equations with an arbitrarily high unsolvability index. Conditions were established guaranteeing the internal structure of the system at hand against the internal structural modifications caused by the perturbations of the matrix coefficients. Under the assumptions of structural persistence, the sufficient conditions for robust stability were obtained, and the values of real stability radii were given.

Stability radii for real linear Hamiltonian systems with perturbed dissipation

We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical systems. We analyze when such a system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the real distance to instability (real stability radius) has been a challenging problem. We analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical stability of a DH system is much more robust than for unstructured perturbations.

Approximating the Real Structured Stability Radius with Frobenius-Norm Bounded Perturbations

We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the $H_\infty$ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost is dominated by the computation of the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequenc...

Optimal decentralized control of large scale systems

This paper presents a new optimized decentralized controller design method for solving the tracking and disturbance rejection problems for large-scale linear time-invariant systems, using only low-order decentralized controllers. To illustrate the type of results which can be obtained using the new optimized decentralized control design method, the control of a large flexible space structure is studied and compared with the standard centralized LQR-observer controller. The order of the resultant decentralized controller is much smaller than that of the standard centralized LQR-observer controller. The proposed controller also has certain fail-safe properties and, in addition, it can be five orders of magnitude more robust than the standard LQR-observer controller based on their real stability radii. The new decentralized controller design method is applied to a large flexible space structure system with 5 inputs and 5 outputs and of order 24.

On the Method by Rostami for Computing the Real Stability Radius of Large and Sparse Matrices

In a recent paper, Rostami [SIAM J. Sci. Comput, 37 (2015), pp. S447--S471] has presented an interesting algorithm for the computation of the real pseudospectral abscissa and the real stability radius (aka the distance to instability) of a square matrix $A \in \mathbb R^{n,n}$ in the spectral norm. The method is particularly well suited for large sparse matrices. The algorithm to compute the stability radius is based on the alternation of an iterative method in the rank-$2$ manifold of real matrices and a Lyapunov inverse iteration. The algorithm shows quadratic convergence in all experiments, but this peculiar property is only conjectured. In this paper, we provide a rigorous proof and propose a modification which replaces the Lyapunov inverse iteration and speeds up the method. Moreover, we interpret the algorithm to compute the real $\varepsilon$-pseudospectral abscissa as a nonstandard discretization of the ODE-based method given in Guglielmi and Lubich [SIAM J. Matrix Anal. Appl., 34 (2013), pp. 40--...

New Algorithms for Computing the Real Structured Pseudospectral Abscissa and the Real Stability Radius of Large and Sparse Matrices

We present two new algorithms for investigating the stability of large and sparse matrices subject to real perturbations. The first algorithm computes the real structured pseudospectral abscissa and is based on the algorithm for computing the pseudospectral abscissa proposed by Guglielmi and Overton [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166--1192]. It entails finding the rightmost eigenvalues for a sequence of large matrices, and we demonstrate that these eigenvalue problems can be solved in a robust manner by an unconventional eigenvalue solver. We also develop an algorithm for computing the real stability radius of a real and stable matrix, which utilizes a recently developed technique for detecting the loss of stability in a large dynamical system. Both algorithms are tested on large and sparse matrices.

A new approach for calculating the real stability radius

We present a new fast algorithm to compute the real stability radius with respect to the open left half plane which is an important problem in many engineering applications. The method is based on a well-known formula for the real stability radius and the correspondence of singular values of a transfer function to pure imaginary eigenvalues of a three-parameter Hamiltonian matrix eigenvalue problem. We then apply the implicit determinant method, used previously by the authors to compute the complex stability radius, to find the critical point corresponding to the desired singular value. This corresponds to a two-dimensional Jordan block for a pure imaginary eigenvalue in the parameter dependent Hamiltonian matrix. Numerical results showing quadratic convergence of the algorithm are given.

A method for the calculation the real stability radius of bidimensional time-invariant systems

A method for the calculation the real stability radius of bidimensional time-invariant systems

Stability radius of implicit dynamic equations with constant coefficients on time scales

This paper deals with the stability radii of implicit dynamic equations on time scales when the structured perturbations act on both the coefficient of derivative and the right-hand side. Formulas of the stability radii are derived as a unification and generalization of some previous results. A special case where the real stability radius and the complex stability radius are equal is studied. Examples are derived to illustrate results.

Robust Control and Time-Domain Specifications for Systems of Delay Differential Equations via Eigenvalue Assignment

An approach to eigenvalue assignment for systems of linear time-invariant (LTI) delay differential equations (DDEs), based upon the solution in terms of the matrix Lambert W function, is applied to the problem of robust control design for perturbed LTI systems of DDEs, and to the problem of time-domain response specifications. Robust stability of the closed-loop system can be achieved through eigenvalue assignment combined with the real stability radius concept. For a LTI system of DDEs with a single delay, which has an infinite number of eigenvalues, the recently developed Lambert W function-based approach is used to assign a dominant subset of them, which has not been previously feasible. Also, an approach to time-domain specifications for the transient response of systems of DDEs is developed in a way similar to systems of ordinary differential equations using the Lambert W function-based approach.

Stability radii for positive linear time-invariant systems on time scales

We deal with dynamic equations on time scales, where we characterize the positivity of a system. Uniform exponential stability of a system is determined by the spectrum of its matrix. We investigate the corresponding stability radii with respect to structured perturbations and show that, for positive systems, the complex and the real stability radius coincide.

On Positive Linear Volterra-Stieltjes Differential Systems

We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra integro-differential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover, to the best of our knowledge, most of them are new.

Stability radii of differential algebraic equations with structured perturbations

This paper deals with a formula for computing stability radii of a differential algebraic equation of the form A X ′ ( t ) − B X ( t ) = 0 , where A , B are constant matrices. A computable formula for the complex stability radius is given and a key difference between the ordinary differential equation (ODEs for short) and the differential algebraic equation (DAEs for short) is pointed out. A special case where the real stability radius and the complex one are equal is considered.

極配置―安定半径改善問題の一解法

This paper presents a robust controller design method for linear time-invariant SISO systems based on an optimization approach. The task is to enlarge real stability radius to counter plant parameter perturbations by using extra degree of freedom of the controller parameters in the pole placement scheme. The formula for the real stability radius is known as an infimum of a certain function and that makes its gradients discontinuous. It is shown that by some manipulations, the considered problem can be converted to a nonlinear optimization problem, to which a gradient-based optimization method becomes applicable. The case of complex stability radius is also considered, which is less sharp as a robust stability index than the real counterpart, but has a much simpler form. Numerical examples for both cases are presented to show that they actually work, and some comparisons are discussed, leading to a suggestion on the effective uses of these two stability radius computation methods.

Stability radii of higher order linear difference systems under multi-perturbations

We study stability radii of higher order linear difference systems under multi-perturbations. A formula for complex stability radius of higher order linear difference systems under multi-perturbations is given. Then, for the class of positive systems, we prove that the complex stability radius and real stability radius of the system under multi-perturbations coincide and they are computed via a simple formula. These are extensions of corresponding results of Hinrichsen and Son, Hinrichsen et al., Ngoc and Son, and Pappas and Hinrichsen. An example is given to illustrate the obtained results.

A parameter space approach to fixed-order robust controller synthesis by quantifier elimination

This paper proposes a new method of parameter space design for robust control synthesis which guarantees the real stability radius constraint, accomplished by using Quantifier Elimination (QE). We aim at practicality by employing the scheme to combine the Sign Definite Condition (SDC) and a special QE algorithm using the Sturm-Habicht sequence. The validity of our approach is confirmed by several design examples.