A novel iterative method to approximate structured singular values

A novel method for approximating structured singular values (also known as mu-values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in structured eigenvalue perturbation theory. Our approach consists of an inner-outer iteration. In the outer iteration, a Newton method is used to adjust the perturbation level. The inner iteration solves a gradient system associated with an optimization problem on the manifold induced by the structure. Numerical results and comparison with the well-known Matlab function mussv, implemented in the Matlab Control Toolbox, illustrate the behavior of the method

Computing µ-values and pseudo-spectra for Airy Operators

Stability analysis play a vital role to design a linear feedback system in system theory. The stability of a feedback system is the direct measure of the roots of characteristic equation of transfer functions. The main objective of this article is to present numerical approximation of bounds of µ-values and computation of pseudospectrum for a class of Airy Operators. The comparison of the bounds of µ-values with the well-known MATLAB routine mussv is investigated which illustrate the behaviour of proposed methodology.Publisher's Versio

Numerical Approximation of Bounds of µ-Values for a Family of Pascal Matrices

In this article, we present numerical approximations to lower bounds of Structured Singular Values (SSV) for a family of Pascal matrices. In mathematics, particularly in matrix theory, Pascal matrix is an infinite matrix containing the binomial coefficients as its elements. The obtained lower bounds of SSV are then compared with the well-known MATLAB routine mussv available in MATLAB Control Toolbox

Computing Lower Bounds of µ-Values for a Class of Rotary Electrical Machines

In this article we present the computations of lower bounds of well-known mathematical quantity in control theory known as structured singular value for a family of structured matrices obtained for a DC Motor, that is an electrical machine. The comparison of lower bounds with the well-known MATLAB function mussv is studied. The structured singular values provide an important tool to synthesize robustness as well as analyze performance and stability of feedback systems

Quadratic stability of non-linear systems modeled with norm bounded linear differential inclusions

In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system. KEYWORDS: quadratic stability; Lyapunov function; gradient system of ODE's; bounded linear differential inclusio

Computing Nearest Correlation Matrix via Low-Rank ODE’s Based Technique

For n-dimensional real-valued matrix A, the computation of nearest correlation matrix; that is, a symmetric, positive semi-definite, unit diagonal and off-diagonal entries between −1 and 1 is a problem that arises in the finance industry where the correlations exist between the stocks. The proposed methodology presented in this article computes the admissible perturbation matrix and a perturbation level to shift the negative spectrum of perturbed matrix to become non-negative or strictly positive. The solution to optimization problems constructs a gradient system of ordinary differential equations that turn over the desired perturbation matrix. Numerical testing provides enough evidence for the shifting of the negative spectrum and the computation of nearest correlation matrix

Computing Structured Singular Values for Sturm-Liouville Problems

In this article we present numerical computation of pseudo-spectra and the bounds of Structured Singular Values (SSV) for a family of matrices obtained while considering matrix representation of SturmLiouville (S-L) problems with eigenparameter-dependent boundary conditions. The low rank ODE’s based technique is used for the approximation of the bounds of SSV. The lower bounds of SSV discuss the instability analysis of linear system in system theory. The numerical experimentation show the comparison of bounds of SSV computed by low rank ODE’S technique with the well-known MATLAB routine mussv available in MATLAB Control Toolbox

MATHICSE Technical Report : A novel iterative method to approximate structured singular values

A novel method for approximating structured singular values (also known as $\mu$- values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in structured eigenvalue perturbation theory. Our approach consists of an inner-outer iteration. In the outer iteration, a Newton method is used to adjust the perturbation level. The inner iteration solves a gradient system associated with an optimization problem on the manifold induced by the structure. Numerical results and comparison with the well-known Matlab function mussv, implemented in the Matlab Control Toolbox, illustrate the behavior of the method

Stability analysis of linear feedback systems in control

This article presents a stability analysis of linear time invariant systems arising in system theory. The computation of upper bounds of structured singular values confer the stability analysis, robustness and performance of feedback systems in system theory. The computation of the bounds of structured singular values of Toeplitz and symmetric Toeplitz matrices for linear time invariant systems is presented by means of low rank ordinary differential equations (ODE’s) based methodology. The proposed methodology is based upon the inner-outer algorithm. The inner algorithm constructs and solves a gradient system of ODE’s while the outer algorithm adjusts the perturbation level with fast Newton’s iteration. The comparison of bounds of structured singular values approximated by low rank ODE’s based methodology results tighter bounds when compared with well-known MATLAB routine mussv, available in MATLAB control toolbox

Computing μ-Values for Real and Mixed μ Problems

In various modern linear control systems, a common practice is to make use of control in the feedback loops which act as an important tool for linear feedback systems. Stability and instability analysis of a linear feedback system give the measure of perturbed system to be singular and non-singular. The main objective of this article is to discuss numerical computation of the μ -values bounds by using low ranked ordinary differential equations based technique. Numerical computations illustrate the behavior of the method and the spectrum of operators are then numerically analyzed