scholarly journals Numerical Computation of Lower Bounds of Structured Singular Values

2018 ◽  
Vol 11 (3) ◽  
pp. 844-868
Author(s):  
M. Fazeel Anwar ◽  
Mutti-Ur Rehman
Keyword(s):  
System Theory ◽  
Linear Algebra ◽  
Lower Bounds ◽  
Robust Stability ◽  
Numerical Approximation ◽  
Singular Values ◽  
Numerical Linear Algebra ◽  
Linear Feedback ◽  
Feedback Systems ◽  
Stability And Instability

In this article we consider the numerical approximation of lower bounds of Structured Singular Values, SSV. The SSV is a wellknown mathematical quantity which is widely used to analyse and syntesize the robust stability and instability analysis of linear feedback systems in control theory. It links a bridge between numerical linear algebra and system theory. The computation of lower bounds of SSV by means of ordinary differential equations based technique is presented. The obtained numerical results for the lower bounds of SSV are compared with the well-known MATLAB function mussv available in MATLAB control toolbox.

scholarly journals On Instability Analysis of Linear Feedback Systems

Computation ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 16
Author(s):  
Mutti-Ur Rehman ◽  
Jehad Alzabut
Keyword(s):  
Differential Equation ◽  
System Theory ◽  
Numerical Simulation ◽  
Lower Bounds ◽  
Numerical Approximation ◽  
Upper Bounds ◽  
Low Rank ◽  
Linear Feedback ◽  
Feedback Systems ◽  
Lower And Upper Bounds

The numerical approximation of the μ -value is key towards the measurement of instability, stability analysis, robustness, and the performance of linear feedback systems in system theory. The MATLAB function mussv available in MATLAB Control Toolbox efficiently computes both lower and upper bounds of the μ -value. This article deals with the numerical approximations of the lower bounds of μ -values by means of low-rank ordinary differential equation (ODE)-based techniques. The numerical simulation shows that approximated lower bounds of μ -values are much tighter when compared to those obtained by the MATLAB function mussv.

In Search of Weaker Conditions to Insure Robust Stability in Linear Feedback Systems

1991 ◽  
Vol 113 (1) ◽  
pp. 168-170
Author(s):  
Yossi Chait ◽  
Nir Cohen ◽  
C. R. MacCluer
Keyword(s):  
Robust Stability ◽  
Approximation Error ◽  
Maximum Modulus ◽  
Linear Feedback ◽  
Sufficient Condition ◽  
Feedback Systems ◽  
Sector Condition ◽  
Plant Uncertainty ◽  
Weaker Conditions

This paper is concerned with the well-known condition for robust stability of systems with plant uncertainty. It is shown that the usual sufficient condition for robust stability, given in terms of the maximum modulus of the plant approximation error, can be relaxed to a sector condition. This sector condition, related to the phase of the uncertain portion of the plant, can increase the range of the allowed variations in the parameters of the uncertain plant sufficient for robust stability.

Communication lower bounds and optimal algorithms for numerical linear algebra

Acta Numerica ◽  
2014 ◽  
Vol 23 ◽  
pp. 1-155 ◽  
Author(s):  
G. Ballard ◽  
E. Carson ◽  
J. Demmel ◽  
M. Hoemmen ◽  
N. Knight ◽  
...  
Keyword(s):  
Linear Algebra ◽  
Lower Bounds ◽  
Krylov Subspace ◽  
Sparse Matrices ◽  
Arithmetic Operation ◽  
Matrix Multiplication ◽  
Direct Methods ◽  
Numerical Linear Algebra ◽  
Theory And Practice ◽  
Large Speed

The traditional metric for the efficiency of a numerical algorithm has been the number of arithmetic operations it performs. Technological trends have long been reducing the time to perform an arithmetic operation, so it is no longer the bottleneck in many algorithms; rather, communication, or moving data, is the bottleneck. This motivates us to seek algorithms that move as little data as possible, either between levels of a memory hierarchy or between parallel processors over a network. In this paper we summarize recent progress in three aspects of this problem. First we describe lower bounds on communication. Some of these generalize known lower bounds for dense classical (O(n3)) matrix multiplication to all direct methods of linear algebra, to sequential and parallel algorithms, and to dense and sparse matrices. We also present lower bounds for Strassen-like algorithms, and for iterative methods, in particular Krylov subspace methods applied to sparse matrices. Second, we compare these lower bounds to widely used versions of these algorithms, and note that these widely used algorithms usually communicate asymptotically more than is necessary. Third, we identify or invent new algorithms for most linear algebra problems that do attain these lower bounds, and demonstrate large speed-ups in theory and practice.

scholarly journals Stability Analysis of Linear Feedback Systems in Control

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1518
Author(s):  
Mutti-Ur Rehman ◽  
Jehad Alzabut ◽  
Muhammad Fazeel Anwar
Keyword(s):  
System Theory ◽  
Stability Analysis ◽  
Linear Time ◽  
Singular Values ◽  
Low Rank ◽  
Gradient System ◽  
Feedback Systems ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Time Invariant

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.

scholarly journals Computing μ-Values for Real and Mixed μ Problems

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 821
Author(s):  
Mutti-Ur Rehman ◽  
Muhammad Tayyab ◽  
Muhammad Fazeel Anwar
Keyword(s):  
Control Systems ◽  
Numerical Computation ◽  
Feedback System ◽  
Feedback Loops ◽  
Linear Control ◽  
Linear Control Systems ◽  
Linear Feedback ◽  
Feedback Systems ◽  
Perturbed System ◽  
Stability And Instability

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.

scholarly journals Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices

2021 ◽  
Author(s):  
Stefano Massei
Keyword(s):  
Computer Science ◽  
Recent Work ◽  
Linear Algebra ◽  
Optimization Problems ◽  
Approximation Error ◽  
Positive Semidefinite ◽  
Numerical Linear Algebra ◽  
Maximum Volume ◽  
Deterministic Algorithms ◽  
Principal Submatrix

AbstractVarious applications in numerical linear algebra and computer science are related to selecting the $$r\times r$$ r × r submatrix of maximum volume contained in a given matrix $$A\in \mathbb R^{n\times n}$$ A ∈ R n × n . We propose a new greedy algorithm of cost $$\mathcal O(n)$$ O ( n ) , for the case A symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by $$(r+1)$$ ( r + 1 ) times the error of the best rank r approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms, which are capable to retrieve a quasi optimal cross approximation with cost $$\mathcal O(n^3)$$ O ( n 3 ) .

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