http://faculty.petra.ac.id/resmana/private/matlab-help/pdf_doc/robust/robust.pdf

Robust Control Toolbox™ User's Guide

The design and analysis of linear control systems: 

$$\dot x(t) = Ax(t) + Bu(t),\,\,\,\,\,\,\,\,\,\,y = Cx(t)$$

(1.1)

and 

$$x_{k + 1} = Ax_k + Bu_k ,\,\,\,\,\,yk = Cx_k$$

(1.2)

give rise to many interesting linear algebra problems. Some of the important ones are: controllability and observability problems, the problem of computing the exponential matrix e At ,the matrix equations problems: (Lyapunov equations, Sylvester equations, the algebraic Riccati equations), the pole-placement problems, stability problems, and frequency response problems. These problems have been very widely studied in the literature. There exists a voluminous work both on theory and computations. Theory is very rich. Unfortunately, the same can not be remarked about computations. Many of the earlier methods were developed before the computer era, and are not based on numerically sound techniques. Fortunately, in the last twenty years or so, numerically effective techniques have been developed for most of these problems, and numerical analysis aspects of these methods (e.g. study of stability by round-off error analysis) and of the problems themselves (e.g. study of sensitivity) have been studied.

Numerical methods for linear control systems

This chapter describes the subroutine library SLICOT that provides Fortran 77 implementations of numerical algorithms for computations in systems and control theory. Around a nucleus of basic numerical linear algebra subroutines, this library builds methods for the design and analysis of linear control systems. A brief history of the library is given together with a description of the current version of the library and the ongoing activities to complete and improve the library in several aspects.

/pdf/slicot-a-subroutine-library-in-systems-and-control-theory-3ass3dtvmq.pdf

SLICOT—A Subroutine Library in Systems and Control Theory

Two new simple, complete, analytical, and restriction-free solutions with complete and explicit freedom of the matrix equation AV+BW=VF are proposed. Here (AB) is known and is controllable, and F is in the Jordan form with arbitrary given eigenvalues. Based on the proposed solutions of this matrix equation, a complete parametric approach for eigenstructure assignment in linear systems via state feedback is proposed, and two new algorithms are presented. The proposed solutions of the matrix equation and the eigenstructure assignment result are generalizations of some previous results and are simpler and more effective. >

Solutions of the equation AV+BW=VF and their application to eigenstructure assignment in linear systems

In this tutorial paper, an overview is given of progress over the past ten to fifteen years towards reliable and efficient numerical solution of various types of Riccati equations. Our attention will be directed primarily to matrix-valued algebraic Riccati equations and numerical methods for their solution based on computing bases for invariant subspaces of certain associated matrices. Riccati equations arise in modeling both continuous-time and discrete-time systems in a wide variety of applications in science and engineering. One can study both algebraic equations and differential or difference equations. Both algebraic and differential or difference equations can be further classified according to whether their coefficient matrices give rise to so-called symmetric or nonsymmetric equations. Symmetric Riccati equations can be further classified according to whether or not they are definite or indefinite.

Invariant Subspace Methods for the Numerical Solution of Riccati Equations

An efficient computational algorithm for pole assignment of linear multiinput systems is proposed. A preliminary stage of the algorithm is a reduction of the system matrices into orthogonal canonical form. The gain matrix elements are then found by orthogonal transformation of the closed-loop system matrix into upper quasi-triangular form whose diagonal blocks correspond to the desired poles. The algorithm is numerically stable, the computed gain matrix being exact for a system with slightly perturbed matrices. It works equally well with real and complex, distinct, and multiple poles and is applicable to ill-conditioned and high-order problems.

A computational algorithm for pole assignment of linear single-input systems

On the numerical properties of the schur approach for solving the matrix Riccati equation

Synthesis of linear systems with desired equivalent form

The sensitivity of the Schur system (the Schur basis and the Schur form) of a general matrix relative to perturbations, is studied. The estimates obtained are nonlocal and sharp. Asymptotic bounds (condition numbers particularly) in the form of power series are derived as a particular case.

N.D. Christov

Papers

A computational algorithm for pole assignment of linear single-input systems

On the numerical properties of the schur approach for solving the matrix Riccati equation

Synthesis of linear systems with desired equivalent form

Nonlocal Perturbation Analysis of the Schur System of a Matrix

Perturbation Analysis of Linear Control Problems