(1983). Large-Scale Systems: Modeling and Control. Journal of the Operational Research Society: Vol. 34, No. 10, pp. 1016-1016.

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Large-Scale Systems: Modeling and Control

A survey of the matrix sign function is presented including some historical background, definitions and properties, approximation theory and computational methods, and condition theory and estimation procedures, Applications to areas such as control theory, eigendecompositions, and roots of matrices are outlined, and some new theoretical results are also given. >

The matrix sign function

Paper: Self-tuning control of nonminimum-phase systems

The quadratic matrix equation AX2+ BX + C = 0in n x nmatrices arises in applications and is of intrinsic interest as one of the simplest nonlinear matrix equations. We give a complete characterization of solutions in terms of the generalized Schur decomposition and describe and compare various numerical solution techniques. In particular, we give a thorough treatment of functional iteration methods based on Bernoulli’s method. Other methods considered include Newton’s method with exact line searches, symbolic solution and continued fractions. We show that functional iteration applied to the quadratic matrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem ({lambda}2A + {lambda}B + C)x = 0.

https://www.maths.manchester.ac.uk/~higham/narep/narep347.pdf

Numerical analysis of a quadratic matrix equation

New theoretical results are presented about the principal matrix pth root. In particular, we show that the pth root is related to the matrix sign function and to the Wiener–Hopf factorization, and that it can be expressed as an integral over the unit circle. These results are used in the design and analysis of several new algorithms for the numerical computation of the pth root. We also analyze the convergence and numerical stability properties of Newton’s method for the inverse pth root. Preliminary computational experiments are presented to compare the methods.

Algorithms for the matrix pth root

Similarity block transformations are developed to transform a class of linear time-invariant MIMO state equations in arbitrary coordinates into block companion forms so that the classical lines of thought for SISO systems can be extended to MIMO systems.

Transformations of a class of multivariable control systems to block companion forms

The paper presents a state-space approach for dealing with self-tuning control problems. A joint algorithm for system-parameter identification and system-state estimation is derived. With the joint algorithm, modern control techniques can be easily applied to systems having stochastic disturbances. The state-space version of the self-tuning control with pole assignment is investigated. Based on the state-space approach, the proposed self-tuning control algorithm can be applied to time-invariant linear systems including the unstable and nonminimum-phase systems.

State-space approach for self-tuning feedback control with pole assignment

This paper proposes an alternate representation of a matrix sign function based on an irrational function described by a continued fraction. The properties of the continued fraction and the truncated continued fractions are investigated. Also, new algorithms for computing the matrix sign function are developed. The matrix sign function is then extended to a generalised matrix sign function for directly solving discrete-time system problems.

Some properties of matrix sign functions derived from continued fractions

A fast state-space self-tuner is developed for suboptimal control of linear stochastic multivariable systems. The suboptimal self-tuner is determined by utilising both the standard recursive-extended-least-squares parameter estimation algorithm and the recently developed matrix sign algorithm, which gives a fast solution of the steady-state discrete Riccati equation. The developed suboptimal state-space self-tuner can be applied to a class of stable/unstable and minimum/non-minimum phase linear stochastic multivariable systems, in which the pair (A, C) is block observable and the pair (A, B) is stablisable. Also, the pair (A, Q?) with Q?Q?T = Q is detectable where A, B and C are system, input and output matrices, respectively, and Q is a weighting matrix in a quadratic performance index.

Fast suboptimal state-space self-tuner for linear stochastic multivariable systems

A matrix sign function in conjunction with a geometric approach is utilized to construct a block modal matrix and a (scalar) modal matrix of a system map, so that the system map can be block-diagonalized and block-triangularized, and that the Riccati-type problems can be solved.

Yih T. Tsay

Papers

Transformations of a class of multivariable control systems to block companion forms

State-space approach for self-tuning feedback control with pole assignment

Some properties of matrix sign functions derived from continued fractions

Fast suboptimal state-space self-tuner for linear stochastic multivariable systems

Block-Diagonalization and Block-Triangularization of a Matrix via the Matrix Sign Function.