Polynomial Matrix Factorization Research Articles

Polynomial Control Toolbox: A New Aid for Teaching Linear Systems in MATLAB

Abstract The Polynomial Control Toolbox is the Matlab Toolbox for polynomials, polynomial matrices and their use for linear control design. It covers basic operations on and among polynomial matrices, special forms of polynomial matrices, various linear equations polynomial matrices and polynomial matrix factorization problems. In addition, procedures for solving both standard and modern control design problems for linear multi-input multi-output systems based on polynomial matrix operations and equations are supported.

Generalized Lyapunov equation and factorization of matrix polynomials

This paper presents an algorithm for the construction of a solution of the generalized Lyapunov equation. It is proved that the polynomial matrix factorization relative to the imaginary axis may be reduced to the successive solution of Lyapunov equations, i.e. the factorization is reduced to the solution of a sequence of generalized Lyapunov equations, not to the solution of generalized Riccati equation.

Generalized Lyapunov Equation and Factorization of Matrix Polynomials

An algorithms for the construction of the solution of the Generalized Lyapunov equation is presented. It is proved that the polynomial matrix factorization relative to the imaginary axis can be reduced to a successive solutions of the Generalized Lyapunov equation.

Discrete generalized algebraic Riccati equations and polynomial matrix factorization

This paper presents an algorithm for solving discrete generalized algebraic Riccati equations with the help of an orthogonal projector. A generalization of the procedure of forming and correcting the orthogonal projector is considered and also that of correcting the proper solution by the Newton-Raphson scheme. The possibility to use the discrete generalized Riccati equations for polynomial matrix factorization with respect to the unit circle is demonstrated. A numerical example is given.

Rational and polynomial matrix factorizations via recursive pole-zero cancellation

We develop a recursive algorithm for obtaining factorizations of the type R(λ)=R 1(λ)R 2(λ) where all three matrices are rational and R 1(λ) is nonsingular. Moreover the factors R 1(λ) and R 2(λ) are such that either the poles of [ R 1(λ)] -1 and R 2(λ) are in a prescribed region Γ of the complex plane, or their zeros. Such factorizations cover the specific cases of coprime factorization, inner-outer factorization, GCD extraction, and many more. The algorithm works on the state-space (or generalized state-space) realization of R(λ) and derives in a recursive fashion the corresponding realizations of the factors.

A Polynomial Matrix Approach to Doubly Coprime Stable Rational Factorizations

This paper gives a new method for computing a doubly coprime factorization of a transfer function matrix. From a polynomial matrix factorization, the method easily derives proper stable rational matrices which give a doubly coprime factorization. This paper also clarifies a relationship between this method and the well-known method which uses state-space representation.

Self-tuning controller design for systems with arbitrary time delays Part 1. Theoretical development

Self-tuning control of multivariable systems with arbitrary time delays has been an active area of research in recent years. The use of an interactor matrix in deriving controllers for such systems has been suggested by various authors. Factorization of the input polynomial matrix has also been proposed for the controller design stage. It is shown in this paper that these two approaches are not, in general, equivalent. A simple and straightforward procedure for the extraction of the input and output time delay matrices is proposed for the input polynomial matrix factorization approach. Conditions for the time delay matrices to exist are also derived for a class of multivariable systems. These conditions provide a theoretical justification for the use of time delay matrices in self-tuning control of such systems.

Proper stable transfer matrix factorizations and internal system descriptions

The exact relations between coprime proper stable factorizations of P(s) and coprime polynomial matrix factorizations are derived, and they directly lead to relations with internal descriptions of the plant in differential operator or state-space form. It is shown that obtaining any right or left proper stable coprime factorization is equivalent to state-feedback stabilization or to designing a full-order full-state observer, respectively. Solving the Diophantine equation is shown to be equivalent to designing a full or reduced-order observer of a linear functional of the state and to designing a stable inverse system; and this suggests new computational methods to solve the Diophantine.

Estimation and solution of linear rational expectations models using a polynomial matrix factorization

A method to solve and estimate multivariate linear rational expectations models is described. The method is based on an iterative factorization of the polynomial matrix that describes the lags and expected leads in the model. Our experience is that the method works well in a variety of applications where other methods are either difficult or expensive to use.

The Linear Servomechanism with Non-Unity Feedback

The paper studies the robust linear time-invariant servomechanism problem with non-unity feedback, using the transfer function set-up and the polynomial matrix factorization method. It is shown that in the scalar case, unity feedback is indeed not necessary for a robust solution with respect to "arbitrary" perturbations of both plant and loop stabilizer. In the multivariable problem however unity feedback becomes necessary when every mode of the signal to be tracked is present in each of its components and when there exists no sensor such that the loop is stabilizable by a stable loop-stabilizer.

High-gain decentralised control of interconnected dynamical systems

It is shown that the class of controllable and observable interconnected dynamical systems that are amenable to high-gain decentralised control can be characterised in terms of the relatively prime polynomial matrix factors of the transfer-function matrices of the disconnected subsystems. Moreover, this characterisation greatly facilitates the synthesis of high-gain decentralised controllers for such amenable systems.

Design of linear multivariable discrete-time tracking systems incorporating fast-sampling error-actuated controllers

Abstract In this paper, the class of linear multivariable continuous-time plants which ire amenable to fast-sampling error-actuated control is characterized in terms of the relatively prime polynomial matrix factors of the transfer function matrices of such plants. It is shown that this characterization greatly facilitates the design of fast-sampling error-actuated controllers, and the resulting synthesis technique is illustrated by designing such a controller for a third-order plant with two inputs and two outputs.

Design of linear multivariable discrete-time tracking systems incorporating fast-sampling decentralized error-actuated controllers

Abstract In this paper, the class of linear multivariable continuous-time plants which are amenable to fast-sampling decentralized error-actuated control is characterized in terms of the relatively prime polynomial matrix factors of the transfer function matrices of such plants. It is shown that this characterization greatly facilitates the design of fast-sampling decentralized error-actuated controllers, and the resulting synthesis technique is illustrated by designing such a controller for a third-order plant. with two inputs and two outputs.

Design of linear multivariable continuous-time tracking systems incorporating high-gain decentralized error-actuated controllers

Abstract In this paper, the class of linear multivariable continuous-time plants which are amenable to high-gain decentralized error-actuated control is characterized in terms of the relatively prime polynomial matrix factors of the transfer function matrices of such plants. It is shown that this characterization greatly facilitates the design of high-gain decentralized error-actuated controllers, and the resulting synthesis technique is illustrated by designing such a controller for a third-order plant with two inputs and two outputs.

Design of linear multivariable continuous-time high-gain output-feedback regulators

In this paper, the class of linear multivariable continuous-time plants which are amenable to regulation by high-gain output-feedback controllers is characterized in terms of the relatively prime polynomial matrix factors of the transfer function matrices of such plants. It is shown that this characterization greatly facilitates the design of high-gain output-feedback regulators, but that the assignment of finite characteristic roots is constrained by the canonical structure of the numerators of transfer function matrices. The resulting synthesis technique is illustrated by designing high-gain output-feedback regulators for two fifth-order plants.

Design of linear multivariable continuous-time tracking systems incorporating error-actuated controllers

In this paper, the class of linear multivariable continuous-time plants which are amenable to high-gain error-actuated control is characterized in terms of the relatively prime polynomial matrix factors of the transfer function matrices of such plants. It is shown that this characterization greatly facilitates the design of high-gain error-actuated controllers, and the resulting synthesis technique is illustrated by designing such a controller for a third-order plant with two inputs and two outputs.