Generalized proper inverse of polynomial matrices and the existenceof infinite decoupling zeros

A necessary and sufficient condition is presented for the existence of a generalized proper rational inverse for nonsquare polynomial matrices. The condition is then proved to be equivalent to the absence of infinite decoupling zeros. This condition provides a simple test for the absence of infinite decoupling zeros in the polynomial fraction form. It can also be used to remove the infinite decoupling zeros in order to achieve a strongly irreducible system, and to provide a new look of the unimodular matrix operation effect at infinity, for the left and right polynomial fraction forms     

Regulator problem with internal stability: A frequency domain solution

This paper considers the general regulator problem with internal stability where the measured outputs are not necessarily the same as the regulated outputs. Using polynomial matrix techniques, necessary and sufficient conditions are obtained in terms of skew-primeness of two polynomial matrices; one of these polynomial matrices represents the disturbance modes, whereas the other is the polynomial system matrix representing the system zeros. Various special cases considered in the literature are also analyzed in terms of these necessary and sufficient conditions.     

On polynomial matrix spectral factorization by symmetric extraction

We revise the 1963 Davis algorithm [2] for the spectral factorization of a para-Hermitian nonnegative polynomial matrixPhi, by symmetric factor extraction: this algorithm is careless about zeros at infinity. By introducing the notion of diagonal reducedness ofPhi, we obtain an easy sufficient test for the absence of zeros at infinity. We show then how to getPhi, diagonally reduced by diagonal excess reduction steps (similar to the Oono and Yasuura steps), removing all zeros at infinity, and then how to remove synunetrically finite zeros while keeping el, diagonally reduced (hence, free of zeros at infinity). This results in a revised symmetric extraction spectral factorization algorithm with monotone degree control. An example shows the didactical conceptual simplicity of the method. Appropriate symmetric extraction is discovered by revising and discovering important particular one-sided factor extraction properties of polynomial matrices.     

Minimal inversion,command matching and disturbance decoupling in multivariable systems

This paper addresses the two basic and related problems of minimal inversion and perfect output control in linear multivariable systems. A simple analytical expression is obtained for the inverse of the transfer-function matrix of a square multivariable system. This expression is then used to construct a minimal-order inverse of the system. It is shown that the poles of the minimal-order inverse are the transmission zeros of the system. As a result, necessary and sufficient conditions for existence and stability of the inverse system are stated simply in terms of the zero polynomial of the original system. Furthermore, the minimal-order inverse is shown to be proper provided the original system has a full rank feedthrough matrix. The related problem of perfect output control, namely command matching and disturbance decoupling, in linear multivariable systems by means of feedforward controllers is also formulated and solved in a transfer-function setting. It is shown that a necessary and sufficient condition for existence of the required controllers is that the plant zero polynomial is not identical to zero or unstable. The order of the required controllers is equal to the number of plant transmission zeros. The control scheme proposed in this paper is composed of a feedback controller to enhance system stability and robustness, a feedforward controller to ensure command matching, and another feedforward controller to achieve disturbance decoupling. The three controllers have no effect on each other and can therefore be designed independently. A number of numerical examples are discussed for illustration.     

Time-relevant stability of 2D systems

For many 2D systems, one of the independent variables plays a distinct role in the evolution of the trajectories; since often this special independent variable is time, we call such systems ‘time-relevant’. In this paper, we introduce a stability notion for time-relevant systems described by higher-order difference equations. We give algebraic tests in terms of the location of the zeros of the determinant of a polynomial matrix describing the system. We also give an LMI characterization of time-relevant stability involving only constant matrices.     

Novel theory for polynomial and rational matrices at infinity. Part1. Polynomial matrices

A new approach for dealing with the structural properties of polynomial and rational matrices at infinity is presented. In this first part of the development, the attention is focused on polynomial matrices. Some fundamental notions and definitions such as the notion of homogeneous form for polynomial matrices are introduced. These enable us to define important concepts such as the cancellation at infinity, coprimeness at infinity, etc. for polynomial matrices in precisely the same way as for their finite counterparts. We demonstrate that our approach allows interesting generalizations of some important conventional concepts.     

New coprimeness over multivariable polynomial matrices and its application to control delay systems

This paper presents a new notion of coprimeness over multivariable polynomial matrices, where a single variable is given priority over the remaining variables. From a characterization through a set of common zeros of the minors, it is clarified that the presented coprimeness is equivalent to weakly zero coprimeness in the particular variable. An application of the presented coprimeness to control systems with non-commensurate delays and finite spectrum assignment is also presented. Because the presented coprimeness is stronger than minor coprimeness, non-commensurate delays are difficult to deal with in algebraic control theory. The “rational ratio condition” between delays, which can reduce non-commensurate delays to commensurate delays, proves to be both powerful and practical concept in algebraic control theory for delay systems.     

System zeros analysis via the Moore-Penrose pseudoinverse and SVDof the first nonzero Markov parameter

A new characterization of system zeros of an arbitrary linear system described by a state-space model S(A, B, C, D) is presented. The transmission zeros are characterized as invariant zeros of an appropriate strictly proper system with a smaller number of inputs and outputs than the original system. The approach is based on singular value decomposition (SVD) of the first nonzero Markov parameter. This result together with characterization of invariant and decoupling zeros, based on the Moore-Penrose inverse of the first nonzero Markov parameter and the Kalman canonical decomposition theorem, provided in the first part of the paper yield a complete characterization of system zeros of an arbitrary multi-input/multi-output system     

A definition of column reduced proper rational matrices

The aim of this work is to develop analogue concepts of column reduced polynomial matrices for proper rational matrices. A definition of column reducedness for a class of proper rational matrices is proposed and the properties of such matrices are studied, in particular reduction to column reduced form by elementary operations over the ring of proper rational functions, and the relationship between the degrees of the invariant factors of a column reduced matrix and the so-defined column indices. The physical significance of such matrices in terms of their finite structure is explained; this interpretation completely complements the physical interpretation of a column reduced polynomial matrix. An application of the properties of column reduced proper rational matrices to the decoupling problem is also presented: the infinite structure which can be obtained while decoupling a linear multivariable system by non-regular static state feedback is completely characterized.     

Singular filtering via spectral interactor matrix

In this paper the notion of spectral interactor matrix (SIM) for a given square transfer matrix is introduced and used in the singular filtering problem. A SIM is defined as an interactor matrix preserving the spectral properties of the underlying system. After studying the class of all spectral interactor matrices associated with a given transfer matrix, an algorithm for the computation of a SIM is proposed. By resorting to the SIM, it is possible to point out the role played by delays (zeros at infinity) and nonminimum phase zeros in the solution of the singular filtering problem     

Approximate decoupling and asymptotic tracking for MIMO systems

This paper presents an algorithm for approximate input-output decoupling of nonlinear MIMO systems that are either numerically ill-posed or exhibit nearly singular behavior in the application of decoupling algorithms. Although the systems considered are regular, so that the exact decoupling algorithms are applicable in this case, they require inversion of an ill-conditioned matrix, and yield high gain feedback solutions that may result in actuator saturation and cancellation of high frequency zeros. The approximate algorithms of this paper are numerically robust, and provide solutions that do not cancel far off right-half plane zeros. This latter characteristic is especially valuable when some of the far off right-half plane zeros are unstable. The algorithms are inspired by and are generalizations of some examples in the flight control literature     

Disturbance decoupling problems via dynamic output feedback

This paper considers the disturbance decoupling problems, with or without internal stability and pole placement, via dynamic output feedback using polynomial and rational matrix techniques. We show that in all three problems considered, the central solvability condition can be expressed as a two-sided matching problemA = BXC, whereA, B, andCare the polynomial system matrices of certain natural subsystems of the system model andXis to be determined over various subrings of the rational functions. This matching problem can in turn be reduced to certain appropriate zero-cancellation conditions on the polynomial system matricesA, B, andC.     

自校正控制非最小相位的电加热系统

本文提出了一种简单的多变量极点配置自校正解耦控制器。该控制器不仅避免了在线求解矩阵方程和矩阵求逆运算而且实现了自适应解耦控制。该自适应控制器已被成功地应用来控制双输入双输出非最小相位的电加热炉。     

A Toeplitz algorithm for polynomial J-spectral factorization

A block Toeplitz algorithm is proposed to perform the J-spectral factorization of a para-Hermitian polynomial matrix. The input matrix can be singular or indefinite, and it can have zeros along the imaginary axis. The key assumption is that the finite zeros of the input polynomial matrix are given as input data. The algorithm is based on numerically reliable operations only, namely computation of the null-spaces of related block Toeplitz matrices, polynomial matrix factor extraction and linear polynomial matrix equations solving.     

On conditions guaranteeing two polynomial matrices possessidentical zero structures

Some sufficient conditions for two polynomial matrices to simultaneously possess identical finite and infinite zero structures are developed. A special case of this problem, that of obtaining conditions under which a polynomial matrix has no infinite zeros, is also considered and certain connections with previous work are established     

线性定常多变量系统的解耦控制和极零点配置

本文利用多项式矩阵方法,讨论了采用输入——输出反馈补偿器和串联补偿器解耦的有关问题。这些问题是:系统可以解耦的充分必要条件,解耦系统的可控性、可观测性和稳定性,以及解耦系统的极——零点配置等问题。最后,将这种解耦方法和状态反馈解耦进行了比较。     

An algebraic method to determine the common divisor,poles and transmission zeros of matrix transfer functions

A purely algebraic method which uses the matrix Routh algorithm and its reverse process of the algorithm is presented to decompose a matrix transfer function into a pair of right co-prime polynomial matrices or left co-primo polynomial matrices. The poles and transmission zeros of the matrix transfer function are determined from a pair of relatively prime polynomial matrices. Also, the common divisor of two matrix polynomials can be obtained by using the matrix Routh algorithm and the matrix Routh array.     

Disturbance decoupling control design for switched Boolean control networks

This paper investigates a kind of disturbance decoupling problems (DDPs) for switched Boolean control networks (BCNs), that is, the disturbance-independent output decoupling problem, via the semi-tensor product of matrices. By converting the dynamics of switched BCNs into an algebraic form and using the redundant variable separation technique, all possible state feedback controllers are designed for the disturbance-independent output decoupling problem. Then, based on the obtained state feedback disturbance decoupling controllers and the system’s output equation, a constructive procedure is proposed to design all possible output feedback disturbance decoupling controllers. The study of two illustrative examples shows that the new results obtained in this paper are effective in designing disturbance decoupling controllers for switched BCNs.