Left Matrix Fraction Description Research Articles

Model specification and selection for multivariate time series

Three major difficulties are identified with an established echelon form approach (see Hannan (1987)) to specifying a Vector Autoregressive Moving Average, VARMA, model for an observed time series. A family of state space representations, valid for each integer, h, is introduced, and collectively referred to as multistep state space representations. This family includes as its special case, with h=0, a state space representation introduced earlier by Akaike (1974), and, with h=1, that introduced by Cooper and Wood (1982). Appropriate generalizations of the notions of minimality, McMillan degree, left matrix fraction description and Kronecker indices, as applicable individually to each member of this family, are presented. The reverse echelon form and state space representation corresponding to the Kronecker indices for each h are derived, and the former illustrated with three examples of standard VARMA processes. The question of how the presence of zero constraints on the coefficients of a reverse echelon form may be detected solely from an inspection of the Kronecker indices is examined. A canonical correlation procedure proposed originally by Akaike (1976) for h=0 is considered for estimating the Kronecker indices with each h. The efficacy of the estimation procedure is investigated by a simulation study. A procedure is suggested for implementing the new approach introduced in this paper with an observed time series, and three different applications of this approach are outlined. This approach is also related to some of its alternatives, including the Kronecker invariants of Poskitt (1992) and the scalar component approach of Tiao and Tsay (1989).

A New control structure for tert-amyl methyl ether production using reactive distillation

The aim of this paper is to introduce new control structures in both centralized and decentralized scheme for tert-amyl methyl ether (TAME) production unit using reactive distillation which developed as a benchmark by Luyben (2005). Steady state and dynamic simulation was performed based on the case study data using Aspen® simulator. Dynamic simulation shows that control performance of the plant has oscillatory behavior using conventional PI controllers. Hence, a proper modification on the process flow sheet was carried out in order to remove oscillatory behavior of the process. Although, the new structure has better dynamic performance than the previous one using PI controller, it is not capable of eliminating the oscillations completely. Non-minimal state space with proportional integral plus controller (NMSS–PIP) controller was applied to dynamic model of the modified process. In this way, the system identification was carried out using refined instrumental variable (RIV) method and state matrices was obtained as left matrix fraction description (LMFD). The control performance of the PI and PIP controller for the modified structure revealed that the NMSS–PIP, completely eliminates the process oscillations while it has smoother responses with better settling time than the PI configuration. Furthermore, comparison of the ITAE criterion between main case and modified process indicates that the ITAE for PI and PIP control decreased by about 20% and 88%, respectively.

New developments for matrix fraction descriptions: A fully-parametrised approach

This article aims at giving a new answer for the challenging problem of the parametrisation of multi-input multi-output matrix fraction descriptions. In order to reach this goal, new parametrisations of matrix fraction descriptions, called fully-parametrised left matrix fraction descriptions (F-LMFD), are first introduced. Their structural properties as well as their suitability for multi-input multi-output model description are more precisely analysed. As any over-parametrised model description, the F-LMFD cannot describe a transfer function uniquely. The structure of the space of equivalent F-LMFD is then investigated through the determination of its basis. The study carried out in this article is the prelude to a computational improvement of the identification of matrix fraction descriptions with gradient-based optimisation methods.

A New Elementary Operation Approach to Multidimensional Realization and LFR Uncertainty Modeling: The MIMO Case

This paper proposes a new elementary operation approach (EOA) to multidimensional (-D) realization and linear fractional representation (LFR) modeling for multi-input and multi-output (MIMO) -D systems, as an extension of the new EOA proposed for the single-input and single-output (SISO) case by the authors recently. It is shown that, due to the substantial differences between the SISO and MIMO systems, the extension is not straightforward and further significant development is necessary. A matrix relation property among the associated matrices under the augmenting and admissible elementary operations is first revealed. Based on this matrix relation property, the realization problem for the MIMO -D case is formulated as an elementary operation problem of a certain -D polynomial matrix, which makes the extension possible. General constructive procedures are then established for the regular realizations based on the right and left matrix fraction descriptions (MFDs) of a given transfer matrix, respectively, such that one can easily implement this approach by a computer program in, e.g., MATLAB or Maple. Numerical and symbolic examples are provided to illustrate the main ideas and the effectiveness of the proposed approach.

Identification of a Flexible Cantilever Beam Equipped in Piezoelements for Vibration Control System

AbstractThe identification of an active vibration damping system with the use of piezoelements glued to the structure is presented in the paper. Lab stand was completed with smart beam, power amplifier and charge amplifier. First, the open‐loop system was identified in the frequency domain in using experimental setup The piezo‐stripes to have worked in the mode of actuators and sensors, respectively. For the frequency range from 1 Hz to 400 Hz we obtained Bode plots with help of the Dynamic Signal Analyzer. Frequency characteristics are a base for further calculations. Second, the mathematical model was obtained by finite elements methods and was adjusted to the experimental results by least square sum approach. Third, observable canonical‐form in state space model was derived by using of the LMFD method (Left Matrix Fraction Description). Finally, a minimal realization of the transfer function was reached by Balance method. The transfer function describes the first three resonances and first three anti‐resonances in considered frequency range. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Extending the SRIV Identification Algorithm to MIMO LMFD Models

In this paper the Simplified Refined Instrumental Variable (SRIV) identification algorithm for SISO systems is extended to MIMO systems described by a Left Matrix Fraction Description (LMFD). The performance of the extended algorithm is compared to the well-known MIMO four-step instrumental variable (IV4) algorithm. Monte Carlo simulations for different signal to noise ratios are conducted to assess the performance of the algorithm. Moreover, the algorithm is applied to a simulated quadruple tank process.

Left and right matrix fraction description conversion by solving a square non-singular system of linear equations

This note presents a method of converting a left matrix fraction description into a right matrix fraction description for a square linear time-invariant multivariable system. The method stands out among current methods in that it requires only the solution of a set of linear, square and non-singular system of equations. The method is simple and the conversion from right to left matrix fraction descriptions is analogous.

A Behavioral Approach to the Pole Structure of One-Dimensional and Multidimensional Linear Systems

We use the tools of behavioral theory and commutative algebra to produce a new definition of a (finite) pole of a linear system. This definition agrees with the classical one and allows a direct dynamical interpretation. It also generalizes immediately to the case of a multidimensional (nD) system. We make a natural division of the poles into controllable and uncontrollable poles. When the behavior in question has latent variables, we make a further division into observable and unobservable poles. In the case of a one-dimensional (1D) state-space model, the uncontrollable and unobservable poles correspond, respectively, to the input and output decoupling zeros, whereas the observable controllable poles are the transmission poles. Most of these definitions can be interpreted dynamically in both the 1D and nD cases, and some can be connected to properties of kernel representations. We also examine the connections between poles, transfer matrices, and their left and right matrix fraction descriptions (MFDs). We find behavioral results which correspond to the concepts that a controllable system is precisely one with no input decoupling zeros and an observable system is precisely one with no output decoupling zeros. We produce a decomposition of a behavior as the sum of subbehaviors associated with various poles. This is related to the integral representation theorem, which describes every system trajectory as a sum of integrals of polynomial exponential trajectories.

On the elementary factors of linear systems over unique factorization domains

The concept of elementary factors introduced by Sule (SIAM J. Control Optim. 32 (1994) 1675–1695) plays an important role in the output feedback stabilizability and stabilization of linear systems over unique factorization domains (UFDs). This paper gives a simple proof that the elementary factors of T and W are identical, where T and W are matrices associated with a right and a left matrix fraction description of a linear system over a UFD. Relationship between the reduced minors and the elementary factors is established. An efficient method for computing the elementary factors based on the reduced minors is also proposed.

Feedback stabilization of MIMO 3-D linear systems

The authors solve the open problem of the existence of double coprime factorizations for a large class of multi-input/multi-output (MIMO) three-dimensional (3-D) linear systems. It is proven that if all the unstable zeros of the contents associated with left and right matrix fraction descriptions of a given feedback stabilizable causal MIMO 3-D plant are simple, then the plant has a double coprime factorization. The authors then give a parameterization of all stabilizing compensators for a MIMO 3-D system in this class. The key result developed in the paper is a novel and constructive technique of "replacing" an unstable polynomial with a stable polynomial step by step. An illustrative example is also provided.

Dynamic Pole Assignment Using Global, Blow Up Linearization: Low Complexity Solutions

The problem of pole assignment by low-order dynamics output feedback controllers with McMillan degree n1 is studied for minimal systems described by a proper transfer function matrix G(s)ɛR m×p (s) with McMillan degree n. The new method is based on an asymptotic linearization of the pole placement map related to the problem. In this context, the problem is reduced to solving a set of linear equations, and this is done without losing any of the degrees of freedom of the controller. In terms of composite left matrix fraction description (MFD) representations, the asymptotic solution of the problem as ɛ→0 is given in closed form in terms of a one-parameter family of feedback compensators A(s)+ɛB(s), where A(s) is the so-called degenerate compensator and B(s) is the solution of the linear equations. The arbitrary pole placement property holds true when the linearization matrix has full rank, and it is shown that this occurs genetically when \(n_1 \geqslant n'_1 \), where \(n'_1 \) is the smallest multiple of min(p,m) satisfying \(n'_1 \left( {m + p} \right) + mp \geqslant n + n'\)

On the Identification of MFD From Input/Output Samples of MIMO Systems

The purpose of this paper is to establish a simple relation between input/output samples of discrete-time multivariable (MIMO) systems and parameters in a corresponding left coprime Matrix Fraction Description (MFD). This relation is used as a basis for developing either off-line or on-line MIMO system identification procedures. The paper concentrates on the off-line identification procedure only. The extension to on-line identification will be treated in subsequent papers. In the presence of noise, the procedure described here behaves similarly to Least Square identification procedures. An included computational example illustrates the feasibility of the suggested procedure.

Calculation of Markov parameters from the transfer function matrix of MIMO systems

This paper presents a simple computational algorithm for determining the Markov parameters of a multivariable linear system specified either by a transfer function matrix or a right matrix fraction description. In addition, the procedure gives the observability/controllability index and the order of the minimal realization. The dual version of the algorithm can be used when the transfer function matrix is given by a left matrix fraction description. Computational examples are given to show the feasibility of the suggested procedure

An algorithm for calculating Markov parameters from transfer function matrices

A simple computational algorithm is presented for determining the Markov parameters of a multivariable linear system specified by a transfer function matrix or right matrix fraction description. In addition, the procedure gives the observability/controllability index and the order of the minimal realization. The dual version of the algorithm can be used when the transfer function matrix is given by a left matrix fraction description. Computational examples are given to show the feasibility of the suggested procedure.

Realization, design and implementation of row-pseudoproper left matrix fraction descriptions with impulsive modes

A novel and minimal realization algorithm is proposed for determining generalized state-space representation from a so-called row-pseudoproper left matrix fraction description (LMFD). The realized system with the state-space representation form is proved to be controllable and observable in the sense of [1,2] if the given row-pseudoproper MFD is left coprime. Besides, the proposed state feedback control law not only satisfies the optimal regional-pole-placement design for the realized generalized dynamical system, but also eliminates the impulsive terms in the state response of the closed-loop system. For practical consideration, an equivalent input-output feedback structure of the designed state-feedback controller is adopted. Based on the cascaded and/or parallel active RC networks with better sensitivity and stability properties, the resulting structure of the equivalent input-output feedback controller can be readily implemented based on proper subsystems.

Coprime Matrix Fraction Description via Sylvester's Resultant Matrix

Two algorithms are presented in this paper in order to compute the coprirne matrix fraction description of a given proper rational transfer function matrix. Generalised Sylvester resultant matrix is formed and coprirne matrix fractions are determined employing Gaussian elimination method. With the help of one of these algorithms both the right and the left coprirne matrix fraction descriptions as well as the greatest common divisor can simultaneously be determined.

Minimal order realization with a special coordinate for matrix fraction descriptions

This paper presents a generalized and minimal realization algorithm for constructively determining generalized state-space representation from a so-called column (row)- pseudoproper or a column(row)-proper or a nonstrictly-proper right(left) matrix fraction description (MFD). The realized state-space representation form with a special coordinate is proved to be controllable and observable in the sense of Rosenbrock and Cobb; besides, the dimension of the system after realization is equal to the determinantal degree of the given right(left) MFD in the normal sense.

Generalized matrix Euclidean algorithms for solving diophantine equations and associated problems

Based on the matrix fraction descriptions of multi-variable control systems, this paper presents the generalized matrix Euclidean algorithms for solving Diophantine equations and associated problems. A chain rule is developed for finding the greatest common right(left) divisor of two square(non-square) polynomial matrices and the irreducible right(left) matrix fraction description from the reducible or irreducible left(right) matrix fraction description. Also, the development of multi-variable pole-assignment controllers are discussed for engineering applications.

Generalized realizations with a special coordinate for matrix fraction descriptions

A generalized and feasible realization algorithm for constructively determining generalized state-space representation from a so-called column (row)-pseudo-proper or a column (row)-proper right (left) matrix fraction description (MFD) is proposed. The realized state-space representation form with a special coordinate is proved to be controllable and observable in the sense of Rosenbrock and Cobb; moreover, the dimension of the system after realization is equal to the deter-minantal degree of the given right (left) MFD in the normal sense.

Partial-fraction expansion in system analysis

Abstract A procedure is given for finding the partial-fraction expansion of strictly proper rational transfer functions that are commonly found in systems analysis. Eight formulae for the expansions are given to include the cases: (a) scalar transfer functions, (b) matrix transfer functions, (c) left and right matrix fraction descriptions, (d) transfer function matrices, and (e) block expansion of left and right matrix fraction descriptions. These formulae are derived from system theory using the observability canonical form and do not require the use of derivatives, the particular nature of the Jordan form, or interpolating polynomials; rather, it is required that the Vandermonde matrix be formed from the roots, latent roots, eigenvalues or solvents of the denominator. The main features of the procedure include that all the residues are found at the same time by simple matrix manipulation and can be used for the repeated and/or non-repeated case. The structured form of the formulae is very simple and fil...