Invariants and Canonical Forms under Feedback.

Relying on the theory behind the Smith-MacMillan form of a rational matrix at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s = \infty</tex> , a necessary and sufficient condition is derived from the solvability of matrix Diophantine equations of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(s)X(s)+B(s)Y(s) = M(s)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(s), B(s)</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M(s)</tex> are given proper rational matrices and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X(s)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y(s)</tex> are unknown proper rational matrices. It is shown that the above result can be used in order to resolve in a new illuminating way the exact model matching problem (EMMP). If a solution to EMMP exists, then the family of all solutions is parametrized.

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.

A new modified state variable diagram method is proposed for determination of positive realizations of linear continuoustime systems with delays in state and input vectors. Using the method, it is possible to find a positive realization with reduced numbers of delays for a given transfer matrix. Sufficient conditions for the existence of positive realizations of given proper transfer matrices are established. The proposed method is demonstrated on numerical examples.

An algorithm for inverting rational matrices

On parametrizations for the minimal partial realization problem

The paper presents a simple method for solving the optimal (LQG) control problem on an infinite time horizon for linear plants described by proper rational transfer function matrices. Since the class of proper plants discussed in the paper is more general than the one commonly used, additional properties of solutions are presented. Two numerical examples illustrate the theoretical considerations. The examples have been performed on an IBM PC using the proposed algorithm. This algorithm is a part of a public ASMCS package developed by the author for analysis, synthesis, and simulations of multi-input, multi-output (MIMO) control systems.

STRICTLY DOUBLY COPRIME FACTORIZATIONS RELATED TO REDUCED ORDER OBSERVERS

The question of achieving a variety of specific design goals (e.g., arbitrary pole placement, decoupling, and exact model matching) for linear multivariable systems subjected to unknown step disturbances on the entire internal state and output is constructively resolved via a synthesis procedure which simultaneously combines integral feedforward compensation with asymptotic state estimation. The technique relies on the ability to express the desired closed-loop transfer matrix of the compensated system as the product of the open-loop transfer matrix and any proper rational transfer matrix, assuming the given system has no zeros at the origin and a stable closed-loop design is sought. The method employs some recent results involving the differential operator approach to multivariable system analysis and synthesis, and the treatment is confined almost exclusively to the frequency domain.

The problem of reducing polynomial matrices to canonical form by using semiscalar equivalent transformations is studied. This problem is wild as a whole. However, it is tame in some special cases. In the paper, classes of polynomial matrices are singled out for which canonical forms with respect to semiscalar equivalence are indicated. We use this tool to construct a canonical form for the families of coefficients corresponding to the polynomial matrices. This form enables one to solve the classification problem for families of numerical matrices up to similarity.

In this article we consider linear systems having real rational transfer matrices that may be improper. We investigate the properties of strictly positive real and weakly strictly positive real systems and their connections. Our first contribution is to establish a necessary and sufficient condition for a rational and possibly improper transfer matrix to be strictly positive real. Beside stability, this condition only involves the properties of the transfer matrix in the imaginary axis and a geometric relation between the feedthrough term and the residue at infinity of the transfer matrix. Our condition connects some of the properties of the system with those of its proper part. This leads to our second contribution: integrate in the improper setting all the several conditions derived in the literature for the proper case that discriminate weakly strictly positive real and strictly positive real systems.

are left prime, that h-1 g is analytic and nonsingular for I z I < 1 and that G is nonsingular. Nevertheless (1) is not uniquely specified. However (Hannan, 1969), if, for prescribed q, r, [A(q) *. B(r)] is of rankp then (1) is uniquely specified. This will be called the rank condition. Some structures (1) cannot be brought to a form where this condition is satisfied (Hannan, 1971), so that the condition is overidentifying. The set, Cq,r, of all structures satisfying the rank condition for given q and r is mapped, by using the elements of A(j) and B(j) and the on and above diagonal elements of C, onto an open set in Euclidean space, if it is required that h-1 g is nonsingular for I z I < 1, and hence constitutes an analytic manifold. For q and r fixed the set of structures (1) not in Cq,r is evidently of lower dimension than Cq,r Tuan (1978, end of ? 1) states a number of objections to the use Of Cq,r and we wish to discuss these, and the general problem of parameterizing (1). For this last purpose Tuan (1978) used a family of canonical forms, called by him the quasiautoregressive-moving average representation. With any structure (1) is associated a set of integers mj (j = 1, ...,p) which determine the form of this representation (Tuan, 1978, p. 101). Thus given any structure (1) there is associated a set of mj and a matrix of polynomials u(z), with unit determinant, such that ug and uh are in the canonical form. Of course the matrix function u is an extremely complicated function of g and h. Using K as a symbol for the m1 (j = 1, .. . p), we may also map the set CK of all structures with these mj into an open set in Euclidean space, if h-' g is nonsingular for I z I < 1. Now there is no overidentification. However, there is a major problem if there are prior constraints imposed on (1) for it seems almost impossible, in general, to translate these constraints into constraints on the canonical forms because u is such a complicated function of g and h. On the other hand, the set of all structures (1) for given q and r is very complicated to parameterize. However, for the reason mentioned at the end of the previous paragraph, almost all of that set is constituted by Cq,r, which is easily parameterized. Tuan (1978) objects to Cq,r because of overidentification and because constraints may cause the rank condition to fail, and for other reasons to be discussed below. It is most unlikely that constraints would make [A (q) .B(r)] identically of rank less than p,

A polynomial matrix method for computing stable rational doubly coprime factorizations

In the present paper, methods and algorithms for numerical solution of spectral problems and some problems in algebra related to them for one- and two-parameter polynomial and rational matrices are considered. A survey of known methods of solving spectral problems for polynomial matrices that are based on the rank factorization of constant matrices, i.e., that apply the singular value decomposition (SVD) and the normalized decomposition (the QR factorization), is given. The approach to the construction of methods that makes use of rank factorization is extended to one- and two-parameter polynomial and rational matrices. Methods and algorithms for solving some parametric problems in algebra based on ideas of rank factorization are presented. Bibliography: 326titles.

There exist two fundamental tools for modeling and simulating wave action of internal combustion engines. The first is a nonlinear time domain solution of the Euler equations using a space–time meshing. The second is a frequency domain solution of the linear wave equations. These two methods exist with a wide range of complexity and sophistication. Hybrid coupled methods also exist; they attempt to bridge the gap between the two techniques while maintaining the overall goal of engine simulation in mind. This work deals with the frequency characterization of a complex intake element, the charge air cooler. First, a transfer matrix for a simple tube is defined and measured. Two identical versions of the previous tube serve for identifying the transfer matrix of the charge air cooler directly on an operating four-cylinder turbocharged engine. Once the transfer matrix is measured, it is coupled to GT-Power as four transfer functions coded into Simulink. The final validation comprises two tubes in GT-Power with measured boundary conditions of pressure and mass flow with the Simulink model in between. Results are presented in the time and frequency domains with future objectives and perspectives as well.

In this paper an in-flight tracking control system minimizing vibratory loads at the frequencies 1/rev and 2/rev using steady deflections of active trailing-edge flaps is examined. The control system is based on the relaxed HHC algorithm and uses the Moore-Penrose pseudoinverse of a symmetrized transfer matrix to calculate the control commands for vibration reduction. The focus of the current investigation is on symmetrization effects of the transfer matrices used for controller design on the performance of the control system. In a theoretical pre-flight analysis bounds for the stationary solution of the state-space representation of the closed loop control system are derived on the basis of the Kalman decomposition controller in the frequency domain is calculated. This analysis shows the vibration reduction by the controller using the symmetrized transfer matrices to be close to the optimally achievable reduction. Subsequent, flight tests on a full-scale experimental helicopter in open and closed loop configuration not only verify the functionality of the in-flight tracking control system, but also validate the insignificance of the degradation in control performance due to transfer matrix symmetrization predicted by the theoretical analysis.