Polynomial Matrices and Matrix Fraction Descriptions

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...

Let [Ctilde](s)[Dtilde](s)−1 be a right coprime matrix fraction description of the (r × m) strictly proper transfer function matrix T(s) of a linear multivariable system Σ = (A, B, C). In this paper we establish a direct relation between (i) the algebraic concept of a right divisor C R(s) of the ‘numerator’ matrix [Ctilde](s), i.e. a (not necessarily square) polynomial matrix C L(s) that satisfies: [Ctilde](s) = C L(s)C R(s) for some polynomial matrix C L(s), and (ii) the geometric concept of an (A, B)-invariant subspace which is contained in the kernel of C. A special class of right divisors C R(s) of C(s) which correspond to (A, B)-invariant subspaces in ker C is characterized by a number of properties and a simple formula is presented which expresses the above correspondence. By generalizing the concept of a greatest right divisor (GRD) of a polynomial matrix to include also possibly non-square polynomial matrices, it is shown that GRDs of the numerator matrix C(s) correspond to the maximal (A...

A simple approach to synthesizing a proportional and derivative state feedback gain for assigning poles to infinity is developed. We also apply the synthesized gain to determine a doubly coprime matrix fraction description (MFD) of a state-space system and solve related problems. It is interesting to note that the MFD and the solutions to the related problems could be obtained in an explicit formula by using the synthesized gain.

Based on the matrix fraction description (MFD) consideration and polynomial matrix algebra, interesting results for the multivariable discrete linear quadratic regulator and optimal estimator are derived. The solutions are obtained by the spectral factorization of polynomial matrices in concise forms. The difficulties of solving the discrete algebraic Riccati equation can thus be avoided. In the derivation, some connections between the state space approach and the frequency domain approach are further investigated.

In this paper we study direct and inverse problems for discrete and continuous skew‐selfadjoint Dirac systems with rectangular (possibly non‐square) pseudo‐exponential potentials. For such a system the Weyl function is a strictly proper rational matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a nonlinear generalized discrete Heisenberg magnet model, extending earlier results for the isotropic case. A large part of the paper is devoted to the related discrete systems of which the pseudo‐exponential potential depends on an additional continuous time parameter. Our technique allows us to obtain explicit solutions for the generalized discrete Heisenberg magnet model and evolution of the Weyl functions.

An algebraic theory of linear periodic discrete-time (LPDT) systems is developed in terms of a periodic polynomial matrix description (PMD) and in a dynamics (module) structure defined over a non-commutative, non-integral and non-principal ideal ring /spl Rscr/ (the ring of periodic polynomials). Various properties of matrices and modules over /spl Rscr/ are explored, and links between algebraic properties of LPDT systems such as reachability, controllability are established directly in the periodic PMD and /spl Rscr/-module descriptions; i.e., without using the technique of lifting a LPDT system to its associated linear time-invariant (LTI) ones. The advantage of such characterizations is to show that LPDT systems are not index-invariant only for a class-that we characterize-of non-reversible time systems and that, except for this class of systems, the parametrization of all stabilizing controllers can be constructed similarly to LTI systems. Furthermore, one shows that for the class of non-reversible time systems above-mentioned, it is not possible to extract a canonical part like for LTI systems.

The problem of inferring the oscillatory behavior of the impulse and step responses of a system from the location of poles and zeros of its transfer function has practical importance. The authors first show that all the induced norms of a LTI system equal the zero frequency gain of its transfer function, if its impulse response is nonnegative. An important step in tackling this problem is to characterize the oscillatory nature of the impulse response of a system, given its minimal representation. We use Bernstein's theorem to show that, for a minimal representation, (A, B, C), the impulse response, Ce/sup At/B /spl ges/ 0 for all t > 0 iff C(/spl lambda/I - A)/sup -k/ B /spl ges/ 0 for all k and for some /spl lambda/ > /spl rho/(A), where /spl rho/(A) is the spectral radius. Additionally, the authors show that the number of sign changes of the impulse response of a non-minimum phase system, ((s - /spl sigma/)/sup 2/ + w/sup 2/) H/spl circ/ (s), where /spl sigma/ > 0, and H/spl circ/ (s) is of relative degree greater than 1, is less than or equal to the number of sign changes of H/spl circ/ (s), whenever w is greater than a w*, which is a function of (H/spl circ/ (s + /spl sigma/)); in other words, the influence of the non-minimum phase zero is not felt by the transfer function ((s - /spl sigma/)/sup 2/ + w/sup 2/) H/spl circ/ (s). We use this result to refine the definition of a weakly non-minimum phase system and provide an example of designing a controller for a non-minimum phase system to achieve non-negative impulse and non-overshooting step responses.

On the equivalence between MFD models and pseudo-observable forms of MIMO systems

This paper presents a one-to-one correspondence between a pseudo-observable form and the matrix fraction description (MFD) of a MIMO system so that by given one form, the other may be derived accordingly. In the development, we consider only a left coprime MFD and pseudo-observable form. The right coprime MFD and pseudo-controllable form can be derived in a dual sense. >

A new solution to the quadratic optimal tracking problem in linear multivariable discrete-time systems is presented. The approach is based on polynomial input-output models. The optimal control law consists of a plant output feedback and a reference feedforward. It is obtained by performing spectral factorization and then solving two matrix polynomial equations. The design procedure is simple and well suited for systems with inaccessible states.

The paper suggests how to compute the order of the minimal realization of the discrete-time nonlinear control system, described by the set of input-output difference equations, or alternatively, by the two polynomial matrices with the entries in the non-commutative skew polynomial ring. For that purpose the Ore determinant is specified further by including certain additional restrictions and proved that under these additional restrictions the degree of the dieudonne determinant is equal to the degree of the Ore determinant. Finally, the system order is shown to be equal to the degree of the determinant of one polynomial matrix. Only the special case of two output systems is considered in this paper.

This paper gives a new criterion for the inverse regulator problem in terms of polynomial matrix fractions. The generalized Kalman equation in polynomial matrix form leads to a necessary and sufficient condition for a stable feedback law to be optimal for some (unknown) quadratic performance index. This criterion consists of (i) a symmetric factorizability of a polynomial matrix, and (ii) a polynomial-type criterion that guarantees detectability. The relationship between the Riccati and Kalman equations is also established. Finally, an example is given to illustrate the new result in contrast with the existing result.

Mixed polynomial matrices are polynomial matrices with two kinds of nonzero coefficients: fixed constants that account for conservation laws and independent parameters that represent physical characteristics. The computation of their maximum degrees of minors is known to be reducible to valuated independent assignment problems, which can be solved by polynomial numbers of additions, subtractions, and multiplications of rational functions. However, these arithmetic operations on rational functions are much more expensive than those on constants.In this paper, we present a new algorithm of combinatorial relaxation type. The algorithm finds a combinatorial estimate of the maximum degree by solving a weighted bipartite matching problem, and checks if the estimate is equal to the true value by solving independent matching problems. The algorithm mainly relies on fast combinatorial algorithms and performs numerical computation only when necessary. In addition, it requires no arithmetic operations on rational functions. As a byproduct, this method yields a new algorithm for solving a linear valuated independent assignment problem.

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a “free” polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.

This paper presents theory for multivariable system identification using matrix fraction descriptions and the matrix continued fraction description approach which, in turn, yields a lattice-type order-recursive structure. Once the matrix continued-fraction expansion has been determined, it is straightforward to obtain solutions to both the left and right coprime factorizations of transfer function estimates and, in addition, solution to problems of state estimation (observer design) and pole-assignment control. An important and attractive technical property is that calculation of transfer functions in the form of right and left coprime factorizations, calculation of state variable observers and regulators all can be made using causal polynomial transfer functions defined by means of matrix sequences of the continued-fraction expansion applied in causal and stable forward-order and backward-order recursions. >