Prime Polynomials Research Articles

Output regulation and tracking in linear multivariable systems

It is shown that the related questions of output tracking and output regulation (with and without input sensor measurement), all with associated internal stability, can be resolved via a single polynomial matrix formulation which employs the recently developed notion of skew prime polynomial matrices. In particular, the primary condition which must be satisfied in all cases is that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(s)</tex> , the plant numerator matrix, and a matrix where zeros represent the modes of the exogenous system, be externally skew prime. This observation is then elucidated in light of a new internal model principle.

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.

Skew prime polynomial matrices

A pair of polynomial matrices, P(s) and Q(s) , is defined to be skew if and only if a solution, M(s), N(s) , to the polynomial matrix equation P(s)M(s)+N(s)Q(s)=I exists. It is shown that P(s) and Q(s) are externally skew prime if and only if Q(s)P(s)= \bar{P}(s)R(s) with Q(s) and \bar{P}(s) relatively left prime and P(s) and R(s) relatively right prime. This observation implies a new constructive procedure for determining M(s) and N(s) where P(s) and Q(s) are found to be externally skew prime and P(s) is nonsingular. A new procedure for obtaining solutions to the more general polynomial matrix equation, P(s)M(s)+N(s)Q(s)= V(s) , based on the notion of skew-prime polynomial matrices is also presented. A characterization of all solutions when V(s)= I is also given, under appropriate assumptions, and then employed to determine a unique solution to this polynomial matrix equation.

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.

Existence criterion for nonessential singularities of the second kind

The stability of a single-input single-output multivariable system is governed by the zeros of the numerator as well as the denominator of its multivariable transfer function. Where the relatively prime numerator and denominator polynomials share a common zero, the transfer function is said to have a nonessential singularity of the second kind. Such a singularity on the stability region boundary mandates more complicated stability tests than would otherwise be needed. Thus, their existence must be known. A criterion for their existence is described in this short paper.

New criteria and system theoretic interpretations for relatively prime polynomial matrices

New criteria as a generalization of the resultant of two polynomials are given for relative primeness of polynomial matrices together with system theoretic interpretations and unification of the existing and new criteria.

Prime Polynomial Sequences

Prime Polynomial Sequences

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.

The differential operator approach to linear system analysis and design

The time domain, state-space approach to control systems analysis and design has dominated the control literature since the early 1960's. More recently, however, there has been a resurgent interest in frequency domain methods. This is due in large part to the development of an alternative (to the state-space approach) time domain technique, namely the differential operator approach for the analysis and synthesis of linear multivariable systems. More specifically, a significant number of investigators have now noted and employed the fact that the (pxm) transfer matrix, T(s), of a linear, time-invariant, multivariable system can be factored as R(s) P-1R(s) [orP-1Q(s) Q(s)]. with R(s) and PR(s) [PQ(s) and Q(s)] relatively right (left) prime polynomial matrices in the Laplace operator's and PR(s) [PQ(s)] column (row) proper. Furthermore, such a factorization directly implies a minimal (i.e. both controllable and observable) time domain realization of T(s) in “differential operator form”, namely PR(D) z(t) = u(t); y(t) = R(D) z(t) [orPQ(D) y(t) = Q(D) u(t)], where PR(D) andR(D) [PQ(D) andQ(D)] are polynomial matrices in the differential operator D = d/dt, withPR(D) [PQ(D)] nonsingular. It might be noted that a differential operator description of the dynamical behavior of a physical system often follows as a direct result of applying well-known physical laws to describe the performance of the system. The primary purpose of this paper is to delineate and unify a number of the more important differential operator techniques which have recently been developed and to compare them to analogous, but perhaps more familiar state-space procedures.

An Equivalence Theorem

The theory of three terminal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> networks is given a new unified formulation based on five relatively prime polynomials. A set of necessary and sufficient mutually independent conditions, which are equivalent to the residue theorem, coefficient conditions, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> admittance and impedance conditions, is given. These conditions are stated by means of the zero distributions of the five polynomials and a multilinear relation among them.

Some properties of relatively prime polynomial matrices

Some necessary and sufficient conditions are given for the polynomial matrices sIn, ? A and B to be relatively prime.

Relatively prime polynomial matrices

Relatively prime polynomial matrices are defined, and some of their properties are deduced and applied. The problem of extending these properties is posed.

An extension of Pólya’s theorem on power series with integer coefficients

This was proved first by Polya [4] for the case in which G is simply connected. A general proof was given later by Polya [5, p. 703]. (It is understood that f(z) is single valued in G.) A more precise conclusion is that f(z) can be expressed in the form P(z)/ Q(z), where P(z) and Q(z) are relatively prime polynomials with integer coefficients, and Q(z) has its leading coefficient equal to 1. Thus the poles off(z) can occur only at sets of conjugate algebraic integers lying in E. By a theorem of Fekete [2], there are only a finite number of such algebraic integers when the transfinite diameter of E is less than 1. Polya's theorem is an extension of a theorem of Carlson [1], according to which a functionf(z) of the same form which is regular for lzl > 1 must either be rational or have the unit circle as a natural boundary. (This was stated originally for power series regular inside the unit circle.) In [4], Polya also proved the converse for the case in which G is simply connected. That is, if the transfinite diameter of E is at least 1, then there exist nonrational functions f(z) whose expansions at x have integer coefficients; indeed, there are a nondenumerable infinity of such functions. I have not seen a proof of the converse in general. One will be given in this paper as part of our discussion of the extended Polya theorem. Fekete [2] proved that the transfinite diameter of E is less than 1 if and only if there exists a polynomial

An example concerning rational convexity

A rat ionally convex domain, G, in C ~, is a domain of ho lomorphy in C n such tha t every analyt ic funct ion on G can be uniformly approximated, on compac t subsets, by rat ional functions of the coordinates. I f these rational functipns can always be taken to be polynomials, then G is a Runge domain. I n 1941, in Domaines d'holomorphie et domaines rationellement eonvexes (Japan. J . Math. 17, 517--521), K. 0KA presented an example of a bounded domain of ho lomorphy in C ~ which is no t rat ionally convex. This domain is no t simply-connected. I n 1960, in Addendum to " A n Example Concerning Polynomial Convexity" (Math. Ann. 140, 322--323), J. WERMER constructed a bounded domain of ho lomorphy in C a which is not a Runge domain, bu t which is analyt ical ly homeomorphic with a polycylinder. I n this note, we show tha t Wermer ' s domain is also no t ra t ional ly convex. Theorem. There is a bounded domain o/ holomorphy in C a which is analytically homeomorphic with the polycylinder Izil max(IR(y) l : y ~ K}. Express R as g/h, where 9 and h are relatively prime polynomials, and set / = g h. Then /(x) = 0, bu t / does no t vanish a t any point of K. Since K is contractible, there is a continuous, single-valued, branch of log (/) defined on K. I f we recall t h a t the circle, T, lies in K, and now apply the argument principle to / on D, we find tha t

On the multiplicative semigroup of commutative ring

There seems to be no approach in the literature to a general theory of the structural restrictions which a semigroup must satisfy to be the multiplicative part of a ring. Johnson has treated the case that, as in Boolean rings, the addition is uniquely determined by the multiplication [1]. I do not know whether the present theorem extends to the noncommutative case. I am indebted to John Rainwater for arousing my interest in the problem and to the referee for suggestions on the arrangement of the proof. Let S denote the direct product of the multiplicative and the additive semigroups of positive integers, i.e. the set of all pairs (m, a) of positive integers with (m, a)(m', a') = (mm', a+a'). Let Xp denote, for each prime p, the multiplicative semigroup of all polynomials over the p-element field with zero constant term. Each Xp contains an isomorphic copy of S, with (m, a) = (q1l ... qan a) corresponding to xaP1 Pa, for a suitable infinite list of prime polynomials Pi. We shall prove the

Necessary Conditions on the Matrix of an RC Grounded Quadripole

Conjectures of Lucal and Darlington on the synthesis of the matrix of an RC three-terminal network are examined in this paper. Suppose each of the three open-circuit impedances, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{11}(s), z_{12}(s)</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{22}(s)</tex> , is specified as the quotient of two relatively prime polynomials; each is then rewritten as a rational fraction with the polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(s)</tex> as denominator, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(s)</tex> is the least common multiple of the three denominators. Lucal and Boxall conjecture that in this representation every numerator coefficient of each impedance is nonnegative, and in addition, each coefficient in the numerator of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{12}(s)</tex> does not exceed the corresponding coefficient in the numerator of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{11}(s)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{22}(s)</tex> . It is observed that these two propositions are equivalent necessary conditions, and a counterexample to the first proposition is exhibited. Darlington conjectures that any realizable RC three-terminal network can be synthesized by the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\pi -T</tex> decomposition method described by Lucal. This counterexample casts some light on Darlington's conjecture. An examination of the above counterexample suggests another condition which may be necessary. It is shown that if the number of nodes of the network does not exceed five, this new condition is indeed necessary; however, it is not known whether it remains valid when the number of nodes exceeds five. In the discussion of these matters a new proof is exhibited of the fact that each root of the determinant of the nodal admittance matrix is a nonpositive real number.

Dimension of Ideals in Polynomial Rings

A well-known theorem asserts that if K is a field, a prime ideal in the polynomial ring S = K[X1, … Xn] and d the transcendence degree of S / over K n = rank + d. In the first half of this paper we extend this result to the case of arbitrary commutative noetherian K, as well as giving a purely homological proof of the classical theorem. In the second half we use our first result to compute the analogue of the dimension of the product and intersection of two affine varieties when K is a Dedekind ring.