Showing papers on "Coprime integers published in 1990"
01 Jan 1990
TL;DR: In this paper, the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by bounded perturbations to the numerator N and denominator A o f the normalized left coprime factorization of a nominal system.
Abstract: The problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by H , bounded perturbations to the numerator Nand denominator A o f the normalized left coprime factorization of a nominal system. This problem can be reduced to a Nehari extension problem directly and gives an optimal stability margin J1ll[N, mil f . All controllers satisfying a suboptimal stability margin are characterized and explicit state-space formulas are given.
231 citations
TL;DR: A necessary and sufficient condition for the existence of a bistable spectral factor (with entries in the distributed proper-stable transfer function algebra) in the context of distributed multivariable convolution systems with no delays was proved in this article.
Abstract: A necessary and sufficient condition is proved for the existence of a bistable spectral factor (with entries in the distributed proper-stable transfer function algebra 𝒜-) in the context of distributed multivariable convolution systems with no delays; a by-product is the existence of a normalized coprime fraction of the transfer function of such a possibly unstable system (with entries in the algebra ℬ of fractions over 𝒜-). We next study semigroup state-space systems SGB with bounded sensing and control (having a transfer function with entries in ℬ) and consider its standard LQ-optimal regulation problem having an optimal state feedback operator K0. For a system SGB, a formula is given relating any spectral factor of a (transfer function) coprime fraction power spectral density to K0; a by-product is the description of any normalized coprime fraction of the transfer function in terms of K0. Finally, we describe an alternative way of finding the solution operator K0 of the LQ-problem using spectral factor...
74 citations
16 Sep 1990
TL;DR: In this paper, two procedures for reduced-order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed, and it is shown that robust stability can be achieved with these controllers.
Abstract: Two procedures for reduced-order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed. It is shown that robust stability can be achieved with these controllers. The results obtained show how to reduce the controller degree systematically. They clearly indicate how the maximum stability margin influences allowable model reduction error. >
66 citations
TL;DR: In this paper, the problem of robustly stabilizing a linear system subject to H∞-bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of infinite-dimensional systems.
Abstract: The problem of robustly stabilizing a linear system subject to H∞-bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of infinite-dimensional systems. This class has possible unbounded, finite-rank input and output operators, which include many delay and distributed systems. The optimal stability margin is expressed in terms of the solutions of the control and filter algebraic Riccati equations.
62 citations
TL;DR: In this paper, two procedures for reduced-order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed, and it is shown that robust stability can be achieved with these controllers.
Abstract: Two procedures for reduced-order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed. It is shown that robust stability can be achieved with these controllers. The results obtained show how to reduce the controller degree systematically. They clearly indicate how the maximum stability margin influences allowable model reduction error. >
61 citations
TL;DR: In this paper, the robust stabilization of a class of perturbed coprime factors of a delay system with transfer function G(s = h2(s)h1(s).
58 citations
TL;DR: A constructive approach to the left coprime fractional representations for a class of (stable and unstable) nonlinear control systems described by certain multi-input/multi-output nonlinear ordinary differential equations is proposed in this article.
37 citations
05 Dec 1990
TL;DR: In this paper, the authors developed a robust stabilization theory for nonlinear plants using the left coprime factorizations of the plant and controller under certain differential boundedness assumptions, and showed that a necessary and sufficient condition for K/sub Q/ to stabilize G/sub S/ with Q, S not necessarily stable, is that S stabilizes Q.
Abstract: The authors describe steps toward the development of a robust stabilization theory for nonlinear plants. An approach using the left coprime factorizations of the plant and controller under certain differential boundedness assumptions is used. Attention is focused on a characterization of the class of all stabilizing nonlinear controllers K/sub Q/ for a nonlinear plant G, parameterized in terms of an arbitrary stable (nonlinear) operator Q. Also considered is the dual class of all plants G/sub S/ stabilized by a given nonlinear controller K and parameterized in terms of an arbitrary stable (nonlinear) operator S. It is shown that a necessary and sufficient condition for K/sub Q/ to stabilize G/sub S/ with Q, S not necessarily stable, is that S stabilizes Q. This robust stabilization result is of interest for the solution of problems in the areas of nonlinear adaptive control and simultaneous stabilization. >
35 citations
TL;DR: In this article, the authors developed a robust stabilization theory for nonlinear plants using the left coprime factorizations of the plant and controller under certain differential boundedness assumptions. But their work is limited to the problem of nonlinear adaptive control and simultaneous stabilization.
34 citations
24 citations
IBM1
TL;DR: A method of combining programs for two relatively prime integers p and q to obtain a program for sampling space containing p .
TL;DR: Algorithms for constructing H@?K and N"G(H) are described, constructed from generalizations of these algorithms and from an orbit-stabilizer algorithm.
05 Dec 1990
TL;DR: In this article, it is shown that a finite-dimensional linear time-varying continuous-time system admits normalized coprime factorizations if and only if it is internally stabilizable via dynamic output feedback.
Abstract: It is shown that a finite-dimensional linear time-varying continuous-time system admits normalized coprime factorizations if and only it admits a stabilizable and detectable realization, or equivalently, if and only if it is internally stabilizable via dynamic output feedback. In the process, a simple proof that stabilizability and detectability are sufficient to ensure the existence of stabilizing solutions to standard continuous-time control and filter Riccati equations is given. State-space formulas for normalized coprime factorizations are given. These are used to define the graph metric for the set of internally stabilizable finite-dimensional linear time-varying plants. As an application a robustness estimate is derived for feedback stability of linear time-varying plants using nonlinear time-varying controllers. >
01 Jan 1990
TL;DR: In this article, a necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of a fixed coprime factor of a plant is given.
Abstract: In this paper, a necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of a fixed coprime factors of a plant is given. The computation of the largest stability margin is discussed via solving a nonsquare El optimal control problem. A new algorithm for obtaining lower approximaions the minimum value of the optimaization problem, p0, is proposed. This, together with the standard algorithm which provides upper approximations, p0 can be computed within any degree of accuracy.
TL;DR: In this paper, the deficiency of direct products of SL(2, 2n) for coprime ni is shown to be bounded by a factor greater than or equal to 2.
Abstract: We give presentations for the groups PSL(2, pn), p prime, which show that the deficiency of these groups is bounded below. In particular, for p = 2 where SL(2, 2n) = PSL(2, 2n), we show that these groups have deficiency greater than or equal to – 2. We give deficiency – 1 presentations for direct products of SL(2, 2n) for coprime ni. Certain new efficient presentations are given for certain cases of the groups considered.
TL;DR: In this paper, a nonsingular antiperiodic automorphism of a Lebesgue probability space is defined and a measurable set B of positive measure is defined such that the relative distribution of return times under σ to B is the given distribution.
TL;DR: In this article, a new method for computing stable rational doubly coprime factorizations from a given transfer matrix is proposed, which makes full use of polynomial matrices, and the whole operation is carried out directly in the frequency domain.
TL;DR: In this article, it is shown how to construct two special kinds of coprime factorizations for it-normalized and all-pass denominator for rational, discrete-time transfer matrices.
Abstract: Given a rational, discrete-time transfer matrix, it is shown how to construct two special kinds of coprime factorizations for it-normalized and all-pass denominator. This is done in two ways. First, by carrying the existing continuous-time construction results to discrete-time via a bilinear transformation. This is shown to be messy and computationally burdensome. Second, direct constructions are derived in discrete time. The derivations point out interesting differences between discrete and continuous time for these constructions. >
01 Jan 1990
TL;DR: In this paper, the problem of robustly stabilizing a linear system subject to H ∞ -bounded perturbations in the numerator and the denominator of its normalized left coprime factorizations is considered for a class of infinite-dimensional systems.
Abstract: The problem of robustly stabilizing a linear system subject to H ∞ -bounded perturbations in the numerator and the denominator of its normalized left coprime factorizations is considered for a class of infinite-dimensional systems. This class has possibly unbounded, finite-rank input and output operators which includes many delay and distributed systems. The optimal stability margin is expressed in terms of the solutions of the control and filter algebraic Riccati equations. The applicability of this theory is demonstrated by a controller design for a flexible beam with uncertain parameters.
TL;DR: In this paper, explicit formulas for doubly coprime matrix fraction descriptions (MFDs) of the transfer matrix of a linear time-invariant state-space system are given in terms of a controllable and observable state space realization of the transferred matrix.
01 Jan 1990
TL;DR: In this paper, the problem of constructing a right coprime factorization of a nonlinear system is considered, which is based on the concept of right coprimeness introduced recently and on a stabilizing state feedback.
Abstract: In this paper, the problem of constructing a right coprime factorization of a system is considered. The approach is based on the concept of right-coprimeness introduced recently and on a stabilizing state feedback. Since our approach does not require construction of solutions to a Bezout identity, the construction of a stabilizing output injection is not needed. Some relationships between the existence of stabilizing output feedbacks and solutions to Bezout identities for (time-varying) linear systems are discussed and a development of left-coprime factorization of a nonlinear system is presented. We also prove that the existence of two coprime factorizations will imply the existence of the other two when the plant and the controller are both linear and the unity feedback system is finite-gain stable.
01 Jan 1990
TL;DR: In this paper, the robust stabilization of a class of perturbed coprime factors of a delay system with transfer function is investigated, and asymptotic estimates are obtained based on recent explicit results on the approximation and stabilization of normalized Coprime Factor.
Abstract: Given a delay system with transfer function \( G(s) = {h_2}(s)/{h_1}(s) \), where w\( {h_1}(s) = \sum\limits_0^{{n_1}} {qi} (s){e^{ - \gamma is}} \), and \( {h_2}(s) = \sum\limits_0^{{n_2}} {qi(s)} {e^{ - \beta is}} \), with \( 0 = {\gamma _0} < \gamma 1 < \ldots < {\gamma _{{n_1}}},0 \leqslant {\beta _0} < \ldots < {\beta _{{n_2}}} \), the p i being polynomials of degree δ i , and δ i < δ 0 for i ≠ 0, and the q i polynomials of degree d i < δ 0 for each i, the robust stabilization of a class of perturbed coprime factors of this system is considered. Asymptotic estimates are obtained based on recent explicit results on the approximation and stabilization of normalized coprime factors.
TL;DR: In this article, a very large infinite-dimensional Banach space of bounded nonlinear operators is suggested as an underlying framework for studies of nonlinear systems control, and all results obtained in Verma [6] can be extended to this much larger family of control systems.
TL;DR: From numerical analyses of phase pictures, trajectories in a real space, winding numbers, and Farey trees, a strong similarity on the topological structure of chaos irrespective of the exchange of the nature of fixed points is found and the results on the symmetry and winding number shed light on the question of the integrability.
Abstract: The dynamics of a three-particle motion under singular potentials such as Yukawa and Coulomb provides a Hamiltonian system leading to the billiard system in a high-energy region, which clarifies the interrelation between these two systems. From numerical analyses of phase pictures, trajectories in a real space, winding numbers, and Farey trees, we find several interesting facts: a strong similarity on the topological structure of chaos irrespective of the exchange of the nature of fixed points; a close relation between the symmetry and the winding numbers, i.e., the ${C}_{3v}$ completely symmetric orbit leads to the coprime rational and the partially symmetric one to the noncoprime one; a classification of the symmetry of orbits on the Farey tree. The results on the symmetry and winding number shed light on the question of the integrability. On the other hand, in a low-energy region, the dynamics is described by the H\'enon-Heiles system and hence the billiard motion contains both features of the ${C}_{3v}$ symmetry and a circle.
TL;DR: In this paper, the existence of effectively computable bounds on the solutions to the diophantine equation ap x + bq y = c+dp z q w is shown, which involves the application of linear forms in both real and p-adic logarithms.
TL;DR: An algorithm for determining the degrees of those irreducible representations of G, which restrict to Z to give the scalar representation, @l, is presented and Clifford's theorem is used as the basic reduction tool.
01 Jan 1990
TL;DR: In this paper, the problem of robustly stabilizing a linear system subject to H ∞-bounded perturbations in the numerator and the denominator of its normalized left coprime factorizations is considered for a class of infinite-dimensional systems.
Abstract: The problem of robustly stabilizing a linear system subject to H ∞-bounded perturbations in the numerator and the denominator of its normalized left coprime factorizations is considered for a class of infinite-dimensional systems. This class has possibly unbounded, finite-rank input and output operators which includes many delay and distributed systems. The optimal stability margin is expressed in terms of the solutions of the control and filter algebraic Riccati equations.
TL;DR: In this article, a pair of factor coprime matrices satisfies the particular Bezout identities, which are useful for feedback stabilization, and the properties of the common minor zeros (common zeros of all the maximum minors) of 2-D transfer function matrices are investigated.
Abstract: This paper is concerned with factor/zero coprimeness of 2-D polynomial matrices, and studies relating ‘Bezout identities’ and some of their properties. First, we show that a pair of factor coprime matrices satisfies the particular ‘Bezout identities’, which are useful for feedback stabilization. Secondly, we investigate the properties of the common minor zeros (common zeros of all the maximum minors) of factor coprime factorizations of 2-D transfer function matrices. Finally, we apply the results to time-delay systems of neutral or retarded type, and provide tests for the spectrally controllability/canonicality.
TL;DR: It is shown that log a n ∼ 2π √6 n log( n) is relatively prime and that λ = {λ1, λ2,…} if and only if gcd(λi, j) = 1 for all i ≠ j.