Showing papers on "Riccati equation published in 1978"
01 Jan 1978
TL;DR: In this article, a new algorithm for solving algebraic Riccati equations (both continuous-time and discrete-time versions) is presented, which is a variant of the classical eigenvector approach and uses instead an appropriate set of Schur vectors.
Abstract: In this paper a new algorithm for solving algebraic Riccati equations (both continuous-time and discrete-time versions) is presented. The method studied is a variant of the classical eigenvector approach and uses instead an appropriate set of Schur vectors thereby gaining substantial numerical advantages. Complete proofs of the Schur approach are given as well as considerable discussion of numerical issues. The method is apparently quite numerically stable and performs reliably on systems with dense matrices up to order 100 or so, storage being the main limiting factor. The description given below is a considerably abridged version of a complete report given in [0].
1,002 citations
TL;DR: In this paper, a quadratic positive definite functional that yields necessary and sufficient conditions for the asymptotic stability of the solutions of the matrix difference-differential equation x (t) = Ax(t) + Bx(t − τ) is constructed and its structure is analyzed.
147 citations
Abstract: The problem of optimal measurement locations for state estimation in linear distributed parameter systems is considered. It has previously been shown that the optimal sensor location problem for distributed systems can be posed as an optimal control problem for a system described by the infinite-dimensional matrix Riccati equation for the filter covariance. A more efficient approach based on an upper bound of the filter covariance is developed in the present study. The relationship between the present approach and that of minimizing a measure of the filter covariance is studied. A detailed example is considered, and the results of the two approaches are compared.
98 citations
TL;DR: New lower bounds on the spectral norms of the positive definite solutions to the continuos and discrete algebraic matrix Riccati and Lyapunov equations are derived.
Abstract: New lower bounds on the spectral norms of the positive definite solutions to the continuos and discrete algebraic matrix Riccati and Lyapunov equations are derived. These bounds are much easier to compute than previously available bounds and appear to be considerably tighter in many cases.
91 citations
TL;DR: A new method of detecting and handling discontinuities in arbitrary functions which form part of an ordinary differential equation set that applies to any predictor corrector integration algorithm with Nordsieck step size control.
82 citations
01 Jan 1978
TL;DR: In this article, the spatially homogeneous model of a high activation energy thermal explosion is studied when the heat loss parameter a is close to the classically defined critical value a = e. The solution properties of this nonlinear equation permit one to define a value of A = a which separates subsequent subcritical and supercritical behaviour.
Abstract: The spatially homogeneous model of a high activation energy thermal explosion is studied when the heat loss parameter a is close to the classically-defined critical value a = e. Asymptotic solutions are developed which describe the time- history of the temperature and reactant depletion. It is shown that there is a critical time period, large with respect to the characteristic conduction time, in which the temperature variation is described by a Riccati equation. The solution properties of this nonlinear equation permit one to define a value of A = a — e which separates subsequent subcritical and supercritical behaviour.
64 citations
TL;DR: In this article, the controllability of linear periodic systems and the existence of periodic solutions for periodic matrix Riccati equations are discussed, and a few problems playing a basic role in periodic control theory are discussed.
Abstract: Referring to recently published results, a few problems apparently playing a basic role in periodic control theory are discussed in this paper. Specifically, the problems dealt with are the controllability of linear periodic systems and the existence of periodic solutions for periodic matrix Riccati equations.
52 citations
TL;DR: In this paper, the authors present necessary and sufficient conditions for the existence of CARATHEODORY solutions of different differentially differential equations, and present a discussion of these conditions.
Abstract: (1978). ON NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF CARATHEODORY SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS. Quaestiones Mathematicae: Vol. 2, No. 4, pp. 507-512.
45 citations
TL;DR: By considering the square root of a matrix as a special case of matrix Riccati-type equations, a fast economical algorithm was developed as a stable generalization of the process given in this paper.
Abstract: By considering the square root of a matrix as being a special case of a matrix Riccati-type equation, a fast economical algorithm is developed as a stable generalization of the process given in [1].
41 citations
TL;DR: The causal estimation of a two-parameter Gaussian random field in the presence of an additive, independent, white Gaussian noise is studied.
Abstract: The causal estimation of a two-parameter Gaussian random field in the presence of an additive, independent, white Gaussian noise is studied. The dynamics of this random field are modeled by partial differential equations from which the recursive filtering equations and the generalized Riccati equation are derived. A specific example is solved in detail.
38 citations
TL;DR: In this paper, the structural properties of the coefficient matrices of a higher-order square-matrix-Riccati differential equation are established, and the structural property of its solution is inferred.
Abstract: Open-loop multilevel Stackelberg strategies in deterministic, sequential decision-making problems for continuous linear systems and quadratic criteria are developed. Characterization of the Stackelberg controls via the solution of a higher-order square-matrix-Riccati differential equation is established; also, the basic structural properties of the coefficient matrices of this differential equation are established, and the basic structural properties of its solution are inferred.
TL;DR: In this article, a representation theorem is obtained for solutions of the nonlinear functional differential equation (l) u'(t) F(u,), t > 0, u(t <.(/), t < 0, as a semigroup of nonlinear operators on a space of initial data X of fading memory type.
Abstract: A representation theorem is obtained for solutions of the nonlinear functional differential equation (l) u'(t) F(u,), t > 0, u(t) <.(/), t < 0, as a semigroup of nonlinear operators on a space of initial data X of \"fading memory type.\" Equation (1) is studied in the abstract setting of a Banach space E. The nonlinear functional F is a uniformly Lipschitz continuous mapping from X to E. The semigroup is constructed by transforming (1) to an abstract Cauchy problem (CP) w'U) + Aw(t) « 0, w(0) , in the space X and applying a generation theorem of M. Crandall and T. Liggett to the operator A in X. The case when (1) is a nonlinear Volterra integrodifferential equation of infinite delay is given special consideration. The semigroup representation is used to obtain finite difference approximations for solutions of (CP) and to study the continuity of solutions of (1) with respect to perturbations of F and $.
01 Jan 1978
TL;DR: It is shown that by introducing a certain 'hereditary operator' F one can characterize more precisely the structure of the solution of the operator Riccati equation, and results in a simplification and in some reduction in the RicCati equation.
TL;DR: In this paper, the stable factorizations of a monic matrix polynomial are characterized in terms of spectral properties, based on the divisibility theory developed by I. Gohberg, P. Lancaster and L. Rodman.
Abstract: The stable factorizations of a monic matrix polynomial are characterized in terms of spectral properties. Proofs are based on the divisibility theory developed by I. Gohberg, P. Lancaster and L. Rodman. A large part of the paper is devoted to a detailed analysis of stable invariant subspaces of a matrix. The results are also used to describe all stable solutions of the operator Riccati equation.
TL;DR: In this article, the solution to the Riccati equation is given in terms of the partition of the transition matrix and matrix differential equations are derived and solved using methods developed in the fields of free vibration theory and aircraft flutter analysis.
Journal Article•
TL;DR: In this article, the problem of designing a regulator, optimal by a quadratic performance criterion, on an infinite time interval is examined for a linear periodic system, where the control plant's motion is described by a system of linear periodic finite-difference equations.
Abstract: The problem of designing a regulator, optimal by a quadratic performance criterion, on an infinite time interval is examined for a linear periodic system. It is assumed that the control plant's motion is described by a system of linear periodic finite-difference equations. Controllable plants whose motion is described by differential and by finite-difference equations on different parts of the period are analyzed as well. The optimal regulator design problem is reduced to the determination of a periodic solution of an appropriate Riccati equation. An algorithm for constructing such a solution is derived. It is noted that this result can be used in periodic optimization problems /1/ and in the design of a stabilization system for a pacing apparatus.
TL;DR: In this article, it was shown that if the original equation is of completely integrable Hamiltonian form there are an infinity of explicit solutions of the linearized equation, and these are almost always linearly independent.
Abstract: This paper treats the problem of solving the linear equation which arises when a nonlinear evolution equation is linearized around some particular solution. It is shown that if the original equation is of completely integrable Hamiltonian form there are an infinity of explicit solutions of the linearized equation. These are almost always linearly independent.
TL;DR: It is shown that the Riccati equation for the linear approximation is solved by simple transformation of the averaging solution, and the numerical procedures are identical, except for the computation of an initial condition.
Abstract: A certain approximation scheme based on "piecewise linear" approximations of L 2 spaces is employed to formulate a numerical method for solving quadratic optimal control problems governed by linear retarded functional differential equations. This piecewise linear method is an extension of the so-called averaging technique. It is shown that the Riccati equation for the linear approximation is solved by simple transformation of the averaging solution. In fact, the numerical procedures are identical, except for the computation of an initial condition. Numerical examples are included.
TL;DR: In this article, a nonlinear ordinary differential equation is presented, which is shown to occur as a particular integral of Einstein's equations of gravitation, and may be used to generate a class of solutions which contains the Kerr, and the Tomimatsu-Sato metrics as particular cases.
Abstract: A nonlinear ordinary differential equation is presented. This equation is shown to occur as a particular integral of Einstein’s equations of gravitation, and may be used to generate a class of solutions which contains the Kerr, and the Tomimatsu–Sato metrics as particular cases.
TL;DR: In this paper, the authors studied the stabilization problem for non-autonomous control processes in Hilbert spaces and proved that a stabilizing feedback exists if and only if an associated Riccati equation has a bounded solution which is symmetric and positive definite.
Abstract: In this paper we study the stabilization problem for non autonomous control processes in Hilbert spaces. We prove that a stabilizing feedback exists if and only if an associated Riccati equation has a bounded solution which is symmetric and positive definite. An application to control processes with delays in control is presented.
TL;DR: The fundamental nature of the generalized partitioned algorithms (GPA) is demonstrated by showing that previous major filtering and smoothing algorithms can be obtained as special cases from the forward and the backward formulation of the GPA.
Abstract: In this paper the generalized partitioned algorithms for the discrete linear estimation problem are presented. These serve as the unifying framework for linear estimation. The fundamental nature of the generalized partitioned algorithms (GPA) is demonstrated by showing that previous major filtering and smoothing algorithms can be obtained as special cases from the forward and the backward formulation of the GPA. A particularly effective algorithm is given for the steady-state solution of the discrete Riccati equation.
TL;DR: In this paper, an analytic solution is obtained for a certain non-linear one-dimensional Boltzmann equation describing the temporal relaxation to equilibrium of a system of particles, and its general features are elucidated.
Abstract: An analytic solution is obtained for a certain non-linear one-dimensional Boltzmann equation describing the temporal relaxation to equilibrium of a system of particles, and its general features are elucidated. A solution is also found for the corresponding linearised problem and for two BGK models, one with the correct energy-dependent and the other with a mean energy-independent relaxation time. The accuracy of the linearised equation is superior to that of the models, even for large displacements from equilibrium, and the gain in accuracy of the energy-dependent BGK model over the energy-independent one may well be offset by the additional computational work involved in using the former. A calculation of the successive time derivatives of the entropy, based on the exact non-linear equation, show that these alternate in sign, at least up to the tenth derivative.