Showing papers on "Riccati equation published in 1974"
TL;DR: In this paper, the Chandrasekhar-type Riccati-type difference equation is replaced by another set of difference equations, which are then used for recursive estimation in constant continuous-time linear systems.
Abstract: Certain recently developed fast algorithms for recursive estimation in constant continuous-time linear systems are extended to discrete-time systems. The main feature is the replacement of the Riccati-type difference equation that is generally used for such problems by another set of difference equations that we call of Chandrasekhar-type. The total number of operations in the new algorithm is in general less than with the Riccati-equation based Kalman filter, with significant reductions being obtained in several important special cases. The algorithms are derived via a factorization of increments of the Riccati equation variable, a method that can be extended to nonsymmetric Riccati equations as well.
215 citations
TL;DR: In this article, limit properties of the Riccati operator difference equation with a quadratic cost functional were investigated. But the limit properties were not investigated in the context of infinite-dimensional control systems.
Abstract: The aim of this paper is to investigate limit properties of the infinite-dimensional control system described by a difference equation with a quadratic cost functional Some related results of independent interest concerning stability and the Riccati operator difference equation are given also
168 citations
TL;DR: A constructive proof is given for easily finding constant feedback gains that stabilize a linear, time-invariant, discrete system.
Abstract: A constructive proof is given for easily finding constant feedback gains that stabilize a linear, time-invariant, discrete system. The results are directly applicable to initializing certain iterative methods that find steady-state gains for the discrete optimal regulator.
98 citations
TL;DR: In this article, a projection operator which reduces quantities to their expectation value is introduced and an exact integro-differential equation for the expectation value of the stochastic process is derived.
96 citations
TL;DR: In this paper, the assumptions of two theorems of Kleinman concerning Newton's method for the Riccati equation can be weakened, and the assumptions are shown to be false.
Abstract: The purpose of this note is to point out that the assumptions of two theorems of Kleinman concerning Newton's method for the Riccati equation can be weakened.
56 citations
Journal Article•
48 citations
TL;DR: In this article, a necessary and sufficient frequency-domain condition for the existence of a non-positive definite solution to the Riccati equation is shown to be necessary but not sufficient.
Abstract: The purpose of this correspondence is to correct a claim of [1]. Reference [1, theorem 4] states a necessary and sufficient frequency-domain condition for the existence of a nonpositive definite solution to the algebraic Riccati equation. This condition is, however, not correct, since the stated condition is necessary but not sufficient. This will be demonstrated by means of a counterexample.
45 citations
TL;DR: In this paper, the existence and uniqueness theorems for mild solutions of the Riccati equation with unbounded operators are established using a sequential approach based on approximating controllers for the linear quadratic cost control problem.
43 citations
TL;DR: In this article, an algorithm for the numerical solution of inhomogeneous linear two-point boundary value problems using the method of invariant imbedding is presented, which handles the case in which the standard Riccati equation fails to have a solution over the entire interval of interest.
38 citations
26 citations
TL;DR: A computational algorithm for solving a system of coupled matrix algebraic equations is presented, which arises in optimal control problems of linear systems with jump parameters.
Abstract: A computational algorithm for solving a system of coupled matrix algebraic equations is presented in this paper. Such equations arise in optimal control problems of linear systems with jump parameters.
TL;DR: In this paper, a procedure is given whereby an analytic solution may sometimes be found for a matrix algebraic equation, such as occurs in deterministic and stochastic optimization problems.
Abstract: A procedure is given whereby an analytic solution may sometimes be found for a matrix algebraic equation, such as occurs in deterministic and stochastic optimization problems.
TL;DR: In this paper, an integro-differential equation (IDE) on a finite closed interval is studied, and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides.
Abstract: In this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.
TL;DR: In this article, it was shown that a Riccati equation of particular structure which arises in a number of singular optimal estimation and control processes can be reduced in order, leading directly to a procedure for the design of a class of minimal order observers, the structure of which can be interpreted as the limiting form of appropriate Kalman estimators with vanishing observation noise.
Abstract: It is shown that a Riccati equation of particular structure which arises in a number of singular optimal estimation and control processes can be reduced in order. This fact leads directly to a procedure for the design of a class of minimal order observers, the structure of which can be interpreted as the limiting form of appropriate Kalman estimators with vanishing observation noise.
TL;DR: A new computational method for calculating the asymptotic solution to the matrix Riccati equation is described, which is fast, efficient and gives all possible solutions to the Matrix quadratic form.
Abstract: This paper describes a new computational method for calculating the asymptotic solution to the matrix Riccati equation This method is fast, efficient and gives all possible solutions to the matrix quadratic form Matrix sign functions are used to find the asymptotic solutions
TL;DR: In this paper, the Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, and it is shown that, subject to a continuity restriction, the RDE is unique among comparable equations in possessing this property.
Abstract: In the domain of real symmetric matrices ordered by the positive definiteness criterion, the symmetric matrix Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, Here is is shown that, subject to a continuity restriction, the Riccati equation is unique among comparable equations in possessing this property.
01 Jan 1974
TL;DR: A solution for un + 1 in terms of u1 and u0 is given in this article, where the elements of the sequence {un} satisfy the linear difference equation two linearly independent solutions of the equation are written as determinants and relations are given which can be used to check the evaluation of these determinants.
Abstract: A solution is given for un + 1 in terms of u1 and u0, where the elements of the sequence {un} satisfy the linear difference equation Two linearly independent solutions of the equation are written as determinants and relations are given which can be used to check the evaluation of these determinants.
01 Jan 1974
TL;DR: In this article, the authors define a class of detection-estimation problems on matrix Lie groups in which the observation noise is multiplicative in nature and derive the relevant likelihood ratio formula and the associated optimal estimation equations for the signal given the observations and the assumption that the signal is present.
Abstract: We define a class of detection-estimation problems on matrix Lie groups in which the observation noise is multiplicative in nature. By examining the differential versions of the hypotheses, which are bilinear, we are able to derive the relevant likelihood ratio formula and the associated optimal estimation equations for the signal given the observations and the assumption that the signal is present. These estimation equations are of interest in their own right, in that they represent a finite dimensional optimal solution to a nonlinear estimation problem and consist of a Kalman-Bucy filter along with the on-line computation of the solution of the associated Riccati equation, which is driven by the observations. The usefulness of these results is illustrated via an example concerning the detection of an actuator failure in a rigid body rotational control system.
01 Nov 1974
TL;DR: In this article, a numerical approximation to the operational Riccati differential equation occurs in the optimal control theory of linear hereditary differential systems with a linear-quadratic cost function, and it is shown that this approximation can be used to obtain a linear approximation of the RDE.
Abstract: We present a numerical approximation to the operational Riccati differential equation which occurs in the optimal control theory of linear hereditary differential systems with a linear-quadratic cost function.
01 Aug 1974
TL;DR: In this article, a model-following controller for the lateral motion of a typical fighter aircraft using unstable model equations is presented, and mathematical conditions are presented which insure a steady-state value of the model following gain matrix regardless of the behavior of the underlying Riccati equation.
Abstract: In the optimal regulator formulation of discrete explicit model following the Riccati equation may fail to reach a steady-state value for model dynamics which are not asymptotically stable. Such conditions often arise in aircraft applications when flying quality criteria based on step inputs are used to define the model equations. Mathematical conditions are presented which insure a steady-state value of the model-following gain matrix regardless of the behavior of the underlying Riccati equation. These results are applied to the design of a model-following controller for the lateral motion of a typical fighter aircraft using unstable model equations.
TL;DR: In this paper, the Riccati equation is used to calculate the reflection function of a given finite order, for a particle that has suffered a specified number of scatterings in the two portions of the total medium.
Abstract: In computing the total reflection function for two media of given scattering properties adjoined together, the usual techniques require a knowledge of reflection and transmission functions of both the media. An alternate method developed here depends only on solving a Riccati equation with suitable initial conditions. By following this procedure, we are able to calculate the reflection function of a given finite order, for a particle that has suffered a specified number of scatterings in the two portions of the total medium. Such calculations may be useful in radiation dosimetry problems where the body to be irradiated is placed on reflecting materials.