Showing papers on "Riccati equation published in 1976"

Filter response of nonuniform almost-periodic structures

Abstract: The filter response of nonuniform, almost-periodic structures, such as corrugated optical waveguides, is investigated theoretically. The filter process, leading to reflection of a band of frequencies near the Bragg frequency, is treated as a contradirectional coupled-wave interaction and shown to obey a Riccati differential equation. The nonuniformity of the structure is represented by a tapering in the coupling strength (e.g., the depth of the corrugation) and by a chirp in the period of the structure. For small reflectivities, the filter response is a Fourier transform of the taper function. For large reflectivities, the Riccati equation was evaluated numerically and plots are given for the response of filters with linear and quadratic tapers and with linear and quadratic chirps.

Linear Quadratic Optimal Stochastic Control with Random Coefficients

Abstract: The purpose of this paper is to apply the methods developed in [1] and [2] to solve the problem of optimal stochastic control for a linear quadratic system.After proving some preliminary existence results on stochastic differential equations, we show the existence of an optimal control.The introduction of an ad joint variable enables us to derive extremality conditions: the control is thus obtained in random “feedback” form. By using a method close to the one used by Lions in [4] for the control of partial differential equations, a priori majorations are obtained.A formal Riccati equation is then written down, and the existence of its solution is proved under rather general assumptions.For a more detailed treatment of some examples, the reader is referred to [1].

Discrete Riccati Equations: Alternative Algorithms, Asymptotic Properties, and System Theory Interpretations

Abstract: Publisher Summary Algorithms have been developed that, while related to the Riccati algorithm, have important computational advantages. This chapter presents a self-contained exposition of the properties of the class of discrete-time Riccati equations that arise in the filtering problem. The point of view adopted is novel, which shows the relationship between various alternative algorithms and the Riccati equation while it connects up the asymptotic theory of such equations with the developments in linear systems theory. The chapter derives the Riccati equation and several related algorithms for the control problem by a novel approach that reveals its linear algebraic nature. It has been shown that the control problem could be reduced to a defined set of linear algebraic equations for which a solution could be found by employing orthogonal transformations. In the time-variable case, the square root version of the Riccati equation that emerges is related to similar algorithms developed in the filtering context.

On the algebraic Riccati equation

Abstract: In this note the matrix equation A + WB + BTW + WCW = 0 is considered. A monotoneity result and an inertia theorem on the location of the eigenvalues of W and B + CW are proved.

A general transfer function approach to linear stationary filtering and steady-state optimal control problems

Abstract: The transfer-function form of the stationary algebraic Riccati equation is investigated. A generalized spectral factorization technique that copes with unstable systems is introduced. This factorization is used to provide an efficient way of solving the Riccati equation and to establish the exact equivalence between the time domain and the transfer-function approaches to the linear stationary filtering and the deterministic optimal control problems. The proposed method is easily extended to cope, in the filtering problem, with coloured signals and the superiority of its computational method for systems having a small number of inputs and outputs is demonstrated. Finally, an application of the transfer-function approach in determining the class of all systems that share the same optimal solution is introduced.

Values and strategies for infinite time linear quadratic games

Abstract: For a class of infinite time linear quadratic games it is shown that an appropriate solution of an algebraic Riccati type equation determines the value of the game but not necessarily any equilibrium strategies. In the case of nonexistence of equilibrium strategies, e-optimal strategies are constructed through the solutions of a differential Riccati equation.

Convergence and Stability Properties of the Discrete Riccati Operator Equation and the Associated Optimal Control and Filtering Problems

Abstract: The convergence properties for the solution of the discrete time Riccati matrix equation are extended to Riccati operator equations such as arise in a gyroscope noise filtering problem. Stabilizability and detectability are shown to be necessary and sufficient conditions for the existence of a positive semidefinite solution to the algebraic Riccati equation which has the following properties (i) it is the unique positive semidefinite solution to the algebraic Riccati equation, (ii) it is converged to geometrically in the operator norm by the solution to the discrete Riccati equation from any positive semidefinite initial condition, (iii) the associated closed loop system converges uniformly geometrically to zero and solves the regulator problem, and (iv) the steady state Kalman–Bucy filter associated with the solution to the algebraic Riccati equation is uniformly asymptotically stable in the large. These stability results are then generalized to time-varying problems; also it is shown that even in infini...

The Infinite-Dimensional Riccati Equation for Systems Defined by Evolution Operators

Abstract: In the paper we consider the linear, quadratic control and filtering problems for systems defined by integral equations given in terms of evolution operators. We impose very weak conditions on the evolution operators and prove that the solution to both problems leads to an integral Riccati equation which possesses a unique solution. By imposing more structure on the evolution operator we show that the integral Riccati equation can be differentiated, and finally by considering an even smaller class of evolution operators we are able to prove that the differentiated version has a unique solution. The motivation for the study of such systems is that they enable us to consider wide classes of differential delay equations and partial differential equations in the same formulation. We derive new results for such a system and show how all of the existing results can be obtained directly by our methods.

Efficient numerical techniques for solving Pocklington's equation and their relationships to other methods

Abstract: It is shown that testing Pocklington's equation with piecewise sinusoidal functions yields an integro-difference equation whose numerical solution is identical to that of the point-matched Hallen's equation when a common set of basis functions is used with each. For any choice of basis functions, the integro-difference equation has the simple kernel, the fast convergence, the simplicity of point-matching, and the adequate treatment of rapidly varying incident fields, but none of the additional unknowns normally associated with Hallen's equation. Furthermore, for the special choice of piecewise sinusoids as the basis functions, the method reduces to Richmond's piecewise sinusoidal reaction matching technique, or Galerkin's method. It is also shown that testing with piecewise linear (triangle) functions yields an integro-difference equation whose solution converges asymptotically at the same rate as that of Hallen's equation. The resulting equation is essentially that obtained by approximating the second derivative in Pocklington's equation by its finite difference equivalent. The authors suggest a simple and highly efficient method for solving Pocklington's equation. This approach is contrasted to the point-matched solution of Pocklington's equation and the reasons for the poor convergence of the latter are examined.

On the matrix Riccati equation for linear systems with random gain

Abstract: Considered is the asymptotic property of the discrete-time matrix Riccati equation arising in the optimal control of linear systems with a random gain. The instability and stability conditions are derived in terms of the degree of stability of the state transition matrix.

Closed-loop structural stability for linear-quadratic optimal systems

Abstract: This paper contains an explicit parametrization of a subclass of linear constant gain feedback maps that will not destabilize an originally open-loop stable system. These results can then be used to obtain several new structural stability results for multi-input linear-quadratic feedback optimal designs.

An Hermitian method for the solution of radiative transfer problems

Abstract: A new method for the solution of the radiative transfer equation is developed. The resulting system, like the ordinary difference equations is tridiagonal in form and easy to use. Numerical examples are given which show the superiority of the new technique to previous differential and integral equation methods.

Scattering theory and linear least squares estimation—Part I: Continuous-time problems

Abstract: The Riccati equation plays as important a role in scattering theory as it does in linear least squares estimation theory. However, in the scattering literature, a somewhat different framework of treating the Riccati equation has been developed. This framework is shown to be appropriate for estimation problems and makes possible simple derivations of known results as well as leading to several new results. Examples include the derivation of backward equations to solve forward Riccati equations, an analysis of the asymptotic behavior of the Riccati equation, the derivation of backward Markovian representations of stochastic processes, and new derivations and new insights into the Chandrasekhar and related Levinson and Cholesky equations.

Polynomial Factorization via the Riccati Equation

Abstract: The paper tackles the problem of factoring a real polynomial $f( z )$, nonzero on $| z | = 1$, as the product of two polynomials $u( z )$ and $v( z )$ with zeros in $| z | 1$. The individual zeros of $f( z )$ are not found. Riccati difference equations are shown to provide a tool for executing the factorization.

Dead-beat control and the Riccati equation

Abstract: Connections between dead-beat control strategies and optimal control policies for linear, time-invariant, discrete-time systems are established. The performance index of the system is quadratic and only the terminal state of the system is penalized. An explicit solution to the singular Riccati equation, associated with this optimization problem, is given. Properties of the time-variable gain matrices, generating the optimal cantrol policy, are presented. In particular, necessary and sufficient conditions for each of these gain matrices to be a time-invariant deadbeat controller are given.

An attempt at the inversion of reflection data

Abstract: The main problem in seismic prospecting is to infer from the observed reflection response the distribution of density and seismic velocity with depth. This process is generally called the inversion of the reflection data. For plane waves propagating through plane parallel stratification, it can be shown that at any depth the ratio between the amplitude of the transmitted and reflected wave satisfies the Riccati equation. Based on this equation we have formulated an iterative inversion method, which is found to be suitable for numerical computations. We have applied this method on synthetic reflection data, and found that it provides a very fast and accurate inversion.

Some reduced-order non-Riccati equations for linear least-squares estimation : the stationary, single-output case†

Abstract: The problem of determining the Kalman—Bucy filter for an n-dimensional single-output model is the topic of this paper. Both the discrete-time case and continuous-time case are considered. The model processes are assumed to be stationary. It is shown that, under certain regularity conditions, only n first-order difference or differential equations are required for determining the error covariance function, and hence also the filter gain, rather than 1/2n(n + 1) equations as with the Riccati approach or 2n as in the previous non-Riccati algorithm. This reduction is achieved by constructing a system of simple integrals for the 2n non-Riccati equations. The reduced-order algorithms have non-trivial steady-state versions, which are equivalent to the algebraic equations obtained by spectral factorization. The stationary and single-output assumptions are for convenience. In fact, the basic method works also in a more general setting.

An overview of polynomic system theory

Abstract: s from Papers Appearing in the Proceedings of the IEEE, Special Issue-JanuaT 1976 An Overview of Polynomic System Theory @ W. A. PORTER Department qf ELectrical Engineering, University of Michigan, Ann Arbor, Michigan ABSTRACT: During the past 5 yr multilinear, multipower and polynomic systems have become a most important area of applications and theoretical development. This paper starts with some typical examples and historical remarks. Using this background, it then proceeds to put the recent literature in perspective and to quantify partially some broad problem categories wh,ere intensive work is underway. During the past 5 yr multilinear, multipower and polynomic systems have become a most important area of applications and theoretical development. This paper starts with some typical examples and historical remarks. Using this background, it then proceeds to put the recent literature in perspective and to quantify partially some broad problem categories wh,ere intensive work is underway. Scattering Theory and Linear Least Squares Estimation Part I: Continuous-time Problems by L. LJUNG, T. KAILATH and B. FRIEDLANDER Information Systems Laboratory Department of Electrical Engineering Stanford University, California ABSTRACT : The Riccati equation plays an equally important role in scattering theory as in linear least-squares estimation theory. However, in the scattering literature, a somewhat cllffererrt ~frarnework of treatgnz,ple rlerl’cations of known results as well as to obtain several new results. Examples include the derivation of backwards equations to solve forwards Riccati equations; an analysis of the asymptotic behaviour of the Riccati equation; the derivation of backwards Markovian representations of stochastic processes; and new derivations and new insights into the Chandrasekhar and related Levinson and Cholesky equatioras. The Riccati equation plays an equally important role in scattering theory as in linear least-squares estimation theory. However, in the scattering literature, a somewhat cllffererrt ~frarnework of treatgnz,ple rlerl’cations of known results as well as to obtain several new results. Examples include the derivation of backwards equations to solve forwards Riccati equations; an analysis of the asymptotic behaviour of the Riccati equation; the derivation of backwards Markovian representations of stochastic processes; and new derivations and new insights into the Chandrasekhar and related Levinson and Cholesky equatioras.

The asymptotic behavior of constant-coefficient Riccati differential equations

Abstract: A simple and self-contained proof is given of a general theorem on the convergence of a constant coefficient Riccati differential equation to a unique limiting value. In particular our result, which includes (strictly) previous results, does not require any analysis of the algebraic Riccati equation.

A Spectral Factorization Approach to the Distributed Stable Regular Problem; the Algebraic Riccati Equation

Abstract: This paper is a study of the discrete-time infinite-dimensional “stable regulator problem” having a cost function which is not necessarily positive. We take a spectral factorization approach to the problem. Also there are results on the algebraic Riccatic equation which are equivalent to results about fixed points for a broad class of symplectic maps.

The linear quadratic optimal control problem for infinite dimensional systems over an infinite horizon: Survey and examples

Abstract: Survey of currently available theory for systems the evolution of which can be described by semigroups of operators of class CO. Connection between the concepts of stabilizability and detectability and the problem of existence and uniqueness of solutions to the operator Riccati equation. Examples and Open problems.

A reformulation of the algebraic Riccati equation problem

Abstract: The algebraic Riccati equation problem is reformulated so as to yield a simple solution when the system has only real roots, as may occur when using a spatially quantized distributed parameter model. A restriction is also placed on the choice of the synthetic output matrix C .

A square root of a matrix approach to obtain the solution to a steady-state matrix Riccati equation

Abstract: By utilizing the square root of a matrix approach, a method of generating the solution to a steady-state matrix Riccati type equation (under certain restrictions) is presented. This approach not only yields a closed form expression for the Riccati solution, but also converts the original Riccati equation into other equations which may have numerical or computational advantages. An example is worked out for a second-order case.

Approximations to Riccati equations having slow and fast modes

Abstract: Problems associated with the optimal linear regulator when one control is penalized much less than the others, and the optimal linear estimator when one measurement is of a much higher quality than the others are considered. Both situations cause the Riccati equation's solution to have transients with radically different speeds. In order to generate solutions with radically different transient speeds, a large number of small numerical integration time steps must be used-small to capture the rapid transient and a large number to generate the slow transient. Approximations are derived to the solutions to these Riccati equations. Although the number of scalar numerical integrations required is reduced by only one, a closed-form approximation is derived for the rapidly varying part of the transient. This allows the use of large numerical integration time steps and results in a considerable decrease in computation time. Measures are derived of both the errors in the approximations and how they affect the resulting regulators and estimators. A numerical example is given comparing the solution of the Riccati equation to its approximation.