Showing papers on "Riccati equation published in 1973"
TL;DR: In this article, the design of a suboptimal terminal guidance system for reentry vehicles with a constraint on the body attitude angle at impact is studied, and the problem is formulated as a linear quadratic control problem.
Abstract: The design of a suboptimal terminal guidance system for reentry vehicles with a constraint on the body attitude angle at impact is studied. Permissible range of the miss distance and the body attitude angle at impact is specified. The problem is formulated as a linear quadratic control problem. The Riccati equation is derived to provide time-varying feedback gains. The desired scheme is suboptimal. The region of initial states for which the system meets the specifications becomes smaller as the initial height of the reentry vehicle at initial time is decreased.
388 citations
TL;DR: This work presents some new algorithms that yield the gain matrix for the Kalman filter directly without having to solve separately for the error-covariance matrix and potentially have other computational benefits.
Abstract: Recursive least-squares estimates for processes that can be generated from finite-dimensional linear systems are usually obtained via an n \times n matrix Riccati differential equation, where n is the dimension of the state space. In general, this requires the solution of n(n + 1)/2 simultaneous nonlinear differential equations. For constant parameter systems, we present some new algorithms that in several cases require only the solution of less than 2np or n(m + p) simultaneous nonlinear differential equations, where m and p are the dimensions of the input and observation processes, respectively. These differential equations are said to be of Chandrasekhar type, because they are similar to certain equations introduced in 1948 by the astrophysicist S. Chandrasekhar, to solve finite-interval Wiener-Hopf equations arising in radiative transfer. Our algorithms yield the gain matrix for the Kalman filter directly without having to solve separately for the error-covariance matrix and potentially have other computational benefits. The simple method used to derive them also suggests various extensions, for example, to the solution of nonsymmetric Riccati equations.
204 citations
TL;DR: In this article, a numerically stable and fast computational method for the solution of the matrix Ricatti differential equation with finite terminal time is given, which is the same as the one presented in this paper.
Abstract: A numerically stable and fast computational method is given for the solution of the matrix Ricatti differential equation with finite terminal time.
105 citations
88 citations
TL;DR: In this paper, an asymptotic expansion method was developed for a singularly perturbed matrix Riccati equation, which can be used in the design of regulator and estimator systems with small parameters.
Abstract: An asymptotic expansion method is developed for a singularly perturbed matrix Riccati equation. The method reduces the system order and avoids difficulties with numerically stiff problems. It can be used in the design of regulator and estimator systems with small parameters.
66 citations
TL;DR: In this paper, bounds on the discrete Riccati equation were given for the stability of discrete-time minimum variance estimators, correcting an error in proving the stability in proving discrete time estimators.
Abstract: This note presents bounds on the discrete Riccati equation, correcting an error in proving the stability of discrete-time minimum variance estimators.
36 citations
TL;DR: In this article, a simplified method for solving the asymmetric, coupled, Riccati-type matrix differential equations occurring in the solution of open-loop Nash controls in linear quadratic differential games is discussed.
Abstract: A simplified method for solving the asymmetric, coupled, Riccati-type matrix differential equations occurring in the solution of open-loop Nash controls in linear quadratic differential games is discussed m this paper. This method offers considerable savings in the computational requirements of the sampled-data Nash solution in time-invariant linear quadratic differential games where the open-loop Nash Riccati equations must be solved repeatedly several times for different boundary conditions. A simple example is presented to illustrate the results
33 citations
TL;DR: In this article, the authors generalized the notion of spectral factorization and showed that solutions of the matrix quadratic equation (BX + XA - XCDX + Q = 0) have a one-to-one relation with factorizations of rational matrix.
Abstract: By generalizing the notion of spectral factorization, solutions X of the matrix quadratic equation $BX + XA - XCDX + Q = 0$ are shown to have a one-to-one relation with factorizations of a rational matrix. By progressive specialization of this factorization, equivalence results are obtained in turn for symmetric solutions, Hermitian solutions, stabilizing solutions and positive definite solutions of the special case of the algebraic Riccati equation $F'X + XF - XGG'X + Q = 0$ .
31 citations
TL;DR: In this paper, the radial factor of a separable solution of the wave equation in Schwarzschild's space-time satisfies a second-order linear differential equation, and the behavior of the solutions near the singular points (the origin, the horizon, and infinity) of the equation is analyzed.
Abstract: The radial factor of a separable solution of the wave equation in Schwarzschild's space‐time satisfies a second‐order linear differential equation. This equation is studied in detail. The behavior of the solutions near the singular points (the origin, the horizon, and infinity) of the equation is analyzed. By an appropriate transformation two simpler differential equations are obtained corresponding to retarded and advanced solutions with characteristic asymptotic expansions. Their properties permit the expression of the general solution of the radial equation in terms of a single contour integral. Finally, through a ``matching'' technique, the behavior of a solution at the singular points is determined from its behavior at a single singular point.
25 citations
TL;DR: Matrix Riccati equations are interpreted as differential equations on Grassman manifolds and have a unique asymptotically stable equilibrium point or periodic solution.
Abstract: Matrix Riccati equations are interpreted as differential equations on Grassman manifolds. Necessary conditions for the Riccati equation to be a Morse-Smale system are given in the autonomous and periodic cases. Under this condition, the equation is structurally stable and has a unique asymptotically stable equilibrium point or periodic solution.
23 citations
TL;DR: In this paper, a characterization of the interrelationship between the recursive Riccati difference equation and its steady-state form is developed via the method of quasilinearization.
TL;DR: In this article, a set of sufficient conditions for a weak minimum is derived for the nonsingular Bolza problem of variational calculus, expressed in control notation, where the initial state and time are assumed fixed.
Abstract: A set of sufficient conditions for a weak minimum is derived for the nonsingular Bolza problem of variational calculus, expressed in control notation The initial state and time are assumed fixed; the terminal state and time are variable, subject to a set of equality constraints Specified terminal time is considered as a special case The sufficient conditions derived are minimal sets for normal problems The existence of a matrix determined by backward integration of a Riccati equation takes the place of the classical conditions involving conjugate points or focal points This condition is easier to implement than the classical conditions in optimization problems requiring numerical solution During the backward integration, feedback gains for deviations in the state variables and terminal constraints are easily computed for use in a neighboring optimum control law The conditions derived are shown to be easier to apply, more concise, and more generally valid than several other recently proposed sets of optimality conditions
TL;DR: In this article, the authors introduced the concept of predictable directions along which the solution goes to zero rather than a nonzero constant, and showed how to convert constant directions to predictable directions and how the concept may be extended to time-variant systems.
TL;DR: In this paper, the Riccati differential equation is discussed from the point of view of dissipativity or conservativity of its solutions, and a survey of results relating to analytic properties of these solutions and to the geometry of the corresponding semigroup of matrix linear fractional transformations is given.
Abstract: The matrix Riccati differential equation is discussed, from the point of view of dissipativity or conservativity of its solutions. A survey is given of results relating to analytic properties of these solutions and to the geometry of the corresponding semigroup of matrix linear fractional transformations; further, a probabilistic interpretation is given of the properties of being dissipative or conservative, and the connection between dissipativity of the solutions of the Riccati equation and the stability of the screw method is studied. Physical and technical applications of the mathematical theory are given.There are 87 references.
01 Dec 1973
TL;DR: In this article, a prefiltering version of the Kalman filter is derived for both discrete and continuous measurements, which consists of determining a single discrete measurement that is equivalent to either a time segment of continuous measurements or a set of discrete measurements.
Abstract: A prefiltering version of the Kalman filter is derived for both discrete and continuous measurements. The derivation consists of determining a single discrete measurement that is equivalent to either a time segment of continuous measurements or a set of discrete measurements. This prefiltering version of the Kalman filter easily handles numerical problems associated with rapid transients and ill-conditioned Riccati matrices. Therefore, the derived technique for extrapolating the Riccati matrix from one time to the next constitutes a new set of integration formulas which alleviate ill-conditioning problems associated with continuous Riccati equations. Furthermore, since a time segment of continuous measurements is converted into a single discrete measurement, Potter's square root formulas can be used to update the state estimate and its error covariance matrix. Therefore, if having the state estimate and its error covariance matrix at discrete times is acceptable, the prefilter extends square root filtering with all its advantages, to continuous measurement problems.
TL;DR: In this paper, a perturbation-expansion method and a local-equilibrium method were proposed to solve the Glauber's equation in a one-dimensional Ising model in a magnetic field.
Abstract: It is of interest to solve the master equation of a one-dimensional Ising model in a magnetic field (Glauber's equation) not only as a mathematical problem, but also for some practical applications. While an exact treatment of Glauber's equation seems impossible, we consider its approximate solutions. We review the applications of the perturbation-expansion method and applications of the mean-field theory to this problem. We also introduce a local-equilibrium method that improves the mean-field theory by taking the short-range correlation into account. The new method yields an analytically soluble differential equation, which, contrary to the preceding two methods, has an exact steady-state solution.
01 Dec 1973
TL;DR: In this paper, a special case of the "structure algorithm" used for characterizing properties of linear systems is presented. But it is not related to the Chandrasekhar-type equations recently introduced by Kailath.
Abstract: A new algorithm for solving discrete time linear-quadratic control problems is given. This algorithm is shown to be a special case of the "structure algorithm" used for characterizing properties of linear systems. It is also shown to be related to the Chandrasekhar-type equations recently introduced by Kailath.
TL;DR: By keeping count of the number of interactions, this article obtained some novel perturbation expansions for the Riccati equation, which were later used in radiative transfer theory by Mingle.
TL;DR: In this paper, the authors discuss the close relations that exist among the three subspaces that arise in three apparently unrelated problem areas: 1) the subspace associated with the error of state vector reconstruction in informationally decentralized control problems; 2) the solution of the Riccati equation; and 3) the null space of the certain observation matrix constructed by the structure algorithm.
Abstract: The close relations are discussed that exist among the three subspaces that arise in three apparently unrelated problem areas: 1) the subspace associated with the error of state vector reconstruction in informationally decentralized control problems; 2) the subspace associated with the solution of the Riccati equation; and 3) the null space of the certain observation matrix constructed by the structure algorithm.
01 Mar 1973
TL;DR: In this paper, the authors investigated the dynamic behavior of an electrode-position control system for an industrial direct electric-arc furnace, which is modelled as an electromechanical system with transfer-function relationships and nonlinear valve characteristics.
Abstract: The dynamic behaviour of an electrode-position control system for an industrial direct electric-arc furnace is investigated. The electrode-positioning servomechanism is modelled as an electromechanical system with transfer-function relationships and nonlinear valve characteristics defining the hydromechanical system. The electrical-power-transmission-system model is based on single-phase circuit equations and incorporates an empirical arc-discharge characteristic. A dynamic representation of the supply system is also used which incorporates a linear model of an equivalent synchronous machine including both transient and subtransient effects. Dynamic performance of the conventionally controlled system is investigated by analogue/hybrid simulation of the nonlinear-system model. Optimal and suboptimal control of a linearised model is also studied, and methods of solution based on dynamic programming and the algebraic matrix Riccati equation are compared. To investigate online calculations of optimal control, various forms of performance functional are considered. Conventional 4-mode control and linear optimal control of arc impedance are applied in real time by process computer to the analogue-simulated nonlinear-system model. The study demonstrates a relative improvement in the dynamic performance of arc current, and hence power, with changing bath conditions and varying arc-discharge characteristics.
01 Dec 1973
TL;DR: In this article, a comprehensive look is taken at the Newton-Raphson method of solving the algebraic matrix Riccati equation, its important numerical properties and computational requirements are also discussed.
Abstract: In this paper, a comprehensive look is taken at the Newton-Raphson method of solving the algebraic matrix Riccati equation. Its important numerical properties and computational requirements are also discussed.
TL;DR: In this paper, an approach for designing a recursive algorithm for finding the root or roots of the equation ǫ(x) = 0, where ǒ is a continuously differentiable function from Rn → Rn, is presented.
Abstract: An approach is presented for designing a recursive algorithm for finding the root or roots of the equation ƒ(x) = 0, where ƒ is a continuously differentiable function from Rn → Rn . A differential equation, or difference equation, is written with ƒ as the dependent variable and a control vector u as the forcing function. Optimal control theory is then used to drive ƒ from a given initial value to zero.
TL;DR: In this paper, the authors extend the usual theory of differential inequalities for first-order systems (Coppel [7]) which applies given the component-wise ordering of vectors, to derive a necessary and sufficient condition for the preservation of order of solutions of DIN, with order being more generally defined.
Abstract: Chapter 1 extends the usual theory of differential inequalities for first-order systems (Coppel [7]) which applies given the component-wise ordering of vectors, to derive a necessary and sufficient condition for the preservation of order of solutions of differential inequalities, with order being more generally defined. The immediate application is to the case of symmetric matrices ordered by positive definiteness, although there are also interesting consequences for the Lorentz ordering, with a right circular positive cone of vectors.
01 Nov 1973
TL;DR: In this paper, it was proved that the elliptic ball criterion is a necessary condition on the matrix M(t) for the linear feedback control equation f(D)x + BM(t),g(D),x = 0 to have a special kind of quadratic Lyapunov function.
Abstract: It is proved that the elliptic ball criterion is a necessary condition on the matrix M(t) for the linear feedback control equation f(D)x + BM(t)g(D)x = 0 to have a special kind of quadratic Lyapunov function. That it is also a sufficient condition has already been proved elsewhere. These two facts lead to a comparison theorem which enables the existence of a quadratic Lyapunov function for one equation to be deduced from that for another equation.
01 Jan 1973
TL;DR: The problem of optimal control of single-input discrete bilinear systems with unbounded control and the cost criterion quadratic in state is investigated in this paper.
Abstract: The problem of optimal control of single-input discrete bilinear systems with unbounded control and the cost criterion quadratic in state is investigated in this paper.
14 Sep 1973
TL;DR: In this paper, a six-engine reaction control jet relay attitude control law with deadband is shown to be a good linear approximation to a weighted time-fuel optimal control law, and several computational algorithms for solving the matrix Riccati equation are investigated with respect to accuracy, computational storage requirements, and computational speed.
Abstract: The conventional six-engine reaction control jet relay attitude control law with deadband is shown to be a good linear approximation to a weighted time-fuel optimal control law. Techniques for evaluating the value of the relative weighting between time and fuel for a particular relay control law is studied along with techniques to interrelate other parameters for the two control laws. Vehicle attitude control laws employing control moment gyros are then investigated. Steering laws obtained from the expression for the reaction torque of the gyro configuration are compared to a total optimal attitude control law that is derived from optimal linear regulator theory. This total optimal attitude control law has computational disadvantages in the solving of the matrix Riccati equation. Several computational algorithms for solving the matrix Riccati equation are investigated with respect to accuracy, computational storage requirements, and computational speed.
01 Jan 1973
TL;DR: In this paper, it was shown that the coupled Maxwell and Bloch equations in the slowly varying envelope approximation have conservation laws in addition to the usual one associated with the conservation of energy, and these higher conservation laws have been used to determine the amplitude of 2π pulses into which a coherent pulse may decompose as it propagates through an absorbing medium.
Abstract: It has been shown that the equations customarily used to describe coherent optical pulse propagation, namely the coupled Maxwell and Bloch equations in the slowly varying envelope approximation, possess conservation laws in addition to the usual one associated with the conservation of energy. These higher conservation laws have been used to determine the amplitude of each of the 2π pulses into which a coherent pulse may decompose as it propagates through an absorbing medium.[1]
TL;DR: The algebraic solution for the discrete Riccati equation may be exploited to obtain nonrecursive solutions for the continuous and discrete smoothed error-covariance matrix.
Abstract: The algebraic solution for the discrete Riccati equation may be exploited to obtain nonrecursive solutions for the discrete smoothed error-covariance matrix. The system matrix required to obtain this result is given, and the corresponding matrix for the continuous case may be found in an analogous manner.