Topic
Riccati equation
About: Riccati equation is a research topic. Over the lifetime, 10428 publications have been published within this topic receiving 210015 citations. The topic is also known as: Riccati's differential equation.
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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: The problem of solving LQR models with system diffusions dependent on both state and control variables, which is left open in part I, is tackled and an optimal control is explicitly constructed.
Abstract: In part I of this paper [S. Chen, X. Li, and X. Zhou, SIAM J. Control Optim., 36 (1998), pp. 1685--1702], an optimization model of stochastic linear quadratic regulators (LQRs) with indefinite control cost weighting matrices is proposed and studied. In this sequel, the problem of solving LQR models with system diffusions dependent on both state and control variables, which is left open in part I, is tackled. First, the solvability of the associated stochastic Riccati equations (SREs) is studied in the normal case (namely, all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite, with at least one positive definite), which in turn leads to an optimal state feedback control of the LQR problem. In the general indefinite case, the problem is decomposed into two optimal LQR problems, one with a forward dynamics and the other with a backward dynamics. The well-posedness and solvability of the original LQR problem are then obtained by solving these two subproblems, and an optimal control is explicitly constructed. Examples are presented to illustrate the results.
385 citations
TL;DR: A numerically stable algorithm is derived to compute orthonormal bases for any deflating subspace of a regular pencil $\lambda B - A$ based on an update of the QZ-algorithm, in order to obtain any desired ordering of eigenvalues in the quasitriangular forms constructed by this algorithm.
Abstract: A numerically stable algorithm is derived to compute orthonormal bases for any deflating subspace of a regular pencil $\lambda B - A$. The method is based on an update of the $QZ$-algorithm, in order to obtain any desired ordering of eigenvalues in the quasitriangular forms constructed by this algorithm. As applications we discuss a new approach to solve Riccati equations arising in linear system theory. The computation of deflating subspaces with specified spectrum is shown to be of crucial importance here.
376 citations
Book•
19 May 2008
TL;DR: In this article, the stability of linear systems variance of linear stochastic systems quadratic performance measure book organization is discussed, as well as robustness and sensitivity of the Lyapunov equation.
Abstract: Part 1 Introduction: stability of linear systems variance of linear stochastic systems quadratic performance measure book organization. Part 2 Continuous algebraic Lyapunov equation: explicit solutions solution sounds numerical solutions. Part 3 Discrete algebraic Lyapunov equation: explicit solutions bounds of solution's attributes numerical solutions. Part 4 Differential and difference Lyapunov equation: explicit solutions bounds of solution's attributes numerical solutions singularly perturbed and weakly coupled systems coupled differential equations. Part 5 Algebraic Lyapunov equation with small parameters: singularly perturbed continuous Lyapunov equation weakly coupled continuous Lyapunov equation singularly perturbed discrete systems recursive methods for weakly coupled discrete systems. Part 6 Robustness and sensitivity of the Lyapunov equation: stability robustness sensitivity of algebraic Lyapunov equation. Part 7 Iterative methods and parallel algorithms: Smith's algorithm ADI iterative method SOR iterative method parallel algorithms parallel algorithms for coupled Lyapunov equations. Part 8 Lyapunov iterations: Kleinman algorithm for Riccati equation Lyapunov iterations for jump linear systems Lyapunov iterations for Nash differential games Lyapunov iterations for output feedback control. Part 9 Concluding remarks: Sylvester equations related topics applications. Appendix: matrix inequalities.
375 citations
TL;DR: A state estimator is designed such that the covariance of the estimation error is guaranteed to be within a certain bound for all admissible uncertainties, which is in terms of solutions of two sets of coupled algebraic Riccati equations.
Abstract: Studies the problem of Kalman filtering for a class of uncertain linear continuous-time systems with Markovian jumping parameters. The system under consideration is subjected to time-varying norm-bounded parameter uncertainties in the state and measurement equations. Stochastic quadratic stability of the above system is analyzed. A state estimator is designed such that the covariance of the estimation error is guaranteed to be within a certain bound for all admissible uncertainties, which is in terms of solutions of two sets of coupled algebraic Riccati equations.
373 citations