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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01 Jan 1975
206 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 traveling-wave solution of the class of equations ∑ p = 1 n 1 α p ∂ p Q ∂ t p + ∑ q = 1 N 2 β q ∂ q Q ∆ x q + ∆ m = 1 M μ m Q m = 0 where α p, β q and μ m are parameters.
203 citations
TL;DR: In this article, a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators is established, which is similar to our framework.
Abstract: This paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators.
202 citations
TL;DR: In this paper, the Riccati equations are applied to the linear-quadratic optimal control problem for hereditary differential systems, and it is shown that, for most such problems, the operator solutions are of trace class (i.e., nuclear).
Abstract: Recent theory of infinite dimensional Riccati equations is applied to the linear-quadratic optimal control problem for hereditary differential systems, and it is shown that, for most such problems, the operator solutions of the Riccati equations are of trace class (i.e., nuclear). With special attention to trace-norm convergence, an abstract approximation theory is developed and applied to a particular approximation scheme. Numerical examples are given.Problems on both finite and infinite time intervals are studied. For both the hereditary system and the approximating systems in the infinite time problem, characteristic equations are derived for the closed-loop eigenvalues, and formulas for the corresponding eigenvectors are given.
201 citations