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.

Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations

Abstract: In this paper, we introduce the doubling transformation, a structure-preserving transformation for symplectic pencils, and present its basic properties. Based on these properties, a unified convergence theory for the structure-preserving doubling algorithms for a class of Riccati-type matrix equations is established, using only elementary matrix theory.

Numerical integration of the differential matrix Riccati equation

Abstract: Two new Bernoulli substitution methods for solving the Riccati differential equation are tested numerically against direct integration of the Riccati equation, the Chandrasekhar algorithm, and the Davison-Maki method on a large set of problems taken from the literature. The first of these new methods was developed for the time-invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric Riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n . This method is fast and accurate when the extremal solutions of the associated algebraic Riccati equation are well separated. The second new method was developed as a means of eliminating the instabilities associated with the Davison-Maki algorithm. By using reinitialization at each time step the Davison-Maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time-invariant and time-dependent problems. From the results of our study we conclude that the modified Davison-Maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the Chandrasekhar algorithm is better.

Constrained Stochastic LQC: A Tractable Approach

Abstract: Despite the celebrated success of dynamic programming for optimizing quadratic cost functions over linear systems, such an approach is limited by its inability to tractably deal with even simple constraints. In this paper, we present an alternative approach based on results from robust optimization to solve the stochastic linear-quadratic control (SLQC) problem. In the unconstrained case, the problem may be formulated as a semidefinite optimization problem (SDP). We show that we can reduce this SDP to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state, thus yielding an approach that is amenable to closed-loop control and analogous to the Riccati equation in our framework. We also consider a tight, second-order cone (SOCP) approximation to the SDP that can be solved much more efficiently when the problem has additional constraints. Both the SDP and SOCP are tractable in the presence of control and state space constraints; moreover, compared to the Riccati approach, they provide much greater control over the stochastic behavior of the cost function when the noise in the system is distributed normally.

Minimax optimal control of uncertain systems with structured uncertainty

Abstract: This paper considers a minimax control problem for an uncertain system containing structured uncertainties. The uncertainties in this system are assumed to satisfy a certain integral quadratic constraint. For a given initial condition, the minimax optimal controller is constructed by solving a parameter-dependent Riccati equation of the game type. This controller leads to a closed-loop uncertain system which is absolutely stable.