On the solution of the Riccati differential equation arising from the LQ optimal control problem

In this paper we consider the matrix Riccati differential equation (RDE) that arises from linear-quadratic (LQ) optimal control problems. In particular, we establish explicit closed formulae for the solution of the RDE with a terminal condition using particular solutions of the associated algebraic Riccati equation. We discuss how these formulae change as assumptions are progressively weakened. An application to LQ optimal control is briefly analysed

Algebraic methods for the solution of some linear matrix equations

The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method

The discrete-time generalized algebraic Riccati equation: Order reduction and solutions’ structure

In this paper we discuss how to decompose the constrained generalized discrete-time algebraic Riccati equation arising in optimal control and optimal filtering problems into two parts corresponding to an additive decomposition X=X0+Δ of each solution X: The first part is trivial, in the sense that it is an explicit expression of the addend X0 which is common to all solutions, so that it does not depend on the particular X. The second part can be – depending on the structure of the considered generalized Riccati equation – either a reduced-order discrete-time regular algebraic Riccati equation whose associated closed-loop matrix is non-singular, or a symmetric Stein equation. The proposed reduction is explicit, so that it can be easily implemented in a software package that uses only standard linear algebra procedures

On solving a linear control problem

The problem of a linear regulator is considered. There is a system of linear differential equations with a quadratic control quality criterion. The method of dynamic programming is applied to the solution of the considered linear problem. As is known, the main difficulty in applying this method is to integrate partial differential equations. In this problem, the obtained optimal control function depends on the solution of the Riccati equation. In [1], conditions were obtained under which there is a solution to such optimal control problems with a quadratic quality criterion. These conditions were obtained along with formulas for minimizing control and for optimal trajectory. But all these statements depended on the ability to solve the matrix Riccati equation under certain boundary conditions given at some time point. To construct a solution to the problem under consideration, a system of 2 n adjoint differential equations is constructed. After splitting the transition matrix of this system into block ones, it is possible to express the state of the system at the time instant t through the state variable and the adjoint variable at the final time instant t 1. A feature of this work is that an example is given, where the solution of the Riccati equation, which determines the optimal solution of the problem, was obtained explicitly

Optimal Covariance Steering for Discrete-Time Linear Stochastic Systems

In this paper we study the optimal control for steering the state covariance of a discrete-time linear stochastic system over a finite time horizon. First, we establish the existence and uniqueness of the optimal control law for a quadratic cost function. Then, we develop efficient computational methods for solving for the optimal control law, which is identified as the solution to a semi-definite program. During the analysis, we also obtain some useful results on a matrix Riccati difference equation, which may be of independent interest

Solvability conditions for indefinite linear quadratic optimal stochastic control problems and associated stochastic Riccati equations

A linear quadratic optimal stochastic control problem with random coefficients and indefinite state/control weight costs is usually linked to an indefinite stochastic Riccati equation (SRE) which is a matrix-valued quadratic backward stochastic differential equation along with an algebraic constraint involving the unknown. Either the optimal control problem or the SRE is solvable only if the given data satisfy a certain structure condition that has yet to be precisely defined. In this paper, by introducing a notion of subsolution for the SRE, we derive several novel sufficient conditions for the existence and uniqueness of the solution to the SRE and for the solvability of the associated optimal stochastic control problem.Comment: 17 page

Conjugate Points and Shocks in Nonlinear Optimal Control

In this paper the authors use the method of characteristics to extend the Jacobi conjugate points theory to the Bolza problem arising in nonlinear optimal control. This yields necessary and sufficient optimality conditions for weak and strong local minima stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation. The same approach allows to investigate as well smoothness of the value function

振動的なﾘｯｶﾁ方程式求解過程の収束性改善

Several methods to solve the Riccati equation are proposed for the optimal regulator problem in modern control theory. In some cases, a solution of the Riccati equation is reached through iterative calculation. But many steps of iteration are sometimes required because of oscillation of iterative process.In this paper, numerical error between left and right sides of matrix-type Riccati equation is treated as a criterion of convergence, and a filtering technique is presented to reduce the number of iterations. Furthermore, the appropriate value of filtering parameter is decided analytically on the basis of the way to get a solution through Lyapunov equation derived from Riccati equation.Finally, the effectiveness of the results of this study is confirmed as well as improvement of efficiency of iterative process by parametric studies of numerical simulation