Explicit Symplectic-Precise Iteration Algorithms for Linear Quadratic Regulator and Matrix Differential Riccati Equation

Efficient, robust and precise algorithms for linear quadratic regulator (LQR) and matrix differential Riccati equation (MDRE) are essential in optimal control. However, there are lack of good algorithms for time-varying LQR problem because of the difficulty of solving the nonlinear time-varying MDRE. In this paper, we proved that the
$n$
-th order LQR problem is equivalent to
$n$
parallel 1-dim Hamiltonian systems and proposed the explicit symplectic-precise iteration method (SPIM) for solving LQR and MDRE. The explicit symplectic-precise iteration algorithms (ESPIA) designed with SPIM have three typical merits: firstly, there are no accumulative errors in the sense of long-term time which inherits from symplectic difference scheme; secondly the stiffness problem due to the inverse of matrix is avoided by the precise iteration method; and finally the algorithmic structure of ESPIA is simple and no extra assumptions are required. Systematic analysis shows that the time complexity of the symplectic algorithms for the
$n$
-th order LQR and MDRE is
$\mathcal {O}(k_{max}n^{3})$
where
$k_{max}$
is the iteration times specified by the time duration. Numerical examples and simulations are provided to validate the performance of the ESPIA.