Low-Rank Second-Order Splitting of Large-Scale Differential Riccati Equations

Abstract

Abstract-We apply first-and second-order splitting schemes to the differential Riccati equation. Such equations are very important in e.g. linear quadratic regulator (LQR) problems, where they provide a link between the state of the system and the optimal input. The methods can also be extended to generalized Riccati equations, e.g. arising from LQR problems given in implicit form. In contrast to previously proposed schemes such as BDF or Rosenbrock methods, the splitting schemes exploit the fact that the nonlinear and affine parts of the problem, when considered in isolation, have closed-form solutions. We show that if the solution possesses low-rank structure, which is frequently the case, then this is preserved by the method. This feature is used to implement the methods efficiently for large-scale problems. The proposed methods are expected to be competitive, as they at most require the solution of a small number of linear equation systems per time step. Finally, we apply our low-rank implementations to the Riccati equations arising from two LQR problems. The results show that the rank of the solutions stay low, and the expected orders of convergence are observed.

Solution formulas for differential Sylvester and Lyapunov equations

“…Other important applications lie in model order reduction [3] or in optimal control of linear time-invariant systems on finite time horizons [30]. Despite its importance, there have been but a few efforts to solve the differential Sylvester / Lyapunov or Riccati equation numerically, see [6,7,16,21,25,26,31,36]. These algorithms are usually based on applying a time discretization and solving the resulting algebraic equations.…”

Numerical solution of the finite horizon stochastic linear quadratic control problem

“…As shown in one study, we can give explicit representations for and , and these can be efficiently evaluated in a low‐rank setting. Unfortunately, there does not seem to be a similarly useful representation for .…”

“…In the rest of this section, we briefly recap how to implement these methods in a low‐rank fashion. For details, we refer to . As previously, let P = Z Z T and be given low‐rank factorizations, and consider first .…”

“…For details, we refer to. 43 As previously, let P = ZZ T and Q = Q 0 Q T 0 be given low-rank factorizations, and consider first  1 (h)P. This can be written as…”

GPU acceleration of splitting schemes applied to differential matrix equations

“…where the integral term may be approximated by high-order quadrature as in [44]. While this does not result in a splitting scheme of the form described above, we still include it in our experiments due to its similarity and efficiency.…”