A Numerical Comparison of Different Solvers for Large-Scale, Continuous-Time Algebraic Riccati Equations and LQR Problems

Abstract

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance. *

The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

“…Many more methods and references can be listed if we bring in more details. Interested readers are encouraged to look through a comparison paper [11] and the references therein. The methods in the former three classes use a lot of shifts in the calculation process, so a shift selection strategy rather than several pre-chosen shifts is needed.…”

Solving differential Riccati equations: A nonlinear space-time method using tensor trains

“…A key ingredient in achieving robust and fast methods is the utilization of the low numerical rank of the solution. For the algebraic counterpart of (2), it is known, see, e.g., [6,5,11,35,37,46] that P ≈ LL T can be well approximated by low-rank factors L ∈ R n×k with k n if at least the control or the observation matrices B and C correspond to finite-dimensional operators. In particular, this holds true for many relevant PDEs of parabolic type (cf.…”

“…In the time-varying case, only a few theoretical results have been obtained. For a recent discussion on low-rank solutions of differential Riccati equations, we refer to [47,6]. Nevertheless, many numerical approaches exist and most of them rely on a time discretization of (2) for an abstract nonlinear matrix-valued ordinary differential equation (ODE) of the forṁ P (t) = f (t, P (t)), P (0) = P 0 .…”

Sparse Grid Approximation of the Riccati Operator for Closed Loop Parabolic Control Problems with Dirichlet Boundary Control

“…A number of methods is available for the approximation of the ARE (see e.g. [5] or [7] for a survey). One basic approach is to approximate the operator ARE directly (cf.…”

“…A number of methods for the solution of non-linear equations, which have been studied for the ARE (see e.g. [3,4,5,34] for a survey), can similarly be implemented for the Riccati-IDE. In this article, we apply Newton's method as suggested in [33].…”