Large-scale Algebraic Riccati Equations Research Articles

On a family of low-rank algorithms for large-scale algebraic Riccati equations

In [3] it was shown that four seemingly different algorithms for computing low-rank approximate solutions Xj to the solution X of large-scale continuous-time algebraic Riccati equations (CAREs) 0=R(X):=AHX+XA+CHC−XBBHX generate the same sequence Xj when used with the same parameters. The Hermitian low-rank approximations Xj are of the form Xj=ZjYjZjH, where Zj is a matrix with only few columns and Yj is a small square Hermitian matrix. Each Xj generates a low-rank Riccati residual R(Xj) such that the norm of the residual can be evaluated easily allowing for an efficient termination criterion. Here a new family of methods to generate such low-rank approximate solutions Xj of CAREs is proposed. Each member of this family of algorithms proposed here generates the same sequence of Xj as the four previously known algorithms. The approach is based on a block rational Arnoldi decomposition and an associated block rational Krylov subspace spanned by AH and CH. Two specific versions of the general algorithm will be considered; one will turn out to be a rediscovery of the RADI algorithm, the other one allows for a slightly more efficient implementation compared to the RADI algorithm (in case the Sherman-Morrision-Woodbury formula and a direct solver is used to solve the linear systems that occur). Moreover, our approach allows for adding more than one shift at a time.

Using $LDL^T$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations

Using $LDL^T$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations

The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

In this paper we derive a Toeplitz-structured closed form of the unique positive semi-definite stabilizing solution for the discrete-time algebraic Riccati equations, especially for the case that the state matrix is not stable. Based on the found form and fast Fourier transform, we propose a new algorithm for solving both discrete-time and continuous-time large-scale algebraic Riccati equations with low-rank structure. It works without unnecessary assumptions, complicated shift selection strategies, or matrix calculations of the cubic order with respect to the problem scale. Numerical examples are given to illustrate its features. Besides, we show that it is theoretically equivalent to several algorithms existing in the literature in the sense that they all produce the same sequence under the same parameter setting.

An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problems

Navier-Stokes equations are well known in modeling of an incompressible Newtonian fluid, such as air or water. This system of equations is very complex due to the non-linearity term that characterizes it. After the linearization and the discretization parts, we get a descriptor system of index-2 described by a set of differential algebraic equations (DAEs). The two main parts we develop through this paper are focused firstly on constructing an efficient algorithm based on a projection technique onto an extended block Krylov subspace, that appropriately allows us to construct a reduced system of the original DAE system. Secondly, we solve a Linear Quadratic Regulator (LQR) problem based on a Riccati feedback approach. This approach uses numerical solutions of large-scale algebraic Riccati equations. To this end, we use the extended Krylov subspace method that allows us to project the initial large matrix problem onto a low order one that is solved by some direct methods. These numerical solutions are used to obtain a feedback matrix that will be used to stabilize the original system. We conclude by providing some numerical results to confirm the performance of our proposed method compared to other known methods.

Efficient solution of large-scale algebraic Riccati equations associated with index-2 DAEs via the inexact low-rank Newton-ADI method

This paper extends the algorithm of Benner et al. (2016) [10] to Riccati equations associated with Hessenberg index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems and the algebraic constraint in the DAE system. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold described by the algebraic constraints. In the algorithm, the equations are never explicitly projected, but the projection is only applied as needed. Numerical experience indicates that the algorithmic choices for the control of inexactness and line-search can help avoid subproblems with matrices that are only marginally stable. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder.

Large-scale algebraic Riccati equations with high-rank constant terms

We consider the numerical solution of large-scale algebraic Riccati equations with high-rank constant terms. The solutions are not numerically low-rank, so the previously successful methods based on low-rank representations are not directly applicable. We modify the doubling algorithm, making use of the low-rank in the input matrix B . We also solve the challenging problems in the estimation of residuals and relative errors, convergence control and the output of the modified algorithm. Illustrative numerical examples are presented.

Nonlinear Least-Squares Approach for Large-Scale Algebraic Riccati Equations

We propose a new optimization approach for solving large-scale continuous-time algebraic Riccati equations with a low-rank right-hand side. First, we project the problem onto a Krylov-type low-dime...

RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations

This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature—all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution.

A POD projection method for large-scale algebraic Riccati equations

The solution of large-scale matrix algebraic Riccati equations is important for instance in control design and model reduction and remains an active area of research. We consider a class of matrix algebraic Riccati equations (AREs) arising from a linear system along with a weighted inner product. This problem class often arises from a spatial discretization of a partial differential equation system. We propose a projection method to obtain low rank solutions of AREs based on simulations of linear systems coupled with proper orthogonal decomposition. The method can take advantage of existing (black box) simulation code to generate the projection matrices. We also develop new weighted norm residual computations and error bounds. We present numerical results demonstrating that the proposed approach can produce highly accurate approximate solutions. We also briefly discuss making the proposed approach completely data-based so that one can use existing simulation codes without accessing system matrices.

An inexact low-rank Newton–ADI method for large-scale algebraic Riccati equations

This paper improves the inexact Kleinman–Newton method for solving algebraic Riccati equations by incorporating a line search and by systematically integrating the low-rank structure resulting from ADI methods for the approximate solution of the Lyapunov equation that needs to be solved to compute the Kleinman–Newton step. A convergence result is presented that tailors the convergence proof for general inexact Newton methods to the structure of Riccati equations and avoids positive semi-definiteness assumptions on the Lyapunov equation residual, which in general do not hold for low-rank approaches. In the convergence proof of this paper, the line search is needed to ensure that the Riccati residuals decrease monotonically in norm. In the numerical experiments, the line search can lead to substantial reduction in the overall number of ADI iterations and, therefore, overall computational cost.

Analysis of the Rational Krylov Subspace Projection Method for Large-Scale Algebraic Riccati Equations

In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse, and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the more established Newton--Kleinman iteration. In spite of convincing numerical experiments, a systematic matrix analysis of this class of methods is still lacking. We derive new relations for the approximate solution, the residual, and the error matrices, giving new insights into the role of the matrix $A-BB^*X$ and of its approximations in the numerical procedure. In the context of linear-quadratic regulator problems, we show that the Riccati approximate solution is related to the optimal value of the reduced cost functional, thus completely justifying the projection method from a model order reduction point of view. Finally, the new results provide theoretical ground for recently proposed modifications of projection methods onto rational Krylov subspaces.

On the solution of large-scale algebraic Riccati equations by using low-dimensional invariant subspaces

This article discusses an approach to solving large-scale algebraic Riccati equations (AREs) by computing a low-dimensional stable invariant subspace of the associated Hamiltonian matrix. We give conditions on AREs to admit solutions of low numerical rank and show that these can be approximated via Hamiltonian eigenspaces. We discuss strategies on choosing the proper eigenspace that yields a good approximation, and different formulas for building the approximation itself. Similarities of our approach with several other methods for solving AREs are shown: closely related are the projection-type methods that use various Krylov subspaces and the qADI algorithm. The aim of this paper is merely to analyze the possibilities of computing approximate Riccati solutions from low-dimensional subspaces related to the corresponding Hamiltonian matrix and to explain commonalities among existing methods rather than providing a new algorithm.

On two numerical methods for the solution of large-scale algebraic Riccati equations

The inexact Newton–Kleinman method is an iterative scheme for numerically solving large-scale algebraic Riccati equations. At each iteration, the approximate solution of a Lyapunov linear equation is required. A specifically designed projection of the Riccati equation onto an iteratively generated approximation space provides a possible alternative. Our numerical experiments with enriched approximation spaces seem to indicate that this latter approach is superior to Newton-type strategies on realistic problems, thus giving experimental grounds for recent developments in this direction. As part of an explanation of why this is so, we derive several matrix relations between the iterates produced by the same projection approach applied to both the (quadratic) Riccati equation and its linear counterpart, the Lyapunov equation.

Optimal Control for Allen-Cahn Equations Enhanced by Model Predictive Control

The Allen-Cahn equation is a simple model of a nonlinear reaction-diffusion process. It is often used to model interface motion in time, e.g. phase separation in alloys. It has applications in many areas including material sciences, biology, geology, as well as image processing. We will consider a simple scalar Allen-Cahn equation subject to distributed control. Here, the nonlinear reaction term is obtained from using the standard double-well potential, leading to a cubic nonlinearity. We will describe a nonlinear feedback control strategy based on the concept of Model Predictive Control (MPC). We also show how to obtain the open-loop trajectory and control using numerical techniques for PDE-constrained optimization. The feedback control scheme is then applied to the spatially semi-discretized nonlinear optimal control problem. For the prediction and control step within the MPC scheme, we apply a linear-quadratic regulator/Gaussian design problem. The arising computational challenge consisting in solving the associated large-scale algebraic Riccati equations has already been shown in the literature to be feasible using reasonably fine discretizations.

Solving large-scale continuous-time algebraic Riccati equations by doubling

We consider the solution of large-scale algebraic Riccati equations with numerically low-ranked solutions. For the discrete-time case, the structure-preserving doubling algorithm has been adapted, with the iterates for A not explicitly computed but in the recursive form Ak=Ak−12−Dk(1)Sk−1[Dk(2)]⊤, with Dk(1) and Dk(2) being low-ranked and Sk−1 being small in dimension. For the continuous-time case, the algebraic Riccati equation will be first treated with the Cayley transform before doubling is applied. With n being the dimension of the algebraic equations, the resulting algorithms are of an efficient O(n) computational complexity per iteration, without the need for any inner iterations, and essentially converge quadratically. Some numerical results will be presented. For instance in Section 5.2, Example 3, of dimension n=20209 with 204 million variables in the solution X, was solved using MATLAB on a MacBook Pro within 45 s to a machine accuracy of O(10−16).