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

Peter Benner,Jens Saak,Patrick Kürschner,Zvonimir Bujanović

doi:10.1007/s00211-017-0907-5

Abstract

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.

Highlights

IntroductionWhile the equation may have many solutions, for such applications one is interested in finding a so-called stabilizing solution: the unique positive semidefinite solution X ∈ Cn×n such that the matrix A − G X is stable (i.e. all of its eigenvalues belong to C−, the left half of the complex plane)
The continuous-time algebraic Riccati equation, A∗ X + X A + Q − X G X = 0, (1) whereQ = C∗C, G = B B∗, A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n, appears frequently in various aspects of control theory, such as linear-quadratic optimal regulator problems, H2 and H∞ controller design and balancing-related model reduction
By setting the quadratic coefficient B in (1) to zero, our method reduces to the low-rank formulation of the Lyapunov alternating directions implicit (ADI) method, see, e.g., [4,7,22,27]

Summary

Introduction

While the equation may have many solutions, for such applications one is interested in finding a so-called stabilizing solution: the unique positive semidefinite solution X ∈ Cn×n such that the matrix A − G X is stable (i.e. all of its eigenvalues belong to C−, the left half of the complex plane). ( A∗, Q∗) is stabilizable), such a solution exists [11,19]. If the pair (A, G) is stabilizable (i.e. rank[A − λI, G] = n, for all λ in the closed right half plane), and the pair (A, Q) is detectable These conditions are fulfilled generically, and we assume they hold throughout the paper. In the case when n is small enough, one can compute the eigen- or Schur decomposition of the associated Hamiltonian matrix

Objectives

Methods

Conclusion