3.2.2 The Inverse-Free Generalized Eigenvector and Schur Methods for the Riccati Equations
The Schur, Newton, and matrix Sign-function methods (described in the following sections) for the CARE and DARE, all require the computation of . Furthermore, the Schur and Newton methods for the DARE require explicit computation of , which may lead to inaccurate solutions when these matrices are singular or nearly singular. The problems can be overcome by using the so-called inverse-free generalized eigenvector and Schur methods, developed by Pappas, Laub, and Sandell (1980). The inverse-free methods replace the Hamiltonian eigenproblem associated with the CARE by the generalized eigenvalue problem given by the following pencil with order
The pencil is further compressed into a pencil using the QR factorization of , without affecting the deflating subspaces (Van Dooren (1981)). The CARE is then solved by construction of either a basis of the generalized eigenvectors associated with the stable eigenvalues (the inverse-free generalized eigenvector method) or an orthonormal basis of the stable deflating subspace (the inverse-free generalized Schur method) of the compressed pencil.
For the DARE , the pencil considered is
This pencil is compressed into a pencil using the QR decomposition of . For details, see Datta (2003).
Inverse-free methods for the Riccati equations.
When the associated matrix pencil has multiple or nearly multiple eigenvalues, the inverse-free generalized Schur methods may perform better than the inverse-free generalized eigenvector methods. In fact, failure of the generalized eigenvector method sometimes occurs because the pencil does not have enough eigenvectors to form a basis (illustrated in the following example).
The generalized eigenvector methods use the function GeneralizedEigensystem and the generalized Schur methods use the function GeneralizedSchurDecomposition, both in Chapter 9, to solve the appropriate generalized eigenvalue problems (see Datta (2003) for details).
Consider solving the DARE with matrices and of the following state-space system of a process control of a paper machine.
The weighting matrices are chosen to be identity matrices of the appropriate dimensions.
This solves the DARE by the inverse-free generalized Schur method.
This auxiliary function computes the residual norm of the solution.