3.2.3 The Matrix Sign-Function Methods for the Riccati Equations
The matrix Sign-function method for the CARE is based on an iterative computation of the matrix Sign-function for the Hamiltonian matrix , where , associated with the CARE. If is an eigenvalue of the matrix , then the corresponding eigenvalue of is when and when .
Starting with , where , the sequence of symmetric matrices
, where ,
is constructed so that it converges to the quadratically. If , then the unique stabilizing solution to the CARE is obtained by solving in the least-squares sense. See Bierman (1984), Byers (1987), or Datta (2003) for details.
Convergence problems may occur due to explicit matrix inversions required by Eq. (3.7) when has some nearly imaginary eigenvalues. However, when convergence is successful, the matrix Sign-function method is potentially useful to solve the Riccati equations of large dimensions.
An analogous method for the DARE would require explicit computation of to form the symplectic matrix . Advanced Numerical Methods implements an alternative approach, suggested by Gardiner and Laub (1986), forming the matrix , where and , and then applying the matrix Sign-function algorithm for the CARE to this matrix.
Solve the Riccati equations via the matrix Sign-function method.
This is a continuous-time state-space model of the vertical-plane dynamics of a flight control system used in Section 3.2.1.
This defines the weighting matrices and .
This finds the solution to the CARE for the flight control system using the matrix Sign-function method.
Here is the relative residual norm of the solution.