3.2.4 The Newton Methods for the Riccati Equations
The Newton method is an iterative method. Starting from a suitable initial guess for the stabilizing solution , a sequence of approximants , which converges to quadratically from onward, is constructed. The matrix should be chosen such that is a stable matrix for the CARE or is a discrete-stable matrix for the DARE (see Datta (2003) for details).
The Newton method is typically used as an iterative refinement technique to improve an approximate solution obtained by a direct method (say, the Schur method or the matrix Sign-function method).
The approximant is constructed as , where the correction matrix is obtained by solving a certain Lyapunov equation.
For the CARE, the matrix is obtained by solving the Lyapunov equation:
For the DARE, the matrix is obtained by solving the discrete Lyapunov equation:
The initial guess is supplied by the user as the value of the option InitialGuess, as shown in the following example. With the option value InitialGuessAutomatic, the initial guess is generated from the Schur method. This is the default option value.
The choice for the initial guess is crucial. If is not close enough to the stabilizing solution , the convergence can be very slow. In this case, even if the convergence does occur, the stabilizing property of the obtained solution cannot be guaranteed.
Since a Lyapunov equation must be solved at each iteration step of the Newton method and there are several options for the function LyapunovSolve, Advanced Numerical Methods allows a list containing these options to be passed to the Lyapunov equation solver through the option LyapunovSolveOptions. By default, the automatic Lyapunov solver is used.
Solve Riccati equations via the Newton method.
This computes the solution of the CARE with the data from the flight control system example using the Schur method.
Here is the relative residual norm of the solution obtained by the Schur method.
The Newton method now is applied with the initial guess .
Here is the relative residual norm computed by the Newton method.