3.2.1 The Schur Methods for the Riccati Equations
The Schur method for the CARE, based on the ordered Schur decomposition of the Hamiltonian matrix , where , computes the stabilizing solution by finding the invariant subspace associated with the stable eigenvalues of . If is such an invariant subspace, then is the stabilizing solution of the CARE. Similarly, the Schur method for the DARE, based on the ordered Schur decomposition of the symplectic matrix , where is , computes the stabilizing solution by finding the invariant subspace of associated with the eigenvalues of moduli less than 1. The Schur method for the CARE and DARE was developed by Laub (1979).
Solve the Riccati equations via Schur decomposition.
Make sure the application is loaded.
In[1]:=
Load the collection of test examples.
In[2]:=
This is a continuoustime statespace model of the verticalplane dynamics of a flight control system.
In[3]:=
Out[3]=
This defines the weighting matrices and .
In[4]:=
This finds the solution to the CARE using the Schur method.
In[5]:=
Out[5]=
This auxiliary function computes the norm of the relative residual of the solution to the CARE.
In[6]:=
Here is the relative residual obtained by the Schur method.
In[7]:=
Out[7]=
This verifies that the solution is positive definite.
In[8]:=
Out[8]=
This verifies that the solution is the stabilizing solution.
In[9]:=
Out[9]=
