blkrsch, cschur (Robust Control Toolbox)

Block ordered real Schur form.

Ordered complex Schur form via complex Givens rotation.

[v,t,m] = blkrsch(a,Type,cut)
[v,t,m,swap] = cschur(a,Type)

Description

Cschur computes a unitary similarity transformation V and a complex upper triangular matrix T for a real or complex matrix A such that

where T has the eigenvalues

_i(A) ordered on the diagonal according to the value of the variable Type:

1: Type = 1 -- .

2: Type = 2 -- .

3: Type = 3 -- eigenvalue real parts in descending order.

4: Type = 4 -- eigenvalue real parts in ascending order.

5: Type = 5 -- modulus of eigenvalues in descending order.

6: Type = 6 -- modulus of eigenvalues in ascending order.

Variable swap records the number of Givens rotations swaps it takes and variable m returns the number of "stable" eigenvalues of A (see rsf2csf or cgivens).

Blkrsch computes a block ordered real Schur form such that the resulting T matrix has four blocks

The input variable cut is the dimension of the square block B₁. If Type is 1, cut is automatically set to m -- the number of eigenvalues of A with negative real parts.

Six options are available:

1: Type = 1 -- .

2: Type = 2 -- .

3: Type = 3 --

.

4: Type = 4 --

.

5: Type = 5 --

.

6: Type = 6 --.

Algorithm

blkrsch and cschur, are M-files in the Robust Control Toolbox. cschur uses schur, rsf2csf and the complex Givens rotation [1] to iteratively swap two adjacent diagonal terms according the style you select. blkrsch projects the resulting complex subspace onto the real.

Limitations

For blkrsch, the matrix A must have zero imaginary part.