Periodic Sylvester equation solvers

pdsylvc(A, B, C; rev = false, isgn = 1, fast = true) -> X

Solve the periodic discrete-time Sylvester equation of continuous-time flavour

A*X + isgn*σX*B = C  for rev = false

or

isgn*A*σX + X*B = C  for rev = true,

where σ is the forward shift operator σX(i) = X(i+1) and abs(isgn) = 1.

The periodic matrices A, B and C must have the same type and commensurate periods. The resulting periodic solution X has the period set to the least common commensurate period of A, B and C and the number of subperiods is adjusted accordingly.

The periodic discrete analog of the Bartels-Stewart method based on the periodic Schur form of the periodic matrices A and B is employed (see Appendix II of [1]). If fast = true, the QR factorization of bordered-almost-block-diagonal (BABD) matrix algorithm of [2] is employed to solve periodic Sylvester equations up to order 2. This option is more appropriate for large periods. If fast = false, the QR factorization of the cyclic Kronecker form for the periodic Sylvester operator is used to to solve periodic Sylvester equations up to order 2.

For the existence of a solution A and B must not have characteristic multipliers α and β such that α +isgn*β = 0.

Reference:

[1] A. Varga. Robust and minimum norm pole assignment with periodic state feedback. IEEE Trans. on Automatic Control, vol. 45, pp. 1017-1022, 2000.

[2] R. Granat, B. Kågström, and D. Kressner, Computing periodic deflating subspaces associated with a specified set of eigenvalues. BIT Numerical Mathematics vol. 47, pp. 763–791, 2007.

pfdsylvc(A, B, C; isgn = 1, fast = true) -> X

Solve the forward-time periodic discrete-time Sylvester equation of continuous-time flavour

A*X + isgn*σX*B = C ,

where σ is the forward shift operator σX(i) = X(i+1) and abs(isgn) = 1.

The periodic matrices A, B and C must have the same type and commensurate periods. The resulting periodic solution X has the period set to the least common commensurate period of A, B and C and the number of subperiods is adjusted accordingly.

prdsylvc(A, B, C; isgn = 1, fast = true) -> X

Solve the reverse-time periodic discrete-time Sylvester equation of continuous-time flavour

isgn*A*σX + X*B = C ,

where σ is the forward shift operator σX(i) = X(i+1) and abs(isgn) = 1.

The periodic matrices A, B and C must have the same type and commensurate periods. The resulting periodic solution X has the period set to the least common commensurate period of A, B and C and the number of subperiods is adjusted accordingly.