Parametrization of lossless matrices
Rational lossless matrices play an important role in system theory because of Douglas-Shapiro-Shields factorization. They also are of independent interest, being the transfer functions of conservative systems: the scattering matrix of a frequency filter, the polyphase matrix of an orthogonal filter bank are lossless. Overall, they lies at the heart of Schur analysis, a rich theory studying interpolation problems for functions satisfying a metric constraint, namely Schur functions.
In a series of papers in collaboration with Bernard Hanzon and Ralf Peeters, we aimed at connecting two types of representations
In discrete-time, realization matrices are obtained as a product of unitary matrices that can be parametrized by a sequence of vectorial Schur parameters. In continuous-time the same results holds true but with a new definition for the product. Moreover, for a canonical choice of interpolation values and interpolation directions in the Schur algorithm, the realization matrix possesses a sub-diagonal pivot structure.
- Atlases of charts derived from interpolation problems. The parameters are thus interpolation values. This is a very efficient way to take into account the Schur metric constraint and subsequently the stability requirement.
- State-space representations and balanced canonical forms. State-space representations involve basically linear algebra techniques and their efficiency for numerical computations has been demonstrated.