Minimal State-Space Realisation of a Transfer Function Matrix

Given the matrices of an LTI system, the corresponding transfer function matrix is unique and can be directly computed. The converse problem, i.e. finding a state-space representation for a given transfer function matrix is less trivial and the result is not unique. For a proper SISO transfer function, a state-space representation in controller canonical form (CCF) can be directly obtained from the coefficients of the transfer function. This direct conversion to CCF can be extended to single-input-multiple-output (SIMO) systems. In the general case of an MIMO system, a state-space realisation for a given transfer function matrix can be be obtained by expressing the MIMO system output as a sum of SIMO system outputs. Constructed state-space realisations of a transfer function matrix are, in general, not of minimal dimension. That is, the number of eigenvalues is higher than the number of the transfer function poles; or in other words, the state-space representation is not both observable and controllable. One option is to transform the state-space model into Kalman decomposition form to find the minimal number of states that are controllable as well as observable. Another option is to apply Gilbert’s minimal realisation method in the case that each entry of the transfer function matrix has distinct poles.