MIMO Uncertainty Model Identification of Time-Delay Systems

A procedure for identifying a linear MIMO model with LFT-type uncertainty is presented. The uncertainty is minimized subject to the requirement that known input-output data do not invalidate the model. The optimization problem is solved stepwise. First, the uncertainty is minimized frequency-by-frequency with respect to the sampled frequency response of a nominal model subject to data-matching constraints, which can be expressed as matrix inequalities. Depending on the uncertainty structure, the matrix inequalities may be linear or bilinear with respect to frequency samples of the nominal model. We show that the bilinear matrix inequalities (BMIs) can be transformed into linear matrix inequalities (LMIs) for a certain type of uncertainty model. The resulting optimization problem is then convex. Next, a state-space model is fitted to the frequency responses subject to the same data-matching constraints. Prior to this fitting, possible time delays in the data are removed by a special technique. In this way, the order or the state-space model can be minimized. Naturally, the time delays are re-inserted in the transfer function calculated from the state-space model. An application to distillation modelling is included.