Developments in LPV state-space model identification

Current state-of-the-art linear parameter-varying (LPV) control synthesis methods presume that an LPV state-space (SS) model of the system with affine dependence on the scheduling variable is available. However, in practice, often available models of the target application based on first principle laws are either not reliable to express the dynamic behaviour of the real system or are too complex for direct synthesis. Hence it is attractive to turn to data-driven modelling methods that are capable to capture the underlying system dynamics in terms of a low complexity LPV state-space model. Unfortunately, many existing LPV-SS identification schemes either suffer heavily from computational issues related to the curse of dimensionality or rely on severe approximations. To overcome these issues, some recent developments of LPV-SS realisation theory and Bayesian identification are presented which made possible a computationally attractive three-step approach for identifying LPV-SS models. In Step 1, the sub-Markov parameters representing the impulse response of the system are estimated in a Bayesian setting, using kernel based Ridge regression with hyper-parameter tuning via marginal likelihood optimisation. Subsequently, in Step 2, an LPV-SS realisation is obtained by using an efficient basis reduced Ho-Kalman like deterministic SS realisation scheme on the identified impulse response. Finally, in Step 3, to reach the maximum likelihood (ML) estimate, the LPV-SS model is refined by applying a Bayesian expectation-maximization (EM) method or a gradient search based prediction error minimisation (PEM). The performance of the proposed 3-step scheme is demonstrated via a Monte-Carlo simulation study