Van Volterra-kernels naar toestandsruimte: op weg naar gestructureerde niet-lineaire blokgeoriënteerde modellering

Mathematical models are essential for understanding, analysing, and predicting the behaviour of real-world systems across a wide range of domains. System identification provides a structured framework for building these models from observed data, enabling accurate predictions and informed decision-making. However, identifying models of complex systems is challenging, particularly in the presence of nonlinearity, noise, or incomplete measurements. Addressing these challenges requires a modelling approach, tailored to the system under study, one that effectively integrates domain knowledge, mathematical insight, and computational tools. While linear approximation techniques are well-established and computationally efficient, they often fail to capture the nonlinear behaviour present in many real-world systems, leading to poor generalisation and loss of interpretability. This work proposes the development of a simple, interpretable nonlinear dynamical model that reduces reliance on expert user input. We present a novel modelling framework for identifying the nonlinear state-space form of block-oriented systems, specifically Hammerstein, Wiener, and Wiener-Hammerstein structures, by exploiting structural insights rather than optimising linear approximations. Major challenges in this context include selecting the appropriate block configuration, determining system order, and addressing time-consuming identifiability issues, particularly in the presence of complex nonlinearities and high output noise. To address these challenges, the Volterra series is used as an intermediate modelling step. Due to its linear-in-the-parameters formulation, the Volterra series provides a tractable yet expressive tool for modelling nonlinear systems. When prior knowledge of the underlying block-oriented structure is available, it can be directly translated into a constrained Volterra kernel structure, resulting in a prior-informed Volterra kernel that significantly reduces model complexity. However, a key limitation of the Volterra framework is the 'curse of dimensionality', which limits memory length and model order, especially for large datasets in Wiener and Wiener-Hammerstein structures. To mitigate this, the estimated Volterra kernels are compressed via low-rank approximations, enabling efficient identification of the block-oriented model. This is followed by regularisation in the reduced parameter space to promote smoothness and decay in the linear dynamics, further improving model interpretability and robustness. From this regularised and optimised block-oriented model, parameters for a Polynomial Nonlinear State-Space (PNLSS) model can be extracted, without requiring an iterative procedure, using standard subspace identification techniques. The proposed approach is validated through a series of simulations, experiments with measurement datasets, and benchmark datasets from the system identification community. Additionally, the algorithm is tested on several nonlinear systems from the process industry that do not typically conform to a block-oriented model structure. Across all test cases, the block-oriented model identified from the optimised and regularised prior-informed Volterra kernel consistently provides a good approximation of the true system, both in terms of output prediction and parameter accuracy. While prediction accuracy decreases with higher noise levels and increased nonlinear complexity, the overall quality of the block-oriented model remains robust. However, the PNLSS model extracted from the block-oriented model yields mixed results. In some cases, it outperforms the baseline PNLSS models obtained from the linear approximation techniques; in other cases, it performs similarly or slightly worse. These variations suggest that while the proposed Volterra-based approach is highly effective up to the block-oriented model stage, the subsequent realisation of a PNLSS model from this block representation may not always lead to improved output approximation. However, the method offers a systematic, interpretable, and data-efficient alternative to traditional identification workflows, particularly valuable when structural insights into the system are available.