Nonlinear system identificationhas proven to be effective in obtaining accurate models from data for complex real-world systems. In particular, recent encoder-based methods with artificial neural network state-space (ANN-SS) models have achieved state-of-the-art performance on various benchmarks, using computationally efficient methods and offering consistent model estimation in the presence of noisy data. However, inclusion of prior knowledge of the system can be further exploited to increase (i) estimation speed, (ii) accuracy, and (iii) interpretability of the resulting models. This paper proposes a model augmentation method that incorporates prior knowledge from first-principles (FP) models in a flexible manner. We introduce a novel linear-fractional-representation (LFR) model structure that allows for the general representation of various augmentation structures including the ones that are commonly used in the literature, and an encoder-based identification algorithm for estimating the proposed structures together with appropriate initialisation methods. The performance and generalisation capabilities of the proposed method are demonstrated on the identification of a hardening mass-spring-damper system in a simulation study and on the data-driven modelling of the dynamics of an F1Tenth electric car using measured data.
As control systems are becoming more complex and performance requirements surge, the need for accurate nonlinear models capable of efficiently capturing complicated behaviours of physical systems is rapidly increasing. It is common practice to derive baseline models using first-principle (FP) methods, e.g., rigid body dynamics [30]; however, these models provide only an approximate system description. Although more accurate FP models can be developed, this is a labour-intensive process, especially when additional physical effects-such as friction or aerodynamic forces-are included. Modelling these phenomena from first principles often requires dedicated experimental campaigns to identify and estimate the associated unknown parameters. Furthermore, the resulting models may become too complex to be handled analytically. In some cases, reliable FP descriptions of the to-be-modelled effects may not even exist, resulting in approximations with varying levels of fidelity.
To overcome these issues, nonlinear system identification (NL-SI) methods offer an alternative option to estimate models directly from measurement data [30]. Black-box models, particularly those that incorporate artificial neural networks (ANNs), have achieved unprecedented accuracy in capturing complex behaviours. In control applications, ANN-based state-space (SS) models have proven to be effective in handling high-order systems and capturing complex nonlinear dynamics [5].
Although black-box methods may result in accurate models, they also have serious downsides. First, flexible function approximators are difficult to interpret, leading to model behaviour that is not well understood. This in turn limits the reliability of the model during, e.g., extrapolation beyond the training data. This is a significant drawback for control applications, where interpretable models are preferred in the design process [10,25]. The second drawback is the significant time spent learning expected behaviour that has already been modelled thoroughly, e.g., FP-based understanding of the rigid-body-dynamics of the system. Physics-informed neural networks [28] and physics-guided neural networks [11] embed the prior knowledge of the physics in the form of equations (algebraic of partial differential) in the cost function, enforcing the learnt functions to fit to known physics behaviour. This leads to more interpretable models as the known physics are enforced, and shows faster learning convergence. These methods, however, require the knowledge of such physics equations and still rely on a black-box model to capture the entire system.
A promising approach is model augmentation, e.g., [16,17,33,35]. This method combines baseline models with flexible function approximators, such as ANNs, in a combined model structure. As a result of this structural combination, the prior knowledge is directly captured in the baseline model and the learning components only need to model unknown dynamics. For control engineering, such a structure is beneficial, as it is clear how a well-understood baseline model is combined with black-box elements.
In the literature, there are a variety of different model augmentation structures, such as parallel [35] and series [16,17,33] interconnections. These interconnections reflect in what form the known baseline model is combined with the learning component that models the unknown behaviour of the system. Although different interconnections may result in an equally accurate model, model complexity and convergence speed also need to be considered, especially when the final model is utilised for control purposes. One interconnection may be equally accurate while having a less complex parameterisation compared to others. It is not trivial to determine which interconnection is the most advantageous for a specific baseline model and data-generating system, as the choice of the optimal interconnection depends on the unknown dynamics of the system. To address this, further research into model augmentation methods is required to develop, e.g., automatic model selection methods. To facilitate this research, a general model augmentation structure is required, such that model augmentation methods can be developed efficiently and compared across different works in literature. Such a general model augmentation structure is currently lacking in the literature.
To solve this problem, a general model augmentation structure based on a Linear Fractional Representation (LFR) is proposed, which has been chosen for its modular and flexible nature, enabling a generalised form for augmenting the FP or the already known dynamics. The formulation of LFRs allows for systematic model augmentation while maintaining a clear separation between the baseline and learning components. The proposed model augmentation structure is able to express a wide range of model augmentation structures used in literature, and thus is a unified representation. Furthermore, LFRs are commonly
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