Nonlinear Modelling from Data

Non-parametric Modelling/Identification Methods

Modern non-parametric statistical methods (e.g. methods based on Gaussian Process Priors) offer many advantages including:

  1. The model is directly based on the data. Contrast this with parametric modelling approaches where a model structure with a small number of parameters is postulated. The parameter values are selected to achieve a good fit to the measured data and in this way the information contained in the measured data is distilled to a small number of parameters. Unfortunately, the reverse procedure is often ill-conditioned with biases and errors in the estimated parameters often leading to unnecessarily poor predictions.
  2. Direct adaptation. Additional data can be directly incorporated into an existing model at little cost.
  3. A wide range of prior knowledge can be accommodated. By suitable choice of interpolation strategy, prior knowledge ranging from almost none to almost complete (e.g. a full prior model of the system) can be supported.
  4. Direct regularisation. Measured data is generally relatively sparse at operating points far from equilibrium. Proper interpolation (based on smoothing or so-called regularisation) can greatly improve the generalisation ability of the model in such operating regions and avoids the numerical ill-conditioning in conventional parametric model associated with the need to estimate parameters from such sparse data.
  5. While non-parametric models offer a number of significant advantages, they are essentially "black box" in nature. Although black box representations are useful for many purposes, their utility for analysis and design is limited. Fortunately, given a non-parametric Bayesian model, the corresponding velocity-based linearisation family can be derived immediately. Velocity-based representations are well-suited to linearisation-based analysis and design and complement non-parametric representations in many ways. The use of a dual non-parametric/velocity-based representation to exploit the considerable synergy which exists between these representations is therefore quite natural and attractive.

By using a dual non-parametric/velocity-based representation, it is in principle possible to directly incorporate prior knowledge of the velocity based linearisation family including the likely linearisations at certain operating points (e.g. equilibrium), the likely scheduling variable, any known decomposition into operating regions and any knowledge of the likely smoothness of system in different operating regions. Conversely, when such information is unknown or requires to be validated, it can be inferred from an identified model. It should be noted that structural information such as knowledge of the scheduling variable is extremely valuable in many contexts, including control design.

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Divide & Conquer Identification

The identification of linear time-invariant (LTI) systems from measured experimental data has received considerable attention over the last thirty years and there exists a wealth of theoretical results relating to issues such as structure identification, parameter estimation, experiment design and model validation testing together with a great deal of accumulated practical experience. However, all systems are in reality nonlinear and identification techniques are less well developed for systems which cannot be accurately approximated by a single LTI system. It is, therefore, often attractive to consider a divide and conquer strategy whereby the analysis/design of a nonlinear system is decomposed into the analysis/design of a collection of LTI systems. Note that the attraction of such an approach extends beyond theoretical considerations: in many situations it is impractical and/or unsafe to collect global test data - a series of experiments providing information on the local behaviour are often much more preferable andnaturally leads to consideration of a divide and conquer approach to identification.  In the context of system identification it is common practice, when faced with the task of modelling a nonlinear system, to initially identify a number of LTI approximations to the system each of which is locally valid. However, in the system identification literature there is notable lack of research relating to the natural next step: namely, attempting to properly reconstruct a nonlinear system from an appropriate family of identified linear systems.  Velocity-based methods appear to have the potential to address precisely this task and thus open up a new paradigm for system identification.

Modular Modelling of Interacting Systems

The common practice, at least in the first instance, is to use linearisation-based approaches to analyse a nonlinear model.   This is reflected, for example, in the ubiquity of tools for trimming and linearising nonlinear simulation models.   However, conventional linearisation-based representations do not readily support modular analysis and design.  Conventional linearisations require knowledge of the equilibrium points of a system which, in a tightly integrated system, are a global property i.e. the equilibria of a sub-system depend on the characteristics of the rest of the system.  Clearly this is at odds with modular analysis and design methodologies which require each sub-system to have a well-defined interface to the rest of the system which is insensitive to the implementation details of the system. (Such modular approaches enable the detailed design and implementation of each sub-system to be carried out separately and are particularly important in projects where sub-contractors are involved). 

Velocity-based linearisation methods can provide a framework which genuinely supports modular analysis and design methods.  Features include: no dependence on detailed equilibrium information, tools which are valid globally (rather than only close to equilibrium operation), analysis and design results obtained with a specific sub-system can be integrated in a direct and transparent manner with those obtained for other sub-systems.  Other important practical advantages include: trimming of simulation models (to determine equilibrium points) and numerical differentiation (linearisation is achieved by 'freezing' rather than differentiating) are not needed.  Both trimming and numerical differentiation are highly non-trivial for the complex large-scale simulation models frequently encountered in industry.