Genetic Programming for System Identification

Introduction

The design of mathematical models able to capture the dynamic behavior of industrial plants represents a very difficult task. As the assumptions imposed by working directly with the physical/chemical laws that govern the inner processes of the targeted systems are commonly not satisfied in the industrial environment, this approach cannot be employed for building analytical models, or at least the model structure, at an adequate level of approximation (Fleming & Purshouse, 2002). Therefore, the models are usually determined by means of system identification - a data driven methodology which constructs the mathematical description of the plant by using a finite number of plant variables’ measurements acquired during a finite period of time. The aim of system identification is to find a valid model able to capture the most relevant properties of the system, without comprising extraneous blocks which are not plant specific. Even if a limited number of observations are used during the design stage, the resulted model must have good generalization capabilities, meaning it has to describe the behavior of the system in a range of operating conditions as large as possible. Additionally, to decrease the risk of inappropriate further model utilization, a reduced sensitiveness of the model output relative to small variations of the model parameters is recommended, together with increased interpretability. This becomes a very challenging problem, especially if the system features complex nonlinearities, whilst scarce information about its dynamics is a priori accessible, and/or large sets of highly dimensional, noisy data are available (Poli et al., 2008).

Note that if the model includes alien parts, it will most likely offer an unsatisfactory approximation on data sets different than those used for model configuration. In this context, in compliance with Occam’s razor principle, the identification procedure has to encourage the selection of the simplest model which can ensure the level of accuracy required by the application. Obviously, no model could be a perfect copy of the real system. The mathematical description involves simplifications which are assumed as acceptable only if the requested model accuracy is satisfied. It is also worth mentioning that the model accuracy cannot exceed the accuracy of data acquisition.

Assuming that a representative data set of plant variables measurements has been experimentally collected during plant operation - in such a way as to reflect as much of the system dynamics as possible, the configuration of the model comprises the selection of relevant inputs, the selection of the model structure and the estimation of the model parameters. These stages can be carried out separately or solved concomitantly, however note that the last alternative provides higher flexibility at the cost of increasing problem complexity. Lastly, the designed model has to be validated on different data sets.

Classical identification approaches deal with only one potential model at a time, restarting the search if the candidate’s features are not satisfactory. By doing so, the identification process tends to become time consuming, and more importantly, there is no guarantee that a certain structure selection will turn out to be better, in terms of end model quality, than the previous ones. Moreover, to ensure faster variables reduction and structure selection, most identification approaches investigate only a reduced number of combinations, by using predefined schemes which do not guarantee the achievement of a convenient solution for any particular application. Additionally, the models parameters are commonly computed via a deterministic optimization procedure that strives to minimize the squared error function. Whilst industrial systems feature complex nonlinearities and work in noisy environments, the estimation of parameters usually implies dealing with nonlinear, discontinuous objective functions, as well as large and non-convex search spaces, therefore the risk of failing into a local optimum point or of generating inconvenient solutions results very high (Fleming & Purshouse, 2002).

Genetic programming (GP) offers solutions to many of the drawbacks inherent to classic identification approaches. Basically, it consists in an evolutionary algorithm designed to evolve graph-based individuals. Therefore, the GP-based identification involves the evolution of a population of potential mathematical descriptions of the dynamic system (called individuals), carried out for numerous iterations (generations) (Nedjah et al., 2006; Poli et al., 2008).