An evolutionary approach to the automatic generation of mathematical models

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Abstract

This paper deals with the development and analysis of an efficient, evolutionary, intelligent, data-based, mathematical modelling method for complex non-linear systems. A new hybrid evolutionary method using genetic programming (GP) and evolutionary programming approaches is proposed. The potentials of the hybrid method for modelling complex, dynamic systems including single and two-link terrestrial manipulator systems are investigated and simulation results are presented.

Introduction

The issue of intelligent, data-based modelling, in the context of evolutionary computation (EC), will be the primary focus of this paper. The potential of EC methods for data-based modelling will be investigated and a new EC-based hybrid method will be developed to address the vital issue of data-based modelling. It will be shown that, in general, genetic programming (GP)-based evolutionary methods for data-based modelling provide a clear understanding of the internal structure of any system. While concentrating on the development of accurate systems models, the demerits of approximate system models will be analysed. Further, it will be argued that the results of this paper can be true in general for any data-based modelling techniques. In addition, it will be shown that the automatic generation of mathematical models from the knowledge of the only input–output dataset with the use of evolutionary methods are not suitable for even two-link manipulator systems. Thus, its applicability will only be limited to very simple single-link manipulator systems.

Neural networks are widely used for data-based modelling that is subsequently used for the control of a system [1], [2], [3], [4]. However, the use of neural networks for the modelling of even simple robotic manipulators has exhibited very limited potential [1]. In addition, the major problem associated with neural modelling is that its non-transparent internal structure inhibits any theoretical understanding of the model obtained. Hence, the analysis of the global and local behaviour of the underlying model is very difficult with a neural modelling approach. This problem of the non-transparent characteristics of the popular neural networks can be addressed with a more transparent symbolic approach like GP. The GP and other EC methods are discussed briefly in Section 2.

Traditionally, dynamic system modelling problems are solved with three distinct steps. Initially, a suitable model structure for the given system is selected. Then, the parameters contained in the assumed model are optimised, and subsequently the dataset is validated. Hence, accurate initial structure selection for a given system is extremely important for proper system modelling. But, in theory, an infinite number of models can be built for a given set of data. This necessitates a judicious development of very effective algorithms that can quickly transform the initially selected model into the optimal model of the system. This problem can be stated as:

For a given input–output dataset, define a μt number of possible initial model structures. Then, find the most appropriate model structure amongst them by manipulating their respective numerical parameters to best fit the given dataset.

The GP-based approaches have the advantage of providing a clear view of the underlying structure of the problem, thus allowing in-depth analysis of the internal structure plasticity during and after learning. This provides a better understanding of the problem. GP-based methods have been used with many reported successes for modelling of moderately complex dynamical systems [5], [6], [7], [8]. The inherent structure of the tree-coded genetic programming methods can be used to represent mathematical expressions in modelling simple non-linear systems. Thus, tree-coded GP methods can be used for the automatic generation of mathematical models for manipulator systems. GP works for any problem by randomly selecting initial tree-structured computer programs that can represent initial models of the problem. Further, with the use of variation operators, these structures change to search for better structures. The power of the GP can further be enhanced by changing the numerical ephemeral values associated with each structure. If a particular structure that can exactly model the given dataset is present, this may perform badly due to unsuitable numerical values being used for that structure, and this may cause that structure to be eliminated altogether from the competition. This can be overcome by using a good optimisation technique to optimise the numerical values of each structure along with its own evolution [9]. The major concern here is the associated computational cost, as such a GP is computationally highly intensive. This has been addressed in this work by updating the numerical values of only the best individual structure to improve its fitness value.

Perhaps the first use of hybrid GP and genetic algorithms (GAs) for data modelling was reported by Howard and D’Angelo [10], who used it to finding a mathematical relationship between physical and biological parameters of a stream ecosystem. In an attempt to model robotic manipulators, Castillo and Melin [11] suggested a hybrid fuzzy–fractal–genetic method with comparatively impressive initial results particularly for single-link manipulators. They used a fuzzy–fractal method for modelling and a fuzzy–genetic method for simulation. Unfortunately, their work does not include sufficient results to make further comments on their proposed method. Cao et al. [9] used GA to optimise the parameters of the tree-structured GP individuals to preserve useful structures in evolving better differential equations to fit a given dataset. Later, they extended this concept to model higher order differential equations for dynamical systems [12]. However, the computational burden is extremely high in all these hybrid methods, which in essence prevents widespread use of this hybridisation philosophy. In order to reduce the computational cost, a new method called Cauchy-guided evolutionary programming (CGEP) method has been developed. To ascertain the potential of the CGEP method, it has been tested on some important benchmark functions. The CGEP method is described in Section 3.

Further improvements in modelling performance are likely to be achieved by a hybrid approach including both GP and CGEP methods. The sole aim of hybridisation is to develop better algorithms for the automatic generation of mathematical models by preserving possibly the best structures in a population pool. As a first step towards the hybridisation of GP and CGEP, the parameters of the GP individuals are fed to the CGEP for further optimisation. However, this increases the execution time tremendously, and it is therefore not suitable for all applications, particularly real time applications. Hence, in the proposed hybrid GP and EP, which is named here as the hybrid genetic evolutionary programming (HGEP) method, the best individual of any GP generation is further optimised by CGEP to better exploit the underlying structure of the best individual tree. However, in order to further minimise the execution time, the CGEP is used only for a few iterations. The HGEP method is described in detail in Section 4.

Section 5 establishes the efficacy of the proposed hybrid evolutionary algorithm for modelling standard symbolic regression problems. In Section 6, a standard model of a simple robotic system is described. Then, the simple robotic manipulator test problems and the details of the experimental set-up are described. The potential results of the experiments on modelling robotic manipulators are illustrated in Section 7. Section 8 discusses the results thoroughly. Finally, in Section 9, conclusions of this paper are presented.

Section snippets

Evolutionary computation methods

In general, evolutionary computation methods are a very rich class of multi-agent stochastic search (MASS) algorithms based on the neo-Darwinian paradigm of natural evolution, which can perform exhaustive searches in complex solution space. These techniques start with searching a population of feasible solutions generated stochastically. Then, stochastic variations are incorporated into the parameters of the population in order to evolve the solution to a global optimum. Thus, these methods

Concepts

Basic evolutionary programming generates one offspring from each parent in a population pool by adding a Gaussian random variable of mean zero and variance proportional to the fitness score of the parent. Then, a stochastic competition selects effective parents for the next generation. In contrast, the CGEP replaces the normally distributed variations of BEP with a Cauchy distributed variation. However, a Cauchy distribution does not posses a finite expected value or standard deviation for

Description of HGEP algorithm

The hybrid genetic evolutionary programming is based on a steady-state GP. In each generation, two potent tree individuals from the population pool consisting of all the tree individuals Pt are selected independently by means of tournament selections with a tournament size tc. In a tournament of size tc, all the individuals compete with each other and finally the winner is selected amongst them. Then, the selected tree individuals generate offspring by using the simple subtree crossover

Test problems

After the development of the HGEP algorithm, its performance on standard problems must be tested before it can be applied to any complex non-linear system. For the verification of the potential of HGEP, two test problems of symbolic regression have been chosen. The first test problem is a single input regression problem, whereas the second one is a two-input regression problem. The second problem was selected to study the effects of multiple inputs on the performance of different GP-based

Test problem 1: one-input symbolic regression

All the results of HGEP have been compared with a simple GP method. The average best and average mean results of 10 independent runs of HGEP and GP are shown in Table 6. The corresponding t-test results are also included in Table 6. The t-test results indicate the statistical significance of the results obtained from the GP and HGEP methods in 10 different trials. When the t-values are negative it suggests that the first of the two methods under test is better than the second one. The

Experimental studies on robot modelling

The dynamic model of a n-link terrestrial manipulator system with all the masses assumed to be point masses at the distal end of each link and gravity terms, can be expressed as follows [24]:T=M(q)q̈+C(q,q̇)+G(q)where T∈3n is the driving torque vector, q, q̇ and q̈∈3n are the joint position, velocity and acceleration vectors, respectively, M(q)∈3n×n is the mass matrix of the manipulator, C(q,q̇)∈3n the centrifugal and coriolis terms and G(q)∈3n the gravity vector. Here, the gravity terms are

Results

It has already been discussed that HGEP outperforms simple GP on both symbolic regression test problems. Hence in this section, both HGEP and GP are considered for the one-link manipulator, whereas only HGEP has been used to extract the model for the complex two-link manipulator system. For all the tests, the best result of the 10 independent runs of HGEP and GP has been chosen as the final model of the system.

Discussion

The results presented in Section 6 can imply that it is necessary for an accurate and precise model generated automatically from a given dataset to have a fitness score that is the same as the original system or the actual model, but it is not a sufficient condition to obtain an exact or a very close approximation of the exact model. This can, otherwise, be stated as that a given model derived from a desired dataset having fitness same as the actual model may not always replicate the actual

Conclusions

This paper has described a hybrid GP and EP (HGEP) method for modelling the task. GP has been used to find an optimal model structure and EP evolves the ephemeral constants contained within a particular GP model structure to make the underlying GP model structure more robust. To speed up the overall process of the model evolution, suitable modifications to the basic EP technique have been performed that use Cauchy distribution instead of the usual normal distribution of BEP. Thus, this yield a

Anjan Kumar Swain, received his bachelors of science degree in electrical engineering in 1988, masters of science in engineering in 1991 from Regional Engineering College, Rourkela, India, and PhD degree from the University of Sheffield, UK in 2001. He worked with Electrical Engineering Department of Regional Engineering College Rourkela, India, from 1988 to 1989 and 1991 to 1992. Subsequently, he worked with Ramco Electronics Division, Madras, India from 1992 to 1993 as a real-time process

References (26)

  • A. Eskandarian et al.

    Dynamic modelling of robotic manipulators using artificial neural network

    J. Robotic Syst.

    (1994)
  • K. Hunt, G. Irwin, K. Warwick, Neural Network Engineering in Dynamic Control Systems, Spinger-Verlag, London,...
  • O. Omidvar, D.L. Elliott, Neural Systems for Control, Academic Press, New York,...
  • D. Psaltis, A. Sideris, A.A. Yamamura, A multilayered neural network controller, IEEE Control Systems Magazine, April...
  • P. Marenbach, K.D. Bettenhausen, S. Freyer, U. Nieken, H. Rettenmaier, Data driven structured modelling of a...
  • P. Marenbach, M. Brown, Evolutionary versus inductive construction of neuro-fuzzy systems for bioprocess modelling, in:...
  • H. Hiden, M. Willis, B. McKay, G. Montague, Non-linear and direction dependent dynamic modelling using genetic...
  • B. McKay, C. Sanderson, M.J. Willis, J. Barford, G. Barton, Evolving a hybrid model of a batch fermentation process,...
  • H. Cao, L. Kang, Z. Michalewicz, Y. Chen, A two-level evolutionary algorithm for modelling system of ordinary...
  • L.M. Howard, J. D’Angelo, The GA-P: a genetic algorithm and genetic programming hybrid, IEEE Expert, June 1995, pp....
  • O. Castillo, P. Melin, A new-fuzzy–fractal–genetic method for automated mathematical modelling and simulation of...
  • H. Cao, L. Kang, Y. Chen, Evolutionary modelling of ordinary differential equations for dynamic systems, in:...
  • D.E. Goldberg, Genetic Algorithms in Search, Optimisation and Machine Learning, Addison-Wesley, Reading,...
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    Anjan Kumar Swain, received his bachelors of science degree in electrical engineering in 1988, masters of science in engineering in 1991 from Regional Engineering College, Rourkela, India, and PhD degree from the University of Sheffield, UK in 2001. He worked with Electrical Engineering Department of Regional Engineering College Rourkela, India, from 1988 to 1989 and 1991 to 1992. Subsequently, he worked with Ramco Electronics Division, Madras, India from 1992 to 1993 as a real-time process control software engineer. After that he joined as a lecturer in the Electrical Engineering Department of Indira Gandhi Institute of Technology, Orissa, India. He has over 50 publications in journals and conferences. He is the recipient of the national best young teacher award for the year 1996 in India. His current research interests include evolutionary computing methods, evolving networks, dynamics and control of multi-arm robotic manipulator systems. He serves as a reviewer of journals and conferences including IEEE Transactions on System, Man and Cybernetics.

    Alan S. Morris was educated in UK and graduated with a BEng degree in electrical and electronic engineering from Sheffield University in 1969. After 5 years of employment with British Steel as a research and development engineer in control system applications, he returned to Sheffield University to carry out research in electric arc furnace control, for which he was awarded the degree of PhD in 1978. Since that time, he has been employed as a lecturer, and more recently senior lecturer, at Sheffield, where he now holds the position of Director of Undergraduate Studies. His main research interests lie in robotics, and he is now the author of over 100 refereed research papers. Professionally, he is a Fellow of the Institute of Measurement and Control and a Member of the Institution of Electrical Engineers, and participates as a member of technical panels and organiser of conferences.

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