Nonlinear rational model identification and control Professor Quan M. Zhu Bristol Institute of Technology University of the West of England Frenchay Campus Coldharbour Lane, Bristol BS16 1QY, UK

Contents 1) Background knowledge 2) Rational models and representations 3) Structure detection and parameter estimation 4) Correlation based validation 5) Controller design 6) Conclusions

Model identification  Input/output data from instrument measurements and expert perceptions  Parametric model structure  Parameter estimation  Validity tests model training/ identification plant

Model validation  Examine residuals  Correlation tests  A valid model’s residuals should be reduced to uncorrelated sequence with zero mean and finite variance model examination/ diagnosis plant _ residual

A general modelling and control structure plant model training control input output residual Target

Rational models (1) --- Expression

Rational models (2) --- Example

Rational models (3) --- Characteristics 1) The model can be much more concise than a polynomial expansion, for example 2) The model can produce large deviations in the output, for example

Rational models (4) --- Errors

Rational models (5.1) --- Representations 1) The polynomial NARMAX models is a special case of RM by setting denominator polynomial b(t) = 1. 2) The model is non-linear in both the parameters and the regression terms, this is induced by the denominator polynomial. 3) Modelling of chemical kinetics, bio dynamics, brain image.

Rational models (5.2) --- Representations 4) Fuzzy systems with centre defuzzifier, product inference rule, singleton fuzzifier, and Gaussian membership function. 5) The normalised radial basis function network is also a type of rational model. When the centres and widths need to be estimated this becomes a rational model parameter estimation problem. 6) Difference in time domain and frequency domain

Structure detection and parameter estimation (1)  Prediction error method  Extended least squares method  Orthogonal structure detection procedure  Recursive least squares method  Back propagation method  Implicit leas squares method

Correlation based validation (1) A basic concept for correlation based model validity tests: that if a model structure is correct and its parameter estimation is unbiased, its residuals should form a random (in theory) / uncorrelated (in practice) sequence with zero mean and finite variance.

Correlation based validation (2)

Controller design (1) 1) Indirect (transformation) method: neural network based design approach (Kumpati Narendra ) using neural network to approach rational models and then design control systems 2) Direct (analytical) method: U-model based design approach there is nothing lost to use U-model to express ration models.

Controller design (2) K. Narendra’s work can be referred from his publications below K.S. Narendra and K. Parthasarathy, Identification and control of dynamic systems using neural networks, IEEE Trans., on Neural Networks, Vol. 1, No. 1, pp. 4-27, J.B.D. Cabrera and K.S. Narendra, Issues in the application of neural networks for tracking based on inverse control, IEEE Trans., on Automatic Control, Vol. 44, No. 11, L.G. Chen and K.S. Narendra, Nonlinear adaptive control using neural networks and multiple models, Automatica, Vol. 37, pp , 2001.

Controller design (3) U-model based NL control system design

 Advantages using rational models 1) Concise and efficient in structure 2) Wider representations  Challenges 1) Model structure detection and parameter estimation 2) State space realisation 3) Model reduction 4) Control system design 5) Stability analysis Conclusions

QM Zhu’s relevant publications (1) S.A. Billings and Q.M. Zhu, Rational model identification using an extended least squares algorithm, Int. J. Control (International Journal of Control), Vol. 54, No. 3, pp , Q.M. Zhu and S.A. Billings, Recursive parameter estimation for nonlinear rational models, Journal of Systems Engineering, No. 1, pp , Q.M. Zhu and S.A. Billings, Parameter estimation for stochastic nonlinear rational models, Int. J. Control, Vol. 57, No. 2, pp , 1993.

QM Zhu’s relevant publications (2) S.A. Billings and Q.M. Zhu, Structure detection algorithm for nonlinear rational models, Int. J. Control, Vol. 59, No. 6, pp , S.A. Billings and Q.M. Zhu, Nonlinear model validation using correlation tests, Int. J. Control, Vol. 60, No. 6, pp , H.Q. Zhang, S.A. Billings, and Q.M. Zhu, Frequency response function for nonlinear rational model, Int. J. Control, Vol. 61, No. 5, pp , 1995.

QM Zhu’s relevant publications (3) S.A. Billings and Q.M. Zhu, Model validity tests for multivariable nonlinear models including neural networks, Int. J. Control, Vol. 62, No. 4, pp , Q.M. Zhu and S.A. Billings, Fast orthogonal identification of nonlinear stochastic models and radial basis function neural networks, Int. J. Control, Vol. 64, No. 5, pp , 1996.

QM Zhu’s relevant publications (4) Q.M. Zhu and L.Z. Guo, A pole placement controller for nonlinear dynamic plants, Proc. Instn. Mech. Enger, Part I: Journal of Systems and Control Engineering, Vol. 216, No. 6, Q.M. Zhu, A back propagation algorithm to estimate the parameters of nonlinear dynamic rational models, Applied Mathematical Modelling, Vol. 27, pp , Q.M. Zhu, An implicit least squares algorithm for nonlinear rational model parameter estimation, Applied Mathematical Modelling, Vol. 29 pp , 2005.

QM Zhu’s relevant publications (5) L.F. Zhang, Q.M. Zhu, and A. Longden, A set of novel correlation tests for nonlinear system variables, Int. J. Systems Science, Vol. 38, pp , Q.M. Zhu, L.F. Zhang, and A. Longden, Development of omni- directional correlation functions for nonlinear model validation, Vol. 43, pp , Automatica, L.F. Zhang, Q.M. Zhu and A. Longden, A correlation tests based validation procedure for identified neural networks, Vol. 20, pp. 1-13, IEEE TNN, 2009.

QM Zhu’s relevant publications (6) Q.M. Zhu, L.F. Zhang, and A. Longden, A correlation test based validity monitoring procedure for online detecting the quality of nonlinear adaptive noise cancellation, Int. J. Systems Science, in print. Q.M. Zhu, An analytical design procedure for control of nonlinear dynamic rational model based systems, (under preparation), 2010.