Non-Parametric Bilinear Approach for Modelling of a Non-Linear System

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1 International Journal of Engineering & echnology IJE-IJENS Vol: 2 No: 62 Non-Parametric Bilinear Approach for Modelling of a Non-Linear System Kahtan I. Aziz PhD, MIE, CEng. College of Engineering and Applied Sciences, Al-Ghurair University, Dubai, UAE Abstract A non-linear modelling technique is presented which uses a non-parametric bilinear representation in the form of a linear and a series of bilinear impulse coefficients. he technique is applied in simulation to the identification of a non-linear continuous stirred tank reactor (CSR) and the resulting model is shown to satisfy correlation based non-linear validity tests. It is also compared with linear modelling techniques and is shown to offer significant advantages. Keywords: bilinear identification, nonparametric, process modelling Bilinear models represent a sub class of the general NARMAX description and have been studied widely both from the theoretical and the practical point of view [6] and [7]. It has been shown that many nonlinear processes can be better approximated and controlled using bilinear models than linear ones and in such cases there may be advantages in considering bilinear modelling. In this paper a non-parametric bilinear modelling technique will be presented and is applied in simulation to a continuous stirred tank reactor (CSR). he results are compared with the linear approach of using an ARX model and validation tests are presented which test the validity of the model. It is shown that bilinear impulse coefficient modelling is capable of accurately capturing the non-linear dynamics of the CSR. All simulations are performed using the MALAB M environment. Introduction Linear system modelling has a major impact in the success of process control. It has matured into a wellestablished collection of basic techniques and algorithms that can be implemented. Perhaps the most common use of linear algorithms is in the linearisation of a non-linear process around its operating point. While linear models provide adequate performance in many applications, there are several instance where they can give misleading results since many systems are inherently non-linear. When linear models are not satisfactory for describing process behaviour, non-linear models can be developed. In recent years interest in non-linear systems theory and its application has grown and the theory for special classes of non-linear systems is well established. Non-linear modelling has received the attention of many researchers over the past few years []-[2]. he NARMAX model [3] is frequently used because it provides a concise presentation for a wide class of non-linear systems. his structure can however lead to excessively complex models due to the potentially large number of terms involved. Modifications involving model structure determination combined with the application of engineering judgement have proved effective in reducing this complexity [4] and [5]. Bilinear Approximation of a non-linear system It is well known that non-linear processes may be represented by the Volterra series which is a multidimensional convolution series in the form of kernels giving the general non-parametric description t t t y = h + h u t d + h2 2 u( t u( t 2 ( ( ( σ) ( σ) σ ( σ, σ ) σ ) σ ) σ ) σ ) () he accuracy of the modelling depends upon the number of kernels considered and the number of terms involved can quickly become unmanageable [8]. Bilinear models may be represented by the following difference equation [9] s s y( k ) = a y( k i ) + b u( k i ) + n y( k i ) u( k j) i i i= i= i= j= s i (2) his model is particularly attractive since it shares several features with linear system models. In particular, if n ij = for all i s and j s, we have the familiar ARX model and the linear part of equation (2) ij

2 International Journal of Engineering & echnology IJE-IJENS Vol: 2 No: 63 can be represented by the impulse coefficients h of the Volterra expansion. he bilinear cross product terms take account of the dependency of the future response of the system to the input and to the historical state of the process. Expansion of equation (2) produces a series of bilinear terms y(k-j-n)u(k-n). hese terms can be regarded as independent inputs to which can be associated a series reassembling impulse response coefficients and which represent the non-linear part of the model []. his is shown in Figure (). he system may now be represented by the following non-parametric form ( ) = HlinU( ) + HnljY( ) U( ) y k k k j k N j= (3) H lin are the linear impulse coefficients [h h 2 h 3....h n ]; H nlj are the j th non-linear impulse coefficients [h j h j2 h j3...h jn ]; U(k-) is the input vector [u(k-) u(k-2) u(k-3)....u(k-n)] ; Y(k-j)U(k-) are the bilinear cross product terms [y(k-j)u(k-) y(k--j)u(k-2) y(k-2-j)u(k-3).. y(k-n-j+)u(k-n)]. For the purposes of identification of the coefficients the following bilinear model representation is used : Y m = D*H (4) Where Y m is the (p-n,) modelled output vector [ y k, y k... y k p n ] Y m = + + D is a (p-n,n(n+)) data matrix D = D D D D [ L nl nl2 nln] H is a (n(n+),) matrix containing the linear and Hlin H nl bilinear impulse coefficients H = H nl2 HnlN Where n is the number of impulse coefficients in each impulse series; N is the number of non-linear impulse series considered; p is the number of data points considered for identification. D L... will contain the input signal u(k), u(k-), u(k-2) Figure () D nl will contain the first nonlinear term y(k)u(k), y(k-)u(k-)... D nl2 will contain the second nonlinear term y(k- )u(k), y(k-2)u(k-)...and so on till N. he impulse coefficients H can now be obtained using the Least-Square method to give [ ] H = D D D Y (5)

3 International Journal of Engineering & echnology IJE-IJENS Vol: 2 No: 64 Modelling a Continuous Stirred ank Reactor (CSR): he continuous stirred tank reactor (CSR) process described by Morningred et al. [] is used to evaluate the modelling technique. CSR processes are widely encountered in the process industry and many first principles models of the process dynamics have been derived, making the CSR an ideal simulation test-bed for identification and control strategies. A schematic of the CSR process is shown in Figure (2) where within a constant volume reactor, a single irreversible exothermic reaction A B is assumed to occur, with cooling provided by a single coolant stream. qc, c CA, q, Figure (2) CA, he process dynamics can be simulated by the following equations: C& a and & = q = V ( C C ) k C q V A ( ) ρc C pc + q c ρc p V A A ( H ) k exp ρc p q c C A ha ρc C p exp exp R (6a) R E ( ) t ( ) c E feed flow rate and q c is the coolant flow rate. he remaining model parameters and the nominal operating conditions are defined in the Appendix. he aim of the identification exercise was to identify a SISO process model of the CSR which related the effects of changes in coolant flow rate q c on the concentration C a. It was assumed that the feed concentration C A, the feed temperature, the process flow rate q and coolant temperature c were held constant. Identification ests he identification tests were carried out about the steady state values given in the Appendix. By changing the amplitude range of the input signal, which is a Gaussian white noise, in order to give an coolant flowrate perturbation of ± lmin -. he perturbations were sufficiently large to excite the non-linear characteristics of the process. he sampling time of all the process measurement was 5 seconds and this was dictated by the process dynamics. A linear model was formed using an eighth order ARX model structure which was chosen by the model selection routine within MALAB M. he MALAB routine will estimate the parameters a i and b i of the ARX model below using the System Identification and Control toolboxes of MALAB. y + a y( t ) anay( t na ) = bu ( t ) bnb u( t nb ) (7) he output of the linear model (y m ) together with the output of the process (y p ) in response to the test input perturbations of amplitude range ± lmin - are shown in Figure (3). (6b) Where C a is the effluent concentration of component A, is the reactor temperature, q is the

4 International Journal of Engineering & echnology IJE-IJENS Vol: 2 No: 65 he bilinear model was formed by inclusion of the linear impulse coefficients and two non-linear series to give H = [ Hlin Hnl Hnl2 ]. wenty coefficients (n=2) were evaluated for each of the impulse series and this choice was made to effectively account for the process response over the settling time whilst keeping the number of impulse coefficients to a minimum. outputs are shown in Figure(4). Model and process im e /m in Figure(3) ARX model output together with the process output y p y m i m e / m i n Figure (4) Bilinear model and process outputs y p y m

5 International Journal of Engineering & echnology IJE-IJENS Vol: 2 No: 66 Validation tests Inspection of figures (3) and (4) give an initial indication of the superiority of the bilinear model. Correlation tests were also performed to include (i) the auto correlation of residuals φ εε ( τ ) and (ii) the cross correlation of residuals and input signal φ εu( τ ) as linear validation tests [2]. It is known that these tests alone are not sufficient for non-linear systems and higher order correlation tests were also performed in an attempt to detect possible missing non-linear terms φεε( τ) [3]. hese tests are (iii) the cross correlation of input squared and residuals φ ( τ ) and (iv) the cross εu 2 correlation of input squared and residuals squared φ ε 2 u 2 ( τ ). Figure(5) and Figure(6) present the normalised correlation results for both the linear and bilinear models with 95% confidence limits (at ± 96. ) marked for reference. N φ ( τ) εu 2 φεu( τ) φ ( τ ) ε 2 u 2 Figure (5) Correlation Based Validation est: ARX Model

6 International Journal of Engineering & echnology IJE-IJENS Vol: 2 No: 67 φ ( εε τ ) φ ( τ) εu 2 φεu( τ) φ ( τ ) u ε 2 2 Figure (6) Correlation Based Validation est: Bilinear-Modelling It is clear that the linear model broadly satisfies linear validation tests but fails the higher order tests indicating the presence of significant and unmodelled non-linear terms within the process itself. Results for the bilinear model show a marked improvement with the model now passing the higher order correlation tests. It was not found necessary to increase the complexity of the bilinear model since inclusion of extra non-linear terms produced only marginal improvement. Conclusion: In this paper a bilinear modelling technique was presented in order to approximate non-linear process behaviour. he technique has been shown to produce excellent modelling using only two bilinear impulse coefficient series. he model captures the non-linear damping and gain characteristics of the process and offers significant improvement in comparison with linear modelling. his indicates the underlying bilinear nature of CSR dynamics and the potential advantages to be gained by using this simple non-linear model structure. he use of a non-parametric model simplifies the identification procedure and also provides a model structure which lends itself to exploitation within a model predictive control framework []. he simulation results presented are encouraging and shows the potential for this method to be applied to real processes. Further test been done on the bilinear modelling technique where the amplitude of the input signal changed to ± 5 lmin -. Furthermore the linear model and the bilinear model been subjected to step response test and the results shows that bilinear model has much improvement over the linear model. References: []. Billings, S A, "Identification of nonlinear systems-a survey", IEE Prc. Part D, 98, 27, pp [2]. Haber, R. and Unbehauen, H, "Structure identification of nonlinear dynamic systems - A survey on input/output approaches", Automatica, 99, 26, pp [3]. Leontaritis, I J and Billings, S A, "Inputoutput parameter models for nonlinear systems. Part: Deterministic nonlinear systems. Part2: Stochastic nonlinear systems", Int. J. Control, 985, 4, 3-34

7 International Journal of Engineering & echnology IJE-IJENS Vol: 2 No: 68 [4]. homson, M. Schooling,S.P. and Soufian,M. he practical application of a non-linear identification methodology. Control Engineering Practice, Vol.4 No. 3,pp295-36,996. [5]. Schooling, S P, homson, M and Mackay, M, "he application of nonlinear identification and control scheme to a continuous stirred tank reactor", IChemE Symp. Advances in Process Control 4, York-UK, 995, [6]. Mohler, R, "Bilinear Control Process", Academic Press, 973. [7]. Mohler, R, "Nonlinear Systems. Vol.2: Applications to Bilinear Control", Prentice Hall, 99. [8]. Fnaiech, F and Ljung, L, "Recursive Identification of bilinear systems", Int. J. Control, 987, 45, pp [9]. Bartee, C F and Georgakis, C, " Identification and control of bilinear systems", Proc. American Control Conference, Chicago, 992, pp []. Mackay, M, homson, M and Soufian, M, "A bilinear non-parametric model based predictive controller", Proc IFAC'96, 996, Vol. F, pp []. Morningred, J.D, Paden, P.E, Seborg, D.E and Mellichamp, D.A, "An Adaptive nonlinear predictive controller", Chem. Eng. Sci., 992, 47, pp [2]. Box, G E P and Jenkins, G M, "ime Series Analysis Forecasting and Control", Holden Day, San Fransisco, 976. [3]. Billings, S A and Zhu, Q M, "Model validation tests for multivariable nonlinear models including neural networks", Int. J. Control, 995, 62, pp Appendix: Nominal CSR parameter values. Process Flow Rate q l min Feed Concentration C A mol /l Feed emperature 35K Inlet Coolant emperature c 35K CSR Volume V l Heat ransfer term ha 7x 5 cal min K Reaction Rate constant k 7.2x min Activation energy term E / R x 4 K Heat of reaction H -2x 5 cal / mol Liquid Densities ρ, ρc x 3 g / l Specific Heats C, C cal g K For a steady state product concentration C A =. mol /l. Reactor emperature K Coolant Flow Rate q c 3.4 l min Dr. Kahtan Ismail Aziz received his B.Sc. from the University of echnology in 987 and Ph.D. in Adaptive Systems from the University of Bristol, UK, in 993. From 994 to 997 he joined the Manchester Metropolitan University as a Lecturer and Research Fellow. In 997 he joined the University of Wales as Senior Lecturer till 2. In 2 he joined NEC (one of the major Japanese company) at their European Research and Development Centre in UK. He worked on the development of the first 3G Mobile phone and was team leader and later he became Engineering Consultant. He joined Al-Ghurair University in 29 as Associate Professor in the area of Wireless and Mobile Networks. Dr. Aziz has published a number of papers in international scientific journals and international conferences. He chaired several scientific session in an international conferences. His current research interests are in the Wireless and Mobile Communication. Dr. Aziz is a member of IE and Chartered Electrical Engineer (CEng). p pc