IDENTIFICATION AND ESTIMATION (EMPIRICAL MODELS)

Transcription

1 Process Control in the Chemical Indstries 8 Introdction IDENTIFICATION AND ESTIMATION (EMPIRICAL MODELS) When the process is mathematically too complex to model from the fndamental physical and chemical laws, empirical models can be obtained from experimental dynamic data. Characterizing and estimating the parameters of this model from its inpt/otpt experimental data is known as process identification.. Process Identification Characterizing a process by an empirical model from its inpt/otpt experimental data is known as process identification. One may jdge from inpt/otpt data if the system needs to be identified by a linear model or non-linear model. Mainly, if the otpt satisfies the sperposition principle, that is, the response of the system to the sm of two inpts is the same as the sm of the response to the individal inpts, then a linear model will be adeqate. Otherwise we need to identify the system by a non-linear model. Many books have been written on the sbject of process identification, see for example, Bendat (990), Box and Jenkins (970), Box and Draper (987), Eykhoff (974), Graps (97), Ljng (980), Mehra and Lainiotis (976), Ray and Lainiotis (987), Sage and Melsa (97), Seinfield and Lapids (974), Sinha and Kszta (988), Soderstrom and Stoica (989) and Unbehaen and Rao (987). In the next sections, we introdce linear and non-linear system identification. Identification of linear Systems The main task is to identify a sitable linear model to represent the data. Having sggested the model, methods of parameter estimation can then be sed and statistical methods are then called for to test the adeqacy of the proposed model. If not adeqate, another model can be sggested. Models sed can be of the form of:. The convoltion integral The general inpt/otpt relation is: t y( t) = g( τ ) ( t τ ) dτ 0 () The above eqation relates the otpt y(t) with the distrbance (t) throgh the implse response g(t). The discrete version of the relation takes the form: y( k) = k i= 0 g( i) ( k i) () Instead of sing implse response fnction g(i), one can se the step response fnction β(i) sch that: y( k) = k i= 0 β( i) ( ( k i) ( k i ) ) (3) King Sad University, 00

2 Process Control in the Chemical Indstries 9 where i β ( i) = g( j) and g ( i) = β( i) β( i ) j= The process of the determination of the implse response fnction from inpt/otpt data is called deconvoltion and involves the soltion of a system of linear eqations (). 4.. State Space Models State space models eqations (35) to (36) in the previos lectre can be sed for process identification Laplace Transfer Fnction Models In this case we se general linear transfer fnction with time delay of the form of eqation (44). It is worth mentioning here that a second order system with time delay approximates the dynamics in many chemical processes. From the otpt data of a process which is sbjected to a certain type of distrbance, one can gess how the empirical model wold look like. The inpt can be chosen to be an implse, a step change or a sine wave. Next we present some possible otpt profiles from certain models sbjected to step inpt change. () models from step change: According to the responses shown in Figre, we can classify the following: (a) Otpt (a) is called Gain only process. It is characterized by the instantaneos response of the otpt and the model is given by: K is called the process gain. K (b) Otpt (b) is sally obtained for a first order system given by: K τs + τ is called time constant of the process. (c) When the otpt responds slowly to the change in the inpt (shown in c), the process can be modeled by a second order over-damped model. It takes the form; K ( τ s + )( τ s ) + (d) When the step response of a process has a decaying oscillation (shown in d), a second order nder-damped model may be sed. It takes the form; King Sad University, 00

3 Process Control in the Chemical Indstries 30 T s K + ζts + where T is the characteristic time, and ζ is the damping ratio which is less than one for nder-damped system. In all above models, if the otpt is delayed for some time ntill it feels the effect of the change in the inpt, all the above models can be mltiplied by which is the transfer fnction of a dead time process and θ is the dead time. e θs Inpt otpt a b c d Figre : Step response of process () Models from sinsoidal response Here the inpt takes the form of a sine wave: ( t) = Asin ωt where A is the amplitde and ω is the freqency. After waiting for all transient to die ot, the otpt cold have different amplitde and the response is delayed by what is called phase lag. The process is sbjected to different sine waves with different freqencies. From the otpt, one can gess the sitable model which represents the process. The following rles are sefl; i) For an n-th order system, the phase lag is less than nπ. King Sad University, 00

4 Process Control in the Chemical Indstries 3 ii) The amplitde ratio between the otpt and the inpt is sally less than. If the amplitde ratio is greater than one for a range of freqency, this indicates an nder-damped model (oscillating). iii) There is no limit for the phase lag in case of dead time. Parameter Estimation in Transfer Fnction Models In most control packages, e.g. CONSYD, there are programs that can estimate optimm parameters sing optimization methods sch as Least Sqare method. Here we present some simple methods to obtain approximately estimates for the parameters. () First order system with dead time θs Ke τs + where K = N M τ is the time constant, which is the time at which the change in the otpt is 63.% of its ltimate vale and θ is the dead time at the end of which the otpt variable starts to change. Inpt M otpt 0.63N N θ τ () Second order over-damped system with dead time θs Ke ( τ s + )( τ s ) + King Sad University, 00

5 Process Control in the Chemical Indstries 3 where K = N M and θ can be obtained easily from the diagram. The step response of second order damped system in the time domain is: τe y ( t) = N ( t θ)/ τ τ τe τ ( t θ) / τ τ and τ can be obtained by solving non-linear eqation in τ, τ for two vales of y(t), e.g. at y(t) = /3 N and y(t) = /3N. otpt N θ (3) Second order nder-damped system with time delay T s θs Ke + ζts + where K and θ can be estimated as before. ζ can be calclated from overshoot = B A = e πζ ζ B A N θ 4. Identification of Nonlinear System We discss here some methods for non-linear model identification. A recent review by Haber and Unbehaen (990) discsses different methods for the identification of nonlinear systems. King Sad University, 00

6 Process Control in the Chemical Indstries Volterra Series Models This is an extension of the convoltion integral of a linear system which is: t y( t) = k( t τ ) ( τ ) dτ 0 (4) The extension is given by: k k ( τ) ( t τ) dτ k ( τ, τ ) ( τ τ )( τ τ ) dτ dτ k ( τ, τ, τ ) ( τ τ )( τ τ )( τ τ ) dτ dτ dτ + L 3 3 (5) where the integral kernels k (τ), k (τ,τ ), k 3 (τ,τ,τ 3 ) are zero when any of their argment, is negative. Practical implementation of Voletrra series is discssed in Seinfeld and Lapids (974). 4.. Neral Networks Model Identification: Here the model is not in a form of explicit algebraic form with parameters to be estimated. Rather the model is in the form of a general strctre consisting of a network of inpt models (nerons) hidden layers and otpt nodes. As shown in Figre (), the inpt to the nodes of each hidden layer has adjstable weights that resemble the nknown parameters in algebraic form. The otpt y i from a hidden layer is given by: y i n = f ( w ), i =,,. n j= ij j (6) Where j s are the inpts, w ji are the weight and f is a simple non-linear fnction, e.g. f ( x) = + e αx (7) Recently some researcher works have been presented for the application of neral networks for dynamic modeling and hence for control prposes. Bhat and Mcavoy (990), Billing et al. (99), Narendra and Parthasarathy (990), Scott and Ray (993a, 993b), S and Sheen (99). El-Hewary (99) gave an accont for the prospects of the neral networks to desalination systems inclding control application NARMAX Models: A general non-linear discrete time system of the form of: y( t) = f ( y( t ), L, y( t n ), ( t ), L, ( t n ), e( t ), L, e( t n ) e( t)) (8) g a e + is called NARMAX model (Non-linear Atoregressive Moving Average with Exogenos inpt). Chen and Billing (989a) describe a recrsive prediction error King Sad University, 00

7 Process Control in the Chemical Indstries 34 parameter estimator. Chen and Billing (989b) expand non-linear terms of the polynomials. Chen et al. (990) compare the se of radial basis fnction with the se of otpt affine models, Polynomial models and rational models. Johansen and Foss (993) sed local ARAMAX models to constrct a global NARMAX model. w w y y wn n y n inpt nodes hidden layers otpt nodes Figre (): A Neral Network Note: One of the workshop ttorials inclde an exercise on modeling inpt-otpt data of a mltivariable process to an ARAMAX model (linear discrete time model). King Sad University, 00

8 Process Control in the Chemical Indstries 35 Reference: Bendat, J.S., Nonlinear System Analysis and Identification from Random Data, Wiely- Inetrscience, New York, 990. Box, G.E.P. and Jenkins. G.M., Time Series Analysis Forecasting and Control, Holden- Day, San Francisco, 970. Box, G.E.P., and Draper, N.R., Empirical Model Bilding and Response Srface, John Wiely, New York, 987. Bhat, N. and MCaAvoy, T. J., "Use of Neral Nets for Dynamic Modeling and Control of Chemical Process Systems", Compter & Chemical Engineering, 4, , 990. Billings, S.A., Jamalddin, H.B. and Chen, S., "Properties Neral Networks with Application to Modeling Non-linear Dynamic Systems", Int. J. Control, 55, 93-4, 99. Chen, S., and Billings, S. A., "A recrsive Prediction Error Estimator for Non-linear Models", Int. J. Control, 49, 03-03, 989a. Chen, S., and Billings, S. A., " Representation of Non-linear Systems: The NARMAX model", Int. J. Control, 49, , 989b. Chen, S., Billings, S. A., Cowan, C.E.N, and Grant, P., "Practical Identification of NARMAX Models Using Radial Basis Fnctions", Int. J, Control, 5, , 990. Eykhoff, P., System Identification, John Wiely, New York, 974. Graps, D., Identification of Systems, Van Nostrand Reinhold Co., New York, 97. Haber, R., and Unbehaen, H., "Strctre Identification of Non-linear Dynamic Systems- A Srvey of Inpt/Otpt Approaches", Atomatica, 6, , 990. Johansen, T. A., and Foss, B. A., "Constrcting NARMAX Models sing ARMAX Models", Int. J. Control, 58, 5-53, 993. Ljng, L., System Identification Theory for the User, Prentice-Hall, Englewood Cliffs, New Jersey, 980. Mehra, R. K. and Lainiotis, D. G., System Identification: Advances and Case Stdies, Academic Press, New York, 976. Narendra, S. and Parthasarathy, K., "Identification and Control of Dynamical Systems Using Neral Networks", IEEE Trans. Neral Networks, NN-, 4-7, 990. Ray, W. H., and Lainiotis, D. G., Distribted Parameter Systems, Identification, Estimation and Control, Mercel Dekker, New York, 987. King Sad University, 00

9 Process Control in the Chemical Indstries 36 Sage, A. P., and Melsa, J.L., System Identification, Academic Press, New York, 97. Scott, G. M., and Ray, W. H., "Creating Efficient Non-linear Neral Network Process Models that Allow Model Interpretation", J. Process Control, 3, 63-78, 993. Sinha, N. K., and Kszta, B., Modeling and Identification of Dynamic Systems, Van Nostrand Rheinhold Co., New York, 983. Soderstrom, T. and Stocia, P., System Identification, Prentice-Hall, Englewood Cliffs, New Jersey, 989. S, Y. T. and Sheen, Y. T., "Neral Networks for System Identification", Int. J. System Sci., 3, 7-86, 99. Unbehaen, H., and Rao, G. P., Identification of Continos Systems, North Holland, Amsterdam, 987. Seinfeld, J. H., and Lapids, L., Mathematical Methods in Chemical Engineering, vol. 3, Process Modeling, Estimation, and Identification, Prentice-Hall, Englewood Cliffs, New Jersey, 974. King Sad University, 00

10 Process Control in the Chemical Indstries 37 Appendix Compter Aided Design Packages Mch of the Programming brden reqired to apply the principles presented in this chapter is no longer needed since there are well developed compter packages that carry ot simlation, identification, parameter estimation, model redction and simplification, analysis and control system design. In the following table we give list of some famos package in the field. Sorces for Control Design Packages Program Sorce CC System Technology Inc. 3766, S. Hawthorne Blvd. Hawthorne, CA 9050, USA CONSYD Prof. H. Ray, Dept. of Chemical Engineering University of Wisconsin 45 Johnson Drive Madison, WI 53708, USA EASY5 Boeing Compter Services POBox 4346 Seattle, WA 984, USA KEDDC Ingenier bero Erble Jahnostr, 73 Grossbettilingen, Germany MATLAB Mathworks Inc. 4 Prince Park Way Natic, MA 0760, USA SIMULINK Mathworks Inc. 4 Prince Park Way Natic, MA 0760, USA Control Station Prof. Doglas Cooper, Chemical Engineering Dept. University of Connectict 9 Aditorim Rd. Storrs, CT sim Controllab Prodcts B.V. Drienerlolaan 5 EL-RT 75 NB Enschede The Netherlands Calerga Calerga Sarl Yves Piget Av. de la Chabliere 35 CH Lasanne Switzerland pidtne EngineSoft P.O. Box 877 Tempe, Arizona King Sad University, 00

11 Process Control in the Chemical Indstries 38 SimApp Visim Besser Engineering Brno Besser, dipl. El.Ing. ETH Wacht 8 CH-8630 Reti ZH, Switzerland Visal Soltions, Incorporated 487 Groton Road Westford, Massachsetts 0886 King Sad University, 00