Use of MATLAB Environment for Simulation and Control of CSTR

Transcription

1 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Ue of MALAB Environment for Simulation and Control of CSR Jiri Vojteek, Petr Dotal Abtract hi contribution preent the uability of the mathematical oftware MALAB (MArix LABoratory) in the field of imulation of the teady-tate, dynamic behaviour and adaptive control of the Continuou Stirred ank Reactor (CSR). hee type of chemical reactor belong to the cla of nonlinear lumped-parameter ytem mathematical model of which i decribed by one or more Ordinary Differential Equation (ODE). he imple iteration method wa ued for teady-tate analyi of the ytem while the Runge-Kutta method wa employed for the numerical olution of the et of ODE. Both method are imple, provide ufficient reult and they are eaily programmable which wa important in our cae. he preented adaptive approach ued for controlling of the ytem provide ufficient reult although the ytem ha negative propertie from the control point of view. he benefit of thi paper can be found in the imulation program made in MALAB with the ue of Graphical Uer Interface (GUI) that provide uer poibilitie to examine imulation without changing of the program code. Keyword Matlab, modeling, imulation, adaptive control, recurive identification. I. INRODUCION HE mot of procee in the real world, not only in the indutry ha nonlinear behaviour. On the other hand, chemical reactor belong to the mot often equipment in the chemical and biochemical indutry [] and that i why thi paper i focued on one particular member of thi family the Continuou Stirred ank Reactor (CSR) with exothermic reaction inide. Specific deign of the controller i uually proceed by few very important tep. Not every property of the controlled ytem i known before we tart and that i why we perform imulation experiment on the ytem. here are two main type of the invetigating of the ytem behaviour () experiment on the real model and () computer imulation. Computer imulation i very often ued at preent a it ha many advantage over an experiment on a real ytem, which Manucript received Augut 9,. J. Vojteek and P. Dotal are with Department of Proce Controll, Faculty of Applied Informatic, oma Bata Univerity in Zlin, Czech Republic (phone: ; fax: ; {vojteek,dotalp}@ fai.utb.cz). i not feaible and can be dangerou, or time and money demanding. Simulation and modelling poibilitie rie with the increaing impact of the digital technology and epecially with the computer technology which grow exponentially every moment. he mathematical model of thi particular CSR i decribed by the et of two nonlinear Ordinary Differential Equation (ODE) which are contructed with the ue of material and heat balance inide. Example for deriving uch mathematical model can be found in []. he teady-tate analyi invetigate behaviour of the ytem in the teadytate which from the mathematical point of view mean numerical olving of the et of nonlinear algebraic method. he imple iteration method [3] wa ued in thi cae becaue the ytem fulfill the convergence condition. he next tep i the dynamic analyi which practically mean numerical olving of the nonlinear et of ODE. A lot of numerical olution method have been developed, epecially for the ODE, uch a Euler method or aylor method [4]. Runge- Kutta method are very popular becaue of their implicity and eay programmability [5]. Although there could be ued everal modern control method [6], [7], the adaptive approach i ued here. he baic idea of adaptive control i that parameter or the tructure of the controller are adapted to parameter of the controlled plant according to the elected criterion [8]. he adaptive approach in thi work i baed on chooing an external linear model (ELM) of the original nonlinear ytem whoe parameter are recurively identified during the control. Parameter of the reulted continuou controller are recomputed in every tep from the etimated parameter of the ELM. he polynomial method introduced by Kucera [9] ued for deigning of the controller here together with the poleplacement method enure baic control requirement uch a tability, reference ignal tracking and diturbance attenuation. he baic control ytem configuration with one degree-of-freedom (DOF) and two degree-of-freedom have been ued. he propoed controller i hybrid becaue polynomial ynthei i made for continuou-time but recurive identification run on the -model, which belong to the cla of dicrete-time model. he MALAB (MArix LABoratory) [] i mathematical oftware often ued for computation and imulation [], []. Although thi oftware ha it own programming language, it alo provide the tool for creating window-like Iue 6, Volume 5, 58

2 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION program which can be ued for teting and imulation without changing of the program and knowledge about thi programming language. hi tool called GUIDE wa ued here for creating of the imulation program a a practical output of thi contribution. he main goal i to provide MALA program which provide baic imulation of one nonlinear ytem. Created program i available for free. People, who are intereted, pleae contact author on hi addre vojteek@fai.utb.cz. II. MODEL OF HE PLAN he propoed control trategy wa teted on the mathematical model of Continuou Stirred ank Reactor (CSR). hi model ha imple exothermic reaction inide the tank which i cooled via cooling coil ee Fig.. Mathematically peaking, thi plant i repreented by the mathematical model which decribe all quantitie i of coure very complex and we need to introduce ome implification. Firt, we expect that reactant i perfectly mixed. hen, we alo aume that volume, heat capacitie and denitie do not change rapidly during the control. Fig. he chematic repreentation of CSR hee aumption reult in the mathematical model repreented by the et of two Ordinary Differential Equation (ODE) [] which are derived from the material and heat balance of the reactant and cooling, i. e. a4 d q a a k c 3 c A a qc e c dt () dca a ca ca k ca dt where a -4 are contant computed a q H c cpc ha a ; a ; a3 ; a4 () V c c V c p p c pc he fixed value of the ytem are hown in able. he nonlinearity of the model i hidden mainly in the computation of the reaction rate, k, which i nonlinear function of the temperature,, and it i computed from Arrheniu law: E R k e k (3) ABLE. FIXED PARAMEERS OF HE REACOR Quantity Reactor volume Reaction rate contant Activation energy to R Reactant feed temperature Inlet coolant temperature Reaction heat Specific heat of the reactant Specific heat of the cooling Denity of the reactant Denity of the cooling Feed concentration Heat tranfer coefficient Symbol and value V = l k = 7. min - E/R = 4 K = 35 K c = 35 K ΔH = - 5 cal.mol - c p = cal.g -.K - c pc = cal.g -.K - ρ = 3 g.l - ρ c = 3 g.l - c A = mol.l - h a = 7 5 cal.min -.K - III. SEADY-SAE AND DYNAMIC ANALYSES he pre-control imulation often include teady tate and dynamic analye which help u with the undertanding, how ytem work in different tate and behave after variou change on the input. A. Steady-tate Analyi he teady-tate analyi how behaviour of the ytem in the teady-tate, i.e. in t and reult in optimal working point in the ene of maximal effectivene and concentration yield. Mathematical meaning of the teady-tate i that derivative with repect to time variable are equal to zero, d( )/dt =. he previou tudie [4] have hown intereting teady-tate feature of thi reactor. It i clear, that the reactant and cooling heat mut be equal in the teady-tate, i.e. Q r = Q c, which mean that the equation () in teady-tate i rewritten to: a4 q a a k c 3 c A a qc e c (4) and reult in relation for thee heat Q r and Q c : Qr a k ca a4 a4 q 3 c q Q 3 c c a a qc e a a q c e If we compute Q r and Q c for variou value of the temperature = <3, 5> K for working point q = l.min - and q c = 8 l.min -, we obtain three teady-tate ee Fig.. A you can clearly ee, thi ytem ha two table teadytate (S and S ) and one untable teady tate (N ). he teady-tate value of the tate variable in thee point are: S : K c A.96 mol. l N : K c A.68 mol. l S : K c A.439 mol. l It i clear, that the econd operating point S ha better efficiency (95.6 % react) for the ame input etting than on the point S (3.8 % react). (5) (6) Iue 6, Volume 5, 59

3 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Q r (),Q c () (K.min - ) 3 - Qr Qc S (K) Fig. Heat balance inide the reactor So the teady-tate model i finally decribed by the et of nonlinear function a4 qc a a k ca a3 qc e a4 qc a a3 qc e (7) a ca ca a k which are eaily olved numerically by the imple iteration method. B. Dynamic Analyi hi analyi mean that we oberve coure of the tate variable in time after the tep change of ome input variable. he tep change of volumetric flow rate q and q c are input variable in our cae and the teady-tate value in Equation (6) are initial condition for the et of ODE (). he Runge- Kutta fourth order method wa ued for numerical olving of the et of ODE. N IV. ADAPIVE CONROL here could be ued everal o called modern technique to control of thi proce uch a robut control, predictive control, fuzzy control etc. In our cae, the adaptive control wa ued mainly becaue of it trong theoretical background and uability for uch kind of procee. he Adaptive control i baed on the quality of real organim which can change behavior according to environmental condition. hi proce i uually called adaptation. here are everal way of ue of the adaptation. It can be done for example by the modification of the controller' parameter by the change of the controller tructure or by generating an appropriate input ignal, which i called adaptation by the input ignal. he adaptive approach in thi work i baed on chooing an external linear model (ELM) of the original nonlinear ytem whoe parameter are recurively identified during the control. Parameter of the reulted continuou controller are recomputed in every tep from the etimated parameter of the ELM. he advantage of thi method i that we do not care about the ytem nonlinearity. Firt we do the dynamic analyi that how u the dynamic behavior of the output S variable which i then ued for the choice of the ELM which decribe the output in the mot accurate way. he poible change of the ELM parameter i taken into account by the recurive identification of ELM during the control. A. External Linear Model (ELM) he ELM could be generally decribed by the tranfer function: Y b G( ) U a (8) he preence of the variable in the Equation (8) indicate continuou-time (C) model. he online identification of uch procee which i neceary in thi cae i not very eay. One way, how we can overcome thi problem i the ue of o called δ model. hee pecial type of model formally belong to dicrete model but it wa proofed for example in [5] that their parameter are cloe to the continuou one for very mall ampling period. he δ model introduce a new complex variable γ ([6]): z (9) z v v Where i a parameter from the interval and v mean a ampling period. It i clear that we can obtain infinite number of -model for variou. A o called forward δ-model for = wa ued and γ operator i then z () v he continuou model (8) i then rewritten to the form a y t b u t () where polynomial a () and b () are dicrete polynomial and their coefficient are different, but for the mall ampling period very cloe to thoe of the C model a() and b(). hee parameter identified recurively, which mean that they are computed by the Recurive Leat Square (RLS) method from differential equation y k k k e k () Where tand for known regreion (data) vector, repreent vector of parameter and e(k) i a general random immeaurable component. B. Recurive Identification A it i written above, the well known and eaily programmable Recurive Leat-Square (RLS) method i ued for the on-line identification. hi method i uually modified with ome kind of forgetting; exponential or directional [7] mainly due to pecific feature of the identified ytem like nonlinearity etc. he baic RLS method i decribed by the et of equation: Iue 6, Volume 5, 53

4 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION k yk k ˆ k k k P k k (3) Lk k P k k P k k k P k P k P k k k k P k k ˆ k ˆ k Lk k Where the forgetting could be affected by the choice of the forgetting factor. he identification method ued in the program are: Without forgetting, e.g. no forgetting factor i inerted. Contant exponential forgetting i for < and =. he value of forgetting factor are from the range <.95;.99>. Parameter λ influence gradual forgetting of the old value and the mot weight i put on the lat value. hi relation can be decribed by criterion k J (4) i k i i hi algorithm can be ued for ytem with changing parameter. Increaing exponential forgetting ha forgetting parameter = and i computed from k k (5) ypical value of the forgetting parameter are.95,.99. he value of thi forgetting factor i aymptotically approaching to, which mean that the old data i forgotten. Changing exponential forgetting ha again the value of forgetting parameter = and exponential forgetting i recomputed in every tep a k K k k (6) where K i a very mall value (e.g..). Directional forgetting. hi algorithm forget information only in the direction from which it come. General decription of thi method can be formulated by the following equation: P P k k k r k k k k r k r k r k Pk k k Pk k P k k r k r k k k k L P for for (7) where can be choen imilarly a in exponential forgetting. - C. Control Sytem Synthei he control ytem configuration with one degree-of-freedom (DOF) wa ued thi work ee Fig. 3. w - e v u Fig. 3 DOF control configuration G in Fig. 3 denote tranfer function (8) of controlled plant, w i the reference ignal (wanted value), v i diturbance, e i ued for control error, u i control variable and y i a controlled output. he feedback part of the controller are deigned with the ue of polynomial ynthei: q Q (8) p where parameter of the polynomial p() and q() are computed by the Method of uncertain coefficient which compare coefficient of individual -power from Diophantine equation [9]: a p b q d (9) he reulted, o called hybrid, controller work in the continuou time but parameter of the polynomial a() and b() are identified recurively in the ampling period v. he feedback controller Q() enure tability, load diturbance attenuation for both configuration and aymptotic tracking. he polynomial d() on the right ide of (9) i an optional table polynomial. Root of thi polynomial are called pole of the cloed-loop and their poition affect quality of the control. hi polynomial could be deigned for example with the ue of Pole-placement method. he degree of the polynomial d() i in thi cae deg d deg a deg p () A choice of the root need ome a priory information about the ytem behaviour. It i good to connect pole with the parameter of the ytem via pectral factorization. he polynomial d() then for our ELM (8) can be rewritten for aperiodical procee to the form deg d deg n d n () where α > i an optional coefficient reflecting cloed-loop pole and table polynomial n() i obtained from the pectral factorization of the polynomial a() * * n n a a () V. USED NUMERICAL MEHODS he main numerical method ued in the program are the Simple iteration method in the teady-tate analyi and the Standard Runge-Kutta method ued for the olving of ordinary differential equation in the dynamic analyi and alo in the hybrid adaptive control where the action value (i.e. output from the controller) i computed alo from the ordinary differential equation for the continuou-time. y Iue 6, Volume 5, 53

5 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION A. Simple Iteration Method Conider nonlinear ytem in the form of tate equation x t f x t, u t with initial condition x x and the vector of tate variable i x x, x, x n. Component of thi vector are unknown and they could be computed by olving of equation of the model in teady-tate: f x, u (3) where u = [u, u, u m ] i vector of aigned (known) input variable which come from baic teady-tate. Unknown variable in the equation (3) are component of the tate vector x, which create n tate variable in baic teady-tate. Computation of the initial condition x i for nonlinear ytem important not only for computation of the dynamic but alo it i ued for creating of a linearized mathematical model of the proce in working point. It i known that parameter of thi model depend on value of tate variable in the working point. Equation (3) can be now rewritten to f x (4) for unknown value of x i for i =,, n. Next tep in the olution i following. he equivalent et of equation to the et (4) i x = x (5) where φ i nonlinear vector function φ = [φ, φ, φ n ] and which lead to iterative equation in the form of k k x x for k =,, (6) he iterative method lead to the exact olution only if it converge. he convergence condition of the iterative proce (6) then can be formulated a: Let the vector function φ i defined in the cloed convex region D n and if x D o D too. Moreover, let function φ ha continuou partial differential derivation of all variable x xn in the region D, then there exit matrix d d d dx dx dx n d d d d x dx dx dxn d (7) x dn dn dn dx dx dx n If matrix (7) complete condition x for any x D, * there are only one olution x D of the equation (6). here could be of coure thouand of iteration during the computation but from practical point of view i convenient to top the computation in the cae that difference between value of actual and previou iteration i ufficiently mall, i.e. condition k k x x (8) i fulfilled for accuracy >, value of which depend on uppoed abolute dimenion of computed variable. B. Standard Runge-Kutta Method he econd main group of the numerical olving i olution of the ordinary differential equation (ODE) which can be found mainly in the dynamic analyi a the numerical olution of the et of ODE () but there are ued in the adaptive control a it i mentioned above too. We uppoed the general differential equation be in the form of y t g x t, u t (9) with the initial condition y t y (3) here are a lot of method which can be ued for numerical olving of thi problem. General diviion i into the two main group one-tep and multi-tep method. he popular Runge-Kutta' tandard method wa ued in thi work. hi method i very often ued becaue of it implicity. Runge-Kutta' method belong to the cla of high-order method, they can be ued for computation of the initial value or for the final reult and they are eaily programmable. he (tandard) fourth-order Runge Kutta' method ue firt four part of the aylor' erie: y k yk g g g3 g4 (3) 6 where coefficient g -4 are computed from: g h f x, y x i n n hi g g hi f xn, y xn hi g g3 hi f xn, y xn g h f x h, y x g 4 i n i n 3 (3) he Runge-Kutta' method are in ome cae build-in function in mathematical oftware. For example in MALAB, which i ued for imulation in thi work, are Runge-Kutta' method in function ode3 (the econd order Runge-Kutta formula) or ode45 (the fourth order Runge- Kutta' formula decribed above). One of advantage of thee method i that they have flexible integration tep h i, which recompute every tep according to the actual computation error. he tandart Runge-Kutta method ha a everal modification like Runge-Kutta-Fehlberk method, Runge- Kutta-Nytröm method etc. VI. SIMULAION PROGRAM he imulation program which deal with the imulation of the teady-tate, dynamic and of coure adaptive control of the CSR wa made in mathematical oftware MALAB (MArix LABoratory), verion 7.. from Mathwork [] Iue 6, Volume 5, 53

6 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION uing Graphical Uer Interface (GUI). he ue of thi tool enable programmer to make program uer-friendly and cloe to the uer who do not know or do not like programming. hey can ue all feature of Matlab a a imulation tool by jut changing of the mot important variable and preing button for computing. he MALAB ha pecial tool for creating of program with GUI. hi tool i called imply by typing of the command guide in the Command window and the ample window i hown in Fig. 4. reactant, c. he next part give uer choice between two table tate S or S which cloely decribed in chapter III. Fig. 5 Property inpector in GUIDE Fig. 4 he MALAB tool GUIDE he uage of thi tool i very imple and undertandable. It ha two main part I. workpace for program ketch and II. toolbar on the left ide which provide all common object ued in the program deign like text and edit boxe, button, radio button, lit boxe etc. Each object could be of coure edited via property inpector where you can edit color, font, ize, poition etc. of the object. he output from the GUIDE are two file e.g. ample.fig where the ketch of the program and all object i aved and ample.m where are defined action to individual object mainly procedure to the button. he program can be tart by the typing the command go in the program directory. It i divided into two main window mainly becaue of the pace. he firt window (Fig. 6) involve imulation of the teady-tate and dynamic of the ytem. he uer can et the working point of the reactor which i defined by volumetric flow rate of the reactant and the cooling, q r and q c, input temperature of the reactant and the cooling, r and c, and input concentration of the Fig. 6 GUI for the imulation of the teady-tate and dynamic of CSR Iue 6, Volume 5, 533

7 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION he third part i dedicated to the teady-tate analyi where two analye could be done the teady-tate analyi for different volumetric flow rate of the reactant q r and different volumetric flow rate of the coolant q c where the tarting and end value could be et in the edit boxe. he teady-tate analyi for both input variable together i repreented by the puh-button Compute 3D and reult in 3D graph. he ample reult of the teady-tate analyi are hown in Fig. 7. part here i the choice of the external linear model and etting for the control where the uer could et the poition of the root, ampling period v, definition of tep change of the reference ignal w(t) which repreent wanted value and time when they occur. 48. [K] c [mol/l] i [-] i [-] [K] q r [l/min] c [mol/l] q r [l/min] Fig. 7 Sample reult of the teady-tate analyi he lat part i focued on the dynamic analyi which could be done for tep change of both input variable q r and q c. Both dynamic analye can be done for more tep change (e.g. ix tep change -6, -4, -,, 4 and 6 a it can be een in Fig. 6). he imulation time and the integration tep in Runge-Kutta method could be et via appropriate edit boxe. Again, the ample reult of the dynamic analyi are hown in Fig. 8. he button in the bottom of the window ued for opening the next window for control (puh-button Control ), diplaying the help to the program (puh-button Help ) and cloing of thi window and all graph (puh-button Cloe ). he econd ub-program called by the preing of the puhbutton Control or by the command go_control from the Matlab command window deal with the imulation of the adaptive control. he window i diplayed in Fig. 9. Fig. 9 GUI for the imulation of the adaptive control of CSR he final imulation time i recomputed according to the number of tep and time for each tep. he lat part in thi ub-window i dedicated to the choice of recurive method (ee Fig. ) for identification. 4. y (t) (- ) [K] t [min] y (t) (c A -c A ) [mol.l - ] t [min] Fig. 8 Sample reult of the dynamic analyi he firt two part related to the working point and the choice of the teady-tate are the ame a in previou cae. he new Fig. he choice of the identification method he button below ha the ame function a in previou cae except the firt puh-button on the left which will call the program for imulating of the teady-tate and dynamic in thi cae. Simulation reult of the adaptive control can be een in Fig.. A a reult of the imulation, program how the final value which indicate what computation wa done and in which Iue 6, Volume 5, 534

8 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION MA-file are data aved in. he name of thi file differ with the computation ee Fig.. w(t),y(t) [mol/l] u(t) [%].5..5 Reult for qc t [min] b' a' a' b' 5 Identified parameter during imulation x t [min] Fig. Sample reult of the adaptive control Fig. GUI with the reult of the computation VII. CONCLUSION hi contribution how procedure which i connected with the modeling and imulation of the ytem behaviour that uually precede the deign of the controller. he teady-tate and dynamic analye uncover nonlinearity of the examined continuou tirred tank reactor. he deigned adaptive controller i baed on the recurive identification of the external linear model a a linear repreentation of the originally nonlinear ytem. hi controller could be tuned via the choice of the parameter where increaing value of thi parameter reult in quicker output repone but more haking coure of the input variable. he main goal of thi contribution i to how uability of the mathematical oftware MALAB for creating imulation program which could help uer to invetigate the behaviour of the nonlinear ytem repreented by the CSR. he reulting program ha two main window the firt provide the imulation of the teady-tate and dynamic of the ytem for different value of input quantitie. Reult are diplayed in the eparate figure and the data were alo aved in the MA-file. he econd program deal with adaptive control of thi ytem and uer can again et different input variable and chooe variou computation. he benefit of thi program can be found in the GUI which provide changing of the mot important value by the edit window intead of the change of the program code. he program i available alo for free at the author . REFERENCES [] P. Dotal, V. Bobal, F. Gazdo, Simulation of nonlinear adaptive control of a continuou tirred tank reactor, International Journal of Mathematic and Computer in Simulation,Volume 5, Iue 4,, Page [] J. Ingham, I. J. Dunn; E. Heinzle, J. E. Přenoil, Chemical Engineering Dynamic. An Introduction to Modeling and Computer Simulation. Second. Completely Revied Edition. VCH Verlaggeellhaft. Weinheim, [3] Y. Saad, Iterative Method for Spare Linear Sytem. Society for Indutrial & Applied, 3 [4] F. L. Severance, Sytem Modeling and Simulation: An Introduction. John Wiley & Son [5] J. H. Mathew, K. K. Fink, Numerical Method Uing Matlab. Prentice- Hall 4 [6] L. Pekar, R. Prokop, Stabilization of a delayed ytem by a proportional controller, International Journal of Mathematical Model and Method in Applied Science 4 (4),, pp. 8-9 [7] D. Samek, D. Mana, Artificial neural network in artificial time erie prediction benchmark, International Journal of Mathematical Model and Method in Applied Science 5 (6),, pp [8] V. Bobal, J. Böhm, J. Fel, J. Machacek, Digital Self-tuning Controller: Algorithm. Implementation and Application. Advanced extbook in Control and Signal Proceing. Springer-Verlag London Limited, 5. [9] V. Kucera, Diophantine equation in control A urvey. Automatica , 993 [] MathWork - MALAB and Simulink for echnical Computing, official webpage. Available: (URL) [] Matuu R. A oftware tool for algebraic deign of interval ytem control, International Journal of Computational Science and Engineering 5 (3-4),, pp [] Brancik, L., Sevcik, B. ime-domain imulation of nonuniform multiconductor tranmiion line in Matlab, International Journal of Mathematic and Computer in Simulation 5 (),, pp [3] R. Gao, A. O dywer, E. Coyle, A Non-linear PID Controller for CSR Uing Local Model Network. Proc. of 4th World Congre on Intelligent Control and Automation. Shanghai. P. R. China ,. [4] J. Vojteek, P. Dotal, Effect of External Linear Model Order on Adaptive Control of CSR. In: Proceeding of the IFAC workhop on Adaptation and Learning in Control and Signal Proceing ALCOSP. Antalya, urkey. [5] D. L. Stericker, N. K. Sinha, Identification of continuou-time ytem from ample of input-output data uing the -operator. Control-heory and Advanced echnology, vol. 9, 993, 3-5 [6] S. Mukhopadhyay, A. G. Patra, G. P. Rao, New cla of dicrete-time model for continuo-time ytem. International Journal of Control, vol.55, 99, 6-87 [7] G. P. Rao, H. Unbehauen, Identification of continuou-time ytem. IEEE Proce-Control heory Application., 5, 5, 85- Jiri Vojteek (Ph.D.) wa born in Zlin, Czech Republic in 979 and tudied at the oma Bata Univerity in Zlin. where he got hi mater degree in chemical and proce engineering in. He ha finihed hi Ph.D. focued on Modern control method for chemical reactor in 7. He now work a a Senior Lecturer at Department of Proce Control, Faculty of Applied Informatic, oma Bata Univerity in Zlin. Petr Dotal (prof.) tudied at the echnical Univerity of Pardubice. He obtained hi PhD. degree in echnical Cybernetic in 979 and he became profeor in Proce Control in. Hi reearch interet are modeling and imulation of continuou-time chemical procee. polynomial method. optimal. adaptive and robut control. He work a a profeor and head of the Department of Proce Control, Faculty of Applied Informatic, oma Bata Univerity in Zlin. Iue 6, Volume 5, 535