NONLINEAR ADAPTIVE CONTROL OF CSTR WITH SPIRAL COOLING IN THE JACKET

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1 NONLINEAR ADAPIVE CONROL OF CSR WIH SPIRAL COOLING IN HE JACKE Jiri Vojteek and Petr Dotal Faculty of Applied Informatic oma Bata Univerity in Zlin Nam. GM, 76 Zlin, Czech Republic KEYWORDS Adaptive control, Polynomial approach, Pole-placement method, Recurive identification, CSR ABSRAC he mot of the procee not only in the indutry ha nonlinear behavior and control of uch procee could be difficult. he controller here conit of linear and nonlinear part where the nonlinear part i derived from the tatic analyi and the linear part decribe nonlinear element in the loop by the External Linear Model (ELM), parameter of which are etimated recurively with the ue of delta (-) model. he control ynthei employ polynomial approach with the pole aignment method. he propoed control method atifie baic control requirement and it wa teted by the imulation on the mathematical model of Continuou Stirred ank Reactor (CSR) with piral cooling in the jacket a a typical member of the nonlinear ytem with lumped parameter. INRODUCION Chemical reactor are tool widely ued not only in the chemical indutry for production of variou product. Although there are everal type from the contruction point of view uch a batch, emi-bath etc. (Ingham et al. ), Continuou Stirred-ank Reactor (CSR) and tubular reactor are the mot uitable for control purpoe. he Continuou Stirred-ank Reactor ued in thi work repreent typical nonlinear plant decribed mathematically by the et of two nonlinear ordinary differential equation (ODE) (Gao et al. ). A it i decribed in (Vojteek and Dotal ), thi ytem ha two table and one untable teady-tate which could lead to very untable or unoptimal output repone with the ue of conventional control method. One way how to overcome thi inconvience i the ue of the adaptive control (Åtröm and Wittenmark 989) which adopt parameter of the controller to the actual tate of the ytem via recurive identification of the External Linear Model (ELM) a a linear repreentation of the originally nonlinear ytem (Bobal et al. ). he reult of the adaptive control on thi concrete mathematical model can be found for example in (Vojteek et al. ). he control method ued here i baed on the combination of the adaptive control and nonlinear control. heory of nonlinear control (NC) can be found for example in (Atolfi et al. 8) and (Vincent and Grantham 997), the factorization of nonlinear model of the plant on linear and nonlinear part i decribed in (Nakamura et al. ) and (Sung and Lee 4). he controller conit of a tatic nonlinear part (SNP) and a dynamic linear part (DLP). he tatic part i obtained from the teady-tate characteritic of the ytem, it inverion, uitable approximation and it derivative. A a reult of thi nonlinear decription, the linear part i then decribed by the external linear model with the ue of delta (-) model (Middleton and Goodwin 4) a a pecial type of dicrete-time model which parameter approache to the continuou one for the mall ampling period (Stericker and Sinha 993). he polynomial approach (Kucera 993) in the control ynthei can be ued for ytem with negative propertie from the control point of view uch a nonlinear ytem, non-minimum phae ytem or ytem with time delay. Moreover, the pole-placed method with pectral factorization atifie baic control requirement uch a diturbance attenuation, tability and reference ignal tracking. All graph hown in thi contribution come from the imulation on the mathematical model and they were done on the mathematical imulation oftware Matlab, verion 6... CONINUOUS SIRRED ANK REACOR he controlled proce under the conideration i the continuou tirred tank reactor (CSR) with the piral cooling in the jacket. he cheme of the ytem can be found in Figure. he complete mathematical decription of the proce i very complex and we mut introduce ome implification. At firt, we expect that reactant i perfectly mixed and react to the final product with the concentration c A (t). he heat produced by the reaction i repreented by the temperature of the reactant (t). Furthermore we alo expect that volume, heat capacitie and denitie are contant during the control. A mathematical model of thi ytem i derived from the material and heat balance of the reactant and cooling. he reulted model i then a et of two Proceeding 6th European Conference on Modelling and Simulation ECMS Klau G. roitzch, Michael Möhring, Ulf Lotzmann (Editor) ISBN: / ISBN: (CD)

2 Ordinary Differential Equation (ODE) (Gao et al. ): a4 d = a + a k c + a3 q e dt dca = a ( ca ca) k ca dt q ( ) c A c ( ) () where a -4 are contant computed a q ΔH ρc cpc ha a = ; a = ; a3 = ; a4 = () V ρ cp ρ cp V ρc cpc variable t in previou equation denote time, i ued for temperature of the reactant, V i volume of the reactor, c A repreent concentration of the product, q and q c are volumetric flow rate of the reactant and cooling repectively. Indexe ( ) denote inlet value of the variable and ( ) c i ued for variable related to the cooling. he nonlinearity of the model can be found in relation for the reaction rate, k, which i computed from Arrheniu law: E R k e k = (3) where k i a reaction rate contant, E denote an activation energy and R i a ga contant. he tatic analyi of thi ytem i decribed in detail for example in (Vojteek and Dotal ). he mot important reult of the teady-tate analyi can be found in the complexity of the ytem, it ha three teady-tate one untable (N ) and two table (S and S ). hi pecial feature i hown in Figure which repreent value of the reactant (Q r ) and cooling (Q c ) heat for the working point repreented by the volumetric flow rate q = l.min - and q c = 8 l.min - and variou value of the temperature = <3, > K. Q r (),Q c () [K.min - ] 3 Q r Q c S [K] Figure : Heat balance inide the reactor N S he teady-tate value of the tate variable in all three teady-tate are: Figure : Continuou Stirred ank Reactor he fixed value of the ytem are hown in able (Gao et al. ). able : Fixed parameter of the reactor Reactant flow rate Reactor volume Reaction rate contant Activation energy to R Reactant feed 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 q = l.min - V = l k = 7. min - E/R = 4 K = 3 K ΔH = - 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 cal.min -.K - S : = 34.3 K c A =.96 moll. N : = 39.4 K c A =.68 moll. S : = 46. K c A =.439 moll. It i clear, that the econd operating point S ha better efficiency (9.6 % react) for the ame input etting than on the point S (3.8 % react). hi i the main reaon why we have choen thi econd teady-tate in thi work. (q c )[K] max min q cl q c,min Operating point q c = 8l.min-, = 46.K q c,max (4) q c [l.min - ] Figure 3: Static analyi of the reactor q cu

3 he tatic analyi for the different volumetric flow rate of the coolant qcmin, qc qc,max () wa done. he q c,min and q c,max denote minimal and maximal value of the volumetric flow rate of the coolant and their value are q c,min = l.min - and q c,max = l.min -. he reult are hown in Figure 3. NONLINEAR CONROLLER A it written above, the controller i divided into a tatic nonlinear part (SNP) and a dynamic linear part (DLP) ee Figure 4. where min repreent the lowet value of the teadytate reactant temperature, i.e. for the volumetric flow rate in the upper bound, q cu, in thi cae. he meaured data on the real model are uually affected by the meaurement error. hee error are here imulated by the random white-noie error. he teady-tate characteritic recomputed to the new coordinate ω and ψ i then hown in Figure ψ [K] 4 Figure 4: he cheme of the nonlinear controller he dynamic part DLP define linear dynamic relation between input to the nonlinear part u (t) and the difference between actual and deired reactant temperature (t), i.e. () () u t =Δ w t (6) he tatic part SNP decribe nonlinear relation between u (t) and correponding change of the input volumetric flow rate of the coolant Δq c (t). he interconnection of the controller and the controller plant can be found in the following Figure ω [-] Figure 6: Simulated characteritic ψ = f(ω) he invere of thi teady-tate characteritic i hown in Figure 7 and the reulted imulated data could be approximated by everal function from the ring of polynomial, exponential, rational etc. function. ω [-] ω(ψ) = 3.393x -4 ψ -.99ψ meaured data approximated data ψ [K] Figure : he control cheme he following chapter will decribe individual block in the Figure. Static Nonlinear Part (SNP) he SNP at it i come from the tatic analyi diplayed in Figure for volumetric flow rate between lower bound q cl = l.min - and upper bound q cu = l.min - and we introduce new x- and y-axi coordinate ω and ψ defined a qc qcl ω = [ ]; ψ = min [ K] (7) q cl Figure 7: Simulated (dotted) and approximated (line) characteritic ω= f(ψ) In our cae, the econd order polynomial wa ued for the approximation of the noied data. he reulted polynomial ha form 4 ωψ ( ) = ψ.99ψ (8) and a it can be een in Figure 7, thi function approximate the data in uitable way. he difference of the input volumetric flow rate of the coolant u(t) = Δq c (t) in the output from the nonlinear part can be computed dγ u() t =Δ qc() t = qcl u () t (9) dψ ψ ( )

4 he derivative dω/dψ in the previou equation i computed for each temperature of the reactant from the derivative of the function (8), i.e. dω 4 = ψ.99 () dψ External Linear Model (ELM) of CSR he dynamic behavior of the ytem how that thi ytem could be repreented by the econd order tranfer function with the relative order one: Y( ) b( ) b + b G( ) = = = () U( ) a( ) + a + a hi ELM belong to the cla of continuou-time (C) model. he identification of uch procee i not very eay. One way, how we can overcome thi problem i the ue of o called model. hi model belong to the cla of dicrete model but it parameter are cloe to the continuou one for very mall ampling period a it proofed in (Stericker and Sinha 993). he model introduce a new complex variable γ computed a (ee (Mukhopadhyay et al. 99)): z γ = () β v z+ ( β) v Where β i an optional parameter from the interval β and v denote a ampling period. It i clear that we can obtain infinite number of -model for variou β. A o called forward -model for β = wa ued and γ operator i then z γ = (3) v he continuou model () i then rewritten to the form a ( ) y( t ) = b ( ) u( t ) (4) where polynomial a () and b () are dicrete polynomial and their coefficient are different from thoe of the C model a() and b(). ime t' i dicrete time. Now we can introduce ubtitution t = k n for k n and Equation (4) then will be yk ( n) = buk ( n) + buk ( n) () a y( k n) a y( k n) which mean that the regreion vector ϕ i then ϕ ( k ) [ y ( k ), y ( k ), u ( k ), u ( k ) ] = (6) and the vector of parameter θ i generally ( k) a, a, b, b θ = (7) he differential equation (4) ha then vector form: y k = θ k ϕ k + e k (8) ( ) ( ) ( ) ( ) where e(k) i a general random immeaurable component. Identification of ELM parameter he Recurive Leat-Square (RLS) method i ued for the parameter etimation in thi work. he RLS method i well-known and widely ued for the parameter etimation. It i uually modified with ome kind of forgetting, exponential or directional. Parameter of the identified ytem can vary during the control which i typical for nonlinear ytem and the ue of ome forgetting factor could reult in better output repone. he baic RLS method i decribed by the et of equation: ε ( k) = y( k) ϕ ( ) ˆ k θ ( k ) ξ ( k) = + ϕ ( k) P( k ) ϕ ( k) (9) L( k) = ξ ( k) P( k ) ϕ ( k) P( k ) ϕ ( k) ϕ ( k) P( k ) P( k) = ( k ) λ ( k P ) λ ( k ) + ϕ ( k) P( k ) ϕ ( k) ˆ θ( k) = ˆ θ ( k ) + L( k) ε ( k) RLS with the changing exp. forgetting i ued for parameter etimation, where the changing forgetting factor λ i computed from the equation λ ( ) ( ) k = K ξ k ε ( k) () Where K i a mall number, in our cae K =.. Dynamic Linear Part (DLP) he DLP i contructed with the ue of polynomial approach (Kucera 993) imilarly a it wa ued in adaptive control decribed in (Vojteek et al. ). Figure 8: DOF control cheme in dynamic linear part he control ytem configuration with one degree-offreedom (DOF) with controller in the feedback part wa ued here and it i diplayed in Figure 8. he tranfer function of the controller Q() i deigned with the ue of polynomial ynthei: ( ) Q ( ) ( ) q = () p where degree of polynomial p () and q() are computed from: deg q( ) = deg a( ) ; deg p ( ) deg a( ) () and parameter of thee polynomial are computed by the Method of uncertain coefficient which compare coefficient of individual -power from the Diophantine equation, e.g. (Kucera 993): a p + b q = d (3) ( ) ( ) ( ) ( ) ( ) he polynomial d() on the right ide of (3) i an optional table polynomial. It i obviou, that the degree of thi polynomial i:

5 deg d( ) = deg a( ) + deg p ( ) + (4) and root of thi polynomial are called pole of the cloed-loop and their poition affect quality of the control. hi polynomial i deigned via well-known Pole-placement method. A choice of root need ome a priory information about the ytem behavior. It i good to connect pole with the parameter of the ytem via pectral factorization. he polynomial d() can be then rewritten to the form deg deg ( ) ( ) ( ) d d = n + α n () where α > i an optional coefficient reflecting cloedloop pole and table polynomial n() i obtained from the pectral factorization of the polynomial a() * * n ( ) n( ) = a ( ) a( ) (6) he Diophantine equation (3), a it i, i valid for tep change of the reference and diturbance ignal which mean that deg f() = in (). hi controller enure tability, load diturbance attenuation and aymptotic tracking of the reference ignal. he order of the polynomial q(), p ( ) and d() for econd order tranfer tranfer function () are: deg q( ) = deg a( ) = deg p ( ) deg a( ) deg p ( ) = (7) deg d( ) = deg a( ) + deg p ( ) + = + + = 4 he tranfer function of the controller i then q() q q q Q + + () = = (8) p () ( + p ) and the polynomial d() could be choen a d( ) = n( ) ( + α ) (9) Parameter of the polynomial n() which are computed from the pectral factorization are defined a: n = a, n = a + n a (3) he control ytem ynthei i done here in continuou time, but recurive identification ue dicrete time tep. he reulted, o called hybrid, controller work in the continuou time but parameter of the polynomial in the ytem tranfer function are identified recurively in the ampling period v. hi aumption reult in the condition, that the parameter of the -model are cloe the continuou one for the mall ampling period. SIMULAION RESULS All tudie were done in the mathematical oftware Matlab, verion 6.. and the common value for all imulation were: the ampling period wa v =.3 min, the imulation time 6 min and 6 different tep change were done during thi time. he initial vector of parameter ued for identification wa [ ] ˆ θ =.,.,.,. and the initial covariance matrix wa P ii = 7 for i =,..,4. he goal of the controller i to control the temperature, inide the reactor by the change of the volumetric flow rate of the coolant, i.e. u(t) = Δq c (t), y(t) = (t). he input variable i limited in the bound ±9 l.min -. he imulation wa done for different value of the poition of the parameter α in (9), α =.6,. and.8, and the reult are hown in Figure 9 and. w(t), y(t) [K] - w(t) y(t) (α =.6) - y(t) (α =.) y(t) (α =.8) Figure 9: Reult of the nonlinear adaptive control - the coure of the reference ignal w(t) and the output variable y(t) for different value of α u(t)[l.min - ] - u(t) (α =.6) - u(t) (α =.) u(t) (α =.8) Figure : Reult of the nonlinear adaptive control - the coure of the input variable u(t) for different value of α a (t),a (t)[-] b (t), b (t)[-] 4 3 a (t) a (t) b (t) b (t) 4 6 Figure : he coure of the identified parameter a, a, b and b for α =.

6 he uability of thi control trategy for uch type of chemical reactor i obviou. Although the tep change are relatively high (about ± K in ome cae), the controller dealt with it without big trouble. he only problem can be found at the very beginning of the control which wa caued by the recurive identification which need ome time for adapting to the right parameter. One ample coure of the identified parameter a, a, b and b for parameter α =. during the control i hown in Figure. Reult hown in previou figure clearly how that the increaing value of the parameter α affect mainly the peed of the output repone increaing value of α reult in quicker output repone. Very deirable from the practical point of view i alo mooth coure of the input variable to the ytem, u(t), which i produced by the controller and repreented by e.g. twit of the valve on the input pipe. CONCLUSION he paper how reult of nonlinear adaptive control of the continuou tirred-tank reactor with the piral cooling in the jacket. hi ytem belong to the cla of nonlinear ytem with lumped parameter and the mathematical model i decribed by the et of nonlinear ordinary differential equation. he novelty of thi method compared to the pure adaptive control can be found in the factorization of the controller into linear and nonlinear part. he nonlinear part i deigned with the ue of imulated teady-tate characteritic and appropriate modification. he nonlinearity of the controlled ytem i approximated via External Linear Model with recurively etimated parameter. he linear part of the controller employ polynomial approach and the output repone could be tuned via choice of the pole in cloed-loop ytem. Propoed controller produce mooth coure of the action value (input to the ytem) and conequently alo good and accurate coure of the controlled output from the ytem. he future work will be focued on the comparion of thi method to the pure adaptive control and application of thi control trategy to next type of technological procee uch a tubular reactor, batch reactor etc. REFERENCES Atolfi, A.; D. Karagianni; and R. Ortega. 8. Nonlinear and adaptive control with application. Springer-Verlag, London. Åtröm, K.J. and B. Wittenmark Adaptive Control. Addion Weley. Reading. MA, ISBN Bobal, V.; J. Böhm; J. Fel; and J. Machacek.. Digital Self-tuning Controller: Algorithm. Implementation and Application. Advanced extbook in Control and Signal Proceing. Springer-Verlag London Limited. ISBN Gao, R.; 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 Ingham, J.; I. J. Dunn; E. Heinzle; and J. E. Prenoil.. Chemical Engineering Dynamic. An Introduction to Modeling and Computer Simulation. Second. Completely Revied Edition. VCH Verlaggeellhaft. Weinheim. ISBN Kucera, V Diophantine equation in control A urvey. Automatica Middleton, R.H. and G. C. Goodwin. 4. Digital Control and Etimation - A Unified Approach. Prentice Hall. Englewood Cliff. ISBN Mukhopadhyay, S.; A. G. Patra; and G. P. Rao. 99. New cla of dicrete-time model for continuo-time ytem. International Journal of Control. vol Nakamura, M.;. Sugi and S. Goto.. "Nonlinear eparation model and control for a complex proce realized by conventional PID controller hardware". In Proceeding of the 4th Aian Control Conference, Singapore, Stericker, D.L. and N. K. Sinha Identification of continuou-time ytem from ample of input-output data uing the -operator. Control-heory and Advanced echnology. vol Sung, S. and J. Lee. 4. "Modeling and control of Wienertype procee". Chemical Engineering Science, 9, -. Vincent,;.L. and W.J. Grantham Nonlinear and optimal control ytem. John Wiley & Son, New York. ISBN Vojteek, J.; P. Dotal. Adaptive Control of Continuou-Stirred ank Reactor in wo Stable Steady- State, In Proceeding of the IFAC Workhop Adaptation and Learning in Control and Signal Proceing, Antalya, urkey, ISBN Vojteek, J.; J. Novak; P. Dotal.. Effect of External Linear Model Order on Adaptive Control of CSR. In Proceeding of he 9th IASED International Conference on Applied Simulation and Modelling (ASM ), p ISBN AUHOR BIOGRAPHIES JIRI VOJESEK wa born in Zlin. Czech Republic 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. Hi contact i vojteek@fai.utb.cz. PER DOSAL 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. You can contact him on addre dotalp@fai.utb.cz.