Effective Hybrid Adaptive Temperature Control inside Plug-flow Chemical Reactor

Transcription

1 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Volume, 6 Effective Hybrid Adaptive emperature Control inide Plug-flow Chemical Reactor Jiri Vojteek, Petr Dotal Abtract he paper deal with two method of hybrid adaptive control of the nonlinear ytem repreented by the plug-flow tubular chemical reactor. he mathematical model of thi type of technological procee i decribed by the et of partial differential equation which were olved numerically by the finite difference method and Runge-Kutta method. he adaptivity of the controller i atified by the recurive identification of the external linear model a a linear repreentation of the originally high nonlinear controlled ytem. he firt method ue well known Pole-placement method and the econd i baed on the more ophiticated LQ approach. he advantage of thee method i that both have tuning parameter which can affect control reult. Although the controlled ytem ha highly nonlinear behavior, ued adaptive controller ha good reult. Keyword Adaptive Control, Pole-placement Method, Recurive Identification, LQ Approach, Plug-flow ubular Chemical Reactor. I. INRODUCION HE controlling of chemical reactor i alway challenging becaue of the complexity of the ytem, hazardou and cot aving. he modeling of uch procee uually end with the complicated et of ordinary or even partial differential equation depending on the type of ytem []. he tubular plug-flow reactor belong to the ring of ytem with continuouly ditributed parameter, mathematical model of which ue partial differential equation (PDE) unfortunately in the nonlinear form []. he mathematical olution of the et of PDE ue Finite difference method which dicretize the equation in the axial variable which mean that the et of PDE i tranformed into the et of ordinary differential equation ODE that can be then olved for example by Runge-Kutta method [3] which i eaily programmable or even build-in function in mathematical oftware. Other numerical method are alo dicued in [4]. Once we have done the imulation of the teady-tate and dynamic behavior, we can continue with the choice of the optimal control trategy. here are everal let ay modern control method which were teted on thi or imilar type of J. Vojteek i with the Department of Proce Control, oma Bata Univerity in Zlin, nam. GM 5555, 76 Zlin, Czech Republic (correponding author to provide phone: ; fax: ; vojteek@fai.utb.cz). P. Dotal i with the Department of Proce Control, oma Bata Univerity in Zlin, nam. GM 5555, 76 Zlin, Czech Republic ( dotalp@fai.utb.cz). ytem the robut control, the predictive control or the adaptive control. he adaptive control [5] ha variou improvement and application. he approach applied in thi work ue reult from the dynamic analyi for the choice of the External Linear Model (ELM) parameter of which are etimated recurively during the control which atifie adaptivity of the controller [6]. Control ynthei ue a polynomial approach [7] which atifie baic control requirement like tability of the control loop, the reference ignal tracking and the diturbance attenuation. Another big advantage of thi method i that it provide not only the tructure of the controller but alo relation for computing of the controller parameter. hi method could ue alo other method like the Pole-placement method [7] and LQ approach. hee two method are dicued in thi work. Other, let u ay, modern control method are robut control [8] and predictive control [9]. Advantage of thee method can be found in better efficiency and verability. All experiment in the work are done by imulation uing mathematical oftware Matlab, verion 7... hee method were teted and can be ued alo for the controlling of real ytem, imilarly a in []. II. MODEL OF UBULAR CHEMICAL REACOR he ytem under the conideration i a tubular chemical reactor [] a typical nonlinear equipment ued in indutry. he reaction inide i a imple exothermic reaction in the liquid phae and the reactant i cooled by the cooling liquid inide the jacket of the reactor. he cheme of the reactor could be found in Fig.. he convection of the liquid in the pipe and the cooling jacket i expected to be plug-flow. hat i why are thee type of reactor called Plug-Flow Reactor (PFR). he mathematical model ue material and heat balance inide the reactor. he PFR diplayed in Fig. offer theoretically two type of cooling from the direction point of view co-current and counter-current cooling. It wa proofed for example in [], that the counter-current cooling, where the direction of the cooling flow i oppoite to the direction of the reactant ha better cooling efficiency. hi type of cooling i conidered in thi work mainly becaue of thi efficiency. ISSN:

2 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Volume, 6 he mathematical decription of uch model i very complex and there mut be introduced implification which reduce the complexity of the ytem: we expect, that all denitie, heat capacitie and heat tranfer coefficient are expected to be contant. Alo, we neglect heat loe and conduction along the metal wall of pipe. On the other hand, the heat tranfer through the wall i conequential for the dynamic tudy. A the pace variable i alo important in the mathematical decription, the mathematical model with all mentioned implification i decribed by the et of five partial differential equation (PDE) ca ca + vr = k ca t z cb cb + vr = k ca k cb t z r r hr 4 U () + vr = ( r w) t z ρr cpr d ρr cpr w 4 = d U ( ) ( ) r w + d U c w t d d ρ c ( ) w c 4 n d U v = t z d n d c pw ( 3 ) ( ) c c w c ρc pc where denote temperature, d are diameter of the pipe d i inner diameter of the pipe, d i outer diameter of the pipe and d 3 denote diameter of the jacket. hen, ρ are ued for denitie, c p for pecific heat capacitie, U denote heat tranfer coefficient, n i ued for number of individual pipe and L i length of the reactor. energie and R a a univeral ga contant. he lat, unmentioned variable in () i a reaction heat h r computed from h = h k c + h k c (5) r A B where h j are reaction enthalpie. Fixed parameter of the reactor [] are hown in the following able : able Fixed parameter of the reactor Parameter Inner diameter of the tube Outer diameter of the tube Inner diameter of the reactor Number of pipe Length of the reactor Volum. flow rate of the reactant Volum. flow rate of the cooling Denity of the reactant Denity of the metal wall Denity of the cooling Heat capacity of the reactant Heat capacity of the metal wall Heat capacity of the cooling Heat tranfer coefficient Heat tranfer coefficient Pre-exponential factor Pre-exponential factor Activation energy /ga contant Activation energy /ga contant Reaction enthalpy Reaction enthalpy Input concentration of comp.a Input temperature of the reactant Input temperature of the cooling Notation and value d =. m d =.4 m d 3 = m n = L = 6 m q r =.5 m 3. - q c =.75 m 3. - ρ r = 985 kg.m 3 ρ w = 78 kg.m 3 ρ c = 998 kg.m 3 c pr = 4.5 kj.kg -.K - c pw =.7 kj.kg -.K - c pc = 4.8 kj.kg -.K - U =.8 kj.m -.K -. - U =.56 kj.m -.K -. - k = k = E /R = 3477 K E /R = 59 K h = kj.kmol - h =.8 4 kj.kmol - c A =.85 kmol.m -3 r = 33 K c = 93 K Since the mathematical model of the ytem () i decribed by the et of nonlinear partial differential equation, we are talking about the nonlinear ditributed-parameter ytem. Fig. Scheme of the plug-flow tubular chemical reactor he variable v r and v c are fluid velocitie computed from the volumetric flow rate q and contant f, e.g. qr qc v = r ; vc f = f () r c Where contant f r and f c are connected to the tructure of reactor π d π f = n ; f = d n d (3) r ( ) c he main nonlinearity of thi ytem can be found in reaction velocitie k and k which are nonlinear function of the rectant temperature r according to the Arrheniu law: E j kj = k j exp, for j =, (4) R r with k j a a pre-exponential factor, E j a a activation III. SEADY-SAE AND DYNAMIC ANALYSES he tatic and dynamic analye are uually the firt tep after the modelling part. he goal of thee tudie i at firt verify propoed mathematical model with meaurement on the real ytem. Sometime implification reduce the accuracy of the mathematical decription and the ue of the mathematical model i unacceptable. he econd reaon why we do thee analye i that we need to know the behavior of the ytem for finding of the optimal working point, limitation etc. he tep repone in the dynamic analyi are alo ued for the choice of the External Linear Model in adaptive control decribed later in thi work. A there are theoretically more input and output variable, the change of the cooling volumetric flow rate, q c, wa choen a a input variable for the reactant temperature, r, a an output variable. he volumetric flow rate a an input wa choen from the practical point of view it i repreented by the twit of the valve in thi cae. On the other hand, the ISSN:

3 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Volume, 6 output temperature i better meaured than the output concentration. A. Steady-tate Analyi he tatic analyi explore the behavior of the ytem in teady-tate, i.e. in the tate when tate variable doe not change. Mathematically peaking, the derivative with repect to time are equal to zero in the teady-tate and the et of partial differential equation () i tranformed to the et of ordinary differential equation with repect to pace variable z. he Finite difference method i employed here for olving of thi problem. Derivative with repect to pace variable are replaced by the firt back difference dx x() i x( i ),for i =,, n (6) dz z= z h i z with x a a general variable and tep ize h z = L/N z. A the ytem ha counter-current cooling, the temperature of the cooling c i decribed in the oppoite coordinate and the lat fifth equation in () ue the firt forward difference dx x( j+ ) x( j), for j = n, n-, (7) dz z= z h j z he teady-tate analyi i then olution of the cycle of dicrete equation for different value of the input variable, in thi cae volumetric flow rate of the cooling q c. he tatic analyi wa done for variou value of the cooling volumetric flow rate q c = <.;.35> m 3. - and value of the teady-tate reactant temperature, r, through the length of the reactor (axial variable z = <; 8> m) are hown in Fig.. r [K] z [m] q c [m 3. - ] Fig. Steady-tate characteritic of the reactant temperature, r, for different volumetric flow rate of the coolant q c through the length of the reactor Reult of the teady-tate analyi clearly how the nonlinearity of the ytem. he optimal working point i defined for the volumetric flow rate of the reactant q r =.5 m 3. - and the volumetric flow rate of the coolant q c =.75 m 3. - and thi working point wa ued later in the dynamic analyi and alo in the adaptive control. B. Dynamic Analyi he dynamic analyi oberve the behavior of the output variable, reactant temperature at the end of the reactor r (L), after the tep change of the input variable, in thi cae tep change of the volumetric flow rate of the coolant, Δq c. he input, u(t), and the output, y(t) variable for both dynamic and control purpoe are then qc( t) qc u() t = [%]; y() t = r ( t, L ) r ( L)[ K] (8) qc where q c i volumetric flow rate at the working point and r (L) i the teady-tate value of the output variable in the working point which i alo initial value for the dynamic tudy. hi mean, that the graph tart from zero. From the mathematical point of view, the dynamic analyi i the numerical olution of the et of partial differential equation (). he numerical olution of PDE i not imple and the combination of the Finite difference method decribed above which tranform the et of PDE to the et of ordinary differential equation (ODE) wa ued here. he et of ODE i then olved numerically with the ue of Runge- Kutta method. here were done everal tep change and reult are hown in the following Fig. 3. y(t) [K] % - % - % + % + % +4 % t [] Fig. 3 Dynamic characteritic for variou tep change of the input variable It i clear, that the poitive change of the input variable reult in decreaing value of the output reactant temperature and converely, the negative change of q c produce poitive change of the output temperature. All coure of the output variable could be decribed by econd order tranfer function which will be ued later in the adaptive control. IV. ADAPIVE CONROL Once we have information about the ytem behavior in the teady-tate and dynamic, we can move on to the controller deign. here are everal control method which can be ued for uch nonlinear proce like predictive control, robut control etc. he adaptive approach wa ued in thi work becaue author have good experience with the uage of thi control method for imilar type of technological ISSN:

4 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Volume, 6 procee like heat exchanger, continuou tirred-tank reactor (CSR), water tank etc. An advantage of thi method can be alo find in the big theoretical background, modification and application. he term Adaptivity come from the nature, where animal and plant adopt their behavior depending on the living environment and condition. Similarly, the adaptive controller could adopt (e.g. change) it parameter or tructure according to the actual tate of the ytem and control requirement. here are, of coure, variou adaptive control trategie. A it i already mentioned, the adaptive approach here i baed on on-line recurive identification of the External Linear Model (ELM) which repreent original, nonlinear, proce. Parameter of the controller depend on parameter of the ELM and change in every identification tep according to the identified parameter of the ELM. A. External Linear Model he choice of the ELM come from the dynamic analyi preented above. he output repone have hown, that the change of the output temperature a the output y(t) to the input variable u(t) in Fig. 3 could be decribed by the continuoutime (C) model b( ) b + b G( ) = = (9) a( ) + a + a On-line identification of the C model i complicated. he dicrete-time (D) model are ued more often. hee model do not decribe the ytem in the very accurate way it depend on the choice od the ampling period v. Compromie could be found in the ue of delta-model a a pecial type of the D model where value of the input and output variable are related to the ampling period and it wa proofed, that parameter of the delta-model approache to the parameter of the C model. he delta model introduce a new complex variable γ [3] z γ = () v he ELM (9) could be then rewritten to the form of the differential equation y ( k) = bu ( k ) + bu ( k ) () a y ( k ) a y ( k ) where b, b, a, a are delta-parameter imilar to thoe in (9) for mall ampling period [4]. Delta value of input and output variable in Equation () can be computed a yk ( ) yk ( ) + yk ( ) y ( k) = v yk ( ) yk ( ) uk ( ) uk ( ) () y( k ) = u( k ) = v v y( k ) = y( k ) u( k ) = u( k ) he regreion vector ϕ and the vector of parameter θ are [ ] θ ( k) a, a, b, b ϕ ( k ) = y ( k ), y ( k ), u ( k ), u ( k ) = (3) and the differential equation () ha then vector form y ( k) = θ ( k) ϕ ( k ) + e( k) (4) where e(k) i a general random immeaurable component and the tak of the identification i to etimate the vector of parameter θ from known data vector ϕ. B. Recurive Identification It wa already mentioned, that adaptivity in thi approach i baed on the on-line parameter identification of the ELM. he recurive identification mathematically mean the etimation of the vector of parameter θ from the differential equation (4). he method ued here i a imple Recurive Leat- Square (RLS) method [5] which can be eaily programmed and alo extended by the additional forgetting technique. Generally, the RLS method ued for etimation of the vector of parameter ˆθ ( k ) could be decribed by the et of equation: ε ( k) = y( k) ϕ ( k) ˆ θ( k ) γ ( k) = + ϕ ( k) P ( k ) ϕ( k) (5) L( k) = γ ( k) P ( k ) ϕ ( k) ( k ) ( k) ( k) ( k ) ( k) P ϕ ϕ P P = P( k ) λ ( k ) λ ( k ) + ϕ ( k) P ( k ) ϕ( k) λ ( k ) ˆ θ( k) = ˆ θ ( k ) + L( k) ε ( k) where ε denote a prediction error, P i a covariance matrix and λ and λ are forgetting factor. For example contant exponential forgetting [5] ue λ = and λ ( ) ( ) k = K γ k ε ( k) (6) where K i a very mall value (e.g. K =.). C. Control Synthei It wa already mentioned, that parameter of the ELM are ued in the computation of the controller. he polynomial ynthei i employed here becaue it provide not only the tructure of the controller but alo relation for computing of the controller parameter. Negligible advantage could be found alo in the fulfillment of the baic control requirement and eaily programmability. he implet one degree-of-freedom (DOF) divide the control loop into two part the tranfer function G() repreenting controlled plant (i.e. the ELM of the ytem) and the tranfer function of the controller Q() ee Fig. 4. Fig. 4 One degree-of-freedom (DOF) control configuration ISSN:

5 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Volume, 6 he ignal w in Fig. 4 repreent reference ignal (i.e. wanted value), u i control input, y controlled output, v denote random error and e i control error e = w y. he tranfer function of the controlled plant G() i known from the recurive identification and the tranfer function of the controller i generally q( ) Q ( ) = (7) p ( ) he parameter of the polynomial p ( ) and q() are computed from the Diophantine equation a( ) p ( ) + b( ) q( ) = d( ) (8) by the Method of uncertain coefficient which compare parameter of individual -power in (8). Polynomial a() and b() are known from the recurive identification and the polynomial d() on the right ide of the (8) i table optional polynomial the choice of which affect mainly the quality of the control. wo method of chooing of thi polynomial are dicued and teted in the next chapter Pole-placement method and LQ approach. D. Pole-placement Method he implet way i to chooe the polynomial d() by the Pole-placement method which divide the polynomial generally to deg d( ) d( ) = ( + α ) (9) i= with the tability condition α >. Degree of polynomial p ( ) and q() from (7) and the polynomial d() in (8) are for thi econd order tranfer function with relative order one (9) deg p ( ) = deg a( ) = deg q( ) = deg a( ) = () deg d( ) = deg a( ) + deg p ( ) + = 4 which mean that the tranfer function of the controller i q + q + q Q( ) = ` () ( + p ) and the polynomial d() ha four root. Diadvantage of thi method i that there i no rule how to chooe there root. We can have one quadruple root, two double root, one ordinary and one triple root or four different root. Our previou experiment have hown that it i good to connect the choice of the polynomial d() with the controlled ytem, for example with the ue of pectral factorization of the polynomial a() in the numerator of the tranfer function G(). Let u introduce new polynomial n() computed from the pectral factorization of the polynomial a(), i.e. * * n ( ) n( ) = a ( ) a( ) () It i clear, that thi polynomial ha the ame degree a the polynomial a() and a it i a part of the polynomial d(), we can rewrite thi polynomial to the form d( ) = n( ) ( + α ) (3) which mean that we have reduced the uncertainty to one double root. he controller deigned with thi method ha one tuning parameter α which could affect the quality of control. E. LQ Approach he econd, let ay a bit ophiticated, method i for deigning of the polynomial d() i the ue Linear-Quadratic (LQ) approach which i baed on the minimization of the cot function { μ () ϕ ()} J = e t + u t dt (4) LQ LQ LQ in the complex domain. Parameter > and μ LQ are weighting coefficient, e(t) i the control error and u ( t) denote the difference of the input variable. If we ue again the pectral factorization of the polynomial a(), imilarly a in previou cae, the polynomial d() i then divided into d( ) = n( ) g( ) (5) where the polynomial i olution of the minimization of (4), mathematically olution of the pectral factorization * * ( a( ) f ( ) ) ϕlq a( ) f ( ) + b ( ) μlq b( ) = (6) * = g g ( ) ( ) Degree of the controller polynomial p ( ) and q() and the polynomial d() on the right ide of Diophantine equation are for the econd order ELM (9) deg p ( ) deg a( ) = deg q( ) = deg a( ) + deg f ( ) = (7) deg d( ) = deg a+ = 5 and the tranfer function of the controller i q + q + q Q( ) = (8) ( + a + p) he LQ adaptive controller ha two tuning parameter, weighting factor and μ LQ but our experiment have hown that i good to fix one parameter and change only the econd one [6]. V. SIMULAION RESULS Both technique were teted by the imulation on the mathematical model (). he control output i the change of the input volumetric flow rate of the coolant in % and the controlled output i the change of the output temperature, imilarly a it i in (8): qc( t) qc u() t = [%]; y() t = r ( t, L ) r ( L)[ K] (9) qc Due to better comparability of thee method are alo imulation parameter the ame. he ampling period wa ISSN:

6 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Volume, 6 v =.5, the imulation time wa 8 and there were done four different tep change to the poitive and negative value during thi time. he firt control imulation wa done for the Poleplacement method and variou value of the parameter α. =.7;. and.. w(t), y(t) [K] w(t) y(t) α =.7 y(t) α =. y(t) α = t [] Fig. 5 he coure of the reference ignal, w(t), and the output variable, y(t), for variou value of α, Pole-placement method u(t)[%] 4 - u(t) α =.7 u(t) α =. u(t) α = t [] Fig. 6 he coure of the input variable, u(t), for variou value of α, Pole-placement method a (t)[-] a (t)[-] α =.7 α =. α = x α =.7 α =. α = Fig. 7 he coure of identified parameter a (t) and a (t) for variou value of α, Pole-placement method Obtained imulation reult in Fig. 5 a 6 have hown that the increaing value of α reult in quicker output repone but overhoot of the output variable y(t). he coure of the control (input) variable u(t) i moother for lower value of α. A it wa already written, adaptive approach here i baed on the recurive identification of the ELM (9). he recurive leat quare method with exponential forgetting wa ued for online identification of parameter a (t), a (t), b (t) and b (t) and reult are hown in Fig. 7 and 8. You can ee that there i only problem with identification at the very beginning of the control where controller doe not have any information about the ytem and tarting value of the vector of parameter i generally θ ( ) [.,.,.,.] =. he controller need ome time for adaptation but the etimation i much moother after initial 5 min for all identified parameter. b (t)[-] b (t)[-] x α =.7 α =. α = x α =.7 α =. α = Fig. 8 he coure of identified parameter b (t) and b (t) for variou value of α, Pole-placement method he econd analyi wa done for LQ approach and different value of weighting parameter =.5;. and. and the reult are hown in Fig. 9 and. Although there are imilar value of the weighting parameter a α in previou cae, the meaning of thi parameter i different. In thi cae, increaing value of parameter reult in lower, more ocillating output repone but moother coure of the input variable which could be ometime good from the practical point view. he ue of LQ approach produce generally more ocillating output repone but both control technique could be ued for controlling of uch trongly nonlinear procee. ISSN:

7 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Volume, 6 w(t), y(t) [K] w(t) y(t) =.5 y(t) =. y(t) = t [] Fig. 9 he coure of the reference ignal, w(t), and the output variable, y(t), for variou value of, LQ approach u(t)[%] a (t)[-] 4 - u(t) =.5 u(t) =. u(t) = t [] Fig. he coure of the input variable, u(t), for variou value of, LQ approach. a (t)[-].. -. =.5 =. = x -4 =.5 =. = Fig. he coure of identified parameter a (t) and a (t) for variou value of, LQ approach Fig. and Fig. repreenting reult of online identification how very imilar reult to thoe mentioned above for previou control approach. We can ay here that initial adaptation i much quicker than for previou cae. b (t)[-] b (t)[-] =.5 =. = x -5 =.5 =. = Fig. he coure of identified parameter b (t) and b (t) for variou value of, LQ approach All reult of identification preented in Fig. 7, Fig. 8, Fig. and Fig. how uability of thi recurive leat-quare method. Moreover, we can ee, that identified parameter do not change dramatically after ome, already mentioned, initial adaptation time. Here rie the quetion: I online recurive identification important here, where parameter doe not change? Of coure, we can ue controller with fixed parameter but what if the control condition change? What if there occur unexpected diturbance. In thee cae i the ue of online identification very good option. hoe controller react to thee change quickly and provide more optimal reult. Obtained reult were dicued only from the viual view until now but it i good to have any mathematical decription of reult for comparion. We can ue for example imple quadratic criterion S u and S y which quantitatively decribe the coure of the output variable, y(t), or it difference from the reference ignal, w(t),repectively and the change of the input variable, u(t): N Su = ( u() i u( i )) [ ] ; i= f, for N = (3) N v Sy = ( w() i y() i ) K i= Obtained value of thee quadratic criterion are hown in able, 3 and following figure 3-7. able Computed value of quadratic criterion S u and S y in control with Pole-placement method S u [-] S y [K ] α = α = α = ISSN:

8 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Volume, 6 able 3 Computed value of quadratic criterion S u and S y in control with LQ method S u [-] S y [K ] = = = Preented value of criterion can help u with the choice of the optimal value of tuning parameter α or. For example, able and graph in Figure 3 and 5 which repreent control approach with Pole-placement method indicate, that from the input point of view are the bet reult for control with α =.. hi i repreented by the lowet value of the criterion S u that um quare of change of the input variable. Alo, the value of the econd criterion S y denoting the um of control error (w y) i alo the lowet for the lat control trategy. We can ay, that thi etting ha the bet reult and could be teted on the real proce S u i alo one of the lowet for thi etting, we can ay that control with =.5 ha the bet reult. If we compare alo thee criterion for both control trategie, Pole-placement method produce better value of the criterion S u then LQ method but thi method ha, on the other hand, better reult of the criterion S y. S y [K ] α =.7 α =. α =. Fig. 5 Value of the quadratic criterion S y for variou value of α, Pole-placement method 3 S u [-] S y [K ] α =.7 α =. α =. Fig. 3 Value of the quadratic criterion S u for variou value of α, Pole-placement method 8 =.5 =. =. Fig. 6 Value of the quadratic criterion S y for variou value of, LQ approach 7 S u [-] =.5 =. =. Fig. 4 Value of the quadratic criterion S u for variou value of, LQ approach On the other hand, reult for the next, LQ, trategy doe not indicate o clear reult. Value of input quadratic criterion S u are very imilar and the criterion S y i the lowet for the firt value of =.5. A the value of the criterion VI. CONCLUSION he paper preent two modification of the adaptive control applied on the control of the reactant temperature inide the tubular chemical reactor a a typical nonlinear ytem with ditributed parameter. he nonlinear ytem i decribed by the external linear model in the general form parameter of which are etimated recurively during the control which fulfill the adaptivity of the ytem. he difference between thee two modification i in the choice of the table polynomial in the Diophantine equation. he firt method ue imple Pole-placement method with pectral factorization and the econd modification i baed on the LQ approach again together with the pectral factorization of the polynomial in the denominator of the ELM. Both method have tuning parameter which can affect the quality of control, mainly the peed of the control and the overhoot. Obtained ISSN:

9 INERNAIONAL JOURNAL OF MAHEMAICS AND COMPUERS IN SIMULAION Volume, 6 imulation reult have hown the uability of the adaptive control for controlling of uch complex nonlinear ytem. Obtained reult were alo dicued and quantified by the quadratic criterion that ummarize change of the input variable and the control error. he choice of the bet controller etting alway depend on the main purpoe of the control, e.g. if the minimal control error or the change of the input variable. hee change of the input variable are important mainly from the practical point of view. he future work could be focued on the verification of the obtained reult on the real chemical reactor. REFERENCES [] J. Ingham, I. J. Dunn, E. Heinzle, J. E. Přenoil, Chemical Engineering Dynamic. An Introduction to Modelling and Computer. Simulation. Second, Completely Revied Edition, VCH Verlaggeellhaft, Weinheim,. [] W. L. Luyben, Proce Modelling, Simulation and Control for Chemical Engineer. McGraw-Hill, New York 989. [3] R. L. Johnton, Numerical Method. John Wiley & Son. 98 [4] R. B. Gnitchogna, A. Atangana. 5. Comparion of two iteration method for olving nonlinear fractional partial differential equation. International Journal of Mathematical Model and Method in Applied Science. Volume 9, 5, Page 5-3, ISSN: [5] K. J. Åtröm and B. Wittenmark 989. Adaptive Control. Addion Weley. Reading. MA. [6] V. Bobál, J. Böhm, J. Fel, J. Macháček, Digital Self-tuning Controller: Algorithm, Implementation and Application. Advanced extbook in Control and Signal Proceing. Springer-Verlag London Limited, 5. [7] V. Kučera, Diophantine equation in control A urvey Automatica, 9, , 993. [8] R. Matuu. 4. Robut tability analyi of dicrete-time ytem with parametric uncertainty: A graphical approach. International Journal of Mathematical Model and Method in Applied Science, Volume 8, Iue, 4, Page 95-, ISSN: [9] V. Bobal, P. Chalupa, P. Dotal, M. Kubalcik.. Deign and imulation verification of elftuning mith predictor. International Journal of Mathematic and Computer in Simulation. Volume 5, Iue 4,, Page 34-35, ISSN: [] D. Honc, F. Dušek.. Novel multivariable laboratory plant. Proceeding - 6th European Conference on Modelling and Simulation, ECMS ; [] P. Dotál, R. Prokop, Z. Prokopová, M. Fikar, Control deign analyi of tubular chemical reactor. Chemical Paper, 5, 95-98, 996. [] J. Vojtěšek, P. Dotál, R. Matušů, Effect of Co- and Counter-current Cooling in ubular Reactor, In: Proc. 7th International Scientific- echnical Conference Proce Control 6. Kouty n. Denou. Czech Republic, 6. [3] S. Mukhopadhyay, A. G. Patra, G. P. Rao, New cla of dicrete-time model for continuou-time ytem, International Journal of Control, vol.55, 6-87,99. [4] 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, 3-5, 993. [5] M. Fikar and J. Mikleš. 8. Proce modelling, optimization and control, Springer-Verlag, Berlin. [6] J. Vojteek and P. Dotal. Effect of Weighting Factor in Adaptive LQ Control. In Notradamu 3: Prediction, Modeling and Analyi of Complex Sytem. Springer-Verlag Berlin, 3. ISSN: