One approach to adaptive control of a tubular chemical reactor

One Approach to Adaptive Control of a Tubular Chemical Reactor PETR DOSTÁL, VLADIMÍR BOBÁL, JIŘÍ VOJTĚŠEK, and ZDENĚK BABÍK Tomas Bata University in Zlin Department of Process Control, Centre of Polymer Systems nam. T.G. Masaryka 5555, 760 01 Zlin CZECH REPUBLIC {dostalp, bobal, vojtesek, babik}@fai.utb.cz http://www.fai.utb.cz/ Abstract: – The paper deals with continuous-time adaptive control of a tubular chemical reactor with the countercurrent cooling as a nonlinear single input – single output process. The mean reactant temperature and the output reactant temperature are chosen as the controlled outputs, and, the coolant flow rate as the control input. The nonlinear model of the reactor is approximated by an external linear model with a structure chosen on the basis of controlled outputs step responses. Its parameters are estimated via corresponding delta model. The control system structure with two feedback controllers is considered. The resulting controllers are derived using polynomial approach. The method is tested on a mathematical model of the tubular chemical reactor. Key-Words: – Nonlinear system, tubular chemical reactor, approximate linear model, parameter identification, polynomial approach, pole assignment. 1 Introduction Tubular chemical reactor are units frequently used in chemical industry. From the system theory point of view, tubular chemical reactors belong to a class of nonlinear distributed parameter systems with mathematical models described by sets of nonlinear partial differential equations (NPDRs). The methods of modelling and simulation of such processes are described e.g. in [1] – [5]. It is well known that the control of chemical reactors, and, tubular reactors especially, often represents very complex problem. The control problems are due to the process nonlinearity, its distributed nature, and high sensitivity of the state and output variables to input changes. Evidently, the process with such properties is hardly controllable by conventional control methods, and, its effective control requires application some of advanced methods. Here, various efficient methods may be used as the predictive control, e.g. [6], [7], [8], the robust control, e.g. [9], the fuzzy nonlinear control, e.g. [10], the model reference control, e.g. [11], or nonlinear control, e.g. [12], [13] and [14]. Some others methods are described in [15]. One possible method to cope with this problem is using adaptive strategies based on an appropriate choice of a continuous-time external linear model (CT ELM) with recursively estimated parameters. These parameters are consequently used for parallel updating of controller‘s parameters. Some results obtained in this field were presented by authors of this paper e.g. in [16] and [17]. For the CT ELM parameter estimation, either the direct method [18] and [19] or application of an external delta model with the same structure as the CT model can be used. The basics of delta models have been described in e.g. [20] and [21]. Although delta models belong into discrete models, they do not have such disadvantageous properties connected with shortening of a sampling period as discrete zmodels. In addition, parameters of delta models can directly be estimated from sampled signals. Moreover, it can be easily proved that these parameters converge to parameters of CT models for a sufficiently small sampling period (compared to the dynamics of the controlled process), as shown in [22]. This paper deals with continuous-time adaptive control of a tubular chemical reactor with a countercurrent cooling as a nonlinear single input – single output process. With respect to practical possibilities of a measurement and control, the mean reactant temperature and the output reactant temperature are chosen as the controlled outputs, and, the coolant flow rate as the control input. The nonlinear model of the reactor is approximated by a CT external linear model with a structure chosen on the basis of computed controlled outputs step responses. The parameters of the CT ELM then are estimated via corresponding delta model. The control structure with two feedback controllers is considered, e.g. [23]. The resulting controllers are derived using the polynomial approach [24] and the pole assignment method (see, e.g. [25]). The method WSEAS TRANSACTIONS on FLUID MECHANICS Petr Dostál, Vladimír Bobál, Jiří Vojtěšek, Zdeněk Babík E-ISSN: 2224-347X 13 Issue 1, Volume 7, January 2012 is tested on a mathematical model of a tubular chemical reactor. 2 Model of the Reactor An ideal plug-flow tubular chemical reactor with a simple exothermic consecutive reaction 12 kk ABC→→ in the liquid phase and with the countercurrent cooling is considered. Heat losses and heat conduction along the metal walls of tubes are assumed to be negligible, but dynamics of the metal walls of tubes are significant. All densities, heat capacities, and heat transfer coefficients are assumed to be constant. Under above assumptions, the reactor model can be described by five PDRs in the form 1 AA rA cc vkc tz ∂∂ +=− ∂∂ (1) 12 BB rAB cc vkckc tz ∂∂ +=− ∂∂ (2) 1 1 4() () () rrr rrw pr pr TTQ U vTT tzcdc ρρ ∂∂ += − − ∂∂ (3) [ ] 11 22 21 22 4() ()() ( wrw pw cw TdU T T tdd c dU T T ρ ∂=−+ ∂− +− (4) 12 2 22 312 4() ()() cc cwc pc TT ndU vTT tz dnd c ρ ∂∂ −= − ∂∂ − (5) with initial conditions (,0) () s AA cz cz=, ( ,0) ( ) s BB cz cz=, ( ,0) ( ) s rr Tz T z=, (,0) () s ww Tz Tz=, ( ,0) ( ) s cc Tz T z= and boundary conditions 0 (0, ) ( ) AA ctct=(kmol/m3), 0 (0, ) ( ) BB ctct=(kmol/m3), 0 (0, ) ( ) rr TtTt=(K), (,) () ccL TLt T t=(K). Here, t is the time, z is the axial space variable, c are concentrations, T are temperatures, v are fluid velocities, d are diameters, ρ are densities, cp are specific heat capacities, U are heat transfer coefficients, n1 is the number of tubes and L is the length of tubes. The subscript (⋅)r stands for the reactant mixture, (⋅)w for the metal walls of tubes, (⋅)c for the coolant, and the superscript (⋅)s for steady-state values. The reaction rates and heat of reactions are nonlinear functions expressed as 0exp j jj r E kk R T − ⎛⎞ =⎜⎟ ⎝⎠ , j = 1, 2 (6) 11 2 2 ()() rrArB QHkcHkc=−Δ +−Δ (7) where k0 are pre-exponential factors, E are activation energies, () r H −Δ are in the negative considered reaction entalpies, and R is the gas constant. The fluid velocities are calculated via the reactant and coolant flow rates as 2 11 4r r q vnd π = , 22 312 4 () c c q vdnd π =− (8) The parameter values with correspondent units used for simulations are given in Table 1. Table 1. Used parameter values L = 8 m n1 = 1200 d1 = 0.02 m d2 = 0.024 m d3 = 1 m ρr = 985 kg/m3 cpr = 4.05 kJ/kg K ρw = 7800 kg/m3 cpw = 0.71 kJ/kg K ρc = 998 kg/m3 cpc = 4.18 kJ/kg K U1 = 2.8 kJ/m2s K U2 = 2.56 kJ/m2s K k10 = 5.61⋅1016 1/s k20 = 1.128⋅1018 1/s E1/R = 13477 K E2/R = 15290 K (-ΔHr1) = 5.8⋅104 kJ/kmol (-ΔHr2) = 1.8⋅104 kJ/kmol From the system engineering point of view, out (,) AA cLt c=, out (,) BB cLt c=, out (,) rr TLt T= and out (0, ) cc TtT= are the output variables, and, () r qt, ( ) c qt, 0() A ct, 0() r Ttand ( ) cL Tt are the input variables. Among them, for the control purposes, mostly the coolant flow rate can be taken into account as the control variable, whereas other inputs entering into the process can be accepted as disturbances. In this paper, the mean reactant temperature given by 0 1 () (,) L mr Tt Tztdz L =∫ (9) and the reactant output temperature out () r Tt are considered as the controlled outputs. 3 Computation Models For computation of both steady-state and dynamic characteristics, the finite diferences method is employed. The procedure is based on substitution of the space interval 0,zL∈< > by a set of discrete WSEAS TRANSACTIONS on FLUID MECHANICS Petr Dostál, Vladimír Bobál, Jiří Vojtěšek, Zdeněk Babík E-ISSN: 2224-347X 14 Issue 1, Volume 7, January 2012 node points { } i zfor i = 1, … , n , and, subsequently, by approximation of derivatives with respect to the space variable in each node point by finite differences. Two types of finite differences are applied, either the backward finite difference 1 (,) ( ,) (,) (,) ( 1,) i ii zz yz t yz t yzt zh yit yi t h − = − ∂≈= ∂ −− = (10) or the forward finite difference 1 (,)(,) (,) (1,) (,) i ii zz yz t yz t yzt zh yi t yit h + = − ∂≈= ∂ +− = . (11) Here, a function ( , )yzt is continuously differentiable in the interval 0,L<> , and, hLn= is the diskretization step. 3.1 Dynamic model Applying the substitutions (10), (11) in (1) – (5) and, omitting the argument t in parenthesis, PDRs (1) – (5) are approximated by a set of ODRs in the form [] 01 0 () () () ( 1) AAA dc i bkicibci dt =− + + − (12) [] 102 0 () () () () () (1) BAB B dc i kic i b kic i dt bc i =−+ + +− (13) 1020 2 () () ( ) () ( 1) () rrrr w dT i bQ i b b T i b T i dt bT i =−+ +−+ + (14) [][] 34 () () () () () wrw cw dT i bTi Ti bTi T i dt =−+− (15) 56 5 6 () ()()(1) () ccc w dT m bbTmbTm dt bT m =− + + + + + (16) for 1,...,in= and 1mni=−+, and, with initial conditions (,0) () s AA ci ci=, (,0) () s BB ci ci=, (,0) () s rr Ti T i=, (,0) () s ww Ti Ti= and ( ,0) ( ) s cc Ti T i= for 1,...,in=. The boundary conditions enter into Eqs. (12) – (14) and (16) for i = 1 . Now, nonlinear functions in Eqs. (12) – (16) take the discrete form 0 () exp () j jj r E ki k R Ti − ⎛⎞ =⎜⎟ ⎝⎠ , j = 1, 2 (17) 11 22 () ( ) () () ( ) () () rrA rB Qi H kic i Hkici =−Δ + +−Δ (18) for i = 1, … , n. The parameters b in Eqs. (12) – (16) are calculated from formulas 0r v bh =, 1 1 () p r bc ρ =, 1 21 4 () p r U bdc ρ =, 11 322 21 4 ()() p w dU bdd c ρ =−, 22 422 21 4 ()() p w dU bdd c ρ =− (19) 5c v bh =, 12 2 622 312 4 ()() p c ndU bdnd c ρ =−. Here, the formulas for computation of Tm and ourt T take the discrete form 1 1 () ( ,) n mri i Tt Tzt n= =∑, out () ( ,) rrn TtTzt= (20) 3.2 Steady-state model Computation of the steady-state characteristics is necessary not only for a steady-state analysis but the steady state values ( ) s yi also constitute initial conditions in ODRs (12) – (16) (here, y presents some of the variable in the set (12) – (16)). The steady-state model can simply be derived equating the time derivatives in (12) – (16) to zero. Then, after some algebraic manipulations, the steady-state model takes the form of difference equations 0 01 () ( 1) () ss AA s b ci ci bki =− + (21) 10 02 1 () () () ( 1) () ssss BAB s ci kici bci bki ⎡ ⎤ =+− ⎣ ⎦ + (22) 10 2 02 1 () () ( 1) () ssss rrrw Ti bQi bTi bTi bb ⎡ ⎤ =+−+ ⎣ ⎦ + (23) 34 34 1 () () () sss wrc Ti bTi bTi bb ⎡ ⎤ =+ ⎣ ⎦ + (24) WSEAS TRANSACTIONS on FLUID MECHANICS Petr Dostál, Vladimír Bobál, Jiří Vojtěšek, Zdeněk Babík E-ISSN: 2224-347X 15 Issue 1, Volume 7, January 2012 56 56 1 () ( 1) () sss ccw Tm bTm bTm bb ⎡⎤ =++ ⎣⎦ + (25) for 1,...,in= and 1mni=−+. Nonlinear functions accordant with a steady-state are 0 () exp () j s jj s r E ki k R Ti ⎛⎞ − =⎜⎟ ⎜⎟ ⎝⎠ , j = 1, 2 (26) 11 22 () ( ) () () ( ) () () sss rrA ss rB Qi H kici Hkici =−Δ + +−Δ (27) Now, the formulas for computation T m and ourt T have the form 1 1() n ss mri i TTz n= =∑, out () ss rrn TTz= (28) 3.3 Steady-state and dynamic characteristics Typical reactant temperature profiles along the reactor tubes computed for 02.85 s A c=, 00 s B c=, 0323 s r T=, 0293 s c T= and 0.15 s r q= for various coolant flow rates are shown in Fig. 1. A presence of a maximum on the reactant temperature profiles is a common property of many tubular reactors with exothermic reactions. 012345678 320 330 340 350 360 Reactant temperature (K) z (m) 1 - qs c = 0.2 2 - qs c = 0.25 3 - qs c = 0.3 1 2 3 Fig. 1 Reactant temperature profiles for various coolant flow rates. A dependences of the reactant mean temperature and the reactant output temperature on the coolant flow rate is shown in Fig. 2. The form of both curves documents a nonlinear relation between supposed controlled outputs and the coolant flow rate which is considered as the control input. Dynamic charakteristics were computed in the neighbourhood of the chosen operating point 3 0.27 m /s s c q=, 334.44K s m T=, out 326.10K s r T= For the dynamic analysis and subsequent control purposes, the controlled outputs are defined as 0.20 0.25 0.30 0.35 0.40 310 315 320 325 330 335 340 345 Ts m Ts r out Ts r out = 326.10 Temperatures (K) qs c (m3/s) qs c = 0.27 Ts m = 334.44 Fig. 2 Dependence of the reactant mean temperature on the coolant flow rates. deviations from steady values 1 2 out out out () () () () () () s mmm s rrr yt Tt Tt T yt TtTtT =Δ = − =Δ = − . (29) Such form is frequently used in the control. The deviation of the coolant flow rate is denoted as () s cc c qqtqΔ= −. (30) The responses of both outputs to the coolant flow rate step changes are shown in Figs. 3, 4. 0 50 100 150 200 -12 -10 -8 -6 -4 -2 0 2 4 6 8 y1 (K) t (s) 1 - Δqc = - 0.05 2 - Δqc = - 0.025 3 - Δqc = 0.025 4 - Δqc = 0.05 1 2 3 4 Fig. 3 Reactant mean temperature step responses. 0 50 100 150 200 -10 -8 -6 -4 -2 0 2 4 y2 (K) t (s) 1 - Δqc = - 0.05 2 - Δqc = - 0.025 3 - Δqc = 0.025 4 - Δqc = 0.05 1 2 3 4 Fig. 4 Reactant output temperature step responses. The above shown responses demonstrate more expressive nonlinear behaviour of the reactant output temperature to input changes than the reactant mean temperature. This fact is evident also from the gain values computed as WSEAS TRANSACTIONS on FLUID MECHANICS Petr Dostál, Vladimír Bobál, Jiří Vojtěšek, Zdeněk Babík E-ISSN: 2224-347X 16 Issue 1, Volume 7, January 2012 () lim stc y t gq →∞ =Δ (31) and presented in Tab. 2. Tab. 2 Gains for various input hanges. Δqc - 0.025 - 0.05 0.025 0.05 Main reactant temperature gs -155.4 -166.2 -263.5 -205.1 Output reactant temperature gs -69.6 -72.0 -194.3 -193.5 This fact predicates better properties of the reactant mean temperature as the controlled output than the reactant output temperature. Moreover, the dynamics of the reactant output temperature is slower in comparison with the dynamics of the reactant mean temperature. 4 CT and Delta ELM For the control purposes, the control input variable are considered in the form () () 10 s cc s c qt q ut q − = (32) This expression enables to obtain control input and controlled output variables of approximately the same magnitude. A choice of the CT ELM structure does not stem from known structure of the model (1) – (5) but from a character of simulated step responses. It is well known that in adaptive control a controlled process of a higher order can be approximated by a linear model of a lower order with variable parameters. Taking into account profiles of curves in Figs. 3 and 4 with zero derivatives in t = 0, the second order CT ELM has been chosen for both controlled outputs in the form of the second order linear differential equation 100 () () () ()yt ayt a yt but++ =   (33) where y = y1 or y = y2 , and, in the complex domain, as the transfer function 0 210 () b Gs s as a =++ . (34) Establishing the δ operator 0 1q T δ − = (35) where q is the forward shift operator and T0 is the sampling period, the delta ELM corresponding to (33) takes the form 2100 () () () ()yt a yt a yt but δδ ′′ ′′′′′ ++= (36) where t′is the discrete time. When the sampling period is shortened, the delta operator approaches the derivative operator, and, the estimated parameters ,ab ′′ reach the parameters a, b of the CT model (33). 5 Delta Model Parameter Estimation Substituting 2tk ′=−, equation (36) can be rewriten to the form 210 0 (2) (2) (2) (2) yk a yk a yk buk δδ ′′ −+ −+ −= ′ =− (37) In the paper, the recursive identification method with exponential and directional forgetting was used. Establishing the regression vector () (1) (2) (2)(2) Tkykykuk δ δ −=− − − − −Φ(38) where 0 (1)(2) (2) yk yk yk T δ −− − −= (39) the vector of delta model parameters () 100 () Tkaab δ ′′′ =Θ (40) is recursively estimated from the ARX model 2(2) ()(1)() T yk k k k δδ δε −= −+ΘΦ (41) where 2 2 0 () 2( 1) ( 2) (2) yk yk yk yk T δ −−+− −= (42) 6 Controller Design The control system with two feedback controllers is depicted in Fig. 5. - - v e wu u 0 y R CT ELM Q Fig. 5. Control system with two feedback controllers In the scheme, w is the reference signal, v denotes the load disturbance, e the tracking error, u0 output of controllers, u the control input and y the WSEAS TRANSACTIONS on FLUID MECHANICS Petr Dostál, Vladimír Bobál, Jiří Vojtěšek, Zdeněk Babík E-ISSN: 2224-347X 17 Issue 1, Volume 7, January 2012 controlled output. The transfer function G(s) of the CT ELM is given by (34). The reference w and the disturbance v are considered as step functions with transforms 0 () w Ws s =, 0 () v Vs s = (43) The transfer functions of both controllers are in forms () () () , () () () rs qs Rs Qs p sps ==   (44) where q , r and p  are coprime polynomials in s fulfilling the condition of properness deg deg rp≤ and deg deg qp≤ . The controller design described in this section appears from the polynomial approach. The general requirements on the control system are formulated as its internal properness and strong stability (in addition to the control system stability, also the controller stability is required), asymptotic tracking of the reference and load disturbance attenuation. The procedure to derive admissible controllers can be performed as follows: Transforms of the basic signals in the closed-loop system take following forms (for simplification, the argument s is in some equations omitted) [] () () () b Ys rWs pVs d =+  (45) [] 1 () ( ) () () E sapbqWsbpVs d =+ −   (46) [] () () () a Us rWs pVs d =+  (47) Here, [ ] () () () () () ()ds as ps bs rs qs=++  (48) is the characteristic polynomial with roots as poles of the closed-loop. Establishing the polynomial t as () () () ts rs qs=+  (49) and substituting (49) into (48), the condition of the control system stability is ensured when polynomials p  and t are given by a solution of the polynomial Diophantine equation () () ()() () asps bsts ds+=  (50) with a stable polynomial d on the right side. With regard to the transforms (43), the asymptotic tracking and load disturbance attenuation are provided by divisibility of both terms ap bq+  and p  in (46) by s. This condition is fulfilled when polynomials p and q have forms () () p ssps= , ( ) ( ) qs sqs= . (51) Subsequently, the transfer functions (44) take forms () () () qs Qs p s =, () () () rs Rs s ps = (52) and, a stable polynomial p(s) in their denominators ensures the stability of controllers. The control system satisfies the condition of internal properness when the transfer functions of all its components are proper. Consequently, the degrees of polynomials q and r must fulfil these inequalities deg deg qp≤, deg deg 1rp≤+. (53) Now, the polynomial t can be rewritten to the form () () () ts rs sqs=+ . (54) Taking into account the solvability of (50) and conditions (53), the degrees of polynomials in (50) and (52) can be easily derived as deg deg deg tra==, deg deg 1qa=−, deg deg 1 p a≥−, deg 2degda≥. (55) Denoting deg a = n, polynomials t, r and q have forms 0 () n i i i ts ts = = ∑ , 0 () n i i i rs rs = = ∑ , 1 1 () n i i i qs qs− = =∑ (56) and, relations among their coefficients are 00 rt=, iii rqt+= for 1,...,in=. (57) Since by a solution of the polynomial equation (50) provides calculation of coefficients ti, unknown coefficients ri and qi can be obtained by a choice of selectable coefficients 0,1 i β ∈ such that iii rt β =, (1 ) iii qt β =− for 1,...,in=. (58) The coefficients i β distribute a weight between numerators of transfer functions Q and R. Remark: If 1 i β =for all i, the control system in Fig. 5 reduces to the 1DOF control configuration (Q = 0). If 0 i β =for all i, and, both reference and load disturbance are step functions, the control system corresponds to the 2DOF control configuration. For the second order model (34) with deg 2 a=, the controller's transfer functions take specific forms 21 0 2 210 0 () () () () () () ( ) qs qs q Qs ps s p rs rs r rs Rs sps s s p + == + ++ == + . (59) where 00 rt=, 111 rt β =, 222 rt β =, WSEAS TRANSACTIONS on FLUID MECHANICS Petr Dostál, Vladimír Bobál, Jiří Vojtěšek, Zdeněk Babík E-ISSN: 2224-347X 18 Issue 1, Volume 7, January 2012 111 (1 )qt β =− , 222 (1 )qt β =− . (60) The controller parameters then result from a solution of the polynomial equation (50) and depend upon coefficients of the polynomial d. The next problem here is to find a stable polynomial d that enables to obtain acceptable stabilizing controllers. In this paper, the polynomial d with roots determining the closed-loop poles is chosen as 2 () ()( )ds ns s α =+ (61) where n is a stable polynomial obtained by spectral factorization () () () ()asas nsns ∗∗ = (62) and α is the selectable parameter. Note that a choice of d in the form (61) provides the control of a good quality for aperiodic controlled processes. The coefficients of n then are expressed as 2 00 na=, 2 1100 22nana=+− (63) and, the controller parameters p0 and t can be obtained from solution of the matrix equation 10 00 0 1000 00 00 000 ab ab b ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ × 0 2 1 0 p t t t ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ = 31 20 1 0 da da d d − ⎛⎞ ⎜⎟ − ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ (64) where 2 31 2 10 22 10100 2, 2 2, dn d nn dnndn αα α αα α =+ = ++ =+ = (65) Now, it follows from the above introduced procedure that tuning of controllers can be performed by a suitable choice of selectable parameters β and α. The controller parameters r and q can then be obtained from (60). The adaptive control system is shown in Fig. 6. Controller Parameter estimation Controlled process Computation of controller parameters T 0 T 0 w u q, p v y b, a - Fig. 6 Adaptive control scheme. 7 Control Simulation Also the control simulations were performed in a neighbourhood of the operating point 3 0.27 m /s s c q=, 334.44K s m T=, out 326.10K s r T=. For the start (the adaptation phase), the P controller with a small gain was used in all simulations. With respect to more expressive nonlinearity and slower dynamics of the reactant output temperature in comparison with the reactant mean temperature, the changes of references as well as the control running time intervals were chosen different for both outputs. The effect of the pole α on the controlled responses is transparent from Figs. 7 and 8. For both outputs, two values of α were selected. The control simulations show sensitivity of controlled outputs to α. The higher values of this parameter speed the control, however, they provide greater overshoots (undershoots). Other here not mentioted simulations showed that a careless selection of the parameter α can lead to controlled output responses of a poor quality, to oscillations or even to the control instability. 0 100 200 300 400 500 600 700 800 -8 -6 -4 -2 0 2 4 6 y1 (K) t (s) 1 - α = 0.075 2 - α = 0.2 1 2 1 2 w Fig. 7 Controlled output y1 responses: effect of α (β1 = 1, β2 = 0.5). 0 200 400 600 800 1000 1200 -2 -1 0 1 2 y2 (K) t (s) 1 - α = 0.05 2 - α = 0.025 w 1 2 1 21 2 Fig. 8 Controlled output y2 responses: effect of α (β1 = 1, β2 = 0). Moreover, an increasing α leads to higher values and changes of the control input as shown in Fig. 9 and 10. This fact can be important in control of real technological processes. The controlled output y1 response for two values β2 is shown in Fig. 11. It can be seen that an effect of this parameterer is insignificant. WSEAS TRANSACTIONS on FLUID MECHANICS Petr Dostál, Vladimír Bobál, Jiří Vojtěšek, Zdeněk Babík E-ISSN: 2224-347X 19 Issue 1, Volume 7, January 2012 0 100 200 300 400 500 600 700 800 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 qc (m3/ s) t (s) 1 - α = 0.075 2 - α = 0.2 1 1 1 2 2 2 Fig. 9 Coolant flow rate responses in control of reactant mean temperature – effect of α (β1 = 1, β2 = 0.5). 0 200 400 600 800 1000 1200 0.22 0.24 0.26 0.28 0.30 qc (m3/ s) t (s) 1 - α = 0.05 2 - α = 0.025 1 2 1 2 1 2 Fig. 10 Coolant flow rate responses in control of reactant output temperature – effect of α (β1 = 1, β2 = 0). 0 100 200 300 400 500 600 700 800 -6 -4 -2 0 2 4 y1 (K) t (s) 1 - β2 = 0 2 - β2 = 1 w 1 2 Fig. 11 Controlled output responses: effect of β2 (α = 0.1, β1 = 1). The controlled output responses documenting an effect of the parameter β1 are in Figs. 12 and 13. In both cases, a higher value of β1 results in greater overshoots (undershoots) whereas its influence on the speed of control is inexpressive. Corresponding control input responses can be seen in Figs. 14 and 15. There, an increasing β1 leads to greater values of inputs, however, it can reduce occured oscilattions, as shown in Fig. 15. Of interest, the evolution of estimated CT ELM parameters in control of the reactant mean temperature is shown in Fig. 16. 0 100 200 300 400 500 600 700 800 -8 -6 -4 -2 0 2 4 6 y1 (K) t (s) 1 - β1 = 0.2 2 - β1 = 1 w 1 2 1 2 Fig. 12 Controlled output responses: effect of β1 (α = 0.15, β2 = 0). 0 200 400 600 800 1000 1200 1400 -2 -1 0 1 2 y2 (K) t (s) 1 - β1 = 0 2 - β1 = 1 w 1 1 1 2 2 2 Fig. 13 Controlled output responses: effect of β1 (α = 0.04, β2 = 0). 0 100 200 300 400 500 600 700 800 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 qc (m3/ s) t (s) 1 - β1 = 0.2 2 - β1 = 1 1 1 1 2 2 2 Fig. 14 Coolant flow rate responses in control of reactant mean temperature – effect of β1 (α = 0.15, β2 = 0). 0 200 400 600 800 1000 1200 1400 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 qc (m3/ s) t (s) 1 - β1 = 0 2 - β1 = 1 1 11 2 2 2 Fig. 15 Coolant flow rate responses in control of reactant output temperature – effect of β1 (α = 0.15, β2 = 0). WSEAS TRANSACTIONS on FLUID MECHANICS Petr Dostál, Vladimír Bobál, Jiří Vojtěšek, Zdeněk Babík E-ISSN: 2224-347X 20 Issue 1, Volume 7, January 2012 0 100 200 300 400 500 600 700 800 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 CT ELM parameters t (s) 1 - b0 2 - 10 x a0 3 - 10-1x a1 1 2 3 Fig. 16 CT ELM parameter evolution (α = 0.15, β1 = 1, β2 = 0). A presence of an integrating part in the controller enables rejection of various step disturbances entering into the process. As an example, step disturbances attenuation for the output y1 is presented. Step disturbances 3 00.15 kmol/ m A cΔ= , 3 0.03 m /s r qΔ=− and 02K r TΔ= were injected into the nonlinear model of the reactor in times 220s v t=, 440s v t= and 640s v t=. The controller parameters were estimated only in the first (tracking) interval t < 200 s. The authors' experiences proved that an utilization of recursive identification using the delta model after reaching of a constant reference and in presence of step disturbances decreases the control quality. From this reason, during interval t ≥ 200 s, fixed parameters were used. The controlled output responses y1 are shown in Fig. 17. 0 100 200 300 400 500 600 700 800 900 0 1 2 3 4 5 6 7 y1 (K) t (s) w Fig. 17 Controlled output in presence of step disturbances (α = 0.15, β1 = 0.5, β2 = 0). To illustrate an effect of an additive random disturbance, the result of the controlled output y1 simulation in a presence of the random signal 0 () () s A A vt c t c=− is shown in Fig. 18. 8 Conclusions In this paper, one approach to continuous-time adaptive control of the mean and output reactant -0.04 -0.02 0.00 0.02 0.04 v (kmol/ m3) 0 100 200 300 400 500 600 700 800 -2 -1 0 1 2 3 4 y1 (K) t (s) w Fig. 18. Controlled output in the presence of random disturbance in 0 A c (α = 0.15). temperatures in a tubular chemical reactor was proposed. The control strategy is based on the preliminary steady-state and dynamic analysis of the process and on the assumption of the temperature measurement along the reactor. The proposed algorithm employs an alternative continuous-time external linear model with parameters obtained through recursive parameter estimation of a corresponding delta model. The control system structure with two feedback controllers is considered. Resulting continuous-time controllers are derived using the polynomial approach and given by a solution of the polynomial Diophantine equation. Tuning of their parameters is possible via closed-loop pole assignment. The presented method has been tested by computer simulation on the nonlinear model of the tubular chemical reactor with a consecutive exothermic reaction. The simulation results demonstrate an applicability of the presented control strategy. Acknowledgments The authors wish to thank to the Ministry of Education of the Czech Republic (MSM7088352101) for financial support. This article was created with support of Operational Programme Research and Development for Innovations co-funded by the European Regional Development Fund (ERDF) and national budget of Czech Republic within the framework of the Centre of Polymer Systems project (reg.number: CZ.1.05/2.1.00/03.0111). WSEAS TRANSACTIONS on FLUID MECHANICS Petr Dostál, Vladimír Bobál, Jiří Vojtěšek, Zdeněk Babík E-ISSN: 2224-347X 21 Issue 1, Volume 7, January 2012