Pole-placement and LQ Hybrid Adaptive Control Applied Isothermal Continuous Stirred Tank Reactor

Transcription

1 WSEAS RANSACIONS on SYSEMS and CONROL Pole-placement and LQ Hybrid Adaptie Control Applied Iothermal Continuou Stirred ank Reactor JIRI VOJESEK, PER DOSAL Department of Proce Control, Faculty of Applied Informatic oma ata Unierity in Zlin nám.. G. Maaryka 5555, 76 Zlin CZECH REPULIC Abtract: - he paper deal with imulation experiment on the nonlinear ytem repreented by the iothermal continuou tirred tank reactor. At firt, the mathematical model deried from the material balance inide the reactor will be introduced and then the teady-tate and the dynamic analye were performed on thi model. A a reult of thee tudie, the optimal working point and the choice of the external linear model for the identification will be obtained. he pectral factorization with pole-placement method and linear-quadratic approach were employed in the controller deign and computation. oth type of adaptie controller hae parameter for tuning of the output repone. Moreoer, controller hae atified baic control requirement uch a the tability, the reference ignal tracking and the diturbance attenuation. Key-Word: - teady-tate, dynamic analyi, polynomial approach, Pole-placement method, LQ, recurie identification, adaptie control Introduction It i known that the great majority of ytem ha nonlinear behaior. he control of thee procee with the conentional controller with fixed parameter could lead to the untable, inaccurate or unwanted output repone when the tate of the ytem change or the diturbance occur. he adaptie control [] i one way how we can ole thee problem. hi control method ue idea from the nature where plant or animal adapt their behaior to the actual tate or enironmental condition. he adaptie controller adapt parameter or the tructure to parameter of the controlled plant according to he elected criterion []. Other ery problematic feature from the control point of iew i time delay. Although thi paper doe not deal with it, the problem of time delay i decribed nicely in [3], [4], [5], [6] and [7]. Other oling method and example could be found in [8] and [9]. Agent baed learning i decribed in []. he adaptie approach here i baed on the choice of the External Linear Model (ELM) a a linear approximation of the originally nonlinear ytem, parameter of which are identified recuriely and parameter of the controller are recomputed according to identified one. he choice and the order of the ELM come from the dynamic analyi. he -model [] ued here are pecial type of dicrete-time (D) model parameter of which are related to the ampling period. It wa proofed, that parameter of the -model approach to parameter of the continuou-time (C) model for the mall ampling period []. he polynomial ynthei [3] together with the pectral factorization, the Pole-placement method and the Linear-Quadratic (LQ) approach [4] were ued for deigning of the controller. he product of thi ynthei i the continuou-time controller which atifie baic control requirement uch a the tability, the reference ignal tracking and the diturbance attenuation. he reulted controller i called hybrid becaue it work in continuou-time but it parameter are recomputed in dicrete time interal together with the -ELM identification. here are eeral type of chemical reactor. he main group are tank reactor and tubular reactor. he continuou tirred tank reactor (CSR) i ideal from the control point of iew it could hae ariety of quantitie which can affect the production. here are two way how we can obere the behaior of the ytem by experiment on the real ytem or it maller real model [4]. hi method produce more realitic reult but it could be dangerou or time and money demanding. he other approach ue modeling technique for creating of a mathematical model a an abtract repreentation of E-ISSN: Iue 3, Volume 8, July 3

2 WSEAS RANSACIONS on SYSEMS and CONROL the ytem. he mathematical model in the form of the et of Ordinary Differential Equation (ODE) i then ubjected to imulation which how the tatic and the dynamic behaior of the ytem. he role of the imulation grow nowaday with the increaing peed and the decreaing price of computer. he contribution i diided into ix main part. he econd part after thi introduction will decribe imulated nonlinear ytem which i the iothermal Continuou Stirred-ank Reactor (CSR) the mathematical model of which i decribed by the et of fie ordinary differential equation [6]. he third part will preent the reult of the teady-tate and dynamic analye, the fourth part gie an oeriew and the theoretical background to ued hybrid adaptie control while the fifth part preent reult of the control. he lat, the ixth, part i concluion. All reult hown in thi contribution come from the imulation on the mathematical model and they were done on the mathematical imulation oftware Matlab. Detailed imulation of imilar chemical reactor can be found for example in [7]. dca q c A c A k c A c dt dc q c c k c A c dt V V k c c k c c X 3 Y dcx q c X c X k c A c k c c X dt dc Y q c Y c Y k c c k c c X 3 Y dt V V dc Z q c Z c Z k c c 3 Y () dt V where q denote olumetric flow rate, V i ued for olume of the reactant, c A, c, c X, c Y and c Z are concentration, k -3 are rate contant and t i time. Nonlinear Sytem he nonlinear ytem here i repreented by the Iothermal Continuou Stirred ank Reactor with complex reaction inide [8]. hi reaction could be decribed by the cheme: A + + X + Y k k k3 X Y Z () and the chematic repreentation of thi reactor i in Fig.. he full mathematical decription of the ytem i of coure ery complex. Introduction of the aumption uually reduce thi complexity. We aume that the reactant inide i perfectly mixed and the olume i contant during experiment. A the reactor i iothermal, the temperature of the reactant i not taken into the account. he mathematical model come from material balance inide the reactor. In thi cae, a we hae fie tate ariable (concentration of the compound A,, X, Y and Z), the mathematical model i repreented by the et of fie ordinary differential equation (ODE): Fig. Scheme of the iothermic Continuou Stirred ank Reactor We can ay, that the ytem i nonlinear mainly becaue of the multiplication of the tate ariable. he fixed parameter of thi reactor are hown in able. 3 Simulation Analye Once we hae the mathematical model, we can ubject it to imulation analye. he teady-tate and the dynamic analye were ued in our cae. ecaue of the length of thi contribution, the concentration of the product X, Y and Z are obered in the imulation tudie 3. Steady-tate analyi he goal of the teady-tate analyi i to obere the behaior of the ytem in the teady-tate, e.g. for time t. E-ISSN: Iue 3, Volume 8, July 3

3 WSEAS RANSACIONS on SYSEMS and CONROL he mathematical olution lay i relatiely imple thi condition mean that all deriatie with repect to time in equation () are equal to zero, i.e. d (3) dt he et of ODE () i then implified to the et of nonlinear algebraic equation which could be oled for example by the Simple iteration method. Name of the parameter Symbol and alue of the parameter Volume of the reactant V = m 3 Rate contant of the reaction k = 5-4 m 3.kmol -. - Rate contant of the reaction k = 5 - m 3.kmol -. - Rate contant of the reaction 3 k 3 = - m 3.kmol -. - Input concentration of the concentration c A c A =.4 kmol.m -3 Input concentration of the concentration c c =.6 kmol.m -3 Input concentration of the concentration c X c X = kmol.m -3 Input concentration of the concentration c Y c Y = kmol.m -3 Input concentration of the concentration c Z c Z = kmol.m -3 able Fixed parameter of the reactor here could be theoretically ix input ariable to the ytem olumetric flow rate of the reactant q and fie initial concentration c A, c, c X, c Y and c Z. On the other hand, practical experience hae hown, that only olumetric flow rate of the reactant q could be ued. c X, c Y [kmol.m-3 ] iteration [-] Fig. Computation of the teady-tate alue of concentration c X and c Y through the iteration c Y c X he reult of the firt teady-tate analyi in Fig. and Fig. 3 how the iteration in computation. It can be clearly een, that the et of nonlinear algebraic equation conerge to the accurate reult relatiely quickly, after 5-7 iteration. c Z [kmol.m-3 ] iteration [-] Fig. 3 Computation of the teady-tate alue of the concentration c Z through the iteration he econd teady-tate analyi compute teady-tate alue of the tate ariable for different alue of the olumetric flow rate q = <;.> m 3. - a an input ariable. c X, c Y (q r ) [K] q r [m 3. - ] cz cx cy Fig. 4 he teady-tate characteritic of concentration c X and c Y for ariou olumetric flow rate q c Z (q r ) [K] q r [m 3. - ] Fig. 5 he teady-tate characteritic of concentration c Z for ariou olumetric flow rate q E-ISSN: Iue 3, Volume 8, July 3

4 WSEAS RANSACIONS on SYSEMS and CONROL Fig. 4 and Fig. 5 hae confirmed what we hae expected the ytem ha nonlinear behaior. One output from the teady-tate analyi i alo an optimal working point of the ytem. In our cae, the olumetric flow rate of the reactant q = -4 wa ued a a working point. he teady-tate alue of all tate ariable in thi point are then c A.47; c.34; c.4; c.57; c.53; X Y Z (4) 3. Dynamic Analyi he dynamic analyi obere behaior of the ytem after the tep change of the input ariable, in our cae again olumetric flow rate of the reactant. he mathematical meaning of thi analyi i numerical olution of the et of ODE (). here are eeral numerical method which can be ued. In our cae, the Runge-Kutta tandard method wa ued for eeral reaon. At firt, it i old method with big theoretical background, accurate enough and at lat but not the leat it i eaily programmable or een more it i build-in function in ariou mathematical oftware uch a Matlab, Mathematica etc. he ix tep chage of the input q = ±3, ±6 and ± % were performed and the reult are hown in Fig. 6 - Fig. 7. c (t) - c [kmol.m-3 ] % 6% 3% -3% -6% -% Fig. 6 he dynamic characteritic of the concentration c after tep change of input olumetric flow rate q Note that all output in Fig. 6 Fig. 9 repreent the difference from it actual alue and teady-tate alue of which i alo an input condition to the dynamic tudy. A a reult, all output tart from zero. hi wa done for better undertanding of the ytem gain in the controller deign. c X (t) - c X [kmol.m-3 ] % 6% 3% -3% -6% -% Fig. 7 he dynamic characteritic of the concentration c X after tep change of input olumetric flow rate q c Y (t) - c Y [kmol.m-3 ] % 6% 3% -3% -6% -% 5 5 Fig. 8 he dynamic characteritic of the concentration c Y after tep change of input olumetric flow rate q c Z (t) - c Z [kmol.m-3 ] % -6% -3% 3% 6% % Fig. 9 he dynamic characteritic of the concentration c Z after tep change of input olumetric flow rate q Some of the output hae hown negatie propertie from the control point of iew ee Fig. 7 and Fig. 8 uch a non-minimum phae behaior, nonlinearity etc. On the other hand, output c and c Z in Fig. 6 and Fig. 9 could be decribed by firt or econd order tranfer function: E-ISSN: Iue 3, Volume 8, July 3

5 WSEAS RANSACIONS on SYSEMS and CONROL G( ) b b a a a b b b G( ) a a a a (5) (6) Intereting thing can be alo find in the lat graph, where poitie change of the input produce negatie coure of the output concentration c Z and reere. 4 Control of the Plant he control trategy here i baed on the Adaptie control. A there are a lot of adaptie approache, the Adaptie control with External Linear Model (ELM) of the originally nonlinear ytem parameter of which are identified recuriely. hi approach atifie that the controller could react immediately to the change inide the ytem caued by the change of the ytem tate, diturbance influence etc. 4. External Linear Model he choice of the ELM i tightly connected with the dynamic analyi. Let u uppoe, that the ELM of the controlled output obtained from the dynamic analyi could be decribed by the econd order tranfer function with relatie order one in the - plain, equation (6) Parameter of polynomial a() and b() are commenurable polynomial and the feaibility deg a degb. condition i fulfilled for he tranfer function i relation of the output from the ytem to the input which mathematically mean that thi continuou-time (C) model (6) could be rewritten to: a( ) y( t) b( ) u( t) (7) where a() and b() are polynomial from (6) and i the differentiation operator. he identification of the C model i not ery imple. On the other hand, dicrete-time (D) identification could be inaccurate. Compromie between thee two method can be found in the ue of o called Delta (-) model. hi model ue a new complex ariable γ defined generally a [9]: z z (8) he optional parameter from the interal then produce infinite number of the model. Parameter denote the ampling period. A forward -model wa ued in thi work. he γ operator i then z (9) and the continuou model (7) could be then rewritten to a y t b u t () 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 the dicrete time and with the new ubtitution t = k n for k n the -model for thi concrete tranfer function would be: y( k n) b u( k n) b u( k n) a y( k n) a y( k n) () he equation () produce both the regreion ector and the ector of parameter φ ( k ) y ( k ), y ( k ), θ, u ( k ), u ( k ) k a, a, b, b () where y and u denote the recomputed output and input ariable to the -model and y ( k) y( k) y( k ) y( k ) y ( k ) y( k ) y( k ) y ( k ) y( k ) u( k ) u( k ) u ( k ) u ( k ) u( k ) (3) he differential equation () ha then the ector form: y k θ k φ k e k (4) where e(k) i a general random immeaurable error. 4. On-line Identification he unknown parameter from the differential equation (4) i the ector of parameter. he regreion ector i contructed from the preiou alue of the meaured input u and output y. One of the controller tak i compute thi ector on-line during the control. E-ISSN: Iue 3, Volume 8, July 3

6 WSEAS RANSACIONS on SYSEMS and CONROL he Recurie Leat-Square (RLS) method wa ued for thi challenge. hi method i widely ued for thi on-line identification becaue it ha big theoretical background and it i eaily programmable. It might be modified with exponential or directional forgetting [] becaue parameter of the identified ytem can ary during the control which i typical for nonlinear ytem. he ue of ome forgetting factor could reult in better output repone. he RLS method with the changing exponential forgetting ued here i decribed by the et of equation: k y k k ˆ k k φ k Pk φ k Lk k Pk φ k k k k P Pk φ k φ k Pk k φ k Pk φ k P k k k k θˆ θˆ L (5) where optional the changing forgetting factor i computed from the equation k K k k (6) and K i mall number, in our cae K = Deign of the Controller he control configuration i diplayed in Fig.. Fig. Control ytem configuration where G() repreent the tranfer function (5) of the controlled output and Q() denote the tranfer function of the controller in the continuou-time, generally: Q q (7) p Polynomial q() and p() are imilarly to ytem polynomial a() and b() commenurable polynomial with the properne condition deg p deg q. he Laplace tranform of the tranfer function G() in (6) i generally: Y G Y G U (8) U where Laplace tranform of the input ignal u i from Fig. Q W Y V U Q E V (9) If we put polynomial a(), b(), p() and q() into (9) intead of Laplace tranform G() and Q(), the equation (8) ha form Y bq W bq a p V a p b q a p () oth fraction ha the ame denominator which i called a characteritic polynomial of the cloed loop and thi polynomial, in thi cae a p b q d () where d() i a table optional polynomial and the whole equation () i called Diophantine equation [3]. he tability of the control ytem i fulfilled if the table polynomial d() on the left ide of the Diophantine equation () i alo table polynomial. he baic control requirement uch a an aymptotic tracking of the reference ignal and the diturbance attenuation i attained if the polynomial p() include the leat common diior of denominator of tranfer function of the reference w and diturbance : p f p () If we expect both thee ignal from the range of the tep function, the polynomial f() =. he Diophantine equation () i then a p b q d (3) and the tranfer function of the feedback controller i Q q p (4) A it i written aboe, the polynomial d() on the right ide of the Diophantine equation (3) i the table optional polynomial. here are eeral way how we can contruct thi polynomial. he imple one i the baed on pole- E-ISSN: Iue 3, Volume 8, July 3

7 WSEAS RANSACIONS on SYSEMS and CONROL placement method where d() i diided into one or more part with double, triple, etc. root, e.g. m m d d / m/ (5) where i > i optional poition of the root. hi ariable affect the reult of the control a you will ee in the reult. he diadantage of the Pole-placement method can be found in the uncertainty. here i no general rule which can help u with the choice of root which i, of coure, different for different controlled procee. One way how we can oercome thi unpleaant feature i to ue pectral factorization. ig adantage of thi method i that it can make table root from eery polynomial, een if it i untable. he polynomial d() i in thi cae d n m d n g (6) where parameter of the polynomial n() are computed from the pectral factorization of the polynomial a() in the denominator of (6), i.e. * * n n a a (7) and the econd polynomial m() i contructed with the ue of Pole-placement method. he degree of polynomial p, q() and d() are in thi cae: a deg p deg a deg p deg q deg deg d deg a deg p 4 (8) he polynomial m() i then of the econd degree becaue deg m( ) deg d( ) deg a( ) m i (9) and we obtain one double root a i. he econd approach which can be ued here employ pectral factorization and LQ approach. he polynomial d() i diided again into two part: d n g (3) Where n() come from the pectral factorization explained aboe and the polynomial g(), i computed with the ue of the Linear Quadratic (LQ) tracking [4] which i baed on the minimizing of the cot function in the complex domain LQ LQ LQ J e t u t dt (3) where φ LQ > and μ LQ are weighting coefficient, e(t) i the control error and u t denote the difference of the input ariable. It practically mean, that parameter of the polynomial g() are computed from the pectral factorization * a f LQ a f * * b b g g (3) LQ Degree of unknown polynomial p, q() and d() are for the fulfilled properne condition generally in thi cae: a a deg p deg a deg p deg q deg a deg f deg d deg 5 deg n deg deg g deg d deg n 3 (33) Parameter of the unknown polynomial polynomial p, q() and d() are computed from the Diophantine equation (3) by the method of uncertain coefficient for both trategie. Polynomial a() and b() of the ELM are known from the recurie identification decribed in part Simulation reult he control trategy decribed aboe i called hybrid becaue the computation of the control input i defined in the continuou time but the identification of the ELM run in the dicrete time with the ue of -model. he imulation parameter for all tudie are diplayed in able. 6. Pole-placement Method he firt imulation tudy wa done for ariou alue of the i a a poition of the root in the Poleplacement method. he tranfer function of the controller Q() i then for the degree of the polynomial computed in (8) i then Q q q q q p p (34) hree alue of were teted e.g. i =.,. and.5. E-ISSN: Iue 3, Volume 8, July 3

8 WSEAS RANSACIONS on SYSEMS and CONROL Name of the parameter Control input Symbol and alue of the parameter qt q u t % q Controlled output 3 yt cz t c Z kmol. m Limitation of the input u(t) = <-%; %> Sampling period = Simulation time f = 6 Number of tep change of w(t) 6 Initial ector of θ parameter.,.,.,. able Simulation parameter w(t), y(t) [kmol.m -3 ] w(t) y(t) i =. y(t) i =. y(t) i = Fig. he coure of the output ariable, y(t), and the reference ignal, w(t), for the control with Poleplacement method u(t) [%] 5-5 u(t) i =. u(t) i =. u(t) - i = Fig. he coure of the input ariable, u(t), for the control with Pole-placement method All coure preented in Fig. and Fig. ha acceptable reult except the ery beginning of the control becaue the identification need ome time to adapt a it tart from the general point (). he alue of the optional parameter i affect mainly the peed of the control and oerhoot decreaing alue of thi parameter produce moother coure of both input and output ariable but without the oerhoot. he controller ha mall problem at the ery beginning of the control becaue of the recurie identification which tart from the general point () in able. ut after ome initialization time, the recurie identification run ery moothly ee Fig. 3 and Fig. 4 which preent the coure of the identified parameter for the i =.. a (t)[-] t [min] a a.3. a (t)[-]. Fig. 3 he coure of the identified parameter a and a for control with Pole-placement method b (t)[-] x -7 x t [min] Fig. 4 he coure of the identified parameter b and b for control with Pole-placement method 6. LQ Approach he LQ trategy ha two tuning parameter weighting factor LQ and LQ. he tranfer function of the feedback controller according degree computed in (33) i in thi cae: Q q q q q p p p a - -4 b (t)[-] (35) Experiment were hown, that effect of thee factor i imilar. A a reult, we fixed factor LQ to LQ = and change only LQ =.5,.5 and.5. he reult are hown in Fig. 5 and Fig. 6. We hae found alue of LQ which hae imilar reult a in preiou imulation tudy in purpoe. In thi cae, decreaing alue of the weighting factor LQ produce quicker output repone with oerhoot. Fig. 7 and Fig. 8 preent the coure of the identified parameter for thi imulation tudy. he weighting factor LQ i.5 and LQ i. he reult are ery imilar to the preiou tudy the E-ISSN: Iue 3, Volume 8, July 3

9 WSEAS RANSACIONS on SYSEMS and CONROL identification ha problem only at the ery beginning and it i relatiely mooth after ome initial time. w(t), y(t) [kmol.m -3 ] w(t) y(t) LQ =.5 y(t) LQ =.5 y(t) LQ = Fig. 5 he coure of the output ariable, y(t), and the reference ignal, w(t), for the control with LQ method u(t) [%] 5-5 u(t) LQ =.5 u(t) LQ =.5 u(t) LQ = Fig. 6 he coure of the input ariable, u(t), for the control with LQ method a (t)[-] t [min] a 3. x-5 Fig. 7 he coure of the identified parameter a and a for control with LQ approach 6.3 Diturbance Attenuation Preiou graph hae hown that firt two control requirement for the tability and the reference ignal tracking were accomplihed. he diturbance attenuation a a lat requirement wa teted for two diturbance the diturbance (t) on the input concentration c and the diturbance (t) on the output concentration c Z from the ytem. Only one tep change of the reference ignal wa performed during the imulation time a (t)[-] f = 3. he firt diturbance (t) = -5% c wa injected to the ytem during the time t = < ; 3 > and the econd one (t) = % c Z through time t = < ; 3 >. he factor i wa in the Pole-placement approach et to i =. and the weighting parameter in the LQ approach were LQ = and LQ =.. b (t)[-] - -4 x -7 x t [min] Fig. 8 he coure of the identified parameter b and b for control with LQ approach w(t), y(t) [kmol.m -3 ] PP LQ Fig. 9 he coure of the output ariable, y(t), and the reference ignal, w(t), for pole-placement (PP) and LQ approache in the diturbance attenuation u(t) [%] 5-5 PP LQ Fig. he coure of the input ariable, u(t), for pole-placement (PP) and LQ approache in the diturbance attenuation Reult in Fig. 9 and Fig. how the uability of both method for controlling of ytem where diturbance could occur. hi hybrid adaptie controller deal with the diturbance in the input and output and it i worth to note that both diturbance affect the ytem from time t = < ; 3 > and with no big problem to the output repone. a b (t)[-] E-ISSN: Iue 3, Volume 8, July 3

10 WSEAS RANSACIONS on SYSEMS and CONROL 6 Concluion he contribution how two control approache to the adaptie control of a nonlinear ytem repreented by the iothermal CSR reactor. he teady-tate and the dynamic analye hae hown that the ytem ha nonlinear behaior the concentration of the product Z, c Z, could be decribed by the econd order tranfer function with relatie order one. hi tranfer function wa ued a an ELM of the ytem for the on-line identification in the control part. oth adaptie approache with Pole-placement LQ include tuning parameter( i, LQ and LQ ) which could affect the coure of the output. Preented reult hae hown uability of thee trategie for the nonlinear ytem and they both atifie baic control requirement including the diturbance attenuation. he future work will lead up to applicability of thee method to the real ytem. Reference: [] Åtröm, K.J.; Wittenmark, Adaptie Control. Addion Weley. Reading. MA, 989, ISN [] obal, V.; öhm, J.; Fel, J.; Machacek, J. 5 Digital Self-tuning Controller: Algorithm. Implementation and Application. Adanced extbook in Control and Signal Proceing. Springer-Verlag London Limited. 5, ISN [3] Prokop, R., Korbel, J., Matuu, R. Autotuning Principle for ime-delay Sytem. WSEAS ranaction on Sytem, Iue, Volume, October, pp 56-57, ISSN [4] Pekar, L. A Ring for Decription and Control of ime-delay Sytem. WSEAS ranaction on Sytem, Iue, Volume, October, pp , ISSN [5] Dotal, P., obal, V., abik, Z. Control of Untable and Integrating ime Delay Sytem Uing ime Delay Approximation. WSEAS ranaction on Sytem, Iue, Volume, October, pp , ISSN [6] obal, V., Chalupa, P., Kubalcik, M., Dotal, P. Identification and Self-tuning Control of ime-delay Sytem. WSEAS ranaction on Sytem, Iue, Volume, October, pp , ISSN [7] Kubalcik, M., obal, V. Predictie Control of Higher Order Sytem Approximated by Lower Order ime-delay Model. WSEAS ranaction on Sytem, Iue, Volume, October, pp 67-66, ISSN [8] Pakzad, W. A., Pakzad, A.: Stability Map of Fractional Order ime-delay Sytem, WSEAS ranaction on Sytem, Iue, Volume, October, pp 54-55, ISSN [9] Pakzad, W. A.: Kalman Filter Deign for ime Delay Sytem. WSEAS ranaction on Sytem, Iue, Volume, October, pp 55-56, ISSN [] Neri, F. Agent aed Modeling Under Partial and Full Knowledge Learning Setting to Simulate Financial Market, AI Communication,5(4), IOS Pre, pp [] Middleton, H.; Goodwin, G. C. 4. Digital Control and Etimation - A Unified Approach. Prentice Hall. Englewood Cliff, 4, ISN [] D. L. Stericker, N. K. Sinha, "Identification of continuou-time ytem from ample of input-output data uing the delta-operator". Control-heory and Adanced echnology, ol. 9, 3-5, 993 [3] V. Kučera, Diophantine equation in control A urey Automatica, 9, , 993 [4] Hunt, K. J.; Kucera, V.; Sebek, M. 99. Optimal regulation uing meaurement feedback. A polynomial approach. IEEE ranaction on Automation Control, 37, no. 5, pp [5] Vojteek, J.; Dotál, P. 8. Adaptie LQ Approach Ued in Conductiity Control inide Continuou-Stirred ank Reactor, In Proceeding of the 7th IFAC World Congre, Soul, 8, p , ISN-ISSN [6] J. Ingham, I. J.Dunn, E. Heinzle, J. E. Přenoil, Chemical Engineering Dynamic. An Introduction to Modelling and Computer. Simulation., VCH Verlaggeellhaft, Weinheim, [7] Zelinka, I.; Vojteek, J.; Oplatkoa, Z. 6. Simulation Study of the CSR Reactor for Control Purpoe. In: Proc. of th European Conference on Modelling and Simulation ESCM 6. onn, Germany, p [8] Ruell,.; Denn, M. M. 97 Introduction to chemical engineering analyi. New York: Wiley, 97, xiii, 5 p. ISN [9] Mukhopadhyay, S.; Patra, A. G.; Rao, G. P. 99 New cla of dicrete-time model for continuo-time ytem. International Journal of Control, ol.55, 99, 6-87 [] Rao, G. P.; Unbehauen, H. 5 Identification of continuou-time ytem. IEEE Proce- Control heory Application, 5, 5, p.85- E-ISSN: Iue 3, Volume 8, July 3