Pseudo Dynamic Model Reference Nonlinear Control of a Continuous Bioreactor with Input Multiplicities

Transcription

1 seudo Dynaic Model Reference Nonlinear Control of a Continuous Bioreactor with Input Multiplicities G.rabhaker Reddy*,V. Raesh Kuar and B.Spandana Abstract In the present work, the odel reference nonlinear controller (MRNC) based on pseudo dynaic odel is designed to a continuous bioreactor which exhibits input ultiplicities in dilution rate on productivity. i.e., two values of dilution rate will give the sae value of productivity. The perforance of MRNC and linear I controller have been evaluated through siulation studies. The optial paraeters of MRNC have been obtained based on integral square error criterion (ISE) through siulations. As the MRNC provides always the two values of Dilution rate for control action and by selecting value nearer to the operating point, it is found to give stable and better response than linear I controller. Linear I controller results in wash out condition or switch over fro initial lower input dilution rate to higher input dilution rate or vice versa. Thus, DMRNC is found to overcoe the control probles of I controller due to the input ultiplicities Index Ters Bioreactor, Input ultiplicities, Instability, Nonlinear control. Abbreviations A. List of Sybols D h -1 dilution rate e error K I g/l substrate inhibition constant K g/l substrate saturation constant K c controller gain g/l product concentration g/l product saturation constant Q g/l/h product cells produced per unit tie S g/l substrate concentration S f g/l substrate feed concentration t h tie ts h settling tie X g/l bioass concentration Y g/lh easured value of productivity Yr reference value for Y Manuscript received Deceber 27, 2008 October 9, G.rabhaker Reddy is with the Dept. of Cheical Eng., University College of Technology, Osania University, Hyderabad, , India. ( hone: ;Fax: ; e-ail: gpreddy_ouct@yahoo.co) V.Raesh Kuar is with the Dept. of Cheical Eng., University College of Technology, Osania University,Hyderabad, ,India. (e-ail: raeshkuar_ou_hyd@yahoo.co) B.Spandana is with the Dept. of Cheical Eng., University College of Technology, Osania University,Hyderabad, ,India. B. Greek Letters α g/g product yield paraeter β h-1 product yield paraeter μ h-1 specific growth rate μ h-1 axiu specific growth rate τ I h integral tie τ D h derivative tie ξ daping coefficient ψ g/g cell ass yield II. INTRODUCTION Input ultiplicities ean ore than one value of an input variable (u) produce the sae value of an output variable (y). The condition for the input ultiplicities for single input and single output nonlinear systes is that the derivative of output with respect to the input ust be zero (i.e., (dy/du)0). In general input ultiplicities occur due to the presence of copeting effects in the nonlinear processes [1]-[4]. The nonlinear cheical processes such as the continuous stirred tank reactors (CSTRs) in series with the exotheric reaction(a B),flash separation unit, the nonlinear processes with recycle structures such as heat exchanger network and the cobine unit of reactor and distillation colun are shown to exhibit input ultiplicities[1],[4]. Also, a single exotheric CSTR with series reversible reaction schee [1]. Control of nonlinear processes with input ultiplicities by a conventional I controller can give several probles including driving the process to a physical liit or to a lesser stable input steady state or to a lesser econoical input steady state or yielding an unconstrained liit cycles[1],[2],[4]. In Reference [4], They have considered the nonlinear systes represented by the stable nonlinear gain followed by a unity gain linear syste for the control studies of the processes with input ultiplicities. roportional-integral (I) controller is considered. The hypothetical probles with input ultiplicities with linear stable dynaic systes are chosen, they have shown that the input steady states having process gain of the sae sign are copatible i.e., stable for the sae set of controller paraeters. They

2 have shown the undesirable phenoenon of driving the syste to the lesser stable input steady state in the presence of teporary disturbance and also closed loop syste exhibits unconstrained liit cycle for certain disturbances. In [5], They have deonstrated experientally the superior perforance nonlinear controller over conventional I controller for bioreactor with input ultiplicities. In the present work, the design of odel reference nonlinear controller for a continuous (MRNC) to a continuous bioreactor with input ultiplicities to overcoe the control probles of conventional I controller due to input ultiplicities. IV. DESIGN OF SUEDO DYNAMIC MODEL REFERENCE NONLINEAR CONTROLLER FOR CONTINUOUS BIOREACTOR WITH INUT MULTILICITIES The Design of Model Reference Nonlinear Controller based on pseudo dynaic odel is briefly presented below. Since, the output variable (y) is to be controlled is Q (D). On differentiation, it gives, dq dd D d and the error is defined as, ( 6 ) e y y (7) r III..MODEL OF A CONTINUOUS BIOREACTOR A variety of ferentation can be described by the following unstructured odel [6]. dx ds d DX μ X (1) D (X f X ) μ X ψ (2) D ( α μ β )X (3) The specific growth rate(μ) odel is allowed to exhibit both substrate and product inhibition: μ 1 S 2 S K μ (4) K S The productivity Q can be defined as the aount of product cells produced per unit tie: QD. The noinal odel paraeters [7] considered are α 2.2 g/g,, β 0.2 h -1, μ 0.48 h -1, 50 g/l,k 1.2 g/l, K I 22 g/l, ψ 0.4 g/g, S f 20 g/l. The Bioreactor syste is characterized by input ultiplicities [5] in the dilution rate (D) on the productivity (i.e., two values of D giving an identical value for productivity). For exaple, Q 3 g/l/h can be obtained under either D l/h or D l/h. At lower value of D, the steady-state gain is positive whereas at the larger value of D the gain is negative and thus it poses a control proble. I Where y r is the reference value of Q and y is the actual value of Q fro the process. Here, reference value and the set point of Q are sae. The pseudo dynaic odel is assued to be given by, dy ( y s τ y ) ( 8 Where, y s is the expression for the steady state value of y given by the solution of corresponding steady state equations. Here, τ is an assued tie constant of the syste. Differential of error gives y y e r (9) By substituting, r s y fro Eq. (8) into Eq. (9), e y [(y y) / τ ] (10) On equating the equation (10) to t -k c [e (1/ τ I ) e 0 By cobining Eqs (10) & (11) ] (11) y s y τ[k c (e (1/τ I ) e ) y r. ] (12) By substitution of the expression for y s fro steady state equation, the control law is obtained as ) b1d 4 (b2-h1)d 3 (b3-h2)d 2 h3d-h40 (13) Where,

3 Q s Ys ; h1 α 2 S f ; h2 α{2s f β-q s [1/ ψ (S f α/ )]}; h3 (α/ ψ) Q s2 -Q s [(2αβS f / ) (β/ψ )] S f β 2 ; h4 (β/ )[(Qs 2 /ψ) - βs f Q s ] ; b1α 2 a/μ, ; b2 (2αβ/μ ) a - (αq s /μ ) b; b3 (β 2 /μ )a (Q s β/μ ) b (Q s2 /μ ) c and a (S f2 /K I ) S f K ; b [12(S f /K I )]/ψ ;c 1/ ( ψ 2 K I ) In the design, the error dynaics is given by second order differential equation [i.e., Eqs (10) & (11)]. The controller paraeters of MRNC are given by [8] as K c 10/t s and τ I 0.4t s ξ 2. Where, ξ and t s are the daping coefficient and settling tie of the error dynaics. These values are to be given by designer. The value of t s is to be given fro the knowledge of the open loop syste dynaics. The value of ξ ranges fro 1.0 to 2.0 for a second order error dynaics. The control law equation (13) gives two positive values of D. The value, which is nearer to the noinal steady state condition, is taken for ipleentation for the anipulation. Thus, the MRNC based on seudo dynaic odel equation perits to select the lower or higher value of the anipulated variable. V. SIMULATION RESULTS AND DISCUSSIONS The perforances of MRNC and Conventional I controller have been evaluated through siulations. Dynaic odel of continuous bioreactor with paraeters presented in section III are used for siulation studies. The control studies have been carried out for the productivity, Q3.0g/lh. Which can be either obtained with dilution rate D h -1 (lower input value) or D0.2278h - 1 (higher input value) with feed substrate concentration, S f 20g/l. The regulatory results have been presented below for both the input dilution rates for the step disturbance in feed substrate concentration. A. At lower Dilution rate (D h -1 ) The regulatory responses of MRNC and Conventional I controller are shown in Fig.1 for a disturbance in feed substrate concentration S f fro 20 to 25 g/l. The MRNC controller gives better response with lower ISE4.8, than I controller with large ISE The settling tie of response with MRNC controller is also lower (100h) than I controller (200h). The corresponding anipulation in dilution rate is shown in Fig.2.. Fig.1 Regulatory Responses of MRNC and I controllers at lower dilution rate. Fig. 2. Manipulation in Dilution rate for the responses shown in Fig.1 The responses of closed loop syste when the controller settings eant for the larger value of D is used for the lower value of D are shown in Fig.3 for both the linear and the nonlinear controllers. The MRNC gives faster and stable response, whereas the linear controller first oves towards the unstable direction but soon changes to a stable condition. However the anipulated variable oves to a higher value. The corresponding anipulation variable is shown in Fig. 4 Fig.3 Regulatory Responses of MRNC and I controllers when I controller designed for higher input is applied for lower input Dilution rate.

4 Fig. 4. Manipulation in Dilution rate for the responses shown in Fig.3 B. At higher Dilution rate (D0.2278h-1) The regulatory responses of MRNC and Conventional I controller are shown in Fig.5 for a disturbance in feed substrate concentration S f fro 20 to 25 g/l. the MRNC controller gives better response than I controller. The corresponding anipulation in dilution rate is shown in Fig.6. Fig.7 Regulatory Responses of MRNC and I controllers when I controller designed for lower input is applied for higher input Dilution rate. Fig.8. Manipulation in Dilution rate for the responses shown in Fig.7 Fig.5 Regulatory Responses of MRNC and I controllers at higher input dilution rate. VI. CONCLUSION Based on the siulations results, it is concluded that for a continuous bioreactor with input ultiplicities in dilution rate, the perforance of odel reference nonlinear controller is uch superior to that of I controller at both dilution rates as MRNC gives stable and faster response and Conventional I controller designed at higher dilution rate becoes unstable at lower dilution rate and vice versa. Model reference nonlinear controller based on pseudo dynaic perits the designer to select the value of input as saller or larger of the input dilution rate. ACKNOWLEDGMENT G.RABHAKER REDDY IS THANKFUL TO UGC AND MHRD, GOVT. OF INDIA FOR THE FINANCIAL SUORT UNDER TEQI TO RESENT THIS WORK IN ICCA-2009 IN HONG KONG. REFERENCES Fig. 6. Manipulation in Dilution rate for the responses shown in Fig.5 The response of closed loop syste when the controller settings eant for the lower value of D is used for the larger value of D are shown in Fig. 7 for both the linear and the nonlinear controllers. The response of nonlinear controller is stable response but the linear controller leads to washout condition. The corresponding anipulation variable is shown in Fig.8. [1] L.B.Koppel Input ultiplicities in nonlinear ultivariable control systes, AIChE Journal,28, No.6, 1982,pp [2] L. B. Koppel Input ultiplicities in process control, Cheical Engineering Education, 1983,pp [3] L.B Koppel, Conditions iposed by process static s on ultivariable process dynaics, AICHE Journal, 31,No.1, [4] K.S.Dash and L.B.Koppel, Sudden destabilization of controlled cheical processes,che.eng.co.84, 1989,pp [5] S.V Sunil Kuar, V.Raesh Kuar and G.rabhaker Reddy, Nonlinear Control of Bioreactors with input ultiplicities- An Experiental work Bioprocess and Biosystes Engineering,, 28, 2005, pp [6] M.A. Henson MA and D.E.Seborg, Nonlinear control strategies for continuous ferentation, Cheical Engineering Science, Vol.47, No.4, 1992, pp

5 [7] G..Reddy and M.Chidabara, Nonlinear control of bioreactors with input ultiplicities in dilution rate, Bioprocess Engineering, 12,1995, pp [8] Y.S.N.M. Rao, J.Bhat, M.Chidabara, Robust controllers for nonlinear cheical process, Coputers and Cheical Engineering, 16, No.1, 1992, pp