The paper presents a robust control design strategy for bioprocesses, which are characterized by strongly nonlinear dynamics. More precisely, we present the H2 methodology in order to compute the controller for a recycle Continuous Stirred Tank Bioreactor (CSTB). We consider a general method of formulating control problem, which makes use of linear fractional transformation as introduced by Doyle (1978). The formulation makes use of the general two-port configuration of the generalized plant with a generalized controller. The H2 norm is the quadratic criterion used in optimal control as LQG. The overall control objective is to minimize the H2 norm of the transfer matrix function from the weighted exogenous inputs to the weighted controlled outputs. The advantage of H2 control technique, which uses the linearized model of the CSTB, is that it is completely automated and very flexible. Finally, we prove that the closed loop control structure has very good inner robustness

Cosmin IONETE

Dan SELISTEANU

Emil PETRE

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THE ANNALS OF "DUNAREA DE JOS" UNIVERSITY OF GALATI FASCICLE III, 2000 ISSN 1221-454X ELECTROTECHNICS, ELECTRONICS, AUTOMATIC CONTROL, INFORMATICS This paper was recomended for publication by S. Caraman 41        ROBUST CONTROL METHODS FOR A RECYCLE BIOREACTOR   Cosmin IONETE, Emil PETRE, Dan SELISTEANU   University of Craiova, Department of Automatic Control, A.I. Cuza Str. No. 13, RO - 1100 Craiova, Romania, E-mail: cosmin@automation.ucv.ro     Abstract: The paper presents a robust control design strategy for bioprocesses, which are characterized by strongly nonlinear dynamics. More precisely, we present the H2 methodology in order to compute the controller for a recycle Continuous Stirred Tank Bioreactor (CSTB). We consider a general method of formulating control problem, which makes use of linear fractional transformation as introduced by Doyle (1978). The formulation makes use of the general two-port configuration of the generalized plant with a generalized controller. The H2 norm is the quadratic criterion used in optimal control as LQG. The overall control objective is to minimize the H2 norm of the transfer matrix function from the weighted exogenous inputs to the weighted controlled outputs. The advantage of H2 control technique, which uses the linearized model of the CSTB, is that it is completely automated and very flexible. Finally, we prove that the closed loop control structure has very good inner robustness.  Keywords: Biotechnology, Nonlinear Systems, Linear Fractional Transform, H2 controller.       1. INTRODUCTION  The implementation of advanced control strategies on real bioprocesses is difficult because of absence of reliable instrumentation. The performances of the control depend on the reliability of the model.  When some kinetic parameters are imprecisely known, it is necessary to design adaptive and/or robust controllers. An interesting approach has been proposed by Bastin and Dochain (1990) who designed adaptive controllers for bioreactors.  The design of stable and convergent nonlinear adaptive controllers for bioprocesses is a complex task.  Another viable alternative can be the robust approach. The advantages are that one fixed controller structure apriori designed can be use for an entire class of plants nearby the nominal plant. Using a robust controller good results are obtained for parametric uncertainties or for variable delay in the plant model, which often characterize the recycle CSTB.  The paper is organized as follows. In Section 2, the general dynamical model of the Continuous Stirred Tank Bioreactor is presented. Based on this model, a general control design architecture is analysed. Section 3 deals with the design and implementation of the H2-controller that makes use of the linearized model of the nonlinear plant model. In Section 4, the behaviour of the proposed controller is analysed. The robustness of the proposed control architecture is emphasized. Finally, Section 5 collects the conclusions.   2. MATHEMATICAL MODEL OF THE CSTB  A bioreactor is a tank in which several biological reactions occur simultaneously in a liquid medium. Bioreactors that operate in the continuous mode are THE ANNALS OF "DUNAREA DE JOS" UNIVERSITY OF GALATI FASCICLE III, 2000 ISSN 1221-454X 42 usually known as Continuous Stirred Tank Bioreactors. In a CSTB, the substrates (the nutrients) are fed to the bioreactor continuously and an effluent stream is continuously withdrawn such that the culture volume is constant. Often, a part of the biomass is recycled. To recycle, the biomass must be separated from the substrate and yield, then travel through pipes after separation. This time of recycle introduce delays in the states and complicates the dynamic. The benefits are that the recycle increases the overall conversion and reduces the costs.  The dynamical state-space model of a biotechnological process in a CSTB expresses the mass balance of the components in the bioreactor (Bastin and Dochain, 1990):  1 1 2 11 D ) , (dtdx x x x mx￿ - ￿ =            (1)       in 2 1 2 1 12 S D D ) , ( kdtd￿ + ￿ - ￿ - = x x x x mx          (2)  where  2 1,x x  represent the biomass and the limiting substrate concentrations [g/l].  Sin is the influent substrate concentration and  D is so-called dilution rate [h-1], i.e. the specific volumetric outflow rate. In (1), (2)  m  is the specific growth rate and k1>0 the yield coefficient.   In the CSTB with recycle stream, a part of the biomass is recycled. If the recycle occurs, then the bioprocess model (1), (2) must be rewritten (Selisteanu and Petre, 2001):  ) r t ( D ) q 1 () t ( D ) t ( )) t ( ), t ( (dt) t ( d11 1 2 11- - ++ - =xx x x x mx           (3) in 2 1 2 1 12 F ) t ( D ) t ( )) t ( ), t ( ( kdt) t ( d+ - - = x x x x mx   (4)  In (3)  D q) - (1 ￿  is the recycle flow rate. The constant q varies from 0 to 1, with zero corresponding to total recycle and 1 to no recycle. The constant r is the recycle delay time and  Fin is the input flow. A compact representation of the state- space model (3), (4) is:   ) ( f x x = &              (5)  where   [ ]T2 1x x x =   is   the   state   vector   and   the function  () ￿ f  is the nonlinear vectorial function [ ]T2 1 2 2 1 1 ) , ( f ), , ( f ) ( f x x x x x =  . The equilibrium states of (3), (4) are of two types:  1. Wash-out equilibrium states (E1), defined by:  (E1)  [ ] [ ]TinT2 s 1 s s D / F 0 = = x x x           (6)  2. Operational equilibrium states (E2), implicitly defined by:  (E2) ￿￿￿= +=in 2 s 1 s 2 s 1 s 12 s 1 sF D ) , ( kqD ) , (x x x x mx x m           (7)  Equilibrium (E1) corresponds to the bioreactor wash-out, therefore only equilibrium (E2) has a technological interest. We suppose that the form of the specific growth rate is the Haldane kinetic model that takes into account substrate inhibition on the growth       i22 2 M20 2 2 1K / K) ( ) , (x xxm x m x x m+ += =    (8)  KM is the Michaelis  - Menten constant,  Ki the inhibition constant and  0 m  the maxim specific growth rate.  From (7), if the specific growth rate is the model (8) we have two possibilities for the equilibrium (E2):  a).  qD kD FqD kD F11 , 2 s in1 , 1 sN12 s in1 sxxxx-= =-= ; 1 , 2 sN2 s x x =                              (9) b). qD kD FqD kD F12 , 2 s in2 , 1 sN12 s in1 sxxxx-= =-= ; 2 , 2 sN2 s x x =         (10)  The case (a) corresponds to a stable equilibrium point (stable node). The case (b) leads to a saddle type for the equilibrium (E2) (see Selisteanu and Petre, 2001).  The phase plane corresponding to the system (3), (4) for the values of the process parameters: , 1 k , l / g 100 K , l / g 10 K , h 61 i M10 = = = =- m  , h 6 . 3 D1 - =  Fin = 540 h-1 g/l, q = 0.8, r = 0.25 h and for different initial conditions is represented in Fig. 1. From this picture it can be seen that when the substrate inhibition appears, the process can exhibit unstable or, maybe worse, the evolution leads to wash-out steady-states, for which the microbial life has disappeared and the reactor is stopped. Furthermore, a bigger value for the delay time can induce a worst behaviour. Moreover, in many cases, the stable equilibrium point corresponding to (a) is not technological operable (requires a big initial amount of biomass). The conclusion is that for the CSTB with recycle stream it is necessary to design a control strategy.   THE ANNALS OF "DUNAREA DE JOS" UNIVERSITY OF GALATI FASCICLE III, 2000 ISSN 1221-454X 43   Fig. 1. Phase plane of CSTB with recycle stream  For control purposes, it is necessary to find the linear approximation of the system (3), (4) around the equilibrium point (E2). The goal of the control strategy is to stabilize the equilibrium point (10), which is interesting from technological point of view.   2. H2 CONTROLLER DESIGN  A general method it is considered for formulating H2 control problem, which makes use of linear fractional transformation as introduced by Doyle and Stein (1981). The formulation makes use of the general two-port configuration of the generalized plant presented in Fig. 2.   P WKw zy u Fig. 2. Two-port representation of control loop  The overall control objective is to minimize the norm of the transfer matrix function from the weighted exogenous inputs  w  to the weighted controlled outputs z.  The H2 optimal problem can be stated as  2zwRH KH inf¥ ˛           (11)  and is suitable for linear systems. Generally, if we consider w = {r, d, n} the exogenous input of the system (r for reference signals, d for disturbances and n for measurement noises), z = {z1, z2, z3} the quality output of the system and u the control input (controller output) and (y = e) the controller input (system error), we have the representation from Fig.3. W1, W2 and W3 are appropriate weighting functions, used in the controller design process (see Skogestad and Postlethwaite, 1995). We consider a general  method of formulating control problem, which makes  r d e u n z1 z2 z3 yp y K  G W1 W2 W3 -    Fig. 3. The H2 controller design architecture  use of linear fractional transformation as introduced by Doyle and Stein (1981). The formulation uses the general two-port configuration of the generalized plant with a generalized controller. The classical control loop from Fig. 2 is equivalent with the so-called two-port representation of the generalized plant from Fig. 3.  In the next sections is used only the linearized model G of the nonlinear CSTB around the unstable saddle point (10), which has a reasonable biomass concentration  1 x . Beginning with this linear model G, we shall derive the generalized plant PW from Fig. 3. (called also the augmented plant) using the following general formulas:   œœœœœßøŒŒŒŒŒºØ--= =œßøŒºØœßøŒºØG IG WWG W W) s ( H ) s ( Pyzuw W321 100          (12)  The design of the H2 controller is based on this augmented plant transfer matrix PW(s). From (12) we can see that the augmented plant can be represented in state-space form as:             œœœßøŒŒŒºØ=22 2112 112 121sWD DD DB BCCA) s ( P          (13)  We can solve the H2-norm optimal control problem by observing that it is equivalent to a conventional Linear-Quadratic Gaussian optimal control problem. The H2 optimal controller K(s) is thus realizable in the usual LQG manner as a full-state feedback Kc and a Kalman filter with residual gain matrix  Kf with following relations: a)  Kalman Filter: ( )( )( )( )( ) ( )T T21 1T21 T21 21T21 1T2T1 T21 21T21 1T2 f22 2 fD B C D D D B C A AD D D B C Ku D x ˆ C y K x ˆ A x ˆ+ + - ++ =- - + =--S S S SS&  b)  Full-state feedback: ( ) ( )( ) ( ) ( ) 1T12T2112T12T1T12T2T1T12T2112T12 ccC D P B D D C D P B PA P AC D P B D D Kx ˆ K u+ + - ++ ==--  THE ANNALS OF "DUNAREA DE JOS" UNIVERSITY OF GALATI FASCICLE III, 2000 ISSN 1221-454X 44 œœßøŒŒºØ-+ - -=0KKcK D K K B C K A) s ( Kf c 22 f c 2 2 f s        (14)    K  -- G W1 W2 z1 z2 x1 x* 1  e1  u1  Fig. 4. Specific H2 design architecture  From (13) and (14) it can be observed that the controller order is the sum of the original plant order and the orders of each weighting functions. The controller and the augmented (generalized) plant have the same order. It is our interest, in order to obtain a low-order controller, to use scalar weighting functions or to use only W1 and W2 as in the specific case from Fig. 4.   4. COMPUTER SIMULATION AND RESULTS  4.1. Stabilization of the unstable saddle point  For design the linearized plant model around the saddle point is used, which is the intermediate point between the wash-out and operational stable node. The saddle point used for linearization has the coordinates:      [ ] [ ] 98.1442    64.8197, = 2 1,x x           (15)  and the operational stable node has the coordinates:      [ ] [ ] 10.1891    174.7636, = 2 1,x x     (16)  The weighting functions have a great importance in the H2 controller design, being in fact the only design parameters. For example, using     2231 10 W ,s10W = =   good robustness behaviour of the controlled system is obtained.  The main purpose of the control system is to conduct the operational point from the initial point to the final point, the stable node. From technological and economical reasons, the initial point has to be near to the origin in the phase plane from Fig. 1. In such case, without a control system, the plant will go into wash-out state. Using the designed H2 controller, we can start from the initial point     [ ] [ ] 98.1442    14.8197, , *2*1 = x x  and obtain the following state response:  Fig. 5. State response of CSTB  It is clear that the plant functions in the unstable saddle point, starting with wash-out initial conditions. From this non-economical operational point, we can conduct the plant in another desirable operational point, applying steps on biomass reference.   Fig. 6. Time evolution to the operational point  4.2. Robustness against parametric uncertainty  In the above simulation results, we considered no parameter uncertainty, so we assumed that the model parameters are identical with the real ones. We have to test the control system behaviour in presence of parametric uncertainty. For example, let’s take a variation of the parameter k1 (the yield coefficient), with nominal value 1. The following evolutions are obtained:   Fig. 7. Time evolution for k1=0.1 THE ANNALS OF "DUNAREA DE JOS" UNIVERSITY OF GALATI FASCICLE III, 2000 ISSN 1221-454X 45  Fig. 8. Concentrations profile for k1=1.2  We may simulate multiple parameters variation, as in the next figure, first with two modified parameters and after with three modified parameters (Fig.9 and Fig. 10).    Fig. 9. Time evolution for 2 modified parameters     Fig. 10. Time evolution for 3 modified parameters   4.3. Robustness against time delay  The above simulations results used relations (1) and (2) for the plant model with no recycle stream. The last simulation results are focused on the time delay compensation, so we will use for simulation the mathematical model (3) and (4) of the CSTB with recycle stream.       Fig. 11. Evolution of CSTB with recycle stream     Fig. 12. Evolution for modified parameters.   5. CONCLUSIONS  In this paper a simple but efficient strategy for a CSTB with recycle stream control is proposed: the H2 methodology for controller design. Starting with the nonlinear second-order model of the CSTB plant, the following design steps are proposed:  -  Obtain the linear model from the set of nonlinearly differential equation around the unstable saddle point [ ] [ ] 98.1442    64.8197, = 2 1,x x   -  Compute the H2 controller using appropriate weighting functions. In our case, we do not use W3 and W2 is chosen a scalar only for methodological reasons.  -  Test the control system behaviour for evaluating robustness performances.  Following these steps, a third order SISO H2 controller is founded, with very good robustness performances, which transformed the unstable saddle point into ordinary operational point. Using successive biomass reference steps, the operational point is conducted from the starting wash-out zone to the final stable node.  THE ANNALS OF "DUNAREA DE JOS" UNIVERSITY OF GALATI FASCICLE III, 2000 ISSN 1221-454X 46 In order  to prove the robustness capabilities, the following tests are proposed:  -  non-zero initial conditions response is stable in a large area around the linearization point (saddle point).  -  The H2 controller can compensate large simultaneous variations for different model parameters.  -  In presence of parametric uncertainties, the H2 controller still ensures a good control for a large time-delay in the recycle stream.  Finally, we have to explain why we choose the H2 design procedure instead of  LTR or  ¥ H , methods (see Stein and Athans, 1987; Chiang and Safonov, 1992) recognized as robust techniques (Fig. 13).  ( )( ) w sw w sw212n1 i 02i22sup Hd H==¥=¥￿￿   Fig. 13. H2 and H ¥  optimization  To understand the difference between  the  H2 and ¥ H , we note that the H2-norm is the Frobenius norm in terms of singular values. We see that minimizing the  ¥ H -norm corresponds to minimizing the peak of the largest singular value, whereas minimizing the H2-norm corresponds to minimizing the square of the sum of all singular values over all frequencies.  Minimizing H2-norm often lead to a decrease of the peak of the largest singular value, which guarantees good robustness of stabilization and performances.  REFERENCES  Bastin, G. and D. Dochain (1990).  On-line Estimation and Adaptive Control of Bioreactors, Elsevier Sc. Pub. Chiang, R.Y. and M.G. Safonov (1992).  Robust Control Toolbox, The MathWorks, Inc. Doyle, J.C. and G. Stein (1981). Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis,  IEEE Trans. on Automat. Control, AC-26, pp. 4-16. Selisteanu, D. and E. Petre (2001). Vibrational Control of a Class of Bioprocesses,  Control Engineering and Applied Informatics, Bucharest, Vol. 3, No. 1, pp. 39-50. Skogestad, S. and I. Postlethwaite (1995). Multivariable Feedback Control, Analysis and design, John Willey & Sons. Stein, G. and M. Athans (1987). The LQG/LTR Procedure for Multivariable Feedback Control Design, IEEE Trans. on Automat. Control, AC-32, pp.105-114.   

Robust Control Methods for a Recycle Bioreactor

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