Implementation of robust control theory to a continuous ... - Springer Link

Bioprocess Engineering 22 (2000) 563±568 Ó Springer-Verlag 2000

Implementation of robust control theory to a continuous stirred tank bioreactor P. G. Georgieva, M. N. Ignatova

Abstract An H-inf robust controller for a continuous stirred tank bioreactor (CSTB) is designed. The implementation of H-inf framework in designing requires a linear plant model plus uncertainty description that has to be rearranged into a Linear Fractional Transformation structure. Therefore, the dynamical behavior of the nonlinear plant is suitably approximated by linear state-space model with bounded parameter variations. The control problem considered is to guarantee robust stability and robust performance of the closed loop system. The performance speci®cations are de®ned through weighting functions. H-inf controller was successfully computed to certain process variations around the operating region. List of symbols X…t† biomass concentration, g/l Ssub …t† substrate concentration, g/l l…Ssub † speci®c growth rate, l/h D…t† dilution rate, l/h S0 external substrate concentration, g/l s Laplace operator W…S†1 ; W2 …S† frequency dependent performance weighting functions K…s† controller S…s† sensitivity function T…s† complimentary sensitivity function …A; B; C; D† state-space model

optimal control, to the more sophisticated self-tuning controllers, adaptive linearizing control and knowledge based (or expert) systems. An ef®cient way to cope simultaneously with different uncertainty reasons which could lead to instability and degradation of the process quality is to design an H-inf robustly performing controller. Though many signi®cant applications of advanced robust control are reported in the area of chemical processes, [7] very few results are available for biological systems. In this paper we restrict our attention on the implementation of the H-inf robust control theory to the class of continuous biotechnological processes. A detailed description of all reactions in the bioreactor would lead to a highorder model. Instead of using such a rigorous model, we adopt a modeling strategy based on a reasonable simpli®cation of the description, which preserves the essential structural properties of the process. The design is based on a linearized process model, which is augmented with a norm bounded uncertainty description to capture the discrepancy between the simpli®ed model of the process and the variation in the process parameters. All performance speci®cations are de®ned in the frequency domain. The design aim is to ®nd the best possible controller structure, which optimizes between the con¯icting objectives (tracking and robustness), assuming the worst case of disturbance action.

2 Process model The process considered is a CSTB in which the growth of one population of microorganisms on a single limiting substrate is the reaction to be controlled, see Fig. 1. The reactor is continuously fed with the substrate in¯uent. The rate of out¯ow is equal to the rate of in¯ow and the volume of culture remains constant. It is considered that the feeding substrate is diluted in the water stream and the dilution rate is used as manipulated variable to control the process. The control signal is con®ned by physical constraints and can only be within a certain range. The output is assumed to be measured directly. The control objective is to maintain substrate concentration at a ®xed value despite variations in Received: 5 July 1999 the feed substrate, which is regarded as unmeasured disP. G. Georgieva (&), M. N. Ignatova turbance. Moreover the controller should track set-point Institute of Control and System Research, step changes in the substrate concentration. Such reaction Bulgarian Academy of Sciences, process describes a large class of real industrial cases. A P.O. Box 79, 1113 So®a, Bulgaria typical representative of this class are activated sludge processes for biological puri®cation of contaminated water, This research was supported by the National Foundation ``Scienti®c Investigations'' by the Bulgarian Ministry of Education where to regulate the substrate concentration is equivalent to control the pollution level in the ef¯uent of the plant. and Science, contract No. TN-715/97. 1 Introduction The biotechnological processes have complex dynamical behavior, mathematically modeled by nonlinear differential equations. Often the parameters or even the structure of the model are not perfectly known or change due to variations in the working conditions. Several control strategies are reported to have been successfully applied in bioprocess engineering [6]: these vary from the classical PI and PID controllers, through the advanced quadratic

563

Bioprocess Engineering 22 (2000)

"

_ X…t† S_ sub …t†

#

    a11 a12 b1 X…t† ˆ ‡ D…t† a21 a22 Ssub …t† b2   X…t† ‡ ‰BŠD…t† ; ˆ ‰AŠ Ssub …t† 

…4†

where all elements of the matrices A and B are supposed to change within certain bounds to represent the nonlinear behavior of the real plant (Eq. (1)): 564

amin  aij  amax bmin  bi  bmax ; ij ij ; i i aij ; bi 2 R; i ˆ 1; 2 j ˆ 1; 2 :

Fig. 1.

The general process model obtained from mass balances and conservation laws has the following nonlinear state-space description, [1]:

…5†

As nominal values we consider the quantities in the middle of the ranges:

amax ‡ amin ij ij

‡ bmin i : …6† 2 2 _ …1a† Each perturbed element is represented by a nominal value X…t† ˆ …l…Ssub † ÿ D…t††X…t† ; S_ sub …t† ˆ ÿk1 l…Ssub †X…t† ‡ D…t†…S0 ÿ Ssub …t†† ; …1b† aij , bi , normalized real-valued perturbation Dij or Di and scaling factor aij or bi where the state variables are: the biomass concentration in aij ˆ aij ‡ aij Dij ; bi ˆ bi ‡ bi Di ; …7† the reactor X…t† and the substrate concentration in the reactor Ssub …t† considered as the controlled output. l…
† where denotes the speci®c growth rate, which is the key param…8† eter for description of bioreactor activity and is modeled jDij j  1; jDi j  1; Dij 2 R; Di 2 R ; by a Monod-type equation and lm Ssub amax ÿ amin bmax ÿ bmin ij ij i i l…Ssub † ˆ : ; b : …9† a ˆ ˆ ij i Ks ‡ Ssub 2 2 lm is the maximum growth rate, Ks is the MichaelisMenten constant and ks , the yield coef®cient. By S0 is denoted external substrate concentration and D…t† is the dilution rate assumed to be the manipulated input. This nonlinear system of differential equations describing the process is linearized using the nominal values of the parameters and the stationary values of the states. It is made not only around an equilibrium operation point but around all points in the operating region, [10]. Therefore:

"

_ X…t† S_sub …t†

2

#

ˆ4

lm S Ks ‡S



ÿ D

ÿk1 lm X  Ks ‡S

3   X…t† 5 ÿk1 Ks lm X  Ssub …t† ÿ D  2 Ks lm X  …Ks ‡S †2

…Ks ‡S †

 ÿX  ‡ D…t† ; S0 ÿ S

…2†

where D ; X  and S are the equilibrium state nominal values of D…t†; X…t† and Ssub …t† respectively:

lm S Ks D  ; S ˆ ; Ks ‡ S lm ÿ D S0 ÿ S S0 …lm ÿ D † ÿ Ks D : ˆ X ˆ k1 k1 …lm ÿ D † D ˆ l ˆ

…3†

aij ˆ

;

max

b bi ˆ i

This procedure of parameter presentation is a direct way to receive the LFT structure of the plant LFTu …M; D† as it is shown in Fig. 2.

3 Performance specifications The control objective is to design a stabilizing controller K such that the system characteristics from the input vector uin to the output vector yout are desirable according to some design speci®cations which will be discussed, see Fig. 3. 3.1 Robust stability Robust stability is the ®rst requirement that has to be satis®ed. This means stability despite model uncertainty due to linearization and large variations in kinetic model parameters. The designed system is said to be internally stable if the closed loop system has only eigenvalues with negative real parts. In this case when uin ˆ 0 the states of both P and K go asymptotically to zero for any initial conditions. The controller is then said to be stabilizing.

3.2 Robust performance In fact, these values correspond to some optimal operating This is the second requirement that has to be satis®ed. It point of productivity. The parameters related to D ; X  means that the behavior of the responses of interest should and S will be considered as nominal parameters. be desirable and within admissible tolerances. For the The linearized model (2) can be rewritten as follows, process considered the following set of performance [3]: speci®cations has to be observed:

P. G. Georgieva, M. N. Ignatova: Implementation of robust control theory

565

Fig. 2.



 AW1 BW1 : CW1 DW1 This is the ®rst requirement to be satis®ed. Generally, W1 and W2 provide a trade-off between control effort and performance. In contrast to W1 ; W2 is normally chosen to be a high-pass ®lter to limit the bandwidth for robustness and to penalize the control signal. W2 is required to be non zero at all frequencies to ensure there are no problems with noise from high gains. It is also a technical requirement for a solution to exist. Very often W2 is just a constant value as it is the case in this paper for a previously chosen weight W1 

Fig. 3.

W2 ˆ kw2 ;

(a) Good tracking of the reference signal and disturbance rejection at low frequencies in the face of model uncertainty. The maximum steady-state tracking error is allowed to be not more than 2%; (b) To stay within the amplitude limit of the actuator; (c) The closed loop transfer function from reference to measured value should be strictly proper to prevent noise being transmitted to the output and have a bandwidth of 5 r/h for robustness [5]. In the H-inf framework the above requirements are speci®ed by frequency dependent weighting functions as it is shown in Fig. 2 [2]. Generally, W1 has to be chosen so that at low frequencies the closed loop (both nominal and perturbed) should considerably reject disturbances at the output. Expressed differently, steadystate tracking error due to a reference step-input should be admissibly small. This performance requirement gets less stringent at higher frequencies. W1 has to shape over frequency the Bode diagram of the sensitivity function S(s), which is the transfer function from the reference signal (v) to the regulation error (e) to meet the mentioned speci®cations. H-inf norm performance measure is used to characterize the worst case effect of uncertainty on the weighted sensitivity function S(s). The regulator should guarantee the following condition:

…S…jx†† < jWÿ1 kW1 Sk1 < 1 or r 1 …jx†j ;

…10†

kw2 2 R :

…11†

The second performance requirement is de®ned as

kW2 KSk1 < 1;

…K…jx†S…jx†† < kÿ1 or r w2 :

…12†

The ®nal LFT model of the weighted perturbed plant has the following structure ready for H-inf design, [3]:

 P…s† ˆ

P11 …s† P21 …s†



2

AA P12 …s†  4 CC1 P22 …s† CC2

BB1 DD11 DD21

3 BB2 DD12 5 : DD22

…13† P…s† is partitioned to conform with the inputs …uin ; u†T and outputs …yout ; e†T . It includes the nominal quantities of the plant model plus all uncertainty weights used to normalize the size of the unknown perturbations D and the performance weighting functions. The normalized perturbation matrix has block-diagonal structure:

 DR ˆ diag D11 ; D12 ; D21 ; D22 ; D1 ; D2 ; Dperf kDperf ka  1 : …14† The state-space description of P(s)is as follows:



  A 0 ; AA ˆ  Aw1 ÿBw1 C    B ; BB2 ˆ  ÿBw1 D



B1 BB1 ˆ ÿBw1 D21

 0 ; Bw1 …15a†

Bioprocess Engineering 22 (2000)

3

2



 AA ÿ jxI BB2 (c) has full column rank for all x; 0 CC1 DD12 7   Dw1 5; AA ÿ jxI BB2 has full row rank for all x. (d) CC1 DD12 0

3

2

0 C1 D11 7 6 6  Cw1 5; DD11 ˆ 4 ÿDw1 D21 CC1 ˆ 4 ÿDw1 C 0 0 0 2 3 D12 6 7 DD12 ˆ 4 ÿDw1 D …15b† 5 ; kw2  0Š; DD21 ˆ ‰ÿD21 IŠ; DD22 ˆ ÿD  ; …15c† CC2 ˆ ‰ÿC 566

where " # " # a11 a21 b11 ˆ ˆ A ; B ; a21 a22 b2 D11 ˆ ‰0Š66 ;

2

a11 6 0 6 6 a21 C1 ˆ 6 6 0 6 4 0 0

D21 ˆ ‰0Š66 ;

3 0 a12 7 7 0 7 7; a22 7 7 0 5 0

" B1 ˆ

3 0 607 6 7 607 7 D12 ˆ 6 6 0 7; 6 7 4 b1 5 b2

1

1 0 0 1 0

0

0 1 1 0 1

(c) and (d) are required to guarantee the controller does not cancel poles or zeros on the imaginary axis. There exists a stabilizing controller such that for a given c value the inequality, (20), is true if and only if: (i) The algebraic Riccati equation:

AA0 X1 ‡ X1 AA ‡ CC01 CC1

# ;

‡ X1 …cÿ2 BB1 BB01 ÿ BB2 BB02 †X1 ˆ 0 ; has a positive semide®nite solution X1  0; (ii) The algebraic Riccati equation:

AA0 Y1 ‡ Y1 AA0 ‡ BB1 BB01

2

‡ Y1 …cÿ2 CC01 CC1 ÿ CC02 CC2 †Y1 ˆ 0 ;

has a positive semide®nite solution Y1  0; (iii) All eigenvalues of X1 and Y1 have negative real parts; (iv) The spectral radius of …X1 Y1 † must be less than or equal to c2 , i.e. kmax …X1 Y1 †  c2 . More details about the consequence of obtaining LFT The design variable K(s) belongs to a set of possible structure could be found in [2]. solutions. All such controllers are parameterized by Once it has been de®ned the particular control tasks the stable real rational matrix Q…s†; kQ…S†k1 < c and the system has been transformed to match the general (see Fig. 4): LFT framework, the efforts are now to synthesize the 2 3 control law which meets all requirements. A1 ÿZ1 L1 Z1 B2

 ˆ0 : D

4 H-inf design In the previous chapter we have received the state-space realization (Eq.(13)) for the generalized plant P(s), i.e.: x ˆ AAx ‡ BB1 uin ‡ BB2 u ;

…16†

yout ˆ CC1 x ‡ DD11 uin ‡ DD12 u ;

…17†

e ˆ CC2 x ‡ DD21 uin ‡ DD22 u ;

…18†

M…S†  4 F1 ÿC2 where F1 ˆ ÿB02 X1 ;

0 I

I 0

5 ;

…21†

L1 ˆ ÿY1 C20 ; Z1 ˆ …1 ÿ cÿ2 Y1 X1 †ÿ1 and

A1 ˆ AA ‡ cÿ2 BB1 BB01 X1 ‡ BB2 F1 ‡ Z1 L1 CC2 :

where the signal vectors have compatible dimensions. The Note that X1 and Y1 depend on c. To ®nd the optimal closed loop transfer function from uin to yout is denoted as solution for K…s† we have to ®nd the minimum value of c. yout ˆ LFT1 …P…s†; K…s††uin , where Then a bisection method is used to iterate on the values of c. The optimal case usually arises when LFT1 …P…s†; K…s†† det…I ÿ cÿ2 X1 Y1 † ˆ 0. ÿ1 ˆ P11 …s† ‡ P12 …s†K…s†…I ÿ P22 …s†K…s†† P21 …s† :…19† c-iteration solution procedure is applied to deal with this constrained optimization problem to obtain the best The robust control task is to ®nd a stabilizing controller possible performance under hard physical constraints. K(s)(if it exists) such that The successful H-inf controller has the same dimension …LFT1 …P…jx†; K…jx††† < c; as the augmented plant P(s). kLFT1 …P…s†; K…s††k1 ˆ sup r x

for some given c > 0 :

…20†

Loop-shifting two Riccati equation approach is used, [4]. The H-inf control algorithm requires some preliminary assumptions about the system P(s): (a) …AA; BB2 † is stabilizable and …CC2 ; AA† is detectable. This is required for existence of stabilizing controller; (b) D21 has full row rank and D12 has full column rank to ensure controller is proper; Fig. 4.

P. G. Georgieva, M. N. Ignatova: Implementation of robust control theory

5 Simulation results Simulation investigations for the designed system, which regulates the substrate concentration in a CSTB, are presented. The control algorithm is realized using the commercially available Robust Control Toolbox in Matlab program system, [8]. The numerical data were chosen as reported in [9]. The external substrate concentration is Fig. 5. S0 ˆ 5 g=l; lm ˆ 0:33 hÿ1 ; Ks ˆ 5 g=l; k1 ˆ 2. The two variables X…t† and Ssub …t† in (2) are strictly positive and bounded functions for any t  0 and the control task has physical sense if the dilution rate is bounded, [1] D