# Robust Control of Continuous Bioprocesses

Authors: Gongxian Xu
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

This paper deals with robust control of continuous bioprocesses. According to the material balance equations of continuous bioprocesses, a uniform framework for mathematical modeling of this class of processes is first presented. Then a robust controller is designed by using theH∞ mixed sensitivity method for the biotechnology processes. The corresponding control objective is described as the development of a robust reference-tracking control structure with the best possible disturbance compensation, able to cope with variations in key process parameters. Finally, the proposed robust control strategy is applied to bio-dissimilation process of glycerol to 1, 3-propanediol. Simulation results are given which show that the designed robust controller makes the system have a favourable robust tracking performance.


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## Body

## 1. Introduction

The goal of bioprocess control is to optimize the performance of processes involving industrially important organisms, biomedically relevant species, and the degradation of pollutants [1]. In general, a mathematical model describing the biotechnological process is first needed to do this task. However, it is difficult to obtain its exact process model due to the intrinsic complexity of biological system. Even if the mathematical model is built up, model parameters will vary with the working conditions. In addition, external disturbance signals also have an important effect on the system. These uncertain factors can deteriorate the performance of a bioprocess and lead to the instability of the process. One efficient approach to solving such problems is to design a robust controller via the robust control theory [2–16]. The robust control approach integrates the uncertainty involved in model parameters and external disturbance to synthesize a control law which maintains real plants to work within desired performance specifications despite the effects of uncertainty on the system.The goals of this work are to represent continuous bioprocesses within an uncertain, linear model framework and to design a robust controller that would perform satisfactorily. The corresponding control objective is described as the development of a robust reference-tracking control structure with the best possible disturbance compensation. Simulation results are given which show that the designed robust controller not only ensures the robust stability of the bioprocess in face of the parametric variations in the model, but also makes the system have a favourable robust tracking performance.In the sequel, we first describe the continuous bioprocesses and present a uniform framework for mathematical modeling of this class of processes. This is followed by a discussion ofH∞ mixed sensitivity approach and selection strategies for weighting functions used to H∞ design. Then continuous bio-dissimilation of glycerol to 1, 3-propanediol is chosen as a case study and is presented in terms of simulation experiments. Finally, brief conclusions are given in Section 5.


## 2. Modeling of Continuous Bioprocesses

### 2.1. Material Balance Equations

The process considered is a continuous stirred tank bioreactor shown in Figure1. The characteristic of this kind of process is that the reactor is continuously fed with the substrate influent. The rate of outflow is equal to the rate of inflow and the volume of culture remains constant.Figure 1
Schematic representation of continuous stirred tank bioreactors.The general process model obtained from material balances and conservation laws has the following description:(2.1)d(XB)dt=μ(CS,CP1,CP2,…,CPn)XB-DXB,d(CS)dt=D(CSF-CS)-qS(CS,CP1,CP2,…,CPn)XB,d(CPi)dt=qPi(CS,CP1,CP2,…,CPn,D)XB-DCPi,i=1,2,…,n,
where CSF is the external substrate concentration; D is the dilution rate; XB, CS, and CPi are the concentrations of biomass, substrate, and product Pi, respectively; μ, qS, and qPi are the specific growth rate of cells, specific consumption rate of substrate, specific formation rate of product Pi, respectively. In general μ and qS are the functions of substrate concentration CS and product concentrations CPi. But for the specific formation rate qPi, its expression is a function of substrate concentration CS, product concentrations CPi, and dilution rate D (e.g., the specific formation rate of product ethanol in bio-dissimilation process of glycerol to 1, 3- propanediol, see Section 4).


### 2.2. Control Model of Continuous Bioprocesses

The process dynamics (2.1) is represented as a linear model with uncertain parameters
(2.2)ẋ=A(θ)x+B(θ)u,y=Cx,
where x=(XB,CS,CP1,CP2,…,CPn)T∈Rn+2 is used for the vector of states, u=D is the control input, y=CS is the measured output, θ=(XB,CS,CP1,CP2,…,CPn,D)T∈Rn+3 is a vector of describing uncertain parameters, and(2.3)A(θ)=[μ(θ2,θ3,…,θn+2)00⋯0-qS(θ2,θ3,…,θn+2)00⋯0qP1(θ2,θ3,…,θn+3)00⋯0⋮⋮⋮⋱⋮qPn(θ2,θ3,…,θn+3)00⋯0],B(θ)=[-θ1CSF-θ2-θ3⋮-θn+2],C=[010⋮0]T.The specific growth rate of cells (μ), specific consumption rate of substrate (qS), and specific formation rate of product Pi(qPi) will change within certain ranges due to variations in the working conditions. In other words, all parameters in A(θ) and B(θ) are accepted to vary within known bounds.Considering all the uncertain parameters inθ, we allow their changes of up to Hk% (0<Hk≤100, k=1,2,…,n+3) around the nominal values, respectively. Then all uncertain parameters can be uniformly denoted as
(2.4)θ=(I+HΔ)θ0,
where θ0=(XB0,CS0,CP10,CP20,…,CPn0,D0)T is the nominal value of vector θ, I is the identity matrix, and H and Δ are the diagonal matrices with the following formulations:(2.5)H=[H1H2⋱Hn+3],Δ=[Δ1Δ2⋱Δn+3],
where |Δk|≤1 (k=1,2,…,n+3).The following Theorem2.1 provides a uniform framework for mathematical modeling of continuous bioprocesses.Theorem 2.1.
The transfer function model for continuous bioprocesses can be formulated uniformly as(2.6)GP(s)=qSXB+(CSF-CS)(s-μ)s(s-μ).Proof.
Forθk∈[θ0k(1-Hk),θ0k(1+Hk)], the transfer function of the process can be derived as
(2.7)GP(s)=C(sI-A(θ))-1B(θ)=[010⋮0]T([sss⋱s]-[μ00⋯0-qS00⋯0qP100⋯0⋮⋮⋮⋱⋮qPn00⋯0])-1[-θ1CSF-θ2-θ3⋮-θn+2]=[010⋮0]T[s-μ(θ2,θ3,…,θn+2)00⋯0qS(θ2,θ3,…,θn+2)s0⋯0-qP1(θ2,θ3,…,θn+3)0s⋯0⋮⋮⋮⋱⋮-qPn(θ2,θ3,…,θn+3)00⋯s]-1[-θ1CSF-θ2-θ3⋮-θn+2]=[010⋮0]T[1s-μ(θ2,θ3,…,θn+2)00⋯0-qS(θ2,θ3,…,θn+2)s(s-μ(θ2,θ3,…,θn+2))1s0⋯0qP1(θ2,θ3,…,θn+3)s(s-μ(θ2,θ3,…,θn+2))01s⋯0⋮⋮⋮⋱⋮qPn(θ2,θ3,…,θn+3)s(s-μ(θ2,θ3,…,θn+2))00⋯1s][-θ1CSF-θ2-θ3⋮-θn+2]=qS(θ2,θ3,…,θn+2)θ1+(CSF-θ2)(s-μ(θ2,θ3,…,θn+2))s(s-μ(θ2,θ3,…,θn+2)).
Replacing θ1 and θ2 with XB and CS, respectively, we have
(2.8)GP(s)=qSXB+(CSF-CS)(s-μ)s(s-μ).
This model describes the transfer functions of continuous bioprocesses for all uncertain parameters θk∈[θ0k(1-Hk),θ0k(1+Hk)].In this paper, we choose the multiplicative form of uncertainty modeling to represent the relative error in the process model(2.9)GP(s)=GP0(s)(1+Δm(s)),
where
(2.10)GP0(s)=qSXB0+(CSF-CS0)(s-μ)s(s-μ)
is the nominal model of the plant, and
(2.11)Δm=GP(s)-GP0(s)GP0(s).


## 2.1. Material Balance Equations

The process considered is a continuous stirred tank bioreactor shown in Figure1. The characteristic of this kind of process is that the reactor is continuously fed with the substrate influent. The rate of outflow is equal to the rate of inflow and the volume of culture remains constant.Figure 1
Schematic representation of continuous stirred tank bioreactors.The general process model obtained from material balances and conservation laws has the following description:(2.1)d(XB)dt=μ(CS,CP1,CP2,…,CPn)XB-DXB,d(CS)dt=D(CSF-CS)-qS(CS,CP1,CP2,…,CPn)XB,d(CPi)dt=qPi(CS,CP1,CP2,…,CPn,D)XB-DCPi,i=1,2,…,n,
where CSF is the external substrate concentration; D is the dilution rate; XB, CS, and CPi are the concentrations of biomass, substrate, and product Pi, respectively; μ, qS, and qPi are the specific growth rate of cells, specific consumption rate of substrate, specific formation rate of product Pi, respectively. In general μ and qS are the functions of substrate concentration CS and product concentrations CPi. But for the specific formation rate qPi, its expression is a function of substrate concentration CS, product concentrations CPi, and dilution rate D (e.g., the specific formation rate of product ethanol in bio-dissimilation process of glycerol to 1, 3- propanediol, see Section 4).


## 2.2. Control Model of Continuous Bioprocesses

The process dynamics (2.1) is represented as a linear model with uncertain parameters
(2.2)ẋ=A(θ)x+B(θ)u,y=Cx,
where x=(XB,CS,CP1,CP2,…,CPn)T∈Rn+2 is used for the vector of states, u=D is the control input, y=CS is the measured output, θ=(XB,CS,CP1,CP2,…,CPn,D)T∈Rn+3 is a vector of describing uncertain parameters, and(2.3)A(θ)=[μ(θ2,θ3,…,θn+2)00⋯0-qS(θ2,θ3,…,θn+2)00⋯0qP1(θ2,θ3,…,θn+3)00⋯0⋮⋮⋮⋱⋮qPn(θ2,θ3,…,θn+3)00⋯0],B(θ)=[-θ1CSF-θ2-θ3⋮-θn+2],C=[010⋮0]T.The specific growth rate of cells (μ), specific consumption rate of substrate (qS), and specific formation rate of product Pi(qPi) will change within certain ranges due to variations in the working conditions. In other words, all parameters in A(θ) and B(θ) are accepted to vary within known bounds.Considering all the uncertain parameters inθ, we allow their changes of up to Hk% (0<Hk≤100, k=1,2,…,n+3) around the nominal values, respectively. Then all uncertain parameters can be uniformly denoted as
(2.4)θ=(I+HΔ)θ0,
where θ0=(XB0,CS0,CP10,CP20,…,CPn0,D0)T is the nominal value of vector θ, I is the identity matrix, and H and Δ are the diagonal matrices with the following formulations:(2.5)H=[H1H2⋱Hn+3],Δ=[Δ1Δ2⋱Δn+3],
where |Δk|≤1 (k=1,2,…,n+3).The following Theorem2.1 provides a uniform framework for mathematical modeling of continuous bioprocesses.Theorem 2.1.
The transfer function model for continuous bioprocesses can be formulated uniformly as(2.6)GP(s)=qSXB+(CSF-CS)(s-μ)s(s-μ).Proof.
Forθk∈[θ0k(1-Hk),θ0k(1+Hk)], the transfer function of the process can be derived as
(2.7)GP(s)=C(sI-A(θ))-1B(θ)=[010⋮0]T([sss⋱s]-[μ00⋯0-qS00⋯0qP100⋯0⋮⋮⋮⋱⋮qPn00⋯0])-1[-θ1CSF-θ2-θ3⋮-θn+2]=[010⋮0]T[s-μ(θ2,θ3,…,θn+2)00⋯0qS(θ2,θ3,…,θn+2)s0⋯0-qP1(θ2,θ3,…,θn+3)0s⋯0⋮⋮⋮⋱⋮-qPn(θ2,θ3,…,θn+3)00⋯s]-1[-θ1CSF-θ2-θ3⋮-θn+2]=[010⋮0]T[1s-μ(θ2,θ3,…,θn+2)00⋯0-qS(θ2,θ3,…,θn+2)s(s-μ(θ2,θ3,…,θn+2))1s0⋯0qP1(θ2,θ3,…,θn+3)s(s-μ(θ2,θ3,…,θn+2))01s⋯0⋮⋮⋮⋱⋮qPn(θ2,θ3,…,θn+3)s(s-μ(θ2,θ3,…,θn+2))00⋯1s][-θ1CSF-θ2-θ3⋮-θn+2]=qS(θ2,θ3,…,θn+2)θ1+(CSF-θ2)(s-μ(θ2,θ3,…,θn+2))s(s-μ(θ2,θ3,…,θn+2)).
Replacing θ1 and θ2 with XB and CS, respectively, we have
(2.8)GP(s)=qSXB+(CSF-CS)(s-μ)s(s-μ).
This model describes the transfer functions of continuous bioprocesses for all uncertain parameters θk∈[θ0k(1-Hk),θ0k(1+Hk)].In this paper, we choose the multiplicative form of uncertainty modeling to represent the relative error in the process model(2.9)GP(s)=GP0(s)(1+Δm(s)),
where
(2.10)GP0(s)=qSXB0+(CSF-CS0)(s-μ)s(s-μ)
is the nominal model of the plant, and
(2.11)Δm=GP(s)-GP0(s)GP0(s).


## 3.H∞ Mixed Sensitivity Method

### 3.1.H∞ Mixed Sensitivity Problem

The H∞ mixed sensitivity problem is formulated as the one of finding a feedback controller that stabilizes the closed-loop system shown in Figure 2 and minimizes the H∞-norm of closed-loop transfer function Tzw from the exogenous input w (w=r) to the regulated outputs z (z=[z1,z2,z3]T), namely,
(3.1)γopt=minK∥Tzw(s)∥∞,
where
(3.2)Tzw(s)=[W1(s)S(s)W2(s)K(s)S(s)W3(s)T(s)]=P11(s)+P12(s)K(s)(I-P22(s)K(s))-1P21(s).
Here S(s)=(I+Gp0(s)K(s))-1, K(s)S(s), and T(s)=Gp0(s)K(s)S(s) are the sensitivity transfer matrix, control sensitivity transfer matrix, and complementary sensitivity transfer matrix, respectively; Gp0(s) is the nominal model that has no imaginary axis zeros and/or poles; the terms W1(s), W2(s), and W3(s) are performance weighting function, control weighting function, and robustness weighting function, respectively; P(s) is the augmented plant and can be denoted as
(3.3)
with state-space realization
(3.4)Figure 2
Mixed sensitivity configuration.For theH∞ optimal control problem (3.1), all assumptions concerning the existence of a solution K(s) are satisfied [2, 3].(a)
The pair(Ap,B2) is stabilizable and (Ap,C2) is detectable.(b)
rank(D12)=dim(u) and rank(D21)=dim(y).(c)
The following matrices must have full rank for(3.5)[Ap-jωIB2C1D12],[Ap-jωIB1C2D21],∀ω∈R.Assumption (a) ensures the stability of a synthesizedH∞ controller. The second assumption guarantees that the designed H∞ controller is a proper and real rational function. The final assumption is a mathematical technicality that enables both P12(s) and P21(s) to have no invariant zeros on the imaginary axis [6].


### 3.2. Weighting Function Selection

The selection of the weighting functionsW1(s), W2(s), and W3(s) keeps mainly to the basic rules as follows.(a)
Choose a low-order weighting function, otherwise a high-orderH∞ controller can be achieved.(b)
As the perturbation bound of the uncertaintyΔm, the choice of robustness weighting function W3(s) depends also on whether the nominal model is strictly proper and real rational function. Usually, W3(s) is chosen to be an improper and real rational function because the most system in the world is strictly proper. Though W3(s) cannot be realized in state-space form, W3(s)GP0(s) has a state-space realization since it is a proper structure. This ensures D12 has a full rank.(c)
The performance weighting functionW1(s) is usually a stable, proper, and real rational function. The 0 dB crossover frequency for the Bode plot of W1(s) should be below the 0 dB crossover frequency for the Bode plot of  W3(s). More precisely, for all ω∈R, we require σ̅(W1-1(jω))+σ̅(W3-1(jω))>1, where σ̅ denotes the maximum singular value of a transfer function.(d)
The control weighting functionW2(s) is normally chosen to be a high-pass filter to penalize the control signal and to ensure that the D12 submatrix of state-space representation of the augmented plant P(s) has full column rank. It is also included in this paper to limit the size of the controller gain.


## 3.1.H∞ Mixed Sensitivity Problem

The H∞ mixed sensitivity problem is formulated as the one of finding a feedback controller that stabilizes the closed-loop system shown in Figure 2 and minimizes the H∞-norm of closed-loop transfer function Tzw from the exogenous input w (w=r) to the regulated outputs z (z=[z1,z2,z3]T), namely,
(3.1)γopt=minK∥Tzw(s)∥∞,
where
(3.2)Tzw(s)=[W1(s)S(s)W2(s)K(s)S(s)W3(s)T(s)]=P11(s)+P12(s)K(s)(I-P22(s)K(s))-1P21(s).
Here S(s)=(I+Gp0(s)K(s))-1, K(s)S(s), and T(s)=Gp0(s)K(s)S(s) are the sensitivity transfer matrix, control sensitivity transfer matrix, and complementary sensitivity transfer matrix, respectively; Gp0(s) is the nominal model that has no imaginary axis zeros and/or poles; the terms W1(s), W2(s), and W3(s) are performance weighting function, control weighting function, and robustness weighting function, respectively; P(s) is the augmented plant and can be denoted as
(3.3)
with state-space realization
(3.4)Figure 2
Mixed sensitivity configuration.For theH∞ optimal control problem (3.1), all assumptions concerning the existence of a solution K(s) are satisfied [2, 3].(a)
The pair(Ap,B2) is stabilizable and (Ap,C2) is detectable.(b)
rank(D12)=dim(u) and rank(D21)=dim(y).(c)
The following matrices must have full rank for(3.5)[Ap-jωIB2C1D12],[Ap-jωIB1C2D21],∀ω∈R.Assumption (a) ensures the stability of a synthesizedH∞ controller. The second assumption guarantees that the designed H∞ controller is a proper and real rational function. The final assumption is a mathematical technicality that enables both P12(s) and P21(s) to have no invariant zeros on the imaginary axis [6].


## 3.2. Weighting Function Selection

The selection of the weighting functionsW1(s), W2(s), and W3(s) keeps mainly to the basic rules as follows.(a)
Choose a low-order weighting function, otherwise a high-orderH∞ controller can be achieved.(b)
As the perturbation bound of the uncertaintyΔm, the choice of robustness weighting function W3(s) depends also on whether the nominal model is strictly proper and real rational function. Usually, W3(s) is chosen to be an improper and real rational function because the most system in the world is strictly proper. Though W3(s) cannot be realized in state-space form, W3(s)GP0(s) has a state-space realization since it is a proper structure. This ensures D12 has a full rank.(c)
The performance weighting functionW1(s) is usually a stable, proper, and real rational function. The 0 dB crossover frequency for the Bode plot of W1(s) should be below the 0 dB crossover frequency for the Bode plot of  W3(s). More precisely, for all ω∈R, we require σ̅(W1-1(jω))+σ̅(W3-1(jω))>1, where σ̅ denotes the maximum singular value of a transfer function.(d)
The control weighting functionW2(s) is normally chosen to be a high-pass filter to penalize the control signal and to ensure that the D12 submatrix of state-space representation of the augmented plant P(s) has full column rank. It is also included in this paper to limit the size of the controller gain.


## 4. Case Study

In this section, we study the robust control of continuous bio-dissimilation of glycerol to 1, 3-propanediol.In the bioconversion of glycerol to 1, 3-propanediol, a number of products may be simultaneously produced, depending on the microorganisms and conditions used. Under proper fermentation conditions mainly 1, 3-propanediol, acetic acid and ethanol are formed. The material balance equations of continuous microbial cultures are written as follows [17]:
(4.1)dXdt=(μ-D)X,dCSdt=D(CSF-CS)-qSX,dCPDdt=qPDX-DCPD,dCHAcdt=qHAcX-DCHAc,dCEtOHdt=qEtOHX-DCEtOH,
where X is the biomass, g/L; D is the dilution rate, 1/h; CSF and CS are the substrate concentration (glycerol) in feed and reactor, respectively, mmol/L; CPD, CHAc, and CEtOH are the concentrations of products 1,3-propanediol, acetic acid, and ethanol, respectively, mmol/L; t is the fermentation time, h; μ,qS,qPD,qHAc and qEtOH are the specific growth rate of cells, specific consumption rate of substrate, specific formation rate of products 1,3-propanediol, acetic acid and ethanol, respectively, mmol/gh, which can be expressed as:
(4.2)μ=μmCSKS+CS(1-CSCS*)(1-CPDCPD*)(1-CHAcCHAc*)(1-CEtOHCEtOH*),(4.3)qS=mS+μYSm+ΔqSmCSCS+KS*,(4.4)qPD=mPD+μYPDm+ΔqPDmCSCS+KPD*,(4.5)qHAc=mHAc+μYHAcm+ΔqHAcmCSCS+KHAc*,(4.6)qEtOH=qS(b1c1+DCS+b2c2+DCS).ForKlebsiella pneunoniae cultivated under anaerobic conditions at 37°C and pH 7.0, the maximum specific growth rate μm and the saturation constant for glycerol present the values of 0.67 1/h and 0.28 mmol/L, respectively. The critical concentrations denoted as C* in glycerol, 1, 3-propanediol, acetic acid, and ethanol are 2039, 939.5, 1026, and 360.9 mmol/L, respectively. In addition, the parameters b1, b2, c1, and c2 in (4.6) are 0.025, 5.18, 0.06, and 50.45 mmol/Lh, respectively, while the ones for (4.3), (4.4) and (4.5) are listed in Table 1.Table 1
Parameters in the models (4.3)–(4.5).
Substrate/productsmYmΔqmK*Glycerol2.200.008228.5811.431, 3-propanediol-2.6967.6926.5915.50Acetic acid-0.9733.075.7485.71The process dynamics (4.1) is represented as a linear model with uncertain parameters:
(4.7)ẋ=Ax+Bu,y=Cx,
where x=(X,CS,CPD,CHAc,CEtOH)T is used for the vector of states, u=D is the control input, y=CS is the measured output,
(4.8)A=[μ0000-qS0000qPD0000qHAc0000qEtOH0000],B=[-XCSF-CS-CPD-CHAc-CEtOH],C=[01000]T.
By Theorem 2.1, the process transfer function can be derived as
(4.9)Gp(s)=(CSF-CS)s+qSX-μ(CSF-CS)s(s-μ).
The initial glycerol concentration CSF is set to be 730.8 mmol/L. The variable numerical data for the example are given in Table 2. The plant’s nominal model in a transfer function form is expressed as(4.10)G0(s)=632.7s-0.2857.Table 2
Variable numerical data.
DataNominal valuesVariation boundsMinMaxD (1/h)0.285702D0X (g/L)2.890.75X01.25X0CS (mmol/L)98.10.4CS01.6CS0CPD (mmol/L)400.10.75CPD01.25CPD0CHAc (mmol/L)116.60.75CHAc01.25CHAc0CEtOH (mmol/L)42.330.75CEtOH01.25CEtOH0Based on the previously-mentioned rules concerning the choice of the weighting function, robustness weighting functionW3(s) can be chosen as W3(s)=s/2 whose crossover frequency is ωc3= 2 rad/s; performance weighting function W1(s) is a second-order filter with
(4.11)W1(s)=β(αs2+2ζ1ωc1αs+ωc12)βs2+2ζ2ωc1βs+ωc12,
where β=206: DC gain of the filter (controls the disturbance rejection); α=0.5: high frequency gain (controls the response peak overshoot); ωc1=0.865 rad/s: filter crossover frequency; ζ1=0.6, ζ2=0.7: damping ratios of the corner frequencies. Obviously, W1-1(0)=1/β is the steady-state tracking error, and lims→∞W1-1(s)=1/α=2 is the corresponding amplification factor of the high-frequency disturbances. The weighting function W2(s) is selected as W2(s)=0.1.By using MATLAB, the augmented plantP(s) has the following state-space realization:
(4.12)Ap=[0.285700-632.7-0.0852-0.003601.00000],B1=[010],B2=[100],C1=[-316.350.81370.746400090.381200],C2=[-632.700],D11=[0.500],D12=[00.1000316.35],D21=1,D22=0.
After 9 iterations, γopt is found to be 0.99. The corresponding H∞ controller is stable and has the same number of states as the augmented plant with transfer function
(4.13)K(s)=0.4511s2+0.2927s+0.05571s3+165.5s2+14.1s+0.6006.
The closed-loop poles are -163.5625, -0.6251 + 0.6028i, -0.6251 - 0.6028i, -0.2843, -0.0426 + 0.0426i, and -0.0426 - 0.0426i, respectively.Figure3 shows the singular value Bode plot of cost function Tzw(s). As shown, the cost function Tzw(s) is all-pass, that is, σ̅(Tzw(jω))=1 hold for all ω∈R. The results of the singular values analysis for the sensitivity function S(s), the complementary sensitivity function T(s), and their associated weighting functions W1-1(s) and W3-1(s) are illustrated in Figures 4 and 5. It can be observed that S is below its upper bound W1-1 at a low frequency whereas T locates below its upper bound W3-1 at a high frequency, that is, σ̅(S(jω))≤W1-1(jω) and σ̅(T(jω))≤W3-1(jω) hold. These results not only indicate that the closed-loop system has a favourable performance of disturbance reduction but also guarantee the robust stability of controlled system in face of the parametric uncertainty in model.Figure 3
Singular values Bode plot of cost functionsTzw.Figure 4
Singular values Bode plots ofS and W1-1(s).Figure 5
Singular values Bode plots ofT and W3-1(s).Tests of controller performance were carried out through simulation of the whole nonlinear system employing MATLAB/SIMULINK. The complete simulation model is shown in Figure6. The numerical integration of the nonlinear equations (4.1) is based on the 5th-order Runge-Kutta method. In the simulation experiments, we consider a reference input as follows:
(4.14)r(t)={98.1(1+0.2),0≤t<50,98.1,50≤t≤100.
Then the dynamic response curve of the substrate concentration is plotted in Figure 7. From Figure 7, it can be seen that the substrate concentration CS tracks favourably the reference input r. The results imply that the H∞ controller K(s) has a good control action on the presented bioprocess. The time trajectories of the dilution rate, the biomass concentration, and the concentrations of 1, 3-propanediol, acetic acid, and ethanol are presented in Figures 8, 9, 10, 11, and 12. The dynamic trajectories of all variable data stay within the operation ranges as specified in Table 2.Figure 6
Simulation model of the nonlinear bioprocess.Figure 7
Output (glycerol concentration) response.Figure 8
Time trajectory of dilution rate.Figure 9
Time trajectory of biomass concentration.Figure 10
Time trajectory of 1, 3-propanediol concentration.Figure 11
Time trajectory of acetic acid concentration.Figure 12
Time trajectory of ethanol concentration.To detect the dynamic tracking performance of the system in the presence of noise measurement, an additive white Gaussian noise with variance0.05CS is added in the simulations. The simulation results are shown in Figures 13, 14, 15, 16, and 17. As shown in Figures 13–17, the substrate concentration tracks favourably the reference signal. While the time trajectories of other variables stay within the operation windows as given in Table 2.Figure 13
Output (glycerol concentration) response in the presence of noise.Figure 14
Time trajectory of biomass concentration in the presence of noise.Figure 15
Time trajectory of 1, 3-propanediol concentration in the presence of noise.Figure 16
Time trajectory of acetic acid concentration in the presence of noise.Figure 17
Time trajectory of ethanol concentration in the presence of noise.


## 5. Conclusions

This paper has presented a uniform modeling framework and robust control design for continuous bioprocesses. By taking into account the uncertainties in the model parameters, we have first developed the uncertain, linear state-space model of continuous bioprocesses. Then a uniform transfer function model is derived. In theH∞ controller design, a scalar weighting matrix W2 on the control input to the plant has been used to limit the size of the controller gain. Our work has demonstrated that the designed robust controller not only ensures the robust stability of the bioprocess in face of the parametric variations in the model, but also makes the system has a favourable robust tracking performance in the presence of set-point variations.



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*Source: 627035-2010-08-09.xml*