Robust Control of Continuous Bioprocesses

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 the 𝐻_∞ 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.

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 of 𝐻_∞ 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 Figure 1. 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: d(𝑋_𝐵)d𝑡=𝜇(𝐶_𝑆,𝐶_𝑃1,𝐶_𝑃2,…,𝐶_𝑃𝑛)𝑋_𝐵−𝐷𝑋_𝐵,d(𝐶_𝑆)d𝑡=𝐷(𝐶_𝑆𝐹−𝐶_𝑆)−𝑞_𝑆(𝐶_𝑆,𝐶_𝑃1,𝐶_𝑃2,…,𝐶_𝑃𝑛)𝑋_𝐵,d(𝐶_𝑃𝑖)d𝑡=𝑞_𝑃𝑖(𝐶_𝑆,𝐶_𝑃1,𝐶_𝑃2,…,𝐶_𝑃𝑛,𝐷)𝑋_𝐵−𝐷𝐶_𝑃𝑖,𝑖=1,2,…,𝑛, (2.1) where 𝐶_𝑆𝐹 is the external substrate concentration; 𝐷 is the dilution rate; 𝑋_𝐵, 𝐶_𝑆, and 𝐶_𝑃𝑖 are the concentrations of biomass, substrate, and product 𝑃_𝑖, respectively; 𝜇, 𝑞_𝑆, and 𝑞_𝑃𝑖 are the specific growth rate of cells, specific consumption rate of substrate, specific formation rate of product 𝑃_𝑖, respectively. In general 𝜇 and 𝑞_𝑆 are the functions of substrate concentration 𝐶_𝑆 and product concentrations 𝐶_𝑃𝑖. But for the specific formation rate 𝑞_𝑃𝑖, its expression is a function of substrate concentration 𝐶_𝑆, product concentrations 𝐶_𝑃𝑖, 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 𝑥̇=𝐴(𝜃)𝑥+𝐵(𝜃)𝑢,𝑦=𝐶𝑥, (2.2) where 𝑥=(𝑋_𝐵,𝐶_𝑆,𝐶_𝑃1,𝐶_𝑃2,…,𝐶_𝑃𝑛)^T∈𝑅^𝑛+2 is used for the vector of states, 𝑢=𝐷 is the control input, 𝑦=𝐶_𝑆 is the measured output, 𝜃=(𝑋_𝐵,𝐶_𝑆,𝐶_𝑃1,𝐶_𝑃2,…,𝐶_𝑃𝑛,𝐷)^T∈𝑅^𝑛+3 is a vector of describing uncertain parameters, and𝐴(𝜃)=𝜇(𝜃₂,𝜃₃,…,𝜃_𝑛+2)00⋯0−𝑞_𝑆(𝜃₂,𝜃₃,…,𝜃_𝑛+2𝑞)00⋯0_𝑃1(𝜃₂,𝜃₃,…,𝜃_𝑛+3𝑞)00⋯0⋮⋮⋮⋱⋮_𝑃𝑛(𝜃₂,𝜃₃,…,𝜃_𝑛+3)00⋯0,𝐵(𝜃)=−𝜃₁𝐶_𝑆𝐹−𝜃₂−𝜃₃⋮−𝜃_𝑛+2010⋮0,𝐶=^T. (2.3)

The specific growth rate of cells (𝜇), specific consumption rate of substrate (𝑞_𝑆), and specific formation rate of product 𝑃_𝑖(𝑞_𝑃𝑖) 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 in 𝜃, we allow their changes of up to 𝐻_𝑘% (0<𝐻_𝑘≤100, 𝑘=1,2,…,𝑛+3) around the nominal values, respectively. Then all uncertain parameters can be uniformly denoted as 𝜃=(𝐼+𝐻Δ)𝜃₀, (2.4) where 𝜃₀=(𝑋_𝐵0,𝐶_𝑆0,𝐶_𝑃10,𝐶_𝑃20,…,𝐶_𝑃𝑛0,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:𝐻𝐻=¹𝐻₂⋱𝐻_𝑛+3Δ,Δ=¹Δ₂⋱Δ_𝑛+3, (2.5) where |Δ_𝑘|≤1 (𝑘=1,2,…,𝑛+3).

The following Theorem 2.1 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 𝐺_𝑃𝑞(𝑠)=^𝑆𝑋_𝐵+(𝐶_𝑆𝐹−𝐶_𝑆)(𝑠−𝜇)𝑠(𝑠−𝜇). (2.6)

Proof. For 𝜃_𝑘∈[𝜃_0𝑘(1−𝐻_𝑘),𝜃_0𝑘(1+𝐻_𝑘)], the transfer function of the process can be derived as 𝐺_𝑃(𝑠)=𝐶(𝑠𝐼−𝐴(𝜃))⁻¹=010⋮0𝐵(𝜃)_T(𝑠𝑠𝑠⋱𝑠−𝜇00⋯0−𝑞_𝑆𝑞00⋯0_𝑃1𝑞00⋯0⋮⋮⋮⋱⋮_𝑃𝑛)00⋯0⁻¹−𝜃₁𝐶_𝑆𝐹−𝜃₂−𝜃₃⋮−𝜃_𝑛+2=010⋮0^T𝑠−𝜇(𝜃₂,𝜃₃,…,𝜃_𝑛+2𝑞)00⋯0_𝑆(𝜃₂,𝜃₃,…,𝜃_𝑛+2)𝑠0⋯0−𝑞_𝑃1(𝜃₂,𝜃₃,…,𝜃_𝑛+3)0𝑠⋯0⋮⋮⋮⋱⋮−𝑞_𝑃𝑛(𝜃₂,𝜃₃,…,𝜃_𝑛+3)00⋯𝑠⁻¹−𝜃₁𝐶_𝑆𝐹−𝜃₂−𝜃₃⋮−𝜃_𝑛+2=010⋮0^T1𝑠−𝜇(𝜃₂,𝜃₃,…,𝜃_𝑛+2)00⋯0−𝑞_𝑆(𝜃₂,𝜃₃,…,𝜃_𝑛+2)𝑠(𝑠−𝜇(𝜃₂,𝜃₃,…,𝜃_𝑛+21))𝑠𝑞0⋯0_𝑃1(𝜃₂,𝜃₃,…,𝜃_𝑛+3)𝑠(𝑠−𝜇(𝜃₂,𝜃₃,…,𝜃_𝑛+201))𝑠𝑞⋯0⋮⋮⋮⋱⋮_𝑃𝑛(𝜃₂,𝜃₃,…,𝜃_𝑛+3)𝑠(𝑠−𝜇(𝜃₂,𝜃₃,…,𝜃_𝑛+21))00⋯𝑠−𝜃₁𝐶_𝑆𝐹−𝜃₂−𝜃₃⋮−𝜃_𝑛+2=𝑞_𝑆(𝜃₂,𝜃₃,…,𝜃_𝑛+2)𝜃₁+(𝐶_𝑆𝐹−𝜃₂)(𝑠−𝜇(𝜃₂,𝜃₃,…,𝜃_𝑛+2))𝑠(𝑠−𝜇(𝜃₂,𝜃₃,…,𝜃_𝑛+2.)) (2.7) Replacing 𝜃₁ and 𝜃₂ with 𝑋_𝐵 and 𝐶_𝑆, respectively, we have 𝐺_𝑃𝑞(𝑠)=^𝑆𝑋_𝐵+(𝐶_𝑆𝐹−𝐶_𝑆)(𝑠−𝜇)𝑠(𝑠−𝜇). (2.8) This model describes the transfer functions of continuous bioprocesses for all uncertain parameters 𝜃_𝑘∈[𝜃_0𝑘(1−𝐻_𝑘),𝜃_0𝑘(1+𝐻_𝑘)].

In this paper, we choose the multiplicative form of uncertainty modeling to represent the relative error in the process model 𝐺_𝑃(𝑠)=𝐺_𝑃0(𝑠)(1+Δ_𝑚(𝑠)), (2.9) where 𝐺_𝑃0𝑞(𝑠)=^𝑆𝑋_𝐵0+(𝐶_𝑆𝐹−𝐶_𝑆0)(𝑠−𝜇)𝑠(𝑠−𝜇) (2.10) is the nominal model of the plant, and Δ_𝑚=𝐺_𝑃(𝑠)−𝐺_𝑃0(𝑠)𝐺_𝑃0(𝑠). (2.11)

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 𝑇_𝑧𝑤 from the exogenous input 𝑤 (𝑤=𝑟) to the regulated outputs 𝑧 (𝑧=[𝑧₁,𝑧₂,𝑧₃]^T), namely, 𝛾_opt=min_𝐾‖𝑇_𝑧𝑤(𝑠)‖_∞, (3.1) where 𝑇_𝑧𝑤𝑊(𝑠)=𝑊¹(𝑠)𝑆(𝑠)₂𝑊(𝑠)𝐾(𝑠)𝑆(𝑠)₃(𝑠)𝑇(𝑠)=𝑃₁₁(𝑠)+𝑃₁₂(𝑠)𝐾(𝑠)(𝐼−𝑃₂₂(𝑠)𝐾(𝑠))⁻¹𝑃₂₁(𝑠). (3.2) Here 𝑆(𝑠)=(𝐼+𝐺_𝑝0(𝑠)𝐾(𝑠))⁻¹, 𝐾(𝑠)𝑆(𝑠), and 𝑇(𝑠)=𝐺_𝑝0(𝑠)𝐾(𝑠)𝑆(𝑠) are the sensitivity transfer matrix, control sensitivity transfer matrix, and complementary sensitivity transfer matrix, respectively; 𝐺_𝑝0(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 the 𝐻_∞ optimal control problem (3.1), all assumptions concerning the existence of a solution 𝐾(𝑠) are satisfied [2, 3].(a)The pair (𝐴_𝑝,𝐵₂) is stabilizable and (𝐴_𝑝,𝐶₂) is detectable.(b)rank(𝐷₁₂)=dim(𝑢) and rank(𝐷₂₁)=dim(𝑦).(c)The following matrices must have full rank for 𝐴_𝑝−𝑗𝜔𝐼𝐵₂𝐶₁𝐷₁₂,𝐴_𝑝−𝑗𝜔𝐼𝐵₁𝐶₂𝐷₂₁,∀𝜔∈𝑅. (3.5)

Assumption (a) ensures the stability of a synthesized 𝐻_∞ 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 functions 𝑊₁(𝑠), 𝑊₂(𝑠), and 𝑊₃(𝑠) keeps mainly to the basic rules as follows.(a)Choose a low-order weighting function, otherwise a high-order 𝐻_∞ controller can be achieved.(b)As the perturbation bound of the uncertainty Δ_𝑚, 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)𝐺_𝑃0(𝑠) has a state-space realization since it is a proper structure. This ensures 𝐷₁₂ has a full rank.(c)The performance weighting function 𝑊₁(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 𝜎(𝑊₁⁻¹(𝑗𝜔))+𝜎(𝑊₃⁻¹(𝑗𝜔))>1, where 𝜎 denotes the maximum singular value of a transfer function.(d)The control weighting function 𝑊₂(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]: d𝑋d𝑡=(𝜇−𝐷)𝑋,d𝐶_𝑆d𝑡=𝐷(𝐶_𝑆𝐹−𝐶_𝑆)−𝑞_𝑆𝑋,d𝐶_PDd𝑡=𝑞_PD𝑋−𝐷𝐶_PD,d𝐶_HAcd𝑡=𝑞_HAc𝑋−𝐷𝐶_HAc,d𝐶_EtOHd𝑡=𝑞_EtOH𝑋−𝐷𝐶_EtOH, (4.1) where 𝑋 is the biomass, g/L; 𝐷 is the dilution rate, 1/h; 𝐶_𝑆𝐹 and 𝐶_𝑆 are the substrate concentration (glycerol) in feed and reactor, respectively, mmol/L; 𝐶_PD, 𝐶_HAc, and 𝐶_EtOH are the concentrations of products 1,3-propanediol, acetic acid, and ethanol, respectively, mmol/L; 𝑡 is the fermentation time, h; 𝜇,𝑞_𝑆,𝑞_PD,𝑞_HAc and 𝑞_EtOH 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: 𝜇=𝜇_𝑚𝐶_𝑆𝐾_𝑆+𝐶_𝑆𝐶(1−^𝑆𝐶^∗_𝑆𝐶)(1−^PD𝐶^∗_PD𝐶)(1−^HAc𝐶^∗_HAc𝐶)(1−^EtOH𝐶^∗_EtOH𝑞),_𝑆=𝑚_𝑆+𝜇𝑌^𝑚_𝑆+Δ𝑞^𝑚_𝑆𝐶_𝑆𝐶_𝑆+𝐾^∗_𝑆𝑞_PD,=𝑚_PD+𝜇𝑌^𝑚_PD+Δ𝑞^𝑚_PD𝐶_𝑆𝐶_𝑆+𝐾^∗_PD𝑞_HAc,=𝑚_HAc+𝜇𝑌^𝑚_HAc+Δ𝑞^𝑚_HAc𝐶_𝑆𝐶_𝑆+𝐾^∗_HAc𝑞_EtOH,=𝑞_𝑆(𝑏₁𝑐₁+𝐷𝐶_𝑆+𝑏₂𝑐₂+𝐷𝐶_𝑆). (4.2)(4.3)(4.4)(4.5)(4.6)

For Klebsiella pneunoniae cultivated under anaerobic conditions at 37^∘C and pH 7.0, the maximum specific growth rate 𝜇_𝑚 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).

The process dynamics (4.1) is represented as a linear model with uncertain parameters: 𝑥̇=𝐴𝑥+𝐵𝑢,𝑦=𝐶𝑥, (4.7) where 𝑥=(𝑋,𝐶_𝑆,𝐶_PD,𝐶_HAc,𝐶_EtOH)^T is used for the vector of states, 𝑢=𝐷 is the control input, 𝑦=𝐶_𝑆 is the measured output, 𝐴=𝜇0000−𝑞_𝑆𝑞0000_PD𝑞0000_HAc𝑞0000_EtOH𝐶0000,𝐵=−𝑋_𝑆𝐹−𝐶_𝑆−𝐶_PD−𝐶_HAc−𝐶_EtOH01000,𝐶=^T. (4.8) By Theorem 2.1, the process transfer function can be derived as 𝐺_𝑝(𝑠)=(𝐶_𝑆𝐹−𝐶_𝑆)𝑠+𝑞_𝑆𝑋−𝜇(𝐶_𝑆𝐹−𝐶_𝑆)𝑠(𝑠−𝜇). (4.9) The initial glycerol concentration 𝐶_𝑆𝐹 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 as𝐺₀(𝑠)=632.7𝑠−0.2857. (4.10)

Table 2 

Variable numerical data.

Based on the previously-mentioned rules concerning the choice of the weighting function, robustness weighting function 𝑊₃(𝑠) can be chosen as 𝑊₃(𝑠)=𝑠/2 whose crossover frequency is 𝜔_𝑐3= 2 rad/s; performance weighting function 𝑊₁(𝑠) is a second-order filter with 𝑊₁(𝑠)=𝛽(𝛼𝑠²+2𝜁₁𝜔_𝑐1√𝛼𝑠+𝜔²_𝑐1)𝛽𝑠²+2𝜁₂𝜔_𝑐1𝛽𝑠+𝜔²_𝑐1, (4.11) where 𝛽=206: DC gain of the filter (controls the disturbance rejection); 𝛼=0.5: high frequency gain (controls the response peak overshoot); 𝜔_𝑐1=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 lim_𝑠→∞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 plant 𝑃(𝑠) has the following state-space realization: 𝐴_𝑝=0.285700−632.7−0.0852−0.003601.00000,𝐵₁=010,𝐵₂=100,𝐶₁=−316.350.81370.746400090.381200,𝐶₂=,𝐷−632.700₁₁=000.5,𝐷₁₂=00.1000316.35,𝐷₂₁=1,𝐷₂₂=0. (4.12) After 9 iterations, 𝛾_opt 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 𝐾(𝑠)=0.4511𝑠²+0.2927𝑠+0.05571𝑠³+165.5𝑠²+14.1𝑠+0.6006. (4.13) 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.

Figure 3 shows the singular value Bode plot of cost function 𝑇_𝑧𝑤(𝑠). As shown, the cost function 𝑇_𝑧𝑤(𝑠) is all-pass, that is, 𝜎(𝑇_𝑧𝑤(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, 𝜎(𝑆(𝑗𝜔))≤𝑊₁⁻¹(𝑗𝜔) and 𝜎(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 functions $T_{zw}.$

Figure 4 

Singular values Bode plots of $S$ and $W_1^{-1}(s).$

Figure 5 

Singular values Bode plots of $T$ and $W_3^{-1}(s).$

Tests of controller performance were carried out through simulation of the whole nonlinear system employing MATLAB/SIMULINK. The complete simulation model is shown in Figure 6. 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: 𝑟(𝑡)=98.1(1+0.2),0≤𝑡<50,98.1,50≤𝑡≤100. (4.14) 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 𝐶_𝑆 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 variance 0.05𝐶_𝑆 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 the 𝐻_∞ 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.