# Adaptive Control of a Fermentation Bioprocess for Lactic Acid Production

Authors: Emil Petre; Dan Selişteanu
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper presents the design and the analysis of an indirect adaptive control strategy for a lactic acid production, which is carried out in continuous stirred tank bioreactors. Firstly, an indirect adaptive control structure based on the nonlinear process model is derived by combining a linearizing control law with a new parameter estimator. This estimator is used for on-line estimation of the bioprocess unknown kinetics, avoiding the introduction of a state observer. Secondly, a tuning procedure of estimator design parameters is achieved by stability analysis of the control scheme. The effectiveness and performance of estimation and control algorithms are illustrated by numerical simulations applied in the case of a lactic fermentation bioprocess for which kinetic dynamics are strongly nonlinear, time varying, and completely unknown, and not all the state variables are measurable.


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## Body

## 1. Introduction

The control of biotechnological processes has been and remains an important problem attracting wide attention, the main engineering motivation being the improvement of the operational stability and production efficiency of such living processes.It is well known that traditional control design involves a complicated mathematical analysis and has many difficulties especially in controlling highly nonlinear and time varying plants. A powerful tool for nonlinear controller design is the feedback linearization technique [1–3], but the use of it requires the complete knowledge of the process. In practice, there are many processes characterized by highly nonlinear dynamics; as a consequence, an accurate model for these processes is difficult to develop. Therefore, in recent years, it has been noticed a great progress in adaptive and robust adaptive control of nonlinear systems, due to their ability to compensate for parametric uncertainties. An important assumption in previous works on nonlinear adaptive control was the linear dependence on the unknown parameters [2, 4, 5].In the modern industry, the development of advanced control strategies for bioprocesses is hampered by two major difficulties [6–9]. The first one is related to kinetics of these processes, which are strongly nonlinear and furthermore the kinetic and process parameters are often partially or completely unknown. Another difficulty lies in the absence, for many processes, of cheap and reliable instrumentation suited for real-time monitoring of the process variables. In order to overcome these difficulties, several strategies were developed, such as exactly linearizing control [6, 7, 10], adaptive approach proposed by Bastin and Dochain and applied in several works [6, 7, 10–12], optimal control, especially for fed-batch bioprocesses [6, 13], neural control [14], sliding mode control [15], robust and robust adaptive control [12, 16], and model predictive control [17]. Some problems which occur in implementation of numerical control of bioreactors are presented and analyzed in [18].The difficulties encountered in the measurement of state variables impose the use of so-called “software sensors”—combinations between hardware sensors and software estimators [6, 7]. Note that these software sensors are used not only for the estimation of concentrations of some components but also for the estimation of kinetic parameters or even kinetic reactions. The interest for the development of software sensors for bioreactors is proved by the big number of publications and applications in this area [6–8, 19].This paper, which is an extended version of Petre et al. [20], presents the design and the analysis of a nonlinear adaptive control strategy, capable to deal with the model uncertainties in an adaptive way, for a lactic fermentation bioprocess that is carried out in two continuous stirred tank bioreactors sequentially connected. In order to avoid the derivation of additional state observers, a new indirect adaptive control algorithm is presented. More exactly, the consumption rates of two limiting substrates are considered completely unknown and summarized in two unknown and time varying parameters, which are estimated by means of an appropriately observer-based estimator. This algorithm could be considered as an extension of the theory proposed by Bastin and Dochain [6] and improved by Ignatova et al. [11], since the substrate consumption rates are considered as completely unknown functions describing the whole kinetics.The present work is focused in two directions. First, an adaptive control structure based on the nonlinear model of the process is designed as a combination of a linearizing control law and of a parameter estimator, used for the on-line estimation of bioprocess unknown kinetics. Second, by using the stability and convergence analysis of the proposed control scheme, a tuning procedure of the kinetic estimator design parameters is achieved.The effectiveness and performance of both estimation and control algorithms are illustrated by simulations applied in the case of a lactic fermentation bioprocess for which kinetic dynamics are strongly nonlinear, time varying and completely unknown, and not all the state variables are measurable.We note that the bioprocess model and the control goal are the same as in the Ben Youssef et al. [21], where an adaptive-multivariable predictive control law was proposed and analyzed. However, in the present work, an indirect multivariable adaptive linearizing controller is developed and analyzed. The control structure is derived by combining a linearizing control law with a new parameter estimator used for on-line estimation of the bioprocess unknown kinetics.In the work by Ignatova et al. [11], by using the observer-based estimator proposed by Petre [22], an indirect adaptive scheme was derived and analyzed for a gluconic acid production bioprocess. In the present work, the multivariable control strategy presented in Petre [22], and further developed by Ignatova et al. [11], is improved and full stability and convergence proof are provided. Finally, a tuning procedure of kinetic estimator design parameters for a lactic acid production bioprocess was obtained.The paper is organized as follows. Section2 is devoted to a brief description and mathematical modeling of a lactic acid fermentation bioprocess. Some nonlinear and adaptive control strategies are proposed in Section 3. From the stability analysis of the adaptive control scheme, a tuning procedure of kinetic estimator design parameters is achieved. Simulations results presented in Section 4 illustrate the performance of the proposed control algorithms and, finally, Section 5 concludes the paper.


## 2. Process Modeling and Control Problem

Lactic acid has traditionally been used in the food industry as an acidulating and/or preserving agent, and in the biochemical industry for cosmetic and textile applications [21, 23, 24]. Recently, lactic acid fermentation has received much more attention because of the increasing demand for new biomaterials such as biodegradable and biocompatible polylactic products. Two major factors limit its biosynthesis, affecting growth, and productivity: the nutrient limiting conditions and the inhibitory effect caused by lactic acid accumulation in the culture broth [21]. It was shown that, in addition to the inhibition effect of lactic acid, the richness of the fermentation medium may have a strong influence on growth dynamics and that the nutritional limitations during culture may interfere with inhibition by acid lactic [21]. A reliable model that explicitly integrates nutritional factor effects on both growth and lactic acid production in a batch fermentation process implementing Lb. casei was developed by Ben Youssef et al. [21] and it is described by the following basic differential equations:(2.1)Ẋ=μX-kdX,Ṗ=νpX,Ṡ=-qsX,
where X, S, and P are, respectively, the concentrations of cells, substrate (glucose) and lactic acid. μ, νp, and qs correspond, respectively, to specific growth rate of the cells, specific rate of lactic acid production, and to specific rate of glucose consumption. kd is the rate of cell death.Since in lactic acid fermentation process the main cost of raw material comes from the substrate and nutrient requirements, in [21] some possible continuous-flow control strategies that satisfy the economic aspects of lactic acid production were investigated. The advantage of implementing continuous-flow process is that the main product, which is also an inhibitor, is continuously withdrawn from the system. Much more, according to microbial engineering theory, for a product-inhibited reaction like lactic acid or alcoholic fermentation [21, 25], a multistage system composed of many interconnected continuous stirred tank reactors, where in the different reactors some physiological states of microbial culture (substrate, metabolites) can be kept constant to some optimal values, may be a good idea.Therefore, the model (2.1) can be extended to a continuous-flow process that is carried out in two continuous stirred tank reactors sequentially connected, as in the schematic view presented in Figure 1.Figure 1
A cascade of two reactors for the lactic acid production.For this two-stage continuous flow bioreactor, the mathematical model is given by the following set of differential equations, each stage being of same constant volumeV.First Stage:
(2.2)Ẋ1=(μ1-kd)X1-D1X1,(2.3)Ṗ1=νp1X1-D1P1,(2.4)Ṡ1=-qs1X1+D11S1in-D1S1,(2.5)α̇1=D12α1in-D1α1,
where D1=D11+D12.Second Stage:
(2.6)Ẋ2=(μ2-kd)X2+D1X1-(D1+D2)X2,(2.7)Ṗ2=νp2X2+D1P1-(D1+D2)P2,(2.8)Ṡ2=-qs2X2+D1S1+D2S2in-(D1+D2)S2,(2.9)α̇2=D1α1-(D1+D2)α2,
where Xi,Si,Pi, and αi, (i=1,2) are, respectively, the concentrations of biomass (cells), substrate, lactic acid, and enrichment factor in each bioreactor. μi,νpi, and qsi(i=1,2) correspond, respectively, to specific growth rate of the cells, specific rate of lactic acid production, and specific rate of glucose consumption in each bioreactor. D11 is the first-stage dilution rate of a feeding solution at an influent glucose concentration S1in. D12 is the first-stage dilution rate of a feeding solution at an influent enrichment factor α1in. D2 is the influent dilution rate added at the second stage and S2in is the corresponding feeding glucose concentration. It can be seen that in the second stage no growth factor feeding tank is included since this was already feeding in the first reactor.In the model (2.2)–(2.9), the mechanism of cell growth, the specific lactic acid production rate and the specific consumption rate in each bioreactor are given by [21]:
(2.10)μi=μ̅max⁡i(K̅PigcK̅Pigc+Pi)(SiK̅Sgc+Si)(1-PiPCgc),νpi=ημi+β(SiK̅Sirc+Si),qsi=νpiYPS,i=1,2,
with μ̅maxi the maximum specific growth rate, K̅Pigc the lactic acid inhibition constant, K̅Sgc the affinity constant of the growing cells for glucose, PCgc the critical lactic acid concentrations,  K̅Sirc is the affinity constant of the resting cells for glucose, and YPS is the constant substrate-to-product conversion yield. The superscript gc denotes the parameters related to growing cells and rc to that of the resting cells, and η and β are positive constants.The kinetic parameters of this model may be readjusted depending on the medium enrichment factorαi as follows [21]:(2.11)μ̅max⁡i=μmax⁡(αi-α0)Kαμ+(αi-α0),K̅Pigc=KPmax⁡gc(αi-α0)KαP+(αi-α0),K̅Sirc=KSmax⁡rc(αi-α0)KαS+(αi-α0),
where α0 is the minimal nutritional factor necessary for growth, Kαμ, KαP and KαS are saturation constants. μmax, KPmaxgc and KSmaxrc correspond to the limit value of each kinetic parameter. Note that the hyperbolic-type expressions in (2.11) quantify the nutritional limitations and are based on the growth model proposed by Bibal [26].For this two-stage reactor configuration, the operating point of the continuous lactic acid fermentation process could be adjusted by acting on at least two control inputs, that is, the glucose feeding flow rates both in the first and in the second reactor. These considerations show the multivariable nature of the control problem.


## 3. Control Strategies

### 3.1. Control Objective

As it was formulated in the previous section, the control objective consists in adjusting the plant’s load in order to convert the glucose into lactic acid via fermentation, which is directly correlated to the economic aspects of lactic acid production. More exactly, considering that the process model (2.2)–(2.9) is incompletely known, its parameters are time varying and not all the states are available for measurements, the control goal is to maintain the process at some operating points, which correspond to a maximal lactic production rate and a minimal residual glucose concentration. After a detailed process steady-state analysis, in [21] it has been demonstrated that these desiderata can be satisfied if the operating point of lactic acid fermentation is kept around the points S1*=3g/L and S2*=5g/L. This choice is the best option in order to simultaneously satisfy the both objectives: maximal lactic production rate and minimal residual glucose concentration [21]. As control variables we select the glucose feeding flow rates both in the first and in the second reactor denoted by F1 and F2, respectively, where F1=(D1-D12)S1in and F2=D2S2in. In this way we obtain a multivariable control problem with two inputs: F1 and F2 and two outputs: S1 and S2.In the following, in Section3.2 we will present the design of an exactly linearizing feedback controller. This controller will be used as a benchmark in order to compare its behavior with the behavior of the indirect multivariable adaptive controller developed in Section 3.3. Also, in this subsection, the parameter estimator used in the indirect multivariable adaptive controller will be designed. In Section 3.4 the tuning procedure of estimator design parameters via stability and convergence analysis of the control scheme will be derived.


### 3.2. Exactly Linearizing Feedback Controller

Firstly, we consider the ideal case where maximum prior knowledge concerning the process (kinetics, yield coefficients, and state variables) is available, and the relative degree of differential equations in process model is equal to 1. Assume that for the two interconnected reactors we wish to have the following first order linear stable closed loop (process + controller) behavior:(3.1)[Ṡ1*-Ṡ1Ṡ2*-Ṡ2]+[λ100λ2][S1*-S1S2*-S2]=0,λ1,λ2>0,
where S1* and S2* are the desired values of S1 and S2.Since (2.4) and (2.8) in the process model have the relative degree equal to 1, these can be considered as an input-output model and can be rewritten in the following form:(3.2)[Ṡ1Ṡ2]=[-qs1X1-qs2X2]-[D10-D1D1+D2][S1S2]+[F1F2].
Then from (3.1) and (3.2) one obtains the following exactly multivariable decoupling linearizing feedback control law:(3.3)[F1F2]=[Ṡ1*Ṡ2*]+[λ100λ2][S1*-S1S2*-S2]+[qs1X1qs2X2]+[D10-D1D1+D2][S1S2].The control law (3.3) applied to the process (3.2) will result in:(3.4)[Ṡ1Ṡ2]=[λ100λ2][S1*-S1S2*-S2]+[Ṡ1*Ṡ2*],
and leads to the following linear error models:(3.5)ė1=-λ1e1,ė2=-λ2e2,
where e1=S1*-S1 and e2=S2*-S2 represent the tracking errors. It is clear that for λ1,λ2>0, the error models (3.5) have an exponential stable point at e1=0 and e2=0.


### 3.3. An Indirect Multivariable Adaptive Controller

In the previous subsection, it was assumed that the functional forms of the nonlinearities as well as the process parameters are known. Since such prior knowledge is not realistic, in this subsection we consider that the kinetics functions are completely unknown. So, the substrate consumption rates-qs1X1 and -qs2X2 in (3.2) can be expressed as follows:(3.6)-qs1X1=ρ1,-qs2X2=ρ2,
where ρ1 and ρ2 are two unknown time-varying parameters. In this case, the control law (3.3) becomes:(3.7)[F1F2]=[Ṡ1*Ṡ2*]+[λ100λ2][S1*-S1S2*-S2]-[ρ1ρ2]+[D10-D1D1+D2][S1S2],
where the unknown parameters ρ1 and ρ2 will be substituted by their on-line estimates ρ̂1 and ρ̂2 calculated by using an observer-based parameter estimator (OBE) [22] applied only for the dynamics of S1 and S2 from (2.4) and (2.8) rewritten as follows:(3.8)Ṡ1=ρ1+F1-D1S1,Ṡ2=ρ2+D1S1+F2-(D1+D2)S2.Under these notations, the algorithm for on-line computation ofρ̂1 and ρ̂2 is particularized as follows:(3.9)Ŝ1̇=ρ̂1+F1-D1S1+ω1(S-Ŝ1),(3.10)Ŝ2̇=ρ̂2+D1S1+F2-(D1+D2)S2+ω2(S2-Ŝ2),(3.11)ρ̂1̇=γ1(S-Ŝ1),(3.12)ρ̂2̇=γ2(S2-Ŝ2),
where γ1,γ2>0 are the gains of the updating laws (3.11) and (3.12), and ω1,ω2>0 are design parameters to control the stability and the tracking properties of the estimator.Remark 3.1.
The parameter error convergence for the estimator (3.9)–(3.12) is assured if some signals in system satisfy the well-known persistency of excitation condition [4, 5]: A piecewise continuous signal vector ϕ:ℜ+→ℜn is referred to as persistent excitation with a level of excitation κ0>0 if there exist constants κ1,T0>0 such thatκ1I≥(1/T0)∫tt+T0ϕ(τ)ϕT(τ)dτ≥κ0I, for all t≥0. Although the matrix ϕ(τ)ϕT(τ) is singular for each τ, this condition requires that ϕ(t) varies in such a way with time that the integral of the matrix ϕ(τ)ϕT(τ) is uniformly positive definite over any time interval [t, t+T0]. In our particular case it is necessary that, for example, D1, D2 or some kinetic parameters to be persistently exciting (which is generically fulfilled).Finally,an indirect multivariable adaptive linearizing controller is obtained by combination of (3.9)–(3.12) and (3.7) rewritten as follows:
(3.13)[F1F2]=[Ṡ1*Ṡ2*]+[λ100λ2][S1*-S1S2*-S2]-[ρ̂1ρ̂2]+[D10-D1D1+D2][S1S2].A block diagram of the designed indirect multivariable adaptive control system is shown in Figure2.Figure 2
The structure of the indirect adaptive control system.Remark 3.2.
Since the on-line measurement of the lactic acid concentrations is not possible, in order to provide on-line evaluation of the current lactic acid production, an asymptotic state observer is derived as follows.
Let us define the auxiliary variablesz1 and z2 as:
(3.14)z1=P1+YPSS1,z2=P2+YPSS2,
which use only the measurements of the outputs S1 and S2 that are available on-line. The dynamics of zi,i=1,2, deduced from model (2.2)–(2.9), are expressed by the following linear stable equations:
(3.15)ẑ̇1=-D1ẑ1+YPSF1,ẑ̇2=D1ẑ1-(D1+D2)ẑ2+YPSF2,
that are independent of the process kinetics. Then, from (3.14) and (3.15), the on-line estimation of P1 and P2 are given by the following relations:
(3.16)P̂1=ẑ1-YPSS1,P̂2=ẑ2-YPSS2.


### 3.4. Tuning Procedure of Estimator Design Parameters via Stability and Convergence Analysis

Let us denote byy=[S1S2]T the vector containing the concentrations of controlled feeding substrates and by u=[F1F2]T the vector of the input control variables, that is, the vector of the mass feeds of substrates S1 and S2, respectively. Then (3.8) can be written as
(3.17)ẏ=ρ-Dy+u,
where ρ=[ρ1ρ2]T is the unknown time-varying parameter vector and D is the dilution rate matrix, whose structure is as follows:(3.18)D=[D10-D1D1+D2].The OBE (3.9)–(3.12) can be represented in a matrix form as follows:
(3.19)ŷ̇=ρ̂-Dy+u+Ω(y-ŷ),ρ̂̇=Γ(y-ŷ),
where ρ̂ is the estimation of unknown parameter ρ, and Ω,Γ∈ℜ2×2 are positive definite diagonal matrices whose entries ω1,ω2>0 and γ1,γ2>0, respectively, represent the estimator design parameters.Under the previous notations, the exactly multivariable decoupling linearizing feedback control law (3.3), respectively, (3.7) can be written as:(3.20)u=ẏ*+Λ(y*-y)-ρ+Dy,
where y*=[S1*S2*]T, and the indirect adaptive linearizing control law (3.13) can be written as:
(3.21)u=ẏ*+Λ(y*-y)-ρ̂+Dy,
where Λ=[λ100λ2]>0.Case 1.
By defining the errorsỹ=y-ŷ and ρ̃=ρ-ρ̂, in the case when the unknown parameter ρ is constant, the following error system can be derived from (3.17) and (3.19):
(3.22)ỹ̇=-Ωỹ+ρ̃,ρ̃̇=-Γỹ.
It can be seen that the error system in (3.22) is a linear time-invariant system that can be rewritten as:
(3.23)[ỹ̇ρ̃̇]=[-ΩI2-Γ0][ỹρ̃],
where I2 is the identity matrix of order two. The stability of the equilibrium (ỹ=0, ρ̃=0) of (3.22) can be proved by using the well-known Hurwitz criterion. So, by using the coefficients of the characteristic polynomial associated to (3.23), given by
(3.24)P(λ)=λ4+(ω1+ω2)λ3+(ω1ω2+γ1+γ2)λ2+(γ2ω1+γ1ω2)λ+γ1γ2=a4λ4+a3λ3+a2λ2+a1λ+a0,after straightforward calculation we obtain that all Hurwitz determinants are positive for any ω1,ω2>0 and γ1,γ2>0. Indeed, by using the coefficients of the above characteristic polynomial we can construct the following Hurwitz determinant:
(3.25)Δ4=[a3a100a4a2a000a3a100a4a2a0]=[ω1+ω2γ2ω1+γ1ω2001ω1ω2+γ1+γ2γ1γ200ω1+ω2γ2ω1+γ1ω2001ω1ω2+γ1+γ2γ1γ2].
Now we construct and calculate all Hurwitz determinants as follows:
(3.26)Δ1=[a3]=ω1+ω2>0,Δ2=[a3a1a4a2]=[ω1+ω2γ2ω1+γ1ω21ω1ω2+γ1+γ2]=ω1ω2(ω1+ω2)+γ2ω1+γ1ω2>0,Δ3=[a3a10a4a2a00a3a1]=[ω1+ω2γ2ω1+γ1ω201ω1ω2+γ1+γ2γ1γ20ω1+ω3γ2ω1+γ1ω2]=ω1ω2((ω1+ω2)(γ2ω1+γ1ω2)+(γ1-γ2)2)>0,Δ4=γ1γ2⋅Δ3>0.
According to Hurwitz criterion [27, 28], the system (3.23) is stable since all Hurwitz determinants are positive if ω1,ω2>0 and γ1,γ2>0. Then, the matrix associated to (3.23) is a Hurwitz matrix and the equilibrium (ỹ=0, ρ̃=0) of (3.22) is uniformly asymptotically stable.Case 2.
In the case when the unknown parameterρ is time-varying, the stability analysis of the proposed indirect multivariable adaptive linearizing controller is carried out under the following realistic assumptions.(A1)
The substrate concentrationsS1 and S2, the components of the dilution rate matrix D, and the substrate feed rates are on-line measured.(A2)
The substrate consumption ratesρ1 and ρ2 are fully unknown.(A3)
The reference signal vectory* and its time derivative are piecewise continuous bounded functions of time, and the measured substrate feed rates are continuous bounded functions of time.(A4)
The measurements of substrate vector, denotedym(t), are corrupted by an additive noise vector w(t)=[w1w2]T: ym(t)=y(t)+w(t), for all t.(A5)
The components of time varying parameter vectorρ(t) are bounded as: 0>ρi(t)≥ρi*, ρi*<0, i=1,2, for all t.(A6)
The time derivatives of the components ofρ(t) are bounded as: |ρ̇i(t)|≤m1i, m1i>0, i=1,2, for all t.(A7)
The measurement noise vector is bounded as:|w(t)|≤m2i, m2i>0, i=1,2, for all t.(A8)
The components of dilution rates matrixD are known and bounded: 0≤Di(t)≤Dimax, i=1,2, 0≤D12(t)≤D12max, for all t.
Under these assumptions, the parameter estimator (3.19) and the adaptive control law (3.21) are rewritten as follows:
(3.27)ŷ̇=ρ̂-Dym+u+Ω(ym-ŷ),(3.28)ρ̂̇=Γ(ym-ŷ),(3.29)u=ẏ*+Λ(y*-ym)-ρ̂+Dym.
From (3.17), (3.27), and (3.28) under the condition (A4), the following error system can be derived as follows:
(3.30)ỹ̇=ẏ-ŷ̇=ρ-Dy+u-ρ̂+Dym-u-Ω(ym-ŷ)=ρ̃+D(-y+y+w)-Ω(y+w-ŷ)=-Ωỹ+ρ̃+(D-Ω)w,ρ̃̇=ρ̇-ρ̂̇=ρ̇-Γ(y+w-ŷ)=-Γỹ-Γw+ρ̇,
which can be rewritten in the following form:
(3.31)ỹ̇1=-ω1ỹ1+ρ̃1+(D1-ω1)w1,ρ̃̇1=-γ1ỹ1-γ1w1+ρ̇1,ỹ̇2=-ω2ỹ2+ρ̃2+(D1+D2-ω2)w2-D1w1,ρ̃̇2=-γ2ỹ2-γ2w2+ρ̇2.Defining the state vectorx as x=[x1Tx2T]T with x1=[ỹ1ρ̃1]T and x2=[ỹ2ρ̃2]T, then (3.31) can be rewritten as:
(3.32)ẋ=Ax+b,
where A=diag{Ai}, i=1,2 and:(3.33)Ai=[-ωi1-γi0],b=[b1b2],b1=[(D1-ω1)w1-γ1w1+ρ̇1],b2=[(D1+D2-ω2)w2-D1w1-γ2w2+ρ̇2].The two eigenvalues ofAi denoted by vi1 and vi2 are related to ωi and γi as follows: vi1+vi2=-ωi<0, vi1vi2=γi>0, i=1,2. It can be seen that v1i,v2i<0. The next assumption is considered for the estimator design parameters.(A9)
The design parametersωi and γi are chosen such that Ai has real distinct eigenvalues [6, 11], for example, vi2<vi1<0, where vi1,2=(-ωi±ωi2-4γi)/2, that implies 0<γi<ωi2/4.The system (3.32) describes the dynamics of the estimation errors of the observer-based estimator including the observations of the measured substrates and the estimations of their unknown consumption rates ρi.We can consider that the estimator (3.27)-(3.28) is a set of quasi-independent second order linear systems because one controlled substrate concentration is associated to one unknown parameter of the vector ρ only. Since the dynamics of estimation errors (3.32) is described also by a linear differential equation with a block diagonal matrix with two 2×2 stable blocks, then, we can have the following stability result, which is an extension of the result presented in [6, 11]:Theorem 3.3.
Under assumptions (A1)–(A9), the estimation errorsỹi and ρ̃i are asymptotically bounded for all t as follows:(i)
(3.34)lim⁡t→∞sup⁡|ỹi(t)|≤2δBi1ωi2-4γi+Bi2γi,i=1,2,(ii)
(3.35)lim⁡t→∞sup⁡|ρ̃i(t)|≤Bi1+ωiBi2γi,i=1,2and then the tracking error y*-y is bounded as follows:(iii)
(3.36)lim⁡t→∞|yi*-yi|≤Miiλi,i=1,2,where B11=m21(D1max+ω1), B21=(D1max+D2max+ω2)m22+D1maxm21, Bi2=m2iγi+m1i, δ=(v1i/v2i)(v1i/(v1i-v2i))-(v1i/v2i)(v2i/(v1i-v2i)), and Mii>0, i=1,2 are constant bounds which will be defined.Proof.
From assumptions (A6)–(A8), it follows that the input vectorb=[b1T,b2T]T of the system (3.32) is bounded. Therefore, (3.32) behaves as a linear time-invariant differential system with a bounded input, whose time response is:
(3.37)xi(t)=Φi(t)xi(0)+∫0tΦi(t-τ)bi(τ)dτ,i=1,2,
where Φi(t) is the state transition matrix associated to Ai defined as:
(3.38)Φi(t)=eAit=[φ11i(t)φ12i(t)φ21i(t)φ22i(t)],with φ11i(t)=(1/(νi1-νi2))(νi1eνi1t-νi2eνi2t), φ12i(t)=(1/(νi1-νi2))(eνi1t-eνi2t), φ21i(t)=-γiφ12i(t), φ22i(t)=ωiφ12i(t)+φ11i(t). Since Ai is a stable matrix, then the state xi is bounded (see [5, Lemma 3.3.5] and [6, Theorem A2.6]). A bound of each component of xi(t) in (3.37) is given by:
(3.39)|xi1(t)|≤|φ11i(t)xi1(0)+φ12i(t)xi2(0)|+∫0t|φ11i(t-τ)||bi1(t)|dτ+∫0t|φ12i(t-τ)||bi2(t)|dτ,|xi2(t)|≤|φ21i(t)xi1(0)+φ22i(t)xi2(0)|+∫0t|φ21i(t-τ)||bi1(t)|dτ+∫0t|φ22i(t-τ)||bi2(t)|dτ.
Then, taking the limit of the above expressions for t→∞ and using Technical Lemma A2.2 from [6], we obtain the following inequalities:
(3.40)lim⁡t→∞sup⁡|xi1(t)|≤2Bi1δωi2-4γi+Bi2γi,lim⁡t→∞sup⁡|xi2(t)|≤Bi1+ωiγiBi2,where Bi1>0 and Bi2>0 are the bounds of bi1 and bi2, that is, |bi1|≤Bi1 and |bi2|≤Bi2, i=1,2, with Bi1 and Bi2 defined in Theorem 3.3. Thus, the boundedness of ỹi and ρ̃i given in (i) and (ii), respectively, is proven.
To prove (iii) we proceed as follows. We can see that the adaptive control law (3.29) with parameter estimator (3.27) and (3.28) applied to system (3.17) leads to the following dynamics of the closed-loop system:
(3.41)(ẏ-ẏ*)+Λ(y-y*)+Λw=ỹ̇+Ω(ỹ+w).
Since Ω is a diagonal stable matrix, then from (3.41) it can be seen that the tracking error y*-y is the output of a linear stable filter driven by the observer error ỹ. From the first equation in (3.30) we can write: ỹ̇+Ω(ỹ+w)=ρ̃+Dw. Now, by using (3.35), and assumptions (A7) and (A8), the following inequalities can be written:
(3.42)ỹ̇1+ω1(ỹ1+w1)=ρ̃1+D1w1≤|ρ̃1|+|D1w1|≤B11+m21D1max⁡+ω1B12γ1=M1,ỹ̇2+ω2(ỹ2+w2)=ρ̃2-D1w1+(D1+D2)w2≤|ρ̃2|+|D1w1|+|(D1+D2)w2|≤B21+ω2B22γ2+m22(D1max⁡+D2max⁡)+m21D1max⁡=M2.
From (3.41) and (3.42) one finds that:
(3.43)(ẏi-ẏi*)+λi(yi-yi*)≤|-λiwi|+Mi=λim2i+Mi=Mii.
From (3.43), after some straightforward calculation it can be obtained that limt→∞|yi*-yi|≤Mii/λi and the proof is complete.


## 3.1. Control Objective

As it was formulated in the previous section, the control objective consists in adjusting the plant’s load in order to convert the glucose into lactic acid via fermentation, which is directly correlated to the economic aspects of lactic acid production. More exactly, considering that the process model (2.2)–(2.9) is incompletely known, its parameters are time varying and not all the states are available for measurements, the control goal is to maintain the process at some operating points, which correspond to a maximal lactic production rate and a minimal residual glucose concentration. After a detailed process steady-state analysis, in [21] it has been demonstrated that these desiderata can be satisfied if the operating point of lactic acid fermentation is kept around the points S1*=3g/L and S2*=5g/L. This choice is the best option in order to simultaneously satisfy the both objectives: maximal lactic production rate and minimal residual glucose concentration [21]. As control variables we select the glucose feeding flow rates both in the first and in the second reactor denoted by F1 and F2, respectively, where F1=(D1-D12)S1in and F2=D2S2in. In this way we obtain a multivariable control problem with two inputs: F1 and F2 and two outputs: S1 and S2.In the following, in Section3.2 we will present the design of an exactly linearizing feedback controller. This controller will be used as a benchmark in order to compare its behavior with the behavior of the indirect multivariable adaptive controller developed in Section 3.3. Also, in this subsection, the parameter estimator used in the indirect multivariable adaptive controller will be designed. In Section 3.4 the tuning procedure of estimator design parameters via stability and convergence analysis of the control scheme will be derived.


## 3.2. Exactly Linearizing Feedback Controller

Firstly, we consider the ideal case where maximum prior knowledge concerning the process (kinetics, yield coefficients, and state variables) is available, and the relative degree of differential equations in process model is equal to 1. Assume that for the two interconnected reactors we wish to have the following first order linear stable closed loop (process + controller) behavior:(3.1)[Ṡ1*-Ṡ1Ṡ2*-Ṡ2]+[λ100λ2][S1*-S1S2*-S2]=0,λ1,λ2>0,
where S1* and S2* are the desired values of S1 and S2.Since (2.4) and (2.8) in the process model have the relative degree equal to 1, these can be considered as an input-output model and can be rewritten in the following form:(3.2)[Ṡ1Ṡ2]=[-qs1X1-qs2X2]-[D10-D1D1+D2][S1S2]+[F1F2].
Then from (3.1) and (3.2) one obtains the following exactly multivariable decoupling linearizing feedback control law:(3.3)[F1F2]=[Ṡ1*Ṡ2*]+[λ100λ2][S1*-S1S2*-S2]+[qs1X1qs2X2]+[D10-D1D1+D2][S1S2].The control law (3.3) applied to the process (3.2) will result in:(3.4)[Ṡ1Ṡ2]=[λ100λ2][S1*-S1S2*-S2]+[Ṡ1*Ṡ2*],
and leads to the following linear error models:(3.5)ė1=-λ1e1,ė2=-λ2e2,
where e1=S1*-S1 and e2=S2*-S2 represent the tracking errors. It is clear that for λ1,λ2>0, the error models (3.5) have an exponential stable point at e1=0 and e2=0.


## 3.3. An Indirect Multivariable Adaptive Controller

In the previous subsection, it was assumed that the functional forms of the nonlinearities as well as the process parameters are known. Since such prior knowledge is not realistic, in this subsection we consider that the kinetics functions are completely unknown. So, the substrate consumption rates-qs1X1 and -qs2X2 in (3.2) can be expressed as follows:(3.6)-qs1X1=ρ1,-qs2X2=ρ2,
where ρ1 and ρ2 are two unknown time-varying parameters. In this case, the control law (3.3) becomes:(3.7)[F1F2]=[Ṡ1*Ṡ2*]+[λ100λ2][S1*-S1S2*-S2]-[ρ1ρ2]+[D10-D1D1+D2][S1S2],
where the unknown parameters ρ1 and ρ2 will be substituted by their on-line estimates ρ̂1 and ρ̂2 calculated by using an observer-based parameter estimator (OBE) [22] applied only for the dynamics of S1 and S2 from (2.4) and (2.8) rewritten as follows:(3.8)Ṡ1=ρ1+F1-D1S1,Ṡ2=ρ2+D1S1+F2-(D1+D2)S2.Under these notations, the algorithm for on-line computation ofρ̂1 and ρ̂2 is particularized as follows:(3.9)Ŝ1̇=ρ̂1+F1-D1S1+ω1(S-Ŝ1),(3.10)Ŝ2̇=ρ̂2+D1S1+F2-(D1+D2)S2+ω2(S2-Ŝ2),(3.11)ρ̂1̇=γ1(S-Ŝ1),(3.12)ρ̂2̇=γ2(S2-Ŝ2),
where γ1,γ2>0 are the gains of the updating laws (3.11) and (3.12), and ω1,ω2>0 are design parameters to control the stability and the tracking properties of the estimator.Remark 3.1.
The parameter error convergence for the estimator (3.9)–(3.12) is assured if some signals in system satisfy the well-known persistency of excitation condition [4, 5]: A piecewise continuous signal vector ϕ:ℜ+→ℜn is referred to as persistent excitation with a level of excitation κ0>0 if there exist constants κ1,T0>0 such thatκ1I≥(1/T0)∫tt+T0ϕ(τ)ϕT(τ)dτ≥κ0I, for all t≥0. Although the matrix ϕ(τ)ϕT(τ) is singular for each τ, this condition requires that ϕ(t) varies in such a way with time that the integral of the matrix ϕ(τ)ϕT(τ) is uniformly positive definite over any time interval [t, t+T0]. In our particular case it is necessary that, for example, D1, D2 or some kinetic parameters to be persistently exciting (which is generically fulfilled).Finally,an indirect multivariable adaptive linearizing controller is obtained by combination of (3.9)–(3.12) and (3.7) rewritten as follows:
(3.13)[F1F2]=[Ṡ1*Ṡ2*]+[λ100λ2][S1*-S1S2*-S2]-[ρ̂1ρ̂2]+[D10-D1D1+D2][S1S2].A block diagram of the designed indirect multivariable adaptive control system is shown in Figure2.Figure 2
The structure of the indirect adaptive control system.Remark 3.2.
Since the on-line measurement of the lactic acid concentrations is not possible, in order to provide on-line evaluation of the current lactic acid production, an asymptotic state observer is derived as follows.
Let us define the auxiliary variablesz1 and z2 as:
(3.14)z1=P1+YPSS1,z2=P2+YPSS2,
which use only the measurements of the outputs S1 and S2 that are available on-line. The dynamics of zi,i=1,2, deduced from model (2.2)–(2.9), are expressed by the following linear stable equations:
(3.15)ẑ̇1=-D1ẑ1+YPSF1,ẑ̇2=D1ẑ1-(D1+D2)ẑ2+YPSF2,
that are independent of the process kinetics. Then, from (3.14) and (3.15), the on-line estimation of P1 and P2 are given by the following relations:
(3.16)P̂1=ẑ1-YPSS1,P̂2=ẑ2-YPSS2.


## 3.4. Tuning Procedure of Estimator Design Parameters via Stability and Convergence Analysis

Let us denote byy=[S1S2]T the vector containing the concentrations of controlled feeding substrates and by u=[F1F2]T the vector of the input control variables, that is, the vector of the mass feeds of substrates S1 and S2, respectively. Then (3.8) can be written as
(3.17)ẏ=ρ-Dy+u,
where ρ=[ρ1ρ2]T is the unknown time-varying parameter vector and D is the dilution rate matrix, whose structure is as follows:(3.18)D=[D10-D1D1+D2].The OBE (3.9)–(3.12) can be represented in a matrix form as follows:
(3.19)ŷ̇=ρ̂-Dy+u+Ω(y-ŷ),ρ̂̇=Γ(y-ŷ),
where ρ̂ is the estimation of unknown parameter ρ, and Ω,Γ∈ℜ2×2 are positive definite diagonal matrices whose entries ω1,ω2>0 and γ1,γ2>0, respectively, represent the estimator design parameters.Under the previous notations, the exactly multivariable decoupling linearizing feedback control law (3.3), respectively, (3.7) can be written as:(3.20)u=ẏ*+Λ(y*-y)-ρ+Dy,
where y*=[S1*S2*]T, and the indirect adaptive linearizing control law (3.13) can be written as:
(3.21)u=ẏ*+Λ(y*-y)-ρ̂+Dy,
where Λ=[λ100λ2]>0.Case 1.
By defining the errorsỹ=y-ŷ and ρ̃=ρ-ρ̂, in the case when the unknown parameter ρ is constant, the following error system can be derived from (3.17) and (3.19):
(3.22)ỹ̇=-Ωỹ+ρ̃,ρ̃̇=-Γỹ.
It can be seen that the error system in (3.22) is a linear time-invariant system that can be rewritten as:
(3.23)[ỹ̇ρ̃̇]=[-ΩI2-Γ0][ỹρ̃],
where I2 is the identity matrix of order two. The stability of the equilibrium (ỹ=0, ρ̃=0) of (3.22) can be proved by using the well-known Hurwitz criterion. So, by using the coefficients of the characteristic polynomial associated to (3.23), given by
(3.24)P(λ)=λ4+(ω1+ω2)λ3+(ω1ω2+γ1+γ2)λ2+(γ2ω1+γ1ω2)λ+γ1γ2=a4λ4+a3λ3+a2λ2+a1λ+a0,after straightforward calculation we obtain that all Hurwitz determinants are positive for any ω1,ω2>0 and γ1,γ2>0. Indeed, by using the coefficients of the above characteristic polynomial we can construct the following Hurwitz determinant:
(3.25)Δ4=[a3a100a4a2a000a3a100a4a2a0]=[ω1+ω2γ2ω1+γ1ω2001ω1ω2+γ1+γ2γ1γ200ω1+ω2γ2ω1+γ1ω2001ω1ω2+γ1+γ2γ1γ2].
Now we construct and calculate all Hurwitz determinants as follows:
(3.26)Δ1=[a3]=ω1+ω2>0,Δ2=[a3a1a4a2]=[ω1+ω2γ2ω1+γ1ω21ω1ω2+γ1+γ2]=ω1ω2(ω1+ω2)+γ2ω1+γ1ω2>0,Δ3=[a3a10a4a2a00a3a1]=[ω1+ω2γ2ω1+γ1ω201ω1ω2+γ1+γ2γ1γ20ω1+ω3γ2ω1+γ1ω2]=ω1ω2((ω1+ω2)(γ2ω1+γ1ω2)+(γ1-γ2)2)>0,Δ4=γ1γ2⋅Δ3>0.
According to Hurwitz criterion [27, 28], the system (3.23) is stable since all Hurwitz determinants are positive if ω1,ω2>0 and γ1,γ2>0. Then, the matrix associated to (3.23) is a Hurwitz matrix and the equilibrium (ỹ=0, ρ̃=0) of (3.22) is uniformly asymptotically stable.Case 2.
In the case when the unknown parameterρ is time-varying, the stability analysis of the proposed indirect multivariable adaptive linearizing controller is carried out under the following realistic assumptions.(A1)
The substrate concentrationsS1 and S2, the components of the dilution rate matrix D, and the substrate feed rates are on-line measured.(A2)
The substrate consumption ratesρ1 and ρ2 are fully unknown.(A3)
The reference signal vectory* and its time derivative are piecewise continuous bounded functions of time, and the measured substrate feed rates are continuous bounded functions of time.(A4)
The measurements of substrate vector, denotedym(t), are corrupted by an additive noise vector w(t)=[w1w2]T: ym(t)=y(t)+w(t), for all t.(A5)
The components of time varying parameter vectorρ(t) are bounded as: 0>ρi(t)≥ρi*, ρi*<0, i=1,2, for all t.(A6)
The time derivatives of the components ofρ(t) are bounded as: |ρ̇i(t)|≤m1i, m1i>0, i=1,2, for all t.(A7)
The measurement noise vector is bounded as:|w(t)|≤m2i, m2i>0, i=1,2, for all t.(A8)
The components of dilution rates matrixD are known and bounded: 0≤Di(t)≤Dimax, i=1,2, 0≤D12(t)≤D12max, for all t.
Under these assumptions, the parameter estimator (3.19) and the adaptive control law (3.21) are rewritten as follows:
(3.27)ŷ̇=ρ̂-Dym+u+Ω(ym-ŷ),(3.28)ρ̂̇=Γ(ym-ŷ),(3.29)u=ẏ*+Λ(y*-ym)-ρ̂+Dym.
From (3.17), (3.27), and (3.28) under the condition (A4), the following error system can be derived as follows:
(3.30)ỹ̇=ẏ-ŷ̇=ρ-Dy+u-ρ̂+Dym-u-Ω(ym-ŷ)=ρ̃+D(-y+y+w)-Ω(y+w-ŷ)=-Ωỹ+ρ̃+(D-Ω)w,ρ̃̇=ρ̇-ρ̂̇=ρ̇-Γ(y+w-ŷ)=-Γỹ-Γw+ρ̇,
which can be rewritten in the following form:
(3.31)ỹ̇1=-ω1ỹ1+ρ̃1+(D1-ω1)w1,ρ̃̇1=-γ1ỹ1-γ1w1+ρ̇1,ỹ̇2=-ω2ỹ2+ρ̃2+(D1+D2-ω2)w2-D1w1,ρ̃̇2=-γ2ỹ2-γ2w2+ρ̇2.Defining the state vectorx as x=[x1Tx2T]T with x1=[ỹ1ρ̃1]T and x2=[ỹ2ρ̃2]T, then (3.31) can be rewritten as:
(3.32)ẋ=Ax+b,
where A=diag{Ai}, i=1,2 and:(3.33)Ai=[-ωi1-γi0],b=[b1b2],b1=[(D1-ω1)w1-γ1w1+ρ̇1],b2=[(D1+D2-ω2)w2-D1w1-γ2w2+ρ̇2].The two eigenvalues ofAi denoted by vi1 and vi2 are related to ωi and γi as follows: vi1+vi2=-ωi<0, vi1vi2=γi>0, i=1,2. It can be seen that v1i,v2i<0. The next assumption is considered for the estimator design parameters.(A9)
The design parametersωi and γi are chosen such that Ai has real distinct eigenvalues [6, 11], for example, vi2<vi1<0, where vi1,2=(-ωi±ωi2-4γi)/2, that implies 0<γi<ωi2/4.The system (3.32) describes the dynamics of the estimation errors of the observer-based estimator including the observations of the measured substrates and the estimations of their unknown consumption rates ρi.We can consider that the estimator (3.27)-(3.28) is a set of quasi-independent second order linear systems because one controlled substrate concentration is associated to one unknown parameter of the vector ρ only. Since the dynamics of estimation errors (3.32) is described also by a linear differential equation with a block diagonal matrix with two 2×2 stable blocks, then, we can have the following stability result, which is an extension of the result presented in [6, 11]:Theorem 3.3.
Under assumptions (A1)–(A9), the estimation errorsỹi and ρ̃i are asymptotically bounded for all t as follows:(i)
(3.34)lim⁡t→∞sup⁡|ỹi(t)|≤2δBi1ωi2-4γi+Bi2γi,i=1,2,(ii)
(3.35)lim⁡t→∞sup⁡|ρ̃i(t)|≤Bi1+ωiBi2γi,i=1,2and then the tracking error y*-y is bounded as follows:(iii)
(3.36)lim⁡t→∞|yi*-yi|≤Miiλi,i=1,2,where B11=m21(D1max+ω1), B21=(D1max+D2max+ω2)m22+D1maxm21, Bi2=m2iγi+m1i, δ=(v1i/v2i)(v1i/(v1i-v2i))-(v1i/v2i)(v2i/(v1i-v2i)), and Mii>0, i=1,2 are constant bounds which will be defined.Proof.
From assumptions (A6)–(A8), it follows that the input vectorb=[b1T,b2T]T of the system (3.32) is bounded. Therefore, (3.32) behaves as a linear time-invariant differential system with a bounded input, whose time response is:
(3.37)xi(t)=Φi(t)xi(0)+∫0tΦi(t-τ)bi(τ)dτ,i=1,2,
where Φi(t) is the state transition matrix associated to Ai defined as:
(3.38)Φi(t)=eAit=[φ11i(t)φ12i(t)φ21i(t)φ22i(t)],with φ11i(t)=(1/(νi1-νi2))(νi1eνi1t-νi2eνi2t), φ12i(t)=(1/(νi1-νi2))(eνi1t-eνi2t), φ21i(t)=-γiφ12i(t), φ22i(t)=ωiφ12i(t)+φ11i(t). Since Ai is a stable matrix, then the state xi is bounded (see [5, Lemma 3.3.5] and [6, Theorem A2.6]). A bound of each component of xi(t) in (3.37) is given by:
(3.39)|xi1(t)|≤|φ11i(t)xi1(0)+φ12i(t)xi2(0)|+∫0t|φ11i(t-τ)||bi1(t)|dτ+∫0t|φ12i(t-τ)||bi2(t)|dτ,|xi2(t)|≤|φ21i(t)xi1(0)+φ22i(t)xi2(0)|+∫0t|φ21i(t-τ)||bi1(t)|dτ+∫0t|φ22i(t-τ)||bi2(t)|dτ.
Then, taking the limit of the above expressions for t→∞ and using Technical Lemma A2.2 from [6], we obtain the following inequalities:
(3.40)lim⁡t→∞sup⁡|xi1(t)|≤2Bi1δωi2-4γi+Bi2γi,lim⁡t→∞sup⁡|xi2(t)|≤Bi1+ωiγiBi2,where Bi1>0 and Bi2>0 are the bounds of bi1 and bi2, that is, |bi1|≤Bi1 and |bi2|≤Bi2, i=1,2, with Bi1 and Bi2 defined in Theorem 3.3. Thus, the boundedness of ỹi and ρ̃i given in (i) and (ii), respectively, is proven.
To prove (iii) we proceed as follows. We can see that the adaptive control law (3.29) with parameter estimator (3.27) and (3.28) applied to system (3.17) leads to the following dynamics of the closed-loop system:
(3.41)(ẏ-ẏ*)+Λ(y-y*)+Λw=ỹ̇+Ω(ỹ+w).
Since Ω is a diagonal stable matrix, then from (3.41) it can be seen that the tracking error y*-y is the output of a linear stable filter driven by the observer error ỹ. From the first equation in (3.30) we can write: ỹ̇+Ω(ỹ+w)=ρ̃+Dw. Now, by using (3.35), and assumptions (A7) and (A8), the following inequalities can be written:
(3.42)ỹ̇1+ω1(ỹ1+w1)=ρ̃1+D1w1≤|ρ̃1|+|D1w1|≤B11+m21D1max⁡+ω1B12γ1=M1,ỹ̇2+ω2(ỹ2+w2)=ρ̃2-D1w1+(D1+D2)w2≤|ρ̃2|+|D1w1|+|(D1+D2)w2|≤B21+ω2B22γ2+m22(D1max⁡+D2max⁡)+m21D1max⁡=M2.
From (3.41) and (3.42) one finds that:
(3.43)(ẏi-ẏi*)+λi(yi-yi*)≤|-λiwi|+Mi=λim2i+Mi=Mii.
From (3.43), after some straightforward calculation it can be obtained that limt→∞|yi*-yi|≤Mii/λi and the proof is complete.


## 4. Simulation Results and Comments

The performance of the designed multivariable adaptive controller (3.9)–(3.13), by comparison to the exactly linearizing controller (3.3) (which yields the best response and can be used as benchmark), has been tested by performing extensive simulation experiments.For a proper comparison, the simulations were carried out by using the process model (2.2)–(2.11) under identical and realistic conditions. The simulations were designed so that several set point changes on the controlled variables S1 and S2 occurred, nearby the operational point S1*=3g/L, S2*=5g/L.The kinetic parameter values used in simulations are presented in Table1 [21], and the initial values of variables considered in simulations in Table 2.Table 1
Parameter values.
ParameterValueParameterValueK̅Sgc0.5 g/Lμmax00.45 h−1KSmaxrc12 g/Lη3.5KSmaxrc12 g/Lβ0.9 h−1KPmaxgc15 g/Lα00.02 g/LKαμ0.2 g/LPCgc95 g/LKαP1.1 g/LYPS0.98 g/gKαS4 g/Lkd0.02 h−1Table 2
Initial conditions.
VariableInitial valueVariableInitial valueX100.02 g/LS10in50 g/LS100.5 g/LS20in200 g/LP100.01 g/Lα10in6 g/Lα103.5 g/LD100.058 h−1X200.02 g/LD200.01 h−1S203.5 g/LD1200.025 h−1P200.01 g/Lρ̂100.1 h−1α202.5 g/Lρ̂200.015 h−1The system’s behavior was analyzed assuming that the dilution rates of the two feeding substrates and the influent enrichment factor act as perturbations of the form:(4.1)D1(t)=D10⋅(1-0.15sin⁡(πt25)),D2(t)=D20⋅(1+0.15cos⁡(πt50)),α1in(t)=α10in⋅(1+0.25sin⁡(πt20)).Also, one of the kinetic parameters—the maximum specific growth rate—is considered time varying:(4.2)μmax⁡(t)=μmax⁡0⋅(1-0.1sin⁡(πt40)).The values of the estimator design parameters, chosen so that to satisfy the tuning conditions of assumption (A9), are:λ1=λ2=0.45, ω1=1.75, ω2=0.75, γ1=0.25, γ2=0.1.The behavior of the indirect adaptive controlled system by comparison to the behavior of closed loop system with exactly linearizing controller is shown in Figure3, the time evolution of the two controlled variables S1 and S2, respectively, and in Figure 4, the control inputs F1 and F2, respectively.Figure 3
Time evolution of outputs—adaptive control (2) by comparison to exactly linearizing control (1).Figure 4
Profiles of control inputs—adaptive control (2) by comparison to exactly linearizing control (1).In order to test the behavior of the indirect adaptive controlled system in more realistic circumstances, we considered that the measurements of both controlled variables (S1 and S2) are corrupted with an additive white noise with zero average (5% from their nominal values). The simulation results in this case, conducted in the same conditions as in previous simulation, are presented in Figures 5 and 6, respectively. The simulation shows that the behavior of controlled variables and of control inputs is comparable with the results obtained in the free noise simulation.Figure 5
Evolution of outputs—adaptive control (noisy data) (2) by comparison to linearizing control (1).Figure 6
Control inputs—adaptive control (noisy data) (2) by comparison to linearizing control (1).The time evolution of the estimates of the unknown functions (3.6) provided by the observer-based estimator is presented in Figures 7 and 8, in both simulation cases.Figure 7
Profile of estimates of unknown functionρ1: (1) without and (2) with noisy measurements.Figure 8
Profile of estimates of unknown functionρ2: (1) without and (2) with noisy measurements.From Figures7 and 8, it can be noticed that the time evolution of estimates for noisy measurements of S1 and S2 is similar with the time profiles in free noise case.The production of lactic acid can be evaluated by using the data provided by the asymptotic observer (3.15) and (3.16). The time evolution of the lactic acid concentration estimates in the two reactors is depicted in Figure 9.Figure 9
Profile of estimates of lactic acid concentration: (1) without and (2) with noisy measurements.From graphics in Figures3–6 it can be seen that the behavior of overall system with indirect adaptive controller, even if this controller uses much less a priori information, is good, being very close to the behavior of closed loop system in the ideal case when the process model is completely known. Note also the regulation properties and ability of the controller to maintain the controlled outputs S1 and S2 very close to their desired values, despite the very high variations for D1, D2, α1in, and μmax, time variation of process parameters and the influence of noisy measurements.In order to compare our proposed strategy with some classical control technique, several simulations were performed. As exemplification, Figures10 and 11 present the outputs and control inputs for the case when a classical PI (proportional-integrative) control is used, with the best tuning of control parameters. As can be noticed, the behavior of the classical control is strongly influenced by the noisy measurements and by the variation of kinetic and process parameters.Figure 10
Evolution of outputs—PI control (noisy data) (2) by comparison to PI (free noise) (1).Figure 11
Control inputs—PI control (noisy data) by comparison to PI (free noise).


## 5. Conclusion

This paper is concerned with the design of an indirect multivariable adaptive control for a lactic acid fermentation process that is carried out in a cascade of two stirred tank fermentation reactors.The controller was achieved by assumption that the consumption rates of two limiting substrates are completely unknown and are summarized in two fully unknown time varying parameters which are estimated by means of appropriately observer-based estimators. A simple tuning method of kinetic estimator design parameters was achieved by stability and convergence analysis of the control scheme.The simulation results showed that the performances of the proposed multivariable adaptive controller are very good. The derived control method can be applied also for a large class of square nonlinear bioprocesses.



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