Modeling-Error Compensation Approach for Extremum-Seeking

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Modeling-error compensation approach for extremum-seeking control of continuous stirred tank bioreactors with unknown growth kinetics Monica Meraz, Carlos Ibarra-Valdez, and Jose Alvarez-Ramirez Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b00735 • Publication Date (Web): 17 Mar 2016 Downloaded from http://pubs.acs.org on March 17, 2016

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Industrial & Engineering Chemistry Research

(Corrected Manuscript No. ID ie-2016-00160b – 3rd Round)

Modeling-error compensation approach for extremum-seeking control of continuous stirred tank bioreactors with unknown growth kinetics

M. Meraz(1), C. Ibarra-Valdez(2) and J. Alvarez-Ramirez*,(3)

(1) Departamento de Biotecnología. Universidad Autónoma Metropolitana-Iztapalapa. Apartado Postal 55-534. Iztapalapa, 09340 México. (2) Departamento de Matemáticas. Universidad Autónoma Metropolitana-Iztapalapa. Apartado Postal 55-534. Iztapalapa, 09340 México. (3) Departamento de Ingeniería de Procesos e Hidráulica. Universidad Autónoma Metropolitana-Iztapalapa. Apartado Postal 55-534. Iztapalapa, 09340 México.

Corresponding Author: J. Alvarez-Ramirez. Phone/Fax: +52-55-58044650. Email address: [email protected].

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Abstract In this paper, an adaptive extremum seeking control scheme for continuous stirred tank bioreactors is presented. Unknown growth kinetics without considering an explicit mathematical expression is assumed. An adaptive learning technique based on modeling error estimation coupled with a proportional compensation is used to construct a feedback control that maximizes a production function (e.g., biogas production) that depends on the growth kinetics. Also, it is assumed that only the production output and the substrate concentration are available from measurements. It is shown that the resulting controller is equivalent to a proportional-integral compensator. The closed-loop analysis based on singular perturbation techniques showed the stability of the controlled bioreactor at about the optimal equilibrium point. Numerical simulations on a simple example and a more realistic anaerobic digestion case are used to illustrate the stability properties. Keywords: Extremum-seeking control; modeling error estimation; bioreactors; stability analysis.


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1. Introduction Bioreactors are commonly intended to operate at optimal conditions where a given production function is maximized or minimized. For instance, substrate inhibition effects lead to a maximum of the growth kinetics, which is taken as the optimal operating condition to maximize the production of, e.g., biogas, biomass, etc. Traditionally, the problem of obtaining the optimal operating conditions has been addressed with off-line optimization techniques for either static or dynamical bioreactor models. The former case can be solved with standard (e.g., gradient) optimization techniques, while the second one is addressed by means of dynamic programming techniques. In either case, the solution is a reference set that should be achieved by means of feedback control techniques. The main drawback with the off-line approach is the lack of robustness guarantee in the face of uncertain models and changing (e.g., exogenous) disturbance conditions. Extremum-seeking feedback control is a self-optimizing strategy that was proposed to find operating sets that optimize a given objective function. Firstly explored by LeBlanc1, the approach has been successfully used to control the optimal operation of engine presses2, anti-lock braking systems3, electromechanical actuators4, photovoltaic systems5, chemical oxidation reactors6, among many other instances. Extremum seeking control methods remained dormant by 30 years or so, until the proof of local stability arrived by the turn of the century7. Also, the widespread of more complex and unreliable models in many applications (combustion, biomedical systems, fluid flow, etc.) motivated the use of reliable model free approaches. These two reasons together raised a renewal of interest in the framework of extremum-seeking feedback at a very strong pace, both theoretically and along many applied areas8-11. Extremum-seeking control has been also proposed for bioreactors. Wang et al.12 used a feedback control strategy to maximize the biomass production. Zhang et al.13 as well as Guay et al.14 assumed limited knowledge of the growth kinetics to propose an adaptive feedback controller to drive the bioreactor operation to optimal operating conditions. Marcos et al.15 provided an adaptive extremum-seeking for continuous stirred tank bioreactors with Haldane’s kinetics. Cougnon et al.16 extended the self-optimizing control strategy to fedbatch bioreactors. However, the proposed learning control approaches are based on parametric adaptation techniques, requiring in this way the structure of the growth 3 ACS Paragon Plus Environment


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kinetics and bounds of the involved parameters. As in traditional adaptive control strategies, a common feature of the aforementioned approaches is the use of dithers for guaranteeing the persistence estimation of objective function gradient. Numerical simulations have shown that dithers can lead to excessively sluggish response with low convergence rates. In many practical cases, the growth kinetics structure and parameters are highly uncertain due to changing environment conditions17. In this regard, Lara-Cisneros et al.18 used modeling error compensation techniques19 coupled with sliding-mode feedback to stabilize the optimal operation of fed-batch bioreactors. However, numerical simulations showed that the control performance can be very poor, exhibiting sluggish and highamplitude oscillations about the optimal equilibrium point. Vargas et al.20 considered supertwisting methods for proposing an adaptive feedback controller for extremum-seeking control of a class of bioreactors. The aim of this work is to study the extremum-seeking control of continuous stirred tank bioreactors when the growth kinetics is unknown. Here, the control objective is to maximize certain production output in the face of non-monotonous growth kinetics. The feedback control strategy is based on a proportional feedback of the controlled output (i.e., the gradient of the production output) and a modeling error estimator to compensate for unknown growth kinetics trajectory. It is shown that the feedback control scheme is equivalent to a proportional-integral compensator. Rigorous stability analysis is provided for the ideal and practical controlled bioreactor systems. The case of estimation of the gradient of the production output in terms of substrate concentration and production output is discussed. A gradient estimation scheme based on low-pass filtering is considered. In the spirit of Lara-Cisneros et al.’s work18, the proposed approach does not make use of dither functions for achieving closed-loop convergence to optimal conditions. The main difference with such work is that the feedback of the regulated output does not rely on variable-structure (e.g., sliding-mode) control, so the closed-loop response is smoother. Numerical simulations with a simple system and an anaerobic digester involving two substrates and two biomass strains are used to illustrate the findings, especially those concerning to stability.


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2. Statement of the problem Consider the following dynamics of a bioreactor with single substrate and strain: ds = D ( s f − s ) − k1 µ ( s ) x dt dx = − Dx + µ ( s ) x dt

(1)

where the state variables are the substrate s and biomass x concentrations. Here, µ (s ) is the growth function, k 1 is a yield coefficient, D is the dilution rate, and s f is the feed substrate concentration. Assume that the production output is given by y p = k 2 µ (s) x

(2)

where k 2 is a yield coefficient. The production output can represent biogas production in anaerobic digestion reactors18, the biomass production in microalgae systems21, and the alcoholic production in fermentation systems22. In some cases, due to inhibition effects, the growth function µ (s ) exhibits a maximum value at a given substrate concentration

smax > 0 . As a consequence, the production output y p = k 2 µ ( s ) x also displays a maximum production value at s = s max . This feature motivates the problem of maximizing the production output y p ( s , x ) via manipulations of the dilution rate D . That is, the control problem can be stated as an optimization problem of the following form:

max y p ( s, x) D

(3)

subject to the dynamics given by Eq. (1) Classically, optimal problem of this kind has been addressed with dynamic programming23 and Pontryagin's maximum principle24 methods. However, these approaches lead to prescribed control strategies that are not endowed with correction mechanisms to confront external disturbances (e.g., variations of feed conditions) and parametric uncertainties, i.e., they are not necessarily robust. In this regard, extremum seeking feedback control has been proposed as a reliable strategy to deal with the optimization problem given by Equation 3 (see references12-14,25). This approach is taken in the present work under the following mild assumptions on the growth function µ (s ) : Assumption A.1. µ (s ) is non-negative and upper bounded by a positive constant mmax for all s ≥ 0 . 5 ACS Paragon Plus Environment


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Assumption A.2. µ (s ) is twice continuously differentiable. Assumption A.3. µ (s ) is strictly concave (i.e., d 2 µ (s) / ds 2 < 0 ) for all s ≥ 0 . This means that the growth function µ (s ) has a unique maximum in s ≥ 0 . Assumption A.4. µ (s ) is unknown. Specifically, no explicit or closed form formula of the growth function µ (s ) is assumed in advance. Assumption A.5. Only the substrate concentration s and the production output y p are available for measurements. Assumptions A.1 and A.3 imply that the production output y p ( s , x ) possesses a steadystate maximum value y p , max ( s max , x ) for a positive substrate concentration s max . In the extremum seeking approach, the optimization problem Equation 3 is addressed by means of a feedback control approach by considering the extremum condition as the regulated output. That is, the feedback control problem can be posed as follows: “Under the assumptions A.1 to A.5, design a feedback control input D = f ( s , y p ) such that the production output y P (t ) converges asymptotically to the maximum value y p , max .” In the lack of structural constraints (i.e., bounded control input), the above problem can be recasted as a standard feedback control problem by introducing the controlled output yc =

∂y p ( s , x )

(4)

∂s

Notice that

∂y p ( s , x ) ∂s

= k2

dµ ( s ) x , so that the controlled output corresponds with the ds

maximum of the growth function µ (s ) (Assumption A.3).In terms of the controlled output

y c the control problem can be re-stated as follows: “Under the assumptions A.1 to A.5, design a feedback control input D = f ( s , y p ) such that the control output y c (t ) converges asymptotically to zero.”By Equation 4 and Assumption A.4, the convergence yc (t ) → 0 guarantees the convergence y P (t ) → y p , max .


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3. Ideal feedback control design As a preliminary step towards the design of feedback controls under assumptions A.1 to A.5, the design problem under exact knowledge of the growth kinetics µ (s ) and measurements of the biomass concentration x (t ) will be considered. To this end, let us introduce the physical domain Σ ph of the biological reaction system:

{

Σ ph ≡ ( s, x) : 0 < s < s f , 0 < x < x * , x * = s f / k1

}

(5)

Here, x * is the maximum achievable steady-state biomass when s = s f .

Proposition 1. Consider the following feedback control D=

1 [k1 µ ( s) x + g c y c ] sf − s

(6)

for any positive constant g c . Then, the controlled bioreactor given by Equation 1 and Equation 6 has a unique positive equilibrium point {s max , x max }∈ Σ ph , which corresponds to maximum production output (i.e., yc = 0 ). Besides, all trajectories starting in Σ ph converge asymptotically to such equilibrium point. Proof. i) Equilibrium points: Since yc = k1 µ´(s) x , where µ´(s ) = dµ ( s ) / ds , the closedloop system is given by ds = g c k1 µ ´(s ) x dt dx 1 =− [k 2 µ ( s ) + g c k1 µ´(s )]x 2 + µ ( s ) x dt sf − s

(7)

It is noted that the horizontal axis Ls = {(s, x) : x = 0} is a focus for equilibrium points of the controlled bioreactor given by Equation 7. In turn, the horizontal axis Ls corresponds to wash-out (i.e., x eq = 0 ) conditions. For equilibrium conditions in the positive quadrant, one has the following conditions:

µ´(s eq ) = 0 −

[

]

1 k 2 µ ( s eq ) + g c k1 µ´(s eq ) x eq + µ ( s eq ) = 0 s f − s eq


(8)


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From Assumption A.4, the first equality in Equation 8 implies that s eq = s max is the unique substrate equilibrium. On the other hand, the second equality is reduced to the following one:

  k 2 xeq + 1 µ ( s max ) = 0 −   s f − s max

(9)

Since µ (smax ) ≠ 0 , one has that − k 2 x eq + s f − s max = 0 . That is, the only equilibrium biomass concentration is

xeq = xmax ≡

s f − s max

(10)

k2

ii) Local stability. The local stability of the equilibrium point (smax , xmax ) is established by computing the Jacobian matrix at this point. The Jacobian is given by g c k1 µ´(s )  g c k1 µ´´(s ) x    2 JF ( s, x) =  [k 2 µ (s) + g c k1 µ´(s)]x + µ (s)  * −  s − s f  

(11)

where ‘*’ denotes an entry that is not relevant for the computations and

µ´´(s) = d 2 µ ( s) / ds 2 . By recalling that µ´(smax ) = 0 , one has that  g c k1 µ´´(s max ) x max  JF ( s max , x max ) =  *  

0   k 2 x max − µ ( s max )   s f − s max 

(12)

Here, Equation 10 was also used. The eigenvalues of JF (smax , xmax ) are given by the diagonal

elements.

Since

µ´´(smax ) < 0

g c k1 µ´´(s max ) xmax < 0 . On the other hand, −

(Assumption s max < s f

A.3),

and

one

has

that

µ (smax ) > 0 , so that

k 2 xmax µ (smax ) < 0 . This means that the equilibrium point (s max , xmax ) is locally s f − s max

asymptotically stable for all g c > 0 . iii) Stability in the physical domain Σ ph . Let us analyze the behavior of the substrate trajectory s(t ; s0 , x0 ) for initial conditions ( s 0 , x0 ) in Σ ph . For s < smax , one has that

µ´(s ) > 0 , so that

ds = g c k1 µ´(s ) x > 0. In contrast, for s > s max , µ´(s ) < 0 and dt 8 ACS Paragon Plus Environment

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ds = g c k1 µ´(s ) x < 0. This means that, regardless of the initial biomass concentration x0 , dt the trajectory s(t ; s0 , x0 ) converges to the vertical line L max = {( s , x ) : s = s max , 0 < x < x * }. Besides, since µ´(smax ) = 0 , the line Lmax is an invariant set under the dynamics of the closed-loop system Equation 7. In this way, one can constrain the analysis to this invariant set. Using Equation 10, the biomass concentration dynamics constrained to Lmax can be expressed as

 x  dx = − µ ( s max ) − 1 x dt  xmax 

(13)

As already shown, xmax is the only equilibrium point in Lmax . Since µ ( s max ) x is positive the biomass concentration trajectories x(t ; x0 ) from Equation 13 converge asymptotically to xmax for all x0 ∈ Lmax . The above arguments imply that the equilibrium point (smax , xmax ) is a global attractor for trajectories starting in the physical domain Σ ph .

Remark 1. The proof of the above proposition illustrates the geometry displayed by the dynamical

behavior

(s(t; s0 , x0 ), x(t; s0 , x0 ))

of

the

controlled

bioreactor.

The

system

trajectories

converge with rate O(g c ) to a neighborhood of the invariant set

{

}

L max = ( s , x ) : s = s max , 0 < x < x * . Subsequently, the trajectories are attracted by the

equilibrium point (smax , xmax ) . For large control gain g c , the convergence to Lmax is so fast that the behavior resembles that of sliding-mode dynamics, with Lmax being the sliding set. Let us illustrate via numerical simulations the performance of the controlled bioreactor. In order to do so, we depart from the model free approach just for a moment, and consider the Haldane growth function

µ (s) =

µ max s

(14)

Ks + s + s2 / KI



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with parameters µ max = 5.0 h-1, K s = 0.75 g⋅L-1 and K I = 5.0 g⋅L-1. The additional system parameters are k1 = 2.5 , k 2 = 0.75 and s f = 25 .0 g⋅L-1. For the Haldane model, the optimum substrate concentration is s max = K s K I = 1.936 g⋅L-1. Figure 1 illustrates the phase-portrait for the control gain value g c = 3.0 . As already proven, the system trajectories converge to a neighborhood of the set Lmax , and subsequently exhibit a type of sliding behavior to converge to the optimal equilibrium point (s max , xmax ) .

4. Practical feedback control design The implementation of the ideal feedback control given by Equation 6 requires knowledge of the growth function µ (s ) and the regulated output y c . The growth function is poorly known in many practical situations. Besides, the biomass concentration and the regulated output are not, in general, available for measurements. In what follows, practical solutions for these problems will be proposed.

4.1. Unknown growth function

The lack of availability of the growth function will be addressed by means of modeling error compensation techniques19. To this end, introduce the following modeling error function:

η ( s, x) = k1 µ (s) x

(15)

Let η e (t ) be an estimate of the modeling error trajectory η (t ) = k1 µ ( s(t )) x(t ) . Then, the ideal feedback controller given by Equation 6 can be modified to obtain D=

1 [η e + g c y c ] sf − s

(16)

The estimate η e (t ) can be obtained from the gradient-like estimator dη e = g e (η − η e ) dt

(17)

where g e is a positive parameter to adjust the convergence rate. It is noted that Equation 17 can be seen as a reduced-order observer for the modeling error trajectory η (t ) . From the


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dynamics

η =−

of

the

substrate

concentration

in

Equation

1,

one

knows

that

ds + D( s f − s) , therefore Equation 17 becomes dt

dη e  ds  = g e  − + D( s f − s ) − η e  dt  dt 

(18)

Introduce the variable z = g e−1η e + s . Then, the use of Equation 16 allows writing the modeling error estimator as follows: dz = D(s f − s) − η e dt η e = g e (z − s )

(19)

In this way, the practical controller is given by the feedback function Equation 16 and the modeling error estimator Equation 19.

Remark 2. The practical controller given by equations 16 and 19 is equivalent to a proportional-integral (PI) compensator. For initial condition z0 = 0 , one has that t

z = g c ∫ y c (t´)dt´ . Then, the feedback function becomes 0

D=

t  1  g g  e c ∫ y c (t´)dt´ + ( g c + g e ) y c  sf − s  0 

(20)

This is a PI compensator with proportional and integral gains given by ( g c + g e ) /( s f − s ) and g e g c /( s f − s ) , respectively.

Proposition 2. There exists a positive constant g e* such that, for all g e > g e* , the controlled bioreactor under the practical feedback function given by equations 16 and 19 is asymptotically stable about the optimal equilibrium point (s max , xmax ) with basin of attraction containing the physical domain Σ ph .

Proof. Let e = η − η e be the estimation error. Then, the control input is given by D=

1 [e + k1 µ ( s) x + g c y c ] sf − s

(21)



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and so the closed-loop equations are given by ds = g c k1 µ ´(s ) x + e dt dx 1 =− [k 2 µ ( s ) + g c k1 µ´(s ) ]x 2 + µ ( s ) x − x e dt sf − s sf − s

(22)

On the other hand, the dynamical behavior of the estimation error is governed by the differential equation

εe

de = −e + ε eϕ e (s, x, e) dt

(23)

where ε e = g e−1 and

ϕ e ( s, x, e) = [k1 g c µ´(s ) x ]2 −

k1 µ ( s ) [k 2 µ ( s) + g c k1 µ´(s )]x 2 + k1 µ (s) 2 x sf −s

k µ (s) x + k1 µ´(s ) xe − 1 e sf − s

(24)

In this case, the controlled system is composed by equations 22 and 23. From Proposition 1, it can be shown that the controlled system has a unique equilibrium point given by s eq = s max , x eq = x max and e eq = 0 . On the other hand, equations 22 and 23 can be seen as a

singularly perturbed system with ε e > 0 taken as the small perturbation parameters. The slow subsystem is given by Equation 2 with e ≡ 0 . In turn, Proposition 1 guarantees that the slow subsystem is asymptotically stable about the optimal equilibrium point (s max , xmax ) with basin of attraction containing the physical domain Σ ph . The fast subsystem is obtained by taking the fast time scale t f = t / ε e in Equation 23 and taking the equality e ≡ 0 , which leads to the exponentially stable linear differential equation

de = −e . The stability of the dt f

controlled bioreactor under the growth function estimator given by Equation 19 is guaranteed for sufficiently small perturbation parameter ε e by following standard arguments of stability of singularly perturbed systems26.

Remark 3. Equation 23 states that g e corresponds to the convergence rate of the nominal estimation error dynamics. This means that faster convergence of the modeling error 12 ACS Paragon Plus Environment

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estimator is achieved when the estimator gain g e is increased. On the other hand, the nominal convergence of the controlled reactor is governed by Equation 7 (i.e., Equation 22 with e = 0 ). Let w = s + k1 x , which corresponds to the “total” mass balance in the reactor. From Equation 1, one obtains that

dw = D(w f − w) , where dt

w f = s f . This means that

regardless of the control input, the convergence rate is limited by the maximum allowable dilution rate Dmax = µ max . The above feature should be useful for proposing tuning guidelines for the extremum-seeking control strategy described above. The numerical example described in the above section is also used for illustrating the behavior of the controlled bioreactor with modeling error estimation. Figure 2 shows the phase-portrait for g c = 3.0 and g e = 8.0 . Despite the lack of knowledge of the growth function µ ( s ) , the PI controller is able to drive the bioreactor trajectories towards the optimal equilibrium point. It is apparent that the sliding geometry induced by the ideal feedback controller is not longer present under the action of the modeling error estimator. In fact, all trajectories converge directly to the optimal equilibrium point without approaching the otherwise invariant set Lmax .

4.2. Non-measured controlled output

The implementation of the feedback function given by Equation 16 requires the availability of the controlled output y c . However, this signal is, in general, not available for measurements27. In many practical cases, only the substrate concentration s and the production output y p are measured. A practical approach to this issue is to obtain an estimate the controlled output yc ,e from the measured signals. In this way, the practical feedback control function is D=

1 [η e + g c y c,e ] sf − s

(25)

To gain insights, let us recall that yc = ∂q( s, x) / ∂s . In a naïve approach, one is tempted to assume that the standard (i.e., non-partial) derivative y c (t ) = dy p / ds is approximately 13 ACS Paragon Plus Environment


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correct. One can obtain an estimate of the controlled output via finite-difference schemes. For instance, for the simplest first-order scheme one has that the estimated controlled output yc ,e (tk ) at time t k is given by

yc,e (t k ) =

y p (t k ) − y p (t k −1 )

(26)

s(t k ) − s(t k −1 )

and t k − t k −1 is the sampling period. This approach, endowed with some filtering scheme, has been already considered18 to estimate the trajectory of the controlled output (i.e., the gradient of the production output). However, Equation 26 can lack stability in the face of, e.g., measurement noise. A more systematic approach consists in estimating the timederivatives dy p / dt and ds / dt with approximate derivation filters. For simplicity in notation, let v p = dy p / dt and v s = ds / dt . Approximate values v p , a and v s , a of these velocities can be obtained by approximate differentiation as follows:  λ  y v p ,a =  τ λ +1 p p    λ   s v s , a =  τ sλ + 1 

(27)

where λ = d / dt is the time-derivative operator, and τ p and τ p are filtering time-constants. Time-domain input/ouput realizations of the above expressions can be expressed in the following form:

dw p dt

=−

wp + y p

τp

, v p ,a =

wp + y p

τp

(28)

dws w +s w +s =− s , v s ,a = s τs τs dt

Here, w p and ws are internal variables. In this way, an estimate of the controlled output is given by

y c ,e (t ) =

v p ,a (t )

(29)

v s ,a (t )

Initial conditions for the velocity estimators in Equation 28 can be chosen as follows. It is satisfied that w p = τ p v p , a − y p and ws = τ s v s ,a − s . Since the approximate velocities v p ,a


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and v s , a are not known in advance, suitable initial conditions are w p (0) = − y p (0) and ws (0) = − s (0) . The use of the filtering scheme Equation 28 to obtain approximate timederivatives is widely used in mechanical systems, leading to good performance in real applications28,29. A block diagram summarizing the signals and operations involved in the extremum-seeking control for the biological reactor is given by Figure 3.

Remark 4. The relationship y c (t ) = dy p / ds is not fully correct in general. In fact, since y p = k 2 µ ( s ) x , one has that

dy p = k 2 µ´(s) xds + k 2 µ (s)dx

(30)

= yc ds + k 2 µ ( s)dx so that yc =

dy p ds

− k 2 µ (s)

dx ds

(31)

This shows that the approximation given by y c (t ) = dy p / ds is accurate only when the differential k 2 µ ( s)dx is “sufficiently small” relative to the term k 2 µ´(s) xds . Regarding this issue, the following points are worth noting: i) Establishing conditions for the general case is not an easy task. Situations where the biomass concentration changes slowly with respect to the substrate concentration are likely to meet the approximate relationship y c ≈ dy p / ds . For instance, the biomass concentration in anaerobic digestion exhibits only small variations over a long operating period30,31. ii) It is interesting to note that the term

k 2 µ´(s) xds cannot be much smaller than k 2 µ ( s)dx in a neighborhood of the optimal operating point. At the operating point one has that µ´(s ) → 0 and µ ( s ) approaches its maximum value. In this way, one concludes that the approximation y c ≈ dy p / ds is arbitrarily invalid near the optimal equilibrium point. To illustrate the behavior of the controlled biological reactor under gradient estimation, Figure 4 shows the phase-portrait of the controlled bioreactor for the same tuning conditions used in Figure 2. In this case, the controlled output is replaced by an approximated signal obtained from the scheme given by equations 28 and 29 with sampling 15 ACS Paragon Plus Environment


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period ∆t = 0.01 h, and τ s = τ s = 0.1 h. The convergence to a very small neighborhood of the optimal conditions is achieved for a diversity of initial conditions. It is noted that, similar to the results shown in Figure 2, the sliding-like behavior at the vertical line Lmax is not longer present. The convergence to the optimal conditions is quite smooth, without the sluggish oscillations introduced by dithers13 or variable-structure feedback18. It should be mentioned that despite the interesting results shown in Figure 4, the general problem of estimating the controlled output from measurements of the regulated output and the substrate concentration has not been rigorously solved. Extended Kalman filters have been proposed as a suitable alternative for gradient estimation32. However, the issue deserves detailed studies that should be reported in the future.

4.3. Tuning guidelines An important issue that is rarely addressed in the design of extremum-seeking feedback controllers is the appropriate selection of the control parameters. In our case, the controller gain g c and the estimator gain

g e should be assigned. A tuning rule for selecting the

estimator gain can be delineated as follows. From Equation 17 one recalls that dη e = g e (η − η e ) , so that the gain g e has the same units as the dilution rate D . A dt

traditional tuning rule states that estimation should not be carried out faster than the dominant time-constant of the continuously stirred tank reactors33. As a heuristic rule, estimation characteristic time scale is commonly assigned as about five to ten times the residence time (i.e., D−1 ) of the reactor. In this way, the rule g e ≈ D / n , with n ≈ 5 is suggested. The case of the controller is a subtler issue. From Equation 16, the control gain

g c can be seen as a proportionality constant between the controlled output y c and the effective substrate flow rate D( s f − s) (i.e., D ( s f − s ) = g c y c ). Since y c is the derivative with respect to the substrate, the gain g c has units of mass concentration (e.g., mol/L, g/L, etc.). A rough approach is selecting g c of the order of the substrate feeding conditions. That is, the control gain can be seen as a tuner of the mass conversion at the optimal conditions. Hence, use values of g c of about 5-10 times smaller than s f . In this way, for 16 ACS Paragon Plus Environment

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the working example used above one had s f = 25.0 and g c = 3.0 . Of course, this tuning guidelines offer only a first approach for choosing the controller parameters and further refinement is required to obtain an acceptable closed-loop performance. For the working example used above, Figure 5 illustrates the effects of the controller gain (Figure 5.a) and the estimator gain (Figure 5.b) in the convergence rate of the production output y p . As already commented, increasing the estimator gain g e provides a faster estimation of the modeling error trajectory, which can be accurately counteracted by the action of the feedback function control (see Equation 16). Similar results were obtained for the controller gain g c , although tight tuning is required to avoid unstable behavior due to excessive tight of the control action.

5. Anaerobic digestion process It should be recognized that the biological reactor model given by Equation 1 is simple to represent the operation of most biological reacting systems. Given its simplicity that retains the main structural characteristics of biological reacting processes, this model has been considered in many works as a benchmark to propose feedback control strategies and to establish stability and control tuning guidelines. In the following, a more involved process with two substrate and two biomass strains will be used for illustrating the feedback control strategy described in the above sections. The idea is to show that the feedback controller based on modeling error estimation is able to accomplish extremum-seeking task despite the fact that stability analysis was carried out for the simple model given by Equation 1. Anaerobic digestion is a technology for reducing organic matter content in municipal and agro-food industries. Biogas is a by-product consisting mainly on methane and carbon dioxide, which has been proposed as source of renewable energy34. The optimization of methane outflow rate is a key issue in the operation of anaerobic digestion processes. It has been pointed out that the optimal operation of anaerobic digesters is hampered by the risk of acidification, which arises as an effect of the inhibition of mathanogenic bacteria growth by accumulation of volatile fatty acids35. The model assumes two main populations composed by acidogenic bacteria x1 and methanogenic bacteria x 2 . The former consumes primary substrates s1

to produce volatile fatty acids s 2 , and the latter consumes this 17 ACS Paragon Plus Environment


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substrate to produce methane and carbon dioxide. The system was considered recently for testing extremum-seeking control strategies for anaerobic digestion processes.18 A simplified version of the biological reaction network is described as follows: µ1

k1 s1 → x1 + k 2 s 2

(32)

µ2

k 3 s 2 → x2 + k 4 CH 4 The system dynamics can be described by the following set of differential equations: ds1 dt dx1 dt ds 2 dt dx 2 dt

= D ( s1, f − s1 ) − k1 µ1 ( s1 ) x1 = − aDx1 + µ1 ( s1 ) x1

(33) = D ( s 2, f − s 2 ) + k 2 µ1 ( s1 ) x1 − k 3 µ 2 ( s 2 ) x 2 = − aDx 2 + µ 2 ( s 2 ) x 2

The parameter a denotes the fraction of the bacterial population that is affected by digestion. In this way, a = 0 and a = 1 correspond respectively to ideal fixed bed reactor and CSTR conditions. The outflow rate of methane is given by

Q = k 4 µ 2 (s2 ) x2

(34)

The growth functions for acidogenic and methanogenioc bacteria are given as follows36:

µ1 ( s1 ) = µ 2 (s 2 ) =

µ1, max s1 K S ,1 + s1

(35)

µ 2,max s 2 K S , 2 + s 2 + s 22 / K I , 2

The term s 22 / K I , 2 in µ 2 ( s 2 ) reflects the inhibition effect by volatile fatty acids to the production of methanogenic bacteria. In line with the problem statement in Section 2, the production output corresponds to the methane production given by Equation 34, while the controlled output to the derivative yc =

dµ 2 ( s 2 ) dQ = k4 x2 ds 2 ds 2

(36)

The ideal feedback function is given by (see Equation 6)


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D=

1 [− k 2 µ1 ( s1 ) x1 + k 3 µ 2 ( s 2 ) x 2 + g c y c ] s 2, f − s 2

and the corresponding closed-loop system is

ds1 = D ( s1, f − s1 ) − k1 µ1 ( s1 ) x1 dt dx1 = − aDx1 + µ1 ( s1 ) x1 dt ds 2 dµ 2 ( s 2 ) = gc k4 x2 dt ds 2

(37)

dx 2 = − aDx 2 + µ 2 ( s 2 ) x 2 dt Establishing the stability of the system 37 is not in general an easy task. The problem relies on obtaining a Lyapunov function that describes the behavior of trajectories around equilibrium points. Rather than involving on the rigorous mathematical proof, the anaerobic digestion system will be used to show the ability of the feedback control strategy depicted in the above section to stabilize more realistic biological reaction systems. In this case, the modeling error function (see Equation 15) is given by

η ( s1 , x1 ) = k 2 µ1 (s1 ) x1 − k 3 µ 2 (s 2 ) x2

(38)

and the practical feedback control becomes D=

1 s 2, f − s

[η e + g c y c ]

(39)

where the estimated modeling error ηe is given by Equation 19. The extremum-seeking problem makes sense since the inhibitory effect introduced by the s 22 / K I , 2 in µ 2 ( s 2 ) . Figure 6 presents the growth function µ 2 ( s2 ) for the following set of parameter values(36): k1 = 41.14 g⋅g-1, k 2 = 116.15 mmol⋅g-1, k3 = 268 mmol⋅g-1,

k 4 = 453.0 mmol⋅g-1, µ1,max = 0.05 h-1, µ 2,max = 0.031 h-1, K S ,1 = 7.1 g⋅L-1, K S ,2 = 9.28 mmol⋅L-1, K I ,2 = 16 mmol⋅L-1, a = 0.5 , S1, f = 10 mmol⋅L-1 and S 2, f = 80 mmol⋅L-1. The optimal substrate concentration is about s 2 ,max = 12.56 mmol⋅L-1, corresponding to an optimal steady-state production of methane of about Qmax = 3.65 mmol⋅L-1⋅h-1. Figure 7.a presents the phase-portrait of s 2 versus Q for the controlled biological reactor with the 19 ACS Paragon Plus Environment


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controller parameters g c = 1.0 and g e = 0.01. The trajectory of the methane production is depicted in Figure 7.b, which correspond to the trajectories in Figure 7.a. Since the initial conditions are close to the feed conditions, the methane production exhibits a transit behavior with an overshoot in methane production, which is damped to converge to the steady-state maximum value displayed in Figure 6. It is noted the smoothness of the trajectory convergence, in contrast to the results shown by Lara-Cisneros et al.18, where convergence is affected by highly unrealistic oscillatory behavior induced by the dither-like strategy for estimation tasks. These results show that the proposed feedback control strategy is able to achieve regulation at maximum steady-state methane production despite the rigorous mathematical proof was performed for a simplified, benchmark model of biological reacting systems.

6. Conclusions The work proposed a modeling error estimation to address the extremum-seeking control design for bioreactors with single substrate and biomass strain. The main idea is to use a gradient-like observation scheme to estimate the trajectory of modeling errors (e.g., growth functions) that is incorporated in a feedback control function. Based on singularly perturbed systems, rigorous stability results about optimal equilibrium conditions were provided. However, such stability results relied on the strong assumption that the controlled output related to the extremum conditions is available for feedback. Actually, this is an unrealistic assumption since, in general, only the production output and the substrate concentration are measured. To the best of our knowledge, rigorous solutions for this problem have not been given in the literature. Instead, some approximation based on heuristics have been reported and proven via numerical simulations. The reliability of these approximations for the extremum-seeking problem was discussed in terms of conditions required for their applicability. Overall, the results in this work provided some insights in the stability of extremum-seeking problem for biological reaction systems, as well in the nature of the controlled output estimation issue. This issue was illustrated with a more involved biological reactor for anaerobic digestion processes.

The authors declare no competing financial interest. 20 ACS Paragon Plus Environment

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References (1) Leblanc, M. Sur L’électrification des chemins de fer au moyen de courants alternatifs de fréquence elevée. Revue Générale de L’électricité 1922, 12, 275. (2) Vasu, G. Experiments with optimalizing controls applied to rapid control of engine pressures with high-amplitude noise signals. Trans. ASME 1957, 481. (3) Drakunov, S.; Özgüner, U.; Dix, P.; Ashrafi, B. ABS control using optimum search via sliding modes. IEEE Trans. Contr. Syst. Tech. 1995, 3, 79. (4) Peterson, K.S.; Stefanopoulou, A.G. Extremum seeking control for soft landing of an electromechanical valve actuator. Automatica 2004, 40, 1063. (5) Leyva, R.; Alonso, C.; Queinnec, I.; Cid-Pastor, A.; Lagrange D.; Martinez-Salamero, L. MPPT of photovoltaic systems using extremum-seeking control. IEEE Trans. Electronic Syst. 2006, 42, 249. (6) Lee, H.C.; Kim, S.; Heo, J.P.; Kim, D.H.; Lee, J. Wiener model and extremum seeking control for a CO preferential oxidation reactor with the CuO-CeO2 catalyst. IFACPapersOnLine 2015, 48, 574. (7) Krstić, M.; Wang, H.H. Stability of extremum seeking feedback for general nonlinear dynamic systems. Automatica 2000, 36, 595. (8) Ariyur, K.B.; Krstic, M. Real-time Optimization by Extremum-seeking Control; John Wiley & Sons: New York, 2003. (9) Zhang, C.; Ordóñez, R. Extremum-seeking Control and Applications: A Numerical Optimization-based Approach; Springer Science & Business Media: New York, 2011. (10) Liu, S.J.; Krstic, M. Stochastic Averaging and Stochastic Extremum Seeking; Springer Science & Business Media: New York, 2012. (11)

Guay,

M.;

Dochain,

D.

A

time-varying

extremum-seeking

control

approach. Automatica 2015, 51, 356. (12) Wang, H.H.; Krstic, M.; Bastin, G. Optimizing bioreactors by extremum seeking. Int. J. Adaptive Control Signal Processing 1999, 13, 651. (13) Zhang, T.; Guay, M.; Dochain, D. Adaptive extremum seeking control of continuous stirred‐tank bioreactors. AIChE J. 2003, 49, 113.



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(14) Guay, M.; Dochain, D.; Perrier, M. Adaptive extremum seeking control of continuous stirred tank bioreactors with unknown growth kinetics. Automatica 2004, 40, 881. (15) Marcos, N.I.; Guay, M.; Dochain, D.; Zhang, T. Adaptive extremum-seeking control of a continuous stirred tank bioreactor with Haldane's Kinetics. J. Process Contr. 2004, 14, 317. (16) Cougnon, P.; Dochain, D.; Guay, M.; Perrier, M. On-line optimization of fedbatch bioreactors by adaptive extremum seeking control. J. Process Contr. 2011, 21, 1526. (17) Dochain, D. (Ed.). Automatic Control of Bioprocesses; John Wiley & Sons: New York, 2013. (18) Lara-Cisneros, G.; Aguilar-López, R.; Femat, R. On the dynamic optimization of methane

production

in

anaerobic

digestion

via

extremum-seeking

control

approach. Comp. Chem. Eng. 2015, 75, 49. (19) Alvarez‐Ramírez, J. Adaptive control of feedback linearizable systems: a modelling error compensation approach. Int. J. Robust Nonlinear Contr. 1999, 9, 361. (20) Vargas, A.; Moreno, J.A.; Wouwer, A.V. Super-twisting estimation of a virtual output for extremum-seeking output feedback control of bioreactors. J. Process Contr. 2015, 35, 41. (21) Deschênes, J.S.; Wouwer, A.V. Dynamic optimization of biomass productivity in continuous cultures of microalgae Isochrysis galbana through modulation of the light intensity. IFAC-PapersOnLine 2015, 48, 1093. (22) Nguang, S. K.; Chen, X. D. Extremum seeking scheme for continuous fermentation processes described by an unstructured fermentation model. Bioprocess Eng. 2000, 23, 417. (23) Peroni, C.V.; Kaisare, N.S.; Lee, J.H. Optimal control of a fed-batch bioreactor using simulation-based approximate dynamic programming. IEEE Trans. Contr. Syst. Tech. 2005, 13, 786. (24) Modak, J. M.; Lim, H.C. Optimal operation of fed-batch bioreactors with two control variables. Chem. Eng. J. 1989, 42, B15. (25) Lara-Cisneros, G.; Femat, R.; Dochain, D. An extremum seeking approach via variable-structure control for fed-batch bioreactors with uncertain growth rate. J. Process Contr. 2014, 24, 663. 22 ACS Paragon Plus Environment

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(26) Hoppensteadt, F. Asymptotic stability in singular perturbation problems. II: Problems having matched asymptotic expansion solutions. J. Differential Equations 1974, 15, 510. (27) Dochain, D.; Perrier, M.; Guay, M. Extremum seeking control and its application to process and reaction systems: A survey. Math. Comp. Simulation 2011, 82, 369. (28) Alvarez-Ramirez, J.; Kelly, R.; Cervantes, I. Semiglobal stability of saturated linear PID control for robot manipulators. Automatica 2003, 39, 989. (29) Chaillet, A.; Loría, A.; Kelly, R. Robustness of PID-controlled manipulators vis-à-vis actuator dynamics and external disturbances. European J. Contr. 2007, 13, 563. (30) Gunaseelan, V. N. Anaerobic digestion of biomass for methane production: a review. Biomass Bioenergy 1997, 13, 83. (31) Braun, R. Anaerobic digestion: a multi-faceted process for energy, environmental management and rural development. In Improvement of Crop Plants for industrial End Uses (pp. 335-416); Springer: Netherlands, 2007. (32) Gelbert, G.; Moeck, J.P.; Paschereit, C.O.; King, R. Advanced algorithms for gradient estimation in one-and two-parameter extremum seeking controllers. J. Process Contr. 2012, 22, 700. (33) Luyben, W.L. Simple method for tuning SISO controllers in multivariable systems. Ind. Eng. Chem. Process Design Dev. 1986, 25, 654. (34) Hess, J.; Bernard, O. Design and study of a risk management criterion for an unstable anaerobic wastewater treatment process. J. Process Contr. 2008, 18, 71. (35) Méndez-Acosta, H. O.; Campos-Delgado, D.U.; Femat, R.; González-Alvarez, V. A robust feedforward/feedback control for an anaerobic digester. Comp. Chem. Eng. 2005, 29, 1613. (36) Bernard, O.; Hadj‐Sadok, Z.; Dochain, D.; Genovesi, A.; Steyer, J. P. Dynamical model development and parameter identification for an anaerobic wastewater treatment process. Biotech. Bioeng. 2001, 75, 424.



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Captions to Figures Figure 1. Phase-portrait of a controlled bioreactor example with Haldane growth function given by Equation 14 and ideal feedback function given by Equation 6. The system parameters are µ max = 5.0 h-1, K s = 0.75 g⋅L-1, K I = 5.0 g⋅L-1, k1 = 2.5 , k 2 = 0.75 and s f = 25 g⋅L-1. The controller gain is g c = 3.0 . The vertical line Lmax is an invariant set containing the optimal equilibrium point (s max , xmax ) .

Figure 2. Phase-portrait of the bioreactor in Figure 1 under the action of the feedback controller given by Equation 16 equipped with the modeling error estimation given by Equation 19. Note that the sliding dynamics along the otherwise invariant set Lmax is not longer present.

Figure 3. Block diagram of the signals involved in the feedback control design. Figure 4. Phase-portrait of the bioreactor in Figure 1 as controlled by the feedback control function given by Equation 25 and the controlled output estimator given by Equation 28 and 29.

Figure 5. Effect of the controller parameters in the convergence of the production output y p . (a) Effect of the controller gain g c for g e = 0.5 , and (b) effect of the estimator

gain g e for g c = 2.0 .

Figure 6. Methanogenic growth function µ 2 ( s 2 ) for the parameters described in the text. The function exhibits a maximum at about s 2,max = 12.6 mmol⋅L.

Figure 7. (a) phase-portrait of s 2 versus Q for the controlled biological reactor with the controller parameters g c = 0.015 and g e = 0.01. (b) Methane production trajectories corresponding to the feedback controlled anaerobic digester.


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12

10

8

6

4

Lmax

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Biomass Concentration, x (g/L)

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2 1

10

Substrate Concentration, s (g/L)

Figure 1


12

Lmax

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10

8

6

4

2 1

10


Figure 2


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Figure 3


12

Lmax

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10

8

6

4

2 1

10


Figure 4


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28

Production Output yp (g/L h)

(a) 24

gc = 1.0

20

gc = 2.0 gc = 3.0

16

12

8 0

5

10

15

20

28

Production Output yp (g/L h)

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(b)

24 20

ge = 0.25 ge = 0.5

16

ge = 1.0 12 8 4 0

5

10

15

Time (h)

Figure 5


20


0.014 0.012

µ2(S2) (mmol/L h)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.010 0.008 0.006 0.004 0.002 0.000 0

10

20

30

40

50

Substrate Concentration, S2 (mmol/L)

Figure 6


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Q (mmol/L h)

12 10

(a)

8 6 4 2 1

10

100

S2 (mmol/L) 12

(b)

10

Q (mmol/L h)

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8 6 4 2 1

10

100

Time (h)

Figure 7



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Abstract Graphics


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Modeling-Error Compensation Approach for Extremum-Seeking

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