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MBW:Nonlinear Control for Algae Growth Models

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Article review by Christine Fanchiang, March 2010.

Article: “Nonlinear control for algae growth models in the chemostat.” Bioprocess & Biosystems Eng., 27(5):319-327, Aug. 2005.

For a continuation of this work please see the following Nonlinear Control For Algae Growth Models Part II.

Summary

The purpose of this article is to demonstrate how adding a nonlinear output feedback controller to variable yield growth models can globally stabilize the system. In English, this means the authors are improving the mathematics of a model for some biological phenomenon, which helps increase the robustness of the model so it can better handle input variations or noise to the system. Specifically, the authors of the article use a phytoplankton algae growth model to test their approach with numerical simulations, and they show how it can benefit biological laboratory experiments. The article begins with a description of the variable yield model of phytoplankton algae growth, and then describes the non-linear controller they developed. They then derive proofs that indicate the solutions to the non-linear controller behave as expected from the biological models, and finally, they close with three numerical simulations of potential scenarios in laboratory experiments and how their closed-loop system handles these scenarios. Their findings show that by adding a non-linear controller, the mathematical models are more robust and demonstrates a globally stabilized variable yield growth model.

History

Scientists are constantly seeking more accurate and robust models to describe biological phenomenon so they can better control their experiments, and thus, ultimately produce higher fidelity results from their data. Accurate modeling of biological phenomenon is not a trivial task. To determine how accurately a model represents a biological phenomenon, experimenters focus on the final steady-state values of the system and how accurately they compare to the real experimental values.

But with laboratory experiments, there are usually randomly introduced external affects to the system that can easily perturb a system. Models that are open-loop (without feedback) are not robust enough to handle the perturbations and often show large deviations from the true experimental data. To ensure a more accurate model to handle these perturbations, a controlled or closed-loop model with feedback control needs to be implemented. This improves the robustness of the system, so it can describe real biological phenomenon more accurately.

Biological Phenomenon

The phenomenon of interest is a non-linear autonomous variable yield system. Systems suitable for this modeling include a wide class of variable yield models for micro-organism growth in continuous bioreactors. Specifically for this study, they look at growth rates and biomass of phytoplankton algae growing in chemostats. They use a variable yield model to describe the biological phenomenon associated with the flow of medium, uptake of substrates, and the growth of the algae.

Figure1.png

Bioreactors are enclosed environments that provide sufficient resources for the micro-organism to thrive and grow until their quota is reached. In this study the bioreactor is a chemostat, which is a growth chamber that keeps a cell culture at a specific volume and continuously adds fresh medium while removing spent culture.

The micro-organism population grows in this medium consuming a nutrient (which is characterized here as the substrate, s). A liquid flow (F) passes through the vessel with the substrate at concentration, Sin, while the outflow contains the same compounds that are floating about inside the chemostat.

The table below describes the variables and parameters used throughout this paper.

Table1-1.png

Mathematical Model

The Variable Yield Model

The model begins with a variable yield model that monitors the rate of change of substrate concentration, s, the cell quota, q, and the biomass concentration, x. (See reference 1 for detailed description of variable yield models as described by Droop et al. and how these are derived).

Eqn1.png

The difficulty in most variable yield models is determining the expressions for the substrate uptake rate of the cell, p(.) and the growth rate of algae, u(.). To circumvent this problem, the authors choose to assume qualitative hypotheses about p(.) and u(.).

The hypothesis states that p(0)=0 and the function p(s) = C1, is increasing and bounded; and u(q) is C1, non-negative, increasing and bounded; and also there exists a qm>0 such that u(qm) = 0, where qm is the minimum cell quota that a cell needs for it to grow. So if the intracellular nutrient per unit of biomass decreases to qm, then the rate of cell growth will be zero. This means if there is substrate available than the cell uptakes it; additionally, the more substrate available the higher the uptake rate. The system is also bounded by real biological evidence of algae growth so when there is insufficient internal nutrient for the cell to grow, qm, then the rate of cell growth will be zero as mentioned earlier.

Though to better exhibit the system of equations, the authors use a change of coordinates to express a new variable z = s + qx, which represents the total amount of intracellular and extracellular nutrient is in the chemostat. The following set of equations is obtained:

Eqn2.png

The behavior of this system is well studied in additional references [1] and [2]. The steady-state solutions are of two different types depending on the value of D compared to sin: there exists a positive equilibrium point which the variables tend to stabilize towards, or the algae population washes out (goes extinct). The purpose then is to create controller that prevents washout from occurring and maintaining the positive equilibrium. Specifically, the biomass concentration must be kept toward a chosen positive value.

Also, uncertainty in the in-flow substrate concentration, sin, may destabilize the system or there could be possible algae stress that may lead to biomass washout. These variations or potential stresses must be taken into account when determining the nonlinear control design.

With these constraints in mind, the next section describes how they design and implement their non-linear output feedback controller.

The Nonlinear Control Design

To control biological systems, the model may be only qualitatively known and that the outputs may be some unknown nonlinear functions of the state variables. Moreover, inputs are considered unconstrained in classical control theory, whereas they usually fulfill some constraints (e.g., positivity) in biological systems.

Because of the high variability of biological phenomena, the authors only focus on a qualitatively known model, qualitative outputs and a constrained input. Therefore, they don’t use the classical linearization technique to describe the model, rather, they have to define the manipulated variable (input), and the measured variable (output). For a chemostat system, the dilution rate, D is easily manipulated (or constrained in this case) as the input of the system. The output is something measurable, like the uptake carbon or the produced oxygen due to algae photosynthesis. Either quantity used to measure output is proportional to cell growth, thus they assume the output expression can be written as: y = u(q)x.

Based on the above assumptions and the hypothesis describing the system, the authors define a nonlinear output feedback control law as the following expression:

D(.)= γy = γu(q)x where γ > qm/sin

Essentially, they go through multiple proofs to show that they can rewrite the system in Equation (2) to include D(.), where D(.) is non-negative:

Eqn3.png

By integration, the solution to Equation (4) gives the following for z(t) and x(t):

Eqn4.png

To prove that these equations to the ODE's converge to the steady states of z = sin and x = 1/γ, the authors show that the integral quantity of D(t) does in fact converge toward positive infinity as time goes to infinity. As the integral of D(t) goes to infinity, the steady-state values for z and x converge toward sin and 1/γ.

Given biological constraints such as initial conditions and additional biological constraints that enforce the solutions must be positive (because you can’t have negative amounts of cells or medium), the authors prove the steady state of the system in equation (4) is locally stable. Also from their proof, they show that the steady-state solutions are globally attractive, which means the values must be positive globally asymptotically stable equilibrium points for the closed loop system. From these proofs, they show with mathematical rigor that this feedback control law globally stabilizes the system described by Equation (1) toward the positive equilibrium points, determined by the value of γ.

Full model with results from the paper

The article uses Dunaliella tertiolecta, a green micro-algae model that has been well-studied to test their new model with their non-linear controller.

They use the following equations for the uptake and growth rates of the cell:

Eqn5.png

To prove the basic function of their new model, they run multiple simulations. The first is to show the normal behavior with added noise. The second simulation shows how it stabilizes even with a variable input, sin. And the last simulation is of algae stress, which corresponds to varying algae growth rate due to experimental design.

Simple Noisy Simulation

They use a simple noisy simulation of the controlled process where the in-flow of substrate is assumed to be equal to 20/sigm.L-1, and the output was corrupt with a relative white noise of 30% amplitude.

The following figures show their simulation of the closed loop system with added noise. The result shows the convergence of the steady-state solutions to their expected value as determined from equation (5).

Figure2.png

Figure 2. Noisy simulation of model.

Varying Substrate Input, Sin

To show how time variations in the substrate concentration, sin, do not change the biomass concentration, x, behavior, the authors used a piecewise constant time-dependant sin for half-day time steps. Their simulated results are shown in Figure 3. And again despite these variations, the behavior of variable x remains the same, converging towards its equilibrium.

Figure3.png

Figure 3. Varying input substrate concentration.

Periodic Algae Stress

With experimental design, usually, the chemostat medium needs to be changed weekly, this change of medium induces stress to the micro-organisms thus affecting their growth rate. To model this they use a time dependant growth rate such that: u(.) = d(t)u(q). They show the difference of how the models react for their closed-loop system versus an open-loop system in Figure 4.

The open-loop system shows the algae population is nearly washed’ out, while the controlled closed-loop system drives the algae concentration back to its desired positive steady state values.

Figure4.png

Figure 4. Simulation of closed loop (__) and open loop (--) processes facing periodic algae stress. (See Parameters section for variable definition and units).

Analysis/interpretation

From the author’s rigorous mathematical proof, they show that adding their nonlinear control feedback to the mathematical model of chemostat dynamics can help globally stabilize the model. Perturbations due to normal experimental design can be accounted for by this additional controller. The addition of a nonlinear controller makes the model more stable and robust to handle variations throughout the experiment process, thus more accurately simulating the environments that the micro-organisms undergo.

External Links

[1] O. Bernard, J.L. Gouze. “Transient behavior of biological loop models with application to the Droop model,” Math Biosci.127:19–43, 1995.

[2] K. Lange, F.J. Oyarzun. “The attractiveness of the Droop equations,” Math Biosci 111:61–278, 1992.

[3] J. P. Gauthier, H. Hammouri, and S. Othman, “A simple observer for nonlinear systems applications to bioreactors,” IEEE Trans. Automat. Contr., vol. 37, pp. 875-880, 1995. [[1]]