Inversion-based feedforward control design for the Droop model

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Abstract

An inversion-based feedforward control for the Droop model for microalgae growth in the chemostat is presented. The dilution rate is manipulated in such a way that a desired biomass trajectory for set point changes is tracked. For this purpose a stable inversion technique is employed exploiting the asymptotic stability of the internal dynamics due to mass-conservation. The impact of unknown feed substrate step changes on the controller performance is analytically delimited using the mass-conservation-based bounded input-bounded output stability property. Numerical results show that the transition time can be significantly reduced by the proposed approach in comparison to a simple step change in the feed flow showing the bounded downgrade due to unknown variations in the feed concentration.

Introduction

The production of high-value compound production like fatty-acids, vitamins, or pigments have great impact in the biotechnology market. Particularly usefull resources for these products with relatively rapid growth are microalgae. Additionally, microalgae are also of interest as sources for hydrogen and biofuel production (Havlik et al., 2013).

Typically process control is used to improve the process response times and robustness with respect to unknown (and unpredictable) perturbations in the feed variables, etc. In several process control application examples (Graichen et al., 2009, Hagenmeyer and Nohr, 2008, Kleinert et al., 2010) it has been numerically and experimentally shown that substantial improvement can be achieved using model-based feedforward control. This motivates the design of a feedforward control strategy for microalgae growth processes to reduce the time requirements for set-point changes. Furthermore, as such processes are typically subject to fluctuations, specially in the feed concentration, the effects of such variations has to be carefully delimited. This taks are addressed in the present paper, by exploiting the stable inversion technique (Chen and Paden, 1996, Devasia, 1999). For this purpose, using mass-conservation arguments and the reactor bifurcation behavior it is rigurously shown that bounded feed perturbation will lead to bounded deviations from the reference output. Numerical simulation results illustrate the obtained performance.

Section snippets

Droop’s microalgae growth model

The Droop model for microalgae growth in the chemostat is given by (Droop, 1968, Lemesle and Mailleret, 2008)b˙=db+μqb,b0=b0q˙=ρsμqq,q0=q0s˙=dsinsρsb,s0=s0y=b,

where b is the biomass concentration, q is the internal nutrient quota (i.e. the intracellular nutrient concentration per biomass unit), s is the extracellular nutrient concentration, sin is the piecewise constant, unknown feed concentration with nominal value s¯in, μ is the growth rate, ρ is the uptake rate, d is the dilution rate

Reactor dynamics

In this section the reactor dynamics are characterized in terms of the input-output behavior including steady-state multiplicity, bifurcation and bounded-input-bounded-output stability properties, which are essential for the subsequent design of a feedforward control strategy. The discussion presented here verifies and complements the one reported in (Lemesle and Mailleret, 2008) and forms the basis of the subsequent characterization of the bounded-input bounded-output stability of the reactor

Feedforward control

In this section a feedforward control is designed to achieve trajectory tracking for operation-point changes in the Droop model. For this puropse, the stable inversion technique (Chen and Paden, 1996, Devasia, 1999) is employed. Having the biomass as output variable, denote the reference trajectory by y* (t) = b* (t) with dynamicsy˙*t=d*ty*t+vθ*ty*t,y*0=y0*.

The requirement of transition between the two steady-states corresponding to y(0) = b0 to y(T) = bT yields four boundary conditions for the

Numerical simulations

In order to verify the theoretic assessment, numerical simulations using MatLab with ode15s have been carried out for the parameter set (taken from (Toroghi et al., 2013)) Ks = 0.105, μ¯=2, kq = 1.8, k0 = 9.3.

In Figure 3 the reactor response for the biomass set-point change from b0 = 52.78 to bT = 27.76 in T = 1 day is illustrated. The nominal trajectory is represented by the thin grey continuous line and the reactor feedforward response by the dashed green line.

For comparison purposes the open-loop

Conclusions

An inversion-based feedforward control for the Droop model for microalgae growth in the chemo-stat is presented. The dilution rate is manipulated in such a way that biomass set point changes are followed in finite time with practical stability with respect to feed perturbations. The asymptotic stability of the internal dynamics due to mass-conservation is exploited to show this property. Numerical results show that the steady-state to steady-state transition time can be significantly reduced by

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