In this chapter, the observation problem for systems (2.4) (and also for (2.1), thanks to their similarities) has been discussed. First, an analysis has been carried out to identify their conditions of observability considering the total biomass and the fluorescent reporter as available measurements. It has been shown that the system is observable with both measurements, but merely detectable if only the first is available.
Second, considering the two available measurements, state observers were proposed for each variable, and the stability properties are discussed for the case in which the two measurements are available. Finally, the advantage of measuring the fluorescent reporter is highlighted, in the sense that it not only enables observability of the system, but also alleviates the dependence on the parameters of the system.
In this chapter, the generic bioreactor model (2.1) and the COSY model (2.4) are reconsidered to address the control design for a co-culture of microbes. Mainly, the interest of designing a controller for a heterogeneous community is to ensure their co-existence, i.e., avoiding that one goes extinct. It has also been shown that an oscillating chemostat [Smith, 1981], or the use of feedback control [Leenheer and Smith, 2003], might induce the coexistence of two different species.
Also, another control objective relates to the stabilization (or the optimization) of the production on the bioreactor. Indeed, controlling production requires, in some sense, avoiding that the producing species are outcompeted and that the required nutrient is depleted from the bioreactor (if there is no nutrient, there is no metabolism).
Therefore, the main questions to be tackled in this chapter are:
Q1- Are these systems controllable (or stabilizable) taking into account the control constraints?
Q2- How can the co-population be (robustly) stabilized at desired concentration levels?
Q3- How can the production of the bioreactor be (robustly) stabilized? Is it possible to cast an optimization problem?
These questions are going to be discussed in the sequel, by considering both the generic model (2.1) and also the COSY model (2.4). Before starting to tackle these objectives, a discussion on the available control inputs are presented in the following.
The available control inputs In both models (2.1) and (2.4), there are two controlled variables, both physically altering the composition of the environment in the bioreactor: 1. D(t): describing the dilution rates, i.e., the rates of transfer (inflow or outflow) of medium in the bioreactor (in unit h−1). This variable is often actuated by an electromechanical pump.
A discussion on controllability
2. Sin(t): describing the concentration of substrates diluted in the inflow of medium (in unit gL−1). In the COSY model (2.4), this variable is represented by Gin and Ain are, respectively, the concentration of glucose and acetate.
It is plausible to have the inflow of different media, for instance, containing or not diluted nutrients (Ds(t) and D0(t), respectively). By considering a continuous bioreactor of constant volume, the outflow is equal to total inflow, i.e., D(t) = Ds(t) + D0(t).
The concentration Sin(t) (or, analogously, Gin(t) and Ain(t)) can be modulated by properly selecting the rates D0 and Ds, which indicates that it also might be actuated by electromechanical pumps, as illustrated in Figure 4.1. Considering that Ds relates to the transfer of medium with a high concentration Smax in , the inflow concentration Sin can be determined by Sin(t) = Smax in Ds(t) Ds(t) + D0(t).
This is interesting since it allows the design of a control considering variables of similar nature. Furthermore, the time constant of a electromechanical device is much smaller than the one of a biological process.
A discussion on controllability
Thanks to its physical meaning, systems (2.1) and (2.4) are time-varying, positive systems. Furthermore, they have constrained controls, since D(t) and Sin(t) (or Gin(t) and Ain(t) for the COSY model (2.4)) are also non-negative and possibly upper bounded. In such a case, deriving a rank condition for controllability is not applicable since it cannot be investigated by utilizing Lie algebra, as discussed for general nonlinear systems in [Hermann and Krener, 1977].
Nevertheless, an analysis can still be performed by profiting the structure of (2.1) and (2.4) and it is possible to evaluate the reachability of such a system, according to the following definition:
Definition 4.1. A state x1 is called reachable if there exist a finite time t1 and an admissible control u(t) on [t0, t1] such that x(t0) can be steered to x1.
Consider the competition model (2.1). If the nutrient inflow concentration Sin can be made sufficiently high, it is possible to prevent the nutrients of being depleted from the bioreactor (i.e., S → 0). Then, the sign of the derivative of each xi can be assigned by a proper selection of D.
Table of contents :
Part 1: Introduction and overview
1 General Introduction
1.1 Background and motivation
1.1.1 The d´efi COSY
1.2 Robust output feedback MPC
1.3 State of the art
1.4 Thesis outline
1.5 Publication list
1.5.1 Peer-reviewed international journals
1.5.2 Peer-reviewed international conferences
Part 2: Control and Observation of Microbial Communities
2 Introduction to the overall framework
2.1.1 The competition model
2.1.2 The COSY model
3 Observation of Heterogeneous Communities
3.2 Estimating microbial co-cultures
3.2.1 Observability analysis
3.2.2 Estimation of H, Bp and Bc
3.2.3 Estimation of G and A
3.2.4 Numerical example
3.2.5 Part conclusion
4 Control of Heterogeneous Communities
4.2 A discussion on controllability
4.3 Controlling competing species
4.3.1 A gentle motivation for a robust approach
4.3.2 Design of the control architecture
4.3.3 Numerical example
4.4 Control of the COSY model
4.4.1 Ensuring coexistence through state feedback
4.4.2 Numerical example
4.5 Part conclusion
Part 3: Robust Output Feedback Model Predictive Control
5 MPC using interval estimators
5.2 OF-MPC for LTI systems
5.2.1 Design of interval estimators
5.2.2 Control design
5.2.3 An extension to linear time-delayed systems
5.2.4 Numerical illustration
5.3 OF-MPC for LPV systems
5.3.1 Design of interval estimators
5.3.2 Control design
5.3.3 Numerical illustration
5.4 Design of the predictive controllers
5.5 Complexity and performance
5.6 Numerical examples
5.6.1 The LTI case
5.6.2 The LPV case
5.7 Part conclusion
Part 4: General Conclusions and Outlooks
6 General Conclusions and Outlooks