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## The COSY model

The COSY model, presented by [Mauri et al., 2020], describes the dynamics of a producer strain that grows and produces a special compound by consuming glucose and, by doing so, excretes acetate (which is toxifying, in the sense that it inhibits its growth). The syntrophic relationship is established by adding a second cleaning strain, that has more affinity in consuming acetate and therefore “detoxifies” the environment and alleviates inhibitory effects on the producer growth (see the pathways illustrated in Figure 2.3 below). The producer and the cleaner strains are henceforth denoted Bp and Bc, respectively. The glucose and acetate are denoted by G and A, respectively, and the nutrient state vector is denoted vec(G,A). The equations of the model have the following form: ˙G = DgGin − DG − r1(G,A)Bp − r2(G,A)Bc.

### Estimating microbial co-cultures

Since the structure of systems (2.1) and (2.4) are similar, the observability analysis presented in this section will be derived only for the latter [Souza et al., 2021a]. Indeed, the observability conditions for the species concentration are identical (by substituting 1 − kh by k1, and μp, μc by μ1, μ2, respectively). The observability condition for the nutrients are different but, nonetheless, follow the same rationale in both cases.

Consider (2.4) decomposed in two sub-systems: the first composed of the strains and products and the second composed of the nutrients. Denoting x1 = vec(Bp,Bc,H, F) and x2 = vec(G,A), the first sub-system can be obtained by rewriting (2.4) .

#### Estimation of H, Bp and Bc

In this subsection, the objective is to develop state observers for system (3.2). The estimation of the nutrients G,A will be addressed in the subsequent section. To construct these observers, the differentiation of the available measurements will be needed, hence it is assumed that the input D is continuously differentiable with a known and bounded derivative. As it will be shown in the following, the first and second time derivatives of y1, as well as the first time derivative of y2, will be needed. Therefore, the following differentiators (see Appendix B) will be implemented: z˙0 = λ12L 1 3 1 |z0 − y1|sign(z0 − y1) + z1.

**Estimation using both measurements**

First, simulation results are presented considered observers (3.6), (3.8) and (3.9). The gains of the differentiators (3.3)–(3.4) are taken as L1 = 0.05 and L2 = 1, respectively. Finally, the initial conditions for the observers are taken as ˆG(0) = 0, ˆ A(0) = 0, ˆBp(0) = 0 and ˆBc(0) = 0.

Figures 3.3 and 3.4 illustrate the real trajectories of Bp,Bc and G,A, respectively, as well as their respective estimates. Finally, Figure 3.2 shows the estimation of the heterologous protein H. Although the estimate ˆH shows more sensitivity to noise than the other estimates, which are obtained through integration, it does not require any knowledge of the rates ri.

**Using only the total biomass**

Now, the same scenario described previously will be considered, but using observers given by (3.15). These observers will be initialized with ˆG(0) = 0, ˆ A(0) = 0, ˆBp(0) = 0, ˆBc(0) = 0 and ˆH (0) = 0. The observer gain will be selected as L = 10. Figures 3.5 and 3.6 illustrate, respectively, the estimation of the nutrients G and A, and biomasses Bp and Bc. As it can be seen, since all parameters are perfectly known, all estimates converge to the true values. Also, since they undergo integration, the influence of measurement noise is attenuated. The same conclusion is drawn for the estimate of H, as shown in Figure 3.7.

**Table of contents :**

R´esum´e

Abstract

Acknowledgments

List of Figures

List of Tables

Notation

**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 Introduction

2.1.1 The competition model

2.1.2 The COSY model

3 Observation of Heterogeneous Communities

3.1 Introduction

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

4.1 Introduction

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

5.1 Introduction

5.1.1 Preliminaries

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

Appendices

Appendix A Preliminaries on input-to-state stability

Appendix B Preliminaries on sliding-mode control

**Bibliography **