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Global Observability Analysis of a Growth Model for Insects Farming

Rania Tafat, Jaime A. Moreno, Stefan Streif

TL;DR

This work develops a reduced three-dimensional nonlinear temperature-driven model for Hermetia illucens growth, focusing on dry biomass $B_{dry}$ and two temperatures $T_{med}$ and $T_{air}$. It derives a state-space representation with outputs $[T_{med},T_{air}]$ and proves a global differential observability result: if all parameters are positive and $k_{7} \ge \left(1 - 2 \frac{k_{3}}{k_{1}}\right) k_{6}$, the mapping from the biomass state to the measured signals is globally injective, enabling biomass estimation via an immersion-based observer. This justifies the feasibility of sensorless state estimation for online control in insect farming and informs future observer design for MPC applications. The analysis bridges biological growth modelling with nonlinear observability theory, providing a theoretical foundation for practical biomass estimation in real-time farming settings.

Abstract

The Hermetia illucens insects or the black soldier fly has been attracting a growing interest in the food and feed industry. For its high nutritional value on the one hand, and because it is an adequate species for insects in controlled environmental agriculture systems, on the other. Therefore, several models describing this larvae's behaviour have been developed in the literature. Due to the complex nature of living organisms, systems of controlled environment agriculture are characterised by their strong nonlinearities. In this paper, we present a three dimensional nonlinear model describing the black soldier fly dry biomass weight dynamic changes due to the temperature's influence. In practice, this biomass weight is not measured in real time. This becomes problematic for applying feedback control strategies that assume full information of the states. Thus, this work investigates the observability of the dry biomass of a Hermetia illucens farming batch. The instantaneous and global observability of the aforementioned model is proven by constructing an injective transformation between the state space and a higher dimensional space where the transformed states are observable.

Global Observability Analysis of a Growth Model for Insects Farming

TL;DR

This work develops a reduced three-dimensional nonlinear temperature-driven model for Hermetia illucens growth, focusing on dry biomass and two temperatures and . It derives a state-space representation with outputs and proves a global differential observability result: if all parameters are positive and , the mapping from the biomass state to the measured signals is globally injective, enabling biomass estimation via an immersion-based observer. This justifies the feasibility of sensorless state estimation for online control in insect farming and informs future observer design for MPC applications. The analysis bridges biological growth modelling with nonlinear observability theory, providing a theoretical foundation for practical biomass estimation in real-time farming settings.

Abstract

The Hermetia illucens insects or the black soldier fly has been attracting a growing interest in the food and feed industry. For its high nutritional value on the one hand, and because it is an adequate species for insects in controlled environmental agriculture systems, on the other. Therefore, several models describing this larvae's behaviour have been developed in the literature. Due to the complex nature of living organisms, systems of controlled environment agriculture are characterised by their strong nonlinearities. In this paper, we present a three dimensional nonlinear model describing the black soldier fly dry biomass weight dynamic changes due to the temperature's influence. In practice, this biomass weight is not measured in real time. This becomes problematic for applying feedback control strategies that assume full information of the states. Thus, this work investigates the observability of the dry biomass of a Hermetia illucens farming batch. The instantaneous and global observability of the aforementioned model is proven by constructing an injective transformation between the state space and a higher dimensional space where the transformed states are observable.
Paper Structure (8 sections, 1 theorem, 34 equations, 3 figures, 1 table)

This paper contains 8 sections, 1 theorem, 34 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Hermetia illucens Temperature model
  3. Dry biomass weight $B_{\mathrm{dry}}$
  4. Temperature in the growing medium $T_{\mathrm{med}}$
  5. Temperature of the air $T_{\mathrm{air}}$
  6. Hermetia illucens temperature model: synthesis
  7. Observability analysis
  8. Conclusion

Key Result

Theorem 1

Assume that all parameters of the model eq:state_space are positive. If the following condition is satisfied then the system is globally and differentially observable for all $x_1 \in \mathbb{R}$.

Figures (3)

  • Figure 1: Evolution of function $\Omega_1$ with respect to $x_1$
  • Figure 2: Evolution of function $\Omega_2$ with respect to $x_1$
  • Figure 3: Behaviour of the curve $\Omega(x_1)$

Theorems & Definitions (8)

  • Remark 1
  • Definition 1: Analytic function ComplexAnalysis
  • Definition 2: Instantaneous observability BERNARD2022
  • Definition 3: Differential observability Bernard2019
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3