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3 min readIn general, mathematical models can take very different forms depending on the system under study, which may range from social, economic and environmental to mechanical and electrical systems. Typically, the internal mechanisms of social, economic or environmental systems are not very well known or understood and often only small data sets are available, while the prior knowledge of mechanical and electrical systems is at a high level, and experiments can easily be done. Apart from this, the model form also strongly depends on the final objective of the modelling procedure. For instance, a model for process design or simulation should contain much more detail than a model used for studying different long-term scenarios.

In particular, for a wide range of applications (e.g. Keesman 2011), models are developed to:

Obtain or enlarge insight in different phenomena, for example, recovering physical or economic relationships.

Analyse process behaviour using simulation tools, for example, process training of operators or weather forecasts.

Estimate state variables that cannot be easily measured in real time on the basis of available measurements, for instance, online process information.

Control, for instance, in the internal model control or model-based predictive control concept or to manage processes.

A critical step in the modelling of any system is to find a mathematical model which adequately describes the actual situation or state. Firstly, the system boundaries and the system variables have to be specified. Then relationships between these variables have to be specified on the basis of prior knowledge, and assumptions about the uncertainties in the model have to be made. Combining this information defines the model structure. Still the model may contain some unknown or incompletely known coefficients, the model parameters, which in case of time-varying behaviour define an additional set of system variables. For a general introduction to mathematical modelling we refer to, for instance, Sinha and Kuszta (1983), Willems and Polderman (1998) and Zeigler et al. (2000).

In this chapter, the modelling of an aquaponic (food) production (AP) system will be described. Figure 11.1 shows a typical example of an AP system, i.e. the so-called decoupled three-loop aquaponic system. As a result of basic principles modelling, using conservation laws and constitutive relationships, mathematical models of all kinds of AP systems are usually represented as a set of ordinary or partial differential equations. These mathematical models are commonly used for design, estimation and control. In each of these specific modelling objectives, we distinguish between analysis and synthesis.

**Fig. 11.1** Decoupled, three-loop aquaponic system with RAS, hydroponic and remineralization subsystems. (Goddek, 2017)

The outline of the chapter is as follows. In Sect. 11.1 some background on mathematical systems modelling is presented. Sections 11.2, 11.3, 11.4 and 11.5 describe the modelling of a recirculating aquaculture system (RAS), anaerobic digestion, hydroponic (HP) greenhouse and a multi-loop AP system, respectively. In Sect. 11.6 modelling tools are introduced and illustrated with some examples. The chapter concludes with a Discussion and Conclusions section.

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