## 2 Model Description, Reference Model and Control Goal  
### Model Description  
The second order nonlinear SISO model of an hydroponic culture described in Appendix A is  
\[\frac{dx_{1}}{dt} =a_{1}x_{2}-a_{2}x_{1}-r_{x1} \tag{1}\] \[\frac{dx_{2}}{dt} =a_{3}\frac{1}{x_{3}}(x_{1}-x_{2})+\frac{a_{4}}{x_{3}}x_{2}+bv\] (2) \[\frac{dx_{3}}{dt} =Q_{e}-Q_{i}-Q_{loss}+v \tag{3}\]  
where  
\[b=\frac{P_{ad}-x_{2}}{x_{3}},\ \ a_{1}=Q_{i}/V_{u},\ \ a_{2}=Q_{e}/V_{ u},\ a_{3}=Q_{e},\ a_{4}=Q_{loss}\] \[x_{1}=P_{e},\ x_{2}=P_{i},\ x_{3}=V_{i},\ v=Q_{ad},\]  
\(P_{e}\) is the nutrient concentration in the upper CSTR, \(P_{i}\) is the nutrient concentration in the lower CSTR, \(V_{i}\) is the volume of the lower CSTR, \(V_{u}\) is the volume of the upper CSTR, and \(Q_{e}\) is the flow that leaves the upper CSTR and enters the lower CSTR; \(Q_{i}\) is the flow that leaves the lower CSTR and enters the upper CSTR; \(Q_{ad}\) is the flow of addition of fresh nutrient solution to the mixing tank; and \(P_{ad}\) is the nutrient concentration of the \(Q_{ad}\) flow. The state \(x_{1}\) is the output to be controlled, whereas the inlet flowrate \(Q_{ad}\) is chosen as control input, it is non-negative and its upper bound is determined by the operational limit of the pump. Therefore, the relationship between the constrained control signal (denoted as \(v\)) and the unconstrained control signal (denoted as \(u\)) is  
\[v=\left\{\begin{array}{ll}u_{max}&\text{if }u>u_{max}\\ u&\text{if }u\in[u_{min}\ u_{max}]\\ u_{min}&\text{if }u<u_{min}\end{array}\right. \tag{4}\]  
The following assumptions are considered:  
**Assumption 1**.: _The state variables \(x_{1}\) and \(x_{2}\), \(x_{3}\) are bounded for \(v\) bounded, and satisfy \(x_{1}\in\mathbb{R}^{+}\), \(x_{2}\in\mathbb{R}^{+}\), \(x_{3}\in\mathbb{R}^{+}\)._  
**Assumption 2**.: \(a_{1}\)_, \(a_{2}\), \(a_{3}\), \(a_{4}\) are constant, \(a_{2}\), \(a_{3}\), \(a_{4}\) are unknown whereas \(a_{1}\) is known._  
**Assumption 3**.: _The reaction rate \(r_{x1}\) satisfies one of the following: (i) it is unknown, non-negative, and \(r_{x1}\leq\bar{\mu}_{1}r_{x1}\), where \(\bar{\mu}_{1}\) is unknown, positive, constant, whereas \(r_{x1}\) is a known continuous function of \(x_{1}\) with well-defined \(dr_{x1}/dx_{1}\); (ii) it is a known continuous function of \(x_{1}\), with well-defined \(dr_{x1}/dx_{1}\); in this case, \(r_{x1}=\bar{\mu}_{1}\bar{r}_{x1}\) with \(\bar{r}_{x1}=r_{x1}\), \(\bar{\mu}_{1}=1\) holds true._  
**Assumption 4**.: _The values of \(x_{1}\), \(x_{2}\) are known, whereas \(x_{3}\) is noisily measured: \(x_{3m}=x_{3}+\delta_{x3}\), where \(x_{3m}\) is the noisy measurement and \(\delta_{x3}\) is the measurement noise._  
**Assumption 5**.: _There is lack of knowledge on the control gain \(b\) according to one of the following conditions: (i) \(b=b_{b}b_{m}\), where \(b_{m}\) is known, and possibly varying, and bounded for \(x_{1}\), \(x_{2}\), \(x_{3}\) bounded, whereas \(b_{\delta}\) is unknown, varying, and satisfies: \(\bar{\mu}_{lb}\leq|b_{\delta}|\leq\bar{\mu}_{ub}\), where \(\bar{\mu}_{lb}\), \(\bar{\mu}_{ub}\) are constant, positive and unknown; (ii) \(b\) is unknown, varying, and satisfies \(\bar{\mu}_{lb}\leq|b|\leq\bar{\mu}_{ub}\), where \(\bar{\mu}_{lb}\), \(\bar{\mu}_{ub}\) are unknown positive constants; in this case, \(b\) can be expressed as \(b=b_{\delta}b_{m}\), \(b_{m}=1\), \(b_{\delta}=b\), so that \(b_{\delta}\) is unknown, varying and satisfies \(\bar{\mu}_{lb}\leq|b_{\delta}|\leq\bar{\mu}_{ub}\). In both conditions, \(b=b_{\delta}b_{m}\), where \(b_{m}\) is known, whereas \(b_{\delta}\) is unknown, varying and satisfies \(\bar{\mu}_{lb}\leq|b_{\delta}|\leq\bar{\mu}_{ub}\); so that \(|b|\leq\bar{\mu}_{ub}|b_{m}|\)._