## 4 Simulation Example  
Consider the aeroponic system described in Appendix A whose model is given by Equations (1)-(3), with input constraint (4), Assumptions 1-5, and control goal and desired output \(y_{d}\) stated in Sections 2.2 and 2.3. The control law, the update laws, the auxiliary system, and their parameters and signals are stated in Remark 5. At what follows, the values of \(x_{1}\), \(x_{2}\) and \(x_{3}\) are generated through the model (1) to (3) with specific parameter values. These values of \(x_{1}\), \(x_{2}\) and \(x_{3}\) are used by the controller, but the model parameters and upper or lower bounds are not. The input saturation values are \(u_{min}=0\) and  
\[u_{max}=\left\{ \begin{array}{ll}350&\mathrm{L/day}\ \mathrm{for}\ t\leq 7 \ \mathrm{days}\\ 10.944&\mathrm{L/day}\ \mathrm{otherwise}\end{array} \right.. \tag{79}\]  
The input value \(Q_{ad}=0\) is used until \(x_{1}\) reaches the value \(70\ \mathrm{mg/L}\), so that the controller is started at \(t=4\) days. The parameters of the reference model (5) are chosen as \(a_{m1}=40\) and  
\[W_{ref}=\left\{ \begin{array}{ll}75\,\mathrm{mg/L}&\mathrm{for}\ t\leq 7 \ \mathrm{days}\\ 80\,\mathrm{mg/L}&\mathrm{otherwise}\end{array} \right., \tag{80}\]  
whereas the desired width of the convergence region is chosen to be \(C_{b}=0.4\). The user-defined parameters of the control law, update laws and auxiliary system are chosen as  
\[k_{1}=\left\{ \begin{array}{ll}8&\mathrm{for}\ t\leq 7\ \mathrm{days}\\ 2&\mathrm{otherwise}\end{array} \right., \tag{81}\]  
\(k_{2}=0.005,k_{\phi 1}=20>1/2,k_{\phi 2}=20>1/2,\gamma_{1,1}=8\times 10^{-4}\), \(\gamma_{1,2}=0.008\), \(\gamma_{rx1}=4\times 10^{-7}\), \(\gamma_{3}=0.004\), \(\gamma_{lib}=4\times 10^{-7}\), \(\gamma_{ub}=4\times 10^{-9}\), \(\gamma_{4}=4\times 10^{-4}\). We consider the measurement noise for \(x_{3}\) in the control gain \(b\), such that \(b\) satisfies the first condition of assumption 2.5, with  
\[b_{m}=\frac{P_{ad}-x_{2}}{x_{3m}},\ \ b_{\delta}=\frac{x_{3m}}{x_{3}}\]  
Due to the controller starting at \(t=4\) days and the change of \(W_{ref}\) at \(t=7\) days, the system behavior is separated in the time intervals \([4\quad 7)\) and \([7\quad\infty)\) days. For \(t\in[4\quad 7)\) days:  
* all the closed loop signals are bounded (see Figures 1-3). * the signal \(z_{1}\) is near \(\Omega_{z1}\) at initial time \((z_{1t0}\approx-0.7)\), it enters \(\Omega_{z1}\) at 6.72 days and it remains inside until \(t=7\) days (Figure 1d). * the updated parameters remain bounded, and its change is not excessive; \(\hat{\theta}_{1,1}\), \(\hat{\theta}_{1,2}\) change when \(z_{1}\
otin\Omega_{z1}\), and remain constant otherwise (Figure 3). * input signal \(v\): for \(t\in[4\quad 5.09]\cup[6.52\quad 6.6]\) days it exhibits reiterated saturation at its lower bound, with only one moment of saturation at its upper bound (at \(t=4.89\) days approx); during other moments it exhibits changing behavior (Figure 2c,d). For \(t\geq 7\) days:
* all the closed loop signals are bounded (see Figures 1-3). * the signal \(z_{1}\) is inside \(\Omega_{z1}\) at \(t=7\) days (\(z_{1t0}\approx-0.39\)), it leaves, it enters \(\Omega_{z1}\) at 7.55 days approx. and it remains inside afterwards (Figure 1d). * the updated parameters remain bounded, \(\hat{\theta}_{1,1}\), \(\hat{\theta}_{1,2}\) are constant when \(z_{1}\in\Omega_{z1}\), and the other updated parameters are constant when \(z_{2}\in\Omega_{z2}\) (Figure 3). * input signal \(v\): for \(t\in[7.0\quad 7.57]\) days, it remains saturated at its upper bound; for \(t\in(7.57\quad 8.44]\) days, it exhibits saturation at its lower bound with some few saturationat its upper bounds; for \(t>8.44\) days, it exhibits reiterated saturation at both its upper and lower bounds (Figure 2c,d). Figure 1: Simulation results for the system signals under the formulated controller. (**a**) Time course of the states \(P_{e}\), \(P_{t}\), desired output \(y_{d}\), and auxiliary signal \(\psi_{1}\). (**b**) Detail of the time course of \(P_{e}\), \(P_{t}\), and \(y_{d}\).