## 5 Validation Case Study  
### AAC Design for a MIMO Vertical Tank System  
The controlled process is a vertical MIMO two-tank system (Figure 1c) built around a three-tank laboratory equipment [47] with the continuous-time state-space equations  
\[ \begin{array}{l}\dot{H}_{1}=\frac{k}{av}u_{1}-\frac{1}{av}\overline{C}_{1}H _{1}^{\alpha 1}(2.5\overline{u}_{2}-0.5),\\ \dot{H}_{2}=\frac{1}{av}\overline{C}_{1}H_{1}^{\alpha 1}(2.5\overline{u}_{2}-0.5)- \frac{1}{cv+\frac{12}{P_{\max}}bw}\overline{C}_{2}H_{2}^{\alpha 2},\\ \overline{u}_{2}=\min(\max(u_{2},0.6),1),\end{array}  \tag{33}\]  
Figure 2: VRFT initial experiment data collection. Red line in (**b**) is the dead-zone threshold for \(u_{2}\). with \(a=0.25\)\([\,\mathrm{m}]\), \(w=0.035\)\([\,\mathrm{m}]\), \(c=0.1\)\([\,\mathrm{m}]\), \(b=0.345\)\([\,\mathrm{m}]\) and \(H_{1\mathrm{max}}=H_{2\mathrm{max}}=0.35\)\([\,\mathrm{m}]\). \(x_{1}=y_{1}=H_{1}\in[0,H_{1\mathrm{max}}]\) and \(x_{2}=y_{2}=H_{2}\in[0,H_{2\mathrm{max}}]\) are the water levels in the two tanks considered as system states and controlled outputs. The control inputs \(u_{1},u_{2}\in[0,1]\) (also expressible in [%]) are the duty cycles of the pump direct current (DC) motor and of the electrically controlled valve \(C_{1}\), respectively. \(k=1.66\cdot 10^{-4}[m^{3}/(s\cdot\%)]\) is the gain from the pump input to the inflow, \(\overline{C_{1}}=5.65\cdot 10^{-5}[m^{3-\alpha 1}/s]\), \(\overline{C_{2}}=8\cdot 10^{-5}[m^{3-\alpha 2}/s]\) are the resistances of the outflow orifices of the first (upper) and second (lower) tank, called \(T_{1}\) and \(T_{2}\), respectively, and \(\alpha_{1}=0.29\), \(\alpha_{2}=0.22\). The third equality in Equation (33) reflects the dead-zone plus saturation in the second control input \(u_{2}\). Features of this process include: no water level setpoint for the tank \(T_{2}\) can be set if the water level in \(T_{1}\) is zero; the electrical valve controlling the outflow from \(T_{1}\) (which is inflow to \(T_{2}\)) is changed by \(\widetilde{u_{2}}(u_{2})\); no setpoint can be tracked for each tank if there is more outflow than inflow; \(T_{1}\)'s outflow has a minimum value and can be zero only when \(H_{1}=0\), as per first equality in (33). \(T_{2}\)'s dynamics is slower than \(T_{1}\)'s. Proper selection of the parameters \(\overline{C_{1}},\overline{C_{2}}\) through manual valves allows feasible control trajectories in the constrained input-state-output space. Discretization of (33) reveals its Markov form. The water level is measured using piezoelectric sensors (\(PS_{1}\) and \(PS_{2}\) in Figure 1c). Protection logic disables the pump voltage when water level exceeds the upper bound. The sampling period used for control experiments is \(T_{s}=0.5\) s. Model (33) is not used for control design. For VRFT-based control design, the ORM is selected as the ZOH discretization of \(\mathbf{M}(s)=diag(M_{1}(s),M_{2}(s))\). \(M_{1}(s)=\omega_{0}^{2}/(s^{2}+2c\omega_{0}s+\omega_{0}^{2})\), with the damping factor \(\varsigma=1.0\) and the natural frequency \(\omega_{0}=0.5 rad/s\) selects the speed and shape of the desired response \(y_{k,1}^{m}\) while similar \(M_{2}(s)\) with \(\varsigma=1.0\) and \(\omega_{0}=0.2 rad/s\) describes \(y_{k,2}^{m}\). For collecting the open-loop input-state-output data \(\left\{\widetilde{\mathbf{u}}_{k},\widetilde{\mathbf{x}}_{k},\widetilde{ \mathbf{y}}_{k}\right\}_{k=0..N-1}\), 16,000 samples have been generated from 16,000 samples of a uniformly random sequence of persistently exciting steps lasting for 20 s \(\widetilde{\mathbf{u}}_{k}=[\widetilde{u}_{k,1}\ \widetilde{u}_{k,2}]^{T}\in[0,0.5]\times[0,1]\), for an experiment of 8000 s. The collected data is displayed in Figure 2 and ensures the exploratory conditions from Assumption _A5_. The controllable canonical state-space realizations (\(\mathbf{A}_{1}\), \(\mathbf{B}_{1}\), \(\mathbf{C}_{1}\), \(\mathbf{D}_{1}\)) and \((\mathbf{A}_{2},\mathbf{B}_{2},\mathbf{C}_{2},\mathbf{D}_{2})\) and of \(M_{1}(z)\) and \(M_{2}(z)\) are, respectively:  
\[\mathbf{A}_{1} =\left(