# Experimental Validation of a Guaranteed Nonlinear Model Predictive Control
## 3 Guaranteed Controller Design via Validated Simulation
### Validated Nonlinear Model Predictive Control
#### 3.2.2 Interval Cost Function Formulation
Nonetheless, the simulated \(y_{2}\) with IA parameters comply as closely as possible with the measured one at different initial conditions. All this is due to the fact that all the system variables are taken here as intervals characterizing different uncertainties. In order to emphasize the potential capabilities of the predicted model, the associated coverage rates can be computed. We evaluate the number of cases where the measurements data are included in the simulated tubes, and then the percentage of inclusion is calculated for each tested scenario. In other words, it is calculated by checking at each time step whether the time variable related to the measurement is included in the simulated time interval \(t_{i}\in[\mathbf{t}_{i}]\). If so, we inspect if the corresponding measure \(y_{i}\) is included in the simulated tube \([\mathbf{y}_{i}]\) given by the Dynl8 library. Then, the coverage rate can be approximated to be \(\frac{N_{q}}{N_{t}}\times 100\), where \(N_{t}\) is the number of times that each sampled time is incorporated in the simulated time interval, and \(N_{q}\) is the one related to the second condition regarding \(y_{i}\). The computed values are recapped in Table 2. By the way, this criterion is too strict to test, which explains the small calculated percentages. This is due to the timing lag between the measurement and the estimated model. Even with this issue, the coverage ratios obtained using IA parameters indicate a good match with the measurements process. It also confirms that the model is estimated with high precision when the uncertainties are considered. Overall, the experimental results provided by the guaranteed identification of inertial and frictional parameters are more relevant than those obtained by the classical LSMI. Indeed, the estimation approach based on IA and set inversion techniques allows us to successfully identify these coefficients as intervals considering all the errors and uncertainties related to the sensors and the system modeling. Despite its proven effectiveness to more closely approximate the model output to the actual system's behavior, it still has some drawbacks. The main ones are related globally to the computation time, which depends mainly on the number of parameters and the initial domain. This issue can be enhanced with a contraction approach [24]. ### NMPC Results

#### 4.2.1 Simulation Results

In the simulations, the parameters of this proposed validated NPMC are tuned as: prediction horizon \(N_{p}=10\), control sampling time \(T_{c}=16\) ms and the final time of the simulation \(T_{f}=0.3\) s. The safety saturation on state variables are: \(y_{1}\in[-2\pi;\ 2\pi]\), \(y_{2}\in[-52.4\ \text{rad}\cdot\text{s}^{-1};\ 52.4\ \text{rad}\cdot\text{s}^{-1}]\), \(y_{3}\in[-2\pi;\ 2\pi]\), \(y_{4}\in[-100\ \text{rad}\cdot\text{s}^{-1};\ 100\ \text{rad}\cdot\text{s}^{-1}]\), and the DC motor's torque constraints are \(\tau\in[-8.05\ \text{N}\cdot\text{m};8.05\ \text{N}\cdot\text{m}]\). The reference interval of the desired pendulum arm is \(y_{r}\in[\pi-0.1;\ \pi+0.1]\). The tolerance parameter applied \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Scenario** & **(a)** & **(b)** & **(c)** & **(d)** \\ \hline With IA parameters & 46\% & 61\% & 38\% & 25\% \\ With LSMI parameters & 41\% & 34\% & 20\% & 21\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Degree of coverage between the model and physical reality. Figure 3: Open-Loop validation of the dynamic model under DynIblex library. Comparison between the real and simulated pendulum angular position (represented as tubes) at different initial conditions: **(a)**\(y_{0}=[0,0,-100^{\circ},0]^{\text{T}}\) and \(u_{0}=0\ \text{rad}\cdot\text{s}^{-1}\); **(b)**\(y_{0}=[0,0,50^{\circ},0]^{\text{T}}\) and \(u_{0}=0\ \text{rad}\cdot\text{s}^{-1}\); **(c)**\(y_{0}=[0,0,-90^{\circ},0]^{\text{T}}\) and \(u_{0}=40\ \text{rad}\cdot\text{s}^{-1}\) and **(d)**\(y_{0}=[0,0,-45^{\circ},0]^{\text{T}}\) and \(u_{0}=25\ \text{rad}\cdot\text{s}^{-1}\). for the bisection procedure is adjusted as \(tol=0.25\).