# 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
```
\([\mathbf{y}],[\mathbf{u}_{1}],[\mathbf{u}_{2}],\ldots,[\mathbf{u}_{N_{p}}],tol\) while\(w([\mathbf{u}_{1}])\geq tol\)do \([\mathbf{U}]_{\text{left}}=[\mathbf{u}_{1}]_{\text{left}}\times[\mathbf{u}_{2}] \times\ldots\times[\mathbf{u}_{N_{p}}]\) \([\mathbf{U}]_{\text{right}}=[\mathbf{u}_{1}]_{\text{right}}\times[\mathbf{u}_ {2}]\times\ldots\times[\mathbf{u}_{N_{p}}]\) if\(J([\mathbf{y}],[\mathbf{U}]_{\text{left}})\geq J([\mathbf{y}],[\mathbf{U}]_{ \text{right}})\)then \([\mathbf{U}]^{\star}=[\mathbf{U}]_{\text{right}}\) else \([\mathbf{U}]^{\star}=[\mathbf{U}]_{\text{left}}\) endif endwhile \(u_{1}=\text{min}\{lb([\mathbf{u}_{1}]),ub([\mathbf{u}_{1}])\}\) \(send(u_{1})\)
```  
**Algorithm 2** Optimization  
To summarize, the proposed approach is focused on two main stages :  
**Algorithm 1**  
``` \([\mathbf{y}],

## 4 Experiment Results

The control framework introduced in this paper is applied for the stabilization of the nonlinear inverted pendulum shown in Figure 1. The pendulum is actuated by a DC motor whose angular speed is the input variable. It is controlled using a Phidget Motor Control. This control card is connected to the laptop via a USB cable to control the motor's velocity, using C language. To evaluate the validated NMPC, we will initially summarize in Section 4.1 the guaranteed identification results and its validation using the identified subsets of parameters. In the second place, we will validate the proposed controller in Section 4.2 through simulation tests and then experiments using the inverted pendulum. ### Guaranteed Dynamic Model Identification

The guaranteed identification of dynamic parameters \(p=[\mu_{1},\mu_{2},\mu_{3},\mu_{4},\mu_{g},f_{\
u_{1}},f_{\
u_{2}}]\in\mathbb{ R}^{n_{p}-7}\) as can be seen by the dynamic model (1) was proposed in [20]. These parameters are estimated as intervals instead of real numbers to account for all the uncertainties and errors related to the sensors measurements and the system modeling. The identification approach is based on the IA and set-inversion techniques (SIVIA algorithm) to determine the feasible sets of coefficients [7]. A brief resume of the SIVIA algorithm principle is given in Appendix A. The feasible sets of all the dynamic parameters are recapped in Table 1.

The DC motor and gearbox are of type _MDP-Maxon_, and their characteristics are provided by the manufacturer. Here is a recap of their properties: the motor inertia \(J_{m}=9.0e^{-7}\) kg\(\cdot\)m\({}^{2}\), its no-load speed \(u_{0}=11\),\(700\) tr\(\cdot\)min\({}^{-1}\), the slope speed-torque \(Rv=34\),\(000\) tr\(\cdot\)min\({}^{-1}\cdot\)N\({}^{-1}\cdot\)m\({}^{-1}\), the motor and gearbox efficiencies \(\eta_{m}=90.6\%\) and \(\eta_{g}=90\%\), and gear ratios \(N_{m}=5.2\) and \(N_{g}=5\). To demonstrate the usefulness of our identification, we can solve in a complete and validated way the IVP-ODEs (5) using the identified parameters. The set of possible tight enclosures \([\mathbf{y}_{1}],\ldots,[\mathbf{y}_{n}]\) at a sequence of time-instants \(t_{1},\ldots,t_{n}\) are calculated. To do so, our library Dynl8 is employed to solve this IVP-ODEs [21]. Furthermore, the guaranteed identification is compared to the traditional least square method identification (LSMI) (where the dynamic parameters are estimated as constant values) to show its efficiency. Figure 3 shows the measured pendulum angle \(y_{3}\) (black lines), as well as the ones calculated with the identified parameters with LSMI (red lines) and IA (blue lines) approaches. All the model validation tests are conducted under the same settings and conditions (e.g., same initial conditions \([\mathbf{y}_{0}]\) and \([\mathbf{u}_{0}]\), same inputs, etc.). The simulation duration is fixed to be 4 s and the precision of \(10^{-7}\). We can notice that the simulated tubes of the pendulum angle \(y_{2}\) given by the sophisticated solver Dynl8, using IA and LSMI parameters, are quite similar to the real measured pendulum angle.