# Experimental Validation of a Guaranteed Nonlinear Model Predictive Control
## 3 Guaranteed Controller Design via Validated Simulation
### Validated Numerical Integration  
The dynamic model given by Equation (1) can simply be written by this nonlinear ODEs:  
\[\dot{y}_{t}=\mathbf{f}(t,y_{t},u_{t}) \tag{4}\]  
At each validated numerical integration, we assume that \(u_{t}\) is constant to compute corresponding tight enclosures of the state variable. We are now considering an Initial Value Problem for ODEs (IVP-ODEs) over the time interval \(t\in[0,T]\),  
\[\left\{\begin{array}{l}\dot{y}_{t}=\mathbf{f}(t,y_{t},u_{t})\\ y_{0}\in[\mathbf{y}_{0}]\subseteq\mathbb{R}^{n}\\ u_{t}\in[\mathbf{u}]\subseteq\mathbb{R}^{m}\end{array}\right. \tag{5}\]  
where the sets \([\mathbf{y}_{0}]\) and \([\mathbf{u}]\), both expressed as boxes, are respectively the initial condition related to the initial state vector and the considering input interval. This IVP-ODE has a unique solution \(y_{t}(t;y_{0},u_{t})\) at \(t>0\) since \(\mathbf{f}:\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow \mathbb{R}^{n}\) is continuous in \(t\) and Lipschitz in \(y_{t}\), but, for our purpose, we assume \(\mathbf{f}\) to be smooth enough, i.e., of class \(C^{k}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Measurement Unit** & **Interval Enclosure** \\ \hline \(\mu_{1}\) & kg-m\({}^{2}\) & \([9.63318e^{-4};\,1.22604e^{-3}]\) \\ \(\mu_{2}\) & kg-m\({}^{2}\) & \([2.19585e^{-3};\,2.79471e^{-3}]\) \\ \(\mu_{3}\) & kg-m\({}^{2}\) & \([7.227e^{-4};\,8.833e^{-4}]\) \\ \(\mu_{4}\) & kg-m\({}^{2}\) & \([1.08271e^{-3};\,1.37799e^{-3}]\) \\ \(\mu_{g}\) & kg-m\({}^{2}\)\(\,\)s\({}^{-2}\) & \([6.24299e^{-2};\,8.44641e^{-2}]\) \\ \(f_{v_{1}}\) & N-m-s-rad\({}^{-1}\) & \([0.043012;\,0.13002]\) \\ \(f_{v_{2}}\) & N-m-s-rad\({}^{-1}\) & \([0.000454;\,0.001174]\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Identified inertial and frictional parameters of the inverted pendulum.  
Compared to the classical numerical integration for the IVP-ODE problem, validated approaches consist of solving this problem in a complete and validated way (i.e., each actual guaranteed solution is rigorously returned and enclosed in a tight interval). These methods are generally based on Taylor series [23] or Runge-Kutta methods [21]. Basically, the main principle of the validated methods is to obtain the tight enclosure of the IVP-ODEs problem. As presented in [23], the purpose of a validated numerical algorithm is to solve Equation (5) so as to get a sequence of boxes \([\mathbf{y}_{0}],\ldots,[\mathbf{y}_{n}]\) at time iterations \(t_{0}=0<\ldots<t_{n}=T\) and at given input boxes \([\mathbf{u}_{0}],\ldots,[\mathbf{u}_{n}]\). It is achieved in such a way that the inclusion function verifies \(\left[\mathbf{y}_{j+1}\right]\supseteq\left[\mathbf{f}\right]\left(t_{j}, \left[\mathbf{y}_{j}\right],\left[\mathbf{u}_{j}\right]\right),\ \forall j\in[0,n]\). Such approaches work in two stages at each integration step to compute the guaranteed solutions. They are summarized as:  
1. Computation of a _prior_ enclosure of the solution \(\left[\hat{\mathbf{y}}_{j+1}\right]\), such that \(\mathbf{f}\left(t_{j},\left[\mathbf{y}_{j}\right],\left[\mathbf{u}_{j}\right] \right)\in\left[\hat{\mathbf{y}}_{j+1}\right]\) for all \(t\) in the time interval \(t\in\left[t_{j},t_{j+1}\right]\). This stage enables proving the existence and the uniqueness of the solution.
2. Computation of a _tight_ enclosure of state variable \(\left[\mathbf{y}_{j+1}\right]\) at time instance \(t_{j+1}\), such that \(\mathbf{f}\left(t_{j+1},\left[\mathbf{y}_{j}\right],\left[\mathbf{u}_{j} \right]\right)\in\left[\mathbf{y}_{j+1}\right]\). It uses the solution \(\left[\hat{\mathbf{y}}_{j+1}\right]\) to bound the _truncation error_, i.e., the distance between the exact and the numerical solutions. The _tight_ and _prior_ enclosures calculated along one integration step between \(t_{j}\) and \(t_{j+1}\) (of size \(h_{j}=t_{j+1}-t_{j}>0\)) can be visualized in Figure 2.