# 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
Figure 6: Experimental validation results starting from \(0^{\circ}\) (**Left**) and from \(150^{\circ}\) (**Right**). ## Conflicts of Interest: The authors declare no conflict of interest. ## Appendix A

In this appendix, we remind readers about some mathematical notions related to IA, used particularly to solve linear and/or nonlinear problems in a guaranteed way. For further details on this subject, we encourage the readers to refer to [4; 7]. ### Interval Analysis: The Main Notations and Definitions

* A scalar interval of \(\mathbb{R}\) is denoted as \([\mathbf{x}]=[\underline{x},\,\overline{x}]\) where \(\underline{x}\) is the lower bound and \(\overline{x}\) is the upper bound. Any interval of \(\mathbb{R}\) is finite, closed, and connected subsets. * An interval vector, or a box \([\mathbf{X}]\) of \(\mathbb{R}^{n}\) is a Cartesian product of \(n\) intervals, i.e., \([\mathbf{X}]=[\mathbf{x}_{1}]\times\ldots\times[\mathbf{x}_{n}]\). The set of all boxes of \(\mathbb{R}^{n}\) is denoted by \(\mathbb{IR}^{n}\). * The width \(w([\mathbf{X}])\) of a box \([\mathbf{X}]\) is the length of its largest side. * The interval function \([f]\) from \(\mathbb{IR}^{n}\) to \(\mathbb{IR}^{m}\), is an inclusion function of \(f\) if \(\forall[\mathbf{X}]\in\mathbb{IR}^{n},\quad[f]([\mathbf{X}])\supseteq\{f(x), x\in[\mathbf{X}]\}\). * A subpaving of \(\mathbb{R}^{n}\) is a list of non-overlapping boxes of \(\mathbb{R}^{n}\), enabling for bracketing any compact set between a list of inner and outer subpavings. ### Set Inversion and SIVIA Algorithm for Parameters Estimation

If \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\) and \(Y\subset\mathbb{R}^{m}\). A set inversion problem is described by the reciprocal image of a set \(Y\) by the function \(f\) that can be written as

\[\mathbb{P}=\{\mathbf{p}\in\mathbb{R}^{n}\mid f(\mathbf{p})\in Y\}=f^{-1}(Y) \tag{1}\]

The set \(\mathbb{P}\) should be computed in a guaranteed way relying on the IA tools. For this purpose, the SIVIA algorithm is an effective set-inversion problem solver via IA [7]. It allows us to get the set \(\mathbb{P}\) in an effective and a guaranteed way, through successive and recursive bisections of the prior interest domain. To obtain the feasible set \(\mathbb{P}\), three kinds of union boxes can be distinguished:

1. The feasible subpavings that belong to the set \(\mathbb{P}\) and satisfy this implication test \([f]([\mathbf{p}])\subset Y\quad\Rightarrow\quad[\mathbf{p}]\subset\mathbb{P}\) (denoted \([\mathbf{p_{i}}]\) in Figure 1);
2. The infeasible subpavings that make the empty intersection with \(\mathbb{P}\), i.e., \([f]([\mathbf{p}])\cap Y=\emptyset\Rightarrow[\mathbf{p}]\cap\mathbb{P}= \emptyset\) (denoted \([\mathbf{p_{o}}]\) in Figure 1);
3. The undetermined subpavings (or penumbra) for which nothing can be decided and will be bisected, except if their width is too small (denoted \([\mathbf{p_{u}}]\) in Figure 1). ## References

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