# Experimental Validation of a Guaranteed Nonlinear Model Predictive Control  
[PERSON]  
1 LISIC Laboratory--EA 4491, University of Littoral Cote d'Opale, 62100 Calais, France 1  
[PERSON]RSON] Abstract  
This paper combines the interval analysis tools with the nonlinear model predictive control (NMPC). The NMPC strategy is formulated based on an uncertain dynamic model expressed as nonlinear ordinary differential equations (ODEs). All the dynamic parameters are identified in a guaranteed way considering the various uncertainties on the embedded sensors and the system's design. The NMPC problem is solved at each time step using validated simulation and interval analysis methods to compute the optimal and safe control inputs over a finite prediction horizon. This approach considers several constraints which are crucial for the system's safety and stability, namely the state and the control limits. The proposed controller consists of two steps: filtering and branching procedures enabling to find the input intervals that fulfill the state constraints and ensure the convergence to the reference set. Then, the optimization procedure allows for computing the optimal and punctual control input that must be sent to the system's actuators for the pendulum stabilization. The validated NMPC capabilities are illustrated through several simulations under the DynIplex library and experiments using an inverted pendulum. validated NMPC; ordinary differential equations; interval analysis; uncertain dynamic model +
Footnote †: journal: _Algorithms_  
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and software parts of the automated device [4]. Indeed, IA tools turn out to be more promising and reliable to handle all the noises and uncertainties in the experimental and theoretical data, contrary to these classical approaches that are generally based on simplified models with deterministic values. IA methods are extremely popular in several research fields due to their efficiency and reliability in order to achieve a robust control in the presence of model uncertainties [5; 6] or for state estimation [7]. For example, a robust controller based on interval arithmetic for constrained discrete-time nonlinear systems with additive uncertainties is presented in [5]. In addition, authors in [6] developed a robust feedback controller synthesis such that all the unknown parameters are assumed to vary between known bounds. This controller guarantees at each time step that the closed-loop system behavior is greater than the lower bound of the set-point interval and is lower than its upper bound. Model-based Predictive Control (MPC) methodology (also known as the receding horizon control or the moving horizon control) has become a powerful control technique widely used in a large number of areas (e.g., the automotive industry and chemical processes [8]). There are several reasons for this: (i) its capability to anticipate the future on the reference trajectory (i.e., future changes in set-point), and (ii) its ability to handle nonlinearities of multi-variable systems as well as all the system's constraints, i.e., input and state limits. The primary purpose of the linear and nonlinear MPC is to apply a plant model to predict its behavior along a receding horizon. At each sampling time, the MPC technique computes the optimal control input that minimizes some cost functions and fulfills all the safety constraints (e.g., actuators position, speed, and acceleration bounds). Overall, it has been proven that the MPC technique is much more practical and sophisticated to solve linear and nonlinear control problems [9; 10; 11; 12; 13]. Amongst the existing works, a real-time constrained MPC with safety and stability constraints is proposed in [9; 10], where all the constraints are expressed as inequalities with respect to the optimization variable. Furthermore, a linear time-varying MPC method is designed in [12; 13].