A Convex Data-Driven Approach for Nonlinear Control Synthesis

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.

A Complete Model of a Two Degree of Freedom Platform Actuated by Three Pneumatic Muscles Elaborated for Control Synthesis

Pneumatic muscles have a high potential in industrial use, as they provide safety, high power over volume ratio, low price and wide range of pulling effort. Nevertheless, their control is quite hard to achieve due to the non linearity and hysteresis phenomena, plus the uncertainties in their behavior. This paper presents the modeling of a two degree of freedom platform actuated by three pneumatic muscles for control purposes. Three servovalves are used to supply airflow inside the muscles. The innovative concept is the modeling of each component including the static and dynamic muscle behavior. The model of the servovalve consists of a look-up table gathering the three variables: airflow, pressure and voltage applied to the servovalve. In addition, a thermodynamic and a mechanical study of the system complete the model. The result is a complete model design having as input the voltage applied to the three servovalves, and as outputs, the two angles of rotation. Simulated and experimental results permit to validate the complete model for high variation in static and dynamic conditions. These results will be helpful for nonlinear control synthesis.