On ANN-enhanced positive invariance for nonlinear flat systems
This work addresses the problem of formal verification of stability and safety for autonomous systems, but it is incremental as it builds on existing methods for flat systems with neural network approximations.
The paper tackles the challenge of characterizing positively invariant sets for nonlinear differentially flat systems with constraints by using a neural network to approximate the linearizing mapping, enabling the derivation of such sets for efficient online control strategies, as validated through numerical simulations.
The concept of positively invariant (PI) sets has proven effective in the formal verification of stability and safety properties for autonomous systems. However, the characterization of such sets is challenging for nonlinear systems in general, especially in the presence of constraints. In this work, we show that, for a class of feedback linearizable systems, called differentially flat systems, a PI set can be derived by leveraging a neural network approximation of the linearizing mapping. More specifically, for the class of flat systems, there exists a linearizing variable transformation that converts the nonlinear system into linear controllable dynamics, albeit at the cost of distorting the constraint set. We show that by approximating the distorted set using a rectified linear unit neural network, we can derive a PI set inside the admissible domain through its set-theoretic description. This offline characterization enables the synthesis of various efficient online control strategies, with different complexities and performances. Numerical simulations are provided to demonstrate the validity of the proposed framework.