Underapproximating Safe Domains of Attraction for Discrete-Time Systems Using Implicit Representations of Backward Reachable Sets

Analyzing and certifying stability and attractivity of nonlinear systems is a topic of research interest that has been extensively investigated by control theorists and engineers for many years. Despite that, accurately estimating domains of attraction for nonlinear systems remains a challenging task, where available estimation approaches are either conservative or limited to low-dimensional systems. In this work, we propose an iterative approach to accurately underapproximate safe (i.e., state-constrained) domains of attraction for general discrete-time autonomous nonlinear systems. Our approach relies on implicit representations of safe backward reachable sets of safe regions of attraction, where such regions can be be easily constructed using, e.g., quadratic Lyapunov functions. The iterations of our approach are monotonic (in the sense of set inclusion), where each iteration results in a safe region of attraction, given as a sublevel set, that underapproximates the safe domain of attraction. The sublevel set representations of the resulting regions of attraction can be efficiently utilized in verifying the inclusion of given points of interest in the safe domain of attraction. We illustrate our approach through two numerical examples, involving two- and four-dimensional nonlinear systems.