Improving Robustness to Unknown Disturbances by Expanding the Region of Attraction Near the Current State

Abstract

Physical systems often experience unknown disturbances that can disrupt their safe operation. For example, consider a drone experiencing a wind gust, a power system experiencing a voltage spike, or an autonomous vehicle encountering a sudden obstacle. The normal or safe operating point of these systems can be represented by an equilibrium point. The RoA is the set of all states for which the system can return to the standard operating point. However, calculating a system's RoA is often very difficult and computationally intractable. Expanding the RoA is an effective method to increase a system's robustness against such disturbances. Expanding the RoA is also very challenging. Because of this, instead of expanding the true RoA, existing methods often expand estimates of the RoA, which can be overly conservative or computationally intractable for higher-dimensional systems. Additionally, these methods are limited to systems and/or controllers that adhere to specific specialized structures, and do not generalize to many nonlinear systems of practical importance. Furthermore, these methods work to expand the RoA in all directions as opposed to near the current state. Expanding the RoA near the current state maximizes the stability of the system given the current conditions, whereas expanding the RoA in all directions imposes unnecessary constraints and could lead to less stability given the current conditions in exchange for more robustness somewhere less relevant.

In this thesis, we first use results from dynamical systems theory to re-express the challenging and abstract problem of expanding the RoA near the current state as a concrete max-min numerical optimization problem. This increases robustness against unknown disturbances under current conditions. To do so, we exploit properties of trajectory sensitivities, which measure how susceptible the system's behaviour is to changes in initial conditions and parameters. Trajectory sensitivities can be efficiently computed numerically. The inner optimization finds the closest point on the RoA boundary from the current state for given parameter values, and the outer optimization varies parameter values so as to maximize the distance to the RoA boundary.

We then develop the ERA algorithm, which solves this problem efficiently using a particular choice of successive approximations that avoids the need for computationally intensive second derivatives, which are required for many common bi-level optimization solvers.

Next, we provide a local convergence guarantee for the ERA algorithm for a large class of nonlinear dynamical systems, using a contraction argument to show convergence. Thus, the proposed ERA algorithm is applicable to a wide variety of practical engineered systems.

Finally, we apply the ERA algorithm to two drone simulations in which the drone must take off, fly to a target destination, and then hover at its target despite unknown wind gusts which can disrupt its flight. The first drone uses an integral backstepping controller, and the second drone uses a cascaded PID controller modelled after the default Crazyflie controller as seen in the firmware. We show that, in simulation, with the nominal parameter values, the wind gust knocks the drone out of flight, whereas after the ERA algorithm is run to select new controller parameter values, the drone is able to fly safely to its destination despite experiencing the same wind gust. We then see this increased robustness against unknown disturbances after running the ERA algorithm, replicated in a physical experiment using a Crazyflie 2.1 and a hairdryer.