Disturbance Vulnerability Analysis and Reduction for Nonlinear Systems using Modes of Instability

Abstract

Engineered systems naturally experience large disturbances which have the potential to disrupt desired operation because the system may fail to recover to a desired exponentially stable equilibrium point (SEP). It is valuable to determine the mechanism of instability when the system is subject to a particular finite-time disturbance, because this information can be used to improve vulnerability detection, and to design controllers capable of mitigating these disturbances to enhance system resilience and ensure reliable performance, thereby reducing vulnerability. For a large class of nonlinear systems there exists a particular unstable equilibrium point (UEP) on the region of attraction (RoA) boundary of the desired SEP such that the unstable eigenvector of the Jacobian at this UEP represents the mode of instability for the disturbance. Unfortunately, it is challenging to find this mode of instability, especially in high dimensional systems, because it is often computationally intractable to compute this particular UEP for a given disturbance. Consider a particular finite time disturbance applied to a nonlinear system which possesses a SEP representing desired behavior. The system is able to recover from the disturbance if its post disturbance initial condition (IC) lies inside the RoA of the desired SEP. In cases where the system fails to recover, the nonlinear mode of instability for the disturbance represents the subset of system dynamics most responsible for this failure to recover. This thesis develops a novel algorithm for numerically computing the mode of instability for parameter-dependent nonlinear systems without prior knowledge of the particular UEP, resulting in a computationally efficient method. The key idea is to first consider the setting where the system recovers, and to average the Jacobian along the system trajectory from the post-disturbance state up until the Jacobian becomes stable. As the system approaches inability to recover, the averaged Jacobians converge to the Jacobian at the particular UEP, and can be used to extract the unstable eigenvector representing the mode of instability. Convergence guarantees are provided for computing the mode of instability, both for the theoretical setting in continuous time, and for the proposed algorithm which relies on numerical integration. Numerical examples illustrate the successful application of the method to identify the mechanism of instability in power systems subject to temporary short circuits. Then a novel approach to control design for reducing disturbance vulnerability of nonlinear systems using knowledge of the mode of instability is developed. The main idea is to tune controller parameter values so as to drive the post-disturbance IC further inwards away from the RoA boundary by driving it in the direction opposite to the mode of instability. To achieve this, the problem is formulated as a nonconvex optimization problem, and an efficient algorithm is developed to solve it. Local convergence guarantees are provided for this method. Numerical examples illustrate the successful application of the method to reduce the vulnerability of power systems subject to temporary short circuits.