# Biblio

To appear

This paper develops methods to efficiently compute the set of disturbances on a power network that do not tip the frequency of each bus and the power flow in each transmission line beyond their respective bounds. For a linearized AC power network model, we propose a sampling method to provide superset and subset approximations with a desired accuracy of the set of feasible disturbances. We also introduce an error metric to measure the approximation gap and design an algorithm that is able to reduce its value without impacting the complexity of the resulting set approximations. Simulations on the IEEE 118-bus power network illustrate our results.

In this paper, we present a fundamental limitation of disturbance attenuation in discrete-time single-input single-output (SISO) feedback systems when the controller has delayed side information about the external disturbance. Specifically, we assume that the delayed information about the disturbance is transmitted to the controller across a finite Shannon-capacity communication channel. Our main result is a lower bound on the log sensitivity integral in terms of open-loop unstable poles of the plant and the characteristics of the channel, similar to the classical Bode integral formula. A comparison with prior work that considers the effect of preview side information of the disturbance at the controller indicates that delayed side information and preview side information play different roles in disturbance attenuation. In particular, we show that for open-loop stable systems, delayed side information cannot reduce the log integral of the sensitivity function whereas it can for open-loop unstable systems, even when the disturbance is a white stochastic process.

Controllability metrics based on the controllability Gramian have been widely used in linear control theory, and have recently seen renewed interests in the study of complex networks of dynamical systems. For example, the minimum eigenvalue and the trace of the Gramian are related to the worst-case and average minimum input energy, respectively, to steer the state from the origin to a target state. This paper explores similar questions that remain unanswered for bilinear control systems. In the context of complex networks, bilinear systems characterize scenarios where an actuator not only can affect the state of a node, but also can affect the strength of the interconnections among some neighboring nodes. Under the assumption that the infinity norm of the input is bounded by some function of the network dynamic matrices, we derive a lower bound on the minimum input energy to steer the state of a bilinear network from the origin to any reachable target state based on the generalized reachability Gramian of bilinear systems. We also provide a lower bound on the average minimum input energy over all target states on the unit hypersphere in the state space. Based on the reachability metrics proposed, we propose an actuator selection method that provides guaranteed minimum average input energy. Finally, we show that the optimal actuator selection can be found by maximizing a supermodular function under cardinality constraints.