Biblio

We consider linear time-invariant networks with unknown interaction topology where only a subset of the nodes, termed manifest, can be directly controlled and observed. The remaining nodes are termed latent and their number is also unknown. Our goal is to identify the transfer function of the manifest subnetwork and determine whether interactions between manifest nodes are direct or mediated by latent nodes. We show that, if there are no inputs to the latent nodes, then the manifest transfer function can be approximated arbitrarily well in the $H_ınfty}$-norm sense by the transfer function of an auto-regressive model. Motivated by this result, we present a least-squares estimation method to construct the auto-regressive model from measured data. We establish that the least-squares matrix estimate converges in probability to the matrix sequence defining the desired auto-regressive model as the length of data and the model order grow. We also show that the least-squares auto-regressive method guarantees an arbitrarily small $H_ınfty$-norm error in the approximation of the manifest transfer function, exponentially decaying once the model order exceeds a certain threshold. Finally, we show that when the latent subnetwork is acyclic, the proposed method achieves perfect identification of the manifest transfer function above a specific model order as the length of the data increases. Various examples 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.