Estimation and control with relative measurements: algorithms and scaling laws

In this dissertation we examine a class of estimation and control problems involving interconnected systems. These problems share the common attribute that, between two component subsystems, noisy measurements of the difference of their states alone is available. The estimation problem is relevant to sensor and actuator networks, and the control problem is relevant to coordination in multi-agent systems. Both classes of problems are defined over a graph that is used to describe the interconnections. In the first part of this dissertation, the estimation problem is examined. The variables correspond to the nodes of a graph, and the measurements of the noisy difference between pairs of variables correspond to its edges. The task is to compute estimates of the node variables with respect to a reference node. We begin by designing distributed algorithms to compute the optimal estimate, which refers to the best linear unbiased estimator (BLUE). We then examine the effect of the graph structure on the minimum achievable estimation error. Specifically, we examine how the optimal estimation error of a node variable grows with its distance from the reference node. A classification of graphs—sparse and dense in 1D, 2D, and 3D—is obtained, which determines the error growth rate: linear, logarithmic, or bounded. In the second part of this dissertation, the control of formations over arbitrary graphs is described. Specifically, we examine how the structure of the interconnection graph affects the stability and sensitivity to measurement noise of the formation. The vehicular platoon problem is investigated in detail—especially the decentralized bidirectional control architecture in which each vehicle uses front and back spacing measurements to compute its control signal. Fundamental limitations in disturbance amplification are established for the symmetric bidirectional architecture. Then we show that arbitrary small asymmetry in the front and back controller gains can lead to an order of magnitude improvement in stability margin. The underlying theme of our investigations is that of performance degradation—and possible amelioration—in interconnected systems as the number of constituent sub-systems increases.