Control over Networks of Unreliable Links: Controller Location and Performance Bounds

We address the optimality of control laws for networked control systems where sensors, actuators and controllers are separated by a network of unreliable links. We begin by considering the problem of controller placement and model the networked environment as an array of potential controller locations separated by identical lossy communication links (erasure channels). Our first result establishes that the controller should be placed on the 'shortest' path between the sensor and actuator nodes. The next result deals with where on a shortest path to place the controller logic. This problem is actually quite difficult since it involves consideration and comparisons of systems with non-classical information patterns which are known to be intractable, following the work of Witsenhausen. To nevertheless make progress, we begin by introducing a convenient 'long-packet' formulation under which each packet can be infinitely long and contain all past information from the sending node. Clearly the optimal cost under this optimistic assumption is a lower bound on any realizable cost. Although not implementable, its purpose is to lower bound cost in order to provide a baseline for comparison of other realizable schemes. Using dynamic programming we proceed to derive such a bound. The key insight is that the long-packets assumption permits a separation type result which features an interesting simplification that the cost to go depends only on a further simplification of the usual hyper-state. Finally, using a currently realizable approach we are able to upper bound the optimal cost and thus bracket the optimal achievable performance. We conduct simulations on several systems from the literature and some randomly constructed systems, as well as simulating currently proposed approaches. It turns out that the bounds are fairly close in many cases and we can deduce that the problem may actually effectively be solved for practical purposes in many systems.