Witsenhausen´s Counter Example Holds in the Presence of Side Information

In this paper, we consider a two-step decentralized control problem with two agents which are not co-located. The first agent is located at the source of a disturbance, it can measure the disturbance with high accuracy and it can perform local actuation, while communicating with the second agent via a Gaussian channel with a prescribed signal to noise ratio. What remains of the disturbance, after the control action of the first agent, will hit the second agent after a fixed delay. The second agent has noisy measurements as well as the information conveyed through the communication channel to decide on the best strategy to reduce the effects of the remaining disturbance. The control design paradigm is to minimize a quadratic cost combining the control power of the first agent and the final effect of the disturbance at the second agent. Such a framework can be viewed as an extension of the well known Witsenhausen´s counter-example which did not include a communication channel. In the absence of a channel, Hans Witsenhausen has shown, by means of an example, that nonlinear strategies outperform the best linear strategy. This result is not obvious because the framework is linear and Gaussian, and the cost is quadratic. In this paper, we give an example illustrating that Witsenhausen´s conclusion holds for our framework where a communication channel is present. We provide numerical results showing that, for a large range of the signal to noise ratio at the channel, nonlinear strategies strongly out-perform the best linear strategies