Demystifying the Witsenhausen Counterexample

Q. In my graduate-level controls course, the instructor mentioned the “Witsenhausen counterexample,” which seems to have something to do with decentralized control and communication. But I must confess that I did not understand what was going on or why it was important. Could someone at IEEE Control Systems Magazine provide a clear and simple explanation? Thanks. Pulkit and Anant: Witsenhausen’s counterexample to a conjecture in decentralized control theory is a deceptively simple decentralized stochastic control problem that has remained unsolved for the last four decades [1] (see “Hans S. Witsenhausen”). This problem, shown in Figure 1, has a scalar state that evolves in discrete time according to linear dynamics. The state is constrained by two controllers, each of which acts only once. The “weak” controller Cw operates first after seeing the initial plant state X0. The resulting control signal Uw modifies the state X0 by means of X15X01Uw. The “blurry” controller Cb acts next after having observed a noisy version X11Z of the state X1. The second control signal Ub modifies the state X1 by means of X25X12Ub. Both the initial state X0 and the observation noise Z are independent zero-mean Gaussian random variables with variances s0 and one, respectively. The goal is to design control laws that minimize the expected quadratic cost E 3k UW 1X2 4, where k is a weight that trades off control costs with terminal state costs. For this problem, the optimal strategy for the second controller Cb is to optimize set Ub to be the minimum mean squared error estimate— the conditional expectation—of X1 given its noisy observation X11Z. This choice minimizes the second term E 3X2 4 in the cost. We thus need to find only the first controller Cw since that is what determines the distribution of X1. If we were to constrain ourselves to a linear controller Cw, then X1 would be Gaussian. In that case, the optimal choice for Cb would also be linear. But surprisingly, in his 1968 paper [1], Witsenhausen provides a nonlinear strategy for the problem that outperforms all linear strategies. In Witsenhausen’s strategy, the weak controller quantizes the initial state X0 by forcing X1 to be the closer of either 1s0 and 2s0. This “discreteness” of X1 greatly reduces the second stage cost E 3X2 4. Furthermore, generalizing from two quantization points to an infinite number of regularly spaced quantization points, the resulting nonlinear control strategies outperform optimal linear strategies by arbitrarily large factors [2]. These nonlinear strategies do significantly better if two conditions are met, a) the observation noise variance is small compared to the initial system state variance, and b) “an ounce of prevention is worth a pound of cure,” that is, the weight parameter k on the weak controller’s cost is much smaller than one. Such small weights on the weak controller’s input Uw allow for investing in a moderately large input Uw to greatly reduce the estimation error X2. At first glance, all this might seem to be an isolated curiosity at best. Why is it important? It turns out that the Witsenhausen counterexample is notable because of its links with computational Demystifying the Witsenhausen Counterexample