  In this paper, we propose a learning approach to analyze dynamic systems with
asymmetric information structure. Instead of adopting a game theoretic setting,
we investigate an online quadratic optimization problem driven by system noises
with unknown statistics. Due to information asymmetry, it is infeasible to use
classic Kalman filter nor optimal control strategies for such systems. It is
necessary and beneficial to develop a robust approach that learns the
probability statistics as time goes forward. Motivated by online convex
optimization (OCO) theory, we introduce the notion of regret, which is defined
as the cumulative performance loss difference between the optimal offline known
statistics cost and the optimal online unknown statistics cost. By utilizing
dynamic programming and linear minimum mean square biased estimate (LMMSUE), we
propose a new type of online state feedback control policies and characterize
the behavior of regret in finite time regime. The regret is shown to be
sub-linear and bounded by O(ln T). Moreover, we address an online optimization
problem with output feedback control policies.
