Performance lower bounds in stochastic robust and adaptive control

The fundamental issues of capability of robust and adaptive control in dealing with uncertainty are investigated in stochastic systems. It is revealed that to capture the intrinsic limitations of adaptive control, it is necessary to use sup types of transient and persistent performance, rather than lim sup types which reflect only asymptotic behavior of a system. For clarity and technical tractability, a simple first-order linear time-varying system is employed as a vehicle to explore performance lower bounds of robust and adaptive control. Optimal performances of nominal, robust and adaptive control are explicitly derived and their implications are discussed in an information framework. An adaptive strategy is scrutinized for its achievable performance bounds. The results indicate that intimate interaction and inherent conflict between identification and control result in a certain performance lower bound which does not approach the nominal performance even when the system varies very slowly. Explicit lower bounds are obtained when disturbances are either normally or uniformly distributed.