Time complexity of closed-loop identification

While it is possible to translate a closed-loop identification problem into an open-loop one by first applying a stabilizing controller and then identifying the closed-loop system, this approach leads to model structures, parameters and probing signals that depend on controllers and adaptation algorithms. To understand fundamental issues of time complexity, it is essential to seek closed-loop identification directly on the original plant. This paper introduces one approach along this direction. We seek identification algorithms and probing inputs that can provide satisfactory identification capability in terms of time complexity and accuracy, and that are independent of the controller structures and parameters as long as they do not destabilize the system. Such findings allow controllers to be designed for their intended goals without any detrimental impact on identification. They also reveal the intrinsic limitations of closed-loop identification. The problem is characterized by the following aspects. (1) We require a persistent identification capability, i.e. the identification errors must be small at all times, rather than merely asymptotically. (2) The ideas of deterministic unmodeled dynamics and stochastic disturbances are employed to overcome the shortcoming of worst-case identification on time complexity and consistency. A stochastic formulation of disturbances can restore consistency and polynomial time complexity in open-loop identification problems. This mixture formulation leads to consistency and a much-reduced time complexity in closed-loop identification.