  How to make a dynamic system unidentifiable is an important but still open
issue. It not only requires that the parameters of the systems but also the
equivalent systems cannot be identified by any identification approaches. Thus,
it is a much more challenging problem than the existing analysis of parameter
identifiability. In this paper, we investigate the problem of dynamic
unidentifiability and design the controller to make the system dynamics
unidentifiable. Specifically, we first define dynamic unidentifiability by
taking both system parameters and equivalent systems into consideration. Then,
we obtain the necessary and sufficient condition for unidentifiability based on
the Fisher Information Matrix. This condition is derived by analysis of the
relationship between the unidentifiable parameters and the Hessian matrix of
the system function. Next, we propose a controller design scheme for ensuring
dynamic unidentifiability under linear system models. We prove that for
controllable and observable linear time-invariant systems, the requirement of
unidentifiability is equivalent to the requirement of a low-rank controller.
Then, the low-rank controller design problem is solved by transforming it into
a model order reduction problem. We demonstrate the effectiveness of our method
by simulation.
