Unidentifiability of System Dynamics: Conditions and Controller Design

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 unidentifiability. In this paper, we focus on the condition for dynamic unidentifiability for an arbitrarily parameterized system and design the controller to make the system dynamics unidentifiable. Specifically, we first define dynamic unidentifiability by taking equivalent systems into consideration. Then, we obtain the necessary and sufficient condition for unidentifiability based on the Fisher Information Matrix (FIM). 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 for linear systems. We prove that for controllable and observable linear time-invariant systems, dynamic unidentifiability is equivalent to the requirement of a low-rank controller. Finally, the controller design problem is solved by transforming it into a model order reduction problem. We demonstrate the effectiveness of our method by simulation.