Parameter Identification of Discrete-time Linear Time-invariant Systems Using State and Input Data

Parameter identification involves two fundamental problems, the problem of identifiability and the problem of designing identification algorithms without using knowledge about system parameters. It is well-known that certain initial conditions can destroy identifiability. To avoid problematic initial conditions, we propose the concept of identifiability regardless of the initial condition for deterministic discrete-time linear time-invariant systems. Analysis shows that such an identifiability notion is equivalent to controllability. Identification of controllable systems with state and input measurements is achieved by proposing an algebraic approach. We observe from system dynamics that system parameters are the solution to a set of linear equations. The solution is unique if a data matrix constructed by snapshots of system state and input is invertible. Under a one-step delayed linear state and input feedback law, the data matrix is invertible if and only if the initial input does not belong to a finite set. A one-step delayed input feedback law incurs input sequences that violate a well-known persistent excitation condition for parameter identification. The motivation to initiate an input that escapes the finite set facilitates the design of an iterative algorithm to identify system parameters in finite time. We have shown that parameter identification can be still achieved without satisfying a persistent excitation condition regarding input sequence design.