A Null-Space-based Technique for the Estimation of Linear-Time Invariant Structured State-Space Representations*

Estimating the order as well as the matrices of a linear state-space model is now an easy problem to solve. However, it is well-known that the state-space matrices are unique modulo a non-singular similarity transformation matrix. This could have serious consequences if the system being identified is a real physical system. Indeed, if the true model contains physical parameters, then the identified system could no longer have the physical parameters in a form that can be extracted easily. The question addressed in this paper then is, how to recover the physical parameters once the system has been identified in a fully-parameterized form. The novelty of our approach is on transforming the bilinear equations arising from the similarity transformation equations as a null-space problem. We show that the null-space of a certain matrix contains the physical parameters. Extracting the physical parameters then requires the solution of a non-convex optimization problem in a reduced dimensional space. By assuming that the physical state-space form is identifiable and the initial fully-parameterized model is consistent, the solution of this optimization problem is unique. The proposed algorithm is presented, along with an example of a physical system.