# Data-Driven Criteria for Controllability of Linear-Time Invariant Systems

Controllability of a dynamical system is an important structural property, which essentially means the ability to switch between any two system’s trajectories. There are several equivalent controllability criteria for Linear Time-Invariant (LTI) systems. One of them is the famous Popov-Belevitch-Hautus (PBH) criterion for state controllability of state-space systems. It says that a state-space system, which is given by a matrix pair (A, B), is controllable if and only if no left eigenvector of A is orthogonal to the columns of B.

In this project, we study the controllability of an LTI system based on measured input/output trajectories. We prove a result similar to PBH criterion, but it is a representation-free and based on measured input/output trajectories and prior knowledge of the order of the system. Our approach is algebraic and the measured trajectories are assembled in the Hankel-structured matrices. We then develop an algorithm based on the singular value decomposition of the Hankel matrix constructed from measured input/output trajectories to determine whether or not the underlying system is controllable. Our result also shows that if we do not know the order of the system, but an upper bound to it, we can still verify the uncontrollability of the system.

So far, we have assumed that the measured trajectories are exact (noise-free). However, they are usually corrupted by some noise. The next step involves developing controllability criteria from measured noisy trajectories.

### Reference

- V. K. Mishra, I. Markovsky, and B. Grossmann. Data-driven tests for controllability. IEEE Control Systems Letters, 5:517-522, 2020.