System identification based on the cross Gramian

In order to make mechanical structures more reliable, research is being carried out into methods of detecting and localizing damage to structures, known as Structural Health Monitoring (SHM). One method for continuously monitoring a structure is called State Projection Estimation Error (SP2E) and was developed at the University of Applied Sciences Leipzig. This method is based on state space systems, which are generated by output-only system identification. It is state-of-the-art to create a stabilization diagram to choose the best model order. Since the monitoring process is continuous, a state space model is created every 10 minutes, and therefore it is not possible to look at a stabilization diagram. Thus, the system identification process has to be automated. A novel approach developed in the SPP 100+ research program takes place via the theory of the cross Gramian. The cross Gramian is defined by the Hankel operator, which maps the past input to the future output. This Gramian indicates the interaction between the states and the input/output of a system and can be seen as the energy capacity of a system. By comparing the power of the measurement with the identified energy capacity, the model order selection can be automated. There are limits to the existence of the cross Gramian for multiple-input, multiple-output (MIMO) systems. The system has to be square, stable and symmetric. The covariance driven system identification never provides a symmetric system. A way to create a symmetric system out of the asymmetric measurement data is presented by the use of a symmetrizer, such that the cross Gramian can always be computed. With this symmetrizer a method to estimate the system matrix A by a stable Lyapunov equation, which is classically computed by a numerically unstable pseudo inverse, is presented.