Imposing stability in subspace identification by regularization

In subspace methods for linear system identification, the system matrices are usually estimated by least squares, based on estimated Kalman filter state sequences and the observed inputs and outputs. For an infinite number of data points and a correct choice of the system order, this least squares estimate of the system matrices is unbiased. However, when using subspace identification on a finite number of data points, the estimated model can become unstable, for a given deterministic system which is known to be stable. In this paper, stability of the estimated model is imposed by adding a regularization term to the least squares cost function. The regularization term used here is the trace of a matrix which involves the dynamical system matrix and a positive (semi-) definite weighting matrix. The amount of regularization needed can be determined by solving a generalized eigenvalue problem. It is shown that the so-called data augmentation method proposed by Chui and Maciejowski (1996) corresponds to adding regularization terms with specific choices for the weighting matrix. The choice of the identity matrix for the weighting matrix is motivated by simulation results