Asymptotic Characteristics of Toeplitz Matrix in SISO Model Predictive Control

The singular value decomposition (SVD) of the Toeplitz matrix in the quadratic performance index of Model Predictive Control (MPC) is studied. It was shown in Rojas et al. (2003, 2004) that for sufficiently long prediction horizons, the eigenvalues of the Hessian matrix converge to the energy density spectrum of the associated system seen by the performance index. In this paper, we extend that work and show that the left and right singular vectors of the Toeplitz matrix provide the phase information of the associated system for sufficiently long prediction and control horizons. A SISO system is used to illustrate the method.