A Rigorous Proof for the Tending-To-Im Theorem on Parameter Sum of the Sampled Model of a Continuous-Time Multivariable System

This paper gives the concrete form of the parameter matrix sum of the sampled model of a n-th order r-input-m-output continuous-time linear dynamic system with sampling period h. Furthermore, it is proved that the parameter matrix sum of a multivariable sampled ¿data system tends to identity matrix Im, as the sampling period h approaches zero. The above tending-to-Im theorem is also valid for multivariable system with different input time-delays. The results obtained here are helpful to the robust self-tuning control and on-line estimation of all parameters of a multivariable plant in closed loop.