The positive (minimal) realization problem for a class of singular discrete-time linear single-input, single-output systems with delays in state and delays in control is addressed. Solvability conditions for the positive (minimal) realization problem are established. It is shown that there exists a positive (minimal) realization of an improper transfer function T(z) = n(z) / d(z) if the coefficients of polynomial n(z) are non-negative and of the polynomial d(z) are non-positive except the leading one, which should be positive. A procedure for computation of the positive (minimal) realization of the transfer function is proposed and illustrated by an example.
In recent years, with the rapid development of digital components, digital electronic computers, especially microprocessors, digital controllers have replaced analog controllers on many occasions. The application of digital controller makes the performance analysis of impulsive system more and more important. This paper considers global exponential stability (GES) of impulsive delayed nonlinear hybrid differential systems (IDNHDS).Through the application of the Lyapunov method and the Razumikhin technique, a series of uncomplicated and useful guiding principles have been obtained. The results of a numerical simulation are presented to demonstrate that the method is correct and effective.