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.

Controlling mechanical systems with position and velocity cascade loops is one of the most effective methods to operate this type of systems. However, when using low-rate sampling electronics, the implementation is not trivial and the resulting performance can be poor. This paper proposes effective tuning rules that only require establishing the bandwidth of the inner velocity loop and an estimation of the inertia of the mechanism. Since discrete-time mechatronic systems can also exhibit unstable behavior, several stability conditions are also derived. By using the proposed methodology, a P-PI control algorithm is developed for a desktop haptic device, obtaining good experimental performance with low sampling-rate electronics.

Affine discrete-time control periodic systems are considered. The problem of global asymptotic stabilization of the zero equilibrium of the closed-loop system by state feedback is studied. It is assumed that the free dynamic system has the Lyapunov stable zero equilibrium. The method for constructing a damping control is extended from time-invariant systems to time varying periodic affine discrete-time systems. By using this approach, sufficient conditions for uniform global asymptotic stabilization for those systems are obtained. Examples of using the obtained results are presented.