Discrete-time formulas of one parameter stabilizing controllers and mixed sensitivity problem, for square systems

A comparison between discrete-time techniques (Tustin´s method, forward and backward Euler´s rules) for explicit formulas of the parametrization of one parameter stabilizing controllers for square systems and mixed sensitivity problem are given. Multi Input Multi Output (MIMO), strictly proper, lumped and Linear Time Invariant (LTI) systems with a stabilizable and detectable realization are considered. It is assumed that the state dimension is even, the input dimension is half the state dimension, and the plant is strongly stabilizable and detectable. It is assumed that the sample rate is too short, such that all the signals are close enough to the continuous-time signals. The separation principle is applied to design a dynamic output control in a discrete-time controller-observer feedback configuration. The results for the discrete-time observer are gotten by duality. For both discrete-time controller and observer, right and left coprime factorizations of the transfer function in terms of the state space realization are proposed, right and left Diophantine equations are solved, and the controller and observer belong to set of all one parameter discrete-time stabilizing controllers. The free parameters of the stabilizing controllers are fixed solving a mixed sensitivity control problem. The results are shown through a simulation of a two cart system.