In the present article, we first introduce the definition of the discrete double-sided quaternionic Fourier transform (DQFT) and obtain its inverse. We give examples how to compute the DQFT. We derive a discrete version of the duality property of the DQFT. We finally present an application of the DQFT to study the two-dimensional discrete linear time-varying systems.

The problem of output tracking for a class of nonlinear systems whose zero dynamics are not necessarily stable is addressed in this paper. To solve the problem, we transform the systems into a normal form which is minimum phase with respect to a virtual output, which is a linear combination of state variables. By applying the modified steepest descent control, the system output track to the desired output.

In this paper, we develop a method to design the input control to track the output of a nonminimum-phase nonlinear systems asymptotically. The design of the control inputs is based on an exact linearization. To perform the exact linearization, the other output should be selected such that its relative degree is equal to the dimension of the system. Furthermore, the desired output of the output which has been selected will be set based on the desired output of the original system. In applying the input control which is obtained via output-input linearization sometimes led to singularity. To overcome this singularity problem, polynomial control is developed around the point of singularity.