Construction and parameterization of all static and dynamic H₂-optimal state feedback solutions for discrete time systems

This paper considers an H₂ optimization problem via state feedback for discrete-time systems. The class of problems dealt with here are general and have a left invertible transfer matrix function from the control input to the controlled output. The paper constructs and parameterizes all the static and dynamic H₂ optimal state feedback solutions. Moreover, all the eigenvalues of an optimal closed-loop system are characterized. All optimal closed-loop systems share a set of eigenvalues which are termed as the optimal fixed modes. Every H₂ optimal controller must assign among the closed-loop eigenvalues the set of optimal fixed modes. This set of optimal fixed modes includes a set of optimal fixed decoupling zeros which shows the minimum absolutely necessary number and locations of pole-zero cancellations present in any H₂ optimal design. It is shown that both the sets of optimal fixed modes and optimal fixed decoupling zeros do not vary depending upon whether the static or the dynamic controllers are used. Most of the results presented here are analogous to but not quite the same as those for continuous-time systems. In fact, there are some fundamental differences between the continuous and discrete-time systems reflecting mainly the inherent nature and characteristics of these systems.