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 has 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 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. 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.