Direct method of constructing H₂-suboptimal controllers: Discrete-time systems

For discrete-time systems, an H₂-suboptimal control problem is defined and analyzed. Then, an algorithm called H₂-suboptimal state feedback gain sequence (Algorithm A1) is developed. Rather than utilizing a perturbation method, which is numerically stiff and computationally prohibitive, Algorithm A1 utilizes a direct eigenvalue assignment method to come up with a sequence of H₂-suboptimal state feedback gains. Also, although the sequence of H₂-suboptimal state feedback gains constructed by Algorithm A1 depends on a parameter ε, the construction procedure itself does not require explicitly the value of the parameter ε. Next, attention is focused on constructing a sequence of H₂-suboptimal estimator-based measurement feedback controllers. Three different estimator structures (prediction, current, and reduced-order estimators) are considered. For a given H₂-suboptimal state feedback gain, a sequence of estimator gains for any of the three estimators considered can be constructed by merely dualizing Algorithm A1. The direct method of constructing H ₂-suboptimal controllers developed here has a number of advantages over the perturbation method, e.g., it has the ability to design all three types of estimator-based controllers while still maintaining throughout the design the computational simplicity of it. This paper is the discrete-time version of Ref. 1. There are some conceptual similarities as well as fundamental differences between the H₂-suboptimal control problems for continuous-time and discrete-time systems. The fundamental differences arise mainly from the fact that, in contrast to continuous-time systems, for discrete-time systems the infimum of the H₂-norm over the class of strictly proper controllers is in general different from the infimum of the H₂-norm over the class of proper controllers.