Disturbance localization in linear systems with simultaneous decoupling, pole assignment, or stabilization

For a class of constant linear dynamic systems with unmeasurable disturbances, geometric necessary and sufficient conditions are derived for the existence of a state-feedback controller that localizes the disturbances and simultaneously assigns the closed-loop poles or decouples the closed-loop system. These conditions are simple, intuitively appealing, and closely related to the known necessary and sufficient conditions for the separate problems to be solvable alone. Furthermore, it is shown that for these combined problems and for simultaneous disturbance localization and stabilization problems no additional flexibility is afforded by using dynamic compensation instead of static compensation except for the combined problem of disturbance localization and decoupling where the additional flexibility is derived from that for the decoupling problem alone. Under a standard rank assumption, it is established that the combined problem of simultaneously localizing the disturbances, assigning the poles and decoupling the system has no solution. Separate necessary and sufficient conditions are given for the existence of a solution to the combined problem of disturbance localization, stabilization, and decoupling.