Degree of disturbance rejection capability for linear marginally stable systems

Controllability is a structural property that carries information useful for control system design such as actuator placement. There are several measures to determine the locations and types of actuators, and one of them is degree of disturbance rejection capability (DODR) which is a quantitative measure representing how controllable the system is under the presence of persistent disturbance. However, it is hard to obtain the steady-state solution of DODR if the system is marginally stable because of several problems such as singularity and intensiveness of computation to obtain the Gramians. Thus, this problem to apply the measure to marginally system still remains. In this paper, therefore, the way to apply DODR to marginally systems with simple calculation is proposed. The definition of the measure is same with existing measure of DODR. We used a novel concept of time averaged gradient of Gramians to calculate the steady-state solution of DODR.