Conventional disturbance observers for discrete-time linear stochastic systems assume that the system states are fully estimable and the disturbance estimate is dependent on the estimated system states, hereafter termed full-order disturbance observers (FODOs). This paper investigates the design of reduced-order disturbance observers (RODOs) when the system state variables are not fully estimable. An existence condition of RODOs is established, which is shown to be more easily satisfied than that of conventional FODOs and consequently it has substantially extended the scope of applications of disturbance observer theory. Then a set of recursive formulae for the RODO is developed for online applications. Finally, it is further shown that the conventional FODOs are a special case of the proposed RODO in the sense that the former reduces to the RODO when the states become fully estimable in the presence of disturbances. Examples are given to demonstrate the effectiveness and advantages of the proposed approach.
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