Robust stability and control effort minimization by disturbance rejection

A goal of the paper is to proof that a control strategy with unknown disturbance rejection reduces the control effort to a minimum. A similar statement appeared in the literature, but without any formal proof. The disturbance to be rejected is completely unknown except for a sectorial bound. The control unit is endowed with a state observer which includes a disturbance dynamics whose state tracks the unknown disturbance to be rejected. Observers of this kind are commonly referred to as extended state observers. The novel contributions of the paper are the following. First, we derive a robust stability condition for the proposed control scheme, holding for all the nonlinearities that are bounded by an estimated maximum slope. Second, we compute bounds on the closed-loop bandwidth of the extended state predictor and of the state feedback. Third we propose a novel approach for designing observer and state feedback gains, which guarantee robust closed-loop stability. Fourth, we show that the designed control system yields, with a minimum control effort, the same control performance as a robust state feedback control, which on the contrary may require a larger command activity. A simulated multivariate case study is presented.