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Solvability conditions for disturbance decoupling problems with static measurement feedback. (English) Zbl 0888.93028
The disturbance decoupling problem is concerned with finding a dynamic or static compensator so that the resulting closed-loop transfer matrix from the disturbance to the controlled output is zero for all frequencies. This paper considers the problem for the case of static or constant output measurement feedback, both with and without internal stability, for linear time-invariant systems. Specifically, for the system: \[ \begin{aligned} \dot x & =Ax+ Bu+Ew \\ y & =C_1x \\ z & =C_2x +D_2u \end{aligned} \] the paper considers the problem of finding \(u=Ky\) so that the transfer function between \(w\) and \(z\) \((H_{zw})\) is identically zero and so that \(A+ BKC_1\) is Hurwitz. First, a solvability condition is given to check the existence of a \(K\) to make \(H_{zw}\) zero. Second, a constructive algorithm is given that parametrizes all \(K\) that solve the problem. Finally, when such matrices \(K\) exist, it is shown how to use the parametrization to solve the zeroing problem with stability.

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