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lec21 - 21 21.1 LOOP TRANSFER RECOVERY Introduction The...

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21 LOOP TRANSFER RECOVERY 21.1 Introduction The Linear Quadratic Regulator(LQR) and Kalman Filter (KF) provide practical solutions to the full-state feedback and state estimation problems, respectively. If the sensor noise and disturbance properties of the plant are indeed well-known, then an LQG design approach, that is, combining the LQR and KF into an output feedback compensator, may yield good results. The LQR tuning matrices Q and R would be picked heuristically to give a reasonable closed-loop response. There are two reasons to avoid this kind of direct LQG design procedure, however. First, although the LQR and KF each possess good robustness properties, there do exist plants for which there is no robustness guarantee for an LQG compensator. Even if one could steer clear of such pathological cases, a second problem is that this design technique has no clear equivalent in frequency space. It cannot be directly mapped to the intuitive ideas of loopshaping and the Nyquist plot, which are at the root of feedback control. We now reconsider just the feedback loop of the Kalman ±lter. The KF has open-loop transfer function L ( s ) = ( s ) H , where δ ( s ) = ( sI A ) 1 . This follows from the estimator evolution equation (Continued on next page)

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106 21 LOOP TRANSFER RECOVERY ^ φ (s) x C y H ˆ ˙ x + Bu + H ( y C ˆ x = A ˆ x ) and the fgure. Note that we have not included the Factor Bu as part oF the fgure, since it does not aﬀect the error dynamics oF the flter. As noted previously, the loop has good robustness properties, specifcally to perturbations at the output ˆ y , and Further is amenable to output tracking. In short, the loop is an ideal candidate For a loopshaping design. Supposing that we have an estimator gain H which creates an attractive loop Function L ( s ), we would like to fnd the compensator C ( s ) that establishes P ( s ) C ( s ) ( s ) H , or (259) ( s )
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lec21 - 21 21.1 LOOP TRANSFER RECOVERY Introduction The...

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