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I have to tune a state space model for certain parameters in Matlab. The transfer function has to be $$T(s)=\frac{k_q(1+0.162s)}{s^2+8.23s+67.7}$$ Using the place command, I have been able to place the poles at $$p_{1,2}=-4.115\pm 7.12i$$ with a full state feedback gain of $-280$ and $-28$. However, I now have to place the zero using a prefilter such that the zero is at $s=-0.6653$ and the system tracks an input signal properly. I fail to see how a prefilter would affect the enumerator such that the location of the zero can be altered. How can I place the zero?

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Remember that zeros are not global system properties but depend on the specifics of the selected/defined inputs and outputs. One possible way of modifying the zeros in your model is to modify the system inputs or outputs by adding filters, transformations, etc.

In this specific case, try adding a prefilter with the desired zero and optionally additional poles to cancel any existing zeros. Probably not the best approach, and not preferable in a real application, but it should work in this theoretical example.

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