A continuous-time filter with a transfer function 
is approximated by a digital filter designed using both the direct substitution and matched z-transform methods. Then the digital filters are used as the forward-path transfer functions 
in two discrete-time feedback systems both with feedback transfer functions 
The gain K is gradually increased from 0 upward. Which system will go unstable at a finite value of K? Explain your answer.
What will be an ideal response?
Since the loop transfer function of the direct substitution filter has one finite pole and no finite zeros, the root locus will eventually exit the unit circle and the system will go unstable at that point.
The loop transfer function for the matched z-transform filter has a pole inside the unit circle and a zero at zero and the root locus will simply move from the pole to the zero inside the unit circle without going outside the unit circle. So this system will not go unstable at any finite value of K.
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