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(Solved):   22. A digital control system is described by its difference equation as stated in the foll ...



22. A digital control system is described by its difference equation as stated in the following equation,
\[
y(k)+5 y(k-1)+6

 

22. A digital control system is described by its difference equation as stated in the following equation, \[ y(k)+5 y(k-1)+6 y(k)=x(k-1)+4 x(k-2) \] It is assumed that \( y(k)=0 \) and \( x(k)=0 \) for \( k<0 \) with input signal \( x(k)=0.5 \delta(k-1) \) a. Determine the transfer function \( G(z)=\frac{Y(z)}{X(z)} \) b. Using inverse z-transform determine the output signal \( y(k) \quad 7 \) marks Determine also \( \mathrm{y}(\mathrm{k}) \) for \( \mathrm{k}=0,1 \) and 2 . c. Using initial value theorem, determine \( y(0) \) and compare with your 3 marks answer in question \( b \).


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Part a: We are given the difference equation y[k]+5y[k?1]+6y[k]=x[k?1]+4x[k?2] This equation can be simplified to: 7y[k]+5y[k?1]=x[k?1]+4x[k?2] Next,
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