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Consider the system x(k+1)=\left(\begin{array}{ll}

0 & 2 \\

3 & 1

\end{array}\right) x(k)+\left(\begin{array}{l}

1 \\

1

\end{array}\right) u(k) y(k)=\left(\begin{array}{ll}

1 & 0

\end{array}\right) x(k) a) Is the system stable? Is the system controllable? Is the system observable? Justify your answers. b) Use z-transform to obtain the transfer function of the system. Write down the input-output difference equation. c) Assume the system is controlled by a proportional controller u(k)=K\left(u_{c}(k)-y(k)\right) Derive the transfer function from the command signal uc(k) to theQutput y(k). d) Apply Jury's stability criterion to determine the range of controller gain, K, such that the closed-loop system is stable. e) Determine the steady-state error, uc – y, when uc is a unit step.

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