(Solved): Problem 1: A closed-loop sampled-data system is shown below. The sample rate is 10Hz. The followi ...

Problem 1: A closed-loop sampled-data system is shown below. The sample rate is $10\mathrm{Hz}$. The following equations represent the digital controller and the continuous time plant respectively: $u_1(k)=u_1(k?1)+K(e(k?1)+e(k))\ddot{y}(t)+2\stackrel{?}{y}(t)=?y(t)+4u_2(t)$ 1. What are $\mathrm{C}(\mathrm{z})$ and ${\mathrm{G}}_{\mathrm{p}}(\mathrm{s})$ ? 2. If the $\mathrm{D}/\mathrm{A}$ converter is $\mathrm{ZOH}$. What is the closed-loop transfer function $M(z)=Y(z)/R(z)$ of the equivalent discrete time system? 3. Suppose that the open loop transfer function is: Find out the range of $\mathbf{K}$ for which the system is stable. 4. For the value of $K$ the system is stable, calculate the steady state errors for unit step. unit ramp and unit parabolic inputs. Problem 2: a) Determine the unit step response of the system shown in Figure, with $K=1$ and $T=1\mathrm{Sec}$. b) Determine the range of $K$ for stability fot $T=1\mathrm{Sec},T=0.1\mathrm{Sec}$., and no sampling. c) For the case that $K=10$, determine the range of the sample period $T$ for which the system is stable.