Q2 (10 marks) In Gradescope, upload no more than 3 A4 pages with clearly labelled responses to the following three unrelated questions: (a) The pole/zero plots of six transfer functions are shown in the top part of Figure 2. Match each of these to a corresponding step response from the bottom part of Figure 2. To this end, reproduce the following table and insert labels from the set { T1, T2, T3, T4, T5, T6 } into the second row. You are not required to justify your answers. [3 marks] pole/zero plot PZ1 PZ2 PZ3 PZ4 PZ5 PZ6 step response (b) **Let G(8) = pre be a real rational transfer function with p(s) = 5+ a s? +bs + c and p(-1) = 0. Find conditions on a, b, and c such that: (i) G(s) is stable; and (ii) the transient components of the step response decay with time constants less than or equal to 1. [4 marks] (c) Consider the standard negative feedback interconnection of plant G(s) = (6+) and con- troller C(s) = K > 0. Explain why the closed-loop is unstable for large K, and find the value of K that puts two of the closed-loop poles on the imaginary axis. [3 marks) $ X 0 R X PZ1 PZ2 PZ3 o X R R o PZ4 PZ5 PZ6 Step Response 3 2 TIHTI (T1 -T2) T3) T4) T5 ) T6) 1 -3 -4 -5 0 2 4 6 Time (seconds) 8 10 Figure 2: Pole/zero and step response plots for Q2(i); poles are denoted by X, and zeros by O. /OVER Q2 (10 marks) In Gradescope, upload no more than 3 A4 pages with clearly labelled responses to the following three unrelated questions: (a) The pole/zero plots of six transfer functions are shown in the top part of Figure 2. Match each of these to a corresponding step response from the bottom part of Figure 2. To this end, reproduce the following table and insert labels from the set { T1, T2, T3, T4, T5, T6 } into the second row. You are not required to justify your answers. [3 marks] pole/zero plot PZ1 PZ2 PZ3 PZ4 PZ5 PZ6 step response (b) **Let G(8) = pre be a real rational transfer function with p(s) = 5+ a s? +bs + c and p(-1) = 0. Find conditions on a, b, and c such that: (i) G(s) is stable; and (ii) the transient components of the step response decay with time constants less than or equal to 1. [4 marks] (c) Consider the standard negative feedback interconnection of plant G(s) = (6+) and con- troller C(s) = K > 0. Explain why the closed-loop is unstable for large K, and find the value of K that puts two of the closed-loop poles on the imaginary axis. [3 marks) $ X 0 R X PZ1 PZ2 PZ3 o X R R o PZ4 PZ5 PZ6 Step Response 3 2 TIHTI (T1 -T2) T3) T4) T5 ) T6) 1 -3 -4 -5 0 2 4 6 Time (seconds) 8 10 Figure 2: Pole/zero and step response plots for Q2(i); poles are denoted by X, and zeros by O. /OVER