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ECE380 problem set 4...

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University of Waterloo Department of Electrical & Computer Engineering ECE 380: Analog Control Systems Problem Set 4 January 31, 2020

Topics • Prototype first and second order systems • Asymptotic stability • Bounded-input bounded-output stability • Steady-state gain

Problem 1 Answer (a), (b) and (c) for each of the following transfer functions 25 Y (s) = , U (s) s + 25

Y (s) 1 = 2 , U (s) s + 21s + 20

Y (s) 600 = 2 . U (s) s + 12s + 85

(a) If u(t) = 5 × 1(t), what is the steady-state value of y(t)? (b) Does the step response exhibit oscillations? (c) If the response is oscillatory, find the frequency of oscillation.

Problem 2 A unit step input is applied to two different systems and the resulting outputs are plotted versus time in Figure 1. (a) Find a system model whose step response approximately matches the one in Figure 1a. (b) Find a system model whose step response approximately matches the one in Figure 1b.

Problem 3 Consider the series RLC circuit in Figure 2. The system TF is (verify!) Y (s) 1 = = 2 s LC + sRC + 1 U (s) s2 +

1

1 LC 1 R L s + LC

.

3 4.5

4

2.5 3.5

2

3

2.5

1.5 2

1

1.5

1

0.5 0.5

0 0

0 0

0.5

1

1.5

2

2.5

3

3.5

4

(a) A step response

2

4

6

8

10

12

(b) A more interesting step response.

Figure 1: A pair of step responses

Figure 2: RLC circuit with voltage across capacitor taken as output.

(a) Draw the region of the s-plane in which the system’s poles must lie so that its step response satisfies %OS ≤ 0.02, Ts ≤ 1, Tp ≤ 3. (b) Choose values for the resistance R, capacitance C and inductance L so that the poles of the system are in the allowable region from part (a).

Problem 4 Consider the control system in Figure 3. The plant in this system is a DC motor whose output is the angular position of the shaft. The feedback controller is an amplifier with gain Ka.

Figure 3: DC motor and amplifier in unity feedback configuration.

(a) A particular DC motor has parameters K = 5, a = 2. Estimate the fastest 2% settling time achievable with this motor using a proportional controller as shown above. 2

(b) Suppose we want the system to have a step response with settling time equal to 0.25 seconds and no overshoot. Find a combination of controller and motor parameters (K, a, Ka) that will approximately achieve these specifications.

Problem 5 Find (i) the eigenvalues and (ii) eAt for each A cally stable system x(t ˙ ) = Ax(t)?  0    0 −4 0 , A= A=  0 5 0 0

matrix below. Which matrices yield an asymptoti-

0 0 0 0

0 0 0 0

 1 0 , 0 0

 0 1 . A= −4 −4 

Problem 6 Consider the system modeled by x˙ = Ax + Bu,

y = Cx + Du

where dim u = dim y = 1 and dim x = 2. Given the initial-state responses   1 x(0) = =⇒ y(t) = e−t − 0.5e−2t 0.5   −1 x(0) = =⇒ y(t) = −0.5e−t − e−2t 1 find the initial state response for x(0) =



2 0.5



.

Problem 7 Let



 1 1 0 A =  −2 −2 0  . 0 0 0

Which of these is true as t tends towards ∞: eAt converges to 0; eAt converges but not to 0; eAt does not converge?

Problem 8 A state-space model of the closed-loop motor control system from Problem 5 is     0 1 0 x(t) ˙ = r(t) x(t) + 1 −KaK −a   y(t) = KaK 0 x(t). Find conditions on (K, a, Ka) so that this system is asymptotically stable.

3

Problem 9 Consider the system 

      0 0 1 x1 (t) x˙ 1 (t) u(t) + = −1 1 0 x2 (t) x˙ 2 (t)     x1 (t) . y(t) = 1 −1 x2 (t)

(a) Is the state-space model asymptotically stable? (b) Find the transfer function Y (s)/U (s). Is this system BIBO stable? (c) Find expressions for the state x(t) and the output of the system y(t) when x(0) = 0 and u(t) = 1(t). Does your expression for y(t) contradict your answers from parts (a) and/or (b)?

Problem 10 Consider the BIBO stable LTI system Y (s) = G(s)U (s) with G(s) =

1 . s+3

Find a number γ ∈ R such that for every bounded input, kyk∞ ≤ γkuk∞ .

Problem 11 An ideal differentiator is a system whose output is the derivative of its input; its TF is s. Show that a differentiator is unstable by constructing a bounded input that produces an unbounded output.

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