Design Using Bode Diagrams

Question 1

Describe the gain diagram of an ideal system’s OLTF.

Answer 1

Infinite low-frequency gain, then breaking towards a low high-frequency gain, with an infinite gain margin.

Question 2

Describe the phase diagram of an ideal system’s OLTF.

Answer 2

0° low-frequency phase shift, linearly decreasing towards 90° until the gain crossover point, where it then asymptotes towards 90°. This results in an > 90° phase margin, and an infinite gain margin (since 180° is never reached).

Question 3

What do low phase and gain margins imply about a closed-loop system’s step response?

Answer 3

That the step response will overshoot, since the system is approaching the boundary of stability (i.e. only a small phase shift or gain increment is necessary to make the system unstable).

Question 4

A plant is found to have adequate gain and phase margins (greater than 2dB and 45° respectively), but a large steady-state error. What controller do you recommend do rectify this problem? Why?

Answer 4

A proportional + integral controller will remove all steady-state error, but will not affect the gain and phase margins of the plant, as required.

Question 5

Outline Method 1 for designing a Proportional-plus-Integral controller.

Answer 5

(i) Set Kp = 1.
(ii) Set Ki/Kp = Wgc/10, where Wgc is the gain crossover frequency.
(iii) Plot the overall system’s open-loop response, and adjust Kp if necessary.

Question 6

Outline Method 2 for designing a Proportional-plus-Integral controller.

Answer 6

(i) Set Kp = 1.
(ii) Use the controller zero to cancel the fastest plant pole.
(iii) Plot the system’s OLTF on Bode diagrams, then adjust Kp until the desired closed-loop behaviour is achieved.

Question 7

When might a Proportional + DFB controller be used?

Answer 7

When a system has adequate low-frequency gain, but needs improved gain and phase margins.

Question 8

Describe how you would design a Proportional + DFB controller in the frequency domain.

Answer 8

(i) Set Kp = 1.
(ii) Let 1/Kd = wgc, where wgc is the gain crossover frequency.
(iii) Adjust Kp until the response is OK.

Question 9

What is a possible disadvantage of using a Proportional + DFB controller?

Answer 9

The derivative feedback will amplify high-frequency noise.

Question 10

A plant is found to suffer from overshoot but no steady-state error. What controller would you recommend?

Answer 10

A Proportional + DFB controller: The overshoot is symptomatic of poor gain and phase margins, but it sounds as though the LFG is OK. A Proportional + DFB controller will improve both margins without reducing the LFG.

Question 11

What is a possible shortcoming of a Proportional-plus-Integral controller?

Answer 11

It may negatively affect the system’s phase margins. If the gain crossover frequency is low relative to the controller’s 90° phase lag, then the phase margins will be reduced, which is undesirable.

Question 12

What is the bandwidth frequency of a system w.r.t. that system’s OLTF magnitude response?

Answer 12

The frequency at which the gain is between -6 and -7.5dB below the low-frequency gain.

Question 13

What is the significance of a system’s gain margin?

Answer 13

It corresponds to the scale factor by which we can increase the system gain before the system becomes unstable.

Question 14

Across what range of damping coefficients eta is the approximation Wb /Wn = 1.85 - 1.19eta appropriate?

Answer 14

For 0.3 < eta < 0.8.

Question 15

What happens to the rise time of a system as its bandwidth increases?

Answer 15

The rise time decreases with an increase in bandwidth.

Question 16

Given a system with a low-frequency gain of 1, what would be the benefit of a large bandwidth?

Answer 16

The system would be able to reproduce an input signal accurately across a large range of input frequencies.

Question 17

For a second-order system, for what value of damping coefficient is the gain response resonance peak pronounced?

Answer 17

For eta < 0.3.

Question 18

What is the relationship between resonance frequency and natural frequency for a second-order system?

Answer 18

Wr = Wn sqrt(1 - 2eta^2), i.e. Wr = Wn for small eta.

Question 19

What is the magnitude at the maximum of a resonance peak?

Answer 19

|Gres| = LFG / 2eta sqrt(1 - eta^2)

Question 20

What controller can be used as an alternative to a P + I controller?

Answer 20

A lag controller, i.e. K(1 + Ts)/(1 + aTs) for a > 1.

Question 21

Describe how you would design a Lag controller.

Answer 21

(i) Set a = 10, and set the HFG to 1 : K = a = 10.
(ii) Set 1/T = Wgc/10, such that the higher break frequency is at Wgc/10.
(iii) Adjust K until the desired CL response is achieved.

Question 22

At what frequency is maximum phase lag/lead achieved for a Lag/Lead controller?

Answer 22

Wm = 1/T sqrt(a)

Question 23

What is a disadvantage and a benefit of a Lag controller relative to a Proportional + Integral controller?

Answer 23

+ A Lag controller does not interfere with a phase margin corresponding to a low Wgc.
- A Lag controller has a finite LFG, whereas a P+I controller has infinite LFG.

Question 24

A P+DFB controller must be replaced because it is found to amplify excessive amounts of high-frequency noise. What controller do you recommend as an alternative? Why?

Answer 24

A Lead controller confers the same benefits as a P + DFB controller (i.e. improved gain and phase margins), but has a finite (as opposed to infinite) high-frequency gain.

Question 25

Describe how you would design a Lead controller in the frequency domain.

Answer 25

(i) Set K = 1, a = 0.1.
(ii) Set 1/T = Wgc.
(iii) Adjust K until the desired closed-loop response is achieved.

Question 26

What are possible benefits of using PID control?

Answer 26

An infinite gain margin, an increased phase margin, and infinite low-frequency gain.