METR4201: Introduction to Control Systems
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METR4201: Introduction to Control Systems
Assignment Aligned to Practical 3
Introduction
In completing this assignment, you should get an improved understanding of:
• How feedback structures can be used to endow a system with specified dynamic properties.
• The physical interpretation of the Proportional (P) and Derivative (D) terms in a PID controller
• The effect of zeros on the response of a system.
This assignment is aligned with the work you will complete Practical 3. The assignment will develop your understanding of the following learning objectives
• 1.7 Construct transfer functions for linear dynamic systems from (i) differential equations and (ii) reduction of block diagrams.
• 2.1 Understand how feedback is used to influence the behaviour of SISO systems.
• 2.2 Analyse the transient response of SISO linear dynamic systems using the system's poles and zeros.
• 3.4 Design and implement proportional, derivative, and integral compensators (and combinations thereof) for SISO systems.
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LO 1.7 |
LO 2.1 |
LO 2.2 |
LO 3.4 |
Task 1 |
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Task 2 |
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Task 3 |
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Task 4 |
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Task 5 |
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Important Notes:
1. Your assignment must electronically readable to allow it to be processed by Turn-it-In. It is recommended that you prepare your solutions in word or LaTex.
2. You must submit your assignment by the due date. Late submissions will not be accepted unless they meet the requirements for acceptable late submission set down in Section 5.3 of the course profile.
Task 1 (5 marks)
Develop transfer functions |
!(#) %(#) |
for the two systems shown below. |
Figure 1
Question 2 (5 marks)
A proportional-integral-derivative (PID) controller is a form of controller widely used in industrial control systems. The PID controller computes three quantities from the feedback error which are added together: a ‘P’ component proportional to the error; an ‘I’ component that is proportional to the accumulation of past errors; and a ‘D’ component that is a prediction of future errors based on a rate of change in the error signal. The standard from of the controller is written:
u (t) = K* e (t) + K- . e (t)dt + K0
where u (t) is the output of the controller and e (t) is the error it acts on.
There are two ‘standard’ ways of implementing PID control. Figure 2(a) below shows the implementation of a PID controller to control the position of the mass with the compensator in the forward path. Figure 2(b) shows a variation on the idea where the with a PI compensator in the forward path and the derivative (D) term in the return path.
Develop transfer functions for these two systems.
Figure 2: A mass whose position is controlled by a PID compensator
Question 3 (5 Marks)
If the integral term K- is zero, the corresponding controller is known as a proportional-derivative (or PD) controller. By setting K- = 0 and equating coefficients, show that the two PD controllers above have equivalence to the mechanical systems to the mechanical systems shown in Figure 1 and from this make interpretations of the functions of the P and D terms of a PID controller.
Question 4 (5 Marks)
Develop Simulink models for both systems in Figure 2. Using values m = 2.8 kg, K9: = 1, K* = 1500, K0 = 50, K- = 0 as a starting point, explore how having the derivative action in the forward path affects the time response. Comment in particular on how having derivative action in the forward path affects the rise time and the overshoot of the response when a step input is applied.
A regulator is a class of control systems that aim to hold the output value at a designated level. The input to a regulator is a set point which changes infrequently. When change occurs, it takes the form of a step. PID controllers used regulate a quantity will typically use the method of Figure 2(b) in preference to that of Figure 2(a). Why?
Question 5 (5 Marks)
Using your PID controller as per Figure 2(a), explore the effect of independently varying the parameters from the nominal values given above. Produce three plots showing the effect of varying the three controller parameters from the nominal values: Kp = 1500, Kd = 50, Ki = 0 . That is, the first plot should show how the response changes as Kp is varied whilst holding Kd and Ki constant. The second plot should show how the response changes as Kd is varied and the third should show response change for varying Ki .
The Wikipedia page for PID controllers gives the following table indicating the type of that occurs when each parameter is varied. Using your plots verify this table is correct.
Effects of increasing a parameter independently
Kp
Ki
Kd
Rise time
Decrease
Decrease
Minor
change
Increase Increase
Decrease
Settling
time
Small
change
Increase
Decrease
Steady-state
error
Decrease
Eliminate
No effect in theory
Degrade
Degrade
Improve if Kd
small
.
Definitions
Rise time – The time for a system to respond to a step input and attain a response equal to the magnitude of the input.
Overshoot – The amount the system output response proceeds beyond the desired
response
Settling time – The time response for the system output to settle within a percentage of the input amplitude.
Stability – A system performance measure. The less stable a system, the greater the amplitude of oscillations in its step response and the longer these oscillations persist.
2023-04-18