1. Objectives
1.1 To determine the open-loop time response of an electromechanical system
1.2 To design a stable closed-loop system that will meet the design specifications to a unit-step input
2. Introduction
A mathematical model of a control system is usually defined to allow us to evaluate two important aspects of the system behaviour. These are
2.1 Stability; and
2.2 Performance, i.e., the ability to meet the desired specifications.
In this experiment, students will be introduced to the MATALB, a high-level programming language, which will help them to model and evaluate the behaviour of an electromechanical system in closed-loop control.
3. Closed-Loop Design Specifications
The control system is designed to meet the following specifications to a unit step input:
3.1 Peak/maximum percentage overshoot, %𝑀 <= 9.95%
3.2 2% error (2% criterion) settling time, 𝑡 <= 2.34 sec
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4.1 Modelling
Figure 1: A schematic diagram of a DC motor
Refer to Figure 1, Nise (7th Ed., section 2.8, pp 77-81) gives an example of a DC motor. The transfer function is given by
𝐺(𝑠)=𝜃 =𝑁(𝑠)= 𝑘
𝐸 𝐷(𝑠) 𝑠(𝑠 + 𝛼)
A potentiometer, 𝑘, is used to measure the motor position, 𝜃, and its signal is compared to a desired angle voltage, as shown in Figure 2. The output of the controller, a non‐inverting amplifier with a gain, 𝑘, is used to drive the DC motor to achieve the desired input angular position, 𝜃 .
Figure 2: A servomotor closed-loop control system
The control issues:
4.1.1 Is the closed-loop system stable? i.e., will the motor keep on turning once it is turned ON?
4.1.2 If the system is stable, what value of 𝑘 will make the motor turn and stop at the desired angle within the specified 𝑡 (sec) and without exceeding the %𝑀?
Note that, the above (4.1.1) and (4.1.2) are for discussion purposes, these are not required to include in the report.
4.2 Open-Loop Response
Figure 3 shows the equivalent open-loop system of Figure 2. The open-loop transfer function is
𝐺(𝑠) = 𝑘𝑘𝐺(𝑠) = 𝑁𝑜(𝑠) = 𝑘𝑘𝑘 𝐷𝑜(𝑠) 𝑠(𝑠 + 𝛼)
Figure 3: The open-loop system
4.3 Closed-Loop Response
Figure 4 shows an equivalent of the original system in Figure 2.
Figure 4: The equivalent closed-loop system The closed-loop transfer function is given as
𝐺(𝑠) = 𝜃(𝑠) = 𝑁𝑐(𝑠) = 𝑘𝑘𝑘
𝜃(𝑠) 𝐷𝑐(𝑠) 𝑠 + 𝛼𝑠 + 𝑘𝑘𝑘
The closed-loop poles or roots, 𝑠 and 𝑠, of the denominator of 𝐺(𝑠) can be obtained from the characteristic equation, which is given by
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𝑠 + 𝛼𝑠 + 𝑘𝑘𝑘 = 0
It must be noted that the parameters, 𝛼, 𝑘, and 𝑘, are positive. If 𝑘(> 0) is selected such that 𝑠 and 𝑠 have negative real parts, the closed-loop system will be stable.
5. Procedure
5.1 MATLAB Commands
Refer to Lecture Notes (a) MATLAB for Control Systems (1) and (b) MATLAB for Control Systems (2)
for MATLAB programming, and control system implementation and analysis.
5.2 Open-Loop Response
Refer to Figure 3, defining 𝑁𝑜(𝑠) and 𝐷𝑜(𝑠) for 𝐺 (𝑠), with 𝑘 = 1 and given system parameter values; 𝑘 = 2.4, 𝛼 = 3.0, and 𝑘 = 5.0,
a) Obtain the unit-step time response, 𝜃(𝑡), of the open-loop system.
b) For a step input of amplitude 2, obtain the time response, 𝜃(𝑡).
Print and save the results (MALTAB plots).
5.3 Closed-Loop Response (Unit-Step Time Response)
Refer to Figure 4, defining 𝑁𝑐(𝑠) and 𝐷𝑐(𝑠) for 𝐺(𝑠), with 𝑘, 𝛼, and 𝑘 given by (5.2),
5.3.1 Find the value of 𝑘 to meet the closed-loop design specifications given by
(3.1) and (3.2),
5.3.2 Find the value of 𝑘 to achieve a critically damped response, and
5.3.3 Find the value of 𝑘 to achieve a response with damping ratio = 0.35.
Examine the step response of the closed-loop system, 𝐺(𝑠), under the above three conditions. Print and save the MATLAB plots.
For each of the above three responses, determine the 0-98% rise time (𝑡), percentage overshoot (%𝑀), peak time (𝑡), settling time (𝑡) (2% criterion),
steady-state value (𝑠𝑠), and type of response (unstable, undamped, underdamped, critically damped, or overdamped).
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Type of Response
6. Results and Discussion
6.1 Open-Loop Response (5.2)
a) From the time response plot, if the input is a step signal of amplitude 2, what is the steady-state magnitude?
b) Explain the step response of 𝜃 vs time, and comment about the stability (stable or unstable) of the open-loop system for controlling the position 𝜃?
6.2 Closed-Loop Response (5.3)
a) What is your designed value of 𝑘 that meets the specifications?
b) What is your designed value of 𝑘 that gives a critically damped response?
c) What is your designed value of 𝑘 that gives a response with = 0.35?
d) Will %𝑀 and 𝑡 increase (or decrease) if 𝑘 is increased?
e) Do you find any values (range) of 𝑘 that will make the closed-loop system
7. Conclusions
Do you have a stable open-loop system for the motor position, 𝜃, to achieve/follow the desired input angular position, 𝜃?
Are you able to design a stable closed-loop position control system that meets the desired specifications? What is the amplifier gain, 𝑘, value?
Show the achieved performance specifications on the simulated unit-step response of the designed closed-loop control system.
8. References
[1] Modern Control Engineering, by Katsuhiko Ogata, 5th ed., Pearson, 2010, ISBN 978- 0-13-713337-6
[2] Nise’s Control Systems Engineering, by Norman S. Nise, Global or 7th ed., Wiley, 2015, ISBN 978-1-119-38297-3
9. Lab Report Requirements
9.1 Include in the introduction section the learning objectives.
9.2 Include in the Results and Discussion section the instructions/procedures,
MATLAB code(s), calculated/measured results (4 decimal places), time response
plots, and comments/explanations
9.3 Provide a summary of your learning experience, difficulties faced, and suggestions
in the Conclusions.