Description

15MH302J – LINEAR AND DIGITAL CONTROL SYSTEMS
EXERCISE 10
PD COMPENSATED QUBE SERVO DC MOTOR
LAB PREREQUISITES:
Exercise 1-9
PREREQUISITE KNOWLEDGE:
Fundamentals of MATLAB programming and Simulink.
OBJECTIVES:
To design a PD compensator for the Qube-Servo 2 model to achieve a peak time of 0.15s and a % overshoot of
2.5%. Servo Model:
The QUBE-Servo 2 voltage-to-position transfer function is

where K = 26:5 rad/(V-s) is the model steady-state gain, τ = 0:155 s is the model time constant, Θm(s) = L [θm(t)] is the motor / disk position, and Vm(s) = L [vm(t)] is the applied motor voltage.
PID CONTROL:
The proportional, integral, and derivative control can be expressed mathematically as follows
The corresponding block diagram is given in Figure 1. The control action is a sum of three terms referred to as proportional (P), integral (I) and derivative (D) control gain. The controller Equation 2 can also be described by the transfer function

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Figure.1. Block diagram of PID control
The integral term will not be used to control the servo position. A variation of the classic PD control will be used: the proportional-velocity control illustrated in Figure 2. Unlike the standard PD, only the negative velocity is fed back (i.e. not the velocity of the error) and a low-pass filter will be used in-line with the derivative term to suppress measurement noise. The combination of a first order low-pass filter and the derivative term results in a high-pass filter H(s) which will be used instead of a direct derivative.

Figure.2. Block diagram of PV control
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The back-emf (electromotive) voltage eb(t) depends on the speed of the motor shaft, wm, and the back-emf constant of the motor, km. It opposes the current flow. The back emf voltage is given by:

Table 1. Qube servo 2 system parameters
The proportional-velocity (PV) control has the following structure

where kp is the proportional gain, kd is the derivative (velocity) gain, r = θd(t) is the setpoint or reference motor /load angle, y = θm(t) is the measured load shaft angle, and u = Vm(t) is the control input (applied motor voltage). The closed-loop transfer function of the QUBE-Servo 2 is denoted Y (s)/R(s) = Θm(s)/Θd(s).
Assume all initial conditions are zero, i.e. θm(0-) = 0 and θm_ (0-) = 0, taking the Laplace transform of
Equation 4 yields
U(s) = kp(R(s) – Y (s)) – kdsY (s);
which can be substituted into Equation 1 to result in

Solving for Y (s)/R(s), we obtain the closed-loop expression

This is a second-order transfer function. Recall the standard second-order transfer function

PROGRAMS, OBSERVATIONS AND INFERENCES
1. Calculate the control gains needed to satisfy these requirements of peak time and percentage overshoot, Using the QUBE-Servo 2 model parameters K and  given above in Background section of this lab.
2. Implement a PV controller with a low pass filter of 100/(s+100)
3. Implement a PV control loop on the QUBE-Servo 2 model transfer function that relates Voltage Vm(s) to Position m(s).
4. Build and run the QUARC controller. Determine the responses – (i) Position (rad) (ii) Voltage
5. Keep the derivative gain at 0 and vary kp between 1 and 4. What does the proportional gain do when controlling servo position?
6. Set kp from calculation and vary the derivative gain kd between 0 and 0:15 V/(rad/s). What is its effect on the position response?

15MH302J
D E P A R T M E N T O F M E C H A T R O N I C S E N G I N E E R I N G EXERCISE 10
RESULTS & INFERENCES:
Evaluation
Component Maximum Marks Marks Obtained
Pre-lab Tasks 10
In-Lab Tasks 20
Post-lab Tasks 10
Bonus Tasks 10
(This page must be the last page of the exercise)