DC Motor Control System: Designing Proportional, Integral, and Derivative Controller

Deriving the Motor Transfer Function

The DC motor is a common actuator in control system applications, providing rotational motion for applications such as mobile robots. The motor’s input voltage   applied to its armature and motor’s output speed   are related by the following equations

 

where   is the armature current and parameters L,R,K_t,J,b,K_e are physical parameters of the motor described in Table 1 below with sample values for purposes of this project.

 

 1. By taking the Laplace transform of equations (1) and (2), derive the motor transfer function For simplicity, we took .  (5 marks)

 

2. Simulate the motor’s step response to an input of 1 volt. Show the response plot. What is the steady state output? Mathematically verify the observed steady state output using the Final Value Theorem. (10 marks)

 

A basic requirement of a motor is that it rotates at a desired reference speed. This shall be the control objective i.e. to design a controlled input voltage   that causes the motor to rotate at a desired reference speed  . In achieving the control objective, it is also important to satisfy some performance requirements, prescribed as follows:

For a fixed reference speed  

 

a. Settling time must be under 3 seconds

b. Overshoot percentage must be under 5%

c. Steady state error must be less than 1%

 

3. Design a Proportional (P) controller that best meets the above requirements. Were the requirements met? Show the control parameter (Kp) and highlight any shortcomings of the P controller. (7 marks)

 

4. Design a Proportional and Integral (PI) controller that best meets the above requirements. Were the requirements met? Show your plot and control parameters. Discuss the effect of the integral action. (8 marks)

 

5. Design a Proportional, Integral and Derivative (PID) Controller that best meets the above requirements. Were the requirements met? Show your plot and control parameters. Discuss any final observations. (10 marks)

 

The purpose is to use Simulink to simulate a (much simplified) model of the longitudinal motion of a fighter aircraft. The "angle of attack", is the angle between the direction a plane is pointing, and the direction in which it actually moves through the air.  For a plane flying at approximately constant altitude, this is equivalent to the "pitch angle as illustrated in Fig. 1.  This angle is important because it produces a lift force perpendicular to the axis of the plane, and hence a "normal acceleration",  , (also shown in the figure)

 

 

The pilot wants to be able to control the pitch angle, and does so ultimately by rotating the front fins, and tail elevators of the aircraft, shown in Fig. 2.  The first task is therefore to model the effect of these movements on the "pitch rate and "normal acceleration

 

 1. Normal Acceleration

 

The acceleration of the aircraft in a direction perpendicular to its axis, (the "normal accel.",  ), is determined mainly by the angle   of the tail elevators of the aircraft shown in Fig. 2.  Indeed aerodynamic modelling shows that this relationship can be described by the differential equation

The rate at which the pitch angle changes, (the "pitch rate", q), is determined mainly by the angle   of the front fins of the aircraft shown in Fig. 2. Indeed aerodynamic modelling shows that this relationship can be described by the differential equation

 

q ?+3.1q ?-8.82q=7.25? ?_2+11.27?_2

 

Convert this relationship into a transfer function form (3 marks)