# ENGRMAE 106 Lecture Notes - Lecture 1: Angular Velocity, Damping Ratio, Step Response

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5 Aug 2016

School

Department

Course

Professor

MAE106 Laboratory Exercises

Lab # 5

PD Control of DC motor position

University of California, Irvine

Department of Mechanical and Aerospace Engineering

Goals

Understand how to implement and tune a PD controller to control the position of a DC

motor.

Explore the frequency response of the PD controller by testing how it responds to

sinusoidal inputs of different frequencies.

Parts & equipment

Qty

Part/Equipment

1

Seeeduino board

1

Motor driver

1

DC motor with encoder

Introduction to PD Control

The most common controller used by engineering designers to control the movement of a

motorized part is the PD (proportional-derivative) controller (sometimes an integral control

term is added to create a PID controller, but we will not explore I control in this lab). In this

lab you will implement a PD position controller. Such controllers are also used in robot

arms, radars, numerically controlled milling machines, manufacturing systems, and control

surfaces on aerospace vehicles. The PD control law is:

where:

desired motor angular position

desired motor torque

desired motor angular velocity

proportional gain

actual motor angular position

derivative gain

actual motor angular velocity

Note that the controller has two terms – one proportional to the position error (the “P” part),

and one proportional to the derivative of position (i.e. velocity, the “D” part). Thus, it is

called a “PD” controller. The implementation of this controller for a DC motor with inertia,

J, is shown in Figure 2.

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Figure 1. Block diagram that you will implement to make the PD controller for the motor. J

is the inertia of the motor shaft.

To understand how the actual system behaves we need to first understand its dynamics.

First, let's look at the dynamical equation that describes how θ evolves with time when the

controller is attached to the motor.

Dynamics of the motor and shaft:

Dynamics of the controller system:

Re-writing to make input-output clear:

This differential equation has similar dynamics to a mass-spring-damper system with

Force as the input and Position as the output. That is, it follows the same equations of

motion. This allows us to use our intuition about mass-spring-damper systems when

designing and tuning a PD controller.

Recall that the differential equation of motion for a mass-spring-damper system is given

by:

and thus using the analogy to the PD controller we have that:

mass-spring-damper system

PD controller

m

mass

J

motor inertia

B

damper

derivative control term

K

spring

proportional control term

F

input force

desired motor torque

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In a mechanical system, if you wanted the system to respond more quickly, you would

increase the natural frequency (ωn) by picking a stiffer spring (higher K). Which variable

would you change in your differential equation for the PD system to make your system

respond more quickly (i.e. increase its natural frequency)?

Note: there is a limit to how big you can make this variable because of the time delays in

this sampled data system.

By adding the derivative gain (Kd) to control the position of the motor we must now take

into account the concept of damping when designing and implementing the controller.

With damping in the controller we can have four types of behaviors:

Undamped (i.e. zero damping): The system oscillates at its natural frequency.

These oscillations are a function of the controller's gain, Kp.

Underdamped: The system will move to its desired position and oscillate about this

position with is oscillations gradually decreasing to zero.

Critically damped: The system will move to its desired position as quickly as

possible without oscillating.

Overdamped: The system will move asymptotically towards its desired position

without oscillating, but at a slower rate than the critically damped case.

Suppose you didn’t want your motor to oscillate too much. This is an important issue! You

usually want your motor to go to a desired value quickly and accurately without oscillating.

You will have to change the derivative gain Kd until you get the system to be critically

damped.

find more resources at oneclass.com

find more resources at oneclass.com