Matlab编程代写、代写Matlab编程、代写Matlab程序、Matlab编程代写、Matlab代写、Matlab语言代写代写

日期：2018-09-22 02:15

Project 3: Balancing Ball

Project 3: Balancing Ball

EE 599 – Fall 2018

Last update: September 18, 2018

Contents

1 Instructions 1

2 Physical system 2

2.1 DC motor control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Ball and beam control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.3 Performance specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Control System Configuration 4

4 Problem: System modeling 4

1 Instructions

In this project, you will model, analyze, and design automatic controllers for balancing a ball on a beam

using an armature-controlled DC motor.

You are provided with:

• This document that describes the project and your tasks. This document, and your tasks, will be

updated throughout the semester along with the class progress.

• Model and code files used to do the project.

THE PRODUCTS OF THIS PROJECT, WHICH WILL BE GRADED, ARE:

1. A typed milestone report and a typed final report.

• Each report must have a proper introduction and answer all the questions in this document. The

milestone report must answer all questions assigned by the time of its due date.

• In what follows, questions that must be answered in writing in the report will be marked with

(Report).

• However, the questions are to guide you through the project. Unlike a homework assignment or

an exam, a proper technical report is not a list of answers to questions. It must discuss different

stages and decisions of the control design (including the reasons for the design decisions). The

report must have a logical flow in its content. Imagine that you are an engineer working for a

company, and this is the technical report for your control project assigned by your boss.

1

• The total length of the milestone report should not exceed 10 pages. The total length of the final

report should not exceed 20 pages.

2. Code files and Simulink models, as instructed in your tasks in this document (marked with (Code)).

To help you identify the required tasks in this project, task descriptions will be clearly marked by

distinct frame and background, as illustrated in this paragraph.

2 Physical system

In this system, a ball is placed on a beam as shown in Figure 1. The ball is only allowed to roll along the

length of the beam. A lever arm is attached to the beam at one end and a servo gear at the other. When the

servo gear turns by an angle θ, the lever tilts the beam by an angle α, which causes the ball to roll along the

beam. The servo gear is driven by an armature-controlled DC motor. A control system will be designed to

manipulate the ball’s position on the beam, i.e., to roll the ball to a desired position and keep it there.

Figure 1: Real ball-beam system on the left and its model on the right.

The control system will consist of two control sub-systems:

• A feedback system to control the position of the DC motor that drives the servo gear, and

• A feedback system to control the position of the ball by providing the desired / reference gear position

to the above motor control loop.

In this project, you will design both the controllers.

2.1 DC motor control system

We will use an armature-controlled DC motor to drive the servo gear. The modeling, analysis, and control

design of this system are discussed in details in the textbook. Students should refer to the textbook during

this project. The DC motor is illustrated in Figure 2, reproduced from the textbook.

The system parameters of the motor are summarized in Table 1. We will use similar notations as in the

textbook.

2

Figure 2: Electrical diagram and sketch of a DC motor (source: textbook).

Table 1: System parameters of the automotive suspension system.

Variable Description Unit

Jm moment of inertia of the rotor kg m2

b motor viscous friction constant N m s

Kb back electromotive-force constant V/rad/sec

Km motor torque constant N m/Amp

Ra electric resistance Ohm

La electric inductance H

va armature voltage V

θ motor position rad

2.2 Ball and beam control system

This is the mechanical system as described in Figure 1. Its system parameters are summarized in Table 2.

Table 2: System parameters of the automotive suspension system.

Variable Description Unit

m mass of the ball kg

R radius of the ball m

d lever arm offset m

g gravitational acceleration m/s2

L length of the beam m

Jb ball’s moment of inertia kg m2

r ball position coordinate m

α beam angle coordinate rad

θ servo gear angle rad

3

2.3 Performance specifications

Ball-beam control system:

• Overshoot < 5%

• Settling time < 5 seconds.

DC motor control system: In order to control the ball’s position with high performance, it is crucial to

be able to position the motor very precisely and very quickly. The steady-state error of the motor position

should be zero. Furthermore, the steady-state error due to a constant disturbance should be zero. Finally,

there should not be excessive overshoot. Under a unit step reference input, the following criteria are desired:

• Settling time less than 40 milliseconds

• Overshoot less than 16%

• No steady-state error, even in the presence of a step disturbance input

3 Control System Configuration

To aid the design process of the system, a block diagram of the control system, including both the motor

control loop and the ball-beam control loop, should be developed. You also need to identify the various

variables in the control system.

Your task:

1. (Report) Draw a block diagram of the overall control system, including both the motor control

loop and the ball-beam control loop. Identify the two processes (or plants) in this system. Identify

the essential (physical) variables in the control system. Write a brief description of the entire

control system and the relationship between the two control loops.

4 Problem: System modeling

A critical task in a control design project is to obtain the models of the various components to be used for

control analysis and design.

Do the following tasks:

1. (Report) Write the differential equations of the ball-beam process (without the DC motor process),

as described in Figure 1. Are the equations linear or nonlinear? If nonlinear, linearize them about

α = 0 and write the linearized equations. Then write the transfer function Gb

(s) of the ball-beam

process from its input (identified above) to the position of the ball. How many integrators are

there in the transfer function?

2. (Report) Write the transfer function Gm(s) of the armature-controlled DC motor, from the armature

voltage to the motor position (you can refer to the textbook but note a slight difference in the

notations). You can make assumptions to simplify the model, but not to the point of a first- or

second-order system.

4

3. (Code) Write a script model.m in Matlab that implements the transfer functions Gb

(s) and Gm(s)

of the processes in the system (using Matlab function tf), where system parameters are variables

not yet known.