This project investigates the design of a nonlinear feedback control system for a ball and beam process using numerical optimization in MATLAB. The document first presents the ball and beam physical system and develops a mathematical model. It then designs the nonlinear system in Simulink and optimizes it using MATLAB functions to minimize settling time. Results show the nonlinear system achieved a settling time of 1.74 seconds, meeting the objective of less than 3 seconds. Optimum PID gains are reported and the step response graph depicts the desirable optimized output. The investigation is concluded as successful in producing favorable optimized results for the nonlinear control system.

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Level 3 Individual Major Project 2012/2013
Project ID: JC2
Nonlinear Control Systems Design using
Numerical Optimization
Dissertation Submitted By
Hammas Amer
Student ID: 101xxx5
Under Supervision of
Dr Jie Chen
School of Engineering and Design

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Brunel University
Mechanical Engineering
Academic Session: 2012-2013
Name of Student: HAMMAS AMER
Supervisor: Prof. Dr. JIE CHEN
Title: Nonlinear Control Systems Design of the Ball and Beam process using Numerical Optimization via MATLAB
optimization methods and Simulink simulations.
Abstract: In the report the classical nonlinear control system of the ball and the beam model is under investigation.
A simple system, yet highly unstable when in open-loop configuration. The ball and beam control system is designed
to be a feedback control system, with a PID controller in place. The first stage converts the physical phenomena into
mathematical form, producing a mathematical model of the ball and beam system based on the lagrangian equation.
Then the nonlinear system will be designed and modeled through Simulink and optimized via the lsqnonlin function
(for least-square curve fitting). The nonlinear model is then to be linearized via the Laplace Transform and optimized
under constraints through the fmincon MATLAB function. The step responses of the two control system are drawn
and the optimum system parameters compared. At first both systems are minimized to have a settling time of less than
3 seconds, later on a different parameter is focused on with a phase-Lead compensator to be optimized via the
fminsearch function under simulation settings. An aerospace application of the control system is also discussed
towards the end of the report, where the linearized model is run under the same process to produce optimum gains for
the PID controller.
Objectives:
 To study and investigate the optimization techniques for nonlinear control systems.
 To produce an optimum nonlinear feedback control system to produce desired output.
 To design a suitable controller for nonlinear control systems that results optimal output.
 Understand and use MATLAB-based tools for numerical optimization of control systems.
Background/Applications: The applications of the work generally come under the field of control systems
engineering, which deals with the study of control systems and the different processes involved in a successful control
system. Control systems are applied to almost every electronic or mechanical being on this planet today. The modern
world can hardly sustain without control systems, from aircraft’s autopilot systems to washing machines to fire alarms,
control systems have become a necessity for modern times.
The work presented in this report is partially a follow-up on previous dissertation, in which the MATLAB coding
have been acquired and modified to best fit the objectives of this project.

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Conclusions:
The investigation has been stamped successful, since the optimum results were drawn from the parameters minimized.
The nonlinear feedback control system gave a settling time of 1.74 seconds, which are favorable results, since the
objective was to have a settling time of less than 3 seconds. Optimum PID gains were concluded to be Kp, Kd and Ki
of 101, -9.04 and 30.6 respectively.
The response graphs produced by the system(s) were favored and depicted desirable outputs.
Results:
The main system under investigation was the nonlinear ball and beam feedback control system. However, for
experimental purposes the model was linearized, for comparison purposes.
Below is the overview of the optimized controller parameters:
Linearized system Nonlinear system
PID parameters Kp Ki Kd Kp Ki Kd
Optimum PID gains 3.05 0.0001 80.1 101 -9.04 30.6
Optimized settling time 2.45 1.74
Table 3.4 Listing optimum PID gain parameters and optimized performance index for comparison
This table can be found on page 28 of the project report. It is drawn after all the optimization is processed and the
desired results found. As can be seen the nonlinear system, optimized via the lsqnonlin function, produces better
results. Whereas the linearized system, though meeting the objective, producing a settling time of less than 3 seconds,
results in a 2.4 seconds of settling time. Here we have a difference of almost 30%.
Also the step response of the nonlinear control system is displayed on the following page:

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Figure 5.5 Feedback response of nonlinear ball beam feedback control system
This step response graph can be found on page 27 of the project report. It ultimately shows the reader that the
optimization proved successful, being stable and having favorable outcomes. The settling time can be seen to be
between 1.5 seconds and 2 seconds.

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Contents
Acknowledgements …………………………………………………… iii
Nomenclature ………………………………………………………….. iv
a. Introduction ………………………………………………................1
Aims and Objectives ……………………….…………..1
1. Literature Review ……………………………………………………... 3
1.1 The Control System ………………………….…3
1.1.1 Open-Loop Control System …...….3
1.1.2 Closed-Loop Control System ..……4
1.2 Linear Systems ………………………………….6
1.3 Nonlinear Systems ………………………..........6
1.4 System Performance ……………………………7
1.5 PID Control ……………………………………….8
1.6 Optimization Techniques ……………………….10
1.7 MATLAB Optimization ………………………….14
2. Methodology ………………………………………………………..…15
2.1 Ball and Beam model (Physical) …………………15
2.2 Ball and Beam model (Mathematical) …………...16
3. Results and Discussion ………………………………………………22
4. Using fminsearch ………………………………………………….…..30
5. Aerospace Application ………………………………………….….…34
6. Conclusion ……………………………………………………….….…38
Bibliography ………………………………………………………….…...40
Appendix A1 ………………………………………………………………41
Appendix B1 ………………………………………………………………47
Appendix C1 ………………………………………………………………52