Report of the essay ball and beam system

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY
FACULTY OF ELECTRICALS AND ELECTRONICS ENGINEERING
_______________________
INSTRUCTOR : Assco.Dr. HUYNH THAI HOANG
GROUP 4
STUDENT STUDENT ID
NGO TAN BINH 41100293
PHAN NHAT NGUYEN 41102309
LUU HUU TRI 41103802
NGUYEN TRUNG DOAN 41100820
COURSE: Fundamentrals Of Control Systems
SEMETER I YEAR : 2013- 2014 
PROBLEM SETUP
• Let s’ watch a following short video 
A BALL is placed on a beam, see figure below, where it is allowed to roll 
with 1 degree of freedom along the length of the beam. A lever arm is attached 
to the beam at one end and a servo gear at the other. As the servo gear turns by 
an angle theta θ, the lever changes the angle of the beam by alpha α. When the 
angle is changed from the horizontal position, gravity causes the ball to roll 
along the beam. A controller will be designed for this system so that the ball's 
position can be manipulated.
THE REALISTIC BALL AND BEAM MODEL
USING PLC TO PROGRAM SYSTEM
PLC ALLEN 
BRADLEY
SIMANTIC S7_1200 PLC CP1H CPU 40 
I_O relay output
PLC mini ZEN-
10C3AR-A Omron
PROBLEM (CON’T)
For this problem, we will assume that the ball rolls without slipping and 
friction between the beam and ball is negligible. The constants and variable 
for this example are defined as follows:
Constant & 
Variable
Descriptions Values
M Mass of the ball 0.11 kg
R Radius of the ball 0.015 m
d Lever arm offset 0.03 m
g Gravitation acceleration 9.8 m/s^2
L Length of the beam 1.0 m
J ball's moment of inertia 9.99 *10^-6 
kg.m^2
r Ball position coordinate
α Angle coordinate of the beam
θ Servo gear angle
The design 
criteria for 
this 
problem 
are: 
+ tss <= 3s
+ POT <= 
5%
SOME OF SPECIALIZED 
TERMINOLOGIES
1. Coordinate (n) tọa độ
2. Criteria (n) tiêu chí
3. Moment of inertia (ph of n) mo-men quán tính
4. Velocity (n) vận tốc
5. Gear (n) bánh răng, thiết bị truyền động
6. Torque (n) mo-men xoắn
7. Radius (n) bán kính
8. Friction (n) ma sát
9. Lever arm (n) cánh tay đòn
10.Derivative (n) vi phân
11.Gravitation acceleration (n) gia tốc trọng trường
WHY BE STATE-FEEDBACK 
METHOD???
• Nowaday, this method is very popular in modern control because of many 
of perfect specifications:
+ The state space representation provides a convenient and compact way to 
model and analyze systems with multiple inputs and outputs
+we would otherwise have to write down Laplace transforms to encode all 
the information about a system. 
+Unlike the frequency domain approach, state space representation is not 
limited to systems with linear components and zero initial conditions. "State 
space" refers to the space whose axes are the state variables. 
+The state of the system can be represented as a vector within that space. And 
we can compute in matrixes with assistances from computer.so on
SOLUTION: SYSTEM EQUATION
• The second derivative of the input angle alpha actually affects the second 
derivative of r. However, we will ignore this contribution. The Lagrangian
equation of motion for the ball is then given by the following:
• The beam angle (α) can be expressed in terms of the angle of the gear (θ).
• Linear angle alpha α, we get approximated equation as follow:
(1)
SYSTEM EQUATION (CON’T)
• Replace α from above equation into (1), we have:
• => This is second of differential equation
that describes the relation between input θ
and position coordinate r of the ball.
TRANSFER FUNCTION
• Laplace transform the last equations in system equation part we obtain new 
equations as follow:
• Rewriting the form of a transfer function between the motor angle θ(s) and 
shifted the ball coordinates R (s), we have the open-loop transfer 
function:
P(s) =
• Then , we start with state-feedback method to design system that satisfy 
full of requests.
STATE-SPACE
• The linearized system equations can also be represented in state-space 
form. This can be done by selecting the ball's position ( r) and velocity (̇) 
as the state variable and the gear angle (θ) as the input. The state-space 
representation is shown below:
̇
̈
=
0 1
0 0

̇
+
0

(


+ )
θ
• The same equation for the ball still applies but instead of controlling the 
position through the gear angle (θ) , we will control the torque of the 
beam.
STATE-SPACE (CON’T)
• The state-space representation of the ball and beam example is given 
below:
• We will design a controller for this physical system that utilizes full-state 
feedback control. 
FULL-STATE FEEDBACK CONTROLLER
• A schematic of this type of system is shown below :
• Unlike the previous methods (PID, 
root locus, frequency response) 
where we controlled the gear's 
angle to control the beam and ball, 
here we are controlling a torque 
applied at the center of the beam
by a motor.
• The characteristic polynomial for this closed-loop system is the 
determinant of (s.I-(A-BK)). For our system the A and B*K matrixes are 
both 4x4. Hence, there should be four poles for our system. For our design 
we desire an overshoot of less than 5% and settling time less than 3 
seconds .Hence, after caculating and choosing poles that make system 
suitable, we will place our poles at p1= -2+2j and p2= -2-2j.And certainly, 
we will place the other poles far to the left, so that they will not effect the 
response too much. They are p3=-20 and p4= -80.
• Now that we have our poles we can use MATLAB to find the controller (K 
matrix) by using the place command :
• The results are exactly computed by Matlab soft below: 
m = 0.111;
R = 0.015;
g = -9.8;
J = 9.99e-6;
H = -m*g/(J/(R^2)+m);
A = [0 1 0 0; 0 0 H 0; 0 0 0 1; 0 0 0 0];
• We can now simulate the closed-loop response to a 0.25m step input by 
using the lsim command.Run code below and we should get the following 
plot:
• Now we want to get rid of the steady-state error. We need to compute what 
the steady-state value of the states should be, multiply that by the chosen 
gain K, and use a new value as our reference for computing the input by 
adding a constant gain N_bar after the reference.
t = 0:0.01:5;
u = 0.25*ones(size(t)); 
sys_cl = ss(A-B*K,B,C,D);
[y,t,x] = lsim(sys_cl,u,t); 
plot(t,y) ;
B = [0;0;0;1];
C = [1 0 0 0];
D = [0]; 
ball_ss = ss(A,B,C,D);
p1 = -2+2i; p2 = -2-2i; p3 = -20; p4 = -80; 
K = place(A,B,[p1,p2,p3,p4])
ADDING A CONSTANT GAIN NBAR
• Nbar can be found using the user-defined function rscale.m
• Then, we draw the plot to view the step response with Nbar added.This
work can be do by using some command in Matlab as follow:
• The result is: ( view in next slide)
• Now, we will practice in 
Matlab using supplied command.
Nbar=rscale(ball_ss,K)
t = 0:0.01:5; 
u = 0.25*ones(size(t)); 
[y,t,x]=lsim(Nbar*sys_cl,u,t); 
plot(t,y)
RESPONSE TO A 0.25M STEP INPUT
• Now the steady-state error has been eliminated and all the design criteria 
are satisfied.
SIMULINK MODEL 
Detail every step to realize “ ball & beam model” by simulink in matlab
Step1: Make completely block diagram as follows from basic blocks.We get results:
• Step2: Input elements for matrixes A, B, C, D, K, initial conditions,N_bar , step 
input with to note that size of C is 4x4.
SSPOT=0.07%,T ~0
• Finally, We have response curve to step input, be satisfied to request of 
problem.
• Let s’ watch a Ball and Beam System Simulator
• Watch again another realistic design
CONCLUSION
• The mathematical model for a ball and beam system has been 
derived successfully. The plant is consists of three main 
components which are servo motor model, angle conversion 
gain, and ball on the beam dynamic equation.
• The seting off - criterias are eventually implemented quite 
exactly with unsignificant error(POT= 0,07% and tss~0 with 
standard of 5 %).
• “Ball and beam” is a simple model about designing a system 
apply automatically control theory. Hence this basic will give 
you some idea to develop realistic application.
REFERENCES
• 
• CD-ROM of translators Assoc. Dr. Dao Van Hiep, Department 
of Machine and Robot Institute military technology.
• Some pictures and clips from the internet.
LET S’ JOIN A SUCCESSFUL 
PRESENTATION TODAY
THE END