Robust Stabilizing and Fuzzy Swing up Control For Inverted Pendulum
By
Arslan Tahir
Thesis submitted to the faculty of Engineering in partial fulfillment of requirements for the Degree of MS Systems Engineering.
Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan October 12, 2012
iii
Department of Electrical Engineering Pakistan Institute of Engineering and Applied Sciences (PIEAS) Nilore, Islamabad 45650, Pakistan
Declaration of Originality
I hereby declare that the work contained in this thesis and the intellectual content of this thesis are the product of my own research. This thesis has not been previously published in any form nor does it contain any verbatim of the published resources which could be treated as infringement of the international copyright law. I also declare that I do understand the terms copyright and plagiarism, and that in case of any copyright violation or plagiarism found in this work, I will be held fully responsible of the consequences of any such violation.
Signature: Name: Arslan Tahir Date: October 12, 2012 Place: PIEAS
iv
Certificate of Approval This is to certify that the work contained in this thesis entitled “Robust Stabilizing and Fuzzy Swing up Control For Inverted Pendulum” was carried out by Arslan Tahir under my supervision and that in my opinion, it is fully adequate, in scope and quality, for the degree of MS Systems Engineering from Pakistan Institute of Engineering and Applied Sciences (PIEAS).
Approved By :
Signature: Supervisor: Dr. Muhammad Abid
Verified By:
Signature: Head, Department of Electrical Engineering Stamp:
v
To my parents.
vi
ACKNOWLEDGEMENTS I would like to thank my creator ALLAH (SWT), who created valuable and helpful sources in rigth time and at right place in completion of my thesis.I also want to thank my supervisor Dr.M.Abid who guided in my thesis. I also got benefit fromm experience of my class fellow, M.ahmed, in field of embedded system.I would also like to convey thanks to the Department personnel for providing laboratory facilities and fiscal funds. And also my parents and family memebrs without them i would be nothing.
vii
Contents
Acknowledgements
vi
List of Figures
x
List of Tables
xii
Abstract
xiii
Nomenclature
xiv
1 INTRODUCTION
1
1.1
Applications Of Inverted Pendulum . . . . . . . . . . . . . . . . . .
1
1.2
Rocket Propulsion
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Robot Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.4
Benchmark For Researchers . . . . . . . . . . . . . . . . . . . . . .
2
2 MECHANICS AND MATHEMATICAL MODELING 2.1
3
Mechanics of IP . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1.1
Mechanical Design Of Inverted Pendulum . . . . . . . . . .
3
2.2
Transfer Function Of Motor . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Transfer Function of Cart Pulley System . . . . . . . . . . . . . . .
6
2.4
Transfer Function of Cart and Pendulum . . . . . . . . . . . . . . .
7
2.5
State Space Realization . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6
NonLinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7
An Insight to Mathematical Modeling . . . . . . . . . . . . . . . . . 14
viii
2.8
Mechanical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 ELECTRONICS OF INVERTED PENDULUM
17
3.1
Choice Of Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2
Choice of Controller
. . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1
Quadrature Encoder Interface(QEI) . . . . . . . . . . . . . . 20
3.2.2
PWM module . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.3
Processing Speed . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3
Choice Of Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4
PCB Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5
3.4.1
Motor Drive Card . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.2
Control Card . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Problems With Solutions . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5.1
Bit Bang I 2 C . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5.2
Bounding All States . . . . . . . . . . . . . . . . . . . . . . 28
3.5.3
PIC18f4431 PGM Pin . . . . . . . . . . . . . . . . . . . . . 28
3.5.4
Friction NonLinearity . . . . . . . . . . . . . . . . . . . . . . 28
3.5.5
Shaft Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 SWING UP FUZZY CONTROL 4.1
Design Using Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 REGULATING θ AND CONTROLLING X 5.1
5.2
30
34
SISO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1.1
PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1.2
Loop Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 36
MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2.1
PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.2
LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
ix
6 GRAPHICAL USER INTERFACE 6.1
42
MATLAB GUIDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 CONCLUSION AND FUTURE WORK
45
7.1
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Appendices
47
Appendix A Mathematical Model
48
References
50
About the Author
52
x
List of Figures 2.1
Mechanical structure of inverted pendulum . . . . . . . . . . . . . .
4
2.2
Mechanical structure of inverted pendulum . . . . . . . . . . . . . .
4
2.3
Servo motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.4
Pulley attached with motor . . . . . . . . . . . . . . . . . . . . . .
6
2.5
Inverted pendulum[1] . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.6
Pole zero map of IP . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1
Othogonal signal generated by optical encoder . . . . . . . . . . . . 18
3.2
Dual interrupt circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3
4 states of optical encoder [2] . . . . . . . . . . . . . . . . . . . . . 20
3.4
QEI block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5
Big picture
3.6
PCB designed for IP . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.7
I 2 c 7bit addressing[3] . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8
I 2 C different conditions [3] . . . . . . . . . . . . . . . . . . . . . . . 27
3.9
Offset problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1
Different region of operation . . . . . . . . . . . . . . . . . . . . . . 30
4.2
Switching Of controllers . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3
Fuzzy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4
θ vs θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1
Inverted pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2
Proposed SISO loop
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
0
. . . . . . . . . . . . . . . . . . . . . . . . . . 35
xi
5.3
Loop for bounded θ . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.4
Respose of SISO PID controller . . . . . . . . . . . . . . . . . . . . 36
5.5
Desired loop shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.6
Disturbance rejection . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.7
MIMO configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.8
PID MIMO arrangment . . . . . . . . . . . . . . . . . . . . . . . . 39
5.9
LQG as H2 synthesis[4]
. . . . . . . . . . . . . . . . . . . . . . . . 41
5.10 Results of LQG controller: Tracking x and regulating θ . . . . . . . 41 6.1
Inverted pendulum GUI . . . . . . . . . . . . . . . . . . . . . . . . 43
xii
List of Tables
xiii
Abstract Inverted Pendulum is one of the most famous laboratory equipment to test different control algorithms. It is used as a bench mark to test control schemes like fuzzy control, non-linear control, PID, robust control etc. In this report, complete inverted pendulum project is discussed from hardware design to implementation of robust control algorithms. Hardware design part explains both electronics and mechanics of inverted pendulum whereas control part explains robust control algorithm and their implementation on hardware. Inclusion of dynamics of motor, actuator and pulley system in the inverted pendulum setup make the design of stabilizing controller really a challenging task. Controllers working on simulations needs extra consumption of brain and out of box thinking to counter practical limitations of hardware, and this report will elaborate all such limitations.
xiv
Nomenclature
Symbol IP
Nomenclature Inverted Pendulum
COG
Center of gravity
P ID
Propotional Integral Diffrential
LQG
Linear Quadratic Guassian
SISO
Single input Single Output
M IM O
Multiple Input Multiple Output
GU I
Graphical User Interface
QEI
Quadrature Encoder Interface
P CB
Printed Circuit Board
1
Chapter 1 INTRODUCTION An inverted pendulum, as name implies, is a pendulum which has mass above its pivot point. An inverted pendulum is a popular structure having applications in control engineering that exists in different forms. The main objective of this setup is to balance a mass on one end of a rod controlled by control system using feedback control theory. The simplest realization of inverted pendulum can be though as balancing a broom on ones hand. Broom balancing(Inverted Pendulum on A Cart) is a non-linear, unstable control problem where we consider a flexible broom on a moving cart. The aim is to stabilize the inverted pendulum such that the position of the carriage on the track is controlled quickly and accurately so that the pendulum is always erected in its inverted position during such movements. The inverted pendulum (IP) is among one of the difficult and challenging systems to control in the field of control engineering. Due to its importance in the field of control engineering, it has been a task of choice to be assigned to Control Engineering students to analyze its model and propose a stabilizing controller according to the different control laws.
1.1
Applications Of Inverted Pendulum
Inverted Pendulum problem pooped out in testing and designing of rocket propulsion, robot balancing systems etc.
2
1.2
Rocket Propulsion
Control of an Inverted Pendulum is a Control Engineering problem based on flight simulation of rocket or missile during initial stages of flight. As a rocket flies through the air, it both translates,rotates and move on a desired trajectory. The rotation occurs about a point called the center of gravity(COG). The COG is the location where average weight of the rocket lies. Its mass is distributed throughout the rocket, and for some reasons, it is important to know the distribution of its mass. But for rocket trajectory and maneuvering, we need to be concerned with only the total weight of rocket and the location of the COG.To force the rocket to track the desire path we need a well established control algorithm(3D inverted pendulum).So the subsection of this part i.e. 2D inverted pendulum is implemented in this project.
1.3
Robot Balancing
Robot balancing is also one of the application of inverted pendulum. In this application we are concerned with the center of gravity of robot to be remained under its foot. This robot balancing mechanism can be thought as an inverted pendulum with multiple degrees of freedom.
1.4
Benchmark For Researchers
It is also used as a benchmark for testing control algorithms PID controllers, neural networks,robust controllers (LQG, H∞ ) ,fuzzy control, genetic algorithms.Although it is quiet easy to simulate a given mathematical model in matlab but to implement this setup practically,from hardware design to implementation of control algorithm, makes this task really challenging.
3
Chapter 2 MECHANICS AND MATHEMATICAL MODELING 2.1
Mechanics of IP
Our inverted pendulum structure is similar to cart pulley system. Choice of this structure was to avoid the complexities of mathematical modeling.The Figure (2.1) show the particular arrangement.The motor is attached with the pulley. Belt is used as a source of connection between cart and pulley . With to and fro rotation of pulley cart moves to and fro and can be used to balance the pendulum in upward direction. 2.1.1
Mechanical Design Of Inverted Pendulum
Hardware that we design consist of cart which has capability to slide on two rods. Cart is attached with motor through belt. And pendulum can rotate having pivot point on cart. Figure(2.2) shows particular hardware assembly. The Mathematical modeling of Inverted Pendulum can be divided in three parts i.e. • The transfer function from applied voltage to angular velocity of motor • The transfer function from angular velocity to force appear on cart
4
Figure 2.1: Mechanical structure of inverted pendulum
Figure 2.2: Mechanical structure of inverted pendulum
5
• The transfer function from force appear on cart to angular position of pendulum θ(s) e(s)
=
2.2
ω(s) µ(s) θ(s) e(s) ω(s) µ(s)
So here we will discuss each, one by one.
Transfer Function Of Motor
Figure 2.3: Servo motor The motor used in our project is a DC servo motor. These motors are widely used in control system projects because of low cost and simple dynamics. The differential equation governing the dynamics of servo motor is as follow 00
0
Jθ + Bθ = Ke(t)
(2.1)
Here we are interested in calculating the transfer function between angular velocity 0
ω and the input voltage e(t) so we substitute ω = θ .where as the ratio τ =
J B
is
called the time constant of the motor.J is the inertia and B is the frictional force. By taking the laplace transform and rearranging the equation we get ω(s) K = e(s) τs + 1
(2.2)
Though there are some nonlinearities present in the dynamics of motor equation.2.2 but these nonlinearities are ignored here for the sake of simplicity.
6
K is steady state angular speed of the motor and τ is the time required to attain 60% of angular speed of motor.These both quantities can be calculated easily with the help of optical encoder attached with the motor.
2.3
Transfer Function of Cart Pulley System
In this part we are interested to find the transfer function of pulley i.e the transfer function between angular velocity applied by the motor and as a result force applied on the cart attached to a belt connected with pulley.
Figure 2.4: Pulley attached with motor Figure(2.4) shows the particular arrangement. As we know from modeling 00
theory that for angular motions Js θ = τ where τ is the torque present in the system.( Js is the moment of inertia of pulley.Here for simplicity it is assumed that moment of inertia of whole system is concentrated in the pulley and for pulley we can write moment of inertia as Js = M r2 ) Or simply it can be said as sum of all 00
tourqes present in the system is equal to Js θ . As here we are interested to find the transfer function between the linear motion parameter µ and the rotation motion parameter ω so we will use the relation τ = F r where F is the force present in the belt and r the radius of the pulley. Our present system is not ideal because of the frictional forces present in the system. F = µ − Fb here µ is the force applied on 0
the cart and Fb = Bx is the reaction because of the frictional force in the system.
7
x = rθ is the linear distance. 00
0
Js θ = (µ − Bx )r 00
(2.3)
0
Js θ = (µ − Brθ )r So our final expression for the transfer function of pulley is equation (2.4) µ(s) = M rs + Br ω(s)
2.4
(2.4)
Transfer Function of Cart and Pendulum
In modeling of system from cart to pendulum two equations are of great importance i.e the function governing the angular position w.r.t the force one is the linear position of the cart w.r.t force
X(s) . µ(s)
θ(s) µ(s)
and second
And also the state space
model of this portion will be shown in next chapter.The Figure (2.4) shows the diagram of inverted pendulum to be modeled.
Figure 2.5: Inverted pendulum[1] Define the angle of the rod from the vertical line as θ. Define also the (x, y) coordinates of the center of gravity of the pendulum rod as (xg ,yg ).Then xg = x + lsinθ and yg =lcosθ. Rotational motion of pendulum around the center of gravity is given as
8
00
Jθ = V lsinθ − Hlcosθ
(2.5)
The horizontal and vertical motion of center of gravity of rod can be written as
H=m
d2 (x + lsinθ) dt2
(2.6)
d2 (lcosθ) dt2
(2.7)
d2 x 0 = µ − Bx − H dt
(2.8)
V = mg + m the motion of cart can be modeled as
M
Linearizing the above equations around origin i.e θ=0.We substitute cosθ=1 , sinθ ' θ After Simplification we get following two very important equations. 00
(M + m)x + mlθ = µ − Bx
00
00
(J + ml2 )θ + mlx = mglθ
0
(2.9)
(2.10)
Our next step is to eliminate x from equation 2.9 .Form equation 2.10 we get
x=
mglθ − (J + ml2 )θs2 ml
(2.11)
Substituting value of x in equation 2.9 we get
(M + m)[
mglθ − (J + ml2 )θs2 mglθ − (J + ml2 )θs2 ] + mlθs2 = µ − Bs ml ml
9 Let A= mgl − (J + ml2 )s2 −mls θ(s) = µ(s) (M + m)As + m2 l2 s3 + BA
(2.12)
Further simplification leads us to the following results
θ(s) −mls = 2 3 µ(s) (M J + M ml + mJ)s + B(J + ml2 )s2 − mgl(M + m)s − Bmgl (2.13) As the friction and moment of inertia of rod around its center of gravity can be neglected. So substituting B=0 and J=0 in equation 2.13. We should make above assumption compatible with real hardware.For J=0 mass of rod should be less than mass of bob.And for B=0 we are using linear bearings to reduce friction as much as possible. θ(s) −1 = µ(s) M ls2 − (M + m)g
(2.14)
Although it will be shown later in report that neglecting friction actually creates misconceptions about model of inverted pendulum rather than reducing the complexity.
As the information of linear position of cart is also of interest for us.And in MIMO design we have to track the linear position of cart. So in order to find the transfer function of
X(s) µ(s)
we will use the equation (2.11).Here we get an expression X(s) mglθ − (J + ml2 )s2 = θ(s) mls2
X(s) g (J + ml2 ) = 2− θ(s) s ml
(2.15)
10
The transfer function of
X(s) µ(s)
can be written as X(s) θ(s) X(s) = µ(s) µ(s) θ(s)
(2.16)
X(s) −mls(mgl − (J + ml2 )s2 ) = µ(s) (mls2 )((M J + M ml2 + mJ)s3 + B(J + ml2 )s2 − mgl(M + m)s − Bmgl) (2.17) As again our assumption to simplify the above design B=0 and J=0 we get the final result as −(g − ls2 ) X(s) = 2 µ(s) s (M ls2 − (M + m)g)
2.5
(2.18)
State Space Realization
The classical control theory consists of methods like root locus and nyquist plots. We have been using them, in class to date, for simple transfer function of the plant to obtain our desired response. These methods do not use any knowledge of the interior structure of the plant, and are valid for SISO systems. Limited types of controllers can only be designed with this approach. Modern control theory solves many of the limitations by using a much richer or good description of the plant dynamics i.e state-space realization of system. MIMO systems can easily be handled with this type of approach easily. Coming toward the inverted pendulum case we have to track the position of cart and also regulate the angular position of the pendulum. So it will be quite easy to design a robust controller based on techniques like LQG, H∞ . In this section we will be presenting the state space realization of our inverted pendulum plant. By using equation (2.9) and equation (2.10) we get the following two important equations. (M + m)(J + ml2 ) − (ml)2 00 0 θ = (M + m)gθ − µ + Bx ml
(2.19)
11
(M + m)(J + ml2 ) − (ml)2 00 (ml)2 gθ 0 x = µ − Bx − (J + ml2 ) J + ml2
0
θ1 0 θ 2 x0 1 0 x2
=
0
1 0
0
mgl(M +m) (M +m)(J+ml2 )−(ml)2
0 0
Bml (M +m)(J+ml2 )−(ml)2
0
0 0
1
−(ml)2 (M +m)(J+ml2 )−(ml)2
0 0
−B(J+ml2 ) (M +m)(J+ml2 )−(ml)2
0 −ml (M +m)(J+ml2 )−(ml)2
0 J+ml2 (M +m)(J+ml2 )−(ml)2
(2.20)
θ1 θ2 + x 1 x2
(2.21)
µ
(2.22)
Our interested states that we want to see at output are the angular position of the pendulum and linear displacement of cart.So we choose x1 and θ1 as output states.
θ 1 y1 1 0 0 0 θ2 = y2 0 0 1 0 x1 x2
(2.23)
The above state space model is without actuator dynamics.No by including both the dynamics of pulley and motor we get the following state space model.By using equation (2.4) we get µ = M rsω , + Brω and from equation(2.2) we get 0
ω =
Ke τ
−
ω τ
µ=
M rKe Brτ − M r +ω τ τ
(2.24)
Ke ω − τ τ
(2.25)
0
ω =
12
using equation(2.19), equation(2.20),equation(2.25) and equation(2.24) we get
θ1 0
0 θ2 0 x = 1 0 x2 0 ω1
0
1 0
0
mgl(M +m) (M +m)(J+ml2 )−(ml)2
0 0
Bml (M +m)(J+ml2 )−(ml)2
0
0 0
1
−g(ml)2 (M +m)(J+ml2 )−(ml)2
0 0
−B(J+ml2 ) (M +m)(J+ml2 )−(ml)2
0
0 0
0
0 ml(Brτ −M r) − τ ((M +m)(J+ml 2 )−(ml)2 )
0 (Brτ −M r)(J+ml2 ) τ ((M +m)(J+ml2 )−(ml)2 ) −1 τ
θ1
0
M rKml θ2 − τ ((M +m)(J+ml2 )−(ml)2 ) x + 0 1 M rK(J+ml2 ) x2 τ ((M +m)(J+ml 2 )−(ml)2 ) K ω1 τ
2.6
(2.26)
θ1
θ2 y1 1 0 0 0 0 0 x1 = 0 0 1 0 0 0 y2 x2 ω1
e
(2.27)
NonLinear Model
High nonlinearity is one of the property of Inverted Pendulum . As far as the normal controllers are concerned the linearized plant is used to design a stabilizing control to track the linear position of the cart and regulate the angular position of pendulum, but to design and to analyze the fuzzy control swing up strategy we need the nonlinear model of inverted pendulum.
00
Jθ = V lsin(θ) − Hlcos(θ) − γθ
00
00
0
x (m + M ) + ml(sin(θ)) = µ − Bx
(2.28)
0
(2.29)
13
Above two are the basic nonlinear equations of inverted pendulum. here γ is the friction between the cart and the pendulum i.e the friction present in the rotary joint. where 00
V = mg + mlcos(θ) 00
H = M x + mlsin(θ)
00
00
00
0
cos(θ) = −sin(θ)θ − (θ )2 sin(θ)
00
00
sin(θ) = cos(θ)θ − (θ. )2 sin(θ) simplifying euuation(2.28) we get 00
00
Jθ00 = mglsin(θ) − ml2 θ − mlcos(θ)x − γθ
00
00
θ =
0
0
mglsin(θ) mlcos(θ)x γθ − − J + ml2 J + ml2 J + ml2
(2.30)
simplifying equation(2.29) we get 00
00
0
0
x (m + M ) + θ (mlcos(θ)) = µ − Bx + mlsin(θ)(θ )2 0
00
x =
0
00
µ Bx mlsin(θ)(θ )2 θ (mlcos(θ)) − + − M +m M +m M +m m+M
(2.31)
equation(2.30) and equation(2.31) are the nonlinear state space equations of the inverted pendulum. These two equations are used to test the fuzzy swing up control with detail analysis. Using equation (2.30) and (2.31) and converting them into standard form.
α = (M + m)(J + ml2 ) − (mlcos(x1 ))2
14
0
θ1 = θ2
0
θ2 = α1 ((M + m)mglsin(θ1 ) − mlcos(θ1 )µ + Bx4 mlcos(θ1 )2
−(ml)2 cos(θ1 )sin(θ1 )θ22 − γθ2 (M + m))
0
x 3 = x4
0
x4 = α1 (µ(J + ml2 ) − Bx4 (J + ml2 ) + (J + ml2 )sin(θ1 )θ22
−(ml)2 gsin(θ1 )cos(θ1 ) + mlcos(θ1 )γθ2 ) The above nonlinear state space equations were implemented in the simulink. And corresponding fuzzy control strategy is discussed in the upcoming chapters
2.7
An Insight to Mathematical Modeling
One interesting thing can be observed in the transfer function of equation(2.17) is that it includes unstable dynamics of angular position of cart. Intuition says that if angular position of pendulum is stable then linear position of cart should also be stable. This point can further be demonstrated with the help of following example. If we give impulse as an input to the system, intuition says that after sometime cart should stop beacuase of frictional forces present in the system but here this transfer function shows that dynamics of cart position are unstable i.e response to impulse is unbounded response. The above misleading dynamics are there, because we are using a linearized version of inverted pendulum at θ = 0. And one other important problem is that If we assume that friction in the IP is zero then pole zero map of IP are as shown in Figure(2.6). If we stick to the rule that there should be no pole zero cancellation in right half plane then figure shows that plant is unable to stabilize through classical control techniques because pole
15
and zero is present in right half plane, and root locus will always lie in right half s-plane .But in reality plant friction is present in system so we cannot neglect this parameter in designing the controller for IP, reason!!, presence of friction moves zero of plant from the origon to left side of s-plane. Web Links are provided in apendix where you can get matlab and pi18f4431 codes in soft form and linear and non linear models of IP in simulink.Plant model is available in appendix too.
Figure 2.6: Pole zero map of IP
2.8
Mechanical Parameters
The actual parameters of the mechanical system are calculated and explained in this section First variables are used in MATLAB simulations and second one are
16
used in mathematical modeling.
Mass of Cart
Matlab Variables Modling Variables Value
Units
Mass of Cart
Mc
M
1.25
Kg
Mass of Pendulum
Mp
m
0.50
Kg
Total Mass
M
Mt
1.5
Kg
Length of Pendulum
L
l
0.24
m
Radius of pulley
r
r
0.0235 m
Gravity
g
g
9.8
N/Kg
Time constant of Motor
t
τ
0.13
s
Inertia of COG of IP
J
J
0
Kgm2
Inertia of pulley
Js
Js
0.0069 Kgm2
17
Chapter 3 ELECTRONICS OF INVERTED PENDULUM Coming toward the electronics of the project We have following problems that need efficient solution • Sensing the position and speed of the pendulum. • Sensing the position and speed of the cart. • Manipulate digital data • Digital Framework to implement the digital controller So we provide solution of above problems in following steps • Choice Of Sensors • Choice Of Controller • Choice Of Actuator • Big Picture
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3.1
Choice Of Sensors
We have two choices to measure the position i.e angular and linear. • Potentiometer • Optical Encoder Potentiometers are actually good choice to design an analog controllers. But for digital controllers we need ADC to extract some useful information from them. They have also some very common draw backs that they are sensitive to temperature, they have limited life .Wear and tear is easy in case of potentiometer.Friction and nonlinearities are also present in them. Where as Optical Encoder gives us direct digital output.They give us high resolution .And can be easily interfaced with the digital controller. So we are using 2500 ppr optical encoder of manufacturer named OMRON.
3.2
Choice of Controller
We need a controller which is capable of extracting data form optical encoder and actuate the system with the desired input. As it was discussed earlier that system is actuated with the help of voltage signal given to motor drive card. Where input to motor drive card is PWM signal. We have different methods to extract info from optical encoder through microcontrollers. Optical encoder give us two orthogonal pulses as shown in the Figure (3.1)
Figure 3.1: Othogonal signal generated by optical encoder
19
One way is to use channel A as interrupt signal to microcontroller and increment/decrement the counter depending upon the 1/0 status of channel B.Through this approach we are getting 1x resolution.Another way is that we use a dedicated circuit for both the channel to get interrupt on both rising and falling edges of channel A shown in Figure(3.2).
Figure 3.2: Dual interrupt circuit And channel B is used to extract the information of direction.This approach give us 2x resolution. But there is very serious problem associated with above two methods. If the equipment attached with the optical encoder undergoes too and fro motion, we observe ±offset in the actual reading i.e. offset can be abservered whenever optical encoder will change its direction. So to avoid above problem we should another technique that takes decision on 4 states of orthogonal channels. Figure(3.3) shows the state diagram of optical encoder.4x resolution is achieved through this method.But to apply the above method in controller is not a good idea. Any delay in its processing can cause skipping of counter increment/decrement operation. Dedicated ICs are available to perform the above operation.HCTL2000 is one to them. But we got something better than that. PIC18f4431 is the controller used in our project. Build in modules are present in the microcontroller to interface with the optical encoder.This controller has built in 4 state operation module. It auto increments counter.To actuate the plant we
20
Figure 3.3: 4 states of optical encoder [2] need PWM signals.So we lock overselves on PIC18f4431 which has built in QEI and PWM module and available in market easily. 3.2.1
Quadrature Encoder Interface(QEI)
The Quadrature Encoder Interface (QEI) decodes speed and motion sensor information. It can be used in any application that uses a quadrature encoder for feedback. The interface implements these features: • Three QEI inputs: two phase signals (QEA and QEB) and one index signal (INDX)
21
• Direction of movement detection with a direction change interrupt (IC3DRIF) • 16-bit up/down position counter • Standard and high-precision position tracking modes • Two position update modes (x2 and x4) • Velocity measurement with a programmable post scaler for high-speed velocity measurement • Position counter interrupt (IC2QEIF in the PIR3 register) • Velocity control interrupt (IC1IF in the PIR3 register) The QEI sub-module has three main components: the QEI control logic block, the position counter and velocity post scaler.The QEI control logic detects the leading edge on theQEA or QEB phase input pins, and generates the count pulse which is sent to the position counter logic. It alsosamples the index input signal (INDX), and generates the direction of rotation signal (up/down) and the velocity event signals. The position counter acts as an integrator for tracking distance traveled. The QEA and QEB input edges serve as the stimulus to create the input clock which advances the Position Counter Register (POSCNT). The register is incremented on either the QEA input edge, or the QEA and QEB input edges, depending on the operating mode. It is reset either by a rollover on match to the Period Register, MAXCNT, or on the external index pulse input signal (INDX). An interrupt is generated on a reset of POSCNT if the position counter interrupt is enabled. The velocity postscaler down-samples the velocity pulses used to increment the velocity counter by a specified ratio. It essentially divides down the number of velocity pulses to one output per so many input, preserving the pulse width in the process..Interested readers can furthers refer the original datasheet of this controller.
22
Figure 3.4: QEI block diagram 3.2.2
PWM module
The Power Control PWM module simplifies the task of generating multiple, synchronized pulse width modulated (PWM) outputs for use in the control of motor controllers and power conversion applications. In particular, the following power and motion control applications are supported by the PWM module: • Three-phase and Single-phase AC Induction Motors • Switched Reluctance Motors • Brushless DC (BLDC) Motors • Uninterruptible Power Supplies (UPS) • Multiple DC Brush MotorsThe PWM module has the following features: • Up to eight PWM I/O pins with four duty cycle generators. Pins can be paired to get a complete half-bridge control. • Up to 14-bit resolution, depending upon the PWM period.
23
• On-the-fly PWM frequency changes. • Edge- and Center-aligned Output modes. • Single-pulse Generation mode. • Programmable dead time control between paired PWMs. • Interrupt support for asymmetrical updates in Center-aligned mode. • Output override for Electrically Commutated Motor (ECM) operation; for example, BLDC. • Special Event comparator for scheduling other peripheral events. • PWM outputs disable feature sets PWM outputs to their inactive state when in Debug mode. The Power Control PWM module supports three PWM generators and six output channels on PIC18F2X31 devices, and four generators and eight channels on PIC18F4X31 devices. 3.2.3
Processing Speed
Maximum available speed of controller is 10 MIPS (Mega instructions per second)through PLL module. That is sufficient to implement complex and high order difference equation of robust controller obtained from the techniques like LQG or H∞ .
3.3
Choice Of Actuators
Hbridge is needed to actuate the system with the dc motor. H bridge was made with BUZ11 mosfets with high current ratings and IR2110 is used to open upper side mosfet gates of Hbridge. Other choice was to use lm298 having embedded hbridge. But its current rating was upto 4 Amps. An our motor maximum current is 5Amps, so making Hbridge of our own was preffered.
24
Figure 3.5: Big picture We are using two controllers in our design.The one is responsible to maintain the communication between computer and the plant through RS232. The second one is used to generate and stabilize the plant by interpreting the data and generating actuating signals.Whereas the two controllers remain updated by I 2 C communication protocol.
3.4
PCB Design
The work on PCB design was sliced in two parts. 1. Motor Drive Card 2. Control Card 3.4.1
Motor Drive Card
Motor Drive Card was designed with the capacity of 5Amps rating. Because of high currents this module was designed separately to avoid EMI[8].
25
3.4.2
Control Card
Control Card consists of two controllers. Two optical encoders connects on control card and actuating signals are also generated and sent to motor drive card.
Figure 3.6: PCB designed for IP
3.5
Problems With Solutions
3.5.1
Bit Bang I 2 C
As shown in the bigpicture of complete hardware setup the two controllers PIC18f4431 were connected with I 2 C interface.It was later found that I 2 C mater module is not present in this controller.In order to counter this problem I 2 C was implemented in the firmware to extract the info of second encoder.A brief intro to this protocol will be discussed here and its implementation in pic18f4431. The I 2 C named as Inter-Integrated-Circuit, was originally designed by Philips Inc, to transfer high speed data between ICs at the hardware(PCB) level. The physical layer of this interface consists of two open collector lines and pull ups can be attached depending on desired speed; one line is used as clock (SCL) and
26
other line is used for data (SDA). The data and clock lines are pulled high with 1k resistors in our PCB design to the VDD rail. The bus can have one Master module and many Slave module or may have multiple master modules attached. The master module is used in generating the clock source in both transmission and reception mode. The I 2 C protocol supports 7/10 bit addressing mode, permitting about 128 or 1024 controllers to be communicating on the single bus, respectively. But In practice, the bus reserves some addresses for its own use, so we have slightly fewer addresses available for our use. For example, the 7-bit addressing mode allows 128 usable addresses. The 7-bit addressing protocol is used in our project to trnasmit HIGH and LOW bytes i.e 16 bit word at a time.[3].
Figure 3.7: I 2 c 7bit addressing[3] All data transfers are initiated by the master module and on every clock edge eight bits are transfered, Most Significant bit first. There are no limits to the total size of data that can be sent form one module to other in single go. After each 8bit word transfer, a ninth clock pulse is sent on the bus by master module. At this time instant, the transmitting deve (i.e either master or slave) on the bus releases the data (SDA) line to get acknowledgment of tranmitted data from the reciever
27
. An ACK condition (SDA held low) is sent if the data reception is successfull, or NACK condition (SDA is high) is sent if rececption is unsuccessful. NACK can also be used to terminate data transfer after the reception of last byte. According to I2C protocol, SDA line should changes its state from 0 or 1, when SCL pin is at low. Two unique conditions can be detected on the bus because of above mentioned restriction; (S) a START sequence and a (P) STOP sequence. START sequence occurs when the master module pulls the SDA line low while the SCL line is high. The START sequence tells all Slave devices on the bus that address bytes are about to be sent. The STOP sequence occurs when the SDA line goes high while the SCL line is high, and it terminates the transmission. Slave devices on the bus should reset their receive logic after the STOP sequence has been detected. Because of lack of availability of dedicated master module the
Figure 3.8: I 2 C different conditions [3] operation of master were written in firmware where as slave was configured by
28
just tuning some of the registers of pic18f4431.Interested readers can further refer to the data sheet of pic18f4431. 3.5.2
Bounding All States
As in the previous semester the controller was designed to sbilize the angular position of theta but realized later that even with the balanced theta the state ”‘linear position of cart” becomes unbounded. i.e because of infinite euiliblium point of ”x” exists with θ = 0. So adding a control loop to ”x” solves our problem. 3.5.3
PIC18f4431 PGM Pin
PGM pin of pic18f4431 is usually very sensaative to noise. It was realized after a very tiring effort that the reason the controller was not working is because of PGM pin.And it was properly pull down to ground and controller starts working properly. 3.5.4
Friction NonLinearity
In practical implementation of controllers on IP setup one other important problem was static friction nonlinearity. Bacause of friction a dead band of ±4 volts was present in the system i.e. motor was unable to respond to in this voltage range, and was causing ocillations in balancing the pendulum. Counter to this problem was done in the software. Second problem associated with the friction was that it was not unifrom throughout the rod, so to couter this problem we have to introduce linear bearing in the system. 3.5.5
Shaft Play
Starting routine of our burnt program in microconteoller is that we assign π angle when pendulum is initially at rest in downward position. And then incremental encoder is used to increment or decrement the angle in the program. But what, if there is some play in shaft or some static friction present there in pivot of pendulum
29
and as we are intialiszing our angle with π but in reality angle of pendulum is π −. So instead of balacing the pendulum and 0 angular position our setup will try to balance it to i.e. it will read origon as And cart will attain a constant velocity and will go out of bound.Controller reads the values of Figure(3.9) π − as π and as 0.
Figure 3.9: Offset problem
30
Chapter 4 SWING UP FUZZY CONTROL To bring the pendulum in the upward direction the region of pendulum flight is divided in to three parts as shown in the Figure(4.1).
Figure 4.1: Different region of operation the standup routine should consist of the following subroutines, in this order: 1. Cart is initiated with quik left move to initiate swinging of pendulum. 2. The the critical point of cart movement where maximum energy can transfered to pendulum should be found. 3. As pendulum starts swinging higher,reduce cart actuation amplitude slowly so that the pendulum approaches upside position with small angular velocity.
31
To apply the above algorithm region I is used to add energy in the system i.e actuating signal is applied in this region.In region II actuating signal is removed and pendulum is allowed to deacceleerate which also adds energy and in region III we wait and repeat same procedure in the opposite direction.[11] As the pendulum approaches the inverted position a method of switching between the swing up controller and stabilization controller is required. Different switching criteria are considered in swing up mode. The stabilizing controller is designed to work around the 0 (vertical) position in which it can stabilize the pendulum with the designed stability margins. This is because the stabilizing controller is based on the linearised system about the 0 point. If the stabilizing controller is switched on when the pendulum is outside this region of attraction the system will not be able to stabilize the pendulum. The pendulum must have an angular velocity below a certain value when the stabilization controller switches on. If the angular velocity is too large it will cause the motor to apply a large force to the cart to try and stabilize the pendulum, causing the cart to run out of track. The best switching moment occurs when the pendulum is approaching the 0 vertical position with a small angular velocity. Figure(4.2) shows how both the controllers are connected with each other.
4.1
Design Using Matlab
Fuzzy Logic Toolbox provides MATLAB functions, graphical tools, and a Simulink block for analyzing, designing, and simulating systems based on fuzzy logic. The product guides you through the steps of designing fuzzy inference systems. Functions are provided for many common methods, including fuzzy clustering and adaptive neurofuzzy learning. Figure(4.3) shows the fuzzy surfaces which gives us the insight of actions to be performed for different level of inputs[12]. Figure(4.4) shows the how the angular velocity of inverted pendulum changes with change in angular position of inverted pendulum.
32
Figure 4.2: Switching Of controllers
Figure 4.3: Fuzzy surface
33
0
Figure 4.4: θ vs θ
34
Chapter 5 REGULATING θ AND CONTROLLING X In this chapter SISO and MIMO control strategy will be discussed. As name implies (single input and single output) inverted pendulum can be converted in SISO arrangement with only single input and single regulating output i.e θ. But before going in details of SISO arrangement let try to understand structure of our plant.
Figure 5.1: Inverted pendulum Figure(5.1) shows plant in block diagram. The first thing that can come in readers mind that why not use only stabilizing controller on angular position of inverted pendulum and leave x open. But the problem with the arrangement shown in Figure(5.2) is that the linear position of cart become unbounded. Or it
35
Figure 5.2: Proposed SISO loop is impossible to control the linear position of cart.From state space model it can be seen that state x1 is unobservable from θ1 .Intuitively it can be said that if we are balancing a stick on our hand we can balance a stick but can also attain constant velocity too, if its assumed that there is negligible air drag. Because of missing air drag dynamics our simulation results on the proposed Figure (5.2) arrangement are able to stabilize the angular position of inverted pendulum but linear position of cart become unbounded.
5.1
SISO
In in order to avoid this situation we arrange our block diagram as shown in Figure(5.3).
Figure 5.3: Loop for bounded θ
36
In this arrangement linear position of cart is introduced as disturbance in the loop having intentions to stabilize the angular position of pendulum. With this approach we are able to bound linear position of cart as well as regulating the angular position of pendulum.The block of controller can be controller of any type i.e. PID, Lead/Lag,LQG and H∞ etc. This arrangement helps us to stabilize angular position of of pendulum with hidden bounding loop on angular position of cart. The Matlab code of plant is provided in the Appendix, and simulink models are available on the download links in Apendix for testing.Here the order of our controller is of order 1x1. 5.1.1
PID
PID controller was designed and tuned for Figure(5.3). Figure(5.4) shows the response of PID stabilizing control.
Figure 5.4: Respose of SISO PID controller
5.1.2
Loop Shaping
One of wonders of modern control design theory is H-infinity loop-shaping. Development of this control design technique is based on methods, such as Bode’s sensitivity integral, with H-infinity optimization techniques to design controllers
37
whose stability and performance properties are good in spite of bounded differences between the nominal plant assumed in design and the true plant encountered in practice. The control system designer mention the desired responses (i.e step,impulse or frequency) and noise-suppression properties by applying different weights to the plant transfer function in the frequency domain; then the resulting ’loop-shape’ is used and ’robustified’ through different optimization technique. Robust methods that are applied usually has less effect at high and low frequencies ranges, but its response near unity-gain crossover is adjusted to obtain and maximize the desired system’s stability margins. H-infinity loop-shaping can be applied to multiple-input multiple-output (MIMO) systems. In our SISO configuration we are using it this technique to get our desired response.[14]
Figure 5.5: Desired loop shape
38
Figure 5.6: Disturbance rejection
5.2
MIMO
Comming towards the MIMO design of inverted pendulum.Here the regulating portion of inverted pendulum is as that of SISO but here additional thing is that we also want to track linear position of cart. The order of controller in MIMO design is of 2x1 i.e. 2 inputs and one output. Inputs to our controller are two error signals and out is the voltage signal to regulate angular position of IP and control the linear position of the cart. PID and LQG controller is discussed in the following sections. 5.2.1
PID
The PID control strategy consists of three separate tunable parameters, or it can be said as a controller having three degree of freedom: P the proportional parameter,I the integral parameter and D is for derivative parameter. These values can also be evaluated in terms of time,P has effects on the present error, past errors can be effected by using the parameter I, and derivative term is used for prediction purpose i.e. D. The sum of these three actions followed by gain K is used stabilize the plant.
39
Figure 5.7: MIMO configuration In MIMO arrangement two loops are tuned separately to achieve the desired response of IP. Figure(5.8) shows the particular loop structure.
Figure 5.8: PID MIMO arrangment
5.2.2
LQG
Consider the following plant 0
x = Ax + Bµ + wd
(5.1)
40
y = Cx + wn
(5.2)
The cost function for the LQG problem can be given in the following form as
if we choose
T
Z
1 J = E{ lim T →∞ T
(xT Qx + uT Ru)}
(5.3)
0
1 2
0 x 1 u R2
Q z= 0 then equation(5.3) becomes as
1 J = E{ lim T →∞ T
T
Z
(z(t)T z(t)dt)}
(5.4)
0
H2 norm has different deterministic interpretations and equation(5.4) is also one of them. So equation(5.4) can be written as kFl (P, K)k22
z(s) = Fl (P, K)w(s)
stochastic process can be represented as
wd W = wn 0
1 2
0 w 1 V2
The generalized plant P can be represented as
A W 1 Q2 0 P = 0 0 C 0
1 2
0 0 0 V
1 2
B 0 1 2 R 0
41
if we are interested to control the the plant i.e. we want to track the position of cart so we have to introduce one more row in the generalized plant.
A 1 Q2 P = 0 C
0 W
1 2
0
0
0
0
0
0
0
I
0
V
1 2
B 0 0 0
Figure 5.9: LQG as H2 synthesis[4]
Figure 5.10: Results of LQG controller: Tracking x and regulating θ
42
Chapter 6 GRAPHICAL USER INTERFACE Graphical user interface (GUI) is a type of user interface that allows users to interact with electronic devices with images rather than text commands. Designing the visual composition and temporal behavior of GUI is an important part of any computer application. Its goal is to enhance the efficiency and ease of use for the underlying logical design of a stored program. Methods of user-centered design are used to ensure that the visual language introduced in the design is well tailored to the tasks. Now coming toward the GUI of inverted pendulum setup we have to test different type of control strategies so programming the micro controllers again and again is really a cumbersum task.To avoid these complexities user fascilitatedd visual language is used to ensure the solution of above complexities. Many problems may arise during hardware testing i.e. communication link may fail, sensors are not working properly.To check these problems again and again with ocillioscope and digital multimeter really boils the head of the subject.So by just giving at a look to gui its very easy to locate the error and can be resolved easily
43
6.1
MATLAB GUIDE
GUIDE, the MATLAB graphical user interface development environment provides a set of tools for creating graphical user interfaces (GUIs). These tools simplify the process of laying out and programming GUIs. Using the GUIDE Layout Editor you can populate a GUI by clicking and dragging GUI componentssuch as axes, panels, buttons, text fields, sliders, and so oninto the layout area. You also can create menus and context menus for the GUI. From the Layout Editor, you can size the GUI, modify component look and feel, align components, set tab order, view a hierarchical list of the component objects and set GUI options.
Figure 6.1: Inverted pendulum GUI GUIDE automatically generates a program file containing MATLAB functions that controls how the GUI operates. This code file provides code to initialize the GUI and contains a framework for the GUI callbacksthe routines that execute when a user interacts with a GUI component. Use the MATLAB Editor to add code to the callbacks to perform the actions you want the GUI to perform.
44
The Figure(6.1) shows the gui of the inverted pendulum setup. This gui consists of display panel, control panel , and setting panel. Display panel shows in which mode this IP is operating i.e. in swing up mode or controlling mode etc It also shows connection status and small 2D animation of IP.It will also be capable of displaying different graphs i.e residues step responses. Control panel is used for online updation of difference equations in the micro controller.It really saves our time and reduces complexity of burning code in the controller again and again.We can either write discrete differential equation up to eighth order obtained from different robust control techniques like LQG ,H∞ etc. or use sliders to adjust the PID gain of the system. In hardware projects the generic work always safe your lot of effort and helps you to achieve maximum with minimum effort.So setting panels is provided that helps you to change any parameter in the code i.e. sampling time, cycles per revolutions, gains etc.
45
Chapter 7 CONCLUSION AND FUTURE WORK 7.1
Conclusion
The work presented in the thesis consist of three parts. 1. Mechanical Design of IP 2. Electronic of IP 3. Controller design In implementation phase we observe that controllers designed in MATLAB were unable to provide good stabilizing margins because of missing dynamics, and assumptions that we made in mathematical model. This project highlights both perspectives of good thesis i.e. theory and its implementation. Availability, Cost and Durability; we used these three methods to shortlist our choices and had to compromise between them. Thats why our mathematical model and hardware were unable to equate each other. But despite of these issues we were able to stabilize the pendulum. In implementation of fuzzy control we were facing out of plane vibrations, thats why this work is still pending.
46
7.2
Future Work
This work can be extended to many practical setups. In double inverted pendulum setup our pendulum has one more pivot point at top of it and second pendulum is attached there. And in same way, triple inverted pendulum can be constructed. Another type is varying mass inverted pendulum. In varying mass inverted pendulum, we have long tube filled with fluid behaving as pendulum and fluid discharge with passage of time. This problem is somewhat more similar to rocket propulsion system where mass of rocket also varies. In all above mentioned setups we can validate our modern control theory and research work, and can have insight to practical problems that we face in implementation of controller.
47
Appendices
48
Appendix A Mathematical Model
clc close all clear all numk=1; denk=1;
%% initializations s=0 M=1.250;
%mass of cart Kg
m=0.250;
%mass of pendulum
Mt=M+m;
%total mass of system
l=0.24;
%length of pendulum
r=0.047/2;
%radius of pully
J=0;
%inertia of pendulum
Jpul=Mt∗rˆ2;
Kg
m
i.e 24cm
m
%inertia of complete systrm concentrated on pulley
t=0.1;
%time constant of motor
g=9.8;
%gravity
B=0;
%friction of cart
K=1; %% state space model DDNN=(M+m)∗(J+m∗lˆ2)−(m∗l)ˆ2;
s
49
AA=[0 1 0 0 0; m∗g∗l∗(M+m)/DDNN
0
0 B∗m∗l/DDNN
−m∗l∗(B∗r∗t−Mt∗r)/(t∗DDNN); 0 0 0 1 0; −g∗(m∗l)ˆ2/DDNN 0 0 −B∗(J+m∗lˆ2)/DDNN (B∗r∗t−Mt∗r)∗(J+m∗lˆ2)/(t∗DDNN);0 0 0 0 −1/t] BB=[0
;−Mt∗r∗K∗m∗l/(DDNN∗t) ;0 ;
Mt∗r∗K∗(J+m∗lˆ2)/(t∗DDNN) ; K/t];
CC=[1 0 0 0 0;0 0 1 0 0];
http://sdrv.ms/Q6XLXI simulink linear,nonlinear models and pic18f4431 codes are available here.
http://www.youtube.com/watch?v=7FVDxNI3NAg
actual hardware and result of controllers can be seen here.
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References [1] K. Ogata and Y. Yang, Modern control engineering, vol. 4. Prentice Hall Upper Saddle River, NJ, USA, 1990. [2] A. Technologies, “HCTL 2000 Datasheet.” http://www.boondog.com/ tutorials/mouse/hctl2000.pdf/, 2010. [3] J.
Artiuch,
“I2C
vs
SPI.”
http://dev.emcelettronica.com/
i2c-or-spi-serial-communication-which-one-to-go/, 2010. [4] S. Skogestad and I. Postlethwaite, Multivariable feedback control: analysis and design, vol. 2. Wiley, 2007. [5] B. Kuo and M. Golnaraghi, Automatic control systems, vol. 4. John Wiley & Sons, 2003. [6] R. Bishop and R. Dorf, Modern Control Systems. Prentice Hall College Division, 2004. [7] J. Slotine, W. Li, et al., Applied nonlinear control, vol. 199. Prentice-Hall Englewood Cliffs, NJ, 1991. [8] D. L. Jones, “PCB design Tutorial.” http://www.alternatezone.com/ electronics/files/PCBDesignTutorialRevA.pdf/, 2010. [9] U. of Michigan, “Inverted Pendulum Analysis and Design.” http://www. engin.umich.edu/group/ctm/examples/pend/invpen.html, 1999.
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[10] MicroChip,
“PIC18f4431
Data
Sheet.”
http://ww1.microchip.com/
downloads/en/devicedoc/39616b.pdf, 1999. [11] J. Yi and N. Yubazaki, “Stabilization fuzzy control of inverted pendulum systems,” Artificial Intelligence in Engineering, vol. 14, no. 2, pp. 153–163, 2000. [12] MATLAB,
“Fuzzy
control
toolbox.”
http://www.mathworks.com/
products/fuzzy-logic/, 1999. [13] Microchip, “C18 libraries.” http://ww1.microchip.com/downloads/en/ devicedoc/mplab_c18_libraries_51297f.pdf/, 2010. [14] Wikipedia, “H-infinity loop shaping.” http://en.wikipedia.org/wiki/ H-infinity_loop-shaping, 2010.
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About the Author
Arslan Tahir, live in Lahore, Pakistan. Graduated from FAST(nuces) Lahore in Telecommunication Engineering, Post Graduated from PIEAS Islamabad in System Engineering, having interests in development of elctro-mechanical systems and their instrumentation, with hobbies like web and graphic designing.
