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EE7207 equation of motion

创建日期：2024-05-16 15:32:16

所属分类：物理Physical

标签： EE7207

equation of motion

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EE7207 CA2

1. In this problem, we develop a simple model of an inverted pendulum system. In this system, illustrated in Figure
1.1, a rod is hinged on top of a cart. The cart is free to move in the horizontal plane, and the objective is to balance
the rod in the vertical position. Without any control action u on the cart, if the rod were initially in the vertical
position then even the smallest external disturbance on the cart would make the rod lose balance and hence the
system is unstable. The objective is to design control action u so that the rod is maintained in the vertical position,
i.e. θ = 0.

Figure 1.1 Inverted Pendulum on a Cart

The following model parameters are defined to develop the mathematical model of the system: M is the mass of
the cart; m is the mass of the pendulum assumed to be located on the pendulum tip, L is the length of the pendulum,
x is the displacement made by the cart, I is the inertial of the pendulum, θ is the angle in radians of the pendulum
measured from the vertical and is positive in clockwise direction, b is the friction of the cart, u is the control force
applied to the cart.

The mathematical model of the system can be derived from the free-body diagrams of the cart and the pendulum
shown in Figure 1.2. Consider first the motion of the cart in the horizontal direction:

mx u H bx= − − (1)

By summing the forces along the horizontal for the pendulum, we obtain the following equation of motion

( )
2
2
2
sin cos sin
d
H m x L mx mL mL
dt
    = + = + − (2)

Substitute Equation (2) into Equation (1), we obtain,

( ) 2cos sinM m x bx mL mL u   + + + − = (3)

Figure 1.2 Free-body diagrams of the cart and the pendulum

The second equation of motion is obtained by summing all the forces in the vertical direction (note that θ decreases
as y increases) to the pendulum.

( )
2
2
2
cos sin cos
d
V mg m L mL mL
dt
    − = − = + (4)
Multiplying Equation (2) by cos θ and Equation (4) by sin θ, and then adding them, we have

sin cos sin cosV H mg mL mx    + − = + (5)

Also, taking moment about the c.g. of the pendulum, we get

sin cosVL HL I  − − = (6)

Substituting Equation (6) into Equation (5) results in

( )2 sin cosmL I mgL mLx  + + = − (7)

Equations (3) and (7) are the equations of motion describing the nonlinear behaviour of the inverted pendulum on
a cart.

For this problem, we shall assume the following parameters:

M = 1.5 kg
m = 0.3 kg
L = 0.3 m
b = 0.05 Ns/m
I = 0.006 kg m2
g = 9.81 m/s2

a) Let us use Mamdani method to design a fuzzy controller for the inverted pendulum on a cart as shown in Figure
1.3. The input to the controller is the pair ( ),  . Let us take the input space to be X x Y, where X and Y are
intervals representing radians for  and radians per second for  . The fuzzy controller output space U for u is
force in unit of Newtons. The plant dynamics is described by Equations (3) and (7).

Figure 1.3 Fuzzy Controller
The linguistic labels are modelled as fuzzy subsets of the spaces X, Y and U by specifying their membership
functions. For example, these linguistic labels correspond to negative big (NB), negative medium (NM), negative
small (NS), positive small (PS), positive medium (PM) and positive big (PB) as shown in Figures 1.4, 1.5 and 1.6.

Figure 1.4 Fuzzy Sets for Angle 

Figure 1.5 Fuzzy Sets for Angular Rate 

Figure 1.6 Fuzzy Sets for Force u

With 6 membership functions for each of X and Y, we can formulate rules of the form:

: If is and is then is
where , and are the fuzzy subsets of , and depicted above.
j j j j
j j j
R A B u C
A B C X Y U
 

Knowledge of the system behaviour allows us to formulate a set of 36 general rules as shown in the below table
called a fuzzy associative memory or decision table:

Table 1.1 Look-Up Decision Table
Design the membership functions in Figure 1.4 to Figure 1.6 (for example, by using the minimum and maximum
 to be +/- π/2; the minimum and maximum  to be +/- 30/180*π, and the minimum and control u to be +/- 10N
and then adjusting it accordingly) and the fuzzy associative memory rules in Table 1.1, design the fuzzy logic
controller in Figure 1.3 to ensure that the pendulum returns to vertical, θ=0 for the following 4 initial conditions:

No. Angle
 (rad)
Angular Velocity
 (rad/s)
Distance
x (m)
Velocity
x (m/s)
1 -5/180*π -0.5/180*π 0 0
2 5/180*π -1/180*π 10 0
3 5/180*π 1/180*π 0 -0.5
4 -10/180*π 0 20 1

Table 1.2 Different Initial Conditions

Try to adjust the membership function to achieve an allowable pendulum overshoot of 20 degrees and it should
target to settle in under 5 seconds. Plot the angle, angular rate, distance, velocity and the fuzzy controller output
for each case.

b) One potential solution to reduce the number of rules in Table 1.1 is to introduce one additional fuzzy set, namely
approximately zero (AZ) for each of the variable , and .u  The resulting reduced 7 rules are as follows:

Table 1.3 Reduced Look-Up Decision Table
Design the appropriate membership functions for the fuzzy associative memory rules in Table 1.3 to generate a
fuzzy controller in Figure 1.3 and show the results for the 4 initial conditions shown in Table 1.2.
2. Clustering is one of the most fundamental issues in pattern recognition. It plays a key role in searching for structures
in data. Given a finite set of data, X, the problem of clustering in X is to find several cluster centres that can properly
characterize relevant classes of X. In classical cluster analysis, these classes are required to form a partition of X
such that the degree of association is strong for data within blocks of the partition and weak for data in different
blocks. However, this requirement is too strong in many practical applications, and it is thus desirable to replace it with
a weaker requirement. When the requirement of a crisp partition of X is replaced with a weaker requirement of a
fuzzy partition or a fuzzy pseudopartition on X, we refer to the emerging problem area as fuzzy clustering. Fuzzy
pseudopartitions are often called fuzzy c partitions, where c designates the number of fuzzy classes in the partition.
There are two basic methods of fuzzy clustering. One of them, which is based on fuzzy c-partitions, is called a fuzzy c-
means (FCM) clustering method. The other method, based on fuzzy equivalence relations, is called a fuzzy
equivalence relation-based hierarchical clustering method.

Let X = {x1, x2,... , xn} be a set of given data. A fuzzy pseudopartition or fuzzy c-partition of X is a family of fuzzy
subsets of X, denoted by  1 2, , , cP A A A= , which satisfies
( )  
1
1 for all 1,2, ,
c
i k
i
A x k n
=
=  and
( )  
1
0 for all 1,2, ,
n
i k
k
A x n i c
=
  

The goal of the fuzzy c-means (FCM) clustering method is to find a fuzzy pseudopartition P that minimizes the
performance index Jm(P) given by

( ) ( )
( )
2
1 1
1
where (13.1)
1 (13.2)
mn c
m i k k i
k i
n
m
k i k
k
i
J P A x x v
m
x A x
v
= =
=
= −  
 
  
=


( )
1
, 1, 2, , are the cluster centres (13.3)
n
m
i k
k
i c
A x
=
=
  

That is, the clustering problem is an optimization problem. The following fuzzy c-means algorithm was developed
by Bezdek [1981] for solving this optimization problem.

( ) ( )
   
( ) ( )1 1
1,2, , , 1,2,
max
t t t t
i k i k
k n i c
P P A x A x
+ +
 
− = −

(a) Apply Bezdek fuzzy c-means (FCM) clustering algorithm, as shown above, for the following set of data with
15 points each with 2 features:

Table 2.1

Assume that we want to determine a fuzzy pseudopartition with two clusters (i.e., c = 2). Assume further that we
choose m = 1.25,  = 0.01; in Equation (13.1) is the Euclidean distance, and the initial fuzzy pseudopartition
is
( )  0 1 2,P A A= with

1 1 8 9 15
2 1 8 9 15
0.8 / 0.8 / 0.2 / 0.2 / ,
0.2 / 0.2 / 0.8 / 0.8 / .
A x x x x
A x x x x
= + + + + +
= + + + + +

Determine the cluster centres 1 2 and v v after convergence of the iteration procedure.

(b) Next, we shall consider the classical hard c-means (HCM) approach.

In classical HCM, let ( )
iij A j
x  denote the jth data point in the ith class cluster. Now define a partition
matrix U with c rows and n columns, comprising elements ( ) 1,2, , ; 1,2, , .ij i c j n = = We define a
hard c-partition space for X as the following matrix set

 
1 1
M U 0,1 , 1, 0
c n
c ij ik ik
i k
n  
= =
 
=  =   
 
  ,

Any matrix U Mc is a hard c-partition. The cardinality (number of distinct U ) of any hard c-partition, Mc , is

( )M
1
1
1
!c
c
c i n
i
c
i
ic

−
=
   
= −    
    
 ,

where the expression
c
i
 
 
 
is the binomial coefficient of c things taken i at a time. One of the more popular HCM
algorithm is known as a within-class sum of squared errors approach using a Euclidean norm to characterize
distance. This algorithm uses an objective function of the following form:

( ) ( )
2
1 1
min U, , (13.4)
n c
ik ik
k i
J v d
= =
=
dik is a Euclidean distance measure given as follows

( )
1/2
2
1
m
ik k i kj ij
j
d x v x v
=
 
= − = − 
 

between the kth data sample
m
kx  and the ith cluster centre
m
iv  , which is given by

1
1
, 1, ,
n
ik kj
k
ij n
ik
k
x
v j m


=
=

= =



The iterative optimization algorithm (Bezdeck 1981) to solve the problem (13.4) is as follows:

Apply the above iterative optimization algorithm to determine the hard c-cluster solution for the dataset given in
Problem 2(a).

(c) There are two popular methods, among many others, to defuzzify the final fuzzy partitions,
1
c
A
U
A
 
 =
 
  
, that
is, for hardening the fuzzy classification matrix. This defuzzification may be required in the ultimate assignment
of data to a particular class. These two methods are called the maximum membership method and the nearest
centre classifier.

In the max membership method, the largest element uik in each column of the partition matrixU is assigned a
membership of unity and all other elements in each column are assigned a membership value of zero. In
mathematical terms, if the largest membership in the kth column is uik, , then xk belongs to class i, that is, if

 
   
1,2, ,
max 1; 0 for
for 1,2, , , 1, 2, ,
ik jk jk
j c
u u u j i
j c k n

= = = 
 

In the nearest centre classifier, each of the data points is assigned to the class that it is closest to; that is, the
minimum Euclidean distance from a given data point and the cluster centres 1 2 and v v dictate the class
assignment of that point.

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BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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