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Applied Mathematical Modelling 48 (2017) 635–654
Contents lists available at ScienceDirect
Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
Takagi–Sugeno fuzzy modelling of some nonlinear problems
using ant colony programming
M.Z.M. Kamali a , N. Kumaresan
b , ∗, Kuru Ratnavelu
b
a Centre for Foundation Studies in Science, University of Malaya, Kuala Lumpur 50603, Malaysia b Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur 50603, Malaysia
a r t i c l e i n f o
Article history:
Received 18 February 2015
Revised 24 February 2017
Accepted 19 April 2017
Available online 28 April 2017
Keywords:
Ant colony programming
Differential equation
Fuzzy modelling
a b s t r a c t
In this paper, the Takagi–Sugeno fuzzy model is derived from the given nonlinear systems.
The objective is to linearize these nonlinear systems into several fuzzy differential equa-
tions according to the Takagi–Sugeno fuzzy rules. The present work implemented the non-
traditional ant colony programming (ACP) method to solve these fuzzy differential equa-
tions. The proposed ACP algorithm manages to give either similar or almost close solutions
to the analytical form. Accuracy of the solution computed by this ACP method is qualita-
tively better when it is compared with other nontraditional approaches such as the genetic
programming (GP) method. Illustrative numerical examples and tables are presented for
Fig. 3. Membership functions M 1 ( z 1 ( t )), M 2 ( z 1 ( t )), N 1 ( z 2 ( t )) and N 2 ( z 2 ( t )).
Since M 1 , M 2 , N 1 and N 2 are fuzzy sets, their values can be calculated by using the following relations
M 1 (z 1 (t)) + M 2 (z 1 (t)) = 1 , (2)
N 1 (z 2 (t)) + N 2 (z 2 (t)) = 1 . (3)
The membership function is named as “Small”, “Big”, “Positive”, “Negative”, respectively and is depicted in Fig. 3 . From
this membership functions, the nonlinear systems can be linearized into the i th rule of continuous T–S fuzzy model of the
following forms. Given the singular non-linear system that can be expressed in the form of T–S fuzzy system:
Model Rule i:
If z 1 ( t ) is M i 1 and z 2 ( t ) is M i 2 , i = 1 , 2 , 3 , 4 , then
E i x (t) = A i x (t) + B i u (t) , x (0) = x 0 , (4)
where M ij indicates the fuzzy set rule of the fuzzy model, r is the number of model rules, the matrix E i is singular, x ( t ) ∈R n is a generalized state space vector and u ( t ) ∈ R m is a control variable. A i ∈ R
n ×n and B i ∈ R
n ×m are known as coefficient
matrices associated with x ( t ) and u ( t ) respectively, x 0 is given initial state vector and m ≤ n . Therefore the nonlinear system
is modelled by the following fuzzy rules where the subsystems are defined as
A 1 =
[0 1
max z 1 (t) max z 2 (t)
], A 2 =
[0 1
max z 1 (t) min z 2 (t)
],
A 3 =
[0 1
min z 1 (t) min z 2 (t)
], A 4 =
[0 1
min z 1 (t) max z 2 (t)
].
If all state variables are measurable, then a linear state feedback control law
u (t) = −R
−1 (B
T i λi (t) + Hx (t)) , (5)
can be obtained to the system (4) and
λi (t) = K i (t) E i x (t) , (6)
where K i (t) ∈ R
n ×n matrix such that K i (t f ) = E T i
SE i . To minimize both state and control signals of the feedback control sys-
tem, a quadratic performance index is minimized:
J =
1
2
∫ t f
t 0
(x T (t) Qx (t) + u
T (t) Ru (t) + 2 u
T (t) Hx (t)) dt, (7)
where the superscript T represents the transpose operator, S ∈ R
n ×n and Q ∈ R
n ×n are symmetric and positive definite
(or semidefinite) weighting matrices for x ( t ), R ∈ R
m ×m is a symmetric and positive definite weighting matrix for u ( t ),
H ∈ R
m ×n is a coefficient matrix. Based on the standard procedure, J can be minimized by minimizing the Hamiltonian
subject to the linear singular fuzzy system R i : If z 1 ( t ) is M i 1 and z 2 ( t ) is M i 2 , i = 1 , 2 , 3 , 4 , then
E i x (t) = A i x (t) + B i u (t) , x (0) = x 0 ,
where the appropriate matrices are substituted in (14)
E i =
[1 0
0 0
], S =
[1 0
0 0
], A 1 =
[0 1
2 1
], A 2 =
[0 1
2 −1
],
A 3 =
[0 1
0 . 1 −1
], A 4 =
[0 1
0 . 1 1
],
B i =
[0
1
], R = 1 , Q =
[1 0
0 0
], H =
[1 0
].
Here, the nonlinear system can be represented by the following fuzzy rules:
Model Rule 1:
If z 1 ( t ) is “Positive” and z 2 ( t ) is “Big”, Then E i x (t) = A 1 x (t) + B i u (t) .
Model Rule 2:
If z 1 ( t ) is “Positive” and z 2 ( t ) is “Small”, Then E i x (t) = A 2 x (t) + B i u (t) .
Model Rule 3:
If z 1 ( t ) is “Negative” and z 2 ( t ) is “Small”, Then E i x (t) = A 3 x (t) + B i u (t) .
Model Rule 4:
If z 1 ( t ) is “Negative” and z 2 ( t ) is “Big”, Then E i x (t) = A 4 x (t) + B i u (t) .
Solution using ACP. In this ACP approach, the construction graph has 18 nodes T = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, t} and F = {+, −,∗, /,(, exp , )}. In the first generation, 50–100 ants are sent to visit 5–10 nodes from any initial nodes randomly until the ants
reach to the limit where the terminal condition is satisfied. Although the value for the fitness function may not be close to
zero, but the path or tour that has been taken by the ants, might lead to the final solution. Therefore after completion of
each generation, a global update of pheromone trail takes place in order to increase the pheromone value on the solution
path. This significant piece of information will be used for the next generation. Thus from 5–10 nodes, the mechanism of
the ACP approach will jump to 6–12 nodes and then if it still does not satisfy the initial conditions and the fitness function,
the process will jump to 7–14 nodes and this process will be repeated several times until the final solution is obtained.
Working on this MRDE problem, the expression is generated randomly up to 16 to 18 nodes, where ρ = 0.5, τ ij (0) = 0.2
and β = 1. Fifty ants are sent out through the graph to find the solution for k 11 ( t ) and another fifty for k 12 ( t ). After the
expression satisfies the terminal condition and the fitness function, thus the solution is obtained. Below we listed down the
trial solutions and the fitness functions obtained in order to find the solution for both k 11 ( t ) and k 12 ( t ).
k 11 :
Tours: k 11 ( t ) = e (2 − t) �⇒ 13th generation, E r = 16
Expressions: k 11 ( t ) = e ( 2 −t )
Tours: k 11 ( t ) = 2 / (e (2 − t) + 1) �⇒ 55 th generation, E r = 6 . 25
Expressions: k 11 ( t ) =
2 e 2 −t +1
Tours: k 11 ( t ) = 3 / (5 ∗ e (4 − 2 ∗ t) − 2) �⇒ 79th generation, E r = 0 . 1111
Expressions: k 11 ( t ) =
3 5 e (4 −2 t) −2
Tours: k 11 ( t ) = 2 / (3 ∗ e (4 − 2 ∗ t) − 1) �⇒ 105 th generation, E r = 0 . 00
Expressions: k 11 ( t ) =
2 3 e (4 −2 t) −1
k 12 :
Tours: k 12 ( t ) = (8 − e (t)) / 7 �⇒ 37th generation, E r = 3 . 781
Expressions: k 12 ( t ) =
8 −e t
7
Tours: k 12 ( t ) = 3 / (e (t) + 2) �⇒ 75th generation, E r = 7 . 554
Expressions: k 12 ( t ) =
3 e t +2
Tours: k 12 ( t ) = 1 − 3 / (5 ∗ e (4 − 2 ∗ t) − 2) �⇒ 97 th generation, E r = 0 . 1111
Expressions: k 12 ( t ) = 1 − 3 5 e (4 −2 t) −2
Tours: k 12 ( t ) = 1 − 2 / (3 ∗ e (4 − 2 ∗ t) − 1) �⇒ 157 th generation, E r = 0 . 00
Expressions: k 12 ( t ) = 1 − 2 3 e (4 −2 t) −1
The parse trees for the solutions k 11 ( t ) and k 12 ( t ) are shown in Figs. 4 and 5 . In Table 2 , the average number of genera-
tions together with the computational time are shown. In comparing the ACP and the genetic programming (GP) methods
[27–30] , in terms of the average number of generations (AVG), we find that the ACP method provides faster solutions com-
pared to the GP method. The numerical solutions are given in Table 3 , whereas the candidate solutions are depicted in
Figs. 6 and 7 . We also depicted the performance of the ACP with the other methods based on their relative errors in Figs. 8
and 9 . Similarly, the MRDE can be solved for the matrices A , A and A .
Fig. 16. Performance of the ACP, GP and RK4 based on the relative error for V ( t ).
R i : If x j is M j , i = 1,...,4 and j = 1,2, then
˙ x (t) = A i x (t) + B i u (t) , x (0) = 0 , t ∈ [0 , t f ] , (22)
where
˙ x (t) =
[
˙ x 1 (t) ˙ x 2 (t) ˙ x 3 (t)
]
=
⎡
⎣
˙ N (t) ˙ Q (t) ˙ S (t)
⎤
⎦ , A i =
[
z 1 0 0
0 μmax 0
0 0 z 2
]
, B i =
[
0
0
1
]
,
z 1 = μmax · Q(t)
1 + Q(t) · S(t) , z 2 = −μmax · Q(t)
(1 + Q(t)) · N ( t)
Y n s
,
R i denotes the i th rule of the fuzzy model, M j is membership function, x ( t ) ∈ R n is a generalized state space vector, u ( t ) ∈R m is a control variable and it takes value in some Euclidean space, A ∈ R
n ×n , B ∈ R
n ×m are known as coefficient matrices
associated with x ( t ) and u ( t ) respectively, and x 0 is given initial state vector and m ≤ n .
The values of x 1 , x 2 and x 3 are taken as x 1 ∈ [0.0, 0.5], x 2 ∈ [0.0, 0.5] and x 3 ∈ [0.0, 0.5]. The value of μmax is given as
9.006 [32] . The min and max values of z 1 and z 2 are calculated as follows:
max z 1 (t) = 0 . 3333 , min z 1 (t) = 0 . 0 ,
max z 2 (t) = 0 . 0 , min z 2 (t) = −0 . 16 6 67 .
From the max and min values, z 1 and z 2 can be represented by
z 1 (t) = M 1 (z 1 (t)) · 0 . 3333 ,
z 2 (t) = N 2 (z 2 (t)) · (−0 . 16 6 67) ,
where
M 1 (z 1 (t)) + M 2 (z 1 (t)) = 1 ,
N 1 (z 2 (t)) + N 2 (z 2 (t)) = 1 .
Therefore the membership functions can be computed
Model Rule 1: If z 1 ( t ) is “Positive” and z 2 ( t ) is “Big”, Then ˙ x (t) = A 1 x (t) + Bu,
Model Rule 2: If z 1 ( t ) is “Positive” and z 2 ( t ) is “Small”, Then ˙ x (t) = A 2 x (t) + Bu,
Model Rule 3: If z 1 ( t ) is ”Negative” and z 2 ( t ) is “Big”, Then ˙ x (t) = A 3 x (t) + Bu,
Model Rule 4: If z 1 ( t ) is “Negative” and z 2 ( t ) is “Small”, Then ˙ x (t) = A 4 x (t) + Bu .
Fig. 19. Performance of the ACP, GP and RK4 based on their relative errors for k 11 ( t ).
the solutions with less average no. of generations and computational time when it was compared to the GP approach. This
shows that the ACP gives faster solutions than the GP algorithm. The performance of the ACP is shown better and very close
to the RK4 method, in terms of the relative errors when it is compared to the GP method.
Acknowledgements
NK and KR would like to acknowledge the funding of this project by the UMRG grant (Account No: RG099/10AFR).
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