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Applied Mathematical Modelling 48 (2017) 635–654

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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

comparative purpose.

© 2017 Elsevier Inc. All rights reserved.

1. Introduction

Traditional numerical methods such as Runge–Kutta, Adam–Bashforth, Euler among others have been widely used in the

numerical studies of ordinary differential equations (ODEs). With the ever increasing computational power, these numerical

schemes have seen much success in various real-life problems. In the 20th century, researchers are looking at biological

behaviour and natural selections to develop new nontraditional algorithms to solve engineering problems. These include ant

colony optimization, bees colony, genetic programming, neural networks, etc.

Today, there is a growing trend to develop nontraditional approaches such as genetic programming, neural networks,

ACP and others to obtain accurate numerical solutions of these ODEs. Koza [1] introduced the idea of solving ODE by using

genetic programming. This was followed later by Burgess [2] , who developed an algorithm that utilized GP in order to find

approximate symbolic solutions to solve arbitrary differential equations. Lagaris et al. [3] suggested the neural network algo-

rithm to solve ordinary and partial differential equations. They employed the optimization procedures, to train the network

systems. Further, Balasubramaniam et al. [4–6] have used neural network method for solving the matrix Riccati differential

equations (MRDE). Besides these search procedures, other searching model such as the ant colony optimization (ACO) is

becoming more popular.

The ant colony optimization (ACO) is one of the nontraditional methods that has been used in many combinatorial

optimization problems. The ACO was initially proposed by Marco Dorigo in his PhD thesis [7,8] and it was used to solve

the famous travelling salesman problem. He was inspired by the behaviour of the biological ants seeking path between

their colony and a source of food. Over the years, there has been several extensions and improvised version of ant colony

∗ Corresponding author.

E-mail addresses: [email protected] (M.Z.M. Kamali), [email protected] , [email protected] (N. Kumaresan).

http://dx.doi.org/10.1016/j.apm.2017.04.019

0307-904X/© 2017 Elsevier Inc. All rights reserved.


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636 M.Z.M. Kamali et al. / Applied Mathematical Modelling 48 (2017) 635–654

optimization (ACO) [9–11] . Moreover, researchers have implemented the ant colony algorithm for learning, approximations,

optimization, symbolic regression and multiplexer problems [12–15] . Boryczka [16] also proposed the implementation of ant

colony algorithm to solve approximation problems. He applied ACP as an alternative method for automatic programming

with two different techniques: the expression and the program approach. In the expression approach, the quest for an

approximating function is constructed in the form of an arithmetic expression. These expressions are in prefix notation. In

the second technique, the expression is built from a sequence of assignment instructions which evaluates the function. Later,

Kumaresan et al. [17,18] reported a number of studies that applied the ACP to solve the optimal control for stochastic linear

quadratic singular fuzzy system. In order to obtain the optimal control, the solution of MRDE is computed by solving the

differential algebraic equation using novel and nontraditional ACP approach.

It has been widely known that the fuzzy systems are useful to simulate nonlinear systems and control. There are mainly

two kinds of rule-based fuzzy models: Mamdani fuzzy model and Takagi–Sugeno (T–S) fuzzy model [19] . In the present

work, the T–S fuzzy model is used since it can approximate complex nonlinear systems with fewer rules and higher mod-

elling accuracy. A general T–S fuzzy model employs an affine fuzzy model with a constant term in the consequence. It is

known that smooth nonlinear dynamic systems can be approximated by affine T–S fuzzy models [20,21] . The main feature

of T–S fuzzy models is to represent the nonlinear dynamics by simple (usually linear) models according to the so-called

fuzzy rules and then to blend all the simple models into an overall single model through nonlinear fuzzy membership func-

tions. Each simple model is called a local model or a sub-model. The output of the overall fuzzy model is calculated as a

gradual activation of the local models by using proper defuzzification schemes. It has been proved that T–S fuzzy models

can approximate any smooth nonlinear dynamic systems.

In this paper, the fuzzy differential equations are solved by using ant colony programming. The ant colony programming

is a stochastic search approach that is inspired by the behaviour of real ant colonies. A space graph which consists of the

nodes as representing the functions, variables and the constants, is considered. Functions are normally referred to math-

ematical operations such as the arithmetic operators, operands and Boolean functions. The convergence of the ant colony

algorithm is discussed in [12] . The main objective of the present work is to generate expressions for searching the analytical

solutions for nonlinear and fuzzy differential equations by using the ACP [22] .

In the present work, both the ACP approach and the T–S fuzzy model will be discussed briefly in Sections 2 and 3 ,

respectively. The application of the method together with numerical examples will be discussed in Section 4 , and finally the

conclusion part in the last section.

2. Ant colony programming

Basically the ant colony programming is a method where the ants are send out to find the source of food from its nest.

The searching algorithm is fully random and the ants are scattered everywhere to find their own route to the source of

food. Once the source of food is located and identified, it will analyse the amount and the quality. Then the ant will carry

some of the food to its nest. In this process, the ant will produce some kind of a chemical pheromone trail on the path. This

will somehow give direction to other ants to the food source. The shortest path which is more popular, with highly dense

quantity of pheromone will be the optimal solution compared to the longest path which is less popular. For more details,

the reader is referred to previous work by Kamali et al. [22] .

Fitness function. The fitness function is one of the important steps in the ACP algorithm. Its task is to filter out the best

solution among available solutions. For example if we obtain the MRDE for the singular linear fuzzy system, it is given as

E T i ˙ K i (t) E i + E T i K i (t) A i + A

T i K i (t) E i + Q − (H

T + E T i K i (t) B i ) R

−1 (H + B

T i K i (t) E i ) = 0 ,

with terminal condition K i (t f ) = E T i

SE i . The matrix E i is singular, A i ∈ R

n ×n , B i ∈ R

n ×m and H ∈ R

m ×n are known as coefficient

matrices, whereas R ∈ R

m ×m is a symmetric and positive definite weighting matrix and m ≤ n . The MRDE is solved for K i ( t )

in order to get the optimal solution. After substituting suitable matrices in the above equation, it becomes a differential

algebraic equation (DAE). The DAE for the above equation can be written as

˙ K i j (t) = φi j (K i j (t)) , K i j (t f ) = A i j (i, j = 1 , 2 ...n − 1) ,

K 1 n (t) = ψ(K i j (t)) , K 1 n (t f ) = A 1 n .

Hence, the fitness function is defined as

E r = (K 1 n (t m

) − ψ(K i j (t m

))) 2 +

n −1 ∑

i, j=1

˙ K i j (t m

) − φi j (K i j (t m

)) 2 ,

where m represents the equidistance points in the relevant range [0, t f ].

Terminal criterion. The terminal criterion works side by side with the fitness function. By having this terminal criterion

in the ACP approach, we can decide either to terminate or continue the program. In this process, a number of ants are

randomly sent through all these terminal and function nodes to find a solution. If one of the ants predicts an expression

which satisfies the initial conditions and fulfils the fitness function equal to zero ( E r = 0), then the program will stop.

M.Z.M. Kamali et al. / Applied Mathematical Modelling 48 (2017) 635–654 637

Symbols

Nodes

0 01 12 23 34 45 56 67 78 89 9t 10+ 11* 12 - 13 \ 14( 15 e 16 ) 17

0

0

1

1

2

2

3..

3..

5..

5..

7

7

t

10

+

11

*..

12..

(..

15..

)

17

densityof the pheromonevaluesincrease

Fig. 1. Pheromone density in the space graph.

Table 1

Possible solutions.

Nodes: 3 12 16 15 5 12 10 17

Symbols: 3 ∗ e ( 5 ∗ t )

Nodes: 4 12 16 15 2 11 10 17

Symbols: 4 – e ( 2 + t )

Nodes: 16 15 5 12 10 17 13 3

Symbols: e ( 5 ∗ t ) – 3

Nodes: 7 12 10 12 16 15 10 17

Symbols: 7 ∗ t ∗ e ( t )

However, if it gives an expression which satisfies the initial conditions and gives the fitness function close to zero ( E r → 0)

(let say E r = 1.0E-2), then the global updating rules will be applied. This process continues to update the pheromone values

in all over the edges in the space graph and this information will be passed to the next generation. The process will be

repeated many times until the generated expression makes the fitness function either equal or close to zero ( E r = 1.0E-4).

2.1. ACP methodology

Artificial ants are sent out randomly to find solutions through the connected graph G (V , E ) where the vertices ( V ) indi-

cate the Functions and Terminals whereas the set ( E ) symbolizes the edges which connected the vertices. The ants travel by

applying a stochastic local decision policy that utilizes the combination of pheromone trails and the heuristic information.

The ants move from one node to another in the space graph, for example from node r to node s at time t by following the

probability law [23] . For more details, the proposed method can be referred in the previous work by Kamali et al. [22] .

Each of this digital ant stores the data of its journey in a working memory which can be represented programmatically by

a parse tree structure. In order to obtain a converged solution, tours that cannot be represented in this parse tree structure

can be eliminated. This will save computational time. After completion of each generation, a global update of pheromone

trail takes place and the level of pheromone is replaced as follows:

τi j (t + g) = (1 − ρ) · τi j (t) + ρ · 1

L ,

where g is the number of generations, edges ( i, j ) belong to the optimal tour found so far, (1 − ρ) , ρ ∈ (0,1], is the

pheromone decay coefficient ( ρ is the concentration of pheromone on edge over time, ρ = 0.5), τ ij is the amount of

pheromone trial on edge ( i, j ) and L is the length of this tour. The aim of the pheromone value update rule is to increase

the pheromone values on the solution path.

In Fig. 1 , we depicted the data for the quantity of the pheromone values in each edge after the ants completed their

tours through 8 nodes. Initially, the pheromone values are equally distributed throughout all the edges ( τi j = 0 . 2 ). As the

ants move throughout all the edges, path which satisfies the initial conditions will be more popular than the others. The

best path is the one which satisfies not only the initial conditions but also the fitness function. We depicted all these by

showing the differences in terms of colours. There are five colours shown in the space graph: red, green, yellow, purple and

blue. Initially, the pheromone is distributed equally through out all the edges in the space graph with the blue colour. Those

paths with red colour show the highest density or the most favourable edges chosen by the ants. The favourable edges are

listed down in Fig. 2 and possible solutions are shown in Table 1 .


Edges

(16,15), (10,17)

(12, 16), (12,10), (10,17)

(15,5), (5,12)

(3,12), (4,12), (7,12)(10,12), (15, 2), (2,11)(11,10), (13,3), (17,13)(15,10)

densityof the pheromonevaluesincrease

Others

Pheromone density

Fig. 2. List of the most favourable edges.

3. Takagi–Sugeno fuzzy model

The main objective of using Takagi–Sugeno (T–S) fuzzy model is to express the local dynamics of each fuzzy implica-

tion (rule) by a linear system model. The T–S fuzzy model was proposed by Takagi and Sugeno [19] . It was well described

by fuzzy IF-THEN rules which represent local input–output relations of a nonlinear system. The overall fuzzy model of the

system is achieved by fuzzy “blending” of the linear system models. There are two different approaches to theoretically

construct a T–S fuzzy model. First is from the local linear approximation, which generates a linear consequent part with a

constant term included in each rule; the other one is through the sector nonlinearity concept which leads to a constant-

free linear consequence for each rule. Both methods demonstrate to be universal approximations to any smooth nonlinear

systems. Basically a fuzzy system consists of linguistic IF-THEN rules that have fuzzy antecedent and consequent parts. It is

a static nonlinear mapping from the input space to the output space. The inputs and outputs are crisp real numbers and

not fuzzy sets. The fuzzification block converts the crisp inputs to fuzzy sets and then the inference mechanism uses the

fuzzy rules to produce fuzzy conclusions or fuzzy aggregations and finally the defuzzification block converts these fuzzy

conclusions into crisp outputs. The fuzzy system with singleton fuzzifier, product inference engine, center average defuzzi-

fier and Gaussian membership functions is called the standard fuzzy system [24] . Two main advantages of fuzzy systems

for the control and modelling applications are (i) overcoming the difficulty to express mathematical model using uncer-

tain or approximate reasoning and (ii) decision making problems with the estimated values under incomplete or uncertain

information [25,26] .

4. Experiments and results

In this section, we discuss and compare the proposed ACP method with other numerical techniques such as the genetic

programming (GP) method [27–29] and the Runge–Kutta 4th order method.

4.1. Takagi–Sugeno fuzzy modelling for solving matrix Riccati differential equation (MRDE)

Given the singular non-linear system as

E x 1 = A (x ) x (t) + Bu (t) , x (0) = x 0 , (1)

where the matrix E is singular, x ( t ) ∈ R n is a generalized state space vector and u ( t ) ∈ R m is a control variable, A ∈ R

n ×n

and B ∈ 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 . In order to derive the T–S fuzzy model from the above equation, the first step is to determine the membership

functions. For simplicity, the matrix A ( x ) is given by

A (x ) =

[0 1

x 1 (t) x 2 (t)

]and let x 1 ∈ [0.1, 2] and x 2 ∈ [ −1 , 1] . The x 1 and x 2 are nonlinear terms and we treat them as fuzzy variables. Generally

they are denoted as z 1 and z 2 , that may be functions of state variables, input variables, external disturbances and/or time.

By calculating the max and min values of z 1 and z 2 , the membership function can be obtained as given below:

max (z 1 (t)) = 2 , min (z 1 (t)) = 0 . 1 ,

max (z 2 (t)) = 1 , min (z 2 (t)) = −1 .

Therefore x 1 and x 2 can be represented for the membership functions M 1 , M 2 , N 1 and N 2 as follows:

z 1 (t) = x 1 (t) = M 1 (z 1 (t)) · max (z 1 (t)) + M 2 (z 1 (t)) · min (z 1 (t)) ,

z 2 (t) = x 2 (t) = N 1 (z 2 (t)) · max (z 2 (t)) + N 2 (z 2 (t)) · min (z 2 (t)) .


0

1

0

1

Negative

min z2(t) max z2(t)max z

1(t)

N1(z2(t ))N2(z2(t ))M1(z1(t ))M2(z1(t ))

Positive Big

min z1(t)

Small

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


equation

H (x (t) , u (t) , λi (t)) =

1

2

x T Qx (t) +

1

2

u

T (t ) Ru (t ) + u

T (t ) Hx (t )

+ λT i (t)[ A i x (t) + B i u (t)] . (8)

The necessary conditions for optimality is

∂H

∂u

(x, u, λi , t) = 0 ,

implies that Ru (t) + Hx (t) + B T i λi (t) = 0 and

Ru (t) = −(H + B

T i K i (t) E i (t)) x (t) ,

u = −R

−1 (H + B

T i (t) K i (t) E i (t)) x (t) ,

∂H

∂x = E T i

˙ λi (t) ,

⇒ E T i ˙ λi (t) = −Qx (t) − H

T u (t) − A

T i λi (t) ,

∂H

∂λ= E i x (t) , (9)

⇒ E i x (t) = A i x (t) + B i u (t) . (10)

From (5) , we have

E i x (t) = A i x (t) − B i R

−1 (B

T i λi (t) + Hx (t)) , (11)

whereas (9) and (11) can be written in a matrix form as follows: [E i 0

0 E T i

][˙ x (t)

˙ λi (t)

]=

[A i − B i R

−1 H −B i R

−1 B

T i

−Q + H

T R

−1 H H

T R

−1 B

T i

− A

T i

][x (t) λi (t)

],

where x (0) = x 0 and E T i λi (t f ) = E T

i SE i x (t f ) . Assuming that | R | � = 0, from (6) we have

˙ λi (t) =

˙ K i (t ) E i x (t ) + K i (t) E i x (t) ,

and

E T i ˙ λi (t) = E T i

˙ K i (t ) E i x (t ) + E T i K i (t)(A i x (t) + B i u (t)) . (12)

By substituting (9) and (11) in (12) , we obtain

[ E T i ˙ K i (t) E i + E T i K i (t) A i + A

T i K i (t) E i + Q −

(H


−1 (H + B

T i K i (t) E i )] x (t) = 0 . (13)

Since (13) holds for all non-zero x ( t ), the term pre-multiplying x ( t ) must be zero. Therefore, we obtain the following MRDE

for the singular linear fuzzy system (4)

E T i ˙ K i (t) E i + E T i K i (t) A i + A

T i K i (t) E i + Q −

(H


−1 (H + B

T i K i (t) E i ) = 0 , (14)

with terminal condition K i (t f ) = E T i

SE i . The derivation given in this paper is based on the previous work done by Balasub-

ramaniam et al. [4] .

Solution of MRDE. The MRDE is solved in order to get the optimal solution for K i ( t ). In the solution matrix K i ( t ), the value

of k 22 ( t ) is free

˙ k 11 = f 1 (t, k 11 , k 12 ) , (15)

˙ k 12 = f 2 (t, k 11 , k 12 ) . (16)

In this paper, the above system will be solved by using the non-traditional ACP method.

Consider the optimal control problem: Minimize

J =

1

2

∫ t f

t

(x T (t) Qx (t) + u

T (t) Ru (t) + 2 u

T (t) Hx (t)) dt,
0


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


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


2 3 e (4 −2 t) −1

k 12 :

Tours: k 12 ( t ) = (8 − e (t)) / 7 �⇒ 37th generation, E r = 3 . 781


8 −e t

7

Tours: k 12 ( t ) = 3 / (e (t) + 2) �⇒ 75th generation, E r = 7 . 554


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 .
2 3 4


/

2

-

1

e

*

t

4

-

*

2

3

Fig. 4. Parse tree for k 11 ( t ).


Table 2

Comparison results for k 11 ( t ) and k 12 ( t ) between ACP and

GP.

k 11 ( t ) Average no. generations Average time (s)

ACP 98 120.53

GP 521 524.23


ACP 105 135.77

GP 545 645.34

4.2. Takagi–Sugeno fuzzy modelling for solving the human immunodeficiency virus (HIV) immunology model

The equations for the HIV immunology model used here is based on Kirschner and Webb [31] . The equations are given

as follows:

dT (t)

dt = s 1 − s 2 V (t)

b 1 + V (t) − μT (t) − kV (t ) T (t ) , T (0) = T 0 , (17)

dV (t)

dt =

gV (t)

b + V (t) − cV (t) T (t) , V (0) = V 0 . (18)

2


Table 3

Results obtained by ACP, GP, RK4-method and exact solutions.

ACP GP RK4-method Exact

t k 11 ( t ) k 12 ( t ) k 11 ( t ) k 12 ( t ) k 11 ( t ) k 12 ( t ) k 11 ( t ) k 12 ( t )

0.0 0.01229 0.98772 0.01185 0.98815 0.01228 0.98771 0.01229 0.98772

0.2 0.01838 0.98162 0.01774 0.98226 0.01839 0.98161 0.01838 0.98162

0.4 0.02755 0.97245 0.02659 0.97341 0.02755 0.97245 0.02755 0.97245

0.6 0.04138 0.95862 0.03996 0.96004 0.04139 0.95861 0.04138 0.95862

0.8 0.06236 0.93764 0.06027 0.93973 0.06237 0.93762 0.06236 0.93764

1.0 0.09449 0.90551 0.09142 0.90858 0.09450 0.90550 0.09449 0.90551

1.2 0.14431 0.85569 0.13988 0.86012 0.14433 0.85567 0.14431 0.85569

1.4 0.22321 0.77679 0.21696 0.78304 0.22324 0.77676 0.22321 0.77679

1.6 0.35232 0.64768 0.34407 0.65593 0.35237 0.64763 0.35232 0.64768

1.8 0.57546 0.42454 0.56655 0.43345 0.57549 0.42451 0.57546 0.42454

2.0 1.0 0 0 0 0 0.0 0 0 0 0 1.0 0 0 0 0 0.0 0 0 0 0 1.0 0 0 0 0 0.0 0 0 0 1.0 0 0 0 0 0.0 0 0 0 0

Fig. 6. Candidate solutions for k 11 ( t ) .

Fig. 7. Candidate solutions for k 12 ( t ).


0.0 0.4 0.8 1.2 1.6 2.0

0.0

1.0x10-2

2.0x10-2

3.0x10-2

4.0x10-2

k11

( t )

Rel

ativ

e E

rror

t

ACP GP RK4

Fig. 8. Performance of the ACP, GP and RK4 based on the relative error for k 11 ( t ).

Fig. 9. Performance of the ACP, GP and RK4 based on the relative error for k 12 ( t ).

Table 4

The parameters and their units used in the HIV immunology model.

Parameters Definitions

T Uninfected CD 4 + T cell population

V HIV population

s 1 = 2 . 0 mm

3 day −1

Source of CD 4 + T cells

s 2 = 1 . 0 mm

3 day −1

Source of HIV cells

μ = 0 . 1 day −1

Death rate of uninfected CD 4 + T cell

k = 0 . 1 mm

3 day −1

Rate CD 4 + cells which get infected by the virus V

g = 2 day −1

mm

3 Input rate of external viral source

c = 0 . 1 mm

3 day −1

Lost rate of virus

b 1 = 2 . 0 mm

3 Half saturation constant

b 2 = 1 . 0 mm

3 Half saturation constant
The parameters of HIV model are given in Table 4 . Substituting these parameters in (17) and (18) , the HIV immunology
can be written as

dT (t)

dt = 2 − V (t)

2 + V (t) − 0 . 1 T (t) − 0 . 1 V (t ) T (t ) ,

dV (t)

dt =

2 V (t)

1 + V (t) − 0 . 1 V (t) T (t) .


Fig. 10. Membership functions M 1 ( z 1 ( t )), M 2 ( z 1 ( t )), N 1 ( z 2 ( t )) and N 2 ( z 2 ( t )).

Thus, we can derive the Takagi–Sugeno fuzzy model for this nonlinear systems. The nonlinear systems can be written in

matrix form as: (ˆ T )

=

[−0 . 1 z 1 (t) 0 z 2 (t)

](T

),

where

z 1 (t) = f (T , V ) =

4 + V

2 V + V

2 − 0 . 1 T ,

z 2 (t) = g(T , V ) =

2

1 . 0 + V

− 0 . 1 T .

The next step is to obtain the membership functions. We take that T ( t ) ∈ [0.1, 1] and V ( t ) ∈ [0.1, 1]. Thus we obtain the

min and max values for z 1 ( t ) and z 2 ( t ).

max z 1 (t) = 19 . 5138 , min z 1 (t) = 1 . 5667 ,

max z 2 (t) = 1 . 8082 , min z 2 (t) = 0 . 9 .

From these max and min values, the z 1 ( t ) and z 2 ( t ) can be represented as

z 1 (t) =

4 + V

2 V + V

2 − 0 . 1 T = M 1 (z 1 (t)) · max z 1 (t) + M 2 (z 1 (t)) · min z 1 (t) ,

z 2 (t) =

2

1 . 0 + V

− 0 . 1 T = N 1 (z 2 (t)) · max z 2 (t) + N 2 (z 2 (t)) · min z 2 (t) .

The membership function is depicted in Fig. 10 . Thus, from this membership function, the nonlinear systems can be lin-

earized into these fuzzy differential equations according to the following continuous T–S fuzzy rules with T (0) = 1 and

V (0) = 1:

Model Rule 1: If z 1 ( t ) is “Positive” and z 2 ( t ) is “Big”, Then

ˆ T = A 1 T .

Model Rule 2: If z 1 ( t ) is “Positive” and z 2 ( t ) is “Small”, Then

ˆ T = A 2 T .

Model Rule 3: If z 1 ( t ) is “Negative” and z 2 ( t ) is “Small”, Then

ˆ T = A 3 T .

Model Rule 4: If z 1 ( t ) is “Negative” and z 2 ( t ) is “Big”, Then

ˆ T = A 4 T .

where the subsystems are defined as

A 1 =

[−0 . 1 max z 1 (t)

0 max z 2 (t)

], A 2 =

[−0 . 1 max z 1 (t)

0 min z 2 (t)

],

A 3 =

[−0 . 1 min z 1 (t)

0 min z 2 (t)

], A 4 =

[−0 . 1 min z 1 (t)

0 max z 2 (t)

],


0.2 0.4 0.6 0.8 1.0

10

20

30

40

50

60

70

80

90

0.6 0.8 1.0

20

30

40

50

t

T20 T35 T39 T67 Exact

T

Fig. 11. Candidate solutions for T ( t ).

which is

A 1 =

[−0 . 1 19 . 5138

0 1 . 8082

], A 2 =

[−0 . 1 19 . 5138

0 0 . 9

],

A 3 =

[−0 . 1 1 . 5667

0 0 . 9

], A 4 =

[−0 . 1 1 . 5667

0 1 . 8082

].

Consider the Model Rule 1: If z 1 ( t ) is “Positive” and z 2 ( t ) is “Big”, then (˙ T ˙ V

)=

[−0 . 1 z 1 (t)

0 z 2 (t)

](T V

),

and the initial values is given as T (0) = 1 and V (0) = 1. In this case, we are solving for: (˙ T ˙ V

)=

[−0 . 1 19 . 5138

0 1 . 8082

](T V

).

The ACP is implemented to solve the above equation.

Solution using ACP. By generating the graph randomly, 80 generations with 50–100 number of ants each are sent out through

the graph with ρ = 0 . 5 and β = 1 . In Figs. 11 and 12 , the evolution of a trial solutions for the above problem is shown. These

trial solutions are compared with the exact solutions given as:

T (t) =

9756905

954091

e 904091 50 0 0 0 0 t − 8802814

954091

e −1 10 t ,

V (t) = e 904091 50 0 0 0 0 t .

At generation 20, with fitness value equal to 0.3437, the intermediate solution is:

T ours : T (t) = 5 ∗ 2 ∗ e (2 ∗ t) − 9 ,

Expressions : T (t) = 10 e 2 t − 9 .

Next, at the 35 th generation, the fitness value is 0.1492. This actually predicted the corresponding candidate solution which

is:

T ours : T (t) = 5 ∗ 2 ∗ e (9 / 5 ∗ t) − 9 ∗ e (0 − 1 / 5 ∗ t) ,


Fig. 12. Candidate solutions for V ( t ).

Expressions : T (t) = 10 e 9 5 t − 9 e

−1 5 t .

Later, at the 39 th generation, the ACP computes the solution with fitness value = 0.0075 and its functional form given as

T ours : T (t) = 5 ∗ 2 ∗ e (9 / 5 ∗ t) − 9 ∗ e (0 − 1 / 6 ∗ t) ,

Expressions : T (t) = 10 e 9 5 t − 9 e

−1 6 t .

Finally, at the 67 th generation, the ACP computes the solution with the fitness function 0.0025. The final solution is given

as

T ours : T (t) = (sqr(5) ∗ 2 + 1) / 5 ∗ e ((sqr(6 ∗ 5) + 4 ∗ t ) / (sqrt (5 ∗ 2) ∗ 5))

−(5 ∗ 9 + 1) / 5 ∗ e (0 − t/ 9) ,

Expressions : T (t) =

51

5

e 904 500 t -

46

5

e −t 9 .

Similarly the ACP method is applied to obtain the solution for V . At generation 15, the fitness value is equivalent to

0.0368, the intermediate solution is

T ours : V (t) = e (2 ∗ t) ,

Expressions : V (t) = e 2 t .

Then another candidate solution is predicted at 23 rd generation with

T ours : V (t) = e (9 / 5 ∗ t) ,

Expressions : V (t) = e 9 5 t ,

where the fitness value is less than 7E −5. The final solution is only achieved when the ACP reach at the 44 th generation

with its functional form as given below:

T ours : V (t) = e ((sqr(6 ∗ 5) + 4 * t ) / (sqrt (5 ∗ 2) ∗ 5)) ,

Expressions : V (t) = e 904 500 t .

The fitness value is given as 4E −8. The parse trees for the solutions T and V are shown in Figs. 13 and 14 . The numerical

solutions are given in Table 5 whereas the average number of generations together with the computational time are depicted

in Table 6 . In the systems of ODE’s, solutions obtained are more than one, therefore the task for searching the solution using

this non-traditional method will be very difficult. From the table, the ACP method succeed in finding the solutions within


Fig. 13. Parse trees for T ( t ).

Fig. 14. Parse trees for V ( t ).

the range of 90–150 average no. of generations but the GP method computes the solutions within the range of 190–260. The

performance of the ACP with the other methods are also shown in Figs. 15 and 16 , based on their relative errors. Similarly

the solution for the HIV for fuzzy model rule 2, 3 and 4 can be obtained by using ant colony programming.

4.3. Takagi–Sugeno fuzzy modelling of S-type microbial growth model for ethanol fermentation process and optimal control

The development of mathematical models in the field of predictive microbiology to describe and predict the microbial

evolution in foods are very vital. Single species microbial growth, whether in a (liquid) food product, normally passes three

distinct phases. In the first phase called lag phase, the microbial cells adapt to their new environment and do not multiply.

The total number of microbial cells remains constant during this phase. During the next phase or the exponential growth

phase, the microbial cells multiply exponentially. Finally, the microbial cells cease multiplying and their total number re-


Table 5


ACP solution GP solution RK4 solution Exact solution

t T V T V T V T V

0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.1 3.12305 1.19818 3.11441 1.20249 3.11865 1.19820 3.11865 1.19820

0.2 5.64556 1.43562 5.63793 1.44597 5.63810 1.43568 5.63810 1.43568

0.3 8.64694 1.72013 8.65359 1.73876 8.63800 1.72023 8.63797 1.72022

0.4 12.22232 2.06102 12.26125 2.09084 12.21373 2.06118 12.21369 2.06117

0.5 16.4 856 8 2.46946 16.58093 2.51420 16.47963 2.46971 16.47956 2.46969

0.6 21.57359 2.95885 21.75701 3.02329 21.57267 2.95920 21.57255 2.95917

0.7 27.64967 3.54522 27.96308 3.63546 27.65698 3.54572 27.65678 3.54567

0.8 34.90999 4.24779 35.40787 4.37159 34.92919 4.24847 34.92890 4.24841

0.9 43.58946 5.08960 44.34238 5.25678 43.62493 5.09052 43.62451 5.09044

1.0 53.96951 6.09824 55.06845 6.32120 54.02653 6.09946 54.02593 6.09935

Table 6

Comparison results for T ( t ) and V ( t ) between the ACP

and GP.

T ( t ) Average no. generations Average time (s)

ACP 146 150.53

GP 257 171.45

V ( t ) Average no. generations Average time (s)

ACP 95 95.77

GP 192 163.23

0.0 0.4 0.8

0.0

5.0x10-3

1.0x10-2

1.5x10-2

2.0x10-2T(t)

Rel

ativ

e E

rror

t

ACP GP RK4

Fig. 15. Performance of the ACP, GP and RK4 based on the relative error for T ( t ).

mains constant at the max population density. This third final phase is called the stationary phase. The S-type microbial

growth model below is given in the paper by Van Impe et al. [32] .

˙ N (t) =

(Q(t)

1 + Q(t)

)· μmax · S(t) · N (t) with N (t = 0) = N 0 , (19)

˙ Q (t) = μmax · Q (t) with Q (t = 0) = Q 0 , (20)

˙ S (t) = −(

Q(t)

1 + Q(t)

)· μmax · S(t)

Y N S

· N(t) with S(t = 0) = N 0 . (21)

The first differential equation describes the evolution of the microbial load in time. It consists of the adjustment func-

tion which describes the lag phase by means of a variable representing the physiological state of the cells Q ( t ), as well

as the inhibition function which is a linear function of the substrate concentration S ( t ). The second differential equation

(20) describes the evolution of Q ( t ), which increases exponentially, whereas the third differential equation (21) represents

the evolution of the substrate concentration S ( t ). The T–S fuzzy model can be derived from the above nonlinear system

using sector nonlinearity approach [33] . Consider the linear dynamical fuzzy system [34] that can be expressed in the form:


0.0 0.4 0.810-6

10-5

10-4

10-3

10-2

V(t)

Rel

ativ

e E

rror

t

ACP GP RK4

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 .

where

S =

[1 0

0 0

], A 1 =

[

0 . 3333 0 0

0 9 . 006 0

0 0 0

]

, A 2 =

[

0 . 3333 0 0

0 9 . 006 0

0 0 −0 . 16 6 67

]

,


Fig. 17. Candidate solutions for k 11 ( t ).

A 3 =

[

0 0 0

0 9 . 006 0

0 0 0

]

, A 4 =

[

0 0 0

0 9 . 006 0

0 0 −0 . 16 6 67

]

,

B i =

[

0

0

1

]

, R = 0 , Q =

[

1 0 0

0 0 0

0 0 0

]

.

By following the standard procedures given in Section 4.1 from Eqs. (5) –(14) , the relative MRDE for the linear T–S fuzzy

system (22) is

˙ K i (t) + K i (t) A i + A

T i K i (t) + Q − K i (t) B i R

−1 B

T i K i (t) = 0 , (23)

with terminal condition(TC) K i (t f ) = S. This equation is to be solved for K i ( t ) in the next section for the optimal solution.

After substituting the appropriate matrices in the above equation, they are transformed into system of nonlinear differential

equations. Therefore solving MRDE is equivalent to solving the system of nonlinear differential equations. The numerical

implementation could be adapted by taking t f = 2 for solving the related MRDE of the above linear system. The appropriate

matrices are substituted in (23) and the MRDE becomes a differential algebraic equation (DAE). The numerical solutions of

MRDE are calculated and displayed in Table 8 . Similarly the solution of the above system with the matrix A 2 , A 3 and A 4 can

be obtained using ACP.

4.3.1. Solution using ACP

By generating the graph randomly, 80 generations with 50–100 number of ants each are sent out through the graph with

ρ = 0 . 5 and β = 1 . In Fig. 17 , the trial solutions are plotted. These trial solutions are compared with the exact solution given

as:

k 11 (t) =

−50 0 0

3333

+

8333

3333

e −3333 50 0 0 t

e −3333 2500

.

At 13 th generation, the ACP method computed a trial solution with the fitness function equal to 0.4 4 4 4 and the expression

is given as:

T ours : k 11 (t) = e (2 − t) ,

Expressions : k 11 (t) = e 2 −t .

After the global pheromone update, the ACP method gives an expression with the fitness function is equal to 0.2172 and its

functional form is given as:

T ours : k 11 (t) = 6 / 5 ∗ e (2 − t) − 1 / 5 ,

Expressions : k 11 (t) =

6

5

e 2 −t − 1

5

,

at 28 th generation. Later at 43 rd generation, the ACP predicted an expression with a fitness function equal to 0.0707, with

its functional form given as:

T ours : k 11 (t) = 7 / 5 ∗ e (2 − t) − 2 / 5 ,

Expressions : k 11 (t) =

7

e 2 −t − 2

.
5 5



Table 7

Comparison results for k 11 ( t ) between the ACP and GP.


ACP 77 120.53

GP 305 182.34

Table 8


ACP solution GP solution RK4 solution Exact solution

t k 11 ( t ) k 11 ( t ) k 11 ( t ) k 11 ( t )

1.0 3.36934 3.27096 3.36915 3.36915

1.1 3.05530 2.96326 3.05515 3.05515

1.2 2.76151 2.67541 2.76139 2.76139

1.3 2.48667 2.40613 2.48658 2.48658

1.4 2.22956 2.15421 2.22949 2.22949

1.5 1.98903 1.91854 1.98897 1.98897

1.6 1.76401 1.69807 1.76397 1.76397

1.7 1.55351 1.49181 1.55348 1.55348

1.8 1.35658 1.29886 1.35656 1.35656

1.9 1.17235 1.11836 1.17234 1.17234

2.0 1.0 0 0 0 0 0.94949 1.0 0 0 0 0 1.0 0 0 0 0

Only after reaching 70 th generation, the final solution is obtained. The fitness function is 4.4E −9 and the expression is

given as:

T ours : k 11 ( t ) = 5 / 2 ∗ e ( (4 − 2 ∗ t) / 3 ) − 3 / 2 ,

Expressions : k 11 ( t ) =

5

2

e 4 −2 t

3 − 3

2

.

The parse trees for the solutions k 11 are shown in Fig. 18 . The comparison between the ACP and GP method is given in

Table 7 . The performance of the ACP, GP and RK4 methods is shown in Fig. 19 based on their relative errors.

5. Conclusion

The nontraditional ACP method has been applied to solve the MRDEs with nonlinear singular fuzzy system together with

some fuzzy modelling problems. By manipulating or controlling the terminal criterion, the ACP method predicts either equal

or approximately close results to the exact solution. This approach is very vital for solving complicated differential equations

which gives or predicts solutions with very good accuracy. Furthermore, the present ACP method succeeded in predicting


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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