INTRODUCTION
For decades, growth in the understanding of several computational intelligence
approaches, such as artificial neural networks (Culibrk
et al., 2007), fuzzy systems (Linfeng et al.,
2009), evolutionary algorithms (Zhihuan et al.,
2010) and artificial immune systems (Yu et al.,
2007), has led to propose a soft computing paradigm. The biological immune
system has properties that have very strong robustness and high selfadaptability
even in the face of uncertain situations and unexpected disturbances. Thus,
it is hoped that this immune system will provide new paradigms appropriate for
dynamic systems under unknown environments (Luh and Liu,
2008).
Basically, bone marrow is a type of soft tissue which can be found in the cavity
of elongated bones. During the differentiation of a blood cell into a Bcell,
bone marrow can produce and exhibit an antibody molecule on its surface. An
antibody molecule has two main functions, one is to bind with an antigen and
the other is to perform an effector function. The primary function occurs when
the immune system first encounters an antigen. As it learns about the antigen,
it prepares the body against further invasion by that antigen, thus creating
memory in the immune system. The secondary function occurs when the same antigen
is encountered. This response exhibits a quicker and more abundant production
of antibodies (Takahashi and Yamada, 1998; Yu
et al., 2007). When a naive Bcell encounters an antigen molecule
through its receptor, the cell is activated and begins dividing rapidly; the
cells from these B cells differentiate into memory B cells and effector B cells
or plasma cells.
Fuzzy logic controls have been successfully applied to a wide variety of domain
areas, such as bilinear systems (Li et al., 2008),
carlike mobile robots, robot manipulators and visual servoing (Linfeng
et al., 2009). The concept of type2 fuzzy sets (T2FSs) was first
introduced by Zadeh (1975) as an extension of the concept
of well known ordinary fuzzy sets, type1 fuzzy sets. Typically, T2FSs have
the characteristics of grades of membership fuzzy themselves. A type2 fuzzy
set is characterized by a fuzzy membership function, i.e., the membership grade
for each element is also a fuzzy set in [0,1] (Hsiao et
al., 2008; Karnik et al., 1999), unlike
a typel fuzzy set, where the membership grade is a crisp number in [0,1]. The
membership functions of type2 fuzzy sets are threedimensional and include
a Footprint of Uncertainty (FOU), which is the new third dimension of type2
fuzzy sets. The footprint of uncertainty provides an additional degree of freedom
to make it possible to directly model and handle uncertainties. The type2 fuzzy
sets are useful especially when it is difficult to determine the exact and precise
membership functions.
The type2 fuzzy logic system was coined by Karnik and
Mendel (1998) and it can be used under uncertain circumstances when the
membership grades can not be determined exactly. For systems with uncertainties
and disturbances, the type2 fuzzy logic system can outperform a conventional
fuzzy logic system (Hsiao et al., 2008; Wu
and Mendel, 2007).
In this study, we propose a novel interval type2 fuzzy immune control for a class of discrete nonlinear systems, which combine type2 fuzzy logic and immune systems. This controller can provide more robustness than that of the conventional FLC and also handle uncertainties and disturbances.
IMMUNE FEEDBACK CONTROL DESIGN
The helper Tcell T_{H} in the immune system acts as a stimulant to
the B cell, while the inhibitory Tcell acts as an inhibitant to the B cell.
Suppose that the kth generation of the antigen is ε(k), the output of
the stimulated helper Tcell from the antigen is T_{H}(k) and the inhibition
of the B cell from the inhibitory Tcell restrained is T_{S}(k), then,
the total stimulation received by the B cell can be expressed as (Takahashi
and Yamada, 1998; Kan et al., 2008): where,
f(S_{B}(k), ΔS_{B}(k)) is a nonlinear function that presents
the relation between the output of the T_{S} cell and the antigen, i.e.,
the magnitude of the inhibition capability of the cell.
If we consider that the total stimulation that the B cells receive is the control
input u(k) and the amount of antigen is ε(k), which is equal to e(k).sign(e(k)+λ(e(k)–e(k–1)/T,
where e(k) = r(k)–y(k),
then the feedback control law can be obtained as follows:
where Δu(k–1) = u(k–1)–u(k–2), K_{H}, K_{S}
and λ are scaling factors, K_{H} is the reactive rate and η
= K_{H}/K_{S} is utilized to control the stabilization effect.
Now, the nonlinear function f(u(k–1), Δu(k–1)) will be approximated
by using a type2 fuzzy logic system.
The selection of f(u(k–1), Δu(k–1)) in Eq. 4 may affect the
performance of the controlled system and the fuzzy system is a universal approximator
for a nonlinear system (Dianyou et al., 2007;
Wang and Mendel, 1992). Here, we adopt the interval
type2 fuzzy system to approximate the nonlinear terms in the immune system.
The control input u(k–1) and its variation Δu(k–1) of the immune
system are also the control inputs of the type2 fuzzy system, which is shown
in Fig. 1. It is not necessary to know the precise model of
immune system when designing the type2 fuzzy controller.
INTERVAL TYPE2 FUZZY IMMUNE CONTROL DESIGN
The design procedures for an interval type2 fuzzy logic controller will be introduced here. The architecture of the type2 fuzzy logic system is very similar to that of the conventional fuzzy logic system (type1 fuzzy logical system), but its antecedent and/or consequent sets are now at least one of these sets and are clarified as type2 fuzzy sets. The major difference is the output processor, which includes the typereducer and the defuzzifier: while the former outputs type1 fuzzy sets, the latter outputs a crisp number. The T2FLCs can be used under uncertain circumstances when the membership grades can not be determined exactly.

Fig. 1: 
The block diagram of the internal type2 fuzzy immune control system 

Fig. 2: 
The architecture of the type2 fuzzy logic system 

Fig. 3: 
The triangular shape interval type2 fuzzy set 
Normally, T2FLCs have characteristics of intensive computation due to the heavy
computational load at the step of the type reducing process, which can be simplified
a lot if their secondary membership functions are chosen as the interval sets.
Figure 2 shows the architecture of the type2 fuzzy logic
system, which contains a fuzzifier, rule table, inference engine, typereducer
and defuzzifier.
Fuzzifier:
The fuzzifier nonlinearly maps the input crisp values into interval type2
fuzzy sets. It is obvious that the type2 fuzzy set is in a region constructed
by a principal type1 fuzzy set as shown in Fig. 3. The dotted
line in Fig. 3 represents the primary membership function,
while the shaded region is the FOU. The type2 fuzzy sets can also be represented
by a collection of many embedded type1 fuzzy sets.
For using a fuzzy system to approximate the nonlinear function f(u(k–1),
Δu(k–1)), the input variables u(k–1), and Δu(k–1) are
mapped into three partitions as N (Negative), Z (Zero) and P (Positive), which
is shown in Fig. 4. For the THENpart, a singleton with uncertain
width membership function is selected and is also partitioned into N (Negative),
Z (Zero) and P (Positive), as shown in Fig. 5.
Rule tables:
The rules for a type2 fuzzy logic system still remain the same as those for type1 fuzzy logic system TIFLC, but at least one of their antecedents and the consequents will be represented by interval type2 fuzzy sets.
Table 1: 
Rule table for T1FIC and IT2FIC 

Based on the biological phenomena, the smaller the T_{S} cell that
receives the stimulus, the greater the suppression is and vice versa. We construct
a 9rule fuzzy table shown in Table 1.
Consider an IT2FLC with two inputs u(k–1) and Δu(k–1) and a
single output f(u(k–1), Δu(k–1)), then the nth rule for IT2FLC
can be written as: If
Then
Inference engine:
The inference engine nonlinearly maps all the fired rules from input T2FS to
output T2FS. Multiple antecedents in each rule are connected by using the Meet
operation. The membership grades in the input sets are combined with those in
the output sets by using the extended supstar composition and multiple rules
are combined by using the join operation.
It is necessary to find out the upper and lower bounds for each firing interval fuzzy set before the defuzzification method. As the membership grade can be obtained by using the MaxMin method in a type1 fuzzy logic system, similarly, we adopt this MaxMin scheme to find the upper and lower bounds of the fired resulting interval sets. The detailed procedures are described as follows.
The upper bound of the interval type2 fuzzy sets and
can
be expressed as:
Its associated upper bound of weighting value for the nth fuzzy rule can be
obtained by:
The upper bound of output for the nth fuzzy rule can be obtained by:
So, the upper bound of the resulting interval set can be expressed as:
Similarly, the lower bound can be expressed as:
Its associated lower bound of weighting value for the nth fuzzy rule can be
obtained by:
The lower bound of output for the nth fuzzy rule can be obtained by:
So, the lower bound of the resulting interval set can be expressed as:
The summation of the inference resulting interval set from each fired rule
can be expressed with the upper and lower bounds as:
The inference process can be illustrated graphically as in Fig.
6.
Typereducer: The output processor includes a typereducer and defuzzifier.
The typereduction method is an extension of type1 defuzzification obtained
by applying the Extension Principle to a specific defuzzification method (Visconti
and Tahayori, 2008). The typereduced set using the Center of Sets (COS)
typereduction can be expressed as:
where, U = [u(k–1) Δu(k–1)]^{T}, is the
input of the IT2FLC. Determination of the interval set is
described follows.
The leftmost minimum :
As the leftmost minimum of
the typereduced set is not just simply to calculate the center of gravity of
the lower bound of the fired membership grades, we need to determine point to
rearrange all the fired interval sets to find the center of gravity, i.e., to
determine the weighting of each membership function.
From the weighting methodology, we take the lefthand side of the test point
p as and
the righthand side of the test point P as ,
respectively, as shown in Fig. 7a. As point P intersects the
first singleton output set, C_{L} can be obtained by:
where, Δx is the span of uncertainty and y_{i} is the ith centroid
of the consequent output set. while point P intersects the second or beyond
the second singleton output set, shown in Fig. 7b, C_{L}
can be calculated by:

Fig. 6: 
Graphic illustration of the interval type2 fuzzy implication 
where, N is the number of triggered membership functions. The parameter k can
be obtained by the following IfThen rule.
If
The iteration procedures to obtain the minimum value of are
described as follows:
Step 1: 
Set n = n+1 
Step 2: 
Compute C_{L}(n) 
Step 3: 
If (1<n) and (n≤M), then C_{L}(n) = min[C_{L}(n–1),
C_{L}(n)] 
Step 4: 
If n = M then stop 
Step 5: 
Return to step 1 
The rightmost maximum :
Calculation of is very similar to that of .
Based on the schemes for evaluating mentioned
in the previous subsection, we take the lefthand side of the test point P as
and
the righthand side of the test point P as ,
respectively, as shown in Fig. 8. As point P intersects the
first output set, shown in Fig. 8a, C_{R} can be obtained
by using the center of gravity approach,

Fig. 7: 
Graphic illustration of computing the leftmost point of the
interval output set with the COS method. (a) Determination of the implication
at the first singleton and (b) Determination of the implication at the second
singleton 

Fig. 8: 
Graphic illustration of computing the rightmost point of
an interval output set with the COS method. (a) Determination of the implication
at the first singleton and (b) Determination of the implication at the second
singleton 
where, Δx is the span of uncertainty and y_{i} is the ith centroid
of the consequent output set.
While point P intersects the second or beyond the second output set, shown
in Fig. 8b, C_{R} can also be calculated by using
the center of gravity approach.
where, N is the number of triggered membership functions. The parameter k can
be obtained by the following IfThen rule
If
Similarly, the iteration procedures to obtain the minimum value of are
described as follows:
Step 1: 
Set n = n+1 
Step 2: 
Compute C_{R}(n) 
Step 3: 
If (1<n) and (n≤M), then C_{R}(n) = max[C_{R}(n–1),
C_{R}(n)] 
Step 4: 
If n = M then stop. 
Step 5: 
Return to step 1 
Defuzzifier:
From the typereduction stage, we have typereduced sets determined by their
leftmost points and rightmost points. We defuzzify the interval set by using
the average among them, i.e., the defuzzified crisp output value is:
RESULTS AND DISCUSSION
To examine the feasibility and validity of the proposed controller, we apply
the developed IT2IC to a third order discrete system and an inverted pendulum
system, respectively. Before performing the simulations, a type2 fuzzy inference
graphic interface is first designed with MATLAB (Fausett,
2007) to monitor the controlled system dynamically. This graphic interface
can show all the information in the inference process via dynamic graphic illustration,
including input variables, membership functions and inference results. The blocks
with shaded color represent the region bounded by its upper and lower bounds
of uncertainty for singleton shape output membership function, which are showed
in Fig. 68.
Example 1: Consider a third order discrete system:
In example 1, the sampling time is selected as 0.002 sec. The triangulartype
membership functions with uncertain width is adopted for the IT2FIC and the
rule tables are ranging form 3x3 for simulations. The rule table of T1FLC and
T2FLC is shown in Table 2. The output responses and the control
input of the system are shown in Fig. 9a and b,
respectively. From the time responses shown in Fig. 9a, one can see that IT2FIC
can obtain the smallest maximum overshoot and give the best performance among
the controllers at the same given conditions. In order to compare the feasibilities
of these four controllers, three commonly used performance indexes; Integral
of the Absolute Error (IAE), Integral of Square Error (ISE) and Integral of
Time Multiplied by Absolute Error (ITAE) are adopted to evaluate the tracking
performance of these controllers. These three Performance Indexes (PIs) are
defined as:
where, N = T_{f}/T, T_{f} is the final time of the simulation
and T is the sampling time.
Table 2: 
Rule table for T1FLC and IT2FLC 

Table 3: 
Performance comparison among four controllers in Example 1 


Fig. 9: 
Time responses of the four closed loop controlled systems.
(a) system output (b) control input 
The index ITAE is usually utilized to evaluate both transient and steadystate
response of the control system. From Table 3, one can find
that the IT2FLC gives the best tracking capability among these four controllers
and possesses the best performance from these three PIs.
Example 2:
An inverted pendulum on a cart, which is also considered by Wang
et al. (1996), is described as follows.

Fig. 10: 
Time responses of the inverted pendulum system. (a) angular
displacement (b) control input 
Table 4: 
Performance comparisons among four controllers in Example
2 

where, x_{1} is the angle of the pendulum, x_{2} is the angular
velocity of the pendulum, the mass of the cart is M = 1 kg, the mass of the
pendulum is m = 0.05 kg, g = 9.8 m sec^{2} is the acceleration due
to gravity, the half length of the pendulum is l = 1 m and u is the control
input. By using Euler’s method, Eq. 27 can be expressed as:
where, the sampling time T of the pendulum system is 0.01 sec.
In example 2, the triangulartype membership functions with uncertain width
and nine rules are exploited for these four controllers; T1FLC, IT2FLC, T1FIC
and IT2FIC. Figure 10a shows the simulation results of the
proposed methods applied to the inverted pendulum on a cart under the initial
condition x(0) = [π/6 0]^{T} and the exogenous disturbance input
is ω(k) = [0.8e^{0.04} cos(k) 0.8e^{0.04} sin(k)]^{T}.
All the states can converge to the equilibrium state [0 0]^{T} after
2 sec as shown in Fig. 10a. The control inputs of the controllers
for the system are shown in Fig. 10b.
The indexes of IAE, ISE and ITAE are also exploited to evaluate the tracking
performance of these controllers. The performance comparisons of T1FLC, IT2FLC,
T1FIC and IT2FIC are shown in Table 4. From Table
4, one can also find that the tracking performance of IT2FLC is the best.
From the simulation results and Table 4, one can find that
the proposed scheme can provide the best tracking performance than that of other
control methods even external disturbance appears.
CONCLUSIONS
In this study, a novel interval type2 fuzzy immune control has been proposed for linear and nonlinear discrete systems. The IT2FIC combines the immune system and type2 fuzzy logic control and possesses two features. One is that the IT2FLC can diminish the effect of disturbance than that of T1FLC. The other is that the structure of immune feedback mechanism can make the system output converge to the desired command. A new typereduction algorithm has been also developed for reducing the computational load. The simulation results demonstrate that the proposed IT2FIC can successfully reduce the disturbance and provide the best tracking performance in comparisons with T1FLC, IT2FLC and T1FIC.