INTRODUCTION
In the recent years, controller design for Large Scale Systems (LSS) and effort
to extend it has attracted much attention. Research in controllers of LSS is
motivated by many emerging applications that employ novel actuation devices
for active control of industrial automation, cooperating robotic systems, power
systems and aerospace processes. Centralized controller for the LSS is usually
impractical due to the requirement of a large amount of information exchanges
between subsystems and the lack of computing capacity (Karimi
et al., 2007).
The tunable structure of the FAC and using the knowledge of experts in the
FAC are reasons to attract many researchers to developed appropriate controllers
for nonlinear systems especially for LSS (Ioannou and Sun,
1996).
In the recent year, FAC has been fully studied as follow:
In the first case, the TS fuzzy systems have been used to model nonlinear systems
and then TS based controllers have been designed with guaranteed stability (Feng,
2002; Fenga et al., 2002). To model affine
nonlinear system and to design stable TS based controllers have been employed
by Hsu et al. (2003). Designing of the sliding
mode fuzzy adaptive controller for a class of multivariable TS fuzzy systems
are presented by Cheng and Chien (2006). The nonaffine
nonlinear function are first approximated by the TS fuzzy systems and then stable
TS fuzzy controller and observer are designed for the obtained model (Goleaa
et al., 2003; Park and Park, 2004). In these studies,
modeling and controller has been designed simply, but the systems must be linearizable
around some operating points.
In the second case, the linguistic fuzzy systems have been used to design
controllers for nonlinear systems.
YingGuo and HuaGuang (1998), Jagannathan
(1998), Tong et al. (1999), Tong
et al. (2000) and Zhang and Bien (2000) have
considered linguistic fuzzy systems to design stable adaptive controller for
affine systems based on feedback linearization and furthermore, Tong
et al. (2000) and Zhang and Bien (2000) has
considered that the zero dynamic is stable. Stable FAC based on sliding mode
is designed for affine systems by Labiod et al. (2005).
Designing of the FAC for affine chaotic systems are presented by Tang
et al. (1999) and Chen et al. (1999).
To design stable FAC and linear observer for class of affine nonlinear systems
are presented by Ho et al. (2005),
Zhang (2006), Tong et al. (2004) and Shaocheng
et al. (2005). Fuzzy adaptive sliding mode controller is presented
for class of affine nonlinear time delay systems by Yu (2004),
Chiang (2005) and Jianga et al. (2005). The
output feedback FAC for class of affine nonlinear MIMO systems is suggested
by Yiqian et al. (2004). The main incompetency
of these studies is those restricted conditions on their nonlinear functions.
Labiod and Guerra (2007) and Tong
et al. (2004) are involved stable FAC for class of nonaffine nonlinear
systems. The deficiencies of these studies are bad performance of the controller
when the controller has not been adjusted. Stable adaptive controller for class
of linear LSS is proposed by Pagilla et al. (2007),
Ioannou and Ponte (1988), Shi and Singh
(1992) and Yousef and Simaan (1991). Chiang
and Lu (2007) delt with designing FAC based on sliding mode for class of
large scale affine nonlinear systems. (Zhang et al.,
2002) presented decentralized sliding mode fuzzy adaptive tracking for a
class of affine nonlinear systems in large scale systems. Wu
(2002) designed FAC for a class of affine nonlinear time delayed systems.
These studies have many restricted conditions on their nonlinear function.
FAC has been never applied to nonaffine nonlinear large scale systems.
In this study, the stable decentralized robust adaptive controller has
been designed based on fuzzy systems for a class of large scale nonaffine
nonlinear systems. The controller is robust against uncertainties, external
disturbances and approximation errors.
PROBLEM STATEMENT
Consider the following large scale nonaffine nonlinear system.
where, is
the state vector of the system which is assumed available for measurement,
u_{i}∈R is the control input, y_{i}∈R
is the system output, f_{i} (x_{i}, u_{i}) is
an unknown smooth nonlinear function, mi (x_{1}, x_{2},…x_{N})
is an unknown interconnection term and d_{i}(t) is bounded disturbance.
The control objective is to design an adaptive fuzzy controller for system
(1) such that the system output y_{i}(t) follows a desired trajectory
y_{d}(t) while all signals in the closedloop system remain bounded.
In this study, the following assumptions have been considered concerning
the system (1) and the desired trajectory y_{d}(t).
Assumption 1: Without loss of generality, it is assumed that the
nonzero function satisfies
the following condition:
where, f_{dm} ∈ R is nonzero, known and constant.
Assumption 2: The desired trajectory and its time derivatives
are all smooth and bounded.
Assumption 3: the interconnection term satisfies the following:
where, ξ_{ij} is an unknown time varying parameters.
Assumption 4: The disturbance in the above equation is bounded
by:
Define the tracking error vector as:
Where:
Taking the n_{i}^{th} derivative of both sides of the
Eq. 6, the following equation can be derived.
Use Eq. 5 to rewrite the above equation as:
where, A_{i0} and b_{i} are defined below:
Consider the vector be
coefficients of and
chosen so that the roots of this polynomial are located in the open lefthalf
plane. This makes the matrix be
Hurwitz. Thus, for any given positive definite symmetric matrix Q_{i},
there exists a unique positive definite symmetric solution P_{i}
for the following Lyapunov equation:
Let v_{i} be defined as:
By adding and subtracting the term from
the righthand side of Eq. 8, the following equation
is obtained.
Using assumption 1, Eq. 11 and the signal v_{i} which is not
explicitly dependent on the control input u_{i}, the following
inequality is satisfied:
Invoking the implicit function theorem, it is obvious that the nonlinear
algebraic equation f_{i} (x_{i}, u_{i})–v_{i}
= 0 is locally soluble for the input u_{i} for an arbitrary (x_{i},
v_{i}). Thus, there exists some ideal controller satisfying
the following equality for a given :
As a result of the mean value theorem, there exists a constant λ
in the range of f
<λ1 , such that the nonlinear function f
_{i}
(x_{i}, u_{i}) can be expressed around as:
Where:
Using Eq. 15 and 14 to rewrite Eq.
12 as follow:
However, the implicit function theory only guarantees the existence of
the ideal controller for
Eq. 14 and does not recommend a technique for constructing solution even
if the dynamics of the system are well known. In the following, a fuzzy
system and classic controller will be used to obtain the unknown ideal
controller.
FUZZY SYSTEMS
Figure 1 shows the basic configuration of the fuzzy
systems considered in this study. Here, fuzzy systems can be considered
as a multiinput, singleoutput: x ∈ U ⊂ R^{n}→y
∈ V ⊂ R. Consider that a multioutput system can be separated
into a group of singleoutput systems.

Fig. 1: 
Configuration of fuzzy system 
The fuzzifier performs a mapping from a crisp input vector x = [x_{1}
,x_{2} ….,x_{n}]^{T} to a fuzzy set, where the
label of the fuzzy set are such as small, medium, large, etc.
The fuzzy rule base is consisted of a collection of fuzzy IFTHEN rules.
Assume that there are M rules and the l^{th} rule is:
where, x = [x_{1} ,x_{2} ….,x_{n}]^{T}
and y are the crisp input and output of the fuzzy system, respectively.
A^{l}_{j} and B^{l} are fuzzy membership
function in U_{j} and V, respectively.
The fuzzy inference performs a mapping from fuzzy sets in U to fuzzy
sets in V, based on the fuzzy IFTHEN rules in the fuzzy rule base.
The defuzzifier maps fuzzy sets in V to a crisp value in V. The configuration
of Fig. 1 shows a general framework of fuzzy systems, because
many different choices are allowed for each block in Fig. 1
and various combinations of these choices will construct different fuzzy systems
(Wang, 1997). Here, the sumproduct inference and the centeraverage
defuzzifier are used for fuzzy system. Therefore, the fuzzy system output can
be expressed as:
where, is
the membership degree of the input x_{i} to fuzzy set A^{l}_{j}
and y^{l} is the point at which the membership function
of fuzzy set B^{l} achieves its maximum value.
The fuzzy systems in the form of Eq. 18 are proven by Wang
and Mendel (1993) to be a universal approximator if their parameters are
properly chosen.
Theorem 1: Suppose f(x) is a continuous function on a compact set U
(Wang, 1997). Then, for any ∈>0, there exists a fuzzy
system like Eq. 18 satisfying:
The output given by Eq. 18 can be rewritten in the following compact
form:
where, θ = [y^{1} y^{2}…y^{M}] is a vector
grouping all consequent parameters and is w(x) = [w_{1}(x) w_{2}(x)…w_{M}(x)]^{T}
a set of fuzzy basis functions defined as:
The fuzzy system (Eq. 18) is assumed to be well defined so that for
all x ∈ U.
FUZZY ADAPTIVE CONTROLLER DESIGN
Here, it has been shown how to develop a fuzzy system to adaptively approximate
the unknown ideal controller.
The ideal controller can be represented as:
where, ,
and
are
consequent parameters and a set of fuzzy basis functions, respectively.
∈_{iu} is an approximation error that satisfies ∈_{iu}≤∈_{max}
and ∈_{max}>0. The u_{pid} is the primary controller
that developed properly to initially control the underlying system and
parameters are
determined through the following optimization.
Denote the estimate of as
θ_{i1} and u_{irob} as a robust controller to compensate
approximation error, uncertainties, disturbance and interconnection term
to rewrite the controller given in Eq. 22 as:
In which u_{irob} is defined below.
In Eq. 23, approximates
the ideal controller, tries
to estimate the interconnection term, u_{icom} compensates for
approximation errors and uncertainties, u_{ir} is designed to
compensate for bounded external disturbances and is
estimation of .
Define error vector and use Eq. 24 and 25
to rewrite the error Eq. 16 as:
Consider the following update laws.
where,
are constant parameters.
In following equation, λ_{min}, λ_{max} and
svd_{max}(.) are the minimum, maximum eigenvalue and maximum singular
value decomposition, respectively.
Lemma 1: The following inequality holds if .
Proof: From assumption 1 and the lemma 1, it is obvious that:
This in turn leads to the following inequality.
After some algebraic manipulations, the following inequality is obtained.
Use Eq. 31 to have the following which completes the proof.
Lemma 2: based on lemma 1 and Eq. 10, the following inequality
holds.
Proof: Using Eq. 10 and after some algebraic
manipulations, the following inequality is obtained.
Using the Eq. 34, the following equation can be derived.
Use Eq. 29 and 35 to have the following
which completes the proof.
Theorem 2: Consider the error dynamical system given in Eq.
26 for the large scale system (1) satisfying assumption 1, interconnection
term satisfying assumption 3, the external disturbances satisfying assumption
4 and a desired trajectory satisfying assumption 2, then the controller structure
given in Eq. 24, 25 with adaptation laws
Eq. 27 makes the tracking error converge asymptotically to
a neighborhood of origin and all signals in the closed loop system be bounded.
Proof: Consider the following lyapunov function.
Where:
The time derivative of the lyapunov function becomes.
Use Eq. 26, to rewrite above equation as:
Using assumption 1 yields and by assumptions 3 and 4, to rewrite Eq.
39 as follow:
Equation 40 can be rewritten as below:
Using Eq. 27, the above inequality rewrites as:
Use the lemma 1, are
satisfied. Using Barbalet’s lemma, it is guaranteed the tracking error
asymptotically to the neighborhood of the origin. Furthermore, the boundedness
of the coefficient parameters is guaranteed. It completes the proof.
Remark 1: The term tanh(.) is a smooth approximation of the discontinuous
term sign(.). The sign(.) function is not used in the study due to avoiding
chattering in the response.
Remark 2: It is very important to select properly the controller
parameters to gain a satisfactory performance. Here, at this stage, number
of the rules and the input membership functions are obtained by trial
and error.
Remark 3: To guarantee the boundedness of the parameters in the
presence of the unavoidable approximation error, the proposed adaptive
laws Eq. 27 is modified it by introducing a σmodification
term as follows:
SIMULATION RESULTS
Here, the proposed decentralized fuzzy model reference adaptive controller
is applied to a twoinverted pendulum problem (Karimi et
al., 2007) in which the pendulums are connected by a spring as shown
in Fig. 2. Each pendulum may be positioned by a torque input
u_{i} applied by a servomotor and its base. It is assumed that the angular
position of pendulum and its angular rate are available and can be used as the
controller inputs. The pendulums dynamics are described by the following nonlinear
equations.
where, y_{1}, y_{2} are the angular displacements of
the pendulums from vertical position. m_{1} = 2 kg, m_{2}
= 2.5 kg are the pendulum end masses j_{1} = 0.5 kg, j_{2}
= 0.62 kg are the moment of inertia, k = 100 N m^{1} is spring
constant, r = 0.5 m is the height of the pendulum, g = 9.81 m sec^{2}
shows the gravitational acceleration, l = 00.5 m is the natural
length of spring, α_{1}, α_{2} = 25 are the
control input gains and b = 0.4 m presents distance between the pendulum
hinges.

Fig. 2: 
Two inverted pendulum connected by a spring 

Fig. 3: 
Performance of the PID controller in first subsystem 
The desired value of the outputs are zero (y_{id }= 0 for i =
1, 2). As discussed earlier, the following primary PI controller are obtained
after some trials and errors.
Figure 3 and 4 show the outputs of
the system where only the controller defined in Eq. 45 is applied to the
system.
Obviously the primary controller by itself is not admissible. Now the
proposed controller defined in Eq. 24, 25
has been applied to mentioned system. Initially the PID controller keeps
the states of system x_{i1}, x_{i2} in the range of [–1,
1], [–5, 5]. Let x_{i} = [x_{i1}, x_{i2}]^{T},
z =[x_{i1}, x_{i2}, v_{i}]^{T} and v_{i}
are defined over [–45, 45]. For each fuzzy system input, it is defined
6 membership functions over the defined sets. Consider that all of the
membership functions are defined by the Gaussian function:
where, c is center of the membership function and δ is its variance.

Fig. 4: 
Performance of the PID controller in second subsystem 

Fig. 5: 
Performance of the proposed controller in first subsystem 

Fig. 6: 
Performance of the proposed controller in second subsystem 
It has been assumed that the initial value of θ_{i1}(0),
θ_{i2}(0), u_{ir}(0), u_{icom}(0) and be
zero. Furthermore, it has been assumed that f_{min} = 1, Γ_{1}
= 10, λ_{ξi0}. = 10, ,
,
.
In Eq. 43 and remark 1, σ = 0.1, ∈ = 0.01 has been considered.

Fig. 7: 
Control input u_{1} 

Fig. 8: 
Control input u_{2} 
The parameters f_{dm}, f_{min} and the vector k_{i} = [k_{i1}, k_{i2},…,k_{i,ni}]^{T} have
been chosen so that the lemma 2 holds.
As shown in Fig. 36, it is obvious that the performance
of the proposed controller is promising. Based on these simulation results,
the controller can stabilized the closed loop system. It can decrease
error estimation and disturbances effect in the output of the subsystems.
Figure 7 and 8 show the total input
of each subsystem. It is shown that ripple in the input controller can
decrease error estimation and disturbances attenuation.
CONCLUSION
A decentralized fuzzy model reference adaptive output tracking controller
is proposed for a class of large scale nonaffine nonlinear systems in
this study. Fuzzy systems used to approximate the knowledge of the experts
in the controller design procedure. It has been shown that the derived
adaptation laws guaranty the Lyapunov’s stability of closedloop system.
Asymptotic convergence of the tracking error to zero is guarantied. Robustness
against external disturbances and approximation errors, relaxing the conditions
and using knowledge of experts are the merits of the proposed controller.