INTRODUCTION
Recently, the adaptive neural control approach based on backstepping design
has been developed for nonlinear uncertain systems without the requirement of
matching conditions. In Kwan and Lewis (2000), Lewis
et al. (2000) and Zhang et al. (2000a),
stable neural controller design schemes were proposed for unknown nonlinear
SISO systems via backstepping design technique. With the backstepping design
technique, neural networks were mostly applied to approximate the unmatched
and unknown nonlinearities and then to implement adaptive control methods using
the conventional control technology.
The advantage of adaptive neural control based on backstepping methodology
is that both the parameters and the nonlinear functions can be unknown and the
uncertainties in systems need not satisfy the matching conditions (Chen
et al., 2007).
Excellent contributions for backstepping control, using NNs, are presented
by He and Jagannathan (2005), Jagannathan
(1996, 2001), Jagannathan and
Lewis, (1996a, b), Jagannathan
et al. (1998), Jagannathan (2001), Lewis
et al. (1998), Hsu et al. (2006), Alanis
et al. (2007), Polycarpou (1996), Lin
et al. (1998), Wang and Wang (1999) and
Lin and Hs (2002, 2003). There, a multilayer NN controller
is designed to deliver a desired tracking performance for the control of a class
of partially unknown nonlinear system in discrete time; it includes a modified
delta rule weight tuning.
In the past decade, backstepping design procedures have been intensively introduced
by Choi and Farrell, (2001), Kuljaca
et al. (2003) and Lin and Hsu, 2005a, b).
The backstepping control is a systematic and recursive design methodology for
nonlinear systems to offer a choice to accommodate the unmodeled nonlinear effects
and the parameter uncertainties. The essence of backstepping design is to select
recursively some appropriate functions of state variables as pseudocontrol inputs
for lower dimension subsystems of the overall system. Each backstepping stage
results in a new pseudocontrol design, expressed in terms of the pseudocontrol
designs from preceding design stages. When the procedure is terminated, a feedback
design for the true control input results, which achieves the original design
objective by virtue of a final Lyapunov function, which is formed by summing
up the Lyapunov functions associated with each individual design stage (Zhang
et al., 2000b).
This study proposes a GRBABC system for a class of norder nonlinear
systems. This control system combines the Gaussian Radial Basis Function
Neural Network (GRBF_{NN}) identification and adaptive backstepping
control techniques.
The neural backstepping controller containing a GRBF_{NN} identifier
is designed in the sense of the backstepping control technique and the
GRBF_{NN} identifier is utilized to online estimate the system
dynamic function.
The adaptive laws of the GRBABC system are derived in the sense of Lyapunov
function. Thus, the system can be guaranteed to be asymptotically stable.
Finally, two chaotic systems (Duffing Oscillator system and lű
system) are provided as the simulation examples to verify that the proposed
GRBABC scheme can achieve favorable tracking performance with regard to
unknown dynamic function.
MATERIALS AND METHODS
Design of ideal backstepping controller: Consider a class of norder
nonlinear systems:
where, x is the state trajectory of the system, which is assumed to be
available for measurement,
is an unknown real continuous function and u is the input of the system.
The control objective is to find a control law so that the state trajectory
x can track a trajectory command closely.
The Eq. 1 can be rewritten as the following state Eq.
Assuming that the parameters of the system Eq. 2 are
known, the design of ideal backstepping controller is described stepbystep
as follows.
Step 1: Define the tracking error:
and the derivative of tracking error is defined as:
The α_{1} can be viewed as a virtual control in the Eq.
4
Define a Lyapunov function as:
Differentiating Eq. 5 with respect to time and using
Eq. 4, it is obtained that:
Let:
Then,
where, c_{1} is a positive constant.
Step k: (2 ≤ k ≤ n1)
Define
and the derivative of e_{k }is defined as:
where, The α_{k} can be viewed as a virtual control in the
Eq. 10
Define a Lyapunov function as:
Differentiating Eq. 11 with respect to time and using
Eq. 10, it is obtained that:
Let:
Then,
where, c_{1}, c_{2},..., c_{k} are positive constant.
Step n:
Define
and the derivative of e_{n} is defined as:
Define a Lyapunov function as:
Differentiating Eq. 17 with respect to time and using
Eq. 16, it is obtained that
Let:
Then,
where, c_{1}, c_{2},..., c_{n} are positive constant.
Therefore, the ideal backstepping controller in Eq. 19 will
asymptotically stabilize the system.
Design of Gaussian radial basis adaptive backstepping controller:
Since the system dynamic function f(x_{1}, x_{2}, x_{3},...,
x_{n}) may be unknown or perturbed in practical application, the
ideal backstepping controller Eq. 19 cannot be precisely
obtained. To solve this problem, a GRBF_{NN} identifier is utilized
to approximate the system dynamic function. The descriptions of the GRBF_{NN}
identifier and the design steps of the control system are described as
follows.
GRBF_{NN} Identifier: The network structure of the GRBF_{NN}
identifier is shown in Fig. 1, which can be considered
as one layer feed forward neural network with nonlinear element. The GRBF_{NN}
output can perform the mapping according to:
where, z = [z_{1}, z_{2},..., z_{n}]^{T}
∈R^{n} is the input vector, G_{j}(z_{j},
m_{j}, σ_{j}) ∈R^{n}, j = 12,..., n
are the Gaussian radial basis function, α_{j}∈R is the
spread of Gaussian function, n is the number of neurons. Each Gaussian
radial basis function can be represented by:
For ease notation, Eq. 21 can be expressed in compact
vector forms as:
where:
w 
= 
[w_{1}, w_{2},..., w_{n}]^{T} 
G 
= 
[G_{1}, G_{2},..., G_{n}]^{T} 
m 
= 
[m_{1}, m_{2},..., m_{n}]^{T} 
σ 
= 
[σ _{1}, σ _{2},..., σ _{n}]^{T} 
By the universal approximation theorem, there exists an ideal GRBF_{NN}
identifier f* such that:
where, Δ denotes the approximation error and is assumed to be bounded.
w*, m* and σ* are the optimal parameter vectors of w, m and σ,
respectively. In fact the optimal parameter vectors that are needed to
best approximate a given nonlinear function are difficult to determine.
Thus, an estimate function is defined as:
where,
and
are the estimated of w*, m* and σ*, respectively. For notational
convenience, denote G* = G(z, m*, σ*) and .
Define
Note that
and
are assumed to be m* and σ*, respectively.
GRBABC System: The proposed GRBABC system is shown in Fig. 2.
The control law of the GRBABC is developed as follows:

Fig. 2: 
GRBABC for nonlinear system 
The neural backstepping controller is chosen as:
where, the GRBF_{NN} identifier
is designed to online estimate the system dynamic function f. Then, Theorems
1 and 2 show the properties of the proposed GRBABC system.
Theorem 1: Consider a nonlinear system represented by Eq.
1.The control system is designed as Eq. 27 where
the neural backstepping controller is designed as Eq. 28,
in which the adaptation law of the GRBF_{NN} identifier is designed
as:
Proof 1:
Define a Lyapunov function as:
Differentiating Eq. 30 with respect to time:
Let:
From Eq. 14 and 16, we have:
Substituting the Eq. 28 into Eq. 33,
Let f = w*^{T} and
Then,
Substituting the Eq. 35 into Eq. 34,
Substituting the Eq. 36 into Eq. 31
IF the adaption law is obtained as follows:
Then the differentiation of Lyapunov function will be negative.
Therefore, the backstepping controller in Eq. 28 will
asymptotically stabilize the system. Also the GRBF_{NN }weights
will converge to optimal values.
Theorem 2: Consider a nonlinear system represented by Eq.
1. The control system is designed as Eq. 27 where
the neural backstepping controller is designed as Eq. 28,
in which the adaptation law of the GRBF_{NN} identifier is designed
as:
where, k»0.
Proof 2:
Define a Lyapunov function as:
Differentiating Eq. 41 with respect to time:
Substituting the Eq. 34 into Eq. 42,
Let
Then,
The adaption law is obtained as:
Then,
Therefore, if ,
Then .
The backstepping controller in Eq. 28 will asymptotically
stabilize the system. Also the weights of GRBF_{NN }would not
diverge to infinity and we have a stable controller.
In general, the neural backstepping controller of Theorem 1 is same as
Theorem 2, but the adaption law for GRBF_{NN} weights training
in Theorem 1 is different from Theorem 2.
In Theorem 1, the GRBF_{NN} weights will converge to optimal
values. Although, in Theorem 2, the weights of GRBF_{NN} would
not diverge to infinity and we have a stable controller.
Note that Theorem 2 is more generalized than Theorem 1 and provided both
theorems properties.
RESULTS AND DISCUSSION
Description of chaotic systems: Dynamic chaos is a very interesting
nonlinear effect which has been intensively studied during the last three
decades.
Chaos control can be mainly divided into two categories (Chen
and Dong, 1998; Feng et al., 2007; Zhang
et al., 2004; Xiau and Jun 2003): one is the
suppression of the chaotic dynamical behavior and the other is to generate or
enhance chaos in nonlinear system. Nowadays, different techniques and methods
have been proposed to achieve chaos control. For instance, entrainment and migration
control, optimal control method, stochastic control method, robust control method,
adaptive control method, variable structure method, neural network control method
and so on (Ueta and Yet, 1999; Chen
and Lu, 2002; Wang and Ge, 2002; Lu
and Zhang, 2001; Chen et al., 2002; Liu
et al., 2003; Feng et al., 2007; Harb
et al., 2007).
Chaotic phenomena can be found in many scientific and engineering fields such
as biological systems, electronic circuits, power converters, chemical systems
and so on (Chen, 1999). Since the pioneering study of Ott
et al. (1990), Park (2006) and Yongguang
and Suochun (2003) proposed the wellknown OGY control method, the control
of chaotic systems has been widely studied. Recently, numerous backstepping
control design procedures have been proposed to achieve chaotic control (Hsu
et al., 2006; Guan and Chen, 2003; Yassen,
2006). The key idea of backstepping design is to select recursively some
appropriate functions of state variables as virtual control inputs for lower
dimension subsystems of the overall system (Krstic et al.,
1995; Wai et al., 2002; Lin
et al., 2005).
Chaotic systems have been known to exhibit complex dynamical behavior.
The interest in chaotic systems lies mostly upon their complex, unpredictable
behavior and extreme sensitivity to initial conditions as well as parameter
variations.
For some chaotic systems, since the dynamic characteristics of the control
system are nonlinear and the precise models are difficult to obtain, the modelbased
control approaches are difficult to be implemented (Peng et
al., 2007).
Simulation results: In this section, the proposed GRBABC technique
is applied to control two nonlinear chaotic systems: a Duffing Oscillator
system (Example 1) and a Lű system (Example 2). It should be emphasized
that development of the GRBABC does not require the knowledge of the system
dynamic function.
Chaotic systems have been known to exhibit complex dynamical behavior. Several
control techniques have been proposed for the chaotic systems (Lian
et al., 2002). However, some of them cannot achieve favorable control
performance and some of them require system dynamic function.
Duffing oscillator system: Consider a secondorder chaotic system such
as well known Duffing’s equation describing a special nonlinear circuit or a
pendulum moving in a viscous medium under control (Lian et
al., 2002).
where, p, p_{1}, p_{2} and q are real constants. t is
the time variable and ω is the frequency.
is the system dynamic function where p = 0.4, p_{1} = 1.1, p_{2}
= 1.0, ω = 1.8 and q = 0.62, q = 1.95 and q = 7. u is the control
effort.

Fig. 3: 
Simulation results of Duffing Oscillator system, Theorem
1, q = 0.62 
The system dynamic function would be online estimated by the GRBF_{NN}
identifier. The structure of GRBF_{NN} is shown in Fig. 1. A GRBF_{NN}
identifier with five hidden nodes is utilized to approach the system dynamic
function of the chaotic system.
In addition, the control parameters are selected as c_{1} = 5
and c_{2} = 60. The trajectory command is set as x_{d}
= cos(t) .
The simulation results of the GRBABC with consider Eq.
29 for q = 0.62, q = 1.95 and q = 7 are shown in Fig.
35.
The simulation results of the GRBABC with consider Eq.
40 for q = 0.62, q = 1.95 and q = 7 are shown in Fig.
68, respectively.
The performance index I is defined as .
The performance index I is shown that the proposed GRBABC can achieve
favorable tracking performance.
Figure 58 are shown that the results have good performance
compare to other papers like (Lian et al., 2002).
These results are converged to desirable trajectory command in 1 sec; however,
the results of other papers are converged to that desirable trajectory command
in 4 sec. Consider that the control effort is limited.
These results are shown that the better tracking performance can be achieved
by using Theorem 1 compare to Theorem 2.

Fig. 4: 
Simulation results of Duffing Oscillator system, Theorem
1, q = 1.95 

Fig. 5: 
Simulation results of Duffing Oscillator system, Theorem
1, q = 7 
Lű system: Consider a third order chaotic system such as well known
Lű equation describing (Tan et al., 2003).
where, a = 36, b = 3, c = 20 and u is the control effort.
The system Eq. 48 can be rewritten as the following:

Fig. 6: 
Simulation results of Duffing oscillator system, Theorem
1, q = 0.62 

Fig.7: 
Simulation results of Duffing oscillator system, Theorem
2, q = 1.95 
And

Fig. 8: 
Simulation results of Duffing oscillator system, Theorem
2, q = 7 
where, a_{1} = 46656, a_{2} = 35136, a_{3} =
1980, a_{4} = 1296 and a_{5} = 36. U is as the following:
The system dynamic function would be online estimated by the GRBF_{NN}
identifier. The structure of GRBF_{NN} is shown in Fig. 1. A GRBF_{NN}
identifier with five hidden nodes is utilized to approach the system dynamic
function of the chaotic system.
In addition, the control parameters are selected as c_{1} = 5,
c_{2} = 5 and c_{3} = 5. The trajectory command is set
as: x_{d} = cos(t).
The simulation results of the GRBABC with consider Eq.
29 are shown in Fig. 9.
The simulation results of the GRBABC with consider Eq.
40 and k = 5 are shown in Fig.10.
The performance index I is defined as .
The performance index I is shown that the proposed GRBABC can achieve
favorable tracking performance.
Figure 9 and 10 are shown that the results
have good performance compare to other papers like that (Tan
et al., 2003). Consider that the control effort is limited.
These results are shown that the better tracking performance can be achieved
by using Theorem 1 compare to Theorem 2.

Fig. 9: 
Simulation results of Lű system, Theorem 1 

Fig. 10: 
Simulation results of Lű system, Theorem 2 
CONCLUSIONS
For some systems, since the dynamic characteristics of the control system
are nonlinear and the precise models are difficult to obtain, the modelbased
control approaches are difficult to be implemented. To overcome this drawback,
a novel GRBABC system has been proposed.
In the neural backstepping controller, a GRBF_{NN} identifier
is utilized to online estimate the system dynamic function. The two adaptive
laws of the GRBABC system are synthesized using the two type Lyapunov
functions so that the asymptotic stability of the control system can be
guaranteed.
Finally, two chaotic systems (Duffing Oscillator and Lű systems)
are simulated to illustrate the effectiveness of the proposed design method.
Simulation results verified that the proposed GRBABC system with adaption
law Eq. 29 can achieve favorable tracking performance of these nonlinear
systems.