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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 2, APRIL 1998 297
Robust Adaptive Sliding-Mode Control Using FuzzyModeling for an Inverted-Pendulum System
Chaio-Shiung Chen and Wen-Liang Chen
AbstractIn this paper, a new robust adaptive control ar-chitecture is proposed for operation of an inverted-pendulummechanical system. The architecture employs a fuzzy system toadaptively compensate for the plant nonlinearities and forcesthe inverted pendulum to track a prescribed reference model.When matching with the model occurs, the pendulum will bestabilized at an upright position and the cart should return to itszero position. The control scheme has a sliding control input tocompensate for the modeling errors of the fuzzy system. The gainof the sliding input is automatically adjusted to a necessary levelto ensure the stability of the overall system. Global asymptoticstability of the algorithm is established via Lyapunovs stabilitytheorem. Experiments on an inverted-pendulum system are given
to show the effectiveness of the proposed control structure.Index Terms Fuzzy system, inverted-pendulum system, Lya-
punovs stability theorem, reference model, robust adaptive con-trol, sliding control.
I. INTRODUCTION
STABILIZATION of an inverted-pendulum system is a
complex and nonlinear problem. It has been extensively
studied by numerous researchers [7], [10]. An understanding
of how to control such a system will allow us to easily solve
the other related control problems, such as single-link flexible
manipulators [11] and stabilization of a rocket booster by its
own thrust vector.A practical problem with regard to control of an inverted
pendulum on a cart is designing a controller to swing the
inverted pendulum up from a pendant position, achieve in-
verted stabilization, and simultaneously position the cart. This
seemingly simple nonlinear control problem is surprisingly
difficult to solve in a systematic fashion. This problem arises
because there are two degrees of freedom, i.e., the pendulum
angle and the cart position, but only one controls force.
Further, when the pendulum has a large inclination, the strong
nonlinearity makes it difficult to treat this problem using
linear theory. Many authors have investigated such a problem.
Matsuura [6] used two kinds of fuzzy tables, one for stabilizing
the inverted pendulum and the other for positioning the cartlocation. Lin and Sheu [5] proposed a hybrid control method
to operate a pendulumcart system, where fuzzy control is
used to swing up the inverted pendulum and linear state
feedback control is used to stabilize the pendulum near its
upright position. Furuta et al. [16], [17] employed linear servo
theory to stabilize a double and a triple inverted pendulum
Manuscript received February 27, 1996; revised August 12, 1997.The authors are with the Department of Power Mechanical Engineering,
National Tsing Hua University, Hsinchu, 30043 Taiwan, R.O.C.Publisher Item Identifier S 0278-0046(98)01569-X.
and position the cart on an inclined rail. Meier et al. [18]
also considered the triple-inverted-pendulumcart system, but
used the optimal proportional state feedback control to solve
such a problem. However, the previous work concentrated on
the linearized model for the inverted-pendulum system, not on
its nonlinear characteristics. Recently, some authors [2], [4],
[19][21] have employed the approaches of neural network
learning and fuzzy reasoning to solve the nonlinear control
problem of an inverted-pendulum system.
Fuzzy systems can be considered as general tools for
modeling nonlinear functions. As indicated by Sugeno [15],
fuzzy modeling is an important issue in fuzzy theory, sincethe linguistic fuzzy rules from human experts often contain
rich information about how the system behaves. Wang [9]
has developed an important adaptive fuzzy control system to
incorporate with the expert information systematically and has
shown the stability of adaptive control algorithms. An adaptive
fuzzy system is a fuzzy logic system equipped with a training
algorithm, where the fuzzy logic system is constructed from a
set of fuzzy IFTHEN rules, and the training algorithm adjusts
the parameters of the fuzzy logic system, so as to reduce
the modeling error. Conceptually, adaptive fuzzy systems
are constructed so that linguistic information from experts
can be directly incorporated through fuzzy IFTHEN rules,
and numerical information from sensors is incorporated bytraining the fuzzy logic system to match the input/output data.
However, the perfect match via an adaptive fuzzy system is
generally impossible. Although the stability of an adaptive
fuzzy control system has been guaranteed in [9], [13], and [14],
the modeling error may deteriorate the tracking performance.
In this paper, we propose a new robust adaptive control
architecture for the inverted-pendulum system. We first apply
two simple control rules to swing up the pendulum from a
pendant position to an upward position by driving the cart
back and forth and then use the proposed adaptive control law
to stabilize the inverted pendulum. The architecture employs
a fuzzy system to adaptively model the plant nonlinearities,which have unknown uncertainties. Based on the fuzzy model,
the adaptive control law is constructed to force the angle of
the pendulum to follow a prescribed stable reference model,
the inputs of which are the position and velocity of the cart
and the output is the desired angle of the pendulum. When
matching with the reference model occurs, the pendulum will
be stabilized at an upright position and the cart should return
to its zero position. In the proposed scheme, the bound of
the modeling error, which results from the error between
the fuzzy system and the actual nonlinear plant, is identified
02780046/98$10.00 1998 IEEE
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298 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 2, APRIL 1998
Fig. 1. The inverted-pendulum system.
adaptively. Using this estimated bound, a sliding control
input is calculated, such that the tracking error is forced
to a predetermined boundary, and the boundedness of all
signals of the closed-loop system is guaranteed in the sense of
Lyapunov. The proposed scheme is also robust to the variation
of the parameters of the inverted-pendulum system and even
a bounded external disturbance. Finally, we demonstrate the
effectiveness of our scheme via experiments on an inverted-pendulum mechanical system.
This paper is organized as follows. In Section II, the
mathematical representation for an inverted system is derived.
In Section III, a detailed design of the control system is
proposed. In Section IV, the experimental results are given to
illustrate the performance of the developed scheme. Finally,
the conclusions of the paper are given in Section V.
II. SYSTEM MODELS
The inverted-pendulum system is a rigid pendulum hinged
on a cart, so that the pendulum is driven to rotate around
the pivot by advance and retreat of the cart, as shown inFig. 1. To get the dynamic equations of the inverted-pendulum
system, we apply Lagranges formulations as follows. First,
the Lagrange scale function is
(1)
where is the acceleration due to gravity, is the mass of
the cart, is the mass of the pendulum, and is the half
length of the pendulum. Lagranges formulations are
where is the applied force, is the friction of the cart on a
track, is the friction of the pendulum on the pivot, and
represents a uniformly bounded disturbance (i.e.,
for all due to the measurement noise and noises in the
power source. We thus obtain
(2)
(3)
Assume that and where and
are coefficients of friction. Rearranging (2) and (3), we have
(4)
(5)
where
(6)
(7)
and has an unknown upper bound
as
(8)
From (4) and (5), we see that the inverted-pendulum system
includes two dynamic equations; (4) is the dynamic transitionfrom the control force to the angle of the pendulum and
(5) is the dynamic relation between the position of the cart
and the angle of the pendulum In the following section,
we will present a robust adaptive control architecture for the
system (4) to force the angle of the pendulum to follow a
prescribed stable reference model, the inputs of which are the
position and velocity of the cart and the output is the desired
angle of the pendulum. When matching with the model occurs,
the overall system is equivalent to the reference model and the
system (5). Then, the inverted pendulum can be stabilized at
an upright position and the cart will return to its zero position.
A detailed description will be given in the next section.
III. CONTROL SYSTEM DESIGN
In this section, we first specify a reference model to stabilize
the system (5) and then present a robust adaptive fuzzy
architecture to force the system (4) to follow this reference
model. We also propose two simple control rules to swing
up the pendulum from the pendant position to the upward
position.
A. The Reference Model
Let us linearize the system (5) around We
thus obtain the following linear model:
(9)
where If the
uncertainties in the parameters and and the neglected
high-order terms are considered, the linear model can be
represented as
(10)
where and for
Choose a reference model as
(11)
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300 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 2, APRIL 1998
Fig. 2. The configuration of a fuzzy system.
relation of the fuzzy system is obtained in [9] as
(23)
where and are
parameter vectors, and are called fuzzy basic functionsand defined by
(24)
Let us define the optimal approximation parameters and
in the fuzzy system as follows:
(25)
where is a compact set of fuzzy input vector in allrules. In the compact set we have the following upper
approximation error bounds:
and
(26)
where and are unknown real numbers.
C. Robust Adaptive Fuzzy Control Law
Next, we define the error metric as
(27)
The equation defines a hyperplane in on which
the tracking error decays exponentially to zero. Using
the error equation (20), the time derivative of the error metric
can be obtained as
(28)
Fig. 3. The block diagram of the control structure.
Fig. 4. Definition of the angle of the pendulum.
(a)
(b)
Fig. 5. The direction of applied force for swinging up the pendulum. (a)Driving the cart from right to left. (b) Driving the cart from left to right.
where
and
From (8) and (26), we have
(29)
where In order to avoid the chattering
of the controller on the switching hyperplane, a boundary of
width is incorporated into the error metric by defining
as [8]
(30)
where is a saturation function.
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CHEN AND CHEN: ROBUST ADAPTIVE SLIDING-MODE CONTROL USING FUZZY MODELING 301
Fig. 6. Experimental setup.
TABLE IPARAMETERS OF THE PLANT
From (28), an adaptive control law can be synthesized by
(31)
where is a positive constant feedback gain, and
is a sliding control gain, which is the estimated linear bound
of in (29), i.e.,
(32)
where and are the estimates of and respectively.Substituting the control law (31) into (28), and then using
(23), yields
(33)
where and An adaptive mechanism
for the parameters of the fuzzy system and the sliding control
gain is chosen as
(34)
TABLE IIPOLE LOCATIONS AND STABILITY MARGINS FOR DIFFERENT
NOMINAL CUTOFF FREQUENCIES
Fig. 7. The initial fuzzy rules for
.
Fig. 8. The initial fuzzy rules for 0
.
where are positive adaptive gains. The control
architecture is shown in Fig. 3. By the adaptive mechanism in
(34), we have the following results.
Theorem 1: Consider the system (4) with the adaptive
control law (31), where is given by (32), and
and are given by (23). Let the and
be adjusted by the adaptive mechanism (34). Then, all statesin the adaptive system will remain bounded, and the tracking
errors asymptotically converge to a neighborhood of zero.
Proof: Consider the Lyapunov function
(35)
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302 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 2, APRIL 1998
(a) (b)
(c) (d)
(e)
Fig. 9. The dynamic responses of , and control voltage with initial conditions m .
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304 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 2, APRIL 1998
(a) (b)
(c) (d)
(e)
Fig. 10. The dynamic responses of , and control voltage with initial conditions 0 , .
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CHEN AND CHEN: ROBUST ADAPTIVE SLIDING-MODE CONTROL USING FUZZY MODELING 305
implemented with a 100-Hz sampling rate via an IBM PC with
an Intel i486DX-33 microprocessor. The PC is interfaced to the
current servo amplifier and the sensors through a custom card
containing two decoders and a one-channel D/A converter for
one channel analog output. The proposed algorithm is written
in C language .
First, we determine the swing-up voltage and the
traveling length of the cart If the swing-up voltage is
set large, then the required time to swing up the pendulum to
the upward position becomes short. However, it will induce the
larger velocity when the pendulum is near the upward position.
On the other hand, the limits the movement distance of
the cart with respect to its zero position to ensure that the cart
moves within the length of the rail. Therefore, the suitable
values of and are chosen to be within 2.03.5 V
and within 0.050.1 m. From experiments, it is is known
that it takes about 4.5 s to swing up the pendulum to the
upward position and change to the adaptive control law when
we select 3.1 V for and 0.08 m for If is chosen
less than 1.8 V, the pendulum cannot be swung up to the
upward position. Further, when the pendulum is pumped tothe upward position, the adaptive control law takes over from
the swing-up stage. The switching conditions are chosen as
and s
Next, we choose the parameters of the reference model in
(11). In order to have a larger stability margin to robustify
the equivalent linear dynamic system (12), we specify a series
of the poles and the stability margin for different nominal
cutoff frequencies , as shown in Table II, based on the
Bessel prototype method [3]. From Table II, we see that a
larger causes a faster dynamic response, but a smaller
stability margin. By a tradeoff between the convergent rate
of dynamic response and the stability margin, the poles are
chosen as and for the nominalcutoff frequencies at rad/s, where
The values of associated parameters are as follows:
, and
The fuzzy system used for both approximations and
are described in (23). The fuzzy rule base is in
the form of (22). For and we define five fuzzy sets
with triangular membership functions to cover the fuzzy input
region. By inspecting (6) and (7), we quantify the observation
into 25 initial fuzzy rules for and as shown in
Figs. 7 and 8.
The parameters of the adaptive fuzzy control law are se-
lected as follows:
and , and thedesired error tolerance is set to 10, which is experimentally
determined so that the control input is smooth. The initial
values of the estimates and are chosen to be zero.
Two cases are presented in the experiment. Fig. 9 shows
the dynamic responses of and control
voltage with the initial condition m and
The proposed adaptive control scheme is
directly used to drive the cart to return to its zero position and
simultaneously stabilize the inverted pendulum. Fig. 10 shows
the dynamic responses of and control
voltage with the initial condition and
The pendulum is first swung up from
a pendulum position to an upward position, and then, the
proposed adaptive control scheme is switched to stabilize
the inverted pendulum. From Figs. 9 and 10, it can be seen
that the proposed control law can successfully stabilize the
pendulum at the upright position and regulate the position
of the cart to a neighborhood of zero. The position of cart
is bounded within 0.008 m and the angle of the pendulum
within 0.4 . The results demonstrate the usefulness of the
proposed controller in handling the unstable nonlinear system
with unknown uncertainties.
V. CONCLUSION
In this paper, the development and implementation of a
robust adaptive control architecture to operate the inverted-
pendulum system has been presented. The architecture em-
ploys a fuzzy system to adaptively compensate for the plant
nonlinearities and forces the angle of the inverted pendulum
to follow a prescribed reference model. When matching with
the model occurs, the overall control system is equivalent to
a stable dynamic system, such that the inverted pendulum
is stabilized at the upright position and the cart is posi-
tioned at zero. The bounds of the fuzzy modeling error are
estimated adaptively using an estimation algorithm and the
global asymptotic stability of the algorithm is established
in the Lyapunov sense, with tracking errors bounded in the
predetermined tolerance. Finally, experimental results have
verified the effectiveness of the proposed control scheme.
Although only the inverted-pendulum system has been studied
in this paper, the proposed control scheme can also be used
to address the conventional problem of a class of nonlinear
control systems.
REFERENCES
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[3] G. F. Franklin, J. D. Powell, and A. Emami-Naeini,Feedback Controlof Dynamic Systems. Reading, MA: Addison-Wesley, 1986.
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