Robust Adaptive Sliding Mode Control Using Fuzzy Modeling for an Inverted Pendulum System

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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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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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(a) (b)

(c) (d)

(e)

Fig. 9. The dynamic responses of , and control voltage with initial conditions m .


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(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

[1] J. Ackermann,Robust Control: Systems with Uncertain Physical Param-eters. London, U.K.: Springer, 1993.

[2] C. W. Anderson, Learning to control an inverted pendulum using neuralnetworks,IEEE Contr. Syst. Mag., vol. 9, pp. 3137, Apr. 1989.

[3] G. F. Franklin, J. D. Powell, and A. Emami-Naeini,Feedback Controlof Dynamic Systems. Reading, MA: Addison-Wesley, 1986.

[4] J. Hao and J. Vandewalle, A rule-base controller for inverted pendulumsystems, Int. J. Contr., vol. 4, no. 1, pp. 5564, 1993.

[5] C. E Lin and Y. R. Sheu, A hybrid-control approach for pendulum-carcontrol,IEEE Trans. Ind. Electron., vol. 39, pp. 208214, June 1992.

[6] K. Matsuura, Fuzzy control of upswing and stabilization of invertedpole including control of cart position, Jpn. J. Fuzzy Theory Syst., vol.5, no. 2, pp. 239253, 1993.

[7] S. Mori, H. Nishihara, and K. Furuta, Control of unstable mechanicalsystem control of pendulum, Int. J. Contr., vol. 23, no. 5, pp. 673692,1976.

[8] R. M. Sanner and J.-J. E. Slotine, Gaussian networks for direct adaptivecontrol,IEEE Trans. Neural Networks, vol. 3, pp. 837863, Nov. 1992.

[9] L. X. Wang,Adaptive Fuzzy Systems and Control: Design and StabilityAnalysis. Englewood Cliffs, NJ: Prentice-Hall, 1994.

[10] T. Yamakawa, Stabilization of an inverted pendulum by a high-speedfuzzy logic controller hardware system, Fuzzy Sets Syst., vol. 32, no.2, pp. 161180, 1989.

[11] K. S. Yeung and Y. P. Chen, Sliding mode controller design of a singlelink flexible manipulator under gravity, Int. J. Contr., vol. 52, no. 1,pp. 101117, July 1990.

[12] M. Sugeno and T. Yasukawa, A fuzzy logic based approach toqualitative modeling, IEEE Trans. Fuzzy Syst., vol. 1, pp. 731, Feb.1993.

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Robust Adaptive Sliding Mode Control Using Fuzzy Modeling for an Inverted Pendulum System

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