Abstract—This paper presents a fuzzy parallel distributed compensation (PDC) control design for balancing a two-wheeled inverted pendulum (TWIP). A Takagi-Sugeno (T-S) fuzzy model can be firstly constructed from the nonlinear system model of the TWIP. Based on the T-S fuzzy model, a PDC controller is designed with the aid of linear matrix inequality (LMI) concept. The stability of the fuzzy balance control can be guaranteed by solving the inequalities of LMI. Finally, one simulation and its equivalent experiment are given to demonstrate the effectiveness and feasibility of the control scheme.

I. INTRODUCTION

two-wheeled inverted pendulum (TWIP) is a self-balancing vehicle with two wheels attached on both

sides of its chassis. The vehicle is with only two wheels (no the other supporting wheel) and keeps balance by itself. When the pendulum falls down due to the gravity, the motors generate torques and the wheels rotate at the same time to keep its balance. Emphatically, the primary challenge of the control is to find the torque of the driving motors such that the TWIP can stand on its upright posture stably.

There have been many researches on the design and study of the TWIP (see [1], [3], [4], [5] and [12]-[14]). Firstly, in Ha and Tuta [5], a dynamic model of a TWIP was presented using Lagrange’s equation and the equations were linearized in the vertical to design an optimal controller for its posture control. Then, the conventional controller was used to be the moving platform of a humanoid robot [12]. In Grasser et al. [4], another dynamic model was derived using a Newtonian approach and the equations were also linearized around an operating point to design a controller. In Pathak et al. [14], a control approach was designed via the system partial feedback linearization. The study [1] concentrates on dynamic modeling and model identification. However, no control law was discussed. In Fiacchini et al. [3], a physical dynamics of a personal pendulum vehicle was proposed, and two controllers (linear and nonlinear) have been developed and applied. In addition, the recent effort [13] brought up a PISMC controller, which was also designed to its

Manuscript received January 12, 2009. W. J. Wang is with the Department of Electrical Engineering, National

Central University, Jhongli, Taoyuan 32001, Taiwan, and also with the Department of Electrical Engineering, National Taipei University of Technology, Taipei 10608, Taiwan (phone: 886-3-4227151, ext. 34456; fax: 886-3-4255830; e-mail: wjwang@ee.ncu.edu.tw).

C. H. Huang is with the Department of Electrical Engineering, National Central University, Jhongli, Taoyuan 32001, Taiwan (e-mail: chhuang@ee.ncu.edu.tw).

quasi-equilibrium state. Overview the former researches, they all used the linearization model and the stability issue was not discussed enough.

In recent years, fuzzy control utilizing the Takagi-Sugeno (T-S) fuzzy model has been used in many applications. For example, the Pendubot (see [2], [7] and [8]) and the trailers (see [15] and [17]). In other words, the fuzzy control has been widely and successfully applied to a lot of nonlinear control processes. In the fuzzy control, many theoretical works using linear matrix inequalities (LMIs) have been discussed (see [6], [9]-[11] and [16]). In the fuzzy control approaches, the nonlinear system is firstly transferred to a T-S fuzzy model. Then the so-called parallel distributed compensation (PDC) (see [18] and [19]) controller is constructed according to the T-S fuzzy model, and the stability issue and the stabilizing fuzzy controller are accomplished by LMIs.

In this paper, we propose a fuzzy PDC control design for balancing a TWIP. The physical dynamics, proposed by [3], are linearized into two state equations to build two local models. Then, the two local models are summarized to be our presented T-S fuzzy model. The fuzzy controller can be carried via the concept of PDC. Meantime, the stability issue and the stabilizing fuzzy controller are accomplished by the theoretical work [6] of LMIs.

This paper is organized as follows. In Section II, the dynamics of a TWIP is described and the equivalent T-S fuzzy model is obtained. In Section III, the fuzzy PDC controller for balancing a TWIP is carried out. Simulation results and experiment results will be demonstrated in Section IV. At the end, the conclusion is given in Section V.

Model-Based Fuzzy Control Application to a Self-Balancing Two-Wheeled Inverted Pendulum

Wen-June Wang, Fellow, IEEE, and Cheng-Hao Huang

A

Fig. 1. The photograph of the two-wheeled inverted pendulum.

2009 IEEE International Symposium on Intelligent ControlPart of 2009 IEEE Multi-conference on Systems and ControlSaint Petersburg, Russia, July 8-10, 2009

978-1-4244-4603-2/09/$25.00 ©2009 IEEE 1158

II. MODEL OF THE TWIP

The frame of a TWIP is shown in Fig. 1. The TWIP is a kind of robot which has two driving wheels attached on the left and right sides, respectively, of its chassis. There is no other supporting wheel for its balance, so the function for balancing control is necessary. The main task is to find the driving torque of both left and right side motors such that the TWIP balance holds. Referring to [3], the dynamics of a TWIP is described as follows. Its coordinate system with geometric parameters of a TWIP is defined and shown in Fig. 2, where is the inclination angle of the pendulum, is the

wheel’s rotation angle, l is the length between the wheel axle and the gravity center of the pendulum, r is the radius of the wheel, and g is the gravity acceleration. The masses of the pendulum and the cart are represented as

PM and CM ,

respectively. Firstly, according to physical concepts, the equilibrium of forces on the x axis can be obtained as follows [3].

xPCP flMrMM cos)( (1)

where xf is the exerted force by wheels along the x axis. And

RLx fff , where Lf and

Rf denotes the forces from left and

right wheels, respectively. Secondly, the equilibrium of moments around the pendulum’s rotating point is focused to yield the following equation.

0sincos 2 glMlMlrM PPP (2)

Three variables, including the angle of the pendulum , the

angular velocity of the pendulum , and the angular velocity

of the wheels , are interested to be the state vector of the TWIP system. So the state vector can be defined as

Ttttt )()()()( x . For the T-S fuzzy modeling, the

inclination angle of the pendulum is assumed as 66)( π,π-t . Meantime, the desired state of the TWIP

is defined as Tddd dx , where d ,

d and

d are the

desired values of )(t , )(t and )(t , respectively. Then the

error between the current state and the desired state can be represented as follows.

.)(

)()()(

)()()()(

dxx

e

t

ttt

tetetet

T

T

ddd

(3)

Furthermore, the control input is designed as the driving torque to motors, i.e., rtftu x )()( . Assuming the TWIP is

in an upright posture ( 0d ), the T-S fuzzy model of the

state error system can be described as a set of If-then rules for the TWIP.

Model rule i: If )(te

is i , then )()()( tutt ii BeAe , )2,1( i . (4)

Here, the system matrices are

,

00

00)(

010

rM

gMlM

gMM

C

P

C

CP1A

,

1

1

0

2

rM

lrM

C

C1B

,

00))1((

00))1((

)(

010

2

2

rMM

gMlMM

gMM

CP

P

CP

CP

2A

,

))1((

1))1((

0

22

2

rMM

lrMM

CP

CP

2B

where )6(

)6sin(

and )6cos( . The membership

functions of the e

are described by (5) and (6) and shown in

Fig. 3.

;otherwise,

)1)((

)()(sin

0)(,1

)(1

te

tete

te

te (5)

.otherwise,

)1)((

)(sin)(

0)(,0

)(2

te

tete

te

te (6)

(a) (b)

Fig. 2. The WIP coordinate with geometric parameters. (a) Slanted view. (b) Side view.

Fig. 3. Membership functions.

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TABLE I GEOMETRIC PARAMETERS AND VALUES OF THE WIP

Item Symbol Value Unit

Mass of the pendulum MP 9.1 [kg]

Mass of the cart MC 25.2 [kg]

Length between the wheel axle and the center of gravity of the

pendulum l 0.5 [m]

Radius of the wheel r 0.1 [m]

Gravity acceleration g 9.8 [m/s2]

Therefore the defuzzification of the above fuzzy model (4) can be carried out as

2

1

)()()()(i

i tuttet ii BeAe . (7)

III. PDC DESIGN FOR BALANCING CONTROL

For coping with the balancing problem of the TWIP, a fuzzy balancing controller is designed to keep the TWIP holding on its upright posture. According to the T-S fuzzy model, a fuzzy balancing controller is constructed via a so-called parallel distributed compensation (PDC) design approach. Base on the concept of PDC [18, 19], each control rule is derived to compensate each model rule of a fuzzy system. For the fuzzy model (4), the following fuzzy controller via the PDC is employed.

Control rule i: If )(te

is i , then )()( ttu eFi , )2,1( i , (8)

where 1F and

2F are the local feedback gains.

Finally, the resulting overall fuzzy controller is represented as

2

1

)()()(i

i ttetu eFi . (9)

The PDC fuzzy controller design is to determine the local feedback gains

iF in the consequent parts. By substituting

equation (9) into equation (7), the fuzzy model can be obtained as follows.

)(2

)()(2)()(

)()()()(

21

2

1i

2

2

1

2

1

ttetette

ttetet

i

i jji

eGG

eG

eFBAe

2112ii

jii

(10)

where jiiij FBAG . To guarantee the stability of the

closed-loop system, the results in Theorem 7 of [6] can be applied to the fuzzy control system (10).

Lemma 1: The equilibrium of the fuzzy control system of (10) is quadratically stable in the large if there exist symmetric matrices P and

ijW such that (11a)-(11d) are satisfied.

0P (11a)

0WPGPG iiiiTii , )2,1( i (11b)

0W2

GGPP

2

GG12

2112

T

2112

(11c)

0WW

WWW

2212

1211

~ (11d)

□

Define three matrices 1 PX , XFN ii and XXWY ijij ,

and then substitute iN and

ijY into the inequalities

(11b)-(11d) to yield the following LMIs (12a)-(12d), respectively.

0X (12a)

0YNBBNXAXA iiiiTi

Tii

Ti , )2,1( i (12b)

XAXAXAXA 2T21

T1

02YNBBNNBBN 1212T2

T121

T1

T2 (12c)

0YY

YY

2212

1211

(12d)

Consequently, the feedback gains iF , a common P , and the

symmetric matrices ijW can be obtained as follows.

1XP (13)

1ii XNF , )2,1( i (14)

1ij

1ij XYXW , )2( ji (15)

from the solutions X , iN and

ijY . By the result of [16], we

have Lemma 2: Suppose that the initial error vector )0(e is

unknown, but the upper bound is known, i.e., )0(e .

Then the control input can be enforced to satisfy the constraint )(tu , if the following LMIs are added to

Lemma 1.

XI 2 (16)

0IN

NX

i

Ti

2

, )2,1( i (17)

where and are predefined positive scalars. □

IV. SIMULATION AND EXPERIMENT RESULTS

The geometric parameters and values of the TWIP are provided in TABLE I.

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Substituting the parameters into the fuzzy model (4) yields their system matrices as follows.

0035.3889-

0026.6778

010

1A,

3.9683

0.7937-

0

1B,

0026.8430-

0023.3660

010

2A,

3.6397

0.6304-

0

2B.

In the fuzzy balancing control, the desired state of the TWIP is initiated on the upright posture, i.e., T000dx .

Because the TWIP is modeled in 66)( π,π-t , so that

the initial error vector is assumed as T00)0()0( e , where

6)0( . Referring to Lemma 2, the constraint of the initial

error vector can be translated as 6)0( e , i.e., the upper

bound of the initial error vector 6 . Besides, the

constraint of the control signal is selected as 40)( tu , i.e.,

40 . According to Lemma 1 and Lemma 2, with setting

6 and 40 , MATLAB LMI toolbox can be used to

solve such LMI conditions (12a)-(12d), (16) and (17). The two local controllers (8) are constructed via PDC, and their local feedback gains

1F and 2F are determined as follows.

0.01329.222046.97811F ,

0.016210.204249.74702 F .

Then, the positive define P and the symmetric matrices ijW

are obtained as

0

0.00000.00040.0022

0.00040.09150.4524

0.00220.45242.2418

P

,

0

0.00000.00060.00340.0000-0.0001-0.0003-

0.00060.11960.62430.0001-0.0145-0.0724-

0.00340.62433.27630.0003-0.0724-0.3628-

0.0000-0.0001-0.0003-0.00000.00090.0041

0.0001-0.0145-0.0724-0.00090.20100.9463

0.0003-0.0724-0.3628-0.00410.94634.4694

~

2212

1211

WW

WWW

.

Therefore the sufficient conditions (12a)-(12d) for ensuring stability are satisfied. The added conditions (16) and (17) are also satisfied for maintaining control constraints.

A. Simulation Results

The robustness of the fuzzy balancing control is verified in the first simulation, shown in Fig. 4. There are three plots selected in a simulation figure, including the inclination angle of the pendulum, )(t , the angular velocity of the pendulum,

)(t , and the angular velocity of the wheel, )(t ,

respectively. The desired state is the upright posture, i.e.,

T000dx . An emphasis is that a pulse disturbance of

torque, )(td , is added in the simulation.

otherwise.,0

0.7;0.5where,]m-Nt[5)(

ttd (18)

Therefore the control torque of the TWIP system is translated as )()()( tdtutu f . In Fig. 4, one can find the

fuzzy balancing control is efficient against the control disturbance, and all states of the TWIP system go back their originals quickly.

B. Experiment Results

An equivalent TWIP is built practically in the laboratory to illustrate effectiveness of the proposed control algorithm. The prototype of the TWIP system is shown in Fig. 1. A tilt sensor and a gyroscope are installed on the chassis to detect the inclination angle and the angular velocity of the pendulum, respectively. Both wheels are driven by 24V-DC rim motors, which are invented by Elebike Company Ltd. The TWIP system has two rotary encoders (2048 counts-per-revolution) to measure wheels’ rotation and their angular velocities. The main controller of the TWIP is a Nios development board, which is a soft-core embedded system and is manufactured by Altera Corporation. The chip of the board is the EP1S10F780 which is the Stratix Series of Altera chip. This chip is a 32-bit RISC processor that provides 8321 gates and operates at the system clock of 50MHz. The proposed control algorithm can be implemented in the Nios development board. Therefore the real-time state, detected by the tilt sensor, the gyroscope, and encoders, can be processed by the Nios development board to compute the control input for the TWIP system.

To verify the feasibility of the proposed PDC control design, the experiment is given with same settings of the simulation. The experiment result is demonstrated in Fig. 5 and its corresponding sequential image stills are shown in Fig. 6. There are three plots selected in Fig. 5. The first plot is the inclination angle of the pendulum, )(t , detected by the tilt

Fig. 4. Simulation results of the balancing control.

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sensor. The second plot is the angular velocity of the pendulum, )(t , detected by the gyroscope. The third plot is

the mean value of the angular velocity of the two wheels, )(t ,

i.e., )()(2

1)( ttt RL , where )(tL and )(tR are the

measured values from the left and right encoders, respectively. In Fig. 6, the TWIP goes to the vertical quickly and keeps its balance automatically. It’s obvious that the fuzzy balancing control applies to the practical TWIP system successfully.

V. CONCLUSION

A fuzzy PDC control design is proposed in this paper. Practical hardware architectures are established to realize a TWIP system with the balancing behavior. Simulation and the equivalent experiment results are given to demonstrate the feasibility that the fuzzy PDC controller can be implemented in a practical TWIP.

ACKNOWLEDGE

This paper is supported by National Science of Council of Taiwan under the Grant NSC 95-2221-E-008-127-MY3.

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Fig. 6. Nine sequential image stills for the real experiment of the balancing control. Image stills 01-03: experiment result of the disturbance effect, the WIP goes back and leans positively; Image stills 04-09: experiment result of the performance of the fuzzy balancing controller, the WIP reaches to the upright posture quickly.

Fig. 5. Experiment results of the balancing control.

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