Design of A Takagi-Sugeno Fuzzy Compensatorfor Inverted Pendulum Control Using Bode Plots

Xiaojun Ban\ X. Z. Gao2, Xianlin Huang3

, and H. S. Lin4

I Department of Control Theory and Engineering, Harbin Institute of Technology,Harbin, China, xiaojun1978@xinhuanet.com.2 Institute of Intelligent Power Electronics, Helsinki University of Technology,Espoo, Finland, gao@cc.hut.fi.3 Department of Control Theory and Engineering, Harbin Institute of Technology,Harbin, China, xlinhuang@hit.edu.cn4 Department of Control Theory and Engineering, Harbin Institute of Technology,Harbin, China, haishenglin318@yahoo.com.cn

Abstract. In our paper, the frequency-response method is combined with thefuzzy control theory to design a hybrid inverted pendulum control system. A Ta­kagi-Sugeno (T-S) fuzzy compensator, which consists of four linear local com­pensators, is constructed by using the Bode plots. To take further advantage of thefrequency-response method, an integrator is cascaded with each of the four localcompensators to eliminate the effect of a constant friction on the cart. Performancecomparison is also made between the T-S fuzzy controller and linear compensa­tors. Simulation results demonstrate the considerable feasibility of our proposedapproach.

Keywords: Fuzzy control, Takagi-Sugeno fuzzy controller, linear compensator,Bode plots, inverted pendulum.

1. Introduction

The Takagi-Sugeno (T-S) fuzzy model [7] is a well-known landmark in thehistory of fuzzy control theory. Numerous fuzzy control problems, such asstability analysis, systematic design, robustness, and optimality, can be ad­dressed within the framework of the T-S model. Especially, given a T-Sfuzzy model, a fuzzy controller design method named Parallel Dis-tributedCompensation (PDC), has been proposed by Sugeno and Kang [5]. Thecorresponding stability analysis of T-S fuzzy systems is widely discussedin [2,4,6,9]. The unique advantage of this technique is that a lot of conven-

48 X. Ban et al.

tional linear controller design solutions oriented from both classical andmodem control theory, e.g., state space optimal control and robust control,can be employed in designing the T-S fuzzy controllers as well, which areactually nonlinear controllers [3].

As we know, the frequency-response method has been well-developedand widely used in industrial applications, which is straightforward andeasy to follow by practising engineers. Worthy of mentioning, the effect ofnoise in a control system can be evaluated by its frequency response. Thisadvantage is very useful for system analysis, since unavoidable noise usu­ally deteriorates the overall control performance [1]. Therefore, fusion ofthe T-S fuzzy model and frequency-response method is of great signifi­cance in the perspective of control engineering.

Inspired by the above idea, in our paper, the frequency-response methodis merged with fuzzy control theory for the inverted pendulum controllerdesign. Four local linear compensators are first designed by using the Bodeplots. A T-S fuzzy compensator is next constructed based on these com­pensators. An integrator is connected with each of the four local compen­sators to combat with the constant friction existing on the cart. Moreover,performance comparison is made between this T-S fuzzy compensator andlinear controllers.

Our paper is organized as follows. In Section 2, four linear local modelsare derived from the original nonlinear differential equations of the in­verted pendulum. As an example, a linear compensator is synthesized inmore details in the next section. Based on the four local compensators de­signed, the T-S fuzzy compensator is constructed in Section 4. Finally,simulations and conclusions are given in Sections 5 and 6, respectively.

2. Mathematical Models of Inverted Pendulum

The nonlinear model of an inverted pendulum is given as the followingdifferential equations:

Xl (t) = X 2 (t)

. () g sin(x I (t)) - amlxi (t) sin(2x I (t)) /2 - a cosrx, (t))u(t) , (1)x t =--~---~-----------

2 4//3 - ami cos' (x,(t))

where Xl (t) represents the vertical angle of the pendulum, X 2 (t) is the an­

gular velocity, g =9.8 m/ S2 is the gravity constant, m is the pendulum

mass, M is the cart mass, 1 is half length of the pendulum, u is the force

(2)

(4)

(5)

(3)

Design of A Takagi-Sugeno Fuzzy Compensator 49

exerted on the cart, and a = I/(m + M). In our simulations, these parameters

are chosen as m =2.0 kg, M =8.0 kg , and I =0.5 m [8].

To design a compensator for the inverted pendulum in the frequencydomain, four linear models are first acquired based on (1). More precisely,when Xl (I) is about zero and X 2 (I) is zero, the nonlinear model of our in-

verted pendulum can be linearized as follows:

Xl (I) =X2 (I)

. () gXI (I) - au(I) .X 1 =----

2 4113-aml

When Xl (I) is about ±!:., ± fC , ± 3fC , and X2(I) is zero, the correspond-

8 4 8ing linear models are shown from (3) to (5), respectively:

Xl (I) =X2 (I)

g!sin(fC )x,(I) - acos(fC )u(/)X (I) = 1f 8 8

2 1f4113 - amicos' (-)

8Xl (I) =x2 (I)

g -±-sin(fC )x,(I) - acos(fC )u(/)X (I) = 1f 4 4'

2 1f4113 - amicos' (-)

4Xl (I) =x 2 (I)

8 . 3;r 3;rg-sln(-)xI (I) - acos(-)U(/)

X (I) = 31l" 8 82 31f

4//3 - amIcos ' (-)8

3. Local Compensator Design Using Bode Plots

In this section, four local linear compensators are designed based on theabove linear models in the frequency domain. When Xl (I) is about zero,

the transfer function can be derived from (2):

G(s)= -31 17S2 _ 294 . (6)

50 X. Ban et al.

Bode Diagram Bode Diagram

- -

II............

<, I

- - I l--r-,

I--- --

~{>--------tt

III "r--'-- '--

~ - -rr ......

- I--- - ...- ~-- - -

J '"~ i-- ~ - - :::=

- f-- 1--

I..__.

,/ b=J~ I.- =-27010-2 10-1 10° 101 102 103 104

Frequency (rad/sec)

C> -135Q)

~Q) -180enttl

.s::.0.. -225

-100-90

50

m~Q)

"0.a'cg> -50~

10° 101

Frequency (rad/sec)

II

--..; ........ II

<, I

""~~ rf------------------ .•..•.. ....... - f"<;;----+--

~I

I I I

II II I

I!

III I !!

-20

m -40~Q)

"0 -60.a'c0>ttl~ -80

-1001

C> 0.5Q)

~Q) 0enttls:0.. -0.5

(a) (b)

Fig. 1. Bode plots of inverted pendulum(a) without the compensator.(b) with the final compensator CI (s) .

It is observed from GI (s) that there is a pole on the right half of the

complex plane, which definitely makes the inverted pendulum unstable.The corresponding Bode plot is shown in Fig. l(a). Employing the Ny­quist's stability criterion, we utilize a lead compensator in (7) to stabilizethe inverted pendulum:

CII (s) = K (ts +1) , (7)

where K =-138, and, =_1_. To eliminate some constant disturbances,2.7

such as existing friction, an integrator is also introduced as shown below:

CI

2 (s) = s + ' 1, (8)

swhere '1 is 0.724, which is ten times less than the crossover frequency. In

addition, a first order compensator in (9) is exploited here to suppress thehigh-frequency noise:

(9)

where '2 is 1/80.Cascading all the above three compensators, a linear compensator is ob­

tained:

Design of A Takagi-Sugeno Fuzzy Compensator 51

1K( 1)( )

-138(-s + 1)(s+0.724)C(s)= ZS'+ S+'l = 2.7 (10)

1 S(1"2s+1) s(~s+l)80

The final Bode plot is shown in Fig. l(b). The phase margin and thecrossover frequency are now 58.7° and 7.24 rad/sec, respectively. It can beconcluded from these two criteria that the performance of the overall in­verted pendulum control system is satisfactory.

Similar with G,(s), when XI (t) is about ±;r , ±;r , ± 3;r , the local corn­848

pensators are designed as follows:

1-141.3(-s + 1)(s+ 0.744)

C2(s) = 2.45

1s(-s +1)

80

1-175.8(-s +1)(s+0.68)

CJ(s) = 2.4

1s(-s +1)

70

1- 285.1(-s +1)(s+0.612)

C4 (s) = 2.2\s(-s +1)

65

4. Design of T-8 Fuzzy Compensator

(11)

(12)

(13)

Based on the above four local compensators, a T-5 fuzzy compensator isdesigned in this section, which can be described by the following fourrules.

If Xl (t) is "about zero", then u(t) should be calculated according to

1-138(-s + 1)(s+ 0.724)

CI(s) = 2.7 1

s(-s +1)80

52 X. Ban et al.

If Xl(t) is "about ± 1[ ", then u(t) should be calculated according to8

1-141.3(-s + l)(s + 0.744)

C2(s) = 2.45

1s(-s+l)

80

If Xl (t) is "about ± 1[ ", then u(t) should be calculated according to4

1-175.8(-s + 1)(s+ 0.68)

C3(s) = 2.4

1s(-s+l)

70

If xJt) is "about ± 31£ ", then u(t) should be calculated according to8

1- 285.1(-s + 1)(s+ 0.612)

C4(s) = 2.2\

s(-s+l)65

Jr Jr 3Jrwhere "about zero", "about ±- ", "about ±- ", and "about ±-" are all

8 4 8linguistic values that are denoted as Ai' i = 1,"'4. In our paper, they are

quantified by the triangular membership functions, as shown in Fig. 2. Theconfiguration of our T-S fuzzy compensator is illustrated in Fig. 3.

p

A4 A3 A2 1 Al A2 A3 A

Jr Jr 0

InvertedPendulum

Fig. 3. Configuration of the T-5 compensator

Design of A Takagi-Sugeno Fuzzy Compensator 53

The final control output is given in (14):

u(t) = PI(xJu I(t) +P2 (xJu 2 (I) +P3 (XI )u3 (t) +P4 (xJuJt) , (14)u, (x.) + ,u2 (x,) + ,u3 (x.) + u, (x.)

where ,ui(X\) , i =1,.··4 are the degrees of membership calculated from Ai'

i =1,.··4, respectively, and Ui(x.}, i =1,.··4 are the local control outputs of

those four local compensators.

5. Simulations

Firstly, performance comparison is made between our T-S fuzzy compen­sator and the local compensator synthesized for the linearized model incase of Xl(t) is "about zero". We focus on both the region of attraction and

maximum outputs of compensators. The results are given in Table 1.

Table 1. Comparison between T-S fuzzy and linear compensators.

Initial angleMaximum output of compensators Stable/unstableT-S Linear T-S Linear

5 367 357 Yes Yes

15 1162 1071 Yes Yes

25 2038 1784 Yes Yes

35 2993 2498 Yes Yes

45 4027 3211 Yes Yes

55 6312 3925 Yes Yes

65 9102 4639 Yes Yes

75 10976 5352 Yes Yes

80 11708 5709 Yes Yes

81 11854 Yes No

82 12001 Yes No

87 12733 Yes No

88 No No

54 x. Ban et al.

Secondly, the T-S compensator is compared with the linear compensator

designed based on the linearized model at the linearization point (± 31T ,0 ).8

The results are given in Table 2.

Table 2. Comparison between T-S fuzzy and linear compensator.

Initial angleMaximum output of compensators StablelUnstableT-S Linear T-S Linear

5 367 732 Yes Yes

15 1162 2195 Yes Yes

25 2038 3659 Yes Yes

35 2993 5122 Yes Yes

45 4027 6586 Yes Yes

55 6312 8049 Yes Yes

65 9102 9513 Yes Yes

75 10976 10976 Yes Yes

80 11708 11708 Yes Yes

81 11854 11854 Yes Yes

82 12001 12001 Yes Yes

87 12733 12733 Yes Yes

88 No No

Finally, simulations are performed when a constant friction is exerted onthe inverted pendulum. The T-S fuzzy compensator and the linearcompensator for the equilibrium point are compared. The results are shownin Figs. 4 and 5, where the initial angles are 50 , 800, respectively. Thesolid line represents the performance of the linear compensator, and thedotted line the T-S fuzzy compensator.

Some remarks and conclusions are drawn as follows.(a) Compared with the linear compensator synthesized for the zero

point, the region of attraction of the T-S fuzzy compensator is larger. Inthis case, the control output of the T-S compensator is also larger than thatof the linear compensator.

Design of A Takagi-Sugeno Fuzzy Compensator 55

(b) When the initial angle is small, the output of the T-S fuzzy compen­sator is much smaller than that of the linear compensator, which is derived

from the linearization point (31Z" ,0).8

(c) The integrator cascaded with the T-S fuzzy compensator can effi­ciently eliminate the negative effects caused by the constant disturbance onthe pendulum cart.

(d) It can be concluded from Figs. 4 and 5 that the performances of theT-S fuzzy compensator and linear compensator are almost the same, whenthe initial angle is small. However, the T-S compensator significantly out­performs the linear one, while the initial angle becomes larger.

6. Conclusions

In our paper, a T-S fuzzy compensator for the inverted pendulum control isdesigned based on the frequency-response method. This fuzzy compensa­tor consists of four linear local compensators, which are synthesized by us­ing the Bode plots. To eliminate the effect of constant friction existing onthe pendulum cart, an integrator is cascaded with each of the local com­pensators. Simulations show that our T-S fuzzy controller is an 'interpola­tion' between the linear compensators for the zero point and linearization

point (31Z" , 0). We can conclude that the classical frequency-response8

method and modem fuzzy control theory can be merged together to de­velop new and feasible controllers with superior performances. The fusiontechnique can certainly benefit from both these two methods.

0.1r-----.-----.,...----------r-------,I I II I I

0.08 - - - - - - I - - - - - - - T - - - - - - -,- - - - - - -

I I I

0.06 - - - - - - -: - - - - - - - +- - - - - - -:- - - - - - -:c : : :~ 0.04 - - - - - - -, - - - - - - - T - - - - - - -1- - - - - - -

CD I I I

g> 0.02 - - - - - - -: - - - - - - - +- - - - - - -:-- - - - - -« I 1 I

-0 002tz=- _i - - - - - - _l 1_ - - _• I I I

I I II

5 10Time(s)

15 20

Fig. 4. Angle trajectory of inverted pendulum, when initial angle is 5° .

56 X. Ban et al.

1.5r---------r------r------.--------,I

I I I______ L L L _

I I II I I

~------~------~------I I I

I II I I

-=- -:::. T ~ ~.,-=-___,..---__r_---___I

II I I

--,------,------,------I I II I I

-1 - - - I- - - - - - - I- - - - - - - I- - - - - - -I I II I I

5 10Time(s)

15 20

Fig. 5. Angle trajectory of inverted pendulum, when initial angle is 80 0

•

Acknowledgments

X. Z. Gao's research work was funded by the Academy of Finland underGrant 201353.

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