2.6 APPLICATION: INVERTED PENDULUM ON A CART
2.6.2 Four-Rule Modeling and Control
Suppose the pendulum on the cart system is built in such a way that the work w x
space of the pendulum is the full circle y, . In this subsection, we w x
extend the results to the range of x1g y except for a thin strip near
"r2. Balancing the pendulum for the angle range of r2- x1 F is referred to as swing-up control of the pendulum. Recall that for x1s"r2
Ž .
the system is uncontrollable. We add two more rules Rules 3 and 4 to the fuzzy model.
Rule 1:
IFx t1Ž .is about 0,
Ž . Ž . Ž .
THEN x˙t sA x1 t qB1u t. Rule 2:
Ž . Ž .
IFx t1 is about"r2 x t1Ž . -r2 ,
Ž . Ž . Ž .
THEN x˙t sA x2 t qB2u t .
Rule 3:
Ž . Ž .
IFx t1 is about"r2 x t1Ž . )r2 ,
Ž . Ž . Ž .
THEN x˙t sA x3 t qB3u t . Rule 4:
IFx t1Ž .is about,
Ž . Ž . Ž .
THEN x˙ t sA x4 t qB4u t .
Here A1, B1, A2, B2 are the same as above and
0 1 0
2g a
A3s 2 0 , B3s 2 ,
Ž4lr3yaml . 4lr3yaml
0 1 0a
A4s , B4s .
0 0
4lr3yaml
The membership functions of this four-rule fuzzy model are shown in Figure 2.19.
w x
Again choose the closed-loop eigenvalues y2,y2 for A3yB F3 3 and A4yB F4 4. We have
w x
F3s 2551.6 764.0 ,
w x
F4s 22.6667 22.6667 .
Fig. 2.19 Membership functions of four-rule model.
It follows that
A3yB F3 3sA4yB F4 4sG and
0 1
G34s .
y220.5230 y67.4675 Note that G34 is Hurwitz.
Ž .
It can be shown that the P of 2.42 satisfies the additional stability conditions
AiyB Fi i4TPqP A iyB Fi i4 -0, is3, 4, Ž2.47. GT34PqPG34-0. Ž2.48. There is no overlap between membership values h1and h3,h1and h4, h2 and h3, and h2 and h4. Hence only G12 and G34 are needed in stability check.
The PDC controller is given as follows:
Rule 1:
IFx t1Ž .is about 0,
Ž . Ž .
THENu t s yF x1 t . Rule 2:
Ž . Ž .
IFx t1 is about"r2 x t1Ž . -r2 ,
Ž . Ž .
THENu t s yF x2 t . Rule 3:
Ž . Ž .
IFx t1 is about"r2 x t1Ž . )r2 ,
Ž . Ž .
THENu t s yF x3 t . Rule 4:
IFx t1Ž .is about,
Ž . Ž .
THENu t s yF x4 t . That is,
u tŽ .s yh x t1Ž 1Ž ..F x1 Ž .t yh2Žx t1Ž ..F x2 Ž .t
yh3Žx t1Ž ..F x3 Ž .t yh4Žx t1Ž ..F x4 Ž .t . Ž2.49.
Fig. 2.20 Angle response using four-rule fuzzy control.
This control law guarantees stability of the fuzzy control system four-ruleŽ fuzzy modelqPDC control . This controller is applied to the original system. Ž2.41 for evaluation of its performance. Simulation results demonstrate that.
Ž .
the controller 2.49 is able to balance the pendulum for all initial angles except when x t1Ž .is in a thin strip 88⬚- x t1Ž . -94⬚. The size of this thin strip can be reduced by adding more rules to the model and controller.
Figure 2.20 illustrates the response of the closed-loop system for initial conditions x1s125⬚, 145⬚, 165⬚, 180⬚and x2s0.
Ž .
Note that the nonlinear controller 2.46 does not apply for r2 F x t1Ž . F.
Some comparisons between the linear, nonlinear, and fuzzy control de- signs are summarized loosely in Table 2.1.
To test the robustness of this controller, the following simulations are
Ž . Ž .
conducted: 1 mis changed from 2.0 to 4.0 kg, 2 M is changed from 8.0 to 4.0 kg, and 3 2Ž . l is changed from 1.0 to 0.5 m. For each case, we simulate
TABLE 2.1 Comparisons of Different Control Designs
Work range Simple? Stability
Ž .
Linear yr4 r4 Yes Local
Ž .
Nonlinear yr2 r2 No Nonlocal
w x
Fuzzy PDC y Yes Nonlocal
Fig. 2.21 Closed-loop angle response with mchanged.
Fig. 2.22 Closed-loop angle response with Mchanged.
Fig. 2.23 Closed-loop angle response with lchanged.
the closed-loop system for the following initial conditions x1s45⬚, 85⬚, 145⬚, 180⬚ and x t2Ž .s0. The results are shown in Figures 2.21, 2.22, and 2.23, respectively, for cases 1, 2, and 3.
Robustness is not considered in this design. Robust fuzzy control design in Chapter 5 is applicable to this system.
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Copyright䊚2001 John Wiley & Sons, Inc.
Ž . Ž .
ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic
CHAPTER 3