Final Report for CS5611 (Fuzzy Sets: Theory and Applications):

Fuzzy Control for an Broom-Balancing System

S. R. Lin (g843914@Oz.nthu.edu.tw)
J. P. Chen (mr854363@cs.nthu.edu.tw)

Table of Contents

Abstract	Problem Definition	Model Description	Our Approach
Simulation Results	Conclusions	Computer Programs	Division of labor
References

Abstract

This report describes our attempt to apply a fuzzy system to the broom-balancing control problem. Results show that with simple fuzzy rules the balancer works pretty well.

Problem Definition

This project tries to design a fuzzy logic controller for an broom balancing problem, which consists of a rigid rod in an upright vertical position. The rod is mounted on a movable cart, and freely pivots at its base connection to the cart. The cart movement is the only means of balanceing the rod. Failure occurs when either the cart reaches the end of its track or the angle of the rod with vertical exceeds some limit.

Model Description

• Dynamics: The following set of differential equations describes the cart-broom dynamics.[Anderson 1989]

£c''(t) = mg sin(£c(t))- cos(£c(t))[F(t)+mbl(£c'(t))²sin(£c(t))]

(4/3)ml -m_blcos²(£c(t))

• £c(t) = broom angle (with vertical) at time t (radians)
• £c'(t) = angular velocity
• x(t) = cart position at time t (meters)
• x'(t) =cart velocity
• F(t) = force aplied to cart at time t (newtons)
• m = comnined mass of cart and broom (1.1 kg)
• l = length of broom (pivot point to center of mass, 0.5 meters)
• m_b = mass of the broom (0.1 kg)
• g = gravitational constant (9.8 meters/sec²)

The state of the cart-broom systems is defined by the four variables x, x', £c, £c'.

• The effects of friction is ignored.
• Faliure occurs either when £c=¡Ó£kor the cart reaches the end of the track (¡Ó2.4 meters).

Our Approach

• We take the input state parameters(£c, £c', x, x') as the antecedent sets of our fuzzy rules; for simplification, we separately use two at a time, ie. £cand £c', x and x'. The force (with direction) to push the cart would be the output.
• For each input, we gave 3 types of membership functions to describe it: N(egative), ZE(ro), P(ositive).
• The Fuzzy Rules can be described as a fuzzy associative memory (FAM) matrix :
• The 2 values given from the antecedent sets mf will be combine into a weight by taking the min of them.
• We assume that NL, NS, ZE, PS, and PL represent specific numerical calues, and then compute the output:

Output = (w_nl*NL+w_ns*NS+w_ze*ZE+w_ps*PS+w_pl*Pl)/(£Uw)

where NL = -10, NS = -5, ZE = 0, PS = +5, PL = +10

eg. x= -1.6, x'= -1

Fn(-1.6)=0.6, Fze(-1.6)=0.4,Fp(-1.6)=0

Fn(-1)=0.3, Fze(-1)=0.8, Fp(-1)=0

w_pl=min(Fn, Fn)=min{0.6, 0.3}=0.3

w_ze1=min(Fn, Fze)=min(0.6, 0.8)=0.6, w_ze2=min(Fn, Fp)=min(0.6, 0)=0,

w_ze3(Fze, Fn)=min(0.4, 0.3)=0.3, w_ze4(Fze, Fze)=min(0.4,0.8)=0.4, w_ze5(Fze, Fp)=min(0.4, 0)=0

w_ze6(Fp,Fn)=min(0,0.3)=0, w_ze7Fp,Fze)=min(0,0.8)=0, w_ze8(Fp,Fp)=min(0,0.3)=0;

Output=(0.3*10)+(1.3*0)/(1.6)=2

Simulation Results

• Running the setup as we described in the Approach, a balancer can only last for 716 iterations. From <figure 1-1,1-2> we observe that while we keep£c under a small number with small force, the cart keep moving toward positive. To introduce a negative force to backforth the cart, we simplify the force paramer into ¡Óconstant. A proper constant would lead the rod in another direction and thus notify the controller to push in contradictional direction.

• Experiment shows that the balancer could last long to 25087 iterations with a way too smooth line of X pos. comparing to the original setting. <figure 2> The force constant applied is 2 (newtons). We notice that the rod almost stays in vertical.

• The positive-toward may also cause the small x' can't draw the attention to the controller since we made the rule that, only x (N) and x'(N) will cast a force(NL). Leaving rules unchanged we modify the membership function of the x fuzzy set. We widen the area of NL to let the cart more easily get involved in x(N) to get pushed back.
• Experiment shows that with x(N) membership function as a (0, 1) right trapezoids<figure 3>, the cart slows down towarding positive <figure 4>. When the system run to iteration 30000 it only reachs position x 1.2 meters.





• Instead of the assigning a weight value to each entry in the FAM matriz by taking the minimum of the membership function values associated with that entry, we replace the "minimum" as "product" to see the impact. Experiment shows that the cart starts to moving backforth!(negatively) By modifying the membership functions of x(P) to (-1, 0, 0, 0) we could slow down the negatively movement. On the other hand, we see the Theta could be limited to a smaller region.<figure 5>

Conclusions

• Using simple fuzzy rules we can keep the broom balanced with £cin a small region, say, the broom will stay almost vertical.
• From observing the performace, we could adjust the parameters such as membership functions, constant force to improve the control of the broom-balancer.
• It's still unclear that why the "min" operation will let the cart moving forth, while the "product" operation won't.

Computer Programs

• Project files for FLBRM system:
• UT.LIB utilities manipulating of files and arrays
• BRM.LIB routines for handling the broom dynamics, graphics display, and the controller system
• UTGRSET.OBJ specifying screen positon in terms of VGAHI resolution (640*480)
• FLPRS.C trapezoid mf and fuzzy system architecture setup
• FLBINT.C Initialization fuzzy rules and membership functions
• FLBRMBAL.C fuzzy logic controller
• FLBRMSYS.CPP main program to trigger the interface and the balancer
• How to run my program:
• To run the program, execute FLBRM.EXE
• Select "Setup(F5)" to modify the default values
• Select "Run" to play the simulation
• Note
• The UI, the simulation, and the fuzzy broom balancer on which we tested and modified are originally written by Stephen T. Welstead in "Nural Network and Fuzzy Logic Applications in C/C++". Thus it's not for free access through this Web page.

Division of Labor

• S. R. Lin: Project presentation, C coding
• J. P. Chen: Report writing, C coding

References

• [Anderson89] Anderson, Charles, "Learning to Control an Inverted Pendulum Using Neural Networks," IEEE Control Systems Magazion (April 1989):31-36.
• [Welstead94] Stephen T. Welstead, "Nural Network and Fuzzy Logic Applications in C/C++," Wiley,1994
• [Jang97] J.-S. R. Jang, C.-T. Sun, and E. Mizutani, `` Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence,'' Prentice Hall, 1997.