Group Technical Paper
Fuzzy Control System Optimization
Submitted To
The 2012 Academic Year NSF AY-REU Program
Part of
NSF Type 1 STEP Grant
Sponsored By
The National Science Foundation
Grant ID No.: DUE-0756921
College of Engineering and Applied Science
University of Cincinnati
Cincinnati, Ohio
Prepared By
Andrew Janson, Senior, Computer Science
Nicklas Stockton, Sophomore, Aerospace Engineering
Report Reviewed By:
_________________________
Dr. Kelly Cohen
REU Faculty Mentor
University of Cincinnati
September 9 – December 5, 2013
Introduction
Fuzzy logic systems are used as control systems that need to handle the vagueness of the real world. Fuzzy logic
can model and control nuances overlooked by the binary logic of conventional computers.1 In fuzzy logic, the truth
of any statement becomes a matter of degree. Take for example, determining the timespan of your weekend. As
seen in Fig. 1, binary logic only allows a yes/no answer to the question: “Is this day part of my weekend?”
However, in the real world we do not consider exactly 12am Saturday to be the start of our weekend. We usually
view most of Friday a part of the weekend, determined by when we decide to quit working. Fuzzy logic allows us to
specify that most of Friday is considered the weekend as well by assigning it a value of less than one but greater
than 0. This distinction gives the fuzzy logic system much more robustness compared to binary logic.
Figure 1. Comparison between binary and multivalued membership of a weekend.1
Many of structural dynamics problems may be represented by a coupled set of second-order dynamic systems.
Coupled rigid-body and flexible body dynamics are sensitive to gust loads and low flutter speeds. The best solution
is to use active control to augment structural dynamics.
The objective for this research project is to develop an effective non-linear active structural control methodology
to provide stability in large flexible structures. This goal also takes into consideration performance and managing
this stability with minimum cost. An emulation of a minimum time controller using a fuzzy logic universal
approximator will be used in order to develop this methodology. The simplified model will consist of two rigid
bodies connected by a spring, which must travel a specified distance in minimum time. This is shown in Fig. 2. The
output of the fuzzy inference system (FIS) is the force on car 2 that accelerates and decelerates the spring-mass
system.
Figure 2. Diagram of two rigid bodies connected by a spring traversing distance L in minimum time.
Materials and Methods
Simulation
A simulated environment of the model depicted in Figure 2 above is necessary to test the efficacy of the proposed
fuzzy inference systems. The constants for the model are the weight (mass) of the cars, the spring constant of the
spring connecting the two cars, and the distance to the wall that must be traversed.
𝑚1 = 1𝑘𝑔
𝑚2 = 2𝑘𝑔
𝐾 = 250 𝑁/𝑚
𝐿 = 100𝑚
The modeled system contains displacement and velocity sensors on each cart. Therefore, the four inputs to the FIS
are the distances traveled and the velocities of both car 1 and car 2. The output of the system is the force to be
applied to car 2, limited to ±1N. At each time instant the FIS will use the four inputs and determine the force that
must be applied to cart 2, governed by a set of fuzzy rules.
Input:
𝑥1 (𝑡)
𝑥2 (𝑡)
𝑦(𝑡) = [
]
𝑥̇1 (𝑡)
𝑥̇ 2 (𝑡)
Output:
|𝐹(𝑡)| ≤ 1𝑁
The initial conditions at time t=0 are that both car 1 and car 2 are at rest and are starting at position zero. The
maximum runtime for the simulation was 500 seconds. The final conditions for the simulation stipulate that the
distance traveled by the two cars should not exceed 100m and they should be at rest without oscillating.
0
< 100
0
< 100
𝑦(0) = [ ] , 𝑦(500) = [
]
0
0
0
0
The acceleration of car 1 is determined by the displacement between the two cars, the spring constant, and the
mass of the car. The acceleration of car 2 is also determined by these factors in addition to the output force
exerted on the car over its mass.
Car 1:
𝑥1̈ =
Car 2:
𝑥2̈ =
𝑘
𝑚1
𝑘
𝑚2
(𝑥2 − 𝑥1 )
(𝑥1 − 𝑥2 ) +
𝑓
𝑚2
MATLAB’s ode45 differential equation solver and Fuzzy Tool Box were utilized to simulate the spring-mass system
and integrate with the fuzzy controller.
Data Evaluation
The efficiency of a FIS’s control of the system will be based upon how fast it traverses the distance to the wall and
how close the cars settle to the wall without hitting it. The cost function J is then defined by the settling time t f, the
time taken to settle the cars within 1m of the wall and the steady state error, the distance between the front car
and the wall. The control system that produces the lowest value of J will be proven to be the best solution.
𝑡𝑓 = |𝐿 − 𝑥𝑖(𝑡𝑓)| ≤ 1𝑚
𝐽=
𝑡𝑓
+ 2[𝐿 − 𝑥2 (500)]
100
In order to provide a frame of reference for the performance of any control system the theoretical limits of the
simulation were calculated to provide a lower bound. This lower bound is the minimum cost that an ideal, optimal
solution can obtain. For a rigid body, the mass of the two cars is 3kg and the maximum possible force exerted on
car 2 is 1N. From these values the maximum average acceleration of the system is defined as:
𝐹𝑜𝑟𝑐𝑒 (𝐹) = 𝑚𝑎𝑠𝑠(𝑚) ∗ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 (𝑎)
𝑎=
𝐹
1𝑁
1
=
= 𝑚/𝑠 2
𝑚
3 𝑘𝑔 3
From the maximum acceleration of the system the minimum time to traverse the distance to the wall, 100m, can
be calculated. There are two different scenarios to consider.
(1) Applying maximum force over the entire distance and instantaneously stopping the cars at the wall. This
scenario is not feasible, but will serve as the absolute minimum time it will take for the cars to approach the wall.
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑑) =
𝑡=
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝑣)
𝑡𝑖𝑚𝑒 (𝑡)
𝑣
𝑑
The velocity is the average velocity of the cars over the distance traversed, 100m. If a constant force of 1N is
applied to the cars and the initial velocity is zero, the traversal time can be calculated.
1
𝑑 = 𝑣𝑖 𝑡 + 𝑎𝑡 2 ,
2
vi = 0
1
𝑑 = 𝑎𝑡 2
2
𝑡=√
2𝑑
2(99𝑚)
=√
= 24.37 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
1
𝑎
𝑚/𝑠 2
3
(2) Applying maximum force over half the distance and then applying maximum negative force in the second half
to slow the velocity of the cars to zero right at the wall.
First Half:
𝑡=√
2𝑑
2(49.5𝑚)
=√
= 17.23 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
1
𝑎
𝑚/𝑠 2
3
Decelerating back to rest over the next fifty meters takes the same amount of time; therefore the total time taken
to reach 100m is 34.46 seconds. However, we want the time taken to reach the 99m mark. The time taken to
travel the last meter is 2.45 seconds. Therefore, the best time to reach the 99m mark and reach zero velocity at
the wall is:
𝑡 = 32.19 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
If the car stops at exactly 100m the cost function evaluates to:
𝐽=
32.19
+ 2[100 − 100] = 0.3219
100
The latter scenario is the more realistic of the two and will be the limit that the control systems will try to achieve.
The actual system is a flexible body structure that will oscillate, so the control systems will not reach this rigid body
limit.
Fuzzy Controller Development
After developing the basic structure for the fuzzy controller, an iterative process was developed which varied each
of six chosen membership function parameters to a small degree in order to observe the effect on controller
performance and cost. The goal was to accumulate enough data to understand which changes would be most
beneficial to lowering the cost function output. Out of 300 iterations, one in particular showed promise of
dramatically reducing the overall cost and was chosen as a platform from which to develop an optimum solution.
Using these new parameters and the intuition gained from studying the data from the other iterations, certain
parameters were varied in a more structured manner to obtain an optimum solution.
Robustness Test
An initial evaluation of each solution’s robustness, ability to perform under varying conditions, was performed. The
three solutions tested were the linear system, the best provided fuzzy system, and the fuzzy system developed
during this project. To test the robustness of the systems the spring constant was varied ±20% from its nominal
value. The simulations were run with this new value of K and the cost function was calculated and compared to the
value obtained at nominal K.
𝐾𝑛𝑜𝑚 = 250
𝑁
𝑁
𝑁
, 𝐾0.8 = 200 , 𝐾1.2 = 300
𝑚
𝑚
𝑚
Results
Analysis of the provided solutions show similar approaches with nearly identical rule bases. The best three
solutions given are:
Table 1 - Solutions from former students were provided as a reference point of study
Student
A1
B2
C3
𝑡𝑓
32.550s
32.366s
42.000s
𝑥2 (500)
99.998766m
99.984220m
99.965200m
𝐽
0.32797
0.35522
0.48960
To better understand the dynamics of the plant model, various means of non-fuzzy control are employed. Finetuning of switch point (between acceleration and deceleration) and damping strength are key factors in reducing
the cost function. The following table shows the linear (non-fuzzy) attempts at a solution:
Table 2 - Non-fuzzy solutions were developed using simple if-else structures
Attempt #
1
2
3
Switch pt.
17.2896875s
17.2999299s
17.2999299s
Damping force
𝑓 = −𝑥̇ 2
𝑓 = −1.9𝑥̇ 2
7.9112
𝑓 = −𝑥̇ 2 (
)
|100 − 𝑥2 |
𝑡𝑓
35.707s
32.811s
32.269s
𝑥2 (500)
99.9988m
99.9606m
99.9983m
𝐽
0.35956
0.40886
0.32590
Using the lessons learned from developing a linear solution, formulation of a Fuzzy Inference System (FIS) begins
by defining three membership functions for both the 𝑥2 and 𝑥̇ 2 inputs, namely {far,close,veryclose} and
{negative,zero,positive} respectively (see figures 1, 2). The output value was constructed of three membership
functions, namely {negative,zero,positive} (see figure 3).
Figure 1 – Membership functions for input 𝑥2
Figure 2 – Membership functions for input 𝑥̇ 2
Figure 3 – Membership functions for output f
The rules for this FIS were obtained using the knowledge gleaned from working with the non-fuzzy model. Due to
the minimal number of membership functions, few rules were necessary as shown below in table 3.
Table 3 - Rule base for FIS Controller
Negative
Far
Close
VeryClose
𝑥2
Positive
Positive
𝑥̇ 2
Zero
Positive
Zero
Zero
The FIS herein demonstrated produced the following results.
Table 4 – Final results from successful FIS
𝑡𝑓
32.303s
𝑥2 (500)
99.999821m
𝐽
0.32339
Figure 4 – Plot of system control force over time. Note the low-amplitude oscillations
Robustness Test Results
Positive
Negative
Negative
Tests were run on each controller by varying the spring constant K by ±20% and ±40%. The linear controller was
adjusted to match each new value, and the fuzzy systems were left as they were initially developed. The linear
system left unadjusted varied up to 258% while both fuzzy systems varied less than 2% from their nominal costs.
Table 5 - Robustness test results. Shown are the cost values obtained from each test and the percent change from
nominal values.
-40%
-20%
Nominal
20%
40%
150 N/m
200 N/m
250 N/m
300 N/m
350 N/m
Linear
1.1551
0.4817
0.3227
0.7813
0.5835
FIS A
0.3337
0.3269
0.3287
0.3292
0.3257
Project FIS
0.3235
0.3286
0.323
0.3256
0.3245
Linear Adjusted
0.4126
0.3263
0.3227
0.3349
0.3694
Percent Change from Nominal
Linear
257.95%
49.27%
0.00%
142.11%
80.82%
FIS A
1.52%
-0.55%
0.00%
0.15%
-0.91%
Project FIS
0.15%
1.73%
0.00%
0.80%
0.46%
27.86%
1.12%
0.00%
3.78%
14.47%
Linear Adjusted
Discussion
Significant time was spent on the linear solution to this problem in order to discern what aspects of the system
contribute to a minimal cost. It was determined by this process that the final position of 𝑥2 far outweighed the
settling time in the cost function and much effort was expended in minimizing the distance of 𝑥2 from the wall;
therefore, the goal became twofold: 1) reach a minimum settling time and 2) minimize final distance from the wall.
Initially, this goal of approaching the wall closely was undertaken by attempting to “tweak” the switch point
between acceleration and deceleration; the thinking was that if an accurate enough switch point was determined,
the damping would bring the system to a stop very near the wall. It was quickly discovered, however, that the
damping force is far more important to the final position of the system in the final meter than the switching point.
The switch point effectiveness is exhibited in the settling time of the system. As seen in table 2, attempt #2 has a
lower settling time due to a more precise switch point, but this improved time does not lead to a lower cost.
It was determined from these trials that a damping force needs be proportional to the velocity of the system, but it
was unknown by what ratio. By means of trial and error a constant was found which settled the system in the
requisite distance, but led to a poor overall cost. From this, it was determined that a proportionality constant
which varied over distance was necessary in order to effectively damp the oscillations in minimum time. This
discovery led to the damping force, 𝑓 = −𝑥̇ 2 (
7.9112
100−𝑥2
) as shown in table 2. This force, which increases in
magnitude as the system approaches the 100m wall, drastically decreased the value of the cost function because it
allows the cart to approach the wall more closely before damping is completed.
Having gained the insight from working with a linear control structure, an attempt was then made to develop a FIS
structure. Using one of the provided solutions2 as a base, changes were made to reduce the number of
membership functions and to reduce the rule base. Because critical damping does not need to occur until the
system is very close to the wall, it was decided to consider the system as a rigid body until it at least reached the
settling envelope of 1m from the wall, thus the switch point from “far” to “close” is a sharp transition much like a
linear solution would produce. The “veryclose” function was given the bounds [99.9 100.1] in order to reduce all of
the damping to the final .1m of the system’s movement.
The membership functions for the data from 𝑥̇ 2 were made to be symmetrical for ease of adjustment; being
centered on zero, only one parameter was tuned. Again by trial-and-error, the cost was quickly reduced to a nearoptimal 0.32339. As can be seen in figure 4, the amplitude of the damping force is kept at a minimum due to these
tuned membership functions.
The robustness test has shown that fuzzy systems accommodate variations in system parameters. Though the
linear solution was finely-tuned to provide a low-cost solution, it proves to be system specific and performs poorly
under varied conditions.
Conclusion
Fuzzy systems are a low-cost, robust control option for active control of flexible structures. It has been shown that
a fuzzy system outperformed the nominal linear solution in every test case and did so with less developmental
effort expended. The fuzzy system provides near-optimal control even under different conditions than were used
in development. This demonstrates the system-independent characteristics of fuzzy control.
The control methodology developed from this research can be summarized as follows:
1.
2.
3.
Determine in which frames the system behaves like a rigid or flexible body
Treat system as flexible body only in those frames where it behaves as such
Tune flexible control mechanism to minimize cost
The optimal solution arose from the combination of a number of factors. First, the FIS developed in this research
provided oscillatory damping in minimal time. This allowed the controller to exert most of its force in linear
fashion, accelerating and decelerating the system as a rigid body. Due to optimal damping control, damping was
exerted only as the system approached the wall. The extent of the damping increased in proportion to the distance
from the wall while factoring in the decrease in velocity of the cart system. Due to the specific application of
damping force, more time is allowed for the system to be accelerated and decelerated, which reduces settling
time. This results in a system which performs optimally.
References
1.
2.
3.
Vick, Tyler. (2013). “Fuzzy Logic,” AEEM 6096 Report, College of Engineering and Applied Sciences,
University of Cincinnati, Cincinnati, Ohio
Walker, Alex. (2013). “Fuzzy Control Systems,” AEEM 6096 Report, College of Engineering and Applied
Sciences, University of Cincinnati, Cincinnati, Ohio
Carson, Aaron. (2013). “Two Mass System: Fuzzy Logic Control,” AEEM 6096 Report, College of
Engineering and Applied Sciences, University of Cincinnati, Cincinnati, Ohio
Nomenclature
a
d
f
F(t)
J
K
L
m1
m2
t
tf
v
vi
x1
x2
𝑥̇ 1
𝑥̇ 2
𝑥̈ 1
𝑥̈ 2
y(t)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
acceleration
distance
force
force at time t
cost function
spring constant
distance to wall
mass of car 1
mass of car 2
time
settling time
velocity
initial velocity
position of car 1
position of car 2
velocity of car 1
velocity of car 2
acceleration of car 1
acceleration of car 2
input to FIS at time t