Supervisory Fuzzy Controller
for Linear Control System
Sławomir Bydoń
MSc.
Department of Process Control
University of Mining and Metallurgy 30059 Kraków, al. Mickiewicza
30, Poland
Abstract  The paper presents a concept of control system with supervisory fuzzy controller.
The fuzzy controller adjusts the sets (Kp, Ki, Kd) of PID controller to the current
parameters values (amplitude A and frequency f) of disturbance signal. The rules for
knowledge base are created using simulation tests. The performance of control system with
supervisory fuzzy controller is compared to system with single feedback loop.
TABLE OF CONTENTS
1 Introduction
2 System structure
3 Creation of fuzzy system
4 Results of simulation tests
5 Conclusions
6 References
1
Introduction
The new control systems require looking for new and better control algorithms.
Neural networks and fuzzy systems are being used more often now due to
development of microprocessors. Acting of numerous controllers is based on fuzzy
algorithms, but it is still not popular enough to use them in any kind of
control process. Designing of fuzzy logic or neural network is often too
complicated and takes too much time to be used by average design engineer.
2
System structure
The control system with supervisory fuzzy controller consists of two parts (two
feedback loops). First one is a standard control system with linear plant and
PID controller. Second part is supervisory system (fig. 1.).
Fig. 1: Control system with supervisory fuzzy controller.
Main feedback loop
PID controller and linear plant (mass m supported by spring k and vibration damper
b, fig. 2.)
are in the main feedback loop. Formula (1) is a transfer function of
the plant. Actuator F having
transfer function (2) is mounted to the mass in parallel with spring
and damper. Linear PID controller can be described by three parameters
K_{p}, K_{i} and K_{d} which are respectively
proportional, integral and differential gain. This part of control
system can act without supervisory controller.
Fig. 2: Control plant with actuator F.

(1) 

(2)

Supervisory feedback loop
This part of control system consists of analyser and supervisory fuzzy (Mamdani)
controller. Analyser calculates actual value of disturbance signal
parameters. Fuzzy controller is build from three blocks (fig. 3.): fuzzyfication, inference and
defuzzyfication. Membership functions of the input signals are in
the first block. Thanks to them numerical values of inputs are changed
into fuzzy values. Rules and output membership functions are in
inference block. The decision about the optimal sets of PID controller
is determined by rules and is taken upon the disturbance signal
parameters (amplitude A and frequency f ). In
the third part of fuzzy system numeric values of output is being count.
Fig. 3: Block diagram of supervisory fuzzy system.
3
Creation of fuzzy system
Following tasks had to be determined during fuzzy system creation:

number, kind and variation range of inputs and
outputs,

input and output membership functions,

rule base.
Number,
kind and variation range of inputs
Two parameters of sinusoidal disturbance signal
(amplitude A and frequency f ) are
inputs to the fuzzy system. Amplitude changes from 0 [m] to 0,06 [m]
and frequency changes form 0 [Hz] to 12 [Hz]. Variation ranges are
chosen arbitrary.
Number, kind and variation range of outputs
Three
sets of PID controller (K_{p}, K_{i} and K_{d})
are outputs from fuzzy system. Variation range can be define after simulation
tests.
Input membership functions
Triangular,
not symmetrical input membership functions were assumed and were named
with the numbers as it is shown in fig. 4.
Fig. 4: Input membership functions of fuzzy system.
Simulation tests
Simulation tests were led to get knowledge about the optimal values of PID sets for
different values of (A, f), fig. 5. Objective function that was minimised is the time
integral squared error (quality coefficient I_{2}) (3).

(3)

where:e  error, t  time.
After every simulation test PID sets were changed in the way
that leads the value of objective function to be minimised. This
way of sets matching lasts very long. Computer routine was made
to create the knowledge base automatically.
Based on simulation tests rules were created and variation range of
output values could be determined. Coefficient K_{p}
varies from 0 to 200, K_{d} varies from 0 to 20.
Coefficient K_{i} was almost constant, so decision not
to include it into fuzzy system was taken.
Fig. 5:Disturbance signal parameters for which PID sets were matched
Output membership functions
Input membership functions can be chosen arbitrary, but output
membership functions depend on not uniformly distributed
simulation results. Maximum values of membership functions were
determined in places where density of the simulation results was
bigger. Fig. 6. shows output membership functions.
Fig. 6: Output membership functions for (a) K_{p} and (b) K_{d}.
Rule Base
Rules in the fuzzy system join inputs with outputs. It is possible to
show dependency between input and output with the help of
surfaces (fig. 7. and 8.). The surfaces can be modified by
manipulating the elements of fuzzy system; rules, membership
functions or mathematical methods.
Fuzzy system was modified in order to ease tuning and make
system more clear. Supervisory controller represents dependency
between input (A, f )
and output (K_{p}, K_{d}):
[K_{p}, K_{d}] = f_{r}(A, f )
where: f_{r}  vector function.
This vector function was separated into two scalar functions:
K_{p}= f_{r1}(A, f),
K_{d}= f_{r2}(A, f).
The separation was done by dividing rules:
IF A is A^{*} AND f is f^{*}
THEN Kp is Kp^{*} AND Kd is Kd^{*} (w), 
(4) 
where:w  weight,
into two different rules with the same antecedent but not the same consequence:
IF A is A^{*}ANDf is f^{*} THEN
K_{p} is K_{p}^{*}, (w_{1}),
IF A is A^{*} AND
f is f^{*} THEN
K_{d} is K_{d}^{*}, (w_{2}). 
(5) 
Such kind of separation makes tuning of fuzzy system easier, because
it is possible to change output value by changing the weight of
every rule. Modifying weight w
in (4) changes both output values (K_{p} and K_{d})
together. Modifying weights w_{1} and w_{2}
separately in (5) only one output is being changed. Tuning of the system
can be done automatically by computer routine, which changes
weight in every rule and checks if the output of system is the
same as results of simulation tests.
Fig. 7: Graphic representation of dependency between inputs (A, f ) and output K_{p}.
Fig. 8: Graphic representation of dependency between inputs (A, f ) and output K_{d}.
4
Results of simulation tests
Exemplary
simulation tests of control system with (fig. 11.) and without (fig. 10.)
supervisory fuzzy controller were presented. Simulation tests were done with a
help of MatlabSimulink. In the case of control system with single feedback loop
PID sets were matched with help of the
same criterion, but for wide range of parameters (A, f
) variation. Disturbance signal (fig. 9.), in both cases, has constant amplitude
A = 0,05 [m] and its frequency f
variation is presented in fig. 12. During simulation fuzzy system was changing
PID sets in the way shown in fig. 14 and 15.
Fig. 9: Disturbance signal.
Fig. 10: Displacement error of control system without supervisory fuzzy
controller.
Fig. 11: Displacement error of control system with supervisory fuzzy controller.
Fig. 12: Frequency f variation of disturbance signal.
Fig. 13: K_{p} PID set vs. time.
Fig. 14. K_{d} PID set vs. time.
5 Conclusions
Presented simulation tests shows that considered control system with supervisory
fuzzy controller has smaller displacement error than one without
PID sets autotuning. Using fuzzy logic it is possible to adapt
linear PID sets to different disturbance signal parameters. The
way of knowledge base creation is universal enough to be used
with different kind of control systems together with nonlinear
plant [5] and different kind of disturbance signal.
6
References
1. 
PIEGAT
A., Modelowanie i sterowanie rozmyte, Wydawnictwo EXIT, Warszawa 1999. 
2. 
DRIAKOV
D., HELLENDOORN H., REINFRANK M., Wprowadzenie do sterowania
rozmytego, Wydawnictwa NaukowoTechniczne, Warszawa 1996. 
3. 
PLUTA
J., SAPIŃSKI B., SIBIELAK M., Simulation Tests of Elektropneumatic Unit for Mechanical Vibration
Damping, Proceedings of International Carpathian Control
Conference, ICCC 2000, Podbanske, Slovak Republik, May 2326,
2000, str. 269272. 
4. 
PLUTA
J., SAPIŃSKI B., SIBIELAK M., Mathematical Model
of Elektropneumatic Unit with Throttling Control, Proceedings of International Carpathian Control
Conference, ICCC
2000, Podbanske, Slovak Republik, May 2326, 2000, str. 265268. 
5. 
BYDOŃ
S., SAPIŃSKI B., SIBIELAK M., Creation of Knowledge Base for Supervisory Expert Control of
Vibration Damping System, Proceedings of International
Scientific Conference of FME, Ostrava, September 57, 2000. 
