11 - Fuzzy Control FUZZY CONTROL CONVENTIONAL CONTROL Example: design a cruise control system After...

1

FUZZY CONTROL

CONVENTIONAL CONTROL

Example: design a cruise control system

After gaining an intuitive understanding of the plant’s dynamics and establishing the design objectives, the control engineer typically solves the cruise control problem by doing the following:

1. Developing a model of the automobile dynamics (which may model vehicle and power train dynamics, tire and suspension dynamics, the effect of road grade variations, etc.).

2. Using the mathematical model, or a simplified version of it, to design a controller (e.g., via a linear model, develop a linear controller with techniques from classical control).

“Fuzzy Control” Kevin M. Passino and Stephen Yurkovich


3. Using the mathematical model of the closed-loop system and mathematical or simulation-based analysis to study its performance (possibly leading to redesign).

4. Implementing the controller via, for example, a microprocessor, and evaluating the performance of the closed-loop system (again, possibly leading to redesign).





Mathematical model of the plant:

– never perfect

– an abstraction of the real system

– “is accurate enough to be able to design a controller that will work.”!

– based on a system of differential equations



Performance Objectives:

1. Disturbance rejection properties (e.g., for the cruise control

problem, that the control system will be able to dampen

out the effects of winds or road grade variations).

2. Insensitivity to plant parameter variations (e.g., for the

cruise control problem, that the control system will be

able to compensate for changes in the total mass of the

vehicle that may result from varying the numbers of

passengers or the amount of cargo).


2



3. Stability (e.g., in the cruise control problem, to guarantee

that on a level road the actual speed will converge to the

desired set-point).

4. Rise-time (e.g., in the cruise control problem, a measure of

how long it takes for the actual speed to get close to the

desired speed when there is a step change in the set-

point speed).

5. Overshoot (e.g., in the cruise control problem, when there is

a step change in the set-point, how much the speed will

increase above the set-point).“Fuzzy Control” Kevin M. Passino and Stephen Yurkovich



6. Settling time (e.g., in the cruise control problem, how much

time it takes for the speed to reach to within 1% of the set-

point).

7. Steady-state error (e.g., in the cruise control problem, if you

have a level road, can the error between the set-point and

actual speed actually go to zero; or if there is a long

positive road grade, can the cruise controller eventually

achieve the set-point).



Design Constraints:

Cost: How much money will it take to implement the controller,

or how much time will it take to develop the controller?

Computational complexity: How much processor power and

memory will it take to implement the controller?

Manufacturability: Does your controller have any extraordinary

requirements with regard to manufacturing the hardware

that is to implement it?

Reliability: Will the controller always perform properly? What is

its “mean time between failures?”



Design Constraints:

Maintainability: Will it be easy to perform maintenance and

routine adjustments to the controller?

Adaptability: Can the same design be adapted to other similar

applications so that the cost of later designs can be

reduced?

Understandability: Will the people that implement it or test it be

able to fully understand it?

Politics: Is your boss biased against your approach? Is your

approach too novel and does it thereby depart too much

from standard practice?“Fuzzy Control” Kevin M. Passino and Stephen Yurkovich


Controller Design:

Proportional-integral-derivative (PID) control: Over 90% of the

controllers in operation today are PID controllers. Often,

like fuzzy controllers, heuristics are used to tune PID

controllers (e.g., the Zeigler-Nichols tuning rules).

Classical control: Lead-lag compensation, Bode and Nyquist

methods, root-locus design, and so on.

State-space methods: State feedback, observers, and so on.



Controller Design:

Optimal control: Linear quadratic regulator, use of Pontryagin’s

minimum principle or dynamic programming, and so on.

Robust control: H2 or H∞ methods, quantitative feedback

theory, loop shaping, and so on.

Nonlinear methods: Feedback linearization, Lyapunov

redesign, sliding mode control, backstepping, and so on.


3


Controller Design:

Adaptive control: Model reference adaptive control, self-tuning

regulators, nonlinear adaptive control, and so on.

Stochastic control: Minimum variance control, linear quadratic

gaussian (LQG) control, stochastic adaptive control, and

so on.

Discrete event systems: Petri nets, supervisory control,

infinitesimal perturbation analysis, and so on.


FUZZY CONTROL

Fuzzy control provides a formal methodology for

representing, manipulating, and implementing a

human’s heuristic knowledge about how to control a

system.


FUZZY CONTROL

Useful cases:

(1) The control processes are too complex to analyze by

conventional quantitative techniques.

(2) The available sources of information are interpreted

qualitatively, inexactly, or uncertainly.

Advantages of FLC:

(1) Parallel or distributed control multiple fuzzy rules – complex

nonlinear system

(2) Linguistic control. Linguistic terms - human knowledge

(3) Robust control. More than 1 control rules – a error of a rule

is not fatal.

http://if.kaist.ac.kr/lecture/cs670/textbook/

FUZZY CONTROL

Four main components of a fuzzy controller:

(1) The fuzzification interface : transforms input crisp values

into fuzzy values

(2) The knowledge base : contains a knowledge of the

application domain and the control goals.

(3) The decision-making logic :performs inference for fuzzy

control actions

(4) The defuzzification interface


FUZZY CONTROL

Step 1: Choice of state and control variables

State variables

– input variables of the fuzzy control system

– state, state error, state error deviation, and state error integral

are often used

Control variables

– output variables of the fuzzy control system

– selection of the variables depends on expert knowledge on the

process


Step 2: Select inference method.

• Mamdani

• Larsen

• Tsukamoto

• TSK (Takagi Sugeno Kang)

• Other methods (such as SAM)

FUZZY CONTROL


4

Inference method: Mamdani– minimum operator for a fuzzy implication

– max-min operator for the composition

FUZZY CONTROL


0 0

0

0 0 0

01

1 1

1 1 1

1

u

u

v

v

w

wmin

α1

α2

A1A′ B′

A′ B′

B1 C1

A2 B2 C2

Inference method: Larsen

– product operator(•) for a fuzzy implication

– max-product operator for the composition

FUZZY CONTROL


0 0

0

0 0 0

01

1 1

1 1 1

1

u

u

v

v

w

wmin

α1

α2

Inference method: Tsukamoto

– the consequent part : fuzzy set with

a monotonic membership function

– The rule base has the form: Ri: if u is Ai and v is Bi, then w is Ci,i = 1, 2, … , n where Ci is a monotonic function.

– the result of rule Ri :

– the aggregated result :

weighted sum – no defuzzification

FUZZY CONTROL


)(ziC

µ )(ziC

µ

)(1 iCi iz αµ −= )(1 iCi iz αµ −=

21

2211

αααα

++

=′zz

z21

2211

αααα

++

=′zz

z

FUZZY CONTROL

Inference method: Tsukamoto

Graphical representation of Tsukamoto method


Min

µ µ µ

u v

v

w

wu

AAAA1111BBBB1111

CCCC1111

AAAA2222BBBB2222

CCCC2222

u0

α1111

w2222

w1

α2222

Weighted AverageWeighted AverageWeighted AverageWeighted Averagev0

21

22110 αα

αα++

=ww

w

FUZZY CONTROL

Inference method: TSK (Takagi-Sugeno-Kang)

- the consequent part is given as a function of input variables.

Ri: if x is Ai and y is Bi then z is fi(x, y)

where z = f(x, y) is a crisp function of input variables x and y.

- When input data are singletons x0 and y0, then the inferred

result of rule Ri is fi(x0, y0).

- The aggregated result : weighted average using the

matching degree αi


21

00220011 ),(),(

αααα

++

=′yxfyxf

z21

00220011 ),(),(

αααα

++

=′yxfyxf

z

FUZZY CONTROL

Inference method: TSK (Takagi-Sugeno-Kang)


Minµ

µ

µ

µ

u v

vu

A1 B1

A2 B2

u0

α1

α2

w1=p1u0+q1v0+r1

w2=p2u0+q2v0+r2

Weighted Average

v0

21

22110 αα

αα++

=ww

w

5

Step 3: Fuzzification. Process of

making a crisp quantity fuzzy

• If it is assumed that input data

do not contain noise of

vagueness, a fuzzy singleton

can be used

• If the data are vague or

perturbed by noise, they

should be converted into a

fuzzy number

1

x0

µF(x)

x

1

x0

µF(x)

xbase

FUZZY CONTROL


Step 3: Fuzzification. Process of

making a crisp quantity fuzzy

Fuzzification function

x = fuzzifier (x0)

- x0 is a observed crisp value

- x is a fuzzy set

Fuzzifier() represents a

fuzzification operator

1

x0

µF(x)

x

1

x0

µF(x)

xbase

FUZZY CONTROL


FUZZY CONTROL

Step 4: Design the knowledge base.

The knowledge base consists of two parts:

1. Data base. Partition of the variable spaces

2. Rule base. Fuzzy control rules.


FUZZY CONTROL

Data base design.

Discretization of the

universe of discourse.


Range Level No.

x ≤ −2.4 −6−2.4 < x ≤ −2.0 −5

−1.6 < x ≤ −0.8 −4

−0.8 < x ≤ −0.4 −3

−0.4 < x ≤ −0.2 −2

−0.2 < x ≤ −0.1 −1

−0.1 < x ≤ +0.1 0

+0.1 < x ≤ +0.2 1

+0.2 < x ≤ +0.4 2

+0.4 < x ≤ +0.8 3

+0.8 < x ≤ +1.1 4

+1.1 < x ≤ +1.4 5

+1.4 < x 6

Example: A universe of discourse is discretized into 13

levels (-6, -5, -4, … , 0, 1, … , 5, 6).

FUZZY CONTROL

Data base design. Normalization

of the universe of discourse.

• A discretization into a

normalized universe.

• The normalized universe

consists of finite number of

segments.

• The scale mapping can be

uniform, non-uniform, or both.


Range Normalized

segments

Normalized

universe

[−6.9, −4.1] [−1.0, −0.5]

[−4.1, −2.2] [−0.5, −0.3]

[−2.2, −0.0] [−0.3, 0.0] [−1.0, +1.0]

[−0.0, +1.0] [0.0, +0.2]

[+1.0, +2.5] [+0.2, +0.6]

[+2.5, +4.5] [+0.6, +1.0]

the universe of discourse [-6.9, +4.5]

is transformed into the normalized

closed interval [-1, 1].

FUZZY CONTROL

Data base design. Fuzzy partition.

• determines how many terms should

exist in a term set.

• to find the number of primary fuzzy

sets (linguistic terms)


+1−1 0

Z

PN

6

FUZZY CONTROL


7 linguistic terms are often used

• NB: negative big

• NM: negative medium

• NS: negative small

• ZE: zero

• PS: positive small

• PM: positive medium

• PB: positive big


−1 0 +1

NM PS PM PBNS ZENB

FUZZY CONTROL


The number of fuzzy terms in the

input space determines the

maximum possible number of

control rules.


NB NS ZO PS PB

NB

NS

ZO

PS

x1

x2

A fuzzy partition in a 2-dimensional input space.

The maximum number of control rules = 20 (5x4)

Practical design principles for designing the

data base:

– Each membership function should overlaps

only with the closest neighboring membership

functions;

– For any possible input, its membership values

in all relevant fuzzy sets should sum to 1 (or

nearly)

* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

FUZZY CONTROL FUZZY CONTROL

A Membership Function that violates the second principle


FUZZY CONTROL

A Membership Function that violates both principles


FUZZY CONTROL

Asymmetric Membership Function that follows the guidelines


7

FUZZY CONTROL

Data base design. Membership

functions of primary fuzzy

sets.

Various types of MFs:

- Triangular

- Trapezoidal

- Gaussian

- Bell

- etc. Example of triangular membership functions

FUZZY CONTROL

Rule base design.

Source of fuzzy control rules:

1. Expert knowledge and control engineering knowledge.

2. Observation of operator’s actions.

3. Fuzzy model of the process: linguistic description of the

dynamic properties

4. Learning: learning from example or self-organizing

learning.


FUZZY CONTROL

Rule base design. Types of fuzzy control rules.

State evaluation fuzzy rules: state variables in the antecedent part, control variables in the consequent part.

– a collection of rules of the form(MISO version)

R1: if x is A1, … and y is B1 then z is C1…

Rn: if x is An, … and y is Bn then z is Cn

where x, … y and z are linguistic variables representing the process state variable and the control variable. Ai, … Bi and Ci are linguistic values of the variables x, … y and z in the universe of discourse U, … V and W, respectively i = 1, 2, … , n.That is, x ∈ U, Ai ⊂ U, …, y ∈ V, Bi ⊂ V, z ∈W, Ci ⊂W


FUZZY CONTROL


General representation of state evaluation fuzzy rules:

Ri: if x is Ai, … and y is Bi then z = fi(x, … y)

- The state evaluation rules evaluate the process state (e.g. state, state error, change of error) at time t and compute a fuzzy control action at time t.

- In the input variable space, the combination of input linguistic term may give a fuzzy rule.


FUZZY CONTROL


A set of fuzzy rules

Ri: if x is Ai, and y is Bi then z is Ci, i = 1, 2, … , n

Can be represented as a rule table


C1 C2 …

C3 C4 C4

Cn C5 C5

C6

A1 A2 … An

B1

B2

Bn

…

y

x

C7

FUZZY CONTROL


Object evaluation fuzzy rules: predictive fuzzy control.

A collection of rules of the form

R1: if (z is C1 → (x is A1 and y is B1)) then z is C1.

…

Rn: if (z is Cn → (x is An and y is Bn)) then z is Cn.

– A control action is determined by an objective evaluation that satisfies

the desired states and objectives.

– x and y are performance indices for the evaluation and z is control command.

– Aiand B

iare fuzzy values such as NM and PS.

– The most likely control rule is selected through predicting the results (x, y) corresponding to every control command C

i, i = 1, 2, … , n.


8

Step 5: Select defuzzification method.

• In many practical applications, a control command is given as a crisp value.

• a process to get a non-fuzzy control action that best represents the possibility distribution of an inferred fuzzy control action.

• no systematic procedure for choosing a good defuzzification strategy

• select one in considering the properties of application case

FUZZY CONTROL


FUZZY CONTROL

Defuzzification methods.

• Centroid

• Mean-of-maximum

• Weighted average


∑=

=k

j

j

k

zz

1

0 ∑=

=k

j

j

k

zz

1

0

Step 6: Test and tuning.

- Adjust data base

- Adjust rule base

- Try different inference methods

- Try different defuzzification methods

FUZZY CONTROL


Step 7: (Optional) Produce a Lookup table.

• Due to the problem of time complexity, it may take a long time to compute the fuzzy inference and defuzzification.

• A lookup table shows relationships between the input variables and control output actions.

• A lookup table can be constructed after making the FLC and identifying the relationships between the input and output variables.

• In general, it is extremely difficult to get an acceptable lookup table of a nonlinear control system without constructing a corresponding FLC.

FUZZY CONTROL


FUZZY CONTROL

Step 7: (Optional) Produce a Lookup table.

The control surface can be represented as a table


FUZZY CONTROL

Review of the design procedure.

Step 1: Determination of state variables and control

variables

Step 2: Selection of the inference method

-Mamdani

-Larsen

-Tsukamoto

-TSK

Step 2: Selection of the fuzzification method


9

FUZZY CONTROL

Review of the design procedure.

Step 4: Determination of the knowledge base

- Discretization and normalization of state variable space

- Partition of variable space.

- Selection of the MF shapes

- Design of the rule base

Step 5: Selection of defuzzification strategy

Step 6: Test and tuning

Step 7: Construction of a lookup table


FUZZY CONTROL

FUZZY CONTROL EXAMPLE-

SERVOMOTORS

• Servomotors are used in many automatic

system including drivers for printers, floppy

disks, tape recorders, and robot manipulations.

• The servomotor process shows nonlinear

properties

• The goal is to apply the fuzzy logic control to the

motor control.

• The task of the control is to rotate the shaft of

the motor to a set point without overshoot.



SERVOMOTORS

Design procedure for servomotor control

1) Determination of state variables and control variable

(1) State variables (input variable of FLC):

- Error : the set point minus the process output (e).

- Change of error (ce) : the error from the process output minus the error from the last process output.

(2) Control variable (output variable of FLC):

- Control input (v) : the voltage applied to the process.



SERVOMOTORS


2) Determination of inference method

- The Mandani inference method is selected for its simplicity

3) Determination of fuzzification method

- fuzzy singleton : measure the state variables without uncertainty



SERVOMOTORS


4) Discretization and normalization

– The shaft encoder of the motor has a resolution of 1000.

– The universes of discourse are as follows:

−1000 ≤ e ≤ 1000 −100 ≤ ce ≤ 100– The servo amplifier has an output range of 30 V and thus the control variables (v) are in the range −30 ≤ v ≤ 30

– We discretize and normalize the input variables in the range [−1, +1] (next slide).

– The control variable v is normalized in the range [−1, +1] with the equation


vv30

1=′ vv30

1=′

10


SERVOMOTORS


4) Discretization and normalization


error (e) error change (ce) quantized level

−1000 ≤ e ≤ −800 −100 ≤ ce ≤ −80 −1.0

−800 < e ≤ −600 −80 < ce ≤ −60 −0.8

−600 < e ≤ −400 −60 < ce ≤ −40 −0.6

−400 ≤<e ≤ −200 −40 < ce ≤ −20 −0.4

−200 < e ≤ −100 −20 < ce ≤ −10 −0.2

−100 < e ≤ 100 −10 < ce ≤ 10 0

100 < e ≤ 200 10 < ce ≤ 20 0.2

200 < e ≤ 400 20 < ce ≤ 40 0.4

400 < e ≤ 600 40 < ce ≤ 60 0.6

600 < e ≤ 800 60 < ce ≤ 80 0.8

800 < e ≤ 1000 80 < ce ≤ 100 1.0


SERVOMOTORS


4) Partition of input and output spaces

- Partition space of each input and output variable into seven regions.

- Each region is associated with a linguistic term

- The maximum number of possible fuzzy rules is 49.


NM NS ZE PS PM PB

−1

NB

0 +1 e cev


SERVOMOTORS


4) Determination of the shapes of fuzzy sets

- Input and output variables are normalized on the interval[-1, +1]

- The input and partitioned into seven linguistic terms

- Select triangular fuzzy sets


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 1.00.4 0.6 0.8

NB NM PBNS PMPSZE

0.5

1


SERVOMOTORS


4) Construction of the rules– We interviewed with an expert of the servomotor control, and we

collect knowledge such as:

“If the error is zero and the error change is positive small, then the control input is negative small”

(1) If e is PB and ce is any, then v is PB.

(2) If e is PM and ce is NB, NM, or NS, then v is PS.

(3) If e is ZE and ce is ZE, PS, or PM, then v is ZE.

(4) If e is PS and ce is NS, ZE, or PS, then v is ZE.

(5) If e is NS and ce is NS, ZE, PS, or PM, then v is NS.

(6) If e is NS or ZE and ce is PB, then v is PS.



SERVOMOTORS


4) Construction of the rules


NB NM PMPSZENS PB

NB

NM

NS

ZE

PS

PM

NB

PB PB

PSPM

ZENS

NM

NSPS

ce

e


SERVOMOTORS


5) Select centroid defuzzification

6) Test and tune. Compare performance to classical control


11


SERVOMOTORS


7) Construction the look-up table



TRUCK BACKER-UPPER

Harder formulation:

Back-up a truck-trailer to a dock

Simple formulation:

Back-up a truck to a dock.

Approach the dock at 90º angle!


TRUCK BACKER-UPPER

Truck-and-trailer

dynamics:


TRUCK BACKER-UPPER

Truck dynamics:

Θ - steering angle

ϕ - truck angle

x,y – truck position


TRUCK BACKER-UPPER

Assumption: The truck

travels at constant

velocity V and truck

length is L.


TRUCK BACKER-UPPER

The truck dynamics can

be represented in

complex form:

Θ=

Θ=

+=

−

sin

)(cos 2

L

V

dt

d

eVdt

dp

jyxp

j

ϕ

πϕ

Θ=

Θ=

+=

−

sin

)(cos 2

L

V

dt

d

eVdt

dp

jyxp

j

ϕ

πϕ

Fuzzy Systems ToolBox, Mark Beale and Howard Demuth

12


TRUCK BACKER-UPPER

In MATLAB:

function ds=trukmod(s,u) % s(1) =x, s(2)=y, s(3)=f, u = q

% Angles are in radians.

% CONSTANTS

L = 1; % Length of truck.

v = 1; % Velocity of truck.

% STATES

pos = s(1) + sqrt(-1)*s(2); angle = s(3);

% DERIVATIVES

dpos = v*cos(u)*exp(j*(angle-pi/2)); dangle = v*sin(u)/L;

ds = [real(dpos); imag(dpos); dangle];



TRUCK BACKER-UPPER

The universe of

discourse:



TRUCK BACKER-UPPER

Fuzzy sets for horizontal position:

- far left, left, center, right, far right



TRUCK BACKER-UPPER

Fuzzy sets for truck angle:

- 0, 60, 120, 180, 240, 300, 360



TRUCK BACKER-UPPER

Fuzzy sets for steering angle:

- hard left, left, soft left, center, soft right, right, hard right



TRUCK BACKER-UPPER

Fuzzy rules:


13


TRUCK BACKER-UPPER

Fuzzy rules:



TRUCK BACKER-UPPER

MATLAB simulation:

% assume time step = 0.2 second

dt = 0.2;

For I=1:100

if rem(I,5) == 1

% Evaluate controller once a second

% (5 time steps *0.2 seconds)

theta = frule(A,C,Z,[s(1);s(3)]);

end

s = s + dt * truckmod (s,theta);

end



TRUCK BACKER-UPPER

MATLAB simulation: trukcon1

Problem: the truck takes too long toget exactly to thecenter line

Reason: coarsedefinition of thehorizontal position

Solution: redefine

fuzzy sets for the horizontal position



TRUCK BACKER-UPPER

Fuzzy sets for horizontal position:

- far left, left, center, right, far right



TRUCK BACKER-UPPER

MATLAB simulation:trukcon2

Redefinition of the fuzzy sets

for the horizontal position

significantly improved

the results



TRUCK BACKER-UPPER

JAVA simulation: FuzzyTruck.html

http://www.iit.nrc.ca/IR_public/fuzzy/FuzzyTruck.html

14

SUMMARY

Classical control does not provide a “perfect” solution

Fuzzy control is a formal methodology for implementing control systems based on human’s heuristic knowledge

Design procedure:

Step 1: Determination of state variables and control variables

Step 2: Selection of the inference method

Step 2: Selection of the fuzzification method

SUMMARY

Step 4: Determination of the knowledge base

- Discretization and normalization of state variable space

- Partition of variable space.

- Selection of the MF shapes

- Design of the rule base

Step 5: Selection of defuzzification strategy

Step 6: Test and tuning

Step 7: Construction of a lookup table

11 - Fuzzy Control FUZZY CONTROL CONVENTIONAL CONTROL Example: design a cruise control system After...

Documents

Transcript of 11 - Fuzzy Control FUZZY CONTROL CONVENTIONAL CONTROL Example: design a cruise control system After...