online adaptif neuro-fuzzy control

On-Line AdaptiveNeuro-Fuzzy Control

On-Line AdaptiveNeuro-Fuzzy Control

Jyh-Shing Roger Jang

>> 1995>> matlab conference

>> 1995>> matlab conference

2

OutlineOutline

Fuzzy sets, membership functions

Fuzzy if-then rules, fuzzy inference systems

ANFIS - Adaptive Neuro-Fuzzy Inference Systems

ANFIS in FLT

Demos• Inverse learning• Feedback linearization

>> 1995>> matlab conference

3

Fuzzy SetsFuzzy Sets

Sets with fuzzy boundaries

Membershipfunction

Heights5’6’’??6’0’’??

.5

.9

Heights5’6’’??

1.0

Crisp set A Fuzzy set A1.0

A = Set of tall people

>> 1995>> matlab conference

4

Fuzzy Sets: Math FormulationFuzzy Sets: Math Formulation

A = Set of tall people in this room

= {(5'8'', 0.5), (5'9'', 0.6), (5'10'', 0.7), ...}

= {(x, mA(x))}

>> 1995>> matlab conference

5

Membership Functions (MFs)Membership Functions (MFs)

About MFs• Subjective measures• Not probability functions

MFs

Heights5’10”

.5

.8

.1

“tall” in Taiwan

“tall” in the US

“tall” in NBA

>> 1995>> matlab conference

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Fuzzy If-Then RulesFuzzy If-Then Rules• Mamdani style

If pressure is high then volume is small

• Sugeno styleIf speed is medium then resistance = 5*speed

high small

mediumresistance = 5*speed

>> 1995>> matlab conference

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Fuzzy Inference System (FIS)Fuzzy Inference System (FIS)

If speed is low then resistance = 2If speed is medium then resistance = 4*speedIf speed is high then resistance = 8*speed

Rule 1: w1 = .3; r1 = 2Rule 2: w2 = .8; r2 = 4*2Rule 3: w3 = .1; r3 = 8*2

Speed2

.3

.8

.1

low medium high

resistance = S(wi*ri) / Swi = 7.12

MFs

>> 1995>> matlab conference

9

First-Order Sugeno FISFirst-Order Sugeno FIS• Rule base

If X is A1 and Y is B1 then Z = p1*x + q1*y + r1If X is A2 and Y is B2 then Z = p2*x + q2*y + r2

• Fuzzy reasoning

A1 B1

A2 B2

x=3

X

X

Y

Yy=2

w1

w2

z1 =p1*x+q1*y+r1

z =

z2 =p2*x+q2*y+r2

w1+w2w1*z1+w2*z2

P

>> 1995>> matlab conference

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Fuzzy ModelingFuzzy Modeling

• Given desired i/o pairs (training data set) of the form (x1, ..., xn; y), construct a FIS to match the i/o pairs

• Two steps in fuzzy modeling structure identification --- subclust.m parameter identification --- anfis.m

Unknown target system

Fuzzy Inference system

y

y*

x1

xn

. . .

>> 1995>> matlab conference

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Neuro-Fuzzy ModelingNeuro-Fuzzy Modeling

Basic approach of ANFIS

Adaptive networks

Neural networks Fuzzy inferencesystems

Generalization Specialization

ANFIS

>> 1995>> matlab conference

12

ANFISANFIS• Fuzzy reasoning

• ANFIS (Adaptive Neuro-Fuzzy Inference System)

A1 B1

A2 B2

w1

w2

z1 =p1*x+q1*y+r1

z2 =p2*x+q2*y+r2

z = w1+w2w1*z1+w2*z2

A1

A2

B1

B2

P

P

S

S

/

x

y

w1

w2

w1*z1

w2*z2

Swi*zi

Swi

z

x y

>> 1995>> matlab conference

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Hybrid Learning Rule in ANFISHybrid Learning Rule in ANFIS

Hybrid learning rule combines• Backpropagation gradient descent• Least-squares estimator (LSE)

Two modes of hybrid learning rule:• Off-line learning

- System identification• On-line learning

- On-line adaptive control

>> 1995>> matlab conference

14

FIS data structure in FLTFIS data structure in FLT

FIS file and FIS matrix

FIS file(on disk)

FIS D ata S tructure

M F 1L ab e l: S m all

Typ e : G au ss ianP aram s : [5 -1 0 ]

M F 2L ab e l: L a rg e

Typ e: Trian g leP aram s : [-5 1 0 2 0 ]

[In p u t 1 ]N am e: P os it ionR an g e : [-1 0 1 0 ]

M F # : 2

M F 1L ab e l: S m allTyp e : G b e ll

P aram s : [5 2 -4 0 ]

M F 2L ab e l: L a rg e

Typ e: SP aram s : [-5 0 0 5 0 ]

[In p u t 2 ]N am e: V e loc ityR an g e : [-5 0 5 0 ]

M F # : 2

M F 1L ab e l: N eg . B ig

Typ e : ZP aram s : [-1 0 -5 0 ]

M F 2L ab e l: N eg . S m all

Typ e : G b e llP aram s : [5 2 -3 ]

M F 3L ab e l: P os , S m all

Typ e : G b e llP aram s : [5 2 3 ]

M F 4L ab e l: P os . B ig

Typ e : SP aram s : [0 5 1 0 ]

[O u tp u t 1 ]N am e: F orce

R an g e : [-1 0 1 0 ]M F # : 4

[R u les ]R u le lis t

R u le w e ig h tsR u le typ es

[S ys tem ]N am e: m am 2 1Typ e: m am d an i

. . .

FIS matrix(in workspace)

readfiswritefis

>> 1995>> matlab conference

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From Data Sets to FISFrom Data Sets to FIS

Flow chart: From data sets to FIS

FLTGUI tools

genfis1.m

genfis2.m

anfis.mTrainingdata

Initial FIS

Trainingdata

Checkingdata

Final FIS

>> 1995>> matlab conference

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ANFIS in FLT - NowANFIS in FLT - Now

Restrictions of ANFIS in FLT v1• Single learning rule• Single learning mode• Single rule output type: Linear outputs

>> 1995>> matlab conference

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ANFIS in FLT - NextANFIS in FLT - Next

ANFIS in FLT v2• Multiple learning rules: Levenberg-Marquardt,

Gauss-Newton, etc. • Two learning modes: Both off-line and on-line• Output types: Both linear and constant outputs• New SIMULINK blocks: On-line ANFIS block

>> 1995>> matlab conference

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On-Line ANFIS Block in SIMULINKOn-Line ANFIS Block in SIMULINK

Function• Take an input/output data pair and do incremental

parameter updates.

Used when• The plant under control is time-varying.• We want to have an interpretable controller.

Theoretical background• Similar to other nonlinear adaptive controllers

>> 1995>> matlab conference

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Demo 1: Inverse LearningDemo 1: Inverse Learning

Plant

ANFIS

Training

Application

ukxk+1 = f(xk, uk)

xk

xk+1

+-

Plant

ANFIS

xk+1xk

ukxk, d

-1z

-1z

unit-delay system

xk+1 = xk,d

>> 1995>> matlab conference

20

Demo 1: Inverse LearningDemo 1: Inverse Learning

Properties of inverse learning• Exact plant model is not needed, but it has to have

an inverse model• Inversibility is controllability for linear plants

Demo• xk+1 = xkuk/(1+xk ) - tan(uk)• Four-rule ANFIS is used to implement the inverse

mapping from (xk, xk+1) to uk.

2

>> 1995>> matlab conference

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Demo 2: Feedback LinearizationDemo 2: Feedback Linearization

xk+1 = f(xk, xk-1) + uk

unknown and time-varying

Feedback linearizable plant:

Feedback law: uk = - F(xk, xk-1) + axk + bxk-1 + rk

Close-loop system: xk+1 = f(xk, xk-1) - F(xk, xk-1) + axk + bxk-1 + rk

ANFIS

close to zero

>> 1995>> matlab conference

22

Demo 2: Feedback LinearizationDemo 2: Feedback Linearization

Properties of feedback linearization• Exact plant model is not needed, but it has to be of

linearizable form.• Global stability is guaranteed when used with

sliding mode control.

>> 1995>> matlab conference

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Demo 2: Feedback LinearizationDemo 2: Feedback Linearization

Demo• xk+1 = cxkxk-1(xk+2.5)/(1+xk+xk-1) + uk

f, unknown and time-varying

• Four-rule ANFIS is used to approximate f

• Desired model: xk+1 = 0.6xk + 0.2xk-1 + uk