Neuro-Fyzzy Methods
for
Modeling and Identification
Part 2 : Examples
Presented by:
Ali Maleki
Presentation Agenda
Introduction
Tools for Fuzzy Identification and Modelling
MATLAB – Fuzzy Logic Toolbox – ANFIS
Fuzzy Modeling and Identification Toolbox
Examples :
Example 1: Hair Dryer
Example 2: Static Function
Example 3: Neutralization Process
Introduction
Control Systems
Competion
Environment requirements
Energy and material costs
Demand for robust, fault-tolerant systems
Extra needs for Effective process modeling techniques
Conventional modeling?
Lack precise and formal knowledg about the system
Strongly nonlinear behavior,
High degree of uncertainty,
Time varying characteristics
Introduction (cont.)
Solution: Neuro-fuzzy modeling
A powerful tool which can facilitate the effective
development of models by combining information from
different source:
Empirical models
Heuristics
Data
Neuro-fuzzy models
Describe systems by means of fuzzy if-then rules
Represented in a network structure
Apply algorithms from the area of Neural Networks
Introduction (cont.)
Zero-order TS fuzzy model
Typical membership function
Input-output equation
Introduction (cont.)
System identification steps:
• Structure identification
• Parameter estimation
•
choice of the model’s structure determines the flexibility of
the model in the approximation of (unknown) systems
•
model with a rich structure can approximate more
complicated functions, but, will have worse generalization
properties
•
Good generalization means that a model fitted to one
data set will also perform well on another data set from
the same process.
Tools for Fuzzy Modeling & Identification
Fuzzy Modelling and Identification Toolbox
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•
•
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Develped by R. Babuska
http://Lcewww.et.tudelft.nl/~Babuska
Installation
Version 3.03 , April 2001
Fuzzy Logic Toolbox for MATLAB
•
•
www.Mathworks.com/products/fuzzylogic
Version 2.1.3 , June 2004
Fuzzy Modeling and Identification Toolbox
Fuzzy Modeling and Identification Toolbox (cont.)
Fuzzy Modeling and Identification Toolbox (cont.)
FMSIM function:
Simulate a MIMO input-output fuzzy model
Fuzzy Modeling and Identification Toolbox (cont.)
Structure of FM (fmstruct function):
Fuzzy Modeling and Identification Toolbox (cont.)
Structure of FM (fmstruct function):
Fuzzy Modeling and Identification Toolbox (cont.)
plotmfs function:
Plot membership functions
rms function:
Root mean square between two signals
vaf function:
Percentile variance accounted for (VAF) between two signals
MATLAB - Fuzzy Toolbox
anfis function:
Training routine for Sugeno-type FIS
anfisedit function:
Open the ANFIS Editor GUI
genfis1 function:
Generate an FIS structure from data without data clustering
genfis2 function:
Generate an FIS structure from data using subtractive clustering
MATLAB - Fuzzy Toolbox (cont.)
ANFIS Editor GUI
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox)
Nonlinear dynamical system identification
With use of ANFIS function in the Fuzzy Logic Toolbox
Data set was obtained from a laboratory device called
Feedback's Process Trainer PT 326,
L. Ljung, "System Identification, Theory for the User", PrenticeHall, 1987 – Chapter 17
The device's function is like a hair dryer:
Input u(k) : Voltage over the mesh of resistor wires
Output y(k) : Outlet air temperature
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox) (cont.)
Input u(k) : Binary random signal shifting between 3.41 and 6.41 V
Output y(k) : Outlet air temperature
Sampling Time : 0.08 sec
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox) (cont.)
Linear ARX model:
y(k)+a1*y(k-1)+...+am*y(k-m)=b1*u(k-d)+...+bn*u(k-d-n+1)
ai and bj are linear parameters to be determined by least-squares methods
This structure is exactly specified by three integers [m, n, d]
Remind:
System Identification : structure selection + parameter estimation
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox) (cont.)
Remove the means from the data
The data set was divided into a training set and a checking set
Training data set : (k = 1 to 300)
Checking data set : (k = 301 to 600)
An exhaustive search was performed to find the best combination [m, n, d]
each of the integer is allowed to changed from 1 to 10 independently
Run through all different models:
V = arxstruc(ze, zv, struc(1:10, 1:10, 1:10));
Find the best model:
nn = selstruc(V, 0);
The best ARX model : [m, n, d] = [5, 10, 2]
Example 1 : Hair Dryer
Training RMSE = 0.1122
(Fuzzy Logic Toolbox) (cont.)
Checking RMSE = 0.0749
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox) (cont.)
Advantage of ARX model:
Rapid model structure selection
Rapid parameter identification
The performance in the above plots appear to be satisfactory.
If a better performance level is desired, we might want to resort to a
nonlinear model.
Neuro-fuzzy modeling approach, ANFIS
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox) (cont.)
Use ANFIS for system identification:
First step: input selection
To determine which variables should be the input arguments to an ANFIS
model.
For simplicity, we suppose that there are 10 input candidates
y(k-1), y(k-2), y(k-3), y(k-4), u(k-1), u(k-2), u(k-3), u(k-4), u(k-5), u(k-6)
Two approaches for input selection:
Sequential search
Exhaustive search
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox) (cont.)
Sequential search for input selection: can be done by the function seqsrch
10 + 9 + 8
=27
Selected inputs : y(k-1), u(k-3), and u(k-4)
Training RMSE = 0.0609
Checking RMSE = 0.0604.
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox) (cont.)
Exhaustive search on all possible combinations of the input candidates
Can be done by function exhsrch
We want to selects 3 inputs from 10 candidates, therefore, the total number
of ANFIS models is
10!  120
3!7!
Fortunately, for dynamical system identification, we do know that the inputs
should not come from either of the following two sets of input candidates
exclusively:
Y = {y(k-1), y(k-2), y(k-3), y(k-4)}
U = {u(k-1), u(k-2), u(k-3), u(k-4), u(k-5), u(k-6)}
A reasonable guess: two inputs from Y and one from U
4!  * 6  36
2! 2! 
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox) (cont.)
Exhaustive search
Selected inputs : y(k-1), y(k-2) , u(k-3)
Training RMSE = 0.0474
Checking RMSE = 0.0485
Example 1 : Hair Dryer
ARX model
(Fuzzy Logic Toolbox) (cont.)
Example 1 : Hair Dryer
ANFIS model
(Fuzzy Logic Toolbox) (cont.)
Example 1 : Hair Dryer
(Fuzzy Logic Toolbox) (cont.)
Comparision
If fast modeling is the goal, then ARX is the right choice,
If precision is the utmost concern, then we can go for ANFIS that is
designed for nonlinear modeling and higher precision
Example 2 - Static Function
ANFIS model with linear consequent function
Number of rules: five rules
Construction of initial model:
Gustafson-Kessel algorithm
Fit of the function with initial model – local models - membership functions
Example 2 - Static Function (cont.)
this initial model can easily be interpreted in terms of the local behavior
It is reasonably accurate (RMS= 0.0258)
ANFIS method, 100 learning epochs
anfis function of the MATLAB Fuzzy Logic Toolbox
Fit of the function with fine-tuned model, local models, membership functions
Example 2 - Static Function (cont.)
RMS error is about 23 times better than the initial model
Initial model
RMS error = 0.0258
Fine-tuned model
RMS error = 0.0011
Example 2 - Static Function (cont.)
after learning, the local models are much further from the true local
description of the function
Initial model
Fine-tuned model
Fine-tuned model are thus less accurate in describing the system locally
Example 3 – pH Neutralization Process
Influent streams
Acid
Buffer
Base
Acid flowrate = cte
Buffer flowrate = cte
Base stream flowrate
Neutralization
tank
Effluent stream
Neutralization
tank
pH in the tank
Example 3 - pH Neutralization Process (cont.)
Identification and validation data sets:
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•
Simulating the model by Hall and Seborg for random change of the
influent base stream flow rate
N = 499 samples with the sampling time of 15 s.
The process is approximated as a first–order discrete-time NARX model
Example 3 - pH Neutralization Process (cont.)
Membership functions
Befor training
After training
Example 3 - pH Neutralization Process (cont.)
Rules:
Initial Rules:
Fine Tuned Rules: after 1000 epochs of hybrid learning using the ANFIS
function of the MATLAB Fuzzy Logic Toolbox
Example 3 - pH Neutralization Process (cont.)
Overtraining Problem:
•
Comparision of RMS ERROR befor and after training
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Prediction befor and after training
References
[1] Robert Babuska, “Neuro-Fuzzy Methods for Modeling and Identification”,
Recent Advances in Intelligent Paradigms and Application, SpringerVerilag, 2002
[2] Robert Babuska, “Fuzzy Modeling and Identification Toolbox User’s Guide
- For Use with MATLAB”, 1998.
[3] MathWorks Inc., “Fuzzy Logic Toolbox – Users Guide – Version 2”, 2004.
[4] L. Ljung, “System Identification, Theory for the User”, Prentice-Hall, 1987.
THANK YOU VERY MUCH
For your
Attention
Presented by:
Ali Maleki
Introduction - appendix
Types of fuzzy models:
(depending on the structure of if-then rules)
Mamdani Model
IF D1 is low and D2 is high THEN D is medium
Takagi-Sugeno Model
IF D1 is low and D2 is high THEN D=k
(zero-
IF D1 is low and D2 is high THEN D=0.7D1+0.2D2+0.1
(first-
order)
order)