prof k p mohandas email:[email protected] 30th dec 2010 ieeemalabar subsection 1

‘MODEL- FREE’ APPROACH TO MODELLING OF SYSTEMS

Prof K P MOHANDASEmail:[email protected]

• 30th Dec 2010• IEEEMalabar subsection

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INTRODUCTIONModels are :

Re-presentations of the available knowledge about the system in a convenient form

There is nothing like ‘the model’ as there can be several models based on

the purpose and manner of presentation


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APPROACHES TO MODELLING 1. MICROSCOPIC APPROACH

.Element – subsystem- systems approach• Dynamic equations of electric circuits

using Kirchhoff’s Current Law or KVL• Dynamic equations for Mechanical

systems using D’ Alembert’s principle• Major Assumption here is:• sum of parts = whole • Very rarely justified for real systems


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2.THE HOLISTIC/ BLACK BOX APPROACH• The system may be : physical or

defined only conceptually• The characterization is in terms of :• inputs ,outputs and a boundary• No a priori knowledge is assumed• Input output data measurable by

appropriate instrumentation


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TYPES OF MODELS

Sl.No Type of systemMathematical models

Continuous time Discrete time

1 Static Algebraic equations Algebraic equations

2 Dynamic Differential equations Difference equations

3 Time invariant Differential equation with constant coefficients.

Difference equation with constant coefficients

4 Lumped Ordinary differential equation Ordinary difference equations

5 Distributed Partial differential equations Partial difference equations

6 Linear Linear differential equation Linear difference equations

7 Nonlinear Nonlinear differential equation Nonlinear difference equations

8 Predictable Deterministic differential equations Deterministic difference equations

9 Uncertain Stochastic differential equations Stochastic difference equations


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CONVENTIONAL MODELS

Differential equations : usually derived from basic theory or experimental laws

Transfer functions / Impulse response:

from differential equations neglecting

initial energy stored State space models derived from differential equations by defining

new ‘state variables’


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MODEL STRUCTURE AND PARAMETERS

When a conventional mathematical model is to be determined

Structural parameter of the model like system order is to be decided Parameters of the assumed model to

be estimated next Model order determination is essential

before parameter estimation


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MODEL ORDER DETERMINATION

No of inputs and outputs are well defined Model order may have to be determined

from the input output data Methods available : Prediction error method Akaike’s Information criteria (AIC) Markov Parameter methods Most of these fail when the data used is

noisy


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NEW APPROACHES FOR

Modelling from Input output data


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1.FRACTIONAL ORDER SYSTEMS

In 1695 , L’Hospital asked Leibnitz why should the order n in the equation:

be an integer? Can it not be a fraction? Since then fractional calculus has been in

use There are many systems and phenomena

that require fractional order equations


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APPLICATIONS OF FRACTIONAL ORDER SYSTEMS

Transmission line theory, Chemical analysis of aqueous solutions, Design of heat-flux meters, Rheology of soils, Growth of inter-granular grooves on

metal surfaces, Quantum mechanical calculations and

dissemination of atmospheric pollutants.


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MORE APPLICATIONS

Description of systems with memory and hereditary properties of materials

Modelling of dynamic systems , biological systems, etc

There are many physical phenomena which have “intrinsic” fractional order description and hence fractional order calculus is necessary for describing such phenomena.


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GENERALIZED OPERATOR

Fractional calculus is a generalization of integration and differentiation operation to non-integer order fundamental operator.

When the order r is positive, the usual differential results and r is negative integral results


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In the general operator r can be positive ,zero or negative : a and t are the limits of operation


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0 r :

0 r : 1 D

0 r : dtd

r

ta

n

r

FRACTIONAL ORDER DIFFERENTIAL EQUATION

A fractional order linear time invariant system can be described by a fractional order differential equation of the form :

Transfer function of the form


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FRACTIONAL ORDER STATE SPACE MODEL


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NEURO-FUZZY APPROACHES


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ARTIFICIAL INTELLIGENCE TECHNIQUES FOR MODELLING

As the systems to be modelled became more complex, the conventional techniques became inadequate to describe real systems

This led to adaptation of other techniques to modelling

These are : Artificial Neural Networks (ANN) Fuzzy Logic Systems Neuro-fuzzy techniques


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NEURO-FUZZY APPROACH

Neural Networks are extensively used for modeling of systems from input- output data No a priori mathematical model is assumed A neural network consists of :

- a set of input nodes - a set of hidden layers and - a set of output nodes


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INFORMATION PROCESSING IN HUMAN BRAIN?

Information processing in the brain is carried out by a network of millions of simple processing units called neurons

The neurons are basically simple processors. Essentially each neuron receives signals from a large

number of other neurons, combines these inputs and send out the signals to large number of other neurons.

It is the pattern of connections between neurons that seems to embody “knowledge” required for carrying out various information processing tasks. Hence the human brains are supposed to do “ connectionist computing”.


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WHAT CAN ANNS DO ?

Artificial neural networks are capable of learning the characteristics of input output data.

An ANN can learn from examples. If the ANNS are given pairs of data in which the first member of

the pair is the given input and the second member is the desired output, an ANN can be ‘trained’ to adjust its weights so that it associates the correct answer from each input.

This capability is important because there are many problems in which you know what should be the correct output, but it is not possible to lay down a precise procedure or set of rules for finding the result. In such cases, providing examples will enable ANN to develop its own implicit rules in terms of correct weights to use, it is certainly advantageous. A digital computer program requires precise rules to produce an output.

They are universal function approximators


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TYPICAL NEURAL NETWORK

Artificial Neural Network x1,x2 :input nodes : y1,y2 :output


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X1

X2

Y1

Y2

DIFFERENT TYPES OF ANNS

Feed forward or back propagation networks

Feedback or recurrent neural networks Partial recurrent networks etc Radial basis function networks Modelling of dynamic systems require

ANNs with feed back, or recurrent networks


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MAJOR PROBLEMS IN USE OF ARTIFICIAL NEURAL NETWORKS

They are computationally expensive.

Convergence takes a long time ( eg.feed forward networks in particular)

No definite guidelines to choose the ANN architecture

Back-propagation type ANNs not effective in modeling dynamic systems


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2.FUZZY SYSTEMS THEORY

Modeling and Control of systems which are Nonlinear and uncertain in behavior

It replaces the deterministic control laws by a set of linguistic ‘if-then’ rules

A Fuzzy Inference Engine develops a control signal to actuate the controller. The available information is not ‘precise’ or exact , but fuzzy


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WHAT FUZZY LOGIC SYSTEMS DO?

It is an attempt to mimic the method of data processing by human beings

It is used as control strategy in many practical situations where mathematical modelling is difficult

The experience and judgment of humans can be used to formulate fuzzy ‘if then’ rules


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NATURE OF FUZZY RULES If the temperature in a voltage controlled

furnace is classified as LOW, MEDIUM, HIGH we can set the voltage also to ranges like VERY SMALL, SMALL, NORMAL, BIG , and VERY BIG.

A control logic statement, then, will look like:

If the temperature is VERY LOW set the voltage to HIGH

If the temperature is MEDIUM set the voltage to NORMAL etc


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WHEN TO USE FUZZY LOGIC CONTROL ?

Several consumer products such as washing machines, cam-coders.etc

Automobile driving mechanisms Braking of suburban railways in Japan

etc An early application in cement kilns

where raw material quality cannot be measured exactly

Where precise control is not required, this has resulted in considerable saving in energy

• IEEEMalabar subsection

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• 30th Dec 2010

PROBLEMS IN FUZZY LOGIC

It is not a numeric tool and hence cannot deal with numeric data directly

Choice of type of membership function and operating ranges of variables have to be done with care

A combination of the dynamic neural networks and fuzzy logic can be used effectively in modeling and control of uncertain systems.


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3.NEURO-FUZZY SYSTEMS

Modeling of systems using Neural Networks and Fuzzy systems involve:

Acquisition and tuning of fuzzy models based on input output data -called :

fuzzy identification


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ANFIS - ADAPTIVE NEURO FUZZY INFERENCE SYSTEM

Expert knowledge available is expressed as a set of ‘ if-then’ rules. This fixes a structure for the model

Parameters in this structure are fine-tuned by input-output data

No a priori knowledge is essential. The extracted rules and membership functions can give an a posteriori interpretation


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STEPS IN ANFIS MODELING

Choice of input-output variables Choice of structure of rules: linguistic,

relational or Takagi Sugeno model Choice of no. and type of membership function Type of inference mechanism (mostly decided

by the structure of fuzzy model) The whole process is made ‘data driven’

with initial sets of parameters which are self-tuned by the program


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4.NONLINEAR TIME SERIES MODELLING

Description of phenomena like chaos, fractals stock markets etc require nonlinear models

Nonlinear time series can be expressed as :

Where f (.) is a nonlinear function of the arguments


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NONLINEAR TIME SERIES METHODS The problem is to find the nonlinear

function that describes the system. Several methods have been proposed for

modelling such as : Markov switching, Threshold auto-regression and Smooth transition auto-

regression. Classical and Bayesian methods have been

proposed for each of these methodsArtificial neural networks have been used


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IN SHORT ,

The inadequacy of conventional models for complex systems has necessitated

New approaches to modelling such as : Fractional oder system models Artificial neural networks Fuzzy logic systems Neuro-fuzzy approached


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HOWEVER

Modeling is even to-day an art, not a science Effectiveness of the modeling depends on : How much you know about the system The more we know about the system or

phenomenon that we study, better will be the model

The lesson thus is : Try to understand how the system behaves before modelling.


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THANK YOU


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prof k p mohandas email:[email protected] 30th dec 2010 ieeemalabar subsection 1

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