‘MODEL- FREE’ APPROACH TO MODELLING OF SYSTEMS

1. IEEEMalabar subsection Prof K P MOHANDAS Email:kpmdas@nitc.ac.in ‘MODEL- FREE’ APPROACH TO MODELLING OF SYSTEMS

2. Models 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 IEEEMalabar subsection Introduction

3. .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 IEEEMalabar subsection Approaches to Modelling 1. Microscopic approach

4. 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 IEEEMalabar subsection 2.The holistic/ black box approach

5. IEEEMalabar subsection Types of models

6. 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’ IEEEMalabar subsection Conventional Models

7. 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 Modelorder determination is essential before parameter estimation IEEEMalabar subsection MODEL structure and parameters

8. 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’sInformation criteria (AIC) Markov Parameter methods Most of these fail when the data used is noisy IEEEMalabar subsection Model order determination

9. IEEEMalabar subsection Modelling from Input output data New approaches for

10. 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 IEEEMalabar subsection 1.FRACTIONAL ORDER SYSTEMS

11. 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. IEEEMalabar subsection Applications of fractional order systems

12. 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. IEEEMalabar subsection More Applications

13. Fractional calculus is a generalization of integration and differentiation operation to non-integer order fundamental operator. When the order ris positive, the usual differential results and r is negative integral results IEEEMalabar subsection Generalized operator

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

15. A fractional order linear time invariant system can be described by a fractional order differential equation of the form : Transfer function of the form IEEEMalabar subsection Fractional order differential equation

16. IEEEMalabar subsection FRACTIONAL ORDER State space model

17. IEEEMalabar subsection Neuro-fuzzy approaches

18. 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 IEEEMalabar subsection Artificial Intelligence techniques for modelling

19. 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 IEEEMalabar subsection Neuro-fuzzy approach

20. 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”. IEEEMalabar subsection Information processing in human brain?

21. 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 IEEEMalabar subsection What CAN ANNs DO ?

22. X1 X2 Y1 Y2 Artificial Neural Network x1,x2 :input nodes : y1,y2 :output IEEEMalabar subsection Typical Neural Network

23. 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 IEEEMalabar subsection Different types of ANNs

24. They are computationally expensive. Convergencetakes a long time ( eg.feed forward networks in particular) No definite guidelines to choose the ANN architecture Back-propagationtype ANNs not effective in modeling dynamic systems IEEEMalabar subsection Major problems IN USE of Artificial Neural Networks

25. 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 IEEEMalabar subsection 2.Fuzzy Systems theory

26. 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 IEEEMalabar subsection WHAT Fuzzy Logic Systems DO?

27. 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 IEEEMalabar subsection NATURE OF Fuzzy rules

28. 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 WHEN TO USE Fuzzy logic control ?

29. It is not a numeric tool and hence cannot deal with numeric data directly Choice of type of membershipfunction and operating ranges of variableshave 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. IEEEMalabar subsection Problems in fuzzy logic

30. Modeling of systems using Neural Networks and Fuzzy systems involve: Acquisition and tuning of fuzzy models based on input output data -called : fuzzy identification IEEEMalabar subsection 3.Neuro-fuzzy Systems

31. 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 IEEEMalabar subsection ANFIS - Adaptive Neuro Fuzzy Inference System

32. 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 IEEEMalabar subsection Steps in ANFIS Modeling

33. 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 IEEEMalabar subsection 4.Nonlinear time series modelling

34. 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 methods Artificial neural networks have been used IEEEMalabar subsection Nonlinear time series methods

35. 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 IEEEMalabar subsection In short ,

36. 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. IEEEMalabar subsection However

37. IEEEMalabar subsection THANK YOU