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Muddu Madakyaru Ph.D’10

Associate Professor.Dept. of Chemical Engineering

Manipal Institute of Technology, Manipal

Models for Computer Control

Digital Computers as Controller:Main Task of Controller:

About computer control System:

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Digital Control: Measured outputs

Computers & Data Acquisition:

Digital Control: Manipulated Inputs

Computers & Control action:

( ); t = ( ) ( ); < t < (k+1)

( 1); t = ( 1)

u k kTu t u k kT T

u k k T

Mathematical Description of Computer Control

Three Major issues raised by the digital Computer for Control

A. Sampling (and Conditioning) of continuous signalsB. Continuous signal reconstructionC. Appropriate mathematical description of sampled-data systems

Sampling and Conditioning of signals:

Measurements are sampled at a constant rate and uniform sampling interval at T sec.

Thus, measurements, y(k), are available at instant tk=kT: k=0,1,2 ,3 ……..

Aliasing Effect:

Effect of aliasing resulting from sampling a sinusoidal function too slowly:

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Signal Reconstruction:Inputs reconstruction with zero order hold: Manipulated inputs are piecewise constant during the sampling interval

u(t)=u(k) for ( 1)t kT t k T

( ); t = ( ) ( ); < t < (k+1)

( 1); t = ( 1)

u k kTu t u k kT T

u k k T

Apparent signal delay induced by the ZOH element;

Hold:

Discrete Dynamic Models

Discrete Dynamic Models:Discrete Dynamic Models:

Integrating Factor

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Discrete Dynamic Models: Discrete Dynamic Models:

Pulse Transfer Function Matrix Pulse Transfer Function Matrix

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Time Domain Difference Equation Practical Difficulties & Remedy

Identification Experiments on 4 Tank System

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Four Tank System: Input Excitation4 Tank System: Input Excitation

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4 Tank System: Measured Outputs Experimental Data for Modeling

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Splitting data for Identification & ValidationSpitting Data for Modeling & Identification

Model Parameter Estimation

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Model Parameter Estimation Model Parameter Estimation

Pros & Cons Summary

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Stability of Linear Dynamic Systems

1

1( )

1

bzg z

pz

1( ) for 1,2,3,...kg k bp k

The pulse transfer function of the process is

Taking inverse transform of the g(z) gives the impulse response of the process

The nature of the impulse response is clearly governed by the location of the pole i.e., ‘p’. Which must be real number.

Stability of Linear Dynamic Systems Stability of Linear Dynamic Systems

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Stability: Discrete time System