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Expand the nonlinear function f(x) into a Taylor Series around the point x o

Linearization Process

If neglect all terms of order two and higher

Tank system- Total mass balance of the tank system yields

- The resulting total mass balance yields a nonlinear dynamic model

- Take Taylor series expansion around a point ho

- Linearize approximate model


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Linearization of systems with many variables- Consider the following dynamic system

LAPLACE TRANSFORMS

Laplace transform converts integral and differential equations into algebraic equations It applies to phasors such as general signal, not just sinusoids. It also handles non steady

state conditions

- Some basic functions of Laplace transforms:(a)Exponential function(b)Trigonometric function(c)Translated function(d)Unit pulse function(e)Unit impulse function

- Types of input signals:(a)Constant(b)Step function(c)Ramp function(d)Rectangular pulse(e)Unit impulse

The Laplace transform f is the function of :

- Laplace transform of F(s) for function f(t)

- The transformation of time domain (t) to laplace domain (s)


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- The laplace transformation is a linear operation

Step function is given as f(t) = A

Ramp Function

- Ramp function is given as f(t) = at


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Type of Input Functions

Exponential Function (Process)

- Exponential function is given as f(t) = e-at

Translated Function (Output)


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Relationship among the three curves


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TRANSFER FUNCTION

Definition: Transfer function is a mathematical representation, in terms of spatial ortemporal frequency, of the relation between the input and output of a linear time-invariant

system with zero initial conditions and zero-point equilibrium

Transfer function = G(s)

It describe completely the dynamic behavior of the output when the corresponding inputchanges are given

For a particular changes in f(t), we can find its transform of f(s) and the system responseis given by :-

Transfer function for process with multiple outputs- Consider a process with two inputs f1(t) and f2(t) and two outputs, y1(t) and y2(t)


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Common Transfer Function

Given :-

Hence the transfer function is the rational function of

- Poles and zero of transfer functioni. Roots of the polynomial a(s) are called poles of the system and the roots of b(s) are called

the zeros of the system

ii. The poles of the transfer function are the eigenvalues of the system matrix A in the statespace model. They depend only on the the dynamics matrix A, which represents the

intrinsic dynamics of the system. The zeros depend on how inputs and outputs are


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- Dynamic Characteristics(i) A first-order process isself-regulating. The process reaches a new steady state.(ii) The ultimate value of the output is K for a unit step change in the input, or KM for a

step of size M. This can be seen from the equation which yields

y KM as t . This characteristic explains the name steady state or static gain

given for the parameter K, since for any step change in the input the resulting change

in the output steady state is given by:

(output) =K(input)

This Equation tells us by how much we should change the value of the input in order

to achieve a desired change in the output, for a process with given K. Thus, to effect

the same change in the output, we need: A small change in the input if K is large

(very sensitive systems); A large change in the input if K is small.

- For pure capacitive system, the transfer function is given as belows:

- A pure capacitive process causes serious control problem because it cannot balance itself.For small change in the input, the output grows continuously. This is known as non-self-

regulatingprocess.


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