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Linearisation for Engineers

This document discusses linearization of nonlinear systems. It explains that while no physical system is perfectly linear, linearization allows greater analytical study by approximating a nonlinear system as linear for a narrow operating range. The linearization process involves taking the tangent line to the nonlinear curve at an operating point to relate output and input changes. Linearization can be done at multiple points along the curve to enable complete analysis. The document also introduces using Taylor series expansion to linearize systems with multiple input variables by approximating higher order terms as zero for small deviations from the operating point.

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Sushmita Kujur
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56 views9 pages

Linearisation for Engineers

This document discusses linearization of nonlinear systems. It explains that while no physical system is perfectly linear, linearization allows greater analytical study by approximating a nonlinear system as linear for a narrow operating range. The linearization process involves taking the tangent line to the nonlinear curve at an operating point to relate output and input changes. Linearization can be done at multiple points along the curve to enable complete analysis. The document also introduces using Taylor series expansion to linearize systems with multiple input variables by approximating higher order terms as zero for small deviations from the operating point.

Uploaded by

Sushmita Kujur
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Linearisation of non-linear system

Linearisation of non-linear system


No physical system is perfectly linear
Powerful analytical tools are available for linear systems
After linearisation, greater degree of analytical studies can be done
Basics of linearisation
A smooth curve differs very little from its tangent when the variable is
close to the point of tangency
If region of operation is restricted to a narrow range, a non-linear system
can be treated as a linear system
A system can be linearised at several points along the curve so that a
complete analysis can also be done

Linearisation of non-linear system: Taylor series

If (x - xo) is small, higher order terms of (x xo) can be neglected. Thus:


y yo = m (x xo), where m is the slope at the operating point

Linearisation of non-linear system: Taylor series

If y depends on several I/P variables


x1, x2, xn, then the approximation
is:

Problem sum on effect of feedback:

Ex 1..1 & 1.2 of Salivaahanan

A negative feedback system has a forward gain of 10 & feedback gain of 1.


Determine the overall gain of the system
A ve feedback system with forward gain of 2 and a feedback gain of 8 is
subjected to an input voltage of 5 V. Determine the overall gain & the output
voltage of the system

Problem sum on sensitivity:

G(s) = 200/s (s + 5), H(s) = 0.2s + 1


- Find sensitivity of T(s) = C(s)/R(s) w.r.t. variations of G(s) & H(s)
(Prob-7.48, 7.49, 7.50: Varmah)

Problem sum on sensitivity:

-Evaluate the sensitivity of the transfer function T(s) w.r.t. variations of K


(Ex. 3.2 Nagrath)

Problem sum on effect of feedback : Ex 3.7- Nagrath


In an unity feedback system G = /s(s + 1) .
- Determine the sensitivity of the closed loop transfer function w.r.t. .
- Evaluate the sensitivity at = 0.1 and = 1 for = 2

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