Linearization is used to approximate nonlinear systems with linear models. It involves taking the Taylor series expansion of the nonlinear functions around an operating point and neglecting higher order terms. This results in a linear differential equation that is a good approximation when the state is close to the operating point. As an example, a tank system with nonlinear outlet flow was linearized by taking the Taylor series of the square root term and neglecting higher orders to obtain a linearized model. Similarly, a system of two nonlinear differential equations was linearized by expanding the functions around an operating point and neglecting higher order terms, resulting in a set of linear differential equations that approximate the original nonlinear system.

1. Linearization
of
Systems
Presented by:-
Chandan Taluja
EE Department, NITTTR, Chandigarh

2. Why do we
need
Linearization?

3. Taylor Series

4. Linearization of systems
with one variable
Consider the following non linear differential
equation modeling of a given process
𝑑𝑥
𝑑𝑡
=f(x)
Now Expanding the non linear function
f(x) into a taylor series around the point
x0 and take

5. If we neglect all terms of the order two or higher ,
we take the following approximation for the value
of f (x)

6. It is well known that error introduced is of
the same order of the magnitude of the term

7. The linear approximation is
satisfactory only when x is very close
to x0 , where the value of I is very
small

8. Example of
linearization

9. Consider the tank system

10. The total mass balance yields
If the outlet flow rate F is not a linear
function of liquid level

11. Then resulting mass balance
equitation yields non linear dynamic
model
Now the only non linear term is β√h ,
so taking Taylor series expansion for the
term around point h0

12. Neglecting higher order terms, we get

13. Now we get the following linearized model as

14. Linearization of systems
with two variables

15. Consider the following dynamic
systems

16. Now expanding the functions into Taylor
series around the point ( X(1,0) , X(2,0) )

17. Neglecting the higher order terms

18. So the last two equations as the linear
differential equations and constitute the
linearized approximate model of the initial
non linear system

19. Thank You!