1. ME375 Handouts - Fall 2002
MESB374 Chapter8 System Modeling and Analysis Time domain Analysis Transfer Function Analysis

2. Transfer Function Analysis
ME375 Handouts - Fall 2002
Transfer Function Analysis
Dynamic Response of Linear Time-Invariant (LTI) Systems
Free (or Natural) Responses
Forced Responses
Transfer Function for Forced Response Analysis
Poles
Zeros
General Form of Free Response
Effect of Pole Locations
Effect of Initial Conditions (ICs)
Obtain I/O Model based on Transfer Function Concept

3. Dynamic Responses of LTI Systems
ME375 Handouts - Fall 2002
Dynamic Responses of LTI Systems
Ex: Let’s look at a stable first order system:
Time constant
Take LT of the I/O model and remember to keep tracks of the ICs:
Rearrange terms s.t. the output Y(s) terms are on one side and the input U(s) and IC terms are on the other:
Solve for the output:
Forced Response
Free Response

4. Free & Forced Responses
ME375 Handouts - Fall 2002
Free & Forced Responses
Free Response (u(t) = 0 & nonzero ICs)
The response of a system to zero input and nonzero initial conditions.
Can be obtained by
Let u(t) = 0 and use LT and ILT to solve for the free response.
Forced Response (zero ICs & nonzero u(t))
The response of a system to nonzero input and zero initial conditions.
Assume zero ICs and use LT and ILT to solve for the forced response (replace differentiation with s in the I/O ODE model).

5. ME375 Handouts - Fall 2002
In Class Exercise
Find the free and forced responses of the car suspension system without tire model:
Take LT of the I/O model and remember to keep tracks of the ICs:
Rearrange terms s.t. the output Y(s) terms are on one side and the input U(s) and IC terms are on the other:
Solve for the output:
Forced Response
Free Response

6. Forced Response & Transfer Function
ME375 Handouts - Fall 2002
Forced Response & Transfer Function
Given a general n-th order system model:
The forced response (zero ICs) of the system due to input u(t) is:
Taking the LT of the ODE:
Forced
Response
Transfer
Function =
Transfer Function
Inputs =
Inputs

7. Transfer Function Given a general nth order system:
ME375 Handouts - Fall 2002
Transfer Function
Given a general nth order system:
The transfer function of the system is:
The transfer function can be interpreted as:
Static gain
u(t)
Input
y(t)
Output
U(s)
Input
Y(s)
Output
Differential
Equation
G(s)
Time Domain
s - Domain

8. Transfer Function Matrix
ME375 Handouts - Fall 2002
Transfer Function Matrix
For Multiple-Input-Multiple-Output (MIMO) System with m inputs and p outputs:
Inputs
Outputs

9. Poles and Zeros Poles Zeros
ME375 Handouts - Fall 2002
Poles and Zeros
Given a transfer function (TF) of a system:
Poles
The roots of the denominator of the TF, i.e. the roots of the characteristic equation.
Zeros
The roots of the numerator of the TF.
m zeros of TF
n poles of TF

10. Examples (1) Recall the first order system:
Find TF and poles/zeros of the system.
(2) For car suspension system:
Find TF and poles/zeros of the system.
Pole:
Pole:
Zero:
Zero:
No Zero

11. System Connections + + - Cascaded System Input Output Parallel System
ME375 Handouts - Fall 2002
Cascaded System
Input
Output
Parallel System
Input
Output +
Feedback System + -
Input
Output

12. General Form of Free Response
ME375 Handouts - Fall 2002
General Form of Free Response
Given a general nth order system model:
The free response (zero input) of the system due to ICs is:
Taking the LT of the model with zero input
(i.e., )
Free Response
(Natural Response)
A Polynominal of s that depends on ICs =
Same Denominator as TF G(s)

13. Free Response (Examples)
ME375 Handouts - Fall 2002
Free Response (Examples)
Ex: Find the free response of the car suspension system without tire model (slinker toy):
Ex: Perform partial fraction expansion (PFE) of the above free response when:
(what does this set of ICs means physically)?
phase: initial conditions
Decaying rate: damping, mass
Frequency: damping, spring, mass
Q: Is the solution consistent with your physical intuition?

14. Free Response and Pole Locations
ME375 Handouts - Fall 2002
Free Response and Pole Locations
The free response of a system can be represented by:
exponential decrease
Real
Img.
constant
exponential increase
decaying oscillation
Oscillation with constant magnitude
increasing oscillation t

15. Complete Response Y(s) U(s) Output Input Complete Response
ME375 Handouts - Fall 2002
Complete Response
U(s)
Input
Y(s)
Output
Complete Response
Q: Which part of the system affects both the free and forced response ?
Q: When will free response converges to zero for all non-zero I.C.s ?
Denominator D(s)
All the poles have negative real parts.

16. Obtaining I/O Model Using TF Concept (Laplace Transformation Method)
Noting the one-one correspondence between the transfer function and the I/O model of a system, one idea to obtain I/O model is to:
Use LT to transform all time-domain differential equations into s-domain algebraic equations assuming zero ICs (why?)
Solve for output in terms of inputs in s-domain to obtain TFs (algebraic manipulations)
Write down the I/O model based on the TFs obtained

17. Example – Car Suspension System
Step 1: LT of differential equations assuming zero ICs x p
Step 2: Solve for output using algebraic elimination method
# of unknown variables = # equations ?
2. Eliminate intermediate variables one by one. To eliminate one intermediate variable, solve for the variable from one of the equations and substitute it into ALL the rest of equations; make sure that the variable is completely eliminated from the remaining equations

18. Example (Cont.) Step 3: write down I/O model from TFs
from first equation
Substitute it into the second equation
Step 3: write down I/O model from TFs