1. Lec 3. System Modeling Transfer Function Model
Model of Mechanical Systems
Model of Electrical Systems
Model of Electromechanical Systems
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2. Transfer Functions of LTI Systems
Output of a linear time-invariant (LTI) system is given by
where ℎ(𝑡) is the impulse response, output under input u(t)=(t)
Take the Laplace transform:
is called the transfer function

3. LTI Systems Given by Differential Equations
Common LTI system models for practical systems:
𝑎 𝑛 𝑦 (𝑛) + 𝑎 𝑛−1 𝑦 (𝑛−1) +⋯+ 𝑎 0 = 𝑏 𝑚 𝑦 (𝑚) + 𝑏 𝑚−1 𝑦 (𝑚−1) +⋯+ 𝑏 0
Assuming zero initial condition, transfer function model is
𝐻 𝑠 = 𝐵(𝑠) 𝐴(𝑠) = 𝑏 𝑚 𝑠 𝑚 + 𝑏 𝑚−1 𝑠 𝑚−1 +⋯+ 𝑏 0 𝑎 𝑛 𝑠 𝑛 + 𝑎 𝑛−1 𝑠 𝑛−1 +⋯+ 𝑎 0
(𝑛≥𝑚 in general)

4. (Rational) Transfer Functions
𝐻 𝑠 = 𝐵(𝑠) 𝐴(𝑠) = 𝑏 𝑚 𝑠 𝑚 + 𝑏 𝑚−1 𝑠 𝑚−1 +⋯+ 𝑏 0 𝑎 𝑛 𝑠 𝑛 + 𝑎 𝑛−1 𝑠 𝑛−1 +⋯+ 𝑎 0
System is called an n-th order system
Zeros of 𝐻(𝑠): roots of 𝐵(𝑠), 𝑧 1 , 𝑧 2 ,…, 𝑧 𝑚
Poles of 𝐻(𝑠): roots of 𝐴(𝑠), 𝑝 1 , 𝑝 2 ,…, 𝑝 𝑛
Pole zero plot:

5. Standard Forms of Rational Transfer Functions
Basic form:
𝐻 𝑠 = 𝑏 𝑚 𝑠 𝑚 + 𝑏 𝑚−1 𝑠 𝑚−1 +⋯+ 𝑏 0 𝑎 𝑛 𝑠 𝑛 + 𝑎 𝑛−1 𝑠 𝑛−1 +⋯+ 𝑎 0
Factored (or product) form:
Sum form (assume poles are distinct):
If all poles are distinct
p1,…,pn are the poles, r1,…,rn are the corresponding residues

6. Example

7. Example: Car Suspension Model
(body) m2
shock absorber
(wheel) m1
Input: road altitude r(t)
Output: car body height y(t)
road surface

8. Suspension System Model
Differential equation model:

9. Translational Mechanical System Models
Identify all independent components of the system
For each component, do a force analysis (all forces acting on it)
Apply Newton’s Second Law to obtain an ODE, and take the Laplace transform of it
Combine the equations to eliminate internal variables
Write the transfer function from input to output

10. Rotational Mechanical Systems: Satellite
gas jet
Suppose that the antenna of the satellite needs to point to the earth
Ignore the translational motions of the satellite
Input: A force F generated by the release of reaction jet
Output: orientation of the satellite given by the angle 

11. Satellite Model
Torque:
In general
gas jet
Newton’s Second Law:

12. Model of Electrical Systems
Basic components
resistor
inductor
capacitor

13. Impedance
Basic components
resistor
inductor
capacitor

14. Circuit Systems +

15. Electromechanical System: DC Motor
Armature resistance
Torque T
Basic motor properties:
Torque proportional to current:
Motor voltage proportional to shaft angular velocity:
Friction B
Input: voltage source e(t)
Output: shaft angular position q(t)
Basic circuit properties (KVL):
Newton law:

16. Nonlinear System: Pendulum
Input: external force F
Output: angle 
Dynamic equation from Newton’s law