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 General LTI Systems
A linear time-invariant (LTI) system:
Its impulse response h(t) is the output y(t) under input u(t)=(t)
For arbitrary input u(t), the output is given by
Take the Laplace transform:
is called the transfer function

3. LTI Systems Given by Differential Equations
Often times, the LTI systems modeling practical systems are
Transfer function can be directly obtain by taking the Laplace transform (assuming zero initial condition)
Transfer function H(s) is given by

4. (Rational) Transfer Functions
Roots of B(s) are called the zeros of H(s): z1,…,zm
Roots of A(s) are called the poles of H(s): p1,…,pn
System is called an n-th order system
Pole zero plot:

5. Standard Forms of Transfer Function
Ratio of polynomial:
Factored (or product) form:
Sum form (assume poles are distinct):
p1,…,pn are the poles, r1,…,rn are the corresponding residues
If all poles are distinct

6. A Geometric Interpretation of Residues
Distance to zeros
Distance to poles
(except itself)
Pole zero plot:
Remark: if a pole is very close to a zero, its residue will be small.
(Approximate pole-zero cancellation)

7. Example

8. Example solve solve Zeros: Matlab code: Poles: Factored form:
Sum form (by PFD):

9. Modified Example Zeros: Poles:
(Compared with previous example, an extra zero very close to the
pole p2=2 is introduced)
Factored form:
Sum form (by PFD):
residue of the pole p2=2 is much smaller than in the previous case
due to zero-pole cancellation.

10. Model of Mechanical Systems Car Suspension Model
(body) m2
shock absorber
(wheel) m1
Input: road altitude r(t)
Output: car body height y(t)
road surface

11. Force Analysis of Mass One
Ignore the gravity forces (x,y are displacements from equilibrium positions)
(body) m2
By Newton’s Second Law
shock absorber
(wheel) m1
(wheel) m1
road surface

12. Force Analysis of Mass Two
Ignore the gravity forces (x,y are displacements from equilibrium positions)
(body) m2
(body) m2
shock absorber
(wheel) m1
By Newton’s Second Law
road surface

13. Suspension System Model

14. Suspension System Model
Differential equation model
Taking Laplace transform:
Compute X(s) from the second equation, plug in the first one:
transfer function from input r(t) to output y(t)

15. 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

16. 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 

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

18. Model of Electrical Systems
Basic components
resistor
inductor
capacitor

19. Impedance
Basic components
resistor
inductor
capacitor

20. Circuit Systems +

21. 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)

22. A Simple Nonlinear Control System
pendulum
Input: external force F
Output: angle 
Dynamic equation from Newton’s law
A nonlinear differential equation!
Linearization: approximate a nonlinear system by a linear one.
When  is small, sin is approximately equal to .
(see Section 3-10 of the textbook for more details)

23. A Simple Nonlinear Control System
pendulum
Input: external force F
Output: angle 
Dynamic equation from Newton’s law
A nonlinear differential equation!