Post on 29-Dec-2015
Lec 3. System Modeling
• Transfer Function Model
• Model of Mechanical Systems
• Model of Electrical Systems
• Model of Electromechanical Systems
• Reading: 3.1-3.3, 3.6-3.8
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
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
(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:
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
A Geometric Interpretation of Residues
Pole zero plot:
Distance to zeros
Distance to poles(except itself)
Remark: if a pole is very close to a zero, its residue will be small.
(Approximate pole-zero cancellation)
Example
Model of Mechanical SystemsCar Suspension Model
Input: road altitude r(t)Output: car body height y(t)
road surface
(wheel)m1
(body)m2
shock absorber
Suspension System Model
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
Rotational Mechanical Systems: Satellite
Output: orientation of the satellite given by the angle Input: A force F generated by the release of reaction jet
Suppose that the antenna of the satellite needs to point to the earth
Ignore the translational motions of the satellite
gas jet
Satellite Model
Newton’s Second Law:
Torque:
gas jet
Model of Electrical Systems
Basic components
resistor
inductor
capacitor
ImpedanceBasic components
resistor
inductor
capacitor
Circuit Systems
+ +
Electromechanical System: DC Motor
Basic motor properties:
Torque proportional to current:
Motor voltage proportional to shaft angular velocity:
Input: voltage source e(t)
Output: shaft angular position (t)
Armature resistance
Friction B
Torque T
A Simple Nonlinear Control System
pendulum Input: external force FOutput: 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)