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)