EE3064: Feedback Control Lecture Notes (Week 3) Laplace Transform
F (s) =
0
f (t)est dt
(1)
Impulse Signals The unit impulse signal (t) is zero for all t = 0, and satisfies  (t)dt = 1. The unit impulse signal can be considered as an infinitely high and
EE3064: Feedback Control Lecture Notes (Week 2.2) Block Diagram
An Example
Use differential equations to describe the motion of the system and take Laplace transforms of the signals, we have La sIa (s) + Ra Ia (s) = Va (s)  Ke sm (s) Jm s2 m (s) + bsm (s
EE3064: Feedback Control Lecture Notes (Week 2)
Laplace Transform The Laplace transform of f (t), denoted by Lcfw_f (t) = F (s), is a function of the complex variable s = + j, where
F (s) :=

f (t)est dt.
(1)
If we just consider a signal f (t) defined f
EE3064
Feedback Control
Tengfei Liu ECE Department, LC004 Polytechnic Institute of NYU
NYUPoly ECE T. LIU 1
Course Goals
Equip students with needed theoretical and handson knowledge about an important engineering field:
Modeling, analysis and feedback
EE3064
Feedback Control
Tengfei Liu ECE Department, LC259 Polytechnic Institute of NYU
NYUPoly ECE T. LIU 1
ProportionalIntegralDerivative (PID) Controller
Controller Plant
+

The Threeterm Controller:
= + +
Integral
Proportional
Derivative
whe
EE3064
Feedback Control
Tengfei Liu ECE Department, LC259 Polytechnic Institute of NYU
NYUPoly ECE T. LIU 1
Tracking Error
()
Tracking error:
 () 
Assume = 0 and = 0. Then,
=
and () =
1 +
1
1 +
NYUPoly ECE T. LIU
2
Steady State Tracking Error
EE3064
Feedback Control
Tengfei Liu ECE Department, LC259 Polytechnic Institute of NYU
NYUPoly ECE T. LIU 1
What we learned
System modeling with transfer functions
From differential equations to transfer functions (Laplace transform) From block diagram
EE3064
Feedback Control
Tengfei Liu ECE Department, LC259 Polytechnic Institute of NYU
NYUPoly ECE T. LIU 1
Stability: BoundedInput BoundedOutput (BIBO)
A system is said to have boundedinput boundedoutput stability if every bounded input result
EE3064
Feedback Control
Tengfei Liu ECE Department, LC259 Polytechnic Institute of NYU
NYUPoly ECE T. LIU 1
Firstorder SystemsStep Response (Timedomain Specifications)
= = 1 = = 1  1
Rise time
ln 0.9  ln 0.1 2.2 = =
Settling time
4.6 =
Q: How a
EE3064
Feedback Control
Tengfei Liu ECE Department, LC259 Polytechnic Institute of NYU
NYUPoly ECE T. LIU 1
Timedomain Response to Inputs
Transfer function:
= where and are Laplace transforms of the output and the input , respectively.
Under zero initi