Week IV: Linear Systems
y
Today s
Todays discussion topics:
What is a linear system?
Solutions and stability of linear timeinvariant
( ) y
(LTI) systems
Linearization of nonlinear systems
Connections between a nonlinear system and
its linearized model
Li
EL6243: System Theory and
Feedback Control
Zhong Ping
ZhongPing Jiang
ECE Dept, NYU Poly, 5MTC, LC214
eeweb.poly.edu/faculty/jiang
b l d /f
lt /ji
Fall 2012
Z.P. Jiang, NYU POLY
1
Course Goals
Equip students with the basic knowledge
in modeling, analyzin
Week V: Controllability and State
Feedback
Todays discussion topics:
Controllability: Definition and tests
Implications of controllability in state
feedback design
Application examples
Fall 2011
Z.P. Jiang, NYU POLY
118
What is controllability?
y
Strongly
Feedback Systems: Solutions Manual  v1.3b
207
Exercise 8.8 Using block diagram algebra, show that the transfer functions from d to y and n to y in
Figure 8.7 are given by
P
1
G yd =
G yd =
.
1 + PC
1 + PC
Solution.
(a) G yd : y = P(d + u) = Pd + PCe = Pd
294
Feedback Systems: Solutions Manual  v1.3b
Exercise 11.4 Consider the springmass system given by (2.14), which has the transfer function
1
.
+ cs + k
Design a feedforward compensator that gives a response with critical damping ( = 1).
P(s) =
ms 2
Solu
250
Feedback Systems: Solutions Manual  v1.3b
Exercise 10.1 (Ideal PID controllers) Consider the systems represented by the block diagrams in Figure 10.1.
Assume that the process has the transfer function P(s) = b/(s + a) and show that the transfer funct
Feedback Systems: Solutions Manual  v1.3b
62
Exercise 4.1 (Timeinvariant systems) Show that if we have a solution of the differential equation (4.1) given
by x(t) with initial condition x(t0 ) = x0 , then x( ) = x(t t0 ) is a solution of the differentia
Week IX  Frequency Domain
Analysis
Review of Bode plot and Nyquist
diagram
Stability margins
Bodes relation
Fall 2011
Z.P. Jiang, NYU POLY
220
Response to exponential signals
p
p
g
u (t ) e
st
LTI
y (t ) ?
(Assume zero initialstate)
Notes:
When s=0, y(
Week X: FrequencyDomain Design
Topics Under Discussion:
What i
Wh t is PID?
Lead and Lag Compensators
Fall 2011
Z.P. Jiang, NYU POLY
283
What is PID?
r
e
C(s)
u
y
P(s)

ki
Frequencydomain: C(s) k p kd s
s
Timedomain: u(t ) k p e(t ) ki e d kd e t
e d
Week XII: Youla Parameterization
and Robustness Analysis
Todays k
T d key points:
i t
D.C. Youla
Parameterization
Robust Stability
Robust Performance
Fall 2011
Z.P. Jiang, NYU POLY
373
What is Youla Parameterization?
The class of all stabilizing controlle
Week XI:
Sensitivity and Performance
Key points in today s lecture:
todays
Sensitivity and S
S
iti it
d Sensitivity F
iti it Functions
ti
Performance Limitations
Feedforward and Feedback Designs
Fall 2011
Z.P. Jiang, NYU POLY
347
Review
1.
2.
2
1.
2.
Fall
Week VIII  Tools for Frequency
Domain Analysis
Transfer function: a frequency domain
frequencydomain
approach for LTI systems
Poles, zeros, polezero cancellations
Bode l t
B d plot
Application examples
pp
p
For simplicity, we only consider SISO linear
Week VI: Observability, Observer
and Output Feedback
Todays di
T d discussion topics:
i t i
Observability: Definition and tests
Implications of observability in observers
and output feedback design
Application examples
Fall 2011
Z.P. Jiang, NYU POLY
160
W
Week III
Stability and Performance
Today s
Todays discussion topics:
Existence and uniqueness of solutions of
dynamic systems
Stability: Notions and basic properties
y
p p
Tests for stability
Performance metrics for (feedback) systems,
e.g. transient and
Week II: Modeling
g
Today s
Todays Lecture:
Introduce modeling techniques and the
concepts of state inputs outputs through
state, inputs, outputs,
some examples in engineering and biology.
Overhead Crane
C i Control System of a Car
Cruise C t l S t
f C