The Nyquist Criterion for Stability
Determine closedloop feedback system
stability using Cauchys Theorem: 1+GH(s) = 0
Consider a contour in the splane that encircles the
entire RHP (and does not pass through any poles
and zeros of GH(s) on the jaxis)
Stability in Frequency Domain
Mapping contours in one complex plane to
another complex plane by a relationship F(s)
Cauchys Theorem
If a contour in the splane encircles Z zeros and P
poles of F(s) (and does not pass through any
poles or zeros of F(s)
The Steadystate Response of LTI System
Due to a Sinusoidal Input
where H ( j ) = H ( s ) s = j = K
0
The Frequency Response of a LTI System
Consider the transfer function
The magnitude is an even symmetric function
The phase is an odd symmetric functi
The Root Locus Procedure (I)
Consider openloop transfer function G(s)H(s)
Starts with finite poles and ends at zeros (including
zeros at )
On real axis: lies to the left of an odd number of
finite poles and finite zeros
Symmetrical w.r.t realaxis fi
The Resonant Frequency and Peak of
Magnitude for a Secondorder System
Consider
2
n
G( s ) = 2
2
s + 2n s + n
The resonant frequency
and the peak value of
magnitude
r = n 1 2 2 , < 0.707
G( jr ) =
1
2 1 2
The Polar Plot
Let G( j ) = G( s ) s = j = Re[G
The Minimum Phase Transfer Function
Minimum phase transfer function
All zeros in LHF (for a continuous time system)
s+z
G1( s ) =
, minimum phase
s+ p
sz
G2 ( s ) =
, non minimum phase
s+ p
Performance of Secondorder Systems (VII)
The effect of damping ratio and natural frequency
For a fixed damping ratio, larger natural frequency
results faster response
For a fixed natural frequency, larger damping ratio
results slower response
Example
Construct the Routh Array
an s n + an 1s n 1 + an 2 s n 2 + + a1s + a0 = 0
s n an an 2
Step 1. Arrange coefficients n 1
s an 1 an 3
an 4
an 5
Step 2. Calculate other entries in the array
s n 2 b1 b2
s n 3 c1 c2
b3
c3
s 0 h1
where b1 =
1 an an 2
The Stability of Linear Feedback Systems
The concept of stability
BIBO (boundedinputboundedoutput) stability
The RouthHurwitz stability criterion
A criterion for checking stability of linear systems
The Concept of Stability
The general concept
R
The Types of Feedback Systems and
Steadystate Error (II)
E ( s ) = R( s ) H ( s )Y ( s ) =
ess = lim [s E ( s )]
1
1 + GC ( s )G( s )H ( s )
s 0
M
GC ( s )G( s )H ( s ) =
K ( s + zi )
i =1
N l
s l ( s + pi )
i =1
where l is the type number
The Types of F
Chapter 2: Mathematical Models of
Systems
Modeling physical systems using differential
equations
The Laplace transform
Transfer function
Solving differential equations
Block diagram and signalflow models
Masons formula
Modeling
The concept of mode
Review for Midterm Exam I
EE302
Winter 2013
General: Midterm exam is on 2/8 (Friday). It is a 50minute, closed book, closed notes
exam; calculator is allowed, but should be only used for numerical calculations. You
may use one cheat sheet (8x11, singles
Example
Consider the following feedback tracking system, find
the value of K that results in zero overshoot to a step
input
Performance of Secondorder Systems (III)
2
n
T( s ) = 2
2
s + 2n s + n
Y(s) : Unit Step Response
Performance of Secondorder Syst
Chapter 5: The Performance of Feedback
Control Systems
Typical Performance indices of secondorder
systems
Consider tracking system with unit step input
Rise time, peak time, percent overshoot
Settling time, steadystate error
The effect of polezero
Types of Compensation
Cascade (series) vs. feedback vs. output (load)
vs. input
Lead Network
Phase lead network
Differentiatortype
network
Provide a phaselead
angle
Increase gain at high
freq. (attenuation at
low freq.)
Increase BW
GC ( s ) =
K( s
Sensitivity Analysis (I)
Let Td(s) = 0, N(s) = 0
For simplicity, let H(s) = 1 (unity feedback), and
only consider a small change on G(s)
+
_
Gc(s)
G(s)
Sensitivity Analysis (II)
System sensitivity (for small incremental changes)
The ratio of the chang
Secondorder System Step Response (I)
Secondorder System Step Response
(II)
Example
Find the transfer function for the following opamp circuit:
Block Diagram Representation
A block diagram may include
Input/output variables
Operational blocks (transf
Chapter 4: Feedback Control System
Characteristics
Advantages of a welldesigned feedback control
system
Reject disturbance and attenuate noise
Reduce system sensitivity to parameter variations
Adjust transient response and reduce steady state
error
Example
Find the transfer function for the following
system using Masons formula:
Chapter 4: Feedback Control System
Characteristics
The feedback control systems in this chapter
Tracking systems
Unity feedback
A welldesigned feedback control system
Transient Response and Steady State
Error
The controller in feedback control system
(Gc(s) can adjust system transient response
Feedback controller can also reduce system
steadystate error
Openloop: E( s )open loop = [1 GC ( s )G( s )] R( s )
Closed
Introduction
Systems
Control systems
Types of control systems
Why closedloop control?
Control system design flow chart
Course overview
Systems (I)
The concept of a system
Can be considered as a processor that deals
with signals (e.g., inputs and
Root Locus Design Example
Find the value of K1 and K2 so the closedloop system
meets the following specifications:
Steadystate error for a unit ramp input 0.35
Settling time (within 2% of final value) 3 sec.
Damping ratio 0.707
PID Controllers (I)
The Inverse Laplace Transform (I)
Definition:
PFE (partial fraction expansion)
X(s) should be strictly proper
Distinct factors:
Repeated factors:
The Inverse Laplace Transform (II)
Find LTI System Response (Solving ODE)
Using Laplace Transform (I)
F
Review for Final Exam
EE302
Winter 2013
General: Final exam is on 3/18 (Monday), 1:10pm 3:00pm for section 2; 3/20
(Wednesday), 4:10pm 6:00pm for section 3. Take the final exam during the time
slot allocated for the section you enrolled. It is a 110minut
Review for Midterm Exam II
EE302
Winter 2013
General: Midterm exam is on 2/27 (Wednesday). It is a 50minute, closed book, closed
notes exam; calculator is allowed, but should be only used for numerical calculations.
You may use two cheat sheets (8x11, si
EE 302
HW #6
Solution



Note: For this problem (P8.2 (a)(c), you need to know (1) how to sketch Bode plot by
straightline approximation, (2) calculate the values of points on the straightline
approximation, and (3) calculate the exact values of poi
EE 302
HW #7
Solution
Note: For all problems that involve Bode plot, you need to know (1) how to sketch Bode
plot by straightline approximation, (2) calculate the values of points on the straightline approximation, and (3) calculate the exact values of p
Department of Electrical Engineering
California Polytechnic State University
Classical Control Systems
EE 302
Winter 2013
Catalog Course Description:
Introduction to feedback control systems. System modeling. Transfer functions. Graphical system
represent