Outline
Generalize the root locus to examine the
effect of varying other parameters.
Rewrite characteristic equation.
Example.
Generalized Root Locus
M. Sami Fadali
Professor EE
University of Nevada
1
Variable Parameter
2
Example: Variable Pole Locatio
EE DEPT.
UNIV. OF NEVADA, RENO
EE 371 Control Systems
Final Exam
Spring 2000
points
(20)
1. For the shown block diagram
R(s) +
K (s + 0.1)
(s + 1)
a)
b)
5
(s + 1)(s + 5)
C(s)
Obtain the closed-loop transfer function.
Find the stable range of K for the clo
EE DEPT.
UNIV. OF NEVADA, RENO
EE 371 Control Systems
Final Exam
Spring 2000
points
(20)
1. For the shown block diagram
R(s) +
K (s + 0.1)
(s + 1)
5
(s + 1)(s + 5)
C(s)
a) Obtain the closed-loop transfer function.
N ( s)
N ( s ) + D( s )
5K (s + 0.1)
5 K
EE DEPT.
UNIV. OF NEVADA, RENO
EE 371 Control Systems
Final Exam
Spring 1999
points
(20)
1.
For the transfer function with unity feedback
G (s) =
K (s + 0.5)
Ks + 0.5 K
=3
(s 1)(s + 8)(s + 10) s + 17 s 2 + 62 s 80
a) Determine the stable range of K.
b) Sk
Time Response: MATLAB
num=[2,4];den=[1,2,4];
g=tf(num,den)
Transfer function:
2s+4
-s^2 + 2 s + 4
step(g)
>g1=zpk([],[-1+j*sqrt(3),-1-j*sqrt(3)],4)
Zero/pole/gain:
4
-(s^2 + 2s + 4)
>hold on, step(g1)
To get the plot characteristics, right click on the wh
EE DEPT.
UNIV. OF NEVADA, RENO
EE 371 Control Systems
Exam II, Spring 1999
Points
(20) 1. Using block diagram manipulation
D(s)
R(s)
+
+
H2
+
C(s)
G1
G2
+
H1
D(s)
R(s)
+
+
G1
+
G2
1 G2 H 2
C(s)
H1
In each case, the transfer function is obtained using the
EE DEPT.
UNIV. OF NEVADA, RENO
EE 371 Control Systems
Exam II, Spring 1999
Solution
Points
(15) 1. Using block diagram manipulation, obtain the transfer function with input D and output C.
G3
R(s)
D(s)
+
+
G1
G2
C(s)
+
+
H
Using the cascade and parallel r
EE DEPT.
UNIV. OF NEVADA, RENO
EE 371 Control Systems
Exam I, Spring 2000
points
(25)
1. a) Write the equations of motion for the system in the time domain.
&
&
m1 &1 + b1 x1 + b( x1 x 2 ) + k ( x1 x 2 ) = f
x
&
&
&
m 2 &2 + b2 x 2 + b( x 2 x1 ) + k ( x 2
Outline
Sketching Bode plots: asymptotes.
Sketching polar plots.
Examples.
Frequency Response
Examples
M. Sami Fadali
Professor of electrical Engineering
University of Nevada
1
Sketching Asymptotic Bode Plots
Type l, n poles, m zeros
Magnitude plot
1.
EE DEPT.
UNIV. OF NEVADA, RENO
EE 371 Control Systems
Final Exam
Spring 1999
points
(20)
2.
For the transfer function with unity feedback
G (s) = K
s + 0.5
(s 1)(s + 8)(s + 10 )
The closed-loop characteristic polynomial is
s 3 + 17 s 2 + (62 + K )s + 0.5
EE DEPT.
UNIV. OF NEVADA, RENO
EE 371 Control Systems
Exam I, Spring 1999
points
(20)
1.
Write the equations of motion of the system, both in the time domain and in the sdomain, then obtain the transfer function with x2 as output and f as input.
x1
x2
b
f
Transfer Function
LTI System
an
dc
d n 1c
d nc
a n 1 n 1 a 1 a 0 c
n
dt
dt
dt
Modeling
bm
G s
M. Sami Fadali
Professor of Electrical Engineering
University of Nevada
dr
d m1r
d mr
bm1 m1 b1 b0 r
m
dt
dt
dt
C s
b s m bm1 s m1 b1 s b0
mn
R s zero IC a
Outline
Lag Design
M. Sami Fadali
Professor of Electrical Engineering
University of Nevada
Why use lag compensation?
Lag transfer function.
Design procedure.
Example.
1
Why use lag compensation?
2
Lag Transfer Function
Attempt to meet specifications using
Outline
Frequency Response
M. Sami Fadali
Prof. of Electrical Engineering
University of Nevada
Steady-state response to sinusoidal input.
Frequency response plots.
Asymptotic behavior of frequency response.
Asymptotic plots.
1
Linear System
Verification:
Block Diagram
.
Root Locus Examples
C (s )
R (s )
K
G( s )
M. Sami Fadali
Professor EE
University of Nevada
1
2
MATLAB: Breakaway & Break-in
Two Poles and a Zero
G ( s)
Root locus is a circle
centered at the zero.
Radius: geometric
mean of distance
bet
Outline
Rules for sketching the root locus.
Examples showing how the rules
are applied.
Root Locus Rules
M. Sami Fadali
Professor of electrical Engineering
University of Nevada
R(s)
K
G(s)
C(s)
H(s)
1
Root Locus Branches
Number of Root Locus Branches
1
What is the root locus?
Plot of the loci of the closed-loop poles of a
system as its gain is varied from 0 to .
Plots are rarely obtained for negative gains.
Plots can be obtained for system parameters
other than the gain (e.g. time constant).
Useful
Outline
Proportional control.
Second-order systems.
Third-order systems.
Analytical Design
M. Sami Fadali
Professor EBME
1
Example 1 : 2nd Order System
R(s)
K
G(s)
2
Equating Coefficients
C(s)
Equate coefficients
Design for given
and
Solution
Closed-lo
Feedback System
.
+
K
Example: A position control system with load
angular position as output and motor armature
voltage as input has the transfer function
G(s)
G ( s) =
K
s( s + a )
Design a proportional controller for the system to
obtain
Proportional c
Outline
Response due to sinusoidal input.
Frequency response of DT systems.
Properties of the frequency response of
DT Systems .
MATLAB commands.
Sampling theorem and the selection of the
sampling period.
Frequency Response
of Discrete-Time Systems
M
Cascade Compensation
R(s) +
Frequency Domain Design
C(s)
Gc(s)
G(s)
M. Sami Fadali
Professor of Electrical Engineering
University of Nevada
Gc(s) = compensator transfer function
Choose compensator to meet design specs.
Phase Margin, Gain Margin, BW, e(
Outline
Frequency Response Design
Lag design.
Lead design.
M. Sami Fadali
Professor EBME
University of Nevada, Reno
1
Lag Design
2
Lag Design (Cont.)
4. Find
(Increasing
reduces the bandwidth of the system).
5. Choose the upper break frequency 1/T in
t
Outline
Review of the
Laplace Transform
M. Sami Fadali
Professor of Electrical Engineering
University of Nevada
Definition of the Laplace transform.
Properties of the Laplace transform.
Final value theorem.
Laplace transform inversion.
Partial fraction ex
Outline
Block Diagrams
M. Sami Fadali
EBME Dept.
University of Nevada
What are block diagrams?
Main rules: cascade, parallel, feedback.
Interchanging: pickoff, summation.
Combining/expanding summing junctions.
Examples.
1
2
Block Diagram Notation
Block Di
Why use the Nyquist Criterion?
Answers the questions:
Q1. Are there any closed-loop poles in the RHP?
Nyquist Stability Criterion
Q2. If the answer to Q1 is yes, then how many?
M. Sami Fadali
Professor of Electrical Engineering
University of Nevada
+
Outline
What are state variables and why use
them?
State-space models for physical systems.
Electrical circuits.
Mechanical systems.
State-space models from transfer
functions.
Modeling in the Time Domain
M. Sami Fadali
Professor of Electrical Engine