root locus for g(s) in unity feedbackee128/fa15/hw/ps7_solutions.pdf · root locus real axis...
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Root Locus for G(s) in Unity Feedback
Root Locus for GC(s)⋅G(s) in Unity Feedback
-15 -10 -5 0 5-10
-8
-6
-4
-2
0
2
4
6
8
10Root Locus
Real Axis (seconds-1)
Imag
inar
y Ax
is (s
econ
ds-1
)
-15 -10 -5 0 5-10
-8
-6
-4
-2
0
2
4
6
8
10Root Locus
Real Axis (seconds-1)
Imag
inar
y Ax
is (s
econ
ds-1
)
Step Response for G(s) in Unity Feedback (k = 41.04)
Step Response for GC(s)⋅G(s) in Unity Feedback (k = 41.08)
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6Step Response
Time (seconds)
Ampl
itude
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Step Response
Time (seconds)
Ampl
itude
Combined Plot of Step Response for Uncompensated System (Blue) and Lag
Compensated System (Red) in Unity Feedback (at specified gains)
Combined Plot of Step Response for Uncompensated System (Blue) and Lag
Compensated System (Red) in Unity Feedback (at specified gains)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8SYS_UnCompSYS_LagComp
Step Response
Time (seconds)
Ampl
itude
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8SYS_UnCompSYS_LagComp
Step Response
Time (seconds)
Ampl
itude
Improved Step Response (TS) by moving Compensator to GLag(s) = (s + 0.4) / (s + 0.1)
CL Poles at -8.2, -1.8 ± 2.5j, -0.27 (if K = 41.08 maintained).
Combined Plot of Step Response for Uncompensated System (Blue) and Improved Lag
Compensated System (Red) in Unity Feedback (at specified gains)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Step Response
Time (seconds)
Ampl
itude
0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Step Response
Time (seconds)
Ampl
itude
Improved Step Response (TS) by moving Compensator to GLag(s) = (s + 0.8) / (s + 0.2)
CL Poles at -8.2, -1.7 ± 2.4j, -0.59 (if K = 41.08 maintained).
Combined Plot of Step Response for Uncompensated System (Blue) and Improved Lag
Compensated System (Red) in Unity Feedback (at specified gains)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Step Response
Time (seconds)
Ampl
itude
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Step Response
Time (seconds)
Ampl
itude
Root Locus for G(s) in Unity Feedback
Root Locus for GPID(s)⋅G(s) in Unity Feedback
-12 -10 -8 -6 -4 -2 0 2
-8
-6
-4
-2
0
2
4
6
8
Root Locus
Real Axis (seconds-1)
Imag
inar
y Ax
is (s
econ
ds-1
)
-12 -10 -8 -6 -4 -2 0 2-8
-6
-4
-2
0
2
4
6
8Root Locus
Real Axis (seconds-1)
Imag
inar
y Ax
is (s
econ
ds-1
)
Step Response for G(s) in Unity Feedback (k = 416.1)
Step Response for GPID(s)⋅G(s) in Unity Feedback (k = 45.4)
0 0.5 1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Step Response
Time (seconds)
Ampl
itude
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (seconds)
Ampl
itude
Step Response for GPID(s)⋅G(s) in Unity Feedback (k = 30)
Combined Plot of Step Response for Uncompensated System (Solid, k = 416.1) and PID
Compensated System (Dotted, k = 45.4 / Dashed, k = 30) in Unity Feedback
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2Step Response
Time (seconds)
Ampl
itude
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
1.4SYS_UnCompSYS_PID_k45SYS_PID_k30
Step Response
Time (seconds)
Ampl
itude
3.d. uncompensated
-20 -15 -10 -5 0 5 10-15
-10
-5
0
5
10
15
Root Locus
Real Axis (seconds-1)
Imagin
ary
Axis
(seconds
-1)
3.d. compensated
-12 -10 -8 -6 -4 -2 0 2 4 6
-8
-6
-4
-2
0
2
4
6
Root Locus
Real Axis (seconds-1)
Ima
gin
ary
Axis
(se
co
nd
s-1
)
3.e. Comparing proportional to the lead-lag compesnator by directly following book, note that the settling time isnot to speci�cation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.2
0.4
0.6
0.8
1
1.2
Proportional
Lead-lag, book
Step Response
Time (seconds)
Am
plit
ude
1
3.e. Re-placing the lead pole to put dominant poles at the desired location (ξ = .5, Ts = 2.1) gives the followingresponse:
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
Proportional
Lead-lag, book
Lead-lag, lead pole placed after lag
Step Response
Time (seconds)
Am
plit
ude
2
4.a
Ma
gn
itu
de
(d
B)
-20
0
20
40
10-2 10-1 100 101 102 103 104
Ph
ase
(d
eg
)
-90
-45
0
Bode Diagram
Frequency (rad/s)
4.b
Magnitude (
dB
)
-120
-100
-80
-60
-40
-20
100 101 102
Phase (
deg)
-270
-225
-180
-135
-90
Bode Diagram
Frequency (rad/s)
4.c
Ma
gn
itu
de
(d
B)
-50
-40
-30
-20
-10
0
10-2 10-1 100 101 102 103 104
Ph
ase
(d
eg
)
-90
-45
0
45
90
Bode Diagram
Frequency (rad/s)
3
Mag
nitu
de (d
B)
-3
-2
-1
0
1
2
3
4
10-2 10-1 100 101
Phas
e (d
eg)
0
5
10
15
20
25
Bode Diagram
Frequency (rad/s)
6.d.
Magnitude (
dB
)-120
-100
-80
-60
-40
-20
0
10-1 100 101 102 103
Phase (
deg)
-225
-180
-135
-90
-45
0
Bode Diagram
Frequency (rad/s)
6.d.-0.05 0 0.05 0.1 0.15 0.2
-0.1
-0.05
0
0.05
0.1
Nyquist Diagram
Real Axis
Imagin
ary
Axis
zoom near origin
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02
×10-3
-12
-10
-8
-6
-4
-2
0
2
4
6
8
Nyquist Diagram
Real Axis
Ima
gin
ary
Axis
1