Automatic Control Laboratory
D-ITET
Prof. Dr. Roy Smith
Spring Term 2015
C ONTROL S YSTEMS II (R EGELSYSTEME II)
S OLUTIONS TO P ROBLEM S ET 4
March 16, 2015
Exercise 1
Cascade Control
Remarks:
• When computing nested transfer functions with products and quotients (Exercises d,f,g,h), use the
Matlab function minreal to remove redundant poles and zeros. This improves the numerical
properties of the step simulation.
a) The system has three dynamic states. The transfer function of θl (s) = Gθl Va (s)Va (s) can be
decomposed as
θl
Gθl Va (s) =
= Gθl ωl (s)Gωl Ia (s)GIa Va (s)
(1)
Va
with the signals
θl (s) = Gθl ωl (s)ωl (s)
(2)
ωl (s) = Gωl Ia (s)Ia (s)
(3)
Ia (s) = GIa Va (s)Va (s)
.
(4)
Using the Laplace transform of the electromechanical equations one finds
1
θl
=
ωl
s
ωl
Km
Gωl Ia (s) =
=
Ia
Js + Bs
Ia
1
GIa Va (s) =
=
Va
La s + Ra + Ka Km
Gθl ωl (s) =
(5)
(6)
.
(7)
b) One can see from the Bode diagram in 1 that Gθl Va (s) is unstable. A lead compensator with the
zero to the left and the pole to the right of the crossover frequency increases the phase around the
cross over frequency. It also reduces the gain at low frequencies to reduce the overshoot. Using
sisotool to adjust the controller parameter in order to satisfy the specifications, one can find
Kθl Va (s) = 0.01 ·
3s + 1
0.03s + 1
.
(8)
The response of the signals after a step in the reference θl,ref is shown in 2. The specifications
of the risetime θl , the bound on Va and the overshoot θl are met.
1 of 10
bode plots using lead compensator
100
Magnitude (dB)
50
0
−50
−100
−150
90
G
K
θV θV
Gθ V
Phase (deg)
0
Kθ V
−90
−180
−270
−2
10
−1
0
10
1
2
10
10
Frequency (rad/sec)
3
10
10
Figure 1: Bode diagram for lead controller design
reference unit step responses using lead compensator
5
Va
Ia
w
4
θl
l
3
2
1
0
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
time [sec]
Figure 2: Reference step responses with lead controller
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0.7
c) The cascade decomposition with two loops is shown in 3.
Figure 3: Cascade decomposition with two loops
d) For the design of the outer loop of the cascade, one assumes the inner loop of 3 to be perfect, i.e.
ωl = ωl,ref . The transfer functions used to plot the step responses in 5 are
Kθl ωl Gθl ωl
θl
=
θl,ref
1 + Kθl ωl Gθl ωl
Kθl ωl
ωl
Rωl ,1 (s) =
=
θl,ref
1 + Kθl ωl Gθl ωl
Rθl ,1 (s) =
(9)
(10)
The peak value of ωl with the lead controller is 4.6, as shown in 2. Choosing a gain Kθl ωl = 4.6
keeps the step response of ωl just below this value, as shown in 5. The bode diagram is shown in
4, showing a bandwidth of ωc = 4.6 rad/s.
bode plots of outer loop design
100
Magnitude (dB)
50
0
−50
−100
0
Gθ wKθ w
Phase (deg)
Gθ w
Kθ w
−45
−90
−2
10
−1
10
0
1
10
10
Frequency (rad/sec)
2
10
Figure 4: Bode diagram for outer loop controller design
3 of 10
3
10
reference unit step responses of outer loop design
5
w
l
θl
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
time [sec]
Figure 5: Reference step responses with outer control loop
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0.7
e) The cascade decomposition with three loops is shown in 6.
Figure 6: Cascade decomposition with three loops
f) For the design of the middle loop of the cascade, one assumes the inner loop of 6 to be perfect,
i.e. Ia = Ia,ref . The transfer functions used to plot the step responses in 8 are
Kθl ωl Tωl ,2 Gθl ωl
θl
=
θl,ref
1 + Kθl ωl Tωl ,2 Gθl ωl
Kθl ωl Tωl ,2
ωl
Rωl ,2 (s) =
=
θl,ref
1 + Kθl ωl Tωl ,2 Gθl ωl
Kθl ωl Kωl Ia
Ia
RIa ,2 (s) =
=
θl,ref
1 + Kωl Ia Gωl Ia + Kθl ωl Kωl Ia Gωl Ia Gθl ωl
Rθl ,2 (s) =
(11)
(12)
(13)
with
Tωl ,2 =
ωl
ωl,ref
=
Kωl Ia Gωl Ia
1 + Kωl Ia Gωl Ia
.
(14)
The peak value of Ia with the lead controller is 0.36, as shown in 2. Choosing a gain Kωl Ia =
0.08 keeps the step response of Ia just below this value, as shown in 8. The bode diagram is
shown in 7, showing a bandwidth of ωc = 16 rad/s.
5 of 10
bode plots of the middle loop design
60
Magnitude (dB)
40
20
0
−20
−40
0
G wIK wI
Phase (deg)
G wI
KwI
−45
−90
−2
10
−1
0
10
1
2
10
10
Frequency (rad/sec)
3
10
10
Figure 7: Bode diagram for middle loop controller design
reference unit step responses of middle loop design
3.5
Ia
wl
θl
3
2.5
2
1.5
1
0.5
0
−0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
time [sec]
Figure 8: Reference step responses with outer and middle control loop
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0.7
cascade
Kθl ωl (outer loop)
Kωl Ia (middle loop)
KIa Va (inner loop)
inital design
4.6
0.08
2.7
adjusted design
7
0.2
0.7
Table 1: Controller gains for cascade design
g) For the design of the middle loop of the cascade, one uses the full block diagram of 6. The
transfer functions used to plot the step responses in 10 are
Kθl ωl Tωl ,3 Gθl ωl
θl
(15)
=
θl,ref
1 + Kθl ωl Tωl ,3 Gθl ωl
Kθl ωl Tωl ,3
ωl
Rωl ,3 (s) =
(16)
=
θl,ref
1 + Kθl ωl Tωl ,3 Gθl ωl
Kθl ωl Kωl Ia TIa ,3
Ia
(17)
RIa ,3 (s) =
=
θl,ref
1 + Kωl Ia TIa ,3 Gωl Ia + Kθl ωl Kωl Ia TIa ,3 Gωl Ia Gθl ωl
Va
RVa ,3 (s) =
(18)
θl,ref
Kθl ωl Kωl Ia KIa ,Va
=
1 + KIa ,Va GIa ,Va + Kωl Ia KIa ,Va GIa ,Va Gωl Ia + Kθl ωl Kωl Ia KIa ,Va GIa ,Va Gωl Ia Gθl ωl
Rθl ,3 (s) =
with
Ia
KIa Va GIa Va
=
Ia,ref
1 + KIa Va GIa Va
Kωl Ia TIa ,3 Gωl Ia
ωl
=
=
ωl,ref
1 + Kωl Ia TIa ,3 Gωl Ia
TIa ,3 =
(19)
Tωl ,3
(20)
The peak value of Va with the lead controller is 1, as shown in 2. Choosing a gain KIa Va = 2.7
keeps the step response of Va just below this value, as shown in 10. The bode diagram is shown
in 9, showing a bandwidth of ωc = 83.5 rad/s.
The step response in 10 does not satisfy the rise time requirements in problem b). One can adjust
the controller gains in the following way:
– increase the gains Kθl ωl and Kωl Ia of the outer and middle loop to meet the rise time
specification
– decrease the gain KIa Va of the inner loop to meet the Voltage bound specification
– repeat this two steps until the result is satisfactory
Since the integrator term of Gθl ωl will ensure a non-zero offset of θl in outer loop even if the
innerloops do not achieve perfect tracking of the references ωl and Ia , the gain KIa Va can be
even reduced to the point where the inner loop has zero bandwidth.
The controller gains and the bandwidth of the three cascade loops are compared in Table 1 and
Table 2.
The step response comparison of the cascade to the lead controller is shown in 11. The cascade
controller (using simple P-controller) has a comparable response to the lead controller (using a
dynamic compensator). Both satisfy the rise time, overshoot and input bound specifications.
7 of 10
bode plots of the inner loop design
10
5
Magnitude (dB)
0
−5
−10
−15
−20
−25
−30
0
G IVK IV
Phase (deg)
G IV
KIV
−45
−90
−2
10
−1
10
0
1
10
10
Frequency (rad/sec)
2
10
3
10
Figure 9: Bode diagram for inner loop controller design
lead
5.8
cascade
outer loop
middle loop
inner loop
inital design
4.6
16
83.5
adjusted design
7
40
0
Table 2: Bandwidth ωc [rad/s]: Comparison of different controller
h) The step response comparison of the cascade to the lead controller with the perturbed system
is shown in 12. Both controllers violate the overshoot specification, but are quite robust to the
perturbation. The cascade controller is more aggressive, with a larger overshoot, but a faster rise
time. The lead controller also has a offset of θl that is disappearing only slowly. The cascade
controller, which is based on multiple measurement values, converges much faster.
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reference unit step responses of inner loop design
3.5
V
a
I
a
3
wl
θl
2.5
2
1.5
1
0.5
0
−0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time [sec]
Figure 10: Reference step responses with outer, middle and inner control loop
reference unit step responses, nominal system
1.2
1
0.8
V with lead compensator
0.6
a
θl with lead compensator
Va with cascade control
θl with cascade control
0.4
0.2
0
−0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time [sec]
Figure 11: Reference step responses with nominal plant: comparison of cascade and lead controller
9 of 10
reference unit step responses, perturbed system
1.4
1.2
1
0.8
Va with lead compensator
0.6
θl with lead compensator
Va with cascade control
θl with cascade control
0.4
0.2
0
−0.2
−0.4
0
0.2
0.4
0.6
0.8
1
time [sec]
1.2
1.4
1.6
1.8
2
Figure 12: Reference step responses with perturbed plant: comparison of cascade and lead controller
10 of 10