APPLICATIONS OF LINEAR PARAMETER-VARYING CONTROL FOR AEROSPACE SYSTEMS By KRISTIN LEE FITZPATRICK A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003
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APPLICATIONS OF LINEAR PARAMETER-VARYING CONTROL FORAEROSPACE SYSTEMS
By
KRISTIN LEE FITZPATRICK
A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE
0 0 0 0 1 0 f 1 e 7453E f 2f 4 e 6971E f 5 2 e 0641E f 4 6 e 2428E0 0 f 1 e 0921E f 2 1 e 0896E f 1 0f 5 e 3709E f 6 1 e 0095E f 3 0 0 0 0 0f 3 e 5754E f 1 6 e 0213E f 1 1 e 8399E4 0 f 3 e 2185E1 3 e 2112E2 0
gihhhhhhhhhhhhhhj(4.6)
Cθ
0 0 0 0 0 C16 lk 1
05 0
0 0 0 0 0 0 C17 mk 0
05
0 0 0 0 0 0 C227 lk 0
05
0 0 0 0 0 C236 lk 0
05 0
0 0 0 0 0 0 0
0 0 0 0 0 C256 lk 0
05 0
(4.7)
D
0 0 0
0 0 0
0 0 0
0 + 7229 0
0 0 0
0 + 3158E4 5
995E5
(4.8)
As seen in the linear parameter-varying matrices above, both the state matrix 4 A 6and the observation matrix 4C 6 change with temperature. It is common for the state
matrix to change as operating parameters change, but it is not common, in traditional
aircraft, for the observation matrix to change. This change in the observation matrix
accounts for the mode shape changes of the hypersonic vehicle.
The modes of the hypersonic model are shown for different temperatures in
Table 4–2 . The table shows the frequency of each of the modes and the damping
30
corresponding to the frequency. The four modes of the open-loop dynamics are (i)
a height mode, (ii) an unstable phugoid-like mode, (iii) an unstable pitch mode and
(iv) the structural mode. As can be seen in the table, the structural mode for the
model at the cold temperature has a higher frequency than the structural mode at the
hot temperature. Minimizing the affect that the temperature has on this mode is the
objective of the inner-loop LPV controller.
Table 4–2: Modes of the Hypersonic Model
Cold HotMode ω
rad < sec ζ ω
rad < sec ζ
i 0.0024 1.00 0.0024 1.00ii 0.1666 1.00 0.1790 1.00
The closed-loop norms are all greater than unity. Intuitively, these magnitudes
imply the controller is not able to achieve the desired performance and robustness
objectives. Realistically, it must be kept in mind that there are twenty inputs and
twenty outputs creating a large number of transfer functions. This fact suggests that
the magnitude of the norms is not unreasonable. The resulting closed-loop properties
are studied in more detail shortly. It is shown that the large norms are caused by
excessive control actuation. Essentially, the controller is not able to achieve the
54
desired disturbance attenuation without exceeding the actuation limits. Fortunately, this
violation is at low frequencies and is not expected to have a dramatic impact on the
closed-loop simulations.
Also, the values in Table 5–1 are interesting in the sense that the induced norms
increase as the level of phase differential increases. Such behavior indicates that the
excitation phase differential does indeed have a large impact on the fluid dynamics.
The increasingly poor performance of the controllers demonstrates that the flow modes
for a phase differential of 210o have properties that are more difficult to control than
those for a phase differential of 150o, for example.
The last entry in Table 5–1 is the norm associated with the LPV controller.
Allowing the phase differential to be time-varying increases the norm as expected.
What is important to note is that this norm did not raise much above the norm
associated with the H∞ controller for the 210o phase differential model. This condition
indicates that the LPV controller is able account for the time-varying nature of the
phase differential without excessive loss of performance.
5.9 Simulation
5.9.1 Open-Loop Simulation
A series of open-loop simulations are performed to demonstrate the fluid qual-
itative response resulting from the disturbance for both full-order and reduced-order
models. These simulations are similar in the sense that the same magnitude of dis-
turbance is used for the boundary conditions on the top of the domain. Conversely,
the simulations involving the reduced-order models differ in that the flow on the bot-
tom boundary has different values of phase lag with respect to the flow on the top
boundary.
A series of plots will be shown to visualize the flow conditions. In each, the value
of horizontal velocity will be shown as a function of time. The plots are 3-dimensional
because the velocity measured at each of the 19 sensors is shown as a function of time.
55
Again, it is important to note that all measurements are non-dimensional. This
characteristic applies to both the time and velocity component so no units are noted for
the simulations.
The open-loop flow for the full-order model is used as a comparison for the
reduced-order model simulations (Figure 5–3). This plot clearly shows the sinusoidal
nature of the flow that results from the top exogenous disturbance changing with the
sine function, ho sin2πt . The flow near the center of the cavity, near point 11,
shows the largest velocity with a magnitude near -0.2 at t 03 to +0.2 at t 0
7.
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint Index
Vel
ocity
Figure 5–3: Open-Loop Flow Velocities for Full-Order Model
The flow for the reduced-order model with a phase differential of 165o is shown
in Figure 5–4. This plot also demonstrates a sinusoidal nature, but has a smaller
open-loop magnitude compared to the full-order flow with the highest velocity being
0.07.
The flow for the reduced-order model with an phase differential of 210o is shown
in Figure 5–5. The flow again demonstrates a sinusoidal nature and the velocities are
slightly larger than those of the full-order model.
A sinusoidal trajectory of phase differentials shown in Figure 5–6 is used in a
simulation which shows the open-loop characteristics of the reduced-order flow as
phase differential changes.
56
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint IndexV
eloc
ity
Figure 5–4: Open-Loop Flow Velocities for Reduced-Order Model with 165o PhaseDifferential
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint Index
Vel
ocity
Figure 5–5: Open-Loop Flow Velocities for Reduced-Order Model with 210o PhaseDifferential
0 0.2 0.4 0.6 0.8 1150
160
170
180
190
200
210
Time
Phas
e D
iffe
rent
ial
Figure 5–6: Trajectory of Phase Differential
57
The flow velocities for the reduced-order model throughout the time-varying phase
trajectory are shown in Figure 5–7. The sinusoidal nature that is apparent in all of the
other open-loop flows is slightly different for this open-loop flow. This difference is
due to the changing of the parameter through the trajectory. The full-order flow does
not have a dependence on phase differential, therefore, the velocities for the full-order
model’s flow over the phase differential trajectory are the same as those plotted in
Figure 5–3.
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint Index
Vel
ocity
Figure 5–7: Open-Loop Flow Velocities for Reduced-Order Model over a Trajectory ofPhase Differentials
An interesting feature to note is that the flow for each reduced-order model
with phase differential has a similar shape but significantly different magnitude. The
maximum velocity measured at the center of the cavity is smaller in magnitude for the
models with phase differentials located at the beginning of the range than the models
with phase differentials near the end of the range. This feature indicates the flow is
indeed strongly dependent on phase differential and should be considered for control
design.
5.9.2 Reduced-Order Closed-Loop Simulation
The closed-loop dynamics are also simulated to demonstrate the performance of
the controller for the reduced-order models, in this section, and the full-order model,
in the next section. The diagram of the closed-loop system for both the reduced-order
58
models and the full-order model can be seen in Figure 5–8. These simulations use the
same open-loop dynamics but include the linear parameter-varying controller that was
synthesized over the range of phase differentials, which contains 24 states. In each
simulation, the flow on the upper boundary is the same, but now the flow on the lower
boundary results only from the commands issued by the controller. In this section,
the controller was tested with reduced-order models for two specific cases of phase
differential and over a time-varying trajectory of phase differentials.
K '&h &
β Pδ & Ψ
R& Φα && Vm S
Figure 5–8: Closed-loop System
The measured velocities for the reduced-order model with a phase differential of
165o in response to the LPV controller with a phase differential of 165o is shown in
Figure 5–9. The comparison of these velocities with the open-loop measurements in
Figure 5–4 demonstrate a reduction of velocity along the center of the cavity, where the
velocity is greatest, of roughly 70%.
The measured velocities for the reduced-order model with a phase differential
of 210o in response to the LPV controller with a phase differential of 210o is shown
in Figure 5–10. The reduction in velocities is apparent by comparing the closed-loop
velocities in Figure 5–10 with the open-loop velocities in Figure 5–5, which shows a
reduction along the center of the cavity of roughly 90%.
The closed-loop simulation of the reduced-order models over the phase differential
trajectory, whcih also effects the controller, is shown in Figure 5–11. The velocity
magnitude shows a clear reduction in magnitude compared to the open-loop simulation
59
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint IndexV
eloc
ity
Figure 5–9: Closed-Loop Flow Velocities for Reduced-Order Model with 165o PhaseDifferential
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint Index
Vel
ocity
Figure 5–10: Closed-Loop Flow Velocities for Reduced-Order Model with 210o PhaseDifferential
of flow over the phase differential trajectory, which was shown in Figure 5–7. The
reduction along the center is roughly 80%.
The disturbance rejection is significant for the LPV controller with the reduced-
order models. These reductions confirm that the LPV controller will work not only
for reduced-order models at specific phase differentials but also over a time-varying
trajectory of phase differentials. The simulations did show some differences between
each of the reduced-order models. In particular, the amount of attenuation was slightly
less for the reduced-order model with a phase differential of 165o but much higher for
the reduced-order model with a phase differential of 210o. This decrease in attenuation
60
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint IndexV
eloc
ity
Figure 5–11: Closed-Loop Flow Velocities for Reduced-Order Model over a Trajectoryof Phase Differentials
seems almost contradictory considering that the open-loop simulations showed a
decrease in flow velocities for the same models.
5.9.3 Full-Order Closed-Loop Simulation
The simulations that were performed for the reduced-order models were repeated
using the full-order model. The reduced-order models are subspaces of this full-order
model so the performance of the controllers on the full-order model is actually of
predominant interest.
The measured velocities in response to an H∞ controller, created specifically for
the full-order model, are shown in Figure 5–12. Clearly, the magnitude of the velocity
is dramatically decreased below the open-loop level. The velocities in Figure 5–12
are several orders of magnitude less than the corresponding open-loop velocities in
Figure 5–3. This response will be used as a comparison for the responses from the
full-order model controlled by the LPV controller.
The velocities for the full-order model in response to the LPV controller with a
phase differential of 165o is shown in Figure 5–13. Though the velocities were not
reduced to the extent of the full-order simulation in Figure 5–12, they were reduced
by an amount comparable to the response shown by the reduced-order model at a 165o
61
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint IndexV
eloc
ity
Figure 5–12: Closed-Loop Flow Velocities for Full-Order Model
phase differential. The reduction in the velocity magnitude is evident along the center
of the cavity and is roughly 80%.
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint Index
Vel
ocity
Figure 5–13: Closed-Loop Flow Velocities for Full-Order Model with Controller Asso-ciated with 165o Phase Differential
The velocity magnitudes for the full-order model in response to the LPV controller
for a phase differential of 210o is shown in Figure 5–14. The reduction in velocity
compared to the open-loop flow of the full-order flow in Figure 5–3 is very clear.
The velocities along the centerline of the cavity were reduced by 60%. Though the
velocities were not as reduced as much as those in the simulation in Figure 5–12, the
velocities were reduced by an amount comparable to the reduced-order model at a 210o
phase differential.
62
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint IndexV
eloc
ity
Figure 5–14: Closed-Loop Flow Velocities for Full-Order Model with Controller Asso-ciated with 210o Phase Differential
The closed-loop simulation of the full-order model controlled by the LPV
controller over the phase differential trajectory is shown in Figure 5–15. The velocities
show a clear reduction in magnitude compared to the open-loop full-order flow in
Figure 5–3. The flow along the center of the cavity is reduced by 66%.
0
0.5
1
510
1520
−0.2
0
0.2
TimePoint Index
Vel
ocity
Figure 5–15: Closed-Loop Flow Velocities for Full-Order Model over a Trajectory ofPhase Differentials
The disturbance rejection is significant for both the reduced-order models and
the full-order model. These reductions confirm that the LPV controller, created for a
phase differential parameter, will work not only for the reduced-order models, which
are dependent on phase differential, but also for the full-order model.
63
5.10 Conclusion
Flow control is an exceedingly difficult challenge because of the nonlinearities
and time variations inherent to flow fields. These inherent difficulties can be avoided
when restricting the flow to creeping Stokes flow within a driven cavity. This project
has introduced a control methodology suitable for such a system. In particular, the
controllers are designed by considering subspaces of the flow field that describe modes
associated with phase differential between exogenous disturbances. The models of
these subspaces are realized as state-space systems and a controller can be designed
using the linear parameter-varying framework. The resulting controller is shown to
significantly decrease the flow velocities within the cavity for both the reduced-order
subspaces and also the full-order flow.
CHAPTER 6CONCLUSION
Practically all mechanical systems that involve motion need to be controlled with a
gain-scheduling technique. Aerospace systems in particular have the possibility to have
very extensive operating domains. Three specific aerospace systems were discussed in
this paper, the longitudinal dynamics of an
F/A-18, the structural dynamics of a hypersonic vehicle and the flow dynamics of a
driven cavity. The parameters that depicted the operating domain of the F/A-18 prob-
lem were altitude and Mach number. The parameter that depicted the operating domain
of the structure of the hypersonic vehicle was temperature and the operating domain
of the driven cavity was depicted by the phase differential within the fluid. This paper
has introduced a gain-scheduled control methodology, which uses H∞ synthesis to
create a linear parameter-varying controller, that is suitable for such systems. The LPV
controller created for the F/A-18 longitudingal dynamics proved to induce a pitch rate
for the aircraft that was similar to a designated target pitch rate. The LPV controller
created for the structural dynamics of a hypersonic aircraft successfully damped out
the vibrations induced by a temperature change. The LPV controller for the fluid
dynamics within a driven cavity significantly decreased the horizontal component of the
flow velocities along the centerline of the cavity for both the reduced-order subspaces
and the full order flow. The results of the control methodology to create proficient
controllers for three very different aerospace applications leads to the conclusion that
this methodology could be useful for other aerospace applications.
64
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BIOGRAPHICAL SKETCH
Kristin Fitzpatrick was born in Blue Hill, Maine on March 26, 1980. Her
family moved to Florida in 1988 after the death of her mother. She received her
high school diploma from the Center for Advanced Technologies, a magnet program
in St. Petersburg, Florida. She then attended the University of Florida and received a
degree in Aerospace Engineering with Honors in December 2002. She has worked with
the aerospace dynamics and control research group under the direction of Dr. Rick
Lind and Dr. Andy Kurdila and is projected to receive her Master of Science degree in
aerospace engineering in December 2003. She will stay at the University of Florida to
pursue a doctorate in aerospace engineering with the focus in dynamics and control.