Hypersonic vehicle control law development using H(infinity) and mu-synthesis is discussed. Airbreathing SSTO vehicles has a mutli-faceted mission that includes orbital operations, as well as re-entry and descent culminating in horizontal landing. However, the most challenging part of the operations is the ascent to orbit. The airbreathing propulsion requires lengthy atmospheric flight that may last as long as 30 minutes and take the vehicle half way around the globe. The vehicles's ascent is characterized by tight payload to orbit margins which translate into minimum fuel orbit as the performance criteria. Issues discussed include: SSTO airbreathing vehicle issues; control system performance requirements; robust control law framework; H(infinity) controller frequency analysis; and mu controller frequency analysis

Chowdhry, Rajiv S.

Gregory, Irene M.

Mcminn, John D.

Shaughnessy, John D.

English

NASA Technical Reports Server

N94- 251O4
Hypersonic VehicleControl Law
Development
Using Hooand l_-Synthesis
Irene M. Gregory, John D. McMinn
and John D. Shaughnessy
NASA Langley Research Center
and
Rajiv S, Chowdhry
Lockheed Engineering Sciences, Co.
NASA Langley GN&C Workshop
March 18-!9, 1993
193
https://ntrs.nasa.gov/search.jsp?R=19940020622 2020-06-16T16:26:09+00:00Z
Hypersonic Vehicle Control Law Development
Using Hoo and i_-Synthesis
Irene M. Gregory, John D. McMinn and John D. Shaughnessy
NASA Langley RNearch Center
and
Rally S. Chowdhry
Lockheed Engineering Sciences, Co,
"r.
Airbreathing SSTO vehicle has a multi facetted mission that includes
orbital operations, as well as re-entry and descent culminating in horizontal
landing. However, the most challenging part of the operations is the ascent
to orbit. The airbreathing propulsion requires lengthy atmospheric flight that
may last as long as 30 minutes and take the vehicle half way around the
globe.
The vehicle's ascent is characterized by tight payload to orbit margins
which translate into minimum fuel to orbit as the performance criteria.
194
SSTO AIRBREATHING VEHICLE ISSUES
airframe_'propulsion/aerothermoalestic
interactions
lengthyflightinthe atmosphere
at highaltitude
changingatatif
atability margins uncertainflyingqualities
propulsionaansitivityto
smallangular changesin
largevariationsin staticand flowpath
oXynamlcharacteristics
NarrowPerformanceMarginsfor Mission Success
The lengthy atmospheric flight and the minimum fuel to orbit
performance requirement lead to a number of issues. Among these issues
are:
• Large variations in static and dynamic vehicle characteristics that result
from large and rapid mass change as well as aerodynamic heating.
• These variations lead to changing static stability margins as the
aerodynamic center of pressure moves significantly with respect to c.g..
• Furthermore, since the undersurface of the vehicle serves as the
compressing inlet and as the nozzle for the propulsion system, this vehicle
experiences unprecedented degree of airframe/propulsion/aerothermoelastic
interactions that lead to multiple and large parametric uncertainty.
• The lengthy atmospheric flight subjects the vehicle and the propulsion
system to atmospheric turbulence which can excite vehicle dynamic modes as
well as degrade propulsion performance through large density variations.
• Propulsionsystem itself is sensitive to small angular changes in the flow
path that may be caused by interactions or atmospheric turbulence.
• Finally, flying qualities for hypersonic flight are not yet established.
All these issues lead to narrow margins for missionsuccess making optimal
vehicle performance absolutely essential.
195 ' ,
CONTROL SYSTEM PERFORMANCE
REQUIREMENTS
- Tight Performance Margins For Reaching Orbital Speeds
- Closely Tracking Optimal Trajectory
- Tight Angle Of Attack Envelope - 0.5 Deg
- Atmospheric Disturbance Rejection
- Robustness To Parameter Uncertainty
In order to address these issues and enhance vehicle performance, the
control system must satisfy the following requirements. It must :
• stabilize the vehicle,
• precisely track optimal fuel trajectory,
• attenuate atmospheric disturbances,
• while minimizing control effort since even moderate elevon deflections
result in very large integrated drag penalty.
• All of these performance requirements must be satisfied in the
presence of parametric uncertainty
And, as with all piloted vehicles, the flying qualities requirements must be
met.
196
ROBUST CONTROL LAW FRAMEWORK
Airframe/Propulsion
Nonlinear Model with Multiple
Modelling Uncertainty
\
Linearlzed Uncertainty Model
\
Robust Control System
Design & Analysis
\
Robust Stability
& Performance
Recent advances in H=/p robust control theory provide a framework for
explicitly including parametric uncertainty in control law design. The
hypersonic vehicle ascent can be characterized in this framework as follows.
• We begin with aero/propulsion nonlinear model that includes multiple
sources of uncertainty.
• This model is translated into a linearized uncertainty model illustrated
by the diagram in the middle of the slide. The linear model itself is contained
in plant P. All the uncertainty, with the physical relationship to the model
preserved, is collected in block A. The controller K is then designed and
analyzed for this linearized uncertainty model.
Once this process is successfully completed, the resulting control law
proy)des robust stability and desired performance in the presence of specified
uncertainty.
197
APPLICATION EXAMPLE
• Flight condition
- 2000 psf q trajectory
- Mach 8
• Plant- P (s)
- linear model of the vehicle
. design specifications
• Uncertainty block-
- uncertainty in control effectiveness of elevon and fuel flow rate
• Exogenous inputs - d
- commanded variables - velocity and altitude
- atmospheric turbulence
- sensor noise
• Performance outputs - •
- velocity and altitude error
- angle of attack
. control effector - elevon and fuel flow rate
To explore this robust controls framework further, an example
representing a number of issues we discussed has been selected. The vehicle
is a conical configuration following a 2000 psf dynamic pressure trajectory
and accelerating through Mach 8 at the design point.
We return to the Linearized Uncertainty Model diagram from the previous
slide to illustrate how this example fits into the framework.
The plant P contains the linear model of the vehicle and the design
specifications, which will be discussed in more detail later. All the uncertainty
in the problem is relegated to the effectiveness of elevon and fuel flow rate,
and is contained in the uncertainty block 4. The physical inputs, d, into the
system include the commanded variables, velocity and altitude, atmospheric
turbulence, and sensor noise. The performance outputs, e, are velocity and
altitude error, angle of attack, due to propulsion performance sensitivity to
this quantity, and control effectors, both deflection and rate for elevon and
fuel flow rate.
198
PROBLEM FORMULATION
• Applicability and effectiveness of H--#Lcontroller
- Performance
- Stablllty
- Changlng flight condltlons
- Changlng vehlcle characterlstics
• Performance metric
- Velocity end altltude
• 5 % steady state error
• 10 % pvershoot
• 40 sec time constant
- Angle of attack
• limit to 0.5 dell peak-to-peak deflection
- Elevon
• limit deflections to 2 deg
We are interested in establishing how effective is each technique, that is
H= and p, in explicitly dealing with changing vehicle characteristics and flight
conditions while providing performance and stability.
Performance on the global level refers to achieving minimum fuel to orbit.
On the more immediate design level it encompasses a metric such as
illustrated on this slide. The time domain response specifications, limits on
the deflection and rate of the control effectors, and atmospheric turbulence
attenuation are all serve as performance specifications.
It is important to point out that unlike other optimal robust control
methods, H= based design results in a controller for the worse case input
combination.
199
DESIGN SPECIFICATIONS
• Time domain specs translated into frequency domain
- steady state error, % overshoot, time constant --> transfer function
• Disturbance attenuation -:, low frequency transfer function gain
• Rate and position limits on actuators --> transfer function
• Allowable uncertainty
- % parameter variation
. frequency dependent transfer function
I like to spend a moment discussing how we fit design specifications
discussed few slides back into the H= context. To fully exploit H= capabilities,
design specifications must reflect the desired performance as closely as
possible. Given the specifications on the time domain response, steady state
error, percent overshoot, and time constant, translate directly into a transfer
function that is utilized in H= context. The same can be said about
performance specifications on alleviating atmospheric turbulence and limiting
rate and position of actuators.
The allowable uncertainty in the system is also specified in frequency
domain as a percent of nominal. It is either a constant across all frequency or
varies depending on type of uncertainty.
At this juncture, we would like to examine how all this relates to a
standard block diagram.
200
SYSTEM BLOCK DIAGRAM
_ hg p
Longitudinol
_e
I K (a) _ V,,
-- ho
The three block general structure that you may recall from previous slides
is expanded into this block diagram for our example. The dashed boxes,
primarily on the right side of the diagram, represent the performance
specifications. The dotted boxes, on the left, represent the uncertainty. The
20 percent uncertainty that was just discussed is expressed by the matrix W_
and the diagonal structure of A reflects that each actuator effectiveness is
independent of the other.
Now with problem formulated we design a controller which is analyzed in
the following slides.
201
Hoo CONTROLLER FREQUENCY ANALYSIS
1.6
.... N_m_}al Performance
1.4 - - _ust st_i_,- ,/_"
0.4
J
0.2 ---- _ /
0 _L • •,H,= , , ,=,,,,, ........
0.01 0.1 1 10
Frequency, rad/sG¢
- Nominal Performance
- performance of the nominal plant model
• Robust Stability
- stability in the presence of allowable uncertainty
• Robust Performance
- performance in the presence of allowable uncertainty
The first to be analyzed is the H- controller. Very briefly to provide you
with a point of reference. 1 delineates the boundary between successfully
passing a given test vs. failing it. We are interested in three metrics for this
controller. The first is nominal performance which tells us whether the desired
performance has been achieved under ideal conditions, in other words, we
have no uncertainty in our system. As you can see from this plot, nominal
performance is less than 1, therefore satisfying our desired performance
requirements.
But since no realistic system model is ideal, we are really interested in its
behavior in the presence of uncertainty. For this example it constitutes 20
percent control effectiveness uncertainty. The initial interest is in stability.
This controller violates robust stability criteria around 4 rad/sec. As
expected, the level of desired performance in the presence of this uncertainty
is also not achieved.
At this point, we have two options - to relax the uncertainty and
performance specifications or to see if p controller can provide the desired
robust performance.
202
/J CONTROLLER FREQUENCY ANALYSIS
_E
1.2 ..........................
I
0.8
0,6
0,4
0.2
0
0.01
..... Nomir_ai Performar.'c¢ _
" "" Robu._t 5t;_bi!ity
f
_J
....... , • , o ..... ,
0.1 1
Frequency, r'ad/sec
10
Control system satisifes performance requirements in the
presence of 20 % control effectiveness uncertainty
The p controller designed to handle 20 percent control effector
uncertainty satisfies all three metrics. The level of desired performance in the
presence of specified uncertainty is achieved with some margin to spare. We
would like to see how much uncertainty can be tolerated and still satisfy
robust performance.
203
p CONTROLLER FREQUENCY ANALYSIS
1.Z ........ , ........ , ........
1
0.8
0.6
0.4
0.Z
0
0.01
"'"" ..... "I-......... \
-- _ ROI_L_t Stab_iity
Robust Performance
0.1 1 10
Frequency, tad/see
Control system satisfies performance requirements upto
40 % uncertainty in the control effectiveness
The maximum level is achieved at 40 percent uncertainty. In fact, this p
controller satisfied robust performance for up to 40 percent uncertainty in
control effectiveness as indicated in this plot.
No analysis is complete without looking at the actual time histories. So to
validate and to augment conclusions from frequency analysis, a sample of
time responses is presented.
204
NOMINAL ELEVON RESPONSE
0.4
O.Z
0
-O,Z
-0.4
-0.6
-0.B
-i
4"
, | | | i ,
10 ZO 30 40 50
Time, mc
Commanded
- Velocity ( 100 ft/sec )
- Altitude (1000 fl )
I like to point out that in all time responses you will see, the vehicle is
commanded to simultaneously increase velocity and altitude while being
subjected to moderate atmospheric turbulence.
The first plot is elevon response for both H= and/J controllers for an ideal
system, i.e. no uncertainty present. Note that for both, initial deflection is
less than one degree. Important fact is that both responses are very similar,
indicating that improved robustness is achieved at a small loss in ideal
performance as measured by the total deflection.
Introducing 20 percent uncertainty into the system drives H= controller
unstable, which leaves us with/.I controller response to consider.
2O5
WORST CASE PERFORMANCE ELEVEN RESPONSE
0,4
0
-0.4
-0.8
-1.Z
-1.6
II 111 -- -- L-J 4u _unce_amw
II II
I I 9 ....... 40 % :j_:e_aJnl'y
J l l i " |" i • •
0 10 20 30 40 $0
Tmlo, "sic
Commanded
- Velocity ( 100 f'dsec )
. Altitude ( 1000 ft )
Eleven and Fuel Flow Rate Uncertainty
If we look at worst case while the desired performance is still achieved, we
get this plot. Recall that our performance specifications were still satisfied for
40 percent uncertainty. The amplitude of the response is dependent on the
positive or negative uncertainty in the control effectiveness. In the worse
case scenario when the actual effectiveness is 40 percent less than ideal, the
elevon deflection is still less than 2 degrees which was the limit.
Well how does this behavior impact other performance variables of
interest.
206
ALPHA NOMINAL AND WORST CASE RESPONSE
0.3
0.25
i 0.2
0.15
0.1
0.05
0
-O.OS
nominal remponso
.... /J controJ_e_
naednat resDonse
P cx)ntrol_er
10 20 30 40 S0
Time,
Commanded
- Velocity ( 100 Wsec )
- Altitude (1000 ft )
The commanded change in altitude which is facilitated by elevon deflection
is unaffected by the uncertainty. In fact, it is faster and more precise for the
p controller than for the ideal Ho, one. But this performance improvement in
altitude does have an adverse effect on another variable of importance - angle
of attack.
The p controller angle of attack peak is somewhat higher than that of the
Ho, controller, though both responses are well within the specified limit of 0.5
degree. The uncertainty again has very small effect on the p controller
response
- 207
CONCLUSIONS
- Vehicle characteristics and control system requirements translate
explicitly into H_ domain specifications
- H_ controller suffers performance degradation with introduction of control
effectiveness uncertainty
- i_-eynthesls results in an improved robust performance over H. controller
So what have we learned from this initial application of H= and p to an
airbreathing SSTO vehicle. The bottom line is that p framework provides a
systematic approach to include parametric uncertainty in design and to
explore tradeoffs between performance and uncertainty robustness. This
initial application of p synthesis and analysis techniques to an airbreathing
SSTO shows much promise.
208


Hypersonic vehicle control law development using H(infinity) and micron-synthesis

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