AlAA
90-2801
Optimal Input Design for Aircraft Parameter
Estimation using Dynamic Programmng
Principles
Eugene A. Morelli
Lockheed Engineering and Sciences Company
NASA Langley Research Center
Hampton, VA
Vladislav Klein
George Washington University
JlAFS
NASA Langley Research Center
Hampton, VA
AlAA Atmospheric Flight Mechanics
Conference
August
20
-
22, 1990
/
Portland,
OR
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
370
L'Enfant Promenade, S.W., Washington,
DC
20024
OPTIMAL INPUT DESIGN FOR AIRCRAFT PARAMETER ESTIMATION
USING DYNAMIC PROGRAMMING PRINCIPLES
Eugene
A.
Morelli* and Vladislav Kleinx*
The George Washington University JIAFS
NASA Langley Research Center
Hampton, Virginia
Abstract
ym(i) measured output vector
P
sideslip angle, rad
A new technique was developed for designing
6,
aileron deflection, rad
optimal flight test inputs for aircraft parameter
$j
Kronecker delta
estimation experiments. The principles of dynamic
Fr rudder deflection, rad
programming were used for the design in the time
At
sampling interval
domain. This approach made it possible to include
Ilk
kth output amplitude constraint
realistic practical constraints on the input and output
variables. A description of the new approach is
pj
jth input amplitude constraint
presented, followed by an example for a multiple
e
p-dimensional parameter vector
input linear model describing the lateral dynamics of
4'
roll angle, rad
a fighter aircraft. The optimal input designs
ok
Cramer-Rao bound for parameter k
produced by the new technique demonstrated
u(i)
gaussian white noise random vector
improved quality and expanded capability relative
5k
Cramer-Rao bound goal for parameter
k
to the conventional multiple input design method.
0
zero vector
dispersion matrix
expectation operator
system dynamics matrix
acceleration due to gravity, m/sec2
control matrix
observation
matrix
cost function
information matrix
total number of sample times
roll rate,
rps
yaw rate, rps
measurement noise covariance matrix
ith discrete sensitivity matrix
m-dimensional control vector at time t
airspeed,
m/sec
n-dimensional state vector at time t
q-dimensional output vector
Introduction
Aircraft flight tests designed specifically for
the purpose of parameter estimation are generally
motivated by one or more of the following objectives:
1.
The desire to correlate aircraft model
parameter estimates from wind tunnel
experiments with estimates obtained from flight
test data.
2.
Refinement of the parameter estimates for
the aircraft model for puposes of control system
analysis and design.
3.
Accurate prediction of the response of the
aircraft using the mathematical model, including
flight simulation.
4.
Aircraft acceptance testing.
The achievement of any of the above
*
Ph.D. student, Member AIAA
objectives involves many factors, including the
"*
Professor, Associate Fellow AIAA
selection of instrumentation and signal conditioning,
flight test operational procedure, input design,
Copyright
o
1990
by Eugene
A.
Morelli. Published by the
aircraft model determination, and the parameter
American Institute
of
Aeronautics and Astronautics, Inc.
estimation algorithm.
with permission.
In this work, a new computational procedure
was developed for the design of optimal flight test
input signals for parameter estimation experiments.
For the most part, the other considerations in the
flight test design manifest themselves in the detail of
the input design problem formulation. The
fundamental principles and procedures regarding the
input design remain unaltered.
Input designs for aircraft parameter
estimation experiments are evaluated by examining
the Cramer-Rao lower bounds on the parameter
standard errors, which are a function of the
information content in the aircraft response to a given
input1. These Cramer-Rao bounds are the theoretical
lower limits for parameter standard errors using an
asymptotically unbiased and efficient estimator, such
as maximum likelihood.
Comparisons using the
Cramer-Rao bounds isolate the merits of the input
design from the merits of the parameter estimation
algorithm used to extract the aircraft model
parameter estimates from the flight data.
Past studies of optimal input design in the
time domain usually formulated the problem as a
fixed time variational calculus problem, using some
norm of the information matrix or its inverse as the
cost function, and imposing
an
energy constraint on the
input to indirectly implement practical input and
output amplitude
constraint^.^,^
An approach by
~ehra~ analyzed the problem in the frequency
domain, but there is some difficulty in properly
translating results in the frequency domain into a
realistic input design in the time domain.
chen3
developed the first time domain design method
which discarded the fixed time assumption, producing
a suboptimal iterative technique using Walsh
functions.
for validity of the aircraft model whose
parameters are to be estimated from the flight
data, and the safety of the test aircraft and pilot
during flight test operations.
4.
Global minimization of the required flight
test time, subject to the conditions of the problem
formulation, so that results from expensive and
limited flight test resources can be maximized.
5.
Single pass solution.
The next section describes the problem
formulation. Following this is a description of the
solution method which uses the principles of dynamic
programming. Several example input designs using
the new technique for the lateral dynamics of a
fighter aircraft are then given. Finally, some
concluding remarks are included.
Problem
Statem&
For aircraft parameter estimation
experiments, typically a linear perturbation model
structure is assumed. The flight test inputs are
perturbations about trim to ensure that the system
response can be adequately modelled by such a
structure. The assumed model is given by
The new approach to optimal input design for
where the measurement noise
v(i) is assumed gaussian
parameter estimation flight tests described here
...,,,
departs from previous approaches by embodying the
WlIll
following capabilities and characteristics as part of
the problem formulation:
E(v(i))
=O
and
E
(
u(i)
uT(j))
=
R
hj
ij4)
1.
Multiple input design capability.
2.
Practical input constraints, including
maximum input amplitudes, control system
dynamics, input spectrum high frequency
limitations, and, in cases where a human pilot
must realize the designed input, pilot
implementation and coordination constraints.
3.
Output amplitude constraints, which can be
specified a priori as a function of considerations
Constraints were imposed on all input amplitudes and
selected output amplitudes. These constraints arise
from the practical considerations that control input
amplitudes are limited by mechanical stops, flight
control software limiters, or linear control
effectiveness. Selected output variable amplitudes
must be limited to ensure validity of the assumed
linear model form and also to ensure safety during the
flight test. The constraints are given by
sinusoidal type inputs for parameter estimation
Iydt)
1s
qk
kt, kc
.
(6)
experiments, largely due to their wider frequency
mtruml.
L~--~-~
where
pj
and
qk
are positive constants. The minimum
For the above reasons, and to make the
optimization problem tractable, input forms were
achievable values for the parameter standard errors
limited to square waves only;
i.e., only full positive,
using
an
as~m~totical'~
unbiased
and
efficient
full negative, or zero amplitude were allowed
for
any
estimator are given by the square root of the diagonal
control
at
any
time,
With
this
restriction,
the
elements of the so-called dispersion matrix,
D.
The
problem
becomes
a
high
order
combinatorial
problem
dispersion matrix is defined as the inverse of the
involving
output
amplitude
constraints,
which
is
information mahix
M,
the latter being a measure of
well-suited
to
solution
by
the
method
of
dynamic
the information content in the data from an
programming, as described next.
experiment. The expressions for these matrices are
where the sensitivities are computed
h.om
ax
aG
qGl=
gx
+
-
+
-u
j=l
2,...,p
aei ae,
(9)
The above equations follow from equations
(1)
-
(2)
and the assumed analyticity of x(t).
Generally, required accuracies for the
parameters can be specified a priori by the end users of
the parameter estimates. These values represent
goals for the Cramer-Rao bounds of each model
parameter. The optimization problem is then
:
choose
the input which minimizes the time to achiere the a
priori desired accuracies on the parameters.
This
approach obviates the need for parameter weighting
required by fixed test time approaches, and also
maximizes the effectiveness of limited and expensive
flight test time.
As posed, this optimization problem is
difficult to solve in general. At this point,
considerations particular to optimal input design for
aircraft parameter estimation problems were invoked
in order to limit the allowable control set to square
wave inputs only. Among these considerations were
analytic work for similar problems which indicated
that the optimal input should be of the "bang-bang"
type3, the input capabilities of human pilots, and
previous flight test evaluations which demonstrated
that square wave type inputs were superior to
For purposes of illustration, assume that two
output amplitudes are constrained. The allowable
output space at any given time then can
be
represented
by a plane region whose borders correspond to the
output amplitude constraints, as shown in figure
1.
The plane region is divided into discrete output space
boxes. Time is divided into discrete steps called
stages. The constrained outputs of the system are
examined at every discrete time, which are separated
by one stage time. Feasible outputs at any time must
be contained in one of the discrete output space boxes.
Starting at the initial condition box in output space,
all possible controls (full positive, full negative, or
zero amplitude) are applied over one stage time and
the consequences of each control possibility are
computed. These consequences include the system
outputs from integrating system dynamic equations of
motion (equations
(1)
and (2)) and a cost associated
with each particular control possibility. In general,
this results in several reachable boxes in feasible
output space at the next time stage. Any control
which takes the output outside feasible output space
is dropped from consideration for inclusion as part of
the optimal control sequence. This implements the
output amplitude constraints. For each time stage, the
reachable output space is computed as the result of all
possible control inputs starting at the reachable
output space boxes found for the preceding time stage.
Thus, one can picture a complicated network of
connections between output space boxes separated by
one time stage. After only a few stages, several boxes
in output space at a particular time can be reached by
more than one input sequence. The preferred sequence
is that which results in the lowest cost. This input
sequence is saved and associated with that particular
output space box at that time, while the other
(inferior) control sequences which reach the same box
in output space at that time are discarded.
The cost function was chosen as the square of
the Euclidean distance between the point in
parameter hyperspace corresponding to the current
Cramer-Rao bounds computed from
tlie information
matrix to the rectangular parallelepiped in
parameter space which represented the goal values of
the parameters. The cost function can be expressed as
An illustration of the cost value for two parameters in
three example cases
A,B,
and C, is shown in figure
2.
The boundaries of the parallelepiped in parameter
space are the goal values for the Cramer-Rao bounds.
The location of points
A,B,
and C would be determined
by the computed Cramer-Rao bounds associated with,
say, three different inputs, or perhaps the same input
sequence at three different times. At any time, the
Cramer-Rao bounds were obtained from a sequential
computation of the additional information (and thus
the additional available parameter accuracy)
resulting from the application of a particular control
possibility over one stage time.
In order to compute the cost for any candidate
conhol at any time, it is necessary to sequentially
compute the dispersion matrix, from which the
Cramer-Rao bounds for each parameter may be
computed as the square root of the corresponding
diagonal elements. First define the discrete
sensitivity matrix
Then the sequential calculation of the dispersion
matrix, due to Chen3, is given by the relation
Only those parameters whose Cramer-Rao
bound goal is not yet achieved contribute to the cost.
Bellman's principle of optimality4 was used to choose
the optimal control sequence to any box in reachable
output space at any time stage, based on the cost
function given above. The first time stage where the
cost became zero for some box in output space (all
Cramer-Rao bound goals attained) was designated the
minimum time solution, and the input sequence to
reach that output space box at that stage was
designated the optimal input sequence.
Several modifications which take advantage
of the structure of the optimal input problem for
aircraft parameter estimation were made to reduce
memory requirements.
In addition, sophistications
were added that adjusted the input possibilities at
certain time stages in order to account for control
system dynamics, limitations on high frequencies in
the input, and practical implementation and pilot
coordination constraints, the latter being especially
important for multiple input designs. Details of these
features and other aspects of the solution method may
be
found in reference
5.
The lateral dynamics of an advanced fighter
aircraft in level flight at 10,000 m altitude and an
airspeed of
179.72
m/sec. may be represented by the
dynamic system and measurement model specified by
equations
(1)
-
(3) where