Document
N90-20936
APPLICATION OF STOCHASTIC ROBUSTNESS TO
AIRCRAFT CONTROL SYSTEMS
Laura E. Ryan Department of Mechanical and Aerospace Engineering Princeton University
INTRODUCTION
Guaranteeing robustness has long been an important design objective of control system analysis. Stochastic robustness is a simple numerical procedure that can be used to measure and gain insight into robustness properties associated with linear control systems. In the realm of aircraft control systems, problems such as the effects of flight condition perturbations and model-order uncertainties on robustness are easily and effectively analyzed using stochastic robustness. The concept of stochastic robustness will be reviewed and examples will be presented demonstrating its use in flight control system analysis.
• Summary of stochastic robustness
• Control system robustness with flight condition perturbations
uncertainties
• Control system robustness with model-order
Actuator dynamics
Aeroelastic effects
• Summary of results
DEFINITIONS
Control system robustness is defined as the ability to maintain satisfactory stability or performance characteristics in the presence of all conceivable parameter variations. A good robustness measure is vital to guarantee and understand control system robustness. Stochastic robustness provides such a measure. It uses the statistics of a plant's variable parameters and Monte Carlo simulation to compute the probability distributions of closed-loop system characteristics. Present research has concentrated on stability robustness as characterized by the closed-loop eigenvalues, although the method can be extended to other closed-loop characteristics. Stochastic robustness is computationally simple. For a single Monte Carlo evaluation, random numbers are generated and shaped to match the parameter statistics, added to the mean parameter vector, and the closed-loop eigenvalues are computed using the modified parameters.
Repeated Monte Carlo evaluations give rise to the stochastic root locus, a plot of the probability distributions of the closed-loop eigenvalues. The probability ofinstabilily, or probability that all of these eigenvalues lie in the open left-half s plane, is the scalar measure of robustness.
Robustness
The ability to maintain satisfactory stability/performance
characteristics in the presence of all conceivable parameter
variations.
Stochastic robustness
A robustness measure based on the probability distributions of
closed-loop characteristics, given the statistics of a plant's variable
parameters.
• Characteristics can be eigenvalues, performance,
control authority, disturbance rejection
• Based on Monte Carlo simulation
• Not limited to Gaussian parameters
Stochastic root locus
Plot of the probability distributions of closed-loop eigenvalues.
Probability of instability
Probability that closed-loop system is unstable - a scalar measure of
stability robustness.
STOCHASTIC ROBUSTNESS
APPLIED TO DEMONSTRATOR AIRCRAFT
As an example of the application of stochastic robustness, three controllers are applied to a fourth-order longitudinal model of an open-loop unstable aircraft. A ten-element parameter vector consisting of elements of the dynamic and control effect matrices is chosen. The three control designs are chosen to reflect increasingly robust controllers. The first two Cases are LQR controllers with low and high control weigttting respectively, and the third controller multiplies the Case (b) controller by a factor of five to restore the closed-loop bandwidth to that of Case (a). These three cases have been chosen not to satisfy any particular flying qualities criteria, but merely to demonstrate the impact of differing generalized design criteria on stochastic robustness.
Fourth-order longitudinal dynamic model
x = F(p) x + G(p) u
u =-C x
Ten-element parameter vector
P = [faa f12 f13 f22 f32 f33 gll g12 g31 g32]
fij, gij are elements of matrices F and G.
Control design ( C matrix) to demonstrate stochastic robustness
Case (a) LQR with low control weighting.
Case (b) LQR with high control weighting.
Case (c) Gain matrix of Case (b) is multiplied by 5 to restore
bandwidth.
Stochastic root loci for demonstrator aircraft with 30%
standard-deviation Gaussian parameters
The results for 30% Gaussian parameters and 10,000 Monte Carlo evaluations reflect the expected increase in robustness between control designs. The stochastic root locus shows the extent to which eigenvalues can vary. The eigenvalue near the origin is least affected by the parameter changes, and its peak dominates the distribution. In Cases (a) and (c), the left-most eigenvalue (not shown) has an enormous variance along the real axis. Interaction of roots around the origin causes instability.
Robustness improves from Case (a) to (b) as control usage is restrained by high control weighting, and the ad hoc robustness recovery technique used in Case (c) gives additional improvement.
pr(X) jto_, lO.O
- I?=0.0711
5.e It prO.)
J_//," le.e
" i _i _""_ Case(b) -____ _ z._ z-_ o 17=0.0169
"' Case (c)
17= 0.0033
o z.5
Stochastic root loci for demonstrator aircraft with 30% uniformly
distributed parameters
For 30% uniformly distributed parameters and 10,000 Monte Carlo evaluations, the probability of instability for all three cases is zero. The stochastic root locus gives a good indication of the effects of Gaussian "tails" on the eigenvalue probability densities.
pr(k)
Case (a)
P=O.O
. r S.e /_-" s.e PRO.)
1_887"5 j_
Case (b)
/2"
F=O.O
S.O ;P.E pr(_)
I?=0.0
J'_ '"' Case (c)
o /-- s.e
CONTROL SYSTEM ROBUSTNESS WITH
FLIGHT CONDITION PERTURBATIONS
Demonstrator Aircraft with Flight Condition Effects
Dynamic pressure variations can be considered separately and included in the parameter vector.
Although velocity (V) and air-density (P) are essentially deterministic, including them as separate parameters gives the ability to look at flight condition perturbations around the nominal and eliminates correlation of the remaining parameters. A twelve-element parameter vector results, p and V are modeled as uniform parameters, giving an indication of stochastic robustness over a range of flight conditions.
Fourth-order longitudinal dynamic model
_: = F(p) x + G(p) u
u=-Cx
Twelve-element parameter vector
P = [P Vfll f12f13f22f32f33 gll g12 g31 g32]
p is the air density (nominal value 0.00152 s/ft 3)
V is the velocity (nominal value 670 ft/sec)
fij, gij are elements of F and G with P and V considered separately.
Model p and V as uniform parameters and apply stochastic
robustness using the same three control designs as previous example.
Stochastic root loci for demonstrator aircraft with 30% Gaussian
parameters and 30% uniform p and V
Next, we examine the stochastic root loci for these three cases: for 30% uniform p and V, and 30%-standard-deviation Gaussian uncertainty on each of the remaining elements of the parameter vector.
The shapes of the root loci are similar to the case with correlated parameters.
If pr(z) j,o ,..o Case (a)
I?=0.0736
,A- S.8 /I pr()_) jo) lo.e
Case (b)
I? = 0.0166
(Y pr(Z.)
18.8
'_ Case (c)
,, 17= 0.006
SUMMARY OF STABILITY ROBUSTNESS RESULTS FOR
DEMONSTRATOR AIRCRAFT FOURTH-ORDER MODEL
Considering a case with non-varying flight condition along with the above results, the probabilities of instability seem to indicate that instability in Case (a) is a stronger function of uncertainties in individual parameters or stability derivatives than 30% V and p variations, while the remaining two cases are sensitive to flight condition variations.
Controller
Case (a) Case (b) Case (c)
Correlated parameters 0.0711 0.0169 0.0033
30% Gaussian variations
0.0736 0.0166 0.0060
Uncorrelated parameters
30% Gaussian variations
30% uniform P and V
0.0746 0.0162 0.0030
Uncorrelated parameters
30% Gaussian variations
no 9 and V variations
CONTROL SYSTEM ROBUSTNESS
WITH ACTUATOR DYNAMICS
Stochastic robustness can be used to quantify the effects on robustness of actuator dynamics.
First-order actuator dynamics are added for each control, resulting in a 14-element parameter vector. A controller is designed with LQR weighting specifications intended to approximate the controller of Case (a), while not pushing the actuator dynamics to unrealistic frequencies.
Fourth-order longitudinal dynamics and first-order actuator dynamics
for each control
i = F'(p') x + G'(p') u
u=-Cx
14-element parameter vector
P'= [I9 W fll f12f13f22f32f33 gll g12 g31 g32 "Cc z't ]
"_c = canard time constant (nominal value 0.1 sec)
z t = thrust time constant (nominal value 1.0 sec)
• Redesign Case (a) controller such that closed-loop longitudinal
eigenvalues are the same as previous example and actuator dynamics
are reasonable.
• Apply stochastic robustness using new controller.
PROBABILITY OF INSTABILITY FOR VARIOUS GAUSSIAN
CONTROL PARAMETER UNCERTAINTIES
Stochastic robustness is applied for different values of the variance associated with each time constant, in order to detail the separate effects of each control lag. As indicated by the first line in the Table, simply including actuator dynamics increases the probability of instability, even if the associated parameters are known perfectly. This is a reason_ole result because actuator dynamics are no longer inf'mitely fast but are allowed to interact with the rigid-body states. Qualitatively, bringing actuator dynamics in from infinity pushes the root-locus closer to instability. Stochastic robustness quantifies the effect. The thrust time constant has a small effect on the probability of instability, while a large increase in the probability of instability is seen as the canard time-constant standard-deviation increases from 30% to 150%.
p and V are 30% uniform parameters.
fij, gij are 30% Gaussian parameters.
I?
standard-deviation of -c t
standard-deviation of z c
0 0 0.O92
0 30 0.0988
3Q 30 0.101
30 150 0.1014
150 30 0.1474
0.0736
No control dynamics
Stochastic root loci for demonstrator aircraft with 30% Gaussian
parameters, 30% uniform p and V, and non-varying "_c and I:t
The stochastic root loci show that a strong coupling due to uncertainties can occur between the control and dynamic states, which tends to push more eigenvalues towards the right-half plane.
j(o 45.e 35.e
Case (a)
_? = 0.092
zs.e ORIGINAL PAGE IS OF POOR QUALITY Is.e :..., S.e 3,': _- 4 ¸¸ , (3 5'.e -46'. e -3s.e -zs.e -15.e - 5.e pr(k) 15 • zs.e -4 _s.e 0
Stochastic root loci for demonstrator aircraft with 30% Gaussian
parameters, 30% uniform P and V, and 30% Gaussian 1:c and "_t
Increasing the standard-deviations associated with the time constants shows that the complex pair of eigenvalues has a small "variance" in the c-direction, and a large variance in the jc0 d direction.
jo)
Case (a)
I?= 0.101
3s.e zs.e 2.
OF POOR QUALITY 15.e 5.e • ;" ( ' ' :,1" : " • _, .. .
• -45.0 -3..d -Z5.0 -15'. e - 5.0 5'.e pr(_.)
55.6
Stochastic root loci for demonstrator aircraft with 30% Gaussian
parameters, 30% uniform 9 and V,
30% Gaussian I:c, and 150% Gaussian x t
The o-direction variance is largely due to the uncertainty associated with the thrust time constant, as illustrated by increasing the standard-deviation on this parameter to 150%. Increasing the uncertainty of "c t has little effect on the probability of instability because it does not cause significant coupling with the dynamic modes.
jto 45.e .:...
Case (a)
., .LT_ •
_? = 0.1014
ZS.e ORIGINAE PAGE IS lS.0
i )i
OF POOR QUALITY "i_2.
S.e G s'.e -45'. 0 -35.0 -ZS'. 0 -lS'.O - S.e i r(%) " -1 / s.e S.e
Stochastic root loci for demonstrator aircraft with 30% Gaussian
parameters, 30% uniform p and V,
150% Gaussian 1: c, and 30% Gaussian 1: t
The variation at constant c and coupling of 6ontroller and dynamic modes is largely due to variation of the canard time constant. Uncertainty in x c causes eigenvalues to migrate to the real axis and split off to form the complex "cloud" of eigenvalues that reaches instability.
jco 45.9
Case (a)
35.e
]?=0.1474
zs.e _w_ov_ _ F_: _ • • , ;:'- "'\: i " 15.8 G_ POOR _,LiTY 5.e t , , )L ,.-45.0 -3<0. e -ZS'. e 5'.e -15.8 - 5.e pr(_.)
dg _/1 jco " -1 .e
Stochastic root loci for demonstrator aircraft with aeroelastic effects
for 30% uniform p and V
For 30% uniform variations in velocity and density alone, using the reduced-order gains, the probabilities of instability are zero. As expected, the closed-loop torsion eigenvalues at s= -0.1 _ 212.5j (not shown in figures) do not change with velocity and do not effect the probabilities of instability.
Bending mode eigenvalues show a definite velocity trend, migrating towards instability as velocity increases. The closed-loop mean eigenvalues of rigid-body modes shift from the reduced-order case because of the presence of the added dynamics. In each case, the open-loop bending mode eigenvalues (-2.95 5: 32.03j) shift towards instability.
j(.o / , _F..e \
Case (a)
15.0
F=0.0
x 0 -ZO. e
(
IS.0
Case (b)
F=O.O
f_"..l o - 2i.i) J Is,0
Case (c)
F=O.O
-zl_,e .0
Stochastic root loci for demonstrator aircraft with aeroelastic effects
for 30% Gaussian parameters and 30% uniform p and V
Next, the stochastic root loci for 30% uniform variations in p and V and 30% Gaussian variations of the parameters are evaluated. The peak near the origin is magnified to bring out the distribution associated with the bending mode eigenvalues.
j(o g
Case (a)
I? = 0.043
4.
t pr(_.)
ORIGINAL PAGE IS OF POOR QUALITY.
lkl i __._(0 i
CONTROL SYSTEM ROBUSTNESS WITH AEROELASTIC
EFFECTS
With second-order, dynamic-pressure-dependent aeroelastic effects representing the wing's first bending and torsional modes, the (8x8) and (8x2) system matrices can be partitioned as shown, where F r = F and Gr = G represent dynamic and control effects for rigid body modes, Fra and Far couple the rigid and flexible modes, and F a, G a represent aeroelastic dynamic and control effects. To separate effects of material properties, the four parameters K b , M b , K t , and M t are assumed to be known perfectly. The aeroelastic matrices introduce 32 additional parameters, and the resulting 44-element parameter vector includes separate P and V effects. For preliminary analysis, 40 parameters were used, although concern for statistical significance and limits on the computational facilities used to date calls for modification of the number of parameters for future studies.
Fourth-order longitudinal dynamics coupled with fourth-order
aeroelastic effects
= F'(p') x + G'(p') u
u=-Cx
F,:[Fr Fral EOrl
Far Fa Ga
F r = F, G r = G represent rigid body modes
Fa, G a represents coupled second-order bending and torsion modes
Fra and Far couple rigid and aeroelastic modes
44-element parameter vector
+ uncertain terms
P'= [I3 V fll f12f13f22f32f33 gll g12 g31 g32
from Fra, Far, F a, and G a ]
J
CONTROL SYSTEM ROBUSTNESS WITH AEROELASTIC
EFFECTS (CONTINUED)
In terms of structural-dimensional derivatives, F a can be represented as given. Material properties dominate the torsional mode, which varies little with dynamic pressure variations. Stochastic robustness is applied to the new system using the reduced-order gains. Coupling of the systems through Fra and Far causes the closed-loop system to be sensitive to a_roelastic terms.
• Assume generalized mass and stiffness of bending and torsion
modes are known perfectly.
m m 0 1 0 0 1 1 (S b-Kb)/Mb S/Ib/M b S rh/Mb S _h/Mb F a -- 0 0 0 1 2 2 2 2 _b]'lVlt S_blqVlt (S rh-Kt)/Mt S ,I,h/Mt_
K b, M b, K t and M t are generalized stiffness and mass for each
mode.
srli are structural dimensional derivatives.
• Apply stochastic robustness to new system using gains established
previously.
c' =[c.0]
Fclose .,oop : [Fr" Ore Fra ]
Far- GaC Fa
Stochastic root loci• for demonstrator aircraft with 30% Gaussian
parameters and 30% uniform p and V
.. j_ _:!!!
Case (b)
?=0.017
.15.e (3 - Z'e.e x B.e pr(k) OF PO0_ QUALI'r_.
s.e (3
Stochastic root loci for demonstrator aircraft with 30% Gaussian
parameters and 30% uniform p and V
Case (c)
I?=0.0415
IS.O L£ :J J ":_-'- i. , " , _ -ZO. e .e jm pr(k) jf.o -----4..
5.0
SUMMARY OF ROBUSTNESS OF DEMONSTRATOR
AIRCRAFT WITH AEROELASTIC EFFECTS
While definite conclusions cannot be reached because of the small sample space, this example illustrates the type of analysis possible using stochastic robustness. Certain trends are evident. The disparity in robustness between Cases (a) and (c) is reduced, and Case (c) shows a considerable decrease in robustness, while the robustness of the first two cases is at least retained or possibly improved.
Application of a reduced-order controller to a higher order system does not guarantee that the robustness margins of the original system are retained, but the robusmess of the system does not always go in the adverse direction. (This is somewhat analogous to the loss of guaranteed stability margins when applying LQG). Stochastic robustness again provides an excellent framework to quantify the effects of applying a reduced-order controller to a higher-order system. In each of these examples, questions concerning the selection of the number of Monte Carlo simulations, confidence limits, and statistical significance of results are issues of future research.
Probability of instability:
without aeroelastic effects
with aeroelastic effects
Case (a) 0.0736 0.0430
Case (b) 0.0166 0.017
Case (c) 0.006 0.0415
Closed-loop eigenvalues:
without aeroelastic effects
with aeroelastic effects
-0.02
Case (a)
-0.02
-3.32, -5.14
-4.96 + 1.27j
-35.0
-35.0
-2.3 + 32.0j
-0.02
Case (b)
-0.02
-1.09
-1.01
-3.36, -5.15
-4.8 + 1.38j
-2.53 + 32.0j
-0.01
Case (c)
-0.02
-3.44, -5.15
-3.6, -5.53
-32.21
-34.1
-1.74 + 32.88j
SUMMARY
Stochastic robustness offers a rigorous yet straightforward altemative to current metrics for control system robustness that is simple to compute and is unfettered by normally difficult problem statements, such as non-Gaussian statistics, products of parameter variations, and structured uncertainty. The approach answers the question, "How likely is the closed-loop system to fail, given limits of parameter uncertainty?" It makes good use of modem computational and graphic tools, and it is easily related to practical design considerations.
The examples presented here illustrate the use of stochastic robustness and its advantage in studying aircraft control systems. The parameters of aircraft stability and control effect matrices (stability derivatives and nominal flight condition parameters) lend themselves to this type of analysis tool. The stochastic robustness of different control system designs can be directly compared. Stochastic robustness can be used to study stability with flighv condition variations. The method is also easily applied to model-order uncertainties in aircraft control systems by adding the uncertain dynamics to the system and assigning appropriate statistics to the new parameters. Quantitative effects of individual parameters or combinations of parameters on robustness can be measured in terms of the probability of instability. The principal difficulty in applying this method to control systems is that it is computationally intensive; however, requirements are well within the capabilities of existing computers. The principal advantage of the approach is that it is easily implemented, and results have direct bearing on engineering objectives.
1. Stochastic robustness can be used to study effects of flight
condition perturbations on robustness.
• By considering flight condition parameters separately, parameters are uncorrelated.
• Can separate flight condition effects on robustness from
parameter uncertainty effects.
2. Stochastic robustness can be used to study effects of model-order
uncertainties on robustness.
• Shows magnitude of actuator dynamics effect on robustness.
• Reveals instability or robustness degredation due to
neglected dynamics.