Aeroservoelastic Analysis of Missile Control Surfaces via Robust Control Methods

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Copyright © IFAC Automatic Control in Aerospace, Saint-Petersburg, Russia, 2004

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AEROSERVOELASTIC ANALYSIS OF MISSILE CONTROL SURFACES VIA ROBUST CONTROL METHODS Alper Akm~el, Mutlu D. Comert 1, BOIent E. Platin 1 1 TVBiTAK-SAGE The Scientific and Technical Research Council of Turkey Defense Industries Research and Development Institute Ankara, Turkey ]Middle East Technical University, Ankara, Turkey

Abstract: in this study the effect of uncertainties on the flutter and instability speeds of the control actuation system of a missile is analyzed. The control surface is modeled as a typical section wing with uncertainties in damping and stiffuess coefficients. The analyses are carried out for incompressible flow with some uncertainties in aerodynamic coefficients. The actuator dynamics of the aeroservoelastic system is modeled as a second order system with a frequency varying uncertainty. Copyright © 2004 IFAC Keywords: Active Control, Robust Stability, Uncertainty, Missiles.

In this study, methods presented by Lind et a/. (1998, 1999) are applied to a missile control actuation system (CAS). The control surface (fin) is modeled as a typical section thin airfoil and the aerodynamics is mode led as steady and in the incompressible subsonic region. In contrast to the above mentioned studies, the control is effected through the input shaft of the fin instead of an aileron. Both aeroelastic and aeroservoelastic analyses are performed for the CAS with both nominal and uncertain plants.

1. INTRODUCTION Flutter is one of the most catastrophic failures suffered by structures that are exposed to airflow. An aircraft is one such structure and since WW I the flutter phenomenon has been observed numerous times in aircraft. The mechanism of potential flow flutter was understood by 1935, and different analysis methods were developed to derive the flutter boundaries of air vehicles. Today aeroelastic analysis can be done via commercial software packages using various methods (P, K, P-K, etc). Some of these packages also have simple controller models for aeroservoelastic analysis. Starting from the late 80's some robust control methods were developed and in the 90's they were applied to flutter analysis. One such method named as the ",-method has been applied to flutter margin analysis in particular. With the use of ",-method, it becomes possible to construct more realistic mathematical models by defining model uncertainties. A study on robust aeroelastic and aeroservoelastic flutter analysis was made by Lind et al. (1998, 1999). As in the typical section model of Lind (1999), in flutter and flutter suppression analysis the typical section model is mostly augmented with an aileron as a control surface, Edwards et al. (1978), Horikawa et al. (1979), Ohta et al. (1989), Hitay et al. (1994), Jeffi'ey et al. (1999).

The constructed ",-method model searches for the critical dynamic pressure at which instability occurs using the density of air as the actual search parameter. An iterative search for the airspeed of instability at a given air density is also performed. Numerical calculations are made by using MATLAB~ (The MathWorks, Inc.) computer software package.

2. MODELING A three degree of freedom aeroservoelastic model . given in Fig. I is obtained by adding a servo actuator to control the angular degree of freedom of the typical section wing, Fig. 2. The typical section parameters are taken from Lind (1999). Additional parameters used in the study are 719

given in Table I . In Fig. 3, the block diagram of the closed loo~ system is given in MA TLAB~ SIMULINK from. The Linear Fractional Transformation (LFT) system for the typical section plant part of the three degree of freedom model is given in Fig. 4. This LFT system is to be used for robust stability analysis in the ~ framework with parameterization over dynamic pressure and uncertainty description.

p shown in Fig. 4 is given in (I), where {z} terms are the additional states due to uncertainties, {w} terms are the effects of uncertainties and [W] terms are diagonal uncertainty weighting matrices.

Table 1. Additional parameters Symbol Wc/• Wc...

kT Im cm N

=

Iq}

Value 0.1 0. 1

Explanation Uncertainty of aerodynamic lift coefficient Uncertainty of aerodynamic moment coefficient 0.2 N.mlA Motor torque constant 2x10' s kg.rn 2 Inertia of motor at wing shaft IxIO') kg.rn.s Damping of motor at wing shaft 10 Transmission ratio

[0... 1

[0... 1

[0,~1

[0,.. 1

P,.. }

[Mt'[ic...l [Mt'(c... l [Mt' ii.[Mt' -[Mt' -[Mt' [Mt'{k:}

t.}) f._} f..} f..}

[1,..1

[o.~1

Iq}

[A,1 [A,1 [0,.. 1 [w_IA,1 [w_IA.1 [0... 1 [w.1 [0, ..1 [0... 1 [w.1 [0...1 [0...1 [1,,,1 [0,,,1 [0,.. 1

[1,.. 1 [0,..1 [0...1 [0,.. 1 [0,~1

[0,.. 1 [0,..1 [0,.. 1 [0,.. 1 [0,.. 1 [0,.. 1 [0...1 [0,.. 1

[0,.. 1

[0,~1

P..,} P..,} P,.. } P,.. } Ph'}

{q}

lil

Iw.)

... _}

....} ....} q.

(I)

where

[A z]=

[ bI] -2h.c,a U 2 2.b .c..a

[A]= [00

[

(2)

0

-2.b] 2.b Z

3

[A.]=

0

- 2.bU~ 0] 2.b2~ 0 U

6, 6,

0

6.

Fig. I . Three degree of freedom aeroservoelastic model

6_ I---

6,. 6.. 0

z

6.. 6~

----

.. ..~

.-

-

P

q {q} Fig. 4. Typical section plant part for Linear Fractional Transformation

"of'""ftl ocity U

Fig. 2. Typical Section Wing

lzDI :

•

Fig. 3. Block Diagram of Aeroservoelastic System for Robust Aeroservoelastic

720

~-method

infinite norm bounded by real scalar a, if and only if .u(P) <

to

to:

F_y(radiInsi.OC:J

spectral radius of the frequency-varying transfer function matrix. However the spectral radius is a discontinuous function of frequency, thus the true solution of Jl can not be guaranteed via search over fmite frequency points.

Fig. 5. High Pass Filter Type Weighting of Motor Uncertainty Closed Loop Model: In order to model the uncertainty of the motor, a frequency varying (3), input multiplicative type uncertainty is used (Fig. 5). Gain· MagSca/e·

S+W

I •t MagSca e S + w t

A simple alternative method is to iteratively search the stability of the system for different perturbations. One of the simplest approaches used for the nominal flutter margin search, the bisection method given by Zhou et al. (1998) and Lind et al. (1998, 1999) is adopted for this purpose.

(3)

In the typical section plant model, an uncertainty was added to the torsional stiffness, the same torsional stiffness also appears in the closed loop system, Fig. 3. This is used to transmit the torque back to the motor. Since there is only one physical spring, in order to be consistent, the same uncertainty is used on both torsional stiffnesses, and they are linked to the same perturbation. The linear fractional transformation system for the closed loop system is given in Fig. 6.

Robust Flutter Analysis: Defme the perturbation matrix for the robust aeroelastic system given in Fig. 4 as tJ.. The uncertainties are defmed for unit perturbation, hence the perturbation matrix should satisfy IItJ.L ~ 1. Thus ~t5q SI, which leads to a search

IL

over a unit perturbation to dynamic pressure, 1 Pa. This is a useless, overlimited search range for dynamic pressure. In order to extend the search range, the perturbation to dynamic pressure is multiplied with a weight, Wqt > 1 .

6, 6, 6••

~q} ~

i---

6.. 6..

~::} I k}

(4)

{W q} {Wc/mal

6••

{Zc/..a}

This weight is included into the analysis through scaling the feedback signals to P . With this scaling a newly scaled plant P is obtained. For robust aeroelastic analysis with extended dynamic pressure range, the plant P of Fig. 4 is replaced by P .

{W.}

L----I'

I

G

Ya .

Here a is the largest perturbation to dynamic pr~ssure for which the nominal aeroelastic system is stable. Since 15-q is a real scalar, the solution is the maximum

11.....4' - -_ _...J

r

L-_---l

Fig. 6. Linear Fractional Transformation System for Robust Stability Analysis of Aeroservoelastic System in the Jl Framework with Parameterization over Dynamic Pressure and Uncertainty Description

(5) (6)

As an introductory study, a simple proportional controller is used to stabilize the system.

The robust flutter margin of the system can be computed by determining the scaling matrix of dynamic pressure, !wq], for which .u(P) = 1 . A simple

3 . INSTABILITY ANALYSIS

method to iteratively search the 3. Jp-Method Analysis

[15.

o

0] ,

15.

.u(P) =

1+

& ,

with accuracy &, given by Lind et al. (1998, 1999) is used for this purpose.

Nominal Flutter Analysis: Eliminating the rows and columns corresponding to uncertainties of stiffness, damping and aerodynamic coefficients, the nominal aeroelastic model P nom is obtained. Using robust stability analysis tools the largest perturbation to dynamic pressure for which the nominal aeroelastic system is still stable is found. P nom is robustly stable with respect to the perturbation set

[WqI for

Nominal Instability Analysis: Aeroservoelastic stability margin is the smallest change in dynamic pressure which causes instability of the aeroservoelastic closed-loop system. On the other hand, the flutter margin is the smallest change in dynamic pressure which causes instability of the aeroelastic open-loop system. In contrast to the aeroelastic systems, in aeroservoelastic systems the instability can be due to the controller or to the

which is

721

From the pitch motion magnitude graphics it can be seen that the effective damping of the closed loop system at the low frequency peak is significantly increased in pitch motion. Besides, the closed loop system suppresses the pitch motion at low frequencies better than the open loop system. In Fig. 9 and Fig. 10 V-g results of the aeroelastic and aeroservoelastic systems are given. Examining the

coupling between the controller and the aeroelastic system. On the other hand, the problem can be transfonned into the Linear Fractional Transfonnation fonn similar to the aeroelastic problem. Thus the problem can be solved in a similar way to the aeroelastic problem. But a small modification is required: The aeroservoelastic instabilities can be encountered at the edges corresponding to the low and high dynamic pressures, whereas the aeroelastic instability is encountered at the edge corresponding to high dynamic pressure only. Hence, it is necessary to modify the search algorithm utilized for nominal flutter analysis. An additional search algorithm for lower bound of nominal aeroservoelastic stability margin devised by Lind et al. (1998, 1999) is used for this purpose. Robust Instability Analysis; Through similar causes given above for nominal instability analysis, besides the calculation of upper dynamic pressure limit of instability, another search is required to fmd the lower dynamic pressure of stability. For this purpose another search algorithm devised by Lind et al. (1998, 1999) is used.

3.2 Results ofRegular AnalYSis In regular analysis, time domain and frequency domain response analyses and V-g flutter analysis are made using MA TLABiIl software.

3.5

Fig. 8. Frequency response of the open-loop (-) and closed-loop (---) systems from noise to measurement

.[::t·· · ··i·:·.···...·.:!-·-·:u· , · · · · r~ !....... -i-.... !......· · ·.l

45

Ttme (S)

E

0 .2

~

0

. - . . ••• ~ .•.. - . -.~. -- -. - • •• - .. . . . - ....

• ..•• . •. : .• .••••• : .. . ••.. m

m

4~

·

..

~

•

~

~

ro

~

~

~

Oyn8fTlic: P,usur. (p.)

Fig. 7. Time response of open-loop (-) and closed-loop ( --- ) systems

Fig. 9. Modal properties of open-loop system

r:f ' ,luu!; ,lu "ui,u1i . . 1

A time domain analysis is made by releasing the fin from 0.1 rad pitch position, Fig. 7. From the results it is seen that the closed loop system suppresses this input faster than the open loop system; however the magnitude of the plunge motion is increased. In Fig. 8 frequency response to disturbance graphics for a dynamic pressure of 22.5 Pa are given. The disturbance is applied to aerodynamic forces on the pitch degree of freedom. From the plunge motion magnitude graphics it can be seen that the peak value for the closed loop system becomes greater than that for the open loop system, which means that the effective damping in plunge motion is decreased.

~:_.m···r· . . ···[m .... r·· ···~m·····~·· . ·. T ~

~

•

~

[)ynomic -

~

ro

~

-j

~

m

(p.)

10' ,""' " . . . .... , ." . . . ... .... ... """ .. " "" . , . • "". , ,. """

f

~

t';

i

~

10

~ ~ ~ ~ ~ ~ ~~ !~~ ~ ~ ~~~~~~! ~ ~~ ~ ~~ ~ ~f! ~~ ~ ~ ~~~ ~i!~ ~ ~ ~~~ ~ ~ !~ ~ ~ ~ ~~ ~~ ~ !~ ~ ~ ~ ~~~ ~ ~ i~~! ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~ ~ ~~ ~ ~~ ~ ~~~~ ~ ~ ~ ~~ ~~ ~I~ ~ ~ ~ ~~! ~ ~ ~ ~ ~~~ ~ I~ ~ ~ ~ ~ ~ ~ ~! ~~ ~ ~ ~ ~~~! ~ ~ ~ ~ ~ I~ ~ ::::::::r:::::::F-::::::f:·· .. ·uf.. ·.... ·!· .. ····· l····.... t··::::::

1O~'::--::'~:---f.40,---:':~--60~---:1~O--::80'::----::!oO"--.-..J 100 Dynom;c Pr...",. (pa)

Fig. 10. Modal properties of closed-loop system 722

4. CONCLUSIONS

graphics, the dynamic pressure of flutter is 55.3 Pa, and the dynamic pressure of instability is 85 .8 Pa.

A mathematical model of a missile control surface taken as a typical section in steady incompressible flow is derived for Il-analysis. The derived model includes the uncertainties of stiffiless, damping, aerodynamic coefficients and actuator dynamics. The dynamic pressures of flutter and instability of the nominal systems are calculated using V-g and Ilanalysis methods and it is observed that the results are in good accordance. The dynamic pressures of robust flutter and robust instability of the systems are calculated. Results reveal that the robust dynamic pressure of flutter and instability can be less than the results obtained by dividing nominal values by common safety factors .

3.3 Results of J.l-Method Analysis The equations for robust analysis are derived for six uncertainties. In order to use MA TLAB~ commands to make the robust aeroelastic analysis, the rows and columns corresponding to the uncertainties with zero weights are eliminated. The results of Il-analysis are given in Table 2. It is seen that Nominal Il-analysis and V-g analysis results are in good accordance. Table 2. Dynamic pressures and frequencies of instability Model NominalAE RobustAE NominalASE RobustASE

Instability Nominal Flutter Robust Flutter Nominal Instability Robust Instability

qbar [Pal 55.3 41.8 85.8 68.7

The Il-analysis gives the values for the density of air at which flutter and instability occur. The airspeed of instability corresponding to a given air density is calculated using an iterative method.

ro[Hz] 1.52 1.41 2.87 3.08

REFERENCES

The ratio between dynamic pressure of nominal and robust flutter is 1.32 and the ratio between dynamic pressure of nominal and robust instability is 1.25 which are both above 1.2 which is a common safety factor used for flutter analysis.

Edwards J.W., 1.V. Breakwell and A.E. Bryson (1978). Active Flutter Control Using Generalized Unsteady Aerodynamic Theory. Journal of Guidance and Control, Volt, pp.32-40. Hitay O. and R.B. Glen (1994). H2lHoo Controller Design for a Two-Dimensional Thin Airfoil Flutte~ Suppression. Journal of Guidance, Control, and Dynamics, Volt7, pp.722-728. Horikawa H. and E.H. Dowell (1979). An Elementary Explanation of the Flutter Mechanism with Active Feedback Controls. Journal of Aircraft, Vo116, pp.725-732. Jeffrey S.V., M.B. Jeffrey, L.C. Robert and 1.B. Gary (1999). Comparison of 11- and HrSynthesis Controllers on an Experimental Typical Section. Journal of Guidance, Control, and Dynamics, Vo122, pp.278-285. Lind, R. and M. Brenner (1998). Robust Flutter Margin Analysis That Incorporates Flight Data. NASAlTP-1998-206543 . Lind, R. and M. Brenner (1999). Robust Aeroservoelastic Stability Analysis. SpringerVerlag, London. Ohta H. , A. Fujimori, P.N. Nikiforuk and M.M. Gupta, (1989). Active Flutter Suppression for Two-Dimensional Airfoils. Journal of Guidance, Volt2, pp.188-194. Zhou, K. and J.C. Doyle (1998). Essentials of Robust Control. Prentice-Hall Inc., New Jersey.

Effect of initial airspeed value; Examining equations (I) and (2), one can see that the equation of motion is a nonlinear function of airspeed, U, which is the reason why these equations are not parameterized around airspeed. Hence, the calculated dynamic pressure gives the density of air at which the instability occurs. An iterative search is performed to calculate the airspeed of instability at a given air density. Starting from a lower airspeed, dynamic pressure of instability is calculated. Then the candidate airspeed of instability is calculated by keeping the density of air fixed (at initial air density). Results of iterative search for airspeeds of instability at sea level are given in Table 3 for three steps. From the three iterations it can be seen that the results are converging. Continuing the iterations it is seen that at the 5th step the changes in the results drop below 1%. Table 3. Results of iterations for the airspeed of instability Vo

Step 2 4 .4447

Step 1 1 qbar [pal

N.F. R.F. N.I. R.I.

Vcol

N.F. R. F. N. I. R.I. Vo V eal

ID [Hzl

qbar [pal

Step 3 10.2515 ID[Hzl

52.8 1.37 1.50 38.4 38.4 15.8 1.23 1.39 86.1 2.88 2.91 91.7 64.4 3. I1 2.19 12.3 10.25 4.48 Nominal Flutter Robust Flutter Nominal Aeroservoelastic instability Robust Aeroservoelastic Instability initial Airspeed Calculated Airspeed of Instability

qbar [pal

64.4 44.4 85.4 69.3 10.64

ID [Hzl

1.61 1.44 2.86 3.09

723