Aircraft Stability & Control - Final Project

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James Carrillo AA 516 Final Project Report | Pg 2 / 9

Figure 1: 3 Views of NASAs Generic Winged-Cone Hypersonic Vehicle

2 Analytical Approach

I began the project with a thorough exploration of the available literature related to nonlinear and linearizedhypersonic vehicle models. I was disappointed to find that several of the more recent X-planes simply re-turned test results and/or lacked detailed values for adequate model buildup. I even contacted a leadingSenior Technical Fellow within the Boeing Co. to get access to the X-51 aerodynamic and engine models. Un-fortunately, due to ITAR regulations and program security clearance requirements, I was unable to ascertain

any X-51 data. The Winged-Cone generic hypersonic vehicle did ultimately provide adequate aerodynamicdata which allowed me to build a rough longitudinal nonlinear model. I found two articles which evaluatedthe model at different trim conditions, Mach 15 at 110,000 ft[10] and Mach 5 at 65,000 ft[3].

Mach 15 at 110,000 ft

The Mach 15 trim condition was a longitudinal model built using an inverse-square-law gravitational modeland the centripetal acceleration for a non-rotating earth. From this article, I was able to construct a nonlinearlongitudinal model with the following state vector:

X=

V h q T

(1)

I made a few key assumptions when building the model that did not coincide with the authors approach.I did not include additive uncertainties within the model and assumed all aerodynamic coefficient functionsreflected true conditions near the trim condition. The authors included these uncertainties to later demon-strate the robustness of their controller. This was not necessary for my model as I did not intend to design acontroller. I also did not include engine/throttle dynamics. The authors assumed the following second-orderdynamics for throttle setting:

= k1+ k2+ k3command (2)

These dynamics were not included as I could not find any values for k1, k2, or k3 and was not comfortableassuming values for them. My final assumption was a rigid and non-diminishing mass vehicle. These


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assumptions resulted in the following dynamic equations:

V =Tcos D

m

sin

r2 (3)

=L + Tsin

mV

( V2r)cos

V r2 (4)

h= V sin (5)= q (6)

q= Myy/Iyy (7)

where

L= 12

V2SCL (8)

D= 12

V2SCD (9)

T= 12

V2SCT (10)

Myy = 1

2V2Sc [CM() + CM(e) + CM(q)] (11)

r= h + RE (12)

The authors simplified the aerodynamic coefficients around the nominal cruising condition (M=15,V=15,060ft/s,=0 deg, and q=0 deg/s) which are illustrated as

CL= 0.6203 (13)

CD = 0.64502 + 0.0043378 + 0.003772 (14)

CT = 0.02576 (15)

CM() = 0.0352 + 0.036617 + 5.326106 (16)

CM(q) = (c/2V)q(6.7962 + 0.3015 0.2289) (17)

CM(e) = 0.0292(e ) (18)

The above equations allowed me to build a nonlinear model of a generic hypersonic aircraft. I then used trim

methods established during Problem Set 3 to linearize the system about the specified trim condition. I thencomputed the linearized system eigenvalues, damping coefficients, and natural frequency. I compared thesecomputed values to the authors referenced values to validate my model. I further simulated the nonlinearsystem response to elevator and throttle step and doublet inputs as shown in the Appendix.

Mach 5 at 65,000 ft

The Mach 5 trim condition was a longitudinal and lateral linearized model that assumed a flat-earth andconstant gravitational acceleration. This article also reported the linearized A,B,C, and D matrices as

A=

0.0016 0.0058 6238 31.2100 0.0194 0 0 0 0 00.1737 0 4851 4851 0 0 0 0 0 00.0220 0.0333 0.1845 0.0110 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0 00.0419 0.0026 12.5000 0 0.0204 0 0 0 0 0

0 0 0 0 0 0 0 0 0 00.0217 0.0036 0.0051 0 0 0 0.1014 0.0062 0.0361 0.9994

0 0 0 0 0 0 0 0 1 0.21380.0050 0.0789 0.0132 0 0 0 2.3000 0 0.0274 0.03310.0284 0.0447 0.0082 0 0 0 0.5095 0 0.0306 0.0241


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B =

36.9200 0.0029 0 0.01590 0 0 0

0.0027 0.0069 0 0.00240 0 0 00 0.0184 0 00 0 0 0

0 0 0.0091 0.00560 0 0 00 0 0.0564 0.00470 0 0.0718 0.0214

C=

0.0034 0.0524 12.4800 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 57.3000 0 0 0 0 0 0 0

D=

0 0.0107 0 0.01070 0 0 0

0 0 0 0

for the flight condition with the corresponding state vector

X=

V h q T s p RT

(19)

From these matrices, I calculated the corresponding eigenvalues, damping coefficients, and natural frequenciesfor comparison to the previous trim condition. The results of these findings are detailed below.

3 Results & Discussion

A stability comparison of the same generic hypersonic vehicle configuration at two distinct trim conditions

yielded interesting results. While one trim condition considered lateral and longitudinal conditions, theother only considered longitudinal dynamics. It has been shown for this configuration that the lateral andlongitudinal dynamics can be decoupled[2] for this configuration. Using this assumption, I generated aPole-Zero map of both systems to compare their stability characteristics. This comparison can be seen inFigure 2. The eigenvalues from the linearized nonlinear model I constructed were similar to those detailedin the corresponding article[10] and are shown in Table 1. It appears that at higher Mach, the vehicle

X Eigenvalue Damping Frequency 8.13e-19 -1.00e+00 8.13e-19 1.78e-07 + 2.95e-02i -6.05e-06 2.95e-02h 1.78e-07 - 2.95e-02i -6.05e-06 2.95e-02q -5.61e-02 + 1.05e+01i 5.36e-03 1.05e+01V -5.61e-02 - 1.05e+01i 5.36e-03 1.05e+01

Table 1: Eigenvalues, Damping Ratios, and Frequency of Linearized HSV, Mach 15, 110,000 ft

becomes slightly unstable in the Phugoid mode. This makes sense as most of the adaptive controls paperswere related to the higher Mach numbers. Both conditions had mildly unstable height (i.e. h) modes whichconsequently diverge during cruise flight and would require the vehicle have a state feedback control tomaintain stability. When comparing the lower Mach flight condition to the F-16 (Fig.3) its apparent thatthe higher speeds have lesser damping effects in the phugoid. One can also notice lateral instabilities inthe hypersonic vehicle case. Obviously this is akin to comparing apples and oranges and difficult to drawconcrete conclusions. It does however convey a general idea of the impact that an order of magnitude Mach


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Figure 2: Pole-Zero map comparing Phugoid trim stability at Mach 15, 110,000ft and Mach 5, 64,000ft

increase can have between aircraft stabilities. Other system responses to elevator and throttle step inputswere also performed and results can be seen in the Appendix. Analysis with respect to MIL-STDs was notconducted for these results, as there was insufficient information regarding pilot location. Also found in theAppendix are Simulink block diagrams detailing internal model structure.

References

[1] Joseph M Hank, James S Murphy, and Richard C Mutzman. The x-51a scramjet engine flight demon-stration program. AIAA Paper, 2540:2008, 2008.

[2] Shahriar Keshmiri. Nonlinear and linear longitudinal and lateral-directional dynamical model of air-breathing hypersonic vehicle. Inproceedings of 15th AIAA International Space Planes and HypersonicSystems and Technologies Conference, Dayton, O-hio, volume 28, 2008.

[3] Shahriar Keshmiri, Richard Colgren, and Maj Mirmirani. Six-dof modeling and simulation of a generichypersonic vehicle for control and navigation purposes. In AIAA Guidance, Navigation and ControlConference, 2006.

[4] Charles R. McClinton, Vincent L. Rausch, Luat T. Nguyen, and Joel R. Sitz. Preliminary x-43 flighttest results. Acta Astronautica, 57(2?8):266 276, 2005. ce:titleInfinite Possibilities Global Realities,Selected Proceedings of the 55th International Astronautical Federation Congress, Vancouver, Canada,4-8 October 2004/ce:title.


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Figure 3: Pole-Zero map comparing stabilities of a Generic Winged-Cone HSV linearized about Mach 5 at110,000ft and Linearized F-16 Model at 502 ft/s and 0 ft


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[5] Paul L. Moses, Vincent L. Rausch, Luat T. Nguyen, and Jeryl R. Hill.{NASA}hypersonic flight demon-strators?overview, status, and future plans. Acta Astronautica, 55(3?9):619 630, 2004. ce:titleNewOpportunities for Space. Selected Proceedings of the 54th International Astronautical FederationCongress/ce:title.

[6] Johnd Shaughnessy, Szane Pinckney, Johnd Mcminn, Christopheri Cruz, and Marie Kelley. Hypersonic

vehicle simulation model: winged-cone configuration. 1990.[7] Brian L Stevens and Frank L Lewis. Aircraft control and simulation. 2003.

[8] Randall T. Voland, Lawrence D. Huebner, and Charles R. McClinton. X-43a hypersonic vehicle tech-nology development. Acta Astronautica, 59(1?5):181 191, 2006. ce:titleSpace for Inspiration of Hu-mankind, Selected Proceedings of the 56th International Astronautical Federation Congress, Fukuoka,Japan, 17-21 October 2005/ce:title xocs:full-nameSpace for Inspiration of Humankind, Selected Pro-ceedings of the 56th International Astronautical Federation Congress, Fukuoka, Japan, 17-21 October2005/xocs:full-name.

[9] Steven H Walker, J Sherk, Dale Shell, Ronald Schena, J Bergmann, and Jonathan Gladbach. Thedarpa/af falcon program: the hypersonic technology vehicle# 2 (htv-2) flight demonstration phase.AIAA Paper, 2539:2008, 2008.

[10] Qian Wang and Robert F Stengel. Robust nonlinear control of a hypersonic aircraft. Journal ofGuidance, Control, and Dynamics, 23(4):577585, 2000.

[11] Daniel Philip Wiese, Anuradha M Annaswamy, Jonathan A Muse, and Michael A Bolender. Adaptivecontrol of a generic hypersonic vehicle. PhD thesis, Massachusetts Institute of Technology, 2013.

Appendix

Figure 4: State responses to elevator step input @ 1 sec.


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Figure 5: State responses to throttle step input @ 1 sec.

Figure 6: State responses to elevator doublet


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Figure 7: Step input response of linearized GHSV model Mach 15 at 110,000 ft


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Figure 8: Differential State Equations


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Figure 9: Aero Coefficients and body forces


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Figure 10: Internal model structure


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Figure 11: State Block containing all the fun

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Aircraft Stability & Control - Final Project