NOV.-DEC. 1990 ENGINEERING NOTES 1163

ConclusionsIn this Note, we consider the stabilization of large, flexible

structures with mode residualization. A new frequency do-main stability condition for a high-order structure controlledby a reduced-order observer-based controller is derived. Aslinear parametric perturbations are involved in the controlledstructure, an additional stability condition that guaranteesrobust stability for the perturbed structure is also presented.

Appendix: Proof of Theorem 1Taking Laplace transform of the state-space representations

of Eq. (5) yields(Ala)(Alb)(Ale)Y(s) = C{Z{(s) + C2X2(s)

where *i(0) and jc2(0) are, respectively, the initial states of z\(t)and x2(t). Substituting Eq. (Alb) into Eq. (Ala), and aftersome manipulations, we get

(A2)

2Bossi, J. A., and Price, G. A., "A Flexible Structure ControllerDesign Method Using Mode Residualization and Output Feedback,"Journal of Guidance, Control, and Dynamics, Vol. 7, 1984, pp.125-127.3Lynch, P. J., and Banda, S. S., "Active Control for VibrationDamping," Large Space Structures: Dynamics and Control, edited byS. N. Atluri, Springer, New York, 1988.

4Vidyasagar, M., Control System Synthesis: A Factorization Ap-proach, MIT Press, Cambridge, MA, 1985.5Desoer, C. A., and Vidyasagar, M., Feedback Systems: Input-Output Properties, Academic, New York, 1975.6Delsarte, P., Genin, Y., and Kamp, Y., "Schur Parametrization ofPositive Definite Block-Toeplitz Systems," SI AM Journal on AppliedMathematics, Vol. 36, 1979, pp. 34-46.7Qiu, L., and Davison, E. J., "New Perturbation Bounds for theRobust Stability of Linear State Space Modes," Proceedings of the25th IEEE Conference on Decision and Control, Athens, Dec. 1986,pp. 751-755.8Bar-Kana, L, Kaufman, H., and Balas, M., "Model ReferenceAdaptive Control of Large Structural Systems," Journal of Guid-ance, Control, and Dynamics, Vol. 6, 1983, pp. 112-118.

Effect of Thrust/Speed Dependenceon Long-Period Dynamics in

Supersonic Flightwhere K(s)^sI-Hn-Hl2(sf-H22)~lH2l. K(s) can be fur-ther expressed as

-Hl2(sI-H22)-lH2l

(where & = HU-H12H22 1H21)

(A3)Substituting Eq. (A3) into Eq. (A2), we obtain

(A4)Since

(sI-H22)-ltS"

by Lemmas 1 and 2, we therefore know that if

a[/co(/a)/-A)-1//12//221(/co/~//22)~1//2i]0holds, then z\(t), the inverse Laplace transform of Z\(s) in Eq.(A4), is asymptotically stable. From Eq. (Alb), since(5/-jF/22)"ISll2x/l2 and z,(f) and x2(t) have been stabilized.Since both z\(t) and x2(t) are stabilized, obviously, y(t) is alsoasymptotically stable.

References^akahashi, M., and Slater, G. L., "Design of a Flutter Mode

Controller Using Positive Real Feedback," Journal of Guidance,Control, and Dynamics, Vol. 9, 1986, pp. 339-345.

A (s)BCD

u

CL(xCmCmCmaCII|Acgh/vkpMnuqSsTuV

jca.67fj,pPha

Gottfried Sachs*Technische Universitat Munchen,

Munich, Germany

Nomenclature= coefficient matrix of the homogenous system- scaling matrix for control inputs= drag coefficient= dCD/d(u/V0)=dCD/da= lift coefficient

= dCL/doi= pitching moment coefficient= pitch damping, = 2dCm /d(qc/ K0)=dCm/da=2dCm/d(ac/V)= mean aerodynamic chord= acceleration due to gravity- altitude= radius of gyration

= Mach number= thrust/speed dependence, =(V0/To)dT/du- dynamic pressure, =(p/2)K2= reference area= Laplace operator= thrust= speed perturbation= airspeed= variable vector= angle of attack= thrust setting= relative mass parameter, =2m/(pSc)= air density= density gradient, =(l/p0) dp/d/z= real part of complex variable

Received April 24, 1989; revision received June 7, 1989. Copyright 1989 by the American Institute of Aeronautics and Astronautics,Inc. All rights reserved.

* Director, Ins t i tu te of Flight Mechanics and Flight Control. Associ-ate Fellow A1AA.

1164 J. GUIDANCE J. GUIDANCE

r = reference time, =c/F0oj = imaginary part of complex variableco,7 = natural frequency

Introduction

IT is well known that thrust changes caused by speed pertur-bations exert a significant influence on the phugoid in sub-sonic flight.1'2 This effect may be stable or unstable, depend-ing on the thrust slope with speed. A positive thrust slopeyields a destabilization whereas the opposite is true for anegative thrust slope. According to the effect of inherentthrust changes with speed, a well-known means for improvingthe dynamic stability of aircraft is the autothrottle, which is abasic control concept using speed feedback to the throttle.1'2This particularly concerns the long-period dynamics and pro-vides the ability for an efficient stabilization of the phugoidmode of motion. In Fig. 1, it is shown in which way thephugoid eigenvalues are changed and how efficient the autothrottle is as a stabilization device for subsonic flight.

In supersonic flight, the long-term stability may show onlymarginal characteristics (e.g., see Ref. 3). This particularlyconcerns the phugoid. Thus, a particular need can exist for astability augmentation system for which an autothrottle maybe considered as an efficient candidate due to the good experi-ence gained with this device. Furthermore, advanced air-breathing propulsion systems for supersonic aircraft showsignificant variations in thrust with Mach number.4 These

The results of this Note may also be of interest as regardsthe experience gained with an autothrottle control system ap-plied to a Mach 3 aircraft.5 The fact that this system has beensuccessfully operated is not in contradiction with the results ofthe present Note because it has been applied in combinationwith other feedback loops. Rather, the effects that an au-tothrottle control provides when operating alone may help oneto understand why it can be applied successfully in combina-tion with other feedback loops. Furthermore, the autothrottlecontrol is an important basic feedback concept the characteris-tics of which should be understood. This particularly holds forthe case considered, which shows a significant reduction ofautothrottle effectiveness for supersonic flight when com-pared with the well-known and efficient autothrottle charac-teristics in subsonic flight.

Aircraft DynamicsIn supersonic flight, aerodynamic forces and moments due

to altitude perturbations (density changes) exert an influenceon dynamic stability.4'5 Accordingly, they have to be ac-counted for in modeling aircraft dynamics. The resulting lin-earized equations of motion for perturbations from a horizon-tal reference flight condition may be expressed as

A(s)x(s)= -BdT(s) (la)where

2CL + CLL L0

Co*- Vi(sT)2(iy/c)2

+ sr(Cmq + CW(.)/2 +

-0.2

OPEN LOOP ZERO

a [1/s]

srCL- fJi(ST)2 + CPhCL

(sr)2Cmq/2

(Ic)

inherent thrust variations are generally considered to have asignificant influence on the phugoid.

It is the purpose of this Note to show that thrust variationswith speed (or Mach number) have practically no effect on thephugoid in supersonic flight. This particularly concerns a pos-itive thrust slope (dT/du >0 or d7YdM>0), which is usuallyconsidered as strongly destabilizing the phugoid. It will beshown that this effect is not existent. Rather, the phugoid isnot influenced at all in supersonic flight. Similarly, it will beshown that an autothrottle provides no means for improvingphugoid stability.

AT/TC

(Id)

Fig. 1 Effect of speed feedback to thrust on long-term dynamics insubsonic flight (M = 0.25).

The effect of thrust/speed dependence is treated by apply-ing the root locus technique. Therefore, thrust changes due tospeed perturbations may be understood as produced by ahypothetical control loop using speed feedback to thrust (asindicated by the block diagram in the upper part of Fig. 3).This reasoning may be applied to engine characteristics show-ing inherent thrust changes with speed or to an autothrottlesystem artificially introducing thrust changes.

The root locus of such a control loop is determined by itsopen-loop poles and zeros. The open-loop poles represent theeigenvalues of an aircraft, showing no thrust changes withspeed. In supersonic flight, there are five open-loop poles,which are associated with three modes of motion. They maybe identified as 1) short-period mode (two complex poles: aa,o>,, ), 2) phugoid (two complex poles of small magnitude:aplwn ), and 3) height mode (real pole of small magnitude:

It is not necessary for the following considerations to pre-sent any details or approximate expressions for the poles.Rather, their relative location to the open-loop zeros is ofinterest. There are two complex pairs of zeros (a\, wni and a2,w,,2) in supersonic flight (see the Appendix). Their relativelocation to the open-loop poles may be expressed as

o\ ~ a(v, ww. co,, (2)

(3)According to Eq. (2), the poles (a(V, a>w

NOV.-DEC. 1990 ENGINEERING NOTES 1165

1.0n

0.5-

/MAXIMUM.MINIMUM

| 0

1166 J. GUIDANCE J. GUIDANCE

AppendixBy expanding the A matrix with the B matrix as its first

column, the following approximate expressions for the zerosmay be derived (neglecting unimportant contributions)

CW.)/2]a, - K0/(2Mc)[CL - (c/iy)\Cm + Cm/2] (Al)

a2 0 (A2)Equations (Al) agree with the well-known approximations forthe short-period mode (aa, un ), thus resulting in Eqs. (2).With regard to Eqs. (A2), the following phugoid approxima-tions resulting from expanding the A matrix may be consid-ered

(A3)

Combining these expressions with Eqs. (A2), the correlationdescribed by Eqs. (2) results.

References'Roskam, J., "Airplane Flight Dynamics and Automatic Flight

Controls," Roskam Aviation and Engineering Corp., Lawrence, KS,Pts. I, II, 1979/82.

2Brockhaus, R., Flugregelung II, R. Oldenbourg Verlag, Munich,1979.

3Powers, B. G., "Phugoid Characteristics of a YF-12 Airplane withVariable-Geometry Inlets Obtained in Flight Tests at a Mach Numberof 2.9," NASA TP 1107, 1977.

4Berry, D. T., "Longitudinal Long-Period Dynamics of AerospaceCraft," Proceedings of the AIAA Atmospheric Flight MechanicsConference, AIAA, Washington, DC, 1988, pp. 254-264.

5Gilyard, G. B., and Burken, J. J., "Development and Flight TestResults of an Autothrottle Control System at Mach 3 Cruise," NASATP 1621, 1980.

Generalized Gradient Algorithm forTrajectory Optimization

Restoration Algorithm (SGRA),1 which can solve general tra-jectory optimization problems. Goh and Teo2 proposed aunified control parameterization approach that converts anoptimal control problem into a parameter optimization prob-lem. These algorithms employ a sequence of cycles, each cyclehaving two phases. One phase improves the performance in-dex while the other reduces the constraint violations.

In this Note, a generalized gradient is found that improvesthe performance index and reduces the constraints at the sametime.

Problem StatementThe general problem that will be considered has the follow-

ing form:

Pmin / = [JC(1),7T] = 0 (3)

S(x, u, TT, T) = 0 (4)where u is the m x 1 control vector, x is the n x 1 state vector,TT is the p x 1 parameter vector, r is the normalized time in(0,1), ^ is the q x 1 terminal constraint vector, and S is ther x 1 path equality constraint vector. A meaningful problemrequires: r