Autonomous Reconfigurable Control Allocation(ARCA) for Reusable Launch Vehicles∗

A. S. Hodel, member AIAA†

Ronnie Callahan‡

Introduction

The role of control allocation (CA) in modernaerospace vehicles is to compute a command vec-tor δc ∈ IRna that corresponding to commandedor desired body-frame moments (torques) τc =[

L M N]T

to the vehicle, compensating for

and/or responding to inaccuracies in off-line nom-inal control allocation calculations, effector fail-ures and/or degradations (reduced effectivness),or effector limitations (rate/position saturation).The command vector δc may govern the behaviorof, e.g., aerosurfaces, reaction thrusters, enginegimbals and/or thrust vectoring. Typically, theindividual moments generated in response to eachof the na commands does not lie strictly in theroll, pitch, or yaw axes, and so a common practiceis to group or gang effectors so that a one-to-one mapping from torque commands τc to effectorcommands δc may be achieved in an off-line com-puted CA function.

We shall assume the existence of an off-linecomputed nominal linear affine CA function

δc = F (x)τc + δ0(x) (1)

where τc is the commanded torque vector, x is avehicle state vector, δ0 is a trim (neutral torque)vector and F (x) is a matrix of nominal controlallocation gains. One may interpret the columnsof F (x) as a set set of gains defining “ganged”

∗This work was supported in part by NASA contractNAS8-01105 and in part by a NASA Graduate StudentResearch Fellowship.

†a.s.hodel@eng.auburn.edu, Dept. Elect. & Comp.Eng., 200 Broun Hall, Auburn University , Auburn, AL36849-5201, corresponding author

‡a.s.hodel@eng.auburn.edu

Copyright c© 2002 by the American Institute of Aeronauticsand Astronautics, Inc. No copyright is asserted in the UnitedStates under Title 17, U.S. Code. The U.S. Government hasa royalty-free license to exercise all rights under the copyrightclaimed herein for Governmental Purposes. All other rights arereserved by the copyright owner.

effectors for each control axis. The resulting nom-inal autopilot/control allocation block diagram isshown in Figure 1. The vector τb in Figure 1 refersto the body torques induced on the vehicle by theeffectors. Ideally, the control allocation matrixF (x) would be chosen to be the pseudo-inverseG(x)† of the Jacobian matrix

G(x) =[

∂τb,i

∂δj

]∣

∣

∣

x∈ IR3×na

where na is the number of effectors; that is, wewish to design the control allocation matrix F (x)such that

G(x)F (x) =

1 0 00 1 00 0 1

(2)

so that the induced body-frame vehicle torquesτb match the commanded body torques τc. How-ever, due to sensor inaccuracy, modeling errors,and data compression in the allocation function,the ideal condition (2) cannot be achieved. Assuch, semidefinite programming techniques1,2 areused to design control allocation matrices F (x)that achieve

‖G(x)F (x)− I‖ < 1− γ(x)

for some positive constant γ to achieve the “bestpossible” allocation given off-line data.

The linear affine nominal control allocation law(1) by itself is inadequate for the control controlallocation problem for four reasons:

1. It fails to respond to torque allocation errorsthat can be detected on-line.

2. It fails to take into account saturation issues.

3. It fails to respond to on-line detected failuresin effectors, and

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Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

τbSMC

autopilotCA actuators

vehicle

EOM

NAV

guidance

commands

attutideangular rates

δcτc

Fig. 1 Nominal autopilot/control allocation block diagram.

4. It fails to provide a framework to work withdiscrete-valued (on-off) effectors such as re-action thrusters.

Dynamic control allocation? may be used to com-pensate for torque allocation errors detected on-line. DCA treats torque allocation error as aunknown additive uncertainty in the system Ja-cobians

Gtrue(x) = G(x) + ∆G(x)

where ∆G is an unknown gain that respects thecondition

‖Gtrue(x)F (x)− I‖ < 1

over all operating conditions. If this condition isnot met, then the CA problem is greatly compli-cated, requiring the use of on-line system identifi-cation.3–6 Discrete-valued effectors may be dealtwith by either using pulse-width modulation toemulate continuous valued effectors or, where ef-fector structure will not permit the use of PWM,one may use these effectors as “back-up” to thecontinuous valued effectors as in.7 We address inthis paper items 2 and 3 above by autonomousreconfigurable control allocation (ARCA).

An optimal solution of a constrained control al-location problem involves the solution of convexprogramming problems.8–10 Some heuristic ap-proximate allocation solutions are presented in11

such as computation of the attainable momentsubset, generalized inverses, and daisy chaining.Alternatively, adaptive control may be used intandem with linear system theory techniques inan attempt to avoid and/or compensate for ef-fector saturation.12 In this paper we build onthe work of,9,11,13 and14 to present an on-line au-tonomous reconfigurable control allocation tech-nique that is computationally tractable and prac-tical for use in closed loop with a robust autopilot

(attitude control law) such as sliding mode con-trol,15,16.17 Our technique makes use of a fastquadratic programming iteration step so that theeffector command δc results in a vehicle bodytorque τb that tracks the commanded torques τcwhen they lie within the attainable moment setor else approximates the commanded torques ina least-squares sense when they are not in theattainable moment set (the underlying quadraticprogramming problem is infeasible).

Autonomous Reconfigurable ControlAllocation (ARCA)

Reconfigurable control refers to the ability ofa control allocation law to continue to maintaintracking of the moment command τc in the face ofeffector failures/degradation. In the short term,effector saturation is indistinguishable from ef-fector failure, since in both cases an additionalconstraint is entered into the control allocationlaw. However, in the case of an effector failure, theconstraint is permanent and thus requires coor-dinated treatment between the control allocationlaw, the attitude control law, and the guidancelaw.

Problem statement

We formally describe the ARCA problem asfollows. We shall denote the sequence of autopi-lot torque commands as τc(k) and the sequence ofCA generated effector command vectors as δc(k).In order to accomodate effector rate limits, weshall compute the effector command vectors re-cursively, i.e.,

δc(k) = δc(k − 1) + δe(k)

where δe(k) is an update to the previous effectorcommand vector δc(k − 1). We shall omit thedependence on the time index k where it is clear

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by context. We shall make use of the followingdefinitions:

Definition 1 The effector status vector δstat(t)has entries in the range of [0, 1] where δstat,i = 0reflects complete failure (effector i has no impacton vehicle body torques) and δstat,i = 1 reflectsnonminal operation of effector i. 2

Remark 1 We shall assume that the the effec-tor status vector δstat is made available to the CAmodule by, e.g., a vehicle health monitoring sys-tem or an on-line system identification module.

Definition 2 The effective effector Jacobian

G(x) is the system Jacobian

G(x)∆=[

∂τb

∂δ

]

evaluated under current operating conditions, in-cluding effects of effector degradation and failure.2

Remark 2 Notice that the effective effector Ja-cobian G(x) is unknown prior to flight time. Ifthe nominal system Jacobians G(x) ≈ F (x)† areavailable, then G(x) = G(x)diag(δstat).

Definition 3 Given a sampling interval T , thecurrent effector position δ and vectors of effectormaximum position values δ+, minimum effectorposition values δ−, and maximum effector ratesδmax, the next-step feasible set ∆f (δc) is definedas

∆f (δc)∆=

{

δe : δ− ≤ δc + δe ≤ δ+

and |δe − δ| ≤ T δmax

}

Remark 3 The set ∆f (δ(kT )) is the set of legalnext-step effector commands δc((k + 1)T ) givencurrent effector position values δ(k).

Denoting δc = δc(k − 1), δe(k), the reconfig-urable control allocation problem can thus be ex-pressed as the quadratic programming problem18

min (δc + δe)TW1 (δc + δe) + δe

TW2δe(1)

subject to G(x)δe = ∆τ

and δe ∈ ∆f (δc)

where W1 is a position penalty on the commandvector δc(k), W2 is a rate penalty on the change

in command vector δe, and ∆τ is a torque com-mand update, usually (but not always) defined as∆τ(k) = τc(k)− τc(k−1). Exceptions to this ruleare discussed below.

In the case where the quadratic programmingproblem (1) is feasible, then the minimizationsearches for an update δe that matches the com-manded torque while reducing the magnitude(cost) of the control command. Conversely, ifthe ideal torque command is not feasible, i.e., nosolution δe ∈ ∆f (δc) exists to the equality con-straint G(x)δe = ∆τ , then it is necessary to relaxthe equality constraint and instead find a ver-tex (extreme point18) of the feasible set ∆f (∆c)

that minimizes the norm ‖τerr(k)‖ where τe∆=

G(x)δe−∆τ . In this case, the next value of ∆τ isselected to reflect both the update to the torquecommand τc and the “unallocated” torque τe fromthe previous iteration.

Brute force application of standard quadraticprogramming techniques may not be desirable inthe ARCA problem for the following reasons.

1. Computation time: solution of the quadraticprogramming problem can requires several it-erations, each requiring solution of a linearsystem of equations Ax = b of dimension upto na × na.

2. Repetition: the quadratic programmingproblem to be solved in our application issolved repeatedly for problems and condi-tions that do not vary greatly from one sam-ple time to the next. For example, duringtimes of highly aggressive maneuvers, it islikely that the same effectors are saturatedfrom one time step to the next. The sim-ilarity of these problems is not exploited bystandard quadratic programming techniques.

3. Singular dual solution: because the numberof constraints is always 4 times the number ofunknowns, the dual problem19 used to solvefor the corresponding Lagrange multiplier ina non-negative least squares problem is sin-gular, which requires at the least an increasein the dimension of the problem solution viathe use of “slack” variables.

Fast QP solution to ARCA

Given the concerns raised above, we proposethe use of a fast QP solution method that differs

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from standard primal problem QP solvers in twoways:

1. The solution (active constraint set18) of theprevious problem is used as a starting pointof the current problem.

2. The number of QP iterations is limited sothat computational burden is reduced.

We consider here the equivalent QP problemusing the notation of.18

minx

J(x)∆= xTQx+ cTx (2)

subject to Ax = b

x− ≤ x ≤ x+

The unknowns x in problem (2) correspond tothe effector command deviations δe in the ARCAalgorithm description. We shall proceed on thefollowing assumptions:

1. x− ≤ 0 ≤ x+; i.e., the previous effectorcommand vector satisfies effector constraintlimits.

This assumption may be violated in the caseof, e.g., initial transients due to engine fail-ure, where additional constraints on differ-ential thrust can be imposed upon failure.Nevertheless, the above assumption will bereasonable in all cases where additional con-straints are not suddenly imposed on effectorcommand behavior.

2. The torque command issued by the autocom-mander is feasible. This condition is can bemet through the use of on-line computationof a local attainable moment set.?

Based on the above assumptions, we may solvethe QP problem (2) as illustrated in Figure 2. Theinitial point x(0) by assumption satisfies the in-equality constraints x− ≤ x(0) ≤ x+, but maynot satisfy the equality constraint Ax = b due tovariations in the (effective) Jacobian and torquecommand from one time step to the next. Thepoint w(0) is computed that minimizes the opti-mization

minw(0)

w(0)TQw(0) + cTw(0) (3)

subject to Aw(0) = b

w(0)

x1

x2

x3

x−1x+1

x−2

x+2

x1 = x−1

Ax = b

x(0)

Fig. 2 Quadratic programming problem (2)solution method for the ARCA problem. Thefeasible solution set is denoted by the rectangleon the interior of the hyperplane Ax = b.

with A = A, b = b. As illustrated in the fig-ure, w(0) will satisfy the equality constraints, butmay not satisfy the inequality constraints. Wetherefore choose our next solution value x(1) =(1−α)x(0) +αw(0) where w(0) is chosen so thatx(1) lies at the edge of the feasible set for the in-equality constraints. We then append a row toA and b corresponding to the new “active con-straint,” x1 = x−1 in the case of the figure, andsolve the minimization (3) again, now constrainedto the intersection of the hyperplanes Ax = b andx1 = x−1 . This process continues until the iteratew(k) is in the feasible set of problem (2). (Such asolution exists by assumption.)

Remark 4 Because the system Jacobians arecontinuous (except in the case of sudden failure),the active constraints from one iteration to thenext will likely not change often. As a result, theabove algorithm may be easily modified to solvethe minimization (3) at most once per control stepwhile evaluating revisions to the active constraintset by projecting the gradient

∂J

∂x= 2Qx+ c

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onto the null space of A

v =(

I −A†A)

(2Qx+ c)

and comparing the signs of the resulting gradi-ent search direction v with the active constraintsimposed on x from the previous iteration.

Simulation Results and Conclusions

Our initial tests of our fast QP solution methodon artificially generated problems are very encour-aging, but raise questions regarding convergencebehavior. Closed-loop results using the ARCA al-gorithm in closed loop with a high-fidelity modelof the X-33 experimental launch vehicle were notavailable at press time for this paper. However,we illustrate the utility of the quadratic program-ming method in the following two simulated ex-amples.

The first simulation example was an artificalproblem with a time varying system Jacobian thatwas generated in octave

∗ as

G(t) = G0 +∆G1 sin(ω1t)−∆G2 sin(ω2t)

where the nominal coefficient values are shown inTable 1. Minimum and maximum effector com-mand values (unitless) were selected as

δ− =[

−1 −1 −2 −2 −2 −2]T

δ+ =[

3 3 5 5 2 2]T

with rate limits

δc,max =[

0.5 0.5 1 1 0.8 0.8]T

Open loop torque commands (unitless) are shownin Figure 3. At t = 10s two faults were artificallyintroduced: effector 6 lost 99% of its effective-ness (i.e., column 6 of G(t) is scaled by 0.01 after10s) and effector 5 experienced a stuck-at fault(column 5 of G(t) is identically 0). The resultingsimulated effector commands are shown in Figure4 with the corresponding allocation error (differ-ence between induced moments and commandedmoments) magnitude shown in Figure 5. Similarbehavior was observed in a number of simulationsbased on a linearized aerodynamic model of theX-33 vehicle.

∗www.octave.org

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sion in Control Allocation,” Proceedings of AIAA Guid-

ance, Navigation, and Control Conference and Exhibit ,American Institute of Aeronautics and Astronautics, Den-ver, Colorado, August 14-17 2000, Paper AIAA-2000-4154.

2Boyd, S., El Ghaoui, L., Feron, E., and Balakrish-nan, V., Linear Matrix Inequalities in System and Control

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“System Identification for Adaptive and ReconfigurableControl,” Journal of Guidance, Control, and Dynamics,Vol. 18, No. 3, May-June 1995, pp. 516–524.

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tion, and Control Conference, 1998.5Chen, H. F. and Guo, L., “Convergence rate of least

squares identification and adaptive control for stochas-tic systems,” International J. of Control , Vol. 44, 1986,pp. 1459–1476.

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on Automatic Control , Vol. 36, No. 10, October 1991,pp. 1163–1176.

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namics, 2002, Submitted for publication.8Buffington, J. M., “Tailless Aircraft Control Alloca-

tion,” Proceedings of the 1997 AIAA Guidance, Naviga-

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Journal of Guidance, Control, and Dynamics, Vol. 16,No. 4, August 1993, pp. 717–725.

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ceedings of the AIAA Guidance, Navigation, and Control

Conference, Boston, 1998, Paper No. 98-4108.11Bordignon, K. A. and Durham, W. C., “Closed-Form

Solutions to Constrained Control Allocation Problem,”Journal of Guidance, Control, and Dynamics, Vol. 18,No. 5, September-October 1995, pp. 1000–1007.

12Chaoui, F. Z., Giri, F., Dugard, L., Dion, J. M., andM’saad, M., “Adaptive Tracking with Saturating Input andController Integral Action,” IEEE Transactions on Auto-

matic Control , Vol. 43, No. 11, Nov. 1998, pp. 1368–1643.13Bodson, M. and Pohlchuck, W. A., “Command Lim-

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location For Systems With Coplanar Controls,” AIAA

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G0 =

1 2 3 4 5 6−5 6 −7 8 −8 9−1 −2 3 4 −5 6

∆G1 =

1.604683 0.962070 −0.085787 0.838649 0.419048 −0.241935−1.270698 0.785712 −0.212199 −0.537871 −3.579044 0.0561890.593470 −0.680706 0.344545 −2.400190 0.519556 1.035582

∆G2 =

−0.974363 −0.196414 1.06404 0.421511 −0.362049 0.8025521.12867 0.370901 0.211152 1.29587 0.731121 0.06175060.528389 0.717545 0.10383 1.2435 0.238306 0.426937

ω1 = 0.55252

ω2 = 0.86151

Table 1 Table of coefficients for quadratic programming control allocation test.

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16

plot of the torque commands

line 1line 2line 3

Fig. 3 Moment commands (units=Nm) for artificial test problem G(t) = G0 + ∆G1 sin(ω1t) −∆G2 sin(ω2t) where nominal coefficient values are given in Table 1. The horizontal axis (time) is inunits of seconds.

.16Boskovic, J. D., Li, S.-M., and Mehra, R. K., “Glob-

ally Stable Adaptive Tracking Control Design for Space-craft under Input Saturation,” Conference on Decision and

Control , December 1999, pp. 1952–1957.

17Shtessel, Y., Hall, C. E., and Jackson, M., “ReusableLaunch Vehicle Control in Multiple Time Scale SlidingModes,” Proceedings of AIAA Guidance, Navigation, and

Control Conference, No. AIAA-2000-4155, AIAA, Denver,Co, August 14-17 2000.

18Luenberger, D. G., Introduction to Linear and Non-

linear Programming , Addison-Wesley, Reading, MA, 2nded., 1984.

19Luenberger, D. G., Optimization by Vector Space

Methods, Wiley and Sons, Inc., New York, NY, 1969.

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-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14 16

plot of effector position 1

effector 1low limit 1

high limit 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16

plot of effector position 2

effector 2low limit 2

high limit 2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10 12 14 16

plot of effector position 3

effector 3low limit 3

high limit 3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16

plot of effector position 4

effector 4low limit 4

high limit 4

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 2 4 6 8 10 12 14 16

plot of effector position 5

effector 5low limit 5

high limit 5

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8 10 12 14 16

plot of effector position 6

effector 6low limit 6

high limit 6

Fig. 4 Effector commands (unitless) in artificial model test. Horizontal axis values are simulatedtime in units of seconds. At t = 10s effector 5 is failed with a stuck-at fault and effector 6 is failedwith a 99% loss of control authority. Observe that effector 6 is driven to its minimum limit valuewhile the stuck-at fault of effector 5 is recognized by the QP problem due to the correspondingzero coefficients in the system Jacobian.

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0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14 16

norm of the error || Gt*x - tau_c ||

alloc err

Fig. 5 Allocation error ‖G(t)δc(t)− τc(t)‖ between the induced moments resulting from effectorcommands δc(t) and the commanded moments τc(t) for the artificial model test. Observe thestartup transient and failure transients at t = 0s, t = 10s. The loss of effector rate authority att = 10s results in an inability to track the moment commands τc as seen in the allocation errorafter approximately t = 11s.

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