JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Vol. 29, No. 3, May–June 2006

Fault-Tolerant Structured Adaptive Model Inversion Control

Monish D. Tandale∗ and John Valasek†

Texas A&M University, College Station, Texas 77843-3141

An adaptive dynamic inversion control formulation is presented that takes advantage of the inherent dynamicstructure of the state-space description of a large class of systems. The formulations impose the exact kinematicdifferential equations, thereby restricting the adaptation process that compensates for model errors to the accel-eration level. The utility of this formulation is demonstrated for the problem of fault tolerance to actuator failureson redundantly actuated systems. The approach incorporates an actuator failure model in the controller formu-lation, so that actuator failure can be identified as a change in the parameters of the failure model. Tracking ofreference trajectories is imposed, and initial error conditions and structured parametric uncertainties are incor-porated explicitly in both the plant parameters and the control influence matrix. A numerical example consistingof a nonlinear model of an F-16 type aircraft with thrust vectoring is presented. Simulation results show thatthe fault-tolerant adaptive controller is capable of simultaneously handling parametric uncertainties, large initialcondition errors, and actuator failures while maintaining adequate tracking performance.

I. Introduction

I N recent years, there has been much interest in the developmentof reconfigurable control systems that can accommodate actua-

tor failures without compromising mission integrity. There has beensubstantial progress in the development of real-time failure detectionand isolation algorithms, system identification after failure, and con-trol reconfiguration techniques in aerospace applications. In Ref. 1,a survey of various reconfigurable flight control methodologies ispresented and it is shown that most traditional reconfiguration flightcontrol approaches rely on failure detection and isolation. The com-plexity of such a system with this feature grows with the increase inthe number of failures, and there tends to be a significant possibilityof false alarms.1,2 A different approach to reconfigurable flight con-trol is based on adaptive control theory, in which the adaptive controlstructure implicitly reconfigures the control law using adaptive es-timates of the altered dynamics after failure.3 In Ref. 3, an adaptivecontrol scheme is presented that uses a linear approximation of theplant model to compute the control, and a neural network basedadaptive control law for flight reconfiguration has been developedand successfully flight tested.4−6 A robust fault-tolerant controllerhas also been developed to reject state-dependent disturbances.7

The approach presented in this paper uses a structured nonlinearadaptive dynamic inversion control methodology. Instead of usingan explicit failure detection and isolation algorithm, this method-ology is based on the adaptive control theory where the controlleris constantly updating itself. This methodology is applicable to ageneral class of nonlinear systems that are affine in the control withuncertain parameters appearing linearly. Fault-tolerance capabilityis introduced by incorporating a failure model in the controller sothat a failure can be identified and compensated for by a change inthe parameters of the failure model.

First, model reference adaptive control, structured model ref-erence adaptive control, and structured adaptive model inversion

Presented as Paper 2002-4437 at the AIAA Guidance, Navigation, andControl Conference, Monterey, CA, 5–8 August 2002; received 22 Decem-ber 2004; accepted for publication 11 January 2005. Copyright c© 2005 byMonish D. Tandale and John Valasek. Published by the American Instituteof Aeronautics and Astronautics, Inc., with permission. Copies of this papermay be made for personal or internal use, on condition that the copier paythe $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose-wood Drive, Danvers, MA 01923; include the code 0731-5090/06 $10.00 incorrespondence with the CCC.

∗Graduate Research Assistant, Flight Simulation Laboratory, AerospaceEngineering Department; monish@neo.tamu.edu. Student Member AIAA.

†Associate Professor & Director, Flight Simulation Laboratory,Aerospace Engineering Department; valasek@aero.tamu.edu. AssociateFellow AIAA.

control are introduced. Then, a failure model that can handle a widerange of failures is developed, followed by the mathematical devel-opment and stability analysis of the fault-tolerant structured adap-tive model inversion (SAMI) controller. A nonlinear six-degree-of-freedom simulation of an F-16 type aircraft with thrust vectoring isdeveloped. Finally, a numerical example is presented that demon-strates fault-tolerant performance of the controller in the presenceof actuator failure, using the simulation model developed.

All dynamic systems that exist in practice have uncertain param-eters, which may be constant with respect to time, or may vary withrespect to the changing environment in which the dynamic systemis functioning. Because of this variation, performance of the systemmay degrade if a fixed parameter controller is used. However, per-formance can be ensured if the controller parameters can be adjustedto adapt to the changes in the plant parameters. The basic premiseof adaptive control is that the controller parameters are variable,and there exists a mechanism for updating the parameters of thesystem real time, based on the signals of the system. In the broadclass of adaptive control there are several variants, some of whichare described here. In model reference adaptive control (MRAC),8,9

a reference model is specified that in turn specifies the desired dy-namics that the plant is supposed to follow. A feedback controllerexists for the plant as usual, but the parameters of this controller arevarying. The adaptation mechanism for the controller parameters isdriven by the error between the output of the actual plant and thereference model (Fig. 1). The adaptation mechanism is usually syn-thesized by the use of Lyapunov stability theory (see Ref. 10). Thebasic theory of MRAC is well developed due to significant contri-butions by Narendra and Annaswamy,11 Sastry and Bodson,12 andIannou and Sun.13

When advantage is taken of the inherent kinematic and dynamicstructure of a system, complexity in the mathematical formulationof the controller can be reduced, resulting in a more efficient controlalgorithm. The dynamics of most mechanical systems can frequentlybe represented in the form of a second-order differential equation,which can be separated into a kinematic part and a dynamic part. Insome other cases, the system is in first-order form, but can naturallybe partitioned into a cascade form with kinematic and dynamicequations. The kinematic equations are accurately known and donot contain uncertain system parameters such as mass, moment ofinertia, etc. The uncertainties exist only in the momentum leveldynamic equations, and for this reason it makes sense to restrictadaptation to the momentum level equations only.

As a simple example, the second-order differential equationF = ma can be written as x = v and v = F/m. The uncertain sys-tem parameter m exists only in the dynamic part, and so adaptationcan be restricted to the second equation only. When the adaptationis restricted to a subspace of the entire state space, the adaptation

635

636 TANDALE AND VALASEK

Fig. 1 MRAC system.

becomes more efficient. Structured MRAC (SMRAC)14 has beenshown to be effective in the presence of large system model errors,15

bounded disturbances,16 and actuator saturation limits.17

SAMI18 is based on the concepts of feedback linearization,9 dy-namic inversion, and SMRAC. SAMI uses dynamic inversion tosolve for the control. In dynamic inversion, the desired trajectoriesthat the system is supposed to follow are specified beforehand. Theactual plant dynamics are constructed using a plant mathematicalmodel, so that the calculated controls simultaneously cancel out theplant dynamics and augment the desired dynamics.

For dynamic inversion to work efficiently, the mathematicalmodel of the plant should be accurate so that the actual dynam-ics of the plant are canceled exactly. However, in practice, an exactcancellation is usually not achieved because the system parametersare not always known accurately. In SAMI, an adaptive controlleris wrapped around the dynamic inversion to account for the un-certainties in the system parameters. This controller is designed todrive the error between the output of the actual plant and that of amodel reference to zero, as in MRAC. The adaptation included inthis framework can be limited to only the momentum level states, sothat the structured flavor is retained in the controller. SAMI has beenshown to be effective for tracking spacecraft19 and for aggressiveaircraft maneuvers.20

II. Actuator FailureIn high-performance dynamic systems, the total number of actu-

ators used may be greater than the number of states to be controlledor tracked. This control redundancy generally exists to achieve opti-mality with respect to control effort. Consider a case where the num-ber of actuators is greater than the number of states to be tracked.Mathematically, there are more variables than the equations to besatisfied, therefore, an infinite number of solutions exist. All of thesesolutions have varying control energy requirements, and so the so-lution with minimum control energy can be selected, and optimalitycan be achieved.

In this case of redundant actuation, it is still possible to trackclosely the desired states, even if some of the actuators fail, as longas the number of active actuators is greater than or equal to thenumber of states to be tracked. Thus, if it is possible to reconfigurethe control after failure, the stability and performance of the systemcan in theory be maintained. Many actuator failure schemes employsome form of failure detection algorithm to detect the failure. Anew control effectiveness matrix is estimated, and the controller isredesigned by recalculating the control gains.21

This approach depends strongly on the efficacy of the failure de-tection algorithm. The algorithm may fail to detect a failure, or maygive false warning when there is no actuation failure. In contrast, theSAMI controller is constantly updating its parameters so that it doesnot need to detect the failure specifically. The failure is implicitlyidentified as a change in the parameters of the failure model, andthe adaptation mechanism adapts to this change. Therefore, SAMIis a good candidate to address the problem of actuator failure.

III. Mathematical Modeling of Actuator FailureThe actuator failures commonly encountered in aircraft include

the following : 1) lock in place, 2) hard-over failure, 3) partial loss of

effectiveness, and 4) float. In lock-in-place type failures, the controlsurface freezes or remains fixed at a position that may or may notbe the neutral position or zero. In this case, the remaining operatingactuators must not only compensate for the lack of the desired con-trol effort of the failed actuator, but must also cancel the undesiredcontrol effect produced if the actuator freezes at any position otherthan zero. In the hard-over failure, the actuator freezes at either oneof the position limits. Partial loss of effectiveness may occur dueto physical damage of the control effector. Float-type failure occurswhen the actuator floats in the airflow and does not produce anycontrol moment.

The preceding control failures can be modeled for each controlby the following mathematical model:

uapp = ducal + e (1)

where d and e are constants for a particular control configuration,but may change and settle to other constant values if a control failureoccurs. Here, ucal is the control demanded by the controller and uapp

is the control that can actually be applied because of the saturationlimit. Because uapp = ucal in the absence of failure, d is initialized toone, and e is initialized as zero. An actuator freeze can be modeledby d going to zero and e going to the constant value at which the con-trol surface has frozen. Thus, this model accounts for lock in placeand hard-over failures. This mathematical model can also modelloss of effectiveness. Consider a case in which a control surface isdamaged/lost such that only one-half of it remains. If it is assumedthat the control effectiveness is reduced by one-half, d should goto 0.5, and e should remain at zero. Float-type failures can also beaccounted for, with d going to zero and e remaining at zero.

For the case in which the control is a vector, the failure modelcan be assembled as follows:⎡⎣ ua1

ua2

ua3

⎤⎦ =

⎡⎣ d1 0 0

0 d2 0

0 0 d3

⎤⎦⎡⎣ uc1

uc2

uc3

⎤⎦ +

⎡⎣ e1

e2

e3

⎤⎦ (2)

Thus,

uapp = Ducal + E (3)

By augmentation of the SAMI formulation with this control failuremodel, a framework is created that accommodates actuator failuresand damage as changes in the parameters of the system.

IV. Mathematical Formulation of Fault-TolerantSAMI Controller

A. Definition of Plant and Reference ModelConsider a nonlinear dynamic system that is affine in the control

and can be split into a structured form consisting of an exactly knownkinematic differential equation and a momentum level equation withuncertain parameters

σ = J (σ)ω (4)

ω = A(σ,ω) + B(σ,ω)uapp (5)

where σ ∈ Rn is the vector of position level coordinates, ω ∈ Rn isthe vector of velocity level coordinates, J ∈ Rn × n is the nonlineartransformation relating σ and ω, A(σ,ω) quantifies the unforcedbehavior of the system, and B(σ,ω) is the control effectivenessmatrix.

The dynamics of the system are assumed to be modeled ac-curately, and only structured parametric uncertainties exist in themodel. Note that the most convenient velocity coordinates may notalways be the derivatives of the position. When the case of an air-craft is considered, the position coordinates are the translationaldisplacements along the Earth-fixed inertial axis. The velocity levelequation is written in the body axis because the moment of inertiaremains constant in the body axis at every instant, but the momentof inertia in the inertial axis changes, depending on the orientation

TANDALE AND VALASEK 637

of the body. Thus, the velocity coordinates are the body-axis ve-locities, and the velocity coordinate and the position coordinate arerelated via a series of coordinate axis rotations. Thus, the term J inEq. (4) is a result of multiplication of a series of coordinate rotationmatrices.

Substituting for uapp from Eq. (3) into Eq. (5), we get

ω = A(σ,ω) + B(σ,ω)Ducal + B(σ,ω)E (6)

The control objective is to track the reference trajectory in termsof σr and ωr where σr and ωr are related by σr = Jr (σr )ωr .

For the case of an unmanned air vehicle, predefined referencetrajectories (σr ,ωr ) can be provided to the controller. If the vehicleis piloted, these reference trajectories can be generated in real time asthe output of a reference model. Consider a reference model havinga structure similar to that of the nonlinear plant

σr = Jr (σr )ωr (7)

ωr = A(σr ,ωr ) + B(σr ,ωr )ur (8)

Thus, the reference model is a representation of the dynamics desiredby the pilot flying the aircraft, and ur are the pilot’s inputs.

B. Calculating Control by Dynamic Inversionof Dynamic Level Equation

Let the error in the position and the velocity level states be s andx, respectively,

s = σ − σr (9)

x = ω − ωr (10)

The control appears only in the velocity level equation and, hence,the corresponding velocity level error. The control affects theposition through the integration of Eq. 5 via the coupling seenin Eq. 4,

x = ω − ωr (11)

x = A + B Ducal + BE − ωr (12)

We want the error between the reference and the plant to go to zero.Therefore, we prescribe the following dynamics for x:

x = Ahx + φ (13)

where Ah is a Hurwitz matrix, that is, all eigenvalues lie in the openleft-half plane so that the velocity error dynamics are stable andthe velocity error goes to zero asymptotically. Ah can be selectedarbitrarily, but a proper choice can specify how fast the velocityerror goes to zero. Here, φ is a forcing function on the velocity errordynamics, which helps in achieving the tracking objective. This isdiscussed in detail later.

When Ahx +φ is added and subtracted on the right-hand side ofEq. (12),

x = Ahx + φ + A + B Ducal + BE − (ωr + Ahx + φ) (14)

We assume that the state variables are measurable, which is a rea-sonable assumption for the aircraft model. Because the quantity inthe brackets is known, let

ψ�= ωr + Ahx + φ (15)

Therefore,

x = Ahx + φ + A + B Ducal + BE − ψ (16)

We calculate the control so that the velocity error x has the desireddynamics as shown in Eq. (13). When dynamic inversion is used tosolve for the control,

ucal = (B D)−1(ψ − A − BE) (17)

If m is equal to n, that is, the number of controls is equal to thenumber of velocity level states, (B D) will be an n ×n matrix of fullrank, and its inverse can be computed. If the system is underactuated(m less than n), (B D) will be rank deficient, and its inverse cannotbe computed. If the system is overactuated (such that actuation isredundant) then m is greater than n, (B D) will be a wide matrix,and the minimum norm solution can be used to calculate ucal. Ifthere is actuation failure, (B D) will lose rank because the diagonalelement corresponding to the failed control goes to zero. Rank (B D)is equal to the number of active controls; hence, the number of activecontrols after failure should be greater than or at least equal to n.For redundant actuation,

ucal = pinv(B D)(ψ − A − BE) (18)

where

pinv(B D) = (B D)T [B D(B D)T ]−1 (19)

C. Definition of Adaptive Learning ParametersEquation (18) requires the exact values of A and B so that the plant

dynamics are canceled exactly. However, the system parameters Aand B are not assumed to be known accurately, and so best guessesfor A and B (Aest and Best) will be used. Let

ucal = pinv(Best D)(ψ − CaAest − BestE) (20)

where Ca is the adaptive learning matrix that will be updated onlineto ensure stability and performance of the system. There exists amatrix C∗

a , which is the ideal value for the learning parameter Ca ,such that

C∗a Aest = A (21)

The D matrix that had been introduced to incorporate the controlfailure can also account for the parametric uncertainty in B, just asthe uncertainty in A is accommodated using Ca . Similarly,

Best D∗ = B D (22)

Best E∗ = B E (23)

From Eq. (20)

ψ = CaAest + Best Ducal + BestE (24)

Substituting Eqs. (21–23) in Eq. (16) results in

x = Ahx + φ + C∗a Aest + Best D

∗ucal + BestE∗

− CaAest − Best Ducal − BestE (25)

When

Ca�= C∗

a − Ca (26)

D�= D∗ − D (27)

E�= E∗ − E (28)

are defined, Eq. (25) now becomes

x = Ahx + φ + CaAest + Best Ducal + BestE (29)

D. Incorporating Position into Tracking ErrorThe control is derived from the dynamic part, and it is assumed

that the adaptive mechanism provides perfect velocity tracking.However, this does not ensure that the position reference will betracked correctly. The initial errors in position or errors in the ve-locity during the transient stage, before perfect velocity tracking isachieved, will cause the position to stray from the reference, and noattempts at correcting this error will be made unless this deviationis identified as an error. Therefore, the tracking error should have a

638 TANDALE AND VALASEK

contribution from the position level state also. Let the total trackingerror be defined as

y�= s + λs (30)

where λ ∈ Rn × n is a positive definite matrix. As t → ∞, if y isdriven to zero, it is ensured that s → 0 and s → 0.

When Eq. (9) is differentiated with respect to time,

s = σ − σr (31)

s = Jω − Jrωr (32)

y = Jω − Jrωr + λs (33)

When Jωr is added and subtracted from the right-hand side,

y = J (ω − ωr ) + Jωr − Jrωr + λs (34)

By definition, x =ω − ωr

y = Jx + Jωr − Jrωr + λs (35)

When Eq. (35) is differentiated to obtain the derivative of the track-ing error,

y = J x + J x + ( J − Jr )ωr + (J − Jr )ωr + λs (36)

When x is substituted for from Eq. (29),

y = J (Ahx + φ + CaAest + Best Ducal + BestE)

+ Jx + ( J − Jr )ωr + (J − Jr )ωr + λs (37)

The quantities Ca , D, and E are not known, whereas all other quan-tities are known. For the tracking error to stabilize, the following dy-namics are prescribed to the known quantities. The uncertain quan-tities will be addressed with a Lyapunov analysis shown later. Wewant the tracking error y to go to zero, and so we select the forc-ing function φ so that the tracking error will have the followingdynamics:

y = Ahy (38)

Because Ah has all eigenvalues in the left-half plane, the trackingerror dynamics will be stable and the tracking error will converge tozero asymptotically. Thus, when the known terms on the right-handside of Eq. (37) are equated to Ahy,

J Ahx + Jφ + Jx + ( J − Jr )ωr + (J − Jr )ωr + λs = Ahy (39)

When the forcing function φ is solved for,

φ = J −1(Ahy − λs − Jω + Jr ωr + Jrωr ) − ωr − Ahx (40)

Finally, by substitution of the value of φ in Eq. (37),

y = Ahy + J (CaAest + Best Ducal + BestE) (41)

E. Lyapunov Analysis and Update Laws for AdaptiveLearning Parameters

Consider the candidate Lyapunov function V . The function V is ameasure of the total error in the system, where y is the total trackingerror, Ca represents the error in the unforced dynamic behavior,D represents the error in the control effectiveness and the failuremodel, and E is the error in the failure model, If P , W1, W2 arepositive definite matrices,

V = yT Py + T r(CT

a W1Ca + DT W2 D) + ET W3E (42)

When the derivative of the Lyapunov function is taken,

V = yT P y + yT Py + 2T r(CT

a W1˙Ca + DT W2

˙D)+ 2ET W3˙E (43)

V = yT P Ahy + yT ATh Py + 2T r

(AT

estCTa J T Py + uT

cal DT BT

est JT Py

)+2ET BT

est JT Py + 2T r

(CT

a W1˙Ca + DT W2

˙D) + 2ET W3˙E(44)

P is selected such that P Ah + ATh P = −Q, where Q is a positive-

definite matrix. The existence of P and Q to satisfy the precedingrelation is guaranteed because Ah is Hurwitz. By the use of theidentity if A and B are row and column vectors, respectively, thenAB = T r(B A):

V = −yT Qy + 2T r[CT

a

(J T PyAT

est + W1˙Ca

)]+ 2T r

[DT

(BT

est JT PyuT

cal + W2˙D)]+ 2ET

(BT

est JT Py + W3

˙E)(45)

When only the negative definite part −yT Qy is retained and all othersign indefinite terms in Eq. (45) are set to zero,

˙Ca = −W −11

(J T PyAT

est

)(46)

C∗a − Ca = −W −1

1

(J T PyAT

est

)(47)

However, C∗a is assumed to be constant, so that

Ca = W −11

(J T PyAT

est

)(48)

Similarly,

D = W −12

(BT

est JT PyuT

cal

)(49)

E = W −13

(BT

est JT Py

)(50)

These are the update equations for the various adaptive learn-ing parameters. A schematic of a fault-tolerant SAMI controlled isshown in Fig. 2.

F. Stability Analysis

The Lyapunov function V is a function of (y, Ca, D, E), and

V = 0 when y = 0, Ca = 0, D = 0, and E = 0. Here, 0 is a null vec-tor or null matrix of appropriate dimension. The derivative of theLyapunov function V is a function of y only. Thus, the derivative ofthe Lyapunov function is zero when y = 0, irrespective of the values

of Ca , D, and E. Hence, V is negative semidefinite.Thus, the adaptive control law (20), along with the update laws

(48–50), ensure global stability. From the properties of V and V juststated, we conclude that y ∈L2 ∩ L∞ and Ca, D, and E ∈L∞.

Fig. 2 Schematic of fault-tolerant SAMI controller.

TANDALE AND VALASEK 639

Fig. 3 Inertial and body axis.

Here, y is defined as y = s + λs. If (s + λs) ∈L2 ∩L∞, thens ∈L2 ∩L∞, and s ∈L2 ∩L∞. Because the reference trajectoriesare bounded, (σr , σr ) ∈L∞. Thus, from the definition of s and s,it can be concluded that (σ, σ) ∈L∞. Because σ= Jω, ω ∈L∞,which implies A(σ,ω) and φ∈L∞. Equation (41) shows that allof the signals in the y are bounded, hence y ∈L∞.

From Barbalat’s lemma (see Ref. 10), we conclude that y → 0as t → ∞. Thus, s → 0 and s → 0, which implies that σ → σr

and ω →ωr as t → ∞. Thus, the states of the plant asymptoticallyconverge to the reference, and perfect tracking can be achieved.

With perfect tracking, the system will follow a trajectory, and thethe tracking error will go to zero as time goes to infinity. However,V is a function of the tracking error only, and so when the track-ing error becomes zero, the system parameters do not update. Thestability analysis guarantees asymptotic tracking of the referencetrajectories. However, the adaptively estimated parameters may notconverge to the actual parameters of the system during the durationof the maneuver. Because it is assumed that parameters like C∗

a , D∗,and E∗ are constants, this formulation works only when the plantparameters are constant with respect to time, or slowly time varying,as compared to the rate of update of the adaptive parameters.

V. Nonlinear Six-Degree-of-Freedom AircraftSimulation Model

The general six-degree-of-freedom equations of motion for anaircraft are derived for the body axis fixed to body with the origin

[φ θ ψ dx d y dz]T =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 Sφ tan(θ) Cφ tan θ 0 0 0

0 Cφ −Sφ 0 0 0

0 Sφ sec θ Cφ sec θ 0 0 0

0 0 0 Cθ Cψ Sφ Sθ Cψ − Cφ Sψ Cφ Sθ Cψ − Sφ Sψ

0 0 0 Cθ Sψ Sφ Sθ Sψ − CφCψ Cφ Sθ Sψ − SφCψ

0 0 0 −Sθ SφCθ CφCθ

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

p

q

r

u

v

w

⎤⎥⎥⎥⎥⎥⎥⎦ (55)

located at the center of gravity.22 The orientation and position isdefined in terms of an inertial reference frame fixed to the Earth(Fig. 3). Let dx, dy, and dz denote the position of the aircraft alongthe Xn , Yn , and Zn axes, respectively. The angular orientation of theaircraft can be described by a 3–2–1 rotation sequence through theEuler angles ψ, θ , and φ, respectively. The vectors � = [p q r ]T

and V = [u v w]T denote the angular velocity and the linear velocityalong the body axes, respectively.

The angle of attack and the sideslip angle are shown in Fig. 4 andcan be calculated in terms of the velocity components as

α = tan−1(w/u) (51)

Fig. 4 Angle of attack α and sideslip angle β.

β = sin−1(v/Vres) (52)

where Vres is the resultant velocity. The variables that completelydefine the state space for this aircraft model are the position levelvector

σ = [φ θ ψ dx dy dz]T (53)

and the velocity level vector

ω= [p q r u v w]T (54)

The aircraft model used in the examples is an F-16 type withthrust-vectoring capability. Following the earlier discussion, it isclear that the number of controls must be greater than the numberof velocity level states. There are eight aerodynamic controls: righthorizontal tail, left horizontal tail, right aileron, left aileron, rudder,and three controls for thrust vectoring: thrust along the X , Y , andZ axes. With a total of eight controls, the control algorithm cantolerate a maximum of two control failures, both of which may failat arbitrary times.

We define the notation Sθ = sin(θ), Cθ = cos(θ), etc., for writingconvenience. The structured model for this example with separatekinematic and dynamic parts can be written as 1) the kinematicpart,

which is in the form of the model, Eq. (4)

σ = J (σ)ω

and 2) the dynamic part, where I is allowed be the moment of inertiaof the aircraft about the body axes and m the mass.The inertia matrix is

I =

⎡⎢⎣Ixx Ixy Ixz

Iyx Iyy Iyz

Izx Izy Izz

⎤⎥⎦ (56)

640 TANDALE AND VALASEK

From Euler’s rigid-body equations,

� = I −1

⎡⎣⎡⎣L

M

N

⎤⎦ − �IΩ

⎤⎦ (57)

V = 1

m

⎡⎣ X − mgSθ

Y + mgCθ Sφ

Z + mgCθ Cφ

⎤⎦ (58)

where �( ) is the matrix representation of the cross product betweenvector Ω and any compatible vector ( ), [X Y Z]T is the force vectorand [L M N]T is the moment vector in the body axes,

� =

⎡⎣ 0 −r −q

r 0 −p

−q p 0

⎤⎦ (59)

The external moments and forces have wing and body contributionsthat are dependent on angle of attack, sideslip angle, and the controls.

Consider the total external pitching moment acting on the aircraft,

M = [12ρV2 S

(Cmmβ

β + Cmm p p + Cmmr r)] + [

Cmmcont u]

(60)

where 12ρV2 is the dynamic pressure; S is the wing planform area;

Cmmβ, Cmm p , and Cmmr are the corresponding stability derivatives;

and Cmmconst is a row vector. The i th element of this vector representsthe contribution to the pitching moment due to unit deflection of thei th control.

Because external force and moment contributions in the first termof Eq. (60) play a role in the unforced dynamic behavior, they con-stitute a part of the A vector. The external force and moment con-tributions in the second term of Eq. (60) show the control influenceand, hence, constitute a part of the B matrix.

The dynamic part of the equation of motion can now be writtenas

[�

V

]=

⎡⎢⎢⎢⎢⎢⎢⎢⎣I −1

⎡⎣⎡⎣L

M

N

⎤⎦unforced

− �I�

⎤⎦1

m

⎧⎨⎩⎡⎣X

Y

Z

⎤⎦unforced

+

⎡⎣−mgSθ

mgCθ Sφ

mgCθ Cφ

⎤⎦⎫⎬⎭ − ωV

⎤⎥⎥⎥⎥⎥⎥⎥⎦+

⎡⎢⎢⎢⎢⎢⎢⎣

Cucont

Cmmcont

Cnncont

Cxcont

Cycont

Czcont

⎤⎥⎥⎥⎥⎥⎥⎦ u

(61)

which has the same form as Eq. (5),

ω = A(σ,ω) + B(σ,ω)uapp

as desired.

VI. Numerical ExampleThe example consists of an aileron actuator failure that occurs

from an initially trimmed flight condition. The flight condition isMach 0.8 at 50,000 ft. The simulations are performed with para-metric errors in the vector A and the B matrix and with initial con-dition errors. Parametric errors of the order of ±10% are introducedin the various parameters such as weight and various inertia terms.The aircraft is given an initial angular velocity of 1 deg/s in all ofthe three body axes, to simulate initial condition errors. For eachcase, the right aileron fails at time equal to 5 s and then settlesdown to a specified constant value. Because of this failure, the air-craft rolls rapidly and without adaptation departs controlled flight.The adaptive controller is expected to adjust its parameters so that itcompensates for this failure and provides a restoring rolling momentby using a differential horizontal tail. Two test cases are presented:aileron failure during steady, level, 1-g flight (case 1), and aileronfailure during a steady level turn (case 2).

A. Case 1: Steady Level 1g FlightThe trim angle of attack is 3 deg, and, on failure, the aileron settles

down to 2 deg. For case 1, the initial reference states are containedin Table 1.

Figure 5 shows the angular states, and Figure 6 shows the trans-lational states. Because of failure of the right aileron, the aircraftbegins to roll rapidly, and, with the nonadaptive controller unableto handle the failure, the states diverge and the aircraft departs fromcontrolled flight. The same failure scenario, but with the adaptivecontroller operating is shown in Fig. 7. Deviations in the states fromthe reference when time is less than 5 s is due to initial conditionerrors and the parametric error in the system matrices. However, thestates settle down to their reference values by the time the failureoccurs at time equals 5 s. All of the states show perturbations at theonset of failure, but the controller adapts, and the states settle downto their reference values. The update parameters (not shown here)stop updating as soon as the tracking error goes to zero. Figure 8shows the controls for both the nonadaptive and adaptive controllers.

Table 1 Reference states for case 1

Variable State

φ, dy, dz, p, q, r , β, ν 0θ = α constantu Vres cos α

w Vres sin α

ψ constantdx (Vres)t where t is timeVres constant

Fig. 5 Time histories of angular states, case 1, steady level flight.

Fig. 6 Time histories of linear states, case 1, steady level flight.

TANDALE AND VALASEK 641

Table 2 Reference states for case 2

Variable State

α constant

ψ ψ tθ constantφ constantdx Vrestdy dx = β = v = 0

p − sin θψ

q cos θ sin φψ

r cos θ cos φφ

u Vres cos α

w Vres sin α

Vres constant

Fig. 7 Time histories of states with adaptive control, case 1, steadylevel flight.

Fig. 8 Time histories of controls, adaptive and nonadaptive, case 1,steady level flight.

For the latter, recovery from failure and the performance shown isachieved with reasonable control deflection limits. The control rates(not shown here) are also within reasonable limits.

B. Case 2: Steady Level TurnTrim angle of attack is 14 deg, bank angle is 52.4 deg, and turn rate

is 3 deg/s. On failure, the aileron settles down to 1 deg. There is noinitial condition error and no uncertainty in the system parameters.The reference states are shown in Table 2.

Fig. 9 Case 2 steady level turn.

Fig. 10 Time histories of angular states, case 2, steady level turn.

Fig. 11 Time histories of linear states, case 2, steady level turn.

The reference states for a steady level turn are a constant turn rateψ and bank angle bank = tan−1(ψV /g) (Fig. 9). The transformationfrom inertial to body axes is a 3–1–2 Euler angle rotation throughψ, bank, and α, respectively. This 3–1–2 rotation is converted to a3–2–1 rotation to obtain ψ, θ , and φ.

Figures 10 and 11 show that the nonadaptive controller providesperfect tracking before failure, but diverges rapidly afterward. InFig. 12, all of the states show perturbations at failure, but the con-troller successfully adapts, and the states settle down to their ref-erence values. The adaptive learning parameters remain constant

642 TANDALE AND VALASEK

Fig. 12 Time histories of states with adaptive control, case 2, steadylevel turn.

Fig. 13 Time histories of controls, adaptive and nonadaptive, case 2,steady level turn.

before failure because there are no initial condition errors and noerrors in system parameters, but they update after failure and settledown to constant values that are different from the initial values.Figure 13 shows that the adaptive controller recovers from the fail-ure with reasonable control effort and within position limits. Thecontrol rates (not shown) are also within limits.

VII. ConclusionsThis paper derives and validates a fault-tolerant SAMI control

law. A mathematical model of control effector failure was addedto the plant model, so that the failure can be identified as a changein the system parameters. The control law ensures global asymp-totic stability of the tracking errors, even though adaptive learningparameters may not converge to the actual parameters.

Based on the results presented in this paper, the following con-clusions are made:

1) For the test cases evaluated, the fault tolerant adaptive con-troller is able to recover from actuator failure and achieve closetracking in cases where the nonadaptive controller loses tracking oris unable to maintain stability.

2) The fault-tolerant adaptive controller formulation presentedcan only handle parametric uncertainties that are constant with re-spect to time, or slowly time varying as compared to the rate of

update of the adaptive learning parameters of the system. Thus, itcan handle actuator failures in which the parameters in the fail-ure model remain constant/piecewise constant after the failure. Thisconstant value may or may not be zero. This conclusion is valid withthe assumption that the system is controllable with the remainingactive controls, such that the control effectiveness of the remainingcontrols is large enough to counter the undesirable effect producedwhen the failed actuator settles to an arbitrary value.

References1Bodson, M., and Groszkiewicz, J., “Multivariable Adaptive Algorithms

for Reconfigurable Flight Control,” IEEE Transactions on Control SystemsTechnology, Vol. 5, No. 2, 1997, pp. 217–229.

2Boskovic, J. D., and Mehra, R. K., “Multiple Model Adaptive FlightControl Scheme for Accommodation of Actuator Failures,” Journal ofGuidance, Control, and Dynamics, Vol. 25, No. 4, 2002, pp. 712–724.

3Ahmed-Zaid, F., Ioannou, P., Gousman, K., and Rooney, R., “Accom-modation of Failures in the F-16 Aircraft Using Adaptive Control,” IEEEControl Systems Magazine, Vol. 11, No. 1, 1991, pp. 73–78.

4“Reconfigurable Systems for Tailless Fighter Aircraft RESTORE,” Sci-entific and Technical Repts., System Design Rept., Contract Data Require-ment List (CDRL) Sequence A007 F33615-96-C-3612, Boeing PhantomWorks, St. Louis, MO, May 1998.

5Brinker, J., and Wise, K., “Reconfigurable Flight Control of a TaillessAdvanced Fighter Aircraft,” AIAA Paper A98-37001, Aug. 1998.

6Calise, A., Lee, S., and Sharma, M., “Direct Adaptive ReconfigurableControl of a Tailless Fighter Aircraft,” Proceedings of the 1998 AIAA Guid-ance, Navigation, and Control Conference, Vol. 1, AIAA, Reston, VA, 1998,pp. 88–97.

7Boskovic, J. D., and Mehra, R. K., “Robust Fault-Tolerant Control De-sign for Aircraft Under State-Dependent Disturbances,” AIAA Paper 2003-5490, Aug. 2003.

8Astrom, K. J., and Wittenmark, B., Adaptive Control, 2nd ed., Prentice–Hall, Upper Saddle River, NJ, 1994.

9Slotine, J., and Li, W., Applied Nonlinear Control, Prentice–Hall, UpperSaddle River, NJ, 1990.

10Khalil, H. K., Nonlinear Systems, 3rd ed., Prentice–Hall, Upper SaddleRiver, NJ, 2001.

11Narendra, K. S., and Annaswamy, A., Stable Adaptive Systems,Prentice–Hall, Upper Saddle River, NJ, 1989.

12Sastry, S., and Bodson, M., Adaptive Control: Stability, Convergence,and Robustness, Prentice–Hall, Upper Saddle River, NJ, 1989.

13Iannou, P. A., and Sun, J., Stable and Robust Adaptive Control, Prentice–Hall, Upper Saddle River, NJ, 1995.

14Akella, M. R., Structured Adaptive Control: Theory and Applica-tions to Trajectory Tracking in Aerospace Systems, Ph.D. Dissertation,Dept. of Aerospace Engineering, Texas A&M Univ., College Station, TX,1998.

15Schaub, H., Akella, M. R., and Junkins, J. L., “Adaptive Realization ofLinear Closed-Loop Tracking Dynamics in the Presence of Large SystemModel Errors,” Journal of Astronautical Sciences, Vol. 48, No. 4, 2000,pp. 537–551.

16Akella, M. R., and Junkins, J. L., “Structured Model Reference Adap-tive Control in the Presence of Bounded Disturbances,” Advances in theAstronautical Sciences, Vol. 99, No. 1, 1998, pp. 98–121.

17Akella, M. R., and Junkins, J. L., “Structured Model Reference AdaptiveControl with Actuator Saturation Limits,” AIAA Paper 98-4472, Aug. 1998.

18Subbarao, K., Structured Adaptive Model Inversion: Theory and Ap-plications to Trajectory Tracking for Non-Linear Dynamical Systems, Ph.D.Dissertation, Dept. of Aerospace Engineering, Texas A&M Univ., CollegeStation, TX, Aug. 2001.

19Subbarao, K., Verma, A., and Junkins, J. L., “Structured Adaptive ModelInversion Applied to Tracking Spacecraft Maneuvers,” AAS/AIAA Space-flight Mechanics Meeting, American Astronautical Society, Paper AAS-00-202, Jan. 2000.

20Subarrao, K., Steinburg, M., and Junkins, J. L., “Structured Adap-tive Model Inversion Applied to Tracking Aggressive Aircraft Maneuvers,”AIAA Paper 2001-4019, Aug. 2001.

21Ward, D. G., Monaco, J. F., Barron, R. L., Bird, R. A., Virnig, J. C., andLanders, T. F., “Self-Designing Controller Design, Simulation, and FlightTest Evaluation,” Air Vehicles Directorate, Tech. Rept. WL-TR-97-3095,U.S. Air Force Research Lab., U.S. Air Force Material Command, WL/FIGC,Wright–Patterson AFB, OH, 1996.

22Roskam, J., Airplane Flight Dynamics and Automatic Flight Control,Vol. 1, Design, Analysis, and Research Corp., Lawrence, KS, 1994, pp. 3–26,65–122.