autonomous guidance and control of airplanes under...

Autonomous Guidance and Control of Airplanes under

Actuator Failures and Severe Structural Damage

Girish Chowdhary∗, Eric N. Johnson†,

Rajeev Chandramohan‡, M. Scott Kimbrell§, Anthony Calise ¶

Georgia Institute of Technology, Atlanta, GA, 30318

This paper presents control algorithms for guidance and control of airplanes under actu-ator failures and severe structural damage. The presented control and guidance algorithmsare validated through experimentation on the Georgia Tech Twinstar twin engine, fixedwing, Unmanned Aerial System. Damage scenarios executed include sudden loss of allaerodynamic actuators resulting in propulsion only flight, 25% left wing missing, suddenloss of 50% right wing and aileron in flight, and injected actuator time delay. A statedependent guidance logic is described that ensures the aircraft tracks feasible commandsin the presence of faults. The commands are used by an outer-loop linear controller togenerate feasible attitude commands. The inner loop attitude control can be achieved byusing either a linear attitude controller or a neural network based model reference adap-tive controller. The results indicate the possibility of using control methods to ensuresafe autonomous flight of transport aircraft through validation on a scaled model that hascharacteristics of a typical transport category aircraft.

I. Introduction

Fixed wing aircraft can be rendered inoperable by in-flight damage. Such damage may include actuatorfailures, battle damage, structural failure, damage due to terrorist attacks, and damage due to high speedimpact. Lifting surface loss and control surface loss are examples of failures that make it extremely hard forpilots to maintain stable flight and ensure that the aircraft is capable of landing successfully. Under suchcircumstances, automatic control systems can be used for autonomously maintaining stable flight and aidingthe pilot in landing.

Commonly used closed loop flight control approaches assume a linearized dynamical model for the dy-namics and use linear control methods (see for example 1, 2). Under severe structural faults, however, thedynamics of the aircraft are significantly altered. For example, under an asymmetric wing failure, aircraftwill experience asymmetric lateral forces coupled with longitudinal motion that cannot be represented bystandard symmetric wing decoupled aircraft models. In order to maintain safe flight, such changes in dy-namics, brought about by system faults, must be handled by the onboard controller. Fault Tolerant Control(FTC) research aims at designing controllers that can mitigate such changes automatically and maintainsafe flight. Several recent reviews and textbooks have been published in this area (see e.g.3,4, 5, 6). Threetypes of failures have been studied: sensor failure, actuator failure and structural damage. Fault tolerantcontrol in the presence of sensor or actuator damage with online identified failure characteristics were studiedin 3, 7. Fault tolerant control in the case of structural damage, including partial loss of wing, vertical tailloss, horizontal tail loss, and engine loss, has also been recently explored recently.8,9, 10 Crider exploredlinear quadratic regulators for partial tail loss through a simulation study on a linear Boeing 747 model. Themethod focused on linear decoupled longitudinal and lateral models with parameter variations.11 Hallouziand Verhaegen used subspace identification techniques along with model predictive control for simulation

∗Research Engineer II, School of Aerospace Engineering, AIAA member†Lockheed Martin Professor of Avionics Integration, School of Aerospace Engineering, AIAA member‡Research Assistant, School of Aerospace Engineering, AIAA student member§Graduate of the School of Aerospace Engineering¶Professor, School of Aerospace Engineering, AIAA member

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American Institute of Aeronautics and Astronautics

study of elevator lock-in-place and rudder-runaway faults based on a linear Boeing 747 model.12 Hitatchiused H∞ robust control design techniques for damage tolerant control using propulsion only control. Sim-ulation results on a linear Boeing 747 model and a nonlinear aircraft model were presented.13,14 Yucelenet al. proposed the derivative-free model reference adaptive control approach for adaptation in presence ofrapidly varying uncertainties.15,16 They validated this approach through simulation on the NASA GenericTransport Model (GTM) with aeroelastic effects.17 Lombartes et al. used online system identification in adynamic nonlinear model inversion scheme with the parameters of the inversion model identified online.18

They validated the method on a Boeing 747 model in simulation for several damage scenarios includingrudder runaway, stabilizer runaway, and rudder loss. Lunze and Steffen use disturbance decoupling methodsto handle identified actuator faults for linear models.19 They assume that the fault only affects the controleffectiveness parameters and that a model of the damaged plant is known. Experimental results on a lab-oratory mounted helicopter-like experimental setup are presented. Boskovic et al. used retro-fit control forfault tolerant control design under bounded fault-induced uncertainty.10 Nguyen et al. used direct-indirectadaptive control and recursive least squares based adaptive control for control of aircraft under simulated30% loss of left wing of the NASA generic transport model.9 Many other authors have also presented faulttolerant control approaches using Model Reference Adaptive Control (MRAC) architecture.20,21,5, 22 Jour-dan et al. demonstrated in flight MRAC based inner-loop attitude control in the presence of severe structuralfaults on a UAV that was a reduced scale model of a F-18.23 Faults considered included up to 60% wingloss.

In most of the previously mentioned approaches, the focus was on inner loop adaptive/reconfigurableattitude control given an attitude command or regulation in the presence of an initial attitude disturbance.The outer-loop control, that is the generation of desired velocity and attitude commands to navigate to awaypoint were exogenous inputs that do not explicitly take into account that failure has occurred, such asinputs provided by a pilot. Under severe faults, the aircraft’s capability to track commands is also severelylimited. Therefore, a key and a relatively unaddressed problem in guaranteeing safe flight in the presenceof damage is to ensure that the aircraft is commanded feasible attitudes and acceleration commands in thepresence of faults. The state dependent outer-loop guidance approach presented in this paper has beenspecifically designed for subsonic aircraft, and uses estimates of the aircraft position, velocity, and attitudemeasurements, along with air data measurements to continually modify the commands to help ensure thatsafe flight is maintained. The guidance logic provides acceleration commands, which are converted to attitudecommands (roll, pitch, yaw). The inner loop attitude controller then generates the appropriate actuatorcommands. A feedback is established between attitude guidance and acceleration guidance that ensures theacceleration commands are modified if the attitude commands exceed heuristically determined limits. Forexample, forward acceleration is modified to maintain level flight if the desired angle of attack commandexceeds a heuristically determined limit that typically avoids stall. This results in a cascaded inner-outerloop architecture as depicted in Figure 1. The attitude controller can either be a linear PID type attitudecontroller or a model reference adaptive controller.

An issue that arises when using adaptive methods, particularly for manned aircraft, concerns the effectthat adaptation might have on any existing stability margins. It is possible that an adaptive controller, de-signed to assist or augment an existing non-adaptive flight controller, might significantly reduce the stabilitymargin of the non-adaptive flight controller in the event of an in-flight failure, and even in the absence offailures in actuation or airframe damage. Calise and Yucelen derive a modification term (namely AdaptiveLoop transfer Recovery (ALR)) that can be used with a wide variety of adaptive approaches that acts topreserve the loop transfer properties of a non-adaptive controller in the absence and in the presence of dam-ages. The principle is based on loop transfer recovery, similar to that employed in robust control design. Inthe context of fault-tolerance, the ALR terms is described and flight-tested here for ensuring safe flight inthe presence of actuator time delays.

In this paper we present the details and flight test results of fault tolerant guidance and control algorithms.All of the algorithms presented have been flight tested on the Georiga Tech Twinstar (GT Twinstar) twinengine, fixed wing, Unmanned Aerial System. Damage scenarios executed include loss of all aerodynamicactuators resulting in propulsion only control, 25% loss of left wing, sudden loss of 50% right wing, andinjected time delay. The contributions of this paper are summarized here:

1. Description and flight-test validation of a state dependent autonomous waypoint following guidanceapproach that establishes a feedback between the outer-loop guidance and inner-loop attitude guidanceto ensure level flight when attitude commands exceed predetermined safe limits due to faults

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Compute

Acceleration

commands

Inner-loop

attitude

controller

Compute

attitude

commands

Waypoints

Air-speed

measurement

zyx aaa ,,

ddd ,,

Figure 1. The presented fault tolerant inner-outer-loop control architecture. The guidance logic provides theaircraft with acceleration commands, which are then converted to desired attitude commands which are usedby the inner loop attitude controller to command appropriate actuator commands. The baseline linear innerloop attitude controller can be replaced with a Model Reference Adaptive Controller.

2. Description and flight-test validation of inner-loop PID attitude controller with the state dependentguidance approach of 1. for maintaining autonomous waypoint following flight in the presence of knowncomplete loss of all aerodynamic effectors and unknown loss of 50% of right wing and all of the rightaileron

3. Description and flight-test validation of an inner-loop model reference adaptive controller with thestate dependent guidance approach of 1. for autonomous waypoint following flight in the presence ofunknown severe structural faults including 25% left wing loss

4. Flight-test validation of the adaptive loop recovery method for mitigating unknown actuator timedelays

The guidance method presented here is based on a combination of the physics and heuristics of flying anaircraft. As such, no theoretical proof of stability is presented for the cascaded outer-loop and inner-loopguidance and control methods. However, the output of the guidance system, namely the desired accelerationsand attitudes, are hard-limited in the software. This indicates that the desired commands passed to thelinear or the MRAC attitude inner-loop controller are guaranteed to be bounded. It is well known that thelocal stability of PID inner-loop attitude controllers with integrator windup protection and bounded desiredcommands can be established with Lyapunov’s second method (see for example Chapter 3.7 of 25). Thestability proofs of the inner-loop MRAC attitude control methods in the presence of parametric uncertaintyand bounded reference commands have been previously presented.26 Therefore, it should be reasonable toexpect local stability of the cascaded closed loop system for an appropriate choice of gains. This intuition isreflected by the positive flight test results. Section III presents the details of the state dependent guidancelogic used. The details of a linear attitude controller are presented in Section IV, and results with thelinear attitude controller are presented in Section V. The details of the MRAC architectures are presentedin Section VI and flight test results with MRAC under structural faults are presented in Section VII. Theadaptive loop transfer recovery method is introduced in Section VIII and flight test results with injectedactuator time delays are presented in Section VIII.A. The paper is concluded in Section IX.

II. The Flight Test Vehicle

Flight testing of fault-tolerant control algorithms for safe flight in the presence of severe structuralfaults has been performed at the UAV Research Facility at Georgia Institute of Technology. These flighttests were performed on the GT Twinstar fixed wing Unmanned Aerial System (UAS). The GT Twinstar(Figure 2) is a foam built, twin engine aircraft based on the Multiplex Twin Star II model airplane. It has

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been equipped with an off-the-shelf autopilot system.The control algorithms discussed in this paper wereimplemented on the autopilot system through custom developed software. The autopilot system comes withan integrated navigation solution that fuses information using an extended Kalman filter from six degreeof freedom inertial measurement sensors, Global Positioning System, air data sensor, and magnetometerto provide accurate state information.27 The available state information includes velocity and position inglobal and body reference frames, accelerations along the body x, y, z axes, roll, pitch, yaw rates and attitude,barometric altitude, and air speed information. These measurements can be further used to determine theaircraft’s velocity with respect to the air mass, and the flight path angle. The Twinstar can communicatewith a Ground Control Station (GCS) using a 900 MHz wireless data link. The GCS serves to displayonboard information as well as send commands to the autopilot. Flight measurements of airspeed andthrottle setting are used to estimate thrust with this model. An elaborate simulation environment has alsobeen designed for the GT Twinstar. This environment is based on the Georgia Tech UAS Simulation Tool(GUST) environment.28 A linear model for the Twinstar in nominal configuration (without damage) hasbeen identified using the Fourier Transform Regression (FTR29) method by the authors.30 A linear modelwith 25% left wing missing has also been identified.31

Figure 2. The Georgia Tech Twinstar UAS. The GT Twinstar is a fixed wing foam-built UAS designed forfault tolerant control work.

III. State Dependent Outer-loop Guidance for Fault-Tolerant Operation

The task of the guidance system is to provide feasible acceleration and attitude commands such that theaircraft can track a desired trajectory despite structural damage. Due to this reason, the guidance logic playsa central role in ensuring safe flight in the presence of damage. In this section we describe a guidance schemedesigned to accommodate significant faults. These faults include structural damage, and partial or complete(known) failures of all aerodynamic effectors (propulsion only control). The guidance system is designedto achieve a desired airspeed vdes, altitude hdes, and a prescribed course. These are typically specified bywaypoint locations with associated speed and altitudes between any two waypoints. Special provision is alsoincluded to fly a holding pattern around a specified point.

The guidance system is designed to calculate acceleration commands axG, ayG, azG in the body x, y, zaxes given the desired waypoints, speed, and altitude using state-dependent guidance laws. The commandsare adapted to current flight condition by scaling them based on the measured airspeed. A feedback isestablished between attitude (inner-loop) guidance and acceleration (outer-loop) guidance that ensures theacceleration commands are modified if the attitude commands exceed heuristically determined limits.

A. Speed Guidance

Speed guidance is concerned with providing an acceleration command axG in the body x axis given a desiredforward speed. One simple approach is to set the forward acceleration to be proportional to the differencebetween the desired speed vdes and the measured airspeed vas: axG = vdes−vas

τspeedwhere τspeed denotes a desired

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time constant for speed regulation. However, this formulation ignores the coupling between the forwardairspeed and the angle of attack of an aircraft. In particular, it ignores the fact that the lift of an aircraft isdependent both on its airspeed and its angle of attack, which in turn is related to azG. To account for thiscoupling, the commanded speed is modified by the vertical guidance through the variable vmod when angleof attack limits are exceeded or due to a failure of longitudinal flight control (elevator). The modificationterm vmod is discussed in Section C. This term forms a feedback between the inner-loop attitude controllerand the outer-loop guidance to ensure that the aircraft is commanded feasible trajectories. This results inthe following expression for the desired forward acceleration command

axG =vdes + vmod − vas

τspeed. (1)

B. Vertical Guidance

The task of the vertical guidance is to provide a vertical desired acceleration command azG given the desiredaltitude. The desired altitude (hdes) and the desired altitude rate (hdes ) commands are found by interpolatingbetween current waypoints with current actual position. Letting τaltitude denote a time constant associatedwith the desired first order altitude response, and h denote the current altitude, the desired vertical speedvsdes

is found based on the following equation

vsdes =hdes − hτaltitude

+ hdes. (2)

Here the term τaltitude is the time constant for a first order command filter on the desired altitude. It isimportant to limit the vertical airspeed between attainable limits. This is achieved by estimating maximumand minimum sustainable vertical speeds for the current airspeed, and by using them to limit vsdes :

vsmax =(δt,max − δt,LPF)

gvasKPaxG

, vsmin =(δt,min−δt,LPF)

gvasKPaxG

, (3)

where the variable KPaxGis a constant gain which relates throttle changes to expected acceleration changes, g

is the acceleration due to gravity, and vas is the measured airspeed. Note that the term gvasKPaxGscales the

limits with the measured airspeed vas. Furthermore, δt denotes the throttle command while δt,LPF denotesthe predicted throttle position to maintain current flight condition. The variable δt,LPF is estimated usingthe following Low Pass Filter (LPF):

δt,LPF =(δt + sxKPaxG − δt,LPF)

τthrustLPF(4)

where sx is the measured specific force in the direction of air-relative motion directly available throughaccelerometer measurements and τthrustLPF is the time constant of the LPF. The LPF reduces the effect ofnoisy accelerometer data or high frequency on throttle changes. Finally, the (positive down) vertical desiredacceleration command azG is computed using a low pass command filter with a time constant of τvs :

azG=− (vsdes−vs)

τvs

. (5)

C. Lateral Guidance

The prescribed course between any two waypoints is assumed to be a straight line. The cross track erroris defined as the distance from the current location to the course, and is defined to be positive to the rightof the course. To intercept the course from the current location, an intercept velocity is found using a firstorder model with a desired time constant τintercept and the cross track error using the following equation

vintercept =1

2

ecrosstrack

τintercept. (6)

A lead angle 4T intercept, which is the difference between the desired intercept path and the actual path ofthe aircraft, is estimated by using an estimate of the ground speed vGS . This lead angle is used to controlthe track angle of the aircraft for intercepting the desired path ( to eliminate cross track error):

4T intercept = sin−1 vintercept

vGS. (7)

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It is important to ensure that the magnitude of 4T intercept is limited from above in order to avoid it frombecoming excessively large (e.g. 30 deg), and limited below, in order to ensure that it does not diminishvery close to the desired waypoint. The desired track angle is:

Tdesired=Tcourse+4T intercept, (8)

Tdesired is then converted to an air-relative track angle. If the aircraft is flying at zero sideslip, then thiswould be the desired heading, otherwise, the desired heading needs to be adjusted by adding the estimateddrift angle 4Tdrift to the desired track angle. The latter handles mind cross-winds if air-data measurementsare used to estimate 4Tdrift. The estimated drift angle is defined as the angle between the direction ofmovement through airmass and movement across ground. The desired track angle is found using

ψdesired = Tdesired+4T drift. (9)

The case when the drift angle exceeds 90 degrees in magnitude relates to the case when the aircraft hasnegative ground speed (going backward), this case requires further modifications that are not discussed inthis paper. A lateral guidance acceleration command ayG is then found by comparing this air-relative trackangle ψdesired with the actual track angle ψ:

ayG= (ψdesired−ψ)2vGS

τintercept |cos4T drift |. (10)

It is important to limit the magnitude of ayG to an experimentally determined reasonable value to avoidexcessive bank angles.

IV. Inner-loop Guidance and Linear Attitude Control

The guidance system provides the inner loop control system with acceleration commands in an air-relativevelocity frame. The task of the inner loop control system is to convert these commands to aileron, elevator,rudder, and two throttle commands (for a twin engine aircraft).

A. Roll Control

Desired roll angle ϕdes should be commanded to achieve the lateral desired acceleration command ayG andthe aircraft flight path angle (γ). However, the control authority of the bank angle control law needs to bescaled with the desired vertical acceleration azG in order to ensure that the aircraft does not loose altitudedue to excessive bank. This is achieved by the following equation:

ϕdes=atan2(ayG−azG+g0cosγ

). (11)

The desired roll angle is used to find the aileron command δa using a Proportional-Integral-Derivative law(PID):

δa=KPϕ (ϕdes − ϕ)−KDϕϕ+δa,I , (12)

where Kpϕ and Kdϕϕ are proportional and derivative gains respectively, and δa,I is the integrator state. Theaileron command is limited between the aileron deflection limits. The integrator state is updated by

δa,I=KIϕ (δa−δa,I) (13)

with the integration performed after limiting the aileron command. This effectively provides an inputproportional to the integral of a sum of bank error and bank angle rate. The combined control law thereforehas built-in integrator windup protection. A similar strategy is used for the other axes. If the aileron is notavailable due to damage, then the same formulation is used to calculate rudder deflection, with different gains(and the computation given below for directional control is ignored). If no aileron or rudder is available, thendifferential thrust command is found by using the same formulation but with different gains. In all othercases the differential thrust command is set to zero. Implementation of this architecture requires that theaircraft control system estimate which actuators have become ineffective. In the flight test results presentedhere with actuator failures (Section V.B) the aircraft control system was assumed to have the knowledgewhether or not aerodynamic actuators are stuck, however, it does not know which actuators are stuck. Inpractice, this can be achieved by monitoring actuation mechanism health, by measuring actuator deflection,or through comparison of the aircraft response to actuator commands to nominal response.

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B. Yaw Control

The yaw control has a PID form with built-in integrator windup protection similar to the roll control logicdescribed above. In the nominal case when the aileron is available, the rudder command is

δr=KPay

v2as

(ayGcosϕ+azGsinϕ−sy) +δr,I (14)

where KPay is a selected proportional gain, and sy is the specific force experienced at the center of gravityof the aircraft along the body y axis; found directly through accelerometer measurements transported to thecenter of gravity. Letting KIay denote a chosen integral gain, the integral term δr,I is updated by:

δr,I = KIay (δr−δr,I) , (15)

with the integration performed after limiting the rudder command. When the aileron is not available, therudder is used for roll control as described in Section A.

C. Pitch Control

The goal of the pitch control is to provide the required elevator deflection to achieve the desired accelerations.Letting KPaz denote the chosen proportional gain, the desired angle of attack command αdes is found bythe following expression

αdes=−KPaz

v2as

(azGcosϕ−ayGsinϕ−sz) +αdes,I . (16)

The integral term αdes,I eliminates the steady state error and is updated using the integral gain KIaz asfollows

αdes,I=KIaz (αdes−αdes,I) . (17)

The angle of attack command is then limited between a minimum and maximum value, based on expectedpositive and negative stall angles. If the command is beyond these limits, then the commanded speed ismodified to alter the desired forward speed by modifying the guidance command axG of (1) by updating thevariable vmod using the following equation

vmod=vas

√azGcosϕ−ayGsinϕ

sz−vdes. (18)

This modification assumes that the lift coefficient will remain constant and changes the commanded speed toachieve desired lift corresponding to requested acceleration. This formulation accounts for stall preventionas well as for the case of elevator failure. In implementing equation 18 care was taken to prevent computingthe square root of a negative number by monitoring the sign of the term inside the square root. It is alsoimportant to note that vmod was limited to an experimentally determined reasonable value, particularly onthe negative side. Elevator command δe is found by using a PID control law with KPα and KDα denotingthe proportional and derivative gains:

δe=KPα (αdes−α)−KDαα+δe,I . (19)

Letting KIα denote the chosen integral gain, the integral term δe,I is updated using

δe,I=KIα (δe−δe,I) . (20)

D. Throttle Control

Symmetric throttle command is found by the following PI control law where KPax and KIax denote theproportional and integral gains:

δt=KPax (axG−vas) +δt,I , (21)

where vas is the online estimated time derivative of the true airspeed. The integral term updated by:

δt,I=KIax (δt − δt,I) . (22)

Differential throttle (described in the section on roll control) and symmetric throttle are combined to getleft engine and right engine throttle. Priority is given to symmetric throttle if they cannot both be satisfiedwithin limits.

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V. Flight Test Results with State Dependent Guidance and Baseline LinearController

A. Asymmetric Structural Damage: Sudden 50% right wing failure

We present flight test results as the aircraft undergoes a 50% right wing loss (along with losing completeright aileron functionality) in autonomous flight with the baseline non-adaptive controller and the state-dependent adaptive guidance logic. Figure 3 shows the GT Twinstar in autonomous flight with 50% rightwing loss, resulting in a severe loss of control authority. The aircraft is put in an oval holding pattern abouta waypoint, the right wing is then ejected in mid-flight at about 2021 seconds using a specifically designedejection mechanism that is triggered from the ground. The aircraft onboard control system is not awareof the fault, and the aircraft is expected to continue to track an oval holding pattern. Accordingly, thecontroller continues to use the same parameters as in the case of no faults. Figure 4 show the ground trackof the aircraft as the aircraft attempts to track a holding pattern about a waypoint. Figure 5 shows therecorded altitude of the aircraft. Figure 6 shows the recorded angular rates. It can be seen that while theaircraft is successful in maintaining flight, the angular rate has an oscillation. Particularly, oscillations inangle of attack and roll are observed, which result in the oscillations in altitude seen in Figure 5. There arealso oscillations in yaw rate, however their time constant is much lower. It was observed that in order toperform turns required to maintain the holding pattern, the guidance system forced the aircraft into stablelimit cycles due to the conflicting requirements of maintaining forward speed, angle of attack, and turn rate.This indicates that the onboard controller was not able to find a set of inputs corresponding to a steadytrimmed state. This is not unreasonable, given the asymmetric nature of the damage and that the rightaileron functionality is missing. Figure 7 shows the recorded control inputs. It can be seen that the rudder,aileron, and throttle often saturate. In particular, the aileron saturates at its minimum value repetitively asthe aircraft tries to maintain level flight. The results indicate that the aircraft is able to maintain autonomousflight in spite of the severe structural damage. The frequency and amplitude of oscillations observed werefound to be within acceptable limits for this type of aircraft and the rather extreme damage condition. Itmay be possible to reduce the oscillations through appropriate choice of angle of attack limits and yaw ratelimits. Furthermore, online detection of fault and adjustment of controller parameters accordingly wouldalso help in reducing the oscillations.

Figure 3. GT Twinstar in autonomous flight after jettisoning 50% right wing.

B. Failure of all aerodynamic effectors: Propulsion only control

The presented algorithm is able to mitigate the complete loss of all aerodynamic effectors by using differentialthrottle for control. This flight condition is termed as propulsion only control. Figure 8 shows the groundtrack of the Twinstar UAS as it tracks an oval pattern in propulsion only control, while Figure 9 shows thealtitude. Figure 10 shows the control commands. At around 1055 seconds into flight, the loss of aileron,elevator, and rudder actuators is simulated by keeping them constant; the control from this point onward isaccomplished by using only differential throttle. The last subfigure in Figure 10 shows that the differential

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0

Ground track of GT Twinstar, aircraft jettisons 50% right wing in mid flight

East ft

Nort

h f

t

Figure 4. Ground track of GT Twinstar; aircraft jettisons 50% right wing in mid flight.

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Figure 5. GT Twinstar altitude; aircraft jettisons 50% right wing at 2021 seconds.

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rate

, ra

d/s

angular rate data corrected for bias

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ate

, ra

d/s

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yaw

rate

, ra

d/s

flight time

Figure 6. Measured angular rate data in autonomous flight; aircraft jettisons 50% right wing at 2021 seconds.

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rudder

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vato

r

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aile

ron

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ttle

Time seconds

Figure 7. Recorded onboard control inputs; aircraft jettisons 50% right wing at 2021 seconds. Rudder,elevator, and aileron inputs are normalized; Throttle input is shown in percentage.

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throttle only gets activated when the aerodynamic effectors are deemed failed. Figure 11 shows the angularrate data. Despite increased oscillations in roll, the control performance of the UAS was found to besatisfactory to enable automated landing.

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Ground track

East ft

Nort

h f

t

Figure 8. Ground track of GT Twinstar in the propulsion only case.

VI. Inner-loop Model Reference Adaptive Control

The inner-loop state-dependent attitude controller can be replaced with a MRAC attitude controller.The adaptive controller is provided with attitude commands calculated using the inner-outer loop guidancedescribed in Sections III and IV. The Euler angle attitude commands are converted to a quaternion repre-sentation. The desired quaternion command is then calculated using techniques described in Section II of 32.The idea is to calculate an error quaternion based on a quaternion product between the desired quaternionand the conjugate of the measured quaternion.33 The desired angular rate command is also calculated usingthe desired velocity and measured acceleration. In the following, we describe the details of the adaptivecontrol architecture that converts the desired attitude and angular rate commands to actuator commands.

Let Dx ⊂ <n be compact, and Let x(t) be the known state vector, let δ ∈ <n denote the control input.Note that it is assumed that the dimension of the control input is same as the dimension of x. This is areasonable assumption for inner-loop attitude control of aircraft, where the angular rates p, q, r correspondto the aileron, elevator, and rudder control inputs. In context of MRAC, this assumption can be viewed asa specific matching condition. Consider the following system that describes the dynamics of an aircraft:

x(t) = f(x(t), x(t), δ(t)), (23)

In the above equation, the function f is assumed to be globally Lipschitz continuous in x, x ∈ Dx, andcontrol input δ is assumed to be bounded and piecewise continuous. Therefore, existence and uniqueness ofpiecewise solutions to (23) are guaranteed. This assumption is easily satisfied by most aircraft systems.

The goal of the Approximate Model Inversion (AMI)-MRAC controller is to track the states of a referencemodel given by:

xrm = frm(xrm, xrm, r), (24)

where frm denotes the reference model dynamics which is assumed to be continuously differentiable inxrm, xrm for all xrm, xrm ∈ Dx ⊂ <n. The command r(t) is assumed to be bounded and piecewise continuous.Furthermore, it is assumed that the reference model is chosen such that it has bounded outputs (xrm, xrm)

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altitude f

t

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Figure 9. Altitude in the propulsion only case. The aircraft begins propulsion only control at 1055 secondsinto flight.

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Figure 10. Actuator commands in the propulsion only case.

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0

0.5

r ra

d/s

time secods

Figure 11. Angular rates in propulsion only case.The aircraft begins propulsion only control at 1055 secondsinto flight.

for a bounded input r(t). Often a linear reference model is desirable, however the theory allows for a generalnonlinear reference model as long as it is bounded-input-bounded-output stable. When the actuators aresubjected to saturation, a nonlinear reference model may be required.26

Since the exact model 23 is usually not available, we choose an approximate model f(x, x, δ) which isinvertible with respect to δ such that

δ = f−1(x, x, ν). (25)

where ν ∈ <n is the pseudo control input, which represents the desired acceleration. Note that the choiceof the inversion model should capture the control assignment structure, however the parameter of thatmapping need not be completely known. This is important in guaranteeing that appropriate control inputsare mapped to appropriate states. For example, the inversion model should capture the fact that a deflectionin aileron affects the pitch rate, although the exact parameters of the relationship need not be known. It isassumed that for every (x, x, ν) the chosen inversion model returns a unique δ. This can be realized if thedimension of the input δ is the same as the dimension of x. In many control problems, and indeed in theaircraft attitude control problem studied here, this is true. Hence, the pseudo control input satisfies

ν = f(x, x, δ). (26)

This approximation results in a model error of the form

x = ν + ∆(x, x, δ) (27)

where the model error ∆ : <2n+k → <n is given by

∆(x, x, δ) = f(x, x, δ)− f(x, x, δ). (28)

A tracking control law consisting of a linear feedback part νpd = −[Kp Kd]T [x x], a linear feedforward

part νrm = xrm, and an adaptive part νad(x) is chosen to have the following form

ν = νrm + νpd − νad. (29)

Define the tracking error e ∈ <2n as e(t) = [xrm(t) − x(t), xrm(t) − x(t)]T . To derive the equation fortracking error dynamics note that e = xrm − x = νrm − (ν + ∆) due to (28). Substituting (29) we see that

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e = νpd − (νad −∆) and then, letting A =

[0 I

−Kp −Kd

], and B =

[0

I

], the tracking error dynamics

are found to be34,26,35,36

e = Ae+B[uad(x, x, δ)−∆(x, x, δ)]. (30)

The baseline full state feedback controller gains Kp,d are chosen such that A is a Hurwitz matrix. Hencefor any positive definite matrix Q ∈ <2n×2n , a unique positive definite solution P ∈ <2n×2n exists to theLyapunov equation

ATP + PA+Q = 0. (31)

Letting x = [x, x, δ] ∈ <2n+k, it is assumed that the uncertainty ∆(x) can be approximated using a SingleHidden Layer (SHL) Neural Network (NN).

A. Model Reference Adaptive Control using a Single Hidden Layer Neural Network

A SHL NN is a nonlinearly parameterized map that can be used for capturing unstructured uncertaintiesthat are known to be continuous and defined over a compact domain.37 Let x denote the input to the SHLNN, the output of a SHL NN can be given as

uad(x) = WTσ(V T x) ∈ <n3 . (32)

In the above equation, W ∈ <(n2+1)×n3 , where n2 is the number of hidden layer neurons and n3 is thedimension of the NN output, is the NN synaptic weight matrix connecting the hidden layer with the outputlayer and has the following form:

W =

Θw,1 · · · Θw,n3

w1,1 · · · w1,n3

.... . .

...

wn2,1 · · · wn2,n3

∈ <(n2+1)×n3 , (33)

where V ∈ <(n1+1)×n2 is the NN synaptic weight matrix, with n1 being the number of inputs to the NN,connecting the input layer with the hidden layer and has the following form:

V =

Θv,1 · · · Θv,n2

v1,1 · · · v1,n2

.... . .

...

vn1,1 · · · wn1,n2

∈ <(n1+1)×n2 , (34)

and x ⊂ <n1+1. x contains the data on which the network trains xin and the constant bias term bv:

x =

(bv

xin

)=

bv

xin1

xin2

...

xinn1

∈ <n1+1. (35)

Typically xin = x, however, the actuator deflections and other states may also be used to train the NN.32Forease in notation, let z = V T x ∈ <n2 , then the vector function σ(z) ∈ <n2+1 is given by:

σ(z) =

bw

σ1(z1)...

σn2(zn2)

∈ <n2+1. (36)

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The elements of σ consist of sigmoidal activation functions, which are given by:

σj(zj) =1

1 + e−ajzj. (37)

SHL NN are known to be universal approximators.37,38 That is, for all x ∈ D, where D is a compact set,there exists a number of hidden layer neurons n2, and an ideal set of weights (W ∗, V ∗) that brings the NNoutput to within an ε neighborhood of the function approximation error. The largest such ε is given by

ε = supx∈D

∥∥∥W ∗T σ(V ∗T

x)−∆(x)∥∥∥ . (38)

Hence the following approximation holds for all x ∈ D

∆(x) = W ∗T

σ(V ∗T

x) + ε(x), (39)

where D is compact, and ε = supx∈D ‖ε(x)‖ can be made arbitrarily small given sufficient number of hiddenlayer neurons.

The NN weight adaptation laws are given by:39,40,32,35

W = −(σ(V T x)− σ′(V T x)V T x)rTΓw − k‖e‖W (40)

V = −ΓV xrTWTσ′(V T x)− k‖e‖V, (41)

where the last term in 40 is Narendra and Annaswamy’s e-modification term which is used to prevent weightdrift in the presence of sensor noise.41 Figure 12 depicts the control architecture for MRAC control discussedin this section. Pseudo Control Hedging34,32 is used for mitigating actuator saturation. It should be notedthat for the above adaptive laws to hold, the reference model and the exogenous reference commands shouldbe constrained such that the desired trajectory does not leave the domain over which the neural networkapproximation is valid. Under these assumptions, the closed loop MRAC architecture used here has beenshown to have guarantees of uniform ultimate boundedness.26,35 Furthermore, ensuring that the stateremains within a given compact set implies an upper bound on the adaptation gain (see for example Remark2 of Theorem-1 in 42). Note also that ∆ depends on νad through the pseudocontrol ν, whereas νad has tobe designed to cancel ∆. Hence the existence and uniqueness of a fixed-point-solution for νad = ∆(x, νad) isassumed. Sufficient conditions for this assumption are also available43.44

VII. Flight Test Results with MRAC

Results from a flight tests are presented as the aircraft tracks an oval pattern while holding altitude at200 ft with 25% of the left wing missing. The GT Twinstar uses the guidance algorithm discussed in SectionIII to ensure that the aircraft can track feasible trajectories even when it has undergone severe structuraldamage. The adaptive control algorithm has a cascaded inner and outer loop design. The state dependentguidance logic described in Section III, commands the desired roll angle, angle of attack, and sideslip angleto achieve desired waypoints. The inner loop ensures that the states of the aircraft track these desiredquantities using the control architectures described in section VI. The SHL MRAC implementation has fivehidden layer neurons. The learning rates in equation 40 are set as Γw = 1 and ΓV = 5 for the SHL MRACimplementation while the e-modification constant κ is set to 0.1.

A. Flight Test Results Without Damage

In this section flight test results are presented as the aircraft operating in nominal conditions tracks fourwaypoints arranged in a rectangular pattern. The ground track of the controller is shown in figure 13. Inthat figure, the circles denote the commanded way points, the dotted line connecting the circles denotes thepath the aircraft is expected to take, except while turning at the way points. While turning at the waypoints, the onboard guidance law smooths the trajectory with circles of 80 feet radius. Figure 14 showsthe altitude tracking performance of the SHL MRAC adaptive controller. The inner loop tracking errorperformance of the SHL MRAC adaptive controller is shown in figure 15. The actuator input required forthe SHL MRAC adaptive controller is shown in figure 16. The inner loop tracking performance and generalinner loop handling was found satisfactory.

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rmx

Reference Model 1ˆ −f Nonlinear Actuator Dynamics f

Adaptation Law

PD compensator

δ

pdν

adν−

Neural Network

rme xδ cmdδ crmν ν+

cx

Figure 12. Neural Network Adaptive Control using Approximate Model Inversion

B. Flight Test Results With Damage

In this section flight test results are presented as the aircraft tracks four waypoints arranged in a rectangularpattern with 25% of the left wing missing. The guidance and control laws remain unaltered in both nominalflight and damaged flight, with the adaptive controller expected to mitigate the effects of the enforceddamage. Figure 17 shows the ground track of the aircraft with 25% left wind missing. Figure 18 shows thatthe altitude tracking performance of the SHL MRAC adaptive controller. The cross-tracking and altitudeperformance of the SHL MRAC controller when flying with damage remained comparable to controllerperformance under nominal conditions (nominal performance was shown in Figures 13,14).The inner looptracking error performance of the SHL MRAC controller is shown in figure 19. The actuator input requiredfor the SHL MRAC controller is shown in figure 20. The inner loop tracking performance and generalinner loop handling of the damaged aircraft was found satisfactory to attempt several successful automatedlandings.

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−300 −200 −100 0 100 200 300 400

−500

−400

−300

−200

−100

0

Ground track

East ft

Nor

th ft

cmdSHL MRAC

Figure 13. Flight recorded ground track for SHL MRAC under nominal conditions.

0 5 10 15 20 25 30 35 40192

194

196

198

200

202

204

206

time seconds

altit

ude

ft

cmdSHL MRAC

Figure 14. Flight recorded altitude tracking performance for SHL MRAC and DFMRAC under nominalconditions on the GT Twinstar UAS. The commanded altitude is at 200 ft.

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0 5 10 15 20

−0.5

0

0.5

time seconds

φ ra

dian

s

0 5 10 15 20

−0.5

0

0.5

α ra

dian

s

time seconds

0 5 10 15 20

−0.5

0

0.5

β ra

dian

s

time seconds

innerloop errors

Figure 15. Inner loop tracking errors for SHL MRAC

0 5 10 15 20 25−0.5

0


rudd

er

0 5 10 15 20 250

0.2

0.4

elev

ator

0 5 10 15 20 25−0.5

0

0.5

aile

ron

0 5 10 15 20 2520

40

60

80

Thr

ottle

Time seconds

Figure 16. Actuator inputs for SHL MRAC

−200 −100 0 100 200 300 400 500−600

−500

−400

−300

−200

−100

0

Ground track

East ft

Nor

th ft

cmdSHL MRAC

Figure 17. Flight recorded ground track for SHL MRAC with 25% left wing missing.

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0 5 10 15 20 25 30 35 40194

196

198

200

202

204

206

208

time seconds

altit

ude

ft

cmdSHL MRAC

Figure 18. Flight recorded altitude tracking performance for SHL MRAC with 25% left wing missing. Thecommanded altitude is at 200 ft.

0 5 10 15 20 25 30 35

−0.5

0

0.5

time seconds

φ ra

dian

s

0 5 10 15 20 25 30 35

−0.5

0

0.5

α ra

dian

s

time seconds

0 5 10 15 20 25 30 35

−0.5

0

0.5

β ra

dian

s

time seconds

innerloop errors

Figure 19. Inner loop tracking errors for SHL MRAC

0 5 10 15 20 25 30 35 400

0.5

1Controller inputs

rudd

er

0 5 10 15 20 25 30 35 400

0.5

1

elev

ator

0 5 10 15 20 25 30 35 400

0.5

1

aile

ron

0 5 10 15 20 25 30 35 400

50

100

Thr

ottle

Time seconds

Figure 20. Actuator inputs for SHL MRAC

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VIII. Adaptive Loop Transfer Recovery for Flight Control under ActuatorTime Delay

Actuator time delays can be induced due to unmodeled actuator dynamics and can worsen over timedue to actuator wear. Traditionally, aircraft control system designers have relied on phase and time delaymargins of linear controllers to guarantee robustness to unmodeled time delays. These concepts do notdirectly extend to adaptive controllers because the closed loop is nonlinear. However, MRAC is designed totrack a linear reference model, which is typically chosen to satisfy the desired closed loop margins (gain andtime delay margins). The goal of the ALR approach is to design an adaptive law such that the states trackthe reference model asymptotically, and the reference model margins are preserved to the maximum degreepossible.

In order to study the stability margins of the closed loop system, linearization about an equilibrium pointx of the adaptive law of (40) is typically required along with the assumption that the external commands

are constant. Such an analysis involves the term WT ∂σ(V T x)∂x , where W , V denote the frozen weights. Figure

21 shows the loop with the linearization and the weights frozen. If W = 0, and if the system were exactlymodeled as a linear system, then the margins calculated with the loop broken at x correspond to the marginsof the reference model. The bottom portion of the diagram shows the effect of the adaptive element in steadystate (that is when W = 0, V = 0). It can be seen that even with e(t) = 0 and the weights constant, theadaptive term modifies the margins of the reference model in an unknown way. Furthermore, when weightsare not constant, the feedback loop becomes nonlinear and it is not possible to use such a representation toanalyze the margins using this representation. However, it may still be possible to use this representation toanalyze the margins of the linearized system if the following constraint were approximately satisfied

WT (t)∂σ(V T x(t))

∂x(t)= 0, (42)

by modifying the adaptive law such that it satisfies asymptotically 42 as the modification gain approachesinfinity.45

Linearized system

𝐾𝑥

𝑊𝑇𝜕

𝜕𝑥𝜎(𝑉𝑇𝑥 )

- +

x

Figure 21. Visualization of the linearized adaptive system dynamics

This is achieved by adding a modification term to the adaptive law of (40). Let J(t) = 12‖W

T (t)σ(V T x(t))‖2F ,

20 of 25


denote a cost function where ‖.‖F denotes the Frobenious norm. Minimization of this cost would mean thatthe constraint in (42) is approximately satisfied. Similar to other modifications for adaptive control, ALRadds a term to the adaptive law in the direction of maximum reduction of this cost. Noting that

∂J(t)

∂W (t)= σ(V T x)σT (V T x)W, (43)

the modified adaptive law for the outer layer weights W has the form:

W = −(σ(V T x)− σ′(V T x)V T x)rT − Γwk‖e‖W − Γaσ(V T x)σT (V T x)W. (44)

In the above adaptive law, the last term is the ALR term with Γa > 0 being the ALR gain. Theorem 1 in

[42] shows that for a system affine in control with matched linearly parameterized uncertainty, if ∂σ(V T x(t))∂x(t)

has full column rank, there exists a Γ∗a such that if Γa > Γ∗A x(t) asymptotically tracks the reference statesxrm. Furthermore, there exists k1 and α such that ‖WTσ(x)‖ ≤ K1e

−αΓat + O(1/Γa). This means thatW(t) is driven into a subspace within which the constraint in (42) is approximately satisfied, exponentiallyin time. The speed with which this happens is proportional to the ALR gain, and the accuracy to which theconstraint is satisfied is inversely proportional to the ALR gain.

The ALR design procedure consists of selecting the ALR gain Γw sufficiently large such that the cost in(43) remains sufficiently small for all t > t1 > 0, where t1 is also made sufficiently small by a sufficientlylarge ALR gain. While a similar modification term can be found for the hidden layer weights V , it was foundthat the approximate satisfaction of 42 can be achieved with only modifying the outer layer weights using(44). It is further shown in [45] that increasing the ALR gain does not amplify the effect that sensor noisehas on the adaptive control law as compared to the effect sensor noise would have had if Γa were equal tozero.

A. Flight Test Results using ALR under Actuator Time Delay

The effectiveness of ALR in handling time delay was tested in flight on the GT Twinstar aircraft. Timedelay was injected in the elevator command until unacceptable performance was observed, which was charac-terized by limit cycles with high frequency and unacceptable magnitude.46 The time delay above which theunacceptable performance was observed is referred to here as the time delay margin. It was found throughflight test results that the time delay margin without ALR was about 0.08 seconds, and with ALR about0.12 seconds. Figure 22 shows the evolution of pitch rate as time delay of 0.11 seconds is added and the ALRterm is toggled. As the time delay is added, we see the onset of undesirable limit cycles in the pitch rate, atabout 31 seconds into flight. When the ALR term is switched on a reduction in undesirable oscillations isobserved. At 65 seconds, the ALR term is switched off again, and we notice a reappearance of the undesirableoscillations.

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0 10 20 30 40 50 60 70 80 90-1

-0.5

0

0.5

1

1.5

pitch rate

pitch

rate,

rad/s

flight time

Time delay of 0.11

seconds addedALR turned ON

Vehicle

performs

a turn

ALR turned

OFF

Pit

ch R

ate

(ra

d/s

ec)

Time (sec)

Figure 22. Evolution of pitch rate with injected time delay.

22 of 25


IX. Conclusion

We described outer-loop guidance and inner-loop attitude control algorithms that ensure safe waypointfollowing autonomous flight of a twin engine aircraft in the presence of severe structural damage or actuatordamage. A specifically designed state dependent guidance law for subsonic aircraft was described and usedto ensure that the aircraft is commanded a feasible trajectory in the presence of damage. The guidancelaw established a feedback between inner-loop guidance and outer-loop guidance to ensure that the aircraftacceleration commands were modified if the attitude commands exceeded predetermined limits. This providesa simple way to ensure that the aircraft commands do not leave the safe-flight envelop. However, since nofault detection algorithm is implemented onboard, the attitude limits must be conservatively chosen to ensurereasonable compromise between performance in nominal conditions and safety with damage. With betterknowledge of the damage, a more accurate estimate of the feasible envelop could be generated online. Thisis expected to improve the performance significantly.

A baseline linear attitude controller with the state dependent guidance logic was tested in flight with 50%of the aircraft’s right wing jettisoned in mid flight. The baseline controller was also tested in a propulsiononly control scenario, in which all aerodynamic effectors of the aircraft were frozen. A single hidden neuralnetwork based model reference adaptive attitude controller along with the state dependent guidance logicwas tested in flight with 25% of left wing missing. The Adaptive Loop Recovery modification to adaptivecontrol was verified in flight to allow the adaptive controller to tolerate larger actuator time delays. Theflight-tests were performed on the GT Twinstar UAV which is a small foam built airplane with aerodynamiccharacteristics that resemble a typical transport category aircraft (twin-engine, propeller driven, un-swepthigh mounted wings). This work highlighted the fact that outer-loop guidance methods that ensure feasibletrajectories are commanded to the aircraft are critical in ensuring safe autonomous flight in the presence offaults. These results indicate the possibility of using autonomous flight control methods for ensuring safeflight, and in some cases safe automated landing, of aircraft with severe structural damage.

Acknowledgments

This work was supported in part by NSF ECS-0238993 and NASA Cooperative Agreement NNX08AD06A.The authors thank Jeong Hur, Research Engineer I at Georgia Tech. Jeong was the lead test-pilot for severalof the flight test results presented.

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