Multiple-Model Adaptive Fault-Tolerant
Control of a Planetary Lander
Jovan D.Bo škovi ć,∗Joseph A.Jackson,†and Raman K.Mehra ‡Scienti fic Systems Company,Inc.,Woburn,Massachusetts 01801
and
Nhan T.Nguyen §
NASA Ames Research Center,Moffet Field,California 94035
DOI:10.2514/1.42719
In this paper we present an approach to fault-tolerant control based on multiple models,switching,and tuning and its implementation to a hardware-in-the-loop simulation of Delta Clipper Experimental dynamics.The Delta Clipper Experimental is characterized by large control input redundancy,which made it an ideal test bed for evaluation of advanced fault-tolerant and adaptive recon figurable control strategies.The overall failure detection,identi fication,and accommod
ation architecture is an upgraded version of our Fast Online Actuator Recon figuration Enhancement (FLARE)system.The FLARE approach is based on representing different possible fault and failure scenarios using multiple observers,such that the case of nominal (no-failure)operation is covered along with the loss-of-effectiveness,lock-in-place,and hardover failures of the flight control effectors.Based on a suitably chosen performance criterion,the FLARE system quickly detects single or multiple failures and recon figures the controls,thus achieving either the original desired performance or graceful performance degradation.In the first stage of the project,the FLARE system was tested on a medium-fidelity simulation of Delta Clipper Experimental dynamics,resulting in excellent performance over a large range of single and multiple faults and failures.Following that,in collaboration with Boeing Phantom Works,the FLARE run-time code was installed at their site and tested on a hardware-in-the-loop test bed consisting of an electromechanical actuator actuating a gimballed engine as a part of a simulation of the Delta Clipper Experimental dynamics.A large number of hardware-in-the-loop simulations were run to cover a dense test-case matrix,including cases of up to 10simultaneous control effector failures.In all cases FLARE was able to quickly and accurately detect the failures and recon figure the controls,resulting in excellent overall system performance.In this paper we describe the Delta Clipper Experimental and its dynamics model,along with the multiple models,switching,and tuning based modi fication of our FLARE system.This is follo
wed by a description of the experimental test bed and a discussion of the results obtained through hardware-in-the-loop testing.
Nomenclature
a =lateral distance from engines ,m
b =longitudinal distance from engines ,m
c =lateral distance from thrusters ,m c 1,c 2=performance index gains
d =longitudinal distanc
e from thrusters ,m F ab =force along a axis due to actuator b ,N G o =nonlinear control derivative matrix
g =acceleration due to gravity,9:81kg m =s 2I j t =performance index of estimator j at time t
J ,J ii =
inertial matrix,elements with J xx ,J yy ,and J zz on the diagonal,kg =m 2
K ,k i =loss-of-effectiveness failure diagonal matrix,i th value m =mass,kg
P i
=magnitude of i th reaction control system thruster,N
r ab
=component along a axis of the b th reaction control
system thruster,m
r
=commands for the reference trajectory,m =s T i
=thrust magnitude from engine i ,N t Fi
=time of failure injection on actuator i ,s u =control inputs,u 1–u 4:engine thrust,N;u 5–u 12:gimbal
angles,rad;u 13–u 16:thrust from reaction control system,N
u c
=commanded control input vector x =system state
x ,y ,z =position coordinates in the inertial frame,m
Ri
=radial angle of i th engine gimbal,rad Ti
=tangential angle of i th engine gimbal,rad =estimator adaptation gain , =normalization matrix,factor
, i =lumped failure parameter diagonal matrix,i th value , i =actuator gain diagonal matrix,i th value,1=s , i =lock-in-place failure diagonal matrix,i th value
ab
=torque about a axis due to actuator b ,Nm !=attitude vector, ; ; T ,rad
I.Introduction
I
N THE recent past there has been much interest in the design and implementation of failure detection,identi fication,and recon-figuration (FDIR)techniques to aerospace vehicles under subsystem and component failures and damages [1–5].Although many of the proposed techniques have been demonstrated as effective control recon figuration strategies for accommodation of different types of failures and damages in aerial vehicles,space vehicles such as plan-etary landers have attracted comparably less attention in the existing literature.
Presented as Paper 7290at the AIAA Guidance,Navigation and Control Conference and Exhibit,Honolulu,HI,18–21August 2008;received 23January 2009;revision received 18August 2009;accepted for publication 20August 2009.Copyright ©2009by Scienti fic Systems Company,Inc..Published by the American Institute of Aeronautics and Astronautics,Inc.,with permission.Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00per-copy fee to the Copyright Clearance Center,Inc.,222Rosewood Drive,Danvers,MA 01923;include the code 0731-5090/09and $10.00in correspondence with the CCC.∗
Intelligent &Autonomous Control Systems Group Leader,500West Cummings Park,Suite 3000;jovan@ssci.Senior Member AIAA.†
Research Engineer,Intelligent &Autonomous Controls Systems,500West Cummings Park,Suite 3000;jjackson@ssci.Member AIAA.‡
President and CEO,500West Cummings Park,Suite 3000;rkm@ssci.Member AIAA.§
Principal Research Scientist,Mail Stop 269-4;Nhan.T.v.Associate Fellow AIAA.
J OURNAL OF G UIDANCE ,C ONTROL ,AND D YNAMICS Vol.32,No.6,November –December 2009
1812
The onboard FDIR problem is particularly important for space exploration vehicles that require minimal downtime for repairs during a mission.Our focus has been on the Delta Clipper Experi-mental (DC-X)[6–10],which was designed by McDonnell Douglas in the early 1990s as a one-third-scale prototype for a proposed vertical takeoff and landing (VTOL)reusable launch vehicle capable of single stage to orbit.¶
This work presents hardware-in-the-loop (HWIL)simulation results using an adaptive recon figurable flight control design for a DC-X control design model consisting of translational and attitude dynamics,
four gimballed engines,four reaction control system (RCS)thrusters for attitude control,and actuator dynamics with position and rate limits.A variety of failures were injected into the model,and our modi fied FDIR system [11,12],introduced in the following paragraph and continued in Sec.IV ,was implemented and tested in hardware-in-the-loop simulations to evaluate the overall system performance [13].Special consideration was given to deter-mining the total number of failures that could be accommodated using the available actuator redundancy.
We have recently modi fied our baseline Fast Online Actuator Recon figuration Enhancement (FLARE)system to include a new failure parametrization [14],as well as a multiple-model failure detection and identi fication (FDI)based upon previous work [15].Central to FLARE,shown in Fig.1,are FDI observers based on the new failure parametrization that describes a large class of failures in terms of a single uncertain parameter.The FLARE system achieves very fast detection and identi fication of failures in flight control actuators and effective control recon figuration in the presence of single or multiple actuator failures and control effector damages even while rejecting external disturbances.The FLARE system combines different FDIR algorithms with a disturbance rejection mechanism within a retro fit control architecture.In collaboration with Boeing Phantom Works,the performance of the previous version of the FLARE system was extensively evaluated usin
g high-fidelity and piloted simulators [11].The FLARE system achieved excellent response in the presence of severe flight-critical control effector failures and received excellent handling quality ratings from the pilot.The FLARE system was used as a basis for FDIR design in the context of the DC-X model.
The sections that follow describe the plant and actuator dynamic model,trajectory design,failure injection,recon figurable control design,and simulation results.
II.Delta Clipper Experimental Model Representation
The DC-X,shown during an early flight test in Fig.2,is actuated by four liquid-propelled engines,which are af fixed to electro-mechanical actuators (EMAs)driving two gimbal angles for each engine,and four RCS thrusters.The basic schematic of the vehicle is shown in Fig.3.
For each engine,three control inputs are available:thrust magni-tude (T ),radial gimbal angle ( R ),and tangential gimbal angle ( T ).The radial gimbal angle is measured positive outward from the craft,whereas the tangential gimbal angle is measured positive counter-clockwise when looking from the nose toward the base of the vehicle.The four engines are numbered,beginning with engine 1along the x axis and engines 2–4proceeding counterclockwise around the perimeter of the vehicle.T
he subscripts e 1 e 4denote the engines,whereas t 1 t 4denote RCS thrusters.
Next we describe a model of DC-X dynamics based upon its detailed simulation provided by the Boeing Phantom Works team.Based upon angular de finitions in Fig.3,the forces acting on the body can be calculated as follows:
Fig.2Delta Clipper Experimental at takeoff (courtesy of NASA).
¶
Data on the McDonnell Douglas DC-X is available online at /wiki/McDonnell_Douglas_DC-X [retrieved 4Sept.2009].
BO ŠKOVI ĆET AL.1813
F xe 1 T 1sin R 1;F ye 1 T 1cos R 1sin T 1
F xe 2 T 2cos R 2sin T 2;F ye 2 T 2sin R 2
F xe 3 T 3sin R 3;
F ye 3 T 3cos R 3sin T 3
F xe 4 T 4cos R 4sin T 4;F ye 4 T 4sin R 4F zei T i cos Ri cos Ti ;
for i 1;2;3;4(1)
where F xe 1de fines the force generated in the x direction by engine 1,and so on.The RCS thrusters are installed so that all resultant forces are applied in the x –y plane of the body frame.Hence,the thrusters are used for attitude control.The resultant thrust vectors from the RCS are designed primarily to give a greater amount of torque to the pitch and roll of the vehicle.The thrusters also produce minimal yaw moment on the vehicle,because the resultant vector from each thruster is not incident upon the vehicle ’s center of gravity.
The contributions of each thruster to the total force acting on the vehicle is calculated as follows:
F xti r xi P i ;
F yti r yi P i ;
i 1;2;3;4
where r xi and r yi are the components of RCS along the body frame x and y axes,respectively,and P i are the thrust magnitudes from each RCS thruster.
For the representation of the plant,the forces are grouped according to the coordinate axis along which they are applied to yield the net force in each direction in body coordinates.Assuming a vertical world-frame orientation,the component force due to gravity is in the negative z direction.The elements of the force vector,F F x F y F z >,are expressed as sums of forces from the engines and RCS thrusters:
F x
X 4i 1
F xei F xti ;F y X 4i 1
F yei F yti
F z
X 4i 1
F zei F zti mg
(2)
Torque is de fined by i r i F i ,with i xi yi zi >.
Given the lateral and longitudinal distances,a and b ,respectively,from the center of gravity to the point at which the engine thrust
exerts forces on the vehicle,the cross products representing the torques from the four engines about the principal axes result in
xe bF ye 1 bF ye 2 aF ze 2 bF ye 3 bF ye 4 aF ze 4 ye bF xe 1 aF ze 1 bF xe 2 bF xe 3 aF ze 3 bF
xe 4
ze aF ye 1 aF xe 1 aF ye 3 aF xe 4
(3a)
Let c denote the lateral distance from the thrusters to the center of gravity,as in Fig.4,and let d denote the longitudinal (out-of-plane)distance from the thrusters to the center of gravity.The sum of the torques from the four thrusters can be computed
as
Fig.3Schematic of DC-X.a)The engines were modeled to be coincident along the x and y axes.As a VTOL vehicle,thrust vectoring intends to maintain vertical orientation during ascent and descent.b)The thrust delivered from each engine can be vectored using a two-axis gimbal mechanism to direct the thrust radially by adjusting R or tangentially by adjusting T .
Fig.4The RCS thrusters are oriented to provide adjustments to the vehicle ’s roll and pitch.Each thruster is characterized by a distance c along the x and y axes from the z axis (z axis assumed coin
cident with the center of gravity)and a distance d along the z axis from the center of gravity.The angles of actuation from the RCS relative to the DC-X are fixed,but the magnitude of the RCS thrust can be changed through commanding.Here E1–E4denote the locations of the engines,whereas RCS1–RCS4denote the locations of the RCS thrusters.
1814BO ŠKOVI ĆET AL.
xt dF yt1 dF yt2 dF yt3 dF yt4
yt dF xt1 dF xt2 dF xt3 dF xt4
zt cF xt1 cF yt1 cF xt2 cF yt2 cF xt3
cF yt3 cF xt4 cF yt4(3b) The overall torque about each body-frame axis is simply the sum of the torque from the engines in Eq.(3a)and the torque from the
thrusters in Eq.(3b).
Let the attitude dynamics be of the form J_! P
,where,due to
small-angle approximation,we assume that! >and that the Coriolis term! J!can be neglected.The system’s position and attitude dynamics are now of the following form:
x 1=m F x; y 1=m F y; z 1=m F z
_ 1=J
xx x;_ 1=J yy y;_ 1=J zz z
(4)
III.Actuator Dynamics Under Failures
The actuator dynamics are assumed to be approximatelyfirst order and are modeled by means of a single gain ai>0.The gains for the engines are an order of magnitude smaller than those of the electro-mechanical actuators.
Three basic actuator fault modes are included in the simulation:
1)lock in place(LIP)(u t is locked at its current position),
2)hardover(u t locks at the position limit),and3)loss of effectiveness(LOE)(actuator gain decreases from k 1to a value of k2 0;1 ).
To simulate engine or gimbal failures,an appropriate failure model is used.First,a LIP matrix is defined as diag 1 2 m ,where i t 1for t<t Fi and i t 0for t t Fi.This matrix is initialized to an n n identity matrix,where n is the number of control inputs.When a LIP failure occurs at the j th input,the value at j;j is set to zero.For convenience,the value at j;j will also be referred to as j.To include LOE within the model,an input effectiveness matrix K is chosen as a diagonal matrix for which the elements describe the effectiveness of each control input.K is also initialized as an n n identity matrix.
Now the actuator dynamics including the failure model is of the form
_u a u K u c I u (5) where u c is the controller output,u is the output of the actuator, a diag a1 a2 am is the matrix of actuator gains,and K diag k1k2 k m ,where k i2 i;1 and i 1.
IV.Failure Detection and Identification
The algorithms for estimating the unknown failure-related parameters associated with DC-X actuator
s are based on the basic FLARE design augmented with the new failure parametrization-based FDI observers and are described below.
A.Estimation of Failure-Related Parameters
1.Observer
The observers for the model[Eq.(5)]are based on the new failure parametrization[14]of the form
_u a u a u c u (6) where diag 1 2 m ,
diag
1
2
m
(7) and0< 1.It is seen that,when i 0,_u i 0.When i
i >0,we have_u i ai u i i u ci ,because u i= i 0.
Hence,this model has the desired properties of covering both the
LIP and LOE cases for a sufficiently small .
Now the observer is chosen in the following form:
_^u
a u a
^ u
c
^ u o~u(8)
where~u ^u u,^ diag ^
1
^
2
^ m ,
^ diag
^
1
^
2
^
m
(9)
and o diag o1 o2 om ,where oi>0.
2.Adaptive Laws
Adaptive laws are of the following form:
_^
i Proj 0;1 f i!i~u i g;^ i 0 1(10)
where Proj f g denotes the projection operator,which keeps the
estimates of^ t inside the interval[0,1]for all time; i>0denote
adaptive gains;and
!i u ci
u i
i 2
As shown recently[14],these adaptive laws result in a stable
estimator.
B.Multiple-Model-Based Failure Detection and Identification
As with many design problems,scaling plays a major role in the
computations of the estimators and adaptation laws.To improve the
performance of the controller,scaling the terms such that the thrust
and angular measurements have a similar order of magnitude is
desirable.Nominal thrusts are of the order of50,000N,whereas the
angles’limits are approximately0.14rad.
An additional issue here is that,due to the size of the thrust signals,
the effect of loss of effectiveness is very different than that of the
lock-in-place failures.It was found that a single FDI observer with a
single adjustable parameter(recall the parametrization described
earlier)is not well suited to cover all the cases of loss of effectiveness
and lock in place.We hence tested a new approach to FDI based on a
multiple models,switching,and tuning(MMST)technique pioneer-
ed by Narendra and Balakrishnan[16].
The concept of MMST is based on the idea of describing the
dynamics of the system using different models for different operating
regimes;such models identify the current dynamics of the system
and are consequently referred to as the identification models.The
basic idea is to set up such identification models and corresponding
controllers in parallel,Fig.5,and to devise a suitable strategy for
switching among the controllers to achieve the desired control
Fig.5Structure of the multiple-model-based controller:outputs of the
parallel observers O1;O2;...;O N are used tofind the observer that is
closest,in some sense,to the corresponding controller from
C1;C2;...;C N.
BOŠKOVIĆET AL.1815
objective.While the plant is being controlled using one of these controllers,the identi fication models are run in parallel to generate some measure of the corresponding identi fication errors and find a model which is,in some sense,closest to the current operating regime of the plant.Once such a model is found,the switching mechanism switches to (or stays at)the corresponding controller,for which the switching interval is a parameter chosen by the designer.
adaptiveThe main feature of this approach is that,in linear time-invariant systems,if the switching interval is chosen to be strictly greater than zero,it results in a stable overall system in which asymptotic convergence of the output error to zero is guaranteed under relatively mild conditions [16].In addition,if adaptive control is used and the controller parameters are adjusted using adaptive algorithms with projection,the overall system is robust to different perturbations,including bounded external disturbances,time variations of plant parameters,and some classes of unmodeled dynamics.As also shown through extensive simulations,the performance of the overall switching system can be dramatically improved as compared with that achieved using a single adaptive controller.Several different structures of the overall switching system are possible.One includes all fixed parallel models and controllers and results in the bounded-ness of the output error provided at least one of the models is suf ficiently close to the model describing the current dynamics of the plant.
Such fixed models and controllers can also be combined with an adaptive controller in the sense that the system first switches to the fixed model closest to the plant and then adapts from there.This results in substantially improved overall performance and in a stable overall system in which asymptotic convergence of the output error to zero is guaranteed.Another possibility is to employ all adaptive models,which yields the same stability result,but is more compu-tationally intensive.
In all these cases,the switching algorithm is implemented by first calculating the following performance indices:I j t c 1k ^e j t k 2 c 2Z t
t o
exp t o k ^e
j k 2d ;j 1;2;...;N (11)
where observer state errors are denoted for the j th observer
by ^e
j ^x j x ,c 1>0,c 2>0,and >0.The scheme is now implemented by calculating and comparing these in
dices every t s instants,finding the minimum,and switching to the corresponding controller.In the case of fixed models and controllers,the overall scheme is guaranteed to result in bounded errors as long as T min s >0,where T min is some minimum-length time interval needed to guarantee error boundedness.
In the context of recon figurable control design in the presence of parametric uncertainties and/or sensor,actuator,and structural fail-ures,the identi fication models (observers)O 1;...;O N from Fig.5correspond to different regions in the parameter space,characterizing different types of failures,whereas C 1;...;C N denote the corre-sponding controllers.
In the context of FDI design for the DC-X model,we used the MMST approach with a model that corresponds to the nominal (no-failure)case and models for LIP and LOE failures.The best estimate at every instant,as selected by the MMST subsystem,is used in the control law.A summary of the equations for each estimator is found in Table 1.
The best estimator is chosen according to the performance indices from Eq.(11),calculated using the errors from each estimator.The
observer with the best (minimum)performance index is used in the
control law.The failure parameter estimate is assigned according to
^ i t (1;if I 1i t is min
k i t if I 2i t is min
i t if I 3i t is min
(12)V.Control Design
The recon figurable control design starts with the design of a baseline controller.The latter is used as a basis for the adaptive controller that at every instant uses estimates generated by FDI observers.We describe the baseline control design next.
A.
Baseline Controller
In the no-failure case,the nonlinear state-space model of the DC-X dynamics can be represented in a compact form as
_x
1 x 2(13)_x
2 G o u 00 g 000 >
(14)
where x 1denotes the states corresponding to the 6deg of freedom in
both translation and rotation,x 2denotes the corresponding rates,and u 2R for the 16actuators.It is seen that this system can be rewritten as
x
1 G o u g (15)
where g
00 g 000 >.
B.
Reference Model Design
A typical test maneuver for DC-X follows a basic climb,x and y
translation,and then a descent path.The desired closed-loop dynamics along the z axis are of the form
z k 2_z
k 1z k 1r z (16)
where z ,_z ,and z represent the desired position,velocity,and acceleration,r z is the command in the z direction,and k i >0.Each of the relationships in Eq.(4)has an associated equation of the form in Eq.(16).The resulting reference model is of the form
_x 1i x
2i
(17)
_x 2i k 1x 1i k 2x
2i k 1r i ;
i 1;2;...;6
(18)
In the no-failure case,the actuator dynamics are of the form
_u
a u u c where a diag a 1 a 2 am and ai >0.
Because G o cannot be inverted analytically,using a previously reported approach [17],we take another derivative of the above equation to obtain:
x :::
1 G u _u
(19)
where G @G o u
@u .
Because the order of the system has increased,the reference model is also changed as
Table 1
Multiple-model-based FDI observers
Actuator failure dynamics
Observer dynamics
Nominal
_u
a u u c _^u a u u c nom ^u 1 u LOE _u
a u K u c _^u
a u diag ^k u c loe ^u u _^k
loe ^e 2u c LIP
_u
a u u c _^u
a diag ^ u u c lip ^u u _^
lip ^e 3
u u c
1816BO ŠKOVI ĆET AL.
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