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FAULT TOLERANT STRUCTURED ADAPTIVE MODEL INVERSION Monish D Tandale*
Aerospace Engineering Texas A & M University, College Station, TX 77843-3141
Abstract
This paper presents a model reference adaptive control formulation that uses the dynamical structure of the state space descriptions of a large class of systems. By taking advantage of the inherent structure, the formulations enable the imposition of exact kinematic differential equation and restrict the adaptation process that compensates for model errors, to the acceleration level. The utility of the resulting adaptive control formulation is studied by considering the problem of fault tolerance to actuator failures on differentially flat systems. Tracking of reference trajectories is imposed while structured parametric uncertainties are incorporated explicitly in both the plant parameters and the control influence matrix. A nonlinear model for the equations of an inverted pendulum with two controls, and linear model of a high performance aircraft are considered. Simulation results are presented which show that the fault tolerant adaptive control is capable of simultaneously handling parametric uncertainties, large initial condition errors, and actuator failures while maintaining adequate tracking performance.
Introduction In recent years, the military has realized the
importance of Uninhabited Aerial Vehicles (UAVs) in intelligence gathering, communications, and force applications.1 There is great appeal in being able to conduct these military missions without placing pilots� lives in danger. Besides saving lives, UAVs can be built cheaper, perform extremely dangerous missions, and are less intrusive than manned flights. To achieve these capabilities, UAVs must be survivable, with respect to both combat damage and system faults and failures. A fault-tolerant robust adaptive flight controller that can accommodate actuator failures, without compromising mission integrity, is one solution that is considered in this paper. After developing the controller, fault tolerant performance in the presence of actuator failure is demonstrated with examples of an inverted pendulum, and a high performance aircraft.
1.1 Adaptive Control All dynamic systems that exist in practice have uncertain parameters that may be constant with respect to time or may be varying with respect to the *Research Assistant. Department of Aerospace Engineering. Student Member AIAA.
changing environment in which the dynamic system is functioning. Because of this variation in the parameters, the performance of the system may degrade, if a fixed parameter controller is used. Performance can be ensured if the parameters of the controller are changed so that it adapts to the change in the plant parameters. Hence, the concept of adaptive control arises.
Therefore, the basic essence of adaptive control is that the controller parameters are variable and there exists a mechanism of updating the parameters of the system online based on the signals of the system. 1.2 Model Reference Adaptive Control (MRAC) In MRAC a reference model is specified which in turn specifies the desired dynamics that the plant is supposed to follow. A feedback controller exists for the plant as usual, but the parameters of this controller are varying. The adaptation mechanism for the controller parameters is driven by the error between the output of the actual plant and the reference model.
Figure 1. Block diagram for Model Reference Adaptive Control (MRAC)
1.3 Structured Model Reference Adaptive Control (SMRAC) The dynamics of most systems can be represented in the form of a second order differential equation. The second order differential equation can be separated into a kinematic part and a dynamic part. The kinematic equations are accurately known, as the uncertain system parameters such as mass,
ym Reference
model
Plant
Adaptation Law
EstimatedParameters
r
u
y
Controller
e
41st Aerospace Sciences Meeting and Exhibit6-9 January 2003, Reno, Nevada
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moment of inertia etc do not enter these equations. These uncertainties exist only in the momentum level dynamic equations. Therefore, it makes sense to restrict the adaptation to these momentum level equations.
Consider the equation for Newton�s second law of motion F=ma.This second order differential equation can be written as
x v=&
F=ma
Fvm
=&
Figure 2. Kinematic and Dynamic Expansion of Newton's Second Law
If the dynamics of the system is modeled
correctly, the kinematic equation is perfectly known. The dynamic part has the unknown parameter �mass� and the accuracy of this equation will depend on how accurately the value of this parameter is known. This unknown parameter can be guessed for and adaptive schemes can be used to update these parameters to achieve stability. In SMRAC, the kinematic and the dynamic parts of a mathematical model for the system are identified and the update is restricted only to the unknown parameters in the dynamic part. As the name implies, SMRAC takes advantage of this structure inherent in dynamical systems. 1.4 Structured Adaptive Model Inversion (SAMI)
Structured Adaptive Model Inversion (SAMI) is based on the concepts of Feedback Linearization, Dynamic Inversion, and Structured Model Reference Adaptive Control (SMRAC). In SAMI, dynamic inversion is used to solve for the control. The dynamic inversion control law assumes perfect knowledge of all the system parameters, as the actual system parameters are required in calculating the control. However, the system parameters are not known accurately. An adaptive controller structure is wrapped about the dynamic inverter to account for the uncertainties in the system parameters. The controller is designed to drive the error between the output of the actual plant and that of a model reference to zero, as in model reference adaptive control. The adaptation included in this framework can be limited to only the momentum level states. Thus, the structured flavor is retained in
the controller. The closed loop is shown to be globally asymptotically stable for trajectories without singularities; however, the adaptively estimated parameters do not converge to the actual parameters of the system. 2.1 The Actuation Failure Problem:
In high performance dynamic systems the total number of actuators used may be greater than the number of states to be closely controlled or tracked. This control redundancy generally exists to achieve optimality with respect to control effort. Consider a case wherein the number of actuators is more than the number of states to be tracked. So mathematically, there are more number of variables than the number of equations to be satisfied. Hence, infinite solutions exist. All these solutions have varying control energy requirements, hence the solution with the minimum control energy can be selected and optimality can be achieved.
In this case of redundant actuation it is still possible to closely track the desired states even if some of the actuators fail as long as the number of active actuators is more than or equal to the number of states to be tracked. So if it is possible to reconfigure the control after failure, the stability and performance of the system can be maintained. 2.2 Common strategy to handle Actuation Failure
Most Actuator Failure schemes employ some form of failure detection algorithm to detect the failure. Then the new control effectiveness matrix is estimated and the controller is redesigned by recalculating the control gains. However, this approach depends strongly on the efficacy of the failure detection algorithm. The algorithm may fail to detect a failure or may give false warning, when there is no actuation failure.
In contrast, the SAMI controller is constantly updating its parameters so it does not need to identify the failure. Therefore, SAMI is a good method to address the problem of actuator failure.
2.3 Mathematical modeling of Actuator freezes The actuator failures commonly encountered in aircraft and re-entry vehicles such as the X-38 are control freezes in which the control surface freezes or remains fixed at a position that may or may not be zero. In such a case, the remaining active actuators must not only compensate for the lack of the desired control effort of the failed actuator, but also cancel the undesired control effect produced if the actuator freezes at any position other than zero. Control freezes can be modeled by the following mathematical model:
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applied calculatedD= ⋅ +u u E (1)
where appliedu & mcalculated R∈u , m mD R ×∈ and
mR∈E Where D is a constant matrix and E is a constant vector for a particular control configuration, but they change if failure occurs.
1 11 1 1
2 22 2 2
3 33 3 3
0 00 00 0
a c
a c
a c
u D u Eu D u Eu D u E
= ⋅ +
(2)
If only control freezes are to be modeled, D should be strictly diagonal. If some actuator fails, the corresponding diagonal term in the D matrix should go to zero and the corresponding element in the E matrix should go to the constant value at which the control surface has frozen.
3. Mathematical formulation of the Fault Tolerant
SAMI controller Consider the mathematical model of the system as follows:
( , ) nR∈G q q& = Vector of non-linear function of the states used to describe the unforced dynamic behaviour,
n mC R ×∈ =Control Influence matrix, mR∈u = Vector of the control inputs. ( m is greater
than n for control redundancy and m is at least equal to n after failure).
The dynamics of the system is assumed to
be modeled accurately, and only parametric uncertainties exist in the model. This second order differential equation can be split up into a kinematic and a dynamic part as
=σ w& (5) ( , ) ( , ) appB= + ⋅w A σ w σ w u& (6)
where nR∈ =σ q = Vector of position level coordinates,
nR∈ =w q& = Vector of velocity level coordinates, 1( , ) M −= ⋅A σ w G and 1( , )B M C−= ⋅σ w
Substituting for appliedu from Eq. 1
( , ) ( , )( , ).
calB DB= + ⋅ ⋅
+w A σ w σ w u
σ w E&
(7)
Consider a reference model for the system:
r r=σ w& (8)
r r r rB= + ⋅w A u& (9) Let the error in the position and the velocity level states be s and x respectively.
r= −s σ σ (10)
r= −x w w (11)
r= −x w w& & & (12)
cal rB D B= + ⋅ ⋅ + ⋅ −x A u E w& & (13) However, the error between the model reference and the plant has to go to zero. Hence,
hA= ⋅ +x x φ& (14) where
hA is a Hurwitz matrix, i.e. all the eigenvalues lie in the open left half plane so that the velocity error dynamics is stable. hA can be selected
arbitrarily, but with proper choice of hA it can be specified how fast the velocity error stabilizes. φ is a forcing function on the velocity error dynamics, which will help in achieving the tracking objective. This is discussed in detail later. Adding and subtracting hA ⋅ +x φ to the right hand side
( )
h cal
r h
A B D BA
= ⋅ + + + ⋅ ⋅ + ⋅− + ⋅ +x x φ A u E
w x φ&
& (15)
Since the quantity in the brackets is known let
r hA= + ⋅ +ψ w x φ& (16)
h calA B D B= ⋅ + + + ⋅ ⋅ + ⋅ −x x φ A u E ψ& (17) Using dynamic inversion to solve for the control
1( ) ( )cal B D B−= ⋅ ⋅ − − ⋅ +u A E ψ (18) If number of controls is equal to number of velocity level states ( m is equal to n ), ( )B D⋅ will be a n n× matrix of full rank and its inverse can be computed. If the system is under-actuated ( m is less than n ), ( )B D⋅ will be rank deficient and its inverse cannot be computed. If the system is over-
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actuated ( m is greater than n ), ( )n m m mB D× ×⋅ will be a n m× (wide) matrix and pseudo inverse of ( )B D⋅ will give a minimum norm solution. If there is actuation failure D will lose rank, as the diagonal element corresponding to the failed control goes to zero. Rank of D is equal to the number of active controls, hence number of active controls after failure should be greater than or equal to n .
11
22
33
0 0
0 00 0
D
DD
( ) ( )cal pinv B D B= ⋅ ⋅ − − ⋅ +u A E ψ (19) 1( ) ( ) ( ( ) )T Tpinv B D B D B D B D −⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ (20)
However, the values of A and B are
unknown, hence guesses for A and B are used, namely estA and estB . Let aC be a gain matrix which will be unity at first, and will be adaptively
updated so that it reaches a value *aC such
that *a estC × =A A . The D matrix that has been
introduced to take care of the control freezes can also account for the parametric uncertainty in B similar to the uncertainty in A discussed earlier.
( )( )cal est a est estpinv B D C B= − ⋅ − +u A E ψ (21)
0a est est cal estC B D B⋅ + ⋅ ⋅ + ⋅ − =A u E ψ (22)
* *
*h a est est cal
est
A C B D
B
φ= ⋅ + + ⋅ + ⋅ ⋅
+ ⋅ −
x x A u
E ψ
& (23)
Substituting for ψ from Eq. 22
*
* *
( )
( ) ( )h a a est
est cal est
A C C
B D D B
= ⋅ + + − ⋅
+ ⋅ − ⋅ + ⋅ −
x x φ A
u E E
& (24)
Let *a a aC C C− = % , *D D D− = % , * − =E E E% (25)
h a est est cal
est
A C B D
B
= ⋅ + + ⋅ + ⋅ ⋅
+ ⋅
x x φ A u
E
% %&
% (26)
Let the tracking error be defined as
λ λ= + ⋅ = + ⋅y s s x s& (27)
where n nRλ ×∈ , is a positive definite matrix. As t → ∞ , if y is driven to zero it is ensured that
0→s and 0→s&
λ∴ = + ⋅y x x& & (28)
h a est est cal
est
A C B D
B λ∴ = ⋅ + + ⋅ + ⋅ ⋅
+ ⋅ + ⋅
y x φ A u
E x
% %&
% (29)
For the error to converge
hA= ⋅y y& where hA is Hurwitz. Gathering all the known parameters and assigning them the value of hA ⋅y
h hA Aλ⋅ + + ⋅ = ⋅x φ x y (30) Which on simplifying gives ( )hAλ= ⋅ ⋅ −φ s x (31)
Consider the error departure function as the candidate lyapunov function. If 1 2 3, , ,P W W W are symmetric positive definite matrices
1
2 3
(
)
T Ta a
T T
V P Tr C W C
D W D W
= ⋅ ⋅ + ⋅ ⋅
+ ⋅ ⋅ + ⋅ ⋅
y y
E E
% %
% % % % (32)
Since E% is a column vector 3
T W⋅ ⋅E E% % will be a scalar, hence there is no need for taking its trace. Taking the derivative of the lyapunov function,
1
2 3
2 (
) 2
T T Ta a
T T
V P P Tr C W C
D W D W
= ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅
+ ⋅ ⋅ + ⋅ ⋅ ⋅
y y y y
E E
&% %& & &
& &% % % % (33)
1
2 3
2(
) 2 (
) 2
T T Th h
T T T T Test cal est
T T Test a a
T T
V PA A P
A C P u D B P
B P Tr C W C
D W D W
= ⋅ ⋅ + ⋅ ⋅
+ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅
+ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅
+ ⋅ ⋅ + ⋅ ⋅ ⋅
y y y y
y y
E y
E E
&
% %
&% %%
& &% % % %
(34)
P is selected such that
Th hP A A P Q⋅ + ⋅ = − (35)
where Q is positive definite. Also using the identity: If A and B are row and column vectors respectively, then ( )Tr⋅ = ⋅A B B A .
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1
2
3
2 (
) 2 (
)
T T Ta est
T T T Ta a est cal
T T Test
T
V Q Tr C P
C W C D B P
D W D B P
W
= − ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅
+ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅
+ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅
+ ⋅ ⋅
y y y A
y u
E y
E E
%&
&% % %
&% % %
&% %
(36)
1
2
3
2 ( ( )
( ))
( )
T
T Ta est a
T T Test cal
T Test
V Q
Tr C P W C
D B P W D
B P W
= − ⋅ ⋅
+ ⋅ ⋅ ⋅ ⋅ + ⋅
+ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅
+ ⋅ ⋅ ⋅ + ⋅ ⋅
y y
y A
y u
E y E
&
&% %
&% %
&% %
(37)
*
a a aC C C= −% cannot be calculated because the values of the actual parameters are not known and
hence *aC is unknown. So the coefficient of aC% must
go to zero. Retaining only the negative definite part T Q− ⋅ ⋅y y and setting all other terms to zero.
11 ( )T
a estC W P y A−∴ = − ⋅ ⋅ ⋅&% * 1
1 ( )Ta a estC C W P−∴ − = − ⋅ ⋅ ⋅y A& & (38)
However, *aC is assumed constant. Hence
11 ( )T
a estC W P−= ⋅ ⋅ ⋅y A& (39) Similarly
12 ( )T T
est calD W B P−= ⋅ ⋅ ⋅ ⋅y u& (40) 1
3 ( )TestW B P−= ⋅ ⋅ ⋅E y& (41)
These are the update equations for the various adaptive learning parameters. If these update equations are imposed, V& is restricted to be negative definite, the system will follow a trajectory so that the tracking error as well as the parametric error reduce. However, V& is a function of the tracking error only, so when the tracking error becomes zero, the system parameters do not update. The closed loop is shown to be globally asymptotically stable for trajectories without singularities; however, the adaptively estimated parameters may not converge to the actual parameters of the system during the duration of the maneuver.
Since the parameters like *aC are assumed to
be constants, this formulation works only when the plant parameters are constant with respect to time or slowly time varying as compared to the rate of update of the adaptive parameters. For the same reason this
controller can only handle failures where the control effort remains constant at any value.
4. Control of the Inverted Pendulum on a Cart The Fault Tolerant SAMI controller can now be tested on the classical inverted pendulum on a cart problem. The inverted pendulum on a cart is a nonlinear 2 degree of freedom system. A nonlinear fault tolerant SAMI controller will be formulated and its performance will be evaluated. This controller can handle parametric uncertainties. It is assumed that the parameters of the system are constant, but unknown. Guesses for these parameters are used and these parameters are adaptively updated so that stability is assured. These parameters may not converge to their actual values as discussed earlier.
A failure is simulated, wherein the force on the cart freezes to a constant value that may or may not be zero. The controller is not supplied with the time of failure and the mathematical formulation of the controller does not change before and after the failure.
Figure 3. Geometry Of Inverted Pendulum On Cart With Two Controls
Mass of the Cart = M Mass of the Pendulum = m.
Controls: Force on the cart = F Torque on the Pendulum = T
The mathematical model for the inverted Pendulum is:
2L
x
Force
Torque
X
Y
θ
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2
2
coscos
sin 0 11 0sin
I ml mlml M m x
mgl FTml
θ θθ
θθ θ
+ = + + ×
&&&&
&
(42)
I is the moment of inertia of the pendulum about its end.
The model is of the form ( , ) ( , )M q q q G q q C u⋅ = + ⋅& && & (43)
The simulation is done with the following parameters:
1. Variation in the parameters: Actual
Parameters Estimated/Guessed Values
1 Mass of the Cart
5 kg 6 kg
2 Mass of the pendulum
1 kg 0.5 kg
3 Length of pendulum
5m 8 m
2. The error in the initial conditions :
Actual values Reference values 1. Angular
Position 87 deg 67 deg
2. Angular Velocity
0 deg/sec 0 deg/sec
3. Simulation of failure: At time t=3 sec, the
actuator corresponding to the force on the cart fails and the force attains a constant value of 10 N.
Simulation results for model inversion with
and without adaptation are presented in Fig. 3, 4 and 5. As expected, Fig. 3 shows that the controller with adaptation is able to effectively track the reference trajectory in spite of the actuator failure. The top portion of Fig. 4 shows that the controller effectively uses the control torque while the force control fails at 10N. Fig. 5 shows the convergence of the system gains.
Figure 4. States of Inverted Pendulum on Cart that
are tracked
Figure 5. States of Inverted Pendulum on Cart that
are not tracked
0 2 4 6 8 10 12 14 16 18-40
-20
0
20Controls
Time
U calculatedU applied
0 2 4 6 8 10 12 14 16 18-200
-100
0
100
Figure 6. Controls of Inverted Pendulum on Cart
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0 2 4 6 8 10 12 14 16 180.8
1
1.2
1.4
1.6
1.8Update of the gain Ca
0 5 10 15 200.5
1
1.5Update of the gain D
0 5 10 15 20-0.5
0
0.5
0 5 10 15 20-0.5
0
0.5
0 5 10 15 200
0.5
1
1.5
0 2 4 6 8 10 12 14 16 18-0.04
-0.02
0
0.02Update of the gain E
0 2 4 6 8 10 12 14 16 18-0.1
0
0.1
Figure 7. Adaptive Learning Gains of Inverted
Pendulum on Cart 5. Control of the longitudinal (linear) dynamics of
an F16-Xl aircraft The fault tolerant SAMI controller can now be tested on the linear model of an F16-Xl aircraft at 0.9 M and 25,000 ft. The aircraft has three controls δae=differential elevon, δee=symmetric elevon and δlef=symmetric leading edge flap.
A simulation is done in which it is attempted to track two states, the velocity (u) and the pitch angular velocity (q), as that of the reference. In this simulation also there are parametric uncertainties in the A and the B matrices of the linear model and the
1
1
1
e
LEF
T
M 0.9H 25,000 feet
4.61q 446.3 psf
0.874.67
54.5%
==
α ==
δ = °δ = − °δ =
AE
u -0.0133 -10.672 -0.712 -32.098 u-0.000076 -0.786 0.987 0
symmetric leading edge flap fails at time=5sec and settles down to a value of �10 degrees. As in the previous example, both adaptive and non-adaptive results are presented. Fig. 6 shows that with adaptation the two reference states are tracked almost perfectly, even with an actuator failure.
Figure 8. F-16XL State Tracking Time Histories Fig. 7 shows that the states which are not being tracked remain bounded and well behaved. The time history of the controls (Fig. 8) indicate that the differential and symmetric elevon are easily able to permit tracking in the presence of a leading edge flap failure, as is to be expected with a multivariable system with multiple redundant controls. Fig. 9, 10 and 11 show the learning of the system and updates of the parameters and gains.
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Figure 9. F-16XL Non-Tracked States Time
Histories
Figure 10. F-16XL Update History of Gain Ca
Figure 11. F-16XL Update History of Parameter D
Figure 12. F-16XL Update History of Parameter D
Figure 13. F-16XL Control Position Time Histories
Conclusions This paper developed a fault tolerant model reference adaptive control formulation that takes advantage of it's inherent structure to impose exact kinematic differential equation constraints upon the adaptation process, that compensates for model errors. The system was simulated for actuator failure cases of an inverted pendulum on a cart, and a high performance aircraft. Based upon the preliminary results presented in this paper, the following conclusions are made:
1. The above controller formulation is able to handle parametric uncertainties, initial error conditions and actuator failures.
2. This controller can handle only those parametric uncertainties that are constant with respect to time or slowly time varying as compared to the rate of update of the adaptive learning parameters of the system.
3. The controller can handle actuator freezes, in which the control effort remains constant. This constant value may or may not be zero. This is with the assumption that the system is controllable with the remaining active controls and the control effectiveness of the remaining controls is large enough to
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compensate for the lack of effort produced by the failed control and to counter the undesirable control produced when the actuator settled to an arbitrary value.
4. For the non-adaptive formulation actuator failure may result in loss of tracking or total instability as seen from the examples, but the adaptive controller is able to achieve close tracking in presence of actuator failure.
5. The dynamic inversion methodology places no restriction on the magnitude of the required control, which result in large control magnitudes and control saturation. Some control scheme needs to be devised to handle this control saturation.
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Vehicles Directorate , AF Research Lab, AF Material command, WL/FIGC, WPAFB, OH 45433-7542.
