A Hierarchical Approach to Adaptive Control for Improved Flight Safety

EVALUATION OF AN ADAPTIVE METHOD FOR LAUNCH VEHICLE FLIGHT CONTROL Matthew D. Johnson*, Anthony J. Calise†, Eric N. Johnson‡ School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150 ABSTRACT Future reusable launch vehicles must be designed to fly a wide spectrum of missions and survive numerous types of failures. The X-33 Reusable Launch Vehicle Technology Demonstrator is used as a simulation platform for testing a neural networkbased model-reference adaptive controller and comparing it to a gain scheduled controller through a series of test cases including engine failures, control failures and mismodeling and dispersion cases. The adaptive controller is found to successfully handle a wide variety of failure modes and modeling errors, typically scoring better in both nominal and failure cases, as well as in dispersion tests. NOMENCLATURE A, B B(⋅) C e f (⋅) h I K n P, Q q

η T

W ,V x y Z

δ

∆(⋅) Γ

* † ‡

Error dynamics matrices Control Matrix Output Matrix Model tracking error System dynamics Dead zone of a backlash nonlinearity Identity matrix Diagonal gain matrix Number of states or neural network nodes Positive definite matrices, Lyapunov equation Quaternion Modified model tracking error Transformation Matrix Neural network input, output weights System states System outputs Neural network weights Plant inputs Model error Diagonal matrix containing neural network learning rates

ε κ ν ω σ (⋅)

Model error reconstruction error E-modification parameter Pseudo-control Angular Rate Sigmoidal function

Subscripts Adaptive control Backlash Baseline Commanded/guidance (outer loop) cmd Actuator control (input to actuator) e Effective Demand h Hedge pd , p, d Proportion-derivative linear control r Robustifying rm Reference model

ad b bl c, g

INTRODUCTION The X-33 was a proposed sub-orbital aerospace vehicle intended to demonstrate technologies necessary for future Reusable Launch Vehicles (RLVs) , shown in Fig 1. It included several key features of Lockheed Martin’s proposed VentureStar RLV, including the linear aerospike engine, vertical take-off, horizontal landing, heat dissipation system, and aerodynamic configuration1. To achieve the cost benefits of an operational RLV, the amount of analysis and testing required per mission must be reduced over that performed for the partially re-usable Space Shuttle. A goal for future RLV flight control is to design and test the flight control system to operate within a prescribed flight envelope and loading margin, and requiring only payload/fuel parameters and “route” to be specified for a given mission. It has been estimated that this level of improvement would save three man-years of labor per RLV mission2.

Graduate Research Assistant. Student Member AIAA. Professor. Fellow AIAA. Lockheed Martin Assistant Professor of Avionics Integration. Member AIAA. 1 American Institute of Aeronautics and Astronautics

been done to introduce adaptation in the outer, guidance, loop8. In this previous work a trivial choice of feedback linearization was used based on an estimated inertia matrix, neglecting aerodynamic moments not due to the control effectors. In addition a method for addressing nonlinear system input characteristics, Pseudo Control Hedging (PCH), was introduced to facilitate correct adaptation in the presence of actuator position and rate limits, time delay, and input quantization.

Figure 1. X-33 Reusable Launch Vehicle technology demonstrator. Launch vehicle flight control is conventionally linearized about a series of operating points and then gain-scheduled. These operating points normally include a range of either Mach number, velocity, altitude, time, or some other parameter used to determine vehicle progress with respect to a nominal trajectory. Separate sets of gain tables are often included for abort cases and failure cases, as is the case with the current X-33 design3. Gain scheduling is a very powerful and successful method, but has a distinct drawback for the RLV: the number of required gains to be designed and scheduled becomes very large. If one also imposes the design constraint that these gains must allow for a range of possible missions, payloads, and anticipated failure modes, then the number of required gains can become prohibitive. In recent years, several theoretical developments have given rise to the use of Neural Networks (NN) that learn/adapt online for nonlinear systems4. The use of NN adaptive flight control has been demonstrated in piloted simulation and flight test on the X-36 aircraft5, and in simulation on a civil transport aircraft6. These tests included failures to the flight control system that necessitated adaptation. The fact that this architecture enables adaptation to an arbitrary nonlinear non-affine plant in real-time makes it an attractive candidate to replace RLV gain tables. This approach has the additional benefit that recovery from a class of vehicle component failures can be shown.

The work described here is a continuation of the study described in Ref. 7. The adaptive controller has been improved for evaluation on a broader range of test cases, and adjusted so as to improve its performance in accordance with test criteria established by NASA Marshall Space Flight Center9. The test matrix includes nominal flight, Power Pack Out (PPO), Thrust Vector Control (TVC) failures and mismodeling, aerosurface failures, and reaction control system (RCS) failures. Dispersion cases are investigated for both nominal and abort situations. Across the test matrix the algorithm was scored on actuator deflection magnitude, duty cycle (a measure of control activity) peak body rates, dynamic pressure profile, steady state error, and how closely the vehicle follows the intended trajectory. The adaptive controller presented here was compared in Ref. 9 to a sliding mode controller10, a trajectory linearization controller11, and a reconfigurable In addition, a hybrid direct-indirect allocator12. adaptive controller13 was tested in ascent only but not compared to the other controllers. Ultimately, NASA will compare the adaptive controller presented here with these other controllers again after further refinement. This paper presents a short overview of the adaptive controller, then discusses more recent results, comparing them to NASA’s baseline controller. Efforts to improve the results relative to NASA's criteria are also documented. CONTROL ARCHITECTURE Fig 2 is an illustration of Model Reference Adaptive Control (MRAC)7,14 with the addition of PCH compensation. The PCH compensator is designed to modify the response of the reference model in such a way as to prevent the adaptation law from seeing the effect of certain controller or plant system characteristics.

This approach has been implemented on the X-337 demonstrating adaptation to failures. Also, work has 2 American Institute of Aeronautics and Astronautics

&x& = ν + ∆(x, x& , δ )

Model Reference Adaptive Control The design of the dynamic inversion controller architecture illustrated in Fig 2 is now described. For simplicity, consider the case of full model inversion, in which the plant dynamics are taken to be of the form &x& = f (x, x& , δ )

where ∆(x, x& , δ ) = f (x, x& ,ν ) − fˆ (x, x& ,ν )

(6)

fˆ is assumed to have a known inverse and obeys the control effectiveness sign condition

(1)

where x, x& , δ ∈ ℜ n . We introduce a pseudo-control input ν such that the dynamic relation between it and the system state is linear &x& = ν

(5)

 ∂fˆ  ∂f  sign   = sign   ∂δ  ∂δ  

   

(7)

Based on this approximation, the actuator command is constructed as

(2)

where

δ cmd = fˆ −1 (x, x& ,ν )

ν = f (x, x& , δ )

(8)

(3) Approximate dynamic inversion produces a model inversion error that will be adaptively compensated using a neural network trained on-line.

The actual system controls δ are obtained by inverting Eq. (3). Since the function f (x, x& , δ ) is usually not known exactly, an approximation is introduced

ν = fˆ (x, x& , δ )

As shown in Figure 2, the total pseudo-control signal is constructed of four components ν = ν rm + ν pd − ν ad + ν r (9)

(4)

which results in a modeling error, and system dynamics described by

where ν rm is the pseudo-control signal generated by the reference model, ν pd is the output of the linear

νh v rm

ν h = ν − fˆ ( x, x&, δˆ )

ν pd

Approximate δ cmd δ Actuator Dynamic Inversion

δˆ

PCH

xc

x Reference rm + Model -

e

P-D Compensator

ν

− ν ad

e

Adaptive Element

δˆ

Adaptation Law

Figure 2. MRAC including an approximate dynamic inversion with PCH.

3 American Institute of Aeronautics and Astronautics

Plant

x

compensator, ν ad is the signal generated by the adaptive element introduced to compensate for the model inversion error, and the robustifying term ν r is not included for clarity. For this second order example, when there is no actuator saturation or failure and hence no PCH, the commanded pseudocontrol signal generated by the reference model equals to a commanded acceleration ν rm .

PCH with Redundant Controls In cases where there exists redundant actuation, care must be taken when calculated the hedging signal. Consider the following affine system: x& = f ( x) + B ( x)δ

(10)

y = Cx

where x is the state vector, u is the vector of control variables and y represents the regulated output variables. It is assumed that dim{δ} > dim {y}. Since there are more control variables than there are regulated output variables, a control allocation matrix is often introduced that relates the control demand associated with each output variable to the actual controls. Letting δ e denote the effective control demand, then

δ = Ta δ e

(11)

To compute the inversion, differentiate the output once and use (10) and (11) to obtain y& = Cx& = C[ f ( x) + BTaδ e ] = ν

(12)

where ν is the required pseudo control. Assuming that CBˆ ( x)Ta is invertible for all x , then the effective control solution reduces to

δ e = (CBˆ ( x)Ta ) −1 (ν − Cfˆ ( x))

(13)

where Bˆ ( x) and fˆ ( x ) denote estimates of B(x ) and f (x ) . Substitution of (13) into (11) provides the actual control that is to be applied to the plant. The PCH signal is next introduced as an additional input into the reference model, forcing it to “move back”. If the reference model update without PCH was of the form &x&rm = f rm (x rm , x& rm , x c )

(14)

where x c is the external command signal, then the reference model update with PCH becomes &x&rm = f rm (x rm , x& rm , x c ) − ν h

(15)

The instantaneous pseudo-control output of the reference model that is used as an input to the linearized plant model is not changed by the use of PCH and remains

ν rm = f rm (x rm , x& rm , x c )

(16)

Hence, the effect of the PCH signal on the pseudocontrol is introduced only through the reference model dynamics. This results from the stability analysis of NN based adaptive control with PCH, detailed in Ref. 7. When implementing PCH, (11) is viewed as defining the commanded control

δ c = Ta δ e

(17)

which may be different from the actual control displacements, due to actuator position and rate limits, and other effects that one might choose to model. For example, with PCH one may also take into account the dynamics of the actuator as well, although if the bandwidth of an actuator is much greater than the design bandwidth set by the command filters, this effect can be ignored. The hedging signal is computed according to the following equation

ν h = ν − C[ fˆ ( x) + Bˆ ( x)δˆ ]

(18)

where the elements of δˆ denote estimates of the actual control positions derived by processing each element of the commanded control obtained from (17) through a model for the corresponding actuator. Linear Compensator Design The ideally linearized plant consists of r pure integrators, where r is the relative degree of the system. The linear compensator is designed so that these r poles are stabilized. This is most often achieved using standard proportional-derivative (PD) compensators, although additional integral action can be incorporated to address steady state performance. The latter could potentially slow down the closed


loop system response in return for improved steady state characteristics. In general, the linear compensator can be designed using any linear control design technique as long as the linearized closed loop system is stable. For the second order system, PD compensation is expressed by

ν pd = [K d

Kp] e

(19)

where the state tracking error is defined by  x& rm − x&  e=   x rm − x 

The

compensator

(20)

gain

matrices

K p ∈ ℜ n and

K d ∈ ℜ n are chosen so that the tracking error dynamics given by e& = Ae + B (ν ad − ∆ )

(21)

are asymptotically stable, i.e., A is Hurwitz. Here, − K d A=  I

− Kp , 0 

I  B=  0

(22)

The linearized closed-loop system is driven by the output of an at least r th - order reference model. The order of the reference model is required for the subsequent stability analysis and the derivation of the adaptation algorithm. The reference model is hedged in the presence of saturation or failure using the pseudo-control hedging methodology that will be presented below. Neural Network for Inversion Error Compensation The dynamic model inversion and thus the nonlinear system linearization are, in general not exact, mainly because the exact nonlinear model is not known or too complex to be implemented. Clearly, simplified inversion functions are advantageous from real-time implementation perspective and thus are often adopted when adequate feedback linearization error compensation is incorporated in the controller design. A nonlinear single hidden-layer (SHL) NN is used to compensate for the inversion error. The SHL NN was chosen because of its universal approximation property.15,16

For an input vector x , the output of the SHL NN is given by

(

ν ad = W T σ V T x

)

(23)

where V and W are the input and output weighting matrices, respectively, and σ is a sigmoid activation function. Although ideal weighting matrices are unknown and cannot be computed, they can be estimated with sufficient accuracy by solving differential equations derived using the Lyapunov stability analysis techniques. The analysis relies on the feedback loop structure presented in Fig 2, and in particular on the linear compensator characteristics, and lead to the following NN weights training rules7:

[(

]

)

W& = − σ − σ ′V T x η + κ e W ΓW

[

V& = −ΓV x ηW T σ ′ + κ e V

]

(24) (25)

where ΓW and ΓV are the positive definite learning rate matrices, σ ′ is the partial derivative of the sigmoids σ with respect to the NN inputs x , and κ is the e-modification parameter. η is defined by

η = e T PB

(26)

Here, P f 0 is a positive definite solution of the Lyapunov equation AT P + PA + Q = 0

(27)

for any positive definite Q f 0 . A and B in the above equations are the tracking error dynamics matrices defined in Eq. (24). SIMULATION RESULTS AND DISCUSSION The control system design is tested over a matrix of 46 cases. There are two nominal test cases, four nominal dispersion cases, as well as two PPO main engine failure cases, as well four PPO failure dispersion cases. There are also four test cases where the TVC system fails (typically command bias is introduced) during ascent and four cases where the RCS system fails (two RCS jets are failed to zero). Also present in the test matrix are seven tests with pure modeling errors, where uncertainty is added to aerodynamic coefficients. There are seventeen aerosurface failure cases representing possible failure


modes, aerosurfaces jamming temporarily or for the duration of the flight, failing to null or to another fixed position, and failing to a 'trailing in the breeze' condition. Finally, there are two dispersion tests where flexible mode filters are added to the rate and attitude error loops. NASA has established failure criteria and a set of scoring criteria for each of these tests. In order for a test run to succeed, all of the interface conditions

δ ascent

 Left Elevon     Right Elevon     Left Rudder     Right Rudder     Left Flap   =  Right Flap      1 Left Throttle      Left Throttle 2     Right Throttle 1    Right Throttle 2

δ entry

 Left Elevon     Right Elevon     Left Rudder    =  Right Rudder      Left Flap     Right Flap    [ RCS Thrusters ]8×1 

1) Absolute value of heading error to HAC (heading angle cone) is less than 10 degrees 2) Range to HAC is 30 +/- 10 nm 3) Altitude is 96000 +/- 7000 ft must be met at the end of the entry phase of flight. Each dispersion test is scored by how many of the individual runs succeeded. Successful tests are then scored based on criteria such as fuel usage, actuator activity, and tracking error. The goal for the control system is to succeed in as many cases as possible, then to fly those cases well with minimum control activity. The X-33 is approximated as the following affine system: x& = Bδ

(28)

where x is the state vector, δ is the vector of control variables and y represents the regulated output variables (3 attitude error angles). Note that the approximation completely ignores the effects f(x) from Equation 10, requiring the adaptive element to approximate all those absent dynamics. The X-33 has separate flight control configurations for ascent, where aerodynamic effectors are complemented by thrust, and descent, where they are complemented by RCS thrusters. As assumed in the example above, dim{δ} > dim {y}. The state variables and control variables as follows x T = [q T , ω T ]

(29)

and

(30)

Control allocation is performed in stages. Using δ e to denote the moment demand, we have

δ = Ta δ e

(31)

where Ta = [T1 ] ⋅ [T2 ] ⋅ [T3 ] ⋅ [TBL ]

(32)

For ascent TBL is the baseline control effectiveness matrix, which is a fixed matrix for our design, T3 is a scaling matrix acting on TBL, and T2 allocates from the commanded axis control to the individual aerodynamic actuators and splits the thrust vector allocation into pitch, roll, and yaw values, and finally T1 allocates from thrust vector pitch, roll, and yaw values to individual throttle commands on the four engines. The matrices are shown below.


1  0  0  0  0 T1 =  0   0  0  0  0

The control estimate δˆ is determined from estimated position and rate limits applied to the command for each actuator.

0 0 0 1 0 0 0 0 0 0 0  1 0 0 1 0 0 0 0 0 0 0  0 1 0 0 1 0 0 0 0 0 0  0 0 1 0 1 0 0 0 0 0 0  0 0 0 0 0 1 0 0 0 0 0  0 0 0 0 0 0 1 0 0 0 0   0 0 0 0 0 0 0 1 0 0 0  0 0 0 0 0 0 0 0 1 0 0  0 0 0 0 0 0 0 0 0 1 0  0 0 0 0 0 0 0 0 0 0 1

The entry phase is largely the same, but there are a few differences. One test criterion for entry is the amount of RCS fuel used. In order to use as little as possible, the aerodynamic surfaces are engaged first, and only after they saturate are the RCS jets engaged. Due to the lack of control power in yaw, the aero surfaces are allocated in a manner that reserves some of their authority for each axis in which they have authority (axis priority). For instance, the flaps are used for yaw control, thus the controller is prohibited from saturating them to achieve roll tracking. Due to a lack of control effectiveness, the rudders are not used at all during entry.

0 0 0 0 0 0 − 1 1   − 1 1 0 0 0 0 0 0   − 1 − 1 0 0 0 0 0 0    1 1 0 0 0 0 0 0   0 0 1 0 0 0 0 0    0 0 −1 0 0 0 0 0    T2 =    0 0 0 −1 −1 0 0 0    0 0 0  0 0 0 −1 1    0 0 0 0 0 −1 0 0     0 0 0 0 0 −1 0 0     0 0 0 0 0 0 − 1 − 1    0 0 0 0 0 0 − 1 1  TVS   0   0   0 T3 =   0   0    0   0

0 0

0

0

0

0

1 0

0

0

0

0

0 1

0

0

0

0

0 0

AS

0

0

0

0 0

0

AS

0

0

0 0

0

0

AS

0

0 0

0

0

0

AS

0 0

0

0

0

0

RCS jet selection is performed by selecting a jet combination that minimizes a cost function that penalizes the deficit in command to achieved torque for tracking, the total number of thrusters fired to reduce fuel usage, and the change from the previous to the current actuator command to penalize thruster on/off firings (activity). However, in test cases where RCS thrusters were failed, it is beneficial to relax these penalties. This will be explored further below.

0   0   0   0   0   0    0   AS 

Where TVS and AS are factors that ‘weight’ the matrix according to dynamic pressure. To implement hedging in this case, Equation 18 becomes:

(

v h = v − Ta ⋅ Ta T ⋅ Ta

(

)

)

−1

δˆ

(33)

−1 is C ⋅ Bˆ from Equation 20. where Ta ⋅ Ta T ⋅ Ta There is a slight difference between the two, as the command used is essentially a moment command, and the desired units for the pseudo-control are angular acceleration. Thus the transformation matrix is simply scaled by the appropriate moment of inertia in each axis, and any other terms are handled by the adaptive term. This also acts as a scaling-based on moment of inertia, since as the fuel burns, the moments of inertia decrease on the order of 30-50%.

The adaptive control system was tested over the entire flight control test matrix. Scoring over that matrix can be seen for both controllers in Figure 3. In the majority of the cases, both the baseline controller and the adaptive controller passed the tests. In most cases where both controllers passed, the baseline controller scored slightly higher than the adaptive controller. Both control methods performed well on cases 1 and 2, the nominal flight cases, and 37 – 40, the nominal dispersion cases. For the PPO cases, 3, 4, and 41-44, the adaptive controller generally shows superior performance. Both systems performed well in the presence of TVC failures 5-8, although the baseline controller scored slightly better. Both control methods were competitive in the aerosurface failure cases 9-25 except tests 16 and 17 (flap nulled), 20 and 21 (flap jammed), and 24 (flap 'trailing in the breeze) which neither controller could pass. Flap failure cases where the failure was temporary or occurred only during ascent could be successfully flown, however. The failed flap case is especially difficult in entry as the X-33 primarily uses differential flap for yaw control authority, and has little excess control in yaw.


1.0 0.9 0.8 0.7

Scores

0.6 0.5 0.4 0.3

NASA PID

0.2

ADAPTIVE

0.1 0.0 0

5

10

15

20

25

30

35

40

45

Figure 3. Test results for Adaptive vs. Baseline controllers. RCS failure cases 26-29 were an area where the baseline performed better than the adaptive algorithm. Of interest is case 26, which is a failure in RCS jets 1 and 10 that leads to loss of yaw. This is a case for which the relaxation of the RCS cost function limitations on fuel use is required in order for the adaptive controller to succeed. Finally, for the Monte Carlo dispersion cases, which have both baseline dispersions and PPO dispersions, a general point can be made that the adaptive controller outperformed the baseline. A comparison of the cases failed by each controller is presented in Table 1, along with short descriptions of these particular test cases. Finally, note that Table 1 shows the overall score for the entire battery of tests for the NASA baseline is 64.5% while the adaptive controller scores a 66.0%. Figs 4 – 6 show command tracking results in entry for test case 26 for the NN adaptive controller, with two choices for the RCS cost function. Results labeled 1 show the responses where the cost function is relaxed assuming RCS thrusters are known to have failed. Only the deficit in torque is penalized. Results labeled 2 show the responses where the cost function penalizes fuel use and the number of thruster firings, as well as deficit in torque. Result 2 shows divergence in roll near 280 seconds, while still maintaining tracking in the angle of attack and sideslip angle tracking until nearly a minute later.

Table 1: Comparison of Success/Failure for the Controllers. Case

Description

Baseline

Adaptive

3

36 sec PPO

Fail

Fail

4

50 sec PPO

Fail

Success

Fail

Fail

Fail

Fail

Fail

Fail

Fail

Fail

Fail

Fail

Fail

Success

64.5%

66.0%

16 17 20 21 24 42

Right flap fails to null at 20 seconds into flight Left flap fails to null at 215 seconds into flight Right flap jams at 170 seconds into flight Left flap jams at 280 seconds into flight Right Flap fails to "trailing in the breeze" 112 sec PPO, 100 dispersion Monte Carlo set Weighted score over the test matrix

While roll tracking is poor, especially after 350 seconds, the system eventually returns to the commanded trajectory. In Result 1, the aircraft maintains the desired angle of attack and sideslip angle trajectories. This choice causes slightly worse scores on the RCS test criteria such as fuel use and total number of thruster firings on tests 26-29, but allows the vehicle to survive test 26. This result


highlights the need for more research in the area of control allocation. Figures 7 – 12 show responses for both the NASA PID baseline controller and the adaptive controller for test case 4, which is a PPO failure at 50 seconds. This failure causes the navigation algorithm to create an abort trajectory to another landing site, which is dependent on the aircraft's state when the PPO occurs. 40

α c α 1 α

35

2

Angle of Attack (deg)

30 25 20 15 10 5 0 -5 200

250

300

350

400

450

500

Time (sec)

Figure 4. Angle of Attack tracking in Entry – with cost function relaxation (α1 ) and without (α 2 ) .

Therefore, both control designs will have different command trajectories to follow, which are plotted separately. Figures 7 and 8 show the angle of attack tracking for both control designs. The adaptive response is more oscillatory between 320 and 370 seconds due to very low dynamic pressure in the range from 15 psf. to 30 psf. After dynamic pressure increases to values over 100 psf., the adaptive controller tracks well. Similarly, the sideslip angle response for the adaptive controller shows more oscillation at low values of dynamic pressure, then tracks better for the remainder of the trajectory. The roll angle plots have very dissimilar trajectories, with the adaptive system getting much larger bank angle commands. Neither method tracks roll command well when the dynamic pressure is low and only RCS jets provide control authority. Though both control systems appear to track the commanded trajectory well, the NASA PID controller fails this test due to it's inability to satisfy test criteria 2. The range to HAC for the baseline controller is 41.3 nm, over a mile outside of the required range of 30 +/- 10 nm. The adaptive controller satisfies this criteria with a recorded value of 29.0 nm. 40

2

β c β 1 β

1.5

α c α

35

2

b

30


Sideslip Angle (deg)

1 0.5 0 -0.5

25 20 15 10

-1

5

-1.5

0

-2 200

-5

250

300

350 Time (sec)

400

450

500

Figure 5. Sideslip Angle tracking in Entry - with cost function relaxation (β1 ) and without (β 2 ) .

300

Figure 7. Baseline.

350

400

450 Time (sec)

500

550

Angle of Attack tracking in Entry –

40 φ c φ 1 φ

100

35 30


2

Roll Angle (deg)

50

0

-50

25 20 15 10 α C α

5

a

0

-100 200

-5

250

300

350

400

450

300

350

500

Time (sec)

Figure 6. Roll Angle tracking in Entry - with cost function relaxation (φ1 ) and without (φ 2 ) .

Figure 8. Adaptive.

400

450 Time (sec)

500

550

Angle of Attack tracking in Entry –


2

10 β C β

1.5

5

b

0

1 0.5

Roll Angle (deg)


-5

0 -0.5

-10 -15 -20 -25

-1 -30

-1.5

φ C φ

-35

b

-2

300

350

400

450

500

-40

550

300

350

400

Time (sec)

Figure 9. Baseline.

Sideslip Angle tracking in Entry –

450 Time (sec)

500

550

Figure 11. Roll Angle tracking in Entry – Baseline. 60 φ C φ

2 β C β

1.5

a

20

Roll Angle (deg)

1


a

40

0.5 0

0

-20

-0.5 -40

-1 -1.5 -2

-60

300

350

400

450

500

550

Time (sec)

Figure 10. Adaptive.

300

350

400

450

500

550

Time (sec)

Figure 12. Roll Angle tracking in Entry – Adaptive.

Sideslip Angle tracking in Entry – CONCLUSIONS

ACKNOWLEDGEMENT

This paper presents a comparison between a neural network-based model-reference adaptive controller and a gain scheduled controller over a matrix of test cases including engine failures, aerosurface and RCS control failures and mismodeling, vibrational mismodeling, and dispersion cases. The adaptive controller is found to successfully handle a wide variety of failure modes and modeling errors. Results for two choices of RCS cost function are considered, one that allows fuel use and thruster firing penalties to be relieved in the case that RCS thrusters have failed, and the more flexible cost function is chosen. The adaptive controller, while generally scoring slightly lower than the baseline on individual tests, scores better across the entire test matrix due to the fact that it passes two of the cases that the baseline does not. Both controllers perform poorly in the presence of engine failure (abort) cases, and future work will be done to the adaptive controller to address the identified areas of potential improvement.

Research supported by NASA Marshall Space Flight Center, under grant number NAG 8-1638. REFERENCES 1

Hanson, J., Coughlin, D., Dukeman, G., Mulqueen, J., and McCarter, J. Ascent, Transition, Entry, and Abort Guidance Algorithm Design for X33 Vehicle. AIAA Guidance, Navigation, and Control Conference, 1998. 2

Hanson, J. of NASA MSFC. Communication. April 3, 2000.

Personal

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Hall, C., Gallaher, M, and Hendrix, N. X-33 Attitude Control System Design for Ascent, Transition, and Entry Flight Regimes. AIAA Guidance, Navigation, and Control Conference, 1998. 4

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