American Institute of Aeronautics and Astronautics

1

Integrated Flight/Structural Mode Control for Very Flexible Aircraft Using L1 Adaptive Output Feedback

Controller

Jiaxing Che*, University of Connecticut, Storrs, CT 06269

Irene M. Gregory† NASA Langley Research Center, Hampton, VA 23681

Chengyu Cao ‡ University of Connecticut, Storrs, CT 06269

This paper explores application of adaptive control architecture to a light, high-aspect ratio, flexible aircraft configuration that exhibits strong rigid body/flexible mode coupling. Specifically, an 1L adaptive output feedback controller is developed for a semi-span wind

tunnel model capable of motion. The wind tunnel mount allows the semi-span model to translate vertically and pitch at the wing root, resulting in better simulation of an aircraft’s rigid body motion. The control objective is to design a pitch control with altitude hold while suppressing body freedom flutter. The controller is an output feedback nominal controller (LQG) augmented by an 1L adaptive loop. A modification to the 1L output feedback is

proposed to make it more suitable for flexible structures. The new control law relaxes the required bounds on the unmatched uncertainty and allows dependence on the state as well as time, i.e. a more general unmatched nonlinearity. The paper presents controller development and simulated performance responses. Simulation is conducted by using full state flexible wing models derived from test data at 10 different dynamic pressure conditions. An 1L adaptive output feedback controller is designed for a single test point and

is then applied to all the test cases. The simulation results show that the 1L augmented

controller can stabilize and meet the performance requirements for all 10 test conditions ranging from 30 psf to 130 psf dynamic pressure.

I. Introduction he next generation of efficient subsonic aircraft will need to be both high aspect ratio and light weight, and the

associated structural flexibility presents both challenges and opportunities. An example of advanced vehicles under consideration is illustrated in Fig. 11. Robust active aeroelastic control and Gust Load Alleviation (GLA) have a potential to substantially reduce weight by relaxing structural strength requirements on the wings. However, current tools such as traditional linear aeroelastic and aeroservoelastic analysis methods are inadequate to reliably predict aeroelastic stability and assess active aeroelastic control effectiveness in this type of vehicle. The type of vehicle represents a nonlinear stability and control problem involving complex interactions among the flexible structure, unsteady aerodynamics, flight control system, propulsion system, the environmental conditions, and vehicle flight dynamics. Furthermore, because of the inherent flexibility of the aircraft, the lower order structural mode frequencies are of the same order as the rigid-body mode frequencies. The close proximity of flexible and rigid-body * Ph.D Student, Department of Mechanical Engineering, University of Connecticut. AIAA member. Jiaxing.Che@engineer.uconn.edu † Senior Research Engineer, Dynamic Systems and Control Branch, MS 308, NASA Langley Research Center, Irene.M.Gregory@nasa.gov, AIAA Assoc. Fellow. ‡ Assistant Professor, Department of Mechanical Engineering, University of Connecticut. Ccao@engr.uconn.edu

T

https://ntrs.nasa.gov/search.jsp?R=20120014594 2020-04-19T08:24:26+00:00Z

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American Institute of Aeronautics and Astronautics

3

the spar, strain gauges at the root and mid-spar, a rate gyro at the wing tip, and a rate gyro and accelerometers at the tunnel attachment point.

The mathematical model is linear and includes rigid body translational and rotational displacements and velocities , , ,z w q , as well as twelve flexible modes. The flexible modes are represented by generalized

displacements, i , and velocities, i . There were 10 test points for the flexible vehicle wind tunnel model, each

corresponding to a different dynamic pressure ranging from 30 psf to 130 psf shown in Table 1, and an associated linear model described above. The simulation consists of linear models with third order actuator dynamics, typical of aeroservoelastic models, included for each of the control surfaces in the simulation.

Table 1. Dynamic pressure for different test points.

Test Point

Index 1 2 3 4 5 6 7 8 9 10

Dynamic Pressure (psf)

30 35 40 45 50 60 70 80 90 130

The general structure of the flexible model is given by

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where rx represents rigid body position and rates, ex represents elastic mode deflections and rates, lagx represents

aerodynamic lag states, and x represents actuator states. For control design purposes, the model is residualized to

eliminate lag states and then is further reduced by eliminating higher frequency flexible modes. Furthermore, the actuator dynamics are neglected, and as a result, the control design model is reduced from 112 to 12 state variables consisting of 2 rigid (half-span) modes and 4 flexible modes. Thus, the model used for design has the format

e rrr r r

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rr e

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(2)

where 1 2 3 4[ , , , , ]T

LE TE TE TE TE and y is the output of sensors described in Ref [10]. In order to improve

tracking and add damping into the system, a proportional plus integral (PI) control structure that tracks vertical displacement is chosen for the baseline controller. In order to create a model structure compatible with 1L output

feedback several of the sensor measurements, combined in such a way as to isolate flexible wing modes, are added to the system in Eq. (2) as states. In addition, the integrator on position is augmented to the system in Eq. (2) resulting in a new system structure given by

m m

m

x A x B u

y C x

(3)

where u is a vector of control inputs (one leading edge and four trailing edges), y is a vector of sensor and

integrator outputs, and 16 16 16 5 8 16, ,m m mA R B R C R are matrices with appropriate dimensions.

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IV. 1L Augmentation Adaptive Controller Design

Consider the following system dynamics

( ) ( ) ( ) ( ) ( )

( ) ( ( ), ( ), )

( ) ( )

z w gx t Ax t z t w t Bu t

z t f x t z t t

y t Cx t

(4)

where ( ) nx t R are system states, ly R are output variables available for feedback, ( ) pu t R are control signals,

( ) mz t R are bounded input bounded output (BIBO) unmodeled system dynamics, ( )gw t is a bounded external

disturbance, , ,A B C are unknown matrices of appropriate dimensions, , ,z wA represent unknown parameters

with given conservative bounds, and f is a Lipchitz continuous unknown function.

The control objective is to design an augmentation adaptive output feedback controller, using only available measurements, ( )y t , such that ( )y t tracks a given bounded reference input ( )r t with desired transient and

asymptotic performance. Note that given a Hurwitz matrix mA such that the triple , ,m m mA B C is controllable and observable, Eq. (4) can

be rewritten as

( ) ( ) ( ) ( )

( ) ( ( ), ( ), )

( ) ( )

m m

m

x t A x t B u t t

z t f x t z t t

y t C x t

(5)

where ( )t is a vector of time varying signals that includes all the uncertainties and the disturbances, i.e

( ) ( ) ( ) ( ) ( ) ( ) ( )m z w g mt A A x t z t w t B B u t (6)

The design process is to use the parameters , ,m m mA B C , from Eq. (3),to design an LQG baseline controller to

achieve the desired tracking performance in the nominal system. Then an 1L output feedback controller is designed

to deal with the model uncertainties and the disturbances in the ( )t term.

A. LQG Baseline Controller Design:

The LQG controller is the combination of a Kalman filter with a linear-quadratic regulator (LQR). The detailed design procedure for an LQG is omitted here, but an overview using the above problem formulation is provided below. The Kalman filter is

ˆ ˆ ˆ( ) ( ) ( ) ( )( ( ))

ˆ ˆ( )m m m

m

x t A x t B u t L t y C x t

y C x t

(7)

where ˆ( )x t is the estimated state and ( )L t is the observer gain for the Kalman filter. Tuning the covariance matrix

of process noise and measurement noise in the LQG problem formulation can change the performance of the Kalman filter.

The LQR controller design uses the estimated state, ˆ( )x t . The baseline control law is defined as

ˆ( )u Kx t (8)

The dynamic performance and control effort is tuned by choosing different performance and control effort matrices Q and R. The overall baseline control law is defined in Eq. (8) with the estimated state, ˆ( )x t , generated by Eq. (7).

There are several limitations to the LQG design. The LQG design is optimal when using the real plant model,

, , , ,m m mA B C A B C . In reality , ,A B C are unknown. The LQR controller has limited stability when , ,A B C

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have large deviations from , ,m m mA B C . The white process noise assumption on the Kalman filter is also not

satisfied in this system. Hence, the LQG design has a limited performance and stability margin due to the invalid white noise assumption.

B. 1L Control Augmentation Design:

1. State Predictor:

In addition to Eq. (7), a vector of adaptive parameters, ̂ , is introduced to the predictor.

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )( ( ))

ˆ ˆ( )m m m

m

x t A x t B u t L t y C x t

y C x t

(9)

2. Adaptive Law:

The adaptive law updates ̂ as piece-wise constant signal such that output tracking, ˆ( ) ( )y t y t is ensured.

Prior to introducing the adaptive law, certain variables need to be defined as follows. Since mA is Hurwitz, there exists 0P P that satisfies the algebraic Lyapunov equation

0m mA P PA Q

Q

(10)

From the properties of P , it follows that there exists a non-singular P such that

( )P P P (11)

Given 1( )mC P , let D be an ( 1)n n matrix that contains the null space of 1( )mC P such that

1( ( ) ) 0mD C P , and further let mC

D P

.

Let T be any positive constant, which can be associated with the sampling rate of the available CPU, and let

(1 ) RT m

m m mn

n mI O 1 be the matrix with first sub-matrix an identity matrix and all other elements zero ( 11

matrices are designed to match the dimensions in Eq. (13) ). Let ˆ( ) ( ) ( )y t y t y t be output tracking error, then the

update law for ˆ ( )t is given by

1

ˆ ˆ( ) ( ), [ , ( 1) )

ˆ ( ) ,

0,1,2,

t iT t iT i T

iT T iT

i

(12)

where ( )T and ( )iT are defined as

1 1( )

10( ) e , ( ) e ( ) , 0,1, 2,3, .m m

T A T A TT d iT y iT i 1 (13)

Letting ( )n n mumB R

be the null space of n m

mB R , so that 0T Tm umB B

and ( )=nm umBr Bank , the adaptive

parameter ˆ ( )t can be further decoupled into matched and unmatched terms

1ˆ ( )ˆ ( )

ˆ ( )m

m umum

iTB B iT

iT

(14)

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In the following control law design, the matched disturbance, ˆ ( )m iT , and unmatched disturbance, ˆ ( )um iT , are

filtered with different filter bandwidths. This approach defining and updating compensation for matched and unmatched uncertainty in the state feedback formulation has been successfully flight tested on NASA GTM aircraft, e.g. see Ref [11].

3. Control law design:

In addition to the baseline LQG control signal, bu , in Eq. (8), there are two augmentation control signals, ( )mu t

and ( )umu t , for matched and unmatched uncertainties respectively. This control law is inspired by a successful

application in Ref [11]. In this paper the control law is further modified to be suitable for highly flexible vehicles. The overall control law is generated as

( ) ( ) ( ) ( )m um bu t u t u t u t (15)

1 ˆ( ) ( ) ( )m mu s C s s (16)

2 ˆ( ) ( ) ( )um umu s C s W s (17)

where 1

1( ) ( )m mH s C sI A B , 12 ( ) ( )m umH s C sI A B , and W is the DC gain matrix of the system

11 2( ) ( )H s H s . 1( )C s , 2 ( )C s are the square low pass filter matrix with unity gain for the diagonal element. 1( )C s is

the filter matrix for the matched channel and 2 ( )C s is the filter matrix for unmatched channel.

The design objective of ( )mu t and ( )umu t is to compensate for the effect of model mismatch. This

decomposition enables filter design of different bandwidths for the matched channel and unmatched channel separately. Compensating the unmatched uncertainties usually leads to stability challenges. By tuning these two filter bandwidths of 1( )C s , 2 ( )C s separately, the stability margin of the controller can be configured to match the

actual range of the system parameter. A detailed analysis is summarized in Ref [12]. An important modification in this paper is that instead of dynamic inversion of 1

1 2( ) ( )H s H s , the DC gain

matrix, W , of the system 11 2( ) ( )H s H s is inverted instead. This modification is necessary because the flexible

vehicle model structure has a high system dimension and high frequency lightly damped modes. Under nominal conditions, dynamic inversion will improve the tracking performance, but the control signal will easily excite the high frequency mode shapes, especially when dealing with high system dimensions with large parameter deviations due to wide range of encompassed flight conditions. The frequencies of the flexible modes are uncertain and the first and second flexible modes have frequencies close to rigid body dynamics and are close to each other making cancellation by inversion completely impractical. In addition, the rank of the controllability matrix is low compared to the real system dimensions, which means many of the high frequency mode shapes are nearly uncontrollable. To avoid exciting the uncertain flexible mode frequencies, the dynamic inversion is changed to a static inversion, thus, only compensating the static error which makes more sense for the flexible structure. This results in a different stability condition for the 1L output feedback controller.

Revisiting the baseline control signal ( )bu s in terms of the system formulation in Eq. (9) is given by

ˆ( ) ( ) ( ( ) ( ) ( ))b g p cu s Kx s K K C s r s y s (18)

where K is the feedback gain from the LQR controller design, ˆ( )x t is the estimated state from Eq. (6), and

1 1K ( )mg CA B . This choice for K g ensures that the diagonal elements of the transfer matrix1[ ( ) K ]m gmC sI A B

have DC gain equal to one, while the off-diagonal elements have zero DC gain. pK is a

proportional design factor used to add damping to the adaptation loop inside the controller and to tune frequency response and robustness margins when there is model parameter deviation. ( )cC s

is a command filter with unity DC

gain. Together with the pK design, the baseline control law, ( )bu t , makes the system behavior close to the

command filter ( )cC s . ( )r s is the reference input signal.

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V. Simulation Results The controller was evaluated in simulation on the original full state linear wind tunnel models derived at 10

different test dynamic pressures. The simulation also includes third order actuator dynamics and time-delay in the sensor feedback loop. The simulation time step and, hence, the sampling time was set at 0.005T . For the purposes of this publication, the magnitude of control deflections and vehicle attitude has been normalized with respect to the trim values.

Let ( , , ), 1,....10ir ir irA B C ir , where 112 112 112 5 8 112, ,A R B R C R represent the full state linear model

derived from the wind tunnel test at a given test dynamic pressure with ir associated with a particular dynamic pressure from Table 1. Let ( , , ), 1, 2,...10mi mi miA B C i represent the reduced order model used for controller design

at a particular test dynamic pressure with i associated with a particular dynamic pressure from Table 1. An LQG controller is designed as the baseline controller for each test point. The particular controller is then

evaluated at different test conditions in addition to the nominal test condition. In other words, a particular baseline control signal ( )bu t , given in Eq. (18), is designed using ( , , ), 10mi mi miA B C i is tested for the plant

( , , ), 1,....10ir ir irA B C ir . The stability results for each LQG baseline controller are given in Table 2 where shaded

cells indicate matched design and evaluation conditions, e.g. LQG controller designed using reduced order model at 130 psf is evaluated on full order linear model at 130 psf (ir=10). As seen from Table 2, in the simulation the baseline controller is limited in its ability to stabilize plant models different from the model used to design the controller. For example, the LQG controller designed for model 10 (130 psf test condition) can stabilize full linear models at 90 and 80 psf (ir=9, 8) as indicated by checkmarks in Table 2, but cannot stabilize models for dynamic pressure below 80 psf (ir=7-1, 70 to 30 psf test conditions). The LQG controller does somewhat better in the middle ranges of tested dynamic pressure by stabilizing the system up to 3 test points below its design condition covering the range of 90 to 60 psf. As dynamic pressure drops towards minimum necessary for flight, the range over which LQG can provide stability also decreases. It is interesting to note that in all case, the baseline LQG stabilizes the vehicle for dynamic pressure at or below its design condition. There are two reasons for the unstable response. One is the large parameter variation with dynamic pressure especially in the control effectiveness. The other challenge is the unmodeled dynamics not captured in the reduced order model used for control law design. In summary, the baseline controller designed with reduced order model i appears to stabilize the system for

, 1, 2,ir i ir i ir i and in some cases for 3ir i , which means when the real plant changes, the baseline controller can work for at most 4 test conditions.

Table 2. Stability of baseline LQG controller designed for test conditions.

Index number of model used for the full state plant (used for evaluation)

1 2 3 4 5 6 7 8 9 10

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In Eq. (18) r R is the command for vertical position in the wind tunnel. Figs. 4 to 10 illustrate the response of the baseline controller designed with ( , , ), 10 (130 )mi mi miA B C i psf when the real plant varies from 130 psf to 50

psf, corresponding to 10ir to 5ir . The discussion is similar to the example describing the results in Table 2. When 10ir i , which means the full state model ( , , )ir ir irA B C is at the same test condition as the design and is

close to the model ( , , )mi mi miA B C used for controller design, the system response in Fig. 4 tracks with no error.

When the real plant is changed to 9,8ir , there are some higher frequency oscillations that become apparent in the

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American Institute of Aeronautics and Astronautics

19

While the results are preliminary since the evaluation models are linear with no significant time delay in the system, they are nevertheless encouraging and warrant further application to more sophisticated fully nonlinear very flexible aircraft models.

VII. Acknowledgments The authors would like to thank John Cooper, Jie Luo in AIM lab at University of Connecticut for their

comments and suggestions.

VIII. References: [1] Del Rosario, R, G. Follen, R. Wahls, “NASA Subsonic Fixed Wing Project Annual Review,” Fundamental Aeronautics

Program Annual Review, 18-19 November 2010. [2] Kharisov, E., N. Hovakimyan, and K. Astrom, Comparison of Several Adaptive Controllers According to Their

Robustness Metrics, In Proceedings of AIAA Guidance, Navigation and Control Conference, Toronto, Canada, AIAA-2010-8047, 2010.

[3] Cao, C. and N. Hovakimyan, “Design and Analysis of a Novel L1 Adaptive Control Architecture with Guaranteed Transient Performance,” IEEE Transactions on Automatic Control, vol.53, No.2, pp. 586-591, 2008.

[4] Cao, C. and N. Hovakimyan, “L1 Adaptive Controller for Systems with Unknown Time-varying Parameters and Disturbances in the Presence of Non-zero Trajectory Initialization Error”, International Journal of Control, Vol. 81, No. 7, 1147–1161, July 2008.

[5] Cao, C. and N. Hovakimyan, “Stability Margins of L1 Adaptive Controller,” IEEE Transactions on Automatic Control, vol. 55, No. 2, pp. 480-487, 2010.

[6] Cao, C. and N. Hovakimyan, “L1 Adaptive Controller for a Class of Systems with Unknown Nonlinearities: Part I,” American Control Conference, Seattle, WA, pp. 4093-4098, 2008.

[7] Cao, C. and N. Hovakimyan, “L1 Adaptive Controller for Nonlinear Systems in the Presence of Unmodeled Dynamics: Part II,” American Control Conference, Seattle, WA, pp. 4099-4104, 2008.

[8] Gregory, I. M. “From Dynamically Scaled High Risk Flight Testing to Learn-to-Fly: Control Enabled Capabilities for Autonomous Flight,”, Aerospace Decision and Control Workshop, Georgia Institute of Technology, 11-12 June 2012.

[9] Cao, C., Naira Hovakimyan,“L1 Adaptive Output-Feedback Controllers for Non-Strictly-Positive-Real Reference Systems: Missile Longitudinal Autopilot Design.” Journal of Guidance, Control, and Dynamics, vol. 32, No. 3, pp.717-726, May-June, 2009.

[10] Gregory, I.M., C. Cao, V. Patel and N. Hovakimyan, “Adaptive Control Laws for Flexible Semi-Span Wind Tunnel Model of High-Aspect Ratio Flying Wing,” in Proceedings of AIAA Guidance, Navigation and Control Conference, AIAA-2007-6525.

[11] Gregory, I. and Xargay, E. and Cao, C. and Hovakimyan, N. “Flight Test of L1 Adaptive Controller on the NASA AirSTAR Flight Test Vehicle,” in Proceedings AIAA Guidance, Navigation and Control Conference 2010.

[12] Hovakimyan, N. and Cao, C. L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation. Siam 2010.