Controllers for Disturbance Rejection for a Linear Input-Varying Class ...

Controllers for Disturbance Rejection for a Linear Input-Varying Class of Morphing Aircraft

Kenneth Boothe, Kristin Fitzpatrick and Rick Lind Department of Mechanical and Aerospace Engineering University of Florida

Control of morphing aircraft is particularly difficult due to an inherent nonlinearity in the closed-loop equations of motion. This nonlinearity arises due to the dependence of the dynamics on the control input. The equations of motion for closed-loop dynamics will thus multiply a state-dependent matrix with the states which, consequently, will be a nonlinear function of the states. This paper introduces the linear input-varying framework to describe such systems. Essentially, this framework represents systems whose dynamics are linear when the input is constant but will vary as the input is varied. A pair of controllers are formulated for these types of systems to provide disturbance rejection. Several examples of span-varying morphing, chord-varying morphing, and camber-varying morphing are used to demonstrate the properties of the resulting closed-loop systems.

I. Introduction The concept of morphing is quite natural for aeronautical systems. Birds and insects routines change their shape during flight for propulsive and aerodynamic reasons. The types and shapes of such morphing is quite extensive and ranges across a broad spectrum of biological systems. These creatures change the shape due to rotation around skeletal joints, translation of bones, and of course muscle contractions. Morphing is a control effector in that the shape is changed to alter the flight dynamcis. As such, a simple form of morphing is wing twist. This approach was used for control on the early Wright flyer where the pilot directly twisted the wing using cables. The idea is also used for control on the Active Aeroelastic Wing whre the wing is twisted in response to moments induced by control surfaces.33 .26 The variations can be extended to include other parameters such as span, chord, camber, area, thickness, aspect ratio, planform, and any other metric related to shape.

Recent programs by DARPA40 and NASA44 have focused on aerodynamic performance41 and materials.43 In particular, these programs have demonstrated clear benefits of morphing as measured by aerodynamic performance related to lift and drag.20 Aircraft have been optimized11 using morphing to improve lift and minimize drag along with fuel consumption,6 range and endurance,14 cost and logistics,7 actuator energy,34 and maneuverability.35 Additionally,

Graduate Student, [email protected] Graduate Student, [email protected], Student Member AIAA Assistant Professor, [email protected], Senior Member AIAA

Copyright c 2005 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

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aeroelastic effects have been often studied relative to maximum roll rate 3, 16, 24 and actuator loads.28 Some investigations have considered control of morphing systems; however, they do not fully address manuevering flight. A study using piezoelectric materials designed control only to achieve roll . 25 A variety of other studies have investigated control but only in the context of acutation energy19 and control authority10 for simple changes in flight condition. Other studies have considered morphing for control surfaces bur not associated control synthesis. 36 37 Also, a study designed nonlinear controllers but its morphing model used simply a distributed set of control effectors rather than a shape with fully time-varying dynamics.39

The issue of control design for morphing systems has partly been overlooked because of the lack of flight-capable testbeds. NASA has constructed a hyper-elliptic wing with morphing camber. 12, 42 A set of smart spars are built that provides many types of morphing.2 Another model uses an adaptive planform to dramatically change shape. 31 Mechanisms are also built such as a telescoping spar to morph aspect ratio 4, 5 and an inflatable actuator to change sweep.29 In each case, the mechanisms are too heavy for a flight system so tests are limited to a static mount in a wind tunnel. An inflatable concept has been taken to flight8, 9 but the morphing is a one-time inflation; hence, it is not used for maneuvering control. Micro air vehicles present an attractive platform because their lightweight structures make morphing relatively easy; hence, several vehicles have been flown that incorporate morphing mechanisms for maneuvering.1, 15, 17, 27 This paper presents an approach for designing controllers for morphing aircraft. The critical element to this approach is a framework within which to represent the flight dynamics. Essentially, the flight dynamics are expressed as linear functions of the shape. The resulting models are described as linear input-varying (LIV) to reflect this functional dependence. A pair of controllers are formulated to provide disturbance rejection for this LIV class of systems. Several examples are presented to demonstrate the LIV framework and the associated control formulations. These examples utilize a common nominal configuration that is based upon an existing family of micro air vehicles. These nominal configurations are then augmented to introduce span-varying morphing, chord-varying morphing, and cambervarying morphing. Controllers are designed for each type of morphing are designed and their effectiveness is evaluated for gust rejection. Also, the examples in this paper are restricted to consideration of the longitudinal dynamics. The LIV framework is actually appropriate for longitudinal and lateral-directional dynamics. In this case, the restriction to longitudinal dynamics is sufficient to investigate issues related to control synthesis. The models are not assumed to be high-fidelty representations of the a complete morphing system; rather, they are assumed to account for a set of representative effects that will be present in a morphing system.

II. Flight Dynamics of Morphing Morphing can be considered in both a dynamical and quasi-static sense. In either case, the aircraft will undergo shape transformations but the rate of that transformation is actually what distinguishes them. The dynamical type of morphing, such as wing twist on the F/A-18 AAW,33 occurs on a short time scale and can be used for maneuvering. The quasi-static type of morphing, such as wing sweep on the F-14, changes more slowly and is used mostly for optimization as the aircraft transitions between operating regimes. A full derivation of the flight dynamics for a morphing aircraft, both dynamical and quasi-static, is quite extensive. These models must account for the variety of time-varying quantities such as mass distribution and aerodynamic shape. Additionally, these models must account for the variations in structural load distribution and aeroelastic effects resulting from the shape variation. Such models are high-fidelity representations but are not immediately amenable to control design; instead, reduced-order models can be used for initial investigation that focus on flight control. A basic model can be expressed that accounts for some characteristics of the dynamics of static morphing. Essentially, 2 of 14 American Institute of Aeronautics and Astronautics

the approach is to identify a set of models where each model describes the flight dynamics at a specific morphing configuration. The parameters in each model will vary with morphing so each can be expressed as a finite-order polynomial function. A morphing model then results by expressing a model whose elements are these funtions of the configuration. The effects of morphing on the rigid-body dynamics must be initially evaluated for this type of approach. These dynamics are typically represented as 12 equations of motion involving forces, moments, attitudes and rates. The morphing actually only directly affects the moments; hence, the moment equations must be augmented to account for shape variations. The nonlinear dynamics relating pitch moment to aircraft parameters can fully be expressed as in Equation 1. This equation is based on an equation that can be found in textbooks; 32 however, they differ dramatically due to the introduction of the morphing dynamics. First, the potential asymmetry requires the inclusion of all moments of inertia. Second, the time-varying mass distribution requires the inclusion of time rate of change of moments of inertia. As such, the equations of motion of a morphing aircraft can not be simplified to a small form like for a traditional aircraft.

!"#$ %& ' (! )*+,.--%!( /0+,1 2 3 (! 04 56 )78

(1)

The equation of motion can then be linearized around a trim condition to express the maneuvering dynamics. The assumption of symmetric morphing, as would be appropriate for longitudinal maneuvering, allows separation of longitudinal and lateral-directional components. The nonlinear relationship in Equation 1 can be linearized as in Equation 2.

:9 9 3 9

(2)

The moment in Equation 2 must be computed to describe the flight dynamics. This term is usually expanded in terms of the states for the linearized model. The resulting expression for the moment can be seen in Equation 3.

9 = ; ; 9 >9 ? ; ; 9 9 ? 9

(3)

Finally, the moment in Equation 3 is combined with the force equations along with attitude and position expressions. to result in the general form in Equation 4. This form uses the standard nomenclature for a rigid-body flight dynamics and maintains the same structure.32 Obviously this model does not account for structural or aeroelastic effects; however, it does include representative issues for flight dynamics. Such a model is suitable for investigating effects of morphing on the flight dynamics and control strategies to account for these representative effects. This aerodynamic and inertial effects, given in Equation 1, are incorporated into the stability derivatives in Equation 4.

ABBB

DFE GH! DFI JGH! N M E @ C E JGH!Q SGHRI !GH! M E JGH! I GH!H MOTI JI GHR J! GH! M I JGH! VU GH!H K K

K W=.P I R JGH! MXU GH! Y

L K K K

Z\[[[

]@

(4)

Models can similarly be formulated to account for the lateral-directional dynamics. This paper is only concerned with the longitudinal dynamics so the lateral-directional formulation is not presented. The procedure for developing such models follows the same procedure as for the longitudinal models. Specifically, the nonlinear moment expressions are expanded to account for time-varying moments of inertia and the inclusion of asymmetric terms; these expressions are then linearized; and finally the state-space representation is formulated with terms that depends on the configuration. 3 of 14 American Institute of Aeronautics and Astronautics

III. Linear Input-Varying Class

Gf`bO^ c: g

^H_

E ^ = `bOc:h `9 ? ] C 9p

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American Institute of Aeronautics and Astronautics

]

Z [[[

(6)

B. Span-Varying Morphing A model of the MAV with span-varying morphing is developed by assuming a pure variation in span. In this case, the span is allowed to vary between 60.96 cm and 106.68 cm while the chord is kept constant at 11.43 cm and the camber is kept constant. The representative model in TORNADO30 resulted in configurations such as those shown in Figure 2 3−D Wing configuration

Wing z−coordinate

3−D Wing configuration

Wing z−coordinate

Wing z−coordinate


0.1 0.05

0.1 0.05

0.1 0.05 0 0.5 0.4

0

0.3

0.4

0.2

0.3

0

0.1

0.2

0.3

0

0.1

0.2

−0.1

0

0.1

0.2 −0.2

Wing y−coordinate

Wing y−coordinate

0.1 −0.3

0

0.4 0.3

−0.3

0.2 −0.3

Wing y−coordinate

0.1 −0.4

Wing x−coordinate

−0.2

0.3

−0.2

0.3

−0.1

0.4

−0.1

0.4

0

0.2

−0.4

0.1

−0.5

0

0

¹

Wing x−coordinate

Wing x−coordinate

Figure 2. Span-Varying MAV Model ( )

The flight dynamics of this vehicle are represented in the LIV framework by noting variations due to span. A set of state-space models are generated to account for the flight dynamics at a set of span values. The state-space matrices are then fit as a function of span to a first-order matrix polynomial. These models do not include the effects of any control surfaces, such as elevator or flaps, so the flight dynamics can be represented as in Equation 7.

º»»» BABB *Ku Ke½oY/y K u y|¾{ K @ ¼ C 2 Y3you {3u K¿|Y/y y ·*Y)o¿ { u K3u ½|¿yY Á 2%Y :y yKou u K3¿:K Á|À K K Y

*¿u ÀY K K K

Z\[[[ BABB *Kou K3K|¾:½ oK uY)¿|t ] C * KKou y3uÂt:K3½3½ Á ·*KeY uu ¿e½|¿3t:À{ K K

K K K K Y3u K3{etÃK K K

Z\[[[ Ä ÅÅÅ ] GÆ @

(7)

The model in Equation 7 satisfies the form needed to synthesize a controller using Lemma 1 and Lemma 2. Also, the constant portion of the LIV dynamics is negative definite which implies the nominal system is stable. A pair of controllers, whose gains are given in Table 1, are thus chosen using the synthesis conditions.

>9 = 9>? 9 9p

w

Lemma 1 0 1 0 -0.1

wP w}

Lemma 2 0

0 1 0 -0.1

Table 1. Gains for Span-Varying Aircraft Disturbance Rejection Controller

Simulations of the span-varying vehicle are shown in Figure 3 to demonstrate the behavior after a wind gust. The closed-loop responses are shown along with the open-loop response to indicate the effectiveness of the control strategy. In each case, the initial disturbance to angle of attack is quickly rejected and the system settles back to a trim condition. An obvious feature in Figure 3 is the inability of the controller to affect the responses. Essentially, the closed-loop responses mimic the open-loop responses. One cause of this similarity is that the controller is not optimized for any 7 of 14 American Institute of Aeronautics and Astronautics

6 5

Lemma 1 Lemma 2 Open−Loop

105

4 100

Span (cm)

Angle of Attack (deg)

110


3 2

95

90

1

85

0 −1 0

0.5

1

1.5

Time (s)

80 0

2

0.2

0.4

0.6

0.8

1

Time (s)

1.2

1.4

1.6

1.8

2

Figure 3. Angle of Attack Response and Associated Span of Aircraft

level of performance but merely guarantees closed-loop stability. Another cause of the similarity is that span does not have an overly pronounced effect on rejecting disturbances to angle of attack. The values of span throughout the simulation clearly show that this parameter was indeed varying even though the angle of attack did not change between closed-loop and open-loop responses. The complete set of states are shown in Figure 4. These responses further demonstrate the inability of the controllers to affect the system. For each state, the open-loop response is nearly indistinguishable from either of the closed-loop responses. 1.2


0.3


1

Vertical Airspeed

Forward Airspeed

0.4

0.2

0.1

0.8 0.6 0.4 0.2 0

0 0

0.5

1

Time (s)

0.1

2

0.5

1

Time (s)

1.5

0.02


−0.1 −0.2

2


0.01

Pitch Angle

0

Pitch Rate

1.5

−0.2 0

0 −0.01 −0.02 −0.03

−0.3 −0.04

−0.4 0

0.5

1

Time (s)

1.5

2

−0.05 0

0.5

1

Time (s)

Figure 4. Responses to Span-Varying Morphing


1.5

2

C. Chord-Varying Morphing Another model of the MAV is generated by assuming a pure variation in chord. This model indicates some properties of the flight dynamics of a chord-varying morphing while the span and camber are kept constant. The chord for this model is chosen to range from 11.43 cm to 22.86 cm while the span remains at 60.96 cm. The representative model in TORNADO30 resulted in configurations such as those shown in Figure 5.

0.1 0.05 0 0.3 0.2


Wing z−coordinate


Wing z−coordinate

Wing z−coordinate


0.1 0.05 0 0.3

0.1 0.05 0 0.3

0.2 0.1

0.2 0.1

0.4

0

0

Wing y−coordinate

0

0.2 −0.2

0.1 −0.3

Wing x−coordinate

0.3

−0.1

0.2 −0.2

0.1 −0.3

0.4

0

0.3

−0.1

0.2 −0.2

Wing y−coordinate

0.1 0

0.3

−0.1

0.4

Wing y−coordinate Wing x−coordinate

¹

0.1 −0.3

0

Wing x−coordinate

Figure 5. Chord-Varying MAV Model ( )

The LIV framework is again used to represent the flight dynamics. The model in Equation 8 indicates a first-order fit to the state-space matrices for the flight dynamics at a set of chord values.

º»»» BABB *Kou K|¾3tÇ*KuvY%yÁ K @ ¼ C · yoY3u ½euÂt:À3{e½ À 72to¾uu ÁÀ3Áet:½À *%Y e{ yyu u K3¿3K {3Á K K eY u KeK

Z\[[[ BABB *Kou {3{eK 2you Ke¿3{ K K ] C * y3to{ou u{|¿eyt|Á ¾È·¿eY/ttouY)u {eÀ{o¾ Y · Y)½eK KouY)½ÉKK K K K K

*¿u ÀY K K K G K w

Z\[[[ Ä ÅÅÅ ] GÆ @

(8)

A pair of controllers are designed using Lemma 1 and Lemma 2 for the model in Equation 8. This model satisfies the conditions of the Lemmas including stability for . The resulting gains for these controllers are given in Table 2.

>9 = 9>? 9 9p

Lemma 1 0 -1 0 0.1

wP w}

Lemma 2 0

0 -1 0 -0.5

Table 2. Gains for Chord-Varying Aircraft Disturbance Rejection Controller

Responses to a gust for the open-loop system and closed-loop systems are shown in Figure 6. The open-loop system returns to trim to demonstrate the expected stability when the chord is not varied from the nominal condition. The closed-loop systems, using Lemma 1 and Lemma 2, are also stable and able to reject the initial disturbance. The responses in Figure 6 indicate that chord-varying morphing can indeed be beneficial to rejecting gusts since each controller returns the vehicle to trim faster than the open-loop system. The increase in rejection is perhaps not overly significant but at least is clearly noticeable. The controller changes the chord which actually has the effect of reducing the moment of inertia and, subsequently, increasing the stability derivatives. As the actuation plot shows, the controller reduces the chord to reject this disturbance but then can quickly increase the chord to return to aerodynamic efficiency. 9 of 14 American Institute of Aeronautics and Astronautics

6

20


5


19

17

Chord (cm)


18 4 3 2

16 15 14

1

13 0

12

−1 0

0.5

1

1.5

Time (s)

2

11 0

0.5

1

1.5

Time (s)

2

Figure 6. Angle of Attack Response and Associated Chord of Aircraft

The ability of the controller to return the vehicle to trim is also evident in the responses of all states as shown in Figure 7. Each state returns to equilibrium although the rates of decay in the oscillations after the initial disturbance clearly vary between responses. 1.2


0.3 0.2 0.1 0

0.8 0.6 0.4 0.2 0

−0.1 0

0.5

1

Time (s)

0.05

1.5

−0.2 0

2

1

Time (s)

1.5

0

Pitch Angle

−0.1 −0.15 −0.2

2


0.005

−0.05

−0.005 −0.01 −0.015 −0.02 −0.025 −0.03

−0.25 −0.3 0

0.5

0.01


0

Pitch Rate


1

Vertical Airspeed

Forward Airspeed

0.4

−0.035

0.5

1

Time (s)

1.5

2

−0.04 0

0.5

1

Time (s)

1.5

2

Figure 7. Responses to Chord-Varying Morphing

The behavior of the controllers, and consequently the closed-loop responses, can be determined from Figure 7 in association with Table 2. In particular, this association can explain the differences between actuation responses in Figure 6. The chord variations using the controller of Lemma 1 show an oscillatory nature whereas the chord variations from the controller of Lemma 2 settles to a constant steady-state value. The different behaviors is caused by the linear and quadratic nature of the controllers. Essentially, a small chord is commanded by scaling small values of the states but an even smaller chord is commanded by scaling quadratic values of those small states.


D. Camber-Varying Morphing A model is also generated to account for a camber-varying type of morphing. This model considers the flight dynamics for a vehicle whose camber varies between 8% and 17% while the span remains at 60.96 cm and the chord remains at 11.43 cm. Plots of the wing shape for the model as generated by TORNADO 30 are shown in Figure 8. 40 30 20

Camber (%)

10 0 −10 −20 −30 −40 0

0.1

0.2

0.3

0.4 0.5 0.6 Chord (normalized)

0.7

0.8

0.9

1

Figure 8. Camber-Varying Wing Model (side)

The flight dynamics of this model are cast into the LIV framework. The resulting equation is given in Equation 9.

º»»» BABB *KouY/t3K oK uÂye¾:{ K @ ¼ C 2 Y3yu À3uÂyÀ3Y/{ K ·*Y%o½ y uÂyu ÀoY%Y%¾ yÇ2Y/y3yyou uK3{eK ½3½ K K Y3u K3K

*¿ou ÀoY K K K

Z [[[ BABB KouÂy:½ey * KouÂt:ÀoY ] C ·y YeuÂy3uÂ¾:y:¿3À K * oK Kou ½:uYÁ|Áe¿ À K K

K K K K *Kou ½ey|¾ÊK K K

Z [[[ Ä ÅÅÅ ] GÆ @

(9)

A pair of controllers are computed to stabilize the dynamics in Equation 9. The gains for these controllers are derived using Lemma 1 and Lemma 2 and described in Table 3.

>9 = 9>? 9 9p

w

Lemma 1 1 100 -0.1 -1

wP w}

Lemma 2 1.5

1 10 -1 -1

Table 3. Gains for Camber-Varying Aircraft Disturbance Rejection Controller

The closed-loop responses to an initial disturbances along with the open-loop response are given in Figure 9. These responses indicate the open-loop system and both closed-loop systems are indeed stable and all responses return to equilibrium. Most importantly, the variation in camber is clearly able to affect the system such that the closed-loop responses reject the disturbance significantly faster then the open-loop response. The entire set of states in response to the initial disturbance is shown in Figure 10. The closed-loop systems are stable and the responses all decay to trim; however, the responses are clearly oscillatory for every state except the vertical velocity. 11 of 14 American Institute of Aeronautics and Astronautics

6 5


16

4

15

3

Camber (%)


17


2

14 13

1

12

0

11

−1 0

0.5

1

1.5

Time (s)

2

10 0

0.2

0.4

0.6

0.8

1 time (s)

1.2

1.4

1.6

1.8

2

Figure 9. Angle of Attack Response and Associated Camber of Aircraft

0.6

0.2 0

0.8 0.6 0.4 0.2 0

0.5

1

Time (s)

0.1

1.5

−0.2 0

2

0.5

1

Time (s)

1.5

0.06


−0.1 −0.2 −0.3

2


0.04

Pitch Angle

0

−0.4 0


1

0.4

−0.2 0

Pitch Rate

1.2


Vertical Airspeed

Forward Airspeed

0.8

0.02 0 −0.02 −0.04

0.5

1

Time (s)

1.5

2

−0.06 0

0.5

1

Time (s)

1.5

2

Figure 10. Responses to Camber-Varying Morphing for Forward Velocity (upper left), Vertical Velocity (upper right), Pitch Rate (lower left) and Pitch Angle (lower right)

The controllers show some interesting behaviors. They clearly have a desirable transient effect by rejecting the disturbance but they also have an undesirable steady-state effect associated with long oscillations. These long oscillations have low amplitude so do not present significant variation in angle of attack; however, the oscillations in the remaining states are more pronounced. These oscillations result from a choice of controllers that strove for performance as determined by angle of attack rather than the other states. A different set of controller gains could be chosen that may reduce the oscillations in all states but possibly also reduce the rate of disturbance rejection.


VI.

Conclusion

The linear input-varying (LIV) framework has been introduced to describe a class of morphing aircraft. This framework admits flight dynamics that affinely depend on the morphing configuration but yet are traditional state-space models for a constant value of morphing. The use of state feedback results in a nonlinear closed-loop system and, consequently, eliminates use of the myriad of linear techniques for control synthesis. A pair of designs are presented that guarantee stability of first-order functions in the LIV framework. Several examples of simplified systems with representative morphing are used to demonstrate that the controllers do not necessarily result in performance but closed-loop stability for gust rejection is certainly achieved.

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References

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