American Institute of Aeronautics and Astronautics 092407

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Formation Flight Control Using Model Predictive Approach

Zhao Weihua1, Tiauw Hiong Go

2, and Eicher Low

3

School of Mechanical and Aerospace Engineering,

Nanyang Technological University, Singapore 639798

A leader-follower formation flight control using Model Predictive Control (MPC)

approach is investigated in this paper. In this formation control scheme, the changes in the

leader motion are considered as measured disturbances and the commands to the wing

aircraft are considered as manipulated variables. A cost function for the formation flight

control problem is obtained and the input and output constraints are included. The control

stability is established by adding a terminal state region to the optimization constraints. In

the closed-loop system, commanded separation trajectories are asymptotically tracked by

each wing aircraft while the lead aircraft is maneuvering. A sliding mode approach is

incorporated in order to compensate the effects generated by the vortex of the adjacent lead

aircraft. The applications of the controller to the formation flight of multiple aircrafts are

then simulated. The results show that the MPC controller can successfully maintain the

formation of the flight with only the leader is commanded by the Ground Control Station

(GCS).

Nomenclature

V = velocity

= heading angle

h = altitude

i = the ith

aircraft in the formation T = denotes transposition of vector

c = subscript denotes commands

= time constant

x = x separation

y = y separation

z = z separation

HP = prediction horizon

I. Introduction

igrating birds fly in regular V-shaped formations almost at the same flight level in order to reduce the power

needed for each following bird. Similar to such formation flying mode, research activity in aircraft formation

flight has increased substantially in the last few years. The aerodynamic benefits of formation flight have been well

documented1. Aircrafts in formation flight can potentially be useful for increased surveillance coverage, better target

acquisition and increased security measures.

Various approaches have been developed for formation flight control. Ref. 2 discussed the formation control of

linear models of aircraft, where it is assumed that Mach-, heading-, and altitude-hold autopilots have been

established in the inner-loops. This assumption is also used in this paper. A linear PID mixer controller has been

developed in Ref. 3. In this case, since the constraints are not taken into account, there is no guarantee on the

performance obtained in the presence of the constraints. In Ref.4, 5, the aerodynamic interaction effects between the

leader and the follower aircrafts is studied and a PI controller has been developed. It is shown that the wingman will

be affected by the vortices produced by the leader. An energy conserving formation maneuver and the second-order

C-130 aircraft model have been discussed in Ref. 6. While the heading and altitude response was significantly

1 Graduate Student

2 Assistant Professor. Senior Member AIAA.

3 Associate Professor. Senior Member AIAA.

M

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-59

Copyright © 2009 by Zhao Weihua, Tiauw Hiong Go, and Eicher Low. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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improved using second-order models, the author found that the velocity response was better represented using a

first-order model with a larger time constant. In Ref. 7, a new structured distributed control method applied to a set

of experimental apparatus representing aerial vehicles flying in close formation for the purpose of induced drag

reduction has been discussed. The performances when the centralized controller, decentralized controller and

distributed controller are applied to the formation flight problem are compared.

Model Predictive Control (MPC) is one of the frequently applied advanced control methods in industry. It is

traditionally applied to plants where the dynamics are slow enough to permit a sampling rate amenable to optimal

input computations between samples. Recently, with the advent of faster modern computers, it has become possible

to extend the MPC method to systems exhibiting relatively fast dynamics. MPC can also handle constraints

relatively easily8, 9

Applying MPC to formation control has been reported in Ref. 10, where a dual mode MPC was used for the

ground robot formation. To ensure the stability, the dual model controller has to switch from an MPC controller to a

terminal state controller. MPC has also been used in the aerial vehicle formation control11,12

. In Ref. 12, the

formation commands (heading angle, speed, etc) are given to each agent and the control algorithm is used to reduce

the pilots' stress during the formation flight. In that work, MPC is used for maintaining the formation given a

specified trajectory in the presence of gusts or other disturbances.

In this paper, the MPC is applied differently. In the aircraft formation flight considered here, it is assumed that

the speed and direction commands from the Ground Control Station (GCS) are only delivered to the leader aircraft.

The MPC is used by the followers to maintain the formation based on the changes of the leader states obtained from

measurements. To our best knowledge, the application of MPC in such fashion has not been covered in the

literature.

The paper is organized as follows. Section II introduces the leader-follower formation model. The MPC

framework is described in section III, in which the manipulated input variables, measured disturbance variables, and

output variables are defined. The framework covers the cost function and the inclusion of the constraints. The

analytical stability proof for the resulting controller will also be addressed. In section IV, The sliding mode approach

is incorporated to make the controller more robust to the coupling effect between the aircrafts. Simulation results are

provided in section V. Finally, the conclusions and future works are discussed in section VI.

II. Decentralized Leader-Follower Formation Model

A centralized controller would impose high costs in computation and communication. Moreover, a centralized

controller can not prevent the collision between wing aircrafts. Hence this work treats the formation control problem

using decentralized scheme which means that each wingman communicates with only one neighboring aircraft in the

formation flight (Fig.1 (a)).

In the mathematical formulation, a formation of ( 1)m aircraft is considered. Each aircraft is modeled as a point

mass. The formation coordinates of the ith

aircraft and its leader is shown in Fig 1(b).Similar to Ref. 2, we assume

that standard autopilots have been installed to each aircraft in the formation flight, and their dynamics can be

Figure 1. (a) Decentralized Scheme Figure 1. (b) Leader-follower model

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expressed using first order models. This simplifies the controller design. In the model, the nonlinear effects of the

vortices on the wingman are included. The autopilot dynamics for heading-, mach- , and altitude-hold are described

by ( 0,1,...,i m ):

( )

( )

( )

•T

i ic i i i i i ii

ψi ψi

•T

i i ic iv i i i i i

vi vi

T

i i2 i i1 i ic ih i i i i ih

1 1ψ = - ψ + ψ x , y ,z ,V

τ τ

1 1V = - V + V x , y ,z ,V

τ τ

h = -k h - k (h - h ) x , y ,z ,V

(1)

where the subscript 0 indicates the variables associated with the overall leader aircraft, i.e. the one receiving

commands from the GCS; the ( 1k )th

aircraft is assumed to follow thekth

aircraft. The effect of vortices is modeled

by T

ik i i i i i(x , y ,z ,V ) . According to Ref. 14, the aerodynamic benefits of flying in formation are mainly realized by

the downstream aircraft, i.e. the vortex created by any aircraft only affects the motion of the adjacent wing aircraft,

so the function 0 , { }k k v,ψ,h are zero. and i in s

i ihR R are vectors of the parameters, and ik are vectors of

nonlinear functions of the indicated arguments. Here the aerodynamic parameter vectors i and ih are unknown.

For the kinematics, similar to Ref. 6, only the planar situation is considered. A rotating reference frame fixed to

follower aircraft and with its x axis aligned with follower‟s instantaneous velocity vector WV , and with its y axis

points along the follower‟s starboard wing is used. This is illustrated in Fig.1 (b). By applying the equation of

Coriolis, we get the kinematic equations:

( 1) ( 1)

( )

( )

( )

( )

•T

i i-1 ie i i ic i ψ ix i i i i i

•T

i ic i ψ i-1 ie iy i i i i ii

T

i 2 i 1 i ic (i-1)c ih i i i i ih

T

i h i-1 i-1 i-1 i-1 i h

x =V cosψ -V -(ψ -ψ )(y / τ ) x , y ,z ,V

y = (ψ -ψ )(x / τ )+V sinψ x , y ,z ,V

z = -k z - k (z - h +h ) x , y ,z ,V

x , y ,z ,V

(2)

Where 1 1 and ie i i i i iz h h are the heading error and vertical separation between the

( 1) and th thi i aircraft, respectively, and , ix i iy iy -x . From Eq. (2), if the vortex effect is neglected, the

vertical motion dynamics are decoupled from the horizontal plane dynamics. In this paper, the vortex effect in the

vertical channel is considered as an unknown but bounded function, and a sliding mode controller will be used to

handle the unknown function. In the following, we will synthesize the MPC controller for the planar formation, i.e.

the x and y channels, and the sliding mode controller for the vertical motion.

III. MPC Framework for Planar Motion

For the formation flight plane motion control, each wingman is assumed to be equipped with an MPC controller

which can generate the control inputs to follow the leader in a certain formation. For each leader-wingman

communication vehicle pair, the inputs to the leader are regarded as measured disturbances by dynamic inversion,

the inputs to the wingman are taken as manipulated variables, and the effects of the vortex are considered as the

unmeasured disturbances.

Using the small angle approximation and linearizing the leader-wingman vehicle pair, from Eq. (1) and (2), we

can take the internal model below for our MPC design:

A B Dis Dis c x C x C u C u y C x (3)

where [ , , , , , ]T

W L W Lx V V y x , [ , ]T

Wc WcV u , [ , ]T

Dis Lc LcV u , and the matrices , , , A B Dis CC C C C can be

obtained from the linearization of Eq. (1) and (2).

By using the “c2d” function in MATLAB®

, we can get the continuous model in discrete form:

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1 1 1

1

( 1) ( ) ( )

( ) ( )

k k k

k k

x A x B u

y C x (4)

Since the predictive control algorithm will in fact produce the changes u rather than u , it is therefore

convenient for many purposes to regard the „controller‟ as producing the signal u , and the „model‟ as having this

signal as its input. One way to change the input to u is to define the state vector 13

:

( )

( )( 1)

kk

k

x

y

Then the augmented nominal model can be written as:

( 1) ( ) ( )

( ) ( )

k k k

k k

A B u

y Cx

(5)

where:

1 1

1

1

, = , =

A 0 BA B C C I

C I 0.

It can be shown that the predictions based on the augmented state space model in vector-matrix notation is

( ) Y k GU

(6)

The cost function for the leader-wingman formation control can be defined as follows:

( ) ( ) T TJ Y W Y W U U

(7)

So the MPC control framework can be solved online as

min ( , ( ), )U

J k Y k U

(8)

subject to the following constraints:

( )

min max

min max

Δ Δ (k +i | k) Δ

(k +i | k)

Y k GU

u u u

y y y

(9)

The MPC control algorithm can be described as follows:

1) Obtain the current model output ( )ky ;

2) Compute the required model input ( )ku according to the cost function (8) and by respecting the input, output

constraints (9);

3) Apply ( )ku to the model.

The whole system can be illustrated in Fig. 2.

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System Stability

Stability of MPC is well defined for various constraint and cost conditions, for linear and nonlinear systems. In

this section we will apply the terminal constraints to ensure the stability of the formation flight system.

Since the cost function, i.e. Eq. (7) is a function of the state variables x and input variable u , it can be expressed

as follows:

1

( ) ( ) ( ( ), ( 1))pH

T T

i

q k i k i

J Y W Y W U U x u

where ( , ) 0q x u , and ( , ) 0q x u only if x 0 and u 0 , and q is a decreasing function, subject to the terminal

constraint ( )pk H x 0 . Let 0 ( )tJ be the optimal value of J which corresponds to the optimal input variable 0u ,

as evaluated at time t. For stability analysis, we assume that for each step, the optimum solution can be found.

Obviously, 0 ( ) 0t J and 0 ( ) 0t J only if ( )t x 0 .

0

1

1

0 0

( 1) min ( ( 1 ), ( ))

min ( ( ), ( 1)) ( ( 1), ( )) ( ( 1 ), ( ))

( ( 1), ( )) ( ) min ( ( 1 ), ( ))

p

p

H

ui

H

p pu

i

p pu

t q t i t i

q t i t i q t t q t H t H

q t t t q t H t H

J x u

x u x u x u

x u J x u

Since we have used the terminal constraint ( ) 0pk H x , we can make ( ) 0pt H u and stay at x=0, which

gives min ( ( 1 ), ( )) 0p pu

q t H t H x u . Because 0( ( 1), ( )) 0q t t x u , so 0 0( 1) ( )t t J J is obtained.

0 ( )tJ can be considered as a Lyapunov function for the system, and hence by Lyapunov‟s stability theory, the

equilibrium point x 0 , u 0 is stable which means the separations between the wingman and the leader will

converge to nominal values, i.e. the wingman can catch up the lead provided each optimization is feasible.

IV. Sliding Mode Vertical Motion Control

For the vertical channel control design, the objective is to maintain the leader and wingman on the same altitude

so that the wingman can get the most energy-saving benefit from the formation flight.

Figure 2 Formation flight control diagram and signals

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The vertical motion dynamics is:

2 1( )wc Lcz k z k z h h F (10)

where F denotes the unknown effects of the vortex. Although one can consider a more general representation of the

vortex forces, for the purpose of illustration, using coefficients From Proud, 3 we can calculate the max9.33F F .

For the derivation of the variable structure control law, we select a switching surface: ( ) 0d

s zdt

, that

leads to: s z z . Since we want to keep the same altitude flight, the reference signal needs not be included in the

switching surface expression. The control law is derived using a Lyapunov function of the form: 2W s , and the

resulting control law is of the form

1

1 2( ) ( ) sat( / )z Lcu z k k z z h k s (11)

Where maxk F and is the sliding boundary layer‟s thickness which can smooth out the control

discontinuity.

Using Eq. (11) as the vertical channel control input, the vertical distance between the leader and wingman is

asymptotically converge to zero, the detail of the stability analysis is referred to Ref.[15].

V. Simulation and results

The simulation is carried out in MATLAB and Simulink. For the horizontal plane control, at each optimization

step, the MPC controller generates a control profile and only the first control signal is applied to the model while the

other control signal is discarded. At the next step, the controller needs to solve the constrained nonlinear

optimization problem again.

In the MPC controller, the predictive control horizon Nu and the predictive horizon 2

N are set as 3Nu and

202

N . The sample time is 0.2s, and the aircraft model used here is an agile model similar in Ref. [5].

Since in the section III we have defined the output as: x separation, the speed difference between the two

aircrafts, y separation and the heading angle difference between the two aircrafts. So the set point is: [ ,0, ,0]x y . The

control signal constrains and the output constraints are set as:

min max

min max

min

min

c

c

V V V

x x

y y

The limits are selected as:

min

max

min

max

min

min

10 ft/s per step

5 ft/s per step

0.28 rad per step

0.28 rad per step

45ft

20 ft,

V

V

x

y

Simulation 1:Circular tracking

From Fig.1 (a), because of the

symmetry of the formation, here we just

have simulated one vehicle platoon:

Figure 3 Formation Trajectories

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leader, wingman1 and wingman3. Initially, the aircrafts are flying with nominal desired separation:

60ft, 23.562ftx y and with speed of 825ft/s, then the leader is commanded to fly a circular path with constant

speed. The resulting aircrafts‟ trajectories are given in Fig. 3, and the separations are in Fig 4 and Fig 5.

Simulation 2:Quick 80 degree heading change

In order to test whether the MPC controller can handle the constraints when dealing with the extreme maneuvers,

here a fast 80 degree heading change maneuver is carried out. The nominal desired separation:

120ft, 70.686ftx y , and the formation pair is flying with speed of 825ft/s, the leader is making a sharp turn of

Figure 4 Separations between Leader and W1

Figure 5 Separations between W1 and W3

Figure 9 Vertical Channel Time Responds

Figure 6 Separations between Leader and W1

Figure 7 Velocity control input

Figure 8 Heading control input

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80 degree in a very short time, the response of the wingman is shown in Fig.6, and the control inputs, i.e. the

command velocity input and the command heading angle input to the wingman are shown in Figs.7 and 8.

In Fig. 6, because of the constraints, there is a large separation between the leader and wingman during the

transition time. If there is no input limit, the separation will be much smaller. If the number of wingmen increases,

the last wingmen maybe lost, i.e. can not track the overall leader. This phenomenon is referred as the string

instability for the leader-follower formation flight. To cope with this, we are now trying to establish the

communication channel between the overall leader and the last wingmen, so that they can stay in the formation even

when the number of wingmen is large.

Simulation 3: Vertical channel control

For the vertical channel simulation, we choose the control parameter to be 0.1, =20 . The simulation

result for initial vertical separation of 100ft between wingman1 and the leader is shown in Fig. 9.

From Fig. 9, we can tell that even with the unknown bounded vortex effect, the vertical channel can maintain the

formation altitude smoothly.

VI. Conclusions

The leader-follower formation control architecture has been discussed and an MPC control method has been

proposed. In the formation model analysis, we divide the inputs into manipulated variables and measured input

disturbances, and also define the outputs of interest. Stability of the formation flight has been primarily analyzed. A

sliding mode control has been proposed to take care of the vortex effects from the leader aircraft. Simulation results

demonstrate that the MPC controller proposed is able to maintain the desired formation geometry within the

constraints posted. As part of the future work, we will concentrate on the MPC formation optimization feasibility

and the inclusion of the obstacle avoidance and crosswind effects into the MPC framework.

References

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[13] Maciejowski J.M., “Predictive control with constraints,” Pearson Education Publishers, 2002

[14] J. M. Fowler and R. D‟Andrea, “A formation flight experiment,” IEEE Control Systems Magazine, vol. 23, pp. 35–43,

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[15] Jean-Jacques E. Slotine, Weiping Li., “Applied nonlinear control,” Prentice Hall, c1991

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[17] Seiler, P., A. Pant and J. K. Hedrick (2004). String instabilities in formation flight: Limitations due to integral constraints.

Journal of Dynamic Systems, Measurement, and Control 126(4), 873–879.