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SICE Annual Conference 2007

Sept. 17-20, 2007, Kagawa University, Japan

Autonomous Flight Control System for Longitudinal Motion of a Helicopter

Atsushi Fujimori1 and Kyohei Miura2

1Department of Mechanical Engineering, University of Yamanashi, Kofu, Japan

(Tel : +81-55-220-8195; E-mail: afujimori@yamanashi.ac.jp)2Department of Mechanical Engineering, Shizuoka University, Hamamatsu, Japan

Abstract: This paper presents a flight control design for longitudinal motion of helicopter to establish autopilot techniquesof helicopter-type unmanned aerial vehicles (UAVs). The characteristics of the linearized equation of the helicopter is

changed during a specified flight mission because the trim values of the nonlinear equation are also varied. In this paper,

the longitudinal motion of the helicopter is modeled by a linear interpolative polytopic model whose varying parameter is

the flight velocity. A flight control systems with a gain scheduling state feedback law is designed to stabilize the vehicle

and track a reference which realizes the flight mission. The effectiveness of the proposed flight control system is evaluated

by computer simulation.

Keywords: Autonomous flight control, unmanned aerial vehicles (UAVs), helicopter, gain scheduling.

1. INTRODUCTION

Recently, unmanned aerial vehicles (UAVs) have been

developed for the purposes of scientific observations, de-tecting disasters, surveillance of traffic and army objec-

tives [1]-[3]. This paper presents a flight control design

for a small autonomous helicopter to give insights for de-

veloping helicopter-type UAVs.

A Helicopter is generally an unstable aircraft. Once

it is stalled, it is not easy to recover its attitude. A con-

trol system is therefore needed to keep the vehicle stable

during flight [4], [5]. This paper presents a flight control

design for longitudinal motion of helicopter to establish

autopilot techniques of helicopters. The flight mission

considered in this paper is that a helicopter hovers at a

start position, moves to a goal position with keeping aspecified cruise velocity and hovers again at the goal. The

characteristic of the linearized equation of the helicopter

is changed during this flight mission because the trim val-

ues of the nonlinear equation are also varied. In this pa-

per, the flight control system is designed as follows. The

longitudinal motion of a helicopter is first modeled by a

linear interpolative polytopic model whose varying pa-

rameter is the flight velocity. A flight control systems

with a gain scheduling (GS) state feedback law [6] is de-

signed to stabilize the vehicle and track a reference which

realizes the flight mission. The effectiveness of the pro-

posed flight control system is evaluated by computer sim-

ulation using Matlab / Simulink.

2. EQUATION OF LONGITUDINALMOTION OF HELICOPTER

Figure 1 shows a helicopter considered in this paper.

(x,y,z) represent the body-fixed-axes whose origin is lo-cated at the center of gravity of the vehicle. The forward

velocity is V whose x- and z-axes elements are u and w,respectively. The main rotor produces the aerodynamic

force which consists of the lift L and the thrust T. It de-pends on the tilting angle of the control plane. The rotor

blades are regarded as cantilevered elastic beams. The re-

stored force is modeled by a rotational spring attached at

tip path plane

horizontal plane

control plane

x

z

V

u wf

f

Vd

c

x

a1

c

Fig. 1 Helicopter in forward flight

the rotor hub.

The longitudinal motion of helicopter consists of the

transitional motion with respect to x- and z-axes and therotational motion around y-axis. It is represented as thefollowing equations:

m(u + qw) = Xmg sin (1)

m( w qu) = Z+ mg cos (2)

Iyy q = M (3)

where m and Iyy are respectively the mass and the mo-ment of inertia of the vehicle. g is the gravity accelera-tion. The external forces X, Z and the moment M aregiven by

X = T sin(c a1)Du

V(4)

Z = T cos(c a1)Dw

V(5)

M = T hR sin(c a1) (6)

where D is the drag. hR is the distance of hub from thecenter of gravity. The induced velocity vi through the

rotor is approximated by a first-order system. Define the

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0 20 40 600

20

40

60u [m/s]

0 20 40 6010

8

6

4

2

0w [m/s]

0 20 40 6010

8

6

4

2

0

f

[deg]

V [m/s]0 20 40 60

0

0.02

0.04

0.06

i

V [m/s]

(a) States in trim xp

0 20 40 606

7

8

9

10

0

[deg]

V [m/s]0 20 40 60

0

1

2

3

4

5

c

[deg]

V [m/s]

(b) Inputs in trim up

Fig. 2 Trim values with respect to forward velocity.

state and the input vectors as

xp= [u w q f i]

T 5 (7)

up

= [0 c]T

2

, (8)

where 0 and c are the collective and the cyclic pitchangles. The equation of longitudinal motion of helicopter

is then written as

xp = fp(xp, up) (9)

Equation (9) is referred as the nonlinear plant Pnl here-after.

Letting xe and he be horizontal and the vertical posi-tions of the helicopter from the start, they are given by

xe = u cos f + w sin f (10)

he = u sin f w cos f. (11)

Defining p as p= [xe he]T, they are compactly given

as

p = gp(xp). (12)

Since this paper considers hovering and forward flight,

the trim condition is given in level flight. Letting xpand up be the state and the input in trim, respectively,f(xp, up) = 0 holds. Figure 2 shows variations ofxpand up with respect to the flight velocity V. Almost all

trim values are changed in the range ofV [0, 60] [m/s].

Vc1

0tc1 tc3tc2 tc4 t

Vc2

Vr

tc5

Fig. 3 Flight velocity profile.

Pnl[E -F]KoutKpgp

Kin zpupvzr p

rxp

+

--+++

Fig. 4 Flight control system.

3. CONSTRUCTION OF FLIGHTCONTROL SYSTEM

Let the start position be the origin of the coordinates

(xe, he). The flight mission is to navigate the helicopterfrom the start (0,0) to the goal, denoted as (xr, hr), withkeeping attitude stable. To design a control system, the

followings are assumed to be satisfied:

(i) The motion in y-axis direction is not taken into ac-count.

(ii) xp is measurable.(iii) The trim values xp and up are known in advance.

To realize the flight mission, this paper constructs a track-

ing control system whose controlled variable is the flight

velocity. The flight region is divided into six phases as

shown in Fig. 3. They are referred as follows.

1. 0 t < tc1: initial hovering phase2. tc1 t < tc2: acceleration phase3. tc2 t < tc3: cruise phase4. tc3 t < tc4: deceleration phase5. tc4 t < tc5: low speed phase6. tc5 t: approach phase

Until the low speed phase, the reference of the flight ve-

locity is given by Vr shown in Fig. 3. In this paper, the

total time of the flight is not cared. But the integratedvalue ofVr for t [0, tc5] must be less than xr not toovertake the goal before the approach phase. In the ap-

proach phase, the reference is generated to meet the travel

distance p = [xe he]T with r = [xr hr]

T.

Taking into consideration the above, a double loop

control system [7], [8] is applied to design a flight control

system in this paper. It is given in Fig. 4. Pnl representsthe nonlinear helicopter dynamics given by Eq. (9), Kinis the inner-loop controller, Kout is the outer-loop con-troller and Kp is a gain. The controlled variable fromthe initial hovering to the low speed phase is given by

zp

= [u w]T

and its reference is given by zr

= [ur wr]T

.

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In the approach phase, another loop is added outside of

(zr zp)-loop, where p= [xe he]

T is the controlled

variable and r= [xr hr]T is its reference.

Kin consists of

Kin = [E F] (13)

where E is a feedforward gain for tracking the reference,while F is a feedback gain for stabilizing the plant. Sincethe trim values are changed as shown in Fig. 2, the char-

acteristics of the linearized plant is also varied. Then, Fis designed by a GS technique in terms of LMI formula-

tion.

The reference zr from the initial hover to the low speedphase is generated by the flight velocity profile shown in

Fig. 3, and zr in the approach phase is derived from thepositional error r p. The switch of the reference isdone at t = tc5.

4. DESIGN OF CONTROL SYSTEM

4.1 Linear interpolative polytopic model

The objective of flight control in this paper is that

the controlled variable is regulated to the specified

trim condition. Linearized models along with the trim

may be therefore used for controller design. Letting

xp(V), up(V) be respectively the state and the input intrim where the flight velocity is V, the perturbed stateand the input are defined as

xp(t)= xp(t)xp(V), up(t)

= up(t)up(V) (14)

The linearized equation of Eq. (9) is then given as

xp(t) = Ap(V)xp(t) + Bp(V)up(t) (15)

Ap(V)=

fp(xp, up)

xTp, Bp(V)

=

fp(xp, up)

uTp.

(16)

Although matrices Ap and Bp are functions with respectto V, it is hard to get their explicit representations be-cause of complicated dependence of V as described inSection 2. Then, Ap and Bp are approximated by inter-polating multiple linearized models in the trim condition.

For the range of the flight velocity V [0, Vu], r points{V1, , Vr}, called as the operating points, are chosenas

0 V1 < < Vr Vu. (17)

The linearized model for V = Vi is a local LTI modelrepresenting the plant near the i-th operating point. Lin-early interpolating them, a global model over the entire

range of the flight velocity is constructed as

xp(t) = Ap(V)xp(t) + Bp(V)up(t)

zp(t) = Cpxp(t) + zp(V)

(18)

where

Ap(V) =r

i=1

i(V)Api, Bp(V) =r

i=1

i(V)Bpi,

Cp =

1 0 0 0 0

0 1 0 0 0

. (19)

i(V) satisfied the following relations.

0 i(V) 1 (i = 1, , r) (20)r

i=1

i(V) = 1 (21)

Equation (18) with Eq. (19) is called as the linear inter-

polative polytopic model in this paper.

4.2 Design ofKin

Under assumption (ii), consider a state feedback law

up(t) = F(V)xp(t) + E(V)v(t) (22)

where v is a feedforward input for tracking zr and is givenby v = zr zp when designing Kin. The closed-loopsystem combining Eq. (22) with Eq. (18) is given by

xp(t) = AF(V)xp(t) + Bp(V)E(V)v(t)zp(t) = Cp(V)xp(t) + Dp(V)E(V)v(t) + zp(V)(23)

AF(V)= Ap(V)Bp(V)F(V).

The steady-state controlled variable is given by

zp() = CpA1

F BpEv + zp. (24)v is then given so as to meet zp() with the reference zr;that is, zp() zr . E is designed as

E = (CpA1F Bp)

1. (25)

Next, F(V) is designed so that the closed loop systemis stable over the entire flight range and H2 cost is glob-ally suppressed [6]. The controlled plant is newly given

by

xp(t) = Ap(V)xp(t) + Bp(V)up(t) + B1(V)w1(t)

z1(t) = C1(V)xp(t) + D1(V)up(t)(26)

where z1 and w1 are respectively the input and the out-put variable for evaluating H2 cost. B1(V), C1(V) andD1(V) are matrices corresponding to z1 and w1. Substi-tuting Eq. (22) without v into Eq. (26), the closed-loopsystem is

xp(t) = AF(V)xp(t) + B1(V)w1(t)

z1(t) = C1F(V)xp(t)(27)

C1F(V)

= C1(V) D1(V)F(V).

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In this paper, F(V) is designed so as to minimize theintegration ofH2 cost over V [0, Vu]. That is, theobjective is to find F(V) such that [9].

infF,P,W

Vu0

trW(V)dV subject to

P(V) P(V)B1(V)() W(V)

> 0, (28a)He(P(V)AF(V)) + V dPdV ()

C1F(V) Iq

< 0 (28b)

He(A) is defined as He(A)= A + AT where ()

means the transpose of the element located at the diag-

onal position. P(V) > 0 is the parameter dependentLyapunov variable. To derive finite number of inequal-

ities from Eq. (28), the following procedures are per-

formed. First define new variables X(V)= P1(V) and

M(V)

= F(V)X(V). Ap(V) and Bp(V) are given bythe polytopic forms Eq. (19). B1(V), C1(V), D1(V),X(V), M(V) and W(V) are also given by similar poly-topic forms. Furthermore, dP/dV are approximated as

dP

dV= X1

dX

dVX1 (29)

dX

dV

Xi+1 XiVi+1 Vi

=

XiVi

(30)

Pre- and post-multiplying diag{X, I} to Eq. (28) and us-ing the polytopic forms, the following LMIs are derived

as a sufficient condition.

infMi,Xi,Wi

r1i=1

tr(WiVi) subject to

Xi B1i

() Wi

> 0 (i = 1, , r), (31a)

He(ApiXi BpiMi) XiVi ()

C1iXiD1iMi Iq

< 0,

(31b)

He(Apj Xj Bpj Mj) XiVi ()C1j Xj D1j Mj Iq

< 0,

(31c)

He(ApiXj BpiMj

+Apj Xi Bpj Mi) 2XiVi

()

C1iXj D1iMj

+C1j Xi D1j Mi2Iq

< 0

(31d)

(i = 1, , r 1, j = i + 1), = ,

IfMi, Xi and Wi (i = 1, , r) satisfying the above

LMIs, F(V) is given by

F(V) =

r

i=1

i(V)Mi

r

i=1

i(V)Xi

1. (32)

4.3 Design ofKout

Since v in Eq. (22) is a feedforward input from theflight velocity reference, the tracking error will be oc-

curred by model uncertainties and/or disturbances. Let usevaluate this in the LTI representation. Let Tzpv(s) be atransfer function from v to zp. zp converges to a constantzr if there are no model uncertainties in Tzpv(s) becauseTzpv(0) = I. IfTzpv(0) is varied as Tzpv(0) = I + due to model uncertainties, we have the following steady-

state error:

e0= zr zp() = zr (33)

To reduce the error, a feedback from zp; that is, an outer-loop is added as shown in Fig. 4. The transfer function

from zr to zp is given by

Tzpzr(s) = (I+Tzpv(s)Kout(s))1Tzpv(s)(I+Kout(s))

(34)

The steady-state error is then

e1= zr zp() = zr lim

s0sTzpzr(s)

1

szr

= (I + Tzpv(0)Kout(0))1zr.

(35)

This means that the steady-state error e1 with the outer-loop is reduced by (I + Tzpv(0)Kout(0))

1. Summariz-

ing the above, the design requirements ofKout are givenas follows:

(i) Kout must stabilize Tzpv .(ii) The amplitude of (I + Tzpv(j)Kout(j))1

should be small in the low frequency region.

5. SIMULATION

To evaluate the proposed flight control system, a flight

simulator is built on MATLAB/Simulink. For design and

discussion hereafter, the notations about plant models are

given as follows: Plpv(V) is a linear parameter varying(LPV) model obtained by linearizing Pnl. Ppoly(V) isthe linear interpolative polytopic model given by Eq. (18)

with Eq. (19). Plti(Vd) is an LTI model where the flightvelocity is fixed at Vd.

Two cases of flight control system with respect to Fare compared in simulation. One is that F was designedby GS where the plant model was Ppoly(V). Another isthat F was designed by LQR where the plant model wasPlti(Vd). The former is called as GS-SF, while the latteris called as Fixed-SF. The parameter values of the flight

velocity profile in Fig. 3 were given as follows:

(xr, hr) = (3000, 0) [m],

Vc1 = 50 [m/s], Vc2 = 15 [m/s],

tc1 = 5 [s], tc2 = 30 [s], tc3 = 60 [s],

tc4 = 80 [s], tc5 = 100 [s].

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Table 1 Operating points of polytopic models.

Model Vci [m/s] GS-SF

Ppoly1 { 0, 50 } Fgs1

Ppoly2 { 0, 25, 50 } Fgs2

Ppoly3 { 0, 10, 15, 40, 50 } Fgs3

Table 2 Design points of LTI models.

Model Vd [m/s] Fixed-SF

Plti1 0 Ff ix1

Plti2 25 Ff ix2

Plti3 50 Ff ix3

5.1 Evaluation of design models

According to Section 4.1, three linear interpolative

polytopic models were obtained. Table 1 shows the op-

erating points chosen for the models. The design pointsVd of three LTI models are shown in Table 2. The -gap metric is one of criteria measuring the model error

in the frequency domain. It had been introduced in ro-

bust control theories associated with the stability margin

[10]. The -gap metric between two LTI models, P1(s)and P2(s), is defined as

(P1, P2)= (I+P2P

2 )1/2(P1P2)(I+P1P

1 )1/2

(36)

The range is [0, 1]. A large means that themodel error is large. The -gap metric is used forevaluating the model Ppoly(V) and Plti(Vd). Figure 5shows -gap metric between Plti(Vd) and Plpv (V) andbetween Ppoly(V) and Plpv(V). (Plti(V), Plpv(Vd))was rapidly increased when V was shifted from Vd. Onthe other hand, the maximum of(Ppoly(V), Plpv (V))was reduced according to the number of the operating

points. It was seen that Ppoly(V) appropriately approxi-mated Plpv (V) over the entire flight range.

5.2 Design ofF and H2 cost

The design parameters for designing F in Eq. (26)were given as follows.

B1 =

6.039 10.977

154.03 49.188

3.954 7.187

0 0

0.38495 0.12395

,

C1 =

0.001I5

025

, D1 =

052

I2

They were used for both of GS-SF and Fixed-SF. GS-

SF was designed according to Section 4.2, while Fixed-

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

V [m/s]

gap metric between Plti

and Plpv

Plti1

Plti2

P

lti3

(a) LTI models

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

V [m/s]

gap metric between Ppoly

and Plpv

Ppoly1P

poly2

Ppoly3

(b) polytopic models

Fig. 5 -gap metric

0 10 20 30 40 501.7

1.8

1.9

2

2.1

2.2

2.3

2.4H

2cost

V [m/s]

Ffix3

Fgs1

Fgs2

Fgs3

Fig. 6 H2 cost

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0 50 100 15020

0

20

40

60

u[m/s]

GSSF 3

0 50 100 150

30

20

10

0

10

w

[m/s]

0 50 100 1505

10

15

20

25

t [s]

0

[deg]

0 50 100 1502

0

2

4

6

t [s]

c

[deg]

u

ur

w

wr

(a) Controlled variables and inputs

0 20 40 60 80 100 120 1400

1000

2000

3000

GSSF 3

t [s]

x[m]

0 20 40 60 80 100 120 140300

200

100

0

100

t [s]

h[m]

(b) Positions

Fig. 7 Time responses using Fgs3

SF was designed by LQR technique whose weights were

given by CT1 C1 and DT1 D1.

Figure 6 shows the H2 cost of the closed-loop sys-tem which the designed F is combined with Eq. (26).The H2 cost by Ff ix3 was minimized at V = 40 [m/s]which was near the design point Vd = 50 [m/s], but wasincreased in the low flight region. The H2 cost by Ff ix1and Ff ix2 showed the similar result. On the other hand,the H2 cost by Fgs2 and Fgs3 was kept small over theentire flight region. The H2 cost by Fgs1 was small in

the low flight velocity but was increased in the high flightregion.

5.3 Tracking evaluation

The flight mission given in Fig. 3 was performed in

Simulink. Figure 7 shows the time histories of the closed

loop system whose F was Fgs3. The responses byFgs3 showed improved tracking and settling propertiescompared to other cases.

6. CONCLUDING REMARKS

This paper has presented a flight control design for

longitudinal motion of helicopter to establish autopilot

techniques of helicopter-type UAVs. The characteristics

of the linearized equation of the helicopter was changed

during a specified flight mission because the trim values

of the nonlinear equation were also varied. In this paper,

the longitudinal motion of the helicopter was modeled by

a linear interpolative polytopic model whose varying pa-

rameter was the flight velocity. The model error was eval-

uated by the -gap metric. The effectiveness of the pro-

posed flight control system was evaluated by computersimulation using MATLAB / Simulink. The model er-

ror of the polytopic model was smaller than that of LTI

models which were obtained at specified flight velocity.

Flight control systems with GS state feedback showed

better control performances than those with fixed-gain

state feedback. The double loop flight control structure

was useful for performing flight mission considered in

this paper.

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