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112 Asian Journal of Control, Vol. 6, No. 1, pp. 112-122, March 2004

ROBUST GAIN-SCHEDULED CONTROL OF A VERTICAL TAKEOFF

AIRCRAFT WITH ACTUATOR SATURATION VIA THE LMI METHODP. C. Chen, Y. F. Jeng, Y. H. Chang, Y. M. Wang, and G. Chen.

ABSTRACT

This paper presents a robust gain-scheduled approach for the control of

a vertical/short takeoff. and landing (V/STOL) aircraft. The nonlinear aircraft

dynamics exhibit non-minimum phase characteristics arising from the para-

sitic coupling effect between the aircraft’s lateral force and rolling moment.

The undesired coupling effect also causes modelling uncertainy of the aircraft

dynamics. The nonlinear aircraft dynamics are considered to be composed of

a nominal linear parameter varying (LPV) system and a linear system with anorm bounded uncertainy matrix multiplied by the parasitic uncertain

non-minimum phase coupling parameter. The nominal LPV system is con-

sidered to be affinely dependent on a measurable varying parameter. The

ranges of the varying parameter and its variation as well as its parasitic in-

duced uncertain matrix are addressed by introducing the parameter-dependent

invariant ellipsoid interpretation for dealing with the issue of affinely quad-

ratic stabilization. In this paper, the relations among the magnitude of

actuator saturation, the maximum achievable relative stability, and the

sustainable coupling uncertainty are investigated for the considered robust

gain-scheduled design.

KeyWords: V/STOL aircraft, flight control, LPV system, LMI, non-mini-mum phase system.

I. INTRODUCTION

The vertical/short takeoff and landing (V/STOL)

aircraft has the capabilities of high mobility and maneu-

verability. Typically, the YAV-8B Harrier, produced by

the McDonnell Aircraft Company [1], is powered by a

single turbo-fan engine with exhaust nozzles equipped in

each side of the fuselage to provide the gross thrust for

the aircraft. The nozzles are capable of rotating together

from the aft position forward approximately 100 degrees.This change in the direction of thrust allows the aircraft

Manuscript received December 31, 2002; revised June 13,

2003; accepted August 26, 2003.

P.C. Chen and Y.F. Jeng are with Department of Aircraft

Engieering, Air Force Institute of Technology, Kaohsiung,

Taiwan, R.O.C.

Y.H. Chang is with Department of Electrical Engieering,

Chang Gung University, Taoyuan, Taiwan, R.O.C.

Y.M. Wang and G. Chen are with Department of Electrical

Engieering, Chung Cheng Institute of Technology, National

Defense University, Taoyuan, Taiwan, R.O.C.

This work was supported by the National Science Council of the Republic of China under contract NSC-92-2212-E-344-003.

to operate in two modes and the transition between them.

In the mode of wing-borne forward flight, the nozzles

are in the aft position, and the aerodynamic forces pro-

duced by the surfaces of the wing and fuselage sustain

the weight of the aircraft. In the mode of jet-borne hov-

ering, the nozzles are directed vertically toward the

ground. Moreover, the transit process between these two

modes enables the aircraft to achieve versatility of ma-

neuvering in terms of the velocity and direction of

movement.For the hovering operation considered in this paper,

the upward thrust produced by the nozzles is manipu-

lated by means of the throttle to produce motion in the

vertical direction. In order to produce motion in the lat-

eral direction, the Harrier is equipped with reaction con-

trol valves in the nose, tails, and wingtips. By using the

high pressure flow from the engine’s compressor, these

valves produce moments along the aircraft center of

mass to change its attitude. This mechanism of attitude

control enables the realization of lateral motion during

the hovering operation. By considering only the motion

in the vertical-lateral plane, we are able to represent theaircraft by means of six-order non-linear dynamics. The

−Brief Paper −

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P.C. Chen et al.: Robust Gain-Scheduled Control of A Vertical Takeoff Aircraft 113

states include the lateral position, vertical position, roll

angle, and their correspoding velocities.

When considering the hovering operation, if the air

from the reaction control valves for producing moment in

attitude control induces an unexpected force componentin the lateral direction, the system dynamics may exhibit

non-minimum phase characteristics, depending on the

direction of the induced force. The reason for the

non-minimum phase arising from the coupling effect be-

tween the lateral force and rolling moment is that in dur-

ing of lateral motion, the aircraft must first turn to a cor-

rect attitude; then the thrust force can move the aircraft so

as to follow the commands. If the moment changing the

aircraft attitude induces a force that moves the aircraft

toward the wrong lateral direction, then the inevitable

time delay will produce the non-minimum phase effect.

Several non-linear controller design approacheshave been investigated for application to these non-linear

non-minimum phase planar V/STOL aircraft dynamics.

One scheme is based on the well known input-output

feedback linearization approach, which employs a nonlin-

ear version of pole-zero cancellation [2-4]. For a

non-minimum phase system, this feedback linearization

will result in internally unstable dynamics, even if the

linearized system is stable in the sense of input-output

stability. To avoid this internal instability, feedback lin-

earization is performed based on approximated minimum

phase dynamics, where the influence of the rolling mo-

ment on the lateral force is neglected [1]. It has been

shown that the desired properties, such as bounded track-

ing and asymptotic stability, for the true system can be

maintained if the neglected parasitic coupling effect is

small. In [5], the non-minimum phase V/STOL dynamics

were modeled as a singularly perturbed model [6], in

which the nominal dynamics were represented by the ap-

proximated minimum phase planar V/STOL system, while

the parasitic coupling effect was denoted as perturbed fast

dynamics. Recently, a robustly nonlinear state-feedback

control law was designed for the planar V/STOL aircraft

based on an optimal control approach [7].

Another approach to V/STOL aircraft control is to

design a family of controllers beforehand according to theoperation envelope of the aircraft system. Then, in the

real-time application, a mechanism for scheduling this

family of controllers is activated to realize the instantane-

ous controller based on the aircraft operation. In [8], based

on the generic V/STOL aircraft model (GVAM), a gain-

scheduled design using the H∞ optimal control technique

was investigated. In [9], the non-linear non-minimum

phase aircraft dynamics were formulated as a linear pa-

rameter varying (LPV) system with the roll angle taken as

the varying parameter. In addition, the set-valued methods

for l1 optimal control have been adopted based on the

LPV system with a couple of indexed parameters [10,11].The control laws were non-linear, static state feedback,

where the approximated minimum phase system was used

and parasitic coupling was not addressed. The achieved

control objectives were the tracking performance of the

aircraft position in the vertical-lateral plane and the con-

straints placed on the control efforts. It was shown that theundesired parasitic coupling caused the designated track-

ing performance to deteriorate, especially under a tight

tracking specification.

In this paper, a robust gain-scheduled controller is

designed via the linear matrix inequality (LMI) method

for planar V/STOL aircraft dynamics with imposed ac-

tuator saturation. The nonlinear aircraft dynamics are

considered to be composed of a nominal LPV system

and a linear system with a norm bounded uncertainy

matrix multiplied by the parasitic uncertain non-mini-

mum phase coupling parameter. The nominal LPV sys-

tem is considered to be affinely dependent on the trigo-nometric functions of the measurable system varying

parameter, i.e., the roll angle. The ranges of the varying

parameter and its variation as well as the magnitude of

the denoted affinely trigonometric functions of the roll

angle and parasitic norm-bounded uncertain matrix are

addressed by introducing the parameter-dependent in-

variant ellipsoid interpretation to deal with the issue of

affinely quadratic stabilization. For this robust gain-

scheduled planar V/STOL aircraft controller design, the

relations among the magnitude of actuator saturation, the

maximum achievable relative stability, and the sustain-

able coupling uncertainty are investigated.

The remainder of this paper is organized as follows.

Section II presents the modeling of the planar V/STOL

aircraft dynamics and their reformulation as an uncertain

LPV system with imposed input and output constraints.

Section III introduces the LMI approach for the robust

parameter-dependent planar V/STOL aircraft control.

Section IV presents the simulation results and a discus-

sion. Section V draws conclusions.

II. PLANAR V/STOL AIRCRAFT DYNAMICS

Consider one type of vertical/short takeoff and

landing (V/STOL) aircraft, the YAV-8B Harrier pro-duced by the McDonnell Aircraft Company [1]. The

aircraft is powered by a single turbo-fan engine. The

exhaust nozzles on the turbo-fan engine can be simulta-

neously rotated from the aft position forward about 100

degrees. Therefore, the aircraft is allowed to maneuver in

conventional wing-borne flight, jet-borne flight, and un-

der even nozzle breaking. The thrust vector produced by

the throttle and nozzle enables two-degrees-of-freedom

control in the roll-yaw plane. In order to allow lateral

maneuverability during jet-borne operation, the aircraft

also has a reaction control system (RCS) to provide a

moment around the aircraft center of mass as shown inFig. 1.

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114 Asian Journal of Control, Vol. 6, No. 1, March 2004

α

α

Fig. 1. The bleed air and moment of reaction control system (RCS).

By restricting the aircraft to jet-borne operation,

i.e., thrust directed toward the bottom of the aircraft, we

have simplified the dynamics which describe the motion

of the aircraft in the vertical-lateral directions, i.e., the

motion of a planar V/STOL (PVTOL) aircraft. The air-craft states are the position of the center of mass, ( X , Y ),the roll angle θ , and the corresponding velocities, ( X ,

Y , θ ). The control input is the thrust directed toward

the bottom of aircraft U 1 and the moment around the

aircraft center of mass U 2. If the bleed air from the reac-

tion control valves produces a force which is not perpen-

dicular to the pitch axis, i.e., the angle α ≠ 0 shown in

Fig. 1, then there will be a coupling effect between the

angle rolling moment and lateral moving force. Let the

amount of lateral force induced by the rolling moment be

denoted by ε0; then, we have the aircraft dynamics writ-

ten as

1 0 2

1 0 2

2

sin cos

cos sin

,

mX U U

mY U U mg

J U

θ θ

θ θ

θ

− = − +− = − + − =

ε

ε (1)

where mg is the gravity force imposed on the aircraft

center of mass and J is the mass moment of inertia

around the axis extending through the aircraft center of

mass and along the fuselage.

To simplify the notation of the PVTOL aircraft

dynamics (1), the first and second equations in (1) are

divided by mg , and the third one by J . Let x := − X / g , y :=

−Y / g , u1 := 1U

mg , u2 := 2U

J , and ε := 0 J

mg

ε; then, we

have the normalized PVTOL aircraft dynamics as shown

in Fig. 2:

1 2

1 2

2

sin cos ,

cos sin 1,

.

x u u

x u u

u

θ θ

θ θ

θ

= − + = + − =

ε

ε (2)

The term “−1” denotes the normalized gravity accelera-

tion. The coefficient “ε” denotes the parasitic coupling

effect between the lateral force and rolling moment,

which results in the non-minimum phase characteristic.

Fig. 2. The dynamics of the planar V/STOL (PVTOL) aircraft.

Note that the possible parasitic yaw/rolling coupling and

aerodynamic effects are neglected for the sake of sim-

plicity.

The nonlinear PVTOL model (2) is rewritten as an

uncertain LPV system with state-space dependence onthe measurable varying roll angle θ and uncertain cou-

pling parameter ε:

1 2( ( ) ( )) , x A x B B u Dυ υ υ θ θ = + + + ε (3)

where xυ = ( x, x , y, y , θ , θ )T

and u = (u1, u2)T . In the

system matrix Aυ , the elements Aυ (1, 2), Aυ (3, 4), and

Aυ (5, 6) are equal to 1, and others are equal to zero. The

parameter-dependent input matrices are

1

2

0 sin 0 cos 0 0( ) and

0 0 0 0 0 10 0 0 0 0 0

( ) ,0 cos 0 sin 0 0

T

T

B

B

θ θ θ

θ θ θ

− =

=

(4)

and the disturbance matrix D = (0, 0, 0, −1, 0, 0)T repre-

sents the gravity acceleration.

It is noted that the state dynamics ( x, x ), ( y, y ),

and (θ , θ ) are decoupled, and that each is characterized

by the block diagonal matrix0 1

0 0

as shown in the

system matrix Aυ . In addition, the nominal input matrix

B1(θ ) corresponding to the lateral position dynamics ( x,

x ) is0 0

0 0

at the equilibrium value θ = 0. Therefore,

the LPV representation of the PVTOL aircraft (3) is not

even controllable at the equilibrium point θ = 0 under the

lateral position dynamics in the case of a nominal air-

craft system with ε = 0.

To remedy this uncontrollability arising from the

LPV representation (3) of the original nonlinear PVTOL

dynamics (2), the following procedures are performed.

Let the variable of thrust u1 be centered around equilib-

rium point “1”, i.e., u1 := 1 +ũ

1. The revised LPV sys-tem of (3) is

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1 2( ( ) ( )) ( ), x A x B B u Dυ υ υ θ θ θ = + + + ε (5)

where ũ = (ũ1, u2)T

and ( ) D θ = (0, −sinθ , 0, −1 + cosθ ,0, 0)

T . Let the vector ( ) D θ be decomposed as ( ) D θ =

Aυ 0 xυ + ˆ ( ) D θ . In the matrix Aυ 0, the only nonzero ele-ment is Aυ 0(2, 5) = −1. In the vector ˆ ( ) D θ = (0, θ −sinθ ,0, −1 + cosθ , 0, 0)

T , the elements “θ −sinθ ” and “−1 +

cosθ ” represent the first order Taylor series approxima-

tion errors of the trigonometric functions “−sinθ ” and

“cosθ ,” respectively. Then, the LPV PVTOL aircraft

dynamics (5) can be rewritten as

0 1 2ˆ( ( ) ( )) ( ), x A x B B u Dυ υ θ θ θ = + + + ε (6)

where A0 := Aυ + Aυ 0, and ˆ ( ) D θ is a gravitational dis-

turbance. Note that the introduced nonzero element of

Aυ 0 enables the lateral position variables ( x, x ) to be

controlled through the (θ , θ ) dynamics.

III. ROBUST GAIN-SCHEDULD CONTROLLER DESIGN

The robust gain-scheduled control for the PVTOL

dynamics (6) needs to maintain a certain degree of rela-

tive stability for all admissible values of the measurable

varying parameter θ and it’s variation rate θ , the ex-

pected magnitude of the uncertain coupling coefficient ε,

and the parameter-dependent gravitional disturbanceˆ ( ) D θ while being subject to physical limitations on the

magnitude of the control efforts, 1u and 2u . Thedesigned composite control law is denoted as û = ũ + ũτ ,

where ũ represents a linear parameter-dependent state-

feedback control law constructed by means of LMI algo-

rithms for the uncertain LPV PVTOL dynamics without

gravitational disturbance ˆ ( ) D θ ,

0 1 2( ( ) ( )) , x A x B B uυ υ θ θ = + + ε (7)

and ũτ is a nonlinear control law introduced to handle

the disturbance effect of ˆ ( ) D θ . In this section, the ro-

bust gain-scheduled control laws obtained via LMI ap-

proaches for the aircraft dynamics (7) without and with

the uncertain coefficient ε are presented in sequence.Then, the construction of the nonlinear control law ũτ

used to cancel out the gravitational disturbance effectˆ ( ) D θ is introduced.

3.1 Gain-scheduled control for a nominal aircraft in

affine LPV representation

The PVTOL aircraft dynamics (7), ignoring the

uncertain parameter ε while imposing magnitude con-

straints on the control effort ũ, the roll angle θ , and the

variation rate of the roll angle θ , are represented by

0 1 ,max max max( ) , , , .i i x A x B u u uυ υ θ θ θ θ θ = + ≤ ≤ ≤ (8)

The parameter-dependent input matrix B1(θ ), as shown

in (4), is denoted as an affine function of the parame-

ter-dependent elements, p s := sinθ and pc := cosθ :

1 0( ): , s s c c B B p B p Bθ = + + (9)

where

0

0 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0, , .

0 0 0 0 1 0

0 0 0 0 0 0

0 1 0 0 0 0

s c B B B

−

= = =

(10)

For the roll angle θ with bounded magnitude θ max, let the parameter ranges be denoted as

max max

max

[ sin , sin ]: [ , ],

[cos , 1]: [ , ].

s s s

c c c

p p p

p p p

θ θ

θ

∈ − =

∈ =

For the linear gain-scheduled state-feedback con-

trol law ũ = K (θ ) xυ used to stabilize the input and output

constrained nominal PVTOL aircraft (8), the positive

parameter-dependent quadratic Lyapunov function V ( xυ ,

θ ) =T xυ P (θ ) xυ , P (θ ) > 0 is assumed. If the negative

change rate of V ( xυ , θ ), i.e., ( , )V xυ θ < 0, is estab-

lished for A0, replaced by Ã 0 := A0 − γ I , where γ < 0, thenthe maximum relative stability achieved as well as the

largest decay rate is β = −γ , which means that

lim ( )t

t e x t β

υ →∞= 0 for any initial condition xυ 0 = xυ (t 0). In

the case of a tracking control problem, we have

0lim ( )t

t e r x t β

υ →∞− = 0 for any tracking command r 0 =

r (t 0) and initial tracking error r 0 − xυ (t 0).The need for ( , )V xυ θ < 0 is equivalent to

0 1

0 1

( ( ) ( )) ( )

( )( ( ) ( )) ( ) 0.

T A B K P

P A B K P

θ θ θ

θ θ θ θ

+

+ + + <

(11)

Denoting P (θ )−1 = Q(θ ), and K (θ )Q(θ ) = L(θ ), and

from the identity

1 1 1( ) ( ) ( ) ( ) ( ), P P P P Qθ θ θ θ θ

− − −= − = − (12)

the condition of the matrix inequality (11) can be rewrit-

ten as

0 0 1

1

( ) ( ) ( ) ( )

( ) ( ) ( ) 0.

T T T Q A A Q L B

B L Q

θ θ θ θ

θ θ θ

+ +

+ − <

(13)

For the affinely parameter-dependent matrix B1(θ ) in (9),

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the matrix variables Q(θ ) and L(θ ) can be assumed to

have the same affine dependence on the varying pa-

rameters p s and pc. We can then proceed with the ap-

proach presented in [12]. However, for the parameter-

independent system matrix Ã 0 and parameter-dependentcontrol law K (θ ) = L(θ )Q(θ )−1, a single parame-

ter-independent matrix variable for L(θ ) is chosen to

reduce the numerical complexity. Thus, we have

0( ) 0, ( ) . s s c cQ Q p Q p Q L Lθ θ = + + > = (14)

Then, the matrix inequality (13) reads as

0 0 1 1( ) ( ) ( ) ( )

( ) 2 ( ) 0.

T T T

c s s c

Q A A Q L B B L

p Q p Q Q

θ θ θ θ

θ γ θ

+ + +

− − − <(15)

Since the matrix inequality (15) is affinely dependent onthe parameters { p s, pc} when θ is fixed and affinely

dependent on θ when { p s, pc} is fixed, (15) is convex

along each direction of { p s, pc} and θ . The feasibility of

(15) can be established through evaluation only for the

extreme values of the scalar varying parameters { p s, pc,

θ } and recast as the generalized eigenvalue minimiza-

tion problem (GEVP) [13] to maximize the achievable

relative stability β = −γ as follows:

max max

0

minimize : (14), (15), for { , , }

{{ , },{ , },{ , }}

with matrix variables , , , .

s c

s s c c

s c

p p

p p p p

Q Q Q L

γ θ

θ θ

=

∈ −

(16)

The objective of the maximized relative stability or

decay rate will tend to cause a controller to exhibit high

gain. The required control effort may exceed the magni-

tude limits of the system actuators. Moreover, the be-

havior of the controlled system with the high gain con-

troller will be sensitive to the values of the varying pa-

rameter of system. Therefore, when designing a control-

ler with the maximum relative stability, we need to con-

sider the system varying parameter and the physical con-

straint on the actuator as well. The magnitude constraints

on the control efforts ũi, the roll angle θ , and the varia-tion rate of the roll angle θ can be addressed by intro-

ducing invariant ellipsoid interpretation [14,15] of the

parameter-dependent quadratic Lyapunov matrix P (θ ) >0 such that 0 0( )

T x P xυ υ θ ≤ 1 for some initial condition

xυ 0, which in turn can be written as the LMI constraint in

the affinely parameter-dependent matrix variable Q(θ ):

0

0

10.

( )

T x

x Qυ

υ θ

≥

(17)

If Q(θ ) is also a stabilizing solution for the GEVP de-

scribed in (16), then0 0

( ) ( ) 1T T x P x x P xυ υ υ υ

θ θ < ≤

. The

magnitude constraint ,maxi iu u≤ can be satisfied if

1 1 1

2

,max

1( ) ( ) ( ) ,

T T T i i

i

x Q L L Q x x Q xu

υ υ υ υ θ θ θ − − −≤

where Li

∈ R

1×6is the i-th row of the matrix L and can

be written as the following LMI condition for the given

ui,max:

2

,max

( )0,

T T i

i i

Q L s

s L u

θ ≥

(18)

where the row vector si ∈ R1×2

with the i-th element is

equal to 1 and the others are equal to zero. Similarly, the

magnitude constraint on the roll angle, θ ≤ θ max, and

variation rate of the roll angle, maxθ θ ≤ , can be satis-

fied if the following LMI conditions hold for the given

θ max and maxθ

:

5 6

2 2

5 max 6 max

( ) ( ) ( ) ( )0, 0,

( ) ( )

T T Q Q r Q Q r

r Q I r Q I

θ θ θ θ

θ θ θ θ

≥ ≥

(19)

where the row vector r j ∈ R1×6

with the j-th element is

equal to 1 and the others are equal to zero.

3.2 Robust gain-scheduled control for an aircraft

with the uncertain coupling factor

Consider the LPV PVTOL dynamics with the un-

certain coefficient ε as shown in (7) and magnitude con-straints imposed on the control efforts ũ, the roll angle θ ,and the variation rate of the roll angle θ :

0 1 2 ,max

max max

( ( ) ( )) , ,

, .

i i x A x B B u u uυ υ θ θ

θ θ θ θ

= + + ≤

≤ ≤

ε(20)

The controller design must be robust to the measurable

varying and bounded parameters θ , θ and the uncertain

parasitic coupling ε. For the linear gain-scheduled state-

feedback control law, ũ = K (θ ) xυ , to robustly stabilize

the PVTOL aircraft dynamics (20), let Ã 0 := A0 − γ I , and

let the parameter-dependent quadratic Lyapunov func-tion V ( xυ , θ ) = ( )

T x P xυ υ θ , P (θ ) > 0; then, the need of

negative change rate of V ( xυ , θ ) is equivalent to

0 1 2

0 1 2

( ( ( ) ( )) ( )) ( )

( )( ( ( ) ) ( )) ( ) 0.

T A B B K P

P A B B K P

θ θ θ θ

θ θ θ θ θ

+ +

+ + + + <

ε

ε

(21)

Denoting P (θ )−1= Q(θ ) and K (θ )Q(θ ) = L(θ ), and

from the identity in (12), the matrix inequality (21) can

be rewritten as

0 0 1 2

1 2

( ) ( ) ( ) ( ( ) ( ))

( ( ) ( )) ( ) ( ) 0.

T T T Q A A Q L B B

B B L Q

θ θ θ θ θ

θ θ θ θ

+ + +

+ + − <

ε

ε (22)

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The input matrix B1(θ ) can be denoted as the affinely

parameter-dependent matrix as shown in (9) and (10).

The input matrix B2(θ ), multiplied by the uncertain co-

efficient ε, is represented by

2 0 20( ) ( ( )) : ( ) , B M N B M N θ θ θ = ϒ + ϒ = + ϒ (23)

where

max

max

max0

max

0

max

0 cos 1 0 0 0 0,

0 0 0 sin 0 0

100 1

cos 1, ,0 1

0 0

cos 10

cos 1 .sin

0sin

T

M

N

θ

θ

θ

θ

θ θ

θ

− =

−= ϒ =

−

− ϒ =

(24)

By the formulation of B2(θ ) in (23), (24), and the as-

sumed matrix variables Q(θ ), L(θ ) shown in (14), the

matrix inequality (22) becomes

0 0 1 20

1 20

( ) ( ) ( ( ) )

( ( ) ) ( ( ) )

( ( ) ) ( ) 0.

T T T

T T

c s s c

Q A A Q L B B

B B L L M N

M N L p Q p Q

θ θ θ

θ θ

θ θ

+ + +

+ + + ϒ

+ ϒ − − <

ε

ε ε

ε

(25)

Note that in the parameter-dependent matrix ϒ(θ ) and

the elements ϒ(θ )(1, 1) ∈ [0, 1], ϒ(θ )(2, 2) ∈ [−1, 1],

and ϒ(θ ) satisfy the norm bounded condition

max( ) ( ) , ,T I θ θ θ θ ϒ ϒ ≤ ∀ ≤ (26)

which yields the following inequality:

( )0,

( )T

I

I

θ

θ

ϒ ≥

ϒ (27)

through the application of Schur complements [14]. For

any non-zero number ξ , we have

( )1

1

( )0

( )

T

T T T

I M M L N

I NL

θ ξ ξ ξ

θ ξ

−−

ϒ ≥ − ϒ −

(28)

and

2 2( ( ) ) ( ) .

T T T T M NL M NL MM L N NLθ θ ξ ξ −ϒ + ϒ ≤ + (29)

Then, we have the sufficient condition of (25):

0 0 1 20 1 10

2 2

( ) ( ) ( ( ) ) ( ( ) )

( ) 0. (30)

T T T

T T T c s s c

Q A A Q L B B B B L

MM L N NL p Q p Q

θ θ θ θ

ξ ξ θ −

+ + + + +

+ + − − <

ε ε

ε ε

which in turn can be written as

2

1 2

( ( ), , )

( ) 2 ( ) 0,

T T T

c s s c

Q L MM L N

p Q p Q Q NL I

θ ξ

θ γ θ ξ

−

Ξ +

− − − < −

ε ε

ε(31)

where

0 0 1 20

1 20

( ( ), , ) ( ) ( ) ( ( ) )

( ( ) ) .

T T T Q L Q A A Q L B B

B B L

θ θ θ θ

θ

Ξ = + + ++ +

ε ε

ε

Similar to the LMI conditions for the nominal aircraft

case, the feasibility of (31) for the given parasitic uncer-

tain level ε can be established by evaluating (31) only for

the extreme values of the scalar varying parameters { p s,

pc, θ

} and recast as the GEVP type LMI to maximizethe achievable relative stability β = −γ as follows:

max max

0

minimize : (14), (31), for { , , }

{{ , },{ , },{ , }}

with variables , , , .

s c

s s c c

s c

p p

p p p p

Q Q Q L

γ θ

θ θ

=

∈ −

(32)

3.3 Gravitational disturbance cancellation

Consider the equivalent LPV representation of the

PVTOL aircraft dynamics shown in (6) with magnitude

constraints imposed on the control efforts ũ, the roll an-

gle θ , and the variation rate of the roll angle θ :

0 1 2

,max max max

ˆ( ( ) ( )) ( ),

, , .i i

x A x B B u D

u u

υ υ θ θ θ

θ θ θ θ

= + + +

≤ ≤ ≤

ε(33)

The composite control law û = ũ + ũτ is assumed, where

ũ = K (θ ) xυ is constructed by means of K (θ ) = LQ(θ )−1

for the case of the nominal aircraft in (8) or the case with

ε considered in (20). The nonlinear control law ũτ is in-

troduced to eliminate the effect of the gravitational

disturbance ˆ ( ) D θ such that a stabilizing control law ũ

designed for system (8) or (20) will still stabilize the

original aircraft system (33) whether the uncertainy ε is present or not. In terms of the parameter-dependent

quadratic Lyapunov function V ( xυ , θ ) = ( )T x P xυ υ θ , P (θ )

> 0, with a negative change rate ( , )V xυ θ < 0, the con-

trol law ũτ needs to satisfy

1 2

1 2

ˆ(( ( ) ( )) ( )) ( )

ˆ( )(( ( ) ( )) ( )) 0.

T

T

B B u D P x

x P B B u D

τ υ

υ τ

θ θ θ θ

θ θ θ θ

+ +

+ + + ≤

ε

ε

(34)

In order not to incur the uncertainty ε and due to the fact

that the first column of the parameter-dependent matrix

B2(θ ) is zero, which is multipiled by the uncertainty ε, the

form of the control law ũτ = (uτ , 0)T is assumed to be such

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that B2(θ )ũτ = 0. Therefore, only an extra control signal

for the thrust is manipulated for ũτ . By substituting the

components of B1(θ ) and ˆ ( ) D θ , and denoting g l as the

l -th element of the on-line computed vector Q(θ )−1 xυ for

the nominal system in (8) or for the uncertainty ε consid-ered in (20), we can rewrite condition (34) as

2 4( sin ( 1)) (cos ( 1) 1) 0.u g u g τ τ θ θ θ − + + + − ≤ (35)

One possibility for the choice of uτ is simply to have uτ =

4 2

4 2

(1 cos ) (sin )

cos sin

g g

g g

θ θ θ

θ θ

− + −−

such that the equality of

(35) holds. A reasonable magnitude limit for uτ can be

imposed, that is, uτ ≤ uτ ,max, in the on-line application.

IV. SIMULATIONS AND DISCUSSION

The gain-scheduled controller designs for the input

and output constrained PVTOL aircraft dynamics ob-

tained via the LMI approach can be numerically con-

structed for the types of GEVP problem as follows.

• Nominal aircraft in the affine LPV representation:

Maximize the relative stability β = −γ for the nominal

aircraft in (8) with the uncertain coefficient ε = 0 by

means of (16) while maintaining various magnitude

constraints imposed on the control efforts ũi by (17),

(18) and on the roll angle variation by (19) for { p s, pc,

θ } ∈ {{ , } s s p p , { , }c c p p , max max{ , }}θ θ − . The ini-tial conditions considered in the parameter-dependent

invariant ellipsoid representation (17) are specified as

xυ 0 = (±1, 0, ±1, 0, 0, 0)T

since the performance objec-

tive is to maintain tracking of the normalized position

command signal in the horizontal and vertical direc-

tions. With the following specified magnitude con-

straints: deviated thrust u1,max = 0.6, moment u2,max =

{0.2, 0.4, 0.6, 0.8, 1.0, 1.2}, roll angle and variation

rate of roll angle θ max = maxθ = π ⋅ u2,max/k , the

achieved maximum relative stability β * = −γ . for k ∈

{3, 4, 5, 6} is that depicted in Fig. 3. The chosen mag-

nitude constraints θ max and maxθ

are proportional tou2,max since the control moment u2 needs a certain

range of θ and θ to manipulate motion in the hori-

zontal plane. However, the quantities of θ max and maxθ

correspond to the extreme values of the scalar varying

parameters { p s, pc, θ } that need to be addressed in

the matrix inequality constraints {(16), (17), (18),

(19)}. It can be seen from Fig. 3 that a looser attitude

maneuverability constraint tends to allow faster posi-

tion tracking when a small magnitude of the control

moment is available, while a tighter attitude maneu-

verability constraint tends to maintain faster position

tracking when a large magnitude of the control mo-

ment is available.

Fig. 3. Maximum relative stability γ for u1,max = 0.6, ε = 0 with various

constraints imposed on u2,max, θ max, and maxθ .

Fig. 4. Maximum relative stability γ for u1,max = 0.6, with various con-

straints imposed on u2,max, ε, and θ max, and maxθ = π ⋅ u2,max/5.

• Aircraft with the uncertain coupling factor: Maximizethe relative stability β = −γ for the aircraft dynamics in

(20) with an estimated tolerable uncertain level ε by

means of (32) while maintaining various magnitude

constraints imposed on the control efforts ũi by (17),

(18) and on the roll angle variation by (19) for { p s, pc,

θ } ∈ {{ , } s s p p , { , }c c p p , max max{ , }}θ θ − . To let

considered { xυ 0, u1,max, u2,max} to have the same values

as for the nominal case and θ max = maxθ = π ⋅ u2,max/5,

the achieved maximum relative stability β * = −γ . for ε

= {0, 1, 2, 3, 4} is depicted in Fig. 4. It can be seen

that the position tracking performance worsens due to

the presence of uncertain coupling, and the large mag-nitude of the control moment tends to magnify the

detrimental nonminimum phase effect induced by the

uncertain coupling.

Simulations of the closed-loop controlled system

were conducted as shown in Fig. 5, where the nonlinear

PVTOL dynamics are from (3) with B(θ , ε) := B1(θ ) +ε B2(θ ) and the command generator is a low-pass filter

with a bandwidth of 100 rad/sec for both the lateral

command xd and vertical command yd . Therefore, the

actual tracking command ( xr , yr ) issued to the controlled

system is generated from the exogenous command signal

( xd , yd ) by r = Aα r + Bα d , where

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Fig. 5. The structure of the simulation of the controlled PVTOL air-

craft.

100 0: , : , : ,0 100

100 0: .

0 100

d r

d r

x xr d A y y

B

α

α

− = = = −

=

The parameter-dependent state-feedback matrix K (θ ) =

L(Q0 + p sQ s + pcQc)−1

is obtained with the specified con-

trol effort constraints u1,max = u2,max = 0.6, θ max = maxθ =

π ⋅ u2,max/5 for both designs whether parasitic uncertainty

ε is present or not. For the nominal aircraft, the achieved

maximum relative stability is β ∗ = 0.2387 with the ma-

trix variable

0.0035 0.0025 0.0859 0.3081 0.0035 0.0007

0.0148 0.0454 0.0008 0.0007 0.1067 0.0467. L

− − − −=

− − −

(36)

For the aircraft dynamics with ε = 2, the achieved rela-

tive stability is β * = 0.2115 with the matrix variable

0.0000 0.0000 0.0905 0.2954 0.0000 0.0000

0.0136 0.1387 0.0000 0.0000 0.0139 0.0623. L

− − −=

− − − −

(37)

Figures 6-17 show the time responses of theclosed-loop system with the parameter-dependent con-

troller K (θ ) constructed for the case of the nominal air-

craft. Figures 6, 8, 10, and 12 show the time responses of

the horizontal and vertical position ( x, y), where the

dashed line “--” denotes the exogenous commands xd and

yd , the dotted line “..” denotes the actual commands xr

and yr , and the solid line “−” denotes the controlled time

response. Figures 7, 9, 11, and 13 show the time re-

sponses of the roll angle and it’s variation rate (θ , θ ),

and the magnitude of the control effort, ũ = (ũ1, u2),

where the solid line “−” denotes the roll angle θ and con-

trol moment u2, and the dotted line “..” denotes thevariation rate of the roll angle θ and the deviation of

Fig. 6. Time response of ( x, y) for the lateral command with ε = 0.

Fig. 7. Time response of (θ , θ ), (ũ1, u2) for the lateral command with

ε = 0.

Fig. 8. Time response of ( x, y) for the vertical command with ε = 0.

Fig. 9. Time response of (θ , θ

), (ũ1, u2) for the vertical commandwith ε = 0.

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Fig. 10. Time response of ( x, y) for the command ( xd , yd ) = (1, 1) with ε

= 0.

Fig. 11. Time response of (θ , θ ), (ũ1, u2) for the command ( xd , yd ) =

(1, 1) with ε = 0.

Fig. 12. Time response of ( x, y) for the lateral command with ε = sin

0.2t .

Fig. 13. Time response of (θ , θ

), (ũ1, u2) for the lateral commandwith ε = sin 0.2t .

Fig. 14. Time response of ( x, θ , θ ) for the lateral command with ε =

{0.1, 0.3, 0.5}.

Fig. 15. Time response of (ũ1, u2) for the lateral command with ε =

{0.1, 0.3, 0.5}.

Fig. 16. Time response of ( x, θ , θ ) for the lateral command with ε =

0.55.

Fig. 17. Time response of (ũ1, u2) for the lateral command with ε =0.55.

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the thrust ũ1. As shown in Figs. 6-11 for the system

without parasitic coupling ε, the position tracking per-

formance is good when the command ( xd , yd ) is issued as

(1, 0), (1, 0), or (1, 1). The quantities of 1u , 2u , θ ,

and θ satisfy the speci.ed constraints as well.When a time-varying uncertainy ε = sin 0.2t is pre-

sent as shown in Figs. 12-13, the positive values of ε

produce the non-minimum phase characteristic and cause

the position tracking response for the lateral command

( xd , yd ) = (1, 0) to deteriorate. It is shown that a larger

attitude change occurs, and that the required control

moment u2 increases as well. Figures 14-15 show the

time response of ( x, θ , θ ) and (ũ1, u2) for the lateral

command ( xd , yd ) = (1, 0) with the uncertainty ε = {0.1,

0.3, 0.5}. It can be seen that large oscillations occur dur-

ing the transition due to the consistently positive values

of ε. Figures 16-17 show the time response of ( x, θ , θ

)and (ũ1, u2) for the lateral command ( xd , yd ) = (1, 0) with

the uncertain parasitic coupling ε = 0.55, where the dot-

ted line “..” denotes the time response with the magni-

tude constraint imposed on the control moment u2, and

the solid line “−” denotes the time response without satu-

ration of the control moment u2. It can be seen that the

resulting oscillations increase and eventually cause

instability.

For the case where the parameter-dependent con-

troller K (θ ) is constructed while the uncertainty level

ε = 2 is explicitly addressed, Figs. 18-20 show the time

responses of ( x, y), (θ , θ ), and (ũ1, u2) for the issued

command ( xd , yd ) = (1, 1) in the presence of parasitic

uncertainty ε = {0, 1, 2, 3}. The simulation results show

that robustness is established, and that the design speci-

fication is well satisfied.

V. CONCLUSIONS

This paper has presented a robust gain-scheduled

control for a PVTOL aircraft via the LMI method. The

design is based on an LPV equivalent representation for

non-linear PVTOL aircraft dynamics subject to an un-

certain non-minimum phase effect without any truncated

linearization or approximation. These LPV PVTOL air-craft dynamics consist of a nominal LPV system and a

linear system, where the norm bounded uncertainy ma-

trix is multiplied by the uncertain parasitic coupling. For

the nominal LPV system, which is considered to be affi-

nely dependent on the trigonometric functions of the

measurable varying roll angle, the ranges of the varying

parameter and its variation as well as the magnitudes of

the denoted affinely trigonometric functions and para-

sitic uncertain matrix have been addressed by introduc-

ing the parameter-dependent invariant ellipsoid interpre-

tation to deal with the issue of affinely quadratic stabili-

zation. The relations among the magnitude of actuator saturation, the maximum achievable relative stability,

Fig. 18. Time response of ( x, y) for the command ( xd , yd ) = (1, 1) with ε

= {0, 1, 2, 3}.

Fig. 19. Time response of (θ , θ ) for the command ( xd , yd ) = (1, 1)

with ε = {0, 1, 2, 3).

Fig. 20. Time response of (ũ

1, u2) for the command ( xd , yd ) = (1, 1)with ε = {0, 1, 2, 3).

and the sustainable coupling uncertainty have been in-

vestigated in this robust gain-scheduled design.

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