44. linear parameter varying control of a quadrotor

Linear Parameter Varying Control of a QuadrotorSamarathunga L.M.D. Rangajeeva, Member, IEEE, and James F. Whidborne, Senior Member, IEEE

Abstract—This paper describes a Linear Parameter Varying(LPV) controller design for a quadrotor vehicle. The controllersynthesis requires the LPV plant model which is obtained bylinearisation of the modelling equations, to be in an affine form.However, the LPV representation of the quadrotor dynamicis not affine; hence it has been transformed into to a convexpolytopic form using Tensor Product (TP) transformation. TheH∞ self gain scheduling control method has been applied toobtain a LPV controller which is tested on a simplified nonlinearmodel of the quadrotor. The LPV controller performance wascompared with a separate H∞ controller. The LPV controllerexhibited significant close tracking capabilities with respect tothe H∞ controller.

Index Terms—Quadrotor vehicle, Linear parameter varying,Tensor product model transformation, self gain scheduled

I. INTRODUCTION

THE quadrotor is an aircraft in which lift is generated byfour rotors symmetrically fixed around its centre. The

required flight-manoeuvres (i.e. yawing, rolling and pitching)and the vertical or lateral flight are realised by indepen-dently varying speeds of the four rotors. Moreover not onlyit is capable of performing Vertical Take-Off and Landing(VTOL), also has a simpler configuration for a compact me-chanical design. Therefore the quadrotor vehicle has becomean attractive candidate for small scale and medium scaleUnmanned Aerial Vehicles (UAVs) for applications such asreconnaissance, search, rescue and surveillance. However thedynamics of this aerial vehicle represents a marginally stablesystem and an active control system is essential to stabiliseit.

Though various control laws both linear and nonlinear havebeen tested out on the quadrotor, a fully autonomous control-ling of an quadrotor poses a tough challenge to control systemdesigners. As nonlinear control laws sliding mode control[1], adaptive back stepping control [2] and inversion method[3] have been studied. These methods have demonstratedvery promising performance and robustness. Nevertheless,there are practical issues in applying those techniques onthe real quadrotor. For an instance, the inversion methodcauses actuator saturation in aggressive manoeuvres and in theback stepping method it assumes quick switch over betweencontrollers in real time. With regard to linear methods, controllaws such as PID, LQR and H∞ have been investigated. It wasfound, the performances are limited to the hover condition,as nominal design point for the controller is hover. Whenthe UAV departs from hover or undergoes large perturbations

S.L.M.D. Rangajeeva is with the Department of Mechanical and Manu-facturing Engineering, Faculty of Engineering, University of Ruhuna, Galle,80000 Sri Lanka. e-mail:[email protected]

J.F. Whidborne is with the Department of Aerospace Sciences, Schoolof Engineering, Cranfield University Bedford, MK43 0AL UK. e-mail:[email protected]

control has been lost [4]. In order to extend the flight envelopea common approach is gain scheduling. In which several linearmodels of the quadrotor are obtained for different trim pointsand then number of Liner Time Invariant (LTI) controllersare derived for each point. As operating conditions vary,the global controller is estimated by interpolating gains ofthe local controllers. Even though this has been successfullyimplemented in many engineering applications, there is noguarantee for the performance, robustness and nominal sta-bility of the control design [5].

The Linear Parameter Varying (LPV) technique has beenintroduced as an alternative gain scheduling process. As itname implies the plant model and the controller are still linear,but the dynamics of the plant model as well as the controllerdepend on some time varying parameters whose values areunknown in advance. However, those parameter’s values canbe measured in real time, thus both the plant and the controllerare changing as a function of operating conditions. This typeof control strategy ensures required performance, robustnessand stability along all possible trajectories of the parametersand it has been proved theoretically [6]. The LPV controllersynthesis requires an affine dynamic system that can be con-verted into a convex polytopic form. However the quadrotorLPV plant model is not affine and as a result an infinitenumber of Linear Matrix Inequalities (LMIs) are required todetermine the controller. In order to overcome this difficulty,recently proposed Tensor Product (TP) model transformationwas applied to convert the nonaffine system into a convexpolytopic form.

This paper presents a designing of a LPV controller with theuse of TP–transformation. Further as a means of comparingthe performance an H∞ controller has also been designed.The new controller has been tested and compared in Simulinkenvironment of MATLAB.

II. DYNAMIC MODEL OF THE QUADROTOR

A simplified non-linear dynamic model for commerciallyavailable “Draganflyer XPro” which is owned by CranfieldUniversity was developed for this study. The basic layout ofthe vehicle is shown in Fig.1. It consists of four equal inlength carbon fibre arms which are connected to the centralbody. At the end of each arm a rotor having radius ra isattached. The front and rear rotors revolve at angular speedsω1 and ω2 in counterclockwise (CCW) direction generatingthrusts of τ1 and τ2; while left and right rotate at ω3 and ω4

in clockwise (CW) direction generating thrusts of τ3 and τ4.In a hover flight all rotors have same angular speeds to

generate same amount of thrust to balance the weight. Sincethe pairs of rotors have opposite sense of rotational directionsthe net imbalance of the reaction torques is zero. In order to

2011 6th International Conference on Industrial and Information Systems, ICIIS 2011, Aug. 16-19, 2011, Sri Lanka

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978-1-61284-0035-4/11/$26.00 ©2011 IEEE

xb

yb

zb

xe

ye

ze

O1

1

2 4

3 Front

Rear Right

Left

p / φ

q / θ

r / ψ

Fig. 1. The Quadrotor

perform a vertical flight from a trimmed hovering condition,the total thrust (Γ) need to be augmented by increasing speedof each rotor by an equal amount. For a forward flight(or a sidewards flight) a certain amount of pitch angle θ(or a bank angle φ) needs to be reached and maintainedit. With regard to attitude control for an example to pitch-up the speed (hence the thrust) of front rotor is increasedwhile that of the rear is reduced by a same amount. Soa pitching moment M is produced while keeping the totalthrust remains unchanged. Rolling also can be performed inanalogous manner by varying the thrust generated by left andright side rotors and creating a rolling moment L. In yawingmanoeuvre or to alter the azimuth angle ψ it is necessaryto create an imbalance of torque while keeping the totalthrust intact. This is materialised by, increasing(or decreasing)angular speeds of the rear rotor and the front rotor whiledecreasing (or increasing) speeds of left and right by a sameamount. This generates a yawing moment N .

A. Equations of Motion

The equations of motion (EoM) of the quadrotor have beendiscussed in number of papers [1], [2], thus a detail discussionis not included here. In order to develop the EoM, a body axissystem (xbybzb) and an earth referential (xeyeze) are set upas shown in Fig. 1. When the context is clear subscript b ore will be omitted. The vehicle has a total mass of mb andthe total weight mbg acts at the centre O1 in ze direction.Further due to the symmetry of the configuration the inertiaabout xb and yb has a same value of Ix, the inertia about zbhas a value of Iz that is equal to 2Ix and all the product ofinertia are zero.

In addition to attitude (i.e. φ , θ , and ψ), angular velocitiesabout body axes: roll rate–p (around xb), pitch rate–q (aroundyb), yaw rate–r (aroundzb), linear velocities with respect toearth axes: Ux (along xe), Uy (along ye), Uz (along ze) andposition: xe , ye , ze in earth referential are required to fullydefine the dynamic behaviour of the quadrotor. Hence, takingp , q , r , φ , θ , ψ , Ux , Uy , Uz , xe , ye , and ze as variablesof the state vector x; four control inputs Γ , L ,M , and N asvariables of the control vector u the equations of motion can

be rewritten in a form as:p = f1(x, u) = k3qr + L,q = f2(x, u) = −k3pr +M,r = f3(x, u) = N,

φ = f4(x, u) = p cosψ − q sinψ,

θ = f5(x, u) =p sinψ

cosφ+q cosψ

cosφ,

ψ = f6(x, u) = p sinψ tanφ+ q cosψ tanφ+ r,

Ux = f7(x, u) = −k1U2x − Γ cosφ sin θ,

Uy = f8(x, u) = −k1U2y − Γ(− sinφ),

Uz = f9(x, u) = g − Γ cosφ cos θ,xe = f10(x, u) = Ux,ye = f11(x, u) = Uy,ze = f12(x, u) = Uz,

(1)

where k1 =ρADCD

2mb, k3 = 1− Iz

Ixand AD = 4×πr2a, CD–

the drag coefficient, ρ–density of air. Also x and u denote theelements of state vector x = (p , q , r , φ, . . . , ye , ze )T , andthe control vector u = (Γ , L ,M ,N )T . In this quadrotordynamic model the output vector y is taken as: y = x.

III. LPV CONTROLLER DESIGN

The state space representation of a LTI system is in a formof:

x(t) = Ax(t) +Bu(t),

y(t) = Cx(t) +Du(t), (2)

where A, B, C, and D denote the State space matrix, the Inputmatrix, the Output matrix, and the Feed forward matrix re-spectively and these matrices are assumed to be time invariant.In general dynamic systems behave in nonlinear manner andabove LTI representation is obtained by linearisation (usingmethods such as Jacobian linearisation, function substitutionor state transformation) of the modelling equations of thesystem at an equilibrium point of the operating range. Linearcontrol methods can be applied to stabilise a LTI system,but performance of such a linear controller deteriorates asthe actual system deviates from the equilibrium point. Thisis primarily due to fact that in reality the system matricesA, B, C, D are not constant and continuously change withtime. Even though the gain scheduling controlling methodcould be applied, there is no theoretical justification.

To take into account such variations, the system matricesare described in terms of a time dependent parameter vectorp(t) which is unknown in advance, but can be measured inreal time. This yields the LPV model of the system as follows[6]:

x(t) = A(p(t))x(t) +B(p(t))u(t),

y(t) = C(p(t))x(t) +D(p(t))u(t). (3)

In addition to incorporate the parameter dependency intothe controller a new parameter dependent LPV controller isdefined by:

xK(t) = AK(p(t))xK(t) +BK(p(t))e(t),

u(t) = CK(p(t))xK(t) +DK(p(t))e(t), (4)


484

where the subscript K stands for the controller and e(t) isthe error signal that is the difference between the referencesignal Rref (t) and the system output y(t). Now the LPVplant model and the controller are parameter dependent, inthe gain scheduling process both systems automatically getupdated depending on the operating conditions. This ensuresstability, performance and robustness of the LPV controllerfor all possible trajectories of p(t) [6]. For convenience,in following sections, when the context does not lead to aconfusion, the dependence of x ,y ,u , e ,p on t will bedropped.

A. Controller Synthesis

The LPV controller is derived using Bounded Real Lemma(BRL) with the notion of quadratic H∞ performance γ [6].Given a closed loop (cl) LPV system (i.e. combination of theLPV controller and the LPV system model) having state spacematrices Acl(p), Bcl(p), Ccl(p), and Dcl(p), the system hasquadratic H∞ performance γ if and only if there exists asingle positive definite matrix P such that:AT

cl(p)P + PAcl(p) PBcl(p) CTcl(p)

BTcl(p)P −γI DT

cl(p)Ccl(p) Dcl(p) −γI

< 0, (5)

for all admissible values of the parameter vector p. Thenthe Lyapunov function V (x) = xTPx establishes global(asymptotic) stability and the L2 gain between the input andoutput is bounded by γ. That is:

‖y‖2 < γ‖u‖2, (6)

for along all possible parameter trajectories of p. Therefore,the quadratic H∞ performance requires an existence of afixed quadratic Lyapunov function for the entire operatingrange [6]. Even though the estimation of the controller islimited resolving the LMI given in (5), it generates an infinitenumber of LMIs to cover all possible trajectories of p. Theconcept of an Affine Polytopic LPV plant has been introducedto overcome this difficulty.

1) Affine Polytopic LPV Systems: As outlined in [5], sup-pose in a LPV system that p(t) = [p1(t), p2(t), . . . , pN(t)] ∈RN be a vector of time-varying real parameters. It is also as-sumed that each parameter pi(t) lies between known extremevalues pi and pi, i.e. pi(t) ∈ [pi, pi] and p(t) lies in a polytopeΘ with vertices Θ1,Θ2, . . . ,Θm, p(t) ∈ Θ, m = 2N. In anaffine LPV model the system matrices S(p), which are linearfunction of the time varying parameters, can be written as aconvex combination of the vertex matrices of the parameterpolytope Θ [6]:

S(p) =

(A(p) B(p)C(p) D(p)

)=

m∑i=1

αi(p)

(A(Θi) B(Θi)C(Θi) D(Θi)

),

(7)m∑i=1

αi(p) = 1.

It can be proved (5) holds for all possible p(t), if andonly if it holds at the vertices Θi for i = 1, . . . ,m [6].

Therefore, the infinite number of LMIs to determine theLPV controller can be eliminated thanks to the convexityof the affine LPV system. The LPV controller synthesis isimplemented in MATLAB as a function “hinfgs”.

B. Linearisation of the EoM

In order to apply the LPV controller synthesis, a linearsystem model of the quadrotor is generated by linearising (1)at an equilibrium point of general trimmed forward flight.In such a flight condition, the vehicle motion is completelydefined by φ , θ , Ux , Uy , Uz, xe , ye , and ze while all bodyrates p , q , r and azimuth angle ψ must be zero. The linearmodel of the quadrotor is in a form of (2). Since the desiredoutput vector y is equal to the x, C is an identity matrixwhile D is a null matrix. The system matrices A and B areobtained by using following partial derivative formulae:

Aij =(

∂fi∂xj

)x=x u=u

,

Bij =(

∂fi∂uj

)x=x u=u

,(8)

where x = ( 0, 0 , 0 , φ , θ , 0 , Ux , Uy , 0, 0 , 0 , 0 )T andu = (Γ , 0 , 0 , 0 )T are the state vector and the control vectorat an equilibrium point. It should be noted that the variablesUz , xe, ye and ze are taken as zero since these are independentof system matrices A and B. Equation (8) gives:

A =

03×3 03×3 03×3 03×3P 03×3 03×3 03×3

03×3 Q R 03×303×3 03×3 I3×3 03×3

,

B =

03×1 I3×303×1 03×3S 03×3

03×1 03×3

,(9)

where with notation s− sin , c− cos and t− tan:

P =

1 0 0

01

cφ0

0 tφ 1

, R =

−2Uxk1 0 00 −2Uyk1 00 0 0

,

Q =

Γsφsθ −Γcφcθ 0Γcφ 0 0

Γcθsφ Γcφsθ 0

, S =

−cφsθsφ−cφcθ

.

Therefore system matrices are function of the state variablesand continuously change with time. In such an occasion; thesystem can be put into the LPV form as:

x = A(p)x +B(p)u,y = x,

(10)

where p = [Ux , Uy , φ , θ ,Γ] is the time varying parametervector. Further, it is evident that A and B are not linearfunctions of time varying parameters, hence the quadrotorLPV model is not affine.


485

C. Tensor Product (TP) Model Transformation

Despite the fact that many control problems can be for-mulated into the affine LPV form, still there are substantialamount of nonaffine systems. With reference to (10) quadrotordynamics falls into the nonaffine form and has to be convertedinto a convex form in order to obtain a LPV controllerwith finite number of LMIs. For this conversion processrecently proposed Tensor Product model transformation hasbeen applied. As explained in [7], this is based on HigherOrder Singular Value Decomposition (HOSVD) of tensors.

The S(p(t)) of a non-affine LPV model can be written asa real (R) matrix:

S(p(t)) =

(A(p(t)) B(p(t))C(p(t)) D(p(t))

)∈ R(n+q)×(n+m), (11)

where n,m, q are the number of states, inputs and outputs ofthe system. Also, let p(t) = [p1(t), p2(t), . . . , pN(t)] ∈ RN bea vector of time-varying real parameters. As pi(t) ∈ [pi, pi],the parameter space Ω is, Ω = [p1, p1]×[p2, p2]×. . .×[pN, pN].Now, the Ω is discretised over a large number of points and agrid LPV model is obtained by expressing the S(p(t)) ateach grid point. Then, the discretised system matrices arestored into a tensor SD ∈ RM1×M2×...×MN×(n+q)×(n+m).The M1,M2, . . . ,MN are the number of grid lines defined oneach direction of the parameter vector p(t). After, HOSVD isapplied on first N dimensions of the tensor SD. In this processit is discarded all zero or small singular values σk and theircorresponding singular vectors in all 1, . . . ,N dimensions [7].This results a tensor S and a set of matrices U1, U2, . . . , UNsuch that SD ≈ S × U1 × U2 × . . .× UN.

Then the LTI vertices Sr ∈ R(n+q)×(n+m) are extractedfrom the tensor S having the size of I1× . . .× IN× (n+q)×(n+m) as:

Sr =

(Ar Br

Cr Dr

)= S(I1,I2,...,IN), (12)

where r = ordering(I1, I2, . . . , IN), r = 1 to R and R =∏N1 In. The matrices U1, U2, . . . , UN contain the continuous

weighting functions for one bounded variable. For an exampleUn ∈ RMn×In is made of continuous weighting functionswn,j(pn(t)) ∈ RMn , j = 1, . . . , In, and In is the numberof weighting functions used in the nth dimension of theparameter vector p(t). The weighting function wn,j(pn(t))is the jth weighting function defined on the nth dimension ofthe Ω and pn(t) is the nth element of the vector p(t). Thenthe weighting functions at each vertex are defined as:

wr(p(t)) =N∏

n=1

wn,j(pn(t)). (13)

The whole point of the TP transformation is to find a LTIvertex system Sr and weighting functions wr(p(t)) such thatthe S(p(t)) in (11) can be expressed as:

R∑r=1

wr(p(t))Sr. (14)

Further, the convex combination of Sr LTI vertex system isguaranteed by the following conditions [7]:

∀n ∈ [1,N], j ∈ [1, In], pn(t) : wn,j(pn(t)) ∈ [0, 1],

∀n ∈ [1,N], pn(t) :

In∑j=1

wn,j(pn(t)) = 1,

∀r ∈ [1,R], p(t) : wr(p(t)) ∈ [0, 1], (15)

∀p(t) :R∑

r=1

wr(p(t)) = 1.

Therefore with the TP method a given LPV system S(p(t))can be transformed into a convex hull of LTI vertex system Sr

for any p(t) ∈ Ω and the LPV controller synthesis is readilyapplicable. Nevertheless the TP type convex polytopic systemrepresentation is an approximation and the error δ is definedas [7]: ∥∥∥∥∥S(p(t))−

R∑r=1

wr(p(t))Sr

∥∥∥∥∥ ≤ δ. (16)

D. TP Transformation of the Quadrotor LPV Model

As described in the previous section TP transformationinvolves four key steps: (i) Defining the parameter space anddiscretised it over a large number of points, (ii) Express theLPV plant S(p(t)) at each grid points to obtain the tensorSD, (iii) Carryout HOSVD on the SD to obtain a tensor Sand a set of matrices U1, U2, . . . , UN, and (iv) Extraction ofLTI vertices Sr and weighting functions wr(p(t)) from theS and the set of matrices respectively.

1) Step (i): As per (9) the quadrotor LPV model is afunction of five time varying parameters. However, in aforward flight without any climb rate in ze (i.e. Uz = 0),the relation f9(x, u) in (1) yields:

Γ =g

cosφ cos θ. (17)

Thus, there are only four independent time varying parametersand their ranges have been specified as: -0.4 rad to + 0.4rad for θ and φ, -5 ms−1 to +5 ms−1 for Ux and Uy .These ranges have been selected considering the fact thatthe quadrotor is not designed to operate linear velocitieshigher than 5 ms−1, as it becomes extremely uncontrollable[8]. Also a moderate size of a parameter space helps tolimit the computational loads associate with the TP trans-formation [7]. So the parameter space Ω is specified as:[−0.4 0.4]× [−0.4 0.4]× [−5 5]× [−5 5], which has beendiscretised with a sampling grid of 300× 300× 200× 200.

2) Step (ii) and Step (iii): An open source MATLAB TP-toolbox [9] has been used to execute both of these steps.During the HOSVD, it has been received; five singular valuesin each φ, θ directions and two singular values in each Ux,Uy directions. Without discarding any of the singular values,the TP-toolbox has given the tensor S with dimensions 5 ×5 × 2 × 2 and the set of matrices.


486

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Angle (rad)

Mag

nit

ud

e o

f w

eig

hti

ng

fu

nct

ion

s

w11

w12

w13

w14

w15

Student Version of MATLABFig. 2. Continuous weighting functions for φ : w1,i(φ(t))

3) Step (iv): A TP type convex polytopic LPV modelwith 100 (5 × 5 × 2 × 2) LTI vertices (Sr) has beenextracted from the S . Then the weighting functions wereextracted from the matrices U1, U2, . . . , UN. In the convexpolytopic representation, there are five weighting functions(w1,i(φ(t)) : i = 1, . . . , 5) for the parameter φ, another fiveweighting functions (w2,j(θ(t)) : j = 1, . . . , 5) for the param-eter θ, two weighting functions (w3,k(Ux(t)) : k = 1, . . . , 2)for the parameter Ux and again two weighting functions(w4,l(Uy(t)) : l = 1, . . . , 2) for the parameter Uy . Fig. 2shows variations of the continuous weighting functions for φover the its range -0.4 rad to +0.4 rad. The plot demonstratesthat the continuous weighting functions meet the convexityrequirements outlined in (15).

Now the original non affine quadrotor LPV model in (10)can be re-written in TP type convex polytopic form as:

x=5∑

i=1

5∑j=1

2∑k=1

2∑l=1

(w1,i.w2,j .w3,k.w4,l).(Ai,j,k,l +Bi,j,k,l),

y=x, (18)

where Ai,j,k,l and Bi,j,k,l are obtained from the LTI verticesSr. This new TP-convex polytopic model has been comparedwith the original LPV model of the quadrotor given in (10)over randomly selected 2000 points in the parameter spaceΩ where the maximum error and the mean error were foundto be 1.632e-005 and 7.71e-006 respectively. Hence, over thedefined parameter space Ω the actual quadrotor LPV modelis approximately equal to (18).

E. LPV Controller determination

Having derived the convex polytopic form of the LPVquadrotor model, the controller now can be determined bysolving the LMI given in (5) at each LTI vertex Sr. In practise,the problem is put into the standard regulator form [10] asshown in Fig. 3 and desired performance requirements of thecontroller are specified through weighting functions W1 andW2. In this controller design, performance boundaries havebeen imposed on the sensitivity function S and the controller

A

BCD

EAFDDC

ACDC

C

CC

E

!

"#

Fig. 3. Standard regulator form

outputs to achieve good tracking capabilities while avoidingactuator saturation.

This is known as the SKS problem formulation and theintention is to find a controller such that [10]:∥∥∥∥ W1S

W2KS

∥∥∥∥∞

< γ. (19)

Also a prefilter has been added to remove the parameterdependency of the matrix B, because it is a requirement ofthe controller synthesis [6]. The following weighting functionsand the prefilter were found to be yielded the intendedperformance:

W1 = 20.7s+1 ,

W2 = 10−5s+0.013×10−6s+0.3 ,

Wprefilter = 1000s+1000 .

(20)

The structure shown in Fig. 3 was built in MATLAB and usingthe “hinfgs” routine, a LPV controller with quadratic H∞performance γ = 76.7 has been obtained. Despite the highervalue of γ, the controller has been demonstrated the desiredperformance when it was test on the simplified nonlinearmodel of the quadrotor. The “hinfgs” routine has giventhe set of LTI controllers Kr for each vertex Sr of the TPtype convex polytopic form of the LPV quadrotor model.Hence, the global LPV controller K(p(t)) during the selfgain scheduling process is estimated by:

K(p(t)) =

5∑i=1

5∑j=1

2∑k=1

2∑l=1

w1,i.w2,j .w3,k.w4,l ×Kr, (21)

where Kr =

(AKi,j,k,l

BKi,j,k,l

CKi,j,k,lDKi,j,k,l

).

IV. SIMULATION RESULTS

A simplified nonlinear quadrotor model based on (1) wasimplemented in Simulink. This has been used to assess thecapabilities of the LPV controller and also to compare witha separate H∞ controller. The controllers have been tested


487

-5 0 5 10 15 20 25-5

0

5

10

15

20

25

x position (m)

y p

osi

tio

n (

m)

Demand

Response

Student Version of MATLAB

Fig. 4. Square flight position response: LPV controller

-5 0 5 10 15 20 25-5

0

5

10

15

20

25

x position (m)

y p

osi

tio

n (

m)

Demand

Response


Fig. 5. Square flight position response: H∞ controller

for a ramp test, a circular flight path and a square flightpath by analysing the quadrotor position response attituderesponse, controller signals and the demands on actuators ofthe quadrotor. For each flight test the full state vector valuehas been given as the reference signal.

The controller’s position responses of the square flight pathare presented in Fig. 4 and Fig. 5. During this manoeuvre thequadrotor model augmented with the controller (either LPVor H∞) was initialised in hover, then a square flight pathhaving a total length of 80 m was demanded. The specifiedflight speed was 2 ms−1. In studying the responses, the LPVcontroller has shown close tracking abilities for the position,than that of the H∞ controller. When the flight directionsare changed at the corners, the LPV controller has given lessperturbation than the H∞ one. Moreover, the LPV controllerhas taken less time to bring the quadrotor into the correctpath. However as shown in Fig. 6 the LPV controller impartsmuch aggressive control signals on the quadrotor. This maylead to actuator saturation of the real vehicle.

0 5 10 15 20 25 30 35 400

50

100

150

200

250

300

Time (sec)

Ro

tati

on

al s

pee

d (

rad

s-1)

Front rotor

Rear rotor


Fig. 6. Square flight actuator demands: LPV controller

V. CONCLUSION

In this work a self schedule LPV controller was designedfor the quadrotor vehicle. The controller synthesis requires anaffine LPV system model and the nonaffine quadrotor modelhas been transformed into the convex form using recentlyproposed TP method. The derived LPV controller was testedwith a simplified nonlinear quadrotor model in Simulink andthe performance of the LPV controller was compared with anH∞ controller.The simulation results exhibits very promisingclose tracking capabilities of the LPV controller against theH∞ controller.

The higher actuator demands of the LPV controller canbe minimised further fine tuning the performance weightingfunctions (i.e. W1 and W2) in the controller design. This isneeded to be investigated in a future work.

REFERENCES

[1] R. Xu and Umit Ozguner, “Sliding mode control of a quadrotorhelicopter,” In proceedings of the 45th IEEE conference on decision& control San Diego, CA, USA, pp. 4957–4962, December 2006.

[2] T. Madani and A. Benallegue, “Control of a quadrotor mini-helicoptervia full state backstepping technique,” In proceedings of the 45th IEEEConference on Decision & Control San Diego, CA, USA, pp. 1515–1520, December 2006.

[3] M. Labadille, Non–linear control of a quadrotor. M.Sc. thesis,Cranfield University, 2007.

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