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From Pid To State Feedback Attitude Stabilization Of A Quadrotor UAV

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The work proposed in this paper aim to design tow controllers for attitude stabilization of a Quadrotor UAV. The first is a classical PID controller; the second is a nonlinear state feedback controller which designed using multiple model approach. We
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  International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3,July2012 DOI:10.5121/ijitca.2012.23011 F ROM PID  TO S  TATE F EEDBACK   A   TTITUDE S  TABILIZATIONOF A  Q UADROTOR  UAV  Fouad Yacef  1 ,Hana Boudjedir 1 ,Hicham Khebbache 2 , andOmar Bouhali 1 1 Automatic Laboratory of Jijel(LAJ),AutomaticControlDepartment,Jijel University,ALGERIA yaceffouad@yahoo.fr, bouhali_omar@yahoo.fr, hana_boudjedir@yahoo.fr 2 Automatic Laboratory of Setif(LAS),Electrical EngineeringDepartment,Setif University,ALGERIA khebbachehicham@yahoo.fr  A  BSTRACT  The work proposed in this paper aim to designtow controllers for attitude stabilization of a Quadrotor UAV.The first is a classical PIDcontroller;the secondis a nonlinearstate feedback controllerwhichdesigned using multiple model approach. We start with the design ofa Takagi-Sugeno (T-S) model for Quadrotor modelling, and then we useLinear Matrix Inequality (LMI), and PDC (Parallel DisturbanceCompensation) techniqueto designanonlinear state feedback controller with pole placement in a pre-specified region of the operating space.The requirements of stability and pole-placement in LMI region are formulated based on the Lyapunov direct method. By recasting these constraints into LMIs, we formulatean LMI feasibility problemtocalculatethe controllergains.The bothcontrollersareappliedto a nonlinear Quadrotor system.Simulation results show that the proposed LMI-based design controller(State feedback) yieldsbetter transientperformancethan those of PID controller.  K   EYWORDS  Linear Matrix Inequality(LMI),Parallel Disturbance Compensation(PDC),PIDController,PolePlacement, Quadrotor, Takagi-Sugeno(T-S)model. 1.I NTRODUCTION Unmanned flying robots or vehicles (UAVs) are gainingincreasing interest because of a widearea of possibleapplications. While the UAV markethas first been drivenby militaryapplications and large expensive UAVs, recentresults in miniaturization, Mechatronics andmicroelectronicsalso offer an enormous potential for small and inexpensiveMicro-UAVs forcommercial use.One typeof aerial vehicle with a strong potential also for indoor flightis therotorcraft and the special class of four-rotor aerialvehicles, also called Quadrotor.This helicopteris considered as one of the most popular UAV platform.This kind of helicopters isdynamicallyunstable, and thereforesuitable control methods wereused to make them stable, as back-steppingand sliding-mode techniques[1] [2].Acomplex problemcanbe solved by many strategy;divide & conquerstrategy is one of them.The problem is divided intosimpler parts, which are solved independently and together yields thesolution to the whole problem. The same strategy can be used for modelling and control of non-linear systems, where the non-linear plant is substituted by locally valid set of linear submodels  International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3,July2012 2 [3]. The model should be simple enough so that it can be easy understood. The accurate modelthat characterizes important aspects of the system being controlled is a necessary prerequisite fordesign of a controller.The idea of approximation basedonMultipleModelApproach(MMA) isnot new. Since the publication Johansen and Foss, the multiple-models approach knew anunquestionable interest.The MultipleModelapproachappears in the literature under manydifferent names, includingTakagi-Sugeno(T-S)model[4], local model networks or operatingregime decomposition.Manyworks have been carried out to investigate the stability analysis and the design of statefeedback controller of Takagi-Sugenosystems. Using a quadratic Lyapunov function and PDC(Parallel Disturbance Compensation)technique, sufficient conditions for the stability andstabilisability have been established[5] [6]. The stability depends on the existence of a commonpositive definite matrix guarantying the stability of all local subsystems. The PDC control is anonlinear state feedback controller. The gain of this controller can be expressed as the solution of a linearmatrix inequality (LMIs) set[7].In thiswork, we presentaT-S modelforQuadrotor modelling. A nonlinear state feedback controller is proposed,based onlinear control theory, and PDC techniquewe formulate an LMIfeasibility problemwhich considered as an optimization problem.A comparison study betweenPID andstate feedback controller is made to proved performances of the proposed controller. 2.Q UADROTOR DYNAMICAL MODEL TheQuadrotor is a Mechatronics system with four propellersin a cross configuration. While thefront and the rear motorrotate clockwise,the left and the right motor rotate counter clockwisewhich nearly cancels gyroscopic effects and aerodynamictorques in trimmed flight. Oneadditional advantageof the Quadrotor compared to a conventional helicopter isthesimplifiedrotor mechanics. Byvarying the speed of thesingle motors, the lift force can be changed andverticaland/or lateral motion can be created. Pitch movement isgenerated by a differencebetween the speed of the front andthe rear motor while roll movement results from differencesbetween the speed of the left and right rotor, respectively. Figure 1. Quadrotor Architecture The dynamics of the Quadrotor is described in the space bysix degrees of freedom according tothe fixed inertial framerelated to the ground. This dynamics is related to thetranslational  X  l 2 F  3 F  4 F   Z    m  R b  R 1 Ω 3 Ω 2 Ω      1 F  Y   International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3,July2012 3 positions  ( , , ) and the attitude described bythe Euler angles (  , , ).These six coordinatesare theabsolute position of the centre of masse. The Euler angles aredefined as follows: • Roll angle:  − 2 ≤ ≤ 2⁄⁄ ; • Pitch angle:  − 2 ≤ ≤ 2⁄⁄ ; • Yaw angle:  − ≤ ≤ . The rotation transformation matrixfrom the inertial fixedframeto the body fixed frameis given by: ccsscsccscss Rscssscccssscssccc                                      − += + −− (1)With  (.) and  (.) represent  sin(.) and  cos(.) respectively. To derive the dynamic model of the Quadrotor, the Newton Euler formalism will be usedon both translation and rotation motions.Inthis work we mainly focus our interest to theattitude dynamics and we consider the reduced dynamical model as follows [8]: ( ) ( ) ( ) 212223 111  yz faxr r  x xxx fay zxr r  yyy y xy faz zz z  II K  I u I  III K  II  I u III  I  II K u II  I              −= − Ω − +−= + Ω − +−= − +           (2) Theinputsof the systemare  , , and Ω as a disturbance, obtaining: ( )( )( ) 2 21 4 22 22 3 12 2 2 23 1 2 3 41 2 3 4 r  u blu blu d               = −= −= − + −Ω = − + −  (3) 3.Q UADROTOR T AKAGI -S UGENO MODEL 3.1. Takagi-Sugeno model A T-S model is based on the interpolation between several LTI (linear time invariant) localmodels as follow: ( ) ( ) ( )  ( ) ( ) ( ) 1 r miiiii  xttAxtBut     = = +∑  (4)  International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3,July2012 4 Whereis the number of sub-models,  ( ) ∈ is the state vector,  ( ) ∈ is the inputvector  ∈  × ,  ∈  × ,and  ( ) ∈ is the decision variable vector.The variable  ( ) may represent measurable states and/or inputs and the form of this variablemayleads to different class of systems: if   ( ) is known functions than the model (4) represents anonlinear system and if there are unknown we consider that this leads to linear differentialinclusion (LDI). This variable can also be a function of the measurable outputs of the system.The normalized activation function  ( ) in relation with the ith sub-model is such that: ( ) ( ) ( ) ( ) 1 10 1 r iii t t       = =∑≤ ≤  (5)According to the zone where evolves the system, this function indicates the more or lessimportant contribution of the local model corresponding in the global model (T-S model).The global output of T-S model is interpolated as follows: ( ) ( ) ( )  ( ) ( ) ( ) 1 r miiiii  yttCxtDut     = = +∑ (6)Where  ( ) ∈ is the output vector and  ∈  × ,  ∈  × .More detail about thistype of representation can be found in [4]. 3.2. Quadrotor Takagi-Sugeno model The behaviour of a nonlinear system near anoperating point(  , ), can be described by a lineartime-invariant system (LTI).Using Taylor series about (  , )and keeping only the linear termsyields: ( ) ( ) ( )  ( ) ( )  ( ) i i i i i i  x t A x t x B u t u f x ,u = − + − +  (7)Which can written as ( ) ( ) ( ) i i i  x t A x t B u t d  = + +  (8)With: ( ) , ii i  f x u A x x xu u ∂=∂ == ,  ( ) , ii i  f x u Bu x xu u ∂=∂ == ,  ( ) ( ) ,  f x u x t  =   ,  ( ) , i i i i i i i d f x u A x B u = − − Aftercalculationwe obtained:  International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3,July2012 5 123212354644567787 010000000000000100000,00000000100000000 ii aaaabbb ABaaaabbbaaab              +      = =   +                  (9)Combinedlocalaffinemodels(8)usingGaussian activation function we describe the dynamicmodel of theQuadrotorbya T-S model: ( ) ( ) ( )  ( ) ( ) ( ) ( ) ( ) 31 m i i m i iim m  x t t A x t B u t d  y t Cx t     = = + +∑=  (10)With: ( ) ( )  ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( )  23,3 21,1 , exp2  j i jii i ji ji j t ct t t t              == −= = −∏∑          -Thevector of decision variables  ( )  T  t       =        -The parameters of activations functions(  ,  ,  , ) are given as: • The centres  , are defined according to the operation point. • The Dispersions  , are defined by optimization of acriterion,which representthequadratic error betweenTakagi-Sugeno model outputsand nonlinear systemoutputs, usingParticleSwarmOptimisationalgorithm(PSO)[9].-The operating points are chosen to cover maximum space of the operating space, with smallnumber of local models. The attitude ofQuadrotor(roll, pitch,and yaw) has a limited bound( − 2 ≤ ≤ 2⁄⁄ , − 2 ≤ ≤ 2⁄⁄ ,  − ≤ ≤ ), for this reason we usethree local modelsto cover this space.Linear local model are defined in this table as follow: Table 1. Operation Points Parameters . N° O.PParameters 1  0.523 rad s     = = = −    2  0  rad s     = = =    3  0.523 rad s     = = =    Tovalidate the synthesized T-Smodel a SBPA (input signal) is used, more detail in [10].
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