Author s Accepted Manuscript

Transcription

1 Author s Accepted Manuscript Super twisting control algorithm for the attitude tracking of a four rotors UAV L.Derafa,A.Benallegue,L.Fridman PII: S16-32(11)282-1 DOI: doi:1.116/j.jfranklin Reference: FI 1431 To appear in: Journal of the Franklin Institute Received date: 25 January 211 Revised date: 19 August 211 Accepted date: 17 October 211 Cite this article as: L. Derafa, A. Benallegue and L. Fridman, Super twisting control algorithm for the attitude tracking of a four rotors UAV, Journal of the Franklin Institute, doi:1.116/j.jfranklin This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 Super Twisting Control Algorithm for the Attitude Tracking of a Four Rotors UAV L. Derafa a,1, A. Benallegue b,2,,l.fridman c,3 a Automatic Control Laboratory EMP, Bordj El Bahri, 16, Algeria. b Engineering Systems Laboratory of Versailles, 1-12 avenue de l Europe, 7814 Velizy, France. c Dept. Control Automatico CINVESTAV-IPN Mexico D.F. AP on leave on Departamento de Ingeniera de Control y Robtica, Divisin de Ingeniera Elctrica, Facultad de Ingeniera UNAM Abstract This paper deals with the design and implementation of a nonlinear control algorithm for the attitude tracking of a four-rotor helicopter known as quadrotor. This algorithm is based on the second order sliding mode technique known as Super-Twisting Algorithm (STA) which is able to ensure robustness with respect to bounded external disturbances. In order to show the effectiveness of the proposed controller, experimental tests were carried out on a real quadrotor. The obtained results show the good performance of the proposed controller in terms of stabilization, tracking and robustness with respect to external disturbances. 1. Introduction Unmanned Aerial Vehicles (UAV) are attracting the interest of many researchers all over the world [1], [2], [3], [4], [5]. This popularity may be attributed to their potential use in many applications such as, search and rescue missions, surveillance, law enforcement, inspection, mapping, and aerial cinematography. The quadrotor is a four-rotor VTOL (Vertical Take-Off and Landing) aircraft which has several advantages over the traditional helicopters in terms of manoeuvrability, motion control and Corresponding author addresses: derafa@lisv.uvsq.fr (L. Derafa), benalleg@lisv.uvsq.fr (A. Benallegue), lfridman@servidor.unam.mx (L. Fridman) 1 L. Derafa is with the Automatic Control Laboratory EMP, Algeria. 2 A. Benallegue is with Engineering Systems Laboratory of Versailles, France. 3 L. Fridman is with Dept. Control Automatico CINVESTAV-IPN Mexico Preprint submitted to Elsevier August 19, 211

3 cost [1]. In order to accomplish high level human-planned missions, robust flight control systems are required to track desired trajectories in the presence of wind or other disturbances. Most of the existing flight control systems have been designed by applying classical synthesis techniques (such as single-loop PID systems, root locus, Bode plots etc.) to an approximate linear model of the vehicle dynamics. But the trend of escalating performance, the increasing manoeuvrability, the unpredictable changes in the environment, the stronger dynamic coupling and nonlinearities necessitate more sophisticated control systems [3]. In practical applications, the position in space of the UAVs is generally controlled by an operator through a remote-control system, while the attitude can be automatically stabilized via an onboard controller. The attitude controller is an important feature since it allows the vehicle to maintain a desired orientation and, hence, prevents the vehicle from flipping over and crashing when the pilot performs the desired maneuvers [4]. A wide class of controllers have been proposed for the attitude control problem (see for instance [6], [7], [8], [4], [5], and the list is not exhaustive). Most of them presented only simulation results, and generally, the control strategies are based on simplified models without compensation of modeling errors and external disturbances. One attempt to address this problem is given in [9]. In this work, only simulation results are given and the controller is based on the estimation of the disturbance which is obtained by filtering the output of a first order sliding mode observer. The main objective of the proposed controller is to overcome the disturbance by using Super-Twisting Algorithm [1], [11], [12], without any a priori estimation. In fact, it has been shown in [11] and [13] that this algorithm, which is based on the second order sliding mode technique, ensures robustness with respect to modeling errors and external disturbances while reducing the chattering phenomenon caused by all first order sliding mode based controllers (for the general problem of chattering see [14] and for chattering problems in quadrotor see [5]). The stability and finite time convergence characteristics of the used algorithm have been recently proved by means of Lyapunov functions [15], [16], [17], [18], so why the stability analysis of the proposed controller 2

4 has been conducted in the same way. In order to show the effectiveness of the controller, experimental tests were carried out on a quadrotor. The obtained results show the good performance of the proposed controller in terms of stabilization, tracking and robustness with respect to wind perturbations. The paper is organized as follows: in Section 2, a dynamic model for a miniature four-rotor helicopter is developed. In Section 3, based on a simplified version of the obtained model, we design a dynamic feedback controller which ensures robustness with respect to modeling errors and external disturbances. In section 4, some experimental tests are carried out, to show the effectiveness of the proposed controller in the presence of disturbances and parametric uncertainties. Both attitude stabilization and trajectory tracking have been presented. Finally, section 5 concludes this work. 2. Quadrotor attitude model The attitude dynamical model of the considered mini quadrotor, shown in figure 1, is described in details by [4], [19], [2], [21] and [22]. F F d F E b F Body-fixed O e b frame 2 e b 1 e b 3 v (Pitch) q Earth-fixed frame θ E a e a 2 O e a 3 e a 1 φ x g (Yaw) w r (Roll) p u y z ψ Figure 1: Quadrotor helicopter [22] The attitude is represented by Euler angles Θ =[φ,θ,ψ] T, corresponding to an aeronautical convention. The attitude angles are respectively called Yaw angle (ψ rotation around z-axis), Pitch angle (θ rotation around y-axis) and Roll angle (φ rota- 3

5 tion around x-axis). The angular velocities and accelerations are given respectively by Θ =[ φ, θ, ψ] and Θ =[ φ, θ, ψ]. The equation of the attitude dynamics of the quadrotor will be: Θ =(JM(Θ)) 1 [T prop JN(Θ, Θ) A T (U) (M(Θ) Θ) (JM(Θ) Θ)] (1) where M(Θ) and N(Θ, Θ) are given by 4 1 S θ M(Θ)= C φ S φ C θ S φ C φ C θ C θ θ ψ N(Θ, Θ)= S φ φ θ +C φ C θ φ ψ S φ S θ θ ψ C φ φ θ S φ C θ φ ψ C φ S θ θ ψ The matrix J is the inertia matrix of the quadrotor. T prop is the torque vector of the propeller system. The aerodynamic functions A T (U) are highly nonlinear and dependent on numerous physical variables such as the angle between airspeed and the body-fixed frame and geometric form of the helicopter. Generally, they are approximated using the non-dimensional coefficients C i as A Ti (U) = 1 2 ρc iu 2 where ρ is the air density [23],[24]. 3. Attitude controller design In this section, we design a controller for the attitude of the aircraft using Super Twisting Algorithm. The objective is to ensure the convergence of the attitude positions defined by Euler angles {φ(t),θ(t),ψ(t)} to the bounded desired trajectories {φ d (t),θ d (t),ψ d (t)}. In this case, let s use equation (1) and define the following attitude state variables: x 1 = Θ; x 2 = Θ 4 the abbreviations S (.) and C (.) denote respectively sin(.) and cos(.). 4

6 Then the state-space form of this model is given by: ẋ 1 = x 2 ẋ 2 = f (x)+g(x)u + w(t) [ ] T where w(t)= w 1 w 2 w 3 is considered as external disturbance vector, the control input is u = T prop =[U φ,u θ,u ψ ] T, the vector f (x) is given by: f (x)= (JM(Θ)) 1 [JN(Θ, Θ) (M(Θ) Θ) (JM(Θ) Θ)] (3) and the matrix g(x) is given by: g(x)=(jm(θ)) 1 One can synthesize the control law forcing the state x 1 to follow the desired trajectory x d 1 (t)=[φ d(t), θ d (t), ψ d (t)] T by using inverse dynamic control technique. To this end, the following assumptions are needed: Assumption 1. The signals Θ, and Θ can be measured or estimated by on-board sensors. Assumption 2. The desired trajectories and their first and second time derivatives are bounded. Assumption 3. The velocity U and the acceleration U of the helicopter with respect to the air are bounded. Assumption 4. The roll, pitch and yaw angles are limited to ( π 2 < φ < π 2 ), ( π 2 < θ < 2 π ) and ( π < ψ < π). It means that the acrobatic behavior is not allowed. According to these assumptions and to the expression of the function given by equation (1), it should be noted that following inequalities are satisfied (i = 1,2,3): (2) ẇ i (t) δ i (4) According to assumption 4, the matrix g is non singular and its inverseis given by: g 1 = JM(Θ) (5) 5

7 The controller is designed in order to obtain the error dynamics in the form of Super-Twisting given in [15], [18]. It s goal is to enforce the sliding mode on the manifold: s = ė + λ e (6) where e = x d 1 x 1,ė = x d 2 x 2 = ẋ d 1 ẋ 1 and λ R 3 3 is a diagonal positive definite matrix. The proposed controller is given by the following : u = g 1 (ẍ d 1 λ ė K 1 s 1 t 2 sgn(s) K 2 sgn(s(τ))dτ f (x)) (7) where the function sgn(.) denotes the usual sign function, the gain matrices K 1 = diag(k 11,k 12,k 13 ) and K 2 = diag(k 21,k 22,k 23 ), with k 1i and k 2i are positive gains chosen as follows: k 2i > δ i k1i 2 > 4k 2i (i = 1,2,3) (8) The closed loop error dynamics is given by: Let us take ṡ = K 1 s 1 t 2 sgn(s) K 2 sgn(s(τ))dτ + w(t) (9) z 1i = s i t z 2i = k 2i sgn(s i (τ))dτ + w i (t) (1) ẇ i (t)=ρ i (t) then equation (9) can be written in scalar form (i = 1,2,3) as: ż 1i = k 1i z sgn(z 1i )+z 2i ż 2i = k 2i sgn(z 1i )+ρ i (t) (11) The proof of finite time convergence to zero of the variables z 1i and z 2i is explicitly given in appendix and this results are taken from [16]. It can be concluded that if the conditions on the gains given by (8) are satisfied, we obtain s i = in finite time. Therefore, it can be concluded, according to equation (6), that lime = andlimė =. t t 6

8 Remark 1. According to equation (1) and the finite time convergence to zero of the signals z 2i (i = 1,2,3), it can be concluded that the perturbations w i (t) are estimated in finite time as follows: t T w i (t)=k 2i sgn(s i (τ))dτ and is, therefore, compensated by the controller. Remark 2. In real applications, the control inputs in (2) are the four rotor forces (F 1, F 2, F 3 and F 4 ) of the aircraft (see figure 1). It is, thus, necessary that the total thrust Fprop= i = 1 4 F i is available in order to use it for the calculation of these forces. One manner to design it, is to control the altitude (z-position) of the quadrotor. However, for the sake of simplicity of implementation in this work, Fprop hasbeenfixedto compensate the gravity force. 4. Experimental results In order to validate the proposed controller, we implemented the control law given by equation (7) on a PC Pentium II at 2MHz, equipped with a dspace DS113 PPC real-time controller board using Matlab and Simulink software as shown in Figure 2. The sampling frequency has been fixed to 1 KHz. The mechanical structure of the quadrotor is that of the four-rotor mini-helicopter manufactured by Draganfly Innovations, Inc. ( The physical parameters of the used quadrotor are givenintable1[25]. Parameter Description Value Unit m Mass.42 kg d Distance 2.5 cm c Proportionality factor 1.8 cm I x Roll Inertia (φ) kg.m 2 I y Pitch Inertia (θ) kg.m 2 I z Yaw Inertia (ψ) kg.m 2 Table 1: Quadrotor physical parameters 7

9 Figure 2: Experimental setup The objective of this experiment is to safely test the proposed attitude controller, so we decided to use a stationary ball joint base, as shown in Figure 2. This base gives the aircraft unrestricted yaw movement and around ±3 of pitch and roll, while restricting the aircraft to a fixed point in the three-dimensional space [4]. To measure the aircraft angles and the angular velocity we used an Inertial Measurement Unit (IMU) XSENS MTI-28A53G35 ( The IMU is connected to the dspace DS113 serial communication port EIA-RS-232. The four DC permanent-magnet mini motors are current amplified with intelligent microcomputer speed controllers of type IMCS 25 anddrivenbypwmsignals. To explore the effectiveness and the robustness of the proposed STA controller, four experiments have been performed on the quadrotor. Experiment 1 involves the aircraft attitude stabilization from some given initial angles. Initial conditions are set for the aircraft angles as φ = 23,θ = 13 and the yaw ψ = 2. In experiment 2, attitude tracking test was carried out. The reference signals are of sinusoidal form with magnitude equal to.1 rad for the roll,.2 rad for the pitch and.3 radforthe yaw. In experiment 3, the attitude is externally disturbed to explore the disturbance rejection performance in stabilization around zero and in tracking case. In experiment 4, the quadrotor is in free flight situation where the attitude is stabilized around zero 8

10 with the altitude controlled at around one meter over the floor by an operator through a remote-control system. For this application the chosen gains are given as : λ i = 3 (i =,1,2), K 1 = 4.5 δ, K 2 = 1.1δ with δ = diag[12,12,8] Stabilization Experiment Figures (3) and (4) show the attitude response, the corresponding controllers outputs and the applied forces. One can notice the effectiveness of the proposed controller. The pronounced frequency oscillations in the control signals are due to sensor noise. 3 2 Roll Pitch Yaw Quadrotor Angles (Deg) Figure 3: Euler angles φ, θ, ψ (stabilization test) 9

11 U φ U θ U ψ Figure 4: Controller outputs (stabilization test) 1

12 4.2. Tracking Experiment Attitude tracking experiment has been carried out. Figures (5) and (6) show the attitude response, the corresponding controllers outputs and the applied forces signals. The obtained results clearly show the effectiveness of the controller. 3 2 Pitch Quadrotor Angles (Deg) 1 1 Roll 2 3 Yaw Figure 5: Euler angles φ, θ, ψ (tracking test) 4.3. Disturbance Rejection Experiment Figures (7) and (8) illustrate the Euler angles response and the corresponding controller outputs. As shown in Figure 7 the disturbances are externally applied at time instants 1 sec, 16 sec for the pitch and 22 sec for the roll and the yaw angle. Figures (9) and (1) show respectively the Euler angles response where the attitude is externally disturbed at time 24sec and 35 sec for the roll and 33 sec for the yaw and the corresponding controller outputs. 11

13 U φ U θ U ψ Figure 6: Controller outputs (tracking test) 12

14 3 2 Pitch Roll Yaw Quadrotor Angles (Deg) Figure 7: Euler angles φ, θ, ψ (disturbance test) 13

15 U φ U θ U ψ Figure 8: Controller outputs (disturbance test) 14

16 3 Pitch 2 Quadrotor Angles (Deg) 1 1 Roll 2 3 Yaw Figure 9: Euler angles φ, θ, ψ (tracking disturbance test) 15

17 U φ U θ U ψ Figure 1: Controller outputs (tracking disturbance test) 16

18 4.4. Free flight stabilization test In order to show the effectiveness of the controller in real situation a free flight stabilization test has been carried out (see figure 11). A disturbance is applied by hand to the yaw angle ψ at time t = 8sec. Figures (12) and (13) show respectively the Euler angles response and the corresponding controller outputs. It can be noticed that the controller is reacting against the disturbance in opposite direction in order to compensate it. Figure 11: Free flight quadrotor 5. Conclusion In this paper, a super twisting controller algorithm has been proposed and successfully implemented on a small scale quadrotor aircraft. The controller has been designed using second order sliding mode approach in order to avoid chattering phenomenon and to ensure robustness with respect to model uncertainties and external disturbances. The experimental results obtained on a quadrotor system clearly show the effectiveness of the proposed controller in the stabilization, tracking and disturbance rejection cases. 17

19 Attitude Stabilization test (Super Twisting Algorithm) φ θ ψ Quadrotor Angles (Deg) Figure 12: Euler angles φ, θ, ψ (Free flight test) 18

20 .1 U φ U θ U ψ Figure 13: Controller outputs (free flight test) 19

21 Appendix.1. Lyapunov stability analysis of the STA [16] Consider the standard Super-Twisting Algorithm (STA) with a perturbation term ż 1 = k 1 z 1 1/2 sgn(z 1 )+z 2 (.1) ż 2 = k 2 sign(z 1 )+ρ(t,z) where z 1 R and z 2 R and the perturbation term ρ is uniformly bounded ( ρ < δ) Let us prove the stability of the equibrium point (z 1,z 2 )=(,). The proof is stated as in [15], [17], [16], [18]. Consider the Lyapunov function presented in [15]: V = ζ T P ζ (.2) [ ] where ζ = z 1 1 T 2 sgn(z 1 ),z 2 and P is a positive definite matrix. Notice that V(ζ,t) is continuous and differentiable except when z 1 =. In fact, when z 1, V exists and is negative definite. However, before arriving at the equilibrium point (z 1,z 2 )=(,), the solution of system (.1) crosses the line z 1 = whenz 2. This means that the derivative of the Lyapunov function exists almost everywhere while V(t) decreases until the system reaches the equilibrium. As presented in [15], V(t) is a strong Lyapunov function for (.1) in the form of (.2). Moreover, this Lyapunov function is positive definite but radially unbounded: λ min (P) ζ 2 V λ max (P) ζ 2 (.3) where ζ 2 2 = z 1 + z 2 2 represents the Euclidian norm of ζ. The construction of suitable positive definite matrices P = P T, provided in [15], is based on the following algebraic LMI equation: AT P + PA + εp+ δ 2 C T C PB < (.4) B T P 1 where : A = 1 2 k ; B = [ ] ; C = 1 k 2 1 with k 1 and k 2 are positive gains. [ ] Using the vector ζ = z 1 1 T 2 sgn(z 1 ),z 2, the system (.1) can be rewritten as: 2

22 ζ 1 = 1 ζ 1 (Aζ + B ρ(t)) (.5) where the transformed perturbation ρ(t,ζ)= ζ 1 ρ(t,z) satisfies ρ(t,ζ) δ ζ 1.As a consequence, ω( ρ,ζ)= ρ 2 (t,ζ)+δ 2 ζ1 2. As k 1 and k 2 are positive gains, the system (A,B,C) is observable and controllable, so we can use the bounded-real lemma [26] to determine the condition on the gains k 1 and k 2. It is shown that the Linear Matrix Inequality (.4) is feasible if and only if the linear system defined by H(s)=δ C(sI A) 1 B is nonexpansive, i.e., max H( jω) < 1 ω This implies the following condition : max G( jω) < 1 ω δ where G(s)=C(sI A) B = s k 1s k 2 The previous equality yields to two conditions on gains. By choosing one of them : max ω G( jω) = 1 k 2 if k 2 1 > 4k 2 we can then deduce conditions on gains k 1 and k 2 as follows: k 2 > δ (.6) k 2 1 > 4k 2 Consider the Lyapunov function defined by (.2). Its derivative writes: V(ζ) = 1 ζ ρ T AT P + PA PB ζ ρ ζ 1 B T P 1 ζ ρ T AT P + PA PB ζ ρ + ω( ρ,ζ) ζ 1 B T P 1 ζ ρ T AT P + PA + δ 2 C T C PB ζ ρ ζ 1 B T P 1 1 ζ ρ T AT P + PA + εp εp + δ 2 C T C PB ζ ρ ζ 1 B T P 1 ε ζ 1 ζ T Pζ 21

23 Finally, V ε ζ 1 ζ T Pζ = ε ζ 1 V(ζ) (.7) From (.3), we deduce the following inequality ζ 1 ζ 2 V 1/2 (ζ) λ 1/2 min {P} We can then conclude that V satisfies: V α V 1/2 (ζ) where: α = ελ 1/2 min {P} (.8) The previous result guarantees the finite time convergence of vector z =[z 1,z 2 ] T to zero. This time is bounded by T = 2V 1/2 (ζ()) α (.9) where ζ() is the initial value of ζ and α is given by equation (.8). References [1] V. Mistler, A. Benallegue, N. K. M Sirdi, Linéarisation exacte et découplage entrées-sorties comparaison entre l hélicoptère standard et l hélicoptère 4 rotors, in: Conférence Internatinale Francophone d Automatique (CIFA 22), Nantes, France, 8-1 juillet 22. [2] M. Chen, M. Huzmezan, A simulation model and h loop shaping control of a quad rotor unmanned air vehicle, in: M. H. Hamza (Ed.), Modelling, Simulation, and Optimization, IASTED/ACTA Press, 23, pp [3] S. Bouabdallah, P. Murrieri, R. Siegwart, Design and control of an indoor micro quadrotor, in: Proc. IEEE International Conference on Robotics and Automation, New Orlean, USA, 24. [4] A. Tayebi, S. McGilvray, Attitude stabilization of a vtol quadrotor aircraft, IEEE Transactions on Control Systems Technology

24 [5] T. Madani, A. Benallegue, Backstepping sliding mode control applied to a miniature quadrotor flying robot, in: Proceedings of the 32nd Annual Conference of the IEEE Industrial Electronics Society (IECON 6), Paris, France, November 7-1, 26. [6] O.-E. Fjellstad, T. I. Fossen, Comments on the attitude control problem, IEEE Transactions on Automatic Control 39. [7] S. M. Joshi, A. G.Kelkar, J. T.-Y.Wen, Robust attitude stabilization of spacecraft using nonlinear quaternion feedback, IEEE Transactions on Automatic Control 4. [8] F. Lizarraide, J. T. Wen, Attitude control without angular velocity measurement: A passivity approach, IEEE Transactions on Automatic Control 41. [9] L. Besnard, Y. B. Shtessel, B. Landrum, Control of a quadrotor vehicle using sliding mode disturbance observer, in: Proceedings of the 27 American Control Conference, New York City, USA, Jul , 27. [1] A. Levant, Sliding order and sliding accuracy in sliding mode control, International Journal Control 58 (6) (1993) [11] A. Levant, Robust exact differentiation via sliding mode technique, Automatica 34 (3) (1998) [12] A. Levant, High-order sliding modes: differentiation and output-feedback control, International Journal Control 76(9 1) (23) [13] J. Davila, L.Fridman, A.Levant, Second order sliding mode observer for mechanical systems, IEEE Transaction on Automatic Control 5,(11). [14] V. I. Utkin, Sliding mode and their application in Variable structure systems, Mir, Moscou, [15] J. A. Moreno, M. Osorio, A lyapunov approach to second-order sliding mode controllers and observers, in: Proceedings of 47th IEEE Conference on Decison and Control (CDC 28), Cancun, Mexico, Dec. 9-11,

25 [16] J. A. Moreno, A linear framework for the robust stability of a generalyzed super-twisting algorithm, in: Proceedings of 6th IEEE Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 29), Toluca, México, Nov. 1-13, 29. [17] A. Dávila, J. A. Moreno, L. Fridman, Optimal lyapunov function selection for reaching time estimation of super twisting algorithm, in: Proceedings of 48th IEEE Conference on Decison and Control (CDC 29), Shangai, P.R. China, December 16-18, 29. [18] A. Dávila, J. A. Moreno, L. Fridman, Global non homogeneus super-twisting controller for the quasi-linear systems with unbounded uncertainties: a lyapunov design, in: Proceedings of the American Control Conference (ACC 21), Baltimore, Maryland, USA, June 3 - July 2, 21. [19] T. Hamel, R. Mahony, R. Lozano, J. Ostrowski, Dynamic modeling and configuration stabilization for an x4-flyer, in: Proceedings of IFAC World Congress, Barcelona, Spain, Jul. 22. [2] G. Cai, B. M. Chen, K. Peng, M. Dong, T. H. Lee, Modeling and control of the yaw channel of a uav helicopter, IEEE transactions on industrial electronics 55 (9). [21] P. McKerrow, Modelling the draganflyer four-rotor helicopter, in: IEEE Int. Conf. on Robotics and Automation (ICRA 4), New Orleans, LA, 24. [22] T. Madani, A. Benallegue, Sliding mode observer and backstepping control for a quadrotor unmanned aerial vehicles, in: Proceedings of American Control Conference, New York City, USA, July 11-13, 27. [23] A. Gessow, G. Myers, Aerodynamics of the helicopter, Frederick Ungar Publishing Co, New York, [24] W. Johnson, Helicopter Theory, Dover, New York,

26 [25] L. Derafa, T. Madani, A. Benallegue, Dynamic modelling and experimental identification of four rotors helicopter parameters, in: Proceedings of the IEEE International Conference on Industrial Technology (ICIT 6), Mumbai, India, Dec , 26. [26] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Vol. 15 of Studies in Applied Mathematics, SIAM, Philadelphia, PA,