A Sliding Mode LCO Regulation Strategy for Dual- Parallel Underactuated UAV Systems Using Synthetic Jet Actuators

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1 Publications A Sliding Mode LCO Regulation Strategy for Dual- Parallel Underactuated UAV Systems Using Syntetic Jet Actuators N. Ramos-Pedroza W. MacKunis Embry-Riddle Aeronautical University, mackuniw@erau.edu M. Reyanoglu Embry-Riddle Aeronautical University, reyanom@erau.edu Follow tis and additional works at: ttp://commons.erau.edu/publication Part of te Navigation, Guidance, Control and Dynamics Commons Scolarly Commons Citation Ramos-Pedroza, N., MacKunis, W., & Reyanoglu, M. (215). A Sliding Mode LCO Regulation Strategy for Dual-Parallel Underactuated UAV Systems Using Syntetic Jet Actuators. Aerospace Engineering, 215(Article ID ). ttps://doi.org/1.1155/215/ Tis Article is brougt to you for free and open access by Scolarly Commons. It as been accepted for inclusion in Publications by an autorized administrator of Scolarly Commons. For more information, please contact commons@erau.edu.

2 Aerospace Engineering Volume 215, Article ID , 7 pages ttp://dx.doi.org/1.1155/215/ Researc Article A Sliding Mode LCO Regulation Strategy for Dual-Parallel Underactuated UAV Systems Using Syntetic Jet Actuators N. Ramos-Pedroza, W. MacKunis, and M. Reyanoglu Department of Pysical Sciences, Embry-Riddle Aeronautical University, Daytona Beac, FL 32114, USA Correspondence sould be addressed to M. Reyanoglu; reyanom@erau.edu Received 11 August 215; Accepted 8 October 215 Academic Editor: Ranjan Ganguli Copyrigt 215 N. Ramos-Pedroza et al. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. A sliding mode control- (SMC-) based limit cycle oscillation (LCO) regulation metod is presented, wic acieves asymptotic LCO suppression for UAVs using syntetic jet actuators (SJAs). Wit a focus on applications involving small UAVs wit limited onboard computational resources, te controller is designed wit a simplistic structure, requiring no adaptive laws, function approximators, or complex calculations in te control loop. Te control law is rigorously proven to acieve asymptotic regulation of bot pitcing and plunging displacements for a class of systems in a dual-parallel underactuated form, were a single scalar control signal simultaneously affects two states. Since dual-parallel underactuated systems cannot be expressed in a strict feedback or cascade form, standard backstepping-based control tecniques cannot be applied. Tis difficulty is mitigated troug careful algebraic manipulation in te regulation error system development, along wit innovative design of te sliding surface. A detailed model of te UAV LCO dynamics is utilized, and a rigorous analysis is provided to prove asymptotic regulation of te pitcing and plunging displacements. Numerical simulation results are provided to demonstrate te performance of te control law. 1. Introduction Limit cycle oscillation (LCO) (flutter) is a self-excited aeroelastic penomenon tat can create difficulties in aircraft tracking control and could potentially result in structural damage and even catastropic failures. Even in te low Reynolds number (low-re) regimes, caracteristic of small unmanned aerial veicle (UAV) systems, LCO can be adversarialinfligtcontrolsystemsandcanleadtounsafeuav operating conditions [1]. Motivated by tese callenges, automatic control metods for LCO regulation systems ave been widely investigated in recent controls literature [2 5]. LCO suppression control systems are usually designed using moving deflection surfaces (e.g., elevators, rudders, and ailerons) to deliver te required control force or moment. However, for applications involving smaller, ligter weigt UAVs,teuseofeavymecanicalactuatorsmigtnot be practical. As tese practical considerations motivate te need for smaller, low power-consumptive control actuators, synteticjetactuators(sjas)aveemergedasapopular alternative to mecanical deflection surfaces for UAV control applications [1, 6 14]. SJAs are practical tools for UAV LCO suppression control systems due to teir low cost, small size, and low power consumption properties. SJAs utilize a vibrating diapragm to create trains of vortices (or jets) of air troug te periodic ejection and suction of air troug a small orifice (see Figure 1). Te resulting trains of air vortices impart linear momentum to a flow system, and tis momentum transfer enables SJAs to deliver an equivalent control force or moment wen implemented in UAV wings. Since SJAs only use te surrounding air of te flow system to generate te vortices, tey can deliver a control force or moment wit zero net mass injectionacrossteflowboundary.tisisakeybenefitin small UAV applications: SJAs do not require space for a fuel supply. Te boundary layer flow field near te surface of a UAV wing can be altered by using te trains of air vortices generated by te SJAs. By utilizing tis SJA-commanded modification of te boundary layer flow field, SJAs can be employed to acieve automatic regulation control of LCO

3 2 Aerospace Engineering Vibrating diapragm d D Vortex rings Orifice Figure1:Scematiclayoutofasynteticjetactuator. in UAV wings. SJA-based control design is complicated, owever, due to te inerent nonlinearity in te equations governing te SJA dynamics. Specifically, te virtual surface deflection delivered by te SJAs is a nonlinear (nonaffine) function of te voltage signal commanded to te SJAs. An additional callenge in control design using SJAs is tat te nonlinear actuator dynamic model includes parametric uncertainty. Tis SJA nonlinear SJA dynamic model was determined empirically troug repeated experiments, and it is well-accepted in SJA-based control literature (e.g., see [1, 12, 14]). In addition to te control design callenges involved in compensating for actuator uncertainties, significant callenges arise in situations were multiple actuators lose control effectiveness. Wen te number of control actuators is less tan te number of degrees of freedom to be controlled, te system becomes underactuated, and control design for underactuated systems presents a nontrivial callenge. Te integrator backstepping tecnique is often utilized to address te callenge of control design for underactuated systems [15]. However, te backstepping-based control design approac can only be applied to systems in strict feedback or cascade form. Backstepping tecniques cannot be utilized for systems in a dual-parallel underactuated form, were a single scalar control input simultaneously affects two states. Several open problems remain in control design for dualparallel underactuated systems. To compensate for te nonlinearity and parametric uncertainty inerent in te SJA dynamic model, recent approaces typically employ adaptive parameter estimation, neural networks (NN), fuzzy logic rule sets, or complex fluid dynamics computations in te feedback loop (e.g., see [1, 14, 16]). Tese types of SJA-based control design approaces ave been sown to acieve good closed-loop performance; owever, control designs involving complex calculations or function approximation metods can incur an increased computational requirement, wic migt not be practical for applications involving small UAVs. An adaptive inverse control metod is presented in [1, 14], wic acieves asymptotic trajectory tracking for an aircraft dynamic model equipped wit SJAs. To acieve te results in [1, 14], a series of adaptive parameter estimation laws are utilized along wit rigorous Lyapunov-based stability analyses. Inspired by tese results and motivated by te desire to investigate a more computationally minimal control strategy, our previous result in [17] was te first SJA-based tracking control approac to acieve asymptotic trajectory tracking for an aircraft using a simple (single-loop) feedback control law witout te use of parameter adaptation or function approximators in te control law. A question tat remains to be answered is as follows: Can a computationally minimal nonlinear SJA-based control law acieve asymptotic regulation of LCO for UAV systems in a dual-parallel underactuated form? Te contribution of tis paper is te development and rigorous analysis of a sliding mode control law, wic acieves asymptotic regulation of bot pitcing and plunging LCO in UAV wings using a single, scalar SJA as te control input (i.e., a dual-parallel underactuated system). To te best of te autors knowledge, tis is te first SJA-based nonlinear control result to rigorously prove asymptotic regulation of bot pitcing and plunging LCO states in a dualparallel underactuated UAV system. Moreover, to address a practical UAV scenario were onboard computational resources are limited, te result presented ere compensates for te inerent SJA actuator nonlinearity and parametric uncertainty witout te use of adaptive laws or function approximators. To acieve te result, a sliding mode control strategy is utilized, wic employs a periodic switcing law. A detailed matematical model of te UAV dynamics is utilized to develop te regulation error dynamics, and a rigorous Lyapunov-based stability analysis is presented to prove asymptotic regulation of te pitcing and plunging displacements. Numerical simulation results are also provided to complement te teoretical development. 2. Dynamic Model and Properties Te equation describing dynamics of LCO in an airfoil can be expressed as [18] M s p+c s p+f(p)p=[ L ], (1) M were te coefficients M s,c s R 2 2 denote te structural mass and damping matrices and F(p) R 2 2 is a nonlinear stiffness matrix. In (1), p(t) [(t) (t)] T R 2,were (t), (t) R denote te plunging and pitcing displacements, respectively. Figure 2 illustrates te pitcing and plunging displacements in a standard airfoil. Also in (1), te structural linear mass matrix M s is defined as [18] M s =[ m mx b ], (2) mx b I

4 Aerospace Engineering 3 L U M K Center of gravity k x Figure 2: Illustration of pitc () and plunge () in an airfoil, including lift (L) and moment (M) definitions and te nonlinear stiffness coefficient (K ). b ab c=2b Elastic axis Midcord Te rigt and side of (1) is explicitly given by [18] L=ρU 2 s p bc l [ + M=ρU 2 s p b 2 c m [ + +ρu 2 s p b 2 c mβ β, b +(1 2 a)b U ]+ρu2 s p bc lβ β, b +(1 2 a)b U ] (6) Figure 3: Diagram of te wing section, sowing geometric parameters and te deflection angle β [18]. were te parameter x R denotes te nondimensional distance measured from te elastic axis to te center of mass, b Riste semicord of te wing [m], m Ris te mass of te wing section, and I R is te mass moment of inertia oftewingaboutteelasticaxis(seefigures2and3). Te structural linear damping matrix is described as C s =[ C ], (3) C were te parameters C,C R are te structural damping coefficient in plunge due to viscous damping [kg/s] and structural damping coefficient in pitc due to viscous damping [(kg m 2 )/s], respectively. Te nonlinear stiffness matrix utilized in tis study is F(p)=[ K ], (4) K were K R is te structural spring constant in plunge [N/m] and K R in [(N m)/rad] is te torsion stiffness coefficient described in terms of a polynomial as K = 2.82 ( ). β (5) were U R is te velocity [m/s], ρ is te density of air [kg/m 3 ], s p is te wing span [m], c l is te lift coefficient per angle of attack, c m is te moment coefficient per control surface deflection, c lβ is te lift coefficient per control surface deflection, c mβ is te moment coefficient per control surface deflection, and a is te nondimensional distance from te midcord to elastic axis. Te term β denotes te surface deflection angle of te wing (see Figure 3). By rearranging (1), te dynamics equations can be expressed as x=a(x) x+bβ, (7) were x(t) [x 1 (t) x 2 (t) x 3 (t) x 4 (t)] T R 4 is te state vector wit x 1 (t) (t), x 2 (t) (t), x 3 (t) (t), and x 4 (t) (t). In(7),A(x) R 4 4 is a nonlinear state matrix (containing nonlinearities due to te torsion stiffness coefficient introduced in (5)), and B R 4 1 is te control input gain matrix. By separating te constant elements from te state-dependent elements of te matrix A(x),tedynamic model(7)canberewrittenas x= 1 1 [ a 1 a 2 a 3 a x+f(x 2 )+ 4 ] [ b u, (8) 1 ] [ c 1 c 2 c 3 c 4 ] [ b 2 ] were a 1,...,a 4 R and c 1,...,c 4 R denote known constant parameters, wic are dependent on te pysical parameters of te wing section. Te explicit definitions of te constant parameters are unwieldy and are omitted ere for brevity. In (8), u(t) R represents te virtual surface deflection delivered by te SJA (or SJA array) (i.e., te term β

5 4 Aerospace Engineering introduced in (7)). Also in (8), te nonlinear vector function F(x 2 ) is defined in terms of te stiffness coefficient K as K F(x 2 )=. (9) m(x a b I /(mx a b)) [ (1/ (x a b)) K ] [ m(x a b I /(mx a b)) ] Te constant control input gain terms b 1,b 2 R introduced in (8) are explicitly defined as b 1 = ρv2 b 2 c mβ s p +(I /(mx a b)) ρv 2 bc lβ s p, m(x a b I /(mx a b)) b 2 = ρv2 bc lβ s p (1/(x a b)) ρv 2 b 2 c mβ s p. m(x a b I /(mx a b)) 3. SJA Dynamics (1) Te dynamics of te SJA are nonlinear and tey contain parametric uncertainty. Figure 1 represents te basic structure of a SJA [1, 14]. Tis SJA as a piezoelectrically driven diapragm in its cavity tat generates time varying pressure gradients across a small orifice. Te periodic ingestion and expulsion of te air troug te small orifice results in te formation of vortex rings wic formulates a steady turbulent jet of air. Based on empirical data, te dynamics of a SJA can be expressed as [1, 14] u=θ 2 θ 1 V, (11) were u(t) R denotes te virtual airfoil surface deflection (i.e., te control input), V(t) = A 2 ppi (t) R denotes te peakto-peak SJA voltage, and θ 1,θ 2 R are uncertain pysical parameters. To compensate for te SJA nonlinearity and parametric uncertainty in (11), a robust inverse control design structure will be utilized for te voltage input signal V(t) [17].Terobust inverse controller can be expressed as V (t) = θ 1 θ 2 u d, (12) were θ 1, θ 2 R denote constant feedforward estimates of te uncertain parameters θ 1 and θ 2 ;andu d(t) R is a subsequently defined auxiliary control signal. 4. Control Development Te objective is to design te scalar control signal u d (t) to asymptotically regulate te plunging and pitcing dynamics (i.e., (t) and (t)) to zero. By leveraging te result in [19], u d (t) will be designed using a sliding mode control law wit a periodic switcing function as u d =M tan {sin [ π t ε (s (t) +λ tan (s (τ)) dτ)]}. (13) Based on te dynamic equations in (8) and te subsequent stability analysis, te sliding surface s(t) R in (13) is designed as s (x) = K d x 2 +k 1 x 1 +k 3 x 3 + a i x i, i=1,...,4, (14) were k 1,k 3 R are positive constant control gains and a 1,...,a 4 are introduced in (8) Stability Analysis Teorem 1. Te robust control law in (13) ensures asymptotic convergence to te sliding manifold s(x) =. Proof. ProofofTeorem1canbefoundin[19]andisomitted ere to avoid distraction from te main contribution of te current result. Teorem 2 (main result). Convergence to te sliding manifold s(x) = resultsinasymptoticregulationofbotpitcingand plunging displacements in te sense tat as t. 4 i=1 s (x) (t), (t) (15) Proof (plunging regulation). It follows directly from te definition of te sliding surface s(x) in (14) and te LCO dynamic equations in (8) tat s (x) x 3 k 1 x 1 k 3 x 3. (16) By using te state definition x 3 (t) x 1 (t), te expression in (16)canbeusedtosowtat,onteslidingmanifolds(x) =,tex 1 (t) dynamics are governed by x 1 +k 3 x 1 +k 1 x 1 =, (17) were k 1, k 3 are introduced in (14). Since k 1,k 3 >,te ODE in (17) is Hurwitz, and (17) can be used to prove tat x 1 (t) (t) and x 3 (t) (t).tus,convergence to te sliding manifold s(x) = directly results in asymptotic (exponential) regulation of te plunging displacement (t) to zero. Proof (pitcing regulation). By substituting (14) into (8) and using te fact tat x 1 (t), x 3 (t),itcanbesowntat

6 Aerospace Engineering 5 convergence to te sliding manifold s(x) = results in te pitcing dynamics x 4 c 2 x 2 c 4 x 4 d 2 d K x 2 =, (18) were c 2 and c 4 are known constant parameters introduced in (8); d 2 1/(x a b), d m(x a b I /(mx a b)); andk is defined in (5). By using te state definition x 4 (t) x 2 (t), te expression in (18) can be rewritten as te second-order nonlinear ODE x 2 c 4 x 2 c 2 x 2 d 2 d K x 2 =. (19) After utilizing te definition of K given in (5), te ODE in (19) can be expressed in te form [2] x 2 +b( x 2 ) +c(x 2 ) =. (2) Noting tat c 4 <,teauxiliaryfunctionsb( x 2 ), c(x 2 ) Rin (2) are explicitly defined as b( x 2 ) c 4 x 2, (21) c(x 2 ) d 2 d x5 2 + d 2 d 858x4 2 d 2 d x3 2 + d 2 d 22.1x2 2 (d 2 d c 2)x 2. (22) Note. Itcanbesowntatc(x 2 ) = x 2 (t) =. Te expressions in (21) and (22) satisfy te following properties: x 2 b( x 2 )> for x 2 R {}, (23) x 2 c (x 2 ) > for x 2 R {}. (24) To prove asymptotic regulation of te pitcing displacement x 2 (t) to zero, consider te positive definition function (i.e., Lyapunov function candidate) [2] V= 1 x 2 x c (ξ) dξ. (25) After taking te time derivative of (25) and utilizing (2), V(t) canbeexpressedas V= x 2 ( b ( x 2 ) c(x 2 )) + c(x 2 ) x 2 = x 2 b( x 2 ). (26) Te Lyapunov derivative in (26) is negative semidefinite based on inequality (23). We now use LaSalle s invariance principle to state tat V(t) as t x 2 (t) as t,andtus, x 2 (t) as t.itfurterfollows tat b( x 2 ) as t from (21). Since b( x 2 ), x 2 (t) as t,(2)canbeusedalongwitproperty(24)toprove tat x 2 (t) as t. Feedback control Figure 4: Case 1: x() = [.2,.2,, ] T ; time evolution of te control input, u(t), regulation error for plunging, (t) in [m], and regulation error for pitcing, (t) in [rad]. Table1:Manuallyselectedgainsforrobustcontrol. k s = 9. ε = 1 M = 1. λ = 17. Table 2: Constant parameters. ρ = kg/m 3 a =.6 c m =.635 m =12.387kg b =.125 m V =13m/s C =.36 (kg m 2 )/s c lβ =3.358 s p =.6 m I =.65 kg m C =27.43kg/s c l = 6.28 K = N/m c mβ =.635 x a = Results. In tis section, a numerical simulation was created to demonstrate te performance of te control law developed in (13) and (14). Te simulation is based on te dynamic model given in (1). Te dynamic parameters utilized in te simulation are summarized in Table 2 and were obtained from [18]. Te control gains k s, ε, M,andλwere manually selected as described in Table 1. Te actual SJA parameters θ 2 and θ 1 and estimates θ 2 and θ 1 were selected as 32.9 and 14.7, respectively. Te control law was simulated for tree different cases of initial conditions: Case 1: x() = [.2.2 ] T (Figures 4and5),Case2:x() = [.5.5] T (Figures 6 and 7), andcase3:x() = [ ] T (Figures 8 and 9). Figures 4, 6, and 8 sow te time evolution of te pitcing and plunging displacements (t) and (t) during closed-loop controller operation for Cases 1, 2, and 3, demonstrating te rapid convergence of te pitcing and plunging displacements to zero. In addition, Figures 5, 7, and 9 sow tat te pitcing and plunging velocities (t) and (t) converge to zero rapidly during closed-loop operation. Furter, te top plots in Figures 4, 6, and 8 sow te control effort u(t) used during closed-loop controller for Cases 1, 2, and 3.

7 6 Aerospace Engineering Figure 5: Case 1: x() = [.2,.2,, ] T ; convergence of te regulation error rate for plunging, (t), and pitcing, (t). Feedback control Figure 6: Case 2: x() = [,,.5,.5] T ; time evolution of te control input, u(t), regulation error for plunging, (t) in [m], and regulation error for pitcing, (t) in [rad] Figure 7: Case 2: x() = [,,.5,.5] T ; convergence of te regulation error rate for plunging, (t), and pitcing, (t). Feedback control Figure 8: Case 3: x() = [.2,.2,.3,.3] T ; time evolution of te control input, u(t), regulation error for plunging, (t) in [m], and regulation error for pitcing, (t) in [rad] Figure 9: Case 3: x() = [.2,.2,.3,.3] T ; convergence of te regulation error rate for plunging, (t), and pitcing, (t). 5. Conclusion In tis paper, a sliding mode control law is presented, wic acieves asymptotic LCO regulation in UAV wings using SJAs. Moreover, te control law acieves asymptotic regulation of bot te pitcing and plunging displacements for UAV LCO dynamics in a dual-parallel underactuated form, were a single scalar control signal simultaneously affects bot displacements. To acieve te result, a sliding mode control strategy is utilized, wic employs a periodic switcing function and a novel sliding surface, wic is derived based on te nonlinear dynamics of te UAV LCO system. Te inerent nonlinearity and parametric uncertainty in te SJA dynamic model is addressed by means of a robust inverse control structure, wic utilizes constant feedforward estimates of te uncertain SJA parameters. Te use of constant estimates as opposed to time varying adaptive parameter estimates is motivated by te desire to develop a practical LCO regulation metod tat can be implemented on small UAVs wit limited onboard computational resources. A rigorous analysis is presented to prove te teoretical result,

8 Aerospace Engineering 7 and numerical simulation results are provided to complement te teoretical development. Conflict of Interests Te autors declare tat tere is no conflict of interests regarding te publication of tis paper. Acknowledgment Tis researc is supported by NSF Award number References [1] Q. Xia, S. Lei, J. Ma, and S. Zong, Numerical study of circular syntetic jets at low Reynolds numbers, International Journal of Heat and Fluid Flow,vol.5,pp ,214. [2] K. V. Sing, Active aeroelastic control wit time delay for targeted flutter modes, Aerospace Science and Tecnology, vol. 43, pp , 215. [3] Z. Sun, S. Hagigat, H. H. T. Liu, and J. Bai, Time-domain modeling and control of a wing-section stall flutter, Sound and Vibration,vol.34,pp ,215. [4] N. Ramos-Pedroza, W. MacKunis, and M. Reyanoglu, Sliding mode control-based limit cycle oscillation suppression for UAVs using syntetic jet actuators, in Proceedings of te InternationalWorksoponRecentAdvancesinSlidingModes (RASM 15), pp. 1 5, Istanbul, Turkey, April 215. [5] M. Dardel and F. Baktiari-Nejad, Limit cycle oscillation controlofwingwitstaticoutputfeedbackcontrolmetod, Aerospace Science and Tecnology, vol.24,no.1,pp , 213. [6] M. Ben Ciek, M. Fercici, J.-C. Béra, and M. Micard, Minimization of time-averaged and unsteady aerodynamic forces on a tick flat plate using syntetic jets, Fluids and Structures,vol.54,pp ,215. [7] Y.-W. Lv, J.-Z. Zang, Y. San, and X.-M. Tan, Numerical investigation for effects of actuator parameters and excitation frequencies on syntetic jet fluidic caracteristics, Sensors and Actuators A: Pysical, vol. 219, pp , 214. [8] T. Persoons, General reduced-order model to design and operate syntetic jet actuators, AIAA Journal,vol.5,no.4,pp , 212. [9] A. Kourta and C. Leclerc, Caracterization of syntetic jet actuation wit application to Amed body wake, Sensors and Actuators, A: Pysical,vol.192,pp.13 26,213. [1] D. Deb, G. Tao, J. O. Burkolder, and D. R. Smit, Adaptive syntetic jet actuator compensation for a nonlinear aircraft model at low angles of attack, IEEE Transactions on Control Systems Tecnology,vol.16,no.5,pp ,28. [11] S. H. Kim and C. Kim, Separation control on NACA2312 using syntetic jet, Aerospace Science and Tecnology, vol.13, no. 4-5, pp , 29. [12]M.Ciuryla,Y.Liu,J.Farnswort,C.Kwan,andM.Amitay, Fligt control using syntetic jets on a cessna 182 model, AIAA Aircraft,vol.44,no.2,pp ,27. [13] M. Ben Ciek, M. Fercici, and J.-C. Béra, Modified flapping jet for increased jet spreading using syntetic jets, International Heat and Fluid Flow, vol. 32, no. 5, pp , 211. [14] D. Deb, G. Tao, J. O. Burkolder, and D. R. Smit, Adaptive compensation control of syntetic jet actuator arrays for airfoil virtual saping, Aircraft, vol. 44, no. 2, pp , 27. [15] M.Krstic,I.Kanellakopoulos, and P.Kokotović, Nonlinear and Adaptive Control Design, Wiley-Interscience, [16] G. Tao, Adaptive Control of Systems wit Actuator and Sensor Nonlinearities, Jon Wiley & Sons, New York, NY, USA, [17] W. MacKunis, S. Subramanian, S. Meta, C. Ton, J. W. Curtis, and M. Reyanoglu, Robust nonlinear aircraft tracking control using syntetic jet actuators, in Proceedings of te 52nd IEEE Conference on Decision and Control (CDC 13), pp , IEEE, Firenze, Italy, December 213. [18] M. R. Elami and M. F. Narab, Comparison of SDRE and SMC control approaces for flutter suppression in a nonlinear wing section, in Proceedings of te American Control Conference (ACC 12), pp , Montreal, Canada, June 212. [19] S. V. Drakunov, Sliding mode control wit multiple equilibrium manifolds, Dynamic Systems, Measurement, and Control,vol.55,no.1,pp.11 18,1994. [2] J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall,EnglewoodCliffs,NJ,USA,1991.

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