Sliding Mode Velocity-observer-based Stabilization of a 3-DOF Helicopter Prototype

Proceedings of the 6th IFAC Symposium on Robust Control Design Haifa, Israel, June 16-18, 2009

Sliding Mode Velocity-observer-based Stabilization of a 3-DOF Helicopter Prototype Yuri Orlov ∗ Marlen Meza-S´ anchez ∗ Luis T. Aguilar ∗∗ CICESE Research Center P.O. BOX 434944, San Diego, CA, 92143-4944, (e-mail: yorlov{marmeza}@cicese.mx) ∗∗ Instituto Polit´ecnico Nacional Avenida del Parque 1310 Mesa de Otay, Tijuana 22510 M´exico (e-mail: [email protected]) ∗

Abstract: Sliding mode control synthesis is developed for an underactuated mechanical system. A laboratory helicopter prototype is used as a test bed. Both state feedback design and dynamic output feedback design with sliding mode velocity observers, running in parallel, are proposed. Performance and robustness issues of the closed-loop system are illustrated in a numerical study. The work is in progress and the conference presentation is going to be supported by experimental results. Keywords: Helicopter, Robust stabilizability, Velocity observer, Sliding-mode control. 1. INTRODUCTION Analysis and synthesis of helicopter models have attracted much attention of researchers (see, e.g., Avila-Vilchis et al. (2003); Isidori et al. (2003) and references therein). Several studies on this type of systems such as Dzul et al. (2004), Xu and Ozguner (2006), Bouadi et al. (2007), Hoffmann et al. (2007), have been developed. In the present paper, the sliding mode control synthesis is developed for a helicopter prototype, manufactured by Quanser Company. The prototype represents an underactuated 3-DOF mechanical system, actuated by two motors. As from practical standpoint, this prototype is expected to operate under uncertainty conditions, the sliding mode approach, which is capable of attenuating external disturbances, looks attractive to stabilize the prototype. Different approaches to the stabilization of the 3-DOF helicopter prototype have been recently proposed. In particular, a constrained predictive control method was documented in Lopez et al. (2006). In Garc´ıa-Sanz et al. (2006), the pitch control, using linear controllers, was considered. In addition, an application of adaptive control is presented Andrievsky et al. (2007) for pitch control. In Ishutkina (2004), a supervisory safety control, ensuring the safety of the system under stressed joints and external disturbances, was designed. Aggressive landing maneuvers for the 3-DOF helicopter prototype were adressed in Bayraktar (2004). Output feedback model reference control of a helicopter, using sliding mode controller and observer, was pursued in Spurgeon et al. (1996). In this work, a sliding mode output feedback synthesis is developed for stabilization of the helicopter prototype around a desired position. For this purpose, the stabilization problem in question is decoupled into two indepen-

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dent subproblems. The system is thus first represented in a canonical form where a controller drives along the elevation axis, and another one acts within the subspace spanned by the direction and rotation axes. Once the system is in the normal form, two second order sliding mode controllers are constructed side by side to internally asymptotically stabilize the system along the elevation axis and, respectively, along the direction and rotation axes. While operating under uncertainty conditions, the controllers are shown to attenuate external disturbances with a priori known norm bounds on their magnitudes. In order to meet practical requirements on available measurements, first order sliding mode observers are then involved into the closed-loop system to estimate the plant velocities. Utilizing these estimates in the subsequent output feedback synthesis yields a general framework of robust stabilization of the helicopter prototype, using position measurements only. Performance issues of the closed-loop system and its robustness are illustrated in a numerical study made for the laboratory prototype. The proposed general framework of the robust output feedback stabilization of the 3-DOF helicopter model and its numerical verification constitute the main contribution of the paper into the existing literature in the area. The paper is organized as follows. The dynamic prototype model and problem statement are given in Section 2. The state feedback design is presented in Section 3. The velocity observers and the output feedback design are collected in Section 4. Performance and robustness issues of the closed-loop system are illustrated in the simulation study in Section 5. Finally, Section 6 presents some conclusions.

10.3182/20090616-3-IL-2002.0044

6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009

˙ ψ1 = ψ, ψ2 = ψ, ˙ and a = Kf Lb J −1 , b = Lb Fg J −1 , φ, e e −1 c = Kf Lh Jd , d = Kp Lb Jt−1 , system (1)-(3) takes the form

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where w1 (t) = Jt−1 wt (t).

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Our objective is to asymptotically stabilize the origin θi , φi , ψi = 0, i = 1, 2 while also attenuating the effect of the external disturbances. The position measurements θ1 (t), φ1 (t), ψ1 (t) are assumed to be the only available information on the system.

Fig. 1. 3-DOF helicopter prototype 2. DYNAMIC MODEL AND PROBLEM STATEMENT The mathematical model of the laboratory prototype of the 3-DOF helicopter drawn from the user’s manual Quanser (2004) is given by Je θ¨ = Kf (Vf + Vb )Lb − Fg Lb + we

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In the above equations, θ(t) ∈ IR is the elevation angle, φ(t) ∈ IR is the direction angle, and ψ(t) ∈ IR is the rotation angle, Je is the moment of inertia of the system about the elevation axis, Jd is the moment of the helicopter inertia about the pitch or directional axis, Jt is the moment of the helicopter inertia about the travel or rotation axis, the manipulated variables used for control are the armature voltages of the front and back DC motors denoted as Vf (t) and Vb (t), respectively, we (t), wd (t), wt (t) are the external disturbances, affecting the system; Kf is the force constant of the motor/propeller combination, Lb is the distance from the pivot point to the helicopter body, Fg is the gravitational force, Lh is the distance from the pitch axis to the either motor, and Kp is the force required to maintain the helicopter in flight. The travel variable ψ is the rotation of the entire system around the vertical axis z. The elevation θ is defined as the movement of the helicopter body, which corresponds to the angular displacement of the main sustentation arm with respect to the horizontal axis y. The pitch movement φ corresponds to the change of attitude of the helicopter body and it represents singularities at the vertical positions φ = ± π2 which is why the validity of the above model is confined to the domain φ ∈ (− π2 , π2 ). The inertia model of the system is simplified to point masses associated to the two motors and to the counterweight. In addition, friction and aerodynamics drag effects are neglected. The force generated by each motor-propeller is assumed to be normal to the propeller plane. For the purpose of decomposing the 3-DOF underactuated system (1)-(3) into two subsystems, actuated independently, let us introduce the control inputs u1 = Vf +Vb and ˙ φ1 = φ, φ2 = u2 = Vf − Vb . Then, setting θ1 = θ, θ2 = θ,

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3. STATE FEEDBACK DESIGN 3.1 Second Order Sliding Mode Control for the Elevation Axis Apparently, equations (4), describing the elevation dynamics, constitute an independent subsystem. In order to globally asymptotically stabilize subsystem (4) the quasihomogeneous control law 1 u1 = [b − hθ1 − pθ2 − α sign(θ1 ) − β sign(θ2 )], (7) a inherited from Orlov (2009), is chosen provided that both the elevation angle θ1 and the angular velocity θ2 are available for measurements. With this control law, the closed-loop system (4), (7) appears to be globally finite time stable if the parameter gains are such that h, p ≥ 0, α − M1 > β > M1 (8) for some M1 > 0. Moreover, this stability remains in force, regardless of whichever external disturbance w1 of bounded magnitude sup |w1 (t)| ≤ M1 (9) t

affects the system. Theorem 1. Let the disturbance-corrupted system (4) be driven by controller (7) subject to the parameter subordination (8) with some positive constant M1 > 0. Then the closed-loop system (4), (7) is globally finite time stable, provided that the external disturbance w1 (t) meets the upper bound (9) with the same constant M1 . Proof. By substituting (7) into (4), the closed-loop system takes the form θ˙1 = θ2 , θ˙2 = −hθ1 − pθ2 − α sign(θ1 ) − β sign(θ2 ) + w1 , (10) and by applying (Orlov, 2009, Theorem 4.2) to system (10) under conditions (8), (9), the validity of Theorem 1 is established. 3.2 Second Order Sliding Mode Control for the Direction and Rotation Axes Our next goal is to asymptotically stabilize the underactuated subsystem (5), (6), while also attenuating the external

6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009

disturbances w2 , w3 . Because of loss of the controllability of the helicopter body at φ1 = ± π2 , the pitch dynamics are prohibited to approach these singular points which is why only the local stabilization of the direction angle is treated. The control strategy consists of two steps. First, an output of the system in question is specified in such a way that the corresponding disturbance-free zero dynamics are globally asymptotically stable whereas the influence of the external disturbances on the zero dynamics are attenuated due to the choice of the system output. After that, an input u2 , which in spite of the presence of the external disturbances, locally drives the system to the zero dynamics manifold in finite time, is constructed. The following input s(φ1 , ψ1 , ψ2 ) = d sin(φ1 ) − c1 ψ1 − c2 ψ2 (11) with some positive parameters c1 and c2 is chosen to impose the desired properties on the zero dynamics. Apparently, the travel variable, while being restricted to the manifold s = 0, is governed by the internally asymptotically stable system ψ˙ 1 = ψ2 , ψ˙ 2 = −c1 ψ1 − c2 ψ2 + w3 , (12) whereas the disturbance w3 is attenuated by a proper choice of the gains c1 and c2 (see, e.g., Doyle et al. (1989) for details). In turn, while evolving within the admissible domain φ ∈ (− π2 , π2 ), the pitch variable meets the same properties on the zero dynamics manifold where d sin(φ1 ) = c1 ψ1 + c2 ψ2 . (13) To ensure attaining manifold (13) in finite time the following control input 1 [sin(φ1 )φ22 − c1 sin(φ1 ) − c2 cos(φ1 )φ2 c cos(φ1 ) −α1 sign(s) − β1 sign(s) ˙ − h1 s − p1 s] ˙ (14) with positive parameters α1 , β1 , h1 , p1 is proposed. The idea behind the proposed synthesis is to bring the projection of the closed-loop system (5), (6), (14) onto the subspace, spanned by the output s, into the so-called quasihomogeneous form s¨ = −d[α1 sign(s) + β1 sign(s) ˙ + h1 s + p1 s] ˙ + w (15) where the external disturbance w3 (t), affecting the travel dynamics, has been assumed to be differentiable and the notation w = d cos(φ1 )w2 − c1 w3 − c2 w˙ 3 has been used. By (Orlov, 2009, Theorem 4.4), equation (15) appears to be globally equiuniformly (in w) finite time stable, provided that supt |w(t)| ≤ M for some constant M > 0 and the parameters are subordinated as follows h1 , p1 ≥ 0, dα1 − M > dβ1 > M. (16) u2 =

Assuming that the magnitudes of the external disturbances w2 (t) and w3 (t) and that of the temporal derivative w4 (t) = w˙ 3 (t) are upper bounded sup |wi (t)| ≤ Mi , i = 2, 3, 4 (17) t

by some positive constants M2 , M3 , M4 , the following result is obtained. Theorem 2. Consider the closed-loop system (5), (6), (14) with the parameter subordination (16) and with the above

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assumption (17) on the external disturbances. Let condition (16) holds with M = dM2 + c1 M3 + c2 M4 . Then the closed-loop system is driven to the zero dynamics manifold (13) in finite time, and after that, it is governed by the sliding mode equation (12). Moreover, the internal dynamics of the closed-loop system (5), (6), (14) are locally asymptotically stable. Proof. Since the projection of (5), (6), (14) onto the subspace, spanned by the output s, is locally described by equation (15) and under the conditions of the theorem, this equation is equiuniformly finite time stable according to (Orlov, 2009, Theorem 4.4), the closed-loop system, starting from a finite time instant T > 0, evolves on the zero dynamics manifold (13). Thus, for t ≥ T , system (5), (6), (14) is governed by the internally asymptotically stable sliding mode equation (12). Coupled to the finite time stability of the stage, preceding the sliding modes, and due to the well-posedness of this stage, this yields the internal asymptotic stability of the closed-loop system. Remark 3. Since the parameters c1 and c2 , which determine the zero dynamics manifold (13), can be assigned arbitrarily large, the influence of the external disturbances on the sliding modes (12) is attenuated to as a low level as desired by a proper choice of the parameters 4. OUTPUT FEEDBACK DESIGN The state feedback controllers u1 and u2 , developed so far, require measurements of both the positions and velocities of the system. In the present section, our study is extended toward output feedback design. For this purpose, proper velocity observers of the nonlinear dynamics of the helicopter model are running in parallel to constitute the dynamic output synthesis. 4.1 Semiglobal Design of Sliding Mode Elevation Velocity Observers To begin with, we present a family of variable structure elevation velocity estimators ˙ θˆ1 = θˆ2 + µθ sign(θ1 − θˆ1 ) ˙ θˆ2 = −b + au1 + νθ sign(θ1 − θˆ1 ), (18) parameterized with µθ , νθ > 0. Clearly, the dynamics of the observation errors e1 = θ1 − θˆ1 , e2 = θ2 − θˆ2 are described by the equations e˙ 1 = e2 − µθ sign e1 , e˙ 2 = −νθ sign e1 + w1 , (19) with discontinuous right-hand side. It is subsequently shown that these equations appear to be internally asymptotically stable. Moreover, while being affected by an external disturbance w1 with an upper bound M1 > 0 on its magnitude, the observation errors are shown to decay asymptotically to the segment µθ M1 Sθ = {(e1 , e2 ) ∈ IR2 : e1 = 0, |e2 | ≤ } (20) νθ with the attraction domain µθ νθ Dθ = {(e1 , e2 ) ∈ IR2 : |e2 | < }, (21) M1

6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009

provided that νθ > max{M1 , µθ M1 }. (22) Theorem 4. Consider the error dynamics (19) subject to the parameter subordination (22) with some M1 > 0. These dynamics appear to be internally globally asymptotically stable. While being initialized within domain (21) and affected by an admissible external disturbance w1 of magnitude less than or equal to M1 , these dynamics steer to the interval Iµθ = {(e1 , e2 )] ∈ IR2 : e1 = 0, |e2 | < µθ } in finite time. After that, there appear sliding modes along the vertical axis, which are localized within the interval Sµθ and decay to the embedded segment (20) as t → ∞. Proof. First, let us note that for the discontinuous error system (19), the condition e1 e˙ 1 = e1 (e2 − µθ signe1 ) ≤ −|e1 |(µθ − |e2 |) < 0) for sliding modes to exist holds for all e1 6= 0 and |e2 | < µθ . After that, let us consider the Lyapunov candidate function V (e1 , e2 ) = νθ |e1 | + 21 e22 . By computing the temporal derivative of this function along the trajectories of the disturbance-free system (19), we arrive at V˙ = νθ signe1 (e2 − µθ signe1 ) − e2 νθ signe1 = −µθ νθ < 0 (23) everywhere but on the vertical axis e1 = 0 where the function V is not differentiable. Apparently, inequality (23) ensures that the trajectories hit the sliding mode interval Iµθ in finite time because otherwise they steer to the origin in finite time. Since the sliding modes on the interval Iµθ are governed by the asymptotically stable equation νθ e˙ 2 = − e2 (24) µθ this yields that the error dynamics (19) are internally globally asymptotically stable. For the convenience of the reader, recall that the sliding mode equation (24) is derived according to the equivalent control method Utkin (1992) by substituting the equivalent value signeq e1 of the commuting function sign e1 , that ensures the identity e˙ 1 = 0 along the sliding modes, into the second equation of the disturbance-free system (19) for sign e1 . As a matter of fact, the equivalent value represents a solution of the algebraic equation e2 − µθ sign e1 = 0 with respect to sign e1 and hence, signeq e1 = µ−1 θ e2 . In turn, for the perturbed dynamics (19), relation (23) is modified to V˙ = −µθ νθ + e2 w1 ≤ −µθ νθ + M1 |e2 |, (25) and for the trajectories, initialized within domain (21), it remains negative definite. Thus, along with the internal dynamics, the trajectories of the perturbed system (19) hit the sliding mode interval Iµθ in finite time Tw1 > 0, which certainly depends on the disturbance w1 . The solutions of the perturbed sliding mode equation νθ e˙ 2 = − e2 + w1 (26) µθ are given by Z t e2 (t) = e−(t−Tw1 ) e2 (Tw1 ) + e−(t−τ ) w1 (τ )dτ, Tw1

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and by taking into account the upper bound M1 on the magnitude of the admissible disturbances w1 , these solutions turn out to approach segment (20) as t → ∞. The proof is completed. In order to interpret the above result in terms of the velocity estimator (18), the following concept is introduced. Definition 5. Let a second order dynamic system η¨ = f (η, η) ˙ (27) and a dynamic system of the form ηˆ˙ 1 = f1 (η, ηˆ1 , ηˆ2 ), ηˆ˙2 = f2 (η, ηˆ1 , ηˆ2 ) (28) with scalar state variables η(t) and η(t), ˙ and respectively, (ˆ η1 (t) and ηˆ2 (t) posses Filippov solutions for arbitrary initial conditions. System (28) is said to constitute an asymptotic velocity observer of system (27) with an attraction domain D ⊂ R2 of initial observation errors and of precision ǫ > 0 iff lim sup |η(τ ) − ηˆ1 (τ )| ≤ ǫ, lim sup |η(τ ˙ ) − ηˆ2 (τ )| ≤ ǫ t→∞ τ ≥t

t→∞ τ ≥t

(29) for all Filippov solutions (η, η) ˙ and (ˆ η1 , ηˆ2 ) of (27) and (28) such that the initial observation errors (η(0)− ηˆ1 (0), η(0)− ˙ ηˆ2 (0)) are in D. Relying on Definition 1, Theorem 3 is reformulated as follows. Theorem 6. Consider the variable structure velocity estimator (18) of the elevation dynamics (4). Let an upper bound M1 > 0 of the magnitude of the external disturbance w1 (t), affecting (4), be known a priori. Then under condition (22), imposed on the estimator parameters µθ , νθ , (18) represents an asymptotic elevation velocity 1 observer of precision µθνM with (21) being an attraction θ domain of the initial estimation errors e1 (0) = θ1 (0) − θˆ1 (0), e2 (0) = θ2 (0) − θˆ2 (0). In addition, (18) is a global asymptotic observer of the disturbance-free system (4). Remark 7. By Theorem 4, the estimator family (18) presents a semiglobal estimation of the perturbed elevation dynamics (4) in the sense that whichever large initial estimation error is admitted, there exist sufficiently high gains µθ and νθ such that these initial errors are in the attraction domain of the corresponding velocity estimator 1 we deal with by (18). Moreover, the lower fraction µθνM θ choosing the gains µθ and νθ the better velocity estimate we get; the perfect estimate such that limt→∞ |θ2 (t)−θˆ2 (t)| is particularly obtained as µνθθ → 0. 4.2 Semiglobal Design of Sliding Mode Pitch – Travel Velocity Observers Similar to the elevation estimators family (18), a family of pitch–travel velocity estimators ˙ φˆ1 = φˆ2 + µφ sign(φ1 − φˆ1 ) ˙ φˆ2 = cu2 + νφ sign(φ1 − φˆ1 ), ˙ ψˆ1 = ψˆ2 + µψ sign(ψ1 − ψˆ1 ) ˙ ψˆ2 = −d sin(φ1 ) + νψ sign(ψ1 − ψˆ1 ), parameterized with µφ , νφ , µψ , νψ > 0, is designed.

(30)

(31)

6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009

The estimation errors φi − φˆi , ψi − ψˆi , i = 1, 2 appear to decay asymptotically to the segments Sφ = {(φ1 − φˆ1 , φ2 − φˆ2 ) ∈ IR2 : µφ M2 } φ1 = φˆ1 , |φ2 − φˆ2 | ≤ νφ Sφ = {(ψ1 − ψˆ1 , ψ2 − ψˆ2 ) ∈ IR2 : µψ M3 ψ1 = ψˆ1 , |ψ2 − ψˆ2 | ≤ } νψ with the attraction domains

(32)

of Theorems 1–5. Since such an analysis is rather lengthy, its details are left to the interested reader. To support the present development performance issues and robustness features of the proposed output feedback synthesis are subsequently illustrated in an experimental study made for a laboratory prototype. 5. SIMULATION RESULTS

(33)

Dφ = {(φ1 − φˆ1 , φ2 − φˆ2 ) ∈ IR2 : µφ νφ |φ2 − φˆ2 | < }, (34) M2 Dψ = {(ψ1 − ψˆ1 , ψ2 − ψˆ2 ) ∈ IR2 : µψ νψ }, (35) |ψ2 − ψˆ2 | < M3 provided that νφ > max{M2 , µφ M2 }, νψ > max{M3 , µψ M3 }. (36) Theorem 8. Consider the variable structure pitch–travel velocity estimators (30) and (31) of the pitch–travel dynamics (5) and (6). Let upper bounds M2 > 0 and M3 > 0 of the magnitudes of the external disturbances w2 (t) and w3 (t), affecting (30) and (31), be known a priori. Then under condition (36), imposed on the estimator parameters µφ , νφ , µψ , νψ , the dynamic systems (30) and (31) represent asymptotic pitch–travel velocity observers of precision µφ M2 µ M and, respectively, ψνψ 3 with (34) and (35) being νφ attraction domains of the initial estimation errors φi (0) − φˆi (0), ψi (0) − ψˆi (0), i = 1, 2. In addition, (30) and (31) are global asymptotic observers of the disturbance-free systems (5) and (6). Proof : The line of reasoning used to establish the validity of Theorem 4 applies here as well. The detailed proof of Theorem 5 is therefore omitted. Remark 9. Remark 2 is readily modified to apply to Theorem 5. 4.3 Output Feedback Synthesis The output feedback controllers u1 and u2 1 ˙ u1 = [b − hθ1 − pθˆ2 − α sign(θ1 ) − β sign(θˆ2 )] (37) a 1 u2 = [sin(φ1 )φˆ22 − c1 sin(φ1 ) − c2 cos(φ1 )φˆ2 c cos(φ1 ) (38) −α1 sign(ˆ s) − β1 sign(sˆ˙ ) − h1 sˆ − p1 sˆ˙ ] are obtained if the corresponding observer variables θˆ2 , φˆ2 , ψˆ2 are substituted into the state feedback law (7), (14) for the state variables θ2 , φ2 , ψ2 . In the above synthesis, sˆ = d sin(φ1 ) − c1 ψ1 − c2 ψˆ2 stands for s(φ1 , ψ1 , ψˆ2 ) and s is given by (11). The stability analysis of the over-all closed-loop system (4)–(6), (18), (30), (31), driven by the output feedback controllers (37), (38) and operating under uncertainty conditions, is technical and combines the details of the proof

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Performance issues of the sliding mode position feedback controller were tested in a simulation study performed with Simulink. The parameters of the helicopter, drown from the Quanser 3-DOF helicopter manual Quanser (2004), are given in table 1. The initial conditions for the 3-DOF helicopter, selected for the simulations, were θ1 (0) = 0.5 [rad], φ1 (0) = ψ1 (0) = 0.8 [rad], whereas all the velocity initial conditions ˙ ˙ ˙ were set to θ(0) = φ(0) = ψ(0) = 0 [rad/s]. We simulated two cases of the closed-loop system, under no disturbances and affected by the harmonic disturbances w1 = w2 = 0.5 cos 40t [rad/s2 ], w3 = 0.2 cos 50t [rad/s2 ]. In the simulation runs, the controller gains in (7) and (14) were set to α = 2, β = 1, h = 5, p = 6, α1 = 2, β1 = 1, h1 = 0, p1 = 0, c1 = 2 and c2 = 3. In addition, the initial conditions for the nonlinear velocity ˆ ˆ ˆ observers were also set to θ(0) = φ(0) = ψ(0) = 0 [rad] ˙ ˙ ˙ ˆ ˆ ˆ and θ(0) = φ(0) = ψ(0) = 0 [rad/s]. Table 1. Parameter values of the experimental 3DOF helicopter Notation Lb Lh Je Jd Jt Kf Kp Fg

Value 0.66 0.177 0.91 0.0364 0.91 0.5 0.686 0.686

Units m m Kg · m2 Kg · m2 Kg · m2 N/V olt N N

Simulation results for the 3-DOF helicopter, driven by the sliding mode position controllers (7) and (14), are depicted in Fig. 2 and 3 for the disturbance-free case and for the perturbed case, respectively. As predicted theoretically, the controllers asymptotically stabilize the disturbancefree dynamics of the over-all system, while also attenuating the external disturbances. 6. CONCLUSION An asymptotic position feedback stabilization problem is studied for a 3-DOF helicopter prototype, operating under uncertainty conditions. A general framework of resolving such a problem is proposed. The framework consists of the problem decomposition and output feedback synthesis, involving the second order sliding mode state feedback design and the first order sliding mode velocity observer design. The numerical verification, made for a laboratory prototype, demonstrates the effectiveness of the developed approach. The work is in progress and its final version is going to be supported by experimental results.

6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009

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Fig. 3. 3-DOF helicopter stabilization in the presence of the harmonic disturbances.

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