Output sliding mode-based stabilization of underactuated 3-DOF helicopter prototype and its experimental verification

Author's Accepted Manuscript

Output Sliding Mode-based Stabilization of Underactuated 3-DOF Helicopter Prototype and Its Experimental Verification Marlen Meza-Sánchez, Luis T. Aguilar, Yury Orlov

www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(15)00028-9 http://dx.doi.org/10.1016/j.jfranklin.2015.01.010 FI2215

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

27 August 2012 3 December 2014 14 January 2015

Cite this article as: Marlen Meza-Sánchez, Luis T. Aguilar, Yury Orlov, Output Sliding Mode-based Stabilization of Underactuated 3-DOF Helicopter Prototype and Its Experimental Verification, Journal of the Franklin Institute, http://dx.doi.org/10.1016/ j.jfranklin.2015.01.010 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.

Output Sliding Mode-based Stabilization of Underactuated 3-DOF Helicopter Prototype and Its Experimental Verification Marlen Meza-S´ancheza , Luis T. Aguilarb , Yury Orlova a CICESE

Research Center, Carretera Ensenada-Tijuana No. 3918, Zona Playitas, Ensenada, B.C. 22860 Mexico. b Instituto Polit´ ecnico Nacional, CITEDI, Avenida Instituto Polit´ ecnico Nacional 1310, Mesa de Otay, Tijuana, B.C. 22510 Mexico

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 a sliding mode velocity observer, running in parallel, are proposed. Performance and robustness issues of the closed-loop system are illustrated in an experimental study. 1. Introduction Analysis and synthesis of helicopter models have attracted much attention of researchers (see, e.g., [1, 10] and references therein). 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 direct current motors. Motivated by practical needs, this prototype is to operate under uncertainty conditions. In this regard, the sliding mode approach, which is capable of reducing the effect of external disturbances, is an appropriate tool 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 by Lopez et al. [11]. Garc´ıa-Sanz et al. [8] solved the pitch control problem using linear controllers. Ishutkina [9] designed a supervisory safety control ensuring the safety of the system under stressed joints and external disturbances. Output feedback model reference control of a helicopter, using sliding mode controller and observer, was pursued by Spurgeon et al. [16]. Other efforts such as Oh and Khalil [12] combined variable structure controller with high-gain observer to solve the output-feedback tracking control problem of nonlinear systems with stable zero dynamics. Recently, Freidovich and Khalil Email addresses: [email protected] (Marlen Meza-S´ anchez), [email protected] (Luis T. Aguilar), [email protected] (Yury Orlov)

Preprint submitted to Journal of The Franklin Institute

January 27, 2015

[7] proposed a method to prove transient performance recovery of the high-gain observer. Ferreira et al. [4] designed a sliding-mode observer to estimate the states and identify the uncertainties to solve the robust regulation problem for the two positions (pitch and yaw) of the Quanser’s 3-DOF helicopter. In this work, a sliding mode output feedback synthesis is developed for stabilization of the helicopter prototype around a desired position counting for all the three components, pitch, roll and yaw. To the best of our knowledge, the successful position feedback regulation to desired pitch, roll and yaw positions remained unattained in the existing literature. The regulation problem in question is solved by decoupling it into two independent subproblems. The system is first represented in a canonical form where a controller component drives along the pitch axis, and another one acts within the subspace spanned by the roll and yaw 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 pitch axis and, respectively, along the roll and yaw axes. While operating under uncertainty conditions, the controllers are shown to reduce the influence of external disturbances with a priori known norm bounds on their magnitudes. The choice of the proposed controllers is motivated by their robustness features coupled to the finite time convergence within the controlled subspace to the zero dynamic manifold such that the closed-loop system is asymptotically stable along this manifold. Both global asymptotic stability of the over-all system and desired robustness features are thus provided. In order to meet practical requirements on available measurements, first order sliding mode observers with a priori fixed attraction domains and arbitrarily small precisions are then involved into the closed-loop system to estimate the plant velocities in the nonlinear setting. Although the design of the second order sliding mode observers seems possible and attractive for achieving finite time convergence, its tuning would significantly complicate the over-all synthesis (cf. that of [14] made for the double integrator). Utilizing the velocity estimates in the subsequent output feedback synthesis yields a general framework of robust stabilization of the helicopter prototype, using position measurements only. Implementation of advanced second order sliding mode observers (e.g., supertwisting observer [2, 6]) could be a benefit however it calls for further investigation whereas the present paper intends to demonstrate the capabilities of the sliding mode approach in the given circumstances and it facilitates the exhibition of the method by focusing on straightforward sliding mode algorithms. Performance issues of the closed-loop system and its robustness are illustrated in an experimental study made for the laboratory prototype. The proposed general framework of the robust output feedback stabilization of the 3-DOF helicopter model and its experimental 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 Sections 4 and 5, respectively. Performance and robustness issues of the closed-loop system are illustrated in an experimental study in Section 6. 2

Finally, Section 7 presents some conclusions. Notation. We let R denote the set of real numbers. The signum function of a variable z ∈ R is defined as ⎧ ⎪ if z > 0 ⎨ 1 sign(z) = [−1, 1] if z = 0 ⎪ ⎩ −1 if z < 0. Given a vector x = (x1 , . . . , xn )T ∈ Rn , the following norm x = |x1 | + · · · + |xn |

(1)

is used throughout. The absolute value of a vector x, denoted by |x|, is defined as |x| = (|x1 |, . . . , |xn |)T .

(2)

1.1. Background Material For later use, recall the following [13]. Consider a system governed by x˙ = ϕ(x, t),

(3)

where x = (x1 , . . . , xn )T , t ∈ R, and the function ϕ = (ϕ1 , . . . , ϕn )T is piecewise continuous. Recall that the function ϕ : Rn+1 → Rn is piece-wise continuous iff Rn+1 is partitioned into a finite number of domains Gj ⊂ Rn+1 , j = 1, . . . , N , with disjoint interiors and boundaries ∂Gj of measure zero such that ϕ is continuous within each of these domains and for all j = 1, . . . , N it has a finite limit ϕj (x, t) as the argument (x∗ , t∗ ) ∈ Gj approaches a boundary point (x, t) ∈ ∂Gj . Throughout, the precise meaning of the differential equation (3) with a piece-wise continuous right-hand side is defined in the sense of Filippov [5]. Definition 1. The equilibrium point x = 0 of the differential equation (3) is said to be globally (uniformly) finite time stable if, in addition to the global (uniform) asymptotical stability, the limiting relation x(t, t0 , x0 ) = 0

(4)

holds for each solution x(·, t0 , x0 ) and all t ≥ t0 + T (t0 , x0 ), where the settling time function T (t0 , x0 ) =

sup

x(·,t0 ,x0 )

inf{T ≥ 0 : x(t, t0 , x0 ) = 0 for all t ≥ t0 + T }

3

(5)

is such that T (t0 , x0 ) < ∞

for all t0 ∈ R and x0 ∈ Rn

(respectively, T (Bδ ) = supt0 ∈R, x0 ∈Bδ T (t0 , x0 ) < ∞ for each δ > 0). For a second order nonlinear system x˙ = y,

y˙ = −a sign(x) − b sign(y) − h x − p y + ω(x, y, t),

(6)

with a piece-wise continuous nonlinear perturbation ω(x, y, t) subject to |ω(x, y, t)| ≤ M

(7)

for all continuity points (x, y, t) and some M > 0, the following result is in order. Let the linear gains h and p be non negative, i.e., h ≥ 0,

p ≥ 0,

(8)

and let the magnitudes a and b of switching exceed the perturbation bound M in such a manner that the relation 0
(9)

holds. Theorem 1. ([13][Theorem 4.2]) Let conditions (9), (8) be satisfied. Then system (6), is globally finite time stable around the origin regardless of whichever admissible perturbation (7) affects the system. The above result will be instrumental in the subsequent position output feedback synthesis of the helicopter prototype. 2. Dynamic Model and Problem Statement The mathematical model of the laboratory prototype of the 3-DOF helicopter drawn from [9] and the user’s manual [15] is given by Je θ¨ = −fθ θ˙ − Kf sin(θ) + cos(φ)(Vf + Vb ) + we Jd φ¨ = −fφ φ˙ − Kd sin(φ) + Kf (Vf − Vb )Lh + wd

(10)

Jt ψ¨ = −fψ ψ˙ − Kp Lb sin(φ) + wt .

(12)

(11)

In the above equations, θ(t) ∈ R is the pitch angle, φ(t) ∈ R is the roll angle, and ψ(t) ∈ R is the yaw angle, Je is the moment of inertia of the system about the 4

Fb

z ψ Lb

Ff

Lh

θ

φ

Figure 1: Quanser’s 3-DOF helicopter prototype.

pitch axis, Jd is the moment of the helicopter inertia about the roll or directional axis, Jt is the moment of the helicopter inertia about the yaw 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; fθ , fφ , and fψ are positive constants denoting the viscous friction level; Kf is the force constant of the motor/propeller combination, Lb is the distance from the pivot point to the helicopter body, 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 yaw variable ψ is the rotation of the entire system around the vertical axis z. The pitch 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 roll 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 assumed to be negligible. 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 (10)– (12) into two subsystems, actuated independently, let us introduce the control ˙ φ1 = inputs u1 = Vf + Vb and u2 = Vf − Vb . Then, setting θ1 = θ, θ2 = θ, −1 −1 −1 ˙ ˙ φ, φ2 = φ, ψ1 = ψ, ψ2 = ψ, a1 = Je fθ , a2 = Je Kf , a3 = Je , b1 = Jd−1 fφ , b2 = Jd−1 Kd , b3 = Jd−1 Kf Lh , c1 = Jt−1 fψ , and c2 = Jt−1 Kp Lb , system (10)–(12)

5

takes the form θ˙1 = θ2 , θ˙2 = −a1 θ2 − a2 sin(θ1 ) + a3 cos(φ1 )u1 + w1 φ˙ 1 = φ2 , φ˙ 2 = −b1 φ2 − b2 sin(φ1 ) + b3 u2 + w2 ψ˙ 1 = ψ2 , ψ˙ 2 = −c1 ψ2 − c2 sin(φ1 ) + w3

(13) (14) (15)

where w1 (t) = Je−1 we (t), w2 (t) = Jd−1 wd (t), and w3 (t) = Jt−1 wt (t). Our objective is to asymptotically neglect the position errors θ˜1 = θ1 − θd ,

φ˜1 = φ1 − φd ,

ψ˜1 = ψ1 − ψd ,

with respect to the desired elevation θd , the direction φd , and the rotation ψd , all being constant positions, while also reducing the effect of external disturbances. The position measurements θ1 (t), φ1 (t), ψ1 (t) are assumed to be the only available information on the system. 3. State Feedback Design 3.1. Second Order Sliding Mode Control for the Pitch Axis The pitch dynamics can be decoupled from the other variables using the control law below. In order to globally asymptotically stabilize subsystem (13) the quasihomogeneous control law u1 =

1 [a2 sin(θ1 ) − h θ˜1 − p θ2 − α sign(θ˜1 ) − β sign(θ2 )], a3 cos(φ1 )

(16)

inherited from [13], is chosen provided that both the pitch angle θ1 and the angular velocity θ2 are available for measurements. With this control law, the closed-loop system (13), (16) appears to be globally finite-time stable if the parameter gains are such that h, p ≥ 0, α − M1 > β > M1

(17)

for some M1 > 0. Moreover, this stability remains in force, regardless of whichever external disturbance w1 affects the system provided that its magnitude sup |w1 (t)| ≤ M1 t

(18)

is bounded by the same constant M1 . Theorem 2. Let the disturbance-corrupted system (13) be driven by controller (16) subject to the parameter subordination (17) with some positive constant

6

M1 > 0. Then the closed-loop system (13), (16) is globally finite-time stable, provided that the external disturbance w1 (t) meets the upper bound (18). Proof. By substituting (16) into (13), the closed-loop system takes the form θ˜˙1 = θ2 , θ˙2 = −h θ˜1 − (p + a1 ) θ2 − α sign(θ˜1 ) − β sign(θ2 ) + w1 ,

(19)

and the validity of Theorem 2 is established by straightforwardly applying Theorem 1 to system (19). 3.2. Second Order Sliding Mode Control for the Roll and Yaw Axes Our next goal is to asymptotically stabilize the underactuated subsystem (14), (15), while also reducing the effect of the external disturbances w2 , w3 . Because of loss of the controllability of the helicopter body at φ1 = ± π2 , the roll dynamics are prohibited to approach these singular points which is why only the local stabilization of the directional 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 reduced 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 output s(φ1 , ψ1 , ψ2 ) = c2 sin(φ1 ) − k1 ψ1 − k2 ψ2

(20)

with some positive parameters k1 , k2 is chosen to impose the desired properties on the zero dynamics. Apparently, the yaw variable, while being restricted to the manifold s = 0, is governed by the internally asymptotically stable system ψ˙ 1 = ψ2 ,

ψ˙ 2 = −k1 ψ1 − k3 ψ2 + w3

(21)

whereas the influence of the disturbance w3 is reduced by a proper choice of the gains k1 and k3 = c1 + k2 (see, e.g., [3] for details). In turn, while evolving within the admissible domain φ1 ∈ (− π2 , π2 ), the roll variable meets the same properties on the zero dynamics manifold where c2 sin(φ1 ) = k1 ψ1 + k2 ψ2 .

(22)

To ensure attaining manifold (22) in finite-time the following control input u2 =

1 ˙ φ, ˙ ψ) ˙ − α1 sign(s) − β1 sign(s) [Γ(θ, φ, ψ, θ, ˙ − h1 s − p1 s] ˙ (23) b3 c2 cos(φ1 ) 7

is proposed where ˙ φ, ˙ ψ) ˙ = Γ(θ, φ, ψ, θ,

(24) c2 cos(φ1 )[b1 φ2 + b2 sin(φ1 )] + c2 sin(φ1 )φ22 +(k1 − k2 c1 )[−c1 ψ2 − c2 sin(φ1 )] − k2 c2 cos(φ1 )φ2

with positive parameters α1 , β1 , h1 , p1 . The idea behind the proposed synthesis is to bring the projection of the closed-loop system (14), (15), (23) onto the subspace, spanned by the output s, into the so-called quasihomogeneous form ˙ + h1 s + p1 s] ˙ +w s¨ = −[α1 sign(s) + β1 sign(s)

(25)

where the notation w = c2 cos(φ1 )w2 − (k2 c1 − k1 )w3 − k2 w˙ 3 has been used and the external disturbance w3 (t), affecting the travel dynamics, has been assumed to be differentiable. Provided that the magnitude of the term w is admitted to be upper bounded supt |w(t)| ≤ M by some constant M > 0 such that the controller gains are subordinated as follows h1 , p1 ≥ 0, α1 − M > β1 > M,

(26)

applying Theorem 1 to equation (25) ensures its global finite-time stability for any admissible perturbation. 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 t

(27)

by some positive constants M2 , M3 , M4 , the following result is obtained. Theorem 3. Consider the closed-loop system (14), (15), (23) with the parameter subordination (26) and with the above assumption (27) on the external disturbances. Let condition (26) holds with M = c2 cos(φ1 )M2 +(k2 c1 −k1 )M3 −k2 M4 . Then the closed-loop system is driven to the zero dynamics manifold (22) in finite-time, and after that, it is governed by the sliding mode equation (21). Moreover, the internal dynamics of the closed-loop system (14), (15), (23) are locally asymptotically stable. Proof. Since the projection of (14), (15), (23) onto the subspace, spanned by the output s, is locally described by equation (25) and according to Theorem 1, this equation proves to be finite-time stable under the conditions of the theorem, the closed-loop system, starting from a finite-time instant T > 0, evolves on the zero dynamics manifold (22), regardless of whichever admissible external disturbances affect the system. Thus, for t ≥ T , system (14), (15), (23) is governed by the internally asymptotically stable sliding mode equation (21). Coupled to the finite-time stability of the stage, preceding the sliding modes, 8

and due to the well-posedness of this stage, this yields the internal asymptotic stability of the closed-loop system. Remark 1. Since the parameters k1 and k2 , which determine the zero dynamics manifold (22), can be assigned arbitrarily large, the influence of the external disturbances on the sliding modes (21) is reduced by a proper choice of the parameters to as a low level as desired. 4. Velocity Observer Design In practice, position measurements for each degree of freedom θ1 (t), φ1 (t), and ψ1 (t), are the only available information on the system. First order sliding mode observers to be developed are in order to meet implementation requirements. For later use, we introduce the following. Definition 2. Let a second order dynamic system η¨ = f (η, η) ˙

(28)

and a dynamic system of the form ηˆ˙ 1 = f1 (η, ηˆ1 , ηˆ2 ), ηˆ˙ 2 = f2 (η, ηˆ1 , ηˆ2 )

(29)

with scalar state variables η(t) and η(t), ˙ and respectively, ηˆ1 (t) and ηˆ2 (t) posses Filippov solutions for arbitrary initial conditions. System (29) is said to constitute an asymptotic velocity observer of system (28) with an attraction domain D ⊂ R2 of initial observation errors and of precision > 0 iff ˙ ) − ηˆ2 (τ )| ≤ lim sup |η(τ ) − ηˆ1 (τ )| ≤ , lim sup |η(τ

t→∞ τ ≥t

t→∞ τ ≥t

(30)

for all Filippov solutions (η, η) ˙ and (ˆ η1 , ηˆ2 ) of (28) and (29) such that the initial observation errors (η(0) − ηˆ1 (0), η(0) ˙ − ηˆ2 (0)) are in D. In this work, the velocity estimation is carried out by a family of velocity observers for the 3-DOF underactuated helicopter (13), (14), (15) defined by ˙ θˆ1 ˙ θˆ2 ˙ φˆ1 ˙ φˆ2 ˙ ψˆ1 ˙ ψˆ2

=

θˆ2 + μθ sign(θ1 − θˆ1 ),

=

−a1 θˆ2 − a2 sin(θ1 ) + a3 cos(φ1 )u1 + νθ sign(θ1 − θˆ1 )

=

φˆ2 + μφ sign(φ1 − φˆ1 ),

=

−b1 φˆ2 − b2 sin(φ1 ) + b3 u2 + νφ sign(φ1 − φˆ1 )

=

ψˆ2 + μψ sign(ψ1 − ψˆ1 ),

=

−c1 ψˆ2 − c2 sin(φ1 ) + νφ sign(ψ1 − ψˆ1 ) 9

(31)

(32)

(33)

where μθ , νθ , μφ , νφ , μψ , νψ > 0. The estimation errors θi − θˆi , φi − φˆi , ψi − ψˆi , i = 1, 2 are guaranteed to decay asymptotically to the segments Sθ

=

 (θ1 − θˆ1 , θ2 − θˆ2 ) ∈ R2 : μθ M 1 θ1 = θˆ1 , |θ2 − θˆ2 | ≤ νθ

Sφ

=

 (φ1 − φˆ1 , φ2 − φˆ2 ) ∈ R2 :

μφ M 2 φ1 = φˆ1 , |φ2 − φˆ2 | ≤ νφ

Sψ

=

(34)





 (ψ1 − ψˆ1 , ψ2 − ψˆ2 ) ∈ R2 :

μψ M 3 ψ1 = ψˆ1 , |ψ2 − ψˆ2 | ≤ νψ



with the attraction domains   μθ ν θ 2 ˆ ˆ ˆ (θ1 − θ1 , θ2 − θ2 ) ∈ R : |θ2 − θ2 | < Dθ = M1   μφ ν φ 2 ˆ ˆ ˆ Dφ = (φ1 − φ1 , φ2 − φ2 ) ∈ R : |φ2 − φ2 | ≤ M2   μψ ν ψ 2 ˆ ˆ ˆ (ψ1 − ψ1 , ψ2 − ψ2 ) ∈ R : |ψ2 − ψ2 | ≤ Dψ = M3

(35)

(36)

(37) (38) (39)

provided that νθ

> max{M1 , μθ M1 },

νφ νψ

> max{M2 , μφ M2 }, > max{M3 , μφ M3 }.

(40)

Theorem 4. Consider the variable structure pitch-roll-yaw velocity estimators (31), (32) and (33) of the pitch-roll-yaw dynamics (13), (14), (15), respectively. Let upper bounds M1 > 0, M2 > 0 and M3 > 0 of the magnitudes of the external disturbances w1 (t), w2 (t) and w3 (t), affecting (31), (32) and (33), respectively, be known a priori. Then, under condition (40), imposed on the estimator parameters μθ , νθ , μφ , νφ , μψ , νψ , the dynamic systems (31), (32) and (33) μ M 1 represent asymptotic pitch-roll-yaw velocity observers of precision μθνM , φνφ 2 , θ and

μψ M3 νψ ,

respectively, with (37), (38) and (39) being attraction domains of the initial estimation errors θi (0) − θˆi (0), φi (0) − φˆi (0), ψi (0) − ψˆi (0), i = 1, 2.

10

Proof : Let x1 = (θ1 , φ1 , ψ1 )T , x2 = (θ2 , φ2 , ψ2 )T , x ˆ1 = (θˆ1 , φˆ1 , ψˆ1 )T , and xˆ2 = T ˆ ˆ ˆ (θ2 , φ2 , ψ2 ) . Hence, the velocity observers are given by the following structure xˆ˙ 1 = x ˆ2 + Lμ Sign(x1 − x ˆ1 ) xˆ˙ 2 = f (x1 , x ˆ2 ) + Lν Sign(x1 − xˆ1 )

(41)

where f (x1 , xˆ2 ) ∈ R3 corresponds to the nominal part of the system, Lμ = diag{μθ , μφ , μψ } and Lν = diag{νθ , νφ , νψ } are diagonal positive definite gain matrices, and Sign(xi ) = [sign(θi ), sign(φi ), sign(ψi )]T (i = 1, 2). The dynamics of the observation errors e1 = x1 − x ˆ1 , e2 = x2 − x ˆ2 are described by the equations e˙ 1 = e2 − Lμ Sign(e1 ),

(42)

e˙ 2 = −Ae2 − Lν Sign(e1 ) + wx with discontinuous right-hand sides, A = diag{a1 , b1 , c1 } is a positive definite matrix, and wx = (w1 , w2 , w3 )T . Then, the condition eT1 e˙ 1 = eT1 (e2 − Lμ Sign(e1 )) ≤ −(λmin {Lμ } − e2 )e1  < 0

(43)

for sliding modes to exist holds for all e1 = 0 and |e2 | < λmin {Lμ }, provided that the vector norm and absolute value are defined according to (1) and (2), respectively. Here, λmin {Lμ } denotes the minimum eigenvalue of the matrix Lμ . Now, let us consider the Lyapunov candidate function 1 V (e1 , e2 ) = Lν |e1 | + eT2 e2 . 2

(44)

By computing the temporal derivative of this function along the trajectories of the disturbance-free system (42), we arrive at V˙ = Sign(e1 )T Lν (e2 − Lμ Sign(e1 )) + eT2 (−Ae2 − Lν Sign(e1 )) = −eT2 Ae2 − Lν Lμ < 0

(45)

everywhere but on the vertical axis e1 = 0 where the function V is not differentiable. Now, according to the equivalent control method [17], the equivalent value represents a solution of the algebraic equation e2 − Lμ Sign(e1 ) = 0 with respect to Sign(e1 ) and hence, Signeq (e1 ) = L−1 μ e2 . Then, 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 disturbancefree system (42) for sign e1 , we can conclude that inequality (45) guarantees that trajectories hit the sliding mode set Ix = {(e1 , e2 )] ∈ R6 : e1 = 0, |e2 | < 11

λmin {Lμ }} in finite time because otherwise they steer to the origin in finite time, and that sliding modes on this set Ix are governed by the asymptotically stable equation e˙ 2 = −(A + L−1 μ Lν )e2

(46)

which yields that the error dynamics (42) are internally globally asymptotically stable. In turn, for the perturbed dynamics (42), relation (45) is modified to V˙ = −Lμ Lν + eT2 wx ≤ −Lμ Lν + λmax {Mx }e2 ,

(47)

where Mx = diag{M1 , M2 , M3 } is a diagonal matrix whose elements are the upper bounds for the magnitude of each external disturbance. Thus, along with the internal dynamics, the trajectories of the perturbed system (42) hit the sliding mode set Ix in finite time Tx > 0, which certainly depends on the disturbance wx . The solutions of the perturbed sliding mode equation e˙ 2 = −(A + L−1 μ Lν )e2 + w

(48)

are given by e2 (t) = e

−(A+L−1 μ Lν )(t−Tx )

e2 (Tx ) +

t Tx

−1

e−(A+Lμ

Lν )(t−τ )

wx (τ )dτ,

and by taking into account the upper bound Mx on the magnitude of the admissible disturbances wx , these solutions turn out to approach segments (34)–(36) as t → ∞. Then, under condition (40) and with known upper bound vector Mi of the magnitude of the external disturbances wi (t), i = 1, 2, 3 affecting each degree of freedom (13), (14), (15), respectively, we can state that each velocity observer comply with lim sup|x1 − xˆ1 | ≤ 0,

t→∞

lim sup|x2 − x ˆ2 | ≤ λmin {L−1 μ Lν Mx }.

t→∞

(49)

In addition, we can establish that the observation error dynamic system (42) possesses Filippov solutions for arbitrary initial conditions with (37), (38), and (39), being the attraction domains such that the time derivative (47) of the Lyapunov function (44), computed along the trajectories initialized within these domains, remains negative definite. Moreover, for the perturbed case, whichever large initial estimation error is admitted, there exist sufficiently high gains μx , νx , such that these initial errors are in the attraction domain of the corresponding velocity estimator for each variable. Thus, by straightforwardly verifying Definition 2, the validity of Theorem 4 is established.  Remark 2. By Theorem 4, the lower the fraction 12

μφ M2 μψ M3 μθ M1 νθ , νφ , νψ

is ob-

tained by appropriately choosing the gains Lμ and Lν , the better velocity estimate is ensured; the perfect estimate such that limt→∞ |x2 − xˆ2 | = 0 are particularly μ μ obtained for each gain relationship as ( μνθθ , νφφ , νψψ ) → 0 ∈ R3 . 5. Output Feedback Synthesis The output feedback controllers u1 and u2 1 [a2 sin(θ1 ) − h θ˜1 − p θˆ2 − α sign(θ˜1 ) − β sign(θˆ2 )], (50) cos(φ1 ) 1 [Γ(θ1 , φ1 , ψ1 , θˆ2 , φˆ2 , ψˆ2 ) − α1 sign(ˆ u2 = s) − β1 sign(sˆ˙ ) − h1 sˆ − p1 sˆ˙ ] b3 c2 cos(φ1 ) (51)

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are obtained if the corresponding observer variables θˆ2 , φˆ2 , ψˆ2 are substituted into the state feedback law (16), (23) for the state variables θ2 , φ2 , ψ2 . In the above synthesis, sˆ = c2 sin(φ1 ) − k1 ψ1 − k2 ψˆ2 stands for s(φ1 , ψ1 , ψˆ2 ) and s is given by (20) as before. The stability analysis of the over-all closed-loop system (13)–(15), (31), (32), (33), driven by the output feedback controllers (50), (51) and operating under uncertainty conditions, is rather technical and combines the details of the proof of Theorems 1–3. Since such an analysis is straightforward but lengthy, its details are left to the interested reader and only experimental evidences are further provided to support the present development. In the sequel, performance issues and robustness features of the proposed output feedback synthesis are illustrated in an experimental study made for a laboratory prototype. 6. Experimental Results 6.1. Experimental Setup Performance issues of the sliding mode position feedback controller were tested experimentally. The parameters of the helicopter, drown from the Quanser 3-DOF helicopter manual [15], are given in Table 1. The controller was implemented using SIMULINK 2007 from MathWorks running on a personal computer with AMD A4-3400, 2.70 GHz, 2GB processor. The PCI Multifunction I/O board from Sensoray 626 was used for the real-time control system and it consists of four channels of 14-bits D/A outputs and six quadrature 24-bit encoders. The encoder resolution for yaw angle is 2048 counts/rev and 1024 counts/rev for pitch and roll encoders. The amplifier of the motor accepts a control input from the D/A converter in the range of ±10 V regardless DC motor operates in a range of ±12 V. For the perturbed case, the experiments were complemented using the PXI Express system from NATIONAL INSTRUMENTS which allow to illustrate the robustness of the closed-loop system against matched disturbances. The 13

Table 1: Parameter values of the experimental 3-dof helicopter

Notation Lb Lh Je Jd Jt Kf Kp fθ fφ fψ

Value 0.66 0.177 0.91 0.0364 0.91 0.5 0.686 0.40 0.013 0.457

Units m m kg · m2 kg · m2 kg · m2 N/V N N · s/m N · s/m N · s/m

PXI is both a high-performance and low-cost deployment PC-based platform for acquisition and real-time control equipped with a multifunction DAQ module which has a 16 analog inputs, 48 digital inputs/outputs, and four analog outputs running in a Intel core i7-820QM processor-based embedded controller, 1.73 GHz computer. LabVIEW software was used to create the real-time system for the hardware-in-the-loop testing. The sampling time for both implementations were set as 0.001 s. 6.2. Experiments In the undisturbed experiment, the initial conditions for the 3-DOF helicopter were set to θ1 (0) = φ1 (0) = 0, ψ1 (0) = 29 deg, whereas all the velocity initial conditions were set to θ2 (0) = φ2 (0) = ψ2 (0) = 0 deg/s. Likewise, the controller gains in (16) and (23) were set to α = 0.005, β = 0.004, h = 1.5, p = 3, α1 = 0.8, β1 = 0.7, h1 = 4, p1 = 3, and k1 = k2 = 1. The parameters of the nonlinear velocity observers (31), (32), and (33) were set to μθ = μφ = μψ = 0.5, νθ = νφ = νψ = 0.49. In addition, the initial conditions for observers were also ˆ ˆ ˆ ˆ˙ ˆ˙ ˆ˙ set to θ(0) = φ(0) = ψ(0) = 0 deg and θ(0) = φ(0) = ψ(0) = 0 deg/s. It should be pointed out that a risk of damage can occurs in the motors, power drivers, and in the mechanical structure of the helicopter due to the presence of chattering in the control input, for that reason, we choose the parameters of the controller ad-hoc to satisfy the conditions (17) and (26) but avoiding the frequency of resonance. Desired positions were set to θd = 5.8, φd = ψd = 0 deg in order to achieve helicopter stabilization. In the disturbed case, same initial and desired positions for the 3-dof helicopter prototype were established as control objectives. Furthermore, controllers and observers gains were set equal to undisturbed case. In addition, the applied matched disturbances were set to wi = sin(50t) + 0.2, i = 1, 2, 3. Experimental results for the 3-DOF helicopter, driven by the sliding mode position controllers (16) and (23), are depicted in Figs. 2 and 3 for the disturbance-free case and for the perturbed case, respectively. It is concluded, from 14

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these Figures, that the controllers asymptotically stabilize the disturbance-free dynamics of the over-all system, while also reducing the effect of the external disturbances that has been predicted theoretically. Integral squared errors (ISE) defined as I = (θ˜12 (t) + φ˜21 (t) + ψ˜12 (t))dt, (52) are shown in Fig. 4 in order to provide a measure of the experimental controlled system performance. This Figure also shows that performance of the closed-loop system are preserved in spite of the variation of the sampling time. Furthermore, some important remarks can be brought to attention. First, the controller is robust. By comparing results from Figs. 2 and 3, notice how control objective is achieved regardless simplicity of the model used for statefeedback design and having decoupled it into two independent subsystems. Secondly, as shown in Fig. 3, the controller is robust against matched disturbances with velocity observers initialized with different initial conditions with respect to the 3-DOF dynamical system. 7. Conclusions An asymptotic position feedback stabilization problem is studied for an underactuated 3-DOF helicopter prototype, operating under uncertainty conditions. A general framework of resolving such a problem is proposed. The 15

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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 experimental verification, made for a Quanser’s 3-DOF helicopter with two actuators, demonstrates the effectiveness of the developed approach.

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Acknowledgments Yury Orlov and Luis T. Aguilar gratefully acknowledge the financial support from CONACYT (Consejo Nacional de Ciencia y Tecnolog´ıa) under Grants 165958 and 127575. [1] Avila-Vilchis, J., Brogliato, B., Dzul, A., Lozano, R., 2003. Nonlinear modelling and control of helicopters. Automatica 39, 1526–1530. [2] Davila, J., Fridman, L., Levant, A., 2005. Second-order sliding-mode observer for mechanical systems. IEEE Transactions on Automatic Control 50 (11), 1785–1789. [3] Doyle, J., Glover, K., Khargonekar, P., Francis, B., 1989. State space solution to standard H2 and H∞ control problems. IEEE Transactions on Automatic Control 34 (8), 831–846. [4] Ferreira, A., Rios, H., Rosales, A., 2012. Robust regulation for a 3-DOF helicopter via sliding-mode observation and identification. Journal of Franklin Institute 349, 700–718. [5] Filippov, A., 1988. Differential Equations with Discontinuous Righthand Sides. Mathematics and Its Applications. Kluwer Academic Publisher, Netherlands. [6] Floquet, T., Barbot, J., 2007. Super twisting algorithm-based step-by-step sliding mode observers for nonlinear systems with unknown inputs. International Journal of Systems Science 38 (10), 803–815. [7] Freidovich, L., Khalil, H., Nov. 2008. Performance recovery of feedbacklinearization-based designs. IEEE Transactions on Automatic Control 53 (10), 2324–2334. [8] Garc´ıa-Sanz, M., Elso, J., Ega˜ na, I., April 2006. Control de a´ngulo de cabeceo de un helic´optero como benchmark de dise˜ no de controladores. Revista Iberoamericana de Autom´atica e Inform´atica Industrial 3 (2), 111– 116. [9] Ishutkina, M., June 2004. Design and implementation of a supervisory safety controller for a 3DOF helicopter. Ph.D. thesis, Massachusetts Institute of Technology. [10] Isidori, A., Marconi, L., Serrani, A., March 2003. Robust nonlinear motion control of a helicopter. IEEE Trans. Autom. Control 48 (3), 413–426. [11] Lopez, R., Galvao, R., Milhan, A., Becerra, V., Yoneyama, T., Oct. 2006. Modelling and constrained predictive control of a 3DOF helicopter. In: XVI Congreso Brasileiro de Automatica. Salvador, Bahia, Brazil, pp. 429–434.

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[12] Oh, S., Khalil, H., Oct. 1997. Nonlinear output-feedback tracking using high-gain observer and variable structure control. Automatica 33 (10), 1845–1856. [13] Orlov, Y., 2009. Discontinuous systems –Lyapunov analysis and robust synthesis under uncertainty conditions. Springer-Verlag, London. [14] Oza, H., Orlov, Y., Spurgeon, S., 2012. Lyapunov-based settling time estimate and tuning for twisting controller. IMA Journal of Mathematical Control and Information 29, 471–490. [15] Quanser, 1998. 3D helicopter system with active disturbance. [available] http://www.quanser.com/choice.asp. [16] Spurgeon, S., Edwards, C., Foster, N., 1996. Robust model reference control using sliding mode controller/observer scheme with application to a helicopter problem. In: Proc. of the IEEE International Workshop on Variable Structure Systems. pp. 36–41. [17] Utkin, V., 1992. Sliding Modes in Control Optimization. Springer-Verlag, Berlin.

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