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Real-time attitude control of 3 DOF quadrotor UAV using modified Super Twisting algorithm
Real-Time Attitude Control of 3 DOF Quadrotor UAV using Modified Super Twisting Algorithm
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Real-Time Attitude Control of 3 DOF Quadrotor UAV using Modified Super Twisting Algorithm Mouad Kahouadji, M Rida Mokhtari, Amal Choukchou-Braham, Brahim Cherki PII: DOI: Reference:
S0016-0032(19)30827-0 https://doi.org/10.1016/j.jfranklin.2019.11.038 FI 4273
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
17 October 2018 25 March 2019 13 November 2019
Please cite this article as: Mouad Kahouadji, M Rida Mokhtari, Amal Choukchou-Braham, Brahim Cherki, Real-Time Attitude Control of 3 DOF Quadrotor UAV using Modified Super Twisting Algorithm, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.038
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journal of franklin Institute journal of franklin Institute 00 (2019) 1–13
Real-Time Attitude Control of 3 DOF Quadrotor UAV using Modified Super Twisting Algorithm Mouad Kahouadji1 , M Rida Mokhtari1,2 , Amal Choukchou-Braham1 and Brahim Cherki1 1
2
Laboratoire d’Automatique de Tlemcen (LAT). Université de Tlemcen, Algeria. École supérieure en sciences appliquées de Tlemcen, Algeria.
Abstract A new scheme for the attitude control of the quadrotor in presence of uncertainties and disturbances is presented. The proposed algorithm is a modified Super Twisting control -MSTW-. The aim of this technique is to solve the main disadvantage of the classical Super Twisting -STW- which is chattering. Moreover, the proposed scheme allows reduced control effort and accurate tracking. The elimination of the chattering phenomena is due to the continuous outputs generated by the MSTW. The finite time convergence and stability analysis of the closed loop system are derived using Lyapunov function techniques. The experimental validation is done to emphasize the good performances of the new scheme in terms of accuracy, robustness, finite-time convergence and chattering elimination with less control effort. Keywords: Modified Super Twisting, robust control, Lyapunov stability, attitude control, Quanser quadrotor.
1. Introduction The design of small flying machines capable of performing quasi-stationary as well as forward flight has now become an important research area. Many researchers are attracted by these vehicles, mainly due to the wide range of military and civil applications like searching, aerial surveillance, information collection and aggressive maneuvers in confined and obstructed environments. Rotorcraft UAVs provide tremendous advantages such as: capability of vertical takeoff and landing -VTOL-, stationary and omnidirectional flight and high maneuverability. There exist various designs of small rotorcraft according to the number of rotors and their mechanisms such as the classical helicopter [1], coaxial helicopter [2], twin rotor [3] and quadrotor [4, 5, 6]. Quadrotor UAV is an interesting alternative to the classical helicopter. It is mechanically simpler than a helicopter since it has propellers with constant pitch and does not require a swash plate. Its maintenance is therefore simpler than that of a classical helicopter. In literature, various approaches have been undertaken to ensure position control of a small rotorcraft UAV. Most of the linear/nonlinear controllers present a trade-off between flight performance and control law complexity. Linear controllers are designed for near-hovering flight based on model linearization around a set of preselected equilibrium conditions or trim points. Among these: PID controller [1] and LQR controller [7]. The main drawback of the linear approach is the trade-off between control performance and the number of required trim points or preselected equilibrium conditions. Nevertheless, due to their complex dynamics, nonlinearities, and high degree of coupling between ∗ Corresponding author. E-mail addresses: [email protected] (M. Kahouadji) [email protected] (M.R. Mokhtari) [email protected] (A. Choukchou-Braham) [email protected] (B. Cherki)
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control inputs and state variables, the development of nonlinear controllers has gained importance. Moreover, due to uncertainties and external disturbances, robust controllers are necessary to ensure positioning with high precision. In this respect, sliding mode controllers are used for small rotorcraft UAV [4, 5]. The design of reliable autonomous attitude controller flight is an important issue since it allows navigation of the vehicle in confined environments, and hence, prevents the vehicle from flipping over and crashing when the pilot performs the desired maneuvers [8, 9]. The attitude control problem can usually be classified into tracking control problem and stabilization problem. In practice, two kinds of uncertainties are widely studied in attitude system; one arises from the effect of wind gusts and gyroscopic moment generated by the rotation of the airframe and the rotors and the other represents model uncertainties existing in inertia matrix of the body that always reduce the control effect and make the attitude control problem more difficult and complicated [10, 11]. Super Twisting control algorithm -STW- [7, 12, 13] is one of the most powerful second order sliding mode control algorithms for systems with a relative degree equal to one. It generates control function that drives the sliding variable and its derivative to zero in finite time in presence of smooth bounded matched disturbances/uncertainties. The main advantages of STW are its attractive characteristics: finite-time convergence, robustness to uncertainties and simplicity of implementation. However, STW algorithm contains a discontinuous function smoothed by integration which reduces significantly the chattering phenomenon but does not eliminate it completely. The main contribution of this paper is to propose a robust modified Super Twisting sliding mode control -MSTW- to solve the problem of attitude tracking of quadrotor subject to disturbances and uncertainties. Classical sliding mode controllers -SMC- and second order sliding mode -SOSM- controllers, including the STW control algorithm, can robustly handle such problem of perturbations. However, the main disadvantage of the classical SMC is the introduction undesired chattering in the control signal, while SOSM controllers are able to attenuate it. In this work we are looking for continuous modified STW algorithm that is able to address this problem via generating continuous control signal and eliminating chattering. The proposed control scheme can deal with the so-called chattering phenomenon, and hence, can prevent actuators from damage. The MSTW is designed by extending the STW in [14] and the theory of finite-time-convergence in [11, 15, 16]. The discontinuous control function is replaced by a continuous one in order to generate a continuous control signal, to eliminate chattering effect, to perform accurate tracking and achieve robustness/sensitivity. A comparative study between STW and MSTW is undertaken. The experimental results show the good performance of the proposed algorithm compared to the classical Super Twisting. This paper is organized as follows. In section 2, the experiment set up and the attitude dynamical model of the quadrotor UAV are described. In section 3, original modified Super Twisting MSTW control law is designed and the stability of the closed-loop system is analyzed. In section 4, experimental results of the proposed control scheme implemented on the 3 DOF quadrotor are given. Finally, some conclusions are presented. 2. System Description The 3 DOF Hover Quanser platform used in this experience is depicted in Fig.1. The quadrotor is considered as a solid body incorporating force and moment generation process. Let I = (Xe , Ye , Ze ) be the inertial frame and B = (Xb , Yb , Zb ) the body-fixed frame attached to the center of gravity of the plateform. The experimental setup consists of a planar frame with four propellers driven by four DC motors where opposite motors rotate in the same direction. The frame is mounted on a 3 DOF pivot joint so that it can rotate about the attitude angles. The attitude is represented by Euler angles Θ = [φ, θ, ψ]T corresponding to the aeronautical convention. The attitude angles are respectively called Roll angle (φ rotation around x-axis), Pitch angle (θ rotation around y-axis), and Yaw angle ˙ = [φ, ˙ ψ] ˙ θ, ˙ T and (ψ rotation around z-axis). The angular velocities and accelerations are given respectively by Θ T ¨ ¨ ¨ ¨ Θ = [φ, θ, ψ] . The thrust forces generated by the back, right, front and left motors are denoted by f1 , f2 , f3 , and f4 , respectively. The thrust forces generated by the front and back motors primarily control the motions about the pitch axis while the right and left motors primarily move the hover about its roll axis. The motion about the yaw axis is caused by the difference in torques exerted by the two clockwise and two counter-clockwise rotating propellers. The Quadrotor attitude is controlled by the three control torque Γ = [τφ , τθ , τψ ]T combined to the forces generated by the four propellers f1 , f2 , f3 and f4 . The main thrust T is used to compensate the gravity force and eventually the 2
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vertical movement for a 6 DOF quadrotor.
Figure 1. Quanser quadrotor platform
2.1. Dynamics Using the Euler-Newton formalism and introducing some assumptions, enumerated below, we obtain a simplified attitude dynamic model of the quadrotor, this simplified model will be used in the controller design. ˙ can be measured or estimated by on-board sensors. A 2.1 The signals Θ and Θ A 2.2 The roll, pitch and yaw angles are limited to (−π/2 < φ < π/2), (−π/2 < θ < π/2) and (−π < ψ < π). It means that the acrobatic behavior is not allowed. A 2.3 Joint friction, air resistance and centrifugal forces are neglected. A 2.4 The thrust force is proportional to the motor voltage, and motors/propellers dynamics are neglected. The model described before in [1, 5], can be written as follow : h i ¨ = [M(Θ)]−1 ΨT (Θ)Γ − C(Θ, Θ) ˙ Θ ˙ + Γext Θ
(1)
where M(Θ) = ΨT (Θ)JΨ(Θ) ∈ R3×3 is an auxiliary positive inertia matrix provided that (θ , kπ/2), J ∈ R3×3 is the total inertia matrix, Γ ∈ R3 and Γext ∈ R3 denote the control torque and external disturbances respectively. ˙ ∈ R3×3 is given by2 : C(Θ, Θ) ˙ = −M(Θ)Ψ ˙ −1 (Θ)Ψ(Θ) − ΨT (Θ)sk[JΨ(Θ)Θ]Ψ(Θ) ˙ C(Θ, Θ)
:
The Euler matrix Ψ(Θ) ∈ R3×3 and the cross-product operator sk[β] for a vector β = [β1 β2 β3 ]T ∈ R3 are given by3
2 Ψ(Θ) ˙
3 The
1 Ψ(Θ) = 0 0
0 −sθ cφ sφcθ −sφ cφcθ
;
0 sk[β] = β3 −β2
˙ −1 (Θ)Ψ(Θ) = −Ψ(Θ)Ψ abbreviations s(.) and c(.) denote sin(.) and cos(.), respectively.
3
−β3 0 β1
β2 −β1 0
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A quadrotor helicopter has four rotors located at the extremities of a cross shaped frame as depicted in Fig.1. By considering the symmetry of the quadrotor system [1, 5], the inertia matrix J can be approximated by J = diag[J xx , Jyy , Jzz ], then the matrix M(Θ) and the control inputs (T, Γ) are given by : M(Θ) =
J xx 0 −J xx sθ
0 Jyy s2 φ + Jzz c2 φ (Jyy − Jzz )cθcφsφ
−J xx sθ (Jyy − Jzz )cθcφsφ J xx s2 θ + c2 θ(Jyy s2 φ + Jzz c2 φ)
,
T K f τ 0 φ = τθ −lK f τψ Kt
Kf −lK f 0 Kt
Kf 0 lK f Kt
Kf lK f 0 Kt
V1 V 2 V2 V2
(2)
l is the distance from the rotors to the center of mass of the platform, V j is the voltage input of rotor ” j” and (K f , Kt ) are respectively the thrust and the drag factors which are positive constants characterizing the propellers aerodynamics. The parameter values used in the experimental set up are given in table 1.
Table 1. Quanser parameters [23]
Parameter m l Kf Kt J xx Jyy Jzz
Description Mass Distance between Pivot to each Motor Thrust factor Drag factor Roll inertia φ Pitch inertia θ Yaw inertia ψ
Value 2.85 0.1969 2.98e-6 1.14e-7 0.0552 0.0552 0.1104
Unit Kg m N/V kgm2 kgm2 kgm2
Let us use (1) and define the following attitude state variables : ˙ x2 = Θ
x1 = Θ, Then the state-space form of this model is given by : ( x˙1 x˙2
= =
x2 f (x, t) + g(x, t)u
The control input is u = Γ, the vector f (x, t) and the matrix g(x, t) are given by h i ˙ Θ ˙ + Γext ; g(x, t) = [JΨ(Θ)]−1 f (x, t) = [M(Θ)]−1 −C(Θ, Θ)
(3)
(4)
3. Control Design In system (3), the vector-field f (x, t) is partially known, the perturbation terms Γext are unknown and assumed bounded. The problem is then to design a continuous modified Super Twisting sliding mode controller Γ that drives the attitude positions defined by {φ(t), θ(t), ψ(t)} to follow the bounded desired trajectories {φd (t), θd (t), ψd (t)} in presence of bounded external perturbations. The following lemma and definitions are used for the stability analysis. Lemma 3.1. [10, 16] Consider the continuous system x˙ = f (x), f (0) = 0, x ∈ Rn
(5)
Suppose there exist a continuous positive definite function V : Rn → R, a real numbers γ > 0 , α ∈ (0, 1) and an open neighborhood U0 ⊆ Rn of the origin such that the following inequality is satisfied : ˙ V(x) + γV α (x) ≤ 0, x ∈ U0 /{0}.
(6)
Then, the origin of the system (5) is a finite time stable equilibrium. If U0 = Rn , then the origin is a globally finite time stable equilibrium. The settling time tR satisfies : 1 V 1−α (0) (7) tR ≤ γ(1 − α)
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Notation 1. we define sigσ (x) = sign(x)|x|σ , where σ > 0, x ∈ R and sign(.) is the standard signum function. If x =
[x1 , x2 , ..., xn ]T is a vector, then
sigσ (x) = [sign(x1 )|x1 |σ , sign(x2 )|x2 |σ , ..., sign(xn )|xn |σ ]T
Notation 2. k ? k denotes the Euclidean norm of (?), λmin (?), λmax (?) are respectively the minimum and maximum eigenvalues of matrix (?). 3.1. Modified Super Twisting Algorithm (MSTW) The aim of the control law is to get a good accuracy in term of position tracking for the desired trajectory in spite of parametric uncertainties and disturbances. In order to control the attitude of the quadrotor, a proposed modified Super Twisting algorithm is applied. This controller is able to generate continuous control signal that eliminates chattering and reduces control effort. Consider the system (3) and assume that : A 3.1 A sliding variable s = s(x, t) ∈ R is designed so that the desired compensated dynamics of the system (3) are achieved in the sliding mode s = s(x, t) = 0. A 3.2 The relative degree of system (3) with the sliding variable s = s(x, t) as output equals one, and the internal dynamics are stable. Therefore, the dynamics of the sliding variable s are : s˙
=
∂s ∂s ∂s + f (x, t) + g(x, t) u ∂t ∂x ∂x | {z } | {z } b(x,t)
ψ(x,t)
=
(8)
ψ(x, t) + b(x, t)u
A 3.3 The function b(x, t) ∈ R is known and is assumed invertible. A 3.4 The function ψ(x, t) ∈ R is uncertain and can be written as
ψ(x, t) = ψnom (x, t) + ∆ψ(x, t)
(9)
where ψnom (x, t) is the nominal part of ψ(x, t) which is known, and it exists hmax positive constant such that the derivative function of ∆ψ(x, t) is bounded ˙ t)| ≤ hmax |∆ψ(x, (10)
The problem is to drive the sliding variable s and its derivative s˙ to zero in finite time in presence of bounded disturbances/uncertainties. We define the sliding surface as follows : s = δ˙ + λδ (11) 3×3 ˙ where δ = x1 − x1d , δ = x2 − x˙1d and λ ∈ R is a diagonal positive definite matrix. As system (3)-(11) admits a relative degree equal to 1 for s with respect to u, one gets : s˙ = ψnom (x, t) + ∆ψ(x, t) + b(x, t)u
(12)
˙ Θ ˙ − x¨1d + λδ, ˙ ∆ψ(x, t) = [M(Θ)]−1 Γext and b(x, t) = [JΨ(Θ)]−1 . The proposed controller for where ψnom (x, t) = −[M(Θ)]−1 C(Θ, Θ) the system is given by : # " Z t sigσ2 (s(τ))dτ (13) Γ = [JΨ(Θ)] −ψnom (x, t) − k1 |s|σ1 sign(s) − k2 0
where (1/2 < σ1 < 1) and (σ2 = 2σ1 − 1). k1 = [k1φ k1θ k1ψ ] and k2 = [k2φ k2θ k2ψ ] are the positive control gains given by : k2i >
h2max kqk2 , k1i > 0, i = {φ, θ, ψ} 2 σ1 k1i
The gain k2 allows to tune up the dynamics of adaptation of %(x, t), defined bellow. According to (13), (12) becomes Z
(14)
t
s˙ = −k1 |s|σ1 sign(s) − k2
sigσ2 (s(τ))dτ + ∆ψ(x, t)
0
5
(15)
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then, (15) can be written for each variable (i = {φ, θ, ψ}) as : s˙i µ˙¯ i
= =
−k1i |si |σ1 sign(si ) + µ¯ i −k2i |si |σ2 sign(si ) + %i (x, t)
(16)
with µ¯ i
=
%i (x, t)
=
−k2i
Z
0
t
sigσ2 (si (τ))dτ + ∆ψi (x, t)
˙ (x, t) ∆ψ i
The proof of finite time convergence to a small region around the origin of the variables si and µ¯ i is explicitly given in the sequel Proof 3.1. In order to analyze the stability of (15), the following state vector is introduced : z = [z1 z2 ]T = s µ¯ T
(17)
So the system (16) can be rewritten as :
z˙1i
=
z˙2i
=
−k1i |z1 |σ1 sign(z1 ) + z2
−k2i |z1 |σ2 sign(z1 ) + %(x, t)
(18)
where %(t) is assumed bounded by |%(x, t)| ≤ hmax , with hmax > 0. It can be observed that : • if z1 , z2 → 0 in finite time, then s, s˙ → 0 in finite time.
In this step of the proof, the stability analysis of system (16) is inspired from Moreno [22], therefore the following Lyapunov function candidate is introduced: k2 2σ1 1 2 1 (19) |z1 | + z2 + (k1 |z1 |σ1 sign(z1 ) − z2 )2 V(z) = σ1 2 2 In order to make the proof more readable, we considered only one dimensional case, which can represent either φ, θ or ψ. This allows dropping the double index from gains. We can put V in this form : k2 k2 V(z) = ( + 1 )|z1 |2σ1 + z22 − k1 z2 |z1 |σ1 sign(z1 ) (20) σ1 2 finally V is written as follow : V(z) = ζ T Pζ (21) The vector ζ and the symmetric positive definite matrix P ∈ R2×2 are selected respectively as : ! ! 2 1 2k + k12 −k1 |z1 |σ1 sign(z1 ) σ1 ζ= , P= z2 2 −k1 2 Note that the function (21) is continuous and differentiable everywhere except on the set z = {(z1 , z2 ) ∈ R2 |z1 = 0}, it is positive definite and radially unbounded if k2 > 0, hence, it follows that V(z) satisfies :
where
λmin (P)kζk2 ≤ V(z) ≤ λmax (P)kζk2
(22)
kζk2 = |z1 |2σ1 + z22
(23)
The derivative of the Lyapunov function candidate (21) is : ˙ V(z)
=
˙ V(z)
=
k2 k2 + 1 )|z1 |2σ1 −1 sign(z1 )˙z1 + 2z2 z˙2 − k1 |z1 |σ1 sign(z1 )˙z2 − σ1 k1 z2 |z1 |σ1 −1 z˙1 σ1 2 3σ1 −1 −2k1 k2 |z1 | + 2k2 z2 |z1 |2σ1 −1 sing(z1 ) − σ1 k13 |z1 |3σ1 −1 + σ1 k12 z2 |z1 |2σ1 −1 sign(z1 ) − 2k2 z2 |z1 |σ2 sign(z1 ) + 2z2 %(x, t)
2σ1 (
+k1 k2 |z1 |σ1 +σ2 − k1 |z1 |σ1 sign(z1 )%(x, t) + σ1 k12 |z1 |2σ1 −1 z2 sign(z1 ) − σ1 k1 z22 |z1 |σ1 −1
To simplify the Lyapunov function derivative, we impose that σ2 = 2σ1 − 1. The derivative becomes : ˙ V(z) = (−k1 k2 − σ1 k13 )|z1 |3σ1 −1 + 2σ1 k12 z2 |z1 |2σ1 −1 sign(z1 ) − σ1 k1 z22 |z1 |σ1 −1 + (−k1 |z1 |σ1 sing(z1 ) + 2z2 )%(x, t) We can rewrite (24) as follow :
˙ V(z) = −|z1 |σ1 −1 ζ T Qζ + %qT ζ
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(24) (25)
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Where Q = k1
k2 + σ1 k12 −σ1 k1
−σ1 k1 σ1
!
;
q=
−k1 2
!
Next, the derivative of the Lyapunov function candidate can be bounded by : ˙ V(z) ≤ −|z1 |σ1 −1 λmin (Q)kζk2 + hmax kqkkζk
(26)
From (23), we can get : From (26) and (27), we get :
Let
|z1 |σ1 −1 ≥ kζk
σ1 −1 σ1
(27)
σ +σ h kqk max kζk 1σ1 2 ˙ V(z) ≤ − λmin (Q) − σ2 kζk σ1 ν1 = λmin (Q) −
with kζk >
(28)
hmax kqk
(29)
! σσ1
(30)
σ2
kζk σ1
hmax kqk λmin (Q)
2
to ensure ν1 > 0. From (22) and (29), it can be easily shown that the expression (28) will be : ˙ V(z) ≤ −γ1 V α (z) (31) −σ λmin (P) 2σ12 σ1 + σ2 > 0, which show that V(z) is a strong Lyapunov , then α ∈]0, 1[, and the positive constant γ1 = ν1 √ with α = 2σ1 λmin (P) function. According to lemma 3.1, the finite time stability can be guaranteed, and hence, the error of the system (16) will reach in finite-time the region : ! σ1 hmax kqk σ2 (32) kζk ≤ ∆1 ; ∆1 = λmin (Q) ! hmax kqk < 1 , because ( σσ12 > 1), the exponential term of (32) Consequently, the region ∆1 can be guaranteed to be small enough if λmin (Q) will greatly reduce the size of ∆1 (i.e., ∆1 1), therefore, we can select k1 and k2 such that λmin (Q) > hmax kqk From (33), we get : k2 >
(33)
h2max kqk2 , k1 > 0 σ1 k12
Now we can easily estimate the finite reaching time. Proposition 3.1.1. The MSTW control law (16) drives the sliding variable s and its derivative to the domain (32) in finite time estimated as : 1 V 1−α (z(0)) tF = γ1 (1 − α) where V(z(0)) is the initial value of V(z). Assuming ∆1 = 0 implies s, s˙ → 0 in finite time tR that can be estimated by tR ≤ 3.1. For ∆1 > 0, s, s˙ → ∆1 in finite time tF ≤ tR .
1 V 1−α (z(0)) γ1 (1−α)
which is obtained by Lemma
Remark 1. We know that kζk is sufficiently small after the settling time tF (i.e., the region ∆1 is very small), then according to equation (16) and the finite-time convergence of z2 , it can be concluded that the disturbances ∆ψ(x, t) are canceled in finite-time Z t ∆ψ(x, t) ≈ k2 sigσ2 (s(τ))dτ ≈ 0, f or t > tF (34) 0
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4. Experimental Results In this section, we use the Quanser platform described in section II. Experimental results are given to show the effectiveness of the proposed algorithm. The validation is done using the platform whose components are : data-acquisition card Quanser Q8, power amplifier system UPM-2405, four DC motors Pitman and three encoders used for the attitude measurement. The proposed technique is implemented via MATLAB/Simulink through the Q8 card that uses real time control software QUARC-Simulink. Many tests have been carried out under disturbances and uncertainties in order to study a the performances of the proposed scheme. A comparison with the classical STW algorithm is done with the aim of illustrating the benefits of the developed technique. The controller’s gains are given in table 2 :
Table 2. Controller gains
k1 k2 λ σ1
Roll φ 6 4 1.5 0.8
Pitch θ 5 4.2 1.7 0.8
Yaw ψ 7 5 3.5 0.8
In order to get good results in terms of accuracy, these gains are chosen by respecting the condition (14) for both laws that is in the sequel the chosen gains are the same for the both controls STW R t and MSTW, the only difference is in σ1 which equal 0.8 for 1 the MSTW and 0.5 for STW ( for the STW u = k1 |s| 2 sign(s) − k2 0 sign(s(τ))dτ). hmax is chosen as hmax = [1.5 1.5 1.5], this choice is justified by the fact that we are considering quasi constant perturbations in these experiments.
4.1. Stabilization experiments This experiment is realized to stabilize the 3 DOF system for stationary flight from some random given initial angles defined as φ = 12◦ , θ = 10◦ and ψ = −22◦ for the MSTW and φ = 8◦ , θ = 15◦ and ψ = −21◦ for the STW. In Fig.2 the attitude responses are plotted, where the red plots are the desired angles, the blue ones are the quadrotor trajectories controlled by MSTW and the green ones are the quadrotor trajectories controlled by STW. In stabilization experiment, the desired trajectories are equal to zero that is, we force the attitude angles to converge to zero. We clearly see from Fig.2 that the stabilization is obtained with both controllers, however, for STW’s trajectories, there are oscillations due to the discontinuity of the control unlike in MSTW where the attitude trajectories are smooth. Furthermore, the input signals are displayed in Fig.3, the red plots are associated with the STW control and the blue ones with the MSTW. The figure shows that the control signals remain acceptable in both cases, although the MSTW signal controls are smaller than the STW ones.
Figure 2. Euler angle trajectories for stabilization case
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Figure 3. Input signals
4.2. Attitude Tracking experiments The following experiments deals with the tracking problem. The desired trajectories are sinusoids of 10, 8 and 12 degrees in magnitude for the roll (φ), pitch (θ) and yaw (ψ) angles respectively and of 25 seconds period. In Fig.4 one can see the Euler angles when the system is driven by the STW and MSTW control laws respectively, the desired trajectories are also plotted. We can see that the MSTW control leads to smooth trajectories in blue color when the STW control displays oscillatory trajectories in green color. It means that the chattering caused by the discontinuity of the STW control is eliminated by MSTW that generates a smooth continuous signal. Fig.5 represents error plots of Euler angles trajectories associated with STW and MSTW. The control signal effort is also reduced as shown in Fig.6.
Figure 4. Tracking performances: Desired and actual attitude angles
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Figure 5. Tracking trajectories errors for STW and MSTW
Figure 6. Voltage inputs for tracking case
For a quantitative evaluation of the error tracking experiments, we summarized in table 3, the mean quadratic errors for each control law :
roll pitch yaw
MSTW 0.2130 0.1180 0.5580
1 T
RT 0
e2 (t)dt,
STW 1.5191 2.0378 0.9615
From a quantitative point of view, table 3 confirms that errors associated with the MSTW controller are smaller than those of the STW one.
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4.3. Robustness and disturbance rejection The first disturbance we applied to the 3 DOF system controlled by the MSTW algorithm consists in disabling one of the 4 motors at instant 10s. From Fig. 7, we see that the system continues to perform satisfactorily, the errors are kept small and the effect of this perturbation is more important on the roll and pitch angles than on the yaw angle. The other kind of disturbance consists in adding a mass of 115 grams on the end of the frame in the x direction at instant 15 s. Fig. 8, shows the effectiveness of the control, the perturbation is rejected and the tracking of the trajectories is kept satisfactorily. Next, we applied a manual external forces in interval of 30s < t < 45s, we see that this new perturbation is also rejected. Hence, the efficiency and the robustness of the proposed control are proved in both tests. Fig. 9 and 10, are related to the same robustness and disturbance rejection tests but for the STW controller. We see here also that the same properties of rejection but with a persisting chattering.
Figure 7. Euler angle trajectories with MSTW for one motor disabled
Figure 8. Euler angle trajectories with MSTW for mass and manual disturbancies added
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Figure 9. Euler angle trajectories with STW for one motor disabled
Figure 10. Euler angle trajectories with STW for mass and manual disturbancies added
5. Conclusion The proposed MSTW algorithm solves the problem of chattering which is considered as the main drawback of the super twisting algorithm. The stability of the closed loop is formally proved and the convergence to the desired trajectories is achieved in finite time. The experimental tests show the effectiveness of the proposed control in both cases stabilization and tracking. Through the experiments we also verified the properties of robustness against the parameter uncertainties and external disturbances. As future research, we intend to study more formally the robustness properties of the proposed control law and also to compare the sizes of the control signals generated by the MSTW control versus the one generated by the STW control.
References References [1] P.E. Pounds, R. Bersak Daniel , M. Dollar Aaron, Stability of small-scale UAV helicopters and quadrotors with added payload mass under PID control. Auton Robot 2012; 33: 129-142. [2] A. Drouot , E. Richard ,M. Boutayeb, Hierarchical backstepping-based control of a Gun Launched MAV in crosswinds: Theory and experiment Control Engineering Practice 2004; 25 : 16-25 [3] O. Salas, H. Castaneda,JD. Morales, Attitude Observer-Based Robust Control For A Twin Rotor System. Kybernetika, 2013; 49(5): 809-828.
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[4] M R. Mokhtari, B. Cherki, Robust Control for Attitude Tracking Problem for a Quadrotor Unmanned Aerial Vehicle. Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, October 29-31, 2013. [5] M R. Mokhtari, B. Cherki, A new robust control for minirotorcraft unmanned aerial vehicles. ISA Trans 2015; 56: 86-101. [6] F. Kendoul, Z. Yu, K. Nonami, Guidance and Nonlinear Control System for Autonomous Flight of Minirotorcraft Unmanned Aerial Vehicles. Journal of Field Robotics 2010; 27(3): 311-334. [7] S. Bouabdallah, P. Murrieri, R. Siegwart, PID vs. LQ control Techniques Applied to an Indoor Micro Quadrotor. In Proc IEEE international conference on intelligent Robots and systems, Japan, 2004. [8] Hu.Qinglei, Bo.Li, Qi.Juntong, Disturbance observer based finite-time attitude control for rigid spacecraft under input saturation. Aerospace Science and Technology 2014; 39 : 13-21. [9] X. Huo, B. Xiao, Z. Zhang, Robust finite-time control for spacecraft attitude stabilization under actuator fault. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 2012; 226(3) : 416-428. [10] S. Zhankui, L. Hongxing, S. Kaibiao, Finite-time control for nonlinear spacecraft attitude based on terminal sliding mode technique, ISA Transactions 2014; 53: 117-124. [11] X. Wang, H. Lin, Design and frequency analysis of continuous finite-time-convergent differentiator. Aerospace Science and Technology 2012; 18 : 69-78. [12] A. Levant, Sliding order and sliding accuracy in sliding mode control. International journal of control, 58(6), 1247-1263,(1993). [13] J. Erdong , S. Zhaowei, Robust controllers design with finite time convergence for rigid spacecraft attitude tracking control. Aerospace Science and Technology 2008; 12(4) : 324-330. [14] Y. Shtessel, C. Edwards, L.Fridman, A.Levant, Sliding mode control and observation. New York: Birkhäuser 2014; 10 : 0-978 [15] W. Xinhua, L. Hai, Design and frequency analysis of continuous finite-time-convergent differentiator. Aerospace Science and Technology 2012; 18(1): 69-78. [16] S P. Bhat, D S. Bernstein, finite time stability of continuous autonomous systems. SIAM Journal on Control and Optimization 2000; 38(3) : 751-766. [17] Y. Shtessel, M. Taleb, F. Plestan, A novel adaptive-gain Super Twisting sliding mode controller: Methodology and application. Automatica 2012; 48: 759-769 [18] H. Xing, H. Qinglei, B. Xia, Finite-time fault tolerant attitude stabilization control for rigid spacecraft. ISA Transactions 2014; 53 : 241-250. [19] S. Wu, Z. Sun, H. Deng, Robust Sliding Mode Controller Design For Globally Fast Attitude Tracking Of Target Spacecraft. International Journal of Innovative Computing, Information and Control 2011; 7 : 2087-2098. [20] Q. Hu, X. Huo, B. Xiao, Z. Zhang, Robust finite-time control for spacecraft attitude stabilization under actuator fault. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 2012; 226(3) : 416-428. [21] Yu. Shuanghe, X. Yu, B. Shirinzade, M. Zhihong, Continuous finite time control for robotic manipulators with terminal sliding mode. Automatica 2005; 41(11) : 1957-1964. [22] J.A. Moreno, M. Osorio, Strict Lyapunov functions for the super-twisting algorithm. IEEE transactions on automatic control 57.4 (2012): 1035-1040. [23] 3-DOF Hover Reference Manual,Quanser Consulting Inc.
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Journal Pre-proof
Real-Time Attitude Control of 3 DOF Quadrotor UAV using Modified Super Twisting Algorithm Mouad Kahouadji, M Rida Mokhtari, Amal Choukchou-Braham, Brahim Cherki PII: DOI: Reference:
S0016-0032(19)30827-0 https://doi.org/10.1016/j.jfranklin.2019.11.038 FI 4273
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
17 October 2018 25 March 2019 13 November 2019
Please cite this article as: Mouad Kahouadji, M Rida Mokhtari, Amal Choukchou-Braham, Brahim Cherki, Real-Time Attitude Control of 3 DOF Quadrotor UAV using Modified Super Twisting Algorithm, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.038
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journal of franklin Institute journal of franklin Institute 00 (2019) 1–13
Real-Time Attitude Control of 3 DOF Quadrotor UAV using Modified Super Twisting Algorithm Mouad Kahouadji1 , M Rida Mokhtari1,2 , Amal Choukchou-Braham1 and Brahim Cherki1 1
2
Laboratoire d’Automatique de Tlemcen (LAT). Université de Tlemcen, Algeria. École supérieure en sciences appliquées de Tlemcen, Algeria.
Abstract A new scheme for the attitude control of the quadrotor in presence of uncertainties and disturbances is presented. The proposed algorithm is a modified Super Twisting control -MSTW-. The aim of this technique is to solve the main disadvantage of the classical Super Twisting -STW- which is chattering. Moreover, the proposed scheme allows reduced control effort and accurate tracking. The elimination of the chattering phenomena is due to the continuous outputs generated by the MSTW. The finite time convergence and stability analysis of the closed loop system are derived using Lyapunov function techniques. The experimental validation is done to emphasize the good performances of the new scheme in terms of accuracy, robustness, finite-time convergence and chattering elimination with less control effort. Keywords: Modified Super Twisting, robust control, Lyapunov stability, attitude control, Quanser quadrotor.
1. Introduction The design of small flying machines capable of performing quasi-stationary as well as forward flight has now become an important research area. Many researchers are attracted by these vehicles, mainly due to the wide range of military and civil applications like searching, aerial surveillance, information collection and aggressive maneuvers in confined and obstructed environments. Rotorcraft UAVs provide tremendous advantages such as: capability of vertical takeoff and landing -VTOL-, stationary and omnidirectional flight and high maneuverability. There exist various designs of small rotorcraft according to the number of rotors and their mechanisms such as the classical helicopter [1], coaxial helicopter [2], twin rotor [3] and quadrotor [4, 5, 6]. Quadrotor UAV is an interesting alternative to the classical helicopter. It is mechanically simpler than a helicopter since it has propellers with constant pitch and does not require a swash plate. Its maintenance is therefore simpler than that of a classical helicopter. In literature, various approaches have been undertaken to ensure position control of a small rotorcraft UAV. Most of the linear/nonlinear controllers present a trade-off between flight performance and control law complexity. Linear controllers are designed for near-hovering flight based on model linearization around a set of preselected equilibrium conditions or trim points. Among these: PID controller [1] and LQR controller [7]. The main drawback of the linear approach is the trade-off between control performance and the number of required trim points or preselected equilibrium conditions. Nevertheless, due to their complex dynamics, nonlinearities, and high degree of coupling between ∗ Corresponding author. E-mail addresses: [email protected] (M. Kahouadji) [email protected] (M.R. Mokhtari) [email protected] (A. Choukchou-Braham) [email protected] (B. Cherki)
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control inputs and state variables, the development of nonlinear controllers has gained importance. Moreover, due to uncertainties and external disturbances, robust controllers are necessary to ensure positioning with high precision. In this respect, sliding mode controllers are used for small rotorcraft UAV [4, 5]. The design of reliable autonomous attitude controller flight is an important issue since it allows navigation of the vehicle in confined environments, and hence, prevents the vehicle from flipping over and crashing when the pilot performs the desired maneuvers [8, 9]. The attitude control problem can usually be classified into tracking control problem and stabilization problem. In practice, two kinds of uncertainties are widely studied in attitude system; one arises from the effect of wind gusts and gyroscopic moment generated by the rotation of the airframe and the rotors and the other represents model uncertainties existing in inertia matrix of the body that always reduce the control effect and make the attitude control problem more difficult and complicated [10, 11]. Super Twisting control algorithm -STW- [7, 12, 13] is one of the most powerful second order sliding mode control algorithms for systems with a relative degree equal to one. It generates control function that drives the sliding variable and its derivative to zero in finite time in presence of smooth bounded matched disturbances/uncertainties. The main advantages of STW are its attractive characteristics: finite-time convergence, robustness to uncertainties and simplicity of implementation. However, STW algorithm contains a discontinuous function smoothed by integration which reduces significantly the chattering phenomenon but does not eliminate it completely. The main contribution of this paper is to propose a robust modified Super Twisting sliding mode control -MSTW- to solve the problem of attitude tracking of quadrotor subject to disturbances and uncertainties. Classical sliding mode controllers -SMC- and second order sliding mode -SOSM- controllers, including the STW control algorithm, can robustly handle such problem of perturbations. However, the main disadvantage of the classical SMC is the introduction undesired chattering in the control signal, while SOSM controllers are able to attenuate it. In this work we are looking for continuous modified STW algorithm that is able to address this problem via generating continuous control signal and eliminating chattering. The proposed control scheme can deal with the so-called chattering phenomenon, and hence, can prevent actuators from damage. The MSTW is designed by extending the STW in [14] and the theory of finite-time-convergence in [11, 15, 16]. The discontinuous control function is replaced by a continuous one in order to generate a continuous control signal, to eliminate chattering effect, to perform accurate tracking and achieve robustness/sensitivity. A comparative study between STW and MSTW is undertaken. The experimental results show the good performance of the proposed algorithm compared to the classical Super Twisting. This paper is organized as follows. In section 2, the experiment set up and the attitude dynamical model of the quadrotor UAV are described. In section 3, original modified Super Twisting MSTW control law is designed and the stability of the closed-loop system is analyzed. In section 4, experimental results of the proposed control scheme implemented on the 3 DOF quadrotor are given. Finally, some conclusions are presented. 2. System Description The 3 DOF Hover Quanser platform used in this experience is depicted in Fig.1. The quadrotor is considered as a solid body incorporating force and moment generation process. Let I = (Xe , Ye , Ze ) be the inertial frame and B = (Xb , Yb , Zb ) the body-fixed frame attached to the center of gravity of the plateform. The experimental setup consists of a planar frame with four propellers driven by four DC motors where opposite motors rotate in the same direction. The frame is mounted on a 3 DOF pivot joint so that it can rotate about the attitude angles. The attitude is represented by Euler angles Θ = [φ, θ, ψ]T corresponding to the aeronautical convention. The attitude angles are respectively called Roll angle (φ rotation around x-axis), Pitch angle (θ rotation around y-axis), and Yaw angle ˙ = [φ, ˙ ψ] ˙ θ, ˙ T and (ψ rotation around z-axis). The angular velocities and accelerations are given respectively by Θ T ¨ ¨ ¨ ¨ Θ = [φ, θ, ψ] . The thrust forces generated by the back, right, front and left motors are denoted by f1 , f2 , f3 , and f4 , respectively. The thrust forces generated by the front and back motors primarily control the motions about the pitch axis while the right and left motors primarily move the hover about its roll axis. The motion about the yaw axis is caused by the difference in torques exerted by the two clockwise and two counter-clockwise rotating propellers. The Quadrotor attitude is controlled by the three control torque Γ = [τφ , τθ , τψ ]T combined to the forces generated by the four propellers f1 , f2 , f3 and f4 . The main thrust T is used to compensate the gravity force and eventually the 2
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vertical movement for a 6 DOF quadrotor.
Figure 1. Quanser quadrotor platform
2.1. Dynamics Using the Euler-Newton formalism and introducing some assumptions, enumerated below, we obtain a simplified attitude dynamic model of the quadrotor, this simplified model will be used in the controller design. ˙ can be measured or estimated by on-board sensors. A 2.1 The signals Θ and Θ A 2.2 The roll, pitch and yaw angles are limited to (−π/2 < φ < π/2), (−π/2 < θ < π/2) and (−π < ψ < π). It means that the acrobatic behavior is not allowed. A 2.3 Joint friction, air resistance and centrifugal forces are neglected. A 2.4 The thrust force is proportional to the motor voltage, and motors/propellers dynamics are neglected. The model described before in [1, 5], can be written as follow : h i ¨ = [M(Θ)]−1 ΨT (Θ)Γ − C(Θ, Θ) ˙ Θ ˙ + Γext Θ
(1)
where M(Θ) = ΨT (Θ)JΨ(Θ) ∈ R3×3 is an auxiliary positive inertia matrix provided that (θ , kπ/2), J ∈ R3×3 is the total inertia matrix, Γ ∈ R3 and Γext ∈ R3 denote the control torque and external disturbances respectively. ˙ ∈ R3×3 is given by2 : C(Θ, Θ) ˙ = −M(Θ)Ψ ˙ −1 (Θ)Ψ(Θ) − ΨT (Θ)sk[JΨ(Θ)Θ]Ψ(Θ) ˙ C(Θ, Θ)
:
The Euler matrix Ψ(Θ) ∈ R3×3 and the cross-product operator sk[β] for a vector β = [β1 β2 β3 ]T ∈ R3 are given by3
2 Ψ(Θ) ˙
3 The
1 Ψ(Θ) = 0 0
0 −sθ cφ sφcθ −sφ cφcθ
;
0 sk[β] = β3 −β2
˙ −1 (Θ)Ψ(Θ) = −Ψ(Θ)Ψ abbreviations s(.) and c(.) denote sin(.) and cos(.), respectively.
3
−β3 0 β1
β2 −β1 0
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A quadrotor helicopter has four rotors located at the extremities of a cross shaped frame as depicted in Fig.1. By considering the symmetry of the quadrotor system [1, 5], the inertia matrix J can be approximated by J = diag[J xx , Jyy , Jzz ], then the matrix M(Θ) and the control inputs (T, Γ) are given by : M(Θ) =
J xx 0 −J xx sθ
0 Jyy s2 φ + Jzz c2 φ (Jyy − Jzz )cθcφsφ
−J xx sθ (Jyy − Jzz )cθcφsφ J xx s2 θ + c2 θ(Jyy s2 φ + Jzz c2 φ)
,
T K f τ 0 φ = τθ −lK f τψ Kt
Kf −lK f 0 Kt
Kf 0 lK f Kt
Kf lK f 0 Kt
V1 V 2 V2 V2
(2)
l is the distance from the rotors to the center of mass of the platform, V j is the voltage input of rotor ” j” and (K f , Kt ) are respectively the thrust and the drag factors which are positive constants characterizing the propellers aerodynamics. The parameter values used in the experimental set up are given in table 1.
Table 1. Quanser parameters [23]
Parameter m l Kf Kt J xx Jyy Jzz
Description Mass Distance between Pivot to each Motor Thrust factor Drag factor Roll inertia φ Pitch inertia θ Yaw inertia ψ
Value 2.85 0.1969 2.98e-6 1.14e-7 0.0552 0.0552 0.1104
Unit Kg m N/V kgm2 kgm2 kgm2
Let us use (1) and define the following attitude state variables : ˙ x2 = Θ
x1 = Θ, Then the state-space form of this model is given by : ( x˙1 x˙2
= =
x2 f (x, t) + g(x, t)u
The control input is u = Γ, the vector f (x, t) and the matrix g(x, t) are given by h i ˙ Θ ˙ + Γext ; g(x, t) = [JΨ(Θ)]−1 f (x, t) = [M(Θ)]−1 −C(Θ, Θ)
(3)
(4)
3. Control Design In system (3), the vector-field f (x, t) is partially known, the perturbation terms Γext are unknown and assumed bounded. The problem is then to design a continuous modified Super Twisting sliding mode controller Γ that drives the attitude positions defined by {φ(t), θ(t), ψ(t)} to follow the bounded desired trajectories {φd (t), θd (t), ψd (t)} in presence of bounded external perturbations. The following lemma and definitions are used for the stability analysis. Lemma 3.1. [10, 16] Consider the continuous system x˙ = f (x), f (0) = 0, x ∈ Rn
(5)
Suppose there exist a continuous positive definite function V : Rn → R, a real numbers γ > 0 , α ∈ (0, 1) and an open neighborhood U0 ⊆ Rn of the origin such that the following inequality is satisfied : ˙ V(x) + γV α (x) ≤ 0, x ∈ U0 /{0}.
(6)
Then, the origin of the system (5) is a finite time stable equilibrium. If U0 = Rn , then the origin is a globally finite time stable equilibrium. The settling time tR satisfies : 1 V 1−α (0) (7) tR ≤ γ(1 − α)
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Notation 1. we define sigσ (x) = sign(x)|x|σ , where σ > 0, x ∈ R and sign(.) is the standard signum function. If x =
[x1 , x2 , ..., xn ]T is a vector, then
sigσ (x) = [sign(x1 )|x1 |σ , sign(x2 )|x2 |σ , ..., sign(xn )|xn |σ ]T
Notation 2. k ? k denotes the Euclidean norm of (?), λmin (?), λmax (?) are respectively the minimum and maximum eigenvalues of matrix (?). 3.1. Modified Super Twisting Algorithm (MSTW) The aim of the control law is to get a good accuracy in term of position tracking for the desired trajectory in spite of parametric uncertainties and disturbances. In order to control the attitude of the quadrotor, a proposed modified Super Twisting algorithm is applied. This controller is able to generate continuous control signal that eliminates chattering and reduces control effort. Consider the system (3) and assume that : A 3.1 A sliding variable s = s(x, t) ∈ R is designed so that the desired compensated dynamics of the system (3) are achieved in the sliding mode s = s(x, t) = 0. A 3.2 The relative degree of system (3) with the sliding variable s = s(x, t) as output equals one, and the internal dynamics are stable. Therefore, the dynamics of the sliding variable s are : s˙
=
∂s ∂s ∂s + f (x, t) + g(x, t) u ∂t ∂x ∂x | {z } | {z } b(x,t)
ψ(x,t)
=
(8)
ψ(x, t) + b(x, t)u
A 3.3 The function b(x, t) ∈ R is known and is assumed invertible. A 3.4 The function ψ(x, t) ∈ R is uncertain and can be written as
ψ(x, t) = ψnom (x, t) + ∆ψ(x, t)
(9)
where ψnom (x, t) is the nominal part of ψ(x, t) which is known, and it exists hmax positive constant such that the derivative function of ∆ψ(x, t) is bounded ˙ t)| ≤ hmax |∆ψ(x, (10)
The problem is to drive the sliding variable s and its derivative s˙ to zero in finite time in presence of bounded disturbances/uncertainties. We define the sliding surface as follows : s = δ˙ + λδ (11) 3×3 ˙ where δ = x1 − x1d , δ = x2 − x˙1d and λ ∈ R is a diagonal positive definite matrix. As system (3)-(11) admits a relative degree equal to 1 for s with respect to u, one gets : s˙ = ψnom (x, t) + ∆ψ(x, t) + b(x, t)u
(12)
˙ Θ ˙ − x¨1d + λδ, ˙ ∆ψ(x, t) = [M(Θ)]−1 Γext and b(x, t) = [JΨ(Θ)]−1 . The proposed controller for where ψnom (x, t) = −[M(Θ)]−1 C(Θ, Θ) the system is given by : # " Z t sigσ2 (s(τ))dτ (13) Γ = [JΨ(Θ)] −ψnom (x, t) − k1 |s|σ1 sign(s) − k2 0
where (1/2 < σ1 < 1) and (σ2 = 2σ1 − 1). k1 = [k1φ k1θ k1ψ ] and k2 = [k2φ k2θ k2ψ ] are the positive control gains given by : k2i >
h2max kqk2 , k1i > 0, i = {φ, θ, ψ} 2 σ1 k1i
The gain k2 allows to tune up the dynamics of adaptation of %(x, t), defined bellow. According to (13), (12) becomes Z
(14)
t
s˙ = −k1 |s|σ1 sign(s) − k2
sigσ2 (s(τ))dτ + ∆ψ(x, t)
0
5
(15)
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then, (15) can be written for each variable (i = {φ, θ, ψ}) as : s˙i µ˙¯ i
= =
−k1i |si |σ1 sign(si ) + µ¯ i −k2i |si |σ2 sign(si ) + %i (x, t)
(16)
with µ¯ i
=
%i (x, t)
=
−k2i
Z
0
t
sigσ2 (si (τ))dτ + ∆ψi (x, t)
˙ (x, t) ∆ψ i
The proof of finite time convergence to a small region around the origin of the variables si and µ¯ i is explicitly given in the sequel Proof 3.1. In order to analyze the stability of (15), the following state vector is introduced : z = [z1 z2 ]T = s µ¯ T
(17)
So the system (16) can be rewritten as :
z˙1i
=
z˙2i
=
−k1i |z1 |σ1 sign(z1 ) + z2
−k2i |z1 |σ2 sign(z1 ) + %(x, t)
(18)
where %(t) is assumed bounded by |%(x, t)| ≤ hmax , with hmax > 0. It can be observed that : • if z1 , z2 → 0 in finite time, then s, s˙ → 0 in finite time.
In this step of the proof, the stability analysis of system (16) is inspired from Moreno [22], therefore the following Lyapunov function candidate is introduced: k2 2σ1 1 2 1 (19) |z1 | + z2 + (k1 |z1 |σ1 sign(z1 ) − z2 )2 V(z) = σ1 2 2 In order to make the proof more readable, we considered only one dimensional case, which can represent either φ, θ or ψ. This allows dropping the double index from gains. We can put V in this form : k2 k2 V(z) = ( + 1 )|z1 |2σ1 + z22 − k1 z2 |z1 |σ1 sign(z1 ) (20) σ1 2 finally V is written as follow : V(z) = ζ T Pζ (21) The vector ζ and the symmetric positive definite matrix P ∈ R2×2 are selected respectively as : ! ! 2 1 2k + k12 −k1 |z1 |σ1 sign(z1 ) σ1 ζ= , P= z2 2 −k1 2 Note that the function (21) is continuous and differentiable everywhere except on the set z = {(z1 , z2 ) ∈ R2 |z1 = 0}, it is positive definite and radially unbounded if k2 > 0, hence, it follows that V(z) satisfies :
where
λmin (P)kζk2 ≤ V(z) ≤ λmax (P)kζk2
(22)
kζk2 = |z1 |2σ1 + z22
(23)
The derivative of the Lyapunov function candidate (21) is : ˙ V(z)
=
˙ V(z)
=
k2 k2 + 1 )|z1 |2σ1 −1 sign(z1 )˙z1 + 2z2 z˙2 − k1 |z1 |σ1 sign(z1 )˙z2 − σ1 k1 z2 |z1 |σ1 −1 z˙1 σ1 2 3σ1 −1 −2k1 k2 |z1 | + 2k2 z2 |z1 |2σ1 −1 sing(z1 ) − σ1 k13 |z1 |3σ1 −1 + σ1 k12 z2 |z1 |2σ1 −1 sign(z1 ) − 2k2 z2 |z1 |σ2 sign(z1 ) + 2z2 %(x, t)
2σ1 (
+k1 k2 |z1 |σ1 +σ2 − k1 |z1 |σ1 sign(z1 )%(x, t) + σ1 k12 |z1 |2σ1 −1 z2 sign(z1 ) − σ1 k1 z22 |z1 |σ1 −1
To simplify the Lyapunov function derivative, we impose that σ2 = 2σ1 − 1. The derivative becomes : ˙ V(z) = (−k1 k2 − σ1 k13 )|z1 |3σ1 −1 + 2σ1 k12 z2 |z1 |2σ1 −1 sign(z1 ) − σ1 k1 z22 |z1 |σ1 −1 + (−k1 |z1 |σ1 sing(z1 ) + 2z2 )%(x, t) We can rewrite (24) as follow :
˙ V(z) = −|z1 |σ1 −1 ζ T Qζ + %qT ζ
6
(24) (25)
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Where Q = k1
k2 + σ1 k12 −σ1 k1
−σ1 k1 σ1
!
;
q=
−k1 2
!
Next, the derivative of the Lyapunov function candidate can be bounded by : ˙ V(z) ≤ −|z1 |σ1 −1 λmin (Q)kζk2 + hmax kqkkζk
(26)
From (23), we can get : From (26) and (27), we get :
Let
|z1 |σ1 −1 ≥ kζk
σ1 −1 σ1
(27)
σ +σ h kqk max kζk 1σ1 2 ˙ V(z) ≤ − λmin (Q) − σ2 kζk σ1 ν1 = λmin (Q) −
with kζk >
(28)
hmax kqk
(29)
! σσ1
(30)
σ2
kζk σ1
hmax kqk λmin (Q)
2
to ensure ν1 > 0. From (22) and (29), it can be easily shown that the expression (28) will be : ˙ V(z) ≤ −γ1 V α (z) (31) −σ λmin (P) 2σ12 σ1 + σ2 > 0, which show that V(z) is a strong Lyapunov , then α ∈]0, 1[, and the positive constant γ1 = ν1 √ with α = 2σ1 λmin (P) function. According to lemma 3.1, the finite time stability can be guaranteed, and hence, the error of the system (16) will reach in finite-time the region : ! σ1 hmax kqk σ2 (32) kζk ≤ ∆1 ; ∆1 = λmin (Q) ! hmax kqk < 1 , because ( σσ12 > 1), the exponential term of (32) Consequently, the region ∆1 can be guaranteed to be small enough if λmin (Q) will greatly reduce the size of ∆1 (i.e., ∆1 1), therefore, we can select k1 and k2 such that λmin (Q) > hmax kqk From (33), we get : k2 >
(33)
h2max kqk2 , k1 > 0 σ1 k12
Now we can easily estimate the finite reaching time. Proposition 3.1.1. The MSTW control law (16) drives the sliding variable s and its derivative to the domain (32) in finite time estimated as : 1 V 1−α (z(0)) tF = γ1 (1 − α) where V(z(0)) is the initial value of V(z). Assuming ∆1 = 0 implies s, s˙ → 0 in finite time tR that can be estimated by tR ≤ 3.1. For ∆1 > 0, s, s˙ → ∆1 in finite time tF ≤ tR .
1 V 1−α (z(0)) γ1 (1−α)
which is obtained by Lemma
Remark 1. We know that kζk is sufficiently small after the settling time tF (i.e., the region ∆1 is very small), then according to equation (16) and the finite-time convergence of z2 , it can be concluded that the disturbances ∆ψ(x, t) are canceled in finite-time Z t ∆ψ(x, t) ≈ k2 sigσ2 (s(τ))dτ ≈ 0, f or t > tF (34) 0
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4. Experimental Results In this section, we use the Quanser platform described in section II. Experimental results are given to show the effectiveness of the proposed algorithm. The validation is done using the platform whose components are : data-acquisition card Quanser Q8, power amplifier system UPM-2405, four DC motors Pitman and three encoders used for the attitude measurement. The proposed technique is implemented via MATLAB/Simulink through the Q8 card that uses real time control software QUARC-Simulink. Many tests have been carried out under disturbances and uncertainties in order to study a the performances of the proposed scheme. A comparison with the classical STW algorithm is done with the aim of illustrating the benefits of the developed technique. The controller’s gains are given in table 2 :
Table 2. Controller gains
k1 k2 λ σ1
Roll φ 6 4 1.5 0.8
Pitch θ 5 4.2 1.7 0.8
Yaw ψ 7 5 3.5 0.8
In order to get good results in terms of accuracy, these gains are chosen by respecting the condition (14) for both laws that is in the sequel the chosen gains are the same for the both controls STW R t and MSTW, the only difference is in σ1 which equal 0.8 for 1 the MSTW and 0.5 for STW ( for the STW u = k1 |s| 2 sign(s) − k2 0 sign(s(τ))dτ). hmax is chosen as hmax = [1.5 1.5 1.5], this choice is justified by the fact that we are considering quasi constant perturbations in these experiments.
4.1. Stabilization experiments This experiment is realized to stabilize the 3 DOF system for stationary flight from some random given initial angles defined as φ = 12◦ , θ = 10◦ and ψ = −22◦ for the MSTW and φ = 8◦ , θ = 15◦ and ψ = −21◦ for the STW. In Fig.2 the attitude responses are plotted, where the red plots are the desired angles, the blue ones are the quadrotor trajectories controlled by MSTW and the green ones are the quadrotor trajectories controlled by STW. In stabilization experiment, the desired trajectories are equal to zero that is, we force the attitude angles to converge to zero. We clearly see from Fig.2 that the stabilization is obtained with both controllers, however, for STW’s trajectories, there are oscillations due to the discontinuity of the control unlike in MSTW where the attitude trajectories are smooth. Furthermore, the input signals are displayed in Fig.3, the red plots are associated with the STW control and the blue ones with the MSTW. The figure shows that the control signals remain acceptable in both cases, although the MSTW signal controls are smaller than the STW ones.
Figure 2. Euler angle trajectories for stabilization case
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Figure 3. Input signals
4.2. Attitude Tracking experiments The following experiments deals with the tracking problem. The desired trajectories are sinusoids of 10, 8 and 12 degrees in magnitude for the roll (φ), pitch (θ) and yaw (ψ) angles respectively and of 25 seconds period. In Fig.4 one can see the Euler angles when the system is driven by the STW and MSTW control laws respectively, the desired trajectories are also plotted. We can see that the MSTW control leads to smooth trajectories in blue color when the STW control displays oscillatory trajectories in green color. It means that the chattering caused by the discontinuity of the STW control is eliminated by MSTW that generates a smooth continuous signal. Fig.5 represents error plots of Euler angles trajectories associated with STW and MSTW. The control signal effort is also reduced as shown in Fig.6.
Figure 4. Tracking performances: Desired and actual attitude angles
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Figure 5. Tracking trajectories errors for STW and MSTW
Figure 6. Voltage inputs for tracking case
For a quantitative evaluation of the error tracking experiments, we summarized in table 3, the mean quadratic errors for each control law :
roll pitch yaw
MSTW 0.2130 0.1180 0.5580
1 T
RT 0
e2 (t)dt,
STW 1.5191 2.0378 0.9615
From a quantitative point of view, table 3 confirms that errors associated with the MSTW controller are smaller than those of the STW one.
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4.3. Robustness and disturbance rejection The first disturbance we applied to the 3 DOF system controlled by the MSTW algorithm consists in disabling one of the 4 motors at instant 10s. From Fig. 7, we see that the system continues to perform satisfactorily, the errors are kept small and the effect of this perturbation is more important on the roll and pitch angles than on the yaw angle. The other kind of disturbance consists in adding a mass of 115 grams on the end of the frame in the x direction at instant 15 s. Fig. 8, shows the effectiveness of the control, the perturbation is rejected and the tracking of the trajectories is kept satisfactorily. Next, we applied a manual external forces in interval of 30s < t < 45s, we see that this new perturbation is also rejected. Hence, the efficiency and the robustness of the proposed control are proved in both tests. Fig. 9 and 10, are related to the same robustness and disturbance rejection tests but for the STW controller. We see here also that the same properties of rejection but with a persisting chattering.
Figure 7. Euler angle trajectories with MSTW for one motor disabled
Figure 8. Euler angle trajectories with MSTW for mass and manual disturbancies added
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Figure 9. Euler angle trajectories with STW for one motor disabled
Figure 10. Euler angle trajectories with STW for mass and manual disturbancies added
5. Conclusion The proposed MSTW algorithm solves the problem of chattering which is considered as the main drawback of the super twisting algorithm. The stability of the closed loop is formally proved and the convergence to the desired trajectories is achieved in finite time. The experimental tests show the effectiveness of the proposed control in both cases stabilization and tracking. Through the experiments we also verified the properties of robustness against the parameter uncertainties and external disturbances. As future research, we intend to study more formally the robustness properties of the proposed control law and also to compare the sizes of the control signals generated by the MSTW control versus the one generated by the STW control.
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