Adaptive prescribed performance control of QUAVs with unknown time-varying payload and wind gust disturbance

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Journal of the Franklin Institute 355 (2018) 6323–6338 www.elsevier.com/locate/jfranklin

Education Article

Adaptive prescribed performance control of QUAVs with unknown time-varying payload and wind gust disturbance Changchun Hua a,∗, Jiannan Chen a, Xinping Guan b b School

a Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China of Electronics, Information and Electric Engineering, Shanghai Jiaotong University, Dongchuan Road 800, Shanghai 200240, China

Received 27 June 2017; received in revised form 9 March 2018; accepted 20 May 2018 Available online 11 July 2018

Abstract In this paper, a new robust adaptive prescribed performance control (PPC, for short) scheme is proposed for quadrotor UAVs (QUAVs, for short) with unknown time-varying payloads and wind gust disturbances. Under the presented framework, the overall control system is decoupled into translational subsystem and rotational subsystem. These two subsystems are connected to each other through common attitude extraction algorithms. For translational subsystem, a novel robust adaptive PPC strategy is designed based on the sliding mode control technique to provide better trajectory tracking performance and well robustness. For rotational subsystem, a new robust adaptive controller is constructed based on backstepping technique to track the desired attitudes. Finally, the overall system is proved to be stable in the sense of uniform ultimate boundedness, and numerical simulation results are presented to validate the effectiveness of the proposed control scheme. © 2018 Published by Elsevier Ltd on behalf of The Franklin Institute.

∗

Corresponding author. E-mail address: [email protected] (C. Hua).

https://doi.org/10.1016/j.jfranklin.2018.05.062 0016-0032/© 2018 Published by Elsevier Ltd on behalf of The Franklin Institute.

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1. Introduction Over the past decades, the control of QUAVs has been a topic of considerable interest. This interest is motivated by its wide applications and theoretic challenges [1–8]. These vehicles can be applied to many fields such as surveillance, agriculture and transportation, etc. The main advantages lie in their ability to fly in any direction, take off and land vertically, and hover at a desired altitude. However, the control design for QUAVs is a challenging task. First, the QUAVs are underactuated in the sense that they are equipped with fewer actuators than degrees-of-freedom [9,10], which leads to strong couplings between the dynamic state variables. Then, some parameters associated with the dynamic model, such as the inertial moments, the payloads and aerodynamic coefficients, cannot be measured or obtained exactly. Furthermore, these vehicles are sensitive to external disturbances such as wind gust due to its small size and weight. To solve those aforementioned problems, various control techniques have been developed. For the underactuated property, a novel robust controller that is based on an integral sliding mode approach is proposed for an underactuated quadrotor in [11]. In [12,13], the dynamic system of the QUAV is decoupled into the inner-loop and the outer-loop subsystem. Such scheme is not difficult for implementation, and the asymptotic stability of the closed-loop system is proved via a theorem of cascaded systems. To compensate for the parametric uncertainties and external disturbances, some works have been done. In [14,15], the multiple uncertainties including inner system uncertainties, state variables delay and outside disturbances are considered. In order to solve such problem, a robust control scheme is proposed based on switch control technique. In [16], an integral adaptive sliding mode control scheme is proposed to deal with sinusoidal wind model. An integral predictive/nonlinear H∞ controller is proposed in [17]. In [18–20], model predictive controllers are implemented to tackle with atmospheric disturbances in both translational and rotational subsystems. In [21–24], the disturbances are assumed to be time-invariant with a known upper bound. In [17,25–30], sliding mode control and observation techniques are introduced to compensate for the model uncertainties and external disturbances. Generally, the uncertainty and disturbance impacts on outputs or important variables aren’t taken into consideration in most of these previous works. However, high performance requirements for QUAVs cannot be satisfied without considering these impacts in civil and military applications. In this paper, in response to the aforementioned control problem for the QUAV, a novel robust adaptive PPC scheme is proposed for the QUAV with unknown time-varying payloads and wind gust disturbances. First, to solve the underactuation and the coupling problems, the control system is decoupled into two subsystems: the translational subsystem and rotational subsystem. Second, an important practical engineering problem that is time-varying payloads is first raised and addressed with adaptive technique and Lemma 1 [31]. Further, another intractable problem consists in high performance requirements. Due to the application fields are more and more extensive, it is important to increase the trajectory tracking performance for better application results. To solve such problem, a novel adaptive prescribed performance control strategy is proposed. Prescribed performance control, which belongs to constraint control, ensures the convergence rate not less than a prescribed value, exhibiting a maximum overshoot less than a sufficiently small constant, and the output or tracking error approaches to an arbitrarily small residual set [32–34].

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Fig. 1. A QUAV for pesticide spraying.

This paper is organized as follows. In Section 2, the system model and necessary preliminaries are given. Section 3 presents a novel trajectory tracking controller for translational subsystem based on slide mode control. A new robust adaptive controller based on backstepping technique is presented in Section 4. In order to validate the effectiveness of the constructed controller, the corresponding simulation results are given in Section 5, and conclusions of this work are presented in the last section. Notations: Let P, H, F, M, M¯ , I¯, V and W represent linear momentum, angular momentum, force, torque, mass, inertia, linear velocity and angular velocity in Section 2.1, and [x, y, z]T , [φ, θ , ψ]T , m and [Ixx , Iyy , Izz ]T stand for position, angle, mass and the inertia moment in the rest of the paper. The notations d¯x , d¯y , d¯z , d¯φ , d¯θ and d¯ψ stand for the disturbances that caused by time-varying payloads, and the wind gust disturbances are denoted as dˆx , dˆy , dˆz , dˆφ , dˆθ and dˆψ . The notations dx , dy , dz , dφ , dθ and dψ represent for the lumped disturbances, and their upper boundaries of absolute values are denoted as Dx , Dy , Dz , Dφ , Dθ and Dψ . The notations τ mx , τ my , τ mz , τ mφ , τ mθ and τ mψ represent for the lower boundaries of τ mx , τ my , τ mz , τ mφ , τ mθ and τ mψ . The upper boundaries of |Tmφ |, |Tmθ | and |Tmψ | are denoted as Tˇmφ , Tˇmθ and Tˇmψ , respectively. 2. System description and preliminaries The QUAV is an underactuated vehicle with two pairs of propellers, as shown in Fig. 1. Two propellers rotate in counter-clockwise, and another two propellers rotate clockwise. Motion of such vehicle is controlled by varying the speed of four rotors. The attitude angles and altitude of QUAVs can be controlled directly, and the movement in horizontal plane can be achieved by properly changing the attitude angles, as shown in Fig. 2. 2.1. Unknown time-varying payload impact on QUAV In this subsection, time-varying payload impacts on QUAVs dynamics are first analyzed. These vehicles are regarded as an ideal rigid-body in three-dimension space. The system dynamics are derived from following classical equations: d P = dtd (M¯ V ) = F dt (1) d H = dtd (I¯W ) = M dt

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Fig. 2. Control structure of QUAV.

where matrices P, H, F and M are linear momentum, angular momentum, force and torque, respectively. M¯ , I¯, V and W represent mass, inertia, linear velocity and angular velocity, respectively. In this paper, the variable M¯ is time-varying owning to the time-varying payloads of QUAV, as shown in Fig. 1. As we know, the inertia matrix I¯ = M¯ r 2 dm, therefore the variable I¯ is also time-varying. The time derivative of Eq. (1) is given as,  ˙¯ M¯ V˙ = F − MV (2) ˙ I¯W˙ = M − I¯W ˙¯ and I˙¯W are additional terms, and there are not these terms It is worth mentioning that MV under the classic conditions. In order to clarify above equation, the scalar equations are presented as follows: ⎧ ⎧ ¯ ˙¯ ¯x ⎪ M 1 1 d ⎪ ⎪W˙ = − I¯zz − I¯yy W W + Mφ + dφ ⎪ φ θ ψ ⎪V˙x = Fx − Vx = Fx + ⎪ ⎪ ⎪ ⎪ I¯xx I¯xx I¯xx ⎪ M¯ M¯ M¯ M¯ ⎪ ⎪ ⎪ ⎨ ⎨ d¯y W˙ = − I¯xx − I¯zz W W + Mθ + d¯θ M˙¯ 1 1 θ φ ψ V˙y = Fy − Vy = Fy + ⎪ I¯yy I¯yy I¯yy ⎪ ⎪ M¯ M¯ M¯ M¯ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ¯ ¯ ⎪ ⎪ ¯ ¯ I − I M d¯ψ xx ψ ⎪ ⎩V˙ = 1 F − M V = 1 F + dz ⎪ ⎪W˙ψ = − yy Wφ Wθ + + ⎩ z z z z I¯zz I¯zz I¯zz M¯ M¯ M¯ M¯ where the variables Vi , Fi , I¯ii , (i = x, y, z) are elements of vectors V, F, I¯, and Wi , Mi , (i = φ, θ , ψ ) are elements of vectors W, M. The terms d¯x , d¯y , d¯z , d¯φ , d¯θ and d¯ψ are used to ˙¯ , −MV ¯˙ y , −MV ¯˙ z , −I¯˙xxWφ , −I¯˙yyWθ and −I˙¯zzWψ , respectively. represent −MV x 2.2. Wind impact on QUAV The wind flow impact on the QUAV is comprised of two parts: momentum drag and moment disturbance. The moment disturbance depends on the distance between propeller disk plane and the gravity center, as shown in Fig. 3. Generally, these impacts can be described as: (i) the force that “push” the QUAV away from desired trajectory, (ii) the moment that “rotate” the QUAV away from desired attitude angles. The force and moment at small roll

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Fig. 3. Wind impact on QUAV.

and pitch angles can be formulated as follows: ⎧ ⎪ dˆx ≈ 4N · cosψ ⎪ ⎪ ⎨ˆ dy ≈ 4N · sinψ ⎪ dˆφ ≈ 4N · sinψ · d ⎪ ⎪ ⎩ˆ dθ ≈ 4N · cosψ · d

(3)

where d is distance between propeller disk plane and gravity center, and the momentum drag T N = mV∞ = k(ρAω0 )V∞ , ω0 = 2ρA . Notation T stands for thrust generated by propellers. A is the area of propeller disk, ρ is air density, V∞ is velocity of freestream. It is worth mentioning that there are no deterministic expressions to describe the wind flow impacts on z and ψ subsystems. In this paper, unknown disturbance terms dˆz and dˆψ are used to represent these impacts. 2.3. QUAV system description According to the above analyses about the unknown time-varying payload and wind impact, the dynamics of uncertain QUAV can be formulated as: ⎧ Iyy − Izz Jp 1 dφ ⎪ ⎪ + θ˙ r + Tφ + ⎪φ¨ = θ˙ψ˙ ⎪ I I I I ⎪ xx xx xx xx ⎪ ⎪ Jp Izz − Ixx 1 dθ ⎪ ⎪ ¨ ˙ ˙ ˙ θ = φψ + φ r + Tθ + ⎪ ⎪ ⎪ Iyy Iyy Iyy Iyy ⎪ ⎪ ⎪ Ixx − Iyy 1 dψ ⎪ ⎪ ¨ ˙ ˙ + Tψ + ⎨ψ = φ θ Izz Izz Izz (4) (cos(φ)sin(θ )cos(ψ ) + sin(φ)sin(ψ ))Fz dx ⎪ ⎪ x ¨ = + ⎪ ⎪ mx mx ⎪ ⎪ ⎪ dy (cos(φ)sin(θ )sin(ψ ) − sin(φ)cos(ψ ))Fz ⎪ ⎪ ⎪ y¨ = + ⎪ ⎪ my my ⎪ ⎪ ⎪ dz cos(φ)cos(θ ) ⎪ ⎪ ⎩z¨ = − Fz + g + mz mz where [x, y, z], [φ, θ , ψ] and m represent position, attitude and the mass of the QUAV, respectively. Notations Ixx , Iyy and Izz represent the QUAV inertias around x, y, z axes. Jp , r and g are inertia moment of propeller, margin of propellers rotation speed and gravitational

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acceleration. The notations mx = my = mz = m are the mass of the QUAV, and di = d¯i + dˆi , for i = (x, y, z, φ, θ , ψ ). Denote the vector [x1 , x2 , x3 , x4 , x5 , x6 ]T = [x, x˙, y, y˙, z, z˙]T and ˙ θ , θ˙, ψ, ψ˙ ]T . [x7 , x8 , x9 , x10 , x11 , x12 ]T = [φ, φ, Assumption 1. In actual application, the time-varying payloads of QUAVs are assumed to be bounded with unknown upper and lower bounds. Therefore, system parameters I1xx , I1yy , I1zz , m1x , 1 my

and m1z are bounded with unknown upper and lower bounds according to the definitions, and the lower bounds of above variables are denoted as τ mφ , τ mθ , τ mψ , τ mx , τ my and τ mz , respectively. Assumption 2. According to Assumption 1, it is reasonable to assume the system parameters Iyy −Izz Izz −Ixx I −I I −I xx , Iyy and xxIzz yy are bounded with unknown upper bounds, i.e., | yyIxx zz | < Tˇmφ , | IzzI−I |< Ixx yy Ixx −Iyy ˇ ˇ Tmθ , | | < Tmψ . Izz

Assumption 3. The lumped disturbances vectors [dx , dy , dz ]T and [dφ , dθ , dψ ]T are bounded, and the upper boundaries of absolute values are known explicitly as [Dx , Dy , Dz ]T and [Dφ , Dθ , Dψ ]T , respectively. Remark 1. The unknown time-varying payload problem is studied in this paper. Owning to the mathematical relation with payload, the variables m1x , m1y , m1z , I1xx , I1yy and I1zz are also timevarying which makes the control designs become much complicated. It is worth mentioning that the assumptions about the above variables are loose and satisfy realistic requirements for actual applications. 2.4. Prescribed performance To address the transient and steady-state performance requirements for the corresponding variables, a positive decreasing smooth function ρ(t ) = (ρo − ρ∞ )e−bt + ρ∞ with ρ o > ρ ∞ > 0 is chosen as the prescribed performance function. Define the tracking error z : + → . The expression of prescribed performance is given by the following inequalities: −σ ρ(t ) < z(t ) < σ ρ(t )

(5)

First, the selected ρ o should satisfy the property ρ o > |z(0)|. The value ρ∞ = limt→∞ ρ(t ) represents the maximum allowable size of the steady error. Moreover, the convergence rate is determined by positive constant b, and σ , σ are also positive constants. Therefore, the appropriate selection of prescribed performance function ρ(t) imposes performance characteristics on the tracking error z, as shown in Fig. 4. Remark 2. In order to reduce the adverse impacts of uncertainties and disturbances on outputs or some important variables, a control strategy that based on PPC technique is designed. Through the designed strategy, the tracking error converges to a predefined arbitrarily small residual set, the convergence rate is no less than a given value, and the maximum overshoot during the transient process is within a preassigned boundary. 2.5. Useful lemma In this subsection, the necessary lemma is presented.

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Fig. 4. Prescribed performance requirement.

Lemma 1. For any constant > 0 and any variable ϖ ∈ R, the following relationship holds [35]: 2 2 −√ ≤ | | − √ < .  2 + 2  2 + 2

3. Translational subsystem In this part, the novel adaptive robust prescribed performance controllers ux , uy and uz will be designed for translational subsystem. Under the constructed control framework, the state variables x1 , x3 , x5 track to the desired trajectories x1d , x3d , x5d with better transient and steady-state performances. The translational subsystem can be rewritten as follows: ⎧ x˙1 ⎪ ⎪ ⎪ ⎪ x˙2 ⎪ ⎪ ⎨ x˙3 x˙4 ⎪ ⎪ ⎪ ⎪ x˙5 ⎪ ⎪ ⎩ x˙6

= x2 = τmx · ux + τmx · dx = x4 = τmy · uy + τmy · dy = x6 = −τmz · uz + τmz · dz + g

(6)

where the vector [τ mx , τ my , τ mz ]T is used to represent [ m1x , m1y , m1z ]T for simplification. Define error variables as follows: z 1 = x1 − x1d

z 3 = x3 − x3d

z 5 = x5 − x5d

To address the transient and steady-state performance requirements of error variables, three positive decreasing smooth functions ρi (t ) = (ρio − ρi∞ )e−hi t + ρi∞ , i = 1, 3, 5 are chosen as prescribed performance functions. The trajectories of error variables zi (t) satisfy −σ i ρi (t ) < zi (t ) < σ i ρi (t ), where σ i and σ i are positive constants. We set

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zi (t ) = ρi (t )Si (εi ) and Si (εi ) =

σ i eεi − σ i e−εi eεi + e−εi

where ρi (t ) = (ρi0 − ρi∞ )e−hi t + ρi∞ , with ρ i0 > ρ i∞ > 0 and hi > 0. The notation si is used to represent ρzii for simplification. Since the function Si (εi ) is strictly monotonic increasing, its inverse function exists as, εi = Si−1



zi ρi

 =

1 si + σ i ln 2 σ i − si

(7)

σ

Denote zi = εi − 21 ln σ ii , and the time derivatives are given as,

 z1 ρ˙1 z˙ 1 = R1 z˙1 − ρ1

 z3 ρ˙3 z˙ 3 = R3 z˙3 − ρ3

z5 ρ˙5 z˙ 5 = R5 (z˙5 − ) ρ5

(8)

1 1 1 where R1 = 2ρ1 1 ( s1 +1σ − s1 −σ ), R3 = 2ρ1 3 ( s3 +1σ − s3 −σ ) and R5 = 2ρ1 5 ( s5 +1σ − s5 −σ ). It is 1 3 5 1 3 5 worth mentioning that R1 , R3 and R5 are positive all the time. The sliding mode surfaces are designed as follows:

S1 = z˙ 1 + k1 z1

S3 = z˙ 3 + k3 z3

S5 = z˙ 5 + k5 z5

(9)

where constants k1 , k3 are k5 are positive constants. First, subsystem [x1 , x2 ]T will be taken as an example to present the detailed design procedures. Then, as for other two subsystems, the controllers will be directly presented for clarity. The controller of subsystem [x1 , x2 ]T is presented as follows: ⎧ ⎪ S1 R1 u2x2 dˆτ2mx ⎪ ⎪ ⎪ u = −D sign(S ) − x 1 ⎨ x S12 R12 u2x2 dˆτ2mx + 12

 ⎪ ⎪ z˙1 ρ˙1 + z1 ρ¨1 z1 ρ˙1 z1 ρ˙1 ⎪ z1 ρ˙1 ρ˙1 −1 −1 ˙ ⎪ u = c R S − x ¨ − + + R R ( z ˙ − ) + k z − ˙ ⎩ x2 1 1 1 1d 1 1 1 1 1 ρ12 ρ1 ρ1 ρ1

(10)

where notation dτ mx represents τ 1mx , and dˆτ mx is the estimated value of dτ mx . The parameters c1 , 1 are positive constants. The adaptive law is designed as, ˙ dˆτ mx = r1 R1 S1 ux2 − δ1 r1 dˆτ mx

(11)

where parameters r1 and δ 1 are two positive constants to be determined later. By the same processes, controllers uy and uz of subsystems [x3 , x4 ]T and [x5 , x6 ]T are designed, respectively. The details are omitted for clarity and simplification. The controller uy is presented as,

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⎧ ⎪ S3 R3 u2y2 dˆτ2my ⎪ ⎪ ⎪ u = −D sign(S ) − y 3 ⎨ y S32 R32 u2y2 dˆτ2my + 32



 (12) ⎪ ⎪ z ρ ˙ + z ρ ¨ z ρ ˙ ρ ˙ ρ ˙ ρ ˙ ˙ z z 3 3 3 3 3 3 3 3 3 3 3 ⎪ −1 −1 ⎪ + k3 z˙3 − + + R3 R˙ 3 z˙3 − ⎩uy2 = c3 R3 S3 − x¨3d − ρ3 ρ3 ρ3 ρ32 and controller uz is presented as, ⎧ ⎪ S5 R5 u2z2 dˆτ2mz ⎪ ⎪ u = D sign(S ) + ⎪ z 5 ⎨ z S52 R52 u2z2 dˆτ2mz + 52



 ⎪ ⎪ z˙5 ρ˙5 + z5 ρ¨5 z5 ρ˙5 ρ˙5 z5 ρ˙5 z5 ρ˙5 ⎪ −1 −1 ˙ ⎪ ˙ + ˙ +g u = c R S − x ¨ − + + R R z − k z − ⎩ z2 5 5 5 5d 5 5 5 5 5 ρ5 ρ5 ρ5 ρ52 (13) where terms dτ my and dτ mz represent τ 1my and τ1mz , respectively. The parameters c3 , c5 are positive constants. The adaptive laws are designed as, ˙ dˆτ my = r3 S3 R3 uy2 − δ3 r3 dˆτ my ˙ dˆτ mz = r5 S5 R5 uz2 − δ5 r5 dˆτ mz where parameters r3 , r5 , δ 3 and δ 5 are positive constants. Theorem 1. For the investigated system (6) with controllers ux , uy , uz and adaptive laws ˙ ˙ ˙ dˆτ mx , dˆτ my , dˆτ mz , we can conclude that the translational subsystem is stable in the sense of uniform ultimate boundedness, and prescribed performance constraints will never be violated. Proof. The time derivative of the sliding mode surface S1 is,

 z1 ρ˙1 ˙ ˙ + k1 z˙ 1 + R1 (τmx (ux + dx ) S1 = R1 z˙1 − ρ1 z˙1 ρ˙1 + z1 ρ¨1 z1 ρ˙1 ρ˙1 −x¨1d − + ) ρ1 ρ12 Choose Lyapunov function Vx = 21 S12 + tive is given as,

1 τ d˜2 , 2r1 mx τ mx

where d˜τ mx = dτ mx − dˆτ mx . The time deriva-

1 ˙ V˙x = S1 S˙1 − τ mx d˜τ mx dˆτ mx r1



 1 z1 ρ˙1 z1 ρ˙1 ˙ + k1 S1 R1 z˙1 − + τ mx d˜τ mx (−dˆτ mx ) = S1 R˙ 1 z˙1 − ρ1 ρ1 r1

 z˙1 ρ˙1 + z1 ρ¨1 z1 ρ˙1 ρ˙1 + S1 R1 τmx ux + τmx dx − x¨1d − + ρ1 ρ12

(14)

By using Lemma 1, the important scaling inequalities are given based on Eqs. (14) and (10). Present them as follows: S1 R1 τmx [−Dx · sign(S1 ) + dx ] ≤ 0

(15)

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S1 R1 u2x2 dˆτ2mx −S1 R1 τmx S12 R12 u2x2 dˆτ2mx + 12 ≤ −τ mx

S12 R12 u2x2 dˆτ2mx S12 R12 u2x2 dˆτ2mx + 12

≤ −τ mx S1 R1 ux2 dˆτ mx + τ mx 1

= −S1 R1 ux2 + S1 R1 ux2 τ mx d˜τ mx + τ mx 1

(16)

It is worth mentioning that the term τ mx and its lower bound τ mx are unknown but positive all the time for the mass of the QUAV can not be negative. Substituting Eqs. (15) and (16) into Eq. (14), we have

 z˙1 ρ˙1 + z1 ρ¨1 z1 ρ˙1 ρ˙1 ˙ Vx ≤ S1 R1 −ux2 − x¨1d − + ρ1 ρ12



 1 z1 ρ˙1 z1 ρ˙1 ˙ˆ ˜ ˙ + τ mx dτ mx (−dτ mx ) + S1 R1 z˙1 − + k1 S1 R1 z˙1 − ρ1 r1 ρ1 ˜ (17) + S1 R1 ux2 τ mx dτ mx + τ mx 1 Substituting Eqs. (10) and (11) into Eq. (17), then we can obtain, V˙x ≤ −c1 S12 + δ1 τ mx d˜τ mx dˆτ mx + 1 τ mx δ1 δ1 ≤ −c1 S12 − τ mx d˜τ2mx + τ mx dτ2mx + 1 τ mx 2 2 ≤ −cxVx + x

(18)

where cx = min{2c1 , δ1 }, and x = δ21 τ mx dτ2mx + 1 τ mx . Obviously, subsystem [x1 , x2 ]T is stable in the sense of uniform ultimate boundedness. By the same principle, we can conclude subsystems [x3 , x4 ]T and [x5 , x6 ]T have the same conclusions. That is the end of the proof.  4. Rotational subsystem In this section, a new tracking control strategy will be designed for rotational subsystem. It is worth mentioning the desired trajectories x7d and x9d are derived from the translational subsystem. The rotational subsystem can be rewritten as follows: ⎧ x˙7 = x8 ⎪ ⎪ ⎪ ⎪ ⎪ x˙8 = x10 x12 · Tmφ + τmφ · (uφ + x10 Jp r + dφ ) ⎪ ⎪ ⎪ ⎨ x˙ = x 9 10 (19) ⎪ x˙10 = x8 x12 · Tmθ + τmθ · (uθ + x8 Jp r + dθ ) ⎪ ⎪ ⎪ ⎪ ⎪ x˙11 = x12 ⎪ ⎪ ⎩ x˙12 = x8 x10 · Tmψ + τmψ · (uψ + dψ ) where

the

vector

[ Tmφ ,

I −I I −I xx [ yyIxx zz , IzzI−I , xxIzz yy , I1xx , I1yy , yy

Tmθ ,

1 T ] Izz

Tmψ ,

τ mφ ,

τ mθ ,

τ mψ ] T

is

used

to

represent

for simplification. Define error variables as follows:

C. Hua et al. / Journal of the Franklin Institute 355 (2018) 6323–6338

z 7 = x7 − x7d z8 = x8 − α7

z 9 = x9 − x9d z10 = x10 − α9

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z11 = x11 − x11d z12 = x12 − α11

The virtual controllers α 7 , α 9 and α 11 are designed as follows: α7 = −c7 z7 + x˙7d

α9 = −c9 z9 + x˙9d

α11 = −c11 z11 + x˙11d

where c7 , c9 and c11 are positive constants to be determined later. First, subsystem [x7 , x8 ]T will be taken as an example to present the detailed design procedures. Then, as for other two subsystems, the controllers will be directly presented for clarity. The controller of subsystem [x7 , x8 ]T is presented as follows: ⎧ z8 u2φ2 dˆτ2mφ ⎪ ⎪ ⎨uφ = −x10 Jp r − Dφ sign(z8 ) − (20) z82 u2φ2 dˆτ2mφ + 72 ⎪ ⎪ ⎩ uφ2 = c8 z8 + z7 + |x10 x12 |Tˆˇmφ sign(z8 ) − α˙ 7 where notation dτ mφ represents τ 1mφ . The terms dˆτ mφ and Tˆˇmφ are the estimated values of dτ mφ and Tˇmφ , respectively. The parameters 7 and c8 are positive constants to be determined later. The adaptive laws are presented as follows: ⎧ ⎨ ˙ˆˇ Tmφ = r8 |x10 x12 z8 | − r8 δ8 Tˆˇmφ (21) ⎩d˙ˆ = r z u − δ r dˆ τ mφ

7 8 φ2

7 7 τ mφ

where the parameters r7 , r8 , δ 7 and δ 8 are positive constants to be determined later. By the same processes, the controllers uθ and uψ for subsystems [x9 , x10 ]T and [x11 , x12 ]T are designed, respectively. The details are omitted for clarity and simplification. The controller uθ is designed as, ⎧ 2 ˆ2 ⎪ ⎪u = −x J  − D sign(z ) − z10 uθ2 dτ mθ ⎨ θ 8 p r θ 10 2 u2 dˆ2 + 2 (22) z10 θ2 τ mθ 9 ⎪ ⎪ ⎩ uθ2 = z9 + c10 z10 + |x8 x12 |Tˆˇmθ sign(z10 ) − α˙ 9 and the controller uψ is presented as, ⎧ z12 u2ψ2 dˆτ2mψ ⎪ ⎪ ⎨uψ = −Dψ sign(z12 ) − 2 u2 dˆ2 + 2 z12 ψ2 τ mψ 11 ⎪ ⎪ ⎩ ˆ ˇ uψ2 = z11 + c12 z12 + |x8 x10 |Tmψ sign(z12 ) − α˙ 11

(23)

1 where notations dτ mθ and dτ mψ represent τ 1mθ and τ mψ , respectively. The parameters 9 , 11 , c10 and c12 are positive constants. The adaptive laws are designed as,

˙ Tˆˇmθ = r10 |z10 x8 x12 | − δ10 r10 Tˆˇmθ , ˙ Tˆˇmψ = r12 |z12 x8 x10 | − δ12 r12 Tˆˇmψ ,

˙ dˆτ mθ = r9 z10 uθ2 − δ9 r9 dˆτ mθ ˙ dˆτ mψ = r11 z12 uψ2 − δ11 r11 dˆτ mψ

where parameters r9 , r10 , r11 , r12 , δ 9 , δ 10 , δ 11 and δ 12 are positive constants.

(24)

(25)

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Theorem 2. For the investigated system (19) with controllers uφ , uθ , uψ and adaptive laws (21), (24), and (25), we can conclude that the system is stable in the sense of uniform ultimate boundedness. Proof. Choose Lyapunov function V = 1 z2 + 1 z2 + 1 T˜ˇ 2 + 1 τ d˜2 , where d˜ = φ

2 7

2 8

2r8 mφ

2r7

mφ τ mφ

dτ mφ − dˆτ mφ , T˜ˇmφ = Tˇmφ − Tˆˇmφ . The time derivative of Vφ is presented as follows:



1 ˙ˆ 1 ˙ˆ ˜ 2 ˜ ˙ ˇ ˇ Vφ = −c7 z7 + z7 z8 + τ mφ dτ mφ −dτ mφ + Tmφ −Tmφ r7 r8 + z8 [x10 x12 Tmφ + τmφ (uφ + x10 Jp r + dφ ) − α˙ 7 ]



1 ˙ 1 ˙ = −c7 z72 + z7 z8 + τ mφ d˜τ mφ −dˆτ mφ + T˜ˇmφ −Tˆˇmφ r7 r8  + z8 x10 x12 Tmφ + τmφ (−x10 Jp r − Dφ sign(z8 ) −

z8 u2φ2 dˆτ2mφ z82 u2φ2 dˆτ2mφ + 72

+ x10 Jp r + dφ ) − α˙ 7

τ mφ

 (26)

By using Lemma 1, the important scaling inequality is given based on Eqs. (20) and (26). Present it as follows: −τmφ

z82 u2φ2 dˆτ2mφ z82 u2φ2 dˆτ2mφ

z82 u2φ2 dˆτ2mφ ≤ −τ mφ + 72 z82 u2φ2 dˆτ2mφ + 72

≤ −τ mφ z8 uφ2 dˆτ mφ + τ mφ 7 = −z8 uφ2 + z8 uφ2 τ mφ d˜τ mφ + τ mφ 7

(27)

where τ mφ and its lower bound τ mφ are unknown but positive all the time for the inertia of the QUAV cannot be negative. Substituting Eqs. (20), (21), and (27) into Eq. (26), we can get, ˙ 1 V˙φ ≤ −c7 z72 − c8 z82 + z8 {x10 x12 Tmφ − |x10 x12 |Tˆˇmφ sign(z8 )} − T˜ˇmφ Tˆˇmφ r8 1 ˙ + τ mφ d˜τ mφ z8 uφ2 − τ mφ d˜τ mφ dˆτ mφ + τ mφ 7 r7 ˙ 1 ≤ −c7 z72 − c8 z82 + |z8 x10 x12 |Tˇmφ − |z8 x10 x12 |Tˆˇmφ − T˜ˇmφ Tˆˇmφ r8 + δ7 τ mφ d˜τ mφ dˆτ mφ + τ mφ 7 ≤ −c7 z72 − c8 z82 − ≤ −cφ V8 + φ

δ7 δ8 2 δ7 δ8 2 τ mφ d˜τ2mφ − T˜ˇmφ + τ mφ dτ2mφ + Tˇmφ + τ mφ 7 2 2 2 2 (28)

2 where parameter cφ = min{2c7 , 2c8 , δ7 , δ8 }, and term φ = 7 τ mφ + δ28 Tˇmφ + δ27 τ mφ dτ2mφ . Obviously, subsystem [x7 , x8 ]T is stable in the sense of uniform ultimate boundedness. By the same principle, we can conclude subsystems [x9 , x10 ]T and [x11 , x12 ]T have the same conclusions. That’s the end of the proof. 

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Fig. 5. 3-D trajectory tracking.

Remark 3. In the investigated system, there exist unknown time-varying coefficients, disturbances and uncertainties. Therefore, it is difficult to design the corresponding controllers for high performance requirements of QUAVs. Fortunately, with Lemma 1 and adaptive method, the time-varying coefficients problem can be solved. Further, with PPC technique, the high performance requirements for QUAVs can be ensured under the impacts of uncertainties and disturbances. Finally, a robust adaptive prescribed performance control scheme is constructed with the above presented methods.

5. Simulation results In this section, numerical simulations are conducted to demonstrate the effectiveness of the proposed control strategy. The construction parameters of the QUAV are selected as: τmx = τmy = τmz = 0.67 + pi (kg ), and τmφ = 33 + 5 pi , τmθ = 33 + 5 pi , τmψ = 25 + 5 pi (kg m2 ), for i ∈ {1, 2, 3, 4}. Gravitational acceleration is chosen as g = 9.8 m/s2 . In order to validate the effectiveness of the designed control strategy, a set of comparative simulations are conducted with four cases of time-varying payloads: p1 = 0.1sin(t ), p2 = 0.1sin(10t ), p3 = 0.1e(−10t ) and p4 = 0.1e(−100t ) . The desired trajectory is given as [x(t ), y(t ), z(t ), ψ (t )]T = [0.5sin(0.1t ), 0.5cos(0.1t ), 0.1t, 0]T , and the wind gust disturbances are chosen as dˆx = dˆy = dˆz = 0.08sin(5t ), dˆφ = dˆθ = dˆψ = 0.08sin(5t ). Some of the controller parameters are given as [c1 , c3 , c5 , k1 , k3 , k5 , 1 , 3 , 5 , r1 , r3 , r5 , δ1 , δ3 , δ5 ]T = [10, 7, 9, 5, 3, 5, 1, 2, 1, 0.1, 0.3, 0.1, 0.1, 0.8, 0.1]T and [c7 , c8 , c9 , c10 , c11 , c12 , 7 , 8 , 9 , 10 ,

11 , 12 , r7 , r8 , r9 , r10 , r11 , r12 , δ7 , δ8 , δ9 , δ10 , δ11 , δ12 ]T = [5, 3, 6, 5, 8, 5, 1, 0.8, 1, 2, 1.5, 1, 0.1, 0.2, 0.5, 0.1, 0.1, 0.7, 0.2, 0.1, 0.3, 0.1, 0.6, 0.1]T . The corresponding numerical simulation results are given from Figs. 5–7. Among them, a 3-D trajectory tracking view and the projection views are presented in Fig. 5. The tracking errors of position variables x, y, z with prescribed performance constraints are illustrated in

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Fig. 6. Tracking error plots of position variables x, y, z with PPC.

Fig. 7. Tracking error plots of angle variables φ, θ , ψ.

Fig. 6, and Fig. 7 shows the tracking errors of attitude angles φ, θ , ψ. Obviously, the well performances of the designed control scheme can be validated from the obtained results. 6. Conclusions In this paper, an important practical engineering problem is first raised and addressed for QUAVs with time-varying payloads and wind-gust disturbances. Under the proposed control strategy, the overall control system is decoupled into translational and rotational subsystem.

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First, a novel adaptive robust prescribed performance controller is designed to address the trajectory tracking problem for translational subsystem. Then, a new robust adaptive controller is constructed based on backstepping technique to stabilize the desired attitudes. Both the unknown time-varying payloads and wind gust uncertainties are considered in these subsystems. Finally, a strict and complete comparative numerical simulations are conducted, and the well performances are validated from the obtained simulation results. Acknowledgments This work was partially supported by National Natural Science Foundation of China (61751309, 61673335, 61603329, 61603330, 61503323), Basic Research Program of Hebei Province (F2016203467), Hebei Province Applied Basis Research Project (15967629D), Top Talents Project of Hebei Province. References [1] S. Bouabdallah, R. Siegwart, Full control of a quadrotor, in: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2007, pp. 153–158, doi:10.1109/IROS.2007.4399042. [2] A. Abdessameud, A. Tayebi, Global trajectory tracking control of VTOL–UAVs without linear velocity measurements, Automatica 46 (2010) 1053–1059. [3] A. Isidori, L. Marconi, A. Serrani, Robust nonlinear motion control of a helicopter, IEEE Trans. Autom. Control 48 (2003) 413–426. [4] E. Frazzoli, M.A. Dahleh, E. Feron, Trajectory tracking control design for autonomous helicopters using a backstepping algorithm, in: Proceedings of the American Control Conference, Illinois, Chicago, 2000, pp. 4102–4107. [5] T.J. Koo, S. Sastry, Output tracking control design of a helicopter model based on approximate linearization, in: Proceedings of the Thirty-seventh IEEE Conference on Decision & Control, Tampa, Florida, USA, 1998, pp. 3635–3640. [6] R. Mahony, T. Hamel, Robust trajectory tracking for a scale model autonomous helicopter, Int. J. Robust Nonlinear Control 14 (12) (2004) 1035–1059. [7] R. Olfati-Saber, Global configuration stabilization for the VTOL aircraft with strong input coupling, IEEE Trans. Autom. Control 47 (11) (2002) 1949–1952. [8] B. Zhou, H. Satyavada, S. Baldi, Adaptive path following for unmanned aerial vehicles in time-varying unknown wind environments, in: Proceedings of the American Control Conference IEEE, 2017, pp. 1127–1132. [9] M.D. Hua, T. Hamel, P. Morin, C. Samson, A control approach for thrust-propelled underactuated vehicles and its application to VTOL drones, IEEE Trans. Autom. Control 54 (8) (2009) 1837–1853. [10] A. Roberts, A. Tayebi, Adaptive position tracking of VTOL UAVs, IEEE Trans. Robot. 27 (1) (2011) 129–142. [11] H. Ramirez-Rodriguez, V. Parra-Vega, A. Sanchez-Orta, O. Garcia-Salazar, Robust backstepping control based on integral sliding modes for tracking of quadrotors, J. Intell. Robot. Syst. 73 (1) (2014) 51–66. [12] E.N. Johnson, S.K. Kannan, Adaptive trajectory control for autonomous helicopters, J. Guid. Control Dyn. 28 (3) (2005) 524–538. [13] F. Kendoul, Nonlinear hierarchical flight controller for unmanned rotorcraft: Design, stability, experiments, J. Guid., Control, Dynam. 32 (6) (2009) 1954–1958. [14] H. Liu, D. Li, Z. Zuo, Robust three-loop trajectory tracking control for quadrotors with multiple uncertainties, IEEE Trans. Ind. Electron. 63 (4) (2016) 2263–2274. [15] H. Liu, W. Zhao, Z. Zuo, Robust control for quadrotors with multiple time-varying uncertainties and delays, IEEE Trans. Ind. Electron. 64 (2) (2017) 1303–1312. [16] J. Escareno, S. Salazar, H. Romero, R. Lozano, Trajectory control of a quadrotor subject to 2d wind disturbances, J. Intell. Robot. Syst. 70 (1) (2013) 51–63. [17] M. Kerma, Nonlinear H∞ control of a quadrotor (UAV) using high order sliding mode disturbance estimator, Int. J. Control 85 (12) (2012) 1876–1885. [18] G.V. Raffo, M.G. Ortega, F.R. Rubio, An integral predictive/nonlinear H∞ control structure for a quadrotor helicopter, Automatica 46 (1) (2010) 29–39.

6338

C. Hua et al. / Journal of the Franklin Institute 355 (2018) 6323–6338

[19] K. Alexis, G. Nikolakopoulos, A. Tzes, Experimental model predictive attitude tracking control of a quadrotor helicopter subject to wind-gusts, in: Proceedings of the 2010 Eighteenth Mediterranean Conference on Control & Automation (MED), 2010, pp. 1461–1466. [20] K. Alexis, G. Nikolakopoulos, A. Tzes, Switching model predictive attitude control for a quadrotor helicopter subject to atmospheric disturbances, Control Eng. Pract. 19 (10) (2011) 1195–1207. [21] K. Alexis, G. Nikolakopoulos, A. Tzes, Model predictive quadrotor control: attitude, altitude and position experimental studies, IET Control Theory Appl. 6 (12) (2012) 1812–1827. [22] Z. Fang, W. Gao, Adaptive integral backstepping control of a micro-quadrotor, in: Proceedings of the 2011 Second International Conference on Intelligent Control and Information Processing (ICICIP), 2011, pp. 910–915. [23] D. Cabecinhas, R. Cunha, C. Silvestre, Experimental validation of a globally stabilizing feedback controller for a quadrotor aircraft with wind disturbance rejection, in: Proceedings of the American Control Conference (ACC), 2013, pp. 1024–1029. [24] Y. Chen, Y. He, M. Zhou, Y. He, M. Zhou, Modeling and control of a quadrotor helicopter system under impact of wind field, Res. J. Appl. Sci. Eng. Technol. 6 (17) (2013) 3214–3221. [25] A. Benallegue, A. Mokhtari, L. Fridman, High-order sliding-mode observer for a quadrotor UAV, Int. J. Robust Nonlinear Control 18 (2008) 427–440. [26] R. Xu, U. Ozguner, Sliding mode control of a quadrotor helicopter, in: Proceedings of the Forty-fifth IEEE Conference on Decision and Control, San Diego, 2007, pp. 4957–4962. [27] F. Sharifi, M. Mirzaei, B.W. Gordon, Y. Zhang, Fault tolerant control of a quadrotor UAV using sliding mode control, in: Proceedings of the Conference on Control and Fault Tolerant Systems, 2010, pp. 239–244. [28] A. Mockhtari, A. Benalegue, Y. Orlov, Exact linearization and sliding mode observer for a quadrotor unmanned aerial vehicle, Int. J. Robot. Autom. 21 (1) (2006) 39–49. [29] L. Besnard, Y.B. Shtessel, B. Landrum, Quadrotor vehicle control via sliding mode controller driven by sliding mode disturbance observer, J. Frankl. Inst. 349 (2) (2012) 658–684. [30] L. Luque-Vega, B. Castillo-Toledo, A.G. Loukianov, Robust block second order sliding mode control for a quadrotor, J. Frankl. Inst. 349 (2) (2012) 719–739. [31] C. Wang, Y. Lin, Decentralized adaptive tracking control for a class of interconnected nonlinear time-varying systems, Automatica 54 (2015) 16–24. [32] C.P. Bechlioulis, G.A. Rovithakis, Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed performance, IEEE Trans. Autom. Control 53 (9) (2008) 2090–2099. [33] C. Hua, L. Zhang, X. Guan, Reduced-order observer-based output feedback control of nonlinear time-delay systems with prescribed performance, Int. J. Syst. Sci. 47 (6) (2016) 1384–1393. [34] C. Hua, Y. Li, Output feedback prescribed performance control for interconnected time-delay systems with unknown Prandtl–Ishlinskii hysteresis, J. Frankl. Inst. 352 (7) (2015) 2750–2764. [35] C. Wang, C. Wen, Y. Lin, Decentralized adaptive backstepping control for a class of interconnected nonlinear systems with unknown actuator failures, J. Frankl. Inst. 352 (3) (2015) 835–850.