Exact docking flight controller for autonomous aerial refueling with back-stepping based high order sliding mode

Mechanical Systems and Signal Processing 101 (2018) 338–360

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Exact docking flight controller for autonomous aerial refueling with back-stepping based high order sliding mode Zikang Su a,b,c, Honglun Wang a,c,⇑, Na Li d, Yue Yu a,c, Jianfa Wu a,b,c a

School of Automation Science and Electrical Engineering, Beihang University, 100191 Beijing, China SHENYUAN Honors College of Beihang University, 100191 Beijing, China c Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100191, China d Unmanned Systems Research Institute, Beihang University, Beijing 100191, China b

a r t i c l e

i n f o

Article history: Received 2 May 2017 Received in revised form 13 July 2017 Accepted 22 August 2017

Keywords: Autonomous aerial refueling Docking control Trajectory tracking Back-stepping Sliding mode control

a b s t r a c t Autonomous aerial refueling (AAR) exact docking control has always been an intractable problem due to the strong nonlinearity, the tight coupling of the 6 DOF aircraft model and the complex disturbances of the multiple environment flows. In this paper, the strongly coupled nonlinear 6 DOF model of the receiver aircraft which considers the multiple flow disturbances is established in the affine nonlinear form to facilitate the nonlinear controller design. The items reflecting the influence of the unknown flow disturbances in the receiver dynamics are taken as the components of the ‘‘lumped disturbances” together with the items which have no linear correlation with the virtual control variables. These unmeasurable lumped disturbances are estimated and compensated by a specially designed high order sliding mode observer (HOSMO) with excellent estimation property. With the compensation of the estimated lumped disturbances, a back-stepping high order sliding mode based exact docking flight controller is proposed for AAR in the presence of multiple flow disturbances. Extensive simulation results demonstrate the feasibility and superiority of the proposed docking controller. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction With the persistently increasing number of unmanned aerial vehicles (UAVs) in modern military mission [1,2], the autonomous aerial refueling (AAR) [3], which enables aircraft to extend endurance and save loiter time on station by transferring fuel from the tanker aircraft to the receiver aircraft, has been an active topic [3,4]. It has drawn more and more significant interests from the research and development community [5–8], especially for the purpose of enabling unmanned aerial vehicles with this critical capability. Generally, there are two major types of aerial refueling in operation [3]: probe-drogue refueling (PDR) and boom receptacle refueling (BRR), and both play important roles in modern civil and military applications. In either case, it would be better if the receiver aircraft were exactly controlled for aerial refueling. In this paper, we focus on the probe-drogue refueling (PDR) [9,10], as shown in Fig. 1. The tanker aircraft trails a flexible refueling hose and the drogue at the end of the hose. The hose-drogue aerial refueling system in the PDR, which dragged by the flying tanker, is simple to be adapted to many existing aircrafts and can simultaneously refuel multiple receiver flights.

⇑ Corresponding author at: School of Automation Science and Electrical Engineering, Beihang University, 100191 Beijing, China. E-mail addresses: [email protected] (Z. Su), [email protected] (H. Wang). http://dx.doi.org/10.1016/j.ymssp.2017.08.036 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

339

V0 Tanker

hose

drogue probe

p bp

xB Xg

pp Zg

Yg Og

pb

yB

Receiver

OB

zB

Barycenter

Fig. 1. The configuration of a PDR system.

In a PDR, the receiver’s probe is conducted to capture the wobbly drogue which is affected by the tanker’s motion and the multiple flow disturbances including tanker trailing vortex, bow wave and atmospheric turbulence [3,9,11]. Besides, the motion of the controlled receiver is much slower than fleetly swinging drogue [3,9]. That makes it more intractable for the receiver to track the transient changing drogue. These particularity facts during the AAR docking pose great challenges on the design of a robust and exact docking controller. Although there have already been some previous works [12–21] discussing docking flight controller design for the receiver aircraft, unfortunately, few kinds of literature focus on the above problems during the controller design process. The existing literature mostly designs receiver trajectory tracking controller with the Linear Quadratic Regulator (LQR) theory [3,12–18]. However, LQR theory uses the linearized nominal plant model to design the controller [13], and the antidisturbance ability for the unknown flow disturbances are not considered during the controller designing. Actually, the receiver’s motion is affected by the stochastic atmospheric turbulence and the tail vortex field generated by the front tanker aircraft [3,5,9]. Moreover, the docking accuracy requirement is very high, and these flow disturbances will definitely affect the receiver’ motion, or even lead to the docking failure. On one hand, the amplitude and direction of the surrounding atmospheric turbulence are unpredictable. On the other hand, the amplitude and direction of the tail vortex which acts on the receiver’s body and wings will be very different due to the large scale of the receiver in the vortex. The position and attitude changing of the receiver will definitely cause considerable changing of the amplitude and direction of the tail vortex that acts on the receiver. These external flow disturbances may pose a serious impact on the LQR controller as it does not possess special anti-disturbance mechanism. This poorer anti-disturbance ability may even cause the failure of the AAR docking if the flow disturbances are strong enough. Another linear model based method known as L1 adaptive control methodology is adopted in AAR [19], but the same problem of lacking satisfied anti-disturbance ability will also be faced. And it will also not ensure the receiver’s high tracking performance to the fast moving drogue. The nonlinear dynamic inversion (NDI) is also tried to be applied to the receiver tracking control in AAR together with some uncertainty compensation technique [20–22]. However, these existing NDI based flight controllers are generally designed only in the attitude control loop. As the flow disturbances directly affect the aerodynamic forces on the receiver, and the aerodynamic forces will directly affect the receiver’s translational dynamics, these controllers cannot well ensure the satisfactory anti-disturbance ability in the receiver’s flight path or position loop. Moreover, the neural network based techniques are generally used as common techniques for the disturbance compensation in the NDI based controller [20–22]. But the complexity of the neural network parameters tuning will also limit its application in AAR. Actually, few papers designed the AAR position tracking controller entirely based on the 6 DOF nonlinear receiver model via a unified nonlinear control method (for instance, the NDI). This is because the 6-DOF nonlinear model of the receiver will be non-affine, coupled and particularly complex when the influence of the multiple flow disturbances is considered [9], especially in translational dynamics of the receiver. This also poses an extra challenge on the receiver docking controller. Although the author’s previous work in [9] tried to design the active disturbance rejection control (ADRC) [23–25] based docking controller via the disturbances or uncertainties compensation technique by the extended state observer (ESO) [26,27], the tracking performance for the fast moving drogue is still urgently needed to be improved. As the controlled receiver’s motion is much slower than the fast changing drogue, the relatively simple control structure of the linear ADRC can still not achieve satisfied tracking performance for the drogue, and the docking success rate is still needed to be improved. Thus,

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the robust and exact docking flight controller design for AAR is still open to be solved with some more excellent control structures and control methods. The sliding mode control (SMC) [28] is a special type of variable structure control which has been successful in controlling uncertain nonlinear systems. In this control methodology, the controller switches between two structures to bring the system states to a previously defined sliding manifold. When the states reach the sliding manifold, the order of the system is reduced and the system becomes immune to any kind of matched uncertainties and disturbances [29,30]. SMC processes the advantages not only in fast response and good transient performance but also in its robustness against a large class of disturbances or model uncertainties. However, conventional SMCs face the undesired phenomenon of the high-frequency oscillations known as chattering which can cause instability and damage to the system. Extensive research is continuing for chattering removal [28,31]. This chattering will be especially dangerous when the SMCs is applied to the aircraft flight system due to the high-frequency deflection of the actuator should be especially avoided in the practice. Higher order sliding mode (HOSM) control [31–34] has established itself as a successful technique to reduce chattering. In HOSM, the control acts on the derivative of the control input and the actual control are obtained by integrating the derivative control. Hence the control input is continuous and chattering is reduced. The HOSM control techniques have been, in general, designed to stabilize systems with a relative degree greater than one preserving the robustness and accuracy of the standard sliding modes. As the 6 DOF nonlinear receiver model considering the flow disturbance is non-affine, particularly coupled and underactuated, it is very intractable to apply the HOSM directly to the receiver’s flight controller. The back-stepping control technique provides a powerful tool to design controllers for the under-actuated flight systems by setting a Lyapunov function and then producing a stabilizing control law [35]. The back-stepping technique makes the design of the feedback control strategy systematic: it consists of a recursive determination of a virtual Lyapunov-based control signal and obtaining the actual control law up to the last step. Thus the back-stepping technique is more flexible in designing controllers for high order nonlinear system models [33,35]. In the last decade, the back-stepping theory has been widely used to solve the transient stabilization problems, such as the flight systems [36–38] and the nano-manipulating systems [39]. Inspired by the above analysis, this paper proposed a back-stepping high order sliding mode based flight controller which solves the AAR docking control problem with high precision in the presence of multiple flow disturbances and uncertainties. The receiver’s translational and rotational dynamics are transformed into several subsystems in affine nonlinear form, and the introduced items which have no linear correlation with the virtual control variables in each transformed subsystems are all taken as the ‘‘lumped disturbance”. These unmeasurable lumped disturbances are then observed and compensated by the specially designed HOSM observer. Then, with these estimated lumped disturbances, the receiver flight dynamics is divided into five independent subsystems/loops (position loop, flight path loop, attitude loop, angular rate loop and ground velocity loop), and the receiver’s exact docking flight controller is established based on the HOSM and modified back-stepping technique. The main contributions of this paper lie in following aspects:
The receiver aircraft nonlinear docking flight controller is established with a uniform anti-disturbance nonlinear method in the presence of multiple flow disturbances;
The receiver nonlinear 6 DOF model considering the multiple flow disturbances is presented in the affine nonlinear form to facilitate the nonlinear controller design, and the unmeasurable lumped disturbances are well estimated via a specially designed HOSM observer (HOSMO) with faster convergence characteristic and higher estimation accuracy;
With the compensation of the estimated lumped disturbances, a back-stepping HOSM based flight controller with higher tracking accuracy and stronger anti-disturbance ability is proposed for the AAR exact docking. The paper is organized as follows. The 6 DOF rigid model of the receiver and problem formulation are presented in Section 2. In Section 3, the detailed overall procedure for the receiver’s integrated flight controller design based on the HOSM and back-stepping is illustrated. The stability analysis of the closed-loop is also presented. Simulation results and analysis are shown in Section 4. The paper ends with a few concluding remarks in Section 5. 2. Receiver dynamics modeling and problem formulation Consider the 6 DOF rigid model for a receiver aircraft including wind effect described by the following translational and rotational dynamics equations [9,40]:

8 > < xp ¼ xb þ ðcos h cos wÞxbp þ ðsin /sinh cos w  cos /sinwÞybp þ ðsin /sinw þ cos / sin h cos wÞzbp yp ¼ yb þ ðcos h sin wÞxbp þ ðsin /sinh sin w þ cos / cos wÞybp þ ðcos / sin h sin w  sin / cos wÞzbp > : zp ¼ zb þ ð sin hÞxbp þ ðsin / cos hÞybp þ ðcos / cos hÞzbp

ð1Þ

8 > < x_ b ¼ V k cos c cos v y_ b ¼ V k cos c sin v > :_ zb ¼ V k sin c

ð2Þ

Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

341

8 _ > < mV k ¼ T cosða þ rÞcosb  D  Cbw þ Law  mg sin c mV k cos cv_ ¼ Tðbk cosl þ ðak þ rÞ sin lÞ þ ðC  Dbw Þcosl þ ðL  Daw Þ sin l > : mV k c_ ¼ Tðbk sin l  ðak þ rÞ cos lÞ þ ðC  Dbw Þ sin l  ðL  Daw Þ cos l þ mg cos c

ð3Þ

8 > < a_ ¼ ðq  ðp cos a þ r sin aÞ sin b  c_ cos l  v_ sin l cos cÞ= cos b b_ ¼ p sin a  r cos a  c_ sin l þ v_ cos l cos c > :_ l ¼ ðp cos a þ r sin a þ c_ sin b cos l þ v_ ðsin c cos b þ sin b sin lcoscÞÞ= cos b

ð4Þ

8 > p_ ¼ I I 1I2 ½ðIy Iz  I2z  I2xz Þrq þ ðIx Ixz  Iy Ixz  Iz Ixz Þpq þ Iz L þ Ixz N  > x z > xz < q_ ¼ I1y ½ðIz  Ix Þpr  Ixz p2 þ Ixz r 2 þ M > > > : r_ ¼ 1 2 ½ðI2  Ix Iy þ I2 Þpq  ðIx Ixz  Iy Ixz  Iz Ixz Þrq þ Ixz L þ Ix N  x xz I I I

ð5Þ

x z

xz

where pb ¼ ½xb ; yb ; zb T is the barycenter position vector in an inertial frame, pp ¼ ½xp ; yp ; zp T is the probe position in inertial frame, pbp ¼ ½xbp ; ybp ; zbp T is the probe position relative to the barycenter in body frame, /; h; w are the Euler angles, V k is the ground velocity, g is the gravity, c; v are the flight-path angle and ground track angle, a; b are the flow angles (angle of attack and sideslip angle), l is the roll angle about the velocity vector, p; q; r are the angular rates of the body-fixed reference frame. L; D; C are the lift, drag, and lateral force, L; M; N are the moments along the axis of the body frame, T is the thrust, and the explicit forms of these variables will be given in Appendix A. The obvious feature of path dynamics equations by (12) are that they are inherent non-affine nonlinear forms due to the fact that the virtual control input vectors ½V k ; c; vT and ½a; b; lT do not appear linearly in the equations directly. And the flight path system (2) is also under-actuated with two control inputs a; l and three outputs V k ; c; v; which further adds difficulty during designing control laws for subsystem (1) and (2). For the above reasons, the ground velocity of the aircraft V k is generally controlled independently by the thrust, and the vertical position z and lateral y are controlled through the actuators. Consequently, for simplicity, we define

2

3

2 3

2

3

a p da     c t1 a sin l 1 6 7 6 7 6 7 ; X2 ¼ ;t ¼ ¼ ; X3 ¼ 4 b 5; X4 ¼ 4 q 5; Uact ¼ 4 de 5; Q ¼ qV 2 X1 ¼ 2 zb v t2 a cos l r l dr 

yb



 

ð6Þ

where q is the atmosphere density, V is the airspeed, t is an intermediate virtual control vector, and Q is the dynamic pressure. Note that Xi ; i ¼ 1 . . . 0:4 and V k are taken as controlled system states. We rewrite Eqs. (25) in the following affine nonlinear form as Eqs. (7) and (8) described.

V_ k ¼ f V k þ BV k dT

ð7Þ

8 X_ 1 ¼ F1 ðX2 ; V k Þ þ B1 ðV k ÞX2 ¼ F1 þ B1 X2 ; > > > > > > X_ 2 ¼ F2 ðX2 ; X3 ; V k Þ þ B2 ðX2 ; V k ; Q Þt ¼ F2 þ B2 t; > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T

X3 ¼ ½signðt2 Þ t21 þ t22 b atanðt1 =t2 Þ ; > > > > > > X_ 3 ¼ F3 ðX_ 2 ; X3 ; X4 Þ þ B3 ðX3 ÞX4 ¼ F3 þ B3 X4 ; > > :_ X4 ¼ F4 ðX4 Þ þ B4 ðX2 ; Q ÞUact ¼ F4 þ B4 Uact ;

ð8Þ

where BV k ; Bi ; i ¼ 1; . . . ; 4 are the input matrixes for each subsystem; f V k ; Fi ; i ¼ 1; . . . ; 4 are the lumped items which have no linear correlation with the virtual control variables (X2 ;t;X4 ;Uact ) and in each subsystem. The explicit forms for f V k ; BV k ; Fi ; Bi ; i ¼ 1; . . . ; 4 are given in Appendix B. The items f V k ; Fi ; i ¼ 1; . . . ; 4 are all taken as the ‘‘lumped disturbances” for each subsystem. These lumped disturbances are composed of the items that reflect the influence of the multiple flow disturbances and other items that have no linear correlation with the virtual control variables. It’s obvious that the lumped disturbances contain parts of the aerodynamic forces and moments which are decided by the flight altitude, flight speed, and the surrounding flows. The AAR surrounding flow disturbances may disturb the aerodynamic forces and moments, and then will finally affect the receiver not only in the translational motion but also in the rotational motion. Considering the fact that these AAR surrounding flow disturbances, the aerodynamic forces and aerodynamic moments are all unmeasurable and unpredictable, these items are also unmeasurable and will definitely disturb the receiver’s motion to some degree. Thus, these items are all regarded as the lumped disturbance for convenience to handling and the disturbance compensation in flight controller design. The geometrical equation of the probe and barycenter position in vector form can be written as

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pp ¼ pb þ RI=B pbp

ð9Þ

where RI=B is the rotation matrix from the receiver body frame to the inertial frame. Based on the above analysis and definition, the control task can be given to designing an exact docking flight controller for receiver described by (7)–(9). 3. Back-stepping high order sliding mode based flight controller for AAR docking Consider the actual receiver flight in AAR docking process, the following assumptions are assumed here before the controller design. Assumption 1. All the flight state variables (Xi ; i ¼ 1; . . . ; 4 and V k ) can be obtained through direct or indirect measurement means. Assumption 2. All the ‘‘lumped disturbances” (Fi ; i ¼ 1; . . . ; 4 and f V k ) are bounded, and they are differentiable, and the differentials F_ i ; i ¼ 1; . . . ; 4; f_ V k are bounded. Namely,

f_ V k ; F_ i 2 C1 ; i ¼ 1; . . . ; 4 (

ð10Þ

kf_ V k k ¼ khV k ðtÞk 6 CV k kF_ i k ¼ khi ðtÞk 6 Ci ; i ¼ 1; ::; 4

ð11Þ

where hV k ðtÞ is the derivative of f V k ; hi ðtÞ is the derivative of Fi ; CV k ; Ci ; i ¼ 1; . . . ; 4 are known positive constants, C1 denotes the set of all differentiable functions whose derivative is continuous. Based on the time-scale separation theory, the proposed receiver docking flight controller can be divided into five subsystems/loops: ground velocity loop, position loop, flight path loop, attitude loop, as well as angular rate loop, where the five loops admit back-steeping configuration, as shown in Fig. 2. In this section, the main procedures of back-stepping HOSM (BHOSM) based receiver docking flight controller integrated with HOSM observer are presented for the AAR docking problem in the presence of multiple complex flow disturbances. The lumped disturbances are then estimated and compensated by HOSMO. Then, based on back-stepping method, the control laws for each loop are designed to suppress the tracking errors in the presence of unknown flow disturbances. The desired ground velocity is ensured by an independent control loop with the HOSMO based feedback linearization (FL). Above all, the command for the receiver barycenter needed to be transformed from the desired drogue position according to the receiver’s current attitude.

(

pb ¼ pp  RI=B pbp  T  X1 ¼ pb ð2Þ pb ð3Þ ¼ yb

zb

T

Fig. 2. The architecture of the proposed back-stepping HOSM based flight control scheme for receiver.

ð12Þ

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Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

Then, we will design the barycenter position tracking controller. But the non-affine nonlinear forms of translational dynamics (2-3) make it difficult to design a position controller and flight path controller for a receiver using the back-stepping technique. Thus, its equivalent affine nonlinear form that appears as the former two equations in (8) will be adopted for the position tracking controller and flight path controller design instead of (2-3). Define the virtual control vectors and tracking error vectors as:

8  > < eVk ¼ V k  V k ; t1 ¼ X2 ;  e1 ¼ X1  X1 ; e2 ¼ X2  t1 ; > : e3 ¼ X3  t2 ; e4 ¼ X4  t3 ;

ð13Þ

where t1 ; t2 ; t3 are the virtual controls of position, flight path, and attitude loop, respectively. According to the design principle of back-steeping design [9,33], and with the compensation of estimated lumped disturbances by the HOSMO, the back-stepping HOSM based receiver flight controller can be designed as following steps. Step 1: As t1 ¼ X2 and e1 ¼ X1  X1 defined in Eq. (13), the virtual control for the position loop is proposed as (14) according to the output FL [41].



_

_

t1 ¼ B1 F 1 þ X_ 1 þ H1 e1 1

 ð14Þ _

_ ¼X _  þ e1 is the first derivative where the Hurwitz matrix H1 provides the rate of the convergence of the position loop, X 1 1  _ ; and e1 is the bounded derivative estimation error according to TD convergence estimation of the command signal vector X 1 _

theory [42], B1 1 is the inversion of B1 ; and F 1 is the estimation of F1 by the HOSMO to be designed. As Eq. (8) presented, in the receiver’s translational and rotational dynamics, Fi ; i ¼ 1; . . . ; 4 and f V k are taken as the lumped disturbances. Take the position loop dynamic (first equation in (8)) for an example, the lumped disturbance F1 is taken as an extended state of the position loop dynamic, and we can obtain the augmented position loop dynamic:

(

X_ 1 ¼ F1 ðX2 ; V k Þ þ B1 ðV k ÞX2 ¼ F1 þ B1 t1 F_ 1 ¼ h1 ðtÞ

ð15Þ

where h1 ðtÞ is the derivative of the lumped disturbance F1 : Then, based on the HOSM theory and super-twisting algorithm, we established the HOSMO (16) for the augmented position loop dynamic (15).

8 1 _ _ 2 _ 1 _ > __ : F 1 ¼ 1:1C1 signðX1  X1 Þ _

k

_

_

ð16Þ

_

_

T

where sigðX1  X1 Þ ¼ ½ jyb  y b jk signðyb  y b Þ jzb  z b jk signðzb  z b Þ  ; and C1 ¼ diagð½ C 11 C 12 T Þ is the observer parameter to be selected. Step 2: Analogously, considering e2 ¼ X2  t1 defined in Eq. (13), the virtual control for flight path loop can be proposed as (17). _

_

T _ t2 ¼ B1 2 ðF 2 þ t 1 þ H2 e2  B1 e1 Þ

ð17Þ _

where the Hurwitz matrix H2 provides the rate of the convergence of the flight path loop, t_ 1 ¼ t_ 1 þ e2 is the first derivative _

estimation of the command signal vector t_ 1 ; e2 is the derivative estimation error vector, B1 2 is the inversion of B2 , and F 2 is the estimation of F2 by HOSMO to be designed. Take the lumped disturbance F2 as an extended subsystem state with the flight path loop dynamic, and we obtain the augmented flight path loop dynamic:

(

X_ 2 ¼ F2 ðX2 ; X3 ; V k Þ þ B2 ðX2 ; V k ; QÞt ¼ F2 þ B2 t2 F_ 2 ¼ h2 ðtÞ

ð18Þ

where h2 ðtÞ is the derivative of the lumped disturbance F2 : Then, we established the HOSMO (19) for the augmented flight path loop dynamic (18)

8 _ 12 _ 1 < __ X2 ¼ B2 t2 þ 1:5C22 sigðX2  X2 Þ þ F 2 _ :_ F 2 ¼ 1:1C2 signðX2  X2 Þ

ð19Þ

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Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360 _

k

_

_

_

_

T

where sigðX2  X2 Þ ¼ ½ jv  v jk signðv  vÞ jc  c jk signðc  c Þ  ; and C2 ¼ diagð½ C 21 C 22 T Þ is the observer parameter to be selected. Step 3: With the definition e3 ¼ X3  t2 in Eq. (13), the virtual control for attitude loop can be proposed as (20). _

_

T _ t3 ¼ B1 3 ðF 3 þ t 2 þ H3 e3  B2 e2 Þ

ð20Þ _

where the Hurwitz matrix H3 provides the rate of the convergence of the attitude loop, t_ 2 ¼ t_ 2 þ e3 is the first derivative _

estimation of the command signal vector t_ 2 ; e3 is the derivative estimation error vector, B1 3 is the inversion of B3 , and F 3 is the estimation of F3 by HOSMO to be designed. Analogously, the lumped disturbance F3 is viewed as an extended subsystem state with the attitude loop dynamic, and we obtain the augmented attitude loop dynamic:

(

X_ 3 ¼ F3 ðX_ 2 ; X3 ; X4 Þ þ B3 ðX3 ÞX4 ¼ F3 þ B3 t3 F_ 3 ¼ h3 ðtÞ

ð21Þ

where h3 ðtÞ is the derivative of the lumped disturbance F3 : Then, the HOSMO (22) is designed for the augmented attitude loop dynamic (21)

8 1 _ _ 2 _ 1 _ > __ : F 3 ¼ 1:1C3 signðX3  X3 Þ _

k

ð22Þ

_

_

T

_ T _ _ where sigðX3  X3 Þ ¼ ½ ja  _ a jk signða  aÞ jb  b jk signðb  b Þ jl  l jk signðl  lÞ  ; and C3 ¼ diagð½ C 31 C 32 C 33  Þ is the observer parameter to be selected. Step 4: Considering e4 ¼ X4  t3 defined in Eq. (13), the virtual control for angular rate loop can be proposed as (23) according to the HOSM control method. Z

_

_

2 _ Uact ¼ B1 4 ðF 4 þ t 3 þ ðbsigðsÞ þ 1

ðasignðsÞÞdtÞ  BT3 e3 Þ

ð23Þ

_

where t_ 3 ¼ t_ 3 þ e4 is the first derivative estimation of the command signal t_ 3 ; e4 is the derivative estimation error vector, a ¼ diagða1 ; a2 ; a3 Þ and b ¼ diagðb1 ; b2 ; b3 Þ are the control parameters of the HOSM controller to be selected, the sliding manifold vector is defined as s ¼ e4 ¼ ½ ep

eq

_

er T ; and then sigðsÞk ¼ sigðe4 Þk ¼ ½ jep jk signðep Þ

jeq jk signðeq Þ

T

1 jer jk signðer Þ  , B4

is the inversion of B4 ; F 4 is the estimation of F4 by HOSMO to be designed. Similarly, the lumped disturbance F4 is viewed as an extended subsystem state with the angular rate loop dynamic, and we obtain the augmented angular rate loop dynamic:

(

X_ 4 ¼ F4 ðX4 Þ þ B4 ðX2 ; Q ÞUact ¼ F4 þ B4 Uact F_ 4 ¼ h4 ðtÞ

ð24Þ

where h4 ðtÞ is the derivative of the lumped disturbance F4 : Then, the HOSMO (25) is established for the augmented angular rate loop dynamic (23)

8 1 _ _ 2 _ 1 _ > __ : F 4 ¼ 1:1C4 signðX4  X4 Þ _

k

_

_

ð25Þ

_

_

_

_

T

where sigðX4  X4 Þ ¼ ½ jp  p jk signðp  p Þ jq  q jk signðq  q Þ jr  r jk signðr  r Þ  ; and C4 ¼ diagð½ C 41 the observer parameter to be selected. Step 5: the independently designed ground velocity control law is given: _

_

_ dT ¼ B1 V k ð f V k þ HV k eV k þ V K Þ

C 42

T

C 43  Þ is

ð26Þ _

where the positive constant HV k provides the rate of the convergence of the ground velocity loop, V_ K ¼ V_ K þ e0 is the first derivative estimation of the command signal vector V_  , e0 is the derivative estimation error vector, B1 is the inversion of K

_

Vk

BV k , and f V k is the estimation of f V k by HOSMO. The lumped disturbance f V k is viewed as an extended subsystem state with the ground velocity loop dynamic, then we obtain the augmented ground velocity loop dynamic:

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(

V_ k ¼ f V k þ BV k dT f_ V k ¼ hV k ðtÞ

ð27Þ

where hV k ðtÞ is the derivative of the lumped disturbance f V k : And the HOSMO (28) is established for the augmented ground velocity loop dynamic (27)

8 1 _ _ _ > < V_ ¼ BV dT þ 1:5C 12 sigðV  V Þ2 þ f V k k k Vk k k > _ : __ f V k ¼ 1:1C V k signðV k  V k Þ _

_

k

ð28Þ

_

where sig ðV k  V k Þ ¼ jV k  V k jk signðV k  V k Þ; and C V k is the observer parameter to be selected. Theorem 1. Let the perturbed system (78) be affected by the disturbances which satisfy the assumption 2. Using the estimated lumped disturbances (16), (19), (22), (24) and (28), the controller (14), (17), (20), (23) and (26) guarantee that the asymptotic convergence of the control output to the command X1 ; V K and that asymptotic stability of the closed-loop system.

Proof 1. We will prove Theorem 1 from the flowing two parts.

(1) In the first part of the proof, we investigate the stability and the finite time convergence of the HOSMOs. Consider Assumption 2 is satisfied by the lumped disturbances, the receiver dynamic (8) with the lumped disturbances as an extended state can be written in the form (15), (18), (21), (24) and (27). Then, consider the HOSMO (16) applied to the dynamic (15), we define the observer estimation error _

_

~1 ¼ F1  F 1 According to (15) and (16), the estimation error dynamics can then be obtained as: ~ 1 ¼ X1  X1 ; F X

8 1 1
ð29Þ

C1 

According to the convergence consequence analysis of the super-twisting sliding mode controller by Levant [31], the esti~ 1; F ~1 converges to zero after a finite-time T 10 transient process, and X ~ 1; F ~1 are bounded throughout the conmation error X vergence process. Analogously, _

the

convergence

consequence

analysis

of

the

estimation

_

errors

_

_

~ k ¼ V k  V k , ~f V ¼ f  f V , V Vk k k

~i ¼ Fi  F i ; i ¼ 2; 3; 4 can be proved in a similar form when the observer (19), (22), (25) and (28) are applied ~ i ¼ Xi  Xi ; F X ~ i; F ~i ; i ¼ 2; 3; 4 converges to zero after the ~ k ; ~f V and X to the subsystem (18), (21), (24) and (27). The estimation error V k finite-time T 00 and T i0 ; i ¼ 2; 3; 4; respectively. And they are bounded throughout the convergence process. It should be specially remarked that the transient of the disturbance estimation process can affect the behavior of the closed loop system, i.e., the control (14), (17), (20), (23), (26) depend on the estimated disturbances. And the dynamics of the HOSMOs (16), (19), (22), (25) and (28) also on the basis of virtual control (14), (17), (20) and actual control (23), (26). Therefore, the stability and the convergence characteristic analysis of the closed-loop flight control system, which will be discussed below, must take the impact of the disturbance estimation process by HOSMO into consideration. _

_

_

_

_

Remark 1. The estimations for the first derivative signals (V_ k ; X_ 1 ; t_ 1 ; t_ 2 ; t_ 3 ) of the command signals V k ; X1 ; t1 ; t2 ; t3 in above control laws are obtained via the tracking differentiator (TD), as shown in Fig. 2. According to the convergence proof in [42], the estimation errors are uniformly convergent to zero, and the estimation errors e0 ; ei ; i ¼ 1; . . . ; 4 are bounded.

(2) In the second part of the proof, the stability and convergence of the back-stepping HOSM based receiver flight controller are discussed. Including the lumped disturbances estimation errors by HOSMOs and the first derivative estimation errors by TD, we continue the following closed-loop stability proof.
As the ground velocity is a relatively independent subsystem with respect to the position controllers, we first consider the following Lyapunov function for this subsystem;

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Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

V 0 ¼ 0:5eV k eV k

ð30Þ

The first derivate is given as

V_ 0

¼ eV k e_ V k ¼ eV k ðf V k þ BV k dT  V_ K Þ

ð31Þ

_

_ _ ¼ eV k ðf V k þ BV k B1 V k ðf V k þ H V k eV k þ V K þ e0 Þ  V K Þ ¼ eV k HV k eV k þ eV k ð~f V k þ e0 Þ

The estimation error of the lumped disturbance ~f V k is will convergence to zero in a finite time T 00 ; according to the convergence consequence of the super-twisting sliding mode control [31]. And it is bounded throughout the convergence process. Then, we get inequality (32)

V_ 0

¼ eV k HV k eV k þ eV k ð~f V k þ e0 Þ |fflfflfflfflfflffl{zfflfflfflfflfflffl} D0

ð32Þ

2

6 kHV k kkeV k k þ keV k kkD0 k ¼ keV k kðkHV k kkeV k k  kD0 kÞ

Noted that D0 is the residual estimation error by HOSMO and TD, and it’s bounded with proper parameters selection of the observer and differentiator. Thus, V_ 0 will be negative if the Hurwitz matrix HV k is selected large enough. And the error eV k converges asymptotically to zero and V k converges asymptotically to V k :
Then, consider the error e1 ¼ X1  X1 , let’s select the following Lyapunov function for the pair ðeV k ; e1 Þ:

V 1 ¼ 0:5eV k eV k þ 0:5e1 eT1

ð33Þ

The first derivate is given by

V_ 1

¼ eV k e_ V k þ eT1 e_ 1 ¼ eV k HV k eV k þ eV k ð~f V k þ e0 Þ þ eT1 ðX_ 1  X_ 1 Þ ¼ eV k HV k eV k þ eV k ð~f V k þ e0 Þ þ eT1 ðF1 þ B1 X2  X_ 1 Þ

Consider that X2 ¼ t1 and Eq. (14), the first derivate of Lyapunov candidate V 1 becomes

V_ 1 ¼ eV k HV k eV k þ eV k D0 þ eT1 ðF1 þ B1 t1  X_ 1 Þ _

_ _ ¼ eV k HV k eV k þ eV k D0 þ eT1 ðF1 þ B1 B1 1 ðF 1 þ X1 þ e1 þ H1 e1 Þ  X1 Þ T ~ ¼ eV HV eV þ eV D0 þ e ðH1 e1 þ F1 þ e1 Þ k

k

k

ð34Þ

1

k

~1 þ e1 ÞD ¼ eV k HV k eV k þ eV k D0 þ eT1 H1 e1 þ eT1 ðF |fflfflfflfflffl ffl{zfflfflfflfflfflffl} 1 where D1 is the bounded residual estimation error by HOSMO and TD in position subsystem. We can get inequality (35),

V_ 1 6 kHV k ke2V k  eT1 e1 kH1 k þ keV k kkD0 k þ keT1 kkD1 k ¼ keV k kðkHV k kkeV k k  kD0 kÞ  keT1 kðke1 kkH1 k  kD1 kÞ

ð35Þ

Thus, V_ 1 may be negative if Hurwitz matrix HV k ; H1 are properly selected, and the error system ðeV k ; e1 Þ will converge asymptotically to zero. Hence, the receiver barycenter position X1 converges asymptotically to the command vector X1 and V k converges asymptotically to V k :
Next, consider the second error coordinate e2 ¼ X2  t1 : The Lyapunov function for the pair ðeV k ; e1 ; e2 Þ is selected

V 2 ¼ 0:5eV k eV k þ 0:5e1 eT1 þ 0:5e2 eT2

ð36Þ

The first derivate of is V 2 get

V_ 2

¼ eV k e_ V k þ eT1 e_ 1 þ eT2 e_ 2 ¼ eV k HV k eV k þ eV k D0 þ eT1 ðF1 þ B1 X2  X_ 1 Þ þ eT2 e_ 2 ¼ eV HV eV þ eV D0 þ eT ðF1 þ B1 ðt1 þ e2 Þ  X_  Þ þ eT e_ 2 k

k

k

k

1

1

2

_

T T_ _ _ ¼ eV k HV k eV k þ eV k D0 þ eT1 ðF1 þ B1 ðB1 1 ðF 1 þ X1 þ e1 þ H1 e1 ÞÞ  X1 Þ þ e1 B1 e2 þ e2 e2 T T ~ T T_ ¼ eV HV eV þ eV D0 þ e H1 e1 þ e ðF1 þ e1 Þ þ e B1 e2 þ e e2 k

k

k

k

1

1

1

2

¼ eV k HV k eV k þ eV k D0 þ eT1 H1 e1 þ eT1 D1 þ eT1 B1 e2 þ eT2 ðF2 þ B2 X3  t_ 2 Þ Consider X3 ¼ t2 and submitting Eq. (17) into (37) yields

ð37Þ

Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

V_ 2

347

¼ eV k HV k eV k þ eV k D0 þ eT1 H1 e1 þ eT1 D1 þ eT1 B1 e2 þ eT2 ðF2 þ B2 t2  t_ 2 Þ ¼ eV k HV k eV k þ eV k D0 þ eT1 H1 e1 þ eT1 D1 þ eT1 B1 e2 _

T _ _ þeT2 ðF2 þ B2 B1 3 ðF 3 þ t1 þ e2 þ H3 e3  B3 e2 Þ  t2 Þ

ð38Þ

¼ eV k HV k eV k þ eV k D0 þ eT1 H1 e1 þ eT1 D1 þ eT2 H3 e3 þ eT2 ðF2 þ e2 ÞD2 |fflfflfflfflfflffl{zfflfflfflfflfflffl} 2 2 X X ¼ eV k HV k eV k þ eV k D0 þ eTi Hi ei þ eTi Di i¼1

i¼1

where D2 is the bounded residual estimation error by HOSMO and TD in flight path subsystem. Then, inequality (39) can be got

V_ 2

6 kHV k ke2V k 

2 2 X X kHi keTi ei þ keV k kkD0 k þ keTi kkDi k i¼1

i¼1

¼ keV k kðkHV k kkeV k k  kD0 kÞ 

2 X

ð39Þ

keTi kðkei kkHi k  kDi kÞ

i¼1

Thus, V_ 2 may be negative if Hurwitz matrix HV k ; Hi ; i ¼ 1; 2 are selected large enough, and the error system ðeV k ; e1 ; e2 Þ will asymptotically converge. Hence, X2 converges asymptotically to the virtual control command t1 ; X1 converges asymptotically to the command signal X1 ; and V k converges asymptotically to V k :
Once again, consider the second error coordinate e3 ¼ X3  t2 : The Lyapunov function for the pair ðeV k ; e1 ; e2 ; e3 Þ is given

V 3 ¼ 0:5eV k eV k þ 0:5e1 eT1 þ 0:5e2 eT2 þ 0:5e3 eT3

ð40Þ

We get the first derivate as

V_ 3

¼ eV k e_ V k þ eT1 e_ 1 þ eT2 e_ 2 þ eT3 e_ 3 ¼ eV k HV k eV k þ eV k D0 þ eT1 H1 e1 þ eT1 D1 þ eT2 ðF2 þ B2 X3  X_ 2 Þ þ eT3 e_ 3 ¼ eV k HV k eV k þ eV k D0 þ eT1 H1 e1 þ eT1 D1 þ eT2 ðF2 þ B2 ðt2 þ e3 Þ  t_ 1 Þ þ eT3 e_ 3 ¼ eV k HV k eV k þ eV k D0 þ eT1 H1 e1 þ eT1 D1 _

T T_ _ _ þeT2 ðF2 þ B2 ðB1 2 ðF 2 þ t1 þ e2 þ H2 e2 ÞÞ  t1 Þ þ e2 B2 e3 þ e3 e3 T T T T ~ ¼ eV HV eV þ eV D0 þ e H1 e1 þ e D1 þ e H2 e2 þ e ðF2 þ e2 Þ þ eT B2 e3 þ eT e_ 3 k

k

k

¼ eV k H V k eV k

1

k

1

2

2

2

ð41Þ

3

2 2 X X þ eV k D0 þ eTi Hi ei þ eTi Di þ eT2 B2 e3 þ eT3 ðF3 þ B3 X4  t_ 2 Þ i¼1

i¼1

Consider X4 ¼ t3 and submitting Eq. (20) into (41) yields

V_ 3

¼ eV k HV k eV k þ eV k D0 þ

2 2 X X eTi Hi ei þ eTi Di þ eT2 B2 e3 þ eT3 ðF3 þ B3 t3  t_ 2 Þ i¼1

¼ eV k H V k eV k

i¼1

2 2 X X þ eV k D0 þ eTi Hi ei þ eTi Di þ eT2 B2 e3 i¼1

i¼1

_

T _ _ þeT3 ðF3 þ B3 B1 3 ðF 3 þ t2 þ e3 þ H3 e3  B3 e2 Þ  t2 Þ 2 2 X X ~ 3 þ e 3 ÞD ¼ eV k HV k eV k þ eV k D0 þ eTi Hi ei þ eTi Di þ eTi Hi ei þ eTi ðF |fflfflfflfflffl ffl{zfflfflfflfflfflffl} 3 i¼1 i¼1

¼ eV k HV k eV k þ eV k D0 þ

ð42Þ

3 3 X X eTi Hi ei þ eTi Di i¼1

i¼1

where D3 is the bounded residual estimation error by HOSMO and TD in attitude loop subsystem. And inequality (43) can be got

V_ 3

6 kHV k ke2V k 

3 3 X X kHi keTi ei þ keV k kkD0 k þ keTi kkDi k i¼1

¼ keV k kðkHV k kkeV k k  kD0 kÞ 

i¼1 3 X i¼1

keTi kðkei kkHi k

ð43Þ  kDi kÞ

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Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

Then, V_ 3 may be negative if Hurwitz matrix HV k ; Hi ; i ¼ 1; 2; 3 are selected large enough, and the error system ðeV k ; e1 ; e2 Þ will converge asymptotically to zero. Then, Xi ; i ¼ 2; 3 converges asymptotically to the virtual control command ti ; i ¼ 1; 2 and X1 converges asymptotically to the command X1 , and V k converges asymptotically to V k :
Finally, consider the second error coordinate e4 ¼ X4  t3 : The Lyapunov function for the pair ðeV k ; e1 ; e2 ; e3 ; e4 Þ is given

V4

¼ 0:5eV k eV k þ 0:5e1 eT1 þ 0:5e2 eT2 þ 0:5e3 eT3 þ 0:5ssT

ð44Þ

¼ 0:5eV k eV k þ 0:5e1 eT1 þ 0:5e2 eT2 þ 0:5e3 eT3 þ 0:5e4 eT4 The first derivate of V 4 can be given as

V_ 4

¼ eV k e_ V k þ eT1 e_ 1 þ eT2 e_ 2 þ eT3 e_ 3 þ eT4 e_ 4 2 2 X X ¼ eV k HV k eV k þ eV k D0 þ eTi Hi ei þ eTi Di þ eT3 ðF3 þ B3 X4  X_ 3 Þ þ eT4 e_ 4 i¼1

¼ eV k HV k eV k

i¼1

2 2 X X þ eV k D0 þ eTi Hi ei þ eTi Di i¼1

i¼1

þeT3 ðF3 þ B3 ðt3 þ e4 Þ  t_ 2 Þ þ eT4 e_ 4 2 2 X X ¼ eV k HV k eV k þ eV k D0 þ eTi Hi ei þ eTi Di i¼1

ð45Þ

i¼1

_

T T_ _ _ þeT3 ðF3 þ B3 ðB1 3 ðF 3 þ t2 þ H3 e3 ÞÞ  t2 Þ þ e3 B3 e4 þ e4 e4 3 3 X X ¼ eV k HV k eV k þ eV k D0 þ eTi Hi ei þ eTi Di þ eT3 B3 e4 þ eT4 e_ 4 i¼1

¼ eV k HV k eV k

i¼1

3 3 X X þ eV k D0 þ eTi Hi ei þ eTi Di þ eT3 B3 e4 þ eT4 ðF4 þ B4 Uact  t_ 3 Þ i¼1

i¼1

Consider X4 ¼ t3 and submitting Eq. (23) into (45) yields

V_ 4

¼ eV k HV k eV k þ eV k D0 þ 0 þeT4 @F4 þ

0

3 3 X X eTi Hi ei þ eTi Di þ eT3 B3 e4 i¼1

i¼1

1 1 þ t_ 3 þ e4  BT4 e3 þ A  t_ 3 A R 1 ðbsigðsÞ2 þ ðasignðsÞÞdtÞ _

@ F 4 B4 B1 4

¼ eV k HV k eV k

3 3 X X R 3 ~ 4 þ e 4 ÞD þ eV k D0 þ eTi Hi ei þ eTi Di  bke4 k2  ake4 k dt þ eTi ðF |fflfflfflfflffl ffl{zfflfflfflfflfflffl} 4 i¼1 i¼1

¼ eV k HV k eV k þ eV k D0 þ

ð46Þ

3 4 X X R 3 eTi Hi ei  bke4 k2  ake4 k dt þ eTi Di i¼1

i¼1

where D4 is the bounded residual estimation error by HOSMO and TD in angular rate subsystem. Then, inequality (47) establishes.

V_ 4

6 kHV k ke2V k 

4 3 X X R 3 kHi keTi ei  bke4 k2  ake4 k dt þ keV k kkD0 k þ keTi kkDi k i¼1

i¼1

3 X R 1 ¼ keV k kðkHV k kkeV k k  kD0 kÞ  keTi kðkei kkHi k  kDi kÞ  ke4 kðbke4 k2 þ a dt  kD4 kÞ

ð47Þ

i¼1

Therefore, V_ 4 may be negative if Hurwitz matrixes HV k ; Hi ; i ¼ 1; 2; 3 and diagonal matrixes b; a are selected properly, and the error system ðeV k ; e1 ; e2 ; e3 ; e4 Þ will asymptotically converge to zero. And Xi ; i ¼ 2; 3; 4 converges asymptotically to the virtual control command ti ; i ¼ 1; 2; 3 and X1 converges asymptotically to X1 ; and V k converges asymptotically to V k : Q.E.D.j 4. Simulation results and comparison To verify the validity of proposed back-stepping HOSM based anti-disturbance flight controller for the AAR docking problem, extensive simulations have been carried out based on the 6 DOF model of receiver described by (7-8), and the receiver in [17] and the hose-drogue assembly in [43] are adopted here. In addition, the ADRC control scheme in [9] (all the five loops are designed with FL) may be an alternative method to make comparisons with the proposed back-stepping HOSM (BHOSM) control scheme in the AAR docking scenario. However, as different disturbance observers will lead different performances of the closed-loop system, the ESO in the ADRC control scheme is replaced with the HOSMO for the comparisons fairness here.

Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

349

The FL controller and TD in ADRC are retained. We take the abbreviation ‘‘FLC” to denote this compared FL control scheme presented as Eq. (48).

8 _  1 _ > > > X2 ¼ t1 ¼ B1 ðF 1 þ H1 e1 þ X1 Þ > > _ > > > X3 ¼ t2 ¼ B1 _ > 2 ðF 2 þ H2 e2 þ t1 Þ > < _  1 X4 ¼ t3 ¼ B3 ðF 3 þ H3 e3 þ t_ 2 Þ > > _ > > > ðF 4 þ H4 e4 þ t_ 3 Þ Uact ¼ B1 > 4 > > > _ > : _ dT ¼ B1 V k ð f V k þ H V k eV k þ V K Þ _

ð48Þ

_

where f V k ; F i ; i ¼ 1; . . . ; 4 are obtained with the same HOSMO in (16), (19), (22), (24) and (28). Actually, one can find that controller (48) with HOSMOs is the same robust FL control scheme proposed in [37]. The simulations are conducted in MATLAB/SIMULINK 2011a and implemented on a personal computer with 3.2 GHz Core i5 CPU. The simulation sampling period is set as 0.02 s. For all numerical simulations presented below, the initial parameters and controller parameters are provided in Table 1. In the simulation study, the light and moderate turbulence refueling environment are verified. Case 1: Suppose that the AAR is conducted in the light turbulence, as shown in Fig. 3(a). The sums of the multiple flow disturbances which receiver is exposed are shown in Fig. 3(b). Assume that the receiver starts to track the drogue from t ¼ 30 s: In this case, the drogue’s trajectory from the 30 s to 100 s by the nonlinear hose-drogue model [43] is shown in Fig. 4. The probe position tracking results for the desired drogue target in Fig. 4 are shown in Fig. 5, together with the tracking errors and the integral of time-weighted absolute error (ITAE). From Fig. 5, it can be seen that the controlled probe trajectories by the proposed BHOSM are more adjacent to the desired drogue positions compared to FLC with HOSMO. The probe position tracking errors in y-axis and z-axis are all constrained below 0.1 m, which is smaller than FLC scheme. The ITAE index results can also directly verify the superiority of the proposed BHOSM to track the transient changing drogue under light turbulence environment. To be more visualized, Fig. 6 gives the statistical results for the probe tracking errors in the YOZ plane. The red cycle denotes the AAR tracking error requirement boundary R0 ¼ 0:3 m, the green cycle is the reachable tracking error boundary for each method under light turbulence. It can be seen that the both control methods hold the tracking errors within the required boundary under light turbulence. Even so, one can still find the advantages of the proposed BHOSM. It ensures the tracking error within the reachable error cycle R 6 0:1 05 m, and the FLC scheme ensures R 6 0:13 m. The proposed method performs better in tracking the rapidly changed drogue. The tracking trajectory by BHOSM is closer to the drogue trajectory, which can be easily got in Fig. 5. Actually, its tracking trajectories are slightly shifted forward to some extent. The 6 DOF nonlinear closed-loop simulation for receiver position tracking and relevant results are shown in Figs. 7–9. In order to demonstrate the advantages of the proposed controller, the results for the position tracking, position tracking errors together with their integral of time-weighted absolute error (ITAE) are presented primarily in Figs. 7 and 8. The tracking for the ground velocity by FLC with HOSMO and the proposed method are similarly as they adopt the same controller. But on can still find the ground velocity tracking error by BHOSM is smaller. The receiver barycenter position tracking errors by the BHOSM are both smaller. The ITAE figures for the tracking errors are also be given to further show the tracking performance of these methods. The ITAE index from 30 s to 100 s for the lateral and vertical position error by BHOSM are always smaller. And they are ensured below 160, which are all much smaller than that of FLC scheme. Moreover, it can also be seen that the tracking trajectories by the BHOSM are shifted forward, which can also reduce the slower dynamic response compared to the rapidly changing drogue under light turbulence refueling environment.

Table 1 Simulation parameters of the compared control schemes. Flight condition and tracking error limitations

Altitude: 7010 m; Ground Velocity: 200 m/s; tracking Error Requirement: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2y þ e2z 6 0:3 m;

0m6R¼ FLC with HOSMO (all loops are designed with FL)

Proposed Back-steeping HOSM (BHOSM)

Controller gain for each loop: HV k ¼ 1:2; H1 ¼ 1  diagð1; 1Þ; H2 ¼ 2  diagð1; 1Þ; H3 ¼ 3  diagð1; 1; 1Þ; H4 ¼ 8  diagð1; 1; 1Þ; C ¼ 0:005diagð1; 1Þ; C2 ¼ 0:008diagð1; 1Þ; HOSMO parameter for each loop: 1 C3 ¼ 0:05diagð1; 1; 1Þ; C4 ¼ 0:2diagð1; 1; 1Þ; C V k ¼ 0:05 r ¼ 5; r1 ¼ diagð1; 1Þ; r2 ¼ diagð4; 4Þ; h ¼ 0:02s; TD parameter for each loop: V k r3 ¼ diagð10; 10; 10Þ; r4 ¼ diagð20; 20Þ; h0 ¼ 3h Controller parameters in (23):

a ¼ 1:1  11  diagð1; 1; 1Þ; Other parameters 0:5

b ¼ 1:5  1  diagð1; 1; 1Þ (HV k ; Hi ; i ¼ 1; 2; 3; C V k ; Ci ; i ¼ 1; . . . ; 4 and the parameters for TDs) are kept the same as the FLC with HOSMO

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Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

(b). Sum of flows Wx (m/s)

0 -1

0

50 t(s)

1 0 -1

0

50 t(s)

Wz(m/s)

Turbz (m/s)

0 0

50 t(s)

0

100

0

50 t(s)

100

0

50 t(s)

100

0

50 t(s)

100

-2 -4 -6

100

1

-1

1

-1

100

Wy (m/s)

Turby (m/s)

Turbx(m/s)

(a). Turbulence 1

4 2 0

Fig. 3. Airflows in Case 1.

Drogue trajectory in Case 1 -6999.45 -6999.5 -6999.55 Start

z(m)

-6999.6 -6999.65 -6999.7 -6999.75 End

-6999.8 -6999.85

-18.1

-18

-17.9

-17.8

-17.7

y(m) Fig. 4. Drogue trajectory in Case 1.

Other results for the receiver’s states, such as the path angles, flow angles, and the Euler angles, are shown in Fig. 9. The results of these angles for the two methods are similar. And compared to FLC with HOSMO, the forward leadings by the proposed method can also be carefully identified in the receiver’s angle states. The results for the control inputs can be seen in Fig. 10. The BHOSM achieves better tracking performance at the cost of the slight greater actuator surface deflections. Case 2: Suppose that the AAR is conducted in the moderate turbulence, as shown in Fig. 11(a). The sums of the multiple flow disturbances that receiver is exposed are given in Fig. 11(b). The receiver is also assumed to starts to track the drogue at the time t ¼ 30 s: In this case, the drogue’s trajectory from the 30 s to 100 s by the nonlinear hose-drogue model is shown in Fig. 12. It can be found that the drogue moves faster and its motion range is bigger under moderate turbulence, which will be more challenging for the receiver docking controller. The probe position tracking results, tracking errors and their ITAE to the desired drogue are presented in Fig. 13. Under the moderate turbulence AAR scenario, the trajectory tracking errors increase as the intensity of the turbulence increases. The tracking errors by FLC scheme has already violated the tracking error limitations. But BHOSM can also control the probe position to be more adjacent to the drogue trajectory. The BHOSM constraints the probe position tracking errors in the y-axis

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Drogue FLC Proposed BHOSM

-17.7

68

-6999.5

70

-17.9 -18

30

40

50

60

70 t(s)

0.2

FLC Proposed BHOSM

94

-6999.6

80 0.1 0.05

90

100

-6999.8 30

0.1

66 67 68 69 ezp(m)

0.1

eyp(m)

92

-6999.7

-18.1

0

40

50

60

70 t(s) 0.1

80

0.05 FLC Proposed BHOSM 91

90

100

90

100

92

0.05 0 -0.05

-0.1 30

40

50

60

70

80

t(s) 200

90

200

X: 100 Y: 152.1

100 50 0 30

40

50

60

70

80

90

40

50

60

70

80

t(s)

X: 100 Y: 180.7

FLC Proposed BHOSM

150

-0.1 30

100

ITAEezp(m)

ITAEeyp,(m)

-6999.55 -6999.6 Drogue -6999.65 FLC Proposed BHOSM

-6999.4

z p(m)

yp(m)

-17.8

-17.8 -17.85 -17.9

X: 100 Y: 166.4

FLC Proposed BHOSM

150

X: 100 Y: 157

100 50 0 30

100

40

50

t(s)

60

70

80

90

0.2

0.3

100

t(s)

Fig. 5. Simulation results for probe position tracking in Case 1.

FLC

Proposed BHOSM

0.3

0.3

0.2

0.2

0

0.1 ez (m)

ez (m)

0.1 R0=0.3

-0.1

R0=0.3 0 -0.1 R=0.105

R=0.13

-0.2

-0.3 -0.2 -0.1

0 ey(m)

0.1

-0.2

0.2

0.3

-0.3 -0.2 -0.1

0

0.1

ey(m)

Fig. 6. Statistical results for the probe position tracking errors in Case 1.

and z-axis below 0.2 m and 0.3 m, respectively. They are smaller compared to FLC scheme. The proposed method still works and ensures the required tracking error limitations even under moderate turbulence. And the maximum of ez has been nearly decreased by 19%, compared to the FLC. The ITAE index results also show the superior tracking errors’ statistical results of the proposed BHOSM to track the transient changing drogue under moderate turbulence and other flow disturbances. The ITAE index from 30 s to 100 s for the lateral is ensured below 280 and vertical position error is ensured below 381. Fig. 14 gives the statistical results for the probe tracking errors in the YOZ plane under this scenario. One can see that the FLC scheme cannot still ensure the tracking errors are always among required red cycle R0 ¼ 0:3 as the errors may cross the required red boundary sometimes. But still, the proposed BHOSM docking flight controller can well ensure the tracking error within the desired area, and on violations

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Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

Drogue FLC Proposed BHOSM

VK(m/s)

200.004 200.002 200 199.998 199.996 30

40

50

60

70 t(s)

-17.8

90

100

-18

-17.9 yb(m)

80 -17.8

64

65

66

-18 -18.1 30

40

50

60

70

80

90

100

70

80

90

100

t(s)

z b(m)

-6999.5 -6999.6 -6999.7 -6999.6

-6999.8

40 -6999.750

30

60 t(s) -6999.8 66 66.5 67 67.5

Fig. 7. Simulation results for ground velocity and barycenter position tracking in Case 1.

-3

FLC Proposed BHOSM

2 0 -2 -4

40

60 t(s) 0.1

98 100

0 -0.1

0.05

0.1 ezb(m)

60 t(s) 0.1

80

40

40

60 t(s)

80

91

92

60 t(s)

80

100

100 X: 100 Y: 180.7

X: 100 Y: 152.1

40

60 t(s)

80

X: 100 Y: 157

100

0

100 X: 100 Y: 166.4

200

0 -0.1

X: 100 Y: 2.829

100

0

100

ITAEezb(m)

40

X: 100 Y: 3.142

2

0

100

FLC Proposed BHOSM

200

0.05 96

0.1 eyb(m)

80

4 ITAEeVkb(m)

x 10

ITAEeyb(m)

eVK(m/s)

4

40

60 t(s)

80

100

Fig. 8. Simulation results for tracking error of ground velocity and barycenter position in Case 1.

353

Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

FLC Proposed BHOSM

0.05

0.05 γ (°)

χ (°)

0.1 0 -0.05

0 -0.05

40

60 t(s)

80

100

40

60 t(s)

80

100

40

60 t(s)

80

100

40

60 t(s)

80

100

40

60 t(s)

80

100

0.2 β (° )

α (° )

2.2 0

2 40

60 t(s)

80

4 2

1.2 φ (° )

μ (° )

-0.2

100

0 -2 40

60 t(s)

80

1 0.8

100

ψ (° )

θ (° )

1.2 1 0.8

4 2 0 -2

40

60 t(s)

80

100

Fig. 9. Simulation results for angles in Case 1.

Proposed BHOSM FLC

4

E

δ (°)

A

δ (° )

2 0 -2 40

60 t(s)

80

-0.5 -1 -1.5 -2 -2.5

100

40

60 t(s)

80

100

40

60 t(s)

80

100

10 δ (%)

0

-10

T

R

δ (° )

0.61

40

60 t(s)

80

100

0.6 0.59

Fig. 10. Simulation results for control inputs in Case 1.

on the requirement red boundary occur. It performs better in tracking the rapidly changed drogue trajectory. The tracking trajectory by proposed BHOSM is closer to the drogue trajectory, which can be easily got in Fig. 13. The 6 DOF nonlinear closed-loop simulation for receiver barycenter position tracking results and relevant results are shown in Figs. 15–17. Although the ground velocity tracking error by BHOSM is slightly greater, there are not obvious influence on other states of the receiver due to the very small magnitude of the ground velocity tracking error (0.02 m/s). On the other hand, obvious differences can still be seen in trajectory tracking. The BHOSM control the receiver barycenter trajectories closer to the desire drogue positions, as shown in Fig. 15. And with the proposed BHOSM, the controlled barycenter trajectories is shifted forward when it compared to FLC with HOSMO. Although the anti-disturbance ability of FLC scheme still

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Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

(a). Turbulence

(b). Sum of flows 2 W x (m/s)

Turbx (m/s)

2 0 -2

0

50 t(s)

-2

100

0

0

50 t(s)

100

0

50 t(s)

100

0

50 t(s)

100

-4

3 W z (m/s)

Turbz (m/s)

50 t(s)

-3

-5

100

2 0

-2

0

-2 W y (m/s)

Turby (m/s)

2

-2

0

0

50 t(s)

100

2 1 0

Fig. 11. Airflows in Case 2.

Drogue trajectory in Case 2 -6999.3 -6999.4

Start

-6999.5 End

z(m)

-6999.6 -6999.7 -6999.8 -6999.9 -7000 -7000.1 -7000.2 -18.3

-18.2

-18.1

-18

-17.9 y(m)

-17.8

-17.7

-17.6

Fig. 12. Drogue trajectory in Case 2.

works, the trajectory tracking errors increase as the intensity of the turbulence increases. The maximum of ez by the proposed BHOSM has been nearly decreased by 19%. The tracking error in y-axis and z-axis are controlled below 0.2 m and 0.3 m, respectively. The barycenter tracking errors ITAEs are obviously below the blue ones. The receiver’s angle states are shown in Fig. 17. The results of these angles by the two methods are similar. The control inputs are given in Fig. 18, and their amplitudes are larger than Case 1 because larger controls are needed to track the fastermoving drogue with the intensity of the turbulence increases. And the slight greater actuator surface deflections can also be found in the BHOSM based receiver docking controller.

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Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

-17.7 -17.8

-6999

-6999.4

Drogue FLC Proposed BHOSM

-6999.6 70

73 74 75 z p(m)

-17.6 yp(m)

-17.6

Drogue FLC Proposed BHOSM

-17.8 -18

72

74

-6999.5

-7000 -18.2 30

40

50

60

70

80

90

100

30

40

50

60

t(s) FLC Proposed BHOSM

0.2

90

100

80

90

100

0.2 ezp(m)

0.1

eyp(m)

80

FLC Proposed BHOSM

0.4

0 -0.1

0 -0.2

-0.2 30

40

50

60

70

80

90

-0.4 30

100

40

50

60

t(s) X: 100 Y: 340

FLC Proposed BHOSM

300

500

X: 100 Y: 277.8

200 100 0 30

40

50

60

70 t(s)

ITAEezp(m)

400 ITAEeyp,(m)

70 t(s)

70

80

90

FLC Proposed BHOSM

400

X: 100 Y: 380.2

300 200 100 0 30

100

X: 100 Y: 427.3

40

50

t(s)

60

70

80

90

0.2

0.3

100

t(s)

Fig. 13. Simulation results for probe position tracking in Case 2.

Proposed BHOSM

FLC 0.3 0.3 0.2 0.2 0.1 ez (m)

ez (m)

0.1 0 -0.1

R0=0.3

-0.1 -0.2

-0.2 -0.3 -0.4

R0=0.3 0

-0.2

0 ey(m)

0.2

0.4

-0.3 -0.2 -0.1

0

0.1

ey(m)

Fig. 14. Statistical results the probe position for tracking errors in Case 2.

5. Conclusion In this paper, back-stepping high order sliding mode based flight controller is proposed for docking control problem in autonomous aerial refueling. The 6 DOF model for the receiver and the probe position model are established in several affine nonlinear forms for the convenient flight controller design purpose. The receiver docking flight controller is designed through five control loops. The influences of the unknown flow disturbances on the receiver in each control loop are taken as the component of the ‘‘lumped disturbances” which are estimated and compensated with specially designed finite time convergence high order sliding mode observer. Then with the disturbance compensations, a back-stepping high order sliding mode based receiver exact docking flight controller is proposed to conduct the probe to track the transient changing drogue.

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Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

Drogue FLC Proposed BHOSM

VK(m/s)

200.02 200.01 200 199.99 199.98 30

40

50

60

70 t(s)

-17.6

80 -17.6 -17.8 -18

90

100

yb(m)

72 73 74 -17.8 -18 -18.2 30

40

50

60

70

80

90

100

90

100

t(s) -6999.5

-6999

-7000 89 90 91

z b(m)

-6999.5 -7000 30

40

50

60

70

80

t(s) Fig. 15. Simulation results for ground velocity and barycenter position tracking in Case 2.

ITAEeVkb(m)

0 -0.02

eyb(m)

-0.04

FLC Proposed BHOSM 40

60 t(s)

80

0.15 0.1 0.05 0.2 60 61 62

60 t(s)

0.2

80

40

60 t(s)

80

84

85

-0.2 60 t(s)

80

100

100 X: 100 Y: 340

300 X: 100 Y: 277.8

200 100 40

60 t(s)

80

600

0

40

X: 100 Y: 12.51

5

0

100

ITAEezb(m)

0.4 0.3 0.2 83

0.4

10

400

0

40

FLC Proposed BHOSM

15

0

100

-0.2

ezb(m)

X: 100 Y: 18.69

20

ITAEeyb(m)

eVK(m/s)

0.02

100 X: 100 Y: 427.3

400 X: 100 Y: 380.2

200 0

40

60 t(s)

80

Fig. 16. Simulation results for tracking error of ground velocity and barycenter position in Case 2.

100

357

Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

FLC Proposed BHOSM

0.2 0

γ (°)

χ (°)

0.2 0.1 0 -0.1

-0.2 40

60 t(s)

80

100

60 t(s)

80

100

40

60 t(s)

80

100

40

60 t(s)

80

100

40

60 t(s)

80

100

60

80

100

80

100

0.2 β (°)

α (°)

2.5

40

2

0 -0.2

40

80

100 1.5 φ (°)

μ (°)

4 2 0 -2 -4

60 t(s)

40

60 t(s)

80

1 0.5

100

ψ (°)

θ (°)

1.5 1 0.5

40

60 t(s)

80

100

4 2 0 -2 -4

Fig. 17. Simulation results for angles in Case 2.

Proposed BHOSM FLC E

0 -5

0 δ (°)

A

δ (°)

5

40

60

80

-2 -4

100

40

t(s)

t(s) 0.62 δ (%)

0

T

R

δ (° )

10

-10

0.6 0.58

40

60

80

100

40

60

t(s)

t(s)

Fig. 18. Simulation results for control inputs in Case 2.

Comparison and extensive simulations are conducted to verify the superiority and feasibility of the proposed docking flight controller under different turbulences.

Acknowledgment This research has been funded in part by the National Natural Science Foundations of China under Grant 61673042 and 61175084. The authors would also like to thank the reviewers and the editor for their comments and suggestions that helped to improve the paper significantly.

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Appendix A A.1. Aerodynamic forces and moments

T ¼ T max dT ; q ¼ q0 ekjzb j ; Q ¼ 2

L

3

2

1 qV 2 ; 2

cL;0 þ caL a þ caL a2 þ cqL cq=ð2VÞ þ cdLe de 2

6 6 7 a a2 2 4 D 5 ¼ QS6 4 cD;0 þ cD a þ cD a C cC;0 þ cbC b þ cdCa da þ cdCr dr 2 3 2 0 3 cL þ caL a cL 6 7 6 c0 þ ca a 7 ¼ QS4 D D 5 ¼ QS4 c D 5 c0C þ cbC b 2

3

3 7 7 5 ;

cC

2 0 3 2 3 3   cL þ cdLa da þ cdLr dr cL;0 þ cdLa da þ cdLr dr þ cbL b þ cpL bp=ð2VÞ þ crL br=ð2VÞ cL 6 7 6 7 6 7 6 7 q d d 4 M 5 ¼ QSl4 cM;0 þ cMe de þ cM cq=ð2VÞ 5 ¼ QSl4 cM 5: 5 ¼ QSl4 c0M þ cMe de p  b da dr da dr 0 r  N cN c þ c da þ c dr cN ;0 þ c da þ c dr þ c b þ c bp=ð2VÞ þ c br=ð2VÞ L

2

N

N

N

N

N

N

N

N

where cL ; cD ; cC are the lift, drag and lateral force coefficients, cL ; cM ; cN are the rolling, yawing and pitching moment coefficients, T max is the maximum available thrust, aK ; bK are the path flow angles (angle of attack and sideslip angle in flight path frame) [9,40], aw ; bw are the appendant flow angles caused by flows. Appendix B. Variables and vectors in Eqs. (7), (8)

f V k ¼ ðD  Cbw þ Law  mg sin cÞ=m; BV k ¼ T max cosða þ rÞcosb=m  F1 ¼ F1 ðX2 ; V k Þ ¼  B1 ¼ B1 ðV k Þ ¼

V k ðcos c sin v  vÞ V k ðsin c  cÞ

Vk

0

0

V k

 ;

 ;

3 ! Tðbk cosl þ ðak þ rÞ sin lÞþ 7 6 ðmV 1cos cÞ 7 6 k ðC  Dbw Þcosl þ ðQSc0L  Daw Þ sin l 7 6 ! 7; F2 ¼ F2 ðX2 ; X3 ; V k ; Q Þ ¼ 6 7 6 Tðb sin l  ð a þ r Þ cos l Þ þ mg cos c þ k k 5 4 1 0 ðmV k Þ ðC  Dbw Þ sin l  ðQScL  Daw Þ cos l 2

" # QScaL = cos c 0 1 B2 ¼ B2 ðX2 ; V k ; Q Þ ¼ ; mV k 0 QScaC 2

3 ðc_ cos l  v_ sin l cos cÞ= cos b 6 7 F3 ¼ F3 ðX_ 2 ; X3 ; X4 Þ ¼ 4 c_ sin l þ v_ cos l cos c 5; _ _ ðc sin b cos l þ vðsin c cos b þ sin b sin lcoscÞÞ= cos b 2

 cos a tan b

6 B3 ¼ B3 ðX3 Þ ¼ 4 sin a

cos a= cos b

1  sin a tan b 0

cos a

0

sin a= cos b

3 7 5;

Z. Su et al. / Mechanical Systems and Signal Processing 101 (2018) 338–360

2

1 Ix Iz I2xz

0

0

½ðIy Iz  I2z  I2xz Þrq þ ðIx Ixz  Iy Ixz  Iz Ixz Þpq þ Iz QSlcL þ Ixz QSlcN 

359

3

6 7 0 6 7 F4 ¼ F4 ðX4 Þ ¼ 6 I1y ½ðIz  Ix Þpr  Ixz p2 þ Ixz r2 þ QSlcM  7; 4 5 0 0 2 2 1 ½ðIx  Ix Iy þ Ixz Þpq  ðIx Ixz  Iy Ixz  Iz Ixz Þrq þ Ixz QSlcL þ Ix QSlcN  I I I2 x z

xz

2

Iz cdLa þIxz cdNa

6 Ix Iz I2xz 6 6 B4 ¼ B4 ðX2 ; Q Þ ¼ QSl6 0 6 4 I cda þI cda xz L

x N

Ix Iz I2xz

0 e cdM Iy

0

Iz cdLr þIxz cdNr Ix Iz I2xz

0 Ixz cdLr þIx cdNr

3 7 7 7 7: 7 5

Ix Iz I2xz

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