Asymmetric error-constrained path-following control of a stratospheric airship with disturbances and actuator saturation

Asymmetric error-constrained path-following control of a stratospheric airship with disturbances and actuator saturation

Mechanical Systems and Signal Processing 119 (2019) 501–522 Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journ...
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Mechanical Systems and Signal Processing 119 (2019) 501–522

Contents lists available at ScienceDirect

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

Asymmetric error-constrained path-following control of a stratospheric airship with disturbances and actuator saturation Tian Chen a, Ming Zhu a, Zewei Zheng b,c,⇑ a

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, PR China School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, PR China c The Seventh Research Division, Beihang University, Beijing 100191, PR China b

a r t i c l e

i n f o

Article history: Received 3 April 2018 Received in revised form 17 August 2018 Accepted 5 October 2018

Keywords: Stratospheric airship Path following control Asymmetric error constraints Vector field

a b s t r a c t This paper addresses the asymmetric error-constrained path-following problem of a stratospheric airship with external disturbances and actuator saturation. A path-following algorithm is proposed based on the theories of tan-type barrier Lyapunov function, vector field guidance, adaptive sliding mode control, and radial basis function neural network (RBFNN). First, to satisfy the asymmetric tracking error-constrained requirements of airship position, an asymmetric error-constrained vector field (AECVF) guidance law is presented, which can navigate the stratospheric airship along the predefined path and guarantee that the tracking error is limited by the error constraints. Second, an adaptive sliding mode attitude controller is introduced to track the desired attitudes calculated using the AECVF with disturbances. Finally, an adaptive velocity controller is added to the control algorithm to maintain an appropriate velocity. Moreover, an RBFNN saturation compensator is introduced to solve the actuator saturation problem caused by the low maneuverability. Stability analysis indicates that all the signals in the closed-loop system are uniformly ultimately bounded. Meanwhile, simulation results demonstrate the effectiveness of the proposed control algorithm. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction The application of stratospheric airships to communication relay, space observation, atmosphere measurement, and military reconnaissance surveillance and monitoring is promising [22,54]. To achieve these applications, the control of the airship is required to ensure that it can fly permanently in a predetermined area with a certain attitude. After several years of scientific studies, the motion control of a stratospheric airship has witnessed significant achievements, such as hovering control [3,36,47,59,48], trajectory tracking [5,54,40,49], and path-following [4,15,55,30,34,58,60]. The hovering control is usually applied to the fixed area parking at a high altitude, where the control objective is to maintain the airship in the predefined area against the wind. Trajectory tracking and path-following control can both be used to implement position tracking control missions, such as search and rescue, surveillance, and reconnaissance. The objective of trajectory tracking control is to track a desired time-referenced trajectory, whereas that of the path-following is to track a predefined path without a desired specified temporal constraint.

⇑ Corresponding author. E-mail addresses: [email protected] (T. Chen), [email protected] (M. Zhu), [email protected] (Z. Zheng). https://doi.org/10.1016/j.ymssp.2018.10.003 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

Guidance law is a significant component of the path-following control algorithm. Some strategies for path-following guidance have been proposed in the literatures, such as target pursuit [1,35,26–28,61], line-of-sight (LOS) [9,6], variants of pursuit and LOS [37,56,18], and vector field (VF) guidance laws [31,14,21,20]. The target pursuit method regards the closest point on the path as the tracking target. With the dynamic model in the Serret--Frenet frame, which consists of the tangent and normal vectors to the trajectory at the projection point, the vehicle path-following control method could eliminate the cross-track error at a property reasonable velocity. Unlike the target pursuit method, the target in the LOS algorithm is chosen at a fixed prescribed distance from the robot. The distance between the vehicle and target along the tangential direction to the path is called the look-ahead distance or along-track error. To track the predefined path, the LOS strategy drives the vehicle heading along the LOS vector, whose direction is regulated from the vehicle toward the target point. Notably, the desired path of the LOS guidance law should be differential with respect to a designed parameter. The VF guidance method presents a VF around the predefined path, in which the vector direction is the desired heading direction of the vehicle. By tracking the vector direction in the field, the predefined path will be followed. Paths can be decomposed into multiple segments of arcs, orbits, and straight lines. Thus, two primitive path types are considered in the development of VF pathfollowing controllers: straight lines and circular orbits. The VF guidance algorithm in literature [20] can track an arbitrary smooth curve. In a previous study [7], the researchers proved that the VF path-following controller performed slightly better than the integral LOS path-following controller. The states of a practical motion system are always limited, and a stratospheric airship path-following control system is no exception [19,17]. Recently, the utilization of barrier Lyapunov function (BLF) for solving the control problem of nonlinear systems with constraints has been an active area of research [42,41,24]. When constructing a BLF, a constraint interval is chosen as its domain of definition. When the state tends to the critical condition, the value of the BLF will tend to infinity to ensure that the state always remains within the constraint interval. BLFs, including log-type [11] and tan-type [16], have been employed to design an adaptive controller for nonlinear strict-feedback systems with constant [32], time-varying [43], symmetric [24], and asymmetric [53,13] output constraints. Apart from a partial state-constrained system, full stateconstrained control problems of nonlinear systems were also studied using BLFs in [2,11,24,45,25,23]. Combined with the dynamic models, BLFs have been used to solve state-constrained motion control problems of surface vessels [10,44,57], robots [11], Spacecrafts [39], robotic manipulators [46], gantry cranes [12], and aircrafts [50]. The study [56] presented a method of tracking control of stratospheric airships with input and output constraints. By introducing the tan-type BLF, an error-constrained LOS method with state constraints was designed. Nevertheless, studies that focused on the asymmetric error-constrained path-following problem of a stratospheric airship including the heading limitation related to the illumination incident angle of the solar array, the pitch limitation related to the communication occlusion, and the position limitation related to the coverage area are rare. Another problem that should not be ignored is the physical limitations of the actuators. Owing to the lack of sufficient carrying capacity, which is related to the thin atmosphere at high attitude, the allowable working range of the actuators of a stratospheric airship is relatively narrow. Thus, a phenomenon called actuator saturation, which occurs when the control signal of the actuators exceeds the allowable working range, is more likely to happen. The actuator saturation may reduce the performance of a control system, possibly leading to system instability. Based on the VF guidance method and the asymmetric BLF, an asymmetric error-constrained vector field (AECVF) pathfollowing guidance method is designed in this study. Combined with six degrees-of-freedom model equations and an attitude tracking algorithm, the path-following controller with asymmetric error limitation, external disturbances, and actuator saturation is designed. In summary, the main contributions and features of this paper are as follows: (1) Compared with the traditional VF path-following methods in [31,51], the proposed method can guarantee that the tracking errors do not exceed a pre-specified performance bound. (2) Compared with the error-constrained LOS path-following method presented in [56,52], the proposed method simplifies the computational complexity of the guidance law by using the VF theory. In addition, we introduce a command filter to estimate the first- and second-order derivatives of the desired yaw angle to simplify the calculation further. (3) An adaptive radial basis function neural network (RBFNN) anti-windup compensator is employed to handle the actuator saturation problem. Compared with the saturation compensator presented in our previous works [56], the actuator output is not necessarily measurable in an RBFNN anti-windup compensator. Therefore, it has a wider application scope. In contrast to the robust RBFNN backstepping control of planar path-following in our previous works [60], for the first time, RBFNN is introduced to compensate actuator saturation with full state equations of a stratospheric airship. (4) It is proven that, by using the proposed controller, all closed-loop signals are uniformly ultimately bounded, despite the presence of input saturation and disturbances. The results of comparison are illustrated to show the advantages of the proposed method. The rest of this paper is organized as follows. Some preliminaries and problem formulation are presented in Section 2. In Section 3, the asymmetric error-constrained path-following controller is designed. Simulation results are presented in Section 4. In Section 5, the conclusions are presented.

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503

2. Preliminaries and problem formulation 2.1. Preliminaries 2.1.1. Notations Throughout this paper, for a vector a 2 Rn ; ai ði ¼ 1; 2;    ; nÞ indicates the corresponding component of a. For any vectors a 2 Rn and b 2 Rn ; a < b ndicates that each component of the vectors satisfies ai < bi . j  j indicates the absolute value of a scalar or the absolute value of each component of a vector, i.e., for a vector x 2 Rn ; j x j¼ ½j x1 j; j x2 j;    ; j xn jT . In addition, k  k represents the Euclidean norm of a vector or the Frobenius norm of a matrix. For a matrix X 2 Rnn ; trðX Þ denotes its   trace with the property tr X T X ¼ kXk2 . 2.1.2. RBFNN approximation Suppose f ðxÞ : Rm ! R is an unknown smooth nonlinear function and it can be approximated over a compact set X # Rm using the following RBFNN: [8]:

f ðxÞ ¼ xT UðxÞ þ  where the node number of the neural network (NN) is l. More nodes indicate greater accuracy of approximation. x 2 Rl represents the optimal weight vector, which is defined by



^ T UðxÞ j x ¼ arg min sup j f ðxÞ  x ^ x



x2X

^ is the estimation of x . UðxÞ ¼ ½/1 ðxÞ; /2 ðxÞ;    ; /l ðxÞT : X ! Rl represents the radial basis function vector, whose where x element is chosen as the Gaussian function

/i ðxÞ ¼ exp 

kx  li k2

e2i

!

;

i ¼ 1;    ; l

where li 2 Rm and e 2 R are the center and spread, respectively. where  is an unknown constant.

 is the approximation error bounded over X, i.e., j  j6 ,

2.1.3. Definitions

Definition 1. For any

. 2 R, a saturation function is defined as

8 > < .max ; .; satð.Þ ¼ > : .min ;

. > .max .min 6 . 6 .max . < .min 



.max ; .min are the magnitude constraints. For any . ¼ .1 ; .2 ;    ; .n T 2 Rn , the saturation function vector is     T satð.Þ ¼ sat .1 ; sat .2 ;    ; sat .n . where

2.2. Stratospheric airship model As shown in Fig. 1, the stratospheric airship investigated in this study consists of an ellipsoidal helium balloon, tails, propulsion propellers, and a gondola. The avionics system, power system, and payloads are equipped in the gondola fixed below the balloon, whereas the solar-cell panels are attached on the surface of the balloon to supply electricity. Four main propulsion propellers and a horizontal vector propeller, which supply the thrust force and steering torque, respectively,are installed at the front of the balloon. In order to facilitate model analysis, some hypotheses are listed as follows: Assumption 1. Take earth-surface coordinate frame as the inertial reference frame;

Assumption 2. [36,60] The stratospheric airship operates in a steady atmospheric environment at an altitude of approximately 20 km and remains in buoyancy-weight balance; Assumption 3. [36,60] The elastic effect of the stratospheric airship is neglected.

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

Fig. 1. Structure of the stratospheric airship.

According to Assumption 1, the earth reference frame (ERF) is fixed to the earth with its origin Og located at a fixed point on the ground. The Oe xe axis points to north, the Og zg axis points to the earth core, and the Og yg axis points to east. The body reference frame (BRF) is attached to the airship with its origin O coinciding with the center of volume (CV) as shown in Fig. 1. The Ox axis points to the head of the airship. The Oz axis is perpendicular to the Ox axis and points downward. The Oy axis is determined by the right-hand rule and points toward the right. The position and attitude of the airship are described by the CV position P ¼ ðx; y; zÞT and Euler angles H ¼ ð/; h; wÞT in

ERF. The airspeed and angular velocity of the airship are defined as v ¼ ½u; v ; wT and X ¼ ½p; q; rT , respectively, in BRF.



The torques of inertia and the products of inertia are described by Ix ; Iy ; Iz and Ixy ; Iyz ; Ixz in BRF, respectively. As the airframe is symmetric about the lateral plane and the center of gravity lies immediatelybelow the CV, the products of inertia

Ixy ; Iyz ; Ixz ¼ 0. According to the knowledge of airship modeling, the motion of a stratospheric airship is described by the following equations:

"

P_ _ H

"

#

¼

K

033

033

R

mE þ M 0 mr 0 C

where

2

mr 0 C

I O þ I 0O



v

ð1Þ

X #

v_

_ X

"

þ

#  # " mE þ M 0 H  v þ mH  H  r0C ðG  F B ÞRT ez þ F a þ F T þ F dv ¼ H  ðI O HÞ þ mr 0C  ðH  v Þ M G þ M B þ M a þ M T þ F dx

chcw shcws/  swc/ shcwc/ þ sws/

6 K ¼ 4 chsw sh

ð2Þ

3

7 shsws/ þ cwc/ shswc/  cws/ 5 chs/ chc/

is the direction cosine matrix of BRF to ERF; and sx and cx denote sinðxÞ and cosðxÞ, respectively.

2

3 1 sin / tan h cos / tan h 6 7 R ¼ 40 cos /  sin / 5 0 sin / sec h cos / sec h

is the Euler rotation matrix; E is a 3  3 identity matrix; m is the whole mass of the airship; M 0 and I 0O are the additional mass

and inertia matrices, respectively. r 0 is the inertial matrix to the C represents a skew symmetric matrix; I O ¼ diag I x ; I y ; I z axis of BRF; ez ¼ ½0; 0; 1T ; G is the gravity of the airship; F B and M B are the buoyance and buoyance torque, respectively; M G is the gravity torque; F add and M add are the additional inertia force and torque, respectively; F a and M a are the aerodynamic force and torque, respectively; F T and M T are the propulsive force and torque, respectively. F dv and F dx are the disturbances. The forces and torques on the stratospheric airship during flight are presented in A. To facilitate the design of the controller, the dynamic equations can be deformed as

T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

(

X_ 1 ¼ f 1 ðX 1 ÞX 2 X_ 2 ¼ f 2 ðX 1 ; X 2 Þ þ Bs

505

ð3Þ

h i  T where X ¼ X T1 ; X T2 is the system state vector, and X T1 ¼ ½x; y; z; /; h; wT , X T2 ¼ ½u; v ; w; p; q; r T ; s ¼ su ; sv ; sw ; sp ; sq ; sr . The details of (3) are presented in B. Assumption 4. The disturbances F dv and F dx are continuous and bounded by unknown constant bounds, namely kF dv k 6 cF dv and kF dx k 6 cF dx , respectively. Assumption 5. The control inputs F T and M T satisfy the saturation constraint: F T;min 6 F T 6 F T;max ; M T;min 6 M T 6 M T;max ,



where F T;min ; F T;max and M T;min ; M T;max indicate the minimum and maximum values of control inputs. By disassembling them to the direction of the BRF coordinate axes, the control input s satisfies the input saturation: s ¼ satðs0 Þ, where s0 is the unconstrained control signal, and ds ¼ s  s0 . 2.3. Problem formulation To track a predefined path, the VF guidance law calculates a field around the path. The vectors in the field indicate the direction towards the path to be followed. Paths can be decomposed into multiple segments of arcs, orbits, and straight lines. Therefore, two primitive path types are considered in the development of path-following controllers—linear and circular path—whose example VFs are shown in Fig. 2. According to Eq. (1), the plane kinematics equation of a stratospheric airship can be expressed as follows:



x_ ¼ u cos w  v sin w y_ ¼ u sin w þ v cos w

ð4Þ

From Fig. 2, the position error dl of the linear path and dc of the circular path can bedescribed as follows:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðx  xl Þ2 þ ðy  yl Þ2 sin arctanðyl  y; xl  xÞ  wp;l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dc ¼ ðx  xo Þ2 þ ðy  yo Þ2  Ro dl ¼

ð5Þ

By combining with Eq. (4), we have

   d_ i ¼ u sin w  wp;i þ v cos w  wp;i ¼ U sin w  wp;i þ b ; where U ¼

i ¼ l; c;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 þ v 2 ; b ¼ arctanðv ; uÞ.

Fig. 2. Guiding geometrical illustration of VF guidance method for linear path (left) and circular path (right).

ð6Þ

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

Remark 1. As shown in Fig. wp;c ¼ arctanðy0  y; x0  xÞ  p=2.

2,

wp;l

determined

by

the

predefined

linear

path

is

a

constant,

and

c ðt Þ are the asymmetric differential and positive time-varying error constraints for pathAssumption 6. [43,16] kc ðt Þ; k c ðt0 Þ. following. The initial tracking error of the airship position satisfies kc ðt0 Þ < d ðt0 Þ < k i

Remark 2. Assumption 6 limits the objective of the study to the condition that the initial position error is within the constraints. The condition error starts from outside the constants region, which was studied in [43], and the same augmented method can be used when the initial errors are not within the constraints. The control objective of the stratospheric airship path-following is to design the control input s in the presence of an asymmetric error constraint, external disturbances, and input saturation, such that. (1) the planar position of the stratospheric airship fx; yg tends to follow the desired path with a small error; c ðtÞ for 8t > t 0 ; (2) the position error satisfies kc ðt Þ < di ðt Þ < k (3) all the other airship states and closed-loop signals remain bounded. 3. Path-following control design As shown in Fig. 3, the proposed controller consists of three parts: asymmetric error-constrained guidance law, adaptive sliding mode controller, and adaptive velocity controller. First, the desired yaw angle wc , which is incorporated into the desired attitude Hc ¼ ½/c ; hc ; wc T , is calculated using the proposed AECVF guidance law. Second, an adaptive sliding mode controller is introduced to track the desired attitude Hc with disturbance f x [38,33]. Owing to the computational complexity of the first- and second-order derivatives of Hc , a command filter is introduced. Third, an adaptive velocity controller is proposed to maintain a reasonable velocity v c with disturbance f v . Moreover, RBFNN anti-windup compensators are presented to handle the input saturation dsv and dsx . Stability analysis is performed for the proposed control design. The detailed design of the path-following controller is described in the following subsections. 3.1. Asymmetric error-constrained VF guidance law According to Section 2.3, we will design the desired heading angle wc using the guidance law to track the desired path with asymmetric constraints in Fig. 2. The tan-type asymmetric BLF is introduced to facilitate the discussion about the constrained requirements on the tracking errors. For the sake of convenience, di is denoted by d, and wp;i is denoted by wp .

V VF ¼ qðdÞ where

 qðdÞ ¼

2 k c

p

tan

pd2 2 2k c

!

þ ð1  qðdÞÞ

k2c

p

tan

pd2

!

2k2c

1; d > 0 0; d 6 0

Considering the first-order derivative with respect to time,

Fig. 3. Block diagram of the path-following controller with three control loops: guidance, attitude control, and velocity control.

ð7Þ

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

T

Fig. 4. Disturbances of stratospheric airship path-following ½f v ; f x  .

0 V_ VF ¼ qðdÞ@

dd_ 

c k _ c 2k

pd2

!

_ c k   kc

!

1

2

d 

A 2 cos2 p2kd2 c 1 ! ! 2 2 _ _ _ k d d 2k k p d d c c c @  2 þ  A þ ð1  qðdÞÞ  tan p 2k2c kc cos2 pd2 cos2 pd 2

cos2 p2kd2 0 c

þ

p

tan

2k2c

 _ cos d_ þ 2kc kc tan ¼ qd d

pd

! 2 

2 2k c

p

2 2k c

_ c k c k

!

ð8Þ

2k2c

! ! ! k_ c 2kc k_ c pd2 _  dcos d þ ð1  qd Þ dcos d þ  tan dcos d p 2k2c kc !

_ VF þ ð1  q ÞV_ VF ; , qd V d cos ¼ where qd indicates qðdÞ, and d

d  ; dcos ¼ cos2

pd2

d . cos2

2 k2 c

pd2 2k2

c  As d ¼ U sin w  wp þ b , the desired yaw can be designed as

wc ¼ wp  b þ arctan 2ðhd ; kd Þ; where

ð9Þ

 þ ð1  q Þk ;  hd þ ð1  qd Þhd ; kd ¼ qd k hd ¼ qd  d d hd ¼ d

2 e k k c d p



 2



 2

 e0 d , a e0 d; h ¼ ke sin pd cos pd2 þ d þ k sin p2kd2 cos p2kd2 þ k d d p 2k2c 2k2c c c k2c



 2



 ;k ;k e ; ke ; k e0 ; ke0 are positive constants. ke0 d , ad þ ke0 d, and k d d

Remark 3. The desired angle in the traditional VF method is calculated as [31]

wc ðdÞ ¼ w1

2

p

arctan 2ðd; kÞ;

where d is the lateral distance of the vehicle from the path; w1 is a predefined yaw angle at infinity; k is a positive constant that influences the rate of transition from w1 to zero. To extend the application scenarios of the VF guidance method, we add the path direction compensation wp and sideslip compensation b. To simplify the guidance parameters, we choose w1 ¼ p=2. Subsequently, using (6) and (9), Eq. (8) can be obtained as

0

V_ VF

2 e k Uk B ¼ qd @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c tan 2 p h2 þ k d

0

d

pd2 2 2k c

!

2 e0  _ Uk cos d þ 2kc kc tan pd  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 2 p 2k 2 h2 þ k c d

2

Uke k2 pd B þ ð1  qd Þ@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c tan 2k2c 2 2 p hd þ kd

!

d

!

1 _ c k cos dC  d A kc

Uke0 2kc k_ c pd2 tan  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dcos d þ p 2k2c h2d þ k2d

!

1 k_ c C  dcos dA kc

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

e0 and ke0 are designed to satisfy To eliminate the irrelevances of ABLF, k

_ e0 Uk cos d þ kc d  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d c cos d ¼ 0; k 2 h2 þ k d d

Thus,

q d ¼

we



have

 2d d 1q

2

Uke0 k_ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dcos d þ c dcos d ¼ 0: kc h2d þ k2d

ð10Þ

    e0 d  q 2  2q 2  2q2 a ke0 d  q2 a þ k2 ¼ 0, 2 ¼ 0; 1  q2 d2 k d k d þ k  2d a  2d a k d e0 e0 d d d d d d

where

k_ c . kc U

_ c k  U k

; qd ¼ c e0 and ke0 are selected as follows: k

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  

 e0 ¼ k

q 2d a d d þ q d kd 1  q 2d d2 þ a 2d  2d d 1q

2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  

;

ke0 ¼

q2d ad d þ qd kd 1  q2d d2 þ a2d

ð11Þ

2

1  q2d d

If the following condition (12) is satisfied, (11) will be established.

(

8 _  >  k 

8  > _ 

  k 

q 2d d2 < 1 < kccUjdj < 1 < kc  <  dc U )  _  )     > q2d d2 < 1 > : k_ c  < kc U : kc jdj < 1 d kc U

ð12Þ

n   o _   _   Therefore, (12) will be satisfied if the condition max k c ; kc  < U is satisfied, as kc < d 6 kc , which will be proven later. Thus,from (10), we have

c k _ c 2k

p

tan

pd2

!

2 2k c

2 2Uke0 k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c tan 2 p h2d þ k d

pd2 2 2k c

Thus, V_ VF satisfies the following inequality:

2  U  kc tan V_ VF < qd qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k e  2ke0 p 2 h2 þ k d d

!

pd2 2 2k c

!

;

2kc k_ c

p

tan

pd2 2k2c

!

! 2Uke0 k2 pd2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c tan : 2k2c h2d þ k2d p

U k2 þ ð1  qd Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðke  2ke0 Þ c tan p h2d þ k2d

pd2 2k2c

! :

ð13Þ

3.2. Attitude control In the guidance loop, the desired heading angle wc is calculated and incorporated into the desired attitude Hc ¼ ½/c ; hc ; wc T based on the proposed asymmetric error-constrained VF (AECVF) guidance method. To track the desired attitude Hc under disturbances f x , an adaptive sliding mode controller will be employed. According to the state-space Eq. (3), the stratospheric airship attitude model is expressed as

(

_ ¼ RX H _ ¼ F x þ B22 sx þ B21 sv þ f x X

ð14Þ

where H ¼ ½/; h; wT and X ¼ ½p; q; rT represent the attitude angle in ERF and angular velocity in BRF, respectively. Thus, we have

_ þ RX € ¼RX _ H _ þ RðF x þ B22 sx þ B21 sv þ f x Þ ¼RX _ þ RF x þ RB22 sx þ RB21 sv þ Rf x ¼RX ¼F x;0 þ Bx;0 sx þ dx _ þ RF x ; Bx;0 ¼ RB22 ; dx ¼ B21 sv þ Rf x . where F x;0 ¼ RX A sliding surface is defined as

_ e þ kx He ; s¼H

ð15Þ

€ e þ kx H _e s_ ¼ H  € H € c þ kx H _ H _c ¼H

ð16Þ

where He ¼ H  Hc ; kx ¼ diag ks;/ ; ks;h ; ks;w is a positive definite diagonal constant matrix. Thus, we have the first derivative of Eq. (15)

 € c þ kx H _ þ Bx;0 sx  H _ c þ dx ¼ F x;0 þ kx H

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

The computations of the first- and second-order derivatives of Hc are extremely intricate. Therefore, a second-order command filter is introduced to obtain the derivatives.

(

N_ 1 N_ 2

¼ N2

ð17Þ

¼ 2Kxn N2  x2n ðN1  Hc Þ



where K ¼ diag K/ ; Kh ; Kw is the damping ratio and xn ¼ diag xn;/ ; xn;h ; xn;w is the damping frequency. By defining ^ ^_ ¼ N ; H € c ¼ N_ 2 , Eq. (16) can be rewritten as H c 2

  ^_ þ d ; €^ c þ kx H _ þ Bx;0 ðsx þ ds Þ  H s_ ¼ F x;0 þ kx H c x x

ð18Þ

   ^_ þ H ^ €c  H _c H € c. where dx ¼ dx þ kx H c Remark 4. Setting a sufficiently large xn and appropriate K can guarantee fast tracking of the commanded attitude signal,     ^_ ^ € € _c H i.e., kx H c þ Hc  Hc is bounded. Thus, dx is bounded by an unknown constant, namely kdx k 6 ddx . An adaptive RBFNN saturation compensator is introduced for the actuator saturation dynamics dsv and dsx as follows: T

c rðxNN Þ þ e: W T rðxNN Þ ¼ W

ð19Þ

h

c¼ W c u; W cv; W c w; W c p; W c q; W cr where W

iT

represents the weights of the NN, r ¼



T

ru ; rv ; rw ; rp ; rq ; rr represents the basis

functions, and xNN ¼ ½/; h; w; p; q; r; u; v ; w; pq; qr; pr; uv ; v w; uw represents the inputs of the NN. For attitude control, the errors of approximation of NN weights are T

f x ¼ Wx  W c x; W

ð20Þ

 T where W x ¼ W p ; W q ; W r . Thus, Eq. (18) can be rewritten as

       ^€ þ k H ^_ fx þ W c T rx ðxNN Þ  H _ þ BX;0 sx þ W s_ ¼ F X;0 þ kx H c x c þ dx x

ð21Þ

The attitude control input and the adaptation law are designed as follows:

8   ^€ ^_ cT ^ s _ > sx ¼ B1 > X;0 ks s  F X;0  kx H þ kx Hc þ Hc  ddx ksk  W x rx ðxNN Þ > > <   _ d^dx ¼ kd;s ksk  kdx ^ddx > > > > :c _ cx W x ¼ sT Cx rðxNN Þ  kW kskCx W

ð22Þ

x

^d is the estimate of dd . ks and C are positive definite diagonal constant matrices. kd;s ; kd , and kW are positive where d x x x x constants.  s is the adaptive sliding mode compensator of dx , which consists of external disturbances f x , control Remark 5. ^ ddx ksk   ^_ ^ € € _c H couplings B21 sv , and error of filter estimation kx H c þ Hc  Hc .

_ c Lemma 1. For the RBFNN (19) and the adaptive law W x in (22), there exists a compact set



XW x ¼

c x : kW c xk 6 W

1x kW x

 ;

c x 2 XW ; 8t > 0 provided that W c x ð0Þ 2 XW . where krx ðxNN Þk < 1x with 1x > 0, such that W x x The proof of Lemma 1 can be found in the reference [29]. Consequently, if we choose the Lyapunov function for attitude dynamics control loop as

VH ¼

  1 T 1 e2 1 f T 1 f d dx þ tr s sþ W x Cx W x 2 2kd;s 2

ð23Þ

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where e d dx ¼ ^ ddx  ddx , then the first-order derivative of V H is

  1 e e_ _ f T C1 W f d dx d dx þ tr W V_ H ¼sT s_ þ x x x kd;s       _ _  ^€ þ k H ^_ þ d þ 1 e T 1 f f _ þ BX;0 ðsx þ ds Þ  H  ^ d W d þ tr W C ¼sT F X;0 þ kx H c x c x x x x x kd;s dx dx       s _  f f T rx ðxNN Þ þ e f T C1 W ¼sT ks s  ^ddx d dx ksk  kdx ^ddx þ tr W þ dx þ W x x x x ksk      _ T f f T C1 W d dx e d dx þ ddx þ tr W d dx ksk  kdx e  s r ð x Þ 6  sT ks s  ^ddx ksk þ ddx ksk þ e x x NN x x

ð24Þ

   1 1 f T Wx  W fx d 2dx þ kdx d2dx þ kW x ksktr W 6  sT ks s  kdx e x 2 2     1 1 fT W f T Wx f x þ kW ksktr W ¼  sT ks s  kdx e d 2dx þ kdx d2dx  kW x ksktr W x x x 2 2 From Lemma 1, we can obtain

    1 1 f x þ kW ksk 1x kW x k þ kW x k2 fT W d 2dx þ kdx d2dx  kW x ksktr W V_ H 6 sT ks s  kdx e x x 2 2 kW x

ð25Þ

3.3. Velocity control The velocity controller is designed to eliminate the velocity tracking error v e ¼ v  v c , where v ¼ ½u; v ; wT is the velocity

of the stratospheric airship in BRF, and v c ¼ ½uc ; v c ; wc T is the desired velocity. According to the state-space Eq. (3), the stratospheric airship velocity model is expressed as

v_ ¼ F v þ B11 sv þ B12 sx þ f v Thus, we have the first-order derivative of

ð26Þ

v e,

v_ e ¼v_  v_ c

¼F v þ B11 sv þ B12 sx þ f v  v_ c 

¼F v þ B11 sv þ dv where dv ¼ B12 sx þ f v  v_ c . 

 Remark 6. According to Assumptions 4 and 5, sx and f v are bounded. It is evident that v_ c is also bounded. Thus, dv is  bounded by an unknown constant, namely, kdv k 6 ddv .

According to Eq. (19), the errors of approximation of NN weights are

f v ¼ Wv  W cv: W

ð27Þ T

where W v ¼ ½W u ; W v ; W w  . The velocity control input and the adaptation law are designed as

  8 cT ^  ve > sv ¼ B1 > 11 kv v e  F v  ddv kv e k  W v rv ðxNN Þ > > <   _ d^dv ¼ kd;v kv e k  kdv ^ddv > > > > :c _ cv W v ¼ v Te Cv rv ðxNN Þ  kW v Cv kv e k W

ð28Þ

where ^ ddv is the estimate of ddv . kv and C are positive definite diagonal constant matrices. kd;v ; kdv , and kW v are positive constants.  Remark 7. ^ ddv kvv ee kis the adaptive compensator of dv , which consists of external disturbances f v , control couplings B12 sx , and derivative of desired velocity v_ c .

_ c Lemma 2. For the RBFNN (19) and the adaptive law W v in (28), there exists a compact set

T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522



XW v ¼

c v : kW cvk 6 W

1v kW v

511

 ;

c v 2 XW ; 8t > 0 provided that W c v ð0Þ 2 XW . where krv ðxNN Þk < 1v with 1v > 0, such that W v v The proof of Lemma 2 can be found in the reference [29]. If we choose the Lyapunov function for position dynamics control loop as

V ve ¼

1 T 1 e2 1 f 1 f d  þ W v ve þ v Cv W v 2 e 2kd;v dv 2

ð29Þ

where e d dv ¼ ^ ddv  ddv , then the first-order derivative of V v e is

  1 e e_ _ f f T C1 W d dv d dv þ tr W V_ v e ¼v Te v_e þ v v v 2kd;v    _  _ f f T C1 W ¼v Te F v þ B11 ðsv þ dsv Þ þ dv þ e d dv ^ddv þ tr W v v v

    _  f f T C1 W d dv kv e k  kdv ^ddv þ v Te rv ðxNN Þ þ tr W ¼  v Te kv v e  ^ddv kv e k þ v Te dv þ e v v v    _ T f f T C1 W d dv ^ddv þ tr W d dv kv e k  kdv e  v r ð x Þ 6  v Te kv v e  ^ddv kv e k þ ddv kv e k þ e v NN v v e v      f T Wv  W fv d dv e d dv  ddv þ kW v kv e ktr W 6  v Te kv v e  kdv e v     f v þ kW kv e ktr W fT W f T Wv d 2dv þ e d 2dv ddv  kW v kv e ktr W ¼  v Te kv v e  kdv e v v v     1 1 f v þ kW kv e k 1v kW v k þ kW v k2 fT W 6  v Te kv v e  kdv e d 2dv þ kdv d2dv  kW v kv e ktr W v v 2 2 kW v

ð30Þ

3.4. Stability analysis We now summarize the main results of this study. Theorem 1. Consider the stratospheric airship model (3) under input saturation ds and external disturbances ff v ; f x g, and suppose that Assumptions 1–6 are satisfied. If the desired yaw is calculated using (9), the controllers are obtained from (28) and (22), and the control parameters satisfy (11) and (12), then the following holds: c ðt Þ, for 8t > t0 is never violated. (1) The error-constrained requirements will be satisfied, kc ðt Þ < dðtÞ < k (2) The path-following and state-tracking errors converge to a sufficiently small neighborhood around zero. (3) All the states and control inputs in the closed-loop control system remain bounded.

Proof. The complete Lyapunov function is assigned as

V ¼ V VF þ V x þ V v ¼ qðdÞ þ

2 k c

p

tan

pd2 2 2k c

! þ ð1  qðdÞÞ

k2c

p

tan

pd2 2k2c

!

  1 1 e2 1 f T 1 f 1 1 e2 þ sT s þ d dx þ tr d  W x Cx W x þ v Te v e þ 2 2kd;s 2 2 2kd;v dv

1 f 1 f W v Cv W v 2

ð31Þ

From Eqs. (13), (24), and (30), the derivative of V satisfies

! U k2c pd2 þ ð1  qd Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðke  2ke0 Þ tan p 2k2c h2d þ k2d     1 1 f x þ kW ksk 1x kW x k þ kW x k2 fT W d 2dx þ kdx d2dx  kW x ksktr W  sT ks s  kdx e x x 2 2 kW x     1 1 2 f v þ kW kv e k 1v kW v k þ kW v k2 fT W d 2dv þ kdv ddv  kW v kv e ktr W  v Te kv v e  kdv e v v 2 2 kW v

2  U  kc tan pd V_
2

!

ð32Þ

e  2k e0 > 0; ke  2ke0 > 0, then, In addition, if k

V_ 6 kV V þ D;

ð33Þ

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

where

( ) 8 9  > > < max q pffiffiffiffiffiffiffiffiffi U U  ðke  2ke0 Þ ; 2kmin ðks Þ; 2kmin ðkv Þ; = ke  2ke0 ; ð1  qd Þ pffiffiffiffiffiffiffiffiffi d 2 h2 þk h2d þk2d kV ¼ min d d > > : ; kdx kd;s ; kdv kd;v ; kW x ; ksk; kW v kv e k and

    1 1 1x 1v D ¼ kdx d2dx þ kdv d2dv þ kW x ksk kW x k þ kW x k2 þ kW v kv e k kW v k þ kW v k2 : 2 2 kW x kW v

1) From (33), we obtain

  D kV t D e V 6 V ð0Þ  þ : kV kV

ð34Þ

Therefore, V is bounded, which implies that the ABLF is also bounded. Moreover, we have

qd

2 k c

p

tan

pd2 2 2k c

! þ ð1  qd Þ

k2c

p

tan

pd2 2k2c

!

  D kV t D e þ : 6 V 6 V ð0Þ  kV kV

  2 2 2 2k 2k 2 , i.e., 0 < d < k c . When d > 0; d 6 pc tan1 pkV2 < pc p2 ¼ k c c   2 2 2k2c 2k When d 6 0; d 6 p tan1 pkV2 < pc p2 ¼ k2c , i.e., kc < d 6 0. c

Thus, we have

c : kc < d < k

ð35Þ

2) Furthermore, we have

! !   2 k 1 2 pd2 k2c pd2 D kV t D c þ ð 1  q ð d Þ Þ 6 V ð 0 Þ  e d 6 qðdÞ tan tan þ ; 2 2 p p 2k2c kV kV 2k c   1 T D kV t D e þ ; s ks s 6 V ð0Þ  2 kV kV   1 T D kV t D v ks v e 6 V ð0Þ  e þ : 2 e kV kV

ð36Þ

It follows that the path-following error d, attitude tracking error s, and velocity tracking error v e will converge eventually to n n n pffiffiffiffiffiffiffiffiffiffiffiffiffiffio pffiffiffiffiffiffiffiffiffiffiffiffiffiffio pffiffiffiffiffiffiffiffiffiffiffiffiffiffio the small sets Xd ¼ d :j d j6 2D=kV ; Xs ¼ s :j s j6 2D=kV ; Xv e ¼ v e :j v e j6 2D=kV . n o c : kW c k 6 1 . The sizes of these sets can According to Lemma 1 and 2, W will eventually converge to the small set XW ¼ W kW be made sufficiently small by appropriately choosing design parameters. 3) From (33), the path-following error d is bounded; therefore, the airship position ½x; yT is bounded. The tracking error _ c ; v c are bounded. Thus, all the states in the closed-loop control system remain bounded. s; v e and the desired states Hc ; H f Moreover, the estimate error W and basis functions r are bounded. Thus, the actuator saturation dynamics and control inputs are bounded.

Remark 8. Practically, the payload carried by a stratospheric airship requires that the airship position be maintained within a certain range. If the constraint is exceeded, it may lead to communication interruption, fixed-point observation failure, or other serious consequences. For example, the position constraints of a stratospheric airship deployed near the national borders for border observation mission are asymmetric. To observe the national borders, the flight path must be designed along the path with a symmetrical position constraint. However, an additional position constraint that the airship cannot exceed the national borders should be added to the side near the borders of the path. Thus, the position constraints would be asymmetric.

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Remark 9. In [56], we proposed an error-constrained LOS guidance method, in which the along-track error and cross-track error were both controlled to an acceptable set. The error-constrained VF guidance method proposed in this paper only addresses cross-track error as the control objective. Therefore, the method can effectively reduce computational complexity; therefore, it has practical military application prospect. Moreover, in [7], the authors demonstrated that the VF pathfollowing guidance law performed slightly better than the integral LOS guidance law. Remark 10. The anti-windup compensator in [56] is based on the disturbance observation theory; hence, the actuator output is required to be measurable. However, the actuator output in the RBFNN anti-windup compensator presented in this paper is not necessarily measurable. Therefore, it has a wider application scope. 4. Simulation To test the performance of the proposed control strategy, simulations on path-following are implemented using MATLAB. We apply the path-following controllers to the 6 degrees-of-freedom stratospheric airship model proposed in Section 2, and obtain some results, as shown in the following figures. All the values of parameters used in the simulations are listed in Table 1. The reference path is organized into four linear segments and four arc segments, as demonstrated in Fig. 5. The desired velocity is v c ¼ ½9; 0; 0m=s, and the desired attitude is Xc ¼ ½0; 0; wc rad. The time-varying external disturbances are given as

Table 1 Parameters of the stratospheric airship and path-following controller.

q m lref g



0:088 kg=m3 9400 kg 38 m 9:74 m=s2

Sref

1134 m2

r

10700 m3

 I x ; I y ; Iz ½xC ; yC ; zC  ½G; B   cp ; cq ; cr a1 ; a2 ; b ks kx kW v ; kW x Cv ; Cx

½2; 5:5; 5:5  106 ½0; 0; 0:5 m ½91556; 91556 N ½1; 10; 17  103 73:5; 62:5; 19 diagf15; 20; 0:5g diagf10; 10; 3g 25; 25 0:6  El

K

xn kd;v ; kd;s  k d

c k e k

0:75  E 30  E 1:5; 0:24 100 100 0.6

 k e0 kd kc ke

600 600 3.6

ke0 kv kdx ; kdv l

0 diagf3; 1:5; 2g 3; 0:3 50

Fig. 5. Trajectory of stratospheric airship path-following ½x; y (from A to B).

0

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2

fv fx

10  ð1 þ 0:1  sinðt=60Þ þ 0:2  cosðt=60ÞÞ

3

7 6 0:25  ðsinðt=60Þ þ 2  cosðt=60ÞÞ 7 6 7 6 7 6 sinðt=60Þ þ 2  cosðt=60Þ 3 7 ¼ 10  6 6 2:5  ð1 þ 0:1  sinðt=60Þ þ 0:2  cosðt=60ÞÞ 7: 7 6 7 6 4 2:5  ð1 þ 0:1  sinðt=60Þ þ 0:2  cosðt=60ÞÞ 5 10  ð1 þ 0:1  sinðt=60Þ þ 0:2  cosðt=60ÞÞ

T

Remark 11. The disturbance items ½f v f x  are the disturbances effect on accelerations and angular accelerations. Due to the huge inertia and relatively small driving forces, the dynamic response of the stratospheric airship is extremely slow. For example, It usually takes 1 or 2 min to increase the yaw angle from 0 to 60 . Thus, their accelerations and angular accelerations are relatively small. The disturbances are relatively small. ^d ð0Þ ¼ d ^d ð0Þ ¼ 0; X 0 ¼ ½0; 1000; 18900; 0; 0; 0; 10; 0; 0; 0; 0; 0T . The initial states are N1 ð0Þ ¼ N2 ð0Þ ¼ 0; d x

v

Fig. 6. Position error of stratospheric airship path-following d.

Fig. 7. Attitude of stratospheric airship path-following H.

T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

Fig. 8. Velocity of stratospheric airship path-following

515

v.

Fig. 9. Angular velocity of stratospheric airship path-following X.

The simulation results of the stratospheric airship trajectory, heading angle, and velocity are shown in Figs. 5–15. Figs. 5– 12 show the simulation results of the proposed method, while Fig. 14 and 15 show the control inputs without RBFNN compensators. Figures 5 and 6 demonstrate the time histories of airship position and tracking error, in which the airship moves from A to B. It is evident that the tracking error never exceeded the specified constraints. Moreover, we observe that the linear and arc path could be seamlessly switched. Remark 12. The linear path tracking error could converge to near zero; however, the arc path tracking error was relatively larger, which might be related to the wp . When we obtained the derivative of tracking error d, we ignored the change in wp over time. wp;l is a constant, whereas wp;c varies depending on the change in the airship position. The change in wp;c will be considered in our future work. Furthermore, the relatively larger error is acceptable in practice for a stratospheric airship over 100 m in length.

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

Fig. 10. Adaptive bound estimation of stratospheric airship path-following ^ ddv and ^ dds .

Fig. 11. Control input of stratospheric airship path-following sv .

Figs. 7–9, which are the time histories of airship attitude, velocity, and angular velocity, demonstrate that the airship could track the desired state values well. The external disturbances f x ; f v , control couplings B21 sv ; B12 sx , and error of filter   ^_ þ H ^ €c  H € c are compensated by the adaptive controller. The results show the effectiveness of the _cH estimation kx H c adaptive attitude and velocity control algorithm. The heading angular velocity r appeared turbulent when the desired path was switched. This was caused by the immature switch algorithm, which will be improved in our future work. ^d and ^ As shown in Fig. 10, d dd are bounded. They are both converged to near zero; however, ^ dd continues to increase v

x

x

when tracking the arc path. The reason may be similar to that of the tracking error d. The control inputs are illustrated in Fig. 11 and 12. The control inputs are large to drive the airship toward the path quickly with a large tracking error. Subsequently, the actuator saturation can be effectively compensated by the RBFNN anti-windup compensators. In addition, Fig. 13 shows the good performance of the RBFNN anti-windup compensators. A comparative simulation is performed to show the effectiveness and advantages of the proposed method. In this simulation, apart from RBFNN saturation compensator, all the conditions and controller parameters are the same. The results are shown in Fig. 14 and 15. Comparing Fig. 12 and 15, we can observe that, without the compensator, the input ss showed actuator saturation phenomenon at approximately t ¼ 10, which lasted for some time. Consequently, by removing the RBFNN

T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

517

Fig. 12. Control input of stratospheric airship path-following ss .

Fig. 13. RBFNN estimation of stratospheric airship path-following ds .

saturation compensator from the proposed method, we can observe that the airship could also track the predefined path; however, the actuator saturation phenomenon was more likely to appear and remain for a longer time. Thus, the effectiveness and robustness of the proposed method to successfully track the predefined path with disturbances and actuator saturation is demonstrated by providing more acceptable control inputs.

5. Conclusion A path-following control algorithm has been presented for a stratospheric airship with asymmetric error constraints, external disturbances, and actuator saturation. The unique feature of this study is that the AECVF guidance method is proposed using the ABLF technique, which guarantees that the error constraints for the airship position are never violated. In addition, a second-order command filter is introduced to estimate the derivative of the desired Euler angle calculated using the guidance law, which is difficult to obtain. An adaptive control method is adopted to track the desired attitude and velocity. The disturbance boundaries are estimated using adaptive algorithms. Moreover, RBFNN saturation compensators are introduced to solve the actuator saturation problem. Stability analysis indicates that all the signals in the closed-loop system are uniformly ultimately bounded. Simulation results demonstrate the effectiveness of the proposed control algorithm.

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Fig. 14. Control input without NN saturation compensator sv .

Fig. 15. Control input without NN saturation compensator

ss .

Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 61503010), the Aeronautical Science Foundation of China (No. 2016ZA51001). Appendix A. Force analysis of stratospheric airship motion According to Archimedes’ principle, the buoyant forces of the airship F B and M B in BRF are described as

(

FB MB

¼ F B ½ sin h; cos h sin /; cos h cos / ¼ ½0; 0; 0

T

T

where F B ¼ qrg; q is the density of local air, r is the volume of the airship, and g is the local gravitational acceleration. Similarly, we can obtain the airship gravity G and gravity torque M B in BRF as follows:

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

8 > G > > > < > MG > > > :

T

¼ G½ sin h; cos h sin /; cos h cos / 2 3 zC cos h sin / þ yC cos h cos / 6 7 ¼ G4 zC sin h  xC cos h cos / 5 xC cos h sin / þ yC sin h

The aerodynamic force on the airship in BRF can be described as follows:

8 > > > > > FA > > > < > > > > > > > MA > :

2

C x ða; bÞ

3

6 7 ¼ 12 qV 2 Sref 4 C y ða; bÞ 5 C z ða; bÞ 2 3 C x ða; bÞ þ C p p 6 7 ¼ 12 qV 2 Sref Lref 4 C y ða; bÞ þ C q q 5 C z ða; bÞ þ C r r

where qV 2 =2 is the dynamic pressure, V is the relative airflow velocity, and Sref ¼ r2=3 and Lref are the reference area and length, respectively. a ¼ arctanðw; uÞ is the attack angle, b ¼ arctanðv cos a; uÞ is the side slip angle, C x ða; bÞ; C y ða; bÞ; C z ða; bÞ; C l ða; bÞ; C m ða; bÞ; C n ða; bÞ are the aerodynamic force and torque coefficients related to a and b, respectively. C p ; C q , and C r are the aerodynamic torque coefficients related to the attitude angle velocity. Owing to the large volume, the additional inertia force and torque of the stratospheric airship cannot be ignored. They are expressed in BRF as follows:

(

F add M add

¼ M 0 ðv_ þ H  v Þ _ ¼ I 0 H O

0

where M ¼ diagfa1 ; a2 ; a3 g ¼ qrdiagfk1 ; 1:2k2 ; k2 g and I 0O ¼ diagfa4 ; a5 ; a6 g ¼ qrdiagf0; k3 ; 1:2k3 g are the additional mass and additional inertia matrices, respectively. From the dynamics analysis of the ellipsoid in the potential flow field, the additional parameters are defined as follows:

ag k1 ¼ ; 2  ag where

bg k2 ¼ ; 2  bg

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b e ¼ 1  2; a

 f ¼ log

 2 2 b  a2 ða  bÞ 1    k3 ¼   2 5 2 b  a2 þ b2 þ a2 ðb  aÞ

 1þe ; 1e

g ¼

1  e2 ; e3

 ag ¼ 2g 

 f e ; 2

bg ¼

1 g f ;  e2 2

a¼

a1 þ a2 2

and a1 ; a2 , and b are the profile parameters of the stratospheric airship. As shown in Fig. 1, the thrusters of the airship are four main propulsion propellers and a vector propeller working in Oxy. F T and M T could be disassembled to the direction of the BRF coordinate axes. Thus,

½F T ; M T T ¼ s ¼



su ; sv ; sw ; sp ; sq ; sr

T

Appendix B. Details of the state-space function

f 1 ðX 1 Þ ¼

X 11

033

033

X 22

2

;

X 11

3 chcw shcws/  swc/ shcwc/ þ sws/ 6 7 ¼ 4 chsw shsws/ þ cwc/ shswc/  cws/ 5; sh

chs/

sx; cx, and tx denote sin x; cos x, and tan x, respectively.

chc/

2

X 22

3 thc/ 7 s/ 5; 0 s/=ch c/=ch

1 6 ¼ 40

ths/ c/

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T. Chen et al. / Mechanical Systems and Signal Processing 119 (2019) 501–522

I y þ a5 ; mIy þ ma5  m2 z2C þ a1 a5 þ a1 Iy Ix þ a4 ¼ ; mIx þ Ix a2  m2 z2C þ ma4 þ a2 a4 1 ¼ ; m þ a3 mzC ¼ ; mIy þ ma5  m2 z2C þ a1 a5 þ a1 Iy mzC ¼ ; mIx þ Ix a2  m2 z2C þ ma4 þ a2 a4 1 ¼ : I z þ a6

mzC ; mIy þ ma5  m2 z2C þ a1 a5 þ a1 Iy mzC ¼ ; mIx þ Ix a2  m2 z2C þ ma4 þ a2 a4

b11 ¼

b15 ¼

b22

b24

b33 b42 b51 b66

m þ a2 ; mIy þ ma5  m2 z2C þ a1 a5 þ a1 Iy m þ a1 ¼ ; mIx þ Ix a2  m2 z2C þ ma4 þ a2 a4

b44 ¼ b55

T

f 2 ð X 1 ; X 2 Þ ¼ ½F v ; F x T þ ½f v ; f x  3 2    Iz þ Iy  Ix mzC pr  mIy  m2 z2C ðwq  v rÞ  mzC M ay þ F ax Iy  ðG  F B ÞIy sh þ mGz2C sh þ f 7 6 u mIy þ qIy a1  m2 z2C 7 6 7 6   7 6  I þ I  I mz qr þ mI  m2 z2 ðwp  ur Þ þ mz M þ F I þ ðG  F ÞI chs/  mGz2 chs/ 7 6 z x y C x C ax ay x B x C C þ fv 7 6 2 z2 mI þ q I a m x x 2 7 6 C  7 6 7 6 mðuq  v pÞ þ mzC p2 þ q2 þ F az þ ðG  F B Þchc/ 7 6 þ f w mþqa3 7 ¼6  7 6  7 6  Iz  Iy ðm þ qa2 Þ þ m2 z2C qr þ ðm þ qa2 ÞMax þ mzC F ay þ mzC ðG  F B Þchs/  ðm þ qa2 ÞGzC chs/ 7 6 þ f p7 6 mIx þqIx a2 m2 z2C 7 6  7 6  ðI  I Þðm þ qa Þ  m2 z2 pr þ ðm þ qa ÞM  mz F þ mz ðG  F Þsh  ðm þ qa ÞGz sh 7 6 x z 1 1 ay C ax C B 1 C C þ f 7 6 2 2 q mIy þqIy a1 m zC 7 6 5 4   Iy  Ix pq þ Maz þ f r Iz References [1] J. Ackermann, J. Guldner, W. Sienel, R. Steinhauser, V.I. Utkin, Linear and nonlinear controller design for robust automatic steering, IEEE Trans. Control Syst. Technol. 3 (1) (1995) 132–143, URL:http://ieeexplore.ieee.org/document/370719/. 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