Fault-tolerant anti-windup control for hypersonic vehicles in reentry based on ISMDO

Accepted Manuscript

Fault-Tolerant Anti-windup Control for Hypersonic Vehicles in Reentry based on ISMDO Yizhen Meng, Bin Jiang, Ruiyun Qi, Jianwei Liu PII: DOI: Reference:

S0016-0032(17)30635-X 10.1016/j.jfranklin.2017.12.004 FI 3255

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

25 August 2016 23 August 2017 7 December 2017

Please cite this article as: Yizhen Meng, Bin Jiang, Ruiyun Qi, Jianwei Liu, Fault-Tolerant Anti-windup Control for Hypersonic Vehicles in Reentry based on ISMDO, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.12.004

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Fault-Tolerant Anti-windup Control for Hypersonic Vehicles in Reentry based on ISMDO

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Yizhen Meng, Bin Jiang, Ruiyun Qi, Jianwei Liu College of Automation Engineering, Nanjing University of Aeronautics and Astronautics. No.29, Jiangjun Road, Jiangning district, Nanjing, 211106

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Abstract

Due to the extreme large flight scale of Hypersonic Vehicle (HSV), the system inevitably possesses strong nonlinearity, coupling, fast time-variability and is also sensitive to disturbance and fault. The method of external anti-windup system combined with the terminal sliding mode control law (TSMC) is presented for the nonlinear control problem under the restriction of control surfaces

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for HSV. It can realize the compensation for the control surface saturation and let the HSV smoothly track the command signals. Then, the improved sliding

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mode disturbance observer (ISMDO) is proposed to estimate unknown parameters and strong external disturbance as well as the unknown actuator fault. This method does not need the information of disturbance and the fault bounds and

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has fewer learning parameters, which makes it suitable for the real-time control. Finally, the simulation test of attitude control for the reentry HSV is conducted,

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and the results show the effectiveness and robustness of the proposed scheme. Keywords: fault-tolerant control, sliding mode control, hypersonic vehicle,

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super-twisting disturbance observer, anti-windup

✩ This work is supported by National Natural Science Foundation (NNSF) of China under Grant (61533009, 61374116), the Six Talent Peaks Project in Jiangsu Province (No. HKHT010) and the Funding of Jiangsu Innovation Program for Graduate Education (No. KYLX160377). ∗ Corresponding author Email address: [email protected] ()

Preprint submitted to Journal of LATEX Templates

December 16, 2017

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1. Introduction Hypersonic Vehicle (HSV) is in a state of unpowered and long-time gliding during reentry, which has a hash challenge for its controller. During reentry,

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the rarefied air, complicated temperature variation, disturbance and the fault

make it a difficult mission for realizing the nice attitude control [1]. The rarefied air combined with the actuator fault will make the deflection angle break the

physical limitations of the rudder deflection angle quickly, moreover, the large

flight envelope also makes the aerodynamic characteristics of HSV present a serious nonlinear feature, following the changes of flight altitude, velocity and the

flight attitude. In addition, as for the uncertainty of aerodynamic parameters,

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un-modeled dynamics and the external strong disturbances also exert the great challenge for the control of HSV [2]. Therefore, an effective anti-saturation nonlinear robust fault-tolerant controller is crucial for HSV to guarantee its safety and achieve the control goals.

Due to the hash flight environment of HSV, the actuator fault is more likely

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to occur, causing serious consequences. Thus, the fault-tolerant scheme is crucial for HSV and must be considered when designing the attitude controller. A

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second-order dynamic terminal sliding mode is applied to handle the problem of near-space vehicle with actuator faults in [3], achieving the well fault-tolerant effect, however, it needs an auxiliary system to improve its tracking effects.

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A second-order system is applied to describe the actuator loss-of-effectiveness (LOE) and stuck failure in [4]. Based on the actuator model, a modified fault

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diagnosis and identification (FDI) and the backstepping controller are designed to handle the problem of near space vehicle with actuator faults. However, the

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actuator model is not easy to get. A Takagi-Sugeno (T-S) fuzzy model is ap-

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plied to describe the near-space vehicle (NSV) attitude dynamics with actuator faults in [5], the designed fault-tolerant controller achieves a good control effect based on the sliding mode observer. However, the proper fuzzy rules are hardly established. Except for the sliding mode control, the adaptive control algorithm

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is also an effective method to handle the fault-tolerant control. The dynamic

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surface control strategy, a modified control algorithm of backstepping control, is proposed in [6] as the basic controller to handle the problem of actuator-fault. The adaptation laws are designed to update parameter estimations for getting

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more accurate approximation of the parameterized uncertain actuator fault. An adaptive output feedback fault-tolerant controller is designed in [7] to deal with

the parameter uncertainties, actuator faults and external disturbances in attitude control of HSV. Adaptive laws are designed for updating the controller

parameters when both the plant parameters and actuator fault parameters are unknown. Based on the above research results, the basic controller combined with the fault-tolerant control subsystem possesses the well control effect.

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As for the deflection saturation of control surfaces, keeping the large angle of attack needs the big deflection angle of control surface during reentry. Besides, the rarefied air makes the situation even worse. Once the actuator fault appears, the physical limitations of the deflection angles of control surfaces will 45

be broken soon, resulting in the control surface stuck. If they are saturated for

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a long time, the performance of control system can be dramatically reduced, or even the system becomes unstable. Some anti-saturation compensation con-

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trollers based on L2 and linear matrix inequality (LMI) are designed for the linearized longitudinal channel of HSV by Di et al in [8]. The velocity, altitude, 50

attitude angle, angular rate, dynamic pressure and other states are all restricted

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in [9], to design the anti-windup controller. However, the model of HSV in [9] is decoupled, while the actual model is seriously coupled. An anti-saturation

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compensation controller based on LMI is designed in [10], guaranteeing the stability of constrained control systems. The input saturation problem is trans-

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formed into the constraint conditions of the system [11], which can be handled

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by optimal control techniques. As for the direct anti-saturation design methods, [10] and [11] are developed with quite desirable control performance , however, these algorithms are complex and require huge computation. Based on small gain theory, the stability of input constrained System with Anti-windup system

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is studied in [12], which greatly improves the performance of controller, avoiding actuator being saturated for a long time. 3

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As is known, the uncertain parameters, the strong disturbances and the actuator fault possess the bad effect on the system control, which should be inhibited as much as possible. Fault detection and identification (FDI), one kind of observer, can be used to estimate the information of those unfavorable

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factors, which is crucial for fault-tolerant control (FTC). There are varieties

of FDI approaches based on the models of linear or nonlinear systems, such as: observer based approaches, parity space and parameter estimation methods

[13]. An nonlinear disturbance observer(NDO) is proposed in [14] to estimate 70

the information of fault, uncertainty and disturbance. Then, the controller

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with the information of NDO achieves the good control effects. However, NDO needs the derivation of disturbance maintaining zero, meaning that disturbances and faults are slowly changing, but disturbance and fault don’t always belong to the slowly changing signal. Sliding mode disturbance observer (SMDO) is 75

able to approach the disturbance and fault perfectly in [15], but the bounds of the disturbances are indispensable. The fuzzy disturbance observer is used to

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estimate the uncertainty and disturbance of the chaotic neural network in [16]. Though it does not need the boundary value of disturbances, the appropriate

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fuzzy rules are hardly to obtain. The adaptive control method based on Neural Network is proposed by [17], which can modify the weights of network online to approximate the time varying disturbance effectively. However, there are

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too many online learning parameters to meet the real-time demands of control system of HSV, which will give rise to the unknown effects on the safety of HSV

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and the goals attainment. In this paper, referring to the methods of anti-saturation control, an exter-

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nal anti-windup compensator combined with the terminal sliding mode control

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(TSMC) is designed for HSV to handle the problem of control surface saturation during reentry. As for the uncertainties and the actuator fault, an improved sliding mode disturbance observer (ISMDO) based on the super-twisting control

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law (STW) in [18], [19] is proposed to estimate those information. ISMDO does not need the prior knowledge of disturbances and actuator faults and has fewer designing parameters, making it suitable for the real time control of HSV. The 4

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simulation results show that the proposed control strategy of this paper is of effect, good robustness and rapidity. The rest sections of this paper are organized as follows. Section 2 presents the

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dynamics of HSV and the details of actuator faults and limitations. In section 3, the anti-windup system and the control purpose are described in details.

The control structure is given in Fig.2. In section 4, the terminal sliding mode fault-tolerant anti-windup control system is designed. Section 5 represents the 100

simulation studies. At last, section 6 is the conclusion.

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2. HSV Reentry Model

The design model of HSV comes from the winged-cone configuration derived by NASA Langley Research Center[20] and [21]. In this paper, we just focus on the attitude control of HSV during its reentry, and the details are provided as follows

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α˙ = q − tan β(p cos α + r sin α) +

1 (−L +M g cos γ cos µ) M V cos β

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1 β˙ = p sin α − r cos α + (Y − M g cos γ sin µ) MV tan β µ˙ = − sec β(p cos α + r sin α) − (L − M g cos γ cos µ) MV tan γ + (L sin µ − Y cos µ) MV

(1) (2)

(3) (4)

q˙ = (Izz − Ixx ) · pr/Iyy + mA /Iyy

(5)

r˙ = (Ixx − Iyy ) · pq/Izz + nA /Izz

(6)

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p˙ = (Iyy − Izz ) · qr/Ixx + lA /Ixx

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where γ represents the flight path angle of HSV. α is the attack angle of HSV.

β represents the sideslip angle, and µ devotes to the roll angle of HSV. p is the angular rate of rolling, and q, r represent the angular rates of pitching, yawing, respectively. M devotes to the mass of HSV. D represents the force of drag for

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HSV, and Y , L represent the lateral force and lift force, respectively. Ixx , Iyy and Izz are the moments of inertia, which are the nonlinear functions of the mass 5

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of HSV. lA , mA and nA represent the pneumatic moments of rolling, pitching and yawing, respectively. g is the gravitational acceleration of the earth. This nonlinear model is composed of six rigid-body state variables [α, β, µ, p, q, r]T ,

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the output states are selected as α, β, µ. The control input u = [δe , δa , δr ]T does not appear explicitly in the equations [22]. In order to ensure the safety of the

reentry flight and meet the limitations of aircraft structure, the limitations of deflection angles of pneumatic control surface are provided as follows [15] −10◦ /s ≤ p, q, r ≤ 10◦ /s, −30◦ ≤ δe , δa , δr ≤ 30◦

(7)

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where δe , δa and δr represent the deflection angles of aileron, elevator and rudder, respectively. 2.1. FTC objective

Due to the hash flight environment, fault is likely to occur and may locate in the actuators, the components or the elevators of HSV. When faults occur,

(8)

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the faulty system can be described as [23]   x˙ = fF (xF , uF (t))  y = h (x)

where fF (·) represents the component faults of the vehicle. The actuator fault

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uF (t) may appear in many forms, such as

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uF i (t) = (1 − λi )u(t)

(9)

where 0 < λi ≤ 1, and λi represents the loss of actuator effectiveness. Because

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the fault does not always exist, Eq.(9) is modified as follows [24] uF i (t) = (1 − %i (t − T )λi )u(t)

  % (t − T ) = 0, i where %i (t − T ) =  % (t − T ) = 1,

t≤T

(10)

. T represents the time that the t≥T fault occurs. when %i (t − T ) = 0, there is no fault with actuator, while, when i

%i (t − T ) = 1, the actuator is out of order. 6

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Besides, both control inputs are supposed to be constrained by a saturation value, expressed by uimin ≤ uF i (t) ≤ uimax

(11)

ration function sat(uF i (t)) is defined as    ui max , ui max > uF i (t)   sat (uF i (t)) = uF i (t) , ui min < uF i (t) < ui max     u , u (t) < u i min

Fi

(12)

i min

Remark 1: The upper bound and the lower bound of the saturation func-

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which means that each of the inputs has a separate saturation limit. The satu-

tion, ui max and ui min are not constrained to be symmetric. Besides, ui max and ui min may be not equal to the real limitation of the vehicle, since actuator may be locked in a small area. Then the upper bound and the lower bound will change according to the real situation.

3. Structure of Fault-Tolerant Control System 3.1. Anti-windup System

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An anti-windup system is a system that can inhibit the decline in performance of the system due to the saturation of actuators caused by their physical limitations, particularly when the actuator fault occurs. The mechanism of antiwindup system is described as follows: under the normal circumstances, it will

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not work. Once the saturation occurs, the anti-windup system will be switched

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on until the saturation disappearing. As shown in Fig.1, the output of the anti-windup system is directly working on the input and output of the norminal controller, without any effects on the norminal controller, seen the relationship between v1 ,v2 and K in Fig.1, which is called the external anti-windup system

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[25]. The module of F is the designed anti-windup compensation system. K is the norminal controller without the actuator constraints. y represents the output feedback signal of the system. v1 , v2 are the outputs of the compensation

system. In this paper, terminal sliding mode control (TSMC) is designed as the 140

nominal controller K. 7

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sat ( u ) - u

F v2 + K

u

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+ v1

y

Figure 1: Basic structure of the external anti-windup system

3.2. Structure of Fault-Tolerant Control System

According to the time-scale separation principle, the attitude control system

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for HSV reentry can be divided into fast and slow control loops, which is shown in Fig.2. The control purpose is to track the expected attitude commands Ωc . 145

At first, Ωc will be mapped into the corresponding commands of control surface δc under the controller. Secondly, the actuator works quickly to respond the commends, δc , then, based on real action of control surface and the dynamic equations of HSV, the real states of HSV can be obtained. After the repeated

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iterations, the attitude angle Ω can track Ωc at eventually.

As for anti-windup system, when the error of δC0 and δC occurs, the error

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is transformed as the control torque compensation MC0 and the state error compensation ωaw by anti-windup controller F . Through the work of fast-loop TSMC and MC0 , the saturation conditions of actuator is relieved and improved.

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Besides, an improved sliding mode disturbance observer(ISMDO) is designed to estimate those uncertainties and deduce the law of compensation.

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Thus, an anti-saturation fault-tolerant controller, consisted by TSMC, antisaturation controller (F ) and ISMDO, is designed in this paper. The specific

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design process will be carried out in the follow-up sections.

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Du

F +

waw es

fast- M C M C 0 control loop allocM1 TSMC ation

slowwc loop TSMC -

-

+

D+F

dC

dC 0

- g -f 1

W

ISMDO

-

W HSV w

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Wc

Dd C

gd

p,q,r (w )

a , b ,s ( W )

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Figure 2: Structure diagram of HSV attitude control system

4. Design of HSV Attitude Controller 160

4.1. Slow-Loop Control Law of HSV

The slow-loop of HSV simulation model (1-3) can be simplified to the affine nonlinear function as follows

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˙ s + ∆fs + gs ω + ∆s , ys = Ω Ω=f

(13)

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where Ω = [α, β, µ] . ∆fs represents the uncertainty part of fs . ∆s represents the external disturbances. In order to facilitate the analysis, we remark Ds =∆fs + ∆s , and suppose that Ds satisfies kDs k ≤ r1 + r2 kΩk, where r1 and T

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r2 are two unknown positive constants. ω = [p, q, r]

represents the fast-loop

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states of HSV. As for fs = [fα , fβ , fµ ] , the details are as follows (14)

fβ = (ˆ q SCY,β β − M g cos γ sin µ)/M V

(15)

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fα = (−ˆ q SCL,α + M g cos γ cos µ)/M V cos β

fµ = −(ˆ q SCL,a − M g cos γ cos µ) · tan β/M V + qˆSCL,a sin µ tan γ/M V − qˆSCY,β β cos µ tan γ/M V  − tan β cos α 1 − tan β sin α   gs =  sin α 0 − cos α  − sec β cos α 0 − sec β sin α 9

    

(16)

(17)

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where qˆ is the dynamic pressure and CL,α represents the lift increment coefficient for basic vehicle. CY,β is the side force with sideslip derivative for basic vehicle. In the attitude control system, the role of slow-loop is to track Ωc in finite

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time and provides the command angular rate ωc for the fast control loop. Referring to the control methods in [26], the terminal sliding mode surface is designed as s = Ωe +

Z

t

(a1 Ωe + b1 Ωe q1 /p1 )dτ

0

(18)

where Ωe = Ω − Ωc represents the tracking error. a1 , b1 , p1 , q1 are the design 165

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parameters, meeting a1 > 0, b1 > 0, p1 > q1 . p1 and q1 are positive odd numbers.

Remark 2: As is known, comparing with the linear sliding surface in SMC, the integral item will improve the transient performance for the SMC system [27]. Besides, by introducing the concept of terminal attractor of neural network [28], (18) is designed to improve the convergence performance of SMC system. When Ωe is far away from zero, a1 Ωe plays a leading role. When Ωe is close to

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zero, b1 Ωe q1 /p1 plays the key role.

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Differentiate the both sides of (18), and we have s˙ = Ω˙ e + a1 Ωe + b1 Ωe q1 /p1 = fs + Ds + gs ω − Ω˙ c + a1 Ωe + b1 Ωe q1 /p1

(19)

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In order to improve the dynamic characteristics of the sliding mode control,

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the reaching law is designed as follows k

s˙ = −l1 s − l2 |s| sgn (s)

(20)

where l1 and l2 are the gain coefficients of reaching law, meeting l1 > 0, l2 > 0.

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k is the design parameter of reaching law, meeting 0 < k < 1. According to

the analysis thought of [29], (20) can be divided into two parts. When s is far

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away from zero, the convergence characteristic of s˙ is up to (−l1 s). When s is k

close to 0, the convergence characteristic of s˙ is determined by (−l2 |s| sgn (s)), which can further improve the convergence characteristics of TSMC.

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Then, the slow-loop controller of HSV is obtained as follows k ωc = gs−1 (−fs + Ω˙ c − a1 Ωe − b1 Ωe q1 /p1 − l1 s − l2 |s| sgn (s)

− (ˆ r1 + rˆ2 kΩk)sgn (s))

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(21)

and the adaptive updating laws of r1 , r2 are provided as follows rˆ˙1 = m−1 ˆ˙2 = m−1 1 ksk , r 2 ksk kΩk

(22)

where rˆ1 , rˆ2 are the estimations of r1 , r2 , respectively. mi is a design parameter, satisfying mi > 0. k·k represents the 2-norm in this paper.

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Theorem 1 Under the control law (21) and the adaptive law (22), the

attitude tracking error can converge to a small region in finite time and the closed loop system is stable. The convergence region is described as follows 1/k

|s| ≤ (δ/l2 )

+δ

(23) T

where δ represents the estimation error of Ds , and δ = [δ1 , δ2 , δ3 ]

combined

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1/k

, |s| ˙ ≤ l1 (δ/l2 )

T

with s = [s1 , s2 , s3 ] , corresponding to the three channels of HSV. δi , i = 1, 2, 3,

δi , δi > 0.

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represents the bound of estimation errors of Dsi , namely, |ˆ ri1 + rˆi2 kΩi k − Dsi | ≤ Proof: A Lyapunov function is chosen as

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V (s, r˜1 , r˜2 ) =

1 1 1 T s s + m1 r˜12 + m2 r˜22 2 2 2

(24)

where r˜1 = rˆ1 − r1 , r˜2 = rˆ2 − r2 .

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Take the derivative of Eq.(24), and we have

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V˙ = sT s˙ + m1 r˜1˜˙r1 + m2 r˜2˜˙r k

= sT (Ds − rˆ1 − rˆ2 kΩk − l1 s − l2 |s| sgn(s)) + m1 r˜1˜˙r1 + m2 r˜2˜˙r2 k

≤ sT (kDs k − rˆ1 − rˆ2 kΩk) + sT δ − l1 sT s − l2 sT |s| sgn(s) + r˜1 ksk + r˜2 ksk kΩk k

≤ (kDs k − rˆ1 − rˆ2 kΩk) ksk + sT δ − l1 sT s − l2 sT |s| sgn(s) + r˜1 ksk + r˜2 ksk kΩk ≤ (r1 − rˆ1 ) ksk + (r2 − rˆ2 ) kΩk ksk + (ˆ r1 − r1 ) ksk + (ˆ r2 − r2 ) ksk kΩk k

− l1 sT s − l2 sT |s| sgn(s) + sT δ 11

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k

≤ −l1 sT s − sT (l2 |s| − δ)

(25) k

Referring to [30] and [31], if l2 |s| ≥ δ , we have V˙ + l1 V ≤ 0

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(26)

and the convergence region is obtained as follows 1/k

|s| ≤ (δ/l2 )

(27)

Substitute Eq.(27) into Eq.(20), and we get k

1/k

|s| ˙ ≤ l1 (δ/l2 )

+δ

1/k

n ok 1/k + l2 (δ/l2 )

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|s| ˙ ≤ l1 |s| + l2 |s| ≤ l1 (δ/l2 )

(28)

According to (20), it can be obtained as follows s+l ˙ 1 s = −l2 sk

(29)

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Then, multiply the both sides of (29) by s−k , and we get ds +l1 s1−k = −l2 dt

(30)

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s−k

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Let λ = s1−k , (30) can be rewritten as follows dλ + (1 − k) l1 λ = − (1 − k) l2 dt

(31)

After solving the first order linear non-homogeneous differential equation (31),

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we have

λ=c1 e

−

R

(1−k)l1 dt

−

l2 e

R

(1−k)l1 dt

l1

e−

R

(1−k)l1 dt

= c1 e(1−k)l1 t −

l2 l1

(32)

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Substitute λ = s1−k into (32), and we have s1−α = c1 e(1−k)l1 t −

l2 l1

(33)

When t = 0 and s = s0 , c1 can be obtained as follows c1 = s1−k 0 12

l2 l1

(34)

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Here, according to (33) and (34), (29) can be solved as follows t=

(35)

Then, the states of (18) can converge to the region (27) within a finite time,

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1 (ln(s1−k + l2 /l1 ) − ln(s1−k + l2 /l1 )) 0 (1 − k)l1

seen in (35).

As we all know, the switching function of the TSMC will bring the dramatic system chattering, which has a great influence on the performance of system.

In order to inhibit the chattering of system, the switching item sgn(s) in (21) is replaced by a continuous function with respect to tracking error Ωe . The details

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are shown as follows sign(s) =

s ksk + ∆ + kΩe k

(36)

where s represents the sliding mode surface. ∆ is a a small enough constant. Ωe represents the tracking error, namely, Ωe = Ω − Ωc . Besides, considering the

adaptive law in (22), a little offset of s will lead to ksk > 0, resulting in rˆ˙ > 0,

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causing rˆ continuously increasing, so does the output of the controller, which will give rise to some bad influence on the tracking effect. In order to effectively

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stabilize changes of rˆ , the boundary layer is brought into the adaptive law and

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the improved adaptive law is as follows   m−1 ksk, ksk > ε 0 1 rˆ˙1 =  0, ksk < ε0   m−1 ksk kΩk ,ksk > ε 0 2 rˆ˙2 =  0, ksk < ε

(37)

(38)

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0

where ε0 > 0 is a small design parameter.

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4.2. Fast-loop Control law of HSV The fast-loop of HSV simulation model (4-6) can be simplified to the affine

nonlinear function. Here, the partial loss-of-efficiency (LOE) fault is brought into the dynamic model, and the details are provided as follows ω=f ˙ f + ∆ff + gf Mc + ∆f , yf = ω 13

(39)

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T

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where ω=[p, q, r] , ∆ff is the uncertainty part of ff . As to ff = [fp , fq , fr ] , the specific expressions are as follows

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fp = (Iyy − Izz ) · qr/Ixx + laero /Ixx fq = (Izz − Ixx ) · pr/Iyy + maero /Iyy fr = (Ixx − Iyy ) · pq/Izz + naero /Izz

(40)

q gf = diag {glp , gm , gnr }

(41)

u(t) = Mc represents the control moment of fast-loop of HSV without fault, ∆f = %i (t − T )λi u(t) represents loss of actuator effectiveness. Thus, the con-

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trol moment with fault (uF i (t)) and the saturation limits (uimax , uimin ) are described in (10) and (12).

Assumption 1 In order to facilitate the expression in fast-control loop, we

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take Df =∆ff + ∆f as the compound uncertainties. Besides, we assume that ˙ Df i ≤ 2i (i = 1, 2, 3) , t ≥ 0, where 2i is unknown positive constant.

4.2.1. Design of Anti-windup Auxiliary System

As shown in Fig.2, the anti-windup system is applied to inhibit the degra-

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dation of tracking performance due to the saturation of control surface, and regain the most control effect of the norminal controller. Referring to [25], the

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anti-windup compensation system is designed as follows ω˙ aw = ff (ωaw ) + gf · (h(ωaw ) + ∆u)

(42)

ν1 = Cωaw , ν2 = h(ωaw ) + gf−1 · ff (ωaw )

(43)

where ωaw represents the states of compensation system, determined by ∆u=MC −

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MC0 , which is specified later. C is the designed gain coefficient, which is a positive constant. ν1 and ν2 are the outputs of the anti-windup system. Here, the function h(ωaw ) is designed when ∆u = 0 (i.e. when saturation

does not occur) to guarantee system ω˙ aw = ff (ωaw ) + gf h(ωaw )

14

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globally stable. For such problem, the dynamic inversion is applied to guarantee the system (42) being globally stable. Here, some instructions should be given firstly. The actuator is simulated by one-order inertia plant, namely, ω˙ aw =

assumption of above, h(ωaw ) is designed as follows

CR IP T

kc (ωaw − ωawc ), where kc represents the bandwidth frequency. Based on the h(ωaw ) = gf−1 [−kc (ωaw − ωawc ) − ff (ωaw ) + ω˙ awc ] 200

(44)

where ωawc is the commanded signal, and the ωaw is the state of anti-windup system.

function is selected as follows

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In order to proof the stability of the anti-windup system, the Lyapunov

Vωaw = where e = ωaw − ωawc .

1 T e e 2

(45)

Then, take the derivation of (45), and we have

M

e˙ = ω˙ aw − ω˙ awc = ff (ωaw ) + gf h(ωaw ) − ω˙ awc

(46)

Substitute (44) into (46), and we have

ED

V˙ ωaw = eT e˙ = eT (ω˙ aw − ω˙ awc ) = eT (ff (ωaw ) + gf h(ωaw ) − ω˙ awc )

PT

= eT (ff (ωaw ) + gf gf−1 [−kc (ωaw − ωawc ) − ff (ωaw ) + ω˙ awc ] − ω˙ awc ) 2

≤ −kc kek2

(47)

CE

2 Here, based on V ≥ 0, V˙ ≤ −kc kek2 ≤ 0, we obtain that V is bounded. Then,

according to (45), we have that e is bounded. Due to e˙ = −kc e, it is obvious

205

that e˙ is bounded. Based on Barbalat lemma, e can converge to zero. Above all,

AC

the anti-windup system is asymptotically stable according to Lyapunov theorem of stability. 4.2.2. Design of TSMC Control System As shown in Fig.2, the terminal sliding mode control is designed as the basic

210

controller for nominal system without considering the ∆ff and ∆f of (39), and 15

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details are provided as follows. The sliding mode surface is designed as follows Z t sf = ωe + (a2 ωe + b2 ωe q2 /p2 )dτ

(48)

CR IP T

0

where ωe = ω − ωc − ωaw is the total error of fast control loop, a2 > 0, b2 > 0,

p2 > 0 and p2 , q2 are the positive odd numbers. Differentiating both sides of the equation (48), and with (39), (without ∆ff and ∆f ) and (48), we can get s˙ f = ω˙ e + a2 ωe + b2 ωe q2 /p2 = ω˙ − ω˙ c − ω˙ aw + a2 ωe + b2 ωe q2 /p2

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= ff + gf MC − ω˙ c − ff (ωaw ) − gf · h(ωaw ) − gf ∆u + a2 ωe + b2 ωe q2 /p2

(49)

where MC is directly applied as the control moment of the HSV, satisfying that MC = gδ δc , where gδ is the control surface distribution matrix and δc represents the deflection angle of control surface.

Due to the physical structure constraints of actuator in this paper, we assume

M

that δc represents the angle of rudder reflection with restriction, and δc0 is the output of the fast-loop controller without limitations, satisfying ∆δc = δc − δc0 .

ED

As shown in Fig.2, ∆u satisfies

∆u = gδ ∆δc = gδ δc − gδ δc0 = MC − MC0

(50)

Then, substitute (50) into (49), and we have

PT

s˙ f = ff + gf MC0 − ω˙ c − ff (ωaw ) − gf h(ωaw ) + a2 ωe + b2 ωe q2 /p2

(51)

Order s˙ f = −kf · sf , kf is the design parameter, and then, the anti-saturation

CE

control law of the fast-loop of HSV is provided as follows

AC

MC0 = −gf−1 (ff − ω˙ c − ff (ωaw ) − gf · h(ωaw ) + a2 ωe + b2 ωe q2 /p2 + kf sf ) = −gf−1 (ff − ω˙ c + a2 ωe + b2 ωe q2 /p2 + kf · sf ) + gf−1 · ff (ωaw ) + h(ωaw ) (52)

In order to proof the stability of the fast-loop of HSV under the controller (52), the Lyapunov function is chosen as V2 =

1 T sf sf 2 16

(53)

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Differentiate the both sides of (53) and we get   V˙ 2 = sTf s˙ f = sTf ff + gf MC0 − ω˙ c − ff (ωaw ) − gf h (ωaw ) + a2 ωe + b2 ωe q2 /p2 (54)

CR IP T

Then, substitute (51) and (52) into (54) and we have   V˙ 2 = sTf ff − ω˙ c − ff (ωaw ) − gf h (ωaw ) + a2 ωe + b2 ωe q2 /p2   + sTf gf −gf−1 (ff − ω˙ c + a2 ωe + b2 ωe q2 /p2 + kf · sf ) + gf−1 · ff (ωaw ) + h(ωaw ) 2

= −kf sTf sf ≤ −kf ksf k2

If and only if sf = 0, V˙ 2 = 0. Based on the Lyapunov theorem, the system is

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215

(55)

asymptotic stability.

So far, the fast-loop controller of HSV has been designed. However, the disturbance, LOE, and uncertainty have the huge influence on the performance of controller. In order to handle such problem, an improved sliding mode observer is designed in next section.

M

220

4.2.3. Design of ISMDO for fast control loop In order to eliminate the adverse effects of Df in equation (39), ISMDO,

ED

based on super-twisting algorithm, is proposed to estimate the complex disturbance in fast control loop. The details are as follows. At the beginning of the 225

proof, the following lemma [19] is given as follows

CE

PT

Lemma 1 Consider the following nonlinear equations   x˙ = −k |x |1/2 sign(x ) + x + ξ(t) 1 1 1 1 2  x˙ = −k sign(x )

namely, x˙ 1 + k1 |x1 |

1/2

2

2

(56)

1

sign(x1 ) + k2

R

sign(x1 )dτ = ξ(t), where the design pa-

AC

rameters k1 > 0, k2 > 0, it can be guaranteed that x1 and x˙ 1 converge to the equilibrium point in finite time. In this paper, ISMDO is constructed as follows    σ =ω+z   z˙ = −ff − gf MC − ν     D ˆ =ν f

17

(57)

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where where ω presents the state of the fast control loop. z is the auxiliary state. ν = [ν1 , ν2 , ν3 ]T are designed as follows 1/2

νi = li1 · |σi |

· sign(σi ) +

Z

li2 sign(σi )dτ

CR IP T


 l˙i1 = ρi kωk2 kσk2 , li2 = εli1 + ε2 + λ 2

(58) (59)

where σi is the auxiliary sliding mode surface. ρi is the gain of adaptive parameter li1 , which meets ρi > 0. λ and ε are the positive real numbers and li1 , li2

are the adaptive gain of vi , satisfying li1 , li2 > 0, i = 1, 2, 3. As we all know that, the sgn function sgn (σi ) in (58) will bring the dramatic system chatter-

AN US

ing, which has a great influence on the performance of system. Therefore, the switching item sign (σi ) is replaced as follows sign (σf ) =

σf |σf | + δf + lf kef k

(60)

Theorem 2 Under Assumption 1, the ISMDO designed in (57)-(60) can asymptotically track the disturbance and the actuator fault. The proof of the-

M

230

orem 2 is as follows.

ED

Proof: Take the derivative of σ in Eq.(57), combined with (39), and we get

PT

σ˙ = ω˙ + z˙ = ff + gf MC + Df − ff − gf MC − ν = Df − ν Z 1/2 = Df − l1 · |σ| · sign(σ) − l2 sign(σ)dτ

CE

According to lemma 1, (61) can be reshaped as follows   σ˙ = −l · |σ |1/2 · sign(σ ) + y i i1 i i i  y˙ = −l · sign(σ ) + D˙

AC

where yi = Df i −

i

i2

i

(61)

(62)

fi

Rt

l sign(σi )dτ . 0 i2

For the convenience of analysis, some variable substitution are defined as

follows 

ςi = 

ςi1 ςi2





=

|σi |

1/2

ςiT ςi = ςi1 2 +ςi2 2 = |σi | +yi 2 18

sign(σi )

yi

 

ACCEPTED MANUSCRIPT

sign(σi ) = sign(ςi1 ), |ςi1 | = |σi |

1/2

(63)

i2

i2

i

fi

(64)

CR IP T

After the simplification, we have  .  ς˙i1 =(−li1 |σi |1/2 sign (σi ) + yi ) (2|σi |1/2 )  ς˙ = −l sign(σ ) + D˙

˜˙ = |σ |1/2 D ˙ f i , and (64) is marked as follows Then, we define that D fi i ς˙i = Ai ςi + Bi˜˙Df i 

1/2





0



(65)



1

T

, Bi =  ,order Ci =   . If ςi1 and ςi2 −li2 0 1 0 can converge to zero in finite time, so do σi and σ˙ i .

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where Ai = 

−li1 /2


 1/2 |σi |

After the variable substitution by ς, the convergence of the system (62)

235

can be transformed into the convergence problem of (64), except for the set S = {(σi , yi )R2 |σi = 0}. Therefore, the proof is divided into two parts, for

M

one, σ 6= 0, for the other, σ = 0.

When σ 6= 0, Lyapunov function is selected as follows



λ + ε2

−ε

(66)



 is symmetric positive definite matrix, λ, σ, κ1 , κ2 −ε 1 ∗ ∗ are any positive real numbers. li1 and li2 represent the positive constant. In

PT

where P = 

1 1 ∗ 2 ∗ 2 (li1 − li1 ) + (li2 − li2 ) 2κ1 2κ2

ED

V = ςiT P ςi +

CE

order to facilitate the proof, (66) is rewritten as follows V = V0 +

1 1 ∗ 2 ∗ 2 (li1 − li1 ) + (li2 − li2 ) 2κ1 2κ2

(67)

AC

where V0 = ςiT P ςi . Derivative the both sides of V0 , combined with (65), and we get V˙ 0 = ς˙iT P ςi + ςiT P ς˙i  1  T ˜˙ B T P ς + δ 2 |σ | − D ˜˙ 2 T ≤ ς (P A + A P )ς + 2 D i i fi i i i i i 2i fi 1/2 |σi |

19

(68)

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2 T T 2 where δ2i ςi C Cςi =δ2i |σi |. Referring to Arithmetic-Geometric Average In˜ ˜˙ 2 + B T P ς
T B T P ς , then, the (68) can be rewritten T ˙ equality, 2Df i Bi P ςi ≤ D i i i i fi

as follows [32]

≤ = 240

1 1/2

|σi | 1

1/2

|σi | 1



˜˙ B T P ς − D ˜˙ 2 2 T T ςiT (P Ai + ATi P )ςi + δ2i ςi C Cςi + 2D i fi i fi

2 T T ςiT (P Ai + ATi P )ςi + δ2i ςi C Cςi + ςiT P Bi BiT P ςi

ς T (P Ai 1/2 i

|σi |

2 + ATi P + δ2i C T C + P Bi BiT P )ςi

2 where Q = −(P Ai + ATi P + δ2i C T C + P Bi BiT P ).




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Then, combined with (65), we get  li1 (λ + ε2 ) − 2li2 ε − δ 2 − ε2 Q= 1 2 li2 − li1 2 ε − 2 (λ + ε ) + ε

li2 −

li1 2 ε



CR IP T

=

− 21 (λ + ε2 ) + ε

ε−1

Here, in order to guarantee Q being positive , we order

 

(70)

(71)

M

 li2 = li1 ε/2 + ε2 2 + λ/2

(69)

ED

Substitute (71) into (70), and we have   li1 λ − ε3 − ε2 − λε − δ 2 − ε/2 ε 1  Q − εI =  2 ε ε/2 − 1

(72)

PT

According to lemma of schur complement, the conditions maintaining Q positive

CE

and its minimum eigenvalue λmin (Q) > ε/2 are as follows     ε3 + ε2 + (0.5 + λ) ε + δ 2
λ,      li1 > max  ε4 −ε3 +(λ+0.5)ε2 −(2λ+1)ε−(2+ε)δ2  λ(ε−2)     ε>2

(73)

AC

According to (72), we get Q − εI/2 > 0. Then, we have

2

.  1 ε kςi k V˙ 0 ≤ −ςi T Qςi |σi | 2 ≤ −εςi T ςi 2 |ςi1 | ≤ − kςi k 2 |ςi1 |

(74)

kςi k2 ≥ |ςi1 |

(75)

2 2 2 Due to kςi k2 = ςi1 + ςi2 = |σi | + ςi2 , we have

20

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Then, ε V˙ 0 ≤ − kςi k2 2

2

CR IP T

Due to the function V0 = ςiT P ςi meets

(76)

2

λmin (P ) kςi k2 ≤ V0 = ςiT P ςi ≤ λmax (P ) kςi k2 Then, we have V0 λmax (P )

Based on (76) and (78), we have

1/2

≤ kςi k2

AN US



1/2 V˙ 0 ≤ −rV0

. 1/2 where r = ε 2λmax (P ).

(77)

(78)

(79)

According to (67) and (79), we have [33]

ED

M

1 1 1 ∗ ˙ ∗ ˙ V˙ ≤ −rV0 2 + (li1 − li1 )li1 + (li2 − li2 )li2 κ1 κ2 1 ρi kωk2 kσk2 ρˆi kωk2 kσk2 ∗ ∗ = −rV0 2 − |li1 − li1 |− |li2 − li2 | κ1 κ2 ρi kωk2 kσk2 1 1 ∗ ˙ ∗ ˙ ∗ + (li1 − li1 )li1 + (li2 − li2 )li2 + |li1 − li1 | κ1 κ2 κ1 ρˆi kωk2 kσk2 ∗ |li2 − li2 | + κ2

(80)

PT

˙ , meeting ρˆ > 0, which has the same where ρˆ is applied to describe the gain of li2

AC

CE

function with ρ. p Based on x2 + y 2 + z 2 ≤ |x| + |y| + |z|, we have √ p 1 2ρi kωk2 kσk2 _ ∗ 2 √ |li1 − li1 | − η V (ζi , li1 , li2 ) ≥ −rV0 − 2κ1 √ 2ˆ ρi kωk2 kσk2 ∗ √ − |li2 − li2 | 2κ2 √  √ _ where η = min r, 2ρi kωk2 kσk2 , 2ˆ ρi kωk2 kσk2 . Then, (80) can be rewritten as follows

p 1 _ ∗ ˙ V˙ ≤ − η V (ζi , li1 , li2 ) + (li1 − li1 )li1 κ1 21

(81)

ACCEPTED MANUSCRIPT

+

ρi kωk2 kσk2 ρˆi kωk2 kσk2 1 ∗ ˙ ∗ ∗ (li2 − li2 |li1 − li1 |li2 − li2 )li2 + |+ | κ2 κ1 κ2

(82)

Assuming that the adaptive law of li1 and li2 can made them bounded, (bounded

CR IP T

∗ ∗ ∗ ∗ < 0(∀t ≥ 0), and then, (82) < 0 and li2 − li2 ), there is li1 − li1 and li2 value: li1

satisfies:

p ρi kωk2 kσk2 1 _ ∗ V˙ ≤ − η V (ζi , li1 , li2 ) − |li1 − li1 | ( l˙i1 − ) κ1 κ1 ρˆi kωk2 kσk2 1 ∗ ) − |li2 − li2 | ( l˙i2 − κ2 κ2

(83)

In order to guarantee the convergence of the system, the second and third

245

AN US

element of (83) need to be zero, then, we can get l˙i1 = ρi kωk2 kσk2 , l˙i2 = ρˆi kωk2 kσk2 . Referring to (71), we get l˙i2 = ε/2 · l˙i1 , namely, l˙i2 = ε/2 · ρi kωk2 kσk2 .

Above all, due to V ≥ 0, combined with V˙ ≤ 0, we can get that V is bounded.

∗ ∗ Then, according to (83), it can be obtained that (li2 − li2 ) and (li2 − li2 ) are also

M

bounded. Based on the adaptive law l˙i1 and l˙i2 , li1 and li2 are monotonically _

_

increasing and will end to stable values l i1 and l i2 . Here , for problem of the over-learning of adaptive parameters and parameters drifting, we constrain li1

ED

and li2 , respectively, to reside inside compact sets Ωli1 and Ωli2 , and they are

PT

defined as follows

Ωli1 = {li1 : li1 ≤ li1 ≤ li1 }

(84)

Ωli2 = {li2 : li2 ≤ li2 ≤ li2 }

(85)

CE

where li1 , li1 , li2 , li2 are the design parameters, then, the adaptive law of li1

AC

and li1 are modified as follows l˙i1 = Proj(li1 , ρi kωk2 kσk2 )

(86)

l˙i2 = Proj(li2 , ερi kωk2 kσk2 /2)

(87)

Here, let Proj(a, b) represent (86) and (87), where a = [li1 , li2 ], b = [l˙i1 =

22

ACCEPTED MANUSCRIPT

(88)

CR IP T

ρi kωk2 kσk2 , l˙i2 = ερi kωk2 kσk2 /2]. The specifics as follows    0 if a = Θ, and b < 0   Proj(a, b) = if a = Θ, and b > 0 0     b

∗ ∗ ∈ Ωli2 . So far, the theorem 2 ∈ Ωli1 , li2 where Θ = [li1 , li2 ], Θ = [li1 , li2 ], li1

250

under the conditions of σ 6= 0 is proved. When σ = 0, V0 is rewritten as follows [34]

According to (74), we can get

1/2

Here, due to ςi1 =|σi |

(89)

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V0 =ςiT P ςi

. 1 V˙ 0 ≤ −ςi T Qςi |σi | 2

(90)

sign(σi ), V (σi ) does not satisfy these condition on the

set S = {(σi , yi )R2 |σi = 0}. But, V (σi ) can still be used as a Lyapunov function, in the same spirit as in the theorem of Zubov [35][Theorem 20.2, p568],

255

M

if we are able to show that it decreases monotonically along the trajectories of the system, and converges zero.

ED

If we are able to show that V (ϕ(t, σ0 )) is an absolutely continuous (AC) function of time, then, V (ϕ(t, σ0 )) is a monotonically decreasing function if and only if V˙ is negative definite almost everywhere [36][p.207]. Note that

260

PT

V (ϕ(t, σ0 )) = V oϕ(t, σ0 ) is the composition of V (σ), an AC function of σ, and ϕ(t, σ0 ), that, by definition of a solution of differential inclusion (62) is an AC

CE

function of time. However, the composition of two AC functions is in general not an AC function [37][p.391]. Recall that the addition and product of two (scalar) AC functions hog can be assured to be AC if either h is Lipschitz or

AC

g is monotone[37][p.391]. In the problematic term to assure that V (ϕ(t, σ0 ))

265

1/2

is AC in the term |σi |

|σi |

1/2

1

sign(σi )oϕ1 (t, σ0 ) = |ϕ1 (t, σ0 )| 2 sign(ϕ1 (t, σ0 ), since

sign(σi ) is not Lipschitz at σi = 0. However, as we will show in the

next paragraph, ϕ1 (t, σ0 ) is monotone when it goes through ϕ1 (t, σ0 ) = 0. This

proves that V (ϕ(t, σ0 )) is indeed an AC function of t, and therefore its derivative is defined almost everywhere and it is given by (90), where it is defined. 23

ACCEPTED MANUSCRIPT

In order to show that ϕ1 (t, σ0 ) is monotonic when crossing zero, suppose

270

that ϕ2 (τ, σ0 ) 6= 0 at the instant t = τ , when ϕ1 (t, σ0 ) crosses zero. Then, from 1/2

the differential inclusion σ˙ ∈ −li1 · |σi |

· sign(σi ) + yi , it follows that ϕ1 (t, σ0 )

CR IP T

will be monotonically increasing or decreasing during an interval containing τ . If ϕ2 (τ, σ0 ) = 0 when ϕ1 (t, σ0 ) crosses zero, then ϕ1 (t, σ0 ) will stay in zero. In 275

both cases V (ϕ(t, σ0 )) is AC.

Since V (ϕ(t, σ0 )) is AC and V˙ 0 is almost negative definite everywhere, it follows that V (ϕ(t, σ0 )) is a monotonically decreasing function [36][p.207]. More2

2

over, combined with λmin {P } kςk2 ≤ V (σi ) ≤ λmax {P } kςk2 and 1/2

−1/2

≤ kςk2 ≤ λmin {P }V 1/2 (σi ), it follows that

AN US

|σi |

1/2

λ {P }λmin {Q} 1/2 V˙ (ϕ(t, σ0 )) ≤ − min V (ϕ(t, σ0 )) λmax {P } . Since an AC function is the integral of its derivative[3], then Z t V˙ (ϕ(τ, σ0 ))dτ V (ϕ(t, σ0 )) − V (ϕ(0, σ0 )) = 0

1/2

λmin {P }λmin {Q} λmax {P }

M ≤−

Z

t

V 1/2 (ϕ(τ, σ0 ))dτ

(91)

0

280

ED

. Based on Bihari’s inequality, seen in [35][p.509], we can get that V (ϕ(t, σ0 )) ≤ (V 1/2 (σ0 ) − (κ/2)t)2 , where κ =

1/2

λmin {P }λmin {Q} , λmax {P }

so that V (ϕ(t, σ0 )) and ϕ(t, σ0 )

converge to zero as we expected, and the theorem 2 is proved.

PT

At last, the interference Df can be accurately estimated by ν, and the comˆ f . So far, the whole pensation control law can be obtained as M1 = −gf−1 D

CE

control law of fast control loop is rewritten as ˆf MC0 = −gf−1 (ff − ω˙ c + a2 ωe + b2 ωeq2 /p2 + kf sf ) + ν2 − gf−1 D ¯ c + ν2 + M 1 =M

AC

(92)

¯ C represents the TSMC control law without considering saturation and where M uncertainty. Kf is the controller design parameter. ν2 is the compensation of

anti-windup system, M1 is the compensation of ISMDO.

24

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285

5. Simulation Results Consider that the HSV is carrying out a hypersonic reentry flight with the speed of 3km/s , and the flight height is 30km. M = 136820kg, x0 = y0 =

CR IP T

1000m, α=1◦ , β = 2.5◦ , µ = 3◦ , p =q = r = 0rad/s are the initial attitudes and angular velocities. The reference signals are αc = 1◦ ∼ 5◦ , βc =0◦ , σc =0◦ in 290

1 ∼ 4s, while in 4 ∼ 15s, the control instruction changes to αc = 5◦ ∼ 11◦ , βc =

0◦ , σc = 0◦ . The disturbances in slow loop and fast loop of HSV are shown as follows ds1 =0.005 · (sin (t) + 1) rad/s, ds2 =0.005 · (cos (t) + 1) rad/s, ds3 =0.01 · sin (t + 1) rad/s, df 1 = 200000 sin (4t + 0.2) N · m,

AN US

df 2 = 400000 sin (11t −0.6) N · m, df 3 = 400000 sin (5t + 0.2) N · m.

According to Eq.(10) and Eq.(12), the information of faults and limitations

295

are shown as follows λ = [−0.3, −0.4, −0.2]T . The aero control surface with

the limitations of ±25◦ is applied to simulate minor faults of actuators, namely,

uimin = −25◦ , uimax = 25◦ .The limitations change to ±20◦ are used to simulate

M

severe faults of the actuators, namely,uimin = −20◦ , uimax = 20◦ .

The control parameters in slow control loop are set as:a1 = 2, b1 = 2, q1 =

300

PT

ED

7, p1 = 9, while in fastcontrol loop, a2 = 1, b2 = 3, q2 = 7, p2 = 9, kf = 104 0 0     diag(10, 10, 10), Paw =  0 104 0 . As for the ISMDO, the constant   0 0 1 parameters are selected as follows δf01 = δf 03 = 0.045, δf 02 = 0.05, lf1 = lf2 =lf3 = 1, λf =0.1, m1 = m2 = 1.5. Simulation results are as follows Tracking effects of the attitude and angular velocities are shown in Fig.3-

CE

305

8. Among them, αc , βc , µc represent the tracking indexes of attitude angles,

and pc , qc and rc represent the indexes of angular velocities. In Fig.3-5, the

AC

αAW , βAW , µAW combined with LEAW , REAW , RU DAW are the attitude angle tracking curves and the deflection curves of control surface with anti-windup

310

system, while, αN AW , βN AW , µN AW and LEN AW , REN AW , RU DN AW repre-

sent the control system without anti-windup system. The results show that the attitude system of HSV is sensitive to the actuator fault and the external disturbance, which hardly meets the harsh requirements of reentry flight, seen 25

ACCEPTED MANUSCRIPT

0.06

α AW

5

5

β NAW

0

α NAW

0

β AW

0.02

10

-0.02

15

0

5

t (s)

10

t (s)

0.06 µc µ AW µ NAW

0.02 0 -0.02

15

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µ (deg)

0.04

CR IP T

αc

0

βc

0.04

10

β (deg)

α (deg)

15

0

5

10

15

t (s)

M

Figure 3: Tracking effect of attitude angles of HSV

0

PT

-0.1

AC

0

5

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LE NAW

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Figure 5: Relation angle of control surface of HSV

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Figure 6: Tracking effect of attitude angles of HSV

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in Fig.3-8. Compared with the Fig.3-4 and Fig.6-7, it is obvious that the more hash

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limitations the fault causes, the more quickly the control performance of the

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system decreases. For example, µN AW , seen in Fig.3 and Fig.6, with the smaller range of deflection rudder, it is obvious that the roll rate vibrates quite seriously even if the fault information is obtained, which decreases the effect of fault320

tolerant control. That is because the range of the rudder surface deflection

becomes smaller, the output of the control law cannot be fully implemented, resulting in a large amplitude oscillations, especially in the event of actuator

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fault. Besides, from the response curve of LEN AW , REN AW , RU DN AW in Fig.5

and Fig.8, we can easily get that once the deflection of the rudder angle being 325

in a hash saturated state, the tracking effects of αN AW , βN AW , µN AW becomes more worse, even leading to the divergence of the system, which largely decreases the fault-tolerant control capability and is not allowed in the process of HSV reentry.

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As shown in Fig.2, the error (∆δc ) between the value of over large control law (δc0 ) and the control surface deflection limitations (umin , umax ) will feedback

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to the anti-windup system, which generates the compensation output, MC0 and ωaw . The addition of anti-windup system will adjust the output of the whole control law, avoiding making control surfaces beyond the limitations for

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a long time, guaranteeing the attitude angles can rapidly and smoothly track the command indexes. Take the curves of LEN AW for example, seen in Fig.5

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and Fig.8. The more hash limitations of actuators under the fault condition gives rise to the more frequent swinging of the control surface. After adding the anti-windup system, compared the curves of LEAW and LEN AW in Fig.5 and

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Fig.8 again, the anti-widup system can greatly smooth the deflection angles of

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control surface, improving the tracking effects of attitude angle rate and attitude angle. Take the tracking effect of pAW and pN AW for example, seen in Fig.4 and Fig.7. As for the tracking effect of attitude angle, comparing the curves of µAW , µN AW in Fig.3 and Fig.6, it is obvious that the tracking effect of µAW is better than that of µN AW , which means that the anti-windup system can 29

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Figure 9: Relation angle of control surface of HSV

improve the mismatching conditions between controller and actuator, inhibiting

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the degradation of system control performance, especially in case of fault. The

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same situation occurs in curves of α and β, seen in Fig.1 and Fig.3. Based on the above analysis, the method proposed in this paper possesses a good fault-tolerant and anti-widup ability. In order to compensate the impact of fault on HSV, the ISMDO is applied to

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the fast loop control to estimate the fault and the external disturbance. From

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the simulation results in Fig.9, it is confirmed that the tracking effect of ISMDO for actuator fault and the external disturbances is very good. Besides, ISMDO has the fewer learning parameters and the good robustness, which can offer more information of fault and disturbance to controller, improving the control effect.

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From the simulation results in Fig.3-9, we can conclude that the anti-windup system combined with ISMDO and the TSMC is a useful control strategy to handle the HSV with the fault and disturbance.

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6. Conclusion This paper deals with anti-saturation control problems for HSV, which are

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caused by the partial actuator faults and disturbances. A nonlinear anti-saturation

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control method is proposed, consisted of the anti-windup system, TSMC and ISMDO. Aiming at dealing with the problem of the actuator faults and the

strong external disturbances, ISMDO is proposed to estimate the disturbance 365

and deduce the compensation control law, which has the fewer adaptive learn-

ing parameters and doesn’t need the boundary value and its derivative of the disturbances. As for the anti-saturation control problem, the external anti-

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windup system is designed to realize the compensation for the control surface saturation and realize the smooth tracking of aircraft. The simulation results 370

have shown that the proposed strategy in this paper can effectively compensate the adverse effects caused by faults and disturbances, achieving the good and smooth tracking effects and realising the anti-saturation fault-tolerant control.

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In the future work, the state constrained control will be considered to realize the directing constrained control of p, q, r, for instance, barrier lyapunov 375

function[38]. Besides, the system with Markovian jumping parameters [39], [40],

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which puts forward the more hash constraints for controller and actuator. Some work will be conducted based on the external anti-saturation control and the

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directing state constrained control.

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