Robust adaptive compensation control for unmanned autonomous helicopter with input saturation and actuator faults

Chinese Journal of Aeronautics, (2019), xxx(xx): xxx–xxx

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

Robust adaptive compensation control for unmanned autonomous helicopter with input saturation and actuator faults Kun YAN a, Mou CHEN a,*, Qingxian WU a, Ronggang ZHU b a b

College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Luoyang Institute of Electro-Optical Equipment of AVIC, Luoyang 471000, China

Received 22 July 2018; revised 20 January 2019; accepted 21 March 2019

KEYWORDS Compensation control; Fault tolerant control; Input saturation; Tracking control; Unmanned autonomous helicopter

Abstract This paper studies a robust adaptive compensation Fault Tolerant Control (FTC) for the medium-scale Unmanned Autonomous Helicopter (UAH) in the presence of external disturbances, actuator faults and input saturation. To improve the disturbance rejection capacity of the UAH system in actuator healthy case, an adaptive control method is adopted to cope with the external disturbances and a nominal controller is proposed to stabilize the system. Meanwhile, compensation control inputs are designed to reduce the negative effects derived from actuator faults and input saturation. Based on the backstepping control and inner-outer loop control technologies, a robust adaptive FTC scheme is developed to guarantee the tracking errors convergence. Under the presented FTC controller, the uniform ultimate boundedness of all closed-loop signals is ensured via Lyapunov stability analysis. Simulation results demonstrate the effectiveness of the proposed control algorithm. Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Over the past few years, the extensive applications in military and civil fields of the UAH has made it become an attractive research topic and numerous scientific achievements have been obtained1,2. Especially for the medium-scale UAH featured * Corresponding author. E-mail address: [email protected] (M. CHEN). Peer review under responsibility of Editorial Committee of CJA.

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with long cruise, large payload, high altitude, fast speed and strong robustness, it plays an irreplaceable role in the military field. However, compared with the fixed-wing Unmanned Aerial Vehicle (UAV), UAH has less hardware redundancy. Meanwhile, the complex flight environment increases the possibility of actuator faults, which may lead to severe performance deterioration and even system instability3,4. As a result, with the increasing demands for safety and reliability in practice, compensation control has become an important FTC strategy to deal with the actuator fault and received considerable attentions5–8. Many results of fault compensation can be found in the published literatures. In Ref.9, a backstepping-based adaptive fault compensation scheme was proposed for satellite attitude

https://doi.org/10.1016/j.cja.2019.06.001 1000-9361 Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: YAN K et al. Robust adaptive compensation control for unmanned autonomous helicopter with input saturation and actuator faults, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.06.001

2 control systems with uncertain actuator faults. In Ref.10, based on the dynamics surface control technique, a novel compensation FTC approach was developed for attitude system of UAV with actuator Loss-of-Effectiveness (LOE) faults. By utilizing a descriptor system approach, the problem of sensor fault estimation and compensation for microsatellite attitude dynamics was investigated in Ref.11. An adaptive actuator fault and disturbance compensation scheme, which consisted of backstepping feedback control law and feedforward actuator fault compensator, was presented for attitude tracking control of spacecraft in Ref.12. In Ref.13, an adaptive fault compensation controller was designed and a direct adaptive approach was developed for nonlinear systems in the presence of unknown actuator faults. However, in addition to the actuator faults, the problem of external disturbances also cannot be ignored in the control design. As we know, disturbances widely exist in industrial systems and bring adverse effects on control performance of the systems14,15. During the flight process of UAH, the airflow can induce bumpiness and thrust fluctuation so that the disturbance rejection is a key objective in the controller design. In order to improve the robustness of the controlled systems, a number of disturbance rejection approaches have been proposed in recent years, such as adaptive control10,16–18, disturbance-observer-based control19,20, extend state observer-based control21, high-gain observer-based control22, etc. Among those disturbance rejection methods, the adaptive control approach is extensively investigated and applied attributing to its simple design process. In Ref.16, the problem of adaptive tracking control for switched nonlinear systems subjected to external disturbances was studied. In Ref.17, sliding mode controllers were designed to force the state variables of a spacecraft with external disturbance to converge to the origin in finite time. A distributed robust adaptive control scheme was developed for multi-agent systems with external bounded disturbances to guarantee the output tracking performance in Ref.18. Unfortunately, due to the existence of actuator faults and external disturbances, more control energy is needed so that it is likely to cause the control input saturation of the system. Input saturation is a significant non-smooth nonlinearity that always exists in a practical system. As a matter of fact, only limited control force and control moment can be provided in practical flight control system. If the problem of input saturation is not taken into consideration during the controller design, the flight control performance may be severely degraded. Moreover, the stability of whole closedloop system may not be guaranteed23. Up to now, many analysis and design methods of nonlinear control system with input saturation have been reported. In Ref.24, by employing the backstepping method, the design process of attitude controller for the near space vehicle with control input saturation was described. A backstepping-based controller which was applicable for the hover flight of an UAV with input saturation was presented in Ref.25. By using an inner-outer loop control structure, the position controller was designed for a quadrotor UAV with state and input constraints in Ref.26. In Ref.27, an adaptive neural network control scheme was developed for a small-scale UAH in the presence of input saturations and output constraints. In Ref.28, the output feedback dynamic gain scheduled control strategy was proposed to stabilize a spacecraft rendezvous system subjected to actu-

K. YAN et al. ator saturation. However, when the external disturbances, actuator faults, and input saturation are considered simultaneously in a medium-scale UAH system, the problem of controller design is still a challenging research topic and worthy of further study. Inspired by the above discussion, a backstepping-based robust adaptive compensation FTC scheme is developed for medium-scale UAH to track the desired trajectory in the presence of external disturbances, actuator faults, and input saturation. The remainder of this paper is organized as follows. Section 2 presents the dynamic model of the medium-scale UAH. The proposed control algorithm is detailed in Section 3. Simulation results are exhibited to verify the effectiveness of the designed controller in Section 4, followed by some concluding remarks in Section 5. Notations: Throughout this paper, k  k denotes the Euclidean norms of matrix and vector, respectively. 2. Problem statement According to the helicopter flight dynamics and aerodynamics, the following model which is composed of translation motion and rotational motion can be used to describe a medium-scale UAH29,30: 8_ n¼s > > > < m_s ¼ RF þ mg1 þ d 1 ð1Þ > _ g ¼ Hd > > : II d_ ¼ d  II d þ R þ d2 where n ¼ ½x; y; zT and s ¼ ½u; v; wT denote the UAH position vector and velocity vector with respect to the inertial frame whose center is fixed to a certain place on the earth, g ¼ ½u; h; wT represents the Euler angle vector (roll, pitch, yaw), d ¼ ½p; q; rT is the angular rate with respect to the body-fixed frame whose origin is located in the center of the UAH. m refers to the mass of UAH, g is the acceleration due to a gravity, 1 ¼ ½0; 0; 1T is a unit vector and II ¼ diagfIx ; Iy ; Iz g is the diagonal inertia matrix. H stands for the attitude kinematic matrix and R is the rotation matrix from body-fixed frame to inertial frame29. F ¼ ½0; 0; Tmr T  T and R ¼ Rx ; Ry ; Rz are the control force and moment applied at the center of UAH31, respectively. Tmr is the thrust generated by the main rotor. d1 and d2 are the unknown external force and moment disturbances. In the practical flight control system, input saturation is a potential problem for actuators and it may severely limit system performance. In such case, the control inputs fi ði ¼ 1; 2; 3; 4Þ can be described as24  signðfi Þuimax jfi j > uimax ð2Þ satðfi Þ ¼ fi jfi j  uimax  T where f ¼ Tmr ; Rx ; Ry ; Rz is the control input vector to be designed, satðfi Þði ¼ 1; 2; 3; 4Þ are the actual system input subject to saturation, uimax are the maximum control input of the ith actuator and satisfy uimax > 0. Obviously, owing to the existence of input saturation nonlinearity, there is a difference value aðfi Þ between the designed control input fi and the actual control input satðfi Þ. Then, we have

Please cite this article in press as: YAN K et al. Robust adaptive compensation control for unmanned autonomous helicopter with input saturation and actuator faults, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.06.001

Robust adaptive compensation control aðfi Þ ¼ satðfi Þ  fi

3 ð3Þ

In addition, the actuator may become faulty and LOE faults is a kind of frequent actuator faults in the practical flight control system. In this case, the actuator faults in UAH system can be expressed as10 ff ¼ Bf

ð4Þ

where B ¼ diagfb1 ; b2 ; b3 ; b4 g, bi ði ¼ 1; 2; 3; 4Þ represent the remaining unknown actuator effectiveness factors. Considering Eqs. (2)–(4), the 6-DOF nonlinear model of the medium-scale UAH with external unknown disturbances, actuator faults and input saturation can be written as 8_ n¼s > > > < s_ ¼ f þ v þ D þ ðb  1Þv þ b G aðT Þ a 1 1 a 1 1 mr 1 ð5Þ > _ g ¼ Hd > > : d_ ¼ f2 þ vb þ D2 þ G2 ðba  IÞR þ G2 ba aðRÞ where f1 ¼ g1, f2 ¼ I1 I d  II d,

G1 ¼ R1=m, D1 ¼ d1 =m, va ¼ G1 Tmr , G2 ¼ I1 , D2 ¼ I1 vb ¼ G2 R, ba ¼ I d2 ,  I T diagfb2 ;b3 ; b4 g, aðRÞ ¼ aðRx Þ; aðRy Þ; aðRz Þ , I ¼ diagf1; 1;1g. The essential issues in the control of a UAH are: (A) What kind of scheme can be employed to control the nominal system with unknown external disturbances; (B) In addition to the above mentioned condition, how UAH can be controlled to achieve a stable system and obtain the satisfactory tracking performance in the presence of actuator faults and input saturation. Therefore, in order to track the desired flight trajectory yd , designing a robust FTC strategy for the medium-scale UAH with unknown external disturbances, actuator faults and input saturation will be the main goal of this work. Before developing the control scheme, the following assumptions are required. Assumption 127. The roll angle u and pitch angle h satisfy inequality constraint p=2 < u < p=2 and p=2 < h < p=2, respectively. Assumption 223. All states of the UAH system (5) are measurable and available. Furthermore, the desired flight trajectory yd and its derivatives y_ d , €yd are bounded. In other words, there exists an unknown positive constant C0 such that M0 :¼ fðyd ; y_ d ; €yd Þ : k yd k2 þ k y_ d k2 þ k €yd k2  C0 g. Assumption 310. The unknown actuator effectiveness factors bi ði ¼ 1; 2; 3; 4Þ are assumed to be bounded such that 0 < v  bi  1, where v is the known lower bound of bi . Assumption 418. The unknown continuous functions D1 and D2 are supposed to satisfy k D1 k  r1 and k D2 k  r2 , r1 and r2 being unknown positive constants. Assumption 524. For the UAH system (5), there exist known continuous function vectors h1 ðn; sÞ 2 R13 and h2 ðg; dÞ 2 R13 making jaðTmr Þj  h1 ðn; sÞs1 and k aðRÞ k  h2 ðg; dÞs2 hold, where s1 2 R3 and s2 2 R3 are unknown constant vectors. Assumption 623. For the practical medium-scale UAH system (5) with input saturation and desired trajectory, there should exist a feasible actual controller f which can make the UAH achieve the tracking objective. Remark 1. Due to the external disturbances, actuator faults and input saturation, the control design of the UAH system

becomes more complicated. In order to promote the controller design, the external disturbances, actuator faults and desired trajectory are assumed to be bounded. In fact, the external disturbances can be largely attributed to the exogenous effects and they have finite energy in practical control system. Furthermore, for a practical UAH system described as Eq. (5), there should exist a feasible control input which can complete the tracking task. If the desired trajectory is unbounded, the controller is not likely to have enough energy to perform the task. Similarly, if the actuator lose too much effectiveness, the whole system may lose the capacity of fault tolerant control. Therefore, it is reasonable for Assumptions 2–4. Remark 2. Apparently, the desired control input may be larger than the actual control energy provided. However, we should note a fact that the difference between them cannot be larger. From the view of a practical flight control system, there should exist a feasible control input and the system controllability should be satisfied whether the input saturation exists or not. Thus, the Assumptions 5–6 are reasonable for the UAH system with input saturation. 3. Design of robust adaptive compensation FTC scheme In this subsection, by using the adaptive and backstepping control techniques, a nominal control input vN1 is first designed for the position equation of the UAH with external unknown disturbances. Then, when the actuator faults and input saturation occur, the compensation control inputs vC1 , vM1 , and vL1 are developed and added to the nominal controller vN1 to reduce their effects on the system. 3.1. Position loop control Without considering actuator LOE faults and input saturation constraints, the position equation involved n and s in Eq. (5) can be written as ( n_ ¼ s ð6Þ s_ ¼ f1 þ va þ D1 Define the tracking errors as e1 ¼ nd  n

ð7Þ

e2 ¼ sd  s

ð8Þ

where nd is the desired position trajectory, and sd is the designed virtual control law. Invoking Eq. (6), the derivative of e1 with respect to time is e_ 1 ¼ n_ d  n_ ¼ n_ d  sd þ e2

ð9Þ

Select an appropriate virtual control law sd as sd ¼ n_ d þ K1 e1

ð10Þ

where K1 is the designed positive definite matrix. By substituting Eq. (10) into Eq. (9), one has e_ 1 ¼ K1 e1 þ e2

ð11Þ

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Taking the time derivative of e2 and invoking Eq. (6), it follows that e_ 2 ¼ s_ d  s_ ¼ s_ d  f1  va  D1

ð12Þ

Here, the dynamic surface control technique is used to overcome the so-called explosion of complexity in the sequel steps, which is caused by the repeated derivation of s_ d . Let sd pass the following first-order filter 12 23: e2 1_ 2 þ 12 ¼ sd ;

12 ð0Þ ¼ sd ð0Þ

ð13Þ

where e2 ¼ diagfe21 ; e22 ; e23 g > 0 is the time constant of the filter. Define t2 ¼ 12  sd . We obtain   @sd € @sd _ t þ   t_ 2 ¼ 1_ 2  s_ d ¼ e1 n e 2 d 1 2 @e1 @ n_ d ¼

e1 2 t2

þ C2 ðn_ d ; e1 Þ

ð14Þ

where C2 ðn_ d ; e1 Þ is sufficiently smooth function vector about M2 ðn_ d ; e1 Þ. Since the set M2 ðÞ is compact, the smooth function C2 ðÞ has a maximum C2m on set M2 ðÞ for the given initial conditions23. Then, it gives k C2 ðÞ k  C2m

ð15Þ

From Eqs. (12) and (13), the nominal control input is proposed as ^1 signðe2 Þ þ e1 þ K2 e2 va ¼ 1_ 2  f1 þ r

ð16Þ

^1 is the estimation of r1 , and K2 is the designed positive where r definite matrix. It is noted that the proposed control law is discontinuous due to the introduction of sign function signðe2 Þ, which may lead to the chattering effect and even make the system unstable in practice. In this work, in order to overcome this problem, a continuous function c1 ðe2 Þ is adopted to substitute for the function signðe2 Þ, and then the nominal control input va can be modified to vN1 , which is given as10 ^1 c1 ðe2 Þ þ e1 þ K2 e2 vN1 ¼ 1_ 2  f1 þ r

ð17Þ

2 and l1 is a positive constant. where c1 ðe2 Þ ¼ ke2 ekþl 1 ^1 is given by The parameter adaptive update law of r   T e2 e2 ^1 ^_ 1 ¼ o1 ð18Þ  c1 r r k e2 k þ l1

where o1 and c1 are designed positive constants. Substituting Eq. (17) into Eq. (12) yields e_ 2 ¼ ^ r1 ðc1 ðe2 Þ þ co Þ  e1  K2 e2  D1 þ

e1 2 t2

 C2 ðÞ

 tT2 ðe1 2 t2  C2 ðÞÞ   1 2  eT1 K1 e1  eT2 K2  ðk e1 2 k þ 3ÞI e2 2 T e2 e2 r1 k eT2 kðk e2 k þ l1 Þ 2   r1 þ  c1 r1  c1 r1 r1 k e2 k þ l1 k e2 k þ l1 1 2 2 1 2 2 2  tT2 ðe1 2  IÞt2 þ C2m þ c 1 r1 þ c 1 r1 2 2   1 1 2  2 T T 1 2  e1 K1 e1  e2 K2  ðk e2 k þ 3ÞI e2  ðc1  c 1 Þr1 2 2 1 2 2 2  tT2 ðe1 ð21Þ 2  IÞt2 þ r1 l1 þ ðc1 þ c 1 Þr1 þ C2m 2 According to the ultimately uniformly bounded theorem19, we can draw a conclusion that the developed nominal controller (17) and adaptive update law (18) can make the output track the desired position trajectory. However, in addition to the external unknown disturbances which are together with system operation, the actuator faults and input constraints always occur in the running of the system in practice. In the following, we extend the above result to cope with the control input saturation and actuator LOE faults problems of the UAH system. Based on the designed nominal controller vN1 , compensation control inputs vC1 , vM1 , and vL1 will be presented to restrain the adverse effects of the input saturation and actuator LOE faults. Hence, the final robust FTC input va of the UAH position equation consists of four parts, namely,10 va ¼ vN1 þ vC1 þ vM1 þ vL1

ð22Þ

where vC1 ¼

e2 k vN1 k2 vðk e2 kk vN1 k þ l2 Þ

ð23Þ

vM1 ¼

e2 k h1 ðn; sÞ kh1 ðn; sÞ^s1 k e2 kk h1 ðn; sÞ k þ l3

ð24Þ

vL1 ¼

e2 k vM1 k2 vðk e2 kk vM1 k þ l4 Þ

ð25Þ

where l2 , l3 and l4 are positive constants, and ^s1 is the estimation of the unknown parameter s1 . It is observed that the design process of virtual control law sd is same to the Eqs. (9)–(11). Then, in terms of Eqs. (5), (12) and (22), we obtain e_ 2 ¼ s_ d  f1  va  D1  ðb1  1Þva  b1 G1 aðTmr Þ

ð19Þ

where co ¼ signðe2 Þ  c1 ðe2 Þ is the approximate error with   k co k  c 1 , and c 1 is a positive constant.  ^1  r1 . Choose the Lyapunov function candiDefine r1 ¼ r date V1 as 1 1 1 2 1 T r þ t t2 V1 ¼ eT1 e1 þ eT2 e2 þ 2 2 2o1 1 2 2

V_ 1 ¼ eT1 ðK1 e1 þ e2 Þ þ eT2 ð^ r1 ðc1 ðe2 Þ þ co Þ  e1  K2 e2  D1   eT2 e2  ^ r þ e1 t  C ðÞÞ þ r  c 1 2 2 1 1 2 k e2 k þ l1

ð20Þ

Differentiating Eq. (20) and invoking Eq. (11), Eqs. (18) and (19), one obtains

¼ s_ d  f1  vN1  D1 þ ð1  b1 ÞvN1  b1 vC1  b1 vM1  b1 vL1  b1 G1 aðTmr Þ

ð26Þ

The parameter adaptive update law of ^s1 is proposed as k e2 k2 k h1 ðn; sÞ khT1 ðn; sÞ ^s_ 1 ¼ o2  c2^s1 k e2 kk h1 ðn; sÞ k þ l3

ð27Þ

where o2 and c2 are designed positive constants.  Define s 1 ¼ ^s1  s1 . Choose the Lyapunov function candidate V2 as

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Robust adaptive compensation control V2 ¼ V1 þ

5

1 T  s s1 2o2 1

ð28Þ

Differentiating Eq. (28) and invoking Eqs. (17), (18), (21), (22) and (26), one has V_ 2 ¼ eT1 ðK1 e1 þ e2 Þ þ eT2 ð_sd  f1  vN1  D1 þ ð1  b1 ÞvN1  b1 vC1  b1 vM1  b1 vL1  b1 G1 aðTmr ÞÞ   1 eT2 e2 ^ 1  tT2 ðe1  c1 r þ r1 o1 2 t2 þ C2 ðÞÞ k e2 k þ l1 o1 ! 2 T T k e2 k k h1 ðn; sÞ kh1 ðn; sÞ  c2^s1 þ s1 k e2 kk h1 ðn; sÞ k þ l3   1 1 2  2 T T 1 2  e1 K1 e1  e2 K2  ðk e2 k þ 3ÞI e2  ðc1  c 1 Þr1 2 2 1 2 2 2  tT2 ðe1 2  IÞt2 þ r1 l1 þ ðc1 þ c 1 Þr1 þ C2m 2 þ ð1  b1 ÞeT2 vN1  b1 eT2 vC1  b1 eT2 vM1  b1 eT2 vL1 ! 2 T T k e2 k k h1 ðn; sÞ kh1 ðn; sÞ T  b1 e2 G1 aðTmr Þ þ s 1  c2^s1 k e2 kk h1 ðn; sÞ k þ l3 ð29Þ For the convenience of writing, we define A1 ¼ ð1  b1 ÞeT2 vN1  b1 eT2 vC1 A2 ¼ b1 eT2 vM1  b1 eT2 vL1 þ

T s1

k e2 k2 k h1 ðn; sÞ khT1 ðn; sÞ  c2^s1 k e2 kk h1 ðn; sÞ k þ l3

According to the previous definition of va and considering Eq. (22), we obtain G1 Tmr ¼ va ¼ ½vax ; vay ; vaz T . Solving this algebraic equation, the desired attitude signals for the attitude loop control and the main rotor control input Tmr can be created as follows based on the given yaw signal wd 32: hd ¼ arctan

vax cos wd þ vay sin wd vaz

ð32Þ

cos hd ðvax sin wd  vay cos wd Þ vaz

ð33Þ

ud ¼ arctan Tmr ¼ 

mvaz cos hd cos ud

ð34Þ

3.2. Attitude loop control Similar to the design process of position loop controller, we first consider the nominal system with external unknown disturbances. Considering the UAH system (5), the attitude equation involved g and d can be extracted as  g_ ¼ Hd ð35Þ d_ ¼ f2 þ vb þ D2 The attitude angle error and angular rate error vectors are defined as

!  b1 eT2 G1 aðTmr Þ

Invoking Eqs. (23)–(25), the following facts can be obtained:

e3 ¼ gd  g

ð36Þ

e4 ¼ dd  d

ð37Þ

where gd is the desired attitude trajectory, and dd can be viewed as a virtual control law.

8 e2 kvN1 k2 T > > > A1  k e2 kk vN1 k  b1 e2 vðke2 kkvN1 kþl2 Þ  l2 > > > > A2 ¼ A2 þ eT2 vM1  eT2 vM1 > > >  2  >  ke2 k kh1 ðn;sÞkhT > 1 ðn;sÞ > ^  c s <  ð1  b1 Þk e2 kk vM1 k  b1 eT2 vL1 þ b1 k e2 kk G1 kh1 ðn; sÞs1  eT2 vM1 þ s T1 ke 2 1 kkh ðn;sÞkþl 2 1 3 > > > > > > > > > > > > > :

eT e kv

k2

2 M1 þ k e2 kh1 ðn; sÞs1   k e2 kk vM1 k  b1 vðke22 kkv M1 kþl4 Þ



þs T1

ke2 k2 kh1 ðn;sÞkhT 1 ðn;sÞ ke2 kkh1 ðn;sÞkþl3

eT e kh ðn;sÞkh1 ðn;sÞ^s1 2 2 1 ke2 kkh1 ðn;sÞkþl3

ð30Þ



 c2 s T1^s1  

 l4 þ l3 k s1 k  12 c2 s T1 s 1 þ 12 c2 k s1 k2

Remark 3. In this study, the medium-scale UAH whose weight between 500–1000 kg is considered as the controlled object. According to 1 ¼ ½0; 0; 1T and the rotation matrix R provided in Ref.29, it is observed that k R1 k  m, which means k G1 k  1 holds. Furthermore, the magnitude of rotary inertia with respect to the medium-scale UAH is much larger than 1, namely, Ix 1, Iy 1 and Iz 1. Hence, we obtain k I1 I k  1. Considering Eqs. (29) and (30), it follows that   1 1 2  2 2 k þ 3ÞI e2  ðc1  c 1 Þr1 V_ 2  eT1 K1 e1  eT2 K2  ðk e1 2 2 2 1 T  1 2 2 T 1  t2 ðe2  IÞt2  c2 s 1 s 1 þ r1 l1 þ C2m þ ðc1 þ c 1 Þr21 2 2 1 þ l2 þ l4 þ l3 k s1 k þ c2 k s1 k2 ð31Þ 2

Invoking Eq. (36), the derivative of e3 can be represented as e_ 3 ¼ g_ d  g_ ¼ g_ d  Hdd þ He4

ð38Þ 23

Similarly, let gd pass the following first-order filter 13 : e3 1_ 3 þ 13 ¼ gd ;

13 ð0Þ ¼ gd ð0Þ

ð39Þ

where e3 ¼ diagðe31 ; e32 ; e33 Þ > 0 is the time constant of the filter. Define t3 ¼ 13  gd . Then, we have t_ 3 ¼ 1_ 3  g_ d ¼ e1 3 t3   @gd @g @g _ @g @g ^ 1  d ^s_ 1  d €t2 þ  e_ 1  d e_ 2  d r @e1 @e2 @^ r1 @^s1 @ t_ 2 1 ^ 1 ; ^s1 ; t_ 2 Þ ¼ e3 t3 þ C3 ðe1 ; e2 ; r

ð40Þ

^1 ; ^s1 ; t_ 2 Þ is sufficiently smooth function vector where C3 ðe1 ; e2 ; r about M3 ðe1 ; e2 ; r ^ 1 ; ^s1 ; t_ 2 Þ. Since the set M3 ðÞ is compact, the

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K. YAN et al.

smooth function C3 ðÞ has a maximum C3m on set M3 ðÞ for the given initial conditions23. Then, it gives k C3 ðÞ k  C3m

ð41Þ

Design the virtual control law dd as 1

dd ¼ H ð_13 þ K3 e3 Þ

ð42Þ

where K3 is the designed positive definite matrix. By substituting Eq. (42) into Eq. (38), the following equation can be obtained: e_ 3 ¼ K3 e3 þ He4 þ e1 3 t3  C3 ðÞ

ð43Þ

e_ 4 ¼ d_ d  d_ ¼ d_ d  f2  vb  D2

ð44Þ 23

Let dd pass the following first-order filter 14 : 14 ð0Þ ¼ dd ð0Þ

ð45Þ

where e4 ¼ diagfe41 ; e42 ; e43 g > 0 is the time constant of the filter. Define t4 ¼ 14  dd . Then, we have   @gd @gd _t4 ¼ 1_ 4  d_ d ¼ e1 € e_ 3  t3 4 t4 þ  @e3 @ t_ 3 _ 3Þ ¼ e1 4 t4 þ C4 ðe3 ; t

ð46Þ

where C4 ðe3 ; t_ 3 Þ is sufficiently smooth function vector about M4 ðe3 ; t_ 3 Þ. Since the set M4 ðÞ is compact, the smooth function C4 ðÞ has a maximum C4m on set M4 ðÞ for the given initial conditions23. Then, it gives k C4 ðÞ k  C4m

ð47Þ

In the light of Eqs. (44) and (45), the nominal control input vb of attitude loop is developed as ^2 signðe4 Þ þ HT e3 þ K4 e4 vb ¼ 1_ 4  f2 þ r

ð48Þ

^ 2 is the estimation of r2 , and K4 is a positive definite where r matrix. Analogously, in order to reduce the chattering effect, a continuous function c2 ðe4 Þ is employed to take the place of the function signðe4 Þ. Therefore, the nominal control input vb can be revised to vN2 , which is given as ^2 c2 ðe4 Þ þ HT e3 þ K4 e4 vN2 ¼ 1_ 4  f2 þ r

ð49Þ

4 where c2 ðe4 Þ ¼ ke4 ekþl , and l5 is designed positive constant. 5 ^2 is given by The parameter adaptive update law of r   T e4 e4 ^_ 2 ¼ o3 ^2 r ð50Þ  c3 r k e4 k þ l5

where o3 and c3 are designed positive constants. Substituting Eq. (49) into Eq. (44) leads to r2 ðc2 ðe4 Þ þ ct Þ  H e3  K4 e4  D2 þ e_ 4 ¼ ^ T

e1 4 t4

ð52Þ

Taking the time derivative of V3 obtains T V_ 3 ¼ eT3 ðK3 e3 þ He4 þ e1 r2 ðc2 ðe4 Þ þ ct Þ 3 t3  C3 ðÞÞ þ e4 ð^

 HT e3  K4 e4  D2 þ e1 4 t4  C4 ðÞÞ   T e4 e4  ^2  tT3 ðe1  c3 r þ r2 3 t3  C3 ðÞÞ k e4 k þ l5  tT4 ðe1 4 t4  C4 ðÞÞ     1 1 T 1  eT3 K3  ðk e1 3 k þ 1ÞI e3  e4 K4  ðk e4 k þ 3ÞI e4 2 2 eT4 e4 eT4 e4   ^2 þ k eT4 kr2 þ r2  c3 r2 r k e4 k þ l5 k e4 k þ l5 1 2 2 1 2 2 2 2 T 1  tT3 ðe1 3  IÞt3  t4 ðe4  IÞt4 þ C3m þ C4m þ c 2 r2 þ c 2 r2 2 2     1 1 T 1 T 1  e3 K3  ðk e3 k þ 1ÞI e3  e4 K4  ðk e4 k þ 3ÞI e4 2 2 1 2  2 T 1  ðc3  c2 Þr2  tT3 ðe1 3  IÞt3  t4 ðe4  IÞt4 þ r2 l5 2 1 2 þ ðc3 þ c2 Þr22 þ C23m þ C24m ð53Þ 2 ^2 r

Then taking the time derivative of (43) yields

e4 1_ 4 þ 14 ¼ dd ;

1 1 1 2 1 T 1 V3 ¼ eT3 e3 þ eT4 e4 þ r2 þ t3 t3 þ tT4 t4 2 2 2o3 2 2

Analogously, we can reach a decision that the desired attitude trajectory can be tracked on the basis of the presented nominal controller (49) and adaptive update law (50). Then, we will design the compensation control inputs vC2 , vM2 , and vL2 for attitude loop control to improve the robustness and FTC capability of the UAH system. Same to the previous analysis, the final robust FTC input vb of the UAH attitude equation with external unknown disturbances, actuator faults and input saturation also consists of four parts, i.e., vb ¼ vN2 þ vC2 þ vM2 þ vL2 where vC2 ¼

e4 k vN2 k2 vðk e4 kk vN2 k þ l6 Þ

ð55Þ

vM2 ¼

e4 k h2 ðg; dÞ kh2 ðg; dÞ^s2 k e4 kk h2 ðg; dÞ k þ l7

ð56Þ

vL2 ¼

e4 k vM2 k2 vðk e4 kk vM2 k þ l8 Þ

ð57Þ

where ^s2 is the estimation of s2 , and li ði ¼ 6; 7; 8Þ are positive constants. Invoking Eqs. (5) and (54), (44) can be rewritten as follows: e_ 4 ¼ d_ d  f2  vb  D2  ðba  IÞvb  ba G2 aðRÞ ¼ d_ d  f2  vN2  D2 þ ðI  ba ÞvN2  ba vC2  ba vM2  ba vL2  ba G2 aðRÞ

 C4 ðÞ ð51Þ

where ct ¼ signðe4 Þ  c2 ðe4 Þ is the approximate error with   k ct k  c 2 , and c 2 is a positive constant.  ^ 2  r2 . Choose the Lyapunov candidate Define r2 ¼ r function V3 as

ð54Þ

ð58Þ

The parameter adaptive update law of ^s2 is proposed as ! 2 T _^s2 ¼ o4 k e4 k k h2 ðg; dÞ kh2 ðg; dÞ  c4^s2 ð59Þ k e4 kk h2 ðg; dÞ k þ l7 where o4 and c4 are designed positive constants.  Define s 2 ¼ ^s2  s2 and consider a Lyapunov function candidate V4 as follows:

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Robust adaptive compensation control V4 ¼ V3 þ

1 T  s s2 2o4 2

7 ð60Þ

V5 ¼ V2 þ V4

Taking the time derivative of V4 and invoking Eqs. (54)– (60), we have   1 T 1 _ V4  e3 K3  ðk e3 k þ 1ÞI e3 2   1 1 2  2  eT4 K4  ðk e1 k þ 3ÞI e4  ðc3  c 2 Þr2 4 2 2 1 2 T 1 2  tT3 ðe1 3  IÞt3  t4 ðe4  IÞt4 þ r2 l5 þ c3 r2 þ C3m 2 1 2 þ C24m þ ðc3 þ c 2 Þr22 þ eT4 ðI  ba ÞvN2  eT4 ba vC2 2 2 T  k e4 k k h2 ðg; dÞ kh2 ðg; dÞ  eT4 ba vM2  eT4 ba G2 aðRÞ þ s T2 k e4 kk h2 ðg; dÞ k þ l7 

 c4 s T2^s2  eT4 ba vL2 þ eT4 vM2  eT4 vM2   1  eT3 K3  ðk e1 k þ 1ÞI e3 3 2   1 1 2  2  eT4 K4  ðk e1 4 k þ 3ÞI e4  ðc3  c 2 Þr2 2 2 1 2 T 1 2  tT3 ðe1 3  IÞt3  t4 ðe4  IÞt4 þ r2 l5 þ c3 r2 þ C3m 2 1 2 þ C24m þ ðc3 þ c 2 Þr22 þ k e4 kk vN2 k 2 kmin ðba ÞeT4 e4 k vN2 k2  þ k e4 kk vM2 k  c4 s T2^s2  vðk e4 kk vN2 k þ l6 Þ þ k e4 kh2 ðg; dÞs2  

Proof. Define the Lyapunov function V5 as follows:

kmin ðba ÞeT4 e4 k vM2 k2 vðk e4 kk vM2 k þ l8 Þ

eT4 e4 k h2 ðg; dÞ kh2 ðg; dÞ^s2 k e4 kk h2 ðg; dÞ k þ l7

2 T  k e4 k k h2 ðg; dÞ kh2 ðg; dÞ þ s T2 k e4 kk h2 ðg; dÞ k þ l7   1  eT3 K3  ðk e1 k þ 1ÞI e3 3 2   1 1 2  2  eT4 K4  ðk e1 4 k þ 3ÞI e4  ðc3  c 2 Þr2 2 2 1 T  T 1  tT3 ðe1 3  IÞt3  t4 ðe4  IÞt4  c4 s 2 s 2 þ r2 l5 2 1 1 2 2 2 2 þ c3 r2 þ C3m þ C4m þ ðc3 þ c 2 Þr22 þ l6 þ l8 2 2 1 2 þ l7 k s2 k þ c4 k s2 k 2

ð62Þ

Combining Eqs. (31) and (61), the time derivative of V5 is   1 1 2  2 T T 1 2 _ V5  e1 K1 e1  e2 K2  ðk e2 k þ 3ÞI e2  ðc1  c 1 Þr1 2 2   1 T  1 T 1  tT2 ðe1 2  IÞt2  c2 s 1 s 1  e3 K3  ðk e3 k þ 1ÞI e3 2 2   1 1 2  2  eT4 K4  ðk e1 4 k þ 3ÞI e4  ðc3  c 2 Þr2 2 2 1   T 1 T 1  t3 ðe3  IÞt3  t4 ðe4  IÞt4  c4 s T2 s 2 þ r1 l1 þ C22m 2 1 1 2 2 þ l2 þ l4 þ ðc1 þ c 1 Þr1 þ l3 k s1 k þ c2 k s1 k2 2 2 1 2 2 2 2 þ r2 l5 þ ðc3 þ c 2 Þr2 þ C3m þ C4m þ l6 þ l8 þ l7 k s2 k 2 1 ð63Þ þ c4 k s2 k2  j1 V5 þ D1 2 where 2

2

j1 ¼ minfkmin ð2K1 Þ; kmin ð2K2  ðk e1 2 k þ 3ÞIÞ; ðc1  c 1 Þo1 ; 1 kmin ð2e1 2  2IÞ; c2 o2 ; kmin ð2K3  ðk e3 k þ 1ÞIÞ; 1 kmin ð2K4  ðk e1 4 k þ 3ÞIÞ; kmin ð2e3  2IÞ; 2

kmin ð2e1 4  2IÞ; ðc3  c 2 Þo3 ; c4 o4 gD1 1 2 ¼ r1 l1 þ C22m þ ðc1 þ c 1 Þr21 þ l2 þ l4 þ l3 k s1 k 2 1 1 2 þ ðc3 þ c 2 Þr22 þ r2 l5 þ c2 k s1 k2 2 2 1 2 2 þ C3m þ C4m þ l6 þ l8 þ l7 k s2 k þ c4 k s2 k2 2 Integration of Eq. (63) yields

D1 D1 expðj1 tÞ þ V5 ð0Þ  0  V5  j1 j1

ð64Þ

We can obtain from Eq. (64) that V5 is bounded. According to the definition of V5 , the following inequality is satisfied:

ð61Þ

3.3. Main results Based on the above analysis, the main results can be summarized in the following theorem along with the control design process for the position and attitude trajectory tracking problems of the medium-scale UAH. Theorem 1. Consider the 6-DOF nonlinear model of UAH system (5) with external unknown disturbances, actuator faults and input saturation. The parameter updating laws are chosen as Eqs. (18), (27), (50) and (59). Based on the robust adaptive FTC schemes Eqs. (22) and (54), the tracking error signals of the closed-loop system are ultimately uniformly bounded.

1 T 1 1 2 1 T 1 T  1 r þ t t2 þ s s 1 þ eT3 e3 e e1 þ eT2 e2 þ 2 1 2 2o1 1 2 2 2o2 1 2 1 T 1 2 1 T 1 T 1 T  þ e4 e4 þ r þ t t3 þ t4 t4 þ s s2 2 2o3 2 2 3 2 2o4 2

D1 D1 expðj1 tÞ  þ V5 ð0Þ  ð65Þ j1 j1 h i Then, we have 12 eT1 e1  Dj11 þ V5 ð0Þ  Dj11 expðj1 tÞ, which implies that the error signal e1 converges to the set rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i  ffi D1 D1 k e1 k  2 j1 þ V5 ð0Þ  j1 expðj1 tÞ . As t ! 1, it is qffiffiffiffiffiffiffi observed that k e1 k  2 Dj11 . In other words, the error signal e1 is bounded. Similarly, it can be proved that error signals     e2 ; r1 ; t2 ; s 1 ; e3 ; e4 ; r2 ; t3 ; t4 ; s 2 are all bounded. Therefore, the ultimate uniform boundedness of the all error signals of the closed-loop system can be guaranteed. This concludes the proof. Remark 4. In this study, the inner-outer loop control technique is utilized to proceed with the control design. As well

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8

K. YAN et al.

known, underactuation and strong coupling are the typical characteristics of the UAH system. By means of this control approach, the whole system is segmented into outer loop which is known as position loop and inner loop which is known as attitude loop to design the controller separately. In this way, the complexity of controller design is reduced and satisfactory tracking performance can be achieved only relying on three desired positions and a yaw angle information. Remark 5. In the actual flight, there are some different types of actuator faults that may occur, such as stuck, bias, LOE, hard-over and out of control. The bias fault indicates that there is a bounded difference value between the designed control signal and the actual control signal. Generally speaking, it can be dealt with in the same way as external disturbance. The hard-over fault means that the actuator is locked at a max/min saturated position. Obviously, to some extent, the input saturation can be regarded as a special type of actuator fault. The stuck and out of control faults are the same action and can be expressed as that the actuator is not affected by the control signal and outputs a bounded  value. In this case, it is equivalent to bi ¼ 0 and fi ¼ f i , where f i is a constant (stuck) or time-varying value (out of control). However, the UAH is a strong coupling system with underactuation. The occurrence of stuck and out of control faults will lead to fewer available control inputs. Therefore, additional conditions are required for handling the stuck and out of control faults, such as the actuator redundancy. According to the process of controller design and stability analysis above, the proposed FTC method in this work can be employed to handle the actuator LOE directly. The design approach for coping with the external disturbance and input saturation can be used as a guidance to tackle the actuator bias and hard-over faults, respectively. In general, depending on the practical physical meaning, different actuator faults should be handled in different ways. If

Fig. 1

multiple faults are considered simultaneously, the novel FTC algorithm should be further explored in the future. 4. Simulation results This section describes the numerical evaluation of the developed robust adaptive compensation FTC algorithm for the medium-scale UAH in the presence of external unknown disturbances, actuator faults and input saturation. The basic physical parameters of the medium-scale UAH are borrowed from Ref.33. In Matlab simulation, the initial states of the UAH are assumed as n0 ¼ ½1; 1; 260T m and g0 ¼ ½0:01; 0:02; 0T rad. The desired trajectory are given by nd ¼ ½4cosð0:5tÞ; 4sinð0:5tÞ; 260  0:4tT m and wd ¼ 0:15 rad. The corresponding parameters of the updating laws and robust FTC schemes are chosen as follows: K1 ¼ diagð1; 1; 0:55Þ, K2 ¼ diagð2; 2; 2Þ, K3 ¼ diagð5; 5; 5Þ, K4 ¼ diagð2; 2; 2Þ, o1 ¼ 5, c1 ¼ 0:001, o2 ¼ 0:1, c2 ¼ 0:001, o3 ¼ 5, c3 ¼ 0:001, o4 ¼ 0:1, c4 ¼ 0:001, l1 ¼ 0:3, l2 ¼ 0:5, l3 ¼ 0:2, l4 ¼ 0:2, l5 ¼ 0:3, l6 ¼ 0:5, l7 ¼ 0:2, l8 ¼ 0:2. In order to test the disturbance rejection capability and FTC capability of the proposed robust adaptive compensation FTC scheme for the medium-scale UAH, the external unknown time-varying disturbances and unknown actuator fault signals are assumed as 2 3 2 3 1000sinð0:2tÞ 400sinð0:3tÞ 6 7 6 7 d1 ¼ 4 900sinð0:2tÞ 5; d2 ¼ 4 800sinð0:2tÞ 5; 800sinð0:3tÞ 650sinð0:3tÞ  diagð0; 0; 0; 0Þ t < 16 s B¼ diagð0:8; 0:4; 0:4; 0:4Þ; 16 s  t  30 s In what follows, simulation results are divided into three cases to investigate the effectiveness of the presented robust FTC controller for the medium-scale UAH, where solid lines

Trajectory tracking results without external disturbances, actuator faults and input saturation.

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Robust adaptive compensation control

9

and dash lines define the desired trajectory and actual output responses, respectively. 4.1. Case 1. Trajectory tracking results with external disturbances Firstly, without considering the external disturbances, actuator faults and input saturation, the tracking results of the position

Fig. 2

Fig. 3

and attitude are shown in Fig. 1. From Fig. 1, we note that the tracking performance is satisfactory for the original system and the tracking error can converge to a boundary promptly. However, as shown in Fig. 2, when the external disturbances are taken into account, the actual output responses deviate from the desired trajectory without dealing with them. Therefore, it can be obtained that the external disturbances have an significant effect on the system stability and the robust control

Trajectory tracking results with external disturbances.

Trajectory tracking results under nominal control input.

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10

K. YAN et al.

method needs be adopted to improve the disturbance rejection capability. 4.2. Case 2. Trajectory tracking results with external disturbances, actuator faults and input saturation Secondly, For solving the disturbance rejection problem presented in Case 1, the robust control inputs vN1 and vN2 are

Fig. 4

acted on the nominal system and the simulation results are given as Fig. 3. Fig. 3 displays the effectiveness of the developed nominal control inputs, which can also guarantee the tracking error convergence. However, the aforementioned analysis are all based on the healthy case of actuators. Moreover, the control input saturation is not considered as well. When the external disturbances, actuator faults and input saturation are taken into consideration simultaneously, as shown

Trajectory tracking results with external disturbances, actuator faults and input saturation.

Fig. 5

Trajectory tracking results under robust adaptive compensation FTC scheme.

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Robust adaptive compensation control

11

Fig. 6

Actual control input curves.

in Fig. 4, the satisfactory tracking performance cannot be ensured only depend on the nominal robust control inputs vN1 and vN2 . Fig. 4 indicates that when the actuator faults are introduced at 16 second, the tracking errors of the attitude angle begin to get larger and the altitude of the UAH even starts to change in the opposite direction after a period of time. Hence, based on the compensation control theory, the robust FTC control inputs va and vb are proposed to cope with the adverse effects of external disturbances, actuator faults and input saturation. 4.3. Case 3. Trajectory tracking results under robust adaptive compensation FTC scheme

For the actuator healthy case, the adaptive control technology has been employed to approximate the upper bound of the external disturbances and the nominal controller has been designed to guarantee the stability of overall system. By utilizing the compensation control theory, an adaptive FTC scheme has been proposed for the faulty UAH system with input saturation. Based on Lyapunov theory, the stability of the closedloop control system has been proven. Simulation results have been given to demonstrate the effectiveness of the developed robust FTC method. In our future work, the novel FTC approach to solve the issues of stuck fault and even multiple faults will be further explored for the medium-scale UAH. Acknowledgements

Finally, Fig. 5 is given to demonstrate the effectiveness of the designed robust adaptive compensation FTC scheme. It can be seen from Fig. 5 that the tracking errors are bounded and can converge to a boundary which can be adjusted rapidly. Furthermore, Fig. 6 reveals the control input command is bounded and convergent. In turn, it implies that the input saturation issue is handled by means of the designed robust adaptive compensation FTC strategy. Based on the above simulation results, we can arrive at a conclusion that the presented robust adaptive compensation FTC control method is valid for the medium-scale UAH with external unknown disturbances, actuator faults and input saturation. 5. Conclusions This study has developed a robust adaptive compensation FTC scheme for the medium-scale UAH to obtain satisfactory tracking performance, subjected to external unknown disturbances, actuator faults and input saturation. The 6-DOF nonlinear model of the medium-scale UAH has been established.

This work was supported in part by the National Natural Science Foundation of China (Nos. 61825302, 61573184), in part by the Jiangsu Natural Science Foundation of China (No. BK20171417), and in part by the Aeronautical Science Foundation of China (No. 20165752049). References 1. Liu T, Dai YT, Hong GX, Wang LP. Dynamic response analysis under atmospheric disturbances for helicopters based on elastic blades. Chin J Aeronaut 2017;30(2):628–37. 2. Liang Y, Ying Z, Shuo Y, Zhu XL, Dong J. Numerical simulation of aerodynamic interaction for a tilt rotor aircraft in helicopter mode. Chin J Aeronaut 2016;29(4):843–54. 3. Zhang YM, Jiang J. Bibliographical review on reconfigurable fault-tolerant control systems. Ann Rev Control 2008;32(2):229–52. 4. Qi X, Qi J, Theilliol D. A review on fault diagnosis and fault tolerant control methods for single-rotor aerial vehicles. J Intell Rob Syst 2014;73(1–4):535–55. 5. Lai G, Wen C, Liu Z. Adaptive compensation for infinite number of actuator failures/faults using output feedback control. Inf Sci 2017;399:1–12.

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