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Wenxu Yan, Jie Huang and Dezhi Xu. Abstract ... Dezhi Xu, Key Laboratory of Advanced Process Control for Light Industry ...... Chen M, Wu Q, Jiang C, et al.
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Adaptive fuzzy tracking control for non-affine nonlinear yaw channel of unmanned aerial vehicle helicopter

International Journal of Advanced Robotic Systems January-February 2017: 1–12 ª The Author(s) 2017 DOI: 10.1177/1729881416678137 journals.sagepub.com/home/arx

Wenxu Yan, Jie Huang and Dezhi Xu

Abstract The article deals with the problem of the yaw control of the unmanned aerial vehicle helicopter which is non-affine nonlinear by use of a novel projection-based adaptive fuzzy control approach. First, principle model and control design model of the yaw channel of the unmanned aerial vehicle helicopter are described. Then, a dynamic approximation technique is introduced to approach the non-affine model of the unmanned aerial vehicle helicopter into an affine model with variable parameters, which is applied to facilitate the design of nonlinear control scheme. Next, in the proposed controller, fuzzy controller is designed to deal with the unknown uncertainties and disturbances. Meanwhile, the projection-based adaptive law applied in fuzzy controller bounds the parameter estimation function and can also guarantee the robustness of the designed control scheme against uncertain disturbances. Moreover, the convergence and stability of the designed controller are proved by Lyapunov stability theory. Finally, the simulation results of the yaw channel of an unmanned aerial vehicle helicopter are performed to illustrate that the designed controller has good tracking performance, stability, and robustness under the condition of the time-vary uncertain disturbances. Keywords Unmanned aerial vehicle helicopter, yaw control, non-affine nonlinear systems, fuzzy control, adaptive control Date received: 23 July 2016; accepted: 9 October 2016 Topic: Special Issue - Intelligent Flight Control for Unmanned Aerial Vehicles Topic Editor: Mou Chen

Introduction An unmanned aerial vehicle helicopter (UAVH) is a kind of autonomous control aircraft, which has significant advantages in the condition of takeoff, hovering, vertical landing, low-altitude flight and multi-attitude flight.1 In the past decades, UAVHs are widely used in the civilian and military applications, such as power lines inspection, disaster rescue, agricultural, surveillance missions, wildlife monitoring, national defense, and so on, due to their versatility and maneuverability.2–5 However, the control of UAVHs is a challenging task, because the dynamics of UAVHs are highly nonlinear, inherently unstable, intensely coupled multivariable.5–9 Since the uncertainties of the parameters and model caused by the complicated dynamics of UAVHs affect the control performance, advanced control strategies

are applied in the research of UAVHs to improve their reliability, stability, and robustness, which is still a hotspot in the control field. At present, the flight control technology is mainly divided into two categories: linear control techniques such as proportional–integral–derivative (PID) control,10 H1

Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Electrical Engineering and Intelligent Equipment, Jiangnan University, Wuxi, China Corresponding author: Dezhi Xu, Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Electrical Engineering and Intelligent Equipment, Jiangnan University, Wuxi 214122, China. Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

2 control,11,12 and H 2 control13; nonlinear control techniques such as fuzzy control, 14–18 feedback linearization control, 19 backstepping control, 20–22 sliding model control,1,4,23 and neural network control.5,24–26 Linear control method is very suitable for practical applications, since its design process is simple and the controller is easy to be implemented, which is also the reason for UAVHs dominated by linear control methods.7–13,24 However, the linear control techniques cannot guarantee the global approximation of the model, which may lead to an undesirable response and even instability for the tracking control when the system is far from equilibrium point. Compared with linear control technology, nonlinear control techniques can maintain the stabilization of the system in a large scale, which have been researched in the control system of UAVHs. 14–16,19–23 However, the nonlinear model of the UAVH is too complicated and generally non-affine, so the practical application of nonlinear controller is still an open and challenging research field. Therefore, the key motivation for this article is not only to simplify the yaw channel model of UAVHs for the nonlinear controller design but also guarantee the basic nonlinear characteristics of UAVHs to ensure the designed controller to be globally stable. Due to the rapid development of nonlinear control theory, fuzzy control is an effective nonlinear control techniques applied in UAVHs.14–17,27 It has been known that the fuzzy control is extremely useful to deal with the problem of uncertainties and unknown disturbances in nonlinear control system. Fuzzy controller is a mathematical system for analyzing the analog input by use of logical variables.28 In the fuzzy controller, solutions to the control problem can be realized by understandable terms and IF/THEN rules of the controller can be designed by human experience. However, the nonlinear, high order, time variation, and random disturbance of the system will cause the imperfection of the control rule and also influence the control performance, so fuzzy controller is usually combined with other control methods to achieve the optimal control effect, such as fuzzy PID controller,6 backstepping adaptive fuzzy control,27 adaptive fuzzy sliding mode control,16 fuzzy neural network control,17 a fuzzy adaptive consensus method combined with prescribed performance control,29 and so on. However, to the best of the authors’ knowledge, the existing controllers are mostly proposed for the linear model system or affine nonlinear model system30–32 and general control method cannot control the yaw system well due to the non-affine nonlinearity. In this article, an adaptive fuzzy controller via projection operator is designed for the yaw channel of the UAVH which is non-affine nonlinear. At the same time, the designed control system can maintain asymptotic stability by the proof of stability using the Lyapunov stability theory. The reminder of the article is organized as follows: the yaw channel of the UAVH and its simplified model are described in section ‘‘Model of the yaw channel of the

International Journal of Advanced Robotic Systems

Figure 1. Helicopter and its frame.

UAVH.’’ Section ‘‘Design of controller for yaw channel’’ presents a dynamic approximation method to approach the non-affine model of UAVH into an affine nonlinear model and a projection-based adaptive fuzzy controller is designed for tracking desired yaw angle of the UAVH. In section ‘‘Simulation results,’’ simulation results are provided to prove the validity of the designed control scheme. In ‘‘Conclusion’’ section, some conclusions are drawn and future works are given.

Model of the yaw channel of the UAVH The yaw channel plays a significant part in motion of UAVHs, and the control of a small UAVH is also an extremely complicated problem,7,8 because the torque of the small UAVHs combined with yaw channel is exceedingly sensitive. So, a more accurate model feature is indispensable to obtain better control performance of yaw channel of UAVHs. In the following paper, a higher fidelity mathematical principle model (PM) is applied specifically for the simulation of the UAVH, and a controller design model (CDM) with reduced complexity is applied for the design of controller and stability analysis.

Principle model The PM of yaw channel of the UAVH used in this article is referenced by Leishman.18 The framework of the PM of an UAVH displayed in Figure 1 is built by use of the rigid body motion equation of the UAVH fuselage. The UAVH is subjected to a lot of aerodynamic forces and moments in the normal flight, which can be calculated by summing up all parts of the UAVH including horizontal stabilizer, vertical fin, fuselage, tail rotor, and main rotor. Therefore, the dynamic model of the yaw channel is given by _ ¼ g Izz g_ ¼ Nmr þ Ntr þ Nfus þ Nhs þ Nvf

(1)

where  represents the yaw angle of the UAVH, g represents angular rate of the UAVH, Izz represents the inertia around z-axis, Nvf represents the torque of the vertical fin,

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Nhs represents the torque of horizontal, Nfus represents the torque of fuselage, Ntr represents the torque of tail rotor, and Nmr represents the torque of main rotor. The key forces and torque of the UAVH are determined by tail rotor and main rotor in condition of low speed flight and hovering.18 Therefore, the rest part of the UAVH can be simplified and the dynamic expression of the yaw channel is reformulated as _ ¼ g Izz g_ ¼ Nmr þ Ftr ltr þ z 1 g þ z 2 

(2)

where z 1 and z 2 are damping constants, ltr represents the distance from tail rotor to z-axis, and Ftr represents the thrust of tail rotor.

Nmr ¼

Then, the torque caused by main rotor can be calculated by use of the blade element method as following  ðR  2 2  r Cl c  2 r 2 Cd c þ Nmr ¼ rdr (3) 2 2 R0 with Cd  Cd0 þ Cd 1  þ Cd2  2 , Cl ¼ a, and  ¼  1 = ðrÞ, where  represents the velocity of main rotor,  represents the angle of attack of blade element, c represents the inflow angle,  represents the chord of blade, r represents the velocity radial distance, a represents the slope of lift curve,  represents the density of air, and  1 represents the induced speed. And we can calculate the equation (3) by use of the mathematical software Maple9 as following

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 4 4 2 Cd 2 c ðR  R0 Þmr þ ½8Cd2  R2 ð2C1 mr þ C 22  C2 C 22 þ 4C 2 mr ÞðR30  R3 Þþ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4a R 2 ð2C1 mr þ C22  C 2 C22 þ 4C 1 mr ÞðR3  R30 Þ þ 6Cd2 C1 ðR2  R 20 Þþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cmr 2 2 2 ½3C C 6aC1 ðR20  R 2 Þ þ 6Cd 1  2 R 2 ðR4  R40 Þ d 2 2 C 2 þ 4C 1 mr ðR 0  R Þþ 48R2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3aC2 C 22 þ 4C1 ðR2  R20 Þ þ 4Cd1  R2 ð2C 1 mr þ C 22  C2 C 22 þ 4C1 mr ÞðR30  R3 Þþ c 6Cd0  2 R2 ðR4  R 40 Þ þ 3aC 22 ðR20  R2 Þ þ 3Cd2 C22 ðR 2  R 20 Þ : 48R 2

with C ¼ 1 abc2 ðR 3  R 30 Þ and C2 ¼ 18 abc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 6 2 2=R 2 ðR  R0 Þ, where b, R, and mr are the number and radial of the rotor and pitch angle of main rotor, respectively. Similarly, we can describe the force caused by the tail rotor as  ð Rtr  1 tr1 Ftr ¼ atr btr ctr tr2 tr rtr2  r drtr (5) 2 tr Rtr0 tr1

sffiffiffiffiffiffiffiffiffiffiffi Ttr ¼ 2Atr

(4)

represents the chord of blade, tr represents the induced velocity of tail rotor, rtr represents the radial distance, tr represents the pitch angle, tr represents the velocity of tail rotor, and Atr represents the area of tail rotor disk. Likewise, we can describe the force caused by the main rotor as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Fmr ¼ C1 mr þ C 2 ðC2 þ C22 þ 4C 1 mr Þ (8) 2

(6)

Substituting equation (6) into equation (5), we have 0 sffiffiffiffiffiffiffiffiffiffiffi 1 ð Rtr 1 Ttr r A @tr r 2  Ftr ¼ atr btr ctr tr2 drtr tr 2A 2 tr tr Rtr0 (7) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ C 3 tr þ C 4 ðC 4 þ C 42 þ 4C 3 tr Þ 2 3 with C 3 ¼ 16 atr btr ctr tr2 ðRtr3  Rtr0 Þ and C4 ¼ 18 atr btr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ctr tr 2=Rtr2 ðRtr  Rtr0 Þ, where atr represents the slope of lift curve, btr represents the number of rotor, ctr

Controller design model We can clearly find that the coupling relationship exists between tail rotor thrust Ftr and main rotor torque Nmr in PM. Equations (4) and (7) further display that the PM is too hard to be applied for controller design. So, a simplified model that is used for controller design and stability analysis has been established to replace the dynamic equation expressed by equations (4) and (7). Although the result of nonlinear yaw dynamic model of the UAVH is still very complicated, the advantage of this simplified model which preserves the relevant dynamic characteristics of the PM is that it can be analyzed theoretically.

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The relationship between mr and Nmr can be approximated with quadratic polynomial by drawing the pitch angle versus torque.7 Nmr ¼

2 cN2 mr

þ cN1 mr þ cN0

(10)

where cF 2 , cF 1 , and cF 0 are mainly determined by the velocity of tail rotor tr and the shape of blades. Combining equations (2), (9), and (10), the non-affine CDM can be calculated as following _ ¼ g 2 þ cN1 mr þ cN0 Þ Izz g_ ¼ z 1 r þ z 2   ðcN2 mr

x_i ¼ xiþ1 ; i ¼ 1; 2; . . . ; n  1 x_n ¼ f ðx; uÞ þ dðtÞ

(9)

where cN2 , cN1 , and cN0 are mainly determined by the speed of main rotor mr and the shape of the blades. So again, we can approximate the lift of tail rotor Ftr as Ftr ¼ cF2 tr2 þ cF 1 tr þ cF 0

First, in general, the non-affine nonlinear mathematical model is considered as

(11)

þðcF2 tr2 þ cF 1 tr þ cF 0 Þltr where z 1 and z 2 are damping constants, and we can learn from the equation (11) that: (1) There are coupling relationships between the tail rotor force and mainly rotor torque. (2) The yaw channel of the UAVH is represented by twoorder time-variant system which is nonlinear in the control input. (3) The input nonlinearity mainly determined by the velocity of tail rotor and main rotor and main rotor collective.

(12)

y ¼ x1 where u 2 R represents the system input vector, y 2 R represents the system output, x ¼ ½x 1 ; . . . ; xn T 2 Rn represents the system state vector, dðtÞ represents the unknown exterior disturbances, and f ðx; uÞ represents a known nonlinear function. The Taylor expansion for known smooth nonlinear function f ðx; uÞ in the non-affine nonlinear system in equation (12) with respect to uðtÞ around uðt  Þ is expressed as follows x_i ¼ xiþ1     x_n ¼ f x; uðt  Þ þ fd x; uðt  Þ u þ Rn þ d (13)   ðx;uÞ where fd x; uðt  Þ ¼ @f @u ju¼uðt Þ and u ¼ uðtÞ 2

is restricted to uðt  Þ, and Taylor remainder Rn ¼ fnn u 2 jRn j 

rn u2 2

2

Design of controller for yaw channel In this section, a dynamic approximation approach is designed to approximate the non-affine system of yaw channel of the UAVH into an affine nonlinear form, then fuzzy control scheme is applied to handle the unknown uncertainties and disturbances in the UAVH. By use of the projection-based adaptive law, the parameters estimation function can be bounded, and it can also guarantee the robustness of the controller against the uncertain disturbances. Meanwhile, Lyapunov stability theory is used to prove that the control system can be maintained asymptotically stable.

Non-affine nonlinear approximation It is an extremely difficult task to control the system with non-affine nonlinear characteristics in the control input vector.33 Especially in tracking control, the traditional linearization make the system into a time-varying linearized model under the condition of the stabilization only around the stable state point, so it is quite hard for the design of controller to track the desired trajectory accurately.33 Therefore, a systematic control approach is needed to solve the condition of the nonlinear model with non-affine characteristics, which can make the nonlinear models more appropriate for tracking the desired trajectories accurately.

with fnn ¼ @ @f ðx;uÞ ju¼ , where is a point between uðtÞ and 2u uðt  Þ, and make 0  jfnn j  rn , where rn is defined as a finite positive number. The variable  > 0 is the updating input, which can be selected as the sampling time. When the function of the known smooth nonlinear system f ðx; uÞ changes fast, a good option of variable  is the sampling time, because the higher variable  can cause imprecise approximation. Then, we can rewrite equation (13) as following x_i ¼ xiþ1     (14) x_n ¼ fn x; uðt  Þ þ fd x; uðt  Þ u þ d ðtÞ       where fn x; uðt  Þ ¼ f x; uðt  Þ  fd x; uðt  Þ uðt  Þ and d ðtÞ ¼ Rn þ dðtÞ which is assumed that jd ðtÞj  dM ðtÞ. Moreover, in order to obtain the approximation model, the control input vector u should meet that @f 0 < j @u j  and juj 2 ½0;  with respect to finite positive numbers and . Meanwhile, for the sake of making the approximation error of model (14) bounded, juj should not be too large. Furthermore, in many practical flight systems, juðtÞj 2 ½0;  is the physics limitation of actuators, since the outputs and states cannot change too quickly due to the inertia of the system. According to the equation (14), we can find that uðt  Þ is in the vicinity of u, and the  should not be chosen too

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high to diminish the accuracy of the affine approximation model. Although the smaller is the  the more accurate is the approximation model,  ¼ 0 is impossible to be implemented to obtain the best affine approximation model, because u is the control law which is needed to be processed. So, the further improvement of the abovementioned method is used to get the accurate time-varying trim point which is expressed as following _ ¼  þ u

(15)

Adaptive fuzzy controller design Under normal circumstances, fnd ðx; Þ and fd ðx; Þ are unknown or cannot be known precisely, so it is hard to design a satisfactory controller. In this article, we consider that fnd ðx; Þ and fd ðx; Þ are replaced with fuzzy system f^nd ðx; Þ and f^d ðx; Þ to realize the adaptive fuzzy tracking control. At first, the fuzzy system f^nd ðx; jn Þ is used to approximate fnd ðx; Þ and can be built in the following two steps:

From the equation (15), we can draw that lim !1 ¼ u, and the ! 1 is just an expression in the mathematical sense. Generally speaking, 2 ½5; 50. In order to facilitate the following variables and functions, uðt  Þ is redefined as ðtÞ, fn ðx; Þ þ d ðtÞ is redefined as fnd ðx; Þ, so equation (14) is reformulated as following x_i ¼ xiþ1 x_n ¼ fnd ðx; Þ þ fd ðx; Þu

l

p fuzzy sets are defined as A ðl ¼ 1; 2; . . . ; p Þ, respectively. ^ Step 2: Fuzzy system Yfnnd ðx; jn Þ are constructed using the following p  p fuzzy rules: i¼1 i

(16)

R : IF x 1 is Al11 and . . . and xn is Alnn and is Al ; THEN

From the equation (16), we can see that the non-affine nonlinear model is simplified as a time-varying affine model with the global approximation by use of the dynamic approximation technique, meanwhile the tracking problem can be solved by use of the the affine nonlinear control approach. p1 X

f^ðx; jn Þ ¼

Step 1: For variables xi ði ¼ 1; 2; . . . ; nÞ and , pi fuzzy sets are defined as Alii ðli ¼ 1; 2; . . . ; pi Þ and



l 1 ¼1

p pn X X

f^nd is El 1 ln l

By use of singleton fuzzifier, the product inference engine and center average defuzzifier, the output of fuzzy system is calculated as n Y

yl 1 ln l

ln ¼1 l ¼1 p1 X



l 1 ¼1

ð17Þ

! Ali ðxi Þ  Al ð Þ

i¼1

p pn X n Y X ln ¼1 l ¼1



i

!

(18)

Ali ðxi Þ  Al ð Þ

i

i¼1

where Ali ðxi Þ is the membership function of xi , Al ð Þ is the membership function of , and yl 1 ln l is the free parameter

i

n Y

which are stored in the set n 2 Ri¼1

pi  p

. By introducing the vector ðxÞ, we can rewrite the equation (18) as f^nd ðx; jn Þ ¼ Tn ðx; Þ

where ðxÞ is

Yn i¼1

(19)

pi  p dimensions vector and expressed as: n Y

ðx; Þ ¼

p1 X l 1 ¼1



l ðxi Þ  Al ð Þ

Aii i¼1 p pn X X

n Y

ln ¼1 l ¼1

i¼1

Similarly, fuzzy system f^d ðx; jd Þ can approximate fd ðx; Þ, which is expressed as f^d ðx; jd Þ ¼ Td ðx; Þ Then, the tracking error can be defined as

(21)



!

(20)

Ali ðxi Þ  Al ð Þ i



 T _ . . . ; eðn1Þ e ¼ y  x ¼ e; e;

(22)

where y is the reference trajectory. For system (16), according to the fuzzy system (19) and (21), we can design the control law as

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International Journal of Advanced Robotic Systems

u ¼ ¼

1 f^d ðx; jd Þ

½f^nd ðx; jn Þ þ y_n þ K T e (23)

1 ½Tn ðx; Þ þ y_n þ K T e Td ðx; Þ

–

T

where K ¼ ðkn ; . . . ; k 1 Þ is root of sn þ k 1 sðn1Þ þ    þ k n ¼ 0, which makes the characteristic equation stable, ðx; Þ and ðx; Þ are the fuzzy vectors, and Td and Tn are adaptive parameter changed according to the adaptive law (24) and (25), which are designed as follows   _ n ¼ g1 Proj n ; eT Pb ðx; Þ (24)   _ d ¼ g2 Proj d ; eT Pbðx; Þu

Figure 2. The flow chart of the projection-based adaptive fuzzy controller for the yaw channel of the UAVH. UAVH: unmanned aerial vehicle helicopter.

Step 1: According to equations (23) and (22), state variable x_n can be computed as x_n ¼ f^d ðx; jd Þu þ f^nd ðx; jn Þ  K T e  e_n

(25)

(26)

From equations (16) and (26), the closed loop dynamic equation of fuzzy control system can be written as

where Projð; Þ is the projection operator (see the Appendix 1 for details) which bounds that jn j   and jd j  . In Figure 2, the entire projection-based adaptive fuzzy tracking controller for the yaw channel of the UAVH is illustrated to obtain a clear idea.

e_n ¼ K T e þ ½f^nd ðx; jn Þ  fnd ðx; Þ þ ½f^d ðx; jd Þ  fd ðx; Þu (27) Then, for the need of the the Lyapunov stability theory, we can reformulate the dynamic equation (27) as the vector form

Stability analysis The proof of stabilization of designed projection-based adaptive fuzzy controller is achieved by use of the Lyapunov function with the state tracking errors (equation (22)) and adaptive parameter estimation errors, which can be deduced as the following three steps: 2

0 60 6 6. . where  ¼ 6 6. 6 40 kn

e_ ¼ e þ bf½f^nd ðx; jn Þ  fnd ðx; Þ þ ½f^d ðx; jd Þ  fd ðx; Þug

1 0 .. . 0

0 1 .. . 0

0 0 .. . 0

 0  0 . . .. . .  0

0 0 .. . 1

kn1

kn2

kn3

   k 2

k 1

Step 2: We define the minimum approximation error as ! ¼ f^nd ðx; jn Þ  fnd ðx; Þ þ ½f^d ðx; jd Þ  fd ðx; Þu (29) where n and d are optimal parameters which are described as n ¼ arg min ½ sup j f^nd ðx; jn Þ  fnd ðx; Þj n 2n x; 2Rn d ¼ arg min ½ sup j f^d ðx; jd Þ  fd ðx; Þj d 2d x; 2Rn

3 7 7 7 7; 7 7 5

(28)

2 3 0 607 6 7 6. 7 . 7 b¼6 6. 7 6 7 405 1

where n and d are sets of n and d . Then, from equations (28) and (29), we have e_ ¼ e þ bf½f^nd ðx; jn Þ  f^nd ðx; jn Þ þ ½f^d ðx; jd Þ  f^ ðx; j Þu þ !g d

d

(30) Substituting equations (19) and (21) into equation (30), we can compute the closed loop dynamic equation as e_ ¼ e þ b½ðn  n ÞT ðx; Þ þ ðd  d ÞT ðx; Þu þ ! ¼ e þ M (31)

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The equation (31) clearly describes the relationship between the adaptive parameters n and d and the tracking error e. The purpose of adaptive law is to provide a regulatory mechanism for n and d to make the tracking error e and parameter error n  n and d  d to achieve the minimum. Step 3: Design the Lyapunov function as 1 1 V ¼ eT Pe þ ðn  n ÞT ðn  n Þ 2 2g1 1 ðd  d ÞT ðd  d Þ þ 2g2

V_

where g1 and g2 are positive and P is a positive definite matrix which meets the following Lyapunov function: T P þ P ¼ Q

(33)

where Q is an arbitrary n  n positive definite matrix. According to the equations (31) and (32), the V_ along the dynamical tracking trajectory of error equations can be computed as

(32)

1 1 1 1 ¼ e_T Pe þ eT Pe_ þ ðn  n ÞT _ n þ ðd  d ÞT _ d 2 2 g1 g2 1 1 1 1 ¼ ðeT T þ M T ÞPe þ eT Pðe þ MÞ þ ðn  n ÞT _ n þ ðd  d ÞT _ d 2 2 g1 g2 1 1 1 1 ¼ eT ðT P þ PÞe þ ðM T Pe þ eT PMÞ þ ðn  n ÞT _ n þ ðd  d ÞT _ d 2 2 g1 g2 1 1 1 ¼  eT Qe þ eT PM þ ðn  n ÞT _ n þ ðd  d ÞT _ d 2 g1 g2 1 1 1 ¼  eT Qe þ eT Pb! þ ðn  n ÞT eT Pb ðx; Þ þ ðd  d ÞT eT Pbðx; Þu þ ðn  n ÞT _ n þ ðd  d ÞT _ d 2 g1 g2 1 1 1 ¼  eT Qe þ eT Pb! þ ðn  n ÞT ½_ n þ g1 eT Pb ðx; Þ þ ðd  d ÞT ½_ d þ g2 eT Pbðx; Þu 2 g1 g2 (34)

Substituting adaptive law (24) and (25) into (34) leads to:   1 V_ ¼  eT Qe þ eT Pb! þ ðn  n ÞT ½ Proj n ; eT Pb ðxÞ þ eT Pb ðx; Þ 2   þðd  d ÞT ½ Proj d ; eT PbðxÞu þ eT Pbðx; Þu

Using the property 1A of the projection operator (see the appendix 1 for details), we can obtain that   ðn  n ÞT ½ Proj n ; eT Pb ðxÞ þ eT Pb ðx; Þ  0

(35)

Since  12 eT Qe  0, and the fuzzy system is selected with very small minimum approximation error !, we can obtain V_  0. Thus, the whole system is proved to be asymptotically stable.

(36)   ðd  d ÞT ½ Proj d ; eT PbðxÞu þ eT Pbðx; Þu  0 (37) Then, according to the above analysis, we can get 1 V_ ¼  eT Qe þ eT Pb! 2

(38)

Simulation results In this part, simulations results are used to demonstrate the validity of the designed nonlinear projection-based adaptive fuzzy tracking controller for the yaw channel of an UAVH. First, the PM for simulation is achieved from the helicopter platform referenced by Zhao and Han.7

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The parameters of nonlinear yaw channel model of an UAVH are summarized as following _ ¼ g g_ ¼ a1 g þ a2  þ a3 tr þ a4 tr2 þ a5 tr þ dðtÞ

(39)

with a1 ¼ 1:38, a2 ¼ 3:33, a3 ¼ 63:09, a4 ¼ 11:65, a5 ¼ 0:14, and  ¼ 1200. We can see from the equation (39) that a3 tr þ a4 tr2 þ a5 tr is a nonlinear function about the controller input tr . Then, the initial conditions are n ¼ 0:1 and d ¼ 0:1, ð0Þ ¼ 5, and rð0Þ ¼ 0, and the filter parameter applied to approach u is selected as 50. The remaining parameters 20 0 , k 1 ¼ 50, k 2 ¼ 100, are selected as Q ¼ ½ 0 20 g1 ¼ 50, and g2 ¼ 0:5, which are chosen to achieve optimal control performance according to the adaptive law and the need of stability of control system. ( dðtÞ ¼

Next, the control law of yaw dynamics of the UAVH is given by u¼

1 ½Tn ðx; Þ T d ðx; Þ

þ x_ 2 þ K T e

  _ n ¼ g1 Proj n ; eT Pb ðxÞ   _ d ¼ g 2 Proj d ; eT PbðxÞu

where, as for ðx; Þ and ðx; Þ, five kinds of membership functions shown in Figure 3 are selected as following: Finally, the following two cases are simulated to demonstrate the robustness of the designed controller for the yaw channel of the UAVH. In these simulations, the unknown uncertain disturbance is designed to change as a function of time, that is

0; deg=s 2

t  10

5 sinðtÞ þ 4cosð0:5tÞ þ 3cosð0:5tÞ sinð0:75tÞ; deg=s 2

t > 10

Case 1: Proposed adaptive fuzzy control scheme (AFCS) is used in the step yaw signal tracking. The tracking command of c is considered as 8 t  20 > < 10; c ¼ 15; 20 < t  40 > : 25; t > 40 Then, a filter which is Fc ¼ d =c ¼ 0:8=ðs þ 0:8Þ is applied for c to obtain the desired smooth trajectory d ¼ yd ¼ 0:8c =ðs þ 0:8Þ. Meanwhile, the classical PID control method is used to compare with the AFCS to demonstrate the validity and robustness of the AFCS. The simulation results of tracking response, the related control effort u, and the filtering are illustrated in Figure 4. As can be seen in the Figure 4, the proposed control scheme for step yaw signal tracking control of the UAVH has satisfactory control performance (a minimum response time, without overshot and minimal tracking error) against the unknown time-varying disturbance. By comparing with PID control method, the proposed AFCS has more robust against unknown disturbances in yaw control, which further demonstrate the effectiveness of the designed controller. Moreover, Figure 5 shows the adaptive fuzzy system f^nd ðx; jn Þ and f^d ðx; jd Þ that are used to approximate fnd and fn are bounded, which also guarantees the robustness of the designed AFCS against the parameter uncertainties and unknown disturbances.

(40)

Case 2: Proposed AFCS is used in the sinusoidal yaw signal tracking. The tracking command of d is considered as: d ¼ 30sinðtÞ. Similarly, the classical PID control method is used to compare with the AFCS in the sinusoidal yaw signal tracking. The simulation results of tracking response, the related control effort u, and the filtering parameter are displayed in Figure 6. From the Figure 6, the proposed control scheme for sinusoidal yaw signal tracking control of the UAVH achieves good asymptotic tracking performance under the condition of the unknown time-varying disturbance. We can obviously draw that proposed AFCS comparing with PID control method has more robust against unknown disturbances in yaw control. Moreover, Figure 7 shows the adaptive fuzzy system f^nd ðx; jn Þ and f^d ðx; jd Þ. By comparing Figure 5 with Figure 7, we can find that the approximation value f^nd ðx; jn Þ and f^d ðx; jd Þ in the two case are similar and around 20 and 3, respectively.

Conclusion A systematic study is carried out for the yaw channel of the UAVH in this article. First, since the yaw channel of the UAVH is non-affine, a dynamic approximation method is developed to approach the non-affine system into an affine nonlinear form. Then, fuzzy control scheme is designed to

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Figure 3. The membership function of the fuzzy controller.

Figure 4. System step response with the AFCS under the condition of the time-varying disturbance compared with PID controllers. AFCS: adaptive fuzzy control scheme; PID: proportional–integral–derivative.

Figure 5. Fuzzy system ^f nd ðx; jn Þ and ^f d ðx; jd Þ in the case of the step response.

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International Journal of Advanced Robotic Systems

Figure 6. System sinusoidal response with the AFCS under the condition of the time-varying disturbance compared with PID controller. AFCS: adaptive fuzzy control scheme; PID: proportional–integral–derivative.

Figure 7. Fuzzy system ^f nd ðx; jn Þ and ^f d ðx; jd Þ in the case of the sinusoidal response.

handle the unknown uncertainties and disturbances in the yaw channel. By use of the projection-based adaptive law, the free parameters estimated function in the fuzzy control scheme can be bounded, and it can also guarantee the robustness of the controller against the uncertain disturbances. Remarkably, the proposed projection-based adaptive fuzzy controller is useful when the explicit model of the UAVH is hard to obtain. Finally, simulation results are illustrated to proved that proposed projection-based AFCS can maintain the progressive tracking of the output for the closed-loop control system of the UAVH and have satisfactory control performance in the case of the unknown uncertainties and time-varying disturbances. Nevertheless, there is a chattering phenomenon in the control u, which

may be caused by the selection of fuzzy rule base. Therefore, the solution to solve the chattering phenomenon and supplementary of input and output constraints are our future works.

Appendix 1 The smoothened projection operator34 is applied in this article to bound the parameter estimation equation. The projection operator is very helpful for robust adaptive controller that needs multiple derivative of adaptive law. Below, we describe the main definition and property from.34

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Definition 1. Parameter estimation vector q which belongs to a convex compact set is defined as following  ¼ fk  k  max g

(1A)

where  max represents the norm bound of parameter vector . Then, for any obtained x 2 Rn , the normalized projection operator is defined as following _ ¼ Projð; xÞ

¼

8 x; > > > > < x;

if f ðÞ < 0 if f ðÞ . . . 0 and rf ðÞT x  0

rf ðÞf ðÞT x > > > x  f ðÞ; otherwise > : k f ðÞk 2 (2A)

where r represents the gradient vector of f ðÞ computed at  and f ðÞ : Rn ! R represents the convex smooth function: f ðÞ ¼

T    max " 2 þ 2" max

(3A)

where " represents an arbitrary positive. The main property of the projection operator34 is given as following Property 1. If we choose the initial value ð0Þ 2  and make that the variable ðtÞ are changed according to the following equation:   _ ¼ Proj ðtÞ; x ; ðt0 Þ 2  ðtÞ (4A) Then ðtÞ 2 

(5A)

For any t . . . t0 , the inequality can be established as ð   ÞT Projð; xÞ  ð   ÞT x

(6A)

Acknowledgment The authors sincerely thank the editor and all the anonymous reviewers for their valuable comments and suggestions.

Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by National Natural Science Foundation of China (61503156, 61403161, and 51405198) and the Fundamental Research Funds for the Central Universities (JUSRP11562 and NJ20150011).

References 1. Xian B, Guo J, Zhang Y, et al. Sliding mode tracking control for miniature unmanned helicopters. Chin J Aeronaut 2015; 28(1): 277–284. 2. Raptis IA and Valavanis KP. Linear and nonlinear control of small-scale unmanned helicopters. Vol. 45. Springer Netherlands, Netherlands: Springer Science & Business Media, 2010. DOI 10.1007/978-94-007-0023-9. 3. Al-Sharman MKS, Abdel-Hafez MF, and Al-Omari MA. Attitude and flapping angles estimation for a small-scale flybarless helicopter using a kalman filter. IEEE Sens J 2015; 15(4): 2114–2122. 4. Zheng EH, Xiong JJ, and Luo JL. Second order sliding mode control for a quadrotor UAV. ISA Trans 2014; 53(4): 1350–1356. 5. Nodland D, Zargarzadeh H, and Jagannathan S. Neural network-based optimal adaptive output feedback control of a helicopter UAV. IEEE Trans Neural Netw Learn Syst 2013; 24(7): 1061–1073. 6. Gautam D and Ha C. Control of a quadrotor using a smart self-tuning fuzzy PID controller. Int J Adv Robot Syst 2013; 10(6): 502–512. 7. Zhao X and Han J. Yaw control of helicopter: an adaptive guaranteed cost control approach. Int J Innov Comput I 2009; 5(8): 2267–2276. 8. Cai G, Chen BM, Peng K, et al. Modeling and control of the yaw channel of a UAV helicopter. IEEE Trans 2008; 55(9): 3426–3434. 9. Xu D Z, Jiang B, and Shi P. Global robust tracking control of non-affine nonlinear systems with application to yaw control of UAV helicopter. Int J Control Autom 2013, 11(5): 957–965. 10. Pounds PEI, Bersak DR, and Dollar AM. Stability of smallscale UAV helicopters and quadrotors with added payload mass under PID control. Auton Robot 2012; 33(1–2): 129–142. 11. Prempain E and Postlethwaite I. Static H1 loop shaping control of a fly-by-wire helicopter. Autom 2005; 41(9): 1517–1528. 12. Kim HC, Dharmayanda HR, Kang T, et al. Parameter identification and design of a robust attitude controller using H1 methodology for the raptor E620 small-scale helicopter. Int J Control Autom 2012; 10(1): 88–101. 13. Wang N, Pei H, He Y, et al. Robust H2 static output feedback tracking controller design of longitudinal dynamics of a miniature helicopter via LMI technique. In: 24th Chinese Control and Decision Conference (CCDC) (ed Yang Guang-Hong), Taiyuan, China, 23–25 May 2012, Vol. 229, pp. 346–350. Institute of Electrical and Electronics Engineers (IEEE). 14. Kadmiry B and Driankov D. A fuzzy gain-scheduler for the attitude control of an unmanned helicopter. IEEE Trans Fuzzy Syst 2004; 12(4): 502–515. 15. Hong SK. Fuzzy logic based closed-loop strapdown attitude system for unmanned aerial vehicle UAV. Sens Actuat A Phys 2003; 107(2): 109–118. 16. Yang Y and Yan Y. Attitude regulation for unmanned quadrotors using adaptive fuzzy gain-scheduling sliding mode control. Aerosp Sci Technol 2016; 54, 208–217.

12 17. Dong X, Zhao Y, Karimi HR, et al. Adaptive variable structure fuzzy neural identification and control for a class of MIMO nonlinear system. J Franklin Inst 2013; 350(5): 1221–1247. 18. Leishman GJ. Principles of helicopter aerodynamics. 2nd ed. Cambridge: Cambridge University Press, 2002. 19. Song B, Liu Y, and Fan C. Feedback linearization of the nonlinear model of a small-scale helicopter. J Control Theory and Appl 2010; 8(3): 301–308. 20. Ahmed B, Pota HR, and Garratt M. Flight control of a rotary wing UAV using backstepping. Int J Robust Nonlin 2010; 20(6): 639–658. 21. Chen M, Wu Q, Jiang C, et al. Guaranteed transient performance based control with input saturation for near space vehicles. Sci China Inform Sci 2014; 57(5): 1–12. 22. Chen M and Jiang B. Robust attitude control of near space vehicles with time-varying disturbances. Int J Control Autom 2013; 11(1): 182–187. 23. Chen M, Ren B, Wu Q, et al. Anti-disturbance control of hypersonic flight vehicles with input saturation using disturbance observer. Sci China Inform Sci 2015; 58(7): 1–12. 24. Peng C, Bai Y, Gong X, et al. Modeling and robust backstepping sliding mode control with Adaptive RBFNN for a novel coaxial eight-rotor UAV. IEEE/CAA J Autom Sin 2015; 2(1): 56–64. 25. Garratt M and Anavatti S. Non-linear control of heave for an unmanned helicopter using a neural network. J Intell Robot Syst 2012; 66(4): 495–504.

International Journal of Advanced Robotic Systems 26. Suresh S and Sundararajan N. An on-line learning neural controller for helicopters performing highly nonlinear maneuvers. Appl Soft Comput 2012; 12(1): 360–371. 27. Basri MAM, Husain AR and Danapalasingam KA. A hybrid optimal backstepping and adaptive fuzzy control for autonomous quadrotor helicopter with time-varying disturbance. Proc Inst Mech Eng G J Aerosp 2015; 229(12): 2178–2195. 28. Lugli A, Neto E, Henriques J, et al. Industrial application control with fuzzy systems. Int J Innov Comput I 2016; 12(2): 665–676. 29. Wang W and Yu Y. Fuzzy adaptive consensus of secondorder nonlinear multi-agent systems in the presence of input saturation. Int J Innov Comput I 2016; 12(2): 533–542. 30. Shi P, Su X, and Li F. Dissipativity-based filtering for fuzzy switched systems with stochastic perturbation. IEEE Trans Autom Control 2016; 61(6): 1694–1699. 31. Su X, Wu L, Shi P, et al. A novel approach to output feedback control of fuzzy stochastic systems. Automatica 2014; 50(12): 3268–3275. 32. Su X, Shi P, Wu L, et al. Fault detection filtering for nonlinear switched stochastic systems. IEEE Trans Autom Control 2016; 61(5): 1310–1315. 33. Boskovic JD, Chen L, and Mehra RK. Adaptive control design for nonaffine models arising in flight control. J Guid Control Dyn 2004; 27(2): 209–217. 34. Cai Z, de Queiroz MS and Dawson DM. A sufficiently smooth projection operator. IEEE Trans Autom Control 2006; 51(1): 135–139.