Tracking error constrained robust adaptive neural ...

Research Article

Tracking error constrained robust adaptive neural prescribed performance control for flexible hypersonic flight vehicle

International Journal of Advanced Robotic Systems January-February 2017: 1–16 ª The Author(s) 2017 DOI: 10.1177/1729881416682704 journals.sagepub.com/home/arx

Zhonghua Wu1, Jingchao Lu1, Jingping Shi1, Qing Zhou2 and Xiaobo Qu1

Abstract A robust adaptive neural control scheme based on a back-stepping technique is developed for the longitudinal dynamics of a flexible hypersonic flight vehicle, which is able to ensure the state tracking error being confined in the prescribed bounds, in spite of the existing model uncertainties and actuator constraints. Minimal learning parameter technique–based neural networks are used to estimate the model uncertainties; thus, the amount of online updated parameters is largely lessened, and the prior information of the aerodynamic parameters is dispensable. With the utilization of an assistant compensation system, the problem of actuator constraint is overcome. By combining the prescribed performance function and sliding mode differentiator into the neural back-stepping control design procedure, a composite state tracking error constrained adaptive neural control approach is presented, and a new type of adaptive law is constructed. As compared with other adaptive neural control designs for hypersonic flight vehicle, the proposed composite control scheme exhibits not only low-computation property but also strong robustness. Finally, two comparative simulations are performed to demonstrate the robustness of this neural prescribed performance controller. Keywords Prescribed performance control, minimal learning parameter, state tracking error constraint, assistant compensation system Date received: 26 June 2016; accepted: 2 November 2016 Topic: Special Issue - Control of Hypersonic Flight Vehicles Topic Editor: Danwei Wang

Introduction Research effort of hypersonic flight vehicle (HFV) has drawn considerable attention during the past several years, because it can provide large time reductions within both civil and military flight activities.1–3 The success of the experimental aircraft NASA’s X-43A has affirmed the feasibility of this technique.3 Unfortunately, the control of HFV is still confronted with a large amount of intractable issues, such as the famous vibrational effects caused by slender geometry and special structures of HFV, intensely coupling between engine system and aerodynamic force and the variation of vehicle characteristics with different flight conditions. 4 Thus, in order to keep the flight stability and safety of

HFV, transient and steady-state performance characteristics are desirable to be ensured through suitable controllers. In the literature, due to lack of experimental date on lateral model of HFV, the control problem of HFV mainly

1

School of Automation, Northwestern Polytechnical University, Xi’an, China 2 Xi’an Aeronautics Computing Technique Research Institute, AVIC, Xi’an, China Corresponding author: Zhonghua Wu, Northwestern Polytechnical University, 127 West Youyi Road, Shaanxi, Xi’an 710129, China. Emails: [email protected]; [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 focuses on the longitudinal channel. The sum of squares/ robust linear matrix inequation method is proposed to design the nonlinear controller for the longitudinal dynamics of HFV with parametric uncertainties.5 Utilizing the TakagiSugeno (T-S) fuzzy modelling technique to approximate the nonlinear dynamics of HFV with modelled and unmodelled disturbance, a robust disturbance observer mixed H 2 =H1 controller is designed for the obtained T-S fuzzy model.6 To provide stable tracking of the velocity and altitude reference trajectories, a high-order extended state observer-enhanced control is adopted to improve the tracking performance.7 By employing the input/output linearization technique to transform the nonlinear model of HFV, a back-stepping technique–based exponential sliding controller is proposed and analysed for the longitudinal dynamics with mismatched uncertainties.8 Moreover, some other sliding mode control schemes, such as super twisting sliding mode control,9 highorder sliding mode control10,11 and recursive terminal sliding mode control12, are also proposed to design the control system of HFV in the presence of parametric uncertainties. In addition to the aforementioned control method, numerous significant approaches have been applied to tackle the control problem of HFV, including linear parameter varying control,13 minimax linear quadratic regulator (LQR) control,14 back-stepping-based method,3,4,15–17 predictive control18 as well as neural/fuzzy approach.19–22 Approximation-based adaptive back-stepping control methods have been widely researched for a nonlinear strict-feedback system,23,24 and the noteworthy problem of ‘explosion of items’ is elegantly overcome by introducing a dynamic surface control (DSC) technique (low-pass filter) in the control design.25–28 More specifically, adaptive neural control with the back-stepping technique has also been extensively developed for HFV control.29,30 Noting another fact that the operation mode of engine system has rigorous demands on actuators as well as HFV states, the constraints should be considered in the controller design from a practical perspective.31 To ensure relatively satisfying control performance of HFV when physical limitations are in effect, a compensation system is first used to avoid actuator constraint problem.31 After that, other kinds of compensation mechanisms have emerged and evolved to deal with this issue.32–35 Similarly, command filter–based adaptive back-stepping control is also investigated for HFV in the presence of constraints on system states and actuators.36,37 Despite the prominent progress in adaptive neural control methods of HFV, a common characteristic of aforementioned outstanding works is that transient performance is often neglected; only the convergence of the altitude and velocity tracking errors to a residual set are established.38–40 Recently, the study of prescribed performance control (PPC) methodology has drawn considerable attention. 39,41,42 The PPC represents that the tracking error should converge to a predefined bound accompanying with the convergence rate no less than a certain value.41,43 By exploiting an output error transformation technique, an adaptive PPC method is first developed for a strict-

International Journal of Advanced Robotic Systems feedback nonlinear system.41 Afterwards, Bu et al.42,44 have proposed a control scheme by integrating a PPC technique and neural network (NN) to cope with the output constraint problem of HFV. The control scheme presented in the studies by Bu et al.42,44 ensures prescribed transient and steadystate performance in the presence of parameter uncertainties. However, those approaches cannot be utilized to tackle the state constraint problem, and the stability analysis of the constrained state needs to be investigated further. Subsequently, a low-complexity approximation-free PPC scheme has been employed to the control of HFV.45 Although the prescribed performance problem has been tackled, the actuator saturation problem is not considered in it, and there indispensably exist the strict assumptions for nonlinear function with this control scheme. To the best of the author’s knowledge, so far, a composite low-computational neural controller, which is capable of handling actuator and state constraints as well as guaranteeing prescribed performance on transient and steady-state behaviour of the output tracking errors, simultaneously, has not been investigated for HFV. Motivated by the aforementioned discussion and the control approach,46 a robust adaptive neural PPC scheme is studied for flexible HFV with model uncertainty and actuator constraints. First, on the basis of the functional decomposition, the longitudinal model of HFV can be divided into altitude and velocity subsystems. NNs are employed to estimate the unknown items, thus the prior information of the aerodynamic parameters is no longer needed. Using the PPC technique, an adaptive neural controller is proposed, which is able to ensure the output tracking errors confined in the prescribed bounds, while the issue of state tracking error constraint is also addressed. In order to deal with the problem of ‘increase of NN’ updating parameters’ and ‘explosion of the items’ in a conventional neural back-stepping method, the minimal learning parameter (MLP) technique and ‘firstorder sliding mode differentiator (FOSD)’ are used in the control design. Consequently, a low-computational control scheme is obtained. Using an assistant compensation system, the problem of actuator constraint is also eliminated. The contributions of this article are shown as follows: 1. In contrast to the approach in the study by Bechlioulis and Rovithakis46, MLP and DSC techniques are incorporated into the controller design, which avoids the problem of increase of NN learning parameters and explosion of the items’, thus deriving a low-computation design. 2. Compared to previous constraint control schemes,36,37 a composite adaptive neural PPC method, which is capable of handling actuator and state constraints as well as guaranteeing prescribed performance on transient and steady-state behaviour of the output tracking errors, is first presented, and a new type of adaptive law is constructed by synthesizing the PPC and MLP technique in the back-stepping design procedures.

Wu et al.

3

3. Rigorous Lyapunov analysis has been conducted to prove the stability of this control scheme. Meanwhile, comparative simulations are performed to highlight the robustness of the proposed neural prescribed performance controller.

h_ ¼ V sing g_ ¼

L þ T sin  mgcosg mV _ ¼ q  g_

(2) (3) (4)

HFV model and some preliminaries HFV model

q_ ¼

The model of HFV used in this study is based on the study by Parker et al.2 The HFV model includes system states (V , h, , g, q), flexible states ( 1 and  2 ) and control inputs (e and ), where V denotes the velocity, h denotes the altitude,  represents the angle of attack, g denotes the flight path angle, q denotes the pitch rate; e and  denote the elevator deflection and the fuel equivalence ratio, respectively4,47,48 D þ T cos  mg sing V_ ¼ m

Myy ~ 1 € 1 ~ 2 € 2 þ þ Iyy Iyy Iyy

€i ¼ 2 i !i _ i  !i2 i þ Ni þ ~ i q; i ¼ 1; 2

(5)

(6)

where m, Iyy and g denote the mass of HFV and the moment of inertia and gravity. T, D, L and MA denote the thrust of the engine, drag force, lift force and pitching moment, respectively.23,47 The related definitions are given as follows

(1)

T ¼ T ðÞ þ T0 ðÞ  ½ 1  þ  2  3 þ ½ 3  þ  4  2 þ ½ 5  þ  6  þ ½ 7  þ  8    h  h0 2 D q SðCD  2 þ CD  þ CD0 Þ; L ¼ CL qS  qSðCL0 þ CL Þ;  ¼  0 exp  hs _

 2 2 þ C   þ C 0 þ C e  Þ; q ¼ 1 V 2 cðCM Myy ¼ MT þ M 0 ðÞ þ Me e  zT T þ qS M e M M 2 2 2 N1 ¼ CN1  2 þ CN1  þ CN0 1 ; N2 ¼ CN2  2 þ CN2  þ CN0 2 þ CNe2 e

The explanation of the other parameters can be referred to the study by Parker et al.2

formulation f 2 ðX2 Þ ¼ ½L þ T sinð  gÞ  mgcosg=ðmV Þ   f 4 ðX4 Þ ¼ ½Myy þ ~ 1 € 1 þ ~ 2 € 2 =Iyy þ e .

Model transformation

Velocity subsystem. Velocity subsystem (1) is transformed into equation (8) shown as follows

Altitude subsystem. Consider x 1 ¼ h, x 2 ¼ g, x 3 ¼ , and x 4 ¼ q, where  ¼  þ g and x ¼ ðx 2 ; x 3 ; x 4 Þ. Therefore, equations (2) to (5) can be converted into the formulation as shown below x_ 1 x_ 2 x_ 3 x_ 4

¼ Vx 2 ¼ f 2 ðX2 Þ þ x 3 ¼ x4 ¼ f 4 ðX4 Þ þ u

(7)

y ¼ x 1 ; u ¼ e where y is the output signal of altitude subsystem (7), f 2 ðX2 Þ and f 4 ðX4 Þ are unknown functions with the

V_ ¼ fV ðXV Þ þ 

and

(8)

where fV ðXV Þ ¼ ðD þ T cos  mg singÞ=m   is an unknown function. Remark 1. Since we only consider the cruise phase in this article, g is quite small so we can take sing  g in equation (2) to simplify the model. Remark 2. Considering the fact that the existence of flexible states  1 and  2 is hard to be measured directly,

4

International Journal of Advanced Robotic Systems we regard  1 and  2 as a part of lumped nonlinear function. Thus, the controller design only depends on system states (V , h, , g, q).

Lemma 2. The FOSD is shown as follows   &_ i1 ¼ 

i1 j& i1  lðtÞj 0:5 sign & i1  lðtÞ þ & i2

Prescribed performance In this section, we will generalize the preliminaries of PPC.43,49 To achieve the control objective, the tracking errors zi ðtÞ; i ¼ 1; 2; 3; 4; V should be confined in the prescribed bounds shown as follows Mi i ðtÞ < zi ðtÞ < i ðtÞ; if zi ð0Þ > 0  i ðtÞ < zi ðtÞ < Mi i ðtÞ; if zi ð0Þ < 0

i ðtÞ ¼ ð i0  i1 Þexpðli tÞ þ i1

(10)

where i0 , i1 and li are design positive constants; i0 ¼ i ð0Þ while i1 ¼ limt!1 i ðtÞ; li denotes the minimum speed of convergence and i1 is the maximum allowed steady error.43,49 To transform the constrained tracking error condition (9) into an equivalent unconstrained one, the following transformation is employed. We have   zi ðtÞ

i ðtÞ ¼ Ri (11) i ðtÞ where i ðtÞ is a transformed error and Ri ðÞ is an increasing transformation function shown as follows43,49 0 1 zi ðtÞ   M þ i B C zi ðtÞ  i ðtÞC Ri ¼ lnB @ A if zi ð0Þ > 0 i ðtÞ 1zi ðtÞ Mi i ðtÞ  Ri

zðtÞ ðtÞ



B ¼ lnB @

 Mi

1

1þzi ðtÞ

i ðtÞ

Mi 

zi ðtÞ

i ðtÞ

C C if zi ð0Þ < 0 A

The derivative of equation (11) is   _ i ðtÞ

_ i ðtÞ ¼ ri z_i ðtÞ  zi ðtÞ i ðtÞ where ri ¼

@Ri z ðtÞ @ i

i ðtÞ

(15)

i2 signð& i2  & i1 Þ &_ i2 ¼   1 and

 2 are the where & i1 and & i2 are the system states,

positive parameters of FOSD, and lðtÞ is an input function. _ to a desirable precision in case the initial &_ i1 can estimate lðtÞ _ 0 Þ are bounded.50 errors & i1 ðt 0 Þ  lðt 0 Þ and &_ i2 ðt0 Þ  lðt

(9)

where 0  Mi  1 and i ðtÞ > 0 named performance function is defined as

0

where  0 is a constant satisfying  0 ¼ eð 0 þ1Þ , that is,  0 ¼ 0:2785.

(12)

Controller design and stability analysis The control objective in this study is to design a composite adaptive neural prescribed performance controller u and  to steer system outputs such as h and V to track their corresponding desired reference signal hd and Vd with their tracking errors confined to the prescribed performance bounds. Assumption 1. Assume that the system states are measurable, and there is no time delay in the signal transformation. Assumption 2. Systems (7) and (8) satisfy 1 þ @f2 =x 3 6¼ 0, 1 þ @f 4 =u 6¼ 0 and 1 þ @fV = 6¼ 0. Obviously, there are ideal weight vectors W 2 , W 4 , and WV , such that 8 T > < f2 ¼ W 2  2 ðX2 Þ þ " 2 j"2 j  " 2M (16) f4 ¼ W 4T  4 ðX4 Þ þ " 4 j"4 j  " 4M > : fV ¼ WVT V ðXV Þ þ "V j"V j  "VM where "i and "iM denote the approximation errors and their bounds, respectively. To decrease the computational burden, the MLP scheme is employed to update NN parameters. Those parameters are defined as ’i ¼ jjWi jj2 ði ¼ 2; 4; V Þ. In the following, we replace i ðÞ with i to simplify the expression.

Altitude controller design. To proceed the design process (13)

43,49 1 i ðtÞ > 0.

Useful function and key lemmas Lemma 1. For any !0 > 0 and  2 R, the following inequality is established23    0  jj   tanh (14)  0 !0 !0

and tackle the actuator saturation problem,25 the assistant system (17) is constructed to generate 1 _ 1 ¼ u  k 1 1

(17)

where k 1 > 0 is a positive parameter, and u ¼ u  ud , u is the actual control input to the altitude subsystem and ud is the control input to be designed.25 The relationship between ud and u can be expressed as follows  signðud Þuþ if jud j > uþ d; d u¼ (18) ud ; else where uþ d > 0 denotes the bound of ud . The coordinate change (19) is built shown as follows

Wu et al.

5 8 z1 > > > < z2 > z3 > > : z4

¼ x 1  yd ¼ x2  1 ¼ x3  2

(19)

¼ x4  3  1

where 1 , 2 and  3 are middle controllers being established at steps 1, 2 and 3, respectively. yd ¼ hd is the reference altitude signal. The control scheme for the altitude subsystem is developed via a back-stepping technique, which contains four-step recursive design procedure. Step 1. The derivative of z 1 ¼ x 1  yd is expressed as follows z_1 ¼ x_ 1  y_d ¼ Vx 2  y_d ¼ V ðz 2 þ  1 Þ  y_d

(20)

Using equations (13) and (20), the derivative of the transformed altitude error 1 ðtÞ is shown as follows     _ _

_ 1 ðtÞ ¼ r 1 z_1  1 z 1 ¼ r 1 Vx 2  h_d  1 z 1 (21) 1 1 where r 1 ¼ @R1 =@ðz 1 = 1 Þ= 1 > r1 min > 0 and 1 ðtÞ ¼ ð 10  11 Þexpðl 1 tÞ þ 11 . The virtual controller  1 is designed as follows     ð r_ _  k 1  12 1  k 12 1 dt þ y_d þ 1 z 1 2r 1 1 1 ¼ (22) V where k 1 and k 12 are the positive parameters. By invoking equations (13) and (22), one has     ð r_

_ 1 ¼ r 1 Vz 2  k 1  12 1  k 12 1 dt 2r 1 Define a positive Lyapunov function ð 2 1 k 12 2



þ dt L1 ¼ 1 1 2 2 2V r 1 2V

(23)

(24)

Considering the following fact 1 2 k 11 2 2 V 2 ð0Þ

þ 2k 11 1 2

where k 11 > 0. Step 2. The differentiation of z 2 is obtained as follows z_ 2 ¼ x_ 2  _ 1 ¼ f 2 þ z 3 þ  2  _ 1

(26)

(28)

Using equations (13) and (28), the derivative of 2 ðtÞ is shown as follows ! ! _ 2 _ 2

_ 2 ðtÞ ¼ r 2 z_2  z 2 ¼ r2 f2 þ z 3 þ  2  _ 1  z 2 2 2 (29) where r 2 ¼ @R2 =@ðz 2 = 2 Þ= 2 > 0 and 2 ðtÞ ¼ ð 20  21 Þexpðl 2 tÞ þ 21 . To avoid the complex computation of _ 1 , an FOSD (30), based on lemma 2, is applied to approximate it &_ 11 ¼  11 j& 11  1 j0:5 signð& 11   1 Þ þ & 12 &_ 12 ¼  12 signð& 12  & 11 Þ

(30)

where & 11 and & 12 are the states of the system (30), and 11 and 12 are the positive design constants. According to equation (30), we have _ 1 ¼ &_ 11 þ 1

where V is the bound of V . According to equation (23) and lemma 2, the derivative of L1 is shown as follows ð 1 k 12 r_ 1 2 L_ 1 ¼ 2 1 _ 1 



1 dt 1 2 2 1 V 2V r 12 V r 1 (25) k 1 2 V ¼ 2 1 þ 2 1 z2 V V

V 1 z2 

Substituting equation (26) into equation (25) results in   1 1 V2 _ k 11 22 ð0Þ

21 þ L1   2 k1  2 2k 11 V  2 V  (27) 1 1 1 2 2   2 k1 

þ k11 2 ð0Þ 2k 11 1 2 V

(31)

where 1 denotes the estimate error. Obviously, we have that j 1 j   1 with  1 > 0. The controller r 2 is designed as follows   _ k2 1 r2 2 2 ^ 2 T2  2  d^2 tanh 2 ¼  2 þ &_ 11  2 ’  z2 2 r2 w2 2 (32) where k 2 and w 2 are the positive design parameters. d2 ¼ " 2  1 is bounded with its supremum satisfied with ^ 2 and d^2 denote the estimations of ’ 2 d2M   1 þ " 2M . ’ and d2M , respectively. The structure of adaptive control laws is expressed as follows  ^ 2Þ ^_ 2 ¼ 21 ðr 2 22 T2  2  221 ’ (33) ’ 2     r2 2 _  22 d^2 (34) d^2 ¼  22 r 2 2 tanh w2 By substituting equation (32) into equation (29), it can be rewritten as follows

6

International Journal of Advanced Robotic Systems 0

0 1 1 k 1 r

2 2 ^ 2 T2 2  d^2 tanh@ 2 A  1 A

_ 2 ¼ r 2 @f 2 þ z 3  2 þ &_ 11  2 ’ 2 r2 w2 0 0 11 k 1 r

2 2 ^ 2 T2  2 þ d2  d^2 tanh@ 2 AA ¼ r 2 @W2T  2 þ z 3  2 þ &_ 11  2 ’ 2 r2 w2

Choosing the candidate Lyapunov function 2 d~2

~2 ’ 1 L2 ¼ 22 þ 2 þ 2 2 21 2 22

(36)

(35)

^ 2 and d~2 ¼ d2M  d^2 . ~ 2 ¼ ’2  ’ where ’ Using equations (33) to (35), the derivative of L2 is shown as follows

1 1 ~ ^_ ~ ’ ^_ L_ 2 ¼ 2 _ 2  ’  21 2 2   d 2 d 2 22

0 1 1 r

2  r 2 2 W 2T  2 þ r 2 2 z 3  k 2 22  r 2 22 ’ 2 T2 2 þ r 2 2 d2  r 2 2 d 2M tanh@ 2 A 2 w2

(37)

~ 2’ ^ 2 þ 22 d~2 d^2 þ 21 ’ Next by considering the following facts 21 2 21 2 1 1 ~ 22  ’ ^ 22 Þ  ~ 22 Þ; r 2 2 W 2T 2  r 2 22 ’ 2 T2  2 þ r 2 ~ 2’ ^2 ¼ 21 ’ ð’2  ’ ð’2  ’ 2 2 2 2

we have

(38)

1 1 1 1 2 23 ð0Þr 22 k 22 2 2 2 2 2  d~2  d^2 Þ  22 d2M  22 d~2 ; r 2 2 z 3 

þ 22 d~2 d^2 ¼ 22 ðd 2M 2 2 2 2k 22 2 2

(39)

    r2 2 r2 2 r 2 2 d 2  r 2 2 d2M tanh  jr 2 2 jd2M  r 2 2 d2M tanh   0 d2M w2 w2 w2

(40)

0

1 1 1 1 1 2 2 A 2  1 21 ’ ~ 22  22 d~2 þ 21 ’22 þ 22 d2M L_ 2  @k 2  2 2k 22 2 2 2 2 1 2 ð0Þr 22 k 22 þ  0 d2M w 2 þ r 2 þ 3 2 2

Step 3. The differentiation of z 3 is obtained as follows z_3 ¼ x_ 3  _ 2 ¼ z 4 þ  3  _ 2

(42)

The derivative of 3 ðtÞ is shown as follows     _ _

_ 3 ðtÞ ¼ r 3 z_3  3 z 3 ¼ r 3 z 4 þ  3  _ 2  3 z 3 3 3 (43) where r 3 ¼ @R3 =@ðz 3 = 3 Þ= 3 > 0 and 3 ðtÞ ¼ ð 30  31 Þexpðl 3 tÞ þ 31 .

(41)

In order to estimate the derivative of  2 , an FOSD is applied the same with step 2. According to lemma 2, we have _ 2 ¼ &_ 21 þ 2

(44)

where 2 denotes the estimate error of FOSD with j 2 j   2 . Thus,  3 is shown as follows 3 ¼ 

k3 _ 3

3 þ &_ 21  z 3 r3 3

where k 3 > 0 is a control gain.

(45)

Wu et al.

7

By substituting equation (45) into equation (43), we have   k3 (46)

_ 3 ¼ r 3 z 4  3  2 r3

    r4 4 _  42 d^4 d^4 ¼  42 r 4 4 tanh w4

(55)

Choosing the following candidate Lyapunov function

where  41 > 0 and  42 > 0 are the positive design parameters. Thus equation (51) can be rewritten as follows

1 L3 ¼ 23 2

   r

k 1 ^ 4 T4 4 þ d 4  d^4 tanh 4 4

_ 4 ðtÞ ¼ r 4 W 4T 4  4 4  4 ’ r4 w4 2

(47)

(56)

Consider the following inequality z 4  2  jz 4 j þ  2  M 3 ; r 3 3 ðz 4  2 Þ 1 2 k 31 2 2 r M

þ  2k 31 3 2 3 3

Considering the candidate Lyapunov function (48)

where M 3 ¼ 4 ð0Þ þ  2 and k 31 > 0. By invoking equation (48), the time derivative of L3 is obtained as follows   k 1 L_ 3 ¼ 3 r 3 ðz 4  k3 3  2 Þ   k 3 

2 þ 31 r 32 M32 2 2k31 3 (49)

Step 4. The actual controller u will be established. The derivative of z 4 is shown as follows z_4 ¼ f 4 þ ud  _ 3 þ k 1 1

(50)

Using equation (13), the derivative of 4 ðtÞ is developed as follows 

_ 4 ðtÞ ¼ r4

_ z_4  4 z4 4



 ¼ r4

_ f4 þ ud  _ 3 þ k 1 1  4 z4 4



(51) where r 4 ¼ @R4 =@ðz 4 = 4 Þ= 4 > 0 and 4 ðtÞ ¼ ð 40  41 Þexpðl 4 tÞ þ 41 . As done previously, an FOSD is applied to estimate _ 3 . Considering lemma 2, we have _ 3 ¼ &_ 31 þ 3

(52)

where 3 denotes the estimation error with j 3 j   3 . According to MLP and PPC techniques, the controller u is designed as follows   r

k 1 ^ 4 T4  4  d^4 tanh 4 4 ud ¼  4 4 þ &_ 41  4 ’ 2 r4 w4 _ 4  z 4  k 1 1 4 (53) where k 4 and w 4 are the positive control constants. d4 ¼ "4  3 is the lump approximation error with ^ 4 and d^4 denote the estimations of ’ 4 and jd4 j  d4M . ’ ^ 4 and d^4 are updated as follows d4M , respectively. ’

 ^_ 4 ¼ 41 r 4 24 T4  4  241 ’ ^4 (54) ’ 2

2 ~2 ’ d~ 1 L4 ¼ 24 þ 4 þ 4 2 2 41 2 42

(57)

^ 4 and d~4 ¼ d4M  d^4 . ~ 4 ¼ ’4  ’ where ’ By invoking equations (54) to (56), results in the time derivative of L 4 are as follows 1 1 ~ _^ ~ 4’ ^_ 4  L_ 4 ¼ 4 _ 4  ’ d 4d 4  41  42 1  k 4 24 þ r 4 4 W4T  4  r 4 24 ’ 4 T4  4 þ r 4 4 d4   2 r

4 4 ~ 4’ ^ 4 þ 42 d~4 d^4  r 4 4 d^4 tanh þ  41 ’ w4 (58) Following facts can be obtained as follows   ~ 24  ’ ^ 24 Þ  41 ð’ 24  ’ ~ 24 Þ; ~ 4’ ^ 4 ¼ 41 ð’ 24  ’ 41 ’ 2 2 2 1 2 42 d~4 d^4  42 ðd4M  d~4 Þ (59) 2   r4 4 r 4 4 d4  r 4 4 d 4M tanh w4   r4 4  jr 4 4 jd4M  r 4 4 d 4M tanh  0 d4M w 4 w4 (60) 1 1 T 2 T r4 4 W4   r4 4 ’4 4 4 þ r4 (61) 2 2 Thus, equation (58) can be rewritten as follows 2 1 1 1 2 ~ 24   42 d~4 þ ð 42 d 4M L_ 4  k 4 24   41 ’ þ  41 ’ 24 þ r 4 Þ 2 2 2

(62) Theorem 1. Considering the altitude subsystem (7), if the adaptive neural PPC laws are chosen as equations (22), (32), (45), and (53), and updated laws as (33), (34), (54), (55) as well as the parameters satisfied (65). ~ i¼2;4 and d~i¼2;4 can Thus, the boundness of i¼1;2;3;4 , ’ be ensured. Proof. We select the candidate Lyapunov function shown as follows L ¼ L1 þ L2 þ L3 þ L4

(63)

8

International Journal of Advanced Robotic Systems

Substitute equations (27), (41), (49) and (62) into the derivative of equation (63), we have     1 1 1 1 2 _ ~ 22 L   2 k1 

1  k2 

22  21 ’  2k 2k 2 11 21 V   2 1 1 1 ~ ~ 24  22 d 2  k 3 

2  k 4 24   41 ’ 2 2k 31 3 2 2 1  42 d~4 þ C2 2 (64) 8 > > > > > > > > > > > > > > > > > >  1 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <  2 > > > > > > > > > > > > > > > > > > > > > > > > > >  3 > > > > > > > > > > > > > > > > > > > > > >  4 > > :



2 þ r2 þ 23 ð0Þr22 k 22 þ where C2 ¼ 12 k11 22 ð0Þ þ 21 ’22 þ 22 d2M

 2 þ 41 ’ 24 þ r 4 þ  0 d 2M w2 þ  0 d 4M w 4 k 31 r32 M 32 þ 42 d 4M

The corresponding design parameters should be chosen such that k 1  0:5=k 11 > 0; k2  0:5=k 21 > 0; k 3  0:5=k 31 > 0; k 4 > 0; ij > 0; i ¼ 2; 4; j ¼ 1; 2 (65) Define

9 8 > > > > > > > > > > > > > > > > > > > > > > > > v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > = < u C2 u 1 ¼ 1 jj 1 j  u0 > u k  0:5 > > > > u@ 1 > > A> > > u > > > > u k > > 11 > > u > > > > t > > > > 2 ; : V 9 8 > > > > > > > > > > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v > > = < u C 2 u0 1 ¼ 2 jj 2 j  u > > u > > > u@k 2  0:5A> > > > > t > > > > k 21 ; : 9 8 > > > > > > > > > > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v > > = < u C 2 u0 1 ¼ 3 jj 3 j  u > > u > > > u@k 3  0:5A> > > > > t > > > > k 31 ; :

8 9 8 sffiffiffiffiffiffiffiffi= < > > 2C 2 > > ’~ ¼ ’ ~ 2 jj~ ’2 j  > > 2 : ;  > 21 > > > > > > 8 9 > sffiffiffiffiffiffiffiffi= > > < > 2C 2 > > > ~ jj~ ’~ 4 ¼ ’ ’ j > > : 4 4 41 ; > < 8 9 > sffiffiffiffiffiffiffiffi= > < > > 2C 2 > > d~2 ¼ d~2 jjd~2 j  > > : >  22 ; > > > > > > 8 9 > > sffiffiffiffiffiffiffiffi= > < > > 2C 2 > ~ ~ > > : d~4 ¼ :d 4 jjd 4 j   42 ;

8 9 sffiffiffiffiffiffi= < C2 ¼ 4 jj 4 j  : k4 ;

L_ will be negative if 12 =  1 , 22 =  2 , 32 =  3 , ~ 22 ~ 42 =  4 , ’ = ’~ 2 , ’ = ’~ 4 , d~22 = d~2 and d~42 = d~4 . There 42 ~ fore, aforementioned errors

,’ and d~i¼2;4 are i¼1;2;3;4

i¼2;4

bounded. Remark 3. Assumption 2 imposes a controllability condition on systems (7) and (8), which is rational and equivalent to most controllability condition of HFV in the literature.17,21,47 Remark 4. By combining sliding mode differentiator, the MLP technique, a composite constrained adaptive

neural PPC schemer, is presented, and a new type of adaptive law is constructed simultaneously. The proposed controller is not only able to ensure the state tracking errors confined in the desired performance sets but also owns low-computation since there is only one parameter online to be adjusted for each NNs.

Velocity controller The velocity tracking error is defined as follows zV ¼ V  Vd  V

(66)

Wu et al.

9

where V is an assistant signal to compensate the saturation effect, and the additional auxiliary system is constructed as follows _ V ¼ kV þ 

(67)

where kV > 0 is a designed assistant parameter, and  ¼   d denotes the error between  (actual input) and d (designed input). The derivative of zV is described as follows z_V ¼ fV þ d  V_ d þ kV

(68)

According to equations (13) and (68), the time derivation of the transformed error V ðtÞ is shown as follows   _

_ V ðtÞ ¼ rV z_V  V z V   _ ¼ rV fV þ d  V_ d  V zV þ kV (69) V where

rV ¼

@RV 1 @ðzV = V Þ V

Considering the following candidate Lyapunov function 1 1 1 ~2 ~ V2 þ ’ d LV ¼ V2 þ 2 2V 1 2V 2 V

(74)

~ V ¼ ’V  ’ ^ V and d~V ¼ dVM  d^V . where ’ Based on equations (71) to (73), the derivative of LV is described as follows 1 1 ~ _^ ~ ’ ^_  L_V ¼ V _ V  ’ dV d V V 1 V V V 2 1 ¼ kV V2 þ rV V WVT V  rV V2 ’V TV V 2   rV V ~V ’ ^V þ rV dV V  dVM V tanh þ V 1 ’ wV þ V 2 d~V d^V (75) Note that the following inequalities hold

> rV min > 0

and

V ðtÞ ¼

ð V 0  V 1 ÞexpðlV tÞ þ V 1 . By employing the MLP technique, the controller  is designed as follows   kV 1 1 rV V ^ V TV V  d^V tanh

V  rV V ’ d ¼  2 rV wV _ V zV  kV þ V_ d þ V (70) where kV 1 > 0 and wV > 0 denote designed control gains. ^ V and d^V denote the estimation of ’V and dVM , respectively. ’ dV ¼ "V is lump approximation error with upper bound dVM . ^ and d^V Consider the following adaptive laws for ’ V

 ^V Þ ^_ V ¼ V 1 ðrV V2 TV V  2V 1 ’ ’ 2     rV V _ d^V ¼ V 2 rV V tanh  V 2 d^V wV

(71) (72)

where V 1 , V 2 , V 1 and V 2 denote the positive design parameters. Theorem 2. Considering velocity subsystem (8), if adaptive neural controller is chosen as equation (70) and updated laws as equations (71) and (72). Then, the boundness of ~ V and d~V in equation (74) is kept. signals V , ’ Invoking equations (69) and (70) yields  k 1 ^ V TV V þ dV

_ V ðtÞ ¼ rV  V V þ WVT V  V ’ 2 rV   r

d^V tanh V V wV (73)

1 1 rV V WVT V  rV V2 ’V TV V þ rV ; 2 2 2  2 V 2 (76) ðdVM  d~V Þ V 2 d~V d^V  2   rV V rV dV V  rV dVM V tanh wV   rV V  dVM jrV V j  dVM rV V tanh   0 wV dVM wV (77) V 1 2 V 1 2 2 2 2 ~V  ’ ^V Þ  ~V Þ ~V ’ ^V ¼ ð’V  ’ ð’V  ’ V 1 ’ 2 2 (78) By considering in equations (76), (77) and (78), equation (75) can be reformulated as follows 2   ~ 2  V 2 d~V þ rV  0 wV dVM L_V  kV V2  V 1 ’ 2 V 2 1 2 þ ðV 1 ’V2 þ V 2 dVM þ rV Þ 2 2   ~ V2  V 2 d~V þ CV 2 (79)  kV V2  V 1 ’ 2 2 2 where CV 2 ¼ rV  0 wV dVM þ 12 ðV 1 ’V2 þ V 2 dVM þ rV Þ. Define the following compact sets 8 9 8 sffiffiffiffiffiffiffiffi= < > C > V2 > >  V ¼ V jj V j  > > : k > V 1; > > > 8 9 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi= > < < CV 2 ~ jj~ (80) ’~ V ¼ ’ ’ j > : V V ð0:5V 1 Þ; > > > > 8 9 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi= > < > > C V2 > ~ ~ > > : d~V ¼ :d V jjd V j  ð0:5V 2 Þ;

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× 104

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Figure 1. Altitude and velocity tracking. 3

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Figure 2. System states and tracking errors.

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Wu et al.

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Figure 3. Flexible states. 30

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Figure 4. Control inputs.

0.02

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Figure 5. Assistant states.

~ V2 If V 2 =  V , ’ = ’~ V , d~V 2 = d~V , L_V is negative. There~ V and d~V in equation fore, the boundness of signals V , ’ (74) is ensured.

Simulations The model parameters of HFV are the same with the study by Parker et al. 2 The initial trim conditions are set as V0 ¼ 7700 ft= s, h0 ¼ 85000 ft,  0 ¼ 1:6325 , g 0 ¼ 0, q0 ¼ 0, 1 ¼ 0:97, _ 1 ¼ 0,  2 ¼ 0:7967 and _ 2 ¼ 0. The parameters of flexible states47 are set as ~ 1 ¼ 363:5003, ~ 2 ¼ 328:8029, !1 ¼ 16:0214, !2 ¼ 13:1082,  1 ¼ 0:05 and  1 ¼ 0:03. The performance functions are shown

as i ðtÞ ¼ ð i0  i1 Þexpðli tÞ þ i1 i ¼ 1; 2; 3; 4; V , where 10 ¼ 50, 11 ¼ 15, 20 ¼ 0:04, 21 ¼ 0:02, 30 ¼ 0:05, 31 ¼ 0:02, 40 ¼ 0:1, 41 ¼ 0:05, V 0 ¼ 3, V 1 ¼ 0:5 and li ¼ 0:08; moreover, Mi is set equal to one. The NN inputs are selected as X2 ¼ ½V ; g; , X4 ¼ ½V ; g; ; q and XV ¼ ½V , and the scope of the input variables is defined as V 2 ½7700 ft= s; 7900 ft= s, g 2 ½1 ; 1 ,  2 ½0; 3  and q 2 ½4 = s; 4 = s. The centres including 50 nodes are equally distributed in their scopes. The control parameters are chosen as k 1 ¼ 25:8, k 12 ¼ 0:6, k 2 ¼ 0:006, k 3 ¼ 0:01, k 4 ¼ 0:06, kV ¼ 6, k 1 ¼ 1 and kV ¼ 1. Gains for the adaptive laws are selected as 21 ¼ 40,  41 ¼ 10, V 1 ¼ 2, 21 ¼ 0:01, 41 ¼ 0:05, V 1 ¼ 0:25, i2 ¼ 1, i2 ¼ 0:1 and wi ¼ 1,

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Figure 6. Tracking errors and control inputs with AP in case 1.

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Figure 7. Tracking errors and control inputs with AP in case 2.

i ¼ 2; 4; V . Reference commands are smoothened via several second-order filters shown as follows hd Vd 0:01 0:01 ¼ ¼ ; hd0 s 2 þ 0:19s þ 0:01 Vd0 s 2 þ 0:19s þ 0:01 (81)

Simulation 1. In the simulation, the initial tracking errors are assumed to be z 1 ð0Þ ¼ 40 ft and zV ð0Þ ¼ 2 ft= s. For comparison purposes, the adaptive PPC scheme (named AP)42 is used. In this article, the control parameters of AP are chosen through a trial method to achieve nearly equal tracking performance with robust adaptive prescribed

Wu et al.

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Figure 8. Tracking errors and control inputs with RAP in cases 1, 2, and 3.

performance controller (RAP). The simulation results are shown in Figures 1 to 6. The superiority of RAP will be further revealed in the next simulation. The output tracking errors for altitude and velocity along with their performance function bounds are presented in Figure 1. Meanwhile, the system states (g, , q) and their accompanying tracking errors (z 2 , z 3 , z 4 ) are depicted in Figure 2. The control inputs (e , ) are shown in Figure 4. Sure enough, the output tracking performance is satisfactory, and all the state tracking errors are confined in their prescribed bounds, despite there exists some shake of (e , ) along with the first 20 s. In addition, the flexible states and auxiliary states are pictured in Figures 3 and 5, respectively. Obviously, the flexible states are both bounded for RAP and AP controller. Compared with AP, the RAP has short convergence time with its flexible states. In all, this simulation has proved the availability of RAP control scheme.

Simulation 2. To ulteriorly illustrate the robustness of the proposed RAP controller, the coefficient variation of the HFV model is taken into consideration in three different cases as shown below. For comparison purpose, the NNs are hold to be unvaried, while the control parameters, the performance functions and the initial values are kept as before. Thus the superiority of RAP is shown in Figures 6 to 8. First, we apply the AP scheme to control the HFV with coefficients variation like cases 1 and 2. As Figure 6 clearly demonstrates, the output performance of AP controller can be ensured with case 1. Unfortunately, if we increase the coefficient to case 2 without changing the control gains, the tracking errors of AP control approach overstep the prescribed performance bounds 1 and V ; meanwhile, the closed-loop system becomes unstable as shown in Figure 7. In contrast to AP control scheme, we apply the RAP control scheme to HFV affected by the same coefficient variation with case 1. As shown in Figure 8, the RAP control scheme

14 also guarantees the output error performance. If we increase the coefficient to case 2, the output tracking prescribed performance and stable closed-loop system behaviour of RAP control scheme are still achieved. It must be pointed out that the RAP control scheme operates successfully even though we further soar the coefficient to case 3. Therefore, compared to AP control scheme, the significant increase in robustness of the proposed RAP is achieved.  (  C 0 1 þ 0:2 sinð0:05tÞ ; if t  60s Case 1 : C ¼ C0 ; else   ( C 0 1 þ 0:3 sinð0:05tÞ ; if t  60s Case 2 : C ¼ C0 ; else  (  C0 1 þ 0:5 sinð0:1tÞ ; if t  60s Case 3 : C ¼ C0 ; else

Conclusion In this study, a guaranteed prescribed performance adaptive neural control scheme has been presented for flexible HFV. Using the MLP, FOSD technique and prescribed performance function, a low-computation state tracking error constrained adaptive neural controller is constructed, wherein the issue of increase of NN learning parameters and explosion of the items’ is removed. With the utilization of an assistant system, the problem of actuator saturation is also eliminated. Compared with other adaptive neural control designs, the proposed controller is not only able to ensure the state tracking errors confined in the desired performance sets but also owns low-computation and better robustness. Finally, two simulations have been performed to demonstrate the robustness of this control scheme. 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 partially supported by the Natural Science Foundation of China (Grant no,61573286, 61374032 ), Aeronautical Science Foundation of China (Grant no, 20140753012).

References 1. Xu B and Shi ZK. An overview on flight dynamics and control approaches for hypersonic vehicles. Sci China Inform Sci 2015; 58(7): 1–19. DOI: 10.1007/s11432-014-5273-7 2. Parker JT, Serrani A, Yurkovich S, et al. Control-oriented modeling of an air-breathing hypersonic vehicle. J Guid Contr Dynam 2007; 30(3): 856–869. DOI: 10.2514/1.27830

International Journal of Advanced Robotic Systems 3. Zong Q, Ji Y, Zeng F, et al. Output feedback back-stepping control for a generic hypersonic vehicle via small-gain theorem. Aero Sci Technol 2012; 23(1): 409–417. DOI: 10.1016/j. ast.2011.09.012 4. Fiorentini L, Serrani A, Bolender MA, et al. Nonlinear robust adaptive control of flexible air-breathing hypersonic vehicles. J Guid Contr Dynam 2009; 32(2): 401–416. DOI: 10.2514/1. 39210 5. Ataei A and Wang Q. Non-linear control of an uncertain hypersonic aircraft model using robust sum-of-squares method. IET Contr Theory Appl 2012; 6(2): 203–215. DOI: 10.1049/iet-cta.2011.0143 6. Wu HN, Feng S, Liu ZY, et al. Disturbance observer based robust mixed H2/H1 fuzzy tracking control for hypersonic vehicles. Fuzzy Set Syst 2017; 306: 118–136. DOI: 10.1016/ j.fss.2016.02.002 7. Zhang Y, Jiang Z, Yang H, et al. High-order extended state observer-enhanced control for a hypersonic flight vehicle with parameter uncertainty and external disturbance. Proc Inst Mech Eng G 2015; 229(13): 2481–2496. DOI: 10.1177/ 0954410015578480 8. Zhang Y, Li R, Xue T, et al. Exponential sliding mode tracking control via back-stepping approach for a hypersonic vehicle with mismatched uncertainty. J Franklin Inst 2016; 353(10): 2319–2343. DOI: 10.1016/j.jfranklin.2016.04.005 9. Zong Q, Dong Q, Wang F, et al. Super twisting sliding mode control for a flexible air-breathing hypersonic vehicle based on disturbance observer. Sci China Inform Sci 2015; 58(7): 1–15. DOI: 10.1007/s11432-015-5350-6. 10. Sun H, Li S and Sun C. Finite time integral sliding mode control of hypersonic vehicles. Nonlinear Dynam 2013; 73(1–2): 229–244. DOI: 10.1007/s11071-013-0780-4 11. Zhang Y, Li R, Xue T, et al. An analysis of the stability and chattering reduction of high-order sliding mode tracking control for a hypersonic vehicle. Inform Sci 2016; 348: 25–48. DOI: 10.1016/j.ins.2016.02.012 12. Wang J, Wu Y and Dong X. Recursive terminal sliding mode control for hypersonic flight vehicle with sliding mode disturbance observer. Nonlinear Dynam 2015; 81(3): 1489–1510. DOI: 10.1007/s11071-015-2083-4 13. Huang Y, Sun C, Qian C, et al. Linear parameter varying switching attitude control for a near space hypersonic vehicle with parametric uncertainties. Int J Syst Sci 2015; 46(16): 3019–3031. DOI: 10.1080/00207721.2014.886743 14. Rehman OU, Fidan B and Petersen IR. Uncertainty modeling and robust minimax LQR control of multivariable nonlinear systems with application to hypersonic flight. Asian J Contr 2012; 14(5): 1180–1193. DOI: 10.1002/asjc.399 15. Bu X, Wu X, Zhang R, et al. Tracking differentiator design for the robust back stepping control of a flexible air-breathing hypersonic vehicle. J Franklin Inst 2015; 352(4): 1739–1765. DOI: 10.1016/j.jfranklin.2015.01.014 16. Zong Q, Wang F, Tian B, et al. Robust adaptive approximate back stepping control design for a flexible air-breathing hypersonic vehicle. J Aero Eng 2015; 28(4): 04014107. DOI: 10.1061/(ASCE)AS.1943-5525.0000438

Wu et al. 17. Sun H, Li S, Yang J, et al. Non-linear disturbance observerbased back-stepping control for air breathing hypersonic vehicles with mismatched disturbances. IET Contr Theory Appl 2014; 8(17): 1852–1865. DOI: 10.1049/iet-cta.2013.0821 18. Tao X, Li N and Li S. Multiple model predictive control for large envelope flight of hypersonic vehicle systems. Inform Sci 2016; 328: 115–126. DOI: s10.1016/j.ins.2015.08.033 19. Xu B, Fan Y and Zhang S. Minimal-learning-parameter technique based adaptive neural control of hypersonic flight dynamics without back-stepping. Neurocomputing 2015; 164: 201–209. DOI: 10.1016/j.neucom.2015.02.069 20. Wu HN, Liu ZY and Guo L. Robust L1-gain fuzzy disturbance observer-based control design with adaptive bounding for a hypersonic vehicle. IEEE T Fuzzy Syst 2014; 22(6): 1401–1412. DOI: 10.1109/TFUZZ.2013.2292976 21. Xu B, Wang D, Sun F, et al. Direct neural control of hypersonic flight vehicles with prediction model in discrete time. Neurocomputing 2013; 115: 39–48. DOI: 10.1016/j.neucom. 2012.12.028 22. Hu X, Wu L, Hu C, et al. Fuzzy guaranteed cost tracking control for a flexible air-breathing hypersonic vehicle. IET Contr Theory Appl 2012; 6(9): 1238–1249. DOI: 10.1049/iet-cta.2011.0065 23. Xu B, Yang C and Pan Y. Global neural dynamic surface tracking control of strict-feedback systems with application to hypersonic flight vehicle. IEEE Trans Neural Netw Learn Syst 2015; 26(10): 2563–2575. DOI: 10.1109/TNNLS.2015. 2456972 24. Tong S, Wang T, Li Y, et al. A combined back stepping and stochastic small-gain approach to robust adaptive fuzzy output feedback control. IEEE T Fuzzy Syst 2013; 21(2): 314–327. DOI: 10.1109/TFUZZ.2012.2213260 25. Gao S, Ning B and Dong H. Fuzzy dynamic surface control for uncertain nonlinear systems under input saturation via truncated adaptation approach. Fuzzy Sets Syst 2016; 290: 100–117. DOI: 10.1016/j.fss.2015.02.013 26. Chen M, Tao G and Jiang B. Dynamic surface control using neural networks for a class of uncertain nonlinear systems with input saturation. IEEE Trans Neural Netw Learn Syst 2015; 26(9): 2086–2097. DOI: 10.1109/TNNLS.2014. 2360933 27. Xu B, Shi Z, Yang C, et al. Composite neural dynamic surface control of a class of uncertain nonlinear systems in strictfeedback form. IEEE Trans Cybernet 2014; 44(12): 2626–2634. DOI: 10.1109/TCYB.2014.2311824 28. Butt WA, Yan L and Kendrick AS. Adaptive dynamic surface control of a hypersonic flight vehicle with improved tracking. Asian J Contr 2013; 15(2): 594–605. DOI: 10.1002/asjc.450 29. Xu B, Wang D, Sun F, et al. Direct neural discrete control of hypersonic flight vehicle. Nonlinear Dynam 2012; 70(1): 269–278. DOI: 10.1007/s11071-012-0451-x 30. Xu B, Zhang Q and Pan Y. Neural network based dynamic surface control of hypersonic flight dynamics using small-gain theorem. Neurocomputing 2016; 173: 690–699. DOI: 10.1016/j.neucom.2015.08.017 31. Xu B, Huang X, Wang D, et al. Dynamic surface control of constrained hypersonic flight models with parameter

15

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

estimation and actuator compensation. Asian J Contr 2014; 16(1): 162–174. DOI: 10.1002/asjc.679 Zong Q, Wang F, Tian B, et al. Robust adaptive dynamic surface control design for a flexible air-breathing hypersonic vehicle with input constraints and uncertainty. Nonlinear Dynam 2014; 78(1): 289–315. DOI: 10.1007/s11071-0141440-z Bu X, Wu X, Tian M, et al. High-order tracking differentiator based adaptive neural control of a flexible air-breathing hypersonic vehicle subject to actuators constraints. ISA Trans 2015; 58: 237–247. DOI: 10.1016/j.isatra.2015.06.004 Bu X, Wu X, Wei D, et al. Neural-approximation-based robust adaptive control of flexible air-breathing hypersonic vehicles with parametric uncertainties and control input constraints. Inform Sci 2016; 346–347: 29–43. DOI: 10.1016/j. ins.2016.01.093 Wang F, Zou Q, Hua C, et al. Disturbance observer-based dynamic surface control design for a hypersonic vehicle with input constraints and uncertainty. Proc Inst Mech Eng I 2016; 230(6): 522–536. DOI: 10.1177/0959651816633390 Su X and Jia Y. Constrained adaptive tracking and command shaped vibration control of flexible hypersonic vehicles. IET Contr Theory Appl 2015; 9(12): 1857–1868. DOI: 10.1049/ iet-cta.2014.0750 Xu B, Wang S, Gao D, et al. Command filter based robust nonlinear control of hypersonic aircraft with magnitude constraints on states and actuators. J Intell Robotic Syst 2014; 73(1–4): 233–247. DOI: 10.1007/ s10846-013-9941-4 Chen M, Liu X and Wang H. Adaptive robust fault-tolerant control for nonlinear systems with prescribed performance. Nonlinear Dynam 2015; 81(4): 1727–1739. DOI: 10.1007/ s11071-015-2102-5 Yang Q and Chen M. Adaptive neural prescribed performance tracking control for near space vehicles with input nonlinearity. Neurocomputing 2016; 174: 780–789. DOI: 10.1016/j.neucom.2015.09.099 Chen M, Wu QX, Jiang CS, et al. Guaranteed transient performance based control with input saturation for near space vehicles. Sci China Inform Sci 2014; 57(5): 1–12. DOI: 10. 1007/s11432-013-4883-9 Bechlioulis CP and Rovithakis GA. Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems. Automatica 2009; 45(2): 532–538. DOI: 10.1016/j.automatica.2008.08.012 Bu X, Wu X, Zhu F, et al. Novel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errors. ISA Trans 2015; 59: 149–159. DOI: 10.1016/j.isatra.2015.09.007 Yang YN, Ge C, Wang H, et al. Adaptive neural network based prescribed performance control for teleoperation system under input saturation. J Franklin Inst 2015; 352(5): 1850–1866. DOI: 10.1016/j.jfranklin.2015.01.032 Bu X, Wu X, Huang J, et al. A guaranteed transient performance-based adaptive neural control scheme with low-complexity computation for flexible air-breathing

16 hypersonic vehicles. Nonlinear Dynam 2016; 84: 2175–2194. DOI: 10.1007/s11071-016-2637-0 45. Bu X, Wu X, Huang J, et al. Robust estimation-free prescribed performance back-stepping control of air-breathing hypersonic vehicles without affine models. Int J Contr 2016; 89: 1–16. DOI: 10.1080/00207179.2016.1151080 46. Bechlioulis CP and Rovithakis GA. A priori guaranteed evolution within the neural network approximation set and robustness expansion via prescribed performance control. IEEE Trans Neural Netw Learn Syst 2012; 23(4): 669–675. DOI: 10.1109/TNNLS.2012.2186152 47. Xu B. Robust adaptive neural control of flexible hypersonic flight vehicle with dead-zone input nonlinearity. Nonlinear

International Journal of Advanced Robotic Systems Dynam 2015; 80(3): 1509–1520. DOI: 10.1007/s11071-0151958-8 48. Gao DX and Sun ZQ. Fuzzy tracking control design for hypersonic vehicles via T-S model. Sci China Inform Sci 2011; 54(3): 521–528. DOI: 10.1007/s11432-011-4190-2 49. Yang Y, Ge C, Wang H, et al. Adaptive neural network based prescribed performance control for teleoperation system under input saturation. Journal of the Franklin Institute 2015; 352: 1850–1866. 50. Chen M, Shi P and Lim CC. Robust constrained control for MIMO nonlinear systems based on disturbance observer. IEEE T Automat Contr 2015; 60(12): 3281–3286. DOI: 10. 1109/TAC.2015.2450891