Adaptive fuzzy finite-time control with prescribed performance for waverider vehicles

This paper proposes an adaptive fuzzy control method with prescribed performance for Waverider Vehicles (WVs), being able to guarantee finite-time convergence and small overshoot for tracking errors. Firstly, we design a new type of performance function that is independent of the initial error, and possess finite-time convergence and small overshoot. Then, we transform the inequality constraints on tracking errors into an unconstrained equation by introducing a transformed error. On this basis, we design a prescribed performance control (PPC) approach to limit the tracking errors within prescribed funnels utilizing the transformed error and fuzzy approximation, which ensures that satisfactory transient performance and steady-state accuracy can be guaranteed for tracking errors. Compared with the existing PPC, the improvement is to assure finite-time convergence of tracking errors with almost zero overshoot. Finally, compared simulations are given to verify the advantage.


Introduction
Waverider Vehicles (WVs) have been the primary development direction for countries around the world because of their advantages of cost-effectiveness such as fast response time, high mobility, long range, and strong penetration ability. [1][2][3][4][5][6] The control system is the core of WVs, enabling they to execute and complete flight missions safely and efficiently. Many scholars and scientific research institutions have developed researches in this field. The motion model established for WVs has a lot of nonlinearity and uncertainty because the configuration of WVs is particularly complicated, and there are many unknown factors in the flight environment. Simultaneously, WVs' high velocity flight also puts forward extremely requirements on the transient performance of the control system. Therefore, the robustness and transient performance of the control system have also been the focus of attention. Sun et al. 7 proposed a control method with prescribed performance for WVs by designing a new prescribed function, while the dependence of the control law on the initial error value was eliminated. Further, the new prescribed function proposed in Bu and Qi 8 was extended to the control problem with unknown direction, and the Nussbaum function was used to estimate the control gain whose sign is unknown. This ensures the satisfactory transient performance and steady-state accuracy of the velocity tracking error and altitude tracking error of the WVs. In order to achieve faster error convergence, some scholars have carried out research on PPC with finitetime convergence. The main idea is to design a piecewise function that meets prescribed performance conditions to achieve finite-time convergence. However, there is still the problem of uncontrollable overshoot. For this reason, the small overshoot PPC was studied in Xiang and Liu, 9 and the convergence of small overshoot or even zero overshoot of the WV tracking error was realized. Besides, to ensure the real-time performance of the control system, scholars mainly carry out work from two aspects: (1) reduce the structural complexity, (2) reduce the amount of online learning. Zheng et al. 10 designed an inversion control law based on finite-time convergent differentiators, which estimated the derivatives of the virtual control inputs. Zheng and Xie 11 exploited the Multi-Layer Perception method, and as a result, only one parameter was needed to be adaptively regulated, which reduces the learning amount of the neural network and ensures the good real-time performance of the controller.
Though the above methods are able to ensure the robustness and transient performance indicators for the control system, there still exist some challenging problems such as relying on the initial value of the tracking error, longer convergence time, and large overshoot. To overcome those shortcomings, this paper exploits a new PPC method for WVs to guarantee the tracking errors with satisfactory prescribed performance. The main contributions are summarized as: (1) Different from the existing studies, the proposed new performance functions are able to guarantee the velocity tracking error and the altitude tracking error with finite-time prescribed performance with almost zero overshoot. (2) A low-complexity control structure is obtained since the complex design procedure of back-stepping is avoided. Moreover, the computational burden is reduced by introducing an improved fuzzy approximation with less online learning parameters.
In (1)2(7), the parameter fitting forms of aerodynamic force and moment are T, D, L, M, N 1 , and N 2 .
The above motion model contains five rigid body states (V, h, g, u, and Q), two flexible states (h 1 and h 2 ) and two control inputs (F and d e ). F and d e are implicit in aerodynamic force and moment (T, D, L, M, N 1 , and N 2 ). Where the flexible states h 1 and h 2 are unpredictable, and there is no corresponding actuator to control the flexible states in actual engineering. Therefore, h 1 and h 2 are regarded as unknown disturbances in the control law design process. Moreover h 1 and h 2 are bounded under the condition that the rigid body states and the control inputs are bounded. 13 WVs require the controller to realize the robust tracking of the velocity V and the altitude h to the respective reference inputs V ref and h ref by adjusting the prescribed performance controllers F and d e based on fuzzy approximation, and moreover the tracking error of each subsystem has satisfactory transient performance and steady-state accuracy.
Remark 1. We consider that the flexible states h 1 and h 2 cannot be measured, and only the rigid body states V, h, g, u, and Q are used for the control design.

Remark 2.
Under normal circumstances, the parameters range of the WVs flight envelope are shown in Table 1.

Preliminaries
Definition 1. If the function h(q) has the following attributes, it will be regarded as a Nussbaum-Type function 14 Lemma 1. Define h(q) and Y 0 (t)50 in ½0, t f ). If the following inequality holds 15 where S 0 is a suitable constant and l 0 6 ¼ 0, then Y 0 (t), h(q) and Lemma 3. Assume that f(x 0 , y 0 ) : R n 3R ! R is differentiable at every point of open set R n 3(a, b), and that it is continuous at the end points of y 0 = a and y 0 = b. 17 Then there must be a point y Ã 0 2 (a, b) such that Lemma 4. For 8m 1 , m 2 2 R, we obtain 18 with e 0 . 0, p 0 . 1, q 0 . 1, (p 0 À 1)(q 0 À 1) = 1. It is proved that the fuzzy system can be used to approximate the continuous function b(y) on the compact set O y with y = ½y 1 , y 2 , :::, y n T . Use single fuzzification, center average defuzzification and product reasoning, and adopt the following fuzzy rules IF y 1 is C i 1 and y 2 is C i 2 and .and y n is C i n , THEN Y is H i (i = 1, 2, :::, N). Then, the output of the fuzzy system is where C i i and H i are fuzzy sets, the fuzzy membership function -C j i (Á) is chosen as a Gaussian function, f j is the point that lets Then we have Lemma 5. For the continuous function b(y) defined on the compact set O y , there exists a constant e M . 0 and a fuzzy logic system, so that 19 New prescribed performance function design We define the following improved prescribed performance The new performance functions L l (t) and L r (t) in equation (17) are designed as where with f 0 , f T , r, T 2 R + , 04d l 41 and 04d r 41.
The function f(t) satisfies (1) f(t) is a positive monotonic decrement function; ( The defined prescribed performance (17) is shown in Figure 1. By the equation (17), the equation (18), and the introduction of symbol sign(e(0)), L l (t), and L r (t) can be adaptively changed according to different symbols of e(0). Further, by selecting appropriate design parameters for L l (t) and L r (t), it is possible to guarantee the error e(t) with small overshoot and even zero overshoot convergence, as shown in Figure 1.
We define the transformed error as with q(t) = e(t)ÀL l (t) L r (t)ÀL l (t) . Then, we get the following theorem: 20 Define the steady state error e ss : Define the prescribed performance as: where e(t) is a tracking error.
When k is small enough, f(t) ! + ' and Àf(t) ! À' can be known from the property (3) of f(t). Then for uncertain but bounded e(0), there is Therefore, e(0) must be within the prescribed range defined by equation (19), which can avoid the same problems as the control singularity caused by improper initial value setting of the traditional performance function. Figure 1 shows the prescribed performance defined by equation (17). L l (T) and L r (T) represent the range of the steady-state value of e(t), that is L l (T) \ e(') \ L r (T), which can ensure that e(t) has an ideal steady-state accuracy. By selecting appropriate L l (T) and L r (T), it is also ensured that e ss has an ideal range, that is L l (T) \ e ss \ L r (T).The maximum overshoot allowed by e(t) is limited by L l (0) and L r (0).The convergence speed of L l (t) and L r (t) is directly affected by r. The smaller r is, the faster the falling speed of L l (t) and L r (t) will be. Remark 3. The control law will be designed based on the equation (20). As long as h(t) is bounded, e(t) can be limited to the prescribed performance defined by equation (10). By selecting appropriate design parameters for L l (t) and L r (t), it can be ensured that e(t) has satisfactory transient performance and steady-state accuracy.

Controller design and stability analysis
Velocity control law design Define velocity tracking error as Using (8), we get _Ṽ A transformed error e 1 (t) is defined as where ; with During the flight of WVs, by considering the problems of parameter perturbation, input limitation and external disturbance, an adaptive control law is introduced to ensure the robustness of the system. The velocity controller F is selected as where k V1 . 0, k V2 . 0 are design parameters;û 1 denotes the estimation of u 1 with with l 1 . 0.
Theorem 2. Consider the closed-loop system consisting of plant (8) with controller (28) and adaptive law (29). Then all the signals involved are semi-globally uniformly ultimately bounded.

Altitude control law design and stability analysis
The control goal for the altitude subsystem (equations (2)-(5)) is to design a prescribed performance control law d e based on neural approximation such that the altitude h tracks its reference input h ref . Moreover, the tracking error is limited to a prescribed area to ensure satisfactory transient performance and steady-state accuracy.
Define the altitude error as Select the track angle reference input as with k g 2 R + .
Define the transformed error h h (t) as where 1). Take the track angle reference input as with k g 2 R + , and Thus h h (t) must be bounded. Furthermore, the control task becomes g ! g d . Next, we will design a low-calculation fuzzy control law, so that g ! g d . Define Then the rest of the WV altitude subsystem (equations (3)-(5)) can be expressed as the following non-affine form where x = ½x 1 , x 2 , x 3 T , and f 1 (x 1 , x 2 ) and f 3 (x, d e ) are continuously differentiable unknown functions. We give the following reasonable assumption.
Assumption 1. For any (x, d e ) 2 O x 3R, the following inequalities hold 21 where O x is a controllable domain.
Remark 5. According to the literature 21 and the value range of the rigid body state of the WV flight envelope (see Table 1), it can be seen that Assumption 1 holds.
In order to avoid the complicated design process of backstepping control, the model (40) is equivalently transformed as follows.
Step 2: Definition z 3 = _ z 2 = f h1 (x). According to equation (40), the first derivative of z 3 with respect to time is After the above model transformation, equation (40) becomes the following non-affine pure feedback model where f h2 (x, d e ) is a continuously differentiable unknown function.
Remark 6. Assumption 1 imposes the global controllability condition on (44), which is also the condition of satisfying Lemma 2. Different from other work, 22 Remark 8. Compared with equation (40), equation (44) is not only simple in form, but also contains only an unknown function. Based on equation (44) to design the control law, the cumbersome backstepping design process is no longer needed.
Define track angle tracking error e 0 and error function E with m 2 R + . Since (s + m) 3 is a Hurwitz polynomial, when E is bounded, e 0 must be bounded.
The first third derivative of e 0 with respect to time is where m 2 R + is the parameter to be designed, F h (x, d e ) = f h2 (x, d e ) À k h d e is a continuously differentiable unknown function. The first derivative of E with respect to time is According to Lemma 2 and Assumption 1, we know that d Ã e satisfies F h (x, d Ã e ) À g (3) d + 3h€ e + 3h 2 _ e + h 3 e = 0. Then (48) becomes According to Lemma 3, we get According to (49) and (50), we have For the unknown term d Ã e , the fuzzy system is applied to approximate it.
where X 2 = ½g, u, Q T 2 R 3 is the input vector, and e 2M . 0 is the approximate error.
Define u 2 = f 2 2 and select the following control where k h is the design parameter,û 2 is the estimated value of u 2 .The adaptive law ofû 2 is selected as with k 1 . 0.
Remark 9. Theorem 3 shows that the proposed altitude controller allows g ! g d to ensure the boundedness of e h (t). According to Theorem 1, we further obtain that the expected performance ofh can be guaranteed.
Remark 10. Note that there is only one fuzzy system and one learning parameterû 2 . Therefore, the computational load of the proposed controller is lower than the existing study. 23

Simulation analysis
In this section, a numerical simulation is given to test the effectiveness of the control. The membership function of the fuzzy system is selected as Take the performance function as L l1 (t) = ½sign(Ṽ(0)) À 0:5f 1 (t) À 0:09sign(Ṽ(0)) L r1 (t) = ½sign(Ṽ(0)) + 0:5f 1 (t) À 0:09sign(Ṽ(0)) L l2 (t) = ½sign(h(0)) À 0:5f 2 (t) À 0:03sign(h(0)) L r2 (t) = ½sign(h(0)) + 0:5f 2 (t) À 0:03sign(h(0)) f 1 (t) = The design parameters are taken as: k V = 0:9, k V1 = 0:3, k V2 = 0:8, k g = 2, k h = 0:9, k h1 = 50, m = 7, r = 7, k 1 = 0:01. The velocity and altitude reference inputs are both given by the second-order reference model shown in Figure 2. In order to test robustness, we assume that all aerodynamic coefficients are uncertain. Define The simulation results of scenario 1 are presented in Figures 3 to 17. It can be seen from Figures 3 to 6 that when the model parameters are perturbed, the PPC method in this paper can ensure that the velocity tracking error and altitude tracking error have better transient performance and better steady-state accuracy, and also achieve the velocity tracking error and the altitude tracking error with small overshoot finite-time convergence. Although the velocity tracking error fluctuates greatly in the first 5 s, it does not exceed the designed envelope, so the phenomenon of control failure will not occur. In addition, the prescribed performance function designed in this paper can adjust the convergence time. If the convergence time is increased, the fluctuation will not occur. The situation will improve. Figures 7 to 13 show that the attitude angle, flexible states and control inputs of the two control methods are relatively smooth, and there is no high-frequency chattering phenomenon. Figures 14 and 15 show the learning effects ofû 1 andû 2 . Figures 16 and 17 show that h V (t) and h h (t) are bounded. Besides, the simulation results of scenario 2, depicted in Figures 8 to 32 also proves the advantage of the proposed PPC in comparison with

668
Measurement and Control 55(7-8)            NBC. From the simulation results, the finite-time prescribed performance control method proposed in this paper has obvious improvements in terms of overshoot and convergence time. 21,22 Conclusions This paper studies the new non-affine PPC method for WVs. By designing a new type of performance function, the control law gets rid of the dependence on the accurate initial value of the tracking error, and it can ensure that all tracking errors have good transient performance and reach a steady state within a limited time. Fuzzy systems are used to approximate the unknown parameters in the altitude control subsystem. The stability of closed-loop control system is proved via Lyapunov method. Finally, the given simulation     results show that the proposed method can ensure the velocity and altitude tracking errors with small overshoot and finite-time convergence.

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 was supported by Young Talent Support Project for Science and Technology (Grant No. 18-JCJQ-QT-007).

Data availability
The experimental data used to support the findings of this study are available from the corresponding author upon request.