Non-fragile tracking control of constrained Waverider Vehicles with readjusting prescribed performance

The purpose of this article is to solve the fragile problem of prescribed performance control (PPC) with application to Waverider Vehicles (WVs). By fragile problem, it denotes that tracking errors may break away from prescribed funnels if actuators are saturated. Firstly, we propose a new type of readjusting prescribed performance which adaptively readjusts its prescribed funnels such that the spurred transient performance and steady-state performance are still guaranteed for tracking errors in the presence of actuator saturation. Then, we extend the finite-time PPC into the non-affine model of WVs, ensuring that tracking errors always evolve within prescribed funnels. Furthermore, the improved back-stepping procedure without the derivation of virtual control law is utilized to develop non-fragile tracking controllers, and specially, there is no need of any fuzzy/neural approximation, avoiding the computational burden problem. Finally, compared simulation results are presented to verify the effectiveness and advantage.


Introduction
Waverider Vehicles (WVs) have been regarded as the most promising technology for feasible and affordable hypersonic flight. The flight control design, which is the key technique to ensure safe flight and successful completion of flight missions, has caused worldwide concerns [1][2][3][4][5]. The special dynamics of WVs, including uncertainty, nonlinearity, non-affine property, saturation and coupling, leads to control challenges. The current focus of flight control for WVs is mainly on pursing excellent robustness and accuracy under certain conditions such as actuator saturation/fault and uncertainties/disturbances via fuzzy/neural-based intelligent control [6][7][8], sliding mode control [9], back-stepping control [10][11][12], and fault-tolerant control [13,14]. However, all of them are unable to guarantee control system with satisfactory transient performance which is crucial to realize hypersonic maneuvering flight for WVs [15,16].
The prescribed performance control (PPC), firstly proposed by Bechlioulis et al. [17,18], is a pioneering methodology which is expected to provide an effective tool to ensure desired transient and steady-state performance. The main ideal of PPC is to devise a type of prescribed funnels called performance func-tions whose shapes can be designed as needed [19][20][21][22]. The developed controllers based on PPC are capable of guaranteeing that tracking errors always evolve within prescribed funnels such that both the transient performance and the stead-state performance are satisfactory. In the existing studies [23,24], prescribed performance guaranteed tracking controllers were exploited for WVs. Though tracking errors satisfy the desired prescribed performance, those methods [23,24] require that the models of WVs must have affine formulations, which cannot be directly obtained owing to the nonlinear characteristics. It is for this reason that non-affinemodel-based PPC schemes were investigated for WVs in [7,15,16]. Furthermore, to reject system uncertainties and disturbances, fuzzy/neural approximations are commonly used methods [7,15,16,23,24] to enhance the robustness performance. However, too much online learning/computation burden caused by fuzzy/neural approximations undoubtedly harms the real-time performance of control system, which is not conducive to realize hypersonic maneuvering flight for WVs. Besides, to improve the operability and practicability, some researchers proposed improved versions of PPC with finite-time convergence [16,22] such that the regulation time can be quantitatively designed as needed.
Despite the above developments of PPC, a facing defect is the fragile problem. By fragile problem, we mean that tracking errors may break away from prescribed funnels if actuators are saturated. The flight airspace of WVs is so high that it easily leads to actuator saturation. The actuator saturation will inevitably result in error increasing such that tracking errors show the trend of breaking away from prescribed funnels, yielding the fragile problem. Unfortunately, the existing PPC strategies [3,8,[16][17][18][19][20][21][22][23][24] did not take into consideration actuator saturation since all of them couldn't deal with the fragile problem. In view of this, we propose a novel non-fragile tracking controller with readjusting prescribed performance for WVs subject to actuator saturation. The addressed controllers are developed based on non-affine models of WVs. Moreover, there is no need of fuzzy/neural approximation, reducing online learning computation burden. The main contributions are listed as follows.
(1) The addressed method avoids the limitations of previous finite-time PPC (FPPC) approaches [23,24] that they are only suitable for affine dynamic systems. In this article, FPPC is extended into the non-affine model of WVs. (2) The developed controller doesn't require neural/fuzzy approximation [7,15,16,23] that may cause lots of online learning parameters. As a result, the computation burden of the design is satisfactory. (3) The proposed approach handles the fragile problem of PPC that the existing PPC strategies [8,[16][17][18][19][20][21][22][23][24] cannot deal with actuator saturation. In this article, by proposing a new kind of readjusting prescribed performance, both desired transient performance and steady-state performance are guaranteed for tracking errors in finite time under actuator saturation.
The rest of this paper is structured as follows. Section 2 presents problem statement. Section 3 shows the control design process. Section 4 provides the simulation results, and Sect. 5 gives the conclusions.

Vehicle model
We consider the following nonlinear model of an WV [25] whose map of geometry and force is shown in Fig. 1.
The vehicle model has five rigid-body states, i.e., velocity (V ∈ >0 ), altitude (h ∈ >0 ), flight-path angle(γ ∈ ), pitch angle (θ ∈ ) and pitch rate (Q ∈ ), and two flexible states (η 1 ∈ and η 2 ∈ ). The control inputs (fuel equivalence ratio ∈ >0 and elevator angular deflection δ e ∈ ) are implied in trust force T , drag force D, lift force L, pitching moment M, and generalized forces N 1 and N 2 [25], given by All the coefficients and variables in the above equations are clearly defined in [25] and also are listed in the Nomenclature (See the Appendix).

The Fragility of prescribed performance
By prescribed performance we mean that the tracking error e(t) : ≥0 → evolves within prescribed funnels ρ L (t) : ≥0 → <0 and ρ R (t) : ≥0 → >0 that are also called performance functions, that is, Fig. 2a. The overshoot of e(t) is no more than max {−ρ L (0), ρ R (0)} and the steady-state value of e(t) is less than max {−ρ L (∞), ρ R (∞)}. Thus, prescribed behaviours including the transient performance (i.e., overshoot and convergence time) and the steady-state performance (i.e., steady-state error) can be guaranteed for e(t).
It is impossible to devise feed-back controllers by directly using the constraint formulation ρ L (t) < e(t) < ρ R (t). In order to facilitate controller design, we introduce the following equivalent form with S e ε (ε e (t)) = e εe (t) 1+e εe (t) : → (0, 1). The transformed error ε e (t) ∈ is derived from (8) as The transformed error ε e (t) ∈ , instead of the initial tracking error e(t), will be utilized for control feedback. Hereon, we stress the fragile problem of the existing PPC methodologies when there exists actuator saturation. When actuator is saturated, the tracking error e(t) will inevitably increase [26], making e(t) close to (even exceed) the prescribed funnels ρ L (t) and ρ R (t), as shown in Fig. 2b. From (9), we know that ε e (t) → −∞ as e(t) → ρ L (t) and ε e (t) → +∞ as e(t) → ρ R (t). The non-convergence of ε e (t) will cause control singularity problem.

Readjusting prescribed performance
To handle the singularity problem, we propose an alternative PPC approach that is able to adaptively readjust its prescribed funnel, namely the Readjusting Prescribed Performance, given by with p e and κ e 2 ∈ >0 , where u d ∈ is the ideal value of the control input u ∈ , and abs (u − u d ) denotes the saturation of actuator u.
Remark 2 It is clearly concluded that p e lower (t) and and p e lower (t) = −p e upper (t).

Control objective
Due to the nonlinearity of vehicle model, (1)-(5) can be rewritten as the following non-affine formulation [3,5].
δ e δ e : 5 → are unknown but continuous functions. The control inputs and δ e are constrained by Control Objective: Develop control laws for and δ e under constraints (14) and (15)   Assumption 1 [16]. The reference commands and their time derivatives are bounded.

Velocity controller
In this subsection, we develop a constrained controller The velocity tracking error e V should satisfy the following readjusting prescribed performance with performance functions p e V lower (t) = −p e V upper (t) and where are design parameters, and T e V FT ∈ >0 means the convergence time.
is the readjusting term that correspondingly adjusts the funnels of p e V lower (t) and p e V upper (t) to avoid the singularity problem caused by actuator saturation. κ e V ,1 ∈ >0 and κ e V ,2 ∈ >0 are two design parameters. (16) is equivalently expressed as with To cope with the actuator saturation, a compensated system is devised aṡ The system state of (2) .
The velocity controller is chosen as with k d V ∈ >0 .
Substituting (23) into (22) and invoking (18), it leads tȯ Due to the boundedness of arguments, the continuous function All V also is bounded, denoted by abs Taking time derivative along (25) and employing (24), we geṫ If abs z e V V >¯ All V /k d V , we haveẆ e V V < 0 and further obtain abs z e V V ≤ min abs z e V V (0) ,¯ Thereby, the boundedness of d is guaranteed, and − d also is bounded. The Lyapunov function is chosen as W s V = 1 2 s 2 V . Then, by utilizing (20), we knoẇ When t) − s V , we conclude that the transformed error ε e V V (t) also is bounded. We choose the upper bound of ε e V V (t) asε e V V ∈ >0 . On the basis of Theorem 1 presented in [16], it is derived . Thereby, the spurred prescribed performance can be guaranteed.

Altitude controller
In this subsection, we will exploit a constrained con- where p e,i 0 (∈ >0 ) > p e,i Invoking (13) and (30),ε e,i h , i = h, γ, θ, Q are given byε with k i h ∈ >0 , i = h, γ, θ. Substituting (33) into (32) and using (30), we obtaiṅ It is difficult to guarantee the stability of altitude subsystem under the saturation condition. Thus, we develop the following compensated system.
where k s h ∈ >0 is a design parameter, and the state s h ∈ is used to modify ε The altitude controller is developed as   (33) and (40), developed utilizing back-stepping, don't contain the time derivatives of virtual control laws, which eliminates the problem of "explosion of terms". Moreover, there is no need of any fuzzy/neural approximation for (33) and (40). Hence, the problem of computational complexity is avoided.
In the simulation, we can select suitable constants for ι V , ι h γ , ι h θ , ι h Q and ι h δ e . p e V 0 ∈ >0 denotes the initial value of the performance function, and its value can be chosen as a constant that satisfies p e V 0 > abs (e V (0)

Simulation results
The purpose of this section is to validate the effect of the proposed method via compared simulation. The       . The proposed readjusting prescribed performance controller is compared with the traditional PPC [17][18][19]. Moreover, to test the robust performance, we consider parameter uncertainty up to 45  [17][18][19] exhibits fragility to the actuator saturation. If the actuator is saturated (See Fig. 3), the tracking error e V increases gradually so that e V reaches the traditional prescribed funnel [17][18][19] (See Fig. 4). As a result, the transformed errors tend to diverge (See Fig. 5). Thereby, as to the traditional PPC [17][18][19], the actuator saturation will result in the singular problem. As a contrast, the proposed method can effectively guarantee tracking errors with desired prescribed performance in the presence of actuator saturation, as shown in Figs. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16. The velocity tracking performance and the altitude tracking performance are presented in Figs. 6 and 7. Figs. 8,9,10,11,12,13 and 14 indicate that the addressed method is capable of adaptively readjusting the prescribed funnels (See Figs. 8, 9, 10, 11 and 12) when actuators are saturated (See Figs. 13 and 14), while avoiding the fragile problem associated with traditional PPC [17][18][19]. Besides, the developed compensated system can provide timely compensations when actuators are saturated (See Fig.  15). Finally, Fig 16 shows the boundedness of transformed errors.

Conclusions
The non-fragile tracking control methodology is exploited for WVs subject to actuator saturation. A new type of PPC with readjusting prescribed performance is proposed to handle the fragile problem. Moreover, the improved back-stepping is used to devised velocity controller and altitude controller, while the problem of "explosion of terms" is avoided. Besides, compensated systems are developed to provide effective compensations for actuator saturations. Finally, the presented simulation results prove the effectiveness and superiority of the proposed design. In our future work, we will try to handle the actuator saturation utilizing the Bessel function.
Funding This work was supported by Young Talent Support Project for Science and Technology (Grant No. 18-JCJQ-QT-007).

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

Conflicts of interest
The authors declare that they have no conflict of interest.
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