Prescribed Performance Adaptive Control for a Nonlinear Aeroelastic System with Input Constraint

A constrained prescribed performance compensation controller is proposed for a nonlinear aeroelastic system in the presence of wind gust, system uncertainties, and input saturation. To deal with the e ﬀ ects of the nonsmooth saturation nonlinearity, an approximate saturation function is introduced into the controller design, which can smoothly approximate the real saturation with arbitrarily prescribed precision. Speci ﬁ cally, by designing the prescribed performance function, a ﬁ xed-time control framework is designed to ensure that the closed-loop system has the prescribed tracking performance. The designed control algorithm can not only compensate the adverse e ﬀ ect caused by disturbances and uncertainties but also restrain the excessive amplitude of control input. Finally, the stability analysis shows that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded via the Lyapunov stability analysis method, and simulation results are presented to demonstrate the feasibility and e ﬀ ectiveness of the proposed method.


Introduction
Recent developments in aeroelasticity research and the aeroelastic system control design have attracted increasing attention and concern among scientists and practitioners [1][2][3][4][5]. In aeroelastic systems, the limit cycle oscillation is an important problem and is usually associated with flutter, which can degrade the performance of aerospace vehicles and dramatically affect flight safety. Therefore, it is particularly important to develop a reliable and effective control method to solve such problems. In previous studies [6][7][8], researchers have analyzed the nonlinear responses of aeroelastic systems, and various advance control approaches have been extensively studied in the field of flutter and limit cycle oscillation suppression.
In order to obtain an effective controller, a full feedback linearization controller based on two control surfaces was designed in [9], and the stability of the closed-loop aeroelastic system was further investigated. A high-order sliding mode controller [10] was proposed for suppressing limit cycle oscillation with a backstepping design. The statedependent Riccati equation method was developed for non-linear control problems and used to design suboptimal control laws of nonlinear aeroelastic systems considering both quasisteady [11,12] and unsteady aerodynamics [13,14]. In aeroelastic systems, parameter uncertainties and external disturbances are unavoidable [15]; if these problems cannot be compensated and dealt with in time, the relevant performances of the aeroelastic system are likely to be greatly affected and even cause system instability. To the best of the authors' knowledge, adaptive nonlinear control approach is an effective way to solve the above problem. In [16], an adaptive decoupled fuzzy sliding-mode controller was employed for aeroelastic system to achieve system tracking stabilization. Wang et al. [17] proposed an output feedback adaptive controller for achieving the fine tracking performance of MEMS. In [18], an adaptive robust control scheme with input constraints was proposed for MEMS to improve the antidisturbance ability. Thereafter, an adaptive supertwisting continuous control method [19] and adaptive sliding mode controller with neural network observer [20] were developed for multi-input and multioutput aeroelastic system with unsteady aerodynamics in the presence of uncertainty and gust loads. Recently, an effective function estimator using the interval type-2 and type-3 fuzzy logic system has been extensively studied for adaptive control. Based on this function estimation technique, some new adaptive nonlinear control strategies [21,22] were devised to tackle the effects of perturbations and estimation errors.
Although the studies mentioned above can achieve the corresponding control objectives, some important control performance cannot be guaranteed to be determined in advance. For special control tasks, the desired control performances (such as convergence speed, overshoot, and steadystate error) are needed to be set in the early stage of control and realized during the control process. The urgent need for preset performance of control systems drives the development of nonlinear control methods. An effective solution strategy called the prescribed performance constraint method was proposed in [23][24][25][26], which is easily designed a predefined prescribed performance bound to characterize the convergence rate and the maximum overshoot of tracking error such that the desired transient performance can be achieved. Based on the prescribed performance strategy, many nonlinear control methods have also been extended and applied to various nonlinear systems [27][28][29][30][31][32][33]. It is worth mentioning that the above-mentioned control methods are derived in the sense of Lyapunov asymptotic stability, which means that the system error reaches a residual set in infinite time rather than fixed time, and moreover, it is not easy to calculate the residual set size due to unknown bounded terms.
Control input saturation is another issue that needs to be considered in aeroelastic systems, which is also a source of performance degradation. In the past few years, some compensation methods have been proposed for solving input saturation problem in various nonlinear systems [34][35][36][37][38]. In [39], a sliding mode controller using the auxiliary system was developed for attitude tracking of quadrotor with input constraints. Unfortunately, the fixed-time prescribed performance cannot be achieved by the proposed control system. In [40], a predefined-time prescribed performance control approach was designed for spacecraft rendezvous with input saturation, but system parameters (the inertia matrix of pursuer) were considered as the nominal parameters.
Despite the progress in the research on anti-input saturation [34][35][36][37][38][39][40], it should be noticed that little attention has been paid on the fixed-time prescribed performance control for coexistence problem of system parameter uncertainties and input constraints. Therefore, it is still a challenging open issue to improve control performance and robustness for adverse factors including parameter uncertainty and input constraints.
Motivated by the above discussion, an input constrained control scheme is proposed for the aeroelastic system with wind gust and system uncertainties in this paper. The main features of this paper are briefly summarized as follows: (1) By designing the fixed-time performance function, a control method with built-in time and error constraint mechanism is proposed to ensure that the tracking errors converge to a prescribed compact set within a fixed time (2) The proposed control law is compatible with the control input saturation suppression algorithm that naturally fulfills the magnitude limits by designing an input updating law. Under the framework of the proposed control strategy, it is proved by theory that all internal signal variables are bounded (3) The designed control strategy does not need to know the precise information of the system model, and the proposed settling time is independent of the initial conditions The rest of this paper is organized as follows. Section 2 introduces the aeroelastic model and gives the problem description. A fixed-time prescribed performance controller with saturation suppression algorithm and the stability analysis are presented in Section 3. The effectiveness of the proposed strategy is verified by simulation in Section 4. Finally, conclusions are given in section 5.

Problem Formulation and Preliminaries
2.1. System Description. The prototype aeroelastic wing with leading and trailing edge surfaces is shown in Figure 1. The aeroelastic system has two degrees of freedom, one is the plunge displacement h, and the other is the pitch angle α.
In the presence of a flow field, the wing at a flight speed U oscillates along the plunge displacement direction and rotates at the pitch angle about the elastic axis [1].
The dynamics of the aeroelastic system is described by [4,5] Here, aerodynamic lifts (L and L g ) and moments (M and M g ) are given as International Journal of Aerospace Engineering Note that γ and β are the trailing edge and the leading edge control surface deflections, respectively.
To facilitate understanding of the specific meanings of system parameters, the relevant symbol descriptions in Equations (1) and (2) are presented in Table 1. Considering that there are many system parameters in (1), some hybrid parameters are defined in Table 2 for the convenience of analysis. It should be pointed out that these hybrid parameters are not fully known; moreover, b ij ði = 1, 2 ; j = 1, 2Þ can be decomposed into nominal b ij and uncertain parts Δb ij , i.e., b ij = b ij + Δb ij . 12 γ, and u 2 = b 21 β + b 22 γ, for i = 1, 2; the aeroelastic system (1) can be written as In practical applications, the control input should be limited within a reasonable range due to the physical constraints. Considering that the trailing edge and leading edge control surface of the aeroelastic wing have maximum deflection limits, therefore, there exist the maximum bounds and β max such that jγj ≤ γ max and jβj ≤ β max . Furthermore, by combining u 1 = b 11 β + b 12 γ and u 2 = b 21 β + b 22 γ, for i = 1, 2, we can obtain that where τ i are needed to be designed, Obviously, u i are the saturation functions which can lead to nonsmooth control action near the saturation limit. Thus, we introduce the following smooth function Γ τ iM ðτ i Þ proposed in [41] to approximate the saturation functions (7). (A comparison of the three saturated nonlinear functions is shown in Figure 2.) where c > 0 and τ iM > 0 are the design parameters.
Lemma 1 (see [41]). The saturation functions u i ði = 1, 2Þ in (7) can be expressed as Proof. See the appendix. With the aid of Lemma 1, it follows from (6) that where D i = f i + Δτ i . To construct the control system, the following assumptions are needed.
where α max , α max , h max , and h max are the upper bound values of the corresponding variables. Assumption 5. The system states x 2i−1 , x 2i , and _ x 2i ði = 1, 2Þ can be measured by physical sensors. Remark 6. The synthetic functions D i ði = 1, 2Þ in (10) contain several parameters related to the aeroelastic system. In fact, these parameters are all bounded, which means that there is also an upper bound for max f|D 1 |,|D 2 | g. Considering the limited energy of the airflow vibration and the physical constraints of the aeroelastic system, hence, α, _ α, h, and _ h are bounded. In addition, the system states can be obtained by physical sensors (for example, x 2i−1 can be measured from the laser positioning sensor, and x 2i and _ x 2i can be measured from gyroscope sensor integrated by a 3-axis MEMS accelerometer and a digital motion processor). Based on the analysis above, Assumptions 2-5 are valid. The similar assumptions can be found in [38]. It should be pointed out that the bounds of these parameters are not involved in the design of the control law.

Preliminaries.
To study the control performance of the closed loop system, the following definition is required.
Definition 7 (see [42,43]). A smooth function ρðtÞ is called the fixed-time performance function if there exist the preassigned time T S and preset precision ρ TS such that (1) ρðtÞ > 0 and _ ρðtÞ ≤ 0 for ∀t ∈ ½0, T S Þ Definition where The change trend of the performance function ρðtÞ is presented in Figure 3. In this paper, ρðtÞ is used to restrain and schedule the convergence trend of the system errors. Notice that the selection of the fixed-time T S in ρðtÞ cannot be arbitrarily small and it should be larger than the sampling time.
In this paper, the control objective is to design a fixedtime prescribed performance control scheme for system (10), such that the system errors converge to a prescribed compact set within a fixed time T S .

Main Results
We now propose the fixed-time prescribed performance control method for the aeroelastic system. The concrete design procedure is given as follows: Step 1. Define the system tracking errors as follows: Here, φ i are the virtual control laws which will be designed later. Combining with (10) and the definition of e 2i , the time derivative of e 2i−1 can be found as Next, the virtual control laws are designed as in which κ 2i−1 are positive design parameters. Substituting φ i into (15) yields Step 2. Define the auxiliary deviation variables as  where τ i are updated by (30); Γ φ iM ð φ i Þ and φ i are given as In (19) and (20), κ 2i and a 1 are the positive design parameters; φ iM = 0:5c −1 ln ððe cð2τ i M −εÞ + e −cð2τ i M −εÞ Þ/ðe −c ε + e c ε ÞÞ with constant ε satisfying τ iM > ε > 0; in addition, b σ i are updated by It should be pointed out that s i (in (18)) are introduced into the control system design to avoid the excessive amplitude of τ i . The elimination mechanism of excessive control amplitude will be given later.

Remark 8.
According to the function properties of (19), we can obtain that Γ φ iM ð φ i Þ are bounded, i.e., |Γ φ iM ð φ i Þ | ≤ φ iM .  International Journal of Aerospace Engineering Therefore, we can further prove that Using the fact φ iM = ð1/2cÞ ln ððe cð2τ i M −εÞ + e −cð2τ i M −εÞ Þ/ ðe −c ε + e c ε ÞÞ, then therefore, we have Combining with the analysis above, we obtain From (14) and (18), the derivative of e 2i and s i is calculated as follows: where Before the main result is given, a compact sets is defined as Obviously, there must be the point corresponding to the supreme value of Ψ i in Ω × Ζ, such that where σ i > 0 are the unknown constants.
Remark 9. Notice that Ψ i contain the uncertain function terms D i , which make it impossible to directly add the uncertain function to eliminate itself in the controller design.
In the stability analysis, the upper bound of Ψ 2 i can be displayed through inequality scaling under the help of (29), and furthermore, the upper bound of Ψ 2 i can be estimated and compensated by designing the online adaption laws _ b σ i in (21). Based on this processing method, the stability of the closed-loop system can be guaranteed by the proposed control strategy.

Lemma 12.
Consider the auxiliary deviation variables s i and _ s i ; if the input updating laws are chosen as (30), then the following inequality can be established: Proof. The following three cases need to be discussed for the proof of Lemma 12. (26), we can obtain s i ≥ 0 and it can be verified that Case 2. If j τ i j < τ iM − ε, this leads to Case 3. If τ i ≤ −τ iM + ε, then we have s i ≤ 0 from (25); it yields From (33) to (35), Lemma 12 is proven. Now, the main results are summarized as follows.
Theorem 13. Consider the dynamics system (10) with Assumptions 2-4, if the virtual control laws (16) and (20) with parameter updating law as specified in Equations (21) are used and the input updating laws (30) are implemented, then for any initial states in Ω, the following control objectives can be achieved.
(1) All signals in the closed-loop system are semiglobal uniformly ultimately bounded (2) The fixed-time prescribed performance is achieved, such that the tracking errors e 2i−1 ði = 1, 2Þ converge to a prescribed compact set Ω e = fe 2i−1 , j je 2i−1 j < ρ TS g for ∀t > T S Proof. Choose the Lyapunov function candidate V as Based on (17), (27), and (29), the following relationships are established in the compact set Ω: With the help of (38), then (37) becomes International Journal of Aerospace Engineering where α g = min fκ 2i , κ 2i−1 , a 1 ξ i g and Δ g = ða 2 1 /2Þ + ∑ 2 i=1 ða 1 σ 2 i /2Þ. Solving (39), we have According to (40), it can be seen that each signal of the closed-loop system is bounded, and meanwhile, we can obtain that Taking exponentials on both sides of (41) yields Therefore, in the light of the properties of ρ i ðtÞ, we obtain je 2i−1 j < ρ i ðtÞ; it implies e 2i−1 ðtÞ < ρ TS for t ∈ ½T S ,∞Þ. Then, one can conclude that tracking control with the fixed-time prescribed performance is achieved. The control system structure and algorithm flow chart are shown in Figures 4 and 5, respectively.
From the above analysis, e 2i−1 remain bounded inside the region B e ≜ f|e 2i−1 |<ρ i ðtÞ, i = 1, 2g for t ≥ T S . It should be pointed that the reciprocal of ∂Γ τ iM ðτ i Þ/∂τ i plays an important role in the designed input updating law (30). In fact, the input updating law singularity may occur since ∂Γ τ iM ðτ i Þ/∂τ i goes to zero as τ i ⟶ ∞. However, from (30), we can see that variable τ i remains bounded inside the region B τ ≜ f|τ i | ≤ τ iM , i = 1, 2g for t ≥ 0. Hence, the sensibility of the reciprocal of ∂Γ τ iM ðτ i Þ/∂τ i can be acceptable, and moreover, ∂Γ τ iM ðτ i Þ/∂τ i has never gone to zero.
Remark 15. The designed method is a recursive design process borrowed from the traditional backstepping control design idea. Because τ i are contained within the function Γ τ iM ðτ i Þ, therefore, we cannot design the control laws τ i in the form of the function Γ τ iM ðτ i Þ. To solve this problem and complete the control law design, s i ði = 1, 2Þ are introduced such that input updating laws _ τ i can be designed by taking the derivative of s i . On the other hand, from (26), s i ði = 1, 2Þ have a "supervisory role" when τ i exceeding . With the help of (33)-(35), we have s i ⋅ _ s i ≤−κ 2i s 2 i − e 2i s i , which can help us get the inequality result _ V ≤ −α g V + Δ g and obtain the stability result of the closed-loop system on this basis. Remark 16. The tuning guidelines for all the designed parameters are summarized as follows: (1) From (39), larger κ 2i , κ 2i−1 , ξ i , and 1/a 1 ði = 1, 2Þ result in smaller system errors ðe 2i and s i Þ but may lead to the large control magnitude and excite the undesired high-frequency dynamics of the controlled system. In this paper, these parameters are set to small positive numbers, and our concern is to adjust them to establish the bound of system error e 2i−1 . Therefore, the process of choosing the control parameters κ 2i , κ 2i−1 , ζ i , and 1/a 1 is significantly simplified (2) The prescribed performance parameter ρ 0 should be chosen large enough such that the initial error boundedness can be satisfied, i.e., max f|α max − x 1d ð0Þ|, | h max − x 3d ð0Þ | g < ρ 0 . The parameter ρ TS determines the final bound of the steady-state error. The parameter T S determines the time for the tracking error to converge to the specified accuracy. Too small T S can lead to a large control amplitude at the beginning of the control. Combined with the actual control tasks, moderate compromise selection of T S and ρ TS can eliminate the large amplitude for control output at the beginning of the control process Remark 17. The prescribed performance control methods [23][24][25][26] can ensure that the designed closed-loop system has good transient and steady-state behaviors and forces the tracking error to converge to a predetermined range when the time tends to infinity. It is worth mentioning that the above-mentioned methods are developed based on traditional performance functions, which means that the designed control systems by these methods are not equipped with the finite-time prescribed performance. Although some new Set initial values i (0), i (0), i = 1,2; and obtain the initial states x 2i-1 (0).x 2i (0),x 2i (0),i = 1,2 with the help of the sensor . Figure 5: The algorithm flow chart.  International Journal of Aerospace Engineering nonlinear control approaches are proposed in [45,46] to achieve finite-time prescribed performance tracking control, the settling time is related to the initial value of the system states. In this paper, the proposed settling time T S is independent of the initial conditions, and therefore, T S can be set according to the requirement of control tasks.

Numerical Simulations
In this section, simulation results are presented to illustrate the effectiveness of our proposed control method. The system parameters satisfy y = y + Δy in which y and Δy represent the nominal part and uncertain part, respectively. The nominal part of system parameters are chosen from experimental date in [4,47]. The initial value of system states are given as αð0Þ = 0:1 rad, hð0Þ = 0:01 m, and _ αð0Þ = _ hð0Þ = € α ð0Þ = € hð0Þ = 0. Since the control objective is to suppress the aeroelastic vibrations, thus the desired signals are set as zero.
To evaluate the performance of the proposed control method, a comparison will be given between the prescribed performance adaptive compensation control scheme (PPAC) 11 International Journal of Aerospace Engineering proposed in this paper and the adaptive sliding mode compensation control scheme (ASMC) in [5] for the same sampling time and model parameters. The control system parameters of PPAC and ASMC are set in Table 3. According to [5], the ASMC controller is devised as where f n is a known function consisting of the system nominal parameter; s 1 and s 2 are the system error vectors; k 1 , k 2 , η, and φ are design parameters; σ = ½σ 1 , and p = diag fp 1 , p 2 g > 0); and b α = diag fb α 1 , b α 2 g is the disturbance estimation.
Firstly, the open-loop responses for sinusoidal gust w G = 0:07 and U = 13:28m/s are shown in Figure 6; in this situation, the system exhibited the limit cycle oscillation. It is obvious that the plunge and pitch motions represent flutter phenomena, which need to be suppressed.
For the purpose of examining the robustness of the proposed control scheme, the following two cases are executed by choosing different disturbances and unknown parameter parts.

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International Journal of Aerospace Engineering Matlab/Simulink). Figures 7-9 show the control effect under the two control methods. Compared with ASMC, the proposed method has a fast convergence rate at the initial stage, and moreover, the proposed method guarantees that the system error converges into a bounded region when t > 2s, i.e., max f | e 1 |, | e 3 | g < 8 × 10 −5 . Obviously, the simulation results are perfectly matched with Theorem 13. The proposed method realizes the preset control precision after a specified time (as stated by Theorem 13, e 2i−1 converge to a prescribed compact set Ω e = fe 2i−1 | | e 2i−1 | <ρ TS ,g for ∀t > T S based on ρ TS = 8 × 10 −5 and T S = 2s). It means that the proposed control strategy is an effective way to suppress the limit cycle oscillation feature even in the presence of parameter uncertainties. As for ASMC, the maximum errors are shown as max f | e 1 |, | e 3 | g < 5:8 × 10 −4 . From Figures 10 and 11, it can be seen that the initial amplitude values of control input τ i and nonlinear function estimation b σ i are large. This is because the proposed method is tuning and does not adapt to the adverse factors at the beginning. And then, the adaptation updating algorithm starts to exert its effectiveness such that the satisfactory control performance is achieved.
Case B. U = 13:28 m/s, w G = 2:5 ⋅ ð1 − e −0:25Ut/b Þ ⋅ rand ðtÞ, and Δy = 40% × rand ðtÞ × y. From Figures 12-16, we see that the proposed method can guarantee that e 2i−1 ði = 1, 2Þ converge into a bounded region max f | e 1 |, | e 3 | g < 8 × 10 −5 for t > 2s, but the errors obtained by ASMC stay in max f | e 1 |, | e 3 | g < 1:8 × 10 −3 for t > 2s. Obviously, the    14 International Journal of Aerospace Engineering simulation results indicate that the control accuracy of the proposed method is still higher than that of ASMC. Compared with Case A, although uncertainties and the disturbance are increased, the proposed control method still achieves the fixed-time prescribed performance, and it means that the system parameter uncertainties and external disturbances are well suppressed. These simulation results show that the good performance can be obtained by the proposed control algorithm.

Conclusion
In this paper, a fixed-time prescribed performance control algorithm for a nonlinear aeroelastic system is presented. In particular, the control performance of the closed-loop system is guaranteed theoretically with prescribed performance constraint and excellent disturbance rejection capability. The performance of the proposed control system is evaluated by digital simulation in the presence of input constraints. Simulation results show that the designed con-trol method suppresses the limit-cycle oscillations, despite parameter uncertainties and wind gust.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.