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

Authors: Zijun Gao; Junjie Zhang; Jiaqi Li; Zhankui Song
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

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 effects 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. Specifically, by designing the prescribed performance function, a fixed-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 effect 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 effectiveness of the proposed method.


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## Body

## 1. Introduction

Recent developments in aeroelasticity research and the aeroelastic system control design have attracted increasing attention and concern among scientists and practitioners [1–5]. 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 [6–8], 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 state-dependent Riccati equation method was developed for nonlinear 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 steady-state 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 [23–26], 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 [27–33]. 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 [34–38]. 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 [34–40], 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 conditionsThe rest of this paper is organized as follows. Section2 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.


## 2. Problem Formulation and Preliminaries

### 2.1. System Description

The prototype aeroelastic wing with leading and trailing edge surfaces is shown in Figure1. 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].Figure 1
Aeroelastic model.The dynamics of the aeroelastic system is described by [4, 5]
(1)Iαmwxαbmwxαbmtα¨h¨+cα00chα̇ḣ+kαα00khαh=M−L+Mg−Lg.Here, aerodynamic lifts (L and Lg) and moments (M and Mg) are given as
(2)Lg=μUbsclαwGtr,(3)Mg=0.5−a⋅b⋅Lg,(4)L=μU2bsclαα+ḣU+12−abα̇U+μU2bsclββ+clγγ,(5)M=μU2b2sclα2+aclα+2cmαα+ḣU+12−abα̇Uclβ2+aclβ+2cmββ+12+aclγ+2cmγγ.Note thatγ 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, biji=1,2;j=1,2 can be decomposed into nominal b¯ij and uncertain parts Δbij, i.e., bij=b¯ij+Δbij.Table 1
Nomenclature.
SymbolDescriptionmwMass of the winghPlunging displacementαPitch angle about the pitch axisIαInertial moment of the wing about the elastic axismtTotal mass of the wing and its support structurexαDimensionless distance between the center of mass and elastic axisaNondimensional distance from airfoil midchord to elastic axiskαPitch stiffnesskhPlunge stiffnessMAerodynamic momentLAerodynamic liftclαDerivatives of aerodynamic lift coefficient (clβ and cly have the similar definitions)MgAerodynamic moment due to gustLgAerodynamic force duet to gustchPlunge dampingcαPitch dampingμAir densitybAirfoil semichordsWing section spanβAngle of trailing edgeγAngle of leading edgeUFree stream velocitywGDisturbance velocitycmαMoment derivatives (cmβ and cmγ have the similar definitions)Table 2
Hybrid parameter definition.
Definitionk1=Iαkh/mtIα−mwxαb2k2=Iαμbsclα+mwxαb3μscmα/mtIα−mwxαb2k3=−mwxαbkh/mtIα−mwxαb2k4=mwμsxαb2clα+mtb2μscmα/mtIα−mwxαb2p1=mwxαbkα/mwxαb2−mtIαp2=mtkα/mtIα−mwxαb2c1=Iαch+μsUbcla+mwμsUxαb3cmα/mtIα−mwxαb2c2=Iα+μsUb2clamwμsUxαb4cmα0.5−a+/mtIα−mwxαb2c3=mwxαbch+xαμUb2clα+μUb2cmα/mwxαb2−mtIαc4=mwxαμUb3clα+mtμUb2cmα0.5−a/mwxαb2−mtIαq1=mwxαbcα/mwxαb2−mtIαq2=mtcα/mtIα−mwxαb2b11=U2μsmwxαb2clβ+mtb20.5+aclβ+2cmβ/mtIα−mwxαb2b12=U2μsmwxαb2clγ+mtb20.5+aclγ+2cmγ/mtIα−mwxαb2b21=U2μsIαbclβ+mwxαb30.5+aclβ+2cmβ/mwxαb2−mtIαb22=U2μsIαbclγ+mwxαb3s0.5+aclγ+2cmγ/mwxαb2−mtIαd1=mtMgmwxαb⋅Lg/Iαmt−mwxαb2d1=IαLgmwxαb⋅Mg/mwxαb2−IαmtLetx1=α, x2=α̇, x3=h, x4=ḣ, u1=b¯11β+b¯12γ, and u2=b¯21β+b¯22γ, for i=1,2; the aeroelastic system (1) can be written as
(6)ẋ2i−1=x2i,ẋ2i=fi+ui,where f1=d1−k3x3−c3x4−k4U2+p2x1+Δb11β−c4+q2x2+Δb12γ and f2=d2−k1x3−c1x4+Δb21β−k2U2+p1x1−c2+q1x2+Δb22γ.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 andβmax such that γ≤γmax and β≤βmax. Furthermore, by combining u1=b¯11β+b¯12γ and u2=b¯21β+b¯22γ, for i=1,2, we can obtain that
(7)ui=signτi⋅uiMifτi≥τiM,τiifτi<τiM,where τi are needed to be designed, τ1M=b¯11βmax+b¯12γmax, and τ2M=b¯21βmax+b¯22γmax.Obviously,ui are the saturation functions which can lead to nonsmooth control action near the saturation limit. Thus, we introduce the following smooth function ΓτiMτi proposed in [41] to approximate the saturation functions (7). (A comparison of the three saturated nonlinear functions is shown in Figure 2.)
(8)ΓτiMτi=12clnecτiMecτi+e−cτiMe−cτie−cτiMecτi+ecτiMe−cτi,where c>0 and τiM>0 are the design parameters.Lemma 1 (see [41]).
The saturation functionsuii=1,2 in (7) can be expressed as
(9)ui=ΓτiMτi+Δτi,and the following properties hold.
P1: ∂ΓτiMτi/∂τi>0.
P2: Δτi are bounded and limc⟶+∞Δτi=0.
P3: ΓτiMτi≤τiM.Proof.
See the appendix.Figure 2
Comparison of saturated nonlinear functions.With the aid of Lemma1, it follows from (6) that
(10)ẋ2i−1=x2i,ẋ2i=Di+ΓτiMτi,where Di=fi+Δτi. To construct the control system, the following assumptions are needed.Assumption 2.
The nonlinear functionsDii=1,2 are bounded by unknown constant δD, i.e., Di≤δD.Assumption 3.
The physical states of system are bounded and remain in a setZ, i.e.,
(11)Ζ=α,α̇,h,ḣα≤αmax,α̇≤α¯max,h≤hmax,h≤h¯max,where αmax, α¯max, hmax, and h¯max are the upper bound values of the corresponding variables.Assumption 4.
The desired command signalsx2i−1di=1,2 and their derivative ẋ2i−1d and x¨2i−1d are bounded.Assumption 5.
The system statesx2i−1, x2i, and ẋ2ii=1,2 can be measured by physical sensors.Remark 6.
The synthetic functionsDii=1,2 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∣D1∣,∣D2∣. Considering the limited energy of the airflow vibration and the physical constraints of the aeroelastic system, hence, α, α̇, h, and ḣ are bounded. In addition, the system states can be obtained by physical sensors (for example, x2i−1 can be measured from the laser positioning sensor, and x2i and ẋ2i 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.


### 2.2. Preliminaries

To study the control performance of the closed loop system, the following definition is required.Definition 7 (see [42, 43]).
A smooth functionρt is called the fixed-time performance function if there exist the preassigned time TS and preset precision ρTS such that
(1)
ρt>0 and ρ̇t≤0 for ∀t∈0,TS(2)
ρt=ρTS>0 for ∀t≥TS
Based on the above definition, the fixed-time performance functionsρii=1,2 [44] are constructed as
(12)ρit=−ρ¯3TS3t3+ρ¯TS2t2−ρ¯TSt+ρ0,fort∈0,TS,ρTS,fort∈TS,∞,where ρ¯=3ρ0−ρTS with positive constant ρ0. From (12), it easily follows that ρ̇it<0 for t∈0,TS, ρit=ρTS>0 for t∈TS,∞, limt⟶TS−ρit−ρiTS/t−TS=limt⟶TS+ρit−ρiTS/t−TS=0, and limt⟶TS−ρit=limt⟶TS+ρit=ρTS. Then, we have
(13)ρ̇it=−ρ¯TS3t2+2ρ¯TS2t−ρ¯TS,fort∈0,TS,0,fort∈TS,∞.The change trend of the performance functionρt is presented in Figure 3. In this paper, ρt is used to restrain and schedule the convergence trend of the system errors. Notice that the selection of the fixed-time TS in ρt cannot be arbitrarily small and it should be larger than the sampling time.Figure 3
The dynamic diversification curves ofρt.
(a)
The dynamic diversification curves with differentTS(b)
The dynamic diversification curves with differentρTSIn this paper, the control objective is to design a fixed-time prescribed performance control scheme for system (10), such that the system errors converge to a prescribed compact set within a fixed time TS.


## 2.1. System Description

The prototype aeroelastic wing with leading and trailing edge surfaces is shown in Figure1. 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].Figure 1
Aeroelastic model.The dynamics of the aeroelastic system is described by [4, 5]
(1)Iαmwxαbmwxαbmtα¨h¨+cα00chα̇ḣ+kαα00khαh=M−L+Mg−Lg.Here, aerodynamic lifts (L and Lg) and moments (M and Mg) are given as
(2)Lg=μUbsclαwGtr,(3)Mg=0.5−a⋅b⋅Lg,(4)L=μU2bsclαα+ḣU+12−abα̇U+μU2bsclββ+clγγ,(5)M=μU2b2sclα2+aclα+2cmαα+ḣU+12−abα̇Uclβ2+aclβ+2cmββ+12+aclγ+2cmγγ.Note thatγ 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, biji=1,2;j=1,2 can be decomposed into nominal b¯ij and uncertain parts Δbij, i.e., bij=b¯ij+Δbij.Table 1
Nomenclature.
SymbolDescriptionmwMass of the winghPlunging displacementαPitch angle about the pitch axisIαInertial moment of the wing about the elastic axismtTotal mass of the wing and its support structurexαDimensionless distance between the center of mass and elastic axisaNondimensional distance from airfoil midchord to elastic axiskαPitch stiffnesskhPlunge stiffnessMAerodynamic momentLAerodynamic liftclαDerivatives of aerodynamic lift coefficient (clβ and cly have the similar definitions)MgAerodynamic moment due to gustLgAerodynamic force duet to gustchPlunge dampingcαPitch dampingμAir densitybAirfoil semichordsWing section spanβAngle of trailing edgeγAngle of leading edgeUFree stream velocitywGDisturbance velocitycmαMoment derivatives (cmβ and cmγ have the similar definitions)Table 2
Hybrid parameter definition.
Definitionk1=Iαkh/mtIα−mwxαb2k2=Iαμbsclα+mwxαb3μscmα/mtIα−mwxαb2k3=−mwxαbkh/mtIα−mwxαb2k4=mwμsxαb2clα+mtb2μscmα/mtIα−mwxαb2p1=mwxαbkα/mwxαb2−mtIαp2=mtkα/mtIα−mwxαb2c1=Iαch+μsUbcla+mwμsUxαb3cmα/mtIα−mwxαb2c2=Iα+μsUb2clamwμsUxαb4cmα0.5−a+/mtIα−mwxαb2c3=mwxαbch+xαμUb2clα+μUb2cmα/mwxαb2−mtIαc4=mwxαμUb3clα+mtμUb2cmα0.5−a/mwxαb2−mtIαq1=mwxαbcα/mwxαb2−mtIαq2=mtcα/mtIα−mwxαb2b11=U2μsmwxαb2clβ+mtb20.5+aclβ+2cmβ/mtIα−mwxαb2b12=U2μsmwxαb2clγ+mtb20.5+aclγ+2cmγ/mtIα−mwxαb2b21=U2μsIαbclβ+mwxαb30.5+aclβ+2cmβ/mwxαb2−mtIαb22=U2μsIαbclγ+mwxαb3s0.5+aclγ+2cmγ/mwxαb2−mtIαd1=mtMgmwxαb⋅Lg/Iαmt−mwxαb2d1=IαLgmwxαb⋅Mg/mwxαb2−IαmtLetx1=α, x2=α̇, x3=h, x4=ḣ, u1=b¯11β+b¯12γ, and u2=b¯21β+b¯22γ, for i=1,2; the aeroelastic system (1) can be written as
(6)ẋ2i−1=x2i,ẋ2i=fi+ui,where f1=d1−k3x3−c3x4−k4U2+p2x1+Δb11β−c4+q2x2+Δb12γ and f2=d2−k1x3−c1x4+Δb21β−k2U2+p1x1−c2+q1x2+Δb22γ.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 andβmax such that γ≤γmax and β≤βmax. Furthermore, by combining u1=b¯11β+b¯12γ and u2=b¯21β+b¯22γ, for i=1,2, we can obtain that
(7)ui=signτi⋅uiMifτi≥τiM,τiifτi<τiM,where τi are needed to be designed, τ1M=b¯11βmax+b¯12γmax, and τ2M=b¯21βmax+b¯22γmax.Obviously,ui are the saturation functions which can lead to nonsmooth control action near the saturation limit. Thus, we introduce the following smooth function ΓτiMτi proposed in [41] to approximate the saturation functions (7). (A comparison of the three saturated nonlinear functions is shown in Figure 2.)
(8)ΓτiMτi=12clnecτiMecτi+e−cτiMe−cτie−cτiMecτi+ecτiMe−cτi,where c>0 and τiM>0 are the design parameters.Lemma 1 (see [41]).
The saturation functionsuii=1,2 in (7) can be expressed as
(9)ui=ΓτiMτi+Δτi,and the following properties hold.
P1: ∂ΓτiMτi/∂τi>0.
P2: Δτi are bounded and limc⟶+∞Δτi=0.
P3: ΓτiMτi≤τiM.Proof.
See the appendix.Figure 2
Comparison of saturated nonlinear functions.With the aid of Lemma1, it follows from (6) that
(10)ẋ2i−1=x2i,ẋ2i=Di+ΓτiMτi,where Di=fi+Δτi. To construct the control system, the following assumptions are needed.Assumption 2.
The nonlinear functionsDii=1,2 are bounded by unknown constant δD, i.e., Di≤δD.Assumption 3.
The physical states of system are bounded and remain in a setZ, i.e.,
(11)Ζ=α,α̇,h,ḣα≤αmax,α̇≤α¯max,h≤hmax,h≤h¯max,where αmax, α¯max, hmax, and h¯max are the upper bound values of the corresponding variables.Assumption 4.
The desired command signalsx2i−1di=1,2 and their derivative ẋ2i−1d and x¨2i−1d are bounded.Assumption 5.
The system statesx2i−1, x2i, and ẋ2ii=1,2 can be measured by physical sensors.Remark 6.
The synthetic functionsDii=1,2 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∣D1∣,∣D2∣. Considering the limited energy of the airflow vibration and the physical constraints of the aeroelastic system, hence, α, α̇, h, and ḣ are bounded. In addition, the system states can be obtained by physical sensors (for example, x2i−1 can be measured from the laser positioning sensor, and x2i and ẋ2i 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.


## 2.2. Preliminaries

To study the control performance of the closed loop system, the following definition is required.Definition 7 (see [42, 43]).
A smooth functionρt is called the fixed-time performance function if there exist the preassigned time TS and preset precision ρTS such that
(1)
ρt>0 and ρ̇t≤0 for ∀t∈0,TS(2)
ρt=ρTS>0 for ∀t≥TS
Based on the above definition, the fixed-time performance functionsρii=1,2 [44] are constructed as
(12)ρit=−ρ¯3TS3t3+ρ¯TS2t2−ρ¯TSt+ρ0,fort∈0,TS,ρTS,fort∈TS,∞,where ρ¯=3ρ0−ρTS with positive constant ρ0. From (12), it easily follows that ρ̇it<0 for t∈0,TS, ρit=ρTS>0 for t∈TS,∞, limt⟶TS−ρit−ρiTS/t−TS=limt⟶TS+ρit−ρiTS/t−TS=0, and limt⟶TS−ρit=limt⟶TS+ρit=ρTS. Then, we have
(13)ρ̇it=−ρ¯TS3t2+2ρ¯TS2t−ρ¯TS,fort∈0,TS,0,fort∈TS,∞.The change trend of the performance functionρt is presented in Figure 3. In this paper, ρt is used to restrain and schedule the convergence trend of the system errors. Notice that the selection of the fixed-time TS in ρt cannot be arbitrarily small and it should be larger than the sampling time.Figure 3
The dynamic diversification curves ofρt.
(a)
The dynamic diversification curves with differentTS(b)
The dynamic diversification curves with differentρTSIn this paper, the control objective is to design a fixed-time prescribed performance control scheme for system (10), such that the system errors converge to a prescribed compact set within a fixed time TS.


## 3. 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:(14)e2i−1=x2i−1−x2i−1d,e2i=x2i−φi,i=1,2.
Here,φi are the virtual control laws which will be designed later. Combining with (10) and the definition of e2i, the time derivative of e2i−1 can be found as
(15)ė2i−1=ẋ2i−1−ẋ2i−1d=e2i+φi−ẋ2i−1d.
Next, the virtual control laws are designed as(16)φi=−κ2i−1⋅e2i−1+ẋ2i−1d+e2i−1ρ̇iρi,in which κ2i−1 are positive design parameters. Substituting φi into (15) yields
(17)ė2i−1=e2i−κ2i−1⋅e2i−1+e2i−1ρ̇iρi.Step 2.
Define the auxiliary deviation variables as(18)si=ΓτiMτi−Γφ¯iMφ¯i,Δφ¯i=Γφ¯iMφ¯i−φ¯i,i=1,2,where τi are updated by (30); Γφ¯iMφ¯i and φ¯i are given as
(19)Γφ¯iMφ¯i=12clnecφ¯iMecφ¯i+e−cφ¯iMe−cφ¯ie−cφ¯iMecφ¯i+ecφ¯iMe−cφ¯i,(20)φ¯i=−σ^i⋅e2i2a12−κ2i⋅e2i−e2i−1ρi2−e2i−12.
In (19) and (20), κ2i and a1 are the positive design parameters; φ¯iM=0.5c−1lnec2τiM−ε+e−c2τiM−ε/e−cε+ecε with constant ε satisfying τiM>ε>0; in addition, σ^i are updated by
(21)σ^̇i=ξi⋅−a1⋅σ^i+e2i22a12ξi>0,σ^i0=0.It should be pointed out thatsi (in (18)) are introduced into the control system design to avoid the excessive amplitude of τi. The elimination mechanism of excessive control amplitude will be given later.Remark 8.
According to the function properties of (19), we can obtain that Γφ¯iMφ¯i are bounded, i.e., ∣Γφ¯iMφ¯i∣≤φ¯iM.Therefore, we can further prove that(1)
ifτi≥τiM−ε, it can be obtained from the properties of ΓτiMτi that ΓτiMτi≥ΓτiMτiM−ε>0. Notice that(22)ΓτiMτiM−ε=12c⋅lnec2τiM−ε+e−c2τiM−εe−cε+ecε,Γφ¯iMφ¯i≤φ¯iM=12c⋅lnec2τiM−ε+e−c2τiM−εe−cε+ecε.Using the factφ¯iM=1/2clnec2τiM−ε+e−c2τiM−ε/e−cε+ecε, then
(23)si=ΓτiMτi−Γφ¯iMφ¯i≥ΓτiMτiM−ε−φ¯iM=12clnec2τiM−ε+e−c2τiM−εe−cε+ecε−φ¯iM=0.(2)
ifτi≤−τiM+ε, it implies that ΓτiMτi≤ΓτiM−τiM+ε<0Since(24)ΓτiM−τiM+ε=12clnecε+e−cεe−c2τiM−ε+ec2τiM−ε,φ¯iM=−12clne−cε+ecεec2τiM−ε+e−c2τiM−ε,therefore, we have
(25)si=ΓτiMτi−Γφ¯iMφ¯i≤ΓτiM−τiM+ε+φ¯iM=12clnecε+e−cεe−c2τiM−ε+ec2τiM−ε+φ¯iM=0.Combining with the analysis above, we obtain(26)si≥0,ifτi≥τiM−ε,si≤0,ifτi≤−τiM+ε.From (14) and (18), the derivative of e2i and si is calculated as follows:
(27)ė2i=ẋ2i−φ̇i=Di+ΓτiMτi−φ̇i=si+φ¯i+Ψi,ṡi=∂ΓτiMτi∂τi⋅τ̇i−∂Γφ¯iMφ¯i∂φ¯i⋅φ¯̇i,where Ψi=Di+Δφ¯i−φ̇i. Before the main result is given, a compact sets is defined as
(28)Ω=e2i−1,e2i,si∑i=12ρi2ρi2−e2i−12+e2i2+si2≤ϑ,ϑ∈ℝ+.Obviously, there must be the point corresponding to the supreme value ofΨi in Ω×Ζ, such that
(29)Ψi2≤σii=1,2,where σi>0 are the unknown constants.Remark 9.
Notice thatΨi contain the uncertain function terms Di, 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 Ψi2 can be displayed through inequality scaling under the help of (29), and furthermore, the upper bound of Ψi2 can be estimated and compensated by designing the online adaption laws σ^̇i in (21). Based on this processing method, the stability of the closed-loop system can be guaranteed by the proposed control strategy.Next, the input updating laws are designed as follows:(30)τ̇i=∂ΓτiMτi∂τi−1⋅ϖi−121+ϖi2,ifτi≥τiM−ε,∂ΓτiMτi∂τi−1⋅ϖi,ifτ<τiM−ε,∂ΓτiMτi∂τi−1⋅ϖi+121+ϖi2,ifτ≤−τiM+ε,where τi0=0 and ϖi=∂Γφ¯iMφ¯i/∂φ¯i⋅φ¯̇−κ2isi−e2i.Remark 10.
In view of (16), we have φ̇i=−κ2i−1ė2i−1+x¨2i−1d+(ė2i−1ρ̇iρi+e2i−1ρ¨iρi−e2i−1. Combining with (12), (21), and Assumption 5, σ^̇i, φ̇i, ρi, ρ̇i, and ẋ2i can be calculated or measured, and therefore, the derivative of φ¯i can be calculated as
(31)φ¯̇i=−σ^̇ie2i2a12−σ^i2a12+κ2iẋ2i−φ̇i−ė2i−1ρi2+e2i−12ė2i−1−2ρiρ̇ie2i−1ρi2−e2i−122.Remark 11.
Equation (30) plays an key role for the boundedness of τ. If τi≥τiM−ε, then τ̇=∂ΓτiMτi/∂τi−1⋅ϖi−0.51+ϖi2. Since ∂ΓτiMτi/∂τi−1>0 and ϖi−0.5⋅1+ϖi2≤0, it leads to τ̇i≤0, and moreover, Equation (30) will prevent τi from getting greater than τiM. If τi≤−τiM+ε, it is easy to obtain τ̇i≥0 due to ∂ΓτiMτi/∂τi−1>0 and ϖi+0.51+ϖi2≥0. Thus, Equation (26) will prevent τi from getting smaller than −τiM. Based on the above analysis, τi can be constrained within a preset interval, i.e., τi∈−τiM,τiM.Lemma 12.
Consider the auxiliary deviation variablessi and ṡi; if the input updating laws are chosen as (30), then the following inequality can be established:
(32)si⋅ṡi≤−κ2isi2−e2isii=1,2.Proof.
The following three cases need to be discussed for the proof of Lemma12.
Case 1. Ifτi≥τiM−ε, from (26), we can obtain si≥0 and it can be verified that
(33)si⋅ṡi=si⋅∂ΓτiMτi∂τi⋅τ̇i−∂Γφ¯iMφ¯i∂φ¯i⋅φ¯̇i=si⋅ϖi−121+ϖi2−∂Γφ¯iMφ¯i∂φ¯i⋅φ¯̇i≤−κ2isi2−e2isi−0.5⋅si⋅1+ϖi2≤−κ2isi2−e2isi.
Case 2. Ifτi<τiM−ε, this leads to
(34)si⋅ṡi=si∂ΓτiMτi∂τiτ̇i−∂Γφ¯iMφ¯i∂φ¯iφ¯̇i≤−κ2isi2−e2isi.
Case 3. Ifτi≤−τiM+ε, then we have si≤0 from (25); it yields
(35)si⋅ṡi=si⋅∂ΓτiMτi∂τi⋅τ̇i−∂Γφ¯iMφ¯i∂φ¯i⋅φ¯̇i=si⋅ϖi+121+ϖi2−∂Γφ¯iMφ¯i∂φ¯i⋅φ¯̇i≤−κ2isi2−e2isi+0.5⋅si⋅1+ϖi2≤−κ2isi2−e2isi.
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 errorse2i−1i=1,2 converge to a prescribed compact set Ωe=e2i−1,e2i−1<ρTS for ∀t>TSProof.
Choose the Lyapunov function candidateV as
(36)V=12⋅∑i=12logρi2ρi2−e2i−12+e2i2+si2+12⋅∑i=12σ~i2ξi,where σ~i=σi−σ^i. The time derivative of V is expressed as
(37)V̇=∑i=1212ρi2−e2i−12ρi2⋅2ρiρ̇iρi2−e2i−12−ρi22ρiρ̇i−2e2i−1ė2i−1ρi2−e2i−122+∑i=12e2i⋅ė2i+si⋅ṡi−∑i=12σ~iσ^̇iξ=∑i=12e2i−1ė2i−1ρi2−e2i−12−ρ̇ie2i−12ρiρi2−e2i−12+e2iė2i+siṡi−∑i=12e2i2σ~i2a12−a1σ~iσ^i.Based on (17), (27), and (29), the following relationships are established in the compact set Ω:
(38)1e2i−1ė2i−1=e2i−1e2i−κ2i−1⋅e2i−12+e2i−12ρ̇iρi,2e2iΨi≤σie2i22a12+a122,3e2iė2i≤e2isi−κ2ie2i2−e2i−1e2iρi2−e2i−12+σ~ie2i22a12+a122,4e2i−12ρi2−e2i−12≥logρi2ρi2−e2i−12,5a1σ~iσ^i≤−a1ξiσ~i22ξi+a1σi22.With the help of (38), then (37) becomes
(39)V̇≤∑i=12−κ2i−1logρi2ρi2−e2i−12−κ2ie2i2−κ2isi2+σ~ie2i22a12+∑i=12−σ~ie2i22a12+a1σ~iσ^i+a12≤∑i=12−κ2i−1logρi2ρi2−e2i−12−κ2ie2i2−κ2isi2−a1ξiσ~i22ξi+∑i=12a1σi22+a12≤−αg⋅V+Δg,where αg=minκ2i,κ2i−1,a1ξi and Δg=a12/2+∑i=12a1σi2/2. Solving (39), we have
(40)0≤Vt≤Δgαg+V0−Δgαg⋅e−αgt.According to (40), it can be seen that each signal of the closed-loop system is bounded, and meanwhile, we can obtain that
(41)12logρi2ρi2−e2i−12≤Δgαg+V0.Taking exponentials on both sides of (41) yields
(42)e2i−12≤ρi21−e−2Δg/αg+V0<ρi2.Therefore, in the light of the properties ofρit, we obtain e2i−1<ρit; it implies e2i−1t<ρTS for t∈TS,∞. 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.Figure 4
The closed-loop system structure.Figure 5
The algorithm flow chart.From the above analysis,e2i−1 remain bounded inside the region Be≜∣e2i−1∣<ρit,i=1,2 for t≥TS. It should be pointed that the reciprocal of ∂ΓτiMτi/∂τi plays an important role in the designed input updating law (30). In fact, the input updating law singularity may occur since ∂ΓτiMτi/∂τi goes to zero as τi⟶∞. However, from (30), we can see that variable τi remains bounded inside the region Bτ≜∣τi∣≤τiM,i=1,2 for t≥0. Hence, the sensibility of the reciprocal of ∂ΓτiMτi/∂τi can be acceptable, and moreover, ∂ΓτiMτi/∂τi has never gone to zero.Remark 14.
To ensure thatρi20>e2i−120 (i=1,2), the design parameter ρ0 (ρ0 is given in (12)) needs to satisfy ρ0>e2i−10. Combining with (40) and (41), we can obtain that e2i−1<ρit for ∀t∈0,∞; it is obvious that the system error constraints are not violated; namely, e2i−1 are always less than the fixed-time performance function.Remark 15.
The designed method is a recursive design process borrowed from the traditional backstepping control design idea. Becauseτi are contained within the function ΓτiMτi, therefore, we cannot design the control laws τi in the form of the function ΓτiMτi. To solve this problem and complete the control law design, sii=1,2 are introduced such that input updating laws τ̇i can be designed by taking the derivative of si. On the other hand, from (26), sii=1,2 have a “supervisory role” when τi exceeding the setting values (i.e., si≥0ifτi≥τiM−ε; si≤0ifτi≤−τiM+ε). With the help of (33)-(35), we have si⋅ṡi≤−κ2isi2−e2isi, which can help us get the inequality result V̇≤−αgV+Δg 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 κ2i, κ2i−1, ξi, and 1/a1i=1,2 result in smaller system errors e2iandsi 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 e2i−1. Therefore, the process of choosing the control parameters κ2i, κ2i−1, ζi, and 1/a1 is significantly simplified(2)
The prescribed performance parameterρ0 should be chosen large enough such that the initial error boundedness can be satisfied, i.e., max∣αmax−x1d0∣,∣hmax−x3d0∣<ρ0. The parameter ρTS determines the final bound of the steady-state error. The parameter TS determines the time for the tracking error to converge to the specified accuracy. Too small TS can lead to a large control amplitude at the beginning of the control. Combined with the actual control tasks, moderate compromise selection of TS and ρTS can eliminate the large amplitude for control output at the beginning of the control processRemark 17.
The prescribed performance control methods [23–26] 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 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 TS is independent of the initial conditions, and therefore, TS can be set according to the requirement of control tasks.


## 4. Numerical Simulations

In this section, simulation results are presented to illustrate the effectiveness of our proposed control method. The system parameters satisfyy=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 α0=0.1rad, h0=0.01m, and α̇0=ḣ0=α¨0=h¨0=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) 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
(43)uASMC=−fn−ks1−∫0tηsignσ+k2s1+s2+φ2σ4+lα∧2+α^4φσdτ,where fn is a known function consisting of the system nominal parameter; s1 and s2 are the system error vectors; k1, k2, η, and φ are design parameters; σ=σ1,σ2T=s2+k×s1+p×∫ots1dτ (k=diagk1,k2>0, l=diagl1,l2>0, and p=diagp1,p2>0); and α^=diagα^1,α^2 is the disturbance estimation.Table 3
The control parameter configuration.
Control methodPPACκ2i−1=κ2i=3,a1=ε=0.1,c=ρ0=0.5,ξi=10,ρTS=8×10−5,Ts=2,τiM=8ASMCk1=k2=5,p1=p2=1,l1=l2=1,εc=0.03,φ=2.5,η=7Firstly, the open-loop responses for sinusoidal gustwG=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.Figure 6
The open-loop responses of the aeroelastic system.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.Case A.
It is assumed thatU=13.28m/s, wG=1−e−0.25Ut/b, and Δy=25%×randt×y¯ (note that randt can generate random numbers in the interval 0,1 by using 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∣e1∣,∣e3∣<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, e2i−1 converge to a prescribed compact set Ωe=e2i−1∣∣e2i−1∣<ρTS, for ∀t>TS based on ρTS=8×10−5 and TS=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∣e1∣,∣e3∣<5.8×10−4. From Figures 10 and 11, it can be seen that the initial amplitude values of control input τi and nonlinear function estimation 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, wG=2.5⋅1−e−0.25Ut/b⋅randt, and Δy=40%×randt×y¯. From Figures 12–16, we see that the proposed method can guarantee that e2i−1i=1,2 converge into a bounded region max∣e1∣,∣e3∣<8×10−5fort>2s, but the errors obtained by ASMC stay in max∣e1∣,∣e3∣<1.8×10−3fort>2s. Obviously, the 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.Figure 7
α responses of the closed-loop system with Case 1.Figure 8
h responses of the closed-loop system with Case 1.Figure 9
s1 and s2 responses with Case 1.Figure 10
σ^1 and σ^2 responses with Case 1.Figure 11
Control input of the proposed method with Case 1.Figure 12
α responses of the closed-loop system with Case 2.Figure 13
h responses of the closed-loop system with Case 2.Figure 14
s1 and s2 responses with Case 2.Figure 15
σ^1 and σ^2 responses with Case 2.Figure 16
Control input of the proposed method with Case 2.


## 5. 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 control method suppresses the limit-cycle oscillations, despite parameter uncertainties and wind gust.



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