# Noncertainty Equivalent Adaptive Backstepping Control for Advanced Fighter Subject to Unsteady Effects and Input Constraints

Authors: Chijun Zhou; Changxin Luo; Jikun Ye; Jihong Zhu; Humin Lei
Journal: International Journal of Aerospace Engineering
 (2021)

---

## Abstract

This paper presents a noncertainty equivalent adaptive backstepping control scheme for advanced fighter attitude tracking, in which unsteady effects, parameter uncertainties, and input constraints are all considered which increase the design difficulty to a large extent. Based on unsteady attitude dynamics and the noncertainty equivalent principle, a new observer is first developed to reconstruct the immeasurable and time-varying unsteady states. Afterwards, the unsteady aerodynamics is compensated in the backstepping controller where the command filter is introduced to impose physical constraints on actuators. In order to further enhance the robustness, the noncertainty equivalent adaptive approach is again used to estimate the uncertain constant parameters. Moreover, stability of the closed-loop system that includes the state observer, parameter estimator, and backstepping controller is proven by the Lyapunov theorem in a unified architecture. Finally, simulation results show that performance of the deterministic control system can be captured when attractive manifolds are achieved. The effectiveness and robustness of the proposed control scheme are verified by the Herbst maneuver.


---

## Body

## 1. Introduction

Flight envelopes are substantially extended by the fourth generation fighter which directly leads to urgent demands for higher performance control laws. Since full envelope dynamics of aircraft is nonlinear and involves wide variations in aerodynamic parameters, traditional linear control methods are incapable of addressing such challenges [1]. Therefore, to ensure adequate stability and tracking performance in extreme flight regimes, a number of nonlinear control techniques have been extensively investigated, such as dynamic inversion [2, 3], fuzzy logic [4], neural network [5], backstepping [6], and sliding mode control [7]. In particular, Lyapunov-based backstepping control is among the most widely studied of these methods. Due to the cascaded structure of aircraft dynamics, many efforts have been made to develop a flight control system via the combination of backstepping theory and other control technologies, such as disturbance observer [8], radial basis function neural network [9], and adaptive control method [10, 11].Generally, adaptive backstepping provides a systematic approach to solve the tracking or regulation problems of uncertain nonlinear systems. However, it must be noted that the adaptive backstepping control methods mentioned above have not considered input constraints which cannot be ignored in practice due to the existence of physical limits on actuators [12]. When aircraft maneuvers at a high angle of attack (AOA), drastic decreases of control authority often causes aerodynamic surface saturation which may severely degrade control performance by giving undesirable inaccuracy or leading to instability [13]. In some applications, this problem is crucial, especially in combination with online approximation-based control, which tends to be aggressive in seeking desired tracking performance [14]. For the purpose of circumventing this dilemma, it is necessary to take the input constraints into account at the level of the control design. In constraint adaptive control, the key issue is how to analyze the constraint effects on a closed-loop system. Toward this end, an auxiliary system of which states are used for feedback control is proposed in [15, 16]. Recently, it was incorporated into backstepping architecture to solve trajectory tracking problems for reentry vehicles and air-breathing hypersonic vehicles in the presence of actuator constraints [17–19]. In addition, the command filter adaptive backstepping control proposed in [14, 20] is also a wise choice to tackle input constraints, in which the command filter is used to calculate the derivatives of virtual controls and impose constraints on states and inputs. This control scheme is presented for the F-16/MATV fighter in [21], where tracking errors in adaptive laws are replaced by compensated errors; hence, the adaption process can be isolated, i.e., not affected by saturation of virtual controls or actuators.Besides model uncertainties and input constraints, unsteady effects are also potential issues that should be considered for an advanced fighter. Unsteady aerodynamic states are immeasurable in practice and may change rapidly during maneuvers which pose additional challenges for control [22]. Due to the difficulties in unsteady aerodynamic modelling, unsteady effects are usually treated as a part of model uncertainties which increase robust requirements for control system design. Based on an unsteady aerodynamic model, an alternative way is to estimate the unsteady effects via state observation and then make compensation in the controller design. Generally, it is not difficult to merely handle state observation, but the complexity increases drastically when uncertainties and input constraints are all considered. In recent years, a novel approach for stabilization and adaptive control of uncertain nonlinear systems based on immersion and invariance (I&I) methodology has been proposed in [23] and then further developed in [24, 25]. This method gives a noncertainty equivalent adaptive (NCEA) law which is different from the traditional certainty equivalent adaptive (CEA) method. The main feature of this approach lies in the construction of an estimator, which is a sum of a partial estimate generated by an update law and a judiciously chosen nonlinear function. The essential idea is to create a manifold in the extended space of states and parameters to which trajectories are attracted. As the trajectory evolves on this manifold, the closed-loop system captures the behavior of a deterministic system [26]. Due to the advantages in prescribing estimate error dynamics and separately synthesizing controller and estimator, this method shows great potential for uncertain nonlinear systems with complex structures. However, it relies on solving a partial differential equation (PDE), which is difficult for multivariable systems [27]. To overcome this difficulty, auxiliary state filter and regressor matrix filter are introduced [28–30]. Recently, its applications to the control of missiles, quad-rotor UAVs, and satellites are studied successively [26, 31, 32]. Apart from flight control, the I&I approach is also applied in aeroelastic control [33] and mechatronic systems [34].In this research, a novel adaptive backstepping control scheme based on noncertainty equivalent principle is proposed for advanced fighter attitude tracking. Compared with our preceding study in [35], the prior hypothesis about the upper bounds of the uncertainties is removed, and the input constraints are dealt with in a simpler way. Moreover, the coupling relationship of the observer and estimator is considered fully in the closed-loop design. Therefore, the main contributions of this paper can be summarized as follows:
(1)
The I&I method is used to estimate not only immeasurable states but also unknown parameters; it is further modified with different forms in these two situations and is closely associated with the characteristics of an advanced fighter(2)
The mutual effects of unsteady states and parameter uncertainties make the closed-loop control design complex. The observer and estimator are coupled together, but the coupling effects can be weakened by selecting appropriate parameters; in other words, the observer and estimator can be designed to satisfy the spectrum separation principle(3)
The proposed control scheme involves a state observer, a constraint backstepping controller, and two parameter estimators which are designed separately, but the stability of the closed-loop system is proven in a unified architectureThe rest of this paper is organized as follows. In the next section, unsteady attitude dynamics is modeled. The third section details the derivation of the state observer. In the fourth section, the constraint adaptive backstepping control design and stability analysis are presented. Simulations and conclusions are given in fifth and sixth sections, respectively.


## 2. Unsteady Attitude Dynamics

The model adopted in this research is a certain advanced fighter which is developed to investigate the flight dynamics during high AOA maneuvers. The fighter attitude dynamics is formulated as(1)V̇tβ̇α̇=0−rsqs−pstanβ+ΛTw/bX¯+Tx+GxY¯+Ty+GyZ¯+Tz+Gz,ṗsq̇sṙs=rsα̇0−psα̇+Ts/bc1r+c2pq+c3LA+LT+c4NA+NTc5pr−c6p2−r2+c7MA+MTc8p−c2rq+c4LA+LT+c9NA+NT,where Λ=diag1/m,1/mVt,1/mVtcosβ, Tw/b,Ts/b represent the rotation matrices from body axis system to wind axis system and stability axis system, respectively, and ci,i=1,2,⋯,9 are inertia parameters defined in [21].The fighter is equipped with two kinds of actuators, i.e., the aerodynamic surfaces and thrust vector engines, all of which are modeled as a first-order filter with limits in magnitude and rate (see Table1). The sketch of the thrust vector control is depicted in Figure 1.Table 1
Physical constraints on actuators.
Control inputsBandwidth (rad/s)Magnitude limit (°)Rate limit (°/s)δa20[-20, 20][-80, 80]δe20[-20, 20][-100, 100]δr20[-25, 25][-120, 120]δyl,δyr,δzl,δzr15[-20, 20][-50, 50]Figure 1
Sketch of the thrust vector control and its relevant variables.The thrust forces and moments are defined as(2)TxTyTz=Tcosδylcosδzl+TcosδyrcosδzrTsinδyl+Tsinδyr−Tcosδylsinδzl−Tcosδyrsinδzr,LTMTNT=TxT0−cosδylsinδzl−cosδyrsinδzr−sinδyl−sinδyr+TyT2cosδylsinδzl−cosδyrsinδzr0cosδylcosδzl−cosδyrcosδzr.The aerodynamic model is extracted from wind tunnel tests which are conducted on sufficiently close points to capture the nonlinear behavior of the aerodynamics. Based on linear superposition principle, the aerodynamic forces and moments are expressed by(3)X¯=QSC¯X,Y¯=QSC¯Y+CYδaδa+CYδrδr+ηY,Z¯=QSC¯Z+CZδeδe+ηZ,LA=QSbC¯l+Clδaδa+Clδrδr+ηl,MA=QSc¯C¯m+Cmδeδe+ηm,NA=QSbC¯n+Cnδaδa+Cnδrδr+ηn,where ηY,ηZ,ηl,ηm,ηn represent the unsteady states, and the expressions of C¯i,i=X,Y,Z,l,m,n, are given in [36]. Unsteady effects on the axial force are neglected since the axial force measured in the test is small in contrast to other aerodynamic forces [36].Usually, the quasisteady model in which unsteady states are equal to zero can be employed to describe the aerodynamics in normal flight condition. However, owing to notable effects caused by separated and vortical flow, it becomes inadequate to describe the aircraft dynamics at high AOA. Thus, unsteady aerodynamics are introduced and modeled in the following form [37, 38]:
(4)η̇=hx−Bση,where η=ηY,ηZ,ηl,ηm,ηnT is the immeasurable unsteady state; hx=aYβ̇,aZα̇,alβ̇,amα̇,anβ̇T; Bσ=diagbY,bZ,bl,bm,bn, bi>0; and ai,bi,i=Y,Z,l,m,n, are fitting polynomials given in [36]. Note that the unsteady effects will converge to zero exponentially as the maneuver ends since (4) is stable.


## 3. Unsteady State Observation

The main objective of this section is to develop a nonlinear state observer which precisely reconstructs the unsteady states in the presence of parameter uncertainties. To this end, the attitude dynamics used for state observation and control design are rewritten as(5)σ̇=f1x+W1xη+G1ω¯+D1σθ1,ω̇=f2x+W2xη+G2σu+D2σθ2,η̇=hx−Bση, where σ=α,βT is the incidence angle, ω=ps,qs,rsT represents the stability axis angular rate, x=σT,ωTT denotes the measurable state, G1=diag1,−1, θ1=ΔCX,ΔCY,ΔCZT and θ2=ΔCl,ΔCm,ΔCnT are constant parameter uncertainties, ω¯=qs,rsT represents the intermediate control variable, u=δa,δe,δr,δyl,δyr,δzl,δzrT denotes the control input, and f1x,f2x,W1x,W2x,G2σ,D1σ,D2σ are vectors or matrices with appropriate dimensions (see appendix).First of all, related assumptions used in the subsequent developments are given below.Assumption 1.
The desired trajectoriesαd,βd,psd and their derivatives are bounded. The compact set is defined as follows:
(6)Ω1=αd,βd,psdαd+βd+psd+α̇d+β̇d+ṗsd<ϖ,where ϖ is a known positive constant and ⋅ stands for the absolute value of a scalar.Assumption 2.
For matricesW1x,W2x,D1σ,D2σ, there exist positive constants τ1i,τ2i,τ3i,ρ1i,ρ2i>0,i=1,2, such that Wix≤τ1i,Ẇix≤τ2i,W¨ix≤τ3i,Diσ≤ρ1i, Ḋiσ≤ρ2i,∀x∈Ω2 with compact subset Ω2 containing the origin, and ⋅means the two norms of a vector or matrix.To achieve precise reconstruction of unsteady states, a state filter is first introduced:(7)σ^̇=f1x+W1xη^+G1ω¯+D1σθ^1−K1z1,ω^̇=f2x+W2xη^+G2σu+D2σθ^2−K2z2,where K1,K2 are positive definite matrices to be designed, η^ denotes the estimate of η, z1=σ^−σ,z2=ω^−ω are filter errors, and θ^1,θ^2 are estimates of θ1,θ2, respectively. According to (5) and (7), dynamics of z1,z2 take the form as
(8)ż1=W1xη~+D1σθ~1−K1z1,ż2=W2xη~+D2σθ~2−K2z2,where η~=η^−η denotes the reconstruction error, θ~1=θ^1−θ1 and θ~2=θ^2−θ2 are estimate errors of uncertain parameters.The state observer based on I&I theory is proposed as follows:(9)η^=φ0+μ0z,Wx,where φ0 denotes the observer state, μ0z,Wx is a smooth nonlinear function, z=z1T,z2TT, Wx=W1Tx,W2TxT. Taking the time derivative of η~ and using (8), we can get
(10)η~̇=φ̇0+∑i=12∂μ0z,Wx∂ziWixη~+Diσθ~i−Kizi+∑i=12∂μ0z,Wx∂WixẆix−hx+Bση.The nonlinear functionμ0z,Wx is selected so that η~ has a stable behavior. So we choose
(11)∂μ0z,Wx∂z1=−γ0W1Tx,∂μ0z,Wx∂z2=−γ0W2Tx,with γ0>0. Therefore, the nonlinear function is selected as
(12)μ0z,Wx=−γ0W1Txz1−γ0W2Txz2.Then, substituting (12) into (10), we obtain
(13)η~̇=φ̇0−γ0∑i=12WiTxWixη~+Diσθ~i−Kizi−∑i=12γ0ẆiTxzi−hx+Bση.In (13), Ẇix contains the immeasurable state η and uncertain parameters θ1,θ2 which cannot be used directly. To overcome these difficulties, a command filter is employed to provide filtered derivative of Wix; the filter dynamics is expressed by
(14)W¯¨ix+2ςωnW¯̇ix+ωn2W¯ix=ωn2Wix,in which the initial condition satisfies W¯ix0=Wix0,W¯̇ix0=0. Based on (13) and (14), the adaptation law is designed as
(15)φ̇0=γ0∑i=12W¯̇iTxzi−WiTxKizi−WiTxzi+hx−Bση^.Substituting (15) into (13), we can get
(16)η~̇=−γ0∑i=12WiTxWixη~+WiTxDiσθ~i−ΔẆiTxzi+WiTxzi−Bση~,in which ΔẆix=W¯̇ix−Ẇix. According to Assumption 2, there exists a positive constant κ which depends on τ21,τ31,τ22,τ32,ς,ωn such that ΔẆ1x≤κ, ΔẆ2x≤κ [39].Consider the following Lyapunov function candidate(17)V1=12γ0−1η~Tη~+12z1Tz1+12z2Tz2.Using (8) and (16), the time derivative of V1 yields
(18)V̇1=−∑i=12η~TWiTxWixη~+η~TWiTxDiσθ~i−η~TΔẆiTxzi+ziTKizi−ziTDiσθ~i−γ0−1η~TBση~.Using Young’s inequality, we can get(19)η~TWiTxDiσθ~i≤a1Wixη~2+14a1Diσθ~i2,η~TΔẆiTxzi≤ziΔẆixη~≤a2zi2+κ24a2η~2,ziTDiσθ~i≤a3zi2+14a3Diσθ~i2,where a1∈0,1,a2,a3>0. Then, (18) is modified as
(20)V̇1≤−1−a1Wxη~2−γ0−1λminBσ−κ22a2η~2−λminK1−a2−a3z12−λminK2−a2−a3z22+14a1+14a3D1σθ~12+14a1+14a3D2σθ~22,with Wxη~2=W1xη~2+W2xη~2 and λmin⋅ is the minimum eigenvalue of matrix. Note if D1σθ~1=D2σθ~2=0 and the following inequalities hold
(21)γ0−1λminBσ−κ22a2>0,minλminK1,λminK2>a2+a3,then V̇1≤0, which implies that the state observation process is asymptotically stable. In the following parameter estimation, θ^1,θ^2 will be determined to regulate D1σθ~1=D2σθ~2=0.Remark 3.
The differences between the preceding studies [26, 27, 29–33] and this research lie in the following two aspects:
(1)
The I&I approach adopted in this section is used to reconstruct the immeasurable and time-varying states(2)
The filtered derivative of the regressor matrix, instead of filtered regressor matrix itself, is employed to construct the observer to avoid solving PDE


## 4. Constraint Adaptive Backstepping Control

In this section, a constraint adaptive backstepping controller based on I&I method is developed. The control objective is to track the predefined trajectories in the presence of input constraints and parameter uncertainties.


### 4.1. Constraint Backstepping Control

The constraint backstepping control design procedure is initiated by defining the following tracking errors:(22)e1=σ−σr,e¯2=ω¯−ω¯r,where σr=αr,βrT is the reference trajectory generated by the desired command σd=αd,βdT and ω¯r=qsr,rsrT is the virtual control law produced by ω¯d=qsd,rsdT.To eliminate the “explosion of term” problem,ω¯d is filtered via a command filter to provide ω¯r and ω¯̇r. The difference between ω¯r and ω¯d is evaluated by an auxiliary filter
(23)ξ̇1=−A1ξ1+G1ω¯r−ω¯d+G¯1ξ2,where A1>0, G¯1=02×1,G1, ξ1 is the filter state, and ξ2 will be defined later. The compensated errors are defined by
(24)ε1=e1−ξ1,ε2=e2−ξ2,where e2=ps−psr,e¯2TT, and psr denotes the reference trajectory of stability axis roll rate generated by psd. Taking time derivative of e1 and using (5), we obtain
(25)ė1=f1x+W1xη+D1σθ1−σ̇r+G1e¯2+G1ω¯r−ω¯d+G1ω¯d.Therefore, the nominal virtual control lawω¯d is given by
(26)ω¯d=G1−1−f1x−W1xη^−D1σθ^1+σ̇r−A1e1.Combining (23), (24), (25), and (26), the dynamics of ε1 takes the form as
(27)ε̇1=−W1xη~−D1σθ~1+G¯1ε2−A1ε1.Similar to (26), the nominal control law ud for angular rate subsystem is designed as
(28)ud=G2+σ−f2x−W2xη^−D2σθ^2+ω̇r−A2e2−G¯1Tε1, where G2+σ is pseudoinverse of G2σ, ω̇r=ṗsr,ω¯̇rTT, ṗsr can be obtained by filtering psd.To impose physical constraints on actuators,ud is filtered via a command filter to provide the physical limited u. The command filter integrated with magnitude and rate limits is described as [14]
(29)u¨=2ςωnSRωn2ςSMud−u−u̇,where SM⋅,SR⋅ are magnitude and rate saturation function, respectively. The impact of (29) can be evaluated by
(30)ξ̇2=−A2ξ2+G2σu−ud,where A2>0. Combining (5), (24), (28), and (30), the dynamics of ε2 is expressed by
(31)ε̇2=−W2xη~−D2σθ~2−G¯1Tε1−A2ε2.Then, the following Lyapunov function candidate is considered:(32)V2=12ε1Tε1+12ε2Tε2.Taking the time derivative ofV2 along (27) and (31) and using Young’s inequality, we have
(33)V̇2=ε1T−W1xη~−D1σθ~1+G¯1ε2−A1ε1+ε2T−W2xη~−D2σθ~2−G¯1Tε1−A2ε2≤−λminA1−b1−b3ε12−λminA2−b2−b4ε22+14b1W1xη~2+14b2W2xη~2+14b3D1σθ~12+14b4D2σθ~22,where b1,b2,b3,b4>0.


### 4.2. Uncertain Parameter Estimation

Now the design of the parameter estimator is considered. This process is completed by deriving the nonlinear functions and adaptation laws for partial estimates.The estimator ofθ1 based on the I&I approach is constructed as
(34)θ^1=φ1+μ1ε1,D1σ,where φ1 is the partial estimate of θ1 and μ1ε1,D1σ denotes the nonlinear function of ε1 and D1σ. Taking the time derivative of θ^1 and combining (27), we can get
(35)θ~̇1=∂μ1ε1,D1σ∂ε1−W1xη~−D1σθ~1+G¯1ε2−A1ε1+∂D1σ∂αα̇+∂D1σ∂ββ̇T∂μ1ε1,D1σ∂D1σ+φ̇1.To ensure thatθ~1 has a stable behavior, the nonlinear function μ1ε1,D1σ is chosen as
(36)μ1ε1,D1σ=γ1D1Tσε1,where γ1>0. In view of (35) and (36), if φ1 is updated by
(37)φ̇1=γ1D1Tσ−G¯1ε2+A1ε1−γ1∂D1σ∂αα̇+∂D1σ∂ββ̇Tε1,then substituting (37) into (35), we can get
(38)θ~̇1=−γ1D1TσD1σθ~1−γ1D1TσW1xη~.For angular rate subsystem, the estimator ofθ2 is given as follows
(39)θ^2=φ2+μ2ε2,D2σ,with
(40)μ2ε2,D2σ=γ2D2Tσε2,(41)φ̇2=γ2D2TσG¯1Tε1+A2ε2−γ2∂D2σ∂αα̇+∂D2σ∂ββ̇Tε2,where γ2>0. Combining (39), (40), and (41), the dynamics of θ~2 can be expressed by
(42)θ~̇2=−γ2D2TσD2σθ~2−γ2D2TσW2xη~.Define the Lyapunov function candidate(43)V3=12γ1−1θ~1Tθ~1+12γ2−1θ~2Tθ~2.DifferentiatingV3 and invoking (38) and (42), we have
(44)V̇3=−∑i=12θ~iTDiTσDiσθ~i+θ~iTDiTσWixη~≤−1−b1D1σθ~12−1−b2D2σθ~22+14b1W1xη~2+14b2W2xη~2.Remark 4.
Assume that the state observer performs perfectly in reconstructing the unsteady states, thenη~=0 can be achieved. Hence, according to (38), the dynamics of θ~1 can be treated as a linear time-varying system given by θ~̇1t=−γ1D1TσtD1σtθ~1t. Note that if at some instant t1, θ~1t1=0, thenθ~1t=0 for t≥t1. That is, the manifold defined by Ω3=x,η,tφ1+μ1ε1,D1σ−θ1=0 is an invariant manifold. Therefore, the parameter estimate φ1+μ1ε1,D1σ remains frozen at its actual value for t≥t1. This invariant characteristic also holds for θ~2 dynamics.Remark 5.
Note that availability ofα̇,β̇ is assumed in the estimator design; otherwise, we will need to solve PDE. Although α̇,β̇ cannot be measured directly, we could attempt to compute via velocity components and incidence angles.Remark 6.
According to (38) and (42), we can find that the state reconstruction error η~ may affect the estimator performance, that is to say, the observer and estimator are coupled together. To weaken the effects between each other, the design parameters of the observer and estimator need to be chosen appropriately. In practice, the reconstruction error η~ should converge much more quickly than the estimate errors θ~1,θ~2 to satisfy the spectrum separation principle.


### 4.3. Stability Analysis

The stability analysis of closed-loop system is initiated by the following theorem.Theorem 7.
Consider the closed-loop system under the foregoing assumptions, with application of the state filter (7), state observer (9), constraint backstepping controller (26) and (28), and the parameter estimator (34) and (39), Wxη~,η~, z1,z2,ε1,ε2,D1σθ~1,D2σθ~2 can be guaranteed to converge to zero.Proof.
The Lyapunov function is chosen as(45)V4=V1+V2+V3.
Combining (20), (33), and (44), the time derivative of V4 yields
(46)V̇4≤−1−a1Wxη~2−γ0−1λminBσ−κ22a2−τ1122b1−τ1222b2η~2−λminK1−a2−a3z12−λminK2−a2−a3z22−λminA1−b1−b3ε12−λminA2−b2−b4ε22−1−b1−14a1−14a3−14b3D1σθ~12−1−b2−14a1−14a3−14b4D2σθ~22.
To ensure stability of the closed-loop system, we select appropriate matricesA1,A2,K1,K2 and adaptation gain γ0 to satisfy the following inequalities:
(47)γ0−1λminBσ>κ22a2+τ1122b1+τ1222b2,λminA1>b1+b3,λminA2>b2+b4,b1+14a1+14a3+14b3<1,b2+14a1+14a3+14b4<1,and inequalities (21), then V̇4≤0 can be achieved. Thus, we can conclude that all closed-loop signals are bounded based on Assumptions 1 and 2. Further, because V4 is lower-bounded and monotonic by the negative-semidefiniteness of V̇4, we know that ∫0∞V̇4tdt exists and is finite, which in turn implies
(48)Wxη~,η~,z1,z2,ε1,ε2,D1σθ~1,D2σθ~2∈L2∩L∞.
SinceWx,D1σ,D2σ and their derivatives are all bounded, according to Barbalat’s lemma, it follows that
(49)limt→∞Wxη~,η~,z1,z2,ε1,ε2,D1σθ~1,D2σθ~2=0.
Thus,Wxη~,η~,z1,z2,ε1,ε2,D1σθ~1,D2σθ~2 can be guaranteed to converge to zero.According to Theorem7, Wxη~,D1σθ~1,D2σθ~2 asymptotically converges to zero. Thus, the manifold defined by
(50)Ω7=Ω1∩Ω2∩Ω4∩Ω5∩Ω6,with
(51)Ω4=Wxφ0+μ0z,Wx−η=0,Ω5=D1σφ1+μ1ε1,D1σ−θ1=0,Ω6=D2σφ2+μ2ε2,D2σ−θ2=0,is an attractive manifold. Along any trajectories evolving on Ω7, the dynamics of ε1,ε2can be described by
(52)ε̇1=G¯1ε2−A1ε1,ε̇2=−G¯1Tε1−A2ε2.Apparently, (52) can be obtained if a controller based on attitude dynamics with measurable unsteady states and known aerodynamic parameters is implemented. Hence, the constraint backstepping controllers (26) and (28) capture the performance of the deterministic control system. From (52), it can be easily seen
(53)V̇2=−ε1TA1ε1−ε2TA2ε2≤−λminA1ε12−λminA2ε22,which implies thatε1,ε2 will converge to zero as expected.Remark 8.
For the closed-loop stability analysis, two scenarios are considered:(1)
When actuator constraints are not in effect, note the fact thatω¯r−ω¯d and u−ud are bounded, then the auxiliary filter states ξ1,ξ2 converge to a sufficient small domain around origin by selecting appropriate natural angular frequency ωn and control matrices A1,A2. Therefore, from (24), it is obvious that the tracking errors e1,e2 can be guaranteed to be bounded. Consequently, the stability of the closed-loop system can be ensured(2)
When input saturation occurs, the primary control object is to maintain the stability of online approximation process [14]. It is obvious that the tracking errors may increase while the compensated errors still converge to zero when saturation happens. Through replacing the tracking errors by compensated errors in the parameter estimation, the stability of the adaptation process can be ensured not affected by the saturationIn conclusion, the proposed control scheme consists of three modules: unsteady states observation, constraint backstepping control, and parameter estimation. The overview of the design architecture is depicted in Figure2.Figure 2
Overview of the control system architecture.


## 4.1. Constraint Backstepping Control

The constraint backstepping control design procedure is initiated by defining the following tracking errors:(22)e1=σ−σr,e¯2=ω¯−ω¯r,where σr=αr,βrT is the reference trajectory generated by the desired command σd=αd,βdT and ω¯r=qsr,rsrT is the virtual control law produced by ω¯d=qsd,rsdT.To eliminate the “explosion of term” problem,ω¯d is filtered via a command filter to provide ω¯r and ω¯̇r. The difference between ω¯r and ω¯d is evaluated by an auxiliary filter
(23)ξ̇1=−A1ξ1+G1ω¯r−ω¯d+G¯1ξ2,where A1>0, G¯1=02×1,G1, ξ1 is the filter state, and ξ2 will be defined later. The compensated errors are defined by
(24)ε1=e1−ξ1,ε2=e2−ξ2,where e2=ps−psr,e¯2TT, and psr denotes the reference trajectory of stability axis roll rate generated by psd. Taking time derivative of e1 and using (5), we obtain
(25)ė1=f1x+W1xη+D1σθ1−σ̇r+G1e¯2+G1ω¯r−ω¯d+G1ω¯d.Therefore, the nominal virtual control lawω¯d is given by
(26)ω¯d=G1−1−f1x−W1xη^−D1σθ^1+σ̇r−A1e1.Combining (23), (24), (25), and (26), the dynamics of ε1 takes the form as
(27)ε̇1=−W1xη~−D1σθ~1+G¯1ε2−A1ε1.Similar to (26), the nominal control law ud for angular rate subsystem is designed as
(28)ud=G2+σ−f2x−W2xη^−D2σθ^2+ω̇r−A2e2−G¯1Tε1, where G2+σ is pseudoinverse of G2σ, ω̇r=ṗsr,ω¯̇rTT, ṗsr can be obtained by filtering psd.To impose physical constraints on actuators,ud is filtered via a command filter to provide the physical limited u. The command filter integrated with magnitude and rate limits is described as [14]
(29)u¨=2ςωnSRωn2ςSMud−u−u̇,where SM⋅,SR⋅ are magnitude and rate saturation function, respectively. The impact of (29) can be evaluated by
(30)ξ̇2=−A2ξ2+G2σu−ud,where A2>0. Combining (5), (24), (28), and (30), the dynamics of ε2 is expressed by
(31)ε̇2=−W2xη~−D2σθ~2−G¯1Tε1−A2ε2.Then, the following Lyapunov function candidate is considered:(32)V2=12ε1Tε1+12ε2Tε2.Taking the time derivative ofV2 along (27) and (31) and using Young’s inequality, we have
(33)V̇2=ε1T−W1xη~−D1σθ~1+G¯1ε2−A1ε1+ε2T−W2xη~−D2σθ~2−G¯1Tε1−A2ε2≤−λminA1−b1−b3ε12−λminA2−b2−b4ε22+14b1W1xη~2+14b2W2xη~2+14b3D1σθ~12+14b4D2σθ~22,where b1,b2,b3,b4>0.


## 4.2. Uncertain Parameter Estimation

Now the design of the parameter estimator is considered. This process is completed by deriving the nonlinear functions and adaptation laws for partial estimates.The estimator ofθ1 based on the I&I approach is constructed as
(34)θ^1=φ1+μ1ε1,D1σ,where φ1 is the partial estimate of θ1 and μ1ε1,D1σ denotes the nonlinear function of ε1 and D1σ. Taking the time derivative of θ^1 and combining (27), we can get
(35)θ~̇1=∂μ1ε1,D1σ∂ε1−W1xη~−D1σθ~1+G¯1ε2−A1ε1+∂D1σ∂αα̇+∂D1σ∂ββ̇T∂μ1ε1,D1σ∂D1σ+φ̇1.To ensure thatθ~1 has a stable behavior, the nonlinear function μ1ε1,D1σ is chosen as
(36)μ1ε1,D1σ=γ1D1Tσε1,where γ1>0. In view of (35) and (36), if φ1 is updated by
(37)φ̇1=γ1D1Tσ−G¯1ε2+A1ε1−γ1∂D1σ∂αα̇+∂D1σ∂ββ̇Tε1,then substituting (37) into (35), we can get
(38)θ~̇1=−γ1D1TσD1σθ~1−γ1D1TσW1xη~.For angular rate subsystem, the estimator ofθ2 is given as follows
(39)θ^2=φ2+μ2ε2,D2σ,with
(40)μ2ε2,D2σ=γ2D2Tσε2,(41)φ̇2=γ2D2TσG¯1Tε1+A2ε2−γ2∂D2σ∂αα̇+∂D2σ∂ββ̇Tε2,where γ2>0. Combining (39), (40), and (41), the dynamics of θ~2 can be expressed by
(42)θ~̇2=−γ2D2TσD2σθ~2−γ2D2TσW2xη~.Define the Lyapunov function candidate(43)V3=12γ1−1θ~1Tθ~1+12γ2−1θ~2Tθ~2.DifferentiatingV3 and invoking (38) and (42), we have
(44)V̇3=−∑i=12θ~iTDiTσDiσθ~i+θ~iTDiTσWixη~≤−1−b1D1σθ~12−1−b2D2σθ~22+14b1W1xη~2+14b2W2xη~2.Remark 4.
Assume that the state observer performs perfectly in reconstructing the unsteady states, thenη~=0 can be achieved. Hence, according to (38), the dynamics of θ~1 can be treated as a linear time-varying system given by θ~̇1t=−γ1D1TσtD1σtθ~1t. Note that if at some instant t1, θ~1t1=0, thenθ~1t=0 for t≥t1. That is, the manifold defined by Ω3=x,η,tφ1+μ1ε1,D1σ−θ1=0 is an invariant manifold. Therefore, the parameter estimate φ1+μ1ε1,D1σ remains frozen at its actual value for t≥t1. This invariant characteristic also holds for θ~2 dynamics.Remark 5.
Note that availability ofα̇,β̇ is assumed in the estimator design; otherwise, we will need to solve PDE. Although α̇,β̇ cannot be measured directly, we could attempt to compute via velocity components and incidence angles.Remark 6.
According to (38) and (42), we can find that the state reconstruction error η~ may affect the estimator performance, that is to say, the observer and estimator are coupled together. To weaken the effects between each other, the design parameters of the observer and estimator need to be chosen appropriately. In practice, the reconstruction error η~ should converge much more quickly than the estimate errors θ~1,θ~2 to satisfy the spectrum separation principle.


## 4.3. Stability Analysis

The stability analysis of closed-loop system is initiated by the following theorem.Theorem 7.
Consider the closed-loop system under the foregoing assumptions, with application of the state filter (7), state observer (9), constraint backstepping controller (26) and (28), and the parameter estimator (34) and (39), Wxη~,η~, z1,z2,ε1,ε2,D1σθ~1,D2σθ~2 can be guaranteed to converge to zero.Proof.
The Lyapunov function is chosen as(45)V4=V1+V2+V3.
Combining (20), (33), and (44), the time derivative of V4 yields
(46)V̇4≤−1−a1Wxη~2−γ0−1λminBσ−κ22a2−τ1122b1−τ1222b2η~2−λminK1−a2−a3z12−λminK2−a2−a3z22−λminA1−b1−b3ε12−λminA2−b2−b4ε22−1−b1−14a1−14a3−14b3D1σθ~12−1−b2−14a1−14a3−14b4D2σθ~22.
To ensure stability of the closed-loop system, we select appropriate matricesA1,A2,K1,K2 and adaptation gain γ0 to satisfy the following inequalities:
(47)γ0−1λminBσ>κ22a2+τ1122b1+τ1222b2,λminA1>b1+b3,λminA2>b2+b4,b1+14a1+14a3+14b3<1,b2+14a1+14a3+14b4<1,and inequalities (21), then V̇4≤0 can be achieved. Thus, we can conclude that all closed-loop signals are bounded based on Assumptions 1 and 2. Further, because V4 is lower-bounded and monotonic by the negative-semidefiniteness of V̇4, we know that ∫0∞V̇4tdt exists and is finite, which in turn implies
(48)Wxη~,η~,z1,z2,ε1,ε2,D1σθ~1,D2σθ~2∈L2∩L∞.
SinceWx,D1σ,D2σ and their derivatives are all bounded, according to Barbalat’s lemma, it follows that
(49)limt→∞Wxη~,η~,z1,z2,ε1,ε2,D1σθ~1,D2σθ~2=0.
Thus,Wxη~,η~,z1,z2,ε1,ε2,D1σθ~1,D2σθ~2 can be guaranteed to converge to zero.According to Theorem7, Wxη~,D1σθ~1,D2σθ~2 asymptotically converges to zero. Thus, the manifold defined by
(50)Ω7=Ω1∩Ω2∩Ω4∩Ω5∩Ω6,with
(51)Ω4=Wxφ0+μ0z,Wx−η=0,Ω5=D1σφ1+μ1ε1,D1σ−θ1=0,Ω6=D2σφ2+μ2ε2,D2σ−θ2=0,is an attractive manifold. Along any trajectories evolving on Ω7, the dynamics of ε1,ε2can be described by
(52)ε̇1=G¯1ε2−A1ε1,ε̇2=−G¯1Tε1−A2ε2.Apparently, (52) can be obtained if a controller based on attitude dynamics with measurable unsteady states and known aerodynamic parameters is implemented. Hence, the constraint backstepping controllers (26) and (28) capture the performance of the deterministic control system. From (52), it can be easily seen
(53)V̇2=−ε1TA1ε1−ε2TA2ε2≤−λminA1ε12−λminA2ε22,which implies thatε1,ε2 will converge to zero as expected.Remark 8.
For the closed-loop stability analysis, two scenarios are considered:(1)
When actuator constraints are not in effect, note the fact thatω¯r−ω¯d and u−ud are bounded, then the auxiliary filter states ξ1,ξ2 converge to a sufficient small domain around origin by selecting appropriate natural angular frequency ωn and control matrices A1,A2. Therefore, from (24), it is obvious that the tracking errors e1,e2 can be guaranteed to be bounded. Consequently, the stability of the closed-loop system can be ensured(2)
When input saturation occurs, the primary control object is to maintain the stability of online approximation process [14]. It is obvious that the tracking errors may increase while the compensated errors still converge to zero when saturation happens. Through replacing the tracking errors by compensated errors in the parameter estimation, the stability of the adaptation process can be ensured not affected by the saturationIn conclusion, the proposed control scheme consists of three modules: unsteady states observation, constraint backstepping control, and parameter estimation. The overview of the design architecture is depicted in Figure2.Figure 2
Overview of the control system architecture.


## 5. Numerical Simulations and Discussions

This section presents several numerical simulations which are carried out to investigate the performance of the proposed control scheme. The comparisons of the NCEA and CEA laws are completed in the first simulation. The second simulation is performed to examine the effectiveness of the proposed control scheme via the Herbst maneuver. Both the simulations are conducted in the MATLAB/Simulink environment, where the step size is set to 5 ms. The parameters of the fighter are given in [36].


### 5.1. Comparisons of NCEA and CEA Laws

In this subsection, comparisons of the constraint filter backstepping controller based on the NCEA and CEA laws are performed to evaluate the adaptation effects on the closed-loop system. The parameters of the proposed controller are chosen asK1=diag8,8,K2=diag20,20,20,A1=diag3,3,A2=diag6,6,6, γ0=5×10−3,γ1=1×103,γ2=2×10−3,ς=0.8,ωn=30rad/s. The initial estimate values of the unsteady states and uncertain parameters are all set to zero. Besides the desired command of AOA, the commands of β,ps are set to zero to neglect the lateral motion for simplicity. Meanwhile, parameter uncertainties are added into CZ,Cm to simulate model errors. The equilibrium states at α=7.50° are chosen as the initial condition.To illustrate the essence of the NCEA law, adaptive backstepping control approach proposed in [14] is adopted for comparison, the CEA laws are given as
(54)η^̇=hx−Bση^−WTxz,θ^̇1=γ3D1Tσε1,θ^̇2=γ4D2Tσε2,where γ3,γ4 are adaptation gains to be assigned. Consider that the transient control performance is highly dependent on adaptation gains, four sets of gains with different adaptation rates are selected, that is,Case 1.
(slow):γ3=30,γ4=0.003.Case 2.
(medium):γ3=100,γ4=0.01.Case 3.
(fast):γ3=500,γ4=0.3.In order to compare the performance of different adaptive laws, the root-mean-square (RSME) of tracking errors and the energy consumption (EC) are chosen as the evaluation indexes which are defined as(55)RSME=1n∑i=1nαi−αri2+βi−βri2+psi−psri2,EC=∫0tfLcp+Mcq+Ncrdt,where the superscript i means the ith sampling point, n denotes the number of samples, tf denotes the final time of simulation, and Lc,Mc,Nc are the commands of total control moment in each channel.Simulation results are shown in Figures3–6 and Table 2. It should be pointed out that the unsteady states are perfectly reconstructed with fast converge rates for both adaptive laws; therefore, the results are not presented herein. From Figures 3 and 4 and Table 2, it is apparent that the NCEA law outperforms the three CEA laws both in command tracking and energy consumption. The track performance of the CEA laws is sensitive to the adaptation gains. When compared with the case 1, the CEA law in case 3 achieves higher estimate precision and faster converging rate at the expense of sacrificing the energy consumption and transient control performance. It can be observed from Figure 5 that D1σθ~1,D2σθ~2 are converged to zero; thus, the closed-loop trajectory eventually confines to the manifold Ω7, and the NCEA law recovers the performance of the deterministic backstepping controller. Figure 6 shows the time history of parameter estimation; it is obvious that the parameters estimated by CEA laws drift from their true values, especially in case 1. For the NCEA law, ΔCm can be precisely estimated while this is not the case for ΔCZ. The reason is that the rank of D2σ equals to θ~2, then D2σθ~2=0 means θ~2=0; therefore, the invariant manifold is achieved and θ2 can be estimated accurately. However, the rank of D1σ is less than θ~1; hence, θ~1=0 cannot be guaranteed via D1σθ~1=0. In addition, it can be seen that θ^1 is frozen after t=15s at which instant the attractive manifold D1σθ~1=0 is achieved.Figure 3
Time history of the command tracking.Figure 4
Time histories of the control inputs.Figure 5
Time histories of the norms.Figure 6
Estimation of the uncertain parameters.Table 2
The performance indexes of the NCEA and CEA laws.
MethodsRMSE (°)Maximum error (°)EC (J)Maximumδe (°)Maximumδz (°)NCEA0.32051.652.53×10512.158.05CEACase 10.81264.182.64×10519.757.72Case 20.44862.102.75×10513.808.50Case 31.1203.455.55×10514.0510.9


### 5.2. Robust Analysis under Parameter Uncertainties and Unsteady Effects

To justify the effectiveness of the proposed control scheme, three different cases are considered for comparison. In the first case, the controller proposed in Section4 is adopted in which the effects of unsteady aerodynamics and uncertain parameters are both compensated. For the second case, the effects of uncertain parameters are not considered in the controller design. The third case is the worst case in which both the unsteady effects and uncertain parameters are not compensated. The design parameters of the controller and initial condition are all kept the same as that in the previous subsection. The uncertainties of aerodynamic forces and moments in longitudinal and lateral channels are all considered in this simulation. Moreover, the Herbst maneuver is implemented to validate the control performance; the commands of αd,βd,psd are defined in [35].The commands tracked in these three cases are compared in Figures7 and 8, and the control performance of the proposed adaptive backstepping controller in the first case is presented in Figures 9–14. In Figure 14, e1i,ε1i,ξ1i,i=1,2, are components of e1,ε1,ξ1, respectively; e2i,ε2i,ξ2i,i=1,2,3 are components of e2,ε2,ξ2, respectively. It can be observed from Figures 7 and 8 that the tracking performance of the first case outperforms the other two cases. It is apparent that smaller steady tracking errors and better transient tracking performances can be achieved since the effects of unsteady aerodynamics and uncertain parameters are both compensated. It also can be concluded that the unsteady aerodynamics seriously affects the aircraft dynamics and the compensation of unsteady effects could significantly improve the control performance. Note that the thrust force presented in Figure 9 increases drastically in the maneuver; it is quite rational for the pilot to increase the throttle to provide a larger thrust force when the fighter operates at high AOA. In addition, the aerodynamic surfaces are all set to zero when α>55° and the attitude control was completed via thrust vector merely in this deep stall area. From Figures 10–12, it can be seen that the tracking errors increase while the compensated errors still converge to zero even though thrust vector nozzles are saturated. Moreover, the proposed observer is effective in state reconstruction with strong robustness against parameter uncertainties, especially in the lateral channel. Meanwhile, the parameter estimator works fairly well in the presence of unsteady effects. The most essential reason can be concluded that the converging rate of state reconstruction error is much larger than the estimate error, thus the mutual effects nearly can be ignored. Figure 13 shows the uncertain parameters in aerodynamic moments are estimated exactly, but steady errors exist in the estimation of aerodynamic force parameters. This is consistent with the discussion in previous subsection. However, as presented in Figure 14, Wxη~,D1σθ~1, and D2σθ~2 all converge to zero after t=25.5s; the attractive manifold is achieved and the norms of the estimated states and parameters are maintained. Therefore, the performance of the deterministic control system is captured by the proposed constraint adaptive backstepping controller.Figure 7
Time history of the longitudinal command tracking.Figure 8
Time histories of the lateral command tracking.Figure 9
Time histories of the control inputs.Figure 10
Related errors in angular subsystem.Figure 11
Related errors in angular rate subsystem.Figure 12
Observation of the unsteady aerodynamic states.Figure 13
Estimation of the uncertain parameters.Figure 14
Time histories of the norms.


## 5.1. Comparisons of NCEA and CEA Laws

In this subsection, comparisons of the constraint filter backstepping controller based on the NCEA and CEA laws are performed to evaluate the adaptation effects on the closed-loop system. The parameters of the proposed controller are chosen asK1=diag8,8,K2=diag20,20,20,A1=diag3,3,A2=diag6,6,6, γ0=5×10−3,γ1=1×103,γ2=2×10−3,ς=0.8,ωn=30rad/s. The initial estimate values of the unsteady states and uncertain parameters are all set to zero. Besides the desired command of AOA, the commands of β,ps are set to zero to neglect the lateral motion for simplicity. Meanwhile, parameter uncertainties are added into CZ,Cm to simulate model errors. The equilibrium states at α=7.50° are chosen as the initial condition.To illustrate the essence of the NCEA law, adaptive backstepping control approach proposed in [14] is adopted for comparison, the CEA laws are given as
(54)η^̇=hx−Bση^−WTxz,θ^̇1=γ3D1Tσε1,θ^̇2=γ4D2Tσε2,where γ3,γ4 are adaptation gains to be assigned. Consider that the transient control performance is highly dependent on adaptation gains, four sets of gains with different adaptation rates are selected, that is,Case 1.
(slow):γ3=30,γ4=0.003.Case 2.
(medium):γ3=100,γ4=0.01.Case 3.
(fast):γ3=500,γ4=0.3.In order to compare the performance of different adaptive laws, the root-mean-square (RSME) of tracking errors and the energy consumption (EC) are chosen as the evaluation indexes which are defined as(55)RSME=1n∑i=1nαi−αri2+βi−βri2+psi−psri2,EC=∫0tfLcp+Mcq+Ncrdt,where the superscript i means the ith sampling point, n denotes the number of samples, tf denotes the final time of simulation, and Lc,Mc,Nc are the commands of total control moment in each channel.Simulation results are shown in Figures3–6 and Table 2. It should be pointed out that the unsteady states are perfectly reconstructed with fast converge rates for both adaptive laws; therefore, the results are not presented herein. From Figures 3 and 4 and Table 2, it is apparent that the NCEA law outperforms the three CEA laws both in command tracking and energy consumption. The track performance of the CEA laws is sensitive to the adaptation gains. When compared with the case 1, the CEA law in case 3 achieves higher estimate precision and faster converging rate at the expense of sacrificing the energy consumption and transient control performance. It can be observed from Figure 5 that D1σθ~1,D2σθ~2 are converged to zero; thus, the closed-loop trajectory eventually confines to the manifold Ω7, and the NCEA law recovers the performance of the deterministic backstepping controller. Figure 6 shows the time history of parameter estimation; it is obvious that the parameters estimated by CEA laws drift from their true values, especially in case 1. For the NCEA law, ΔCm can be precisely estimated while this is not the case for ΔCZ. The reason is that the rank of D2σ equals to θ~2, then D2σθ~2=0 means θ~2=0; therefore, the invariant manifold is achieved and θ2 can be estimated accurately. However, the rank of D1σ is less than θ~1; hence, θ~1=0 cannot be guaranteed via D1σθ~1=0. In addition, it can be seen that θ^1 is frozen after t=15s at which instant the attractive manifold D1σθ~1=0 is achieved.Figure 3
Time history of the command tracking.Figure 4
Time histories of the control inputs.Figure 5
Time histories of the norms.Figure 6
Estimation of the uncertain parameters.Table 2
The performance indexes of the NCEA and CEA laws.
MethodsRMSE (°)Maximum error (°)EC (J)Maximumδe (°)Maximumδz (°)NCEA0.32051.652.53×10512.158.05CEACase 10.81264.182.64×10519.757.72Case 20.44862.102.75×10513.808.50Case 31.1203.455.55×10514.0510.9


## 5.2. Robust Analysis under Parameter Uncertainties and Unsteady Effects

To justify the effectiveness of the proposed control scheme, three different cases are considered for comparison. In the first case, the controller proposed in Section4 is adopted in which the effects of unsteady aerodynamics and uncertain parameters are both compensated. For the second case, the effects of uncertain parameters are not considered in the controller design. The third case is the worst case in which both the unsteady effects and uncertain parameters are not compensated. The design parameters of the controller and initial condition are all kept the same as that in the previous subsection. The uncertainties of aerodynamic forces and moments in longitudinal and lateral channels are all considered in this simulation. Moreover, the Herbst maneuver is implemented to validate the control performance; the commands of αd,βd,psd are defined in [35].The commands tracked in these three cases are compared in Figures7 and 8, and the control performance of the proposed adaptive backstepping controller in the first case is presented in Figures 9–14. In Figure 14, e1i,ε1i,ξ1i,i=1,2, are components of e1,ε1,ξ1, respectively; e2i,ε2i,ξ2i,i=1,2,3 are components of e2,ε2,ξ2, respectively. It can be observed from Figures 7 and 8 that the tracking performance of the first case outperforms the other two cases. It is apparent that smaller steady tracking errors and better transient tracking performances can be achieved since the effects of unsteady aerodynamics and uncertain parameters are both compensated. It also can be concluded that the unsteady aerodynamics seriously affects the aircraft dynamics and the compensation of unsteady effects could significantly improve the control performance. Note that the thrust force presented in Figure 9 increases drastically in the maneuver; it is quite rational for the pilot to increase the throttle to provide a larger thrust force when the fighter operates at high AOA. In addition, the aerodynamic surfaces are all set to zero when α>55° and the attitude control was completed via thrust vector merely in this deep stall area. From Figures 10–12, it can be seen that the tracking errors increase while the compensated errors still converge to zero even though thrust vector nozzles are saturated. Moreover, the proposed observer is effective in state reconstruction with strong robustness against parameter uncertainties, especially in the lateral channel. Meanwhile, the parameter estimator works fairly well in the presence of unsteady effects. The most essential reason can be concluded that the converging rate of state reconstruction error is much larger than the estimate error, thus the mutual effects nearly can be ignored. Figure 13 shows the uncertain parameters in aerodynamic moments are estimated exactly, but steady errors exist in the estimation of aerodynamic force parameters. This is consistent with the discussion in previous subsection. However, as presented in Figure 14, Wxη~,D1σθ~1, and D2σθ~2 all converge to zero after t=25.5s; the attractive manifold is achieved and the norms of the estimated states and parameters are maintained. Therefore, the performance of the deterministic control system is captured by the proposed constraint adaptive backstepping controller.Figure 7
Time history of the longitudinal command tracking.Figure 8
Time histories of the lateral command tracking.Figure 9
Time histories of the control inputs.Figure 10
Related errors in angular subsystem.Figure 11
Related errors in angular rate subsystem.Figure 12
Observation of the unsteady aerodynamic states.Figure 13
Estimation of the uncertain parameters.Figure 14
Time histories of the norms.


## 6. Conclusions

This paper is devoted to the constraint adaptive backstepping control design for the advanced fighter, where robust tracking of the predefined trajectories has been achieved. In particular, unsteady effects, parameter uncertainties, and input constraints are all considered which brings forward a great challenge for the control design. The proposed control scheme involves a state observer, a constraint backstepping controller, and two parameter estimators, which are designed separately. It should be pointed out that both the state observer and parameter estimator are designed based on the novel noncertainty equivalent principle which renders the closed-loop system with deterministic control performance. Moreover, stability of the closed-loop system can be guaranteed even though the saturation occurs. It can be concluded from simulation results that the backstepping controller integrated with the NCEA law outperforms the traditional adaptive backstepping controller. The effectiveness and robustness of the proposed control scheme are also verified by the Herbst maneuver. This paper mainly focuses on the theoretical aspect; our future work will be devoted to the practical applications of the flight control of scaled aircraft.



---

*Source: 6669380-2021-04-07.xml*