# Projection-Based Adaptive Backstepping Control of a Transport Aircraft for Heavyweight Airdrop

Authors: Ri Liu; Xiuxia Sun; Wenhan Dong; Guangzhi Xu
Journal: International Journal of Aerospace Engineering
 (2015)

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## Abstract

An autopilot inner loop that combines backstepping control with adaptive function approximation is developed for airdrop operations. The complex nonlinear uncertainty of the aircraft-cargo model is factorized into a known matrix and an uncertainty function, and a projection-based adaptive approach is proposed to estimate this function. Using projection in the adaptation law bounds the estimated function and guarantees the robustness of the controller against time-varying external disturbances and uncertainties. The convergence properties and robustness of the control method are proved via Lyapunov theory. Simulations are conducted under the condition that one transport aircraft performs a maximum load airdrop task at a height of 82 ft, using single row single platform mode. The results show good performance and robust operation of the controller, and the airdrop mission performance indexes are satisfied, even in the presence of ±15% uncertainty in the aerodynamic coefficients, ±0.01 rad/s pitch rate disturbance, and 20% actuators faults.


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## Body

## 1. Introduction

Heavyweight airdrop is an essential capability of a large transport aircraft, and it is critical to the success of many military tasks, such as precision delivery of heavyweight equipment and supplies [1, 2]. To perform these tasks with accurate allocation of the payload and to also guarantee flight safety, highly stable aircraft dynamics are required. However, the continuous movement and sudden delivery of the heavy cargo can exert large disturbances on the aircraft, thus leading to considerable deviation of the aircraft dynamics from the trim position [3–5]. Therefore, the design of an aircraft controller that is compatible with heavyweight airdrop tasks is necessary, and the large and sudden disturbances, strong coupling between the cargo and aircraft dynamics, and multiple uncertainties make this a challenging task [6–11].Over recent years, various meaningful achievements have been reported in developing advanced aircraft controllers that are compatible with airdrop tasks. Several control methods that use a linearized model at a given operating point have been proposed in the literature, includingL1 adaptive control [6], robust control [7], and active disturbance rejection control [12, 13]. Although these approaches can improve various aspects of system performance, one shortcoming is that satisfactory performance and robustness are difficult to achieve in the event that the cargo becomes increasingly heavy. In such an event, the aircraft dynamics can change rapidly and deviate far from the operating point, which yields a highly nonlinear system. Many nonlinear control approaches have been developed to handle systems with strong nonlinearities. The theoretically established feedback linearization method is the one that is most widely applied [14–18].The nonlinear system can be decoupled via exact state transformations rather than linear approximations. However, to perform perfect linearization, accurate knowledge of the plant dynamics must be available. This is not the case for airdrop flight controller design, as the complex nonlinear aerodynamic characteristics are very difficult to ascertain and model precisely [4, 6, 7, 10]. Moreover, aerodynamic data obtained from wind tunnel tests, augmented by computational fluid dynamics results, always contain a certain degree of uncertainty. The problem of model deficiencies can be dealt with by closing the control loop with robust controllers, for instance, combining feedback linearization with sliding mode control [8–10] or backstepping sliding mode control [11]. These methods devise control laws based on the knowledge of the bounds on the relevant uncertainties. However, the bounds of the complex nonlinear uncertainties, which are composed of aircraft-cargo dynamics coupled with aerodynamic perturbations, are difficult to obtain in advance. Thus, the control gains need to be set large enough to operate correctly under a variety of conditions, which is generally a very conservative strategy. This approach might also lead to severe chattering phenomena and could damage actuators and systems [19–21].In these cases, nonlinear adaptive control methods are called for. Adaptive backstepping control [22–26], which allows uncertainties of both matched and mismatched type, has been widely applied to flight control projects [23–26]. To be able to estimate the nonlinear uncertainties of the system, it is possible to employ neural networks within the parameter update laws of the adaptive backstepping controller [22, 23]. The neural networks are used to parameterize the nonlinear uncertainties, and this allows the update laws to adapt to the network weights. In spite of their obvious conceptual appeal, the complicated computations are time-consuming and the stability analysis is tedious. These considerations might limit the application of this method in the design of airdrop flight controllers from a purely practical perspective. One interesting technique is to separate the uncertain parameters from the complex nonlinearities and use update laws to adapt the uncertain parameters directly [26]. The design procedure, as well as the performance analysis of such an approach, is relatively easy when compared with that using the neural networks method.The main motivation for the current work is to propose a simplified controller design for the airdrop mode that can accommodate large changes in aircraft dynamics and reject uncertainties of both constant and time-varying types, as well as matched and unmatched types. The contributions of this paper are (1) a flight controller design that inherits the merits of the backstepping approach, thus solving the unmatched control problem of cargo airdrop; (2) the introduction of adaptation theories to estimate the system uncertainties, which overcomes the conservative drawbacks of [8–11] as discussed above; (3) the formation of adaptation laws using the projection operator to bound the estimated functions and theoretically guarantee the robustness of the controller against time-varying disturbances and uncertainties while avoiding singularity of the control law; and (4) a proof of the convergence properties and robustness of the control method based on Lyapunov theory.The structure of this paper is as follows. The aircraft-cargo model with cargo extraction is presented in Section2. The control law for the airdrop mode is derived in Section 3, along with a discussion of the stability properties. Section 4 presents simulation results that verify the correct operation of the proposed controller, and conclusions are presented in Section 5.


## 2. Aircraft-Cargo Model with Cargo Extraction

As stated in the previous section, this paper studies the design of a flight control law for the airdrop operations. The governing equations of motions are recalled from [10] as(1)V˙=Tcos⁡α-D-mbgsin⁡γ+Fcxmb,(2)γ˙=Tsin⁡α+L-mbgcos⁡γ+FczmbV,(3)q˙=My+McIy,(4)θ˙=q,where mb is the mass of the aircraft; V is the airspeed; α is the angle of attack (AOA); γ is the climb angle and θ=γ+α with θ being the pitch angle; q is the pitch rate; Iy is the pitch moment of inertia; T, D, L, and My are the engine thrust, drag, lift, and pitch aerodynamic moment, respectively; and Fcx, Fcz, and Mc are the disturbance forces and moment, respectively, that the cargo imposes on the aircraft.The pitch aerodynamic moment is obtained as(5)My=q-ScACm0+Cmαα-α0+CmqqcA2V+Cmδeδe,where q- is the dynamic pressure; S is the wing area; δe is the elevator deflection; cA is the mean aerodynamic chord; and Cm∗ are the pitch moment coefficients. The drag and lift are found by(6)D=q-SCD0+CDαα-α0+CDδeδe,L=q-SCL0+CLαα-α0+CLδeδe,where CD∗ and CL∗ are the drag and lift coefficients, respectively. The engine thrust is(7)T=Tmδp,where δp is the throttle opening ranging from 0 to 100% and Tm is the maximal thrust.F
c
x, Fcz, and Mc are obtained as(8)Fcx=mcgcos⁡θ-2mcqr˙csin⁡α-Fp-mcrcsin⁡αq˙-mcV˙-mcgsin⁡θ+mcq2rc-mcr¨ccos⁡α,Fcz=-mcgcos⁡θ-2mcqr˙ccos⁡α-mcVγ˙+mcrccos⁡αq˙-mcgsin⁡θ+mcq2rc-mcr¨csin⁡α,Mc=mcrcgcos⁡θ-Fprcsin⁡α-mcrcV˙sin⁡α-Vγ˙cos⁡α+q˙rc+2qr˙c,where mc is the mass of the cargo; Fp is the extraction force where Fp=mcgλ with λ denoting the extraction ratio; and rc is the position of the cargo with respect to the center of gravity (CG) of the aircraft. The cargo dynamics are found by(9)r¨c=V˙cos⁡α+Vsin⁡αγ˙+gsin⁡θ-μgcos⁡θ+μFpsin⁡αmc+rcq2+Fpcos⁡αmc+μV˙sin⁡α-Vγ˙cos⁡α+q˙rc+2qr˙c,where μ is the friction coefficient of rolling between the cargo and the roller on the floor.Remark 1.
It is observed from (1)–(3) and (8) and (9) that the aircraft-cargo dynamics form a strongly nonlinear system subject to the coupling of the aircraft states and the cargo parameters. The system may be further complicated by various uncertainties, such as aerodynamic data perturbation and uncertain machinery faults. Readers can refer to [10] for detailed discussions about the model.From (1)–(9), together with the consideration of uncertainties, the aircraft-cargo model can be rewritten in the following form:(10)x˙1t=σt+Bx2t,x˙2t=Ft+Gtωut+ΔFt,where x1=θ, x2=V,qT, u=δe,δpT, B=[0,1], F=f1,f2T, and G=g11g12g21g22; ΔF=Δf1,Δf2T is the uncertainty function; σ(t)∈R is the unknown time-varying disturbance; and ω∈R2×2 is the unknown input gain matrix. fi and gij (i=1,2; j=1,2) are found by(11)f1=-mcrcsin⁡αf2+T0cos⁡α-mbgsin⁡γ-Λ1cos⁡α+Λ2sin⁡α-q-SCD0+CDαα-α0-Fpmb+mc,f2=q-ScACm0+Cmαα-α0+CmqqcA/2VΛ3+rcΛ2Λ3-rcFpsin⁡αΛ3+mcrc-Λ2+Fpsin⁡α+mbgsin⁡γsin⁡α-mbgcos⁡γcos⁡α+q-Ssin⁡αCD0+CDαα-α0+q-Scos⁡αCL0+CLαα-α0mb+mcΛ3,g11=-mcrcsin⁡αg21+q-SCDδemb+mc,g12=Tmcos⁡αmb+mc,g21=mcrcq-SCDδesin⁡α+CLδecos⁡αmb+mcΛ3+q-ScACmδeΛ3,g22=0,with Λi(i=1,2,3) being defined as follows:(12)Λ1=mcgsin⁡θ+mcq2rc-mcr¨c,Λ2=mcgcos⁡θ-2mcqr˙c,Λ3=Iy+mcrc2-mc2rc2mb+mc.The uncertainty function Δfi(i=1,2) introduced by the aerodynamic data perturbation is obtained as(13)Δf1=-mcrcsin⁡αΔf2-q-SΔCD0+ΔCDαα-α0mb+mc,Δf2=q-ScAΔCm0+ΔCmαα-α0+ΔCmqqcA/2VΛ3+mcrcq-Ssin⁡αΔCD0+ΔCDαα-α0+cos⁡αΔCL0+ΔCLαα-α0mb+mcΛ3.Here, ΔCL∗, ΔCD∗, and ΔCm∗ are the perturbation of the lift, drag, and pitch moment coefficients, respectively. We introduce the following notations:(14)E11=-q-Smc2rc2sin⁡αcos⁡αmb+mc2Λ3,E12=-q-Smb+mc-q-Smc2rc2sin2⁡αmb+mc2Λ3,E13=-q-ScAmcrcsin⁡αmb+mcΛ3,E21=q-Smcrccos⁡αmb+mcΛ3,E22=q-Smcrcsin⁡αmb+mcΛ3,E23=q-ScAΛ3,Et=E11E12E13E21E22E23·1α-α000000001α-α000000001α-α0qcA2V,Pt=ΔCL0,ΔCLα,ΔCD0,ΔCDα,ΔCm0,ΔCmα,ΔCmqT.Then, ΔFt can be written as(15)ΔFt=EtPt.We will use the following assumptions throughout this analysis.Assumption 2 (uniform boundedness ofσ(t) and P(t)).
Here,σ(t)≤ϕ and P(t)∈Θ, where ϕ>0 is a known bound of σ(t) and Θ is a known compact set.Assumption 3 (partial knowledge of the input gain matrix).
ω is a constant diagonal matrix defined as ω=diag⁡(ω11,ω22) with 0<ω11, ω22≤1. There exists a known compact set Ω such that ω∈Ω⊂R2×2.Assumption 4 (uniform boundedness of the rate of variation of parameters).
σ
(
t
) and P(t) are continuously differentiable with uniformly bounded derivatives; that is, σ˙(t)≤dσ<∞ and P˙(t)≤dP<∞ with (·) denoting the 2-norm of the vector.


## 3. Control Law and Stability Properties

The overall control system is designed using three feedback loops, as shown in Figure1. The third loop (outer loop) uses an altitude-hold controller designed using the regular PID control law. This loop generates a pitch angle command input θd for the angular control loop in the second layer. The inner loop contains two controlled variables that are [q,V]T. The pitch rate command qd is generated by the angular control loop, and the velocity command Vd is the trim value.Figure 1
Autopilot control architecture with three layers of feedback.


### 3.1. Adaptive Backstepping Control Law

The steps in the adaptive backstepping control law are described below.Step 1.
Consider the first equation in system (10):(16)x˙1t=σt+Bx2t.The desired pitch angle is x1d(t)=θd, and design the virtual control law of (16) as(17)x2dt=-BTk1x1t-x1dt+σ^t-x˙1dt+Vd0T,where k1>0 and σ^(t) is the estimation of σ(t), which is governed by(18)σ^˙t=Γ·Proj⁡σ^t,x1t-x1dt,where Γ>0 is the adaptation gain and Proj⁡(·,·) is the projection operator (see the appendix for details) which ensures that σ^≤ϕ.Step 2.
Consider the second equation in system (10):(19)x˙2t=Ft+Gtωut+EtPt.The control law is designed as(20)ut=-Gtω^t-1k2x2t-x2dt+ΕtP^t+Ft+BTx1t-x1dt-x˙2dt,where k2>0 is a constant matrix represented by k2=diag⁡(k21,k22,k23). ω^(t) and P^(t) are governed by(21)ω^˙t=Γ·Proj⁡ω^t,GTtx2t-x2dtuTt,P^˙t=Γ·Proj⁡P^t,ETtx2t-x2dt,where the projections are confined to the sets Ω and Θ, respectively; that is, ω^(t)∈Ω and P^(t)∈Θ.Remark 5.
The projection-based adaptation law ensures thatω^(t) is nonsingular. For instance, we can set(22)Ω=0.5,10,0.010,0.010.5,1.From ω^(t)∈Ω, it follows that ω^(t) is always nonsingular. For -2/π<α<2/π, we have g11g22≪g12g21 [8, 10], which, together with the condition that ω^(t) is nonsingular, implies that G(t)ω^(t) is nonsingular.


### 3.2. Stability Analysis

The proof of stability of the control law is achieved by augmenting Lyapunov functions for the state tracking errors and parameter estimation errors. We define(23)σ~t=σ^t-σt,ω~t=ω^t-ω,P~t=P^t-Pt,x~1t=x1t-x1dt,x~2t=x2t-x2dt.Taking the time derivative of x~1t yields(24)x~˙1t=x˙1t-x˙1dt=σt+Bx2t-x˙1dt=σt-k1x1t-x1dt-σ^t+Bx~2t=-k1x~1t-σ~t+Bx~2tand the time derivative of x~2t is(25)x~˙2t=x˙2t-x˙2dt=Ft+Gtωut+EtPt-x˙2dt=Ft+EtPt-k2x~2t+ΕtP^t+Ft+BTx~1t-x˙2dt-Gtω~ut-x˙2dt=-EtP~t-k2x~2t-BTx~1t-Gtω~ut.Theorem 6.
Given the system in (10) controlled by (20) with the adaptation laws defined via (18) and (21), the state tracking errors x~it, i=1,2, are uniformly bounded:(26)x~it≤2lΓ,∀t≥0,where l=1/kmin⁡(dσϕ+maxP∈Θ⁡PdP)+2(ϕ2+maxω∈Ω⁡tr⁡(ωTω)+maxP∈Θ⁡P2), kmin=min⁡{k1,k2′}, and k2′=min⁡{k21,k22,k23}.Proof.
Consider the Lyapunov function candidate:(27)υt=12∑i=12x~iTx~i+12Γσ~2+tr⁡ω~Tω~+P~TP~.First, we prove that υ(t)≤l/Γ. Since the aircraft should maintain a straight and level flight condition before the cargo is unlocked [6–11], we have x~i(0)=0. Then, we can verify that(28)υ0≤12Γ4ϕ2+4maxω∈Ω⁡tr⁡ωTω+4maxP∈Θ⁡P2≤lΓ.The time derivative of υ(t) along the solutions of (24) and (25) is(29)υ˙t=-k1x~1t2-x~1tσ~t+x~1tBx~2t-x~2Ttk2x~2t-x~2TtEtP~t-x~2TtBTx~1t-x~2TtGtω~ut+1Γσ~tσ^˙t+P~TtP^˙t+tr⁡ω~Ttω^˙t-1Γσ~tσ˙t+P~TtP˙t≤-k1x~1t2-k2′x~2t2-1Γσ~tσ˙t+P~TtP˙t+1Γtr⁡ω~Ttω^˙t-ΓGTtx~2tuTt+1ΓP~TtP^˙t-ΓETtx~2t+1Γσ~tσ^˙t-Γx~1t.Substituting (18) and (21) into (29) leads to(30)υ˙t≤-k1x~1t2-k2′x~2t2-1Γσ~tσ˙t+P~TtP˙t+1Γtr⁡ω~TtΓ·Proj⁡ω^t,GTtx~2tuTt-ΓGTtx~2tuTt+1ΓP~TtΓ·Proj⁡P^t,ETtx~2t-ΓETtx~2t+1Γσ~tΓ·Proj⁡σ^t,x~1t-Γx~1t.Using Property A.2 of the projection operator (see the appendix for details), we can obtain that(31)ω~TtΓ·Proj⁡ω^t,GTtx~2tuTt-ΓGTtx~2tuTt≤0,P~TtΓ·Proj⁡P^t,ETtx~2t-ΓETtx~2t≤0,σ~tΓ·Proj⁡σ^t,x~1t-Γx~1t≤0,which implies that(32)υ˙t≤-k1x~1t2-k2′x~2t2-1Γσ~tσ˙t+P~TtP˙t≤k1x~1t2-k2′x~2t2+1Γσ~tσ˙t+P~TtP˙t.From σ^(t)≤ϕ, P^(t)∈Θ, and Assumption 4, we have(33)1Γσ~tσ˙t+P~TtP˙t≤2Γdσϕ+maxP∈Θ⁡PdP,(34)maxt≥0⁡1Γσ~2t+tr⁡ω~Ttω~t+P~TtP~t≤4Γϕ2+maxω∈Ω⁡tr⁡ωTω+maxP∈Θ⁡P2.If there exists t′>0 such that υ(t′)>l/Γ, it then follows from (27) and (34) that(35)∑i=12x~iTt′x~it′>2kminΓdσϕ+maxP∈Θ⁡PdPwhich further leads to(36)k1x~1t′2+k2′x~2t′2≥kmin∑i=12x~iTt′x~it′>2Γdσϕ+maxP∈Θ⁡PdP.Thus, if υ(t′)>l/Γ, then from (32), (33), and (36), we obtain(37)υ˙t′<0.Since υ(0)≤l/Γ, υ(t)≤l/Γ holds for every t≥0. Notice that if 1/2x~i(t)2≤υ(t), then 1/2x~i(t)2≤l/Γ. This proves the bounds in (26).Remark 7.
Such a projection-based adaptation law can bound the estimated function, and this theoretically guarantees the robustness of the controller against time-varying disturbances and uncertainties. However, the stability properties of systems using conventional adaptation laws are provable only under the assumption of constant disturbances and uncertainties [23, 26].Remark 8.
It follows from (26) that one can prescribe the arbitrary desired tracking performance by increasing the adaptation gain Γ. However, a large gain requires more control power and can also lead to control signal oscillations. Thus, the adaptation gain needs to be tuned in a trial-and-error way.


## 3.1. Adaptive Backstepping Control Law

The steps in the adaptive backstepping control law are described below.Step 1.
Consider the first equation in system (10):(16)x˙1t=σt+Bx2t.The desired pitch angle is x1d(t)=θd, and design the virtual control law of (16) as(17)x2dt=-BTk1x1t-x1dt+σ^t-x˙1dt+Vd0T,where k1>0 and σ^(t) is the estimation of σ(t), which is governed by(18)σ^˙t=Γ·Proj⁡σ^t,x1t-x1dt,where Γ>0 is the adaptation gain and Proj⁡(·,·) is the projection operator (see the appendix for details) which ensures that σ^≤ϕ.Step 2.
Consider the second equation in system (10):(19)x˙2t=Ft+Gtωut+EtPt.The control law is designed as(20)ut=-Gtω^t-1k2x2t-x2dt+ΕtP^t+Ft+BTx1t-x1dt-x˙2dt,where k2>0 is a constant matrix represented by k2=diag⁡(k21,k22,k23). ω^(t) and P^(t) are governed by(21)ω^˙t=Γ·Proj⁡ω^t,GTtx2t-x2dtuTt,P^˙t=Γ·Proj⁡P^t,ETtx2t-x2dt,where the projections are confined to the sets Ω and Θ, respectively; that is, ω^(t)∈Ω and P^(t)∈Θ.Remark 5.
The projection-based adaptation law ensures thatω^(t) is nonsingular. For instance, we can set(22)Ω=0.5,10,0.010,0.010.5,1.From ω^(t)∈Ω, it follows that ω^(t) is always nonsingular. For -2/π<α<2/π, we have g11g22≪g12g21 [8, 10], which, together with the condition that ω^(t) is nonsingular, implies that G(t)ω^(t) is nonsingular.


## 3.2. Stability Analysis

The proof of stability of the control law is achieved by augmenting Lyapunov functions for the state tracking errors and parameter estimation errors. We define(23)σ~t=σ^t-σt,ω~t=ω^t-ω,P~t=P^t-Pt,x~1t=x1t-x1dt,x~2t=x2t-x2dt.Taking the time derivative of x~1t yields(24)x~˙1t=x˙1t-x˙1dt=σt+Bx2t-x˙1dt=σt-k1x1t-x1dt-σ^t+Bx~2t=-k1x~1t-σ~t+Bx~2tand the time derivative of x~2t is(25)x~˙2t=x˙2t-x˙2dt=Ft+Gtωut+EtPt-x˙2dt=Ft+EtPt-k2x~2t+ΕtP^t+Ft+BTx~1t-x˙2dt-Gtω~ut-x˙2dt=-EtP~t-k2x~2t-BTx~1t-Gtω~ut.Theorem 6.
Given the system in (10) controlled by (20) with the adaptation laws defined via (18) and (21), the state tracking errors x~it, i=1,2, are uniformly bounded:(26)x~it≤2lΓ,∀t≥0,where l=1/kmin⁡(dσϕ+maxP∈Θ⁡PdP)+2(ϕ2+maxω∈Ω⁡tr⁡(ωTω)+maxP∈Θ⁡P2), kmin=min⁡{k1,k2′}, and k2′=min⁡{k21,k22,k23}.Proof.
Consider the Lyapunov function candidate:(27)υt=12∑i=12x~iTx~i+12Γσ~2+tr⁡ω~Tω~+P~TP~.First, we prove that υ(t)≤l/Γ. Since the aircraft should maintain a straight and level flight condition before the cargo is unlocked [6–11], we have x~i(0)=0. Then, we can verify that(28)υ0≤12Γ4ϕ2+4maxω∈Ω⁡tr⁡ωTω+4maxP∈Θ⁡P2≤lΓ.The time derivative of υ(t) along the solutions of (24) and (25) is(29)υ˙t=-k1x~1t2-x~1tσ~t+x~1tBx~2t-x~2Ttk2x~2t-x~2TtEtP~t-x~2TtBTx~1t-x~2TtGtω~ut+1Γσ~tσ^˙t+P~TtP^˙t+tr⁡ω~Ttω^˙t-1Γσ~tσ˙t+P~TtP˙t≤-k1x~1t2-k2′x~2t2-1Γσ~tσ˙t+P~TtP˙t+1Γtr⁡ω~Ttω^˙t-ΓGTtx~2tuTt+1ΓP~TtP^˙t-ΓETtx~2t+1Γσ~tσ^˙t-Γx~1t.Substituting (18) and (21) into (29) leads to(30)υ˙t≤-k1x~1t2-k2′x~2t2-1Γσ~tσ˙t+P~TtP˙t+1Γtr⁡ω~TtΓ·Proj⁡ω^t,GTtx~2tuTt-ΓGTtx~2tuTt+1ΓP~TtΓ·Proj⁡P^t,ETtx~2t-ΓETtx~2t+1Γσ~tΓ·Proj⁡σ^t,x~1t-Γx~1t.Using Property A.2 of the projection operator (see the appendix for details), we can obtain that(31)ω~TtΓ·Proj⁡ω^t,GTtx~2tuTt-ΓGTtx~2tuTt≤0,P~TtΓ·Proj⁡P^t,ETtx~2t-ΓETtx~2t≤0,σ~tΓ·Proj⁡σ^t,x~1t-Γx~1t≤0,which implies that(32)υ˙t≤-k1x~1t2-k2′x~2t2-1Γσ~tσ˙t+P~TtP˙t≤k1x~1t2-k2′x~2t2+1Γσ~tσ˙t+P~TtP˙t.From σ^(t)≤ϕ, P^(t)∈Θ, and Assumption 4, we have(33)1Γσ~tσ˙t+P~TtP˙t≤2Γdσϕ+maxP∈Θ⁡PdP,(34)maxt≥0⁡1Γσ~2t+tr⁡ω~Ttω~t+P~TtP~t≤4Γϕ2+maxω∈Ω⁡tr⁡ωTω+maxP∈Θ⁡P2.If there exists t′>0 such that υ(t′)>l/Γ, it then follows from (27) and (34) that(35)∑i=12x~iTt′x~it′>2kminΓdσϕ+maxP∈Θ⁡PdPwhich further leads to(36)k1x~1t′2+k2′x~2t′2≥kmin∑i=12x~iTt′x~it′>2Γdσϕ+maxP∈Θ⁡PdP.Thus, if υ(t′)>l/Γ, then from (32), (33), and (36), we obtain(37)υ˙t′<0.Since υ(0)≤l/Γ, υ(t)≤l/Γ holds for every t≥0. Notice that if 1/2x~i(t)2≤υ(t), then 1/2x~i(t)2≤l/Γ. This proves the bounds in (26).Remark 7.
Such a projection-based adaptation law can bound the estimated function, and this theoretically guarantees the robustness of the controller against time-varying disturbances and uncertainties. However, the stability properties of systems using conventional adaptation laws are provable only under the assumption of constant disturbances and uncertainties [23, 26].Remark 8.
It follows from (26) that one can prescribe the arbitrary desired tracking performance by increasing the adaptation gain Γ. However, a large gain requires more control power and can also lead to control signal oscillations. Thus, the adaptation gain needs to be tuned in a trial-and-error way.


## 4. Simulation Analysis

To verify the proposed controller design, a 24,955 kg transport aircraft performing an airdrop of cargo weighing 8,000 kg is simulated. The cargo is initially locked at the CG of the aircraft. The aircraft is trimmed at the following conditions:H0=82 ft, V0=229 ft/s, and α0=θ0=5.9813deg. with δp=34.1%, δe=0deg., the horizontal stabilizer δh=-5.4093deg., and the flap δf=25deg.To satisfy the requirements of mission completeness and flight safety, the performance indexes for the heavyweight airdrop are given as follows [8]: (1) ΔH≤45 ft and Hmin>20 ft; (2) ΔV≤0.13V0; (3) Δθ≤5deg. and θ>2deg.; and (4) αmax≤0.7αs with αs denoting the stalling AOA.The performance and robustness of the controller are first tested in the presence of constant disturbances, actuators faults, and uncertainties. The following three cases are simulated and compared.Case 1.
In the first case,σ=0, P=0000000T, and ω=diag⁡(1,1).Case 2.
In the second case,σ(t)=0.01 rad/s, P=15%CL0CLαCD0CDαCm0CmαCmqT, and ω=diag⁡(0.8,0.8).Case 3.
In the third case,σ(t)=-0.01 rad/s, P=-15%CL0CLαCD0CDαCm0CmαCmqT, and ω=diag⁡(0.8,0.8).The compact sets can be conservatively set toϕ=0.3, Θ=b=(bi)7×1∈R7×1:bi∈-2,2, and Ω is set as in Remark 5 (in Section 3.1). After experimental tuning, we select k1=8, k2=diag⁡(3,5,1), and Γ=20. The outer-loop altitude-hold PID controller parameters are set as Kp=0.05, KI=0.033, and KD=0.009. Figure 2 shows simulation results of the drop process for these three cases, and we can see that the criteria for a successful drop have all been met. In all three cases, the altitude increment is controlled in the range of 2 ft. After the cargo is dropped out, the altitude and the velocity are well maintained at the predefined trim position and fully stabilized within 12 seconds. Owing to the loss of heavy weight, the final AOA and pitch angle become smaller when compared to that of the trim position, especially in Case 2. The observed change in value is less than 5 deg., and the final pitch angle is greater than 2 deg., which meets the mission performance indexes. The elevator deflection and the throttle opening are within the magnitude limits, and there is no severe chattering problem. To summarize, the pitch-up motion of the aircraft by the release of the cargo is suppressed effectively through appropriately configuring the elevator and the throttle, even in the presence of ±15% aerodynamic coefficients uncertainty, ±0.01 rad/s pitch rate disturbance, and 20% actuators faults.Figure 2
Aircraft responses of the dropping process in the presence of constant disturbances and uncertainties (Cases1, 2, and 3).Next, we test the performance and robustness of the control system for the condition of time-varying disturbances and uncertainties, without retuning the parameters of the controller. Note that no actuators faults are considered in the following test (i.e.,ω=diag⁡(1,1)). The external disturbances and uncertainty function are set as follows.Case 4.
In this case,σ(t)=0.01sin⁡t rad/s, P=0000000T.Case 5.
In this case,σ=0, P(t)=15%sin⁡tCL0CLαCD0CDαCm0CmαCmqT.Case 6.
In this case,σ(t)=0.01sin⁡t rad/s, P(t)=15%sin⁡(t)CL0CLαCD0CDαCm0CmαCmqT.Figure3 shows that the flight altitude is controlled in the range of [81ft,83ft], the velocity increment is less than 0.3 ft/s, and the pitch angle as well as the AOA converges to within [4deg.,5.5deg.]. The responses of the aircraft can meet the mission performance indexes for Cases 4–6, which verifies the robustness of the control method against time-varying disturbances and uncertainties.Figure 3
Aircraft responses of the dropping process in the presence of time-varying disturbances and uncertainties (Cases4, 5, and 6).


## 5. Conclusions

This paper focused on the problem of designing an aircraft controller compatible with heavyweight airdrop operations. To achieve good stability and robust characteristics, a novel flight controller combining backstepping control with adaptive function approximation was developed for pitch attitude and velocity control. This method uses projection-based adaptation strategies to achieve robustness against uncertainties. Lyapunov-based analysis shows that the controller ensures uniformly bounded steady-state tracking errors in the presence of constant actuators faults, time-varying external disturbances, and aerodynamic uncertainties. The performance of the controller was evaluated in a maximum load airdrop mission. Simulation results verified that the controller performance satisfies the airdrop mission performance indexes in the presence of pitch rate disturbances, aerodynamic uncertainties, and actuators faults. The application of this research can be used to achieve higher levels of performance and safety in practical airdrop missions.



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*Source: 538901-2015-12-31.xml*