# Full-State-Constrained Adaptive Control for a Class of UAVs Suffering from Coupled Uncertainties Using the HOBLF

Authors: Xiaofei Chang; Kexuan Wang; Kang Chen; Wenxing Fu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Nowadays, the practical tasks of UAVs are becoming more and more complicated and diversified. In the practical flight process, the large-scale changes of the flight environment, the modeling errors, and the external disturbances may induce the instability of the UAV flight system. Meanwhile, the constraints of the UAV attitudes also have to be guaranteed during the flight process. However, most existing control methods still have limitations in handling the constraints and the multisource disturbances simultaneously. To address this problem, in this paper, we focus on the actual output tracking control for the UAV systems with full-state constraints and multisource disturbances. Firstly, a high-order tan-type barrier Lyapunov function (HOBLF) has been constructed for the UAV to maintain the full-state constraints. Secondly, by combining the adaptive backstepping technique and the fuzzy logic systems, the modeling errors and the unknown nonlinearities of the UAV attitude control system can be handled. Moreover, by properly constructing several adaptive laws, the time-varying disturbances existing in the UAV attitude control system can be suppressed. Finally, the full-state-constrained antidisturbance controller is formed, ensuring that the tracking error approaches arbitrarily to small neighborhood and does not violate the given constraints. The simulation results illustrate the feasibility and the advantages of the proposed method.


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## Body

## 1. Introduction

Unmanned aerial vehicles (UAVs) have attracted research interest on account of their simple structures, low price, high mobility, and high flexibility. Due to their significant advantages, they have broad application prospects in military [1] and civilian [2]. However, the nonlinear characteristics and strong coupling of UAVs increase the difficulty of UAV control [3]. The aerodynamic parameter perturbations, modeling errors, and external interference caused by changes in flight conditions may cause instability of the flight system or even failure [4].In order to tackle the control problem of UAVs [5], several approaches have been proposed, such as robust control [6], adaptive control [7], neural networks [8], fuzzy control [9], sliding mode control [10–12], genetic algorithms [13], and linear parameter-varying (LPV) method [14]. Recently, various advanced control methods have been presented for the attitude control of the UAVs. In [15], a nonlinear model predictive control (NMPC) is used to design a high-level controller for a fixed-wing unmanned aerial vehicle. In [16], this method has been improved, and it improved the optimality of the MPC controllers. In [17], H-loop-shaping method is used to provide the stabilizer mode while successfully improving the robustness of the UAV. The LPV model reference method [14] has been further developed in [18] to achieve agile and high-performance tracking objectives of UAVs. Chen et al. [19] used a disturbance observer-based control (DOBC) scheme with transient performance design for UAVs to tackle the mismatched disturbance problem. Besides the disturbance observer, another effective way to account for the unknown dynamics in the UAV is employing active disturbance rejection control (ADRC) [20]. Moreover, by a combination of backstepping and ADRC, an integrated controller is devised for the quadrotor UAV with multiple uncertainties in [21]. Based on filtered dynamic inversion, the controller in [22] enabled the fixed-wing UAV to achieve effective altitude command following in the presence of turbulent wind. In [23], an adaptive second-order sliding mode control based on an extended observer is proposed for UAVs to improve performance under different operating conditions. To achieve an ideal convergence performance of the UAV, prescribed performance controllers [24] and finite-time stable controllers [25–27] were designed.Due to the limitations of actual physical devices, system performance requirements, and safety requirements, most practical systems will have constraints on the input, state, and output, such as nonuniform gantry crane [28], flexible crane systems [29], and robotic manipulator systems [30]. A series of constraint control methods have been proposed, including reference governors [31], model predictive control [32], extremum-seeking control [33], and the notion of set invariance [34, 35]. In recent years, BLF-based constraint control methods have been widely used to deal with state constraints and output constraints because they do not require precise solutions of the system. The synthesis of the barrier Lyapunov function (BLF) with the constraint function is proposed firstly in dealing with the Brunovsky constraint system [36]. Since then, many scholars have focused on studying the adaptive control for uncertain nonlinear systems based on the barrier Lyapunov function. In [37], by the combination of the backstepping method and BLF, an adaptive constrained controller was proposed for partial-state-constrained systems. In [38, 39], nonlinear pure-feedback systems and stochastic systems with full-state constraints have been investigated by constructing barrier Lyapunov functions (BLFs). For the MIMO full-state-constrained nonlinear systems, the opportune integral BLFs were established to prevent violation of the state constraints in [40]. Tee et al. [41] extended the constant constraint in [41] to the time-varying constraint. Furthermore, BLF has also been successfully applied in practical systems, such as electrostatic microactuator [42] and HSV [43, 44]. In [45], a higher-order barrier Lyapunov function (HOBLF) has been firstly proposed while achieving constraint satisfaction without solving the displayed solution of the system, especially for the higher-order uncertain nonlinear system. Given the influence of the external aerodynamic change and strong interference on the UAV in the flight environment, the attitude control of the UAV becomes an important content related to its performance. Therefore, certain state constraints should be applied to its three attitude angles to avoid the occurrence of large deviations. The high-order barrier Lyapunov function can just solve the problem of UAV attitude constraints. Therefore, this paper introduces [45] to solve the problem of UAV state constant constraints. And as far as the authors know, there is no relevant literature that has studied the state constraints of UAVs.In spite of the process, the HOBLF-based full-state-constrained antidisturbance control has never been investigated for the UAVs. According to Sun et al. [45], using the HOBLF can achieve practical output tracking control for a category of high-order uncertain nonlinear systems with full-state constraints. Although Tan and Guo [24] considered the full-state-constrained control for the UAVs, the HOBLF has never been utilized. Moreover, the influence of external aerodynamic changes and strong disturbances of UAVs has to be taken into consideration.Based on the above observations, for UAV systems with full-state constraints,(i)
This paper introduces a high-order tan-type BLF and applies it to the UAV system, whose three attitude angles have state constraints. In addition, when constraints disappear or the high-order system is reduced to a traditional nonlinear system in the form of strict feedback, this BLF is still effective.(ii)
For the time-varying interference that the UAV may encounter during the flight, an adaptive law is constructed to enable the UAV to track the desired signal stably even under interference and has a strong adaptability.(iii)
Compared with the traditional adaptive, DOBC, and general BLF [36], computational burden has been decreased, and the proposed control method has a better tracking effect on account of the HOBLF and fuzzy system.This paper is organized as follows. In Section2, the problem formulation and related knowledge are given, while in Section 3, the high-order tan-type BLF, the design procedures, the controllers, and the adaptive law are designed. Section 4 presents the simulation results and comparison, and Section 5 concludes this paper.


## 2. Problem Formulation and Related Knowledge

### 2.1. System Model

In this paper, we use the nonlinear dynamics of the UAV built in [46]; the details are shown as follows:(1)ϑ˙=ωysinγ+cosγωz+dϑ,ψ˙=ωycosγ−ωzsinγcosϑ+dψ,γ˙=ωx−tanϑωycosγ−ωzsinγ+dγ,Ω˙I=I−1M−ΩI×I∗ΩI+dΩI,where ϑ,ψ,andγ, respectively, represent the pitching, yaw, and rolling attitude angle of the UAV relative to the inertial coordinate system. ωx,ωy,andωz represent the angular velocity of each degree of freedom, respectively, where ΩI=ωx,ωy,ωzT. I is the rotational inertia matrix of the UAV. dii=θ,ψ,γ,ΩI are unknown unmodeled dynamics of the plant. The control scheme is depicted in Figure 1, and the details and expressions of I and M are shown as follows:(2)I=Ixx0Ixz0Iyy0Izx0Izz,M=q¯SbCxq¯Sc¯Cyq¯SbCz,IΔΔ is the product of inertia in both directions, q¯=ρVT2/2 is the dynamic pressure of air, VT is the total airspeed of the UAV, ρ is the air density, S is the total area of the wing, b is the wingspan, and c¯ is the average chord length. The rolling, pitching, and yaw moments of the UAV are mainly controlled by Cx,Cy,andCz, which can be given by(3)Cx=Cxδa1δa1+Cxδa2δa2+Cxδb1δb1+Cxδb2δb2+Cxββ+Δx,Cy=Cyδyδy+Cyββ+Δy,Cz=Cz0+Czδa1δa1+Cxδa2δa2+Czδb1δb1+Czδb2δb2+Czαα+Δz,where δa1,δa2 are the right and left aileron rudder deviation. δb1,δb2 are the deviation of the left and right lifter rudder; δv is the direction of the wing rudder deviation; Cx is the drag coefficient with Cxii=δa1,δa2,δb1,δb2,β being the coefficient of i to Cx; Cv is the drag coefficient with Cyii=δy,β being the coefficient of i to Cv; Cz is the drag coefficient with Czii=δa1,δa2,δb1,δb2,α being the coefficient of i to Cz; α,β are the angle of attack and sideslip angle.Figure 1
The structure of the proposed HOBLF control algorithm.In order to build a model of the UAV attitude system, definex1=ϑ,φ,γT,x2=ωx,ωy,ωzT, and u=δa1δa1δa2δb1δb2δyT. Then, the UAV attitude dynamic equation with time-varying interference can be expressed as(4)x˙1=Atx2+d1t,x˙2=Btut+fx1,x2+Δfx1,x2+d2t,y=x1,where At∈ℝ3×3,d1t∈ℝ3×1,Bt∈ℝ3×5,andfx∈ℝ3×1. They are denoted by(5)At=0sinγcosγ0cosγcosϑ−sinγcosϑ1−tanϑcosγtanϑsinγ,Bt=q¯SbI−1Cxδa1Cxδa2Cxδb1Cxδb200000CyδyCzδa1Czδa2Czδb1Czδb20fx1,x2=I−1q¯SbCxββCyββCz0+Czαα+ΩI×I∗ΩIΔfx1,x2=q¯SbI−1ΔxΔyΔz.All states are bound to the compact set(6)Ωx≔xi∈R,xit≤bci,i=1,2,where bci is a known positive constant.Remark 1.
In fact, it is difficult to develop an exact model for the UAV since we cannot reproduce the complex flight environment of the UAV in a wind tunnel. So, the unmodeled dynamicsdi and uncertain parameters Δi are inevitable.


### 2.2. Assumptions and Lemmas

The purpose of this paper is to design a controlleru: (1) for any initial state, there is a finite time T>0 so that the output tracking error meets yt−ydt≤ε,∀t≥T, where ydt is the desired reference trajectory; (2) all closed-loop signals are bounded; (3) state constraint requirements are not violated.In order to design the controller, analyze the stability of the system, and achieve the above control objectives, the following assumption and lemmas are introduced.Assumption 1.
For a continuously differentiable expected trajectoryydt, there exists known positive y0 such that ydt≤y0,y˙dt≤y0, and y0<bci.Lemma 1.
According to Young’s inequality, given anya,b∈ℝ^3×3,p,q>1 and 1/p+1/q=1,(7)ab≤εap+cεbq,where ε is arbitrarily small and cε is arbitrarily large, and cε=pε−q/pq.Lemma 2.
According to Young’s inequality, for given positive integersm,n and for each x∈ℝ,y∈ℝ, the following inequality holds:(8)xmyn≤1γ1mm+nγ2−n/mxm+n+γ2nm+nγ1m/nym+n,where γ1>0 and γ2>0 can be any constant or real-valued function.Lemma 3.
For any given constantε>0 and z∈ℝ, there is(9)0≤z−ztanhzε≤κε,where κ is a constant that satisfies κ=e−κ+1; for instance, κ=0.2785.


### 2.3. Fuzzy Logic Theory

The specific form of the fuzzy logic system is as follows, which is composed of single point fuzzy set, product inference rule, center average weighted nonfuzzy set and Gaussian membership function:(10)yx=∑i=1NΦi∏j=1nμFjixj∑i=1N∏j=1nμFjixj.Let(11)ψix=∏j=1nμFjixj∑i=1N∏j=1nμFjixj.Ψx=ψ1x,ψ2x,…,ψNxT and Φ=Φ1,…,ΦNT. Then, the fuzzy system is(12)yx=ΦTΨx.Lemma 4.
There is a fuzzy systemΦTΨx; for any given constant ε>0, there is(13)supx∈Ωfx−ΦTΨx≤ε,where fx is a continuous function defined on the compact set Ω.


## 2.1. System Model

In this paper, we use the nonlinear dynamics of the UAV built in [46]; the details are shown as follows:(1)ϑ˙=ωysinγ+cosγωz+dϑ,ψ˙=ωycosγ−ωzsinγcosϑ+dψ,γ˙=ωx−tanϑωycosγ−ωzsinγ+dγ,Ω˙I=I−1M−ΩI×I∗ΩI+dΩI,where ϑ,ψ,andγ, respectively, represent the pitching, yaw, and rolling attitude angle of the UAV relative to the inertial coordinate system. ωx,ωy,andωz represent the angular velocity of each degree of freedom, respectively, where ΩI=ωx,ωy,ωzT. I is the rotational inertia matrix of the UAV. dii=θ,ψ,γ,ΩI are unknown unmodeled dynamics of the plant. The control scheme is depicted in Figure 1, and the details and expressions of I and M are shown as follows:(2)I=Ixx0Ixz0Iyy0Izx0Izz,M=q¯SbCxq¯Sc¯Cyq¯SbCz,IΔΔ is the product of inertia in both directions, q¯=ρVT2/2 is the dynamic pressure of air, VT is the total airspeed of the UAV, ρ is the air density, S is the total area of the wing, b is the wingspan, and c¯ is the average chord length. The rolling, pitching, and yaw moments of the UAV are mainly controlled by Cx,Cy,andCz, which can be given by(3)Cx=Cxδa1δa1+Cxδa2δa2+Cxδb1δb1+Cxδb2δb2+Cxββ+Δx,Cy=Cyδyδy+Cyββ+Δy,Cz=Cz0+Czδa1δa1+Cxδa2δa2+Czδb1δb1+Czδb2δb2+Czαα+Δz,where δa1,δa2 are the right and left aileron rudder deviation. δb1,δb2 are the deviation of the left and right lifter rudder; δv is the direction of the wing rudder deviation; Cx is the drag coefficient with Cxii=δa1,δa2,δb1,δb2,β being the coefficient of i to Cx; Cv is the drag coefficient with Cyii=δy,β being the coefficient of i to Cv; Cz is the drag coefficient with Czii=δa1,δa2,δb1,δb2,α being the coefficient of i to Cz; α,β are the angle of attack and sideslip angle.Figure 1
The structure of the proposed HOBLF control algorithm.In order to build a model of the UAV attitude system, definex1=ϑ,φ,γT,x2=ωx,ωy,ωzT, and u=δa1δa1δa2δb1δb2δyT. Then, the UAV attitude dynamic equation with time-varying interference can be expressed as(4)x˙1=Atx2+d1t,x˙2=Btut+fx1,x2+Δfx1,x2+d2t,y=x1,where At∈ℝ3×3,d1t∈ℝ3×1,Bt∈ℝ3×5,andfx∈ℝ3×1. They are denoted by(5)At=0sinγcosγ0cosγcosϑ−sinγcosϑ1−tanϑcosγtanϑsinγ,Bt=q¯SbI−1Cxδa1Cxδa2Cxδb1Cxδb200000CyδyCzδa1Czδa2Czδb1Czδb20fx1,x2=I−1q¯SbCxββCyββCz0+Czαα+ΩI×I∗ΩIΔfx1,x2=q¯SbI−1ΔxΔyΔz.All states are bound to the compact set(6)Ωx≔xi∈R,xit≤bci,i=1,2,where bci is a known positive constant.Remark 1.
In fact, it is difficult to develop an exact model for the UAV since we cannot reproduce the complex flight environment of the UAV in a wind tunnel. So, the unmodeled dynamicsdi and uncertain parameters Δi are inevitable.


## 2.2. Assumptions and Lemmas

The purpose of this paper is to design a controlleru: (1) for any initial state, there is a finite time T>0 so that the output tracking error meets yt−ydt≤ε,∀t≥T, where ydt is the desired reference trajectory; (2) all closed-loop signals are bounded; (3) state constraint requirements are not violated.In order to design the controller, analyze the stability of the system, and achieve the above control objectives, the following assumption and lemmas are introduced.Assumption 1.
For a continuously differentiable expected trajectoryydt, there exists known positive y0 such that ydt≤y0,y˙dt≤y0, and y0<bci.Lemma 1.
According to Young’s inequality, given anya,b∈ℝ^3×3,p,q>1 and 1/p+1/q=1,(7)ab≤εap+cεbq,where ε is arbitrarily small and cε is arbitrarily large, and cε=pε−q/pq.Lemma 2.
According to Young’s inequality, for given positive integersm,n and for each x∈ℝ,y∈ℝ, the following inequality holds:(8)xmyn≤1γ1mm+nγ2−n/mxm+n+γ2nm+nγ1m/nym+n,where γ1>0 and γ2>0 can be any constant or real-valued function.Lemma 3.
For any given constantε>0 and z∈ℝ, there is(9)0≤z−ztanhzε≤κε,where κ is a constant that satisfies κ=e−κ+1; for instance, κ=0.2785.


## 2.3. Fuzzy Logic Theory

The specific form of the fuzzy logic system is as follows, which is composed of single point fuzzy set, product inference rule, center average weighted nonfuzzy set and Gaussian membership function:(10)yx=∑i=1NΦi∏j=1nμFjixj∑i=1N∏j=1nμFjixj.Let(11)ψix=∏j=1nμFjixj∑i=1N∏j=1nμFjixj.Ψx=ψ1x,ψ2x,…,ψNxT and Φ=Φ1,…,ΦNT. Then, the fuzzy system is(12)yx=ΦTΨx.Lemma 4.
There is a fuzzy systemΦTΨx; for any given constant ε>0, there is(13)supx∈Ωfx−ΦTΨx≤ε,where fx is a continuous function defined on the compact set Ω.


## 3. Main Results

Next, introduce the following transformations:(14)e1=x1−yd,e2=x2−x2c,where e1,e2∈ℝ3×1and x2c∈ℝ3×1, respectively, represent virtual tracking errors and virtual control signal. Furthermore,x2c<x20∗ is constructed, where x20∗>0 is a constant. The following high-order tan-type BLFs are introduced to solve the problem of system state constraints of the fixed-wing UAV system:(15)Vij∗=2bi5−paπ5−pitanπeij5−pi2bi5−pi+D˜ij22ηij.For eachk=1,2,…,n, define Vk=∑i=1kVi∗, where pi∈Rodd≥1:=p/q|p,q are odd positive integers and p≥q,bi=bci−αi0∗, i is the order of the system, j is the dimension of the system, and αi0∗>0 is a constant that will be given later. For the second-order and three-dimensional UAV system in this article, p1=1andp2=3.Remark 2.
Tan-type BLFs (15) are proposed to handle full-state constraints for high-order systems, where bi=bci−αi0∗. Because bci⟶∞ means bi⟶∞,tanπeij5−pi/2bi5−pi∼πeij5−pi/2bi5−pi when bi⟶∞. From this, the formula can be further inferred as follows:(16)limbi⟶∞2bi5−piaπ5−pitanπei5−pi2bi5−pi=ei5−pia5−pi.
Therefore, the high-order tan BLF introduced in this paper will be transformed into a general Lyapunov function, and the design method is also effective for unconstrained high-order systems.The parameters used in the subsequent control design are defined as follows:(17)φij=cos2πeij5−pi2bi5−pi,i=1,2.,l1j=εaφ1j,where a and ε are adjustment parameters.(18)V˙1j=e1j3aφ1jx2j+d1jt−y˙jd+D˜1jD^˙1jη1j=−k1jtanπe1j42b14−l1je1j4+e1j3aφ1jx2j+d1jt+f¯1j+D˜1jD^˙1jη1j,where f¯1j=−y˙jd−aφ1jl1je1j+ak1sinτcosτ/e1j3,j=1,2,3,τ=πe1j4/2b14.Using the fuzzy systemΦ1jTΨ1jX1j, it is easy to get(19)f¯1j=Φ1jTΨ1jX1j+τ1jX1j,where τ1jX1j≤ε1j are the approximation errors. This comes together with Lemma 2:(20)e1j3aφ1jΦ1jTΨ1jX1j+τ1jX1j≤e1j3aφ1jΦ1jΨ1j+ε1j≤1Γ14ς1j−3Φ1ja4+ς1j34Γ1/3Ψ1jφ1j4/3e1j4+1Γ14ς1j−3ε1ja4+ς1j34Γ1/31φ1j4/3e1j4≤ξ1jΓ+ρ1j⋅e1j4,where Γ>0andς1j>0 are arbitrary constants. ξ1j=4ζ1j−3Φ1j/a4+ε1j/a4,ρ1j⋅=ς1j3/4Γ1/3Ψ1j/φ1j4/3+1/φ1j4/3.Substitute (20) into (18) to get(21)V˙1j≤−k1jtanπe1j42b14+e1j4aφ1je2j+d1jt+x2jc+ρ1j⋅e1j4+ξ1jΓ−l1je1j4+D˜1jD^˙1jη1j,where e2j=x2j−x2jc. Define(22)D1j=supt≥0d1jt.Then, according to Lemma3,(23)e1p−p1+1aφ1d1t≤e1p−p1+1aφ1d1t≤e1j3aφ1jD1j≤D1je1j3aφ1jtanhe1j3aφ1jε1j+κD1jε1j.x2jc is designed as(24)x2jc=−aφ1jρ1j⋅e1j−D^1jtanhe1j3aφ1jε1j.In this way,(25)e1j3aφ1jd1jt+x2jc≤D1je1j3aφ1jtanhe1j3aφ1jε1j+κD1jε1j−e1j3aφ1jaφ1jρ1j⋅e1j−e1j3aφ1jD^1jtanhe1j3aφ1jε1j≤−e1j4ρ1j⋅−D˜1je1j3aφ1jtanhe1j3aφ1jε1j+κD1jε1j.Substituting (25) into (21), we have(26)V˙1j≤−k1jtanπe1j42b14+e1j3aφ1je2j+ξ1jΓ−l1je1j4−D˜1je1j3aφ1jtanhe1j3aφ1jε1j+κD1jε1j+D˜1jD^˙1jη1j.The adaptive law is as follows:(27)D^˙1j=η1je1j3aφ1jtanhe1j3aφ1jε1j−σ1jD^1j.So, it can be deduced that(28)V˙1j≤−k1jtanπe1j42b14+e1j3aφ1je2j+ξ1jΓ−l1je1j4−σ1jD˜1jD^1j+κD1jε1j.Given−D˜1jD^1j≤−D˜1j2/2+D1j2/2, (28) can be rewritten as(29)V˙1j≤−k1jtanπe1j42b14+e1j3aφ1je2j−l1je1j4−σ1jD˜1j22+ξ1jΓ+σ1jD1j22+ξ¯1j,where ξ¯1j=κD1jε1j. Then, the asymptotic bounded convergence of the system is proved.Remark 3.
The negative term−l1je1j4 designed in (29) supplies an adequate stable domain to stabilize the positive terms containing e1j which will appear in the next design step.Next, consider the design ofu.(30)V˙2j=V˙1j+V˙2j∗=V˙1j+e2jaφ2juj+fx+d2t−x˙2jc+D˜2jD^˙2jη2j.By substituting (29) into (30), it can be obtained that(31)V˙2j≤−k1jtanπe1j42b4+e1j3aφ1je2j+e2jaφ2juj+fx+d2t−x˙2jc+ξ1jΓ−l1je1j4+ξ¯1j+D˜2jD^˙2jη2j.In accordance with Lemma1,(32)e1j3aφ1je2j≤1aφ1jεe1j3m+Cεe2jn≤ρ21j⋅e2jn+l1je1j3m,where m=4/3,n=4,Cε=nε−n/mn,ρ21j⋅=Cε/aφ1j,andl1j=ε/aφ1j.Substituting (32) into (31),(33)V˙2j≤−k1jtanπe1j42b14+ρ21j⋅e2jn+ξ1jΓ+ξ¯1j+e2jaφ2juj+fx+d2jt−x˙2jc+D˜2jD^˙2jη2j.Further deformation of the above equation can be obtained:(34)V˙2j≤−k1jtanπe42b14−k2jtanπe2j22b22+ρ21j⋅e2jn+ξ1jΓ+ξ¯1j−σ1jD˜1j22+σ1jD1j22−σ1jD˜1j22+σ1jD1j22+e2jaφ2juj+fx+d2jt+e2jaφ2jf¯2j+D˜2jD^˙2jη2j,where f¯2j=−x˙2jc+ak2jsinτcosτ/e2j.Use the fuzzy systemΦ2jTΨ2jX2j to get(35)f¯2j=Φ2jTΨ2jX2j+τ2jX2j,where τ2jX2j≤ε2j is the approximation error; through Lemma 2, we get(36)e2jaφ2jΦ2jTΨ2j+τ2j≤e2jaφ2jΦ2jΨ2j+ε2j≤1Γ34ζ22−1/3Φ2ja4/3+ς2214Γ3Ψ2jφ2j4e2j4+1Γ34ζ2j−1/3ε2ja4/3+ς2j14Γ31φ2j4e2j4=ξ2jΓ+ρ2j⋅e2j4,where ξ2j=3/4ζ2j−1/3Φ2j/a4/3+ε2j/a4/3,ρ2j=ς2j1/4Γ3Ψ2j/φ2j4+1/φ2j4, Γ>0, and ς2j>0, being adjustment parameters. Substituting (36) into (34) results in(37)V˙2j≤−k1jtanπe1j42b14−k2jtanπe2j22b22+ρ21j⋅e2jn+ρ2j⋅e2j4+ξjΓ+ξ¯1j+e2jaφ2jmj+d2jt+D˜2jD^˙2jη2j−σ1jD˜1j22+σ1jD1j22,where ξj=ξ1j+ξ2j.Same as above, according to Lemma3, define D2j=supt≥0d2jt so that(38)e2jaφ2jd2jt≤e2jaφ2jd2jt≤e2jaφ2jD2j≤D2je2jaφ2jtanhe2jaφ2jε2j+κD2jε2j.Designuj as(39)uj=−aφ2je2j3ρ21j+ρ2j−D^2jtanhe2jaφ2jε2j.Then,(40)e2jaφ2jd2jt+mj≤D2je2jaφ2jtanhe2jaφ2jε2j+κD2jε2j−e2jaφ2jaφ2je2j3ρ21j+ρ2j−e2jaφ2jD^2jtanhe2aφ2jε2j≤−e2j4ρ21j+ρ2j−D˜2je2aφ2jtanhe2jaφ2jε2j+κD2jε2j.This together with (37) yields(41)V˙2j≤−k1jtanπe1j42b14−k2jtanπe2j22b22+ξjΓ+ξ¯1j−D˜2je2jaφ2jtanhe2jaφ2jε2j+κD2jε2j+D˜2jD^˙2jη2j−σ1jD˜1j22+σ1jD1j22.Design the adaptive law as(42)D^˙2j=η2je2jaφ2jtanhe2jaφ2jε2j−σ2jD^2j.So, it can be deduced that(43)V˙2j≤−k1jtanπe142b14−k2jtanπe2j22b22+ξjΓ+ξ¯1j+κD2jε2j−σ2jD˜2jD^2j−σ1jD˜1j22+σ1jD1j22.In the same way,−D˜2jD^2j≤−D˜2j2/2+D2j2/2. Therefore, (43) can be rewritten as(44)V˙2j≤−k1jtanπe1j42b14−k2jtanπe2j22b22+ξjΓ+ξ¯1j+ξ¯2j−σ2jD˜2j22+σ2jD2j22−σ1jD˜1j22+σ1jD1j22.Then, the asymptotic bounded convergence of the system can be proved.Because the system is a second-order three-dimensional system,V1j, V2j are asymptotically bounded convergences, and V1=V11+V12+V13, V2∗=V21+V22+V23. So, the system is also asymptotically bounded. According to Vk=∑i=1kVi∗, we know that V2=V1+V2∗. Then,(45)V˙2≤−∑j=13k1jtanπe1j42b14+k2jtanπe2j22b22+σ2jD˜2j22−σ2jD2j22−σ1jD˜1j22+σ1jD1j22+ξ¯1j+ξ¯2j+ξjΓ≤−cV2+ξnΓ+ξ¯n,where ξn=∑j=13ξj,ξ¯n=∑j=13ξ¯ij,i=1,2,andc=min1≤i≤21≤j≤3akijπp−pi+2/2bip−pi+2.Proof.
If the system with full-state constraints meets the assumptions, the trace controlleru can accomplish the following:(i)
All states of a closed-loop system are bounded(ii)
It will not violate the complete state constraints(iii)
The tracking errorse1,e2 converge to any small neighborhood of zero(iv)
The system is stable under time-varying disturbance
ConsideringΓ>1, according to formula (45), we can get V˙2≤−cV2+ξn+ξ¯n. Therefore, we can infer that V2≤V20−ξn+ξ¯n/ce−ct+ξn+ξ¯n/c. Hence, V2 is constrained; it also represents that tanπeip−pi+2/2bip−pi+2 is bounded. Therefore, ei<bi. According to the Assumption 1, we can know that x1≤e1+yd<b1+y0=bc1. On the contrary, it can be inferred from the continuity of x2c and the boundedness of x1,yd,D1j^ that x2c is also constrained. Hence, there is a finite constant x2c0∗ that makes x2c≤x2c0∗. From the boundedness and error transformation of e2 and x2c, the conclusion of x2≤e2+x2c<b2+x2c0=bc2 can be drawn. As a result, all states are bounded and meet the constraints.Remark 4.
For the possible disturbancedit, it is replaced by supt≥0dit, which greatly reduces the computation. If the adaptive parameters are used to cancel the disturbance, the adaptive parameters should be designed for dit, respectively, so that the computation burden will be increased.


## 4. Simulation Experiment

### 4.1. Initial Value Selection

In order to verify the effectiveness and superiority of the HOBLF method proposed in this paper under the influence of time-varying interference, this section studies the simulation effect under the influence of strong time-varying interference and weak time-varying interference, respectively, to verify its validity. Under the same conditions, the simulation effects of the proposed method and DOBC, general adaptive control, and low-order general BLF were compared to verify its superiority.Now, suppose that the UAV rises at a fixed attitude angle in the air, and the UAV needs to change from the rising stage to the flat flying stage. The square wave signal filtered by the second-order link is selected as the expected signal with an amplitude of 1 and a period of 20 s. Related parameters have been listed in Tables1 and 2.Table 1
Related parameters of the fixed-wing UAV.
VariableImplicationValueVm/sVelocity10mkgWeight28bmWingspan3.1Sm2Wing area1.8ρkg/m3Air density1.29Table 2
The initial values of the system states.
VariableImplicationValueϑradPitch angle0.2ψradYaw angle0.1γradRoll angle0.1ωxrad/sPitch rate0ωyrad/sYaw rate0ωzrad/sRoll rate0D^0Adaptive parameter0.001η1,η2Gain of the adaptive law1, 2σ1,σ2Other parameters of the adaptive law5


### 4.2. Simulation Comparison

The effectiveness of the HOBLF is verified by performing the simulation under three cases.Case 1.
First, consider the case of using the HOBLF for control under weak time-varying interference. For pitch angleϑ, yaw angle ψ, and roll angle γ, the time-varying interferences are 0.01sint,0.02sint, and 0.03sint, respectively.Case 2.
Then, consider the case of controlling with the HOBLF under strong time-varying interference. For pitch angleϑ, yaw angle ψ, and roll angle γ, the time-varying interferences are 0.1sint, 0.2sint, and 0.3sint, respectively.The simulation results are shown in Figures2–5. It can be analyzed from Figures 2–5 that, under the influence of strong or weak time-varying interference, the controlled states ϑ,ψ, and γ can track the expected signal ϑd,ψd, and γd smoothly without violating the state constraint. Therefore, the effectiveness and superiority of the HOBLF can be verified.Figure 2
Variation curves of pitch angle in case 1 and case 2.Figure 3
Variation curves of yaw angle in case 1 and case 2.Figure 4
Variation curves of roll angle in case 1 and case 2.Figure 5
Tracking error of pitch angle, yaw angle, and roll angle in case 1 and case 2 andD^ in case 1 and case 2.Case 3.
high-order tan-type BLF is selected. The tangent Lyapunov function can not only deal with the adaptive control of nonlinear systems with state constraints but also can be regarded as a quadratic Lyapunov function without constraints. Therefore, the tangential Lyapunov function can deal with constrained and unconstrained systems. The state constraintbc1 on x1 is 1.5.
The design of inside loop controllerx2c is as follows:(46)φ1=cos2πe142b14,ρ1j⋅=ς1134Γ1/3Ψ1jφ1j4/3+1φ1j4/3,njc=−aφ1jρ1j⋅e1j−D^1jtanhe1j3aφ1jε1jx2jc=A−1njc=A−1−aφ1jρ1j⋅e1j−D^1jtanhe1j3aφ1jε1j.
The design of outside loop controlleru is as follows:(47)Cε=nε−n/mn,ε=0.5,ρ21j⋅=Cεaφ1j,ρ2j=ς22Γ34Ψ2jφ2j4+1φ2j4,mj=−aφ2je2j3ρ21j+ρ2j−D^2jtanhe2jaφ2jε2j,u=BTBTBM−fx.
Other relevant parameters are set as follows:K=40,ς1=1,Ψ1=2,a=1,ς2=1,andΨ2=2.To show the advantages, the TAC (traditional adaptive control), DOBC, and general BLF are applied to the simulation.Using the traditional adaptive method, it first applies backstepping design technology to constructively give the design steps of the state feedback adaptive controller and then combines Barbalat’s lemma to analyze the stability of the closed-loop system, making the closed-loop system state globally asymptotically stable. The remaining closed-loop signals are uniformly bounded. The selected Lyapunov function isV=0.5e2. The relevant parameters are designed as k1=diag1021,andk2=diag 1021.DOBC (disturbance-observer-based control): by designing a disturbance observer to cancel the influence of disturbance on the servo system, the disturbance is regarded as a part of the system state, and the reduced-order state observer is used to estimate and cancel the disturbance. The design of the disturbance observer isΔC^=z+Lx2 which is an auxiliary variable, and the relevant parameters are designed as L=1,k1=diag 1021,andk2=diag 1021.Using the general BLF method, the symmetric positive definite log-type BLF is as follows:(48)Vi=14logkbi4kbi4−ei4+σi2λiμ˜r,iTμ˜r,i.It can guarantee that every step in the backstepping method does not violate the state constraint and guarantee the stability of the system, whereσi and λi are positive design parameters, kbi=kci−αi0, and ur,j is an adaptive parameter. The relevant parameters are designed as γ1=0.5,kb1=1,kb2=2,andγ2=0.2.HOBLF, DOBC, general adaptive, and general BLF were, respectively, used for simulation. The expected signal tracked was the square wave signal after second-order filtering, with an amplitude of 1 and a period of 20 s.The simulation results of the four methods are shown in Figures6–11, where Figures 6 and 9 show the tracking trajectory and error curves of the pitch angle, respectively. Figures 7 and 10 show the yaw angle tracking track and error curve, respectively. Figures 8 and 11, respectively, show the tracking track and error curve of the roll angle. According to the analysis in Figures 6–8, the states in the traditional adaptive method and DOBC will exceed the state constraint and have a large deviation. The tracking trajectory of the high-order BLF and general BLF in three directions is always between the upper and lower limits of the state constraint. However, for the tracking with a constant expected signal, high-order BLF is more stable and has less error. It can be easily known from Figures 9–11 that compared with the high-order BLF method, the traditional adaptive method, DOBC, and general BLF have a larger tracking trajectory and error.Figure 6
Tracking trajectory of pitch angle.Figure 7
Tracking trajectory of yaw angle.Figure 8
Tracking trajectory of roll angle.Figure 9
Tracking error of pitch angle.Figure 10
Tracking error of yaw angle.Figure 11
Tracking error of roll angle.


## 4.1. Initial Value Selection

In order to verify the effectiveness and superiority of the HOBLF method proposed in this paper under the influence of time-varying interference, this section studies the simulation effect under the influence of strong time-varying interference and weak time-varying interference, respectively, to verify its validity. Under the same conditions, the simulation effects of the proposed method and DOBC, general adaptive control, and low-order general BLF were compared to verify its superiority.Now, suppose that the UAV rises at a fixed attitude angle in the air, and the UAV needs to change from the rising stage to the flat flying stage. The square wave signal filtered by the second-order link is selected as the expected signal with an amplitude of 1 and a period of 20 s. Related parameters have been listed in Tables1 and 2.Table 1
Related parameters of the fixed-wing UAV.
VariableImplicationValueVm/sVelocity10mkgWeight28bmWingspan3.1Sm2Wing area1.8ρkg/m3Air density1.29Table 2
The initial values of the system states.
VariableImplicationValueϑradPitch angle0.2ψradYaw angle0.1γradRoll angle0.1ωxrad/sPitch rate0ωyrad/sYaw rate0ωzrad/sRoll rate0D^0Adaptive parameter0.001η1,η2Gain of the adaptive law1, 2σ1,σ2Other parameters of the adaptive law5


## 4.2. Simulation Comparison

The effectiveness of the HOBLF is verified by performing the simulation under three cases.Case 1.
First, consider the case of using the HOBLF for control under weak time-varying interference. For pitch angleϑ, yaw angle ψ, and roll angle γ, the time-varying interferences are 0.01sint,0.02sint, and 0.03sint, respectively.Case 2.
Then, consider the case of controlling with the HOBLF under strong time-varying interference. For pitch angleϑ, yaw angle ψ, and roll angle γ, the time-varying interferences are 0.1sint, 0.2sint, and 0.3sint, respectively.The simulation results are shown in Figures2–5. It can be analyzed from Figures 2–5 that, under the influence of strong or weak time-varying interference, the controlled states ϑ,ψ, and γ can track the expected signal ϑd,ψd, and γd smoothly without violating the state constraint. Therefore, the effectiveness and superiority of the HOBLF can be verified.Figure 2
Variation curves of pitch angle in case 1 and case 2.Figure 3
Variation curves of yaw angle in case 1 and case 2.Figure 4
Variation curves of roll angle in case 1 and case 2.Figure 5
Tracking error of pitch angle, yaw angle, and roll angle in case 1 and case 2 andD^ in case 1 and case 2.Case 3.
high-order tan-type BLF is selected. The tangent Lyapunov function can not only deal with the adaptive control of nonlinear systems with state constraints but also can be regarded as a quadratic Lyapunov function without constraints. Therefore, the tangential Lyapunov function can deal with constrained and unconstrained systems. The state constraintbc1 on x1 is 1.5.
The design of inside loop controllerx2c is as follows:(46)φ1=cos2πe142b14,ρ1j⋅=ς1134Γ1/3Ψ1jφ1j4/3+1φ1j4/3,njc=−aφ1jρ1j⋅e1j−D^1jtanhe1j3aφ1jε1jx2jc=A−1njc=A−1−aφ1jρ1j⋅e1j−D^1jtanhe1j3aφ1jε1j.
The design of outside loop controlleru is as follows:(47)Cε=nε−n/mn,ε=0.5,ρ21j⋅=Cεaφ1j,ρ2j=ς22Γ34Ψ2jφ2j4+1φ2j4,mj=−aφ2je2j3ρ21j+ρ2j−D^2jtanhe2jaφ2jε2j,u=BTBTBM−fx.
Other relevant parameters are set as follows:K=40,ς1=1,Ψ1=2,a=1,ς2=1,andΨ2=2.To show the advantages, the TAC (traditional adaptive control), DOBC, and general BLF are applied to the simulation.Using the traditional adaptive method, it first applies backstepping design technology to constructively give the design steps of the state feedback adaptive controller and then combines Barbalat’s lemma to analyze the stability of the closed-loop system, making the closed-loop system state globally asymptotically stable. The remaining closed-loop signals are uniformly bounded. The selected Lyapunov function isV=0.5e2. The relevant parameters are designed as k1=diag1021,andk2=diag 1021.DOBC (disturbance-observer-based control): by designing a disturbance observer to cancel the influence of disturbance on the servo system, the disturbance is regarded as a part of the system state, and the reduced-order state observer is used to estimate and cancel the disturbance. The design of the disturbance observer isΔC^=z+Lx2 which is an auxiliary variable, and the relevant parameters are designed as L=1,k1=diag 1021,andk2=diag 1021.Using the general BLF method, the symmetric positive definite log-type BLF is as follows:(48)Vi=14logkbi4kbi4−ei4+σi2λiμ˜r,iTμ˜r,i.It can guarantee that every step in the backstepping method does not violate the state constraint and guarantee the stability of the system, whereσi and λi are positive design parameters, kbi=kci−αi0, and ur,j is an adaptive parameter. The relevant parameters are designed as γ1=0.5,kb1=1,kb2=2,andγ2=0.2.HOBLF, DOBC, general adaptive, and general BLF were, respectively, used for simulation. The expected signal tracked was the square wave signal after second-order filtering, with an amplitude of 1 and a period of 20 s.The simulation results of the four methods are shown in Figures6–11, where Figures 6 and 9 show the tracking trajectory and error curves of the pitch angle, respectively. Figures 7 and 10 show the yaw angle tracking track and error curve, respectively. Figures 8 and 11, respectively, show the tracking track and error curve of the roll angle. According to the analysis in Figures 6–8, the states in the traditional adaptive method and DOBC will exceed the state constraint and have a large deviation. The tracking trajectory of the high-order BLF and general BLF in three directions is always between the upper and lower limits of the state constraint. However, for the tracking with a constant expected signal, high-order BLF is more stable and has less error. It can be easily known from Figures 9–11 that compared with the high-order BLF method, the traditional adaptive method, DOBC, and general BLF have a larger tracking trajectory and error.Figure 6
Tracking trajectory of pitch angle.Figure 7
Tracking trajectory of yaw angle.Figure 8
Tracking trajectory of roll angle.Figure 9
Tracking error of pitch angle.Figure 10
Tracking error of yaw angle.Figure 11
Tracking error of roll angle.


## 5. Conclusion

In this work, an adaptive fuzzy controller has been proposed for a fixed-wing UAV control system of full-state constraints and unknown external disturbances. The high-order BLF (HOBLF) is firstly introduced for the UAV systems to realize the state constraints. Meanwhile, the unknown nonlinear functions have been compensated by the fuzzy system and backstepping control method to ensure the stability of the system. According to simulation results and comparison, we have made the conclusion that the proposed control is able to track a desired trajectory with an excellent performance without violating the state constraints. Moreover, further research based on the current results of this paper is how to verify the control feasibility of the UAV system with existing time-varying state constraint issues.



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