# Sliding Mode Control Based on High-Order Linear Extended State Observer for Near Space Vehicle

Authors: Ouxun Li; Ju Jiang; Li Deng; Shutong Huang
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

Aiming at the uncertainty and external disturbance sensitivity of the near space vehicles (NSV), a novel sliding mode controller based on the high-order linear extended state observer (LESO) is designed in this paper. In the proposed sliding mode controller, the double power reaching law is adopted to enhance the state convergence rate, and the high-order LESO is designed to improve the antidisturbance ability. Moreover, the appropriate observer bandwidth and extended order are selected to further reduce or even eliminate the disturbance by analyzing their influences on the observer performance. Finally, the simulation demonstrations are given for the NSV control system with uncertain parameters and external disturbances. The theoretical analyses and simulation results consistently indicate that the proposed high-order LESO with carefully selected extended order and observer bandwidth has better performance than the traditional ones for the nonlinear NSV system with parametric uncertainty and external disturbance.


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## Body

## 1. Introduction

Near space vehicle (NSV) has many excellent characteristics, such as fast speed, strong survival and penetration ability, high flight altitude, wide application range, and strong precision strike ability. Despite the great challenges of NSV control considering the nonlinearity, strong coupling and uncertainty of NSV, in addition with their susceptibility to external disturbances [1], the control technology based on modern control theory has been widely studied. Morden control methods such as the robust control, predictive control, sliding mode control, and intelligent control have been applied to hypersonic vehicle control and achieved certain results. On the other hand, the single control method has its own advantages and disadvantages in the NSV control application, which might not meet the multiple requirements of flight control. Therefore, the combination of different control methods to give full play to their respective advantages has become one of the research focus.In practical applications, uncertainties such as modeling errors and external disturbances are always existent, which make the system output unstable or asymptotically track the desired target. Therefore, different robust control strategies have been proposed. Thereinto, the sliding mode control has been widely used in the flight control system due to its simple structure, fast response, and its insensitivity to external disturbances [2, 3]. In [4], a finite-time attitude control is developed to ensure that the required thrust vector is met exactly at the prescribed time. In [5], a novel recursive singularity free fast terminal sliding mode strategy for finite-time tracking control of nonholonomic systems is proposed. Specially, the sliding mode control with disturbance observation compensation is quite suitable for uncertain nonlinear systems with external disturbances and parameter perturbations [6, 7]. In [8], a novel nonsingular fast terminal sliding mode control method based on disturbance observer is proposed for the stabilization of the uncertain time-varying and nonlinear third-order systems. In [9], the uncertainties are effectively solved by the combination of the sliding mode control and the interval type-2 Takagi-Sugeno-Kang (TSK) fuzzy control. The double power reaching law sliding mode control is adopted in [10] to track the large range command of the climbing stage of the NSV for better performance. In the aspect of anti-interference, the extended state observer can estimate the total disturbance in real time according to the input and output system information and can be eliminated in the feedback control, so that the closed-loop dynamic system has better control performance. Moreover, there is a set of mature empirical formula for the parameter tuning of the observer, which is more convenient for engineering application. In [11], a robust adaptive finite-time fast terminal sliding mode controller is proposed to achieve the desired formation in the presence of model uncertainties and external disturbances. In [12], a novel sliding mode control approach is proposed for the control of a class of underactuated systems which are featured as in cascaded form with external disturbances. Authors in [13] proposed a combination of finite-time robust-tracking theory and composite nonlinear feedback approach for the finite-time and high-performance synchronization of the chaotic systems in the presence of the external disturbances, parametric uncertainties, Lipschitz nonlinearities, and time delays. In [14], an adaptive controller is proposed by employing an event-triggered control and an extended-state-observer, where a simple strategy to tune the observer parameters is provided. Authors in [15] compensate the disturbance by combining the sliding mode control with the autodisturbance rejection control. A novel antisaturation controller is designed in [16] for the air breathing hypersonic vehicle, using the observer to compensate the mismatched disturbance. A state observer with adaptive extension and a continuous sliding mode controller with disturbance observer are proposed and applied to the hypersonic vehicle system in [17, 18], respectively. The nonlinear disturbance observer is applied to the robust flight control of the air breathing hypersonic vehicle in [19], which can significantly improve the robustness and antidisturbance ability of the system. Authors in [20, 21] have pointed out that the high-gain error feedback can guarantee the fast convergence of the observation error and the sufficient estimation accuracy. The performance analyses of the LESO and its higher order form for the second-order system in [22] indicate that the high-order form has faster response speed and better low-frequency characteristics.From the above recent works, it can be seen that the sliding mode control has good robustness in the NSV control system. The disturbance observation compensation can reduce or even eliminate the disturbance and improve the antidisturbance ability. However, considering the high-order, uncertainty, strong coupling of the NSV control system and other factors in the practical application, the traditional LESO could not meet the performance requirements of NSV control system. Given that, a novel sliding mode controller based on the high-order LESO is proposed in this paper. Based on the traditional LESO method, the high-order LESO is used to estimate and compensate the external disturbance. The sliding mode control method with disturbance compensation is used to improve the stability and anti-interference ability of the system. In this paper, we have proven that compared with the traditional LESO, the high-order LESO has better low-frequency characteristics and anti-disturbance ability for the NSV control system. The main contributions of our work can be concluded as follows.(1)
The feedback linearization equivalent model of NSV is established. According to the equivalent model, a sliding mode controller with high-order disturbance compensator is designed to improve the system stability and anti-interference ability. The convergence region of high-order disturbance observer under the condition of lumped disturbance is analyzed, and the system stability is also proved by Lyapunov method(2)
The sliding mode controller with high-order LESO compensation is proposed, and the design method of the high-order LESO is also provided(3)
The transfer functions from the lumped disturbance to the disturbance estimate and the disturbance estimate error in high-order form are deduced, respectively. Moreover, the influence of the observer bandwidth and the extended order on the observer performance are analyzed specifically in the time domain and frequency domain, respectively(4)
The nonlinear NSV model with uncertain parameters and external disturbance has been proven, and the simulation results also show that the high-order LESO has better robustness and antidisturbance ability by selecting appropriate order and observation bandwidthThe remainder of this paper is organized as follows. The theoretical bases are presented in Section2, including the NSV modeling, the sliding mode controller, and the traditional observer design method. The proposed high-order LESO is detailed in Section 3. The aircraft case simulations and discussions are provided in Section 4, followed by the conclusion in Section 5.


## 2. Preliminaries

### 2.1. NSV Model

According to [9] and [23–26], the motion model of NSV longitudinal channel is defined as follows:
(1)v̇=Tcosα−Dm−μsinγr2,γ̇=L+Tsinαmv−μ−v2rcosγvr2,q̇=MyIy,α̇=q−γ̇,ḣ=vsinγ,β¨=−2ξωβ̇−ω2β+ω2βc, where v, γ, q, α, and h indicate the velocity, track angle, pitch rate, angle of attack, and altitude, respectively; β, ω, and ξ are severally the throttle setting, engine natural frequency, and damping coefficient, followed by L, D, T, m, μ, r, My, and Iy as the lift, drag, thrust, mass, gravitational constant, distance from the center of mass to the center of gravity, pitching moment, and the moment of inertia, respectively. Thereinto, the lift, drag, and thrust can be expressed as [9]
(2)L=12ρv2sCL,D=12ρv2sCD,T=12ρv2sCT,Cmδe=0.0292δe−α,My=12ρv2sc¯Cm+Cmδe,CT=0.02576β,β<1,0.0224+0.00336β,β≥1, where CL, CD, CT, and Cm are coefficients for the lift, drag, thrust, and pitch moment, respectively; the control input signals include the throttle setting βc and elevator deflection angle δe.Note that the NSV model in Equation (1) is nonlinear. However, according to the nonlinear theory, the input-output linearization can be adopted first; then, v and h are derived three times and four times, respectively. Thus, the linearized model can be obtained as follows [9]:
(3)v̇=Tcosα−Dm−μsinγr2=f1x,v¨=∂f1x∂xẋ=ω1ẋ,v⃛=ẋT∂ω1∂xẋ+ω1x¨=ω1x¨+ẋTω2x.̇(4)ḣ=vsinγ,h¨=v̇sinγ+vγ̇cosγ,h⃛=v¨sinγ+2v̇γ̇cosγ−vγ̇2sinγ+vγ¨cosγ,h4=v⃛sinγ+3v¨γ̇cosγ−3v̇γ2sinγ+3v̇γ¨cosγ−vγ̇3cosγ−3vγ̇γ¨sinγ+vcosγγ,⃛ where the first, second, and third derivatives of γ are obtained according to Equation (1) as
(5)γ̇=L+Tsinαmv−μ−v2rcosγvr2=f2x,γ¨=∂f2x∂xẋ=π1ẋ,γ⃛=ẋT∂π1∂xẋ+π1x¨=π1x¨+ẋTπ2x,̇where ω1=∂f1x/∂x, ω2=∂ω1/∂x,π1=∂f2x/∂x, andπ2=∂π1/∂x.Note that in Equation (3) and Equation (5), the flight state vector x=vγqαhT is available for measurement. Accordingly, the linearized model can be described as follows:
(6)v⃛=FV+b11βc+b12δe,h4=Fh+b21βc+b22δe.(7)Fv=ω1x¨0+ẋTω2ẋm,Fh=3v¨γ̇cosγ−3v̇γ2sinγ+3v̇γ¨cosγ−vγ̇3cosγ−3vγ̇γ¨sinγ+ω1x¨0+ẋTω2ẋmsinγ+vcosγπ1x¨+ẋTπ0ẋ,b11=ρv2sCβω22mcosαb12=−ρv2sc¯∙ce2mIyTsinα+Dα,b21=ρv2sCβω22msinα+γ,b22=ρv2sc¯∙ce2mIyTcosα+γ+Lαcosγ−Dαsinγ..When the external disturbances are considered, the linearized model can be uniformly described as [27, 28].
(8)v⃛h4=FvFh+BU+d1td2t,whereU=U1U2, and βc and δe in Equation (6) are substituted by U1 and U2 as the control inputs. This linearized model considering external disturbances of Equation (8) is defined under the hypotheses as follows [28]:Assumption 1.
The external disturbancedit and its first-order differential are bounded as dit≤δ, ḋit≤δ¯, where δ and δ¯ are known normal numbers.Assumption 2.
The matrixB is nonsingular. In the whole branch envelope, it is reasonable to assume that the matrix B is nonsingular because the flight path angle of the aircraft satisfies γ≠±π/2.Note that Assumption1 is only suitable for the traditional state expansion observer. The high-order form of observer will be discussed in the following sections.


### 2.2. Design of the Sliding Mode Controller

The design process of the sliding mode controller can be described as the following two steps.(1)
Select the integral sliding surfaceS=SvSh as(9)S=Sv=ddt+λv3∫0tevτdτ,Sh=ddt+λh4∫0tehτdτ,where evt=vt−vdt and eht=ht−hdt are the tracking errors for the velocity channel and the altitude channel, respectively. λv and λh are the positive constants. The derivatives of Equation (9) can be obtained as
(10)Ṡ=ṠV=v⃛−v⃛d+3λve¨v+3λv2ėv+λv3ev,Ṡh=h4−hd4+4λhe⃛h+6λh2e¨h+4λh3ėh+λh4eh.Letψv=−v⃛d+3λve¨v+3λv2ėv+λv3ev, and ψh=−hd4+4λhe⃛h+6λh2e¨h+4λh3ėh+λh4eh. By introducing the model Equation (6) into Equation (10), we can get
(11)Ṡ=Ṡv=Fv+b11U1+b12U2+ψv,Ṡh=Fh+b21U1+b22U2+ψh.(2)
Define the sliding mode approaching rate as(12)Ṡ=−αv1Svβv1sgnSv−αv2Svβv2sgnSv,−αh1Shβh1sgnSh−αh2Shβh2sgnSh,where αv1>0, αv2>0,αh1>0,αh2>0,βv1>1,βh1>1,0<βv2<1, and0<βh2<1. When the system state is far away from the sliding mode, αv1Svβv1sgnSv and αh1Shβh1sgnSh play a dominant role, while αv2Svβv2sgnSv and αh2Shβh2sgnSh play a dominant role when the system state is close to the sliding mode. According to Equations (8), (11), and (12), the continuous-time controller can be obtained as follows:
(13)U=U1U2=B−1−Fv−ψv−Fh−ψh+B−1−αv1Svβv1sgnSv−αv2Svβv2sgnSv−d¯1t−αh1Shβh1sgnSh−αh2Shβh2sgnSh−d¯2t,where
(14)B=b11b12b21b21.The sliding mode controller can make the system state converge into the neighborhood of the equilibrium zero point quickly, if there are uncertain external disturbances in the system [29]. To ensure the convergence of system state, the observer can be used to compensate the bounded lumped disturbance caused by the system modeling error and bounded external disturbances.


### 2.3. Design of Traditional Extended State Disturbance Observer

Considering the following third-order system with external disturbances and uncertainties, the state space form is described as(15)ẋ1=x2,ẋ2=x3,ẋ3=ft,x,w+B∅y,u,y=x1, where ft,x,w is the sum of system uncertainties and external disturbances. Equation (15) can be rewritten as follows:
(16)ẋ1=x2,ẋ2=x3,ẋ3=x4+B∅y,uẋ4=d,y=x1,, where d is the derivative of the system lumped disturbance. Then, the traditional extended state observer can be designed as follows [25, 26]:
(17)x¯̇1=x¯2+a1ω0y−x¯1,x¯̇2=x¯3+a2ω02y−x¯1,x¯̇3=x¯4+∅y,u+a3ω03y−x¯1,x¯̇4=a4ω04y−x¯1, where x¯̇1, x¯̇2, x¯̇3, and x¯̇4 are observer states, a1, a2, a3, and a4 are positive real numbers, and ω0 is the observation bandwidth.According to the above analyses, the state space model of the velocity channel can be expressed as(18)ẋ=A1x+B1∅1y,u+w1d1y=C1xwhere
(19)A1=0100001000000100,B1=0010,C1=1000,w1=0001,∅1y,u=FV+b11U1+b12U2.The disturbance observer can be described as(20)x¯̇v=A1x¯v+B1∅1y,u+L1y−C1x¯v,where L1=av1ωvav2ωv2av3ωv3av4ωv4T, avi=4!/i!4−i!,i=1,2,3,4, avi is selected by the bandwidth-based configuration method. More generally, in traditional LESO for any order system, the adjustable parameter only has the observation bandwidth ωv.Similarly, the altitude channel can be described as(21)x¯̇h=A2x¯h+B2∅2y,u+L2y−C2x¯h,where
(22)A2=0100000100000000000100010,B2=00010,C2=10000T,∅2y,u=Fh+b21U1+b22U2,L2=ah1ωhah2ωh2ah3ωh3ah4ωh4ah5ωh5T,ahi=5!i!5−i!,i=1,2,3,4,5,and the adjustable parameter only has the observation bandwidthωh.Combined Equation (20) with Equation (21), the aircraft disturbance observer can be described as
(23)x¯̇=Ax¯+B∅y,u+Ly−Cx¯,where
(24)A=010000100000010000000000000000000000000000000000000000000100000100000000000100010,B=001000000000000010,C=100000000000010000T,∅y,u=FV+b11U1+b12U2Fh+b21U1+b22U2,L=av1ωvav2ωv2av3ωv3av4ωv4000000000ah1ωhah2ωh2ah3ωh3ah4ωh4ah5ωh5T.


## 2.1. NSV Model

According to [9] and [23–26], the motion model of NSV longitudinal channel is defined as follows:
(1)v̇=Tcosα−Dm−μsinγr2,γ̇=L+Tsinαmv−μ−v2rcosγvr2,q̇=MyIy,α̇=q−γ̇,ḣ=vsinγ,β¨=−2ξωβ̇−ω2β+ω2βc, where v, γ, q, α, and h indicate the velocity, track angle, pitch rate, angle of attack, and altitude, respectively; β, ω, and ξ are severally the throttle setting, engine natural frequency, and damping coefficient, followed by L, D, T, m, μ, r, My, and Iy as the lift, drag, thrust, mass, gravitational constant, distance from the center of mass to the center of gravity, pitching moment, and the moment of inertia, respectively. Thereinto, the lift, drag, and thrust can be expressed as [9]
(2)L=12ρv2sCL,D=12ρv2sCD,T=12ρv2sCT,Cmδe=0.0292δe−α,My=12ρv2sc¯Cm+Cmδe,CT=0.02576β,β<1,0.0224+0.00336β,β≥1, where CL, CD, CT, and Cm are coefficients for the lift, drag, thrust, and pitch moment, respectively; the control input signals include the throttle setting βc and elevator deflection angle δe.Note that the NSV model in Equation (1) is nonlinear. However, according to the nonlinear theory, the input-output linearization can be adopted first; then, v and h are derived three times and four times, respectively. Thus, the linearized model can be obtained as follows [9]:
(3)v̇=Tcosα−Dm−μsinγr2=f1x,v¨=∂f1x∂xẋ=ω1ẋ,v⃛=ẋT∂ω1∂xẋ+ω1x¨=ω1x¨+ẋTω2x.̇(4)ḣ=vsinγ,h¨=v̇sinγ+vγ̇cosγ,h⃛=v¨sinγ+2v̇γ̇cosγ−vγ̇2sinγ+vγ¨cosγ,h4=v⃛sinγ+3v¨γ̇cosγ−3v̇γ2sinγ+3v̇γ¨cosγ−vγ̇3cosγ−3vγ̇γ¨sinγ+vcosγγ,⃛ where the first, second, and third derivatives of γ are obtained according to Equation (1) as
(5)γ̇=L+Tsinαmv−μ−v2rcosγvr2=f2x,γ¨=∂f2x∂xẋ=π1ẋ,γ⃛=ẋT∂π1∂xẋ+π1x¨=π1x¨+ẋTπ2x,̇where ω1=∂f1x/∂x, ω2=∂ω1/∂x,π1=∂f2x/∂x, andπ2=∂π1/∂x.Note that in Equation (3) and Equation (5), the flight state vector x=vγqαhT is available for measurement. Accordingly, the linearized model can be described as follows:
(6)v⃛=FV+b11βc+b12δe,h4=Fh+b21βc+b22δe.(7)Fv=ω1x¨0+ẋTω2ẋm,Fh=3v¨γ̇cosγ−3v̇γ2sinγ+3v̇γ¨cosγ−vγ̇3cosγ−3vγ̇γ¨sinγ+ω1x¨0+ẋTω2ẋmsinγ+vcosγπ1x¨+ẋTπ0ẋ,b11=ρv2sCβω22mcosαb12=−ρv2sc¯∙ce2mIyTsinα+Dα,b21=ρv2sCβω22msinα+γ,b22=ρv2sc¯∙ce2mIyTcosα+γ+Lαcosγ−Dαsinγ..When the external disturbances are considered, the linearized model can be uniformly described as [27, 28].
(8)v⃛h4=FvFh+BU+d1td2t,whereU=U1U2, and βc and δe in Equation (6) are substituted by U1 and U2 as the control inputs. This linearized model considering external disturbances of Equation (8) is defined under the hypotheses as follows [28]:Assumption 1.
The external disturbancedit and its first-order differential are bounded as dit≤δ, ḋit≤δ¯, where δ and δ¯ are known normal numbers.Assumption 2.
The matrixB is nonsingular. In the whole branch envelope, it is reasonable to assume that the matrix B is nonsingular because the flight path angle of the aircraft satisfies γ≠±π/2.Note that Assumption1 is only suitable for the traditional state expansion observer. The high-order form of observer will be discussed in the following sections.


## 2.2. Design of the Sliding Mode Controller

The design process of the sliding mode controller can be described as the following two steps.(1)
Select the integral sliding surfaceS=SvSh as(9)S=Sv=ddt+λv3∫0tevτdτ,Sh=ddt+λh4∫0tehτdτ,where evt=vt−vdt and eht=ht−hdt are the tracking errors for the velocity channel and the altitude channel, respectively. λv and λh are the positive constants. The derivatives of Equation (9) can be obtained as
(10)Ṡ=ṠV=v⃛−v⃛d+3λve¨v+3λv2ėv+λv3ev,Ṡh=h4−hd4+4λhe⃛h+6λh2e¨h+4λh3ėh+λh4eh.Letψv=−v⃛d+3λve¨v+3λv2ėv+λv3ev, and ψh=−hd4+4λhe⃛h+6λh2e¨h+4λh3ėh+λh4eh. By introducing the model Equation (6) into Equation (10), we can get
(11)Ṡ=Ṡv=Fv+b11U1+b12U2+ψv,Ṡh=Fh+b21U1+b22U2+ψh.(2)
Define the sliding mode approaching rate as(12)Ṡ=−αv1Svβv1sgnSv−αv2Svβv2sgnSv,−αh1Shβh1sgnSh−αh2Shβh2sgnSh,where αv1>0, αv2>0,αh1>0,αh2>0,βv1>1,βh1>1,0<βv2<1, and0<βh2<1. When the system state is far away from the sliding mode, αv1Svβv1sgnSv and αh1Shβh1sgnSh play a dominant role, while αv2Svβv2sgnSv and αh2Shβh2sgnSh play a dominant role when the system state is close to the sliding mode. According to Equations (8), (11), and (12), the continuous-time controller can be obtained as follows:
(13)U=U1U2=B−1−Fv−ψv−Fh−ψh+B−1−αv1Svβv1sgnSv−αv2Svβv2sgnSv−d¯1t−αh1Shβh1sgnSh−αh2Shβh2sgnSh−d¯2t,where
(14)B=b11b12b21b21.The sliding mode controller can make the system state converge into the neighborhood of the equilibrium zero point quickly, if there are uncertain external disturbances in the system [29]. To ensure the convergence of system state, the observer can be used to compensate the bounded lumped disturbance caused by the system modeling error and bounded external disturbances.


## 2.3. Design of Traditional Extended State Disturbance Observer

Considering the following third-order system with external disturbances and uncertainties, the state space form is described as(15)ẋ1=x2,ẋ2=x3,ẋ3=ft,x,w+B∅y,u,y=x1, where ft,x,w is the sum of system uncertainties and external disturbances. Equation (15) can be rewritten as follows:
(16)ẋ1=x2,ẋ2=x3,ẋ3=x4+B∅y,uẋ4=d,y=x1,, where d is the derivative of the system lumped disturbance. Then, the traditional extended state observer can be designed as follows [25, 26]:
(17)x¯̇1=x¯2+a1ω0y−x¯1,x¯̇2=x¯3+a2ω02y−x¯1,x¯̇3=x¯4+∅y,u+a3ω03y−x¯1,x¯̇4=a4ω04y−x¯1, where x¯̇1, x¯̇2, x¯̇3, and x¯̇4 are observer states, a1, a2, a3, and a4 are positive real numbers, and ω0 is the observation bandwidth.According to the above analyses, the state space model of the velocity channel can be expressed as(18)ẋ=A1x+B1∅1y,u+w1d1y=C1xwhere
(19)A1=0100001000000100,B1=0010,C1=1000,w1=0001,∅1y,u=FV+b11U1+b12U2.The disturbance observer can be described as(20)x¯̇v=A1x¯v+B1∅1y,u+L1y−C1x¯v,where L1=av1ωvav2ωv2av3ωv3av4ωv4T, avi=4!/i!4−i!,i=1,2,3,4, avi is selected by the bandwidth-based configuration method. More generally, in traditional LESO for any order system, the adjustable parameter only has the observation bandwidth ωv.Similarly, the altitude channel can be described as(21)x¯̇h=A2x¯h+B2∅2y,u+L2y−C2x¯h,where
(22)A2=0100000100000000000100010,B2=00010,C2=10000T,∅2y,u=Fh+b21U1+b22U2,L2=ah1ωhah2ωh2ah3ωh3ah4ωh4ah5ωh5T,ahi=5!i!5−i!,i=1,2,3,4,5,and the adjustable parameter only has the observation bandwidthωh.Combined Equation (20) with Equation (21), the aircraft disturbance observer can be described as
(23)x¯̇=Ax¯+B∅y,u+Ly−Cx¯,where
(24)A=010000100000010000000000000000000000000000000000000000000100000100000000000100010,B=001000000000000010,C=100000000000010000T,∅y,u=FV+b11U1+b12U2Fh+b21U1+b22U2,L=av1ωvav2ωv2av3ωv3av4ωv4000000000ah1ωhah2ωh2ah3ωh3ah4ωh4ah5ωh5T.


## 3. Proposed High-Order Linear Extended State Observer

### 3.1. Design of High-Order Linear Extended State Observer

For the sake of generality, the followingm-th order system with external disturbances and uncertainties are considered. The state space form can be described as
(25)ẋ1=x2,ẋ2=x3,⋮ẋm=ft,x,w+B∅y,uẋm+1=ḟt,x,w,y=x1,, where its extended first-order state observer (traditional LESO) is given as
(26)x¯̇1=x¯2+a1ω0y−x¯1,x¯̇2=x¯3+a2ω02y−x¯1,⋮x¯̇m=x¯m+1+B∅y,u+amω0my−x¯1,x¯̇m+1=am+1ω0m+1y−x¯1.The following hypothesis proposed in [22] is given asAssumption 3.
The derivatives of the lumped disturbanceft,x,w of the system exist and are bounded as dnft,x,w/dtn≤θ.
Accordingly, the form of state equation and extendedn+1-th order state observer can be presented as
(27)ẋ1=x2,ẋ2=x3,⋮ẋm=ft,x,w+B∅y,u,ẋm+1=xm+2,⋮ẋm+n=xm+n+1,y=x1.(28)x¯̇1=x¯2+a1ω0y−x¯1,x¯̇2=x¯3+a2ω02y−x¯1,⋮x¯̇m=x¯m+1+B∅y,u+amω0my−x¯1,x¯̇m+1=x¯m+2+am+1ω0m+1y−x¯1,x¯̇m+2=x¯m+3+am+2ω0m+2y−x¯1,⋮x¯̇m+n=x¯m+n+1+am+nω0m+ny−x¯1,x¯̇m+n+1=am+n+1ω0m+n+1y−x¯1,where xm+1 is the lumped disturbance ft,x,w, xm+2, xm+3, ⋯, xm+n+1 are the first, second, ⋯, and n-th order derivatives of the lumped disturbance, respectively. x¯̇m+1 is the estimate of lumped disturbance ft,x,w, and x¯m+2, x¯m+2, ⋯, x¯m+n+1 are severally the estimates of the first, second, ⋯, and n-th order derivatives of the lumped disturbance. The parameter ai=m+n+1!/i!m+n+1−i!i=1,2,⋯,m+n+1 is selected by the bandwidth-based configuration method, and ω0 is the observation bandwidth, and n=0 indicates the traditional LESO.


### 3.2. Convergence Analysis

In this section, the convergence analysis is presented. First, the definition of the expansion ofn+1-th order state observer for the m-th order system is given below
(29)eit=xit−x¯it,ηit=eiεtεm+n+1−i,i=1,2,⋯,m+n+1,(30)η=η1t,η2t,η3t,⋯,ηmt,ηm+1t,⋯,ηm+nt,ηm+n+1tT.The differential form ofηit can be described as
(31)η̇1t=η2−g1η1t,η̇2t=η3−g2η1t,⋮η̇m+nt=ηm+n+1−gm+nη1t,η̇m+n+1t=−gm+n+1ω1t+ε∆t,η10=e10εm+n,η20=e20εm+n−1,⋮ηm+n0=em+n0ε,ηm+n+10=em+n+10,where 1/ε is the observation bandwidth.For clear illustration, the following hypothesis and theorem proposed in [22] are provided first.Assumption 4.
There are certain constants ofλii=1,2,3,4, k, and the positive definite continuous differentiable functions V and W: ℝm+n+1⟶ℝ, which make
(32)λ1∥ω∥2≤Vω≤λ2∥ω∥2,λ3∥ω∥2≤Wω≤λ4∥ω∥2,∑i=1m+n+1∂V∂ηiηi+1−giη1−∂V∂ηm+n+1gm+n+1η1t≤−Wη,∂V∂ηm+n+1≤kη,where ∙ is the Euclid norm.Theorem 5.
If Assumption3 and Assumption 4 are satisfied, then for the system Equation (28), the following conclusion is true.
(33)SUPxit−x¯it≤Οεm+n+2−i,∀t∈tu,∞0<ε<1.
According to Assumption3 and Assumption 4, it can be concluded that
(34)ddtVωt=∑i=1m+n+1∂V∂ηiηi+1−giη1−∂V∂ηm+n+1gm+n+1η1t+∂V∂ηm+n+1ε∆t≤−Wη+εθkη≤−λ3λ2Vη+λ1λ1εθkVη.
Considering the relationship betweenVη and Vη, there is
(35)ddtVηt≤−λ3λ2Vηt+λ12λ1εθk.
Then, according to Gronwall-Bellman inequality, we can obtain(36)ηt≤Vηtλ1≤λ1Vη0λ1e−λ3t−t02ελ2+εθk2λ1∫0te−λ32λ2t−t0−sds.
Finally, the inequality ofeit can be concluded from the relationship of ηt and et as
(37)eit=ηitεεm+n+1−i≤ηtεεm+n+1−i≤εm+n+1−iλ1Vη0λ1e−λ3t−t02ελ2+εθk2λ1∫t0εtεe−λ32λ2t−t0ε−sds≤Οεm+n+2−i,∀t∈tu,∞,where tu=maxt0−2λ2/λ3m+n+1εlnε,t0.
Accordingly, when the perturbation parameters are small enough, the estimated state of the system Equation (28) is sufficiently close to the state of the system Equation (27). The estimation error converges to Οεm+n+2−i and decreases with the increase of the expansion order.


### 3.3. Stability Analysis

Theorem 6.
For the nonlinear dynamic model of NSV control system show in Equation (1), the continuous time controller of Equation (13) is chosen as the disturbance compensation controller, considering the uncertainty and external disturbance. The system is asymptotically stable when the following conditions are satisfied.
(38)Sv≤minDvαv21/βv2,Dvαv11/βv1,Sh≤minDhαh21/βh2,Dhαh11/βh1.
The Lyapunov function is defined as(39)L=12STS,(40)L̇=STṠ=SvṠv+ShṠh=−αv1Svβv1sgnSv−αv2Svβv2sgnSv−αh1Shβh1sgnSh−αh2Shβh2sgnSh+Svb11d~1t+b12d~2t+Shb21d~1t+b22d~2t≤−αv1Svβv1+1−αv2Svβv2+1−αh1Shβh1+1−αh2Shβh2+1+SvDv+ShDh,where d~it=dit−d¯it, d~it≤k, i=1,2, dit is the lumped disturbance, d¯it is the estimate of lumped disturbance, αv1>0, αv2>0,αh1>0,αh2>0,
(41)b11d~1t+b12d~2t≤b11+b12k=Dv,b21d~1t+b22d~2t≤b21+b22k=Dh.
The inequality (40) can be described as the following two forms:
(42)L̇≤−Svαv2Svβv2−Dv−Shαh2Shβh2−Dh−αv1Svβv1+1−αh1Shβh1+1,(43)L̇≤−Svαv1Svβv1−Dv−Shαh1Shβh1−Dh−αv2Svβv2+1−αh2Shβh2+1.
From inequality (42), we can obtain
(44)L̇≤−αv1Svβv1+1−αh1Shβh1+1≤0,
when(45)Sv≥Dvαv21/βv2,Sh≥Dhαh21/βh2.
From inequality (43), we can obtain
(46)L̇≤−αv2Svβv2+1−αh2Shβh2+1≤0,
when(47)Sv≥Dvαv11/βv1,Sh≥Dhαh11/βh1.
In conclusion, the statesSv and Sh converge to the following regions in the finite time.
(48)Sv≤minDvαv21/βv2,Dvαv11/βv1,Sh≤minDhαh21/βh2,Dhαh11/βh1.


## 3.1. Design of High-Order Linear Extended State Observer

For the sake of generality, the followingm-th order system with external disturbances and uncertainties are considered. The state space form can be described as
(25)ẋ1=x2,ẋ2=x3,⋮ẋm=ft,x,w+B∅y,uẋm+1=ḟt,x,w,y=x1,, where its extended first-order state observer (traditional LESO) is given as
(26)x¯̇1=x¯2+a1ω0y−x¯1,x¯̇2=x¯3+a2ω02y−x¯1,⋮x¯̇m=x¯m+1+B∅y,u+amω0my−x¯1,x¯̇m+1=am+1ω0m+1y−x¯1.The following hypothesis proposed in [22] is given asAssumption 3.
The derivatives of the lumped disturbanceft,x,w of the system exist and are bounded as dnft,x,w/dtn≤θ.
Accordingly, the form of state equation and extendedn+1-th order state observer can be presented as
(27)ẋ1=x2,ẋ2=x3,⋮ẋm=ft,x,w+B∅y,u,ẋm+1=xm+2,⋮ẋm+n=xm+n+1,y=x1.(28)x¯̇1=x¯2+a1ω0y−x¯1,x¯̇2=x¯3+a2ω02y−x¯1,⋮x¯̇m=x¯m+1+B∅y,u+amω0my−x¯1,x¯̇m+1=x¯m+2+am+1ω0m+1y−x¯1,x¯̇m+2=x¯m+3+am+2ω0m+2y−x¯1,⋮x¯̇m+n=x¯m+n+1+am+nω0m+ny−x¯1,x¯̇m+n+1=am+n+1ω0m+n+1y−x¯1,where xm+1 is the lumped disturbance ft,x,w, xm+2, xm+3, ⋯, xm+n+1 are the first, second, ⋯, and n-th order derivatives of the lumped disturbance, respectively. x¯̇m+1 is the estimate of lumped disturbance ft,x,w, and x¯m+2, x¯m+2, ⋯, x¯m+n+1 are severally the estimates of the first, second, ⋯, and n-th order derivatives of the lumped disturbance. The parameter ai=m+n+1!/i!m+n+1−i!i=1,2,⋯,m+n+1 is selected by the bandwidth-based configuration method, and ω0 is the observation bandwidth, and n=0 indicates the traditional LESO.


## 3.2. Convergence Analysis

In this section, the convergence analysis is presented. First, the definition of the expansion ofn+1-th order state observer for the m-th order system is given below
(29)eit=xit−x¯it,ηit=eiεtεm+n+1−i,i=1,2,⋯,m+n+1,(30)η=η1t,η2t,η3t,⋯,ηmt,ηm+1t,⋯,ηm+nt,ηm+n+1tT.The differential form ofηit can be described as
(31)η̇1t=η2−g1η1t,η̇2t=η3−g2η1t,⋮η̇m+nt=ηm+n+1−gm+nη1t,η̇m+n+1t=−gm+n+1ω1t+ε∆t,η10=e10εm+n,η20=e20εm+n−1,⋮ηm+n0=em+n0ε,ηm+n+10=em+n+10,where 1/ε is the observation bandwidth.For clear illustration, the following hypothesis and theorem proposed in [22] are provided first.Assumption 4.
There are certain constants ofλii=1,2,3,4, k, and the positive definite continuous differentiable functions V and W: ℝm+n+1⟶ℝ, which make
(32)λ1∥ω∥2≤Vω≤λ2∥ω∥2,λ3∥ω∥2≤Wω≤λ4∥ω∥2,∑i=1m+n+1∂V∂ηiηi+1−giη1−∂V∂ηm+n+1gm+n+1η1t≤−Wη,∂V∂ηm+n+1≤kη,where ∙ is the Euclid norm.Theorem 5.
If Assumption3 and Assumption 4 are satisfied, then for the system Equation (28), the following conclusion is true.
(33)SUPxit−x¯it≤Οεm+n+2−i,∀t∈tu,∞0<ε<1.
According to Assumption3 and Assumption 4, it can be concluded that
(34)ddtVωt=∑i=1m+n+1∂V∂ηiηi+1−giη1−∂V∂ηm+n+1gm+n+1η1t+∂V∂ηm+n+1ε∆t≤−Wη+εθkη≤−λ3λ2Vη+λ1λ1εθkVη.
Considering the relationship betweenVη and Vη, there is
(35)ddtVηt≤−λ3λ2Vηt+λ12λ1εθk.
Then, according to Gronwall-Bellman inequality, we can obtain(36)ηt≤Vηtλ1≤λ1Vη0λ1e−λ3t−t02ελ2+εθk2λ1∫0te−λ32λ2t−t0−sds.
Finally, the inequality ofeit can be concluded from the relationship of ηt and et as
(37)eit=ηitεεm+n+1−i≤ηtεεm+n+1−i≤εm+n+1−iλ1Vη0λ1e−λ3t−t02ελ2+εθk2λ1∫t0εtεe−λ32λ2t−t0ε−sds≤Οεm+n+2−i,∀t∈tu,∞,where tu=maxt0−2λ2/λ3m+n+1εlnε,t0.
Accordingly, when the perturbation parameters are small enough, the estimated state of the system Equation (28) is sufficiently close to the state of the system Equation (27). The estimation error converges to Οεm+n+2−i and decreases with the increase of the expansion order.


## 3.3. Stability Analysis

Theorem 6.
For the nonlinear dynamic model of NSV control system show in Equation (1), the continuous time controller of Equation (13) is chosen as the disturbance compensation controller, considering the uncertainty and external disturbance. The system is asymptotically stable when the following conditions are satisfied.
(38)Sv≤minDvαv21/βv2,Dvαv11/βv1,Sh≤minDhαh21/βh2,Dhαh11/βh1.
The Lyapunov function is defined as(39)L=12STS,(40)L̇=STṠ=SvṠv+ShṠh=−αv1Svβv1sgnSv−αv2Svβv2sgnSv−αh1Shβh1sgnSh−αh2Shβh2sgnSh+Svb11d~1t+b12d~2t+Shb21d~1t+b22d~2t≤−αv1Svβv1+1−αv2Svβv2+1−αh1Shβh1+1−αh2Shβh2+1+SvDv+ShDh,where d~it=dit−d¯it, d~it≤k, i=1,2, dit is the lumped disturbance, d¯it is the estimate of lumped disturbance, αv1>0, αv2>0,αh1>0,αh2>0,
(41)b11d~1t+b12d~2t≤b11+b12k=Dv,b21d~1t+b22d~2t≤b21+b22k=Dh.
The inequality (40) can be described as the following two forms:
(42)L̇≤−Svαv2Svβv2−Dv−Shαh2Shβh2−Dh−αv1Svβv1+1−αh1Shβh1+1,(43)L̇≤−Svαv1Svβv1−Dv−Shαh1Shβh1−Dh−αv2Svβv2+1−αh2Shβh2+1.
From inequality (42), we can obtain
(44)L̇≤−αv1Svβv1+1−αh1Shβh1+1≤0,
when(45)Sv≥Dvαv21/βv2,Sh≥Dhαh21/βh2.
From inequality (43), we can obtain
(46)L̇≤−αv2Svβv2+1−αh2Shβh2+1≤0,
when(47)Sv≥Dvαv11/βv1,Sh≥Dhαh11/βh1.
In conclusion, the statesSv and Sh converge to the following regions in the finite time.
(48)Sv≤minDvαv21/βv2,Dvαv11/βv1,Sh≤minDhαh21/βh2,Dhαh11/βh1.


## 4. Results and Discussion

In this simulation parts, we first analyze the influence of the extended order and observation bandwidth on the observer performance; then, we further verify the performance of proposed observer in the NSV control system.


### 4.1. Parameter Analysis

According to Equation (27) and Equation (28), we can deduce the transfer function from lumped disturbances to disturbance estimate and disturbance estimate error as follows:
(49)f¯sfs=s+ω0m+n+1−sm+n+1−∑i=1maiω0ism+n+1−is+ω0m+n+1,(50)Q¯sfs=sm+n+1+∑i=1maiω0ism+n+1−is+ω0m+n+1,where Q¯s is the lumped disturbance estimate error, m is the system order, n+1 is the expansion order, ω0 is the observation bandwidth, and ai=m+n+1!/i!m+n+1−i!i=1,2,⋯,m+n+1.Furthermore, the performance analysis of the expansion order and the observation bandwidth is also provided in the time domain and the frequency domain over the third-order velocity channel and the fourth-order altitude channel, respectively. According to Equation (49), the high-order LESO has the following characteristics.As can be seen from Figure1, when the observation bandwidth is a constant, the higher the expansion order, the faster the response speed, but the overshoot will increase. When the expansion order is a constant, the bigger the observation bandwidth, the faster the response speed, and the overshoot remains unchanged. It can be seen from the overall effect that the higher the system order, the greater the overshoot. Therefore, it is particularly important to choose the appropriate observation bandwidth and expansion order.Figure 1
Step response of high-order LESO disturbance estimates: (a) system order is 3; (b) system order is 4.
(a)(b)Figure2 shows the low-frequency characteristics of high-order LESO lumped disturbance estimates, where the frequency range is 0.1,1. Moreover, the reference frequencies are selected as 0.2, 0.4, and 0.8, respectively. The amplitude frequency characteristics and phase frequency characteristics are shown in Tables 1 and 2, respectively. Compared with the traditional LESO, when the expansion order is n+1=2, the amplitude errors increase slightly, but the phase errors decrease greatly. When the expansion order is n+1=3, the amplitude errors and phase errors are smaller than n+1=1 and n+1=2. When the expansion order is a constant, the larger the observation bandwidth, the smaller of magnitude errors and phase errors. It has been shown in Tables 1 and 2 that the estimation error is also related to the system order. Both of the magnitude errors and the phase errors will increase as the system order becomes higher. Generally, in the low-frequency band, the estimation errors of high-order LESO are smaller than those of the traditional LESO.Figure 2
Low-frequency characteristics of high-order LESO disturbance estimates: (a) system order is 3; (b) system order is 4.
(a)(b)Table 1
Magnitude errors of high-order LESO in the low-frequency band.
System orderExpansion orderObservation bandwidthMagnitude errors at 0.2 rad/s (dB)Magnitude errors at 0.4 rad/s (dB)Magnitude errors at 0.8 rad/s (dB)315-0.0277-0.111-0.443250.1350.5061.63350.002320.03610.4853110-0.00695-0.0277-0.11132100.03460.1350.50633100.0001480.002320.0361415-0.0347-0.139-0.554250.2010.7362.184350.004610.0710.9024110-0.0087-0.0348-0.13942100.05160.2020.73543100.0002920.004620.0711Table 2
Phase errors of high-order LESO in the low-frequency band.
System orderExpansion orderObservation bandwidthPhase errors at 0.2 rad/s (deg)Phase errors at 0.4 rad/s (deg)Phase errors at 0.8 rad/s (deg)315-9.14-18.3-36.4325-0.143-1.07-6.813350.07150.5262.773110-4.58-9.14-18.33210-0.0182-0.143-1.0733100.009110.07150.526415-11.4-22.9-45.5425-0.248-1.81-10.074350.1250.8843.934110-5.72-11.5-22.94210-0.0317-0.249-1.8143100.01590.1240.886According to Equation (50), it can be seen from Figures 3 and 4 that, at the same observer bandwidth, increasing the order of the extended state can significantly enhance the disturbance suppression in the low-frequency band, which is conducive to better tracking the system state and disturbance. However, in the intermediate-frequency band, the high-order LESO has larger peak amplitudes, which would lead to bigger overshoots or oscillations of the step response, which is not helpful to the system stability. Furthermore, the observation bandwidth only affects the rapidity of high-order LESO, and the overshoot is not affected by the bandwidth, but only related to the expansion order.Figure 3
Intermediate-frequency and high-frequency characteristics of high-order LESO disturbance estimates: (a) system order is 3; (b) system order is 4.
(a)(b)Figure 4
High-order LESO disturbance suppression frequency response: (a) system order is 3; (b) system order is 4.
(a)(b)In general, both increasing the observation bandwidth and the expansion order can improve the low-frequency performance of observer. However, the higher the expansion order, the bigger the peak amplitude of intermediate frequency, which is prone to lead a larger overshoot and even oscillation; the lager the observation bandwidth, the worse the intermediate-frequency and high-frequency characteristics. Furthermore, the higher the order of the system, the greater the estimation errors. Therefore, for high-order system, we use high-order LESO by choosing appropriate observation bandwidth and expansion order, which has better low-frequency characteristics than traditional LESO.


### 4.2. Numerical Simulations

In order to verify the effectiveness of the proposed method, the mathematical model of the NSV is built on the MATLAB/Simulink platform, and the designed controller is also simulated and verified with MATLAB.In this paper, based on the flight conditions of NSV cruise state, the uncertain parameters are adopted as additional variables. The main parameters are set as follows:m0=50200kg, v0=4590m/s, h0=33528m, s0=369m2, c0=28m, and Iy0=8466900kg∙m2. The parametric uncertainty is defined as
(51)m=m01+∆m,∆m≤0.03,Iy=Iy01+∆Iy,∆Iy≤0.02,s=s01+∆s,∆s≤0.01,c=c01+∆c,∆c≤0.03,ce=ce01+∆ce,∆ce≤0.02,ρ=ρ01+∆ρ,∆ρ≤0.03.The expansion orders ofn+1=1, n+1=2, and n+1=3 in addition with observation bandwidth of ω0=5, ω0=10, and ω0=20 are adopted in the simulations, respectively. Thereinto, n+1=1 indicates the traditional LESO, while n+1=2 and n+1=3 are the recommended LESO in this paper. Under the premise of system stability, the parameters of controller and observer are basically unchanged. Moreover, due to the coupling between the velocity and altitude channels, the observer output needs to be decoupled. The step signal for velocity channel and altitude channel are given as 50m/s and 50m, respectively. The external disturbance d1t=0.02sin0.4t and d2t=0.04sin0.2t are introduced in the 30th second.The simulation results are shown in Figures5–8, where the curves with parameter of n+1=1 indicate the algorithm in [27], the others with parameter of n+1=2 and n+1=3 represent our proposed algorithm, and the compared nonlinear extended state observer (NESO) is proposed in [30]. Figure 5 shows the time response of compared NSV observers over the velocity channel and altitude channel. It can be seen in Figure 5 that both the controllers with LESO and NESO with appropriate parameters can make the system stable. As to the overall effect, the steady-state accuracy of the proposed high-order LESO is obviously better than traditional LESO. When the expansion order parameter is selected as n+1=2, the steady-state accuracy of proposed observer is similar to that of NESO. Furthermore, when the expansion order parameter is selected as n+1=3, the steady-state accuracy of proposed observer is better than NESO. Figure 6 shows the time responses of the observer errors over the velocity channel and altitude channel, where the disturbance estimation errors of the proposed high-order LESO are obviously smaller than traditional LESO and NESO. Figure 7 shows the time responses of the actuator on throttle setting and elevator deflection. It can be concluded from Figure 7 that with the increase of the order and observer bandwidth, the control signal will generate peak amplitude and high-frequency oscillations, which are deleterious to the actuator. However, the curve of NESO is smoother than LESO. The higher the system order and the lager the observer bandwidth, the more serious the situation is. Figure 8 shows the time responses of the attack angle and the track angle. The smaller the observer error is, the smaller the change of attitude angle is, and the better the aircraft stability is. The proposed high-order LESO can improve the system antidisturbance ability by selecting an appropriate order and observer bandwidth. Therefore, for the NSV nonlinear model with uncertain parameters and external disturbance, the simulation results show that the proposed high-order LESO with extended order of 3 and observer bandwidth of 10 is recommendable as the disturbance compensator, which is better than the traditional LESO and the NESO.Figure 5
The response curves of NSV: (a) velocity channel; (b) altitude channel.
(a)(b)Figure 6
Disturbance estimation errors: (a) velocity channel; (b) altitude channel.
(a)(b)Figure 7
Actuator response curve: (a) throttle setting; (b) elevator deflection.
(a)(b)Figure 8
Parametric response curve: (a) attack angle; (b) track angle.
(a)(b)


## 4.1. Parameter Analysis

According to Equation (27) and Equation (28), we can deduce the transfer function from lumped disturbances to disturbance estimate and disturbance estimate error as follows:
(49)f¯sfs=s+ω0m+n+1−sm+n+1−∑i=1maiω0ism+n+1−is+ω0m+n+1,(50)Q¯sfs=sm+n+1+∑i=1maiω0ism+n+1−is+ω0m+n+1,where Q¯s is the lumped disturbance estimate error, m is the system order, n+1 is the expansion order, ω0 is the observation bandwidth, and ai=m+n+1!/i!m+n+1−i!i=1,2,⋯,m+n+1.Furthermore, the performance analysis of the expansion order and the observation bandwidth is also provided in the time domain and the frequency domain over the third-order velocity channel and the fourth-order altitude channel, respectively. According to Equation (49), the high-order LESO has the following characteristics.As can be seen from Figure1, when the observation bandwidth is a constant, the higher the expansion order, the faster the response speed, but the overshoot will increase. When the expansion order is a constant, the bigger the observation bandwidth, the faster the response speed, and the overshoot remains unchanged. It can be seen from the overall effect that the higher the system order, the greater the overshoot. Therefore, it is particularly important to choose the appropriate observation bandwidth and expansion order.Figure 1
Step response of high-order LESO disturbance estimates: (a) system order is 3; (b) system order is 4.
(a)(b)Figure2 shows the low-frequency characteristics of high-order LESO lumped disturbance estimates, where the frequency range is 0.1,1. Moreover, the reference frequencies are selected as 0.2, 0.4, and 0.8, respectively. The amplitude frequency characteristics and phase frequency characteristics are shown in Tables 1 and 2, respectively. Compared with the traditional LESO, when the expansion order is n+1=2, the amplitude errors increase slightly, but the phase errors decrease greatly. When the expansion order is n+1=3, the amplitude errors and phase errors are smaller than n+1=1 and n+1=2. When the expansion order is a constant, the larger the observation bandwidth, the smaller of magnitude errors and phase errors. It has been shown in Tables 1 and 2 that the estimation error is also related to the system order. Both of the magnitude errors and the phase errors will increase as the system order becomes higher. Generally, in the low-frequency band, the estimation errors of high-order LESO are smaller than those of the traditional LESO.Figure 2
Low-frequency characteristics of high-order LESO disturbance estimates: (a) system order is 3; (b) system order is 4.
(a)(b)Table 1
Magnitude errors of high-order LESO in the low-frequency band.
System orderExpansion orderObservation bandwidthMagnitude errors at 0.2 rad/s (dB)Magnitude errors at 0.4 rad/s (dB)Magnitude errors at 0.8 rad/s (dB)315-0.0277-0.111-0.443250.1350.5061.63350.002320.03610.4853110-0.00695-0.0277-0.11132100.03460.1350.50633100.0001480.002320.0361415-0.0347-0.139-0.554250.2010.7362.184350.004610.0710.9024110-0.0087-0.0348-0.13942100.05160.2020.73543100.0002920.004620.0711Table 2
Phase errors of high-order LESO in the low-frequency band.
System orderExpansion orderObservation bandwidthPhase errors at 0.2 rad/s (deg)Phase errors at 0.4 rad/s (deg)Phase errors at 0.8 rad/s (deg)315-9.14-18.3-36.4325-0.143-1.07-6.813350.07150.5262.773110-4.58-9.14-18.33210-0.0182-0.143-1.0733100.009110.07150.526415-11.4-22.9-45.5425-0.248-1.81-10.074350.1250.8843.934110-5.72-11.5-22.94210-0.0317-0.249-1.8143100.01590.1240.886According to Equation (50), it can be seen from Figures 3 and 4 that, at the same observer bandwidth, increasing the order of the extended state can significantly enhance the disturbance suppression in the low-frequency band, which is conducive to better tracking the system state and disturbance. However, in the intermediate-frequency band, the high-order LESO has larger peak amplitudes, which would lead to bigger overshoots or oscillations of the step response, which is not helpful to the system stability. Furthermore, the observation bandwidth only affects the rapidity of high-order LESO, and the overshoot is not affected by the bandwidth, but only related to the expansion order.Figure 3
Intermediate-frequency and high-frequency characteristics of high-order LESO disturbance estimates: (a) system order is 3; (b) system order is 4.
(a)(b)Figure 4
High-order LESO disturbance suppression frequency response: (a) system order is 3; (b) system order is 4.
(a)(b)In general, both increasing the observation bandwidth and the expansion order can improve the low-frequency performance of observer. However, the higher the expansion order, the bigger the peak amplitude of intermediate frequency, which is prone to lead a larger overshoot and even oscillation; the lager the observation bandwidth, the worse the intermediate-frequency and high-frequency characteristics. Furthermore, the higher the order of the system, the greater the estimation errors. Therefore, for high-order system, we use high-order LESO by choosing appropriate observation bandwidth and expansion order, which has better low-frequency characteristics than traditional LESO.


## 4.2. Numerical Simulations

In order to verify the effectiveness of the proposed method, the mathematical model of the NSV is built on the MATLAB/Simulink platform, and the designed controller is also simulated and verified with MATLAB.In this paper, based on the flight conditions of NSV cruise state, the uncertain parameters are adopted as additional variables. The main parameters are set as follows:m0=50200kg, v0=4590m/s, h0=33528m, s0=369m2, c0=28m, and Iy0=8466900kg∙m2. The parametric uncertainty is defined as
(51)m=m01+∆m,∆m≤0.03,Iy=Iy01+∆Iy,∆Iy≤0.02,s=s01+∆s,∆s≤0.01,c=c01+∆c,∆c≤0.03,ce=ce01+∆ce,∆ce≤0.02,ρ=ρ01+∆ρ,∆ρ≤0.03.The expansion orders ofn+1=1, n+1=2, and n+1=3 in addition with observation bandwidth of ω0=5, ω0=10, and ω0=20 are adopted in the simulations, respectively. Thereinto, n+1=1 indicates the traditional LESO, while n+1=2 and n+1=3 are the recommended LESO in this paper. Under the premise of system stability, the parameters of controller and observer are basically unchanged. Moreover, due to the coupling between the velocity and altitude channels, the observer output needs to be decoupled. The step signal for velocity channel and altitude channel are given as 50m/s and 50m, respectively. The external disturbance d1t=0.02sin0.4t and d2t=0.04sin0.2t are introduced in the 30th second.The simulation results are shown in Figures5–8, where the curves with parameter of n+1=1 indicate the algorithm in [27], the others with parameter of n+1=2 and n+1=3 represent our proposed algorithm, and the compared nonlinear extended state observer (NESO) is proposed in [30]. Figure 5 shows the time response of compared NSV observers over the velocity channel and altitude channel. It can be seen in Figure 5 that both the controllers with LESO and NESO with appropriate parameters can make the system stable. As to the overall effect, the steady-state accuracy of the proposed high-order LESO is obviously better than traditional LESO. When the expansion order parameter is selected as n+1=2, the steady-state accuracy of proposed observer is similar to that of NESO. Furthermore, when the expansion order parameter is selected as n+1=3, the steady-state accuracy of proposed observer is better than NESO. Figure 6 shows the time responses of the observer errors over the velocity channel and altitude channel, where the disturbance estimation errors of the proposed high-order LESO are obviously smaller than traditional LESO and NESO. Figure 7 shows the time responses of the actuator on throttle setting and elevator deflection. It can be concluded from Figure 7 that with the increase of the order and observer bandwidth, the control signal will generate peak amplitude and high-frequency oscillations, which are deleterious to the actuator. However, the curve of NESO is smoother than LESO. The higher the system order and the lager the observer bandwidth, the more serious the situation is. Figure 8 shows the time responses of the attack angle and the track angle. The smaller the observer error is, the smaller the change of attitude angle is, and the better the aircraft stability is. The proposed high-order LESO can improve the system antidisturbance ability by selecting an appropriate order and observer bandwidth. Therefore, for the NSV nonlinear model with uncertain parameters and external disturbance, the simulation results show that the proposed high-order LESO with extended order of 3 and observer bandwidth of 10 is recommendable as the disturbance compensator, which is better than the traditional LESO and the NESO.Figure 5
The response curves of NSV: (a) velocity channel; (b) altitude channel.
(a)(b)Figure 6
Disturbance estimation errors: (a) velocity channel; (b) altitude channel.
(a)(b)Figure 7
Actuator response curve: (a) throttle setting; (b) elevator deflection.
(a)(b)Figure 8
Parametric response curve: (a) attack angle; (b) track angle.
(a)(b)


## 5. Conclusions

In this paper, a sliding mode controller with high-order LESO compensation is designed according to the NSV characteristics, which provides a new way to solve uncertain systems with external disturbances. The proposed high-order LESO is suitable for any order system. The transfer function from the lumped disturbance to the disturbance estimates and disturbance estimates error of any order system with any extended order is derived. It can be found that a carefully selected extended order and observer bandwidth can achieve remarkable observer performance gains, more stable system response, and higher steady-state accuracy. However, there are still some other problems need to be solved in the future work, such as the peak phenomenon of intermediate frequency, the high-frequency antidisturbance ability, the different order coupling, and the design of finite-time sliding mode controller. The future work will be focused on how to design the switching control law between the LESO and the NESO, in order to solve the stability problem of multimode switching for near space variable wing vehicle.



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*Source: 6657281-2021-02-10.xml*