# Robust Adaptive Neural Control of Morphing Aircraft with Prescribed Performance

Authors: Zhonghua Wu; Jingchao Lu; Jingping Shi; Yang Liu; Qing Zhou
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This study proposes a low-computational composite adaptive neural control scheme for the longitudinal dynamics of a swept-back wing aircraft subject to parameter uncertainties. To efficiently release the constraint often existing in conventional neural designs, whose closed-loop stability analysis always necessitates that neural networks (NNs) be confined in the active regions, a smooth switching function is presented to conquer this issue. By integrating minimal learning parameter (MLP) technique, prescribed performance control, and a kind of smooth switching strategy into back-stepping design, a new composite switching adaptive neural prescribed performance control scheme is proposed and a new type of adaptive laws is constructed for the altitude subsystem. Compared with previous neural control scheme for flight vehicle, the remarkable feature is that the proposed controller not only achieves the prescribed performance including transient and steady property but also addresses the constraint on NN. Two comparative simulations are presented to verify the effectiveness of the proposed controller.


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## Body

## 1. Introduction

Morphing aircraft has received considerable interest, since it possesses distinct advantages, which is capable of altering autonomously its aerodynamic configuration to obtain optimal flight performance, adapting different flight environments and high efficiency executing multiple types of missions [1]. As regards the control issue of morphing vehicle, the key point is to design a flight control system that has capability to guarantee the stability of the aircraft [2, 3]. The difficulty associated with the control design of such system arises from the fact that morphing aircraft manifests time-varying characteristic of aerodynamic forces, moments, and mass distribution as well as strong nonlinear nature [4, 5]. In the literature, several effective methods have been presented to tackle the control problem of folding-wing and swept-back wing aircraft. A multiloop control structure comprised of linear inner-loop and outer-loop controller is proposed for a kind of folding-wing aircraft [4]; similarly, the idea can also be found in [1] depending on gain self-scheduled H∞ technique. Subsequently, a finite-time boundedness control approach [2] and a switching linear parameter varying method [6] are investigated for swept-back wing aircraft, respectively. The common feature in [1, 2, 4, 6] is that the controllers are capable of ensuring the aircraft flight steady subject to the wing shape changes, but those designs are highly dependent on the precise prior knowledge of the dynamic model. However, the aerodynamic forces and moments are also quite difficult to model accurately. Moreover, general aircraft dynamics possess strong nonlinearities and uncertainties, which have necessitated the use of nonlinear control methods. Therefore, designing a nonlinear control method independent of prior knowledge of the aerodynamic model for the morphing aircraft is still an interesting yet challenging problem.Adaptive back-stepping method has been widely studied in tracking control designs for nonlinear systems in strict-feedback or pure-feedback form, because it owns capability of systematically manipulating mismatched uncertainties [7–10]. Lately, some significant works regarding adaptive neural/fuzzy control for nonlinear systems with totally unknown or nonlinearly parameterized nonlinearities are investigated [11–13], and the noticeable problem of “explosion of items” is elegantly avoided by virtue of dynamic surface control or nonlinear differentiator technique [7, 14–16]. More specifically, adaptive neural/fuzzy approaches are also extensively utilized to the control problem of flight vehicles accompanied with aerodynamic parameters uncertainty in [17–20]. Although the previous adaptive neural control methods have the ability to guarantee the steady-state performance converging to a small residual set, few results consider the transient performance related to overshoot and convergence rate [21]. More recently, the prescribed performance control (PPC) scheme which denotes that the tracking error should converge to a predefined bound with convergence rate no less than a certain value has been an active research area [21–30]. Based on an error transformation technique that incorporates the desirable performance, Bechlioulis and Rovithakis firstly presented an adaptive PPC method for strict-feedback nonlinear systems [22]. Afterwards, an observer based fuzzy adaptive prescribed performance tracking approach is investigated for nonlinear stochastic systems subject to input saturation [26]. Subsequently, the PPC approach has been extended to deal with MIMO state and output feedback nonlinear control problems [27–30]. Moreover, this tool has also been successfully applied to flight control area [24, 31].Despite the recent progress in the neural networks control of unknown nonlinear systems, certain issues still remain open. In practice, even though the stability analysis of aforementioned adaptive neural control schemes is proven, it relies on the condition that the approximation ability of NN must be effective all the time (i.e., the NN should be permanently working in the neural active region). Therefore, the deterioration of the tracking performance or even instability may happen provided that the transient states overstep the neural active region. Additionally, such a condition is also difficult to verify beforehand in real applications [32]. This difficulty can be naturally eliminated in case the invariance of the neural active region is guaranteed. In [33], by introducing n error transformation functions into the back-stepping design, a priori guaranteed evolution within the NN approximation set methodology is proposed to conquer this issue. However, the method developed in [33] constitutes an overparameterized solution which is hard to implement in practical application. Another alternative approach is designing a smooth switching function, which has the capability of switching the normal neural controller to a robust controller for pulling back the escaped transient into the neural effective regions. Several smooth switching function based neural adaptive back-stepping control schemes are proposed to solve aforementioned problem, where each virtual controller contains a neural controller and a robust controller effectively working inside or outside the active region, respectively [32, 34–37]. However, numerous adaptive parameters still need to be updated online and the transient performance problem is also omitted in those methods.Motivated by the aforementioned discussion, a switching strategy based composite adaptive neural control scheme is proposed for a swept-wing morphing aircraft. NNs are employed to approximate unknown functions; thus a priori knowledge of the aerodynamic parameters is not necessary. It is proven that all the signals in the closed-loop systems are bounded. The main contributions of this work are shown as follows:(1)
Different from the works [32, 34–37] which completely neglect the transient performance related to overshoot and convergence rate, by introducing an error transformation, the proposed controller is capable of allowing attributes such as a lower bound on the convergence rate and steady error to be specified. The MLP and FOSD are incorporated into the neural back-stepping design to reduce the online updating parameters of NN and to conquer the problem of “explosion of the items,” thus deriving a low-computational scheme.(2)
In contrast to traditional neural control schemes [21–24, 26–30] whose stability analysis relies on the condition that the NNs should always be kept within the neural active region, a smooth switching function based composite control scheme, presented to manipulate the exchange of control authorities between normal neural controller working in the active region and a robust controller out of this scope, is constructed to relax this constraint, and a new kind of adaptive laws is proposed. Note that it is also the first low-computational neural control scheme which can efficiently address the prescribed performance issue and relax the constraint on NN simultaneously.


## 2. Model Dynamics and Problem Formulation

### 2.1. Morphing Aircraft Model

The longitudinal dynamics of a morphing aircraft considered in this study are derived from [5, 38]. This model includes state variables (V,h,α,γ,q) and control inputs (δe,T), where V denotes the velocity, h is the altitude, α denotes angle of attack, γ represents the flight path angle (FPA), and q is the pitch rate; δe and T represent elevator deflection and thrust, respectively.(1)V˙=-D+Tcos⁡α-mgsin⁡γ+FIxm,(2)h˙=Vsin⁡γ,(3)γ˙=L+Tsin⁡α-mgcos⁡γ-FIKzmV,(4)α˙=-L-Tsin⁡α+mgcos⁡γ+FIzmV+q,(5)q˙=-I˙yq-Sxgcos⁡θ+MA+TZT+MIyIy,(6)FIx=Sxq˙sin⁡α+q2cos⁡α+2S˙xqsin⁡α-S¨xcos⁡α,FIz=FIkz=Sxq˙cos⁡α-q2sin⁡α+2S˙xqcos⁡α+S¨xsin⁡α,MIy=SxV˙sin⁡α+Vα˙cos⁡α-Vqcos⁡α,where D, L, and MA denote drag force, lift force, and pitch moment, respectively. m, Iy, and g denote the mass of aircraft, moment of inertia about pitch axis, and gravity constant. FIx, FIz, FIkz, and MIy are the inertial force and moment caused by morphing process. ZT is the position of engine in the body axis. Sx is the static moment caused by wing sweep. The related definitions are given as follows:(7)CDζ=CD0ζ+CDαζα+CDα2ζα2,Sxζ≈2m1r1x+m3r3x,L=CLζQSwζ,D=CDζQSwζ,Cmζ=Cm0ζ+Cmαζα+Cmδeζδe+CmqqcAζ2V,Q=12ρhV2;CLζ≈CL0ζ+CLαζα,MA=CmζQSwζcAζ,where ζ is the sweep angle; the detailed explanation of the other parameters can be found in [5].


### 2.2. Model Transformation and Control Objective

ThrustT mainly affects velocity V, and elevator deflection δe has a dominant contribution to altitude h change; thus the dynamics model is reasonably decomposed into two subsystems including altitude and velocity subsystems.


#### 2.2.1. Altitude Subsystem

Definex1=h, x2=γ, x3=θ, and x4=q, where θ=α+γ and x-=(x2,x3,x4). Therefore, the altitude subsystem can be converted into the following formulation:(8)x˙1=Vx2,x˙2=f2x2,x3+x3,x˙3=x4,x˙4=f4x-,δe+u,y=x1,u=-δe,where y is the output signal of altitude subsystem (8); f2(x2,x3)=[L+Tsin⁡(θ-γ)-mgcos⁡γ]/(mV)-θ and f4(x-,u)=-I˙yq+-Sxgcos⁡θ+MA+TZT+MIy/Iy+δe are unknown functions.


#### 2.2.2. Velocity Subsystem

Velocity subsystem is transformed into the following formulation:(9)V˙=fVxV,T+T,where fV(xV,T)=-D+Tcos⁡α-mgsin⁡γ+FIx/m-T is an unknown function.Remark 1.
In order to transform the altitude system into pure-feedback system,FIkz in (3) is regarded as an unmodeled term. Since we only consider the cruise phase in this paper, γ is quite small and we can take sin⁡γ≈γ in (2) to simplify the system.
Control Objective. The control objective in this study is to design adaptive controllers δe and T such that(1)
the altitude and velocity can track the desired trajectoryhd and Vd, while guaranteeing that all the signals in the closed loop are bounded;(2)
the corresponding altitude and velocity tracking errors achieve prescribed transient and steady-state performance.


### 2.3. Some Preliminaries

#### 2.3.1. Prescribed Performance

To achieve the control objective, the tracking errorzi(t),i=1,V should satisfy the following prescribed performance bounds [22, 39]:(10)-λit<zit<λit,where λi(t)>0 named performance function is defined as (11)λit=λi0-λi∞exp⁡-lit+λi∞,where λi0, λi∞, and li are positive constants; λi0=λi(0) and λi∞=limt→∞⁡λi(t); li denotes the minimum speed of convergence and λi∞ is the maximum steady-state error.To transform the constrained tracking error condition (10) into an equivalent unconstrained one, the following state transformation is employed. So we have(12)μit=Rizitλit,where μi(t) is the transformed error and Ri(·) is an increasing transformation function shown as follows:(13)Riztλt=ln⁡1+zit/λit1-zit/λit.The derivative of (12) is shown as(14)μ˙it=riz˙it-λ˙itλitzit,where ri=∂Ri/∂zi(t)/λi(t)1/λi(t).Using (13), the following inequalities can be obtained:(15)ri>rimin>0,where rmin=2/(λi(0)).


#### 2.3.2. Useful Function and Key Lemmas

Definition 2 (see [35]).
The boundaries of the compact subsetsΩi are defined by several prescribed constants 0<ri1<ri2,i=2,4; meanwhile some useful switching functions are described as (16)mix-i≜∏k=1iBkxk,Bkxk≜1,ifxk<rk1,rk22-xk2rk22-rk12e-rk22-xk2/rk22-rk12ω-k2b1,ifrk1≤xk≤rk2,0,ifxk>rk2, where ω-k>0 and b1>0 are positive constants.Lemma 3 (see [40]).
The following inequality holds for anyω0>0 and η∈R:(17)0≤η-ηtanh⁡ηω0≤κ0ω0,where κ0 is a constant satisfying κ0=e-(κ0+1); that is, κ0=0.2785.Lemma 4 (see [41]).
The “first-order sliding mode differentiator (FOSD)” is designed as(18)ς˙0=-μ-0ς0-lt0.5sign⁡ς0-lt+ς1,ς˙1=-μ-1sign⁡ς1-ς0,where ς0 and ς1 are the states of system (18), μ-0 and μ-1 are the designed parameters of FOSD, and l(t) is an input function. ς˙0 can estimate l˙(t) to an arbitrary precision in case the initial values ς0-l(t0) and ς˙0-l˙(t0) are bounded.


## 2.1. Morphing Aircraft Model

The longitudinal dynamics of a morphing aircraft considered in this study are derived from [5, 38]. This model includes state variables (V,h,α,γ,q) and control inputs (δe,T), where V denotes the velocity, h is the altitude, α denotes angle of attack, γ represents the flight path angle (FPA), and q is the pitch rate; δe and T represent elevator deflection and thrust, respectively.(1)V˙=-D+Tcos⁡α-mgsin⁡γ+FIxm,(2)h˙=Vsin⁡γ,(3)γ˙=L+Tsin⁡α-mgcos⁡γ-FIKzmV,(4)α˙=-L-Tsin⁡α+mgcos⁡γ+FIzmV+q,(5)q˙=-I˙yq-Sxgcos⁡θ+MA+TZT+MIyIy,(6)FIx=Sxq˙sin⁡α+q2cos⁡α+2S˙xqsin⁡α-S¨xcos⁡α,FIz=FIkz=Sxq˙cos⁡α-q2sin⁡α+2S˙xqcos⁡α+S¨xsin⁡α,MIy=SxV˙sin⁡α+Vα˙cos⁡α-Vqcos⁡α,where D, L, and MA denote drag force, lift force, and pitch moment, respectively. m, Iy, and g denote the mass of aircraft, moment of inertia about pitch axis, and gravity constant. FIx, FIz, FIkz, and MIy are the inertial force and moment caused by morphing process. ZT is the position of engine in the body axis. Sx is the static moment caused by wing sweep. The related definitions are given as follows:(7)CDζ=CD0ζ+CDαζα+CDα2ζα2,Sxζ≈2m1r1x+m3r3x,L=CLζQSwζ,D=CDζQSwζ,Cmζ=Cm0ζ+Cmαζα+Cmδeζδe+CmqqcAζ2V,Q=12ρhV2;CLζ≈CL0ζ+CLαζα,MA=CmζQSwζcAζ,where ζ is the sweep angle; the detailed explanation of the other parameters can be found in [5].


## 2.2. Model Transformation and Control Objective

ThrustT mainly affects velocity V, and elevator deflection δe has a dominant contribution to altitude h change; thus the dynamics model is reasonably decomposed into two subsystems including altitude and velocity subsystems.


### 2.2.1. Altitude Subsystem

Definex1=h, x2=γ, x3=θ, and x4=q, where θ=α+γ and x-=(x2,x3,x4). Therefore, the altitude subsystem can be converted into the following formulation:(8)x˙1=Vx2,x˙2=f2x2,x3+x3,x˙3=x4,x˙4=f4x-,δe+u,y=x1,u=-δe,where y is the output signal of altitude subsystem (8); f2(x2,x3)=[L+Tsin⁡(θ-γ)-mgcos⁡γ]/(mV)-θ and f4(x-,u)=-I˙yq+-Sxgcos⁡θ+MA+TZT+MIy/Iy+δe are unknown functions.


### 2.2.2. Velocity Subsystem

Velocity subsystem is transformed into the following formulation:(9)V˙=fVxV,T+T,where fV(xV,T)=-D+Tcos⁡α-mgsin⁡γ+FIx/m-T is an unknown function.Remark 1.
In order to transform the altitude system into pure-feedback system,FIkz in (3) is regarded as an unmodeled term. Since we only consider the cruise phase in this paper, γ is quite small and we can take sin⁡γ≈γ in (2) to simplify the system.
Control Objective. The control objective in this study is to design adaptive controllers δe and T such that(1)
the altitude and velocity can track the desired trajectoryhd and Vd, while guaranteeing that all the signals in the closed loop are bounded;(2)
the corresponding altitude and velocity tracking errors achieve prescribed transient and steady-state performance.


## 2.2.1. Altitude Subsystem

Definex1=h, x2=γ, x3=θ, and x4=q, where θ=α+γ and x-=(x2,x3,x4). Therefore, the altitude subsystem can be converted into the following formulation:(8)x˙1=Vx2,x˙2=f2x2,x3+x3,x˙3=x4,x˙4=f4x-,δe+u,y=x1,u=-δe,where y is the output signal of altitude subsystem (8); f2(x2,x3)=[L+Tsin⁡(θ-γ)-mgcos⁡γ]/(mV)-θ and f4(x-,u)=-I˙yq+-Sxgcos⁡θ+MA+TZT+MIy/Iy+δe are unknown functions.


## 2.2.2. Velocity Subsystem

Velocity subsystem is transformed into the following formulation:(9)V˙=fVxV,T+T,where fV(xV,T)=-D+Tcos⁡α-mgsin⁡γ+FIx/m-T is an unknown function.Remark 1.
In order to transform the altitude system into pure-feedback system,FIkz in (3) is regarded as an unmodeled term. Since we only consider the cruise phase in this paper, γ is quite small and we can take sin⁡γ≈γ in (2) to simplify the system.
Control Objective. The control objective in this study is to design adaptive controllers δe and T such that(1)
the altitude and velocity can track the desired trajectoryhd and Vd, while guaranteeing that all the signals in the closed loop are bounded;(2)
the corresponding altitude and velocity tracking errors achieve prescribed transient and steady-state performance.


## 2.3. Some Preliminaries

### 2.3.1. Prescribed Performance

To achieve the control objective, the tracking errorzi(t),i=1,V should satisfy the following prescribed performance bounds [22, 39]:(10)-λit<zit<λit,where λi(t)>0 named performance function is defined as (11)λit=λi0-λi∞exp⁡-lit+λi∞,where λi0, λi∞, and li are positive constants; λi0=λi(0) and λi∞=limt→∞⁡λi(t); li denotes the minimum speed of convergence and λi∞ is the maximum steady-state error.To transform the constrained tracking error condition (10) into an equivalent unconstrained one, the following state transformation is employed. So we have(12)μit=Rizitλit,where μi(t) is the transformed error and Ri(·) is an increasing transformation function shown as follows:(13)Riztλt=ln⁡1+zit/λit1-zit/λit.The derivative of (12) is shown as(14)μ˙it=riz˙it-λ˙itλitzit,where ri=∂Ri/∂zi(t)/λi(t)1/λi(t).Using (13), the following inequalities can be obtained:(15)ri>rimin>0,where rmin=2/(λi(0)).


### 2.3.2. Useful Function and Key Lemmas

Definition 2 (see [35]).
The boundaries of the compact subsetsΩi are defined by several prescribed constants 0<ri1<ri2,i=2,4; meanwhile some useful switching functions are described as (16)mix-i≜∏k=1iBkxk,Bkxk≜1,ifxk<rk1,rk22-xk2rk22-rk12e-rk22-xk2/rk22-rk12ω-k2b1,ifrk1≤xk≤rk2,0,ifxk>rk2, where ω-k>0 and b1>0 are positive constants.Lemma 3 (see [40]).
The following inequality holds for anyω0>0 and η∈R:(17)0≤η-ηtanh⁡ηω0≤κ0ω0,where κ0 is a constant satisfying κ0=e-(κ0+1); that is, κ0=0.2785.Lemma 4 (see [41]).
The “first-order sliding mode differentiator (FOSD)” is designed as(18)ς˙0=-μ-0ς0-lt0.5sign⁡ς0-lt+ς1,ς˙1=-μ-1sign⁡ς1-ς0,where ς0 and ς1 are the states of system (18), μ-0 and μ-1 are the designed parameters of FOSD, and l(t) is an input function. ς˙0 can estimate l˙(t) to an arbitrary precision in case the initial values ς0-l(t0) and ς˙0-l˙(t0) are bounded.


## 2.3.1. Prescribed Performance

To achieve the control objective, the tracking errorzi(t),i=1,V should satisfy the following prescribed performance bounds [22, 39]:(10)-λit<zit<λit,where λi(t)>0 named performance function is defined as (11)λit=λi0-λi∞exp⁡-lit+λi∞,where λi0, λi∞, and li are positive constants; λi0=λi(0) and λi∞=limt→∞⁡λi(t); li denotes the minimum speed of convergence and λi∞ is the maximum steady-state error.To transform the constrained tracking error condition (10) into an equivalent unconstrained one, the following state transformation is employed. So we have(12)μit=Rizitλit,where μi(t) is the transformed error and Ri(·) is an increasing transformation function shown as follows:(13)Riztλt=ln⁡1+zit/λit1-zit/λit.The derivative of (12) is shown as(14)μ˙it=riz˙it-λ˙itλitzit,where ri=∂Ri/∂zi(t)/λi(t)1/λi(t).Using (13), the following inequalities can be obtained:(15)ri>rimin>0,where rmin=2/(λi(0)).


## 2.3.2. Useful Function and Key Lemmas

Definition 2 (see [35]).
The boundaries of the compact subsetsΩi are defined by several prescribed constants 0<ri1<ri2,i=2,4; meanwhile some useful switching functions are described as (16)mix-i≜∏k=1iBkxk,Bkxk≜1,ifxk<rk1,rk22-xk2rk22-rk12e-rk22-xk2/rk22-rk12ω-k2b1,ifrk1≤xk≤rk2,0,ifxk>rk2, where ω-k>0 and b1>0 are positive constants.Lemma 3 (see [40]).
The following inequality holds for anyω0>0 and η∈R:(17)0≤η-ηtanh⁡ηω0≤κ0ω0,where κ0 is a constant satisfying κ0=e-(κ0+1); that is, κ0=0.2785.Lemma 4 (see [41]).
The “first-order sliding mode differentiator (FOSD)” is designed as(18)ς˙0=-μ-0ς0-lt0.5sign⁡ς0-lt+ς1,ς˙1=-μ-1sign⁡ς1-ς0,where ς0 and ς1 are the states of system (18), μ-0 and μ-1 are the designed parameters of FOSD, and l(t) is an input function. ς˙0 can estimate l˙(t) to an arbitrary precision in case the initial values ς0-l(t0) and ς˙0-l˙(t0) are bounded.


## 3. Controller Design

In order to process the derivation, motived by [8, 12], filtered signals are used to circumvent algebraic loop problems encountered in the following design; thus we define (19)Δf2=f2x2,x3-f2x2,x3f,Δf4=f4x-,u-f4x-,uf,ΔfV=fVxV,T-fVxV,Tf, where x3f, uf, and Tf are the filtered signals defined by [8]: (20)x3f=HLsx3≈x3,uf=HLsu≈u,Tf=HLsT≈T,where HL(s) is a Butterworth low-pass filter. The corresponding filter parameters of Butterworth filters can be obtained in [8].Assumption 5.
In this paper, we assume that all of the system states are measurable.Assumption 6.
The functionsfi(·), i=2,4, are unknown and are bounded by fi(·)≤fiu(·), where fiu(·) are known nonnegative smooth functions. Meanwhile, it is also assumed that Δfi are bounded.
Obviously, there exist ideal weight vectorsW2∗, W4∗, and WV∗ such that(21)f2x2,x3f=W2∗TΦ2x2,x3f+ε2,ε2≤ε2M,f4x-,uf=W4∗TΦ4x-,uf+ε4,ε4≤ε4M,fVxV,Tf=WV∗TΦVxV,Tf+εV,εV≤εVM,where εi and εiM denote the approximation errors and their upper bounds, respectively. Wi∗ is the weight of NN. Φi(·)=ϕ1·,…,ϕn· is the basis function vector with ϕi(·)=exp⁡(-(·-ci)T(·-ci)/bi2), wherein ci and bi are the centers and widths of ϕi(·). Obviously, the ideal weights W2∗, W4∗, and WV∗ are completely unknown. Thus, the MLP technique is employed to estimate the norm of W2∗, W4∗, and WV∗ to reduce the computation burden. Those parameters are defined as φi=Wi∗2i=2,4,V. In the following, we replace φi(·) with φi to simplify the expression.


### 3.1. Velocity Controller Design

Define velocity tracking error as(22)zV=V-Vd.The time derivative of zV can be described as (23)z˙V=fV+T-V˙d.According to (14) and (23), the time derivation of the transformed error μV(t) is shown as (24)μ˙Vt=rVz˙V-λ˙VλVzV=rVfV+T-V˙d-λ˙VλVzV,where rV=∂RV/∂(zV/λV)1/λV>rVmin>0 and λV(t)=λV0-λV∞exp⁡(-lVt)+λV∞.By employing MLP technique, the controllerTd is designed as (25)Td=-kV1-r^˙V2rV2μV-12μVφ^VΦVTΦV-d^Vtanh⁡μVωV1+V˙d+λ˙VλVzV,where kV1 and ωV1 are positive design parameters. φ^V and d^V denote the estimation of φV and dVM, respectively. dV=ΔfV+εV is the lump approximation error with dV≤dVM and r^˙V denotes the estimation of r˙V by means of sliding mode differentiator. According to Lemma 4, we can easily obtain r^˙V-r˙V≤l- with l->0.Consider the following adaptive laws forφ^V and d^V: (26)φ^˙V=ρV12μV2ΦVTΦV-2σV1φ^V,d^˙V=ρV2μVtanh⁡μVωV1-σV2d^V,where ρV1, ρV2, σV1, and σV2 denote positive design parameters.Theorem 7.
Suppose that the velocity subsystem (9) satisfies Assumption 5; if the adaptive controller is selected as (25) and updating laws are selected as (26), the signals including μV, φ~V, and d~V are ensured to be bounded.Remark 8.
The velocity design is partially derived from [24]. Note that the FOSD is used to estimate unknown item r˙V. By introducing 0.5r^˙VμV/rV2 in (25), the stability analysis problem in [24] is overcome.


### 3.2. Altitude Controller Design

The following coordinate change is constructed to facilitate the control design:(27)z1=x1-yd,z2=x2-α1,z3=x3-α2,z4=x4-α3,where α1, α2, and α3 are the virtual controllers to be designed at Steps 1, 2, and 3, respectively. yd=hd is the reference signal. The control scheme for the altitude subsystem is developed in the framework of back-stepping technique, which contains 4-step recursive design procedure.Step 1.
The time derivative ofz1=x1-yd is expressed as (28)z˙1=x˙1-y˙d=Vx2-y˙d=Vz2+α1-y˙d.
By using (15) and (28), the time derivative of the transformed altitude error μ1(t) is shown as follows:(29)μ˙1t=r1z˙1-λ˙1λ1z1=r1Vx2-y˙d-λ˙1λ1z1,where r1=∂R1/∂(z1/λ1)1/λ1>r1min>0 and λ1(t)=λ10-λ1∞exp⁡(-l1t)+λ1∞.
The virtual controllerα1 is designed as (30)α1=-k1-r˙1/2r12μ1+y˙d+λ˙1/λ1z1V,where k1 is positive parameter. It is worth noticing that r˙1 can be easily obtained via system states.
Invoking (28) and (30), one has(31)μ˙1=r1Vz2-k1-r˙12r12μ1.
In order to avoid the tedious computation ofα˙1, the following FOSD is adopted to estimate it: (32)ς˙10=-μ10ς10-α10.5sign⁡ς10-α1+ς11,ς˙11=-μ11sign⁡ς11-ς10,where ς10 and ς11 are the states of FOSD (32) and μ10 and μ11 are the positive design constants.
Then, we have(33)α˙1=ς˙10+τ1,where τ1 is the estimation error of the FOSD with τ1≤τ-1.Step 2.
The differentiation ofz2 is obtained as follows:(34)z˙2=x˙2-α˙1=f2x2,x3+z3+α2-α˙1.
The virtual controllerα2 is designed as(35)α2=-k2z2+ς˙10-m2u2N-1-m2u2rwith (36)u2N=12z2φ^2Φ2TΦ2+d^2tanh⁡z2ω21,u2r=f2utanh⁡z2f2uω22,where k2, ω21, and ω22 are positive design parameters. d2=Δf2+ε2 is bounded with d2≤d2M. φ^2 and d^2 denote the estimations of φ2 and d2M, respectively.
The structure of adaptive control laws is expressed as follows:(37)φ^˙2=ρ212m2z22Φ2TΦ2-2σ21φ^2,d^˙2=ρ22m2z2tanh⁡z2ω21-σ22d^2.
Substituting (35) into (34), (34) can be rewritten as(38)z˙2=z3-k2z2+ς˙10-α˙1+m2f2-u2N+1-m2f2-u2r=z3-k2z2+ς˙10-α˙1+m2W2∗TΦ2-12z2φ^2Φ2TΦ2+d2-d^2tanh⁡z2ω21+1-m2f2-f2utanh⁡z2f2uω22.
The following FOSD is adopted to estimateα˙2:(39)ς˙20=-μ20ς20-α20.5sign⁡ς20-α2+ς21,ς˙21=-μ21sign⁡ς21-ς20, where ς20 and ς21 are the states of system (32) and μ20 and μ21 are positive design constants.
From (39) and Lemma 3, we have (40)α˙2=ς˙20+τ2,where τ2 is the estimation error with τ2≤τ-2.Step 3.
The differentiation ofz3 is obtained as follows:(41)z˙3=x˙3-α˙2=z4+α3-α˙2.
The virtual control lawα3 is designed as (42)α3=-k3z3+ς˙20-z2,where k3 is a positive design parameter.
Substituting (42) into (41) yields(43)z˙3=z4-z2-k3z3+ς˙20-α˙2=z4-z2-k3z3-τ2.
As done previously, the following FOSD is employed to estimateα˙3: (44)ς˙30=-μ30ς30-α30.5sign⁡ς30-α3+ς31,ς˙31=-μ31sign⁡ς31-ς30,where ς30 and ς31 are the states of the system and μ30 and μ31 are the positive design constants.
Thus, we have(45)α˙3=ς˙30+τ3,where τ3 is an estimation error with τ3≤τ-3.Step 4.
In this step, the actual controlleru will be developed. The differentiation of z4 can be obtained as follows:(46)z˙4=f4x-,u+u-α˙3=m4f4x-,u+1-m4f4x-,u+u-α˙3.
The controlleru is designed as(47)u=-k4z4+ς˙30-z3-m4u4N-1-m4u4rwith(48)u4N=12z4φ^4Φ4TΦ4+d^4tanh⁡z4ω41,u4r=f4utanh⁡z4f4uω42,where k4, ω41, and ω42 are the positive design constants. d4=Δf4+ε4 is the lump approximation error with d4≤d4M. φ^4 and d^4 denote the estimations of φ4 and d4M, respectively. φ^4 and d^4 are updated as(49)φ^˙4=ρ412m4z42Φ4TΦ4-2σ41φ^4,d^˙4=ρ42m4z4tanh⁡z4ω41-σ42d^4.
Thus, (46) can be rewritten as(50)z˙4=-k4z4-z3-τ3+m4W4∗TΦ4-12z4φ^4Φ4TΦ4+d4-d^4tanh⁡z4ω41+1-m4f4-f4utanh⁡z4f4uω42.Theorem 9.
Consider the altitude subsystem (8) with Assumptions 5 and 6; if the switching adaptive neural prescribed performance control scheme is selected as (30), (35), (42), and (47), adaptive laws are selected as (37) and (49), and FOSD is selected as (32), (39), and (44), the signals μ1, zi=2,3,4, φ~i=2,4, and d~i=2,4 in the closed-loop system are bounded.Remark 10.
The altitude controller, composed of a normal adaptive neural controller working in the neural active region, a robust controller being in charge outside the neural approximation region, and a switching strategy supervising the exchange of the former two controllers, is constructed.Remark 11.
In this paper, in order to estimate the derivative of virtual controllersα1, α2, and α3, the FOSD (first-order sliding mode differentiator) is employed. Using (33) as an example, ς˙10 is the estimation of a˙1 and τ1 is the estimation error between actual a˙1 and ς˙10. It must be noted that τ1 is not used in the controller design but is just employed for stability analysis (please see (B.6)).


## 3.1. Velocity Controller Design

Define velocity tracking error as(22)zV=V-Vd.The time derivative of zV can be described as (23)z˙V=fV+T-V˙d.According to (14) and (23), the time derivation of the transformed error μV(t) is shown as (24)μ˙Vt=rVz˙V-λ˙VλVzV=rVfV+T-V˙d-λ˙VλVzV,where rV=∂RV/∂(zV/λV)1/λV>rVmin>0 and λV(t)=λV0-λV∞exp⁡(-lVt)+λV∞.By employing MLP technique, the controllerTd is designed as (25)Td=-kV1-r^˙V2rV2μV-12μVφ^VΦVTΦV-d^Vtanh⁡μVωV1+V˙d+λ˙VλVzV,where kV1 and ωV1 are positive design parameters. φ^V and d^V denote the estimation of φV and dVM, respectively. dV=ΔfV+εV is the lump approximation error with dV≤dVM and r^˙V denotes the estimation of r˙V by means of sliding mode differentiator. According to Lemma 4, we can easily obtain r^˙V-r˙V≤l- with l->0.Consider the following adaptive laws forφ^V and d^V: (26)φ^˙V=ρV12μV2ΦVTΦV-2σV1φ^V,d^˙V=ρV2μVtanh⁡μVωV1-σV2d^V,where ρV1, ρV2, σV1, and σV2 denote positive design parameters.Theorem 7.
Suppose that the velocity subsystem (9) satisfies Assumption 5; if the adaptive controller is selected as (25) and updating laws are selected as (26), the signals including μV, φ~V, and d~V are ensured to be bounded.Remark 8.
The velocity design is partially derived from [24]. Note that the FOSD is used to estimate unknown item r˙V. By introducing 0.5r^˙VμV/rV2 in (25), the stability analysis problem in [24] is overcome.


## 3.2. Altitude Controller Design

The following coordinate change is constructed to facilitate the control design:(27)z1=x1-yd,z2=x2-α1,z3=x3-α2,z4=x4-α3,where α1, α2, and α3 are the virtual controllers to be designed at Steps 1, 2, and 3, respectively. yd=hd is the reference signal. The control scheme for the altitude subsystem is developed in the framework of back-stepping technique, which contains 4-step recursive design procedure.Step 1.
The time derivative ofz1=x1-yd is expressed as (28)z˙1=x˙1-y˙d=Vx2-y˙d=Vz2+α1-y˙d.
By using (15) and (28), the time derivative of the transformed altitude error μ1(t) is shown as follows:(29)μ˙1t=r1z˙1-λ˙1λ1z1=r1Vx2-y˙d-λ˙1λ1z1,where r1=∂R1/∂(z1/λ1)1/λ1>r1min>0 and λ1(t)=λ10-λ1∞exp⁡(-l1t)+λ1∞.
The virtual controllerα1 is designed as (30)α1=-k1-r˙1/2r12μ1+y˙d+λ˙1/λ1z1V,where k1 is positive parameter. It is worth noticing that r˙1 can be easily obtained via system states.
Invoking (28) and (30), one has(31)μ˙1=r1Vz2-k1-r˙12r12μ1.
In order to avoid the tedious computation ofα˙1, the following FOSD is adopted to estimate it: (32)ς˙10=-μ10ς10-α10.5sign⁡ς10-α1+ς11,ς˙11=-μ11sign⁡ς11-ς10,where ς10 and ς11 are the states of FOSD (32) and μ10 and μ11 are the positive design constants.
Then, we have(33)α˙1=ς˙10+τ1,where τ1 is the estimation error of the FOSD with τ1≤τ-1.Step 2.
The differentiation ofz2 is obtained as follows:(34)z˙2=x˙2-α˙1=f2x2,x3+z3+α2-α˙1.
The virtual controllerα2 is designed as(35)α2=-k2z2+ς˙10-m2u2N-1-m2u2rwith (36)u2N=12z2φ^2Φ2TΦ2+d^2tanh⁡z2ω21,u2r=f2utanh⁡z2f2uω22,where k2, ω21, and ω22 are positive design parameters. d2=Δf2+ε2 is bounded with d2≤d2M. φ^2 and d^2 denote the estimations of φ2 and d2M, respectively.
The structure of adaptive control laws is expressed as follows:(37)φ^˙2=ρ212m2z22Φ2TΦ2-2σ21φ^2,d^˙2=ρ22m2z2tanh⁡z2ω21-σ22d^2.
Substituting (35) into (34), (34) can be rewritten as(38)z˙2=z3-k2z2+ς˙10-α˙1+m2f2-u2N+1-m2f2-u2r=z3-k2z2+ς˙10-α˙1+m2W2∗TΦ2-12z2φ^2Φ2TΦ2+d2-d^2tanh⁡z2ω21+1-m2f2-f2utanh⁡z2f2uω22.
The following FOSD is adopted to estimateα˙2:(39)ς˙20=-μ20ς20-α20.5sign⁡ς20-α2+ς21,ς˙21=-μ21sign⁡ς21-ς20, where ς20 and ς21 are the states of system (32) and μ20 and μ21 are positive design constants.
From (39) and Lemma 3, we have (40)α˙2=ς˙20+τ2,where τ2 is the estimation error with τ2≤τ-2.Step 3.
The differentiation ofz3 is obtained as follows:(41)z˙3=x˙3-α˙2=z4+α3-α˙2.
The virtual control lawα3 is designed as (42)α3=-k3z3+ς˙20-z2,where k3 is a positive design parameter.
Substituting (42) into (41) yields(43)z˙3=z4-z2-k3z3+ς˙20-α˙2=z4-z2-k3z3-τ2.
As done previously, the following FOSD is employed to estimateα˙3: (44)ς˙30=-μ30ς30-α30.5sign⁡ς30-α3+ς31,ς˙31=-μ31sign⁡ς31-ς30,where ς30 and ς31 are the states of the system and μ30 and μ31 are the positive design constants.
Thus, we have(45)α˙3=ς˙30+τ3,where τ3 is an estimation error with τ3≤τ-3.Step 4.
In this step, the actual controlleru will be developed. The differentiation of z4 can be obtained as follows:(46)z˙4=f4x-,u+u-α˙3=m4f4x-,u+1-m4f4x-,u+u-α˙3.
The controlleru is designed as(47)u=-k4z4+ς˙30-z3-m4u4N-1-m4u4rwith(48)u4N=12z4φ^4Φ4TΦ4+d^4tanh⁡z4ω41,u4r=f4utanh⁡z4f4uω42,where k4, ω41, and ω42 are the positive design constants. d4=Δf4+ε4 is the lump approximation error with d4≤d4M. φ^4 and d^4 denote the estimations of φ4 and d4M, respectively. φ^4 and d^4 are updated as(49)φ^˙4=ρ412m4z42Φ4TΦ4-2σ41φ^4,d^˙4=ρ42m4z4tanh⁡z4ω41-σ42d^4.
Thus, (46) can be rewritten as(50)z˙4=-k4z4-z3-τ3+m4W4∗TΦ4-12z4φ^4Φ4TΦ4+d4-d^4tanh⁡z4ω41+1-m4f4-f4utanh⁡z4f4uω42.Theorem 9.
Consider the altitude subsystem (8) with Assumptions 5 and 6; if the switching adaptive neural prescribed performance control scheme is selected as (30), (35), (42), and (47), adaptive laws are selected as (37) and (49), and FOSD is selected as (32), (39), and (44), the signals μ1, zi=2,3,4, φ~i=2,4, and d~i=2,4 in the closed-loop system are bounded.Remark 10.
The altitude controller, composed of a normal adaptive neural controller working in the neural active region, a robust controller being in charge outside the neural approximation region, and a switching strategy supervising the exchange of the former two controllers, is constructed.Remark 11.
In this paper, in order to estimate the derivative of virtual controllersα1, α2, and α3, the FOSD (first-order sliding mode differentiator) is employed. Using (33) as an example, ς˙10 is the estimation of a˙1 and τ1 is the estimation error between actual a˙1 and ς˙10. It must be noted that τ1 is not used in the controller design but is just employed for stability analysis (please see (B.6)).


## 4. Simulations

In this section, two comparative cases are presented to illustrate the effectiveness of the switching functions based adaptive neural control for longitudinal model of the morphing aircraft. The aerodynamic coefficients and model parameters are the same as [5]. The initial conditions are set as X0=γ0,α0,q0,h0,V0=0∘,0.99512∘,0∘/s,1000m,30m/s. The control parameters are selected as k1=0.3, k2=0.3, k3=0.05, k4=3.75, and kV1=3. Gains for the adaptive laws are set as ρ21=40, σ21=0.01, ρ22=10, σ22=0.1, ρ41=250, σ41=0.01, ρ42=100, σ42=0.01, ρV1=10, σV1=0.1, ρV2=100, σV2=0.1, ω21=1, ω22=1, ω41=1, and ω42=5. The aforementioned transient and steady output error bounds are prescribed by the performance functions λi(t)=λi0-λi∞exp⁡(-lit)+λi∞,i=1,V, where λ10=0.5, λ1∞=0.2, λV0=0.35, λ1∞=0.1, and li=0.05. The corresponding neural active regions are defined as V∈30m/s40m/s, x2∈-4deg4deg, x3∈-6deg6deg, and x4∈-6deg/s6deg/s. The centers such as c2, c4, and cV including 50 nodes are evenly spaced in their bounds. The widths of Gaussian functions are chosen as bi2=2, bi4=1.5, and biV=4. The parameters of switching function are selected as r21=4deg, r22=7deg, r31=6deg, r32=10deg, r41=5deg/s, r42=10deg/s, ω-i=1,2,3=10, and b1=1. Reference commands are smoothened via several second-order filters which are given in (51).(51)hdhd0=0.04s2+0.4s+0.04,VdVd0=0.04s2+0.4s+0.04,ζdζd0=1s2+2s+1.Case 1.
In this simulation, we assume that the aircraft is cruising at trim states, and only the morphing process is considered. The initial tracking errors are assumed to bez1(0)=0.1m and zV(0)=0.3m/s. For comparison purposes, the switching function based adaptive neural control (SAN which means the PPC technique is not employed in the control design) and adaptive neural prescribed performance control (PPC described in Remark 10) are utilized; meanwhile the control gains are kept fixed to the values used in the proposed control scheme (SPPC). Simulation results are presented in Figures 1–4. Specifically, the output tracking errors are presented in Figure 2. Moreover, the required control input δe, T, system states, and the evolution of NNs’ weight are provided in Figures 3 and 4. Notice that during the morphing process the input states (x2,x3,x4) of NNs always stay in the active region which can be explained in Figure 1(b); thus the proposed SPPC scheme is equivalent to PPC scheme. As expected, output tracking with prescribed performance as well as states boundedness is achieved with reasonable control effort. In contrast to PPC or SPPC approach, although the SAN control scheme can keep the stability of the aircraft, it owns relatively poor system performance as shown in Figure 2. Moreover, the superiority of the proposed SPPC scheme will be further revealed in Case 2 simulation.Figure 1
Sweep reference and switch signal.
(a)
(b)Figure 2
Altitude and velocity tracking errors.
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(b)Figure 3
System states and NN weights.
(a)
(b)
(c)
(d)
(e)
(f)Figure 4
Control inputs.
(a)
(b)Case 2.
In this simulation, the control gains are kept the same as Case1; meanwhile the SAN and PPC approaches are also used as comparison. The upper bounds of f2u and f4u are set as f2u=0.5 and f4u=1. Simulation results are presented in Figures 5–9. The output tracking for SAN and SPPC is presented in Figures 5(a) and 5(c) and the corresponding tracking errors are presented in Figures 5(b) and 5(d) along with their performance bounds. Particularly, the control inputs and system states as well as the value of the switch functions are provided in Figures 6, 7, and 8. As expected, output tracking with prescribed performance and states boundedness are both achieved by using the proposed SPPC scheme. Unfortunately, the altitude and velocity tracking errors of SAN transcend the prescribed bounds λ1 and λV. It is worth noting that, different from Case 1, the switch values are not always equal to one as depicted in Figure 8, which means that the NNs are out of the neural approximation regions. The curve of δe is obtained as shown in Figure 7(a). After about 11 seconds, the robust controller pulls back the escaped transient to the neural effective regions. Some corresponding responses for PPC scheme are pictured in Figure 9. From this figure, we can conclude that the altitude error oversteps the prescribed bounds without the utilization of switching strategy, thus leading to the instability of the system. In all, compared with simulation results, the superiority of SPPC scheme is obvious.Figure 5
Altitude and velocity tracking.
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(d)Figure 6
System states and NN weights.
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(f)Figure 7
Control inputs.
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(b)Figure 8
Switching functions.
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(b)Figure 9
Partial response of PPC scheme.
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(d)


## 5. Conclusion

A composite switching neural prescribed performance control scheme has been proposed for the longitudinal dynamic model of the morphing aircraft. In the control design, by using neural networks to approximate the unknown functions, the prior information of the aerodynamic parameters is unnecessary. By introducing the performance function, the proposed controller is able to permit attributes such as a lower bound on the convergence rate and maximum allowable steady error to be specified. A switching mechanism supervising the exchange of control authorities between the normal neural controller and a robust controller is used to relax the constraint that NN should be kept in the active regions all the time. Two comparative simulations have revealed the superiority of this control scheme.



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*Source: 1401427-2017-05-17.xml*