# Predictive Sliding Mode Control for Attitude Tracking of Hypersonic Vehicles Using Fuzzy Disturbance Observer

Authors: Xianlei Cheng; Guojian Tang; Peng Wang; Luhua Liu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We propose a predictive sliding mode control (PSMC) scheme for attitude control of hypersonic vehicle (HV) with system uncertainties and external disturbances based on an improved fuzzy disturbance observer (IFDO). First, for a class of uncertain affine nonlinear systems with system uncertainties and external disturbances, we propose a predictive sliding mode control based on fuzzy disturbance observer (FDO-PSMC), which is used to estimate the composite disturbances containing system uncertainties and external disturbances. Afterward, to enhance the composite disturbances rejection performance, an improved FDO-PSMC (IFDO-PSMC) is proposed by incorporating a hyperbolic tangent function with FDO to compensate for the approximate error of FDO. Finally, considering the actuator dynamics, the proposed IFDO-PSMC is applied to attitude control system design for HV to track the guidance commands with high precision and strong robustness. Simulation results demonstrate the effectiveness and robustness of the proposed attitude control scheme.


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## Body

## 1. Introduction

Near space is the airspace of Earth altitudes from 20 km to 100 km, which has shown strategy and spatial superiority with the prosperous development of aerospace technology [1]. The near space hypersonic vehicle (HV) which has many outstanding advantages such as large flight envelope, high maneuverability, and fast response ability compared with the ordinary aircraft has attracted a growing worldwide interest [2, 3]. Due to the hypersonic velocity and changeable flight environment, the HV possesses some distinct dynamic characters of highly coupled control channels, serious nonlinearity, and strong uncertainty, which all contribute to the design difficulty of the attitude control system with remarkable precision and strong robustness. Due to the poor performance of the traditional control approaches in addressing the nonlinear and uncertain problem, advanced control methods should be employed in the attitude control system design for the HV, which is still an open problem.The sliding mode control (SMC) is insensitive to system uncertainties and disturbances. It is one of the most important approaches to the control of systems with modeling imprecision, and it has been widely applied to the flight control system design [4–11]. References [12, 13] incorporate the sliding mode surface into the quadratic performance index of the predictive control and then the predictive sliding mode control law which has an explicitly analytical form is derived by minimizing the performance index. By using this method, the undesirable chattering phenomenon is attenuated and the large online computational issue of the predictive control is avoided. The predictive sliding mode control (PSMC) takes merits of the strong robustness of sliding model control and the outstanding optimization performance of predictive control, which appears to be a very promising control method in control engineering.For the design of control system for the HV, many advanced control methods mainly focus on the stability or robust stability rather than considering the system uncertainties and disturbances explicitly in the controller design [14–18]. They may encounter some unexpected problems and the flight control system may even become unstable in the presence of strong disturbances. Thus, it is of profound significance to adopt new strategies to eliminate the influence of the composite disturbances and improve the precision and robustness of the flight control system. Fuzzy disturbance observer (FDO) is a particular type of disturbance observer, which combines the distinct merits of fuzzy control with disturbance observer technology. These advantages make it an appropriate candidate for robust control of uncertain nonlinear systems [19–25]. Nevertheless, the robust flight control scheme based on FDO needs to be further researched for the HV to improve the control ability, which needs to duel with the changeable flight environment and problems caused by system uncertainties.Motivated by the precise and robust attitude control demand of the HV with uncertain model and external disturbances, this paper considers the composite disturbances in flight control system design of the HV to improve the robust control performance. Three major contributions are presented as follows.(1)
We propose a predictive sliding mode control based on fuzzy disturbance observer (FDO-PSMC) method for a class of uncertain affine nonlinear systems with system uncertainties and external disturbances. The composite disturbances are considered, which result in poor performance and instability of the control system.(2)
To address the problems which are brought by the composite disturbances, an improved FDO (IFDO) is proposed through utilizing the special properties of a hyperbolic tangent function to compensate for the approximate error of FDO. By using IFDO, the composite disturbances can be approximated effectively.(3)
Considering the actuator dynamics, we apply the improved fuzzy disturbance observer based predictive sliding mode control (IFDO-PSMC) scheme to the attitude control system design for the HV. Numerical simulation verifies the surpassing performance of the proposed control scheme.The rest of this paper is organized as follows. In Section2, the control-oriented hypersonic flight model during cruise phase is presented. In order to approximate the composite disturbances, Section 3 presents a FDO-PSMC method and the FDO is improved for a better performance for a class of uncertain affine nonlinear systems with external disturbances. In Section 4, the proposed control scheme is applied to the attitude control system design for the HV in consideration of the actuator dynamics. Simulation results are presented in Section 5 to validate the effectiveness of the designed flight control system. Finally, Section 6 provides some conclusions of this paper.


## 2. HV Modeling

Suppose that the fuel slosh is not considered and the products of inertia are negligible [26]. Based on the hypothesis of an inverse-square-law gravitational model, the centripetal acceleration for the nonrotating Earth, the mathematical model of HV during the cruise phase can be described as
(1)v˙=-μr3C1-Dm+Tmcos⁡αcos⁡β,θ˙=1mv-mμr3C2+Lcos⁡γV-NsinγV+TC4,σ˙=-1mvcos⁡θ-mμr3C3+LsinγV+Ncos⁡γV+TC5,(2)φ˙=ωysinγ+ωzcos⁡γ,ψ˙=ωycos⁡γ-ωzsinγsecφ,γ˙=ωx-ωycos⁡γ-ωzsinγtanφ,ω˙x=Jx-1Mx+Jx-1Jy-Jzωzωy,ω˙y=Jy-1My+Jy-1Jz-Jxωxωz,ω˙z=Jz-1Mz+Jz-1Jx-Jyωyωx,(3)sinβM=sinψ-σcos⁡γ+sinφsinγcos⁡ψ-σcos⁡θMM-cos⁡φsinγsinθ,sinαcos⁡β=cos⁡ψ-σsinφcos⁡γcos⁡θM-sinψ-σsinγcos⁡θM-cos⁡φcos⁡γsinθ,sinγVcos⁡βM=cos⁡ψ-σsinφsinγsinθMM+sinψ-σcos⁡γsinθ+cos⁡φsinγcos⁡θ,(4)C1=xcos⁡θcos⁡σ+y+Resinθ-zcos⁡θsinσ,C2=-xsinθcos⁡σ+y+Recos⁡θ+zsinθsinσ,C3=xsinσ+zcos⁡σ,C4=sinαcos⁡γV+cos⁡αsinβsinγV,C5=sinαsinγV-cos⁡αsinβcos⁡γV,
where α,β, and γV are the angle-of-attack, sideslip angle, and bank angle, respectively; φ,ψ, and γ are the pitch, yaw, and rolling angles, respectively; v,θ, and σ are the flight velocity, flight path angle, and trajectory angle, respectively; ωx,ωy, and ωz are pitch, yaw, and roll rates, respectively; x,y,z are the three position coordinates in the ground coordinate frame; Re,r are radial distance from the Earth’s center and the mean radius of the spherical Earth, respectively; Mx,My, and Mz are control moments including roll, yaw, and pitch control moments; L,N, and T are the lift, side, and thrust forces, respectively; Jx,Jy, and Jz are the main moments of inertia of the three axes, respectively.Remark 1.
For the convenience of control system design,φ,ψ,γT is selected as the state variables. In view of the aim of attitude control system design for HV which is to track the guidance commands αc,βc,γVcT precisely, it is necessary to transform αc,βc,γVcT into φc,ψc,γcT by means of (3) which is obtained by the coordinate system transformation.


## 3. Predictive Sliding Mode Control Based on Fuzzy Disturbance Observer

### 3.1. Predictive Sliding Mode Control for Systems with Uncertainties and Disturbances

To develop the predictive sliding mode control, we consider the following MIMO uncertain affine nonlinear system with external disturbances:(5)x˙=fx+Δfx+gx+Δgxu+d,y=hx,
where x∈Rn is the state vector of the uncertain nonlinear system; u∈Rm is the control input vector; y∈Rm is the output vector of the uncertain nonlinear system; fx∈Rn and gx∈Rn×m are the given function vector and control gain matrix, respectively; Δfx and Δgx are the internal uncertainty and modeling error; and d is the external time-varying disturbance.Define(6)Dx,u,d=Δfx+Δgxu+d,
where Dx,u,d denotes the system uncertainties and external disturbances.Without loss of generality, the equilibrium of uncertain nonlinear system (5) is supposed to be x0=0. In accordance with the Lie derivative operation [27] in differential geometry theory, the vector relative degree of MIMO system can be defined as follows.Definition 2 (see [27]).
The vector relative degree of system (5) is {ρ1,ρ2,…,ρm} at x0 under the following conditions, where ρ1,ρ2,…,ρm denote the relative degree of each channel, respectively.(1)
For allx in a neighborhood of x0, LgjLfkhi(x)=0 (1≤i, j≤m, 0<k<ρi).(2)
The followingm×m matrix is nonsingular at x0=0:
(7)Ax=Lg1Lfρ1-1h1xLg2Lfρ1-1h1x⋯LgmLfρ1-1h1xLg1Lfρ2-1h2xLg2Lfρ2-1h2x⋯LgmLfρ2-1h2x⋮⋮⋱⋮Lg1Lfρm-1hmxLg2Lfρm-1hmx⋯LgmLfρm-1hmx,where gj is thejth row vector of gx and hix is theith output of system (5). Similarly, the disturbance relative degree at x0 can be defined as {τ1,τ2,…,τm}.To facilitate the process of control system design, the following reasonable assumptions are required before developing predictive sliding mode control of the uncertain MIMO nonlinear system (5).Assumption 3.
f
x and hx are continuously differentiable, and gx is continuous.Assumption 4.
All states are available; moreover, the output and reference signals are also continuously differentiable.Assumption 5.
The zero dynamics are stable.Assumption 6.
The vector relative degree is{ρ1,ρ2,…,ρm}, and the disturbance relative degree of composite disturbances is {τ1,τ2,…,τm}, where τi≥ρi (i=1,…,m).According to Assumption6, we can obtain
(8)yiρi=Lfρihix+LgLfρi-1hixu+LDLfρi-1hixMi=1,…,m.Associate them equations; the nonlinear system (5) can be written as
(9)yρ=αx+βxu+Δx,u,D,
where
(10)αx=α1,α2⋯αmT∈Rmβx=β1,β2⋯βmT∈Rm×mΔx,u,D=δ1,δ2⋯δmT∈Rm,
where αi=Lfρihix, βi=LgLfρi-1hi(x), δi=LDLfρi-1hixi=1,…,m, and Δ(x,u,D) is related to the composite disturbances.Furthermore, the sliding surface vector is defined as(11)st=s1,s2⋯smT∈Rm,
where each sliding surface is the combination of the tracking error and its higher order derivatives, which can be depicted as
(12)si=eiρi-1+ki,ρi-1eiρi-2+⋯+ki,0ei,
where ei=yi-yci (i=1,…,m) and ki,j(j=0,1,…,ρi-1) must make the polynomial (12) Hurwitz stable. Differentiating (12), we have
(13)s˙i=eiρi+ki,ρi-1eiρi-1+⋯+ki,0e˙i=yiρi-yciρi+ki,ρi-1eiρi-1M+⋯+ki,0e˙iMi=1,…,m.For the sake of simplified notation, definezi as
(14)zi=ki,ρi-1eiρi-1+⋯+ki,0e˙iMi=1,…,m.Then the vectorz can be written as
(15)z=z1,z2⋯zmT∈Rm.Consequently, differentiating the sliding surface vector, we obtain(16)s˙t=yρ-ycρ+z,
where yc(ρ)=yc1(ρ1),yc2(ρ2)⋯ycm(ρm)T∈Rm.Within the moving time frame, the sliding surfaces^t+T at the time T is approximately predicted by
(17)s^t+T=st+Ts˙t.For the derivation of the control law, the receding-horizon performance index at the timeT is given by
(18)Jx,u,t=12s^Tt+Ts^t+T.The necessary condition for the optimal control to minimize (18) with respect to u is given by
(19)∂J∂u=0.According to (19), substitute (17) into (18); then, the predictive sliding mode control law can be proposed as
(20)u=-T-1βx-1·st+Tαx+Δx,u,D-ycρ+z.Remark 7.
IfD=0, the control law given by (20) is the nominal predictive sliding mode control law. If D≠0, Δ(x,u,D) cannot be directly obtained due to the immeasurable unknown composite disturbances D. The performance of flight control system may become worse, even unstable, when D is big enough. To handle this problem, a fuzzy observer would be designed to estimate the composite disturbances.Remark 8.
For the convenience of notation,Δ(x,u,D) would be denoted by Δ(x) in the rest of this paper.


### 3.2. Fuzzy Disturbance Observer

#### 3.2.1. Fuzzy Logic System

The fuzzy inference engine adopts the fuzzy if-then rules to perform a mapping from an input linguistic vectorX=X1,X2,…,XnT∈Rn to an output variable Y∈R. Theith fuzzy rule can be expressed as [19]
(21)Ri:IfX1isA1i,…,XnisAni,.thenYisYi,
where A1i,…,Ani are fuzzy sets characterized by fuzzy membership functions and Yi is a singleton number.By adopting a center-average and singleton fuzzifier and the product inference, the output of the fuzzy system can be written as(22)YX=∑i=1lYi∏j=1nμAjiXj∑i=1l∏j=1nμAjiXj=θ^TϑX,
where l is the number of fuzzy rules, μAji(Xj) is the membership function value of fuzzy variable Xj(j=1,2,…,n), θ^=Y1,Y2,…,YlT is an adjustable parameter vector, and ϑ(x)=ϑ1,ϑ2,…,ϑlT is a fuzzy basis function vector. ϑi can be expressed as
(23)ϑi=∏j=1nμAjiXj∑i=1l∏j=1nμAjiXj.Lemma 9 (see [28]).
For any given real continuous functiongx on the compact set U∈Rn and arbitrary ε>0, there exists a fuzzy system p(X) presented as (22) such that
(24)sup⁡x∈UpX-YX<ε.According to Lemma9, a fuzzy system can be designed to approximate the composite disturbances Δ(x) by adjusting the parameter vector θ^ online. The designed fuzzy system can be written as
(25)Δ^x∣θ^ΔT=δ^1,δ^2⋯δ^mT=θ^ΔTϑΔx,
where
(26)θ^ΔT=diag⁡θ^1T,θ^2T,…,θ^mT,θ^iT=Yi1,Yi2,…,YilT,ϑΔx=ϑ1xT,ϑ2xT,…,ϑmxTT,ϑx=ϑi1,ϑi2,…,ϑilT.Letx belong to a compact set Ωx; the optimal parameter vector θΔ*T can be defined as
(27)θΔ*T=argmin⁡θ^ΔTsup⁡x∈ΩxΔx-Δ^x∣θ^ΔT,
where the optimal parameter matrix θΔ*T lies in a convex region given by
(28)Mθ=θ^Δ∣θ^Δ≤mθ,
where · is the Frobenius norm and mθ is a design parameter with mθ>0.Consequently, the composite disturbancesΔ(x) can be written as
(29)Δx=θΔ*TϑΔx+ε,
where ε is the smallest approximation error of the fuzzy system. Apparently, the norm of θΔ*T is bounded with
(30)ε≤ε-,
where ε- is the upper bound of the approximation error ε.Through adjusting the parameter vectorθ^ online, the composite disturbances Δ(x) can be approximated by the fuzzy system; hence, the performance of the control system with composite disturbances can be improved. As the performance of the control system is closely related to the selected fuzzy system, the control precision will be reduced when the approximation ability of the selected fuzzy system is dissatisfactory. In order to improve the approximation ability, a hyperbolic tangent function is integrated with the fuzzy disturbance observer to compensate for the approximate error.


#### 3.2.2. Fuzzy Disturbance Observer

Consider the following dynamic system:(31)μ˙=-χμ+px,u,θ^Δ,
where p(x,u,θ^Δ)=χyρ-1+αx+βxu+Δ^(x∣θ^ΔT), χ>0 is a design parameter, and Δ^(x∣θ^ΔT)=θ^ΔTϑΔ(x) is utilized to compensate for the composite disturbances.Define the disturbance observation errorζ=y(ρ-1)-μ; invoking (9) and (31) yields
(32)ζ˙=αx+βxu+Δx+σμ-σyρ-1-αx-βxu-Δ^x∣θ^ΔT=-σζ+θ~ΔTϑΔx+ε,
where θ~Δ=θΔ*-θ^Δ is the adjustable parameter error vector.Then, the FDO-PSMC is proposed as(33)u=-T-1βx-1·st+Tαx+θ^ΔTϑΔx-ycρ+z.Invoking (16) and (33), we have
(34)s˙=yρ-ycρ+z=αx+βxu+Δx-ycρ+z=-T-1s+θ~ΔTϑΔx+ε.Invoking (32) and (34), the augmented system is obtained as
(35)ς˙=Aς+Bθ~ΔTϑΔx+ε,
where ς=(ζT,sT)T,A=diag⁡(-χIm,-T-1Im), and B=(Im,Im)T.Theorem 10.
Assume that the disturbance of (9) is monitored by the system (32), and the system (9) is controlled by (33). If the adjustable parameter vector of FDO is tuned by (36), then the augmented error ς is uniformly ultimately bounded within an arbitrarily small region:
(36)θ^˙Δ=ProjλϑΔxςTB=λϑΔxςT-IθλςTBθ^ΔTϑΔxθ^Δ2θ^Δ,
where
(37)Iθ=0,ifθ^Δ<mθ,orθ^Δ=mθ,ςTBθ^ΔTϑΔx≤01,ifθ^Δ=mθ,ςTBθ^ΔTϑΔx>0.Proof.
Consider the Lyapunov function candidate:(38)V=12ςTς+12λθ~ΔTθ~Δ.
Invoking (35) and (36), the time derivative of V is
(39)V˙=ςTς˙+1λθ~ΔTθ~˙Δ=ζTζ˙+sTs˙+1λθ~ΔTθ~˙Δ=ζT-χζ+θ~ΔTϑΔx+ε+sT-T-1s+θ~ΔTϑΔx+εM-1λθ~ΔTProjλφΔxςTB+1ηϑ~ϑ~˙=ζT-χζ+θ~ΔTϑΔx+ε+sT-T-1s+θ~ΔTϑΔx+εM-θ~ΔTϑΔxςT+θ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ.
Considering the fact(40)θ~ΔTθ^Δ=θΔ*Tθ^Δ-θ^ΔTθ^Δ≤12θΔ*2+θ^Δ2-θ^Δ2≤12θΔ*2-mθ2≤0Mwhenθ^Δ=mθ
and invoking (37), we have
(41)θ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ=0orWθ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ<0.
Consequently, we obtain(42)V˙≤ζT-χζ+θ~ΔTφΔx+εM+sT-T-1s+θ~ΔTφΔx+ε-θ~ΔTφΔxςT≤ζT-χζ+θ~ΔTφΔx+ε+sT-T-1s+θ~ΔTφΔx+εM-θ~ΔTφΔxζ-θ~ΔTφΔxs≤ζT-χζ+ε+sT-T-1s+ε≤-χζ2+ζTε-T-1s2+sTε≤-χ2ζ2+12χε2-12Ts2+T2ε2.
Ifζ>ε-/χ or s>Tε-, V˙<0. Therefore, the augmented error ς is uniformly ultimately bounded.Remark 11.
In order to make sure that the FDOΔ^(x∣θ^ΔT) outputs zero signal in case of the composite disturbances D=0, the consequent parameters should be set to be 0:
(43)θ^Δ0=0.Remark 12.
A projection operator (36) is used in the tuning method of the adjustable parameter vector; then, θ^Δ can be guaranteed to be bounded [29].Remark 13.
The adjustable parameter vector of the FDO (36) includes the observation error ζ and the sliding surface s. In view of the fact that the sliding surface s which is the linearization combination of the tracking error e is Hurwitz stable, the observation error ζ and tracking error e are both uniformly ultimately bounded, which ensures stability of the control system.


### 3.3. Improved Fuzzy Disturbance Observer

Lemma 14 (see [30]).
For allc>0 and w∈Rm, the following inequality holds:
(44)0<w-wTtanhwc≤mκc,
where κ is a constant such that κ=e-(κ+1); that is, κ=0.2785.The hyperbolic tangent function is incorporated to compensate for the approximate error and improve the precision of FDO. Consider the following dynamic system:(45)μ˙=-χμ+px,u,θ^Δ,
where p(x,u,θ^Δ)=σy(ρ-1)+αx+βxu+Δ^(x∣θ^ΔT)-υ^tanh(ζ/c), χ>0 is a design parameter, and Δ^(x∣θ^ΔT)=θ^ΔTϑΔ(x) is utilized to compensate for the composite disturbances.Define the disturbance observation errorζ=y(ρ-1)-μ; invoking (9) and (45) yields
(46)ζ˙=αx+βxu+Δx+σμ-σyρ-1-αx-βxu-Δ^x∣θ^ΔT=-χζ+θ~ΔTϑΔx+ε-υ^tanhζc,
where θ~Δ=θΔ*-θ^Δ is an adjustable parameter vector.The IFDO-PSMC is proposed as(47)u=-T-1βx-1M·st+Tstcαx+θ^ΔTϑΔxMMMMMM0-ycρ+z+υ^tanhstc.Invoking (16) and (47), we have
(48)s˙=yρ-ycρ+z=αx+βxu+Δx-ycρ+z=-T-1s+θ~ΔTϑΔx+ε-υ^tanhsc.Invoking (46) and (48), the augmented system is obtained as
(49)ς˙=Aς+Bθ~ΔTϑΔx+ευ^tanhςTBc,
where ς=(ζT,sT)T,  A=diag⁡(-χIm,-T-1Im), and B=(Im,Im)T.Theorem 15.
Assume that the disturbance of the system (9) is monitored by the system (49), and the system (9) is controlled by (47). If the adjustable parameter vector of IFDO is tuned by (50) and the approximate error compensation parameter is tuned by (51), the augmented error ς is uniformly ultimately bounded within an arbitrarily small region:
(50)θ^˙Δ=ProjλϑΔxςTB=λϑΔxςT-IθλςTBθ^ΔTϑΔxθ^Δ2θ^Δ,(51)υ^˙=ηςTB,
where
(52)Iθ=0,ifθ^Δ<mθ,orθ^Δ=mθ,ςTBθ^ΔTϑΔx≤01,ifθ^Δ=mθ,ςTBθ^ΔTϑΔx>0.Proof.
Consider the Lyapunov function candidate:(53)V=12ςTς+12λθ~ΔTθ~Δ+12ηυ~2,
where υ~=υ^-ε-. Invoking (49), (50), and (51), the time derivative of V is
(54)V˙=ςTς˙+1λθ~ΔTθ~˙Δ+1ηυ~υ~˙=ζTζ˙+sTs˙+1λθ~ΔTθ~˙Δ+1ηυ~υ~˙=ζT-χζ+θ~ΔTϑΔx+ε-υ^tanhζcM+sT-T-1s+θ~ΔTϑΔx+εM-υ^tanhsc-1λθ~ΔTProjλϑΔxςTB+1ηυ~υ~˙=ζT-χζ+θ~ΔTϑΔx+ε-υ^tanhζcM+sT-T-1s+θ~ΔTϑΔx+εM-υ^tanhsc-θ~ΔTϑΔxςTM+θ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ+1ηυ~υ~˙≤ζT-χζ+θ~ΔTϑΔx+ε-υ^tanhζcM+sT-T-1s+θ~ΔTϑΔx+ε-υ^tanhscM-θ~ΔTϑΔxςT+1ηυ~υ~˙≤ζT-χζ+θ~ΔTϑΔx+ε-υ^tanhζcM+sT-T-1s+θ~ΔTϑΔx+ε-υ^tanhscM-θ~ΔTϑΔxζ-θ~ΔTϑΔxs+1ηυ~υ~˙≤ζT-χζ+ε+sT-T-1s+ε-ζTυ^tanhζcM-sTυ^tanhsc+υ~ζ+υ~s.According to Lemma14, we have
(55)-ζTυ^tanhζc≤-υ^ζ+mκc,-sTυ^tanhsc≤-υ^s+mκc.Consequently, we obtain(56)V˙≤-σζ2+ζTε-T-1s2+sTε-υ^ζM+mκc-υ^s+mκc+υ~ζ+υ~s≤-σζ2+ζε--T-1s2+sε--υ^ζM+υ~ζ-υ^ζ+υ~s+2mκc≤-σζ2-T-1s2+2mκc.Ifζ>2mκc/χ or s>2mκcT, V˙<0. Therefore, the augmented error ς is uniformly ultimately bounded.Remark 16.
The composite disturbances can be approximated effectively by adjusting the parameter law (50) and (51). If the fuzzy system approximates the composite disturbances accurately, the approximate error compensation will have small effect on compensating the approximate error, and vice versa. Based on this, the approximate precision of FDO can be improved through overall consideration.


## 3.1. Predictive Sliding Mode Control for Systems with Uncertainties and Disturbances

To develop the predictive sliding mode control, we consider the following MIMO uncertain affine nonlinear system with external disturbances:(5)x˙=fx+Δfx+gx+Δgxu+d,y=hx,
where x∈Rn is the state vector of the uncertain nonlinear system; u∈Rm is the control input vector; y∈Rm is the output vector of the uncertain nonlinear system; fx∈Rn and gx∈Rn×m are the given function vector and control gain matrix, respectively; Δfx and Δgx are the internal uncertainty and modeling error; and d is the external time-varying disturbance.Define(6)Dx,u,d=Δfx+Δgxu+d,
where Dx,u,d denotes the system uncertainties and external disturbances.Without loss of generality, the equilibrium of uncertain nonlinear system (5) is supposed to be x0=0. In accordance with the Lie derivative operation [27] in differential geometry theory, the vector relative degree of MIMO system can be defined as follows.Definition 2 (see [27]).
The vector relative degree of system (5) is {ρ1,ρ2,…,ρm} at x0 under the following conditions, where ρ1,ρ2,…,ρm denote the relative degree of each channel, respectively.(1)
For allx in a neighborhood of x0, LgjLfkhi(x)=0 (1≤i, j≤m, 0<k<ρi).(2)
The followingm×m matrix is nonsingular at x0=0:
(7)Ax=Lg1Lfρ1-1h1xLg2Lfρ1-1h1x⋯LgmLfρ1-1h1xLg1Lfρ2-1h2xLg2Lfρ2-1h2x⋯LgmLfρ2-1h2x⋮⋮⋱⋮Lg1Lfρm-1hmxLg2Lfρm-1hmx⋯LgmLfρm-1hmx,where gj is thejth row vector of gx and hix is theith output of system (5). Similarly, the disturbance relative degree at x0 can be defined as {τ1,τ2,…,τm}.To facilitate the process of control system design, the following reasonable assumptions are required before developing predictive sliding mode control of the uncertain MIMO nonlinear system (5).Assumption 3.
f
x and hx are continuously differentiable, and gx is continuous.Assumption 4.
All states are available; moreover, the output and reference signals are also continuously differentiable.Assumption 5.
The zero dynamics are stable.Assumption 6.
The vector relative degree is{ρ1,ρ2,…,ρm}, and the disturbance relative degree of composite disturbances is {τ1,τ2,…,τm}, where τi≥ρi (i=1,…,m).According to Assumption6, we can obtain
(8)yiρi=Lfρihix+LgLfρi-1hixu+LDLfρi-1hixMi=1,…,m.Associate them equations; the nonlinear system (5) can be written as
(9)yρ=αx+βxu+Δx,u,D,
where
(10)αx=α1,α2⋯αmT∈Rmβx=β1,β2⋯βmT∈Rm×mΔx,u,D=δ1,δ2⋯δmT∈Rm,
where αi=Lfρihix, βi=LgLfρi-1hi(x), δi=LDLfρi-1hixi=1,…,m, and Δ(x,u,D) is related to the composite disturbances.Furthermore, the sliding surface vector is defined as(11)st=s1,s2⋯smT∈Rm,
where each sliding surface is the combination of the tracking error and its higher order derivatives, which can be depicted as
(12)si=eiρi-1+ki,ρi-1eiρi-2+⋯+ki,0ei,
where ei=yi-yci (i=1,…,m) and ki,j(j=0,1,…,ρi-1) must make the polynomial (12) Hurwitz stable. Differentiating (12), we have
(13)s˙i=eiρi+ki,ρi-1eiρi-1+⋯+ki,0e˙i=yiρi-yciρi+ki,ρi-1eiρi-1M+⋯+ki,0e˙iMi=1,…,m.For the sake of simplified notation, definezi as
(14)zi=ki,ρi-1eiρi-1+⋯+ki,0e˙iMi=1,…,m.Then the vectorz can be written as
(15)z=z1,z2⋯zmT∈Rm.Consequently, differentiating the sliding surface vector, we obtain(16)s˙t=yρ-ycρ+z,
where yc(ρ)=yc1(ρ1),yc2(ρ2)⋯ycm(ρm)T∈Rm.Within the moving time frame, the sliding surfaces^t+T at the time T is approximately predicted by
(17)s^t+T=st+Ts˙t.For the derivation of the control law, the receding-horizon performance index at the timeT is given by
(18)Jx,u,t=12s^Tt+Ts^t+T.The necessary condition for the optimal control to minimize (18) with respect to u is given by
(19)∂J∂u=0.According to (19), substitute (17) into (18); then, the predictive sliding mode control law can be proposed as
(20)u=-T-1βx-1·st+Tαx+Δx,u,D-ycρ+z.Remark 7.
IfD=0, the control law given by (20) is the nominal predictive sliding mode control law. If D≠0, Δ(x,u,D) cannot be directly obtained due to the immeasurable unknown composite disturbances D. The performance of flight control system may become worse, even unstable, when D is big enough. To handle this problem, a fuzzy observer would be designed to estimate the composite disturbances.Remark 8.
For the convenience of notation,Δ(x,u,D) would be denoted by Δ(x) in the rest of this paper.


## 3.2. Fuzzy Disturbance Observer

### 3.2.1. Fuzzy Logic System

The fuzzy inference engine adopts the fuzzy if-then rules to perform a mapping from an input linguistic vectorX=X1,X2,…,XnT∈Rn to an output variable Y∈R. Theith fuzzy rule can be expressed as [19]
(21)Ri:IfX1isA1i,…,XnisAni,.thenYisYi,
where A1i,…,Ani are fuzzy sets characterized by fuzzy membership functions and Yi is a singleton number.By adopting a center-average and singleton fuzzifier and the product inference, the output of the fuzzy system can be written as(22)YX=∑i=1lYi∏j=1nμAjiXj∑i=1l∏j=1nμAjiXj=θ^TϑX,
where l is the number of fuzzy rules, μAji(Xj) is the membership function value of fuzzy variable Xj(j=1,2,…,n), θ^=Y1,Y2,…,YlT is an adjustable parameter vector, and ϑ(x)=ϑ1,ϑ2,…,ϑlT is a fuzzy basis function vector. ϑi can be expressed as
(23)ϑi=∏j=1nμAjiXj∑i=1l∏j=1nμAjiXj.Lemma 9 (see [28]).
For any given real continuous functiongx on the compact set U∈Rn and arbitrary ε>0, there exists a fuzzy system p(X) presented as (22) such that
(24)sup⁡x∈UpX-YX<ε.According to Lemma9, a fuzzy system can be designed to approximate the composite disturbances Δ(x) by adjusting the parameter vector θ^ online. The designed fuzzy system can be written as
(25)Δ^x∣θ^ΔT=δ^1,δ^2⋯δ^mT=θ^ΔTϑΔx,
where
(26)θ^ΔT=diag⁡θ^1T,θ^2T,…,θ^mT,θ^iT=Yi1,Yi2,…,YilT,ϑΔx=ϑ1xT,ϑ2xT,…,ϑmxTT,ϑx=ϑi1,ϑi2,…,ϑilT.Letx belong to a compact set Ωx; the optimal parameter vector θΔ*T can be defined as
(27)θΔ*T=argmin⁡θ^ΔTsup⁡x∈ΩxΔx-Δ^x∣θ^ΔT,
where the optimal parameter matrix θΔ*T lies in a convex region given by
(28)Mθ=θ^Δ∣θ^Δ≤mθ,
where · is the Frobenius norm and mθ is a design parameter with mθ>0.Consequently, the composite disturbancesΔ(x) can be written as
(29)Δx=θΔ*TϑΔx+ε,
where ε is the smallest approximation error of the fuzzy system. Apparently, the norm of θΔ*T is bounded with
(30)ε≤ε-,
where ε- is the upper bound of the approximation error ε.Through adjusting the parameter vectorθ^ online, the composite disturbances Δ(x) can be approximated by the fuzzy system; hence, the performance of the control system with composite disturbances can be improved. As the performance of the control system is closely related to the selected fuzzy system, the control precision will be reduced when the approximation ability of the selected fuzzy system is dissatisfactory. In order to improve the approximation ability, a hyperbolic tangent function is integrated with the fuzzy disturbance observer to compensate for the approximate error.


### 3.2.2. Fuzzy Disturbance Observer

Consider the following dynamic system:(31)μ˙=-χμ+px,u,θ^Δ,
where p(x,u,θ^Δ)=χyρ-1+αx+βxu+Δ^(x∣θ^ΔT), χ>0 is a design parameter, and Δ^(x∣θ^ΔT)=θ^ΔTϑΔ(x) is utilized to compensate for the composite disturbances.Define the disturbance observation errorζ=y(ρ-1)-μ; invoking (9) and (31) yields
(32)ζ˙=αx+βxu+Δx+σμ-σyρ-1-αx-βxu-Δ^x∣θ^ΔT=-σζ+θ~ΔTϑΔx+ε,
where θ~Δ=θΔ*-θ^Δ is the adjustable parameter error vector.Then, the FDO-PSMC is proposed as(33)u=-T-1βx-1·st+Tαx+θ^ΔTϑΔx-ycρ+z.Invoking (16) and (33), we have
(34)s˙=yρ-ycρ+z=αx+βxu+Δx-ycρ+z=-T-1s+θ~ΔTϑΔx+ε.Invoking (32) and (34), the augmented system is obtained as
(35)ς˙=Aς+Bθ~ΔTϑΔx+ε,
where ς=(ζT,sT)T,A=diag⁡(-χIm,-T-1Im), and B=(Im,Im)T.Theorem 10.
Assume that the disturbance of (9) is monitored by the system (32), and the system (9) is controlled by (33). If the adjustable parameter vector of FDO is tuned by (36), then the augmented error ς is uniformly ultimately bounded within an arbitrarily small region:
(36)θ^˙Δ=ProjλϑΔxςTB=λϑΔxςT-IθλςTBθ^ΔTϑΔxθ^Δ2θ^Δ,
where
(37)Iθ=0,ifθ^Δ<mθ,orθ^Δ=mθ,ςTBθ^ΔTϑΔx≤01,ifθ^Δ=mθ,ςTBθ^ΔTϑΔx>0.Proof.
Consider the Lyapunov function candidate:(38)V=12ςTς+12λθ~ΔTθ~Δ.
Invoking (35) and (36), the time derivative of V is
(39)V˙=ςTς˙+1λθ~ΔTθ~˙Δ=ζTζ˙+sTs˙+1λθ~ΔTθ~˙Δ=ζT-χζ+θ~ΔTϑΔx+ε+sT-T-1s+θ~ΔTϑΔx+εM-1λθ~ΔTProjλφΔxςTB+1ηϑ~ϑ~˙=ζT-χζ+θ~ΔTϑΔx+ε+sT-T-1s+θ~ΔTϑΔx+εM-θ~ΔTϑΔxςT+θ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ.
Considering the fact(40)θ~ΔTθ^Δ=θΔ*Tθ^Δ-θ^ΔTθ^Δ≤12θΔ*2+θ^Δ2-θ^Δ2≤12θΔ*2-mθ2≤0Mwhenθ^Δ=mθ
and invoking (37), we have
(41)θ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ=0orWθ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ<0.
Consequently, we obtain(42)V˙≤ζT-χζ+θ~ΔTφΔx+εM+sT-T-1s+θ~ΔTφΔx+ε-θ~ΔTφΔxςT≤ζT-χζ+θ~ΔTφΔx+ε+sT-T-1s+θ~ΔTφΔx+εM-θ~ΔTφΔxζ-θ~ΔTφΔxs≤ζT-χζ+ε+sT-T-1s+ε≤-χζ2+ζTε-T-1s2+sTε≤-χ2ζ2+12χε2-12Ts2+T2ε2.
Ifζ>ε-/χ or s>Tε-, V˙<0. Therefore, the augmented error ς is uniformly ultimately bounded.Remark 11.
In order to make sure that the FDOΔ^(x∣θ^ΔT) outputs zero signal in case of the composite disturbances D=0, the consequent parameters should be set to be 0:
(43)θ^Δ0=0.Remark 12.
A projection operator (36) is used in the tuning method of the adjustable parameter vector; then, θ^Δ can be guaranteed to be bounded [29].Remark 13.
The adjustable parameter vector of the FDO (36) includes the observation error ζ and the sliding surface s. In view of the fact that the sliding surface s which is the linearization combination of the tracking error e is Hurwitz stable, the observation error ζ and tracking error e are both uniformly ultimately bounded, which ensures stability of the control system.


## 3.2.1. Fuzzy Logic System

The fuzzy inference engine adopts the fuzzy if-then rules to perform a mapping from an input linguistic vectorX=X1,X2,…,XnT∈Rn to an output variable Y∈R. Theith fuzzy rule can be expressed as [19]
(21)Ri:IfX1isA1i,…,XnisAni,.thenYisYi,
where A1i,…,Ani are fuzzy sets characterized by fuzzy membership functions and Yi is a singleton number.By adopting a center-average and singleton fuzzifier and the product inference, the output of the fuzzy system can be written as(22)YX=∑i=1lYi∏j=1nμAjiXj∑i=1l∏j=1nμAjiXj=θ^TϑX,
where l is the number of fuzzy rules, μAji(Xj) is the membership function value of fuzzy variable Xj(j=1,2,…,n), θ^=Y1,Y2,…,YlT is an adjustable parameter vector, and ϑ(x)=ϑ1,ϑ2,…,ϑlT is a fuzzy basis function vector. ϑi can be expressed as
(23)ϑi=∏j=1nμAjiXj∑i=1l∏j=1nμAjiXj.Lemma 9 (see [28]).
For any given real continuous functiongx on the compact set U∈Rn and arbitrary ε>0, there exists a fuzzy system p(X) presented as (22) such that
(24)sup⁡x∈UpX-YX<ε.According to Lemma9, a fuzzy system can be designed to approximate the composite disturbances Δ(x) by adjusting the parameter vector θ^ online. The designed fuzzy system can be written as
(25)Δ^x∣θ^ΔT=δ^1,δ^2⋯δ^mT=θ^ΔTϑΔx,
where
(26)θ^ΔT=diag⁡θ^1T,θ^2T,…,θ^mT,θ^iT=Yi1,Yi2,…,YilT,ϑΔx=ϑ1xT,ϑ2xT,…,ϑmxTT,ϑx=ϑi1,ϑi2,…,ϑilT.Letx belong to a compact set Ωx; the optimal parameter vector θΔ*T can be defined as
(27)θΔ*T=argmin⁡θ^ΔTsup⁡x∈ΩxΔx-Δ^x∣θ^ΔT,
where the optimal parameter matrix θΔ*T lies in a convex region given by
(28)Mθ=θ^Δ∣θ^Δ≤mθ,
where · is the Frobenius norm and mθ is a design parameter with mθ>0.Consequently, the composite disturbancesΔ(x) can be written as
(29)Δx=θΔ*TϑΔx+ε,
where ε is the smallest approximation error of the fuzzy system. Apparently, the norm of θΔ*T is bounded with
(30)ε≤ε-,
where ε- is the upper bound of the approximation error ε.Through adjusting the parameter vectorθ^ online, the composite disturbances Δ(x) can be approximated by the fuzzy system; hence, the performance of the control system with composite disturbances can be improved. As the performance of the control system is closely related to the selected fuzzy system, the control precision will be reduced when the approximation ability of the selected fuzzy system is dissatisfactory. In order to improve the approximation ability, a hyperbolic tangent function is integrated with the fuzzy disturbance observer to compensate for the approximate error.


## 3.2.2. Fuzzy Disturbance Observer

Consider the following dynamic system:(31)μ˙=-χμ+px,u,θ^Δ,
where p(x,u,θ^Δ)=χyρ-1+αx+βxu+Δ^(x∣θ^ΔT), χ>0 is a design parameter, and Δ^(x∣θ^ΔT)=θ^ΔTϑΔ(x) is utilized to compensate for the composite disturbances.Define the disturbance observation errorζ=y(ρ-1)-μ; invoking (9) and (31) yields
(32)ζ˙=αx+βxu+Δx+σμ-σyρ-1-αx-βxu-Δ^x∣θ^ΔT=-σζ+θ~ΔTϑΔx+ε,
where θ~Δ=θΔ*-θ^Δ is the adjustable parameter error vector.Then, the FDO-PSMC is proposed as(33)u=-T-1βx-1·st+Tαx+θ^ΔTϑΔx-ycρ+z.Invoking (16) and (33), we have
(34)s˙=yρ-ycρ+z=αx+βxu+Δx-ycρ+z=-T-1s+θ~ΔTϑΔx+ε.Invoking (32) and (34), the augmented system is obtained as
(35)ς˙=Aς+Bθ~ΔTϑΔx+ε,
where ς=(ζT,sT)T,A=diag⁡(-χIm,-T-1Im), and B=(Im,Im)T.Theorem 10.
Assume that the disturbance of (9) is monitored by the system (32), and the system (9) is controlled by (33). If the adjustable parameter vector of FDO is tuned by (36), then the augmented error ς is uniformly ultimately bounded within an arbitrarily small region:
(36)θ^˙Δ=ProjλϑΔxςTB=λϑΔxςT-IθλςTBθ^ΔTϑΔxθ^Δ2θ^Δ,
where
(37)Iθ=0,ifθ^Δ<mθ,orθ^Δ=mθ,ςTBθ^ΔTϑΔx≤01,ifθ^Δ=mθ,ςTBθ^ΔTϑΔx>0.Proof.
Consider the Lyapunov function candidate:(38)V=12ςTς+12λθ~ΔTθ~Δ.
Invoking (35) and (36), the time derivative of V is
(39)V˙=ςTς˙+1λθ~ΔTθ~˙Δ=ζTζ˙+sTs˙+1λθ~ΔTθ~˙Δ=ζT-χζ+θ~ΔTϑΔx+ε+sT-T-1s+θ~ΔTϑΔx+εM-1λθ~ΔTProjλφΔxςTB+1ηϑ~ϑ~˙=ζT-χζ+θ~ΔTϑΔx+ε+sT-T-1s+θ~ΔTϑΔx+εM-θ~ΔTϑΔxςT+θ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ.
Considering the fact(40)θ~ΔTθ^Δ=θΔ*Tθ^Δ-θ^ΔTθ^Δ≤12θΔ*2+θ^Δ2-θ^Δ2≤12θΔ*2-mθ2≤0Mwhenθ^Δ=mθ
and invoking (37), we have
(41)θ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ=0orWθ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ<0.
Consequently, we obtain(42)V˙≤ζT-χζ+θ~ΔTφΔx+εM+sT-T-1s+θ~ΔTφΔx+ε-θ~ΔTφΔxςT≤ζT-χζ+θ~ΔTφΔx+ε+sT-T-1s+θ~ΔTφΔx+εM-θ~ΔTφΔxζ-θ~ΔTφΔxs≤ζT-χζ+ε+sT-T-1s+ε≤-χζ2+ζTε-T-1s2+sTε≤-χ2ζ2+12χε2-12Ts2+T2ε2.
Ifζ>ε-/χ or s>Tε-, V˙<0. Therefore, the augmented error ς is uniformly ultimately bounded.Remark 11.
In order to make sure that the FDOΔ^(x∣θ^ΔT) outputs zero signal in case of the composite disturbances D=0, the consequent parameters should be set to be 0:
(43)θ^Δ0=0.Remark 12.
A projection operator (36) is used in the tuning method of the adjustable parameter vector; then, θ^Δ can be guaranteed to be bounded [29].Remark 13.
The adjustable parameter vector of the FDO (36) includes the observation error ζ and the sliding surface s. In view of the fact that the sliding surface s which is the linearization combination of the tracking error e is Hurwitz stable, the observation error ζ and tracking error e are both uniformly ultimately bounded, which ensures stability of the control system.


## 3.3. Improved Fuzzy Disturbance Observer

Lemma 14 (see [30]).
For allc>0 and w∈Rm, the following inequality holds:
(44)0<w-wTtanhwc≤mκc,
where κ is a constant such that κ=e-(κ+1); that is, κ=0.2785.The hyperbolic tangent function is incorporated to compensate for the approximate error and improve the precision of FDO. Consider the following dynamic system:(45)μ˙=-χμ+px,u,θ^Δ,
where p(x,u,θ^Δ)=σy(ρ-1)+αx+βxu+Δ^(x∣θ^ΔT)-υ^tanh(ζ/c), χ>0 is a design parameter, and Δ^(x∣θ^ΔT)=θ^ΔTϑΔ(x) is utilized to compensate for the composite disturbances.Define the disturbance observation errorζ=y(ρ-1)-μ; invoking (9) and (45) yields
(46)ζ˙=αx+βxu+Δx+σμ-σyρ-1-αx-βxu-Δ^x∣θ^ΔT=-χζ+θ~ΔTϑΔx+ε-υ^tanhζc,
where θ~Δ=θΔ*-θ^Δ is an adjustable parameter vector.The IFDO-PSMC is proposed as(47)u=-T-1βx-1M·st+Tstcαx+θ^ΔTϑΔxMMMMMM0-ycρ+z+υ^tanhstc.Invoking (16) and (47), we have
(48)s˙=yρ-ycρ+z=αx+βxu+Δx-ycρ+z=-T-1s+θ~ΔTϑΔx+ε-υ^tanhsc.Invoking (46) and (48), the augmented system is obtained as
(49)ς˙=Aς+Bθ~ΔTϑΔx+ευ^tanhςTBc,
where ς=(ζT,sT)T,  A=diag⁡(-χIm,-T-1Im), and B=(Im,Im)T.Theorem 15.
Assume that the disturbance of the system (9) is monitored by the system (49), and the system (9) is controlled by (47). If the adjustable parameter vector of IFDO is tuned by (50) and the approximate error compensation parameter is tuned by (51), the augmented error ς is uniformly ultimately bounded within an arbitrarily small region:
(50)θ^˙Δ=ProjλϑΔxςTB=λϑΔxςT-IθλςTBθ^ΔTϑΔxθ^Δ2θ^Δ,(51)υ^˙=ηςTB,
where
(52)Iθ=0,ifθ^Δ<mθ,orθ^Δ=mθ,ςTBθ^ΔTϑΔx≤01,ifθ^Δ=mθ,ςTBθ^ΔTϑΔx>0.Proof.
Consider the Lyapunov function candidate:(53)V=12ςTς+12λθ~ΔTθ~Δ+12ηυ~2,
where υ~=υ^-ε-. Invoking (49), (50), and (51), the time derivative of V is
(54)V˙=ςTς˙+1λθ~ΔTθ~˙Δ+1ηυ~υ~˙=ζTζ˙+sTs˙+1λθ~ΔTθ~˙Δ+1ηυ~υ~˙=ζT-χζ+θ~ΔTϑΔx+ε-υ^tanhζcM+sT-T-1s+θ~ΔTϑΔx+εM-υ^tanhsc-1λθ~ΔTProjλϑΔxςTB+1ηυ~υ~˙=ζT-χζ+θ~ΔTϑΔx+ε-υ^tanhζcM+sT-T-1s+θ~ΔTϑΔx+εM-υ^tanhsc-θ~ΔTϑΔxςTM+θ~ΔTIθςTBθ^ΔTϑΔxθ^Δ2θ^Δ+1ηυ~υ~˙≤ζT-χζ+θ~ΔTϑΔx+ε-υ^tanhζcM+sT-T-1s+θ~ΔTϑΔx+ε-υ^tanhscM-θ~ΔTϑΔxςT+1ηυ~υ~˙≤ζT-χζ+θ~ΔTϑΔx+ε-υ^tanhζcM+sT-T-1s+θ~ΔTϑΔx+ε-υ^tanhscM-θ~ΔTϑΔxζ-θ~ΔTϑΔxs+1ηυ~υ~˙≤ζT-χζ+ε+sT-T-1s+ε-ζTυ^tanhζcM-sTυ^tanhsc+υ~ζ+υ~s.According to Lemma14, we have
(55)-ζTυ^tanhζc≤-υ^ζ+mκc,-sTυ^tanhsc≤-υ^s+mκc.Consequently, we obtain(56)V˙≤-σζ2+ζTε-T-1s2+sTε-υ^ζM+mκc-υ^s+mκc+υ~ζ+υ~s≤-σζ2+ζε--T-1s2+sε--υ^ζM+υ~ζ-υ^ζ+υ~s+2mκc≤-σζ2-T-1s2+2mκc.Ifζ>2mκc/χ or s>2mκcT, V˙<0. Therefore, the augmented error ς is uniformly ultimately bounded.Remark 16.
The composite disturbances can be approximated effectively by adjusting the parameter law (50) and (51). If the fuzzy system approximates the composite disturbances accurately, the approximate error compensation will have small effect on compensating the approximate error, and vice versa. Based on this, the approximate precision of FDO can be improved through overall consideration.


## 4. Design of Attitude Control of Hypersonic Vehicle

### 4.1. Control Strategy

The structure of the IFDO-PSMC flight control system is shown in Figure1. Considering the strong coupled, nonlinear, and uncertain features of the HV, predictive sliding mode control approach which has distinct merits is applied to the design of the nominal flight control system. To further improve the performance of flight control system, an improved fuzzy disturbance observer is proposed to estimate the composite disturbances. The adjustable parameter law of IFDO can be designed based on Lyapunov theorem to approximate the composite disturbances online. According to the estimated composite disturbances, the compensation controller can be designed and integrated with the nominal controller to track the guidance commands precisely. In order to improve the quality of flight control system, three same command filters are designed to smooth the changing of the guidance commands in three channels. Their transfer functions have the same form as
(57)xrxc=2x+2,
where xr and xc are the reference model states and the external input guidance commands, respectively.Figure 1
IFDO-PSMC based flight control system.


### 4.2. Attitude Controller Design

The states of the attitude model (2) are pitch angle φ, yaw angle ψ, rolling angle γ, pitch rate ωx, yaw rate ωy, and roll rate ωz, and the outputs are selected as pitch angle φ, yaw angle ψ, and rolling angle γ, which can be written as Θ=φ,ψ,γT and ω=ωx,ωy,ωzT, and the output states are selected as y=φ,ψ,γT; then, the attitude model (2) can be rewritten as
(58)Θ˙=Fθωω˙=J-1Fω+J-1u,
where
(59)Fθ=0sinγcos⁡γ0cos⁡γsecφ-sinγsecφ1-tanφcos⁡γtanφsinγ,u=MxMyMz,Fω=Jy-Jzωzωy-J˙xωxJz-Jxωxωz-J˙yωyJx-Jyωxωy-J˙zωz,J=Jx000Jy000Jz.According to the PSMC approach, (58) can be derived as
(60)y¨=αx+βxu,
where
(61)αxM=0sinγcos⁡γ0cos⁡γsecφ-sinγsecφ1-tanφcos⁡γtanφsinγMM·Jx-1000Jy-1000Jz-1MM·Jy-Jzωzωy-J˙xωxJz-Jxωxωz-J˙yωyJx-Jyωxωy-J˙zωzMM+γ˙ωycos⁡γ-ωzsinγφ˙ωycos⁡γ-ωzsinγtanφsecφM-γ˙ωysinγ+ωzcos⁡γsecφγ˙ωysinγ+ωzcos⁡γtanφM-φ˙ωycos⁡γ-ωzsinγsec2φ,βx=0sinγcos⁡γ0cos⁡γsecφ-sinγsecφ1-tanφcos⁡γtanφsinγJx-1000Jy-1000Jz-1.Then, the sliding surface can be selected as(62)st=e˙+Ke,
where e=y-yc is the attitude tracking error vector, yc is attitude command vector φc,ψc,γcT transformed from the input attitude command vector αc,βc,γVcT, and K is the parameter matrix which must make the polynomial (62) Hurwitz stable. Define z as
(63)z=Ke˙.
In addition,
(64)Ycρ=φ¨c,ψ¨c,γ¨cT.Substituting (61), (62), (63), and (64) into (20), the nominal predictive sliding mode flight controller which omits the composite disturbances Δ(x) is proposed as
(65)u0=-T-1βx-1st-Tαx-Ycρ+z.In order to improve performance of the flight control system, the adaptive compensation controller based on IFDO is designed as(66)u1=-T-1βx-1Δ^x,
where Δ^(x) is the approximate composite disturbances for Δ(x).Accordingly, a fuzzy system can be designed to approximate the composite disturbancesΔ(x). First, each state is divided into five fuzzy sets, namely, A1i (negative big), A2i (negative small), A3i (zero), A4i (positive small), and A5i (positive big), where i=1,2,3 represent φ, ψ, and γ. Based on the Gaussian membership function, 25 if-then fuzzy rules are used to approximate the composite disturbances δ^1, δ^2, and δ^3, respectively.Considering the adaptive parameter law (50) and (51), the approximate composite disturbances can be expressed as
(67)Δ^x=δ^1,δ^2,δ^3T=θ^ΔTϑΔx,
where θ^ΔT=diag⁡{θ^1T,θ^2T,θ^3T} is the adjustable parameter vector and ϑΔ(x)=ϑ1(x)T,ϑ2(x)T,ϑ3(x)TT is a fuzzy basis function vector related to the membership functions; hence, the HV attitude control based on the IFDO can be written as
(68)u=u0+u1.Theorem 17.
Assume that the HV attitude model satisfies Assumptions3–6 and the sliding surface is selected as (62), which is Hurwitz stable. Moreover, assume that the composite disturbances of the attitude model are monitored by the system (49) and the system is controlled by (68). If the adjustable parameter vector of IFDO is tuned by (50) and the approximate error compensation parameter is tuned by (51), then the attitude tracking error and the disturbance observation error are uniformly ultimately bounded within an arbitrarily small region.Remark 18.
Theorem17 can be proved similarly as Theorem 15. Herein, it is omitted.Remark 19.
By means of the designed IFDO-PSMC system (68), the proposed HV attitude control system can not only track the guidance commands precisely with strong robustness but also guarantee the stability of the system.


### 4.3. Actuator Dynamics

The HV attitude system tracks the guidance commands by controlling actuator to generate deflection moments; then, the HV is driven to change the attitude angles. However, there exist actuator dynamics to generate the input momentsMx, My, and Mz, which are not considered in most of the existing literature. In order to make the study in accordance with the actual case, consider the actuator model expressed as
(69)xcsxcds=ωn2s2+2ζnωn+ωn2SAsSvs,
where ζn and ωn are the damping and bandwidth, respectively, SA(s) is the deflection angle limiter, and Sv(s) is the deflection rate limiter. Figure 2 shows the actuator model, where xc0, xc, and x˙c are the deflection angle command, virtual deflection angle, and deflection rate, respectively.Figure 2
Actuator model.


## 4.1. Control Strategy

The structure of the IFDO-PSMC flight control system is shown in Figure1. Considering the strong coupled, nonlinear, and uncertain features of the HV, predictive sliding mode control approach which has distinct merits is applied to the design of the nominal flight control system. To further improve the performance of flight control system, an improved fuzzy disturbance observer is proposed to estimate the composite disturbances. The adjustable parameter law of IFDO can be designed based on Lyapunov theorem to approximate the composite disturbances online. According to the estimated composite disturbances, the compensation controller can be designed and integrated with the nominal controller to track the guidance commands precisely. In order to improve the quality of flight control system, three same command filters are designed to smooth the changing of the guidance commands in three channels. Their transfer functions have the same form as
(57)xrxc=2x+2,
where xr and xc are the reference model states and the external input guidance commands, respectively.Figure 1
IFDO-PSMC based flight control system.


## 4.2. Attitude Controller Design

The states of the attitude model (2) are pitch angle φ, yaw angle ψ, rolling angle γ, pitch rate ωx, yaw rate ωy, and roll rate ωz, and the outputs are selected as pitch angle φ, yaw angle ψ, and rolling angle γ, which can be written as Θ=φ,ψ,γT and ω=ωx,ωy,ωzT, and the output states are selected as y=φ,ψ,γT; then, the attitude model (2) can be rewritten as
(58)Θ˙=Fθωω˙=J-1Fω+J-1u,
where
(59)Fθ=0sinγcos⁡γ0cos⁡γsecφ-sinγsecφ1-tanφcos⁡γtanφsinγ,u=MxMyMz,Fω=Jy-Jzωzωy-J˙xωxJz-Jxωxωz-J˙yωyJx-Jyωxωy-J˙zωz,J=Jx000Jy000Jz.According to the PSMC approach, (58) can be derived as
(60)y¨=αx+βxu,
where
(61)αxM=0sinγcos⁡γ0cos⁡γsecφ-sinγsecφ1-tanφcos⁡γtanφsinγMM·Jx-1000Jy-1000Jz-1MM·Jy-Jzωzωy-J˙xωxJz-Jxωxωz-J˙yωyJx-Jyωxωy-J˙zωzMM+γ˙ωycos⁡γ-ωzsinγφ˙ωycos⁡γ-ωzsinγtanφsecφM-γ˙ωysinγ+ωzcos⁡γsecφγ˙ωysinγ+ωzcos⁡γtanφM-φ˙ωycos⁡γ-ωzsinγsec2φ,βx=0sinγcos⁡γ0cos⁡γsecφ-sinγsecφ1-tanφcos⁡γtanφsinγJx-1000Jy-1000Jz-1.Then, the sliding surface can be selected as(62)st=e˙+Ke,
where e=y-yc is the attitude tracking error vector, yc is attitude command vector φc,ψc,γcT transformed from the input attitude command vector αc,βc,γVcT, and K is the parameter matrix which must make the polynomial (62) Hurwitz stable. Define z as
(63)z=Ke˙.
In addition,
(64)Ycρ=φ¨c,ψ¨c,γ¨cT.Substituting (61), (62), (63), and (64) into (20), the nominal predictive sliding mode flight controller which omits the composite disturbances Δ(x) is proposed as
(65)u0=-T-1βx-1st-Tαx-Ycρ+z.In order to improve performance of the flight control system, the adaptive compensation controller based on IFDO is designed as(66)u1=-T-1βx-1Δ^x,
where Δ^(x) is the approximate composite disturbances for Δ(x).Accordingly, a fuzzy system can be designed to approximate the composite disturbancesΔ(x). First, each state is divided into five fuzzy sets, namely, A1i (negative big), A2i (negative small), A3i (zero), A4i (positive small), and A5i (positive big), where i=1,2,3 represent φ, ψ, and γ. Based on the Gaussian membership function, 25 if-then fuzzy rules are used to approximate the composite disturbances δ^1, δ^2, and δ^3, respectively.Considering the adaptive parameter law (50) and (51), the approximate composite disturbances can be expressed as
(67)Δ^x=δ^1,δ^2,δ^3T=θ^ΔTϑΔx,
where θ^ΔT=diag⁡{θ^1T,θ^2T,θ^3T} is the adjustable parameter vector and ϑΔ(x)=ϑ1(x)T,ϑ2(x)T,ϑ3(x)TT is a fuzzy basis function vector related to the membership functions; hence, the HV attitude control based on the IFDO can be written as
(68)u=u0+u1.Theorem 17.
Assume that the HV attitude model satisfies Assumptions3–6 and the sliding surface is selected as (62), which is Hurwitz stable. Moreover, assume that the composite disturbances of the attitude model are monitored by the system (49) and the system is controlled by (68). If the adjustable parameter vector of IFDO is tuned by (50) and the approximate error compensation parameter is tuned by (51), then the attitude tracking error and the disturbance observation error are uniformly ultimately bounded within an arbitrarily small region.Remark 18.
Theorem17 can be proved similarly as Theorem 15. Herein, it is omitted.Remark 19.
By means of the designed IFDO-PSMC system (68), the proposed HV attitude control system can not only track the guidance commands precisely with strong robustness but also guarantee the stability of the system.


## 4.3. Actuator Dynamics

The HV attitude system tracks the guidance commands by controlling actuator to generate deflection moments; then, the HV is driven to change the attitude angles. However, there exist actuator dynamics to generate the input momentsMx, My, and Mz, which are not considered in most of the existing literature. In order to make the study in accordance with the actual case, consider the actuator model expressed as
(69)xcsxcds=ωn2s2+2ζnωn+ωn2SAsSvs,
where ζn and ωn are the damping and bandwidth, respectively, SA(s) is the deflection angle limiter, and Sv(s) is the deflection rate limiter. Figure 2 shows the actuator model, where xc0, xc, and x˙c are the deflection angle command, virtual deflection angle, and deflection rate, respectively.Figure 2
Actuator model.


## 5. Simulation Results Analysis

To validate the designed HV attitude control system, simulation studies are conducted to track the guidance commandsαc,βc,γVcT. Assume that the HV flight lies in the cruise phase with the flight altitude 33.5 km and the flight mach number 15. The initial attitude and attitude angular velocity conditions are arbitrarily chosen as α0=β0=γV0=0 and ωx0=ωy0=ωz0=0; the initial flight path angle and trajectory angle are θ0=σ0=0; the guidance commands are chosen as αc=5°, βc=0°, and γVc=5°, respectively.The dampingζn and bandwidth ωn of the actuator are 0.707 and 100 rad/s, respectively; the deflection angle limit is [−20, 20]°; the deflection rate limit is [−50, 50]°/s.To exhibit the superiority of the proposed scheme, the variation of the aerodynamic coefficients, aerodynamic moment coefficients, atmosphere density, thrust coefficients, and moments of inertia is assumed to be −50%, −50%, +50%, −30%, and −10%, which are much more rigorous than those considered in [18]. On the other hand, unknown external disturbance moments imposed on the HV are expressed as
(70)d1t=105sin2tN·md2t=2×105sin2tN·md3t=106sin2tN·m.The simulation time is set to be 20 s and the update step of the controller is 0.01 s. The predictive sliding mode controller design parameters are chosen asT=0.1 and K=diag⁡(2,50,4); the IFDO design parameters are chosen as λ=275, χ=15, η=95, and c=0.25. In addition, the membership functions of the fuzzy system are chosen as Gaussian functions expressed as (71). The simulation results are shown in Figures 3, 4, and 5. Consider
(71)μAj1xj=exp⁡-xj+π/6π/242,μAj2xj=exp⁡-xj+π/12π/242,μAj3xj=exp⁡-xjπ/242,μAj4xj=exp⁡-xj-π/12π/242,μAj5xj=exp⁡-xj-π/6π/242.Comparison of tracking curves and control inputs under SMC and PSMC in nominal condition.
(a)
(b)
(c)
(d)
(e)
(f)Comparison of tracking curves and control inputs under SMC and PSMC in the presence of system uncertainties.
(a)
(b)
(c)
(d)
(e)
(f)Comparison of tracking curves under PSMC, FDO-PSMC, and IFDO-PSMC in the presence of system uncertainties and external disturbances.
(a)
(b)
(c)
(d)
(e)
(f)Figure3 presents the simulation results under PSMC and SMC without considering system uncertainties and external disturbance. Figures 3(a)~3(c) show the results of tracking the guidance commands under PSMC and SMC. As shown, the guidance commands of PSMC and SMC can be tracked both quickly and precisely. Figures 3(d)~3(f) show the control inputs under PSMC and SMC. It can be observed that there exists distinct chattering of SMC while the results of PSMC are smooth owing to the outstanding optimization performance of the predictive control. The simulation results in Figure 3 indicate that the PSMC is more effective than SMC for system without considering system uncertainties and external disturbances.Figure4 shows the simulation results under PSMC and SMC with considering system uncertainties, where the results are denoted as those in Figure 4. From Figures 4(a)~4(c), it can be seen that the flight control system adopted SMC and PSMC can track the commands well in spite of system uncertainties which proves strong robustness of SMC. Meanwhile, it is obvious that SMC takes more time to achieve convergence and the control inputs become larger than that in Figure 4. However, PSMC can still perform as well as that in Figure 3 with the settling time increasing slightly and the control inputs almost remain the same. In accordance with Figures 4(d)~4(f), we can observe that distinct chattering is also caused by SMC while the results of PSMC are smooth. It can be summarized that PSMC is more robust and more accurate than SMC for system with uncertain model.Figure5 gives the simulation results under PSMC, FDO-PSMC, and IFDO-PSMC in consideration of system uncertainties and external disturbances. Figures 5(a)~5(c) show the results of tracking the guidance commands. It can be observed that PSMC cannot track the guidance commands in the presence of big external disturbances while FDO-PSMC and IFDO-PSMC can both achieve satisfactory performance, which verifies the effectiveness of FDO in suppressing the influence of system uncertainties and external disturbances. Figures 5(d)~5(f) are the partial enlarged detail corresponding to Figures 5(a)~5(c). As shown, the IFDO-PSMC can track the guidance commands more precisely, which declares stronger disturbance approximate ability of IFDO when compared with FDO. To sum up, the results of Figure 5 indicate that IFDO is effective in dealing with system uncertainties and external disturbances. In addition, the proposed IFDO-PSMC is applicative for the HV which has rigorous system uncertainties and strong external disturbances.It can be concluded from the above-mentioned simulation analysis that the proposed IFDO-PSMC flight control scheme is valid.


## 6. Conclusions

An effective control scheme based on PSMC and IFDO is proposed for the HV with high coupling, serious nonlinearity, strong uncertainty, unknown disturbance, and actuator dynamics. PSMC takes merits of the strong robustness of sliding model control and the outstanding optimization performance of predictive control, which seems to be a very promising candidate for HV control system design. PSMC and FDO are combined to solve the system uncertainties and external disturbance problems. Furthermore, we improve the FDO by incorporating a hyperbolic tangent function with FDO. Simulation results show that IFDO can better approximate the disturbance than FDO and can assure the control performance of HV attitude control system in the presence of rigorous system uncertainties and strong external disturbances.



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*Source: 727162-2015-02-22.xml*