# Nonlinear Disturbance-Observer-Based Sliding Mode Control for Flexible Air-Breathing Hypersonic Vehicles

Authors: Na Wang; Xiu-Ming Yao; Wen-Shuo Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper investigates a tracking problem for flexible air-breathing hypersonic
vehicles (FAHVs) with composite disturbance. The composite disturbance produced
by flexible effects, parameter uncertainties, and external interferences is modeled
as a kind of unknown derivative-bounded disturbance in this paper. Then a novel composite
control strategy is presented for the nonlinear FAHV model with the composite
disturbance, which combines a nonlinear disturbance-observer-based compensator (NDOBC)
and a dynamic-inversion-based sliding mode controller (DIBSMC). Specifically,
the NDOBC is constructed to estimate and compensate for the composite disturbance,
and the DIBSMC is designed to track desired trajectories of velocity and flight path
angle. Moreover, the uniformly ultimate boundedness of the composite system can
be guaranteed by using Lyapunov theory. Finally, simulation results on a full nonlinear
model of FAHVs demonstrate that the proposed nonlinear disturbance-observer-based
sliding mode controller is more effective than the traditional DIBSMC. Specifically, it is
shown that the chattering of traditional DIBSMC in presence of composite disturbances
can be attenuated with the NDOBC.


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## Body

## 1. Introduction

Flight control of air-breathing hypersonic vehicles (AHVs) is very important and difficult. AHVs use scramjet engines integrated with the airframe, so the vehicle dynamics display strong interactions between the elastic airframe, the propulsion system, and the structural dynamics (see [1–3] and references therein). In addition, significant flexible effects cannot be neglected in the control design due to the slender geometries and light structures of AHVs (see [4–6] and references therein). It has been shown that these strong couplings, undesired flexible effects, large parametric uncertainties, and various external disturbances may cause degradation of the performance of flight control systems. Thus, the desired control scheme should be robust enough to overcome these factors.In the past few years, many effective robust control laws have been presented (see, e.g., [7–13]) for a nonlinear AHV model developed at NASA Langley Research Center. For example, an observer-based passive fault-tolerant control scheme was successfully applied for this AHV model with both parameter uncertainty and actuator faults in [13]. But only rigid modes were considered in the nonlinear AHV model. Thus a new FAHV model was developed in [6], which includes not only the interactions between the propulsion system and the airframe dynamics but also the strong flexible effects. After that, many nonlinear controllers have been applied effectively to this nonlinear model developed by Parker et al. and Fiorentini et al. (see, e.g., [14–18]). An effective adaptive sliding mode controller was designed for the linearized FAHV model in [15]. But the capability of the linear model to represent the dynamics and the coupling effects is limited. So nonlinear control methods were concerned for the nonlinear FAHV model in [16, 17]. However, it was based on control-design models without any consideration of rigid-flexible coupling dynamics. Due to these couplings, rigid-body state and control input will be affected by flexible modes. In fact, it has been shown that the undesired flexible effects may cause degradation to the performance of flight control systems (see [6, 18]). Therefore, the flexible effects were studied and transformed into elastic-mode-related terms that appeared in forces and moment in [18]. However, it may lead to a large amount of online calculations because many coefficients have to be estimated using adaptive methods. To avoid these problems, a novel nonlinear disturbance-observer-based sliding mode control strategy was proposed to achieve control objective and attenuate the composite disturbance produced by flexible effects, parameter uncertainties, and external interferences in this paper.Recently, the disturbance-observer-based control (DOBC) method has attracted considerable attention as a robust control scheme. Many effective DOBC schemes have been developed for spacecraft, missiles, and hypersonic vehicles (see, e.g., [19–23]). Chen firstly designed a dynamic inversion controller based on disturbance observer for missile systems in [20]. An antidisturbance PD control scheme was successfully applied for the attitude control of flexible spacecrafts in [21]. Chen et al. proposed a robust control law using disturbance observer and neural networks for the linearized AHV model in [22]. Li et al. presented a composite controller based on disturbance observer for the nonlinear AHV model in [23]. However, the study for the nonlinear FAHV model with composite disturbance using DOBC strategy is still insufficient. In this paper, a novel composite control strategy, which combines a nonlinear disturbance-observer-based compensator (NDOBC) and a dynamic-inversion-based sliding mode controller (DIBSMC), is presented for the nonlinear FAHV model with composite disturbance.In conclusion, the main contributions in this paper can be summarized as follows. (1) The composite disturbance produced by flexible effects, parameter uncertainties, and external interferences is formulated as a kind of unknown disturbance, which is considered in the control-oriented model and estimated by a nonlinear disturbance observer. (2) Combining a NDOBC with a DIBSMC, a nonlinear disturbance-observer-based sliding mode controller (NDOBSMC) is proposed to make velocity and flight-path angle track desired signals and reject the composite disturbance. (3) The stability of composite closed-loop system which includes two tracking-error equations and a disturbance estimation error equation is analyzed.The remainder of this paper is organized as follows. In Section2, the nonlinear FAHV model with composite disturbance is introduced. In Section 3, the NDOBC and the NDOBSMC are designed, and stability of the composite system is analyzed. In Section 4, the effectiveness of the proposed NDOBSMC is confirmed through numerical simulations. Conclusions are provided in Section 5.


## 2. Problem Formulation

A nonlinear model for the longitudinal dynamics of flexible air-breathing hypersonic vehicles, as given by Bolender and Doman in [6], is described by the following nonlinear equations:(1)h˙=Vsinθ-α,V˙=Tcos⁡α-Dm-gsinθ-α,α˙=-L+TsinαmV+Q+gVcos⁡θ-α,IyyQ˙=M+ψ~1η¨1+ψ~2η¨2+d¯o,K1η¨1=-2ζ1ω1η˙1-ω12η1+N1-ψ~1MIyy-ψ~1ψ~2η¨2Iyy,K2η¨2=-2ζ2ω2η˙2-ω22η2+N2-ψ~2MIyy-ψ~1ψ~2η¨1Iyy,where ψ~1=∫-Lf0m^fϕf(ξ)dξ, ψ~2=∫0Lam^aϕa(ξ)dξ, K1=1+ψ~12/Iyy, and K2=1+ψ~22/Iyy, in which ϕf(ξ) and ϕa(ξ) are the mode shapes and m^f and m^a are mass densities. It is noted that a pitching disturbance moment, d¯o, is considered in this paper, which may originate from parameter uncertain, sensor noise, actuator error, and external disturbance.The approximations of the forces and moment given by Parker et al. in [17] are described as follows:(2)L=q¯SCLαα+CL0+CLδeδe,D=q¯SCDα2α2+CDαα+CD0+CDδe2δe2+CDδeδe,T=CTΦα3α3+CTΦα2α2+CTΦαα+CTΦ0Φ+CTα3α3+CTα2α2+CTαα+CT0,M=zTT+q¯Sc¯CMα2α2+CMαα+CM0+CMδeδe,N1=CN1α2α2+CN1αα+CN10,N2=CN2α2α2+CN2αα+CN20+CN2δeδe.Coefficients that appear in the forces and moment (i.e.,CL(·), CD(·), CM(·), CT(·), CTΦ(·), and CNi(·)) are subject to uncertainty in this paper. Denote CL0(·), CD0(·), CM0(·), CT0(·), CT0Φ(·), and CNi0(·) as nominal values of the coefficients. The uncertainty is modeled as an additive variable Δ to the nominal values used for control design; then the coefficients can be expressed as CL(·)=CL0(·)(1+ΔCL(·)), CD(·)=CD0(·)(1+ΔCD(·)), CM(·)=CM0(·)(1+ΔCM(·)), CT(·)=CT0(·)(1+ΔCT(·)), CTΦ(·)=CT0Φ(·)(1+ΔCTΦ(·)), and CNi(·)=CNi0(·)(1+ΔCNi(·)). The forces and moment are given by L=L0+ΔL, D=D0+ΔD, T=T0+ΔT, M=M0+ΔM, and Ni=Ni0+ΔNi, where the detailed expressions of L0, D0, T0, Ni0, M0, ΔL, ΔL, ΔD, ΔT, ΔNi, and ΔM are given in the appendix.In this paper, the control design problem is to select control inputs,δe and Φ, that force the outputs, V and γ, to track commanded values, Vd and γd, in the presence of the composite disturbance produced by the couplings, uncertainties, and external interferences.


## 3. Composite Controller Design

This section presents a novel composite controller with a hierarchical architecture for the nonlinear FAHV model. Firstly, a nonlinear control-oriented model (COM) with composite disturbance is provided. Secondly, a nonlinear disturbance observer (NDO) based compensator (NDOBC) is constructed to estimate and compensate for the composite disturbance. Then a NDO-based sliding mode controller (NDOBSMC) is designed to track the desired trajectories in the presence of composite disturbance. Finally, stability of the composite system is analyzed.


### 3.1. COM with Composite Disturbance

In this part, a novel COM with composite disturbance is obtained from the FAHV nonlinear model (1) by removing the altitude dynamic and flexible modes and setting the weak elevator couplings to zero. The new COM is given by the following set of equations:(3)V˙=T0cos⁡α-D¯0m-gsinγ,α˙=-L¯0+T0sinαmV+Q+gVcos⁡γ,γ˙=L¯0+T0sinαmV-gVcos⁡γ,I¯yyQ˙=M0+ψ~1N10+ψ~2N20+d¯,Φ¨=-2ζωΦ˙-ω2Φ+ω2Φc,where I¯yy=Iyy-ψ~12-ψ~22, L¯0=q¯S(CL0αα+CL00), and D¯0=q¯S(CD0α2α2+CD0αα+CD00). Besides, the second-order dynamic equation for fuel equivalence ratio, Φ, is added to obtain the COM with full vector relative degree. In addition, the composite disturbance, d¯, is considered in this COM, which contains d¯o and a pitching disturbance moment, d¯f, produced by the couplings between flexible and rigid modes. Denote d=d¯/I¯yy in order to simplify the following design of disturbance observer.Remark 1.
It is different from [17] that the composite disturbance, d¯, produced by parameter uncertainties, external disturbances, and couplings between rigid and flexible modes is considered in the proposed COM.


### 3.2. NDOBC Design

The nonlinear COM described by (3) is a special case of the generic muti-input/multioutput (MIMO) nonlinear system:(4)x˙=fx+g1xu+g2xd,y=hx,where state x=VαγQΦΦ˙T, control input u=δeΦcT, output y=VγT, and composite disturbance d=d¯/I¯yy. Nonlinear functions f, g1, and g2 are sufficiently smooth with respect to x∈R6, and the concrete forms of f, g1, and g2 are given by(5)fx=f1xf2xf3xf4xf5xf6xT,g1x=g11xg12x,g2x=000100T,in which(6)f1x=T0cos⁡α-D¯0m-gsinγ,f2x=-L¯0+T0sinαmV+Q+gVcos⁡γ,f3x=L¯0+T0sinαmV-gVcos⁡γ,f4x=zTT0+q¯Sc¯CM0α2α2+CM0αα+CM00+ψ~1N10+ψ~2N¯20I¯yy,f5x=Φ˙,f6x=-2ζωΦ˙-ω2Φ,g11x=000q¯Sc¯CM0δe+ψ~2CN20δeI¯yy00T,g12x=00000ω2T,N¯20=N20-CN20δeδe.For the MIMO nonlinear system (4), a NDO is designed to estimate the unknown composite disturbance d according to [20], which is given by(7)d^=z+px,z˙=-lxg2xz-lxg2xpx+fx+g1xu,where d^ and z are the estimate of the unknown disturbance and the internal state of the nonlinear observer, respectively. Besides, p(x) is a nonlinear function to be designed and the nonlinear observer gain l(x) is defined by(8)lx=∂px∂x=l1xl2xl3xl4xl5xl6x.Define the estimation error by ed(t)=d-d^. The error dynamic is described as(9)e˙dt+lxg2xedt=d˙,where d˙ is the first derivative of composite disturbance d. It is assumed that |d˙|≤σ in this paper, where σ is a known positive constant.Substitutingg2(x) and l(x) into (9), the differential equation (9) can be rewritten as(10)e˙dt+l4xedt=d˙,where l4(x)=∂p(x)/∂Q. An appropriate fourth element of observer gain, l4(x), needs to be designed such that ed(t) converge to zero.The following result provides a design scheme for the disturbance observer (7) with guaranteed uniformly ultimate boundedness of system (9) or (10).Lemma 2 (for details see [24]).
LetD⊂Rn be a domain that contains the origin and let W:[0,∞)×D→R be a continuously differentiable function such that(11)α1χ≤Wt,χ≤α2χ,∂W∂t+∂W∂χξt,χ≤-Qχ,χ≥μ>0,for all t≥0 and for all χ∈D, where α1 and α2 are class K functions and Q(χ) is a continuous positive definite function. Then, there is T≥0 (dependent on χ(t0) and μ) such that the solution of the system(12)χ˙=ξt,χsatisfies(13)χt≤α1-1α2μ,∀t≥t0+T.Moreover, if D=Rn and α1 belongs class K∞, then (13) holds for any initial state χ(t0), with one restriction on how large μ is. Note that inequality (13) shows that χ(t) is uniformly ultimately bounded (UUB) with the ultimate bound α1-1(α2(μ)).Theorem 3.
For the estimation error system (10), if there exists a constant l0 satisfying(14)l0>121+M_,where the fourth element of observer gain, l4(x), is chosen as l0(1+M(x)) and M(x) is a bounded nonlinear function with respect to x satisfying 0≤M_≤M(x)≤M¯, then the solution of system (10) is UUB.Proof.
Denote the Lyapunov function(15)Wdt=ed2t.Computing the derivative of Wd(t) along the trajectories of (10), it can be obtained that (16)W˙dt=2edte˙dt=2edt-l01+Mxedt+d˙=-2l01+Mxed2t-ed2t-2edtd˙+d˙2+ed2t+d˙2≤-2l01+Mx-1ed2t+σ2.Thus it can be concluded from Lemma 2 that the solution of estimation error system (10) is uniformly ultimately bounded, if l0>1/[2(1+M_)]. Obviously, there exits a positive constant β such that (1/(1+β))ed2(t)≤Wd(t)≤(1+β)ed2(t). And W˙d(t) is a negative definite matrix when |ed(t)|≥σ>0 hold. According to Lemma 2, the ultimate bound of ed(t) is (1+β)σ; that is, edt≤(1+β)σ.Remark 4.
The NDO (7) reduces to a linear disturbance observer (LDO) if M(x)=0. Obviously, the performance of NDO is better than LDO when LDO takes the same constant l0 as NDO.


### 3.3. NDOBSMC Design

The nonlinear COM described by (3) can be linearized completely using the input/output linearization technique of full state feedback, so the linearized model is developed by repeated differentiation of V and γ as follows:(17)V⃛=σ1x¨1+x˙1TΣ2x˙1m,γ⃛=π1x¨1+x˙1TΠ2x˙1,where x1=[VαγΦ]T, σ1=∂f1(x)/∂x1, Σ2=∂σ1/∂x1, π1=∂f3(x)/∂x1, and Π2=∂π1/∂x1. The detailed expressions of σ1, Σ2, π1, and Π2 are given in the appendix.The right-hand sides of (17) involve second derivatives of α and Φ. The expression of the second derivatives for α and Φ consists of three parts: the first part is control relevant, the second part is disturbance relevant, and the third part is neither control relevant nor disturbance relevant. Consider(18)α¨=α¨0+q¯Sc¯CM0δe+ψ~2CN20δeI¯yy+d,Φ¨=Φ¨0+ω2Φc,where α¨0=[zTT0+q¯Sc¯(CM0α2α2+CM0αα+CM00)+ψ~1N10+ψ~2N¯20]/I¯yy-γ¨ and Φ¨0=-2ζωΦ˙-ω2Φ.Whenx10=Vα0γΦ0T is defined, the output dynamics of V and γ can be written as follows:(19)V⃛γ⃛=fVfγ+Bx1δeΦc+cVx1cγx1d+dVdγ,where control inputs, δe and Φc, and composite disturbance, d, appear explicitly. Disturbances, dV and dγ, are produced by parameter uncertainties. The concrete forms of fV, fγ, B(x1), cV(x1), and cγ(x1) are given by(20)fV=σ1x¨1+x˙1TΣ2x˙1m,fγ=π1x¨1+x˙1TΠ2x˙1,Bx1=b11x1b12x1b21x1b22x1,cVx1=1m∂T0∂αcos⁡α-T0sinα-∂D¯0∂α,cγx1=1mV∂T0∂αsinα+T0cos⁡α-∂L¯0∂α,in which(21)b11x1=q¯Sc¯CM0δe+ψ~2CN20δemIyy000000000×∂T0∂αcos⁡α-T0sinα-∂D¯0∂α,b12x1=ω2m∂T0∂Φcos⁡α,b21x1=q¯Sc¯CM0δe+ψ~2CN20δemVIyy000000000×∂T0∂αsinα+T0cos⁡α-∂L¯0∂α,b22x1=ω2mV∂T0∂Φsinα.Two decoupled sliding mode surfacessV(t) and sγ(t) are defined by sV(t)=(d/dt+λV)3∫0t‍eV(τ)dτ and sγ(t)=(d/dt+λγ)3∫0t‍eγ(τ)dτ, where eV(t)=V-Vd is tracking error of velocity, eγ(t)=γ-γd is tracking error of flight path angle, and λV and λγ are strictly positive constants defining the bandwidth of the tracking error dynamics. The sliding mode surfaces sV(t) and sγ(t) represent linear differential equations where their solutions ∫eV(τ) and ∫eγ(τ) converge to zero exponentially with the time constants 2/λV and 2/λγ, respectively.DifferentiatingsV(t) and sγ(t), we have(22)s˙Vts˙γt=υVυγ+fVfγ+Bx1δeΦc+cVx1cγx1d+dVdγ,where υV=-V⃛d+3λVe¨V(t)+3λV2e˙V(t)+λV3eV(t) and υγ=-γ⃛d+3λγe¨γ(t)+3λγ2e˙γ(t)+λγ3eγ(t).With (22), recall that estimation value of the disturbance d is developed by the NDO (7) and the NDOBSMC can be designed as(23)δeΦc=-B-1x1fVfγ-B-1x1υV+kVsgnsVtυγ+kγsgnsγt-B-1x1cVx1cγx1d^,where kV and kγ are sliding mode gains of the NDOBSMC (23), d^ is the disturbance estimation value obtained by the NDO (7), and B(x1) is assumed to be invertible.Remark 5.
When the output of disturbance observerd^=0, that is, there is no disturbance observer, the NDOBSMC (23) obviously reduces to a DIBSMC, which is written as(24)δeΦc=-B-1x1fVfγ-B-1x1υV+kVsgnsVtυγ+kγsgnsγt.As discussed in [25], the control laws (23) and (24) may result in control chattering because of the discontinuity across the sliding surfaces. As a practical matter, chattering is undesirable because it involves very high control action and may excite high frequency dynamics neglected in the modeling. The discontinuity in the control law can be dealt with by defining two thin boundary layers of widths ΦV and Φγ around the sliding mode surfaces, that is, replacing sgn(sV(t)) and sgn(sγ(t)) with continuous saturation functions sat⁡(sV(t)/ΦV) and sat⁡(sγ(t)/Φγ), where(25)sat⁡xs=1,ifxs>1,xs,ifxs≤1,-1,ifxs<-1.So the NDOBSMC (23) and the DIBSMC (24) are modified as follows:(26)δeΦc=-B-1x1fVfγ-B-1x1υV+kVsat⁡sVtΦVsVtΦVΦγΦγυγ+kγsat⁡sγtΦγ-B-1x1cVx1cγx1d^,(27)δeΦc=-B-1x1fVfγ-B-1x1υV+kVsat⁡sVtΦVsVtΦVΦγΦγυγ+kγsat⁡sγtΦγ.Considering (10) and substituting (26) into (22), the augmented closed-loop system can be obtained as follows:(28)s˙Vt=-kVsat⁡sVtΦV+cVx1edt+dV,s˙γt=-kγsat⁡sγtΦγ+cγx1edt+dγ,e˙dt=-l4xedt+d˙.Next, uniform ultimate boundedness (UUB) of the composite system (28) will be considered.Theorem 6.
For the composite system (28), if the sliding mode gains, kV and kγ, satisfy(29)kV>Max⁡0.25ΦVβV1-2σV12+βV2-2,1+βσσV1+σV2,kγ>Max⁡0.25Φγβγ1-2σγ12+βγ2-2,1+βσσγ1+σγ2and the fourth element of observer gain, l4(x), is chosen as l0(1+M(x)) with the constant l0 satisfying(30)l0>1+βV12+βγ1221+M_,then the solutions of composite system (28) are UUB, where positive constants βV1, βγ1, β, βV2, and βγ2 are tuning parameters to obtain the appropriate values of control gains, and positive constants σV1, σγ1, σ, σV2, and σγ2 are the upper bounds of cV(x1), cγ(x1), d˙, dV, and dγ; that is, |cV(x1)|≤σV3, |cγ(x1)|≤σγ3, |d˙|≤σ, |dV|≤σV2, and |dγ|≤σγ2.Proof.
A Lyapunov function candidate for the system (28) is chosen as W(t)=Wd(t)+Ws(t), where Wd(t)=ed2(t) and Ws(t)=1/2(sV2(t)+sγ2(t)). According to Theorem 3, W˙d(t)≤-[2l0(1+M(x))-1]ed2(t)+σ2.
Taking the derivative ofWs(t) along the trajectories of (28), we have (31)W˙st=sVts˙Vt+sγts˙γt=sVt-kVsat⁡sVtΦV+cVx1edt+dVsVtΦVΦγΦγ+sγt-kγsat⁡sγtΦγ+cγx1edt+dγ.When |sV(t)/ΦV|≤1 and |sγ(t)/Φγ|≤1, the derivative of Ws(t) can be expressed as(32)W˙st=-kVΦVsV2t+cVx1edtsVt+dVsVt-kγΦγsγ2t+cγx1edtsγt+dγsγt=-kVΦV-0.25βV1-2cV2x1-0.25βV2-2sV2t-kγΦγ-0.25βγ1-2cγ2x1-0.25βγ2-2sγ2t-0.5βV1-1cVx1sVt-βV1edt2-0.5βV2-1sVt-βV2dV2-0.5βγ1-1cγx1sγt-βγ1edt2-0.5βγ2-1sγt-βγ2dγ2+βV12+βγ12ed2t+βV22dV2+βγ22dγ2.Furthermore, the derivative of W(t) along the trajectories of (28) can be given by(33)W˙t≤-kVΦV-0.25βV1-2σV12-0.25βV2-2sV2t-kγΦγ-0.25βγ1-2σγ12-0.25βγ2-2sγ2t-2l01+Mx-1+βV12+βγ12ed2t+βV22σV22+βγ22σγ22+σ2.When |sV(t)/ΦV|>1 and |sγ(t)/Φγ|>1, the derivative of Ws(t) can be given by(34)W˙st=-kVsVt+cVx1edtsVt+dVsVt-kγsγt+cγx1edtsγt+dγsγt≤-kV-cVx1edt-dVsVt-kγ-cγx1edt-dγsγt.By now, the derivative of W(t) along the trajectories of (28) becomes(35)W˙t≤-kV-1+βσσV1-σV2sVt-kγ-1+βσσγ1-σγ2sγt-2l01+Mx-1ed2t+σ2.Thus it can be concluded from Lemma 2 that the solutions of the augmented closed-loop system (28) are uniformly ultimately bounded, if conditions (29) and (30) are satisfied.Remark 7.
The result of Theorem6 (l0>(1+βV2+βγ2)/[2(1+M_)]) is different from the result of Theorem 3 (l0>1/[2(1+M_)]) because of the couplings between tracking errors and disturbance estimate error. So it is necessary to analyze the stability of the composite system (28).


## 3.1. COM with Composite Disturbance

In this part, a novel COM with composite disturbance is obtained from the FAHV nonlinear model (1) by removing the altitude dynamic and flexible modes and setting the weak elevator couplings to zero. The new COM is given by the following set of equations:(3)V˙=T0cos⁡α-D¯0m-gsinγ,α˙=-L¯0+T0sinαmV+Q+gVcos⁡γ,γ˙=L¯0+T0sinαmV-gVcos⁡γ,I¯yyQ˙=M0+ψ~1N10+ψ~2N20+d¯,Φ¨=-2ζωΦ˙-ω2Φ+ω2Φc,where I¯yy=Iyy-ψ~12-ψ~22, L¯0=q¯S(CL0αα+CL00), and D¯0=q¯S(CD0α2α2+CD0αα+CD00). Besides, the second-order dynamic equation for fuel equivalence ratio, Φ, is added to obtain the COM with full vector relative degree. In addition, the composite disturbance, d¯, is considered in this COM, which contains d¯o and a pitching disturbance moment, d¯f, produced by the couplings between flexible and rigid modes. Denote d=d¯/I¯yy in order to simplify the following design of disturbance observer.Remark 1.
It is different from [17] that the composite disturbance, d¯, produced by parameter uncertainties, external disturbances, and couplings between rigid and flexible modes is considered in the proposed COM.


## 3.2. NDOBC Design

The nonlinear COM described by (3) is a special case of the generic muti-input/multioutput (MIMO) nonlinear system:(4)x˙=fx+g1xu+g2xd,y=hx,where state x=VαγQΦΦ˙T, control input u=δeΦcT, output y=VγT, and composite disturbance d=d¯/I¯yy. Nonlinear functions f, g1, and g2 are sufficiently smooth with respect to x∈R6, and the concrete forms of f, g1, and g2 are given by(5)fx=f1xf2xf3xf4xf5xf6xT,g1x=g11xg12x,g2x=000100T,in which(6)f1x=T0cos⁡α-D¯0m-gsinγ,f2x=-L¯0+T0sinαmV+Q+gVcos⁡γ,f3x=L¯0+T0sinαmV-gVcos⁡γ,f4x=zTT0+q¯Sc¯CM0α2α2+CM0αα+CM00+ψ~1N10+ψ~2N¯20I¯yy,f5x=Φ˙,f6x=-2ζωΦ˙-ω2Φ,g11x=000q¯Sc¯CM0δe+ψ~2CN20δeI¯yy00T,g12x=00000ω2T,N¯20=N20-CN20δeδe.For the MIMO nonlinear system (4), a NDO is designed to estimate the unknown composite disturbance d according to [20], which is given by(7)d^=z+px,z˙=-lxg2xz-lxg2xpx+fx+g1xu,where d^ and z are the estimate of the unknown disturbance and the internal state of the nonlinear observer, respectively. Besides, p(x) is a nonlinear function to be designed and the nonlinear observer gain l(x) is defined by(8)lx=∂px∂x=l1xl2xl3xl4xl5xl6x.Define the estimation error by ed(t)=d-d^. The error dynamic is described as(9)e˙dt+lxg2xedt=d˙,where d˙ is the first derivative of composite disturbance d. It is assumed that |d˙|≤σ in this paper, where σ is a known positive constant.Substitutingg2(x) and l(x) into (9), the differential equation (9) can be rewritten as(10)e˙dt+l4xedt=d˙,where l4(x)=∂p(x)/∂Q. An appropriate fourth element of observer gain, l4(x), needs to be designed such that ed(t) converge to zero.The following result provides a design scheme for the disturbance observer (7) with guaranteed uniformly ultimate boundedness of system (9) or (10).Lemma 2 (for details see [24]).
LetD⊂Rn be a domain that contains the origin and let W:[0,∞)×D→R be a continuously differentiable function such that(11)α1χ≤Wt,χ≤α2χ,∂W∂t+∂W∂χξt,χ≤-Qχ,χ≥μ>0,for all t≥0 and for all χ∈D, where α1 and α2 are class K functions and Q(χ) is a continuous positive definite function. Then, there is T≥0 (dependent on χ(t0) and μ) such that the solution of the system(12)χ˙=ξt,χsatisfies(13)χt≤α1-1α2μ,∀t≥t0+T.Moreover, if D=Rn and α1 belongs class K∞, then (13) holds for any initial state χ(t0), with one restriction on how large μ is. Note that inequality (13) shows that χ(t) is uniformly ultimately bounded (UUB) with the ultimate bound α1-1(α2(μ)).Theorem 3.
For the estimation error system (10), if there exists a constant l0 satisfying(14)l0>121+M_,where the fourth element of observer gain, l4(x), is chosen as l0(1+M(x)) and M(x) is a bounded nonlinear function with respect to x satisfying 0≤M_≤M(x)≤M¯, then the solution of system (10) is UUB.Proof.
Denote the Lyapunov function(15)Wdt=ed2t.Computing the derivative of Wd(t) along the trajectories of (10), it can be obtained that (16)W˙dt=2edte˙dt=2edt-l01+Mxedt+d˙=-2l01+Mxed2t-ed2t-2edtd˙+d˙2+ed2t+d˙2≤-2l01+Mx-1ed2t+σ2.Thus it can be concluded from Lemma 2 that the solution of estimation error system (10) is uniformly ultimately bounded, if l0>1/[2(1+M_)]. Obviously, there exits a positive constant β such that (1/(1+β))ed2(t)≤Wd(t)≤(1+β)ed2(t). And W˙d(t) is a negative definite matrix when |ed(t)|≥σ>0 hold. According to Lemma 2, the ultimate bound of ed(t) is (1+β)σ; that is, edt≤(1+β)σ.Remark 4.
The NDO (7) reduces to a linear disturbance observer (LDO) if M(x)=0. Obviously, the performance of NDO is better than LDO when LDO takes the same constant l0 as NDO.


## 3.3. NDOBSMC Design

The nonlinear COM described by (3) can be linearized completely using the input/output linearization technique of full state feedback, so the linearized model is developed by repeated differentiation of V and γ as follows:(17)V⃛=σ1x¨1+x˙1TΣ2x˙1m,γ⃛=π1x¨1+x˙1TΠ2x˙1,where x1=[VαγΦ]T, σ1=∂f1(x)/∂x1, Σ2=∂σ1/∂x1, π1=∂f3(x)/∂x1, and Π2=∂π1/∂x1. The detailed expressions of σ1, Σ2, π1, and Π2 are given in the appendix.The right-hand sides of (17) involve second derivatives of α and Φ. The expression of the second derivatives for α and Φ consists of three parts: the first part is control relevant, the second part is disturbance relevant, and the third part is neither control relevant nor disturbance relevant. Consider(18)α¨=α¨0+q¯Sc¯CM0δe+ψ~2CN20δeI¯yy+d,Φ¨=Φ¨0+ω2Φc,where α¨0=[zTT0+q¯Sc¯(CM0α2α2+CM0αα+CM00)+ψ~1N10+ψ~2N¯20]/I¯yy-γ¨ and Φ¨0=-2ζωΦ˙-ω2Φ.Whenx10=Vα0γΦ0T is defined, the output dynamics of V and γ can be written as follows:(19)V⃛γ⃛=fVfγ+Bx1δeΦc+cVx1cγx1d+dVdγ,where control inputs, δe and Φc, and composite disturbance, d, appear explicitly. Disturbances, dV and dγ, are produced by parameter uncertainties. The concrete forms of fV, fγ, B(x1), cV(x1), and cγ(x1) are given by(20)fV=σ1x¨1+x˙1TΣ2x˙1m,fγ=π1x¨1+x˙1TΠ2x˙1,Bx1=b11x1b12x1b21x1b22x1,cVx1=1m∂T0∂αcos⁡α-T0sinα-∂D¯0∂α,cγx1=1mV∂T0∂αsinα+T0cos⁡α-∂L¯0∂α,in which(21)b11x1=q¯Sc¯CM0δe+ψ~2CN20δemIyy000000000×∂T0∂αcos⁡α-T0sinα-∂D¯0∂α,b12x1=ω2m∂T0∂Φcos⁡α,b21x1=q¯Sc¯CM0δe+ψ~2CN20δemVIyy000000000×∂T0∂αsinα+T0cos⁡α-∂L¯0∂α,b22x1=ω2mV∂T0∂Φsinα.Two decoupled sliding mode surfacessV(t) and sγ(t) are defined by sV(t)=(d/dt+λV)3∫0t‍eV(τ)dτ and sγ(t)=(d/dt+λγ)3∫0t‍eγ(τ)dτ, where eV(t)=V-Vd is tracking error of velocity, eγ(t)=γ-γd is tracking error of flight path angle, and λV and λγ are strictly positive constants defining the bandwidth of the tracking error dynamics. The sliding mode surfaces sV(t) and sγ(t) represent linear differential equations where their solutions ∫eV(τ) and ∫eγ(τ) converge to zero exponentially with the time constants 2/λV and 2/λγ, respectively.DifferentiatingsV(t) and sγ(t), we have(22)s˙Vts˙γt=υVυγ+fVfγ+Bx1δeΦc+cVx1cγx1d+dVdγ,where υV=-V⃛d+3λVe¨V(t)+3λV2e˙V(t)+λV3eV(t) and υγ=-γ⃛d+3λγe¨γ(t)+3λγ2e˙γ(t)+λγ3eγ(t).With (22), recall that estimation value of the disturbance d is developed by the NDO (7) and the NDOBSMC can be designed as(23)δeΦc=-B-1x1fVfγ-B-1x1υV+kVsgnsVtυγ+kγsgnsγt-B-1x1cVx1cγx1d^,where kV and kγ are sliding mode gains of the NDOBSMC (23), d^ is the disturbance estimation value obtained by the NDO (7), and B(x1) is assumed to be invertible.Remark 5.
When the output of disturbance observerd^=0, that is, there is no disturbance observer, the NDOBSMC (23) obviously reduces to a DIBSMC, which is written as(24)δeΦc=-B-1x1fVfγ-B-1x1υV+kVsgnsVtυγ+kγsgnsγt.As discussed in [25], the control laws (23) and (24) may result in control chattering because of the discontinuity across the sliding surfaces. As a practical matter, chattering is undesirable because it involves very high control action and may excite high frequency dynamics neglected in the modeling. The discontinuity in the control law can be dealt with by defining two thin boundary layers of widths ΦV and Φγ around the sliding mode surfaces, that is, replacing sgn(sV(t)) and sgn(sγ(t)) with continuous saturation functions sat⁡(sV(t)/ΦV) and sat⁡(sγ(t)/Φγ), where(25)sat⁡xs=1,ifxs>1,xs,ifxs≤1,-1,ifxs<-1.So the NDOBSMC (23) and the DIBSMC (24) are modified as follows:(26)δeΦc=-B-1x1fVfγ-B-1x1υV+kVsat⁡sVtΦVsVtΦVΦγΦγυγ+kγsat⁡sγtΦγ-B-1x1cVx1cγx1d^,(27)δeΦc=-B-1x1fVfγ-B-1x1υV+kVsat⁡sVtΦVsVtΦVΦγΦγυγ+kγsat⁡sγtΦγ.Considering (10) and substituting (26) into (22), the augmented closed-loop system can be obtained as follows:(28)s˙Vt=-kVsat⁡sVtΦV+cVx1edt+dV,s˙γt=-kγsat⁡sγtΦγ+cγx1edt+dγ,e˙dt=-l4xedt+d˙.Next, uniform ultimate boundedness (UUB) of the composite system (28) will be considered.Theorem 6.
For the composite system (28), if the sliding mode gains, kV and kγ, satisfy(29)kV>Max⁡0.25ΦVβV1-2σV12+βV2-2,1+βσσV1+σV2,kγ>Max⁡0.25Φγβγ1-2σγ12+βγ2-2,1+βσσγ1+σγ2and the fourth element of observer gain, l4(x), is chosen as l0(1+M(x)) with the constant l0 satisfying(30)l0>1+βV12+βγ1221+M_,then the solutions of composite system (28) are UUB, where positive constants βV1, βγ1, β, βV2, and βγ2 are tuning parameters to obtain the appropriate values of control gains, and positive constants σV1, σγ1, σ, σV2, and σγ2 are the upper bounds of cV(x1), cγ(x1), d˙, dV, and dγ; that is, |cV(x1)|≤σV3, |cγ(x1)|≤σγ3, |d˙|≤σ, |dV|≤σV2, and |dγ|≤σγ2.Proof.
A Lyapunov function candidate for the system (28) is chosen as W(t)=Wd(t)+Ws(t), where Wd(t)=ed2(t) and Ws(t)=1/2(sV2(t)+sγ2(t)). According to Theorem 3, W˙d(t)≤-[2l0(1+M(x))-1]ed2(t)+σ2.
Taking the derivative ofWs(t) along the trajectories of (28), we have (31)W˙st=sVts˙Vt+sγts˙γt=sVt-kVsat⁡sVtΦV+cVx1edt+dVsVtΦVΦγΦγ+sγt-kγsat⁡sγtΦγ+cγx1edt+dγ.When |sV(t)/ΦV|≤1 and |sγ(t)/Φγ|≤1, the derivative of Ws(t) can be expressed as(32)W˙st=-kVΦVsV2t+cVx1edtsVt+dVsVt-kγΦγsγ2t+cγx1edtsγt+dγsγt=-kVΦV-0.25βV1-2cV2x1-0.25βV2-2sV2t-kγΦγ-0.25βγ1-2cγ2x1-0.25βγ2-2sγ2t-0.5βV1-1cVx1sVt-βV1edt2-0.5βV2-1sVt-βV2dV2-0.5βγ1-1cγx1sγt-βγ1edt2-0.5βγ2-1sγt-βγ2dγ2+βV12+βγ12ed2t+βV22dV2+βγ22dγ2.Furthermore, the derivative of W(t) along the trajectories of (28) can be given by(33)W˙t≤-kVΦV-0.25βV1-2σV12-0.25βV2-2sV2t-kγΦγ-0.25βγ1-2σγ12-0.25βγ2-2sγ2t-2l01+Mx-1+βV12+βγ12ed2t+βV22σV22+βγ22σγ22+σ2.When |sV(t)/ΦV|>1 and |sγ(t)/Φγ|>1, the derivative of Ws(t) can be given by(34)W˙st=-kVsVt+cVx1edtsVt+dVsVt-kγsγt+cγx1edtsγt+dγsγt≤-kV-cVx1edt-dVsVt-kγ-cγx1edt-dγsγt.By now, the derivative of W(t) along the trajectories of (28) becomes(35)W˙t≤-kV-1+βσσV1-σV2sVt-kγ-1+βσσγ1-σγ2sγt-2l01+Mx-1ed2t+σ2.Thus it can be concluded from Lemma 2 that the solutions of the augmented closed-loop system (28) are uniformly ultimately bounded, if conditions (29) and (30) are satisfied.Remark 7.
The result of Theorem6 (l0>(1+βV2+βγ2)/[2(1+M_)]) is different from the result of Theorem 3 (l0>1/[2(1+M_)]) because of the couplings between tracking errors and disturbance estimate error. So it is necessary to analyze the stability of the composite system (28).


## 4. Simulation Studies

The controller derived in the previous section will be tested in simulation using the full nonlinear model of FAHVs. Two case studies will be presented in this section. Firstly, the flexible effects on the pitching moment and the estimating performance of the disturbance observer are studied for the nominal model without parameter uncertainties and external disturbances. Secondly, the tracking performance and robustness of the NDOBSMC (26) and the DIBSMC (27) are compared for the FAHV model with composite disturbance.In the first case study, the nonlinear functionp(x) is taken as l0(Q+cQ3), where c≥0. Thus the NDO gains l(x)=000l0(1+3cQ2)00. The sliding mode gains of NDOBSMC are chosen as kV=10 and kγ=0.01, while the bigger sliding mode gains of DIBSMC are chosen as kV=16 and kγ=0.02 in order to attenuate the pitching disturbance moment, d¯f, which is produced by couplings between flexible and rigid modes. Figure 1 shows the simulation results of V, γ, eV, and eγ using NDOBSMC and DIBSMC. The simulation results confirm the fact that the disturbance d¯f affects the tracking performance of velocity and FPA. And it can be observed that the tracking errors of velocity and FPA under DIBSMC are greater than the ones under NDOBSMC, even if the sliding mode gains of DIBSMC are bigger than the ones of NDOBSMC. It is caused that the NDOBC is able to estimate and compensate for this disturbance d¯f, as shown in Figure 5. Furthermore, it can be seen from Figures 2, 3, 4, and 5 that the changes of Φc, δe, η1, η2, η˙1, η˙2, Q, and α under DIBSMC are bigger than the ones under NDOBSMC. In addition, it is observed from Figures 2 and 3 that the chattering of Φc, δe, η1, η2, η˙1, and η˙2 using DIBSMC can be reduced using NDOBSMC. Finally, we can conclude that the effects produced by the disturbance d¯f could be reduced using the NDOBSMC.Figure 1
Case study 1: (a) velocityV and desired signal Vd, (b) tracking error of velocity eV, (c) flight path angle γ and desired signal γd, and (d) tracking error of flight path angle eγ.
(a)
(b)
(c)
(d)Figure 2
Case study 1: (a) fuel equivalence ratioΦ, (b) elevator deflection δe.
(a)
(b)Figure 3
Case study 1: (a) flexible modeη1, (b) flexible mode η2, (c) η˙1(t), and (d) η˙2(t).
(a)
(b)
(c)
(d)Figure 4
Case study 1: (a) pitch rateQ, (b) angle of attack α.
(a)
(b)Figure 5
Case study 1: composite disturbanced, estimation value and error of d.The second case study considers a maneuver task, that is, acceleration at constant dynamic pressure at timet=50 s. Thus the concrete forms of command signals can be described as Vd(t)=600Vd0(t) and γd(t)=arcsin(2hsV˙d(t)/Vd2(t)), where the condition q¯=const yields γd(t)=arcsin(2hsV˙d(t)/Vd2(t)) (details in [18]), Vd0(t)=1-(1/1-ζ2)e-ζωntsin(wdt+β), ωd=ωn1-ζ2, β=arccosζ, and ζ and ωn are chosen as 0.95 and 0.025, respectively. Furthermore, the pitching disturbance moment, d¯o, which originates from the following parameter uncertainties and external disturbances, is added in this case.(1)
Whent∈[0s,400s], parameter uncertainties ΔCL(·)=ΔCD(·)=ΔCM(·)=ΔCT(·)=ΔCTΦ(·)=ΔCNi(·)=20%, i=1,2, are added in the simulation.(2)
Whent∈[0s,250s], pitching disturbance moment, 4.6×104(sin(0.25t)), which is produced by external interferences, is added in the simulation. Recall that d=(d¯f+d¯o)/I¯yy, where d¯o is the pitching disturbance moment produced by the above parameter uncertainties and external interferences. Thus tracking performance and robustness of the NDOBSMC and the traditional DIBSMC are compared for the system with the composite disturbance d in this case. In order to attenuate the composite disturbance, the sliding mode gains of DIBSMC are chosen as kV=20 and kγ=0.18, while the sliding mode gains of NDOBSMC are chosen as kV=14 and kγ=0.05. In addition, the nonlinear function p(x) and NDO gain l(x) are the same as the first case. Figures 6 to 9 show the simulation results of V, γ, eV, eγ, Φc, δe, η1, η2, η˙1, η˙2, Q, and α using NDOBSMC and DIBSMC. It can be seen from Figure 6 that the tracking ability of velocity and FPA using NDOBSMC are better than the ones using traditional DIBSMC because the NDOBC can estimate and compensate for the composite disturbance d shown in Figure 10. And it can be observed from Figures 7 and 8 that the chattering of traditional DIBSMC in presence of composite disturbances can be attenuated with the NDOBC. Furthermore, the changes of η1η2, η˙1, η˙2, Q, and α are smaller using NDOBSMC, as shown in Figures 8 and 9. In a word, the tracking performance and robustness of NDOBSMC are better than those of DIBSMC.Figure 6
Case study 2: (a) velocityV and desired signal Vd, (b) tracking error of velocity eV, (c) flight path angle γ and desired signal γd, and (d) tracking error of flight path angle eγ.
(a)
(b)
(c)
(d)Figure 7
Case study 2: (a) fuel equivalence ratioΦ, (b) elevator deflection δe.
(a)
(b)Figure 8
Case study 2: (a) flexible modeη1, (b) flexible mode η2, (c) η˙1(t), and (d) η˙2(t).
(a)
(b)
(c)
(d)Figure 9
Case study 2: (a) pitch rateQ, (b) angle of attack α.
(a)
(b)Figure 10
Case study 2: composite disturbanced, estimation value and error of d.


## 5. Conclusion

In this paper, a nonlinear disturbance observer (NDO) based sliding mode controller (NDOBSMC) is designed for the nonlinear longitudinal model of flexible air-breathing hypersonic vehicles, and it is proved that the composite system is uniformly ultimately bounded when the controller gains and disturbance observer gains meet specified conditions. Simulation results show that the NDO based compensator can estimate and compensate for the composite disturbance produced by the couplings between flexible and rigid modes, parameter uncertainties, and external disturbances. Furthermore, tracking performance and robustness can be improved by using NDOBSMC, compared with the traditional dynamic inversion based sliding mode controller. One of the future research topics is to study the control problem for flexible air-breathing hypersonic vehicles based on the data-driven framework [26–28].



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