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Fuzzy disturbance observer based dynamic surfacecontrol for air-breathing hypersonic vehicle with variablegeometry inletsDOI:10.1049/iet-cta.2017.0742

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Citation for published version (APA):Doua, L., Sua, P., Zong, Q., & Ding, Z. (2018). Fuzzy disturbance observer based dynamic surface control for air-breathing hypersonic vehicle with variable geometry inlets. IET Control Theory and Applications.https://doi.org/10.1049/iet-cta.2017.0742

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IET Research Journals

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Fuzzy disturbance observer based dynamicsurface control for air-breathing hypersonicvehicle with variable geometry inlets

ISSN 1751-8644doi: 0000000000www.ietdl.org

Liqian Dou1, Peihua Su1, Qun Zong1, Zhengtao Ding2

1School of Electrical and Information Engineering, Tianjin University, 92 Weijin Road, Tianjin, 300072, China2School of Electrical and Electronic Engineering, The University of Manchester, Manchester, M13 9PL, U.K.

E-mail: zhengtao.ding@manchester.ac.uk

Abstract: For an air-breathing hypersonic vehicle with a variable geometry inlet (AHV-VGI), a movable translating cowl is usedto track the shock on lip conditions to capture enough air mass flow, which can extend the velocity range and be favorableto the acceleration and maneuvering flight. We firstly establish longitudinal dynamics for AHV-VGI, and consider the lumpeddisturbances which include the unknown external disturbance, the parameter uncertainties and the uncertainty parts introducedby translating cowl. Then, the dynamic surface control (DSC) strategy based on fuzzy disturbance observer (FDO) is proposed forAHV-VGI control. The control process for AHV-VGI is divided into two subsystems. For each subsystem, a sliding mode controlleris designed, and FDOs are adopt to compensate the lumped disturbances, which can render the disturbance estimate errorsconvergent. Numerical simulations are presented to illustrate the effectiveness of the proposed method and the advantages oftranslating cowl.

1 Introduction

As a reliable and more cost-efficient way to access space, air-breathing hypersonic vehicle (AHV) has been investigated by manyresearchers in recent decades[1]. This type of vehicle has a uniquedesign, incorporating a supersonic combustion scramjet enginelocated beneath the fuselage, which enable it quick response andglobal reach[2]. In practice, the AHV always fly within a wideflight envelope (range of flight conditions). However, for the AHVwith fixed geometry inlet(AHV-FGI), once it is running at a lowMach number, the shockwave would deviate away from the scram-jet lip. This leads to the scramjet engine could not get sufficient airstream, so that the thrust would be insufficient. In order to solvethe above issue, AHV with variable-geometry inlet(AHV-VGI) arepopularly studied. For example, NASA investigates a rotary lipVGI for the X-43A hypersonic aircraft[3]. The Space and Astro-nautical Science institution of Japan developed a variable geometryaxisymmetric inlet for the ATREX engine[4]. Besides, the Russianscholar Kuranov investigated the Magneto Hydrodynamic controlledinlet[5].

In the past decades, lots of efforts have also been put into flightcontrol of AHV. Feedback linearization is a powerful tool to dealwith the intrinsic nonlinear system[6]. The robust control of AHV isstudied to improves the tracking control performance effectively[7].To overcome the problem of system uncertainty, fuzzy logic sys-tem [8] and neural network [9] are employed due to their powerfulability of approximation for the smooth nonlinearities. Recently,transient performance-based control design [10][11] has become animportant method for the research of uncertain nonlinear systems.Ref.[12] proposed a novel estimation-free prescribed performancenon-affine control strategy for AHV to guarantee tracking error islimited to a predefined arbitrarily small residual set. Beside, Var-ious techniques have been applied to hypersonic vehicles to dealwith parametric uncertainties or bounded uncertainties and unmod-eled system dynamics, such as the sliding model controller[13][14]and the l1 adaptive controller[15], etc.

However, in the context of the aforementioned literatures, AHV-FGI model was adopted for control design. In recent years, inves-tigators have carried on many researches on VGI characteristic of

AHV. In Ref.[16], a new methodology using gas dynamic rela-tions has been developed to obtain optimal geometry of scramjetinlet at different Mach numbers. Ref.[17] compares the propertiesof three kinds of VGI using the low-order control-oriented model,and designs a kind of inlet used in a wide range of Mach numbers.Ref.[18] designs a 2-D hypersonic VGI with movable lip along theflow direction, carries out three dimensional CFD numerical sim-ulations. Comparison of the aerodynamic characteristics was madebetween VGI and FGI. Ref.[19] presents a design method of high-performance VGI, and obtains the adjusting rules and performancevariation of VGI in various conditions. Although there are manyresearches on the configurature of AHV-VGI, the control systemdesigns for AHV-VGI are investigated in the initial phase. The mainchallenge is the complex structure of VGI system which is difficultto control well. A VGI scheme was proposed by [20]. By movingthe translating cowl along the flow direction, the internal contrac-tion ratio could be enlarged, and the propulsion efficiency could beimproved. This VGI scheme is easy to operate, which only needs toadjust the translating cowl to ensure that the shock wave can projectright on the inlet lip. Thus, it provides a feasible idea for us to studythe control design for AHV-VGI.

This VGI configuration can extend the velocity range and befavorable to the acceleration control of AHV. However, the move-ment of translating cowl will induce the uncertain changes of theaerodynamic forces and moment, especially, the thrust will bechanged greatly. And these changes are so large that cannot beregarded as small disturbances[21]. Moreover, some unknown dis-turbances for AHV such as wind gust, uncertain structural dynamics,actuator failure, and sensor faults, bring about great uncertaintiesto the system performance[22]. In this case, the traditional robustcontrol may be too conservative because it can only achieve agood performance level of disturbance attenuation, not cancel theeffect of the disturbance. It is well known that the disturbanceobserver based control (DOBC) approach has been widely usedas an effective robust method to compensate the disturbance andparameter variations from both environment and system[23][24].Many effective DOBC schemes have been developed for space-craft, missiles, and hypersonic vehicles, and the DOBC approachcan enhance the disturbance attenuation ability and the robustness

IET Research Journals, pp. 1–11c© The Institution of Engineering and Technology 2015 1

of the controllers [25]. Ref.[26] designed a dynamic inversion con-troller based on disturbance observer for missile systems. A robustcontrol law using DOB and neural networks for the linearized AHVmodel was adopted in [27]. Ref.[28] presented a robust flight con-trol problem for the longitudinal dynamics of a generic AHV undermismatched disturbances via a nonlinear disturbance observer-basedcontrol (NDOBC) method. A robust backstepping approach basedon NDOBC was designed for a flexible AHV[29]. Ref.[30] pro-posed sliding mode disturbance observer to deal with mismatcheddisturbances in AHV system. Ref.[31] presented a new high-ordernonlinear disturbance observer for AHV, which can guarantees theconvergence of the estimated error for a nonlinear system with a fasttime-varying disturbance

For AHV-VGI, there exit stronger aerodynamic uncertaintiesintroduced by VGI with the translating cowl, and the composite dis-turbances from different sources. And it is hard to compensate forthe more general form of both aerodynamic uncertainties and distur-bances. FDO is a particular type of disturbance observer, which canbe applied for various types of disturbances[32][33]. Moreover, theFDO method is more effective and low cost of hardware implemen-tation because the FDO method only uses one fuzzy system. Thus, inthis paper, a fuzzy disturbance observer (FDO) method is presentedto achieve the rejection of multiple disturbances and an dynamicsurface control (DSC) based on FDO is developed to improve thetracking performance of the control system. Currently, many flightcontrol schemes based on FDO have been researched for the AHVto improve the control ability[34]. Ref.[35] presented a novel com-posite controller design method base on backstepping and nonlineardisturbance observer, which can guarantee system outputs asymptot-ically track the refernce signals. In [36], novel robust fuzzy DOBCdesign was developed to achieve the rejection of multiple distur-bances, where the DOBC strategy is employed to compensate forthe effect of the multiple source disturbances.

In this paper, we investigate the control problem for AHV-VGI. Atranslating cowl is used to track shock on lip conditions for captur-ing off-design flow. By allowing the cowl to move with the changeof the shock wave position, proper mass flow can be taken intothe engine without any flow spillage. However, the movement ofthe cowl leads to the unknown disturbances of the aerodynamicforces, moment and thrust. These aerodynamic disturbances andthe external disturbances may degrade the control performance, andeven induce instability of the control system. Sliding mode con-trol (SMC) has been widely studied for many years and extensivelyemployed in industrial applications due to its conceptual simplicity,and in particular powerful ability to reject disturbances and plantuncertainties[37][38]. Therefore, the SMC method is adopted todesign the basic controller, and FDO is used to estimate the unknownfactors including the uncertainties and disturbances. The proposedcontroller possesses several advantages including strong robustnessand disturbance rejection ability. etc.

This paper is organized as follows. Section 2 states the basicprinciple of VGI and tranalating cowl, and establishes the mathe-matical expression of translating cowl and the longitudinal model ofAHV-VGI system. Section 3 states the basic principle of adaptivefuzzy disturbance observer. Section 4 designs the tracking controllerand proves the uniformly stability of the AHV-VGI control system.Section 5 gives the simulation results in three cases. The conclusionof the paper is drawn in Section 6.

2 Model of AHV-VGI with Translating Cowl

2.1 Basic principle of AHV-VGI with translating cowl

As shown in Fig. 1, the AHV-VGI has a scramjet inlet with trans-lating cowl which can be adjusted forward or backward with therequirements of flight control. When the AHV operates in a rel-atively high Mach number, the forebody oblique shock wouldoccur[39][40]. As shown in Fig. 2(a), The α, τ1l, hi, Lf and θsdenote angle of attack (AOA), the lower forebody angle, the heightof engine inlet, the forebody length and the shock wave angle respec-tively. δs = α+ τ1l is the flow turn angle. The blue dotted line

corresponds to the oblique shock wave and the red solid lines denotethe free stream which hit against the oblique shock wave and thenturn parallel to the lower forebody. In this case, the shock waveangle θs is small and the free stream can all captured by inlet, sothe translating cowl is not necessary to be adjusted. In this moment,the scramjet can capture all the mass flow as long as the obliqueshock wave can be sealed by the scramjet cowl. The capture area Dcan be calculated as

D = Lfsin(τ1l + α)

cos(τ1l)+ hi cos(α) (1)

where τ1l is 6.2deg. The hi is 3.5ft, Lf is 47ft.

Fig. 1: The structure of AHV-VGI

(a)

(b)

Fig. 2: (a)AHV-VGI without flow spillage occur; (b)AHV-VGI withflow spillage occur

However, when the AHV operates in a relatively low Mach num-ber, the shock wave angle θs will increase. Hence, as shown inFig. 2(b), if the cowl of the scramjet engine is fixed, the shock wavewould not be sealed by the cowl. Consequently, that will cause theflow spillage (area D2), and the scramjet engine would not obtainmass flow sufficiently. In this case, if the cowl can be elongateda distance L1, the oblique shock wave can be sealed by the cowlagain. And the inlet of scramjet will capture all mass flow of areaD = D1 +D2, the air mass flow through the engine will obviouslylarger. The capture area can be calculated as

D1 =hi sin(θs) cos(τ1l)

sin(θs − α− τ1l)

D2 =hi + Lf tan(τ1l) sin(θs)

sin(θs − α)(2)

We focus on the second case and study the influence introduced bytranslating cowl. The thrust produced by the engine is given by

T = ma(ve − v∞) + (Pe − P∞)Ae − (P1 − P∞)A1 (3)

IET Research Journals, pp. 1–112 c© The Institution of Engineering and Technology 2015

where ve is the exit flow velocity, v∞ is free stream flow velocity,P1 is the pressure at the engine inlet entrance, Pe is the pressure atthe engine exit plane, P∞ is the free stream pressure,A1 andAe arethe engine inlet area and exit area, respectively, and ma means theair mass flow through the engine as follows

ma = P∞M∞

√λ

RT∞D (4)

where M∞ and T∞ are free stream Mach and temperature respec-tively, λ is heat ratio and R is the gas constant for air.

Obviously, the thrust of AHV was influenced by the elongationdistance. By using the movable translating cowl, we can dramaticallyadjust the amount of air mass flow captured by inlet. It is helpful toreach a more powerful thrust. Besides, the aerodynamic forces andmoments of the nacelle bottom will be influenced and ultimately thepitching moment of AHV will are changed[21][41].

Fig. 3 is the comparison chart of thrust-drag ratio when thetranslating cowl used or not. We can see that the translating cowlhas a great influence on the thrust-drag ratio. The translating cowlimproves the maximum thrust-drag ratio from 1.035 to 1.207. WhenT/D = 1, AHV reaches its limiting speed. The faster AHV flies,the easier the limiting speed reaches. And the AHV will remain lim-iting speed until lacking of thrust and beginning to deceleration. TheAHV driven by scramjet can achieve powered flight throughout thecruise segment, thereby increasing its response performance whenthe flight trajectory needs to be changed.

−5 0 50.4

0.6

0.8

1

1.2

1.4

AOA(deg)

Thr

ust−

drag

rat

io

AHV−FGIAHV−VGI

Fig. 3: The comparison of thrust-drag ratio

Defination 1. The optimal elongation distance of translating cowl isthe position where the inlet of scramjet can capture all mass flow ofarea D.

Assumption 1. The translating cowl can be infinitely fast adjustedto the optimal elongation distance with the flight state of AHV (thesetting time is approximated to zero).

In Fig. 2, we can obtain the following geometrical expression ofelongation distance L1

L1 = Lf − (Lf tan τ1l + hi) cot(θs − α) (5)

Thus, L1 is a function of shock wave angle θs and angle of attack(AOA) α. Shock wave angle θs = θs(Ma,α) can be found as themiddle root of the following shock angle polynomial:

sin6θs + bsin4θs + csin2θs + d = 0 (6)

where the expressions of b, c and d were defined in the [42].In this paper, the estimated value of the elongation distance is pre-

sented by curve fitted approximation[43]. The expression of fittingelongation distance l is shown in (7). The detailed parameters valuecan be found in the Table 1.

l ≈ Cαl α+ Cα2

l α2 + CMal Ma+ CMa2

l Ma2 + C0l (7)

Fig. 4 shows the fitting error is very small. This error is withinthe allowable range from the perspective of control design. And we

Table 1 Fitting coefficients in elongation distance

Parameters Values Parameters Values

Cαl -0.2416 Cα2

l 0.0633CMal -5.2380 CMa2

l 0.1598C0l 37.5193

can see, the translating cowl moves back with the increase of veloc-ity. When the velocity increases to a certain extent, the translatingcowl moves back completely, and the variable geometry inlet willno longer work.

67

Velocity (Mach)

891011654AOA α (deg)

3210-1

12

10

8

6

2

4

0-2

0

2

4

6

8

10

realfit

Fig. 4: The optimal elongation length of translating cowl

Table 2 Fitting coefficients in aerodynamic forces

Parameters Values Parameters Values

CαL 0.0157 CMaL 5.45e-05

CδeL 0.0066 C0L 0.0046

CαD 1.28e-04 Cα2

D 3.59e-04

CδeD -1.97e-10 Cδ2eD 4.37e-05

CαδeD 9.78e-05 CMaD -5.32e-04

C0D 0.0133 CαT 0.0328

CMaT 0.0026 C0

T -0.152CφαT 0.3152 CφMa

T -0.703CφT 8.9227 CαM 0.0064CMaM -0.0022 CδeM -0.014

C0M 0.051 CαL,l 9.02e-05

CMaL,l -1.317e-04 C0

L,l 0.0012CαD,l 1.844e-05 CMa

D,l -1.0037-05C0D,l 5.121e-05 CαT,l 0.0025

CMaT,l -0.0045 C0

T,l 0.0596CαM,l 1.529e-05 CMa

M,l 4.303e-05C0M,l -2.53e-04

2.2 The longitudinal model of AHV-VGI

The dynamic model adopted in this paper is on the basis of Bolen-der and Doman for the longitudinal rigid model of AHV[20]. Thelongitudinal dynamics of the AHV-VGI can be described by a setof differential equations for velocity V , altitude h, fligh path angle

IET Research Journals, pp. 1–11c© The Institution of Engineering and Technology 2015 3

(FPA) γ, AOA α, and pitching rate Q as

V = (T cosα−D)/m− g sin γ + dV (8)

h = V sin γ (9)

γ = (T sinα+ L)/(mV )− g cos γ/V + dγ (10)

α = Q− γ + dα (11)

Q = M/Iyy + dQ (12)

where m, Iyy , g are the mass, moment of inertia, and accelera-tion of gravity respectively. di(i = V, γ, α,Q) denote the unknownexternal disturbances. L, D, T , M are lift, drag, thrust, and pitch-ing moment, respectively. The influence of the translating cowl ofthe AHV-VGI model, which causes change of aerodynamic forces,needs to be considered. We use the curve fitting method to refit theaerodynamic forces as follows

L ≈ qS(CL + CL)

T ≈ q(CT + CφT · φ+ CT )

D ≈ qS(CD + CD)

M ≈ qSc(CM + CM ) + zTT (13)

where, CL, CD , CM and CT are the aerodynamic lift coefficients,drag coefficients, pitching moment coefficients and thrust coeffi-cients respectively, q = 1/2ρV 2 denotes the dynamic pressure, ρ,S, c, zT are air density, reference area, aerodynamic chord, andthrust moment arm respectively. Particularly, the additional termsCL, CD , CT and CM express the influences caused by translatingcowl, which is fitted as functions of Mach number Ma and angle ofattack (AOA) α according to equation (7). The expressions of aero-dynamic coefficients are shown in follows. The detailed parametersvalue can be found in the Table 2.

CL = CαLα+ CMaL Ma+ CδeL δe + C0

L

CD = CαDα+ Cα2

D α2 + CδeD δe + Cδ2eD δ2

e + CαδeD αδe

+ CMaD Ma+ C0

D

CT = CαTα+ CMaT Ma+ C0

T

CφT = CφαT α+ CφMaT Ma+ CφT

CM = CαMα+ CMaM Ma+ CδeM δe + C0

M

CL = ClL · l = (CαL,lα+ CMaL,l Ma+ C0

L,l) · l

CD = ClD · l = (CαD,lα+ CMaD,lMa+ C0

D,l) · l

CT = ClT · l = (CαT,lα+ CMaT,l Ma+ C0

T,l) · l

CM = ClM · l = (CαM,lα+ CMaM,lMa+ C0

M,l) · l (14)

2.3 The strict feedback form

The longitudinal model of VGI-AHV can be rewritten as a nonlinearfeedback form as

V = gV · φ+ fV + ΩV (15)

h = V sin γ (16)

γ = gγ · α+ fγ + Ωγ (17)

α = gα ·Q+ fα + Ωα (18)

Q = gQ · δe + fQ + ΩQ (19)

where

gV = q · CφT · cosα/m

fV = (q · CT · cosα− qS · CD)/m− g sin γ

gγ = qS · CαL/

(mV )

fγ = qS · (CMaL ·Ma+ CδeL · δe + C0

L − CαL · γ)/(mV )

+ T · sinα/(mV )− g · cos γ/V

gα = 1; fα = −gγ · α− fγ

gQ = qSc · CδeM/Iyy

fQ = [qSc · (CαM · α+ CMaM ·Ma+ C0

M ) + zT q · (CT

+ CφT · φ)]/Iyy

Assumption 2. The states x = [V, h, γ, α,Q]T and inputs uc =[φ, δe]

T are assumed to be available for measurement.Assumption 3. The functions gi and its differentials gi are bounded.Let 0 < |gi| ≤ gi , and 0 < |gi| < gdi , i = V, γ, α,Q, where gi andgdi are the upper bounds. This assumption is to ensure the controlsignals are nonsingular and bounded.

In this paper, the nonlinear parts introduced by movable cowl,the parameter uncertainties introduced by coefficients fitting errorsand the unknown external disturbances are combined and treated aslumped disturbances Ωi, i = V, γ, α,Q as

ΩV = ∆gV · φ+ ∆fV + (f lV + ∆f lV ) · l + dV

Ωγ = ∆gγ · α+ ∆fγ + (f lγ + ∆f lγ) · l + dγ

Ωα = ∆fα + (f lα + ∆f lα) · l + dα

ΩQ = ∆gQ · δe + ∆fQ + (f lQ + ∆f lQ) · l + dQ (20)

where f lV , f lγ , f lα, f lQ are the uncertainties introduced by movablecowl, and ′∆′ denotes the perturbation of aerodynamic coefficients.

f lV = (qClT cosα− qSClD)/m; f lγ = qSClL/(mV )

f lQ = q(ScClM + zTClT )/Iyy; f lα = −f lγ

3 Fuzzy Disturbance Observer

3.1 Fuzzy logic system (FLS)

The basic configuratuion of a FLS consists of the fuzzifier, the fuzzyinference engine working on fuzzy rules and the defuzzifier. Thefuzzy inference engine employs fuzzy rules to perform a mappingfrom an input linguistic vector x = [x1, · · · , xn]T ∈ Rn to an out-put linguistic variable y ∈ R. A fuzzy system is a collection of fuzzyIF-THEN rules of the form

R(j) : IF xi is Aj1 and · · · xn is A

jn . THEN y is Bj (21)

where Aj1, · · · , Ajn are fuzzy sets, and Bj is the fuzzy singleton

for the output in the jth fuzzy rule. Then the resulting FLS can berepresented as

y(x) =

∑mj=1 hj

(∏ni=1 µAji

(xi))

∑mj=1

(∏ni=1 µAji

(xi)) (22)

where µAji

(xi) is fuzzy member function, and hj is the point in R,m is the number of rules. By introducing the concept of fuzzy basis

IET Research Journals, pp. 1–114 c© The Institution of Engineering and Technology 2015

function vector ϕ(x), (22) can be rewritten as

y(x) = λTϕ(x) (23)

where λ = (h1, h2, · · · , hm)T is adjustable parameter vector,ϕ(x) = (ϕ1(x), · · · , ϕm(x))T and ϕj(x) are the fuzzy basis func-tions, which can be defined as

ϕj(x) =

∏ni=1 µAji

(xi)∑mj=1

(∏ni=1 µAji

(xi)) (24)

Assumption 4. Let f(x) be a continuous function defined on a com-pact set Mx. Then, for any constant ε > 0, there exists a fuzzy logicsystem (23) such that sup

x∈Mx

∣∣∣f(x)− λTϕ(x)∣∣∣ < ε.

3.2 Fuzzy disturbance observer

In this subsection, the fuzzy distrubance observer is presented.Consider the following nonlinear system

x = f(x) + g(x)u+ Ω(x, u) (25)

where x is the state which is assumed to be available for mea-surement; f(x) and g(x) are system functions; u is the controlinput; Ω(x, u) is the lumped disturbance including uncertainties andexternal disturbance.

For system (25), the adaptive FDO is designed as follows

z = −σz + σx+ f(x) + g(x)u+ Ω(x, u∣∣∣λ ) (26)

where z is an observer internal state, Ω(x, u∣∣∣λ ) is the estimation

of Ω(x, u). Define the disturbance observation error ζ = x− z. Toconstruct Ω(x, u

∣∣∣λ ), which is guaranteed to monitor an unknown

Ω(x, u), we design adaptative law to tune the parameter vector λ.Firstly, the dynamics of the disturbance observer errors can be givenby

ζ = x− z = −σζ + Ω(x, u)− Ω(x, u∣∣∣λ ) (27)

Following the Assumption 4 and the universal approximationcapability of the fuzzy system, the unknown disturbances Ω(x, u)can be described as follows

Ω(x, u) = Ω(x, u∣∣λ∗ ) + ε (28)

where λ∗ denotes the optimal value of the adjustable parameter vec-tor, ε denotes approximation error of fuzzy system. Ω(x, u |λ∗ ) andΩ(x, u

∣∣∣λ ) can be expressed by

Ω(x, u∣∣λ∗ ) = λ∗Tϕ(x, u)

Ω(x, u∣∣∣λ ) = λTϕ(x, u) (29)

By substituting (28), (29) into (27), the dynamics of the distur-bance observer error become

ζ = −σζ + Ω(x, u∣∣λ∗ )− Ω(x, u

∣∣∣λ ) + ε

= −σζ + λTϕ(x, u) + ε (30)

where λ = λ∗ − λ is parameter estimation error, and ϕ is the fuzzybasis function. Assume the upper bound of ε is ε. Now we presentadaptive update law for λ of the estimator Ω(x, u

∣∣∣λ ).

Theorem 1. If the adjustable parameter vector of FDO is tuned by

˙λ = µ1(ϕζ − µ2λ) (31)

where µ1 and µ2 are positive constants, then the disturbanceobserver error ζ is uniformly ultimately bounded within an arbitrar-ily small region.

Proof: Consider the following Lyapunov function candidate

VF =1

2ζ2 +

1

2µ1λT λ (32)

Differentiating (32) and substituting (30) into derivative of (32)yield

VF = −σζ2 + ζλTϕ+ ζε− 1

µ1λT

˙λ

= −σζ2 + λT[ζϕ− 1

µ1

˙λ

]+ ζε (33)

Substituting (31) into (33), we have

VF = −σζ2 + µ2λT λ+ ζε (34)

Considering the following inequalities

ζε ≤ 1

2σζ2 +

1

2σε2

µ2λT λ ≤ −µ2

2λT λ+

µ2

2λ∗Tλ∗ (35)

Then, we can obtain the following inequality

VF ≤ −1

2σζ2 − µ2

2λT λ+

1

2σε2 +

µ2

2λ∗Tλ∗

≤ −min σ, µ2VF +1

2σε2 +

µ2

2λ∗Tλ∗ (36)

Under the assumption that λ∗ are bounded, we can choose thesuitable parameters σ and µ2 to guarantee VF is negative. Thus, thedisturbance observation error is uniformly ultimately bounded.

4 Control System Design

4.1 Control system structure

The overall control system of AHV with variable geometry inletcan be seen in Fig. 5. The inputs of FDO include flight statesx = [V, γ, α,Q], the two control values uc = [φ, δe] and elonga-tion distance l. The output of FDO is the observed disturbancesΩ = [ΩV ,Ωγ ,Ωα,ΩQ].

Sliding Mode

Controller

Fuzzy

Disturbance

Observer

AHV with Variable

Geometry Inlet

Estimate l

[ , ]ef d=u

l

[ , , , ]V Qd d d dg a=d

[ , , , ]V Qg a=x

[ , ]d dV h [ , ]V h[ , ]V hS S

Ω

Fig. 5: The block diagram of AHV control system

IET Research Journals, pp. 1–11c© The Institution of Engineering and Technology 2015 5

4.2 Design of DSC based FDO control

4.2.1 Velocity controller design: Define the velocity slidingmode surface SV =V − Vd, and its derivative is

SV = gV · φ+ fV + ΩV − Vd (37)

Design the FDO of velocity subsystem as

zV = −σV zV + σV V + fV + gV φ+ ΩV (38)

where zV is the observer internal state; σV is a positive designparameter; ΩV = λTV ϕ

(V )(ZV ) is the estimation of ΩV , ZV =[V, γ, α].

Chose the reaching law as SV = −kV 1SV − kV 2SV /(|SV |+σs). Then, the control signal FER φ can be determined as

φ = g−1V [−kV 1SV − kV 2SV /(|SV |+ σs)− fV

− λTV ϕ(V )(ZV ) + Vd] (39)

where σs is a very small positive constant; kV 1 and kV 2 are positiveconstants.Remark 1. The continuous function SV /(|SV |+ σs) is slidingmode switch item, which can ensure the control law is smooth andderivable.

The adaptive law of estimated parameter λV can be given

˙λV = µV 1[ϕ(V )(ZV )(SV + ζV )− µV 2λV ] (40)

where µV 1 and µV 2 are positive design parameters; ζV = V − zVis disturbance observation error of FDO.

4.2.2 Altitude controller design: Eq.(16) shows that the alti-tude h and FPA γ have a one to one relationship. So we transform thealtitude instruction hd into FPA instruction γd, and kh is a positivedesign constant.

γd =(−kh(h− hd) + hd

)/V (41)

Different from the traditional backstepping design method, theDSC method introduce the low pass filter to avoid the "explosion ofterms" problem exist in backstepping method. The design procedureis shown as follows.Step 1: Define the sliding mode surface of FPA Sγ = γ − γd. Thetime derivative of Sγ is

Sγ = γ − γd = gγα+ fγ + Ωγ − γd (42)

Design the FDO of FPA subsystem as

zγ = −σγzγ + σγγ + fγ + gγα+ Ωγ (43)

where zγ is the observer internal state; σγ is a positive design param-eter; Ωγ = λTγ ϕ

(γ)(Zγ) is the estimation of Ωγ , Zγ = [V, γ, α].Chose the reaching law as Sγ = −kγ1Sγ − kγ2Sγ/(|Sγ |+

σs). Define the virtual control input as follows

αd = g−1γ [−kγ1Sγ − kγ2Sγ/(|Sγ |+ σs)− fγ

− λTγ ϕ(γ)(Zγ) + γd] (44)

where kγ1 and kγ2 are positive design constants. The adaptive lawof estimated parameter λγ is

˙λγ = µγ1[ϕ(γ)(Zγ)(Sγ + ζγ)− µγ2λγ ] (45)

where µγ1 and µγ2 are positive design parameters; ζγ = γ − zγ isdisturbance observation error of FDO. Introduce a new variable αdand let αd pass through a first-order filter with time constant τα toobtain αd and αd as

αd = τααd + αd, αd(0) = αd(0) (46)

Step 2: Define the sliding mode surface of AOA: Sα = α− αd.Differentiating Sα and substituting (18) into derivative of Sα yield

Sα = α− αd = Q+ fα + Ωα − αd (47)

Design the FDO of AOA subsystem as

zα = −σαzα + σαα+ fα + gαQ+ Ωα (48)

where zα is the observer internal state; σα is a positive designparameter; Ωα = λTαϕ

(α)(Zα) is the estimation of Ωα, Zα =[V, γ, α].

Chose the reaching law as Sα = −kα1Sα − kα2Sα/(|Sα|+σs). Define the virtual control input as follows

Qd = −gγSγ − kα1Sα − kα2Sα/(|Sα|+ σs)− fα

− λTαϕ(α)(Zα) + αd (49)

where kα1 and kα2 are positive design constants. The adaptive lawof estimated parameter λα is

˙λα = µα1[ϕ(α)(Zα)(Sα + ζα)− µα2λα] (50)

where µα1 and µα2 are positive design parameters; ζα = α− zα isdisturbance observation error of FDO. We introduce a new variableQd and let Qd pass through a first-order filter with time constant τQto obtain Qd and Qd as

Qd = τQQd +Qd, Qd(0) = Qd(0) (51)

Step 3: Define the sliding mode surface of pitch angle rate: SQ =Q−Qd. Differentiating SQ and substituting (19) into derivative ofSQ yield

SQ = Q− Qd = gQδe + fQ + ΩQ − Qd (52)

Design the FDO of pitch angle rate subsystem as

zQ = −σQzQ + σQQ+ fQ + gQδe + ΩQ (53)

where zQ is the observer internal state; σQ is a positive designparameter; ΩQ = λTQϕ

(Q)(ZQ) is the estimation of ΩQ, ZQ =[V, γ, α,Q].

Chose the reaching law as SQ = −kQ1SQ − kQ2SQ/(∣∣SQ∣∣+

σs). Then, the control signal δe is designed as

δe = g−1Q [−Sα − kQ1SQ − kQ2SQ/(

∣∣SQ∣∣+ σs)− fQ

− λTQϕ(Q)(ZQ) + Qd] (54)

where kQ1 and kQ2 are positive design constants. The adaptive lawof estimated parameter λQ is

˙λQ = µQ1[ϕ(Q)(ZQ)(SQ + ζQ)− µQ2λQ] (55)

where µQ1 and µQ2 are positive design parameters; ζQ = Q− zQis disturbance observation error of FDO.

IET Research Journals, pp. 1–116 c© The Institution of Engineering and Technology 2015

4.3 Stability Analysis

Theorem 2. Consider the AHV colsed-loop system that includes theAHV-VGI with the control inputs (39) and (54), the virtual controls(41), (44) and (49), the command filters (46) and (51), and the FDOsystem (38), (43), (48) and (53). Then, the AHV-VGI control systemis uniformly stable.

Proof: Define the estimation errors λi = λ∗i − λi, i = V, γ, α,Qand filter errors yα = αd − αd, yQ = Qd − Qd. Substituting φ,αd, Qd and δe into SV , Sγ , Sα and SQ respectively, we have

SV = −kV 1SV − kV 2SV /(|SV |+ σs) + λ∗TV ϕ(V )(ZV )

− λTV ϕ(V )(ZV ) + εV

= −kV 1SV − kV 2SV /(|SV |+ σs) + λTV ϕ(V )(ZV ) + εV

(56)

Sγ = gγ(Sα + yα + αd) + fγ + λ∗Tγ ϕ(γ)(Zγ) + εγ − γd= gγSα + gγyα − kγ1Sγ − kγ2Sγ/(|Sγ |+ σs)

+ λTγ ϕ(γ)(Zγ) + εγ (57)

Sα = SQ + yQ − gγSγ − kα1Sα − kα2Sα/(|Sα|+ σs)

+ λTαϕ(α)(Zα) + εα (58)

SQ = −Sα − kQ1SQ − kQ2SQ/(∣∣SQ∣∣+ σs)

+ λTQϕ(Q)(ZQ) + εQ (59)

where εi, i = V, γ, α,Q denote approximation errors of fuzzysystem, assume the upper bounds of εi is εi. Besides

yα = −yατα

+Bα(·) (60)

yQ = −yQτQ

+BQ(·) (61)

where

Bα(·) = g−2γ [(kγ1Sγ + kγ2

σs|Sγ |+ σs

+ fγ +˙λTγ ϕ

(γ)(Zγ)

+∂

˙λTγ ϕ

(γ)(Zγ)

∂VV +

∂˙λTγ ϕ

(γ)(Zγ)

∂γγ +

∂˙λTγ ϕ

(γ)(Zγ)

∂αα

− γd) · gγ + (−kγ1Sγ − kγ2Sγ

|Sγ |+ σs− fγ

− λTγ ϕ(γ)(Zγ) + γd) · gγ ] (62)

BQ(·) = gγSγ + gγ Sγ + kα1Sα + kα2σs

|Sα|+ σs+ fα

+˙λTαϕ

(α)(Zα) +∂

˙λTαϕ

(α)(Zα)

∂VV +

∂˙λTαϕ

(α)(Zα)

∂γγ

+∂

˙λTαϕ

(α)(Zα)

∂αα− αd (63)

Assumption 5. The FPA reference signal γd and its derivatives(γd, γd) are smooth and bounded functions.

Considering Assumption 3 and Assumption 5, Bα(·) and BQ(·)are continuous functions on compact sets Bα(Sγ , Sα, λγ , yα, γ, γ,γ) = pα andBQ(Sγ , Sα, SQ, λγ , λα, yα, yQ, γ, γ, γ) = pQ respec-tively. Therefore, the continuous functions |Bα(·)| and

∣∣BQ(·)∣∣

have maximums on pα and pQ respectively, say |Bα(·)| ≤Mα,∣∣BQ(·)∣∣ ≤MQ, with Mα,MQ > 0.

We consider the following Lyapunov function candidate

W = WV +Wγ +Wα +WQ (64)

where

WV =1

2S2V +

1

2ζ2V +

1

2µV 1λTV λV

Wγ =1

2S2γ +

1

2ζ2γ +

1

2µγ1λTγ λγ +

1

2y2α

Wα =1

2S2α +

1

2ζ2Q +

1

2µα1λTα λα +

1

2y2Q

WQ =1

2S2Q +

1

2ζ2Q +

1

2µQ1λTQλQ (65)

Differentiating WV and referring to the Eq (33) − (36) yield

WV = SV SV + ζV ζV −1

µV 1λTV

˙λV

≤ −kV 1S2V − kV 2

S2V

|SV |+ σs− σV ζ2

V + SV λTV ϕ

(V )(ZV )

+ SV εV + ζV λTV ϕ

(V )(ZV ) + ζV εV −1

µV 1λTV

˙λV

(66)

Noticing that

ζV εV ≤1

2σV ζ

2V +

1

2σVε2V

− kV 2S2V

|SV |+ σs+ SV εV ≤

(|SV |+ σs)ε2V

4kV 2(67)

Subsituting (40) and (67) into (66), we have

WV ≤ −kV 1S2V −

1

2σV ζ

2V −

µV 2

2λTV λV +

1

2σVε2V

+(|SV |+ σs)ε

2V

4kV 2+µV 2

2λ∗TV λ∗V

≤ −rVWV +mV (68)

where rV = min 2kV 1, σV , µV 1µV 2mV = 1

2σVε2V +

(|SV |+σs)ε2V4kV 2

+ µV 22 λ∗TV λ∗V

The time derivative of Wγ along the system trajectories is

Wγ = Sγ Sγ + ζγ ζγ −1

µγ1λTγ

˙λγ + yαyα

≤ SγgγSα + Sγgγyα − kγ1S2γ −

1

2σγζ

2γ +

(|Sγ |+ σs)ε2γ

4kγ2

−µγ2

2λTγ λγ +

µγ2

2λ∗Tγ λ∗γ +

1

2σγε2γ −

y2α

τα+ |yαBα|

(69)

Using Young’s inequality ab ≤ a2/2 + b2/2, we have

Sγgγyα ≤1

2S2γ +

1

2g2γy

2α

|yαBα| ≤y2αM

2α

2ω+ω

2(70)

IET Research Journals, pp. 1–11c© The Institution of Engineering and Technology 2015 7

where ω is a positive constant. And we can find a set of constants(ω, ω0) satisfied

1

τα=

1

2g2γ +

M2α

2ω+ ω0 (71)

then

1

2g2γy

2α −

y2α

τα+ |yαBα| ≤ −(

M2α

2ω+ ω0)y2

α +y2αM

2α

2ω+ω

2

≤ −ω0y2α +

ω

2(72)

we can obtain

Wγ ≤ SγgγSα − (kγ1 −1

2)S2γ −

1

2σγζ

2γ −

µγ2

2λTγ λγ − ω0y

2α

+µγ2

2λ∗Tγ λ∗γ +

1

2σγε2γ +

(|Sγ |+ σs)ε2γ

4kγ2+ω

2

≤ −rγWγ +mγ + SγgγSα (73)

whererγ = min2kγ − 1, σγ , µγ1µγ2, 2ω0

mγ = 12σγ

ε2γ +(|Sγ |+σs)ε2γ

4kγ2+µγ22 λ∗Tγ λ∗γ + ω

2

Similarly, we can obtain the following inequality

Wα ≤ SαSQ − SγgγSα − (kα1 −1

2)S2α −

1

2σαζ

2α

− µα2

2λTα λα − (υ0 −

1

2)y2Q +

µα2

2λ∗Tα λ∗α

+1

2σαε2α +

υ

2+

(|Sα|+ σs)ε2α

4kα2

≤ −rαWα +mα + SαSQ − SαgγSγ (74)

whererα = min2kα − 1, σα, µα1µα2, 2υ0 − 1m2 = 1

2σαε2α +

(|Sα|+σs)ε2α4kα2

+ µα22 λ∗Tα λ∗α + υ

2

where υ and υ0 are positive constants and satisfied

1

τQ=M2Q

2υ+ υ0 (75)

and

WQ ≤ −SαSQ − kQ1S2Q −

1

2σQζ

2Q −

µQ2

2λTQλQ

+1

2σQε2Q +

(∣∣SQ∣∣+ σs)ε

2Q

4kQ2+µQ2

2λ∗TQ λ∗Q

≤ −rQWQ +mQ − SαSQ (76)

where rQ = min2kQ, σQ, µQ1µQ2]

mQ = 12σQ

ε2Q +(|SQ|+σs)ε2Q

4kQ2+µQ2

2 λ∗TQ λ∗Q

Then, from (64), (68), (73), (74) and (76), we can obtain

W = WV + Wγ + Wα + WQ ≤ −rW +m (77)

where 0 < r < minrV , rγ , rα, rQ, m = mV +mγ +mα +mQ > 0. It is implied that on W (Si, ζi, λi, yα, yQ) = p, i =

V, γ, α,Q, W ≤ −rp+m. If r ≥ m/p, it follows that W ≤ 0 onW = p. Accordingly, W ≤ p is an invariant set, if W (0) ≤ p, thenW (t) ≤ p for all t > 0. Therefore, the errors Si, ζi, λi, yα, yQ aresemi-globally uniformly bouned in the following compact set

Υ = Si, ζi, λi, yα, yQ |W ≥ m/r (78)

That means the compact set Υ can be kept arbitarily small byadjusting the design parameters. So, using the designed control law,the AHV-VGI control system is stable.

0 50 100 150 200

7

7.1

7.2

7.3

Time(s)

Vel

ocity

(Mac

h)

ReferenceSMC with FDOBC without FDO

0 50 100 150 200 250−5

0

5x 10

−3

Err

or(f

t/s)

(a)

0 50 100 150 200

7.51

7.53

7.55

x 104

Time(s)

Alti

tude

(ft)

ReferenceSMC with FDOBC without FDO

0 50 100 150 200 250−1

0

1

Err

or(f

t)

(b)

0 50 100 150 200

0.1

0.2

0.3

0.4

Time(s)

FE

R φ

SMC with FDOBC without FDO

(c)

0 50 100 150 2000

1.5

3

4.5

6

7.5

Time(s)

Ele

vato

r de

flect

ion

δ e (de

g)

SMC with FDOBC without FDO

(d)

Fig. 6: Curves of system outputs in the presence of lumped distur-bances. (a)velocity; (b)altitude; (c)FER; (d)elevator deflection

5 Simulation Results

In this section, simulation results are provided to demonstratethe effectiveness of the proposed control scheme and illustratethe superiority of VGI with translating cowl. The initial state atV (0) = 7.0Mach, h(0) = 75000ft, γ(0) = 0deg, α(0) = 0deg

IET Research Journals, pp. 1–118 c© The Institution of Engineering and Technology 2015

and Q(0) = 0deg/s. Considering the following constraints: φ ∈[0, 1], δe ∈ [−15, 15]deg, γ ∈ [−0.5, 0.5]deg, α ∈ [−5, 15]deg.The parameters of control system are shown in Table 3. In all cases,a nonlinear filter is used to generate the differentiable commands asfollows

WV (s) =ωn1ω

2n2

(s+ ωn1)(s2 + 2ξ1ωn2s+ ω2n2)

(79)

where ωn1 = 0.1, ωn2 = 0.3, ξ1 = 30.

0 50 100 150 200−1

1

3

5

7

9

Time(s)

Dis

turb

ance

ΩV

real valuesFDO values

(a)

0 50 100 150 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time(s)

Dis

turb

ance

Ωγ

real valuesFDO values

(b)

0 50 100 150 200

−0.3

−0.2

−0.1

0

Time(s)

Dis

turb

ance

Ωα

real valuesFDO values

(c)

0 50 100 150 200

−0.3

0

0.3

0.6

0.9

Time(s)

Dis

turb

ance

ΩQ

real valuesFDO values

(d)

Fig. 7: Curves of lumped disturbances and FDO estima-tions. (a)Disturbance ΩV ; (b)Disturbance Ωγ ; (c)Disturbance Ωα;(d)Disturbance ΩQ

Case 1. To illustrate the effectiveness and superiority of SMCwith FDO method, both the disturbance rejection performance androbustness are consider. In addition, the baseline control (BC) with-out FDO is employed for performace comparsion analysis. To havea fair comparsion, first the control inputs of the two control schemesare adjusted to the same range of variation, second, the parametersof each control algorithm are regulated such that all the two sys-tems achieve relatively good performance. In this case, the velocitycommand varies from 7Mach to 7.3Mach and the altitude commandvaries from 75000ft to 75500ft.

The external disturbances are taken as dV = 2ft/s at120 < t ≤ 150s, dV = 2 cos(0.2t)ft/s at 170 < t ≤ 250s anddγ = 0.01deg, dα = 0.02deg, dQ = 0.2deg/s at 120 < t ≤150s, dγ = 0.01 sin(0.2t)deg, dα = 0.02 sin(0.2t)deg, dQ =0.2 sin(0.3t)deg/s at 170 < t ≤ 250s for AHV-VGI system. Andthe perturbation of aerodynamc coefficients are assumed as −5%(i.e., ∆ = −0.05) of their normal values. The simulation results areshown in Fig. 6 and Fig. 7.

0 50 100 150 200

7

7.5

8

8.5

9

Time(s)

Vel

ocity

(Mac

h)

ReferenceAHV−VGIAHV−FGI

0 50 100 150 200−20

0

20

Err

or(f

t/s)

(a)

0 50 100 150 200

0.2

0.4

0.6

0.8

1

1.2

Time(s)

FE

R φ

AHV−VGIAHV−FGI

(b)

0 50 100 150 2002000

4000

6000

8000

10000

12000

14000

Time(s)

Thr

ust(

lbf)

AHV−VGIAHV−FGI

(c)

0 50 100 150 2002

3

4

5

6

7

8

9

10

Time(s)

Elo

ngat

ion

leng

th(f

t)

(d)

Fig. 8: Response to a 2Ma velocity command without externaldisturbance. (a)velocity; (b)FER; (c)thrust; (d)elongation length

Fig. 6 shows the curves of AHV-VGI system outputs under twocontrollers. The SMC with FDO method and BC without FDOmethod have different curves during the first 120s because of theexistence of uncertainties introduced by translating cowl and fittingparameters, and the tracking errors of SMC with FDO method aresmaller than the BC without FDO method. When the external dis-turbances act, the SMC with FDO method obtains better robustnessperformance and lumped disturbances rejection properties, while theBC without FDO method results in undesirable control performance.

IET Research Journals, pp. 1–11c© The Institution of Engineering and Technology 2015 9

Fig. 7 shows the curves of the lumped disturbances ΩV , Ωγ , Ωα,and ΩQ and their estimated values obtained by FDO, which demon-strated that the FDO can estimate the lumped disturbances effectiveand had a fast convergence speed.

Case 2. In this case, the velocity command varies from 7Mach to9.2Mach and the altitude command varies from 75000ft to 75500ft.In this case, we do not take into account external disturbances, thatis di = 0, i = V, γ, α,Q. We notice that this velocity commandchanges quickly and has a larger variation range. We compare theresponse results of AHV-VGI and AHV-FGI systems under this sit-uation. In the fast acceleration process, the FER φ is large in orderto provide the required thrust. However, considering φ ∈ [0, 1], theFER of the AHV-FGI will reach its saturated state during the timeof 10-40s (shown in Fig. 8(b)), which causes the velocity trackingerror increased suddenly (shown in Fig. 8(a)) and the response curveof the thrust is distorted (shown in Fig. 8(c)). However, the AHV-VGI control system can provide the required thrust by adjusting themovable cowl, and in a smaller FER value, which greatly reducesthe possibility of the FER saturation.

The elongation length is shown in Fig. 8(d). It is graduallydecreased from 9.5836 to 2.593. With the increasing of the velocity,the shock wave angle decreases, and the oblique shock wave gen-erated by AHV frontbody is assembled to the body, so the movablecowl moves back.

Table 3 Control Parameters

Parameters Values Parameters Values

kV 1, kV 2 0.5, 1.4 kh 3.5kγ1, kγ1 8, 12 kα1, kα2 0.15, 1.4kQ1, kQ1 0.8, 0.4 τα, τQ 0.1, 0.1σs 0.01 σV , σγ 0.2, 0.04σα, σQ 0.1, 0.1 µV 1, µV 2 1.2, 0.8µγ1, µγ2 0.2, 1.7 µα1, µα2 0.5, 0.5µQ1, µQ2 1.2, 1.5

6 Conclusion

In this paper, we established the longitudinal model of AHV-VGI,and the lumped disturbances include the parameter uncertainties, theuncertainties introduced by translating cowl and the external distur-bance are considered. We presented a DSC based on FDO controlscheme for the AHV-VGI with translating cowl. The simulationresults shows the effectiveness of this control scheme and illustratesthe superiority of VGI with translating cowl. The work assumedthe translating cowl can be adjusted infinitely fast. But in fact, thecowl need a setting time. Therefore, how to realize the coordinationcontrol of flight and cowl can be the focus of future work.

7 Acknowledgments

This works was partially supported by the Natural Science Founda-tion of China (NSFC) under Grant (61273092,61673294,61773279)and the Natural Science Foundation of Tianjin under Grant12JCZDJC30300.

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