Wei－Ming Li，Rui－Sheng Sun，Hong－Yang Bai，Peng－Yun Liu

(School of Energy and Power Engineering，Nanjing University of Science and Technology，Nanjing 210094，China)

1 Introduction

Morphing-wing technology has led to a series of breakthroughs in a wide variety of applications with the potential to produce large increments in maneuverability， affordability， and environmental adaptability.Deemed as one instance of the morphingwing applications，the cruise missiles with variableswept wings(VSW)have many advantages，such as high lift/drag ratio，low-energy，rapid maneuvering and mission-optimization.Namely，a“VSW”is a morphing-wing able to drastically change planner shape——about 85%in aspect ratio，580%in area，and 50°variation in sweepback during a flight.The VSW will vary configurations to increase the performance of the metrics associated with each mission segment for accomplishing surveillance，search and interception preferably.Whereas，the optimization of aerodynamic performance and the enhanced maneuverability profited by wings morphing arouse large-scale parameter perturbations and additional nonlinearcross-coupling among the pitch-yaw-roll channels，in the meantime，which raise intractable conflicts and challenges in the design of BTT autopilot.

In order to achieve strong adaptability and adequate performanceover theentireenvelope of operating conditions，the BTT autopilot of the cruise missiles must be nonlinear in the process of morphing.The nonlinearity can arise through the direct application of anonlinearcontroltechnique to the problem.Nonlinear control methods which have been used for BTT autopilot design include dynamic inversion［1］，intelligent control［2］，optimal control［3］and feedback linearization theory［4－5］.Whereas the above nonlinear control approaches have not been commonly used in real implementation on the flight control systems on account of the complex computing process.Moreover，feedback linearization requires the complete knowledge ofsystem parameters.During the actualflight，aerodynamic derivatives related to varieties of flight conditions and configuration parametersaccompany some degree of time-variation characteristic and perturbations.Considerable researches have been done for the design of robust autopilot in the presence of parametric perturbations based on the sliding mode control(SMC)theory［6－7］.This approach provides a discontinuous control law using the knowledge of the bounds on uncertain functions，and requires certain constraints on the norm of the uncertain portion of input matrices.Although SMC that is insensitive to parameter perturbations and externaldisturbances［8－10］is an efficient control method with good robustness properties to stabilize systems with nonlinearity and uncertainty features，pure sliding mode control presents drawbacks including large control requirements and chattering［11－12］.To eliminate the chattering and get better time domain performances，we present an adaptive sliding mode controll(ASMC)method which applies a parametric adaptation law to the SMC design.Under such control scheme， the trending parameters are adjusted adaptively and the control inputs are verified to be appropriate.Keum［13］had proposed an ASMC system for climbing turn and helical path following control of a simplified F/A－18 model in spite of large parameter uncertainties.Rong-jun Y［14］had studied a robust adaptive controller to enhance the large-space combat capability for guided projectile which combined a parameter estimation scheme with SMC.

The contribution of this paper lies in the design of an ASMC for BTT autopilot of cruise missiles with VSW.The roll angle，normal overloads and angular rates are used as state variables of the autopilot，and a parametric sliding mode controller is obtained via feedback linearization. Nonlinear time-varying parameter perturbations caused by VSW are estimated on-line through a proposed parametric adaptation law.The simulation results for the roll angle and overload commands tracking in different configuration schemes are presented.The results indicate that the proposed controlleris robust with the large aerodynamic perturbations engendered by wings morphing，and has excellentdynamic tracking performances with no control chattering.

2 Mathematical DynamicEquationsofBTT Autopilot

When the cruise missiles with VSW are simultaneously in both rapid roll and overload vertical plane maneuver， the aerodynamic， inertial and kinematics coupling among the pitch-yaw-roll channels should be considered in designing the BTT autopilot.For the convenient analysis，supposing angle-of-attack α and sideslip angle β are both small，and the missile speed V is constant during the short period，neglecting the second order smallquantities and actuators dynamics，the dynamic equations of BTT autopilot in aerodynamic coefficients form are described as follows

with the pitching aerodynamic coefficients

the yawing

and the rolling

where ωz，ωyand ωxare the pitch，yaw and roll rate of the body coordinate with respectto the inertial coordinate，respectively;γ is the roll angle and ny，nzindicate the normal overloads of the pitch and yaw channel.m，S，L and q denote its mass，reference surface， reference chord and dynamic pressure severally.Jiand δiare momentsofinertia and deflections of each channel;ηRD=180/π is the scale factor concerting from radians to degrees.and，the static or dynamic aerodynamic derivatives， are implicit nonlinear functions of the related variables such as α，ωiand δi，in particular the sweepbackχand mach number Ma.The fluctuation of nonlinear aerodynamic characteristics is obvious under various wing configurations.The reference values of aerodynamic derivatives acquired through the data from wind-tunnel flow test owing to the change rule is partly unknown.andare obtained by nonlinear interpolation using actual sampling value ofχmeasured by the sensors in rotary actuator，approximatively，during a flight.Synchronously，the controlparameterscan achieve on-line adjustment.

3 Feedback Linearization Model of SMC

The purpose of the BTT autopilot design is to achieve the system outputs possessing desired dynamic characteristic and track overload commands accurately.For the nonlinear system Eq.(1)with bounded uncertainties，invariance property ofsliding mode motion can ensure the system taking on an antiinterferences performance to some extent when sliding manifold is asymptotically stable.

Considering Eq.(1)as a nonlinear MIMO system with input-affine and non-square，and regarding some related deviation items as the perturbations and external disturbances with a general form，then，the equations can be written as

where，x=［γ，ny，nz，ωx，ωz，ωy］Tis the state vector; u=［δx，δz，δy］Tdenotes the control vector of a nonsquare system and y=［γr，nyr，nzr］Tis the output vector.^f(x)，^G(x)are，respectively，the estimated function vector and estimated gain matrices with x of the actual system model，concretely.

The parameter perturbation items are represented by Δf(x)and ΔG(x)，obviously，

and unknown vector d indicates external disturbances and modeling errors.Δai|χ，Δbi|χ(i=1，…，5)and Δcj|χ(j=1，2，3)are aerodynamic coefficient errors caused by measuring and changing ofχ.

It is apparent that the relative order of system Eq.(2)is ρ=［2，2，2］T.If the system satisfies the following conditions:LΔfh=0，LΔGh=0;‖Δf(x)‖and‖ΔG(x)‖ with separated uncertainty bounds; equivalent external disturbance term ‖~d‖ =‖LdL^fh‖ ＜D，(D＞0)，hence，Eq.(2)can be converted into the following linearized form

with

and nonlinear decoupling matrix

and

and

The key to SMC design is to acquire an appropriate sliding surface which should ensure the stabilization of sliding mode asymptotically and the tracking error approaches zero，synchronously.The sliding surface is defined as

with a given reference command yr=yc－e and a positive definite diagonal gain matrix λ=diag(λγ，λny，λnz).e is the tracking output error and s=［sγ，sny，snz］Tis so-called sliding mode variable.

In the presence of uncertainty，a discontinuous control law is used for accomplishing sliding motion.The derivative of Eq.(8)leads to the following

where，

K=diag(Kγ，Kny，Knz)，Ki＞0 is a positive diagonal matrix and ΔF=LΔfL^fh.Moreover，，ΔF andare used for judging the asymptotic stability of the sliding mode.

So that，the control vector u can be expressed as

Taking Lyapunov function as

By substituting Eqs.(8)，(9)and(10)into Eq.(11)，we have the following convergence conditions of sliding mode

4 Design of ASMC for Missiles with VSW

For cruise missiles with VSW，which fly in full envelope with variable geometry and obvious aerodynamic nonlinear characteristics，a SMC autopilot is able to response quickly and robust to fast variable perturbations and external disturbances.Whereas，the sliding mode cannotoccur in switching surface accurately caused by switching delay of the real system，and switching chattering may excite the high frequency uncertainties，ineluctably.A desired BTT autopilotshould be adaptive with various flight conditions，therefore，we smooth the discontinuity of control variable adopting a sliding mode boundary layer theory and present a parametric adaptation law with adjusting control parameters in real time via estimating aerodynamic parameter perturbations online from metrical informations.

Defining μ and φ as the thickness and width of boundary layer and replacing sign(*)with the continuous saturation function sat(*)，Eq.(10)can be rewritten as

where，

Forguaranteeing the system robustnessin perturbation and disturbance， the magnitude of reaching mode should be taken into account.Therewith，we redefine the sliding surface as

Substituting Eqs.(13)and(14)into Eq.(3)，we have

A vectorform of γ integrating inputand parameter-error vector can be expressed as

where，the parameter-error vectorthe prior aerodynamic parameters vector υ=［c，a，b］Twith c=［c3，c1，c2］T，a=［a3，a1，a2，a4］T，b=［b3，b1，b2，b4］T.is the estimated value of υ.The nonlinear term matrix M(x)=［Mc，Ma，Mb］with

where，

It seems clear that~s=0 within the boundary layer but on the outside，

Retaking Lyapunov function as

with a positive diagonal weight coefficient matrix Γυ.

Differentiating the above function gives

A parametric adaptation law with on-line estimating nonlinear parameter perturbations is presented as

where，Γυ=diag(τi)，(i=1，2，…11)is a weight coefficients matrix;denotes an adaptive gain of corresponding aerodynamic parametric estimateis an internal estimate and σ is an adaptive factor，σ＞0.

It is readily proved that a parametric adaptation law like Eq.(20)can insure(η＞0)under the control condition of using Eq.(13).That is，the tracking errors can guaranteed to converge within boundary layer region according to the Barlalat lemma［15］.The parametric adaptation law presented in this paper regulates τithrough altering factor σ，appropriately.A less σ might lead to overlarge transient tracking errors，while a larger σ maybe bring about a parametric chattering phenomenon and even instability，inversely.To ensure aerodynamic parameters vector υ converging to the true value，an adaptive updating strategy applicable for bounded c，a and b is adopted: stop renovating the parametric adaptation law ifor，while persist in initiating it on the contrary.

5 Simulation and Analysis

To demonstrate the effectiveness and robustness of the proposed method，several numerical contrastive simulations of tracking performances in two different wings configuration schemes are conducted.One case is to compare those results obtained by the traditional SMC approach and the proposed ASMC method on invariable flight conditons and fixed configuration in the process of simulation.The other case is to compare those response performances and robust stability to aerodynamic perturbations in the given morphing scenario.

Static parameters ofthe missile and flight conditions at a certain feature point for the simulation are listed in Tables 1 and 2，severally.It should point out that the parameters listed in Table 1 are constant about wings morphing in order to facilitate simulation，merely.

Table 1 Static parameters of the cruise missile

Furthermore，we consider the actuator as a fistorder dynamic model and limit deflection angle within ±30°so as to describe the dynamic behavior of the proposed controller.

Table 2 Flight conditions of the simulation

In practice，the measurement errors of sensors and wings morphing would affect the aerodynamic derivatives in various disturbances during a flight.To validate the robustness of the proposed ASMC for parameter perturbations， referential aerodynamic derivatives acquired from wind-tunnel flow test corresponding to differentχare provided in Table 3.

Table 3 Referential aerodynamic derivatives in different configurations

5.1 Fixed Configuration Case

For the command tracking performances assessment，keep the sweepback equivalent to 55°in this section，unvaryingly.It is assumed that there is a 5°－10°constant error in measurements of the servo mechanism and aerodynamic coefficients ofeach channel are perturbed independently.For the static or dynamic aerodynamic derivatives correspond to ai，biand cj，the bounds of uncertain from nominal values are supposed asandin this simulation.

The regulation parameters of the proposed ASMC are chosen as follows:switch gain ε=0.25，boundary layer’s thickness μ=0.005，adaptive factor σ=0.6，sliding mode gain matrix λ=diag(2，3，3)and trending rate matrix K=diag(4，5，5).The initial values of ny，nzand γ are given by 0.5 g，0 g and 0°.The tracking performances in response to a sinusoidal commands nycand a zero-input response to nzcare shown in Fig.1，and the roll angle command tracking response is indicated in Fig.2.The nominalperformance ofthe two controllersis similarin tracking accuracy，with acceptable responses.Because here we select small μ in order to avoid conspicuous tracking errors and enhance control effect.This，however，comes at the expense of inevitable control saturation chattering.

Fig.1 Normal overload commands tracking responses in fixed configuration case

Fig.2 Roll angle command tracking responses in fixed configuration case

The corresponding angle of attack and sideslip are plotted in Fig.3.As can be seen from Fig.3，the sideslip angle in the ASMC can be kept within±3° during the maneuver and its trend is more stable than SMC.In addition，the angular rate of each channel is shown in Fig.4.There is a certain chattering in the response curves of SMC，especially in ωz，owing to a small μ secleted before.Acceptable small fluctuations only occur with abrupt changes of commands in the ASMC.

Figs.5 and 6 denote the defection angel outputs of SMC and ASMC， respectively. As previously mentioned，the secletion of μ can cause the conflict between tracking accuracy and input amplitude.Besides，the defection output with high-frequency chattering is difficult to achieve in real systems with switching delay.Essentially，the proposed ASMC can eliminate the high-frequency chattering of the control signals，and the above ofwhich exemplify that aerodynamic parameter errors can be adaptive estimated and fully indemnified effectively.

Fig.3 Angle of attack and sideslip histories in fixed configuration case

Fig.4 Angular rate histories in fixed configuration case

Fig.5 Defection angel outputs using SMC in fixed configuration case

Fig.6 Defection angel outputs using ASMC in fixed configuration case

The comparison between the two sliding surface curves is drawn in Figs.7 and 8.It is clearly observed that the oscillation occurred in each sliding mode variable and controlinputsbeyond the boundary layer’s thickness cannot be realized smooth in the SMC.ASMC makes the tracking errors converge within the boundary layer during sliding phase which eliminates the chattering phenomenon，and guarantees the stability of the system.

Fig.7 Sliding surface curves in SMC in fixed configuration case

Fig.8 Sliding surface curves in ASMC in fixed configuration case

5.2 Variational Sweepback Case

This section presents the simulation results for contrasting the robust to aerodynamic perturbations and disturbances during the morphing.It is supposed that the measuring errors still exist and actual motion of VSW driven by the shaft in uniform driving mode is given.The change curves ofχin 5s is depicted in Fig.9.The reference points correspond to the values of χprovided in Table 3.The regulation parameters of the proposed ASMC are chosen as follows:switch gain ε= 0.35，boundary layer’s thickness μ=0.05，adaptive factor σ=0.75，sliding mode gain matrix λ=diag(3，5，5)and trending rate matrix K=diag(5，8，8).The initial values of ny，nzand γ are the same as those in Section 5.1.The tracking performances of normal overloads are shown in Fig.10，and the roll angle command tracking response is indicated in Fig.11.

Fig.9 Actual sweepback history in 5 s

Fig.10 Normal overload commands tracking responses in variational sweepback case

Within the morphing phase，the nominal performance of the ASMC is significantly better than the SMC which exhibits an unacceptable tracking error.There are some degree of saltus steps in the responses for both controllers on account of the discontinuity of sampling values at the reference switching points.But then，the presented ASMC can allow the reponse to track the command signals fleetly.The illustrations show that the purposed design method can maintain robust stability of aerodynamic perturbations affected by the wings morphing during a flight.

Fig.11 Rollanglecommand tracking responsesin variational sweepback case

Asshown in Fig.12，itwhich showsthe corresponding angle of attack and sideslip，and the sideslip angle in the ASMC can be kept within±3° during the changing planner shape of VSW，which satisfies the design requirements of BTT autopilot.However，the sideslip angle in the SMC exceeds 5° and even more through most of morphing process.The angular rate of each channel is shown in Fig.13 and chattering still occurs in these angular state variables for the response of SMC.

Fig.12 Angle of attack and sideslip histories in variational sweepback case

The control defections during changing the shape of wings are shown in Figs.14 and 15.These plots clearly show thatthe SMC ismore sensitive to aerodynamic perturbations in each channel than the ASMC.In the SMC，parametric perturbations and boundary layer constraint of sliding surface cause significant high-frequency variation difficult to ensure the full implementation.The proposed ASMC can provide a much smoother and less input amplitude control signal to obtain a better performance results.

Fig.13 Angular rate histories in variational sweepback case

Fig.14 Defection angel outputs using SMC in variational sweepback case

Fig.15 Defection angel outputs using ASMC in variational sweepback case

Figs.16 and 17 illustrate that control inputs beyond the boundary layer’s thickness cannot be realized smooth in SMC within the morphing phase and the oscillation occurred simultaneous with the angular rates. Nevertheless， the presented parametric adaptation law estimates uncertain aerodynamic parameters in ASMC which makes the tracking errors converge within the boundary layer during sliding phase，after adaptive gains were adjusted steadily.Thus，the robustness to perturbations and disturbances can be guaranteed while the wings drastically change planner shape.

Fig.16 Sliding surface curves in SMC in variational sweepback case

Fig.17 Sliding surface curves in ASMC in variational sweepback case

Figs.18－20 plot the curves of aerodynamic estimation values.Sampling estimates are basically matched the reference values at the reference points and the overall estimated trend meets the varying regular of each aerodynamic coefficient.All of above show the correctness of the parametric adaptation law ever designed with good stability and quick convergence during the wings morphing.

Fig.18 On-line estimated parameter vector c

Fig.19 On-line estimated parameter vector a

Fig.20 On-line estimated parameter vector b

6 Conclusions

An adaptive sliding mode control method with online updated unmodeled dynamics and uncertain aerodynamic parameters had been developed for the BTT autopilot such a nonlinear multi-variable coupling system.We chose the roll angle，normal overloads and angular rates as state variables，and a parametric sliding mode controller was obtained via feedback linearization procedure.The parametric adaptation law was designed based on Lyapunov stability theory in order to estimate the nonlinear parameter perturbations and uncertainties in real time.Numerical simulations had been performed for tracking commands.These methodologies were compared on a variety of issues，primarily achievable performance， robustness to uncertainty in the aerodynamic coefficients.The performance ofthe proposed method had been demonstrated by applying it to a fixed and a variational sweepback scheme.The results indicated that the proposed controllerwasrobustto the large-scale aerodynamic parametric perturbations engendered by wings morphing，and had the excellentdynamic tracking performance with no large deflection angle and switch chattering.Recommendations for future research include a more detailed analysis of the robustness properties of the ASMC design.

［1］Ismail H，Abdulrahman H B.Nonlinear generalized dynamic inversion aircraft control.Proceedings of the AIAA Guidance，Navigation and Control Conference.Washington DC:AIAA，2011.6638.

［2］Neha G，Akhilesh J，Jeffrey M，et al.Intelligent Control of a Morphing Aircraft.Proceedings of the 48thAIAA/ASME/ ASCE/AHS/ASC Structures，Structural Dynamics，and Materials Conference on Decision and Control.Washington DC:AIAA，2007.1716.

［3］Mickael L，F(xiàn)rank J.Optimal control applied to aircraft longitudinal axis for energy trajectory recovery.Proceedings of the AIAA Guidance，Navigation and Control Conference.Washington DC:AIAA，2011.6257.

［4］Jason T P，Andrea S，Stephen Y.Approximate feedback linearization of an air-breathing hypersonic vehicle. Proceedings of the AIAA Guidance，Navigation and Control Conference and Exhibit.Washington DC:AIAA，2006.6556.

［5］Prasenjit M，Steven L W.Direct adaptive feedback linearization for quadrotor control.Proceedings of the AIAA Guidance，Navigation and ControlConference.Washington DC: AIAA，2012.4917.

［6］Shkolinikov I A，Shtessel Y B.Aircraft non-minimum phase control in dynamic sliding manifolds.Journal of Guidance，Control，and Dynamic，2001，24(3):566－567.

［7］Singh S N，Steinberg M L，Page A B.Nonlinear adaptive and sliding mode flight path control of F/A－18 model.IEEE Transactions on Aerospace and Electronic Systems，2003，39(4):1250－1261.

［8］Shtessel Y，Hall C J.Reuable launch vehicle control in multiple-time-scale sliding modes.Journal of Guidance，Control，and Dynamic，2000，23(6):1013－1020.

［9］Shtessel Y，Buffington J，Banda S.Multiple timescale flight control using reconfigurable sliding modes.Journal of Guidance，Control，and Dynamic，1999，22(6):873－883.

［10］Zarchan P.Tactical and Strategic Missile Guidance.Washington DC:AIAA，1997.345－356.

［11］Hall C E，Shtessel Y B.Sliding Mode Control System Using Sliding Mode Observers and Gain Adaption.Washington DC:AIAA，2003.5437.

［12］Koorondi P，Young D，Hasimoto H.Sliding mode based disturbance observer for motion control.Proceedings of the 37thIEEE Conference on Decision and Control.Piscataway: IEEE，1998.1926－1927.

［13］Lee K W.Adaptive sliding mode 3－D trajectory control of F/A－18 model via SDU decomposition.Proceedings of the AIAA Guidance，Navigation and Control Conference and Exhibit.Washington DC:AIAA，2008.6460.

［14］Yang Rongjun，Yang Hua，Liang Xudong，et al.Control system design for guidance projectile based on adaptive sliding mode control.Journal of Ballistics，012，4(3):75－79.

［15］Slotine J J E，Li W.Applied Nonlinear Control.Beijing: China Machine Press，2004.