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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 560785, 10 pageshttp://dx.doi.org/10.1155/2013/560785Research ArticleNonlinear Adaptive Equivalent Control Based onInterconnection Subsystems for Air-Breathing HypersonicVehiclesChaofang Hu and Yanwen LiuSchool of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, ChinaCorrespondence should be addressed to Chaofang Hu; cfhu@tju.edu.cnReceived 14 May 2013; Accepted 22 July 2013Academic Editor: Tao ZouCopyright 2013 C. Hu and Y. Liu. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.For the nonminimum phase behavior of the air-breathing hypersonic vehicle model caused by elevator-to-lift coupling, a nonlinearadaptive equivalent control method based on interconnection subsystems is proposed. In the altitude loop, the backstepping strategyis applied, where the virtual control inputs about flight-path angle and attack angle are designed step by step. In order to avoid theinaccurately direct cancelation of elevator-to-lift coupling when aerodynamic parameters are uncertain, the real control inputs,that is, elevator deflection and canard deflection, are linearly converted into the equivalent control inputs which are designedindependently. The reformulation of the altitude-flight-path angle dynamics and the attack angle-pitch rate dynamics is constructedinto interconnection subsystems with input-to-state stability via small-gain theorem. For the velocity loop, the dynamic inversioncontroller is designed. The adaptive approach is used to identify the uncertain aerodynamic parameters. Simulation of the flexiblehypersonic vehicle demonstrates effectiveness of the proposed method.1. IntroductionHypersonic vehicles have a promising prospect in bothmilitary and commercial applications as its flight speed canbe more than 5 times of the speed of sound. However, sincethe model of hypersonic vehicle is nonlinear, multivariable,uncertain, and coupling [1], it is unstable and extremely sensitive to changes in flight condition and parameters. This bringsa great challenge to controller design [2]. At present, mostresearches focus on dealing with nonlinearity and uncertaintyof hypersonic vehicles. For example, linear control methodsare attempted according to linearized hypersonic vehiclemodels, such as pole placement techniques [3], LQR method[4], linear output feedback control [5], and LPV control [6].In addition, nonlinear control strategies are widely used aswell, such as feedback linearization approach [7], slidingcontrol [8, 9], and backstepping technique [10]. For uncertainty of hypersonic vehicles, besides adaptive approaches[11], robust strategies are common tools, for example, 𝜇synthesis, 𝐻 control [12], stochastic robustness control [13],and nonlinear disturbance observer-based robust control[14]. Although these methods are proven to be effective, theydo not usually consider the coupling problems existing inhypersonic vehicles. These problems lead to more difficultiesin the flight controller design. In an air-breathing hypersonicvehicle, it is known that there are structural dynamics, flexibleeffect, elevator-to-lift coupling, and the coupling betweenthrust and pitch moment, where elevator-to-lift coupling isnot neglectable, and it will generate unstable zero dynamicsexponentially, that is, the nonminimum phase behavior inpitch rate model, if the controller is designed directly by theinversion.With regard to the elevator-to-lift coupling problem,some strategies have been tried. The basic method usuallyignores this coupling, and then the nonminimum phase canbe removed from the model during the controller design [2],where this coupling is only regarded as unmodeled dynamics.However, this manner cannot ensure the stability of thecontrol system. The other common approach is to offset theinfluence of the coupling. For example, a canard is adoptedto cancel the influence of elevator on lift, and an adaptiverobust controller based on nonlinear sequential loop-closure

2Journal of Applied Mathematicsapproach is developed [15, 16]. Nevertheless, the changes ofthe uncertain parameters are not considered. This may resultin the inaccurate cancellation, which means elevator and liftare not decoupled completely. Simultaneously, this approachhas an adverse influence on the pitch rate dynamics sinceits inputs also consist of elevator and canard. In addition,for thermal protection problem resulted from the canard,only the elevator is taken as aerodynamic control surfacein reference [17]. The system model is transformed intothe interconnection of systems in feedback and feedforwardforms to eliminate the nonminimum phase. But the robustness with regard to uncertainty of the hypersonic vehiclemodel is not addressed totally.From the analysis, we know that adding canard controlsurface is an effective and simple way to suppress the nonminimum phase behavior, even though the strict cancellationof the elevator-to-lift coupling cannot be realized actually.In this paper, the flexible air-breathing hypersonic vehiclemodel is considered. For the tracking requirement of altitudeand velocity, a nonlinear adaptive equivalent control methodbased on interconnection subsystems is proposed by incorporating canard. Firstly, in the altitude loop, the virtual controlinputs about flight-path angle and attack angle are designedstep by step according to the backstepping strategy. Secondly,the terms about the real control inputs, that is, the elevatorand canard deflection in the flight-path angle dynamicsand the pitch rate dynamics, are linearly converted into theequivalent control inputs instead of direct cancelation ofthe elevator-to-lift coupling. By designing the new inputsindependently, the altitude control loop is reformulated. Andthe adaptive technique is used to identify the uncertain aerodynamic parameters. Then the interconnection subsystemsincluding the altitude-flight-path angle dynamics and theattack angle-pitch rate dynamics are constructed. Via thesmall-gain method, the system is proven to be input-to-statestable. In the velocity loop, the adaptive dynamic inversioncontroller is designed. Simulation results show the power ofour approach.In Section 2, the air-breathing hypersonic vehicle modelis presented. The nonlinear adaptive equivalent control basedon interconnection subsystems is introduced in Section 3.Section 4 presents the simulation. The conclusion is drawnin Section 5.𝛼̇ 𝑄 𝛾,̇𝑀,𝑄̇ 𝐼𝑦𝑦2𝜂𝑖 𝑁𝑖 ;𝜂𝑖̈ 2𝜍𝑚 𝜔𝑚,𝑖 𝜂𝑖̇ 𝜔𝑚,𝑖(1)where 𝑚, 𝐼𝑦𝑦 , 𝑔 represent mass of the aircraft, moment ofinertia, gravitational acceleration; damping ratio and naturalfrequency of the flexible motion are denoted by 𝜍𝑚 and 𝜔𝑚,𝑖 ,respectively; 𝑇, 𝐷, 𝐿, and 𝑁𝑖 and 𝑀 are thrust, drag, lift,generalized forces and moment𝐿 𝑞𝑆𝐶𝐿 ,𝑇 𝑞 (𝐶𝑇,𝜙 𝜙 𝐶𝑇 ) ,𝐷 𝑞𝑆𝐶𝐷,In this study, the flexible air-breathing hypersonic vehiclemodel [18] is considered. This model is composed of fiverigid-body states, that is, velocity 𝑉, altitude ℎ, flight-pathangle 𝛾, attack angel 𝛼, pitch rate 𝑞, and six flexible states,that is, 𝜂1 , 𝜂1̇ , 𝜂2 , 𝜂2̇ , 𝜂3̇ , and 𝜂3̇ . The equations of motion arewritten as𝑇 cos 𝛼 𝐷𝑉̇ 𝑔 sin 𝛾,𝑚ℎ̇ 𝑉 sin 𝛾,𝛾̇ 𝑇 sin 𝛼 𝐿 𝑔 cos 𝛾 ,𝑚𝑉𝑉(2)𝑀 𝑧𝑇 𝑇 𝑞𝑆𝑐𝐶𝑀,𝑁𝑖 𝑞𝐶𝑁𝑖 .The aerodynamic parameters in the above formulation aredescribed as follows:𝛿𝛿Δ𝜏Δ𝜏𝐶𝐿 𝐶𝐿𝛼 𝛼 𝐶𝐿𝑒 𝛿𝑒 𝐶𝐿𝑐 𝛿𝑐 𝐶𝐿0 𝐶𝐿 1 Δ𝜏1 𝐶𝐿 2 Δ𝜏2 ,𝛿Δ𝜏Δ𝜏𝛼𝛿𝑐0𝐶𝑀 𝐶𝑀𝛼 𝐶𝑀𝑒 𝛿𝑒 𝐶𝑀𝛿𝑐 𝐶𝑀 𝐶𝑀 1 Δ𝜏1 𝐶𝑀 2 Δ𝜏2 ,𝛿𝛿Δ𝜏Δ𝜏𝛼0𝐶𝑁𝑖 𝐶𝑁𝛼 𝐶𝑁𝑒𝑖 𝛿𝑒 𝐶𝑁𝑐𝑖 𝛿𝑐 𝐶𝑁 𝐶𝑁𝑖 1 Δ𝜏1 𝐶𝑁𝑖 2 Δ𝜏2 ,𝑖𝑖(𝛼 Δ𝜏1 )2𝐶𝐷 𝐶𝐷2(𝛼 Δ𝜏1 )(𝛼 Δ𝜏1 ) 𝐶𝐷𝛿2𝛿𝛿𝛼𝛿Δ𝜏(𝛼 Δ𝜏1 ) 𝐶𝐷 2 Δ𝜏22𝛼𝛿𝛿𝑐 2 𝐶𝐷𝑒 𝛿𝑒2 𝐶𝐷𝑒 𝛿𝑒 𝐶𝐷 𝑒 𝛼𝛿𝑒 𝐶𝐷𝛿𝑐0 𝐶𝐷𝑐 𝛿𝑐 𝐶𝐷 𝑐 𝛼𝛿𝑐 𝐶𝐷,𝛼𝑀 2𝑀 2𝛼 2 20𝛼 𝐶𝑇,𝜙 𝛼𝑀 𝐶𝑇,𝜙 𝑀 𝐶𝑇,𝜙𝐶𝑇,𝜙 𝐶𝑇,𝜙Δ𝜏2𝛼Δ𝜏Δ𝜏 𝐶𝑇,𝜙 1 𝛼Δ𝜏1 𝐶𝑇,𝜙1 Δ𝜏12 𝐶𝑇,𝜙1 Δ𝜏1 ,𝑀 22. Air-Breathing Hypersonic Vehicle Model𝑖 1, 2, 3,𝐴Δ𝜏 2 𝐶𝑇 𝑑 𝐴 𝑑 𝐶𝑇 1 Δ𝜏1 𝐶𝑇0 ,𝐶𝑇 𝐶𝑇𝛼 𝛼 𝐶𝑇 𝑀 (3)where the control inputs are fuel-to-air ratio 𝜙, elevator deflection 𝛿𝑒 , and canard deflection 𝛿𝑐 ; 𝑞, 𝑆, 𝑧𝑇 , 𝑐, 𝑀 denote dynamic pressure, reference area, thrust moment arm,mean aerodynamic chord, and Mach number; Δ𝜏1 and Δ𝜏2are the forebody turn angle and the aftbody vertex anglewhich are linear mapping of elastic states 𝜂𝑖 .In (3), the elevator-to-lift coupling orients from that 𝐶𝐿includes the term of 𝛿𝑒 , which leads to the nonminimumphase behavior. If 𝛿𝑒 is designed by the dynamic inversiondirectly, the pitch rate dynamics will become a hyperbolicsaddle equilibrium. This unstable zero dynamic brings greatdifficulties to the controller design.

Journal of Applied Mathematics3where 𝐶1 and 𝛽1 are the terms containing the uncertainaerodynamic parameters. They can be expressed as thefollowing equations:3. Nonlinear Adaptive EquivalentControllers DesignIn order to track the altitude and velocity command signalsℎref and 𝑉ref , two controllers will be designed independentlyfor the altitude loop and the velocity loop. During thecontroller design, the flexible motion is viewed as externalperturbation, and its influence on aerodynamic model (3) isneglected.3.1. Altitude Controller. In the altitude loop, the controller isdesigned according to the backstepping approach. Then thevirtual control inputs about flight-path angle and attack angleare determined, respectively.For the altitude dynamics, let ̃ℎ ℎ ℎref ; then its errordynamics is written in the following:̇ℎ̃ 𝑉 sin 𝛾 ℎ̇ ref 𝑉𝛾 ℎ̇ ref .𝛽1 𝜃1𝑇 𝜉2 𝑘̃ℎ ̃ℎ ℎ̇ ref,𝑉𝑇 sin 𝛼 𝐿 𝑔 cos 𝛾 𝛾𝑑̇ .𝛾̃̇ 𝑚𝑉𝑉𝜃1 , 𝜃2 are vectors of the uncertain parameters𝛼𝑀 2𝑀 2𝛿𝐶𝑇0 ). As the variation range of the attack angle issmall, (6) will be expanded around the final expectation 𝛼 .To handle the nonminimum phase problem, the MIMOequivalent method is applied in this paper, which is differentfrom the previous research results [17]. The terms about theelevator and canard deflection are linearly equivalent to thecontrol input vector U [𝑈1 , 𝑈2 ]. The error model of theflight-path angle (6) can be rewritten as𝛿𝐶 𝑒 𝛿𝑒𝐶 𝐶 𝛿𝐶 𝑇𝛼 sin 𝛼 𝑇0 sin 𝛼 𝑞𝑆𝐶𝐿𝛼 𝛼̃̇ 𝑞𝑆 𝐿 𝑞𝑆 𝐿𝛾𝑚𝑉𝑚𝑉𝑚𝑉𝑞𝑆𝐶𝐿0 𝑚𝑔 cos 𝛾 𝑚𝑉𝛾𝑑̇ 𝑚𝑉𝛿𝛿𝐶 𝑒 𝛿𝑒 𝐶𝐿𝑐 𝛿𝑐 𝑇 sin 𝛼 𝑇𝛼 cos 𝛼 𝑇0 cos 𝛼 𝑞𝑆𝐶𝐿𝛼𝛼 𝑞𝑆 𝐿 𝑚𝑉𝑚𝑉𝛿and 𝜉1 , 𝑖 1 . . . 3 are regressors𝜉1 𝑞[(sin 𝛼 𝛼 cos 𝛼 ) 𝜙; (sin 𝛼 𝛼 cos 𝛼 ) ;𝑚𝑉 2 2(sin 𝛼 𝛼 cos 𝛼 ) 𝑀 𝜙; cos 𝛼 𝑀 𝜙; 2cos 𝛼 𝜙; cos 𝛼 𝑀 ; cos 𝛼 𝐴 𝑑 ; cos 𝛼 ; 𝑆; 0] ,𝜉2 𝑞222 2𝜙;[ 𝛼 cos 𝛼 𝜙; 𝛼 cos 𝛼 ; 𝛼 cos 𝛼 𝑀 𝑚𝑉 2(sin 𝛼 𝛼 cos 𝛼 ) 𝑀 𝜙; (sin 𝛼 𝛼 cos 𝛼 )𝜙; 2(sin 𝛼 𝛼 cos 𝛼 ) 𝑀 ; (sin 𝛼 𝛼 cos 𝛼 )𝐴 𝑑 ;(sin 𝛼 𝛼 cos 𝛼 ) ; 0; 𝑆] ,𝜉3 𝑞𝑆[𝛿 ; 𝛿 ] .𝑚𝑉 𝑒 𝑐(10)Therefore the dynamics (7) is reformulated as𝑔 cos 𝛾𝛾̃̇ 𝜃1𝑇 𝜉1 𝛼 𝜃2𝑇 𝜉3 𝜃1𝑇 𝜉2 𝛾𝑑̇ .𝑉𝑈1(11)Then the virtual command of the attack angle is chosen as𝛼𝑑 𝛼 𝛾̃.Let 𝛼̃ 𝛼 𝛼𝑑 ; the error dynamic of the attack angle isformulated as̃̇ 𝑄 𝛾̇ 𝛼̇𝑑 𝑄 𝛾𝑑̇ .𝛼𝑞𝑆𝐶𝐿0 𝑚𝑔 sin 𝛾 𝑚𝑉𝛾𝑑̇𝑇 sin 𝛼 ( 𝑇𝛼 cos 𝛼 𝑇0 cos 𝛼 ) 𝛼 0 𝑚𝑉𝑚𝑉 𝐶1 𝛼 𝑈1 𝛽1 ,𝐴(9)𝑀 2𝐶1𝑀 2𝜃2 [𝐶𝐿𝑒 ; 𝐶𝐿𝑐 ] ,Here, the thrust is described as the function about the attack𝛼𝑀 2 2angle. Define 𝑇 𝑇𝛼 𝑇0 , where 𝑇 𝑞(𝐶𝑇,𝜙 𝑀 𝜙 𝛿𝑀 2𝛼0; 𝐶𝑇𝛼 ; 𝐶𝑇,𝜙 ; 𝐶𝑇,𝜙 ; 𝐶𝑇,𝜙; 𝐶𝑇 ; 𝐶𝑇 𝑑 ; 𝐶𝑇0 ; 𝐶𝐿𝛼 ; 𝐶𝐿0 ] ,𝜃1 [𝐶𝑇,𝜙(6)𝛼 20 2𝜙 𝐶𝑇𝛼 ) and 𝑇0 𝑞(𝐶𝑇,𝜙 𝜙𝑀 𝐶𝑇,𝜙𝜙 𝐶𝑇 𝑀 𝐶𝑇,𝜙(8)𝑈1 𝜃2𝑇 𝜉3 .(5)where 𝑘̃ℎ 0 is the design parameter for ̃ℎ.Let 𝛾̃ 𝛾 𝛾𝑑 ; the error dynamic of flight-path angle ispresented as follows:𝐴𝐶𝑇 𝑑 𝐴 𝑑𝑔 cos 𝛾 𝛾𝑑̇ ,𝑉(4)So the flight-path angle command 𝛾𝑑 is designed into thefollowing equation:𝛾𝑑 𝐶1 𝜃1𝑇 𝜉1 ,(12)A new variable 𝑍 is defined as 𝑍 𝑄 𝛾𝑑̇ 𝑘𝛼̃ 𝛼̃, where 𝑘𝛼̃ 0is a design parameter for 𝛼̃. Then (12) is rewritten as𝛽1(7)̃̇ 𝑍 𝑘𝛼̃ 𝛼̃.𝛼(13)

4Journal of Applied MathematicsUsing the equivalent control method, the time derivative of 𝑍can be formulated with the new input 𝑈2 . It includes the pitchrate dynamics𝑧 𝑇 𝑞𝑆𝑐𝐶𝑀̃̇ 𝑘𝛼̃ 𝛼𝑍̇ 𝑇𝐼𝑦𝑦𝛿𝑈2𝐶2̂2 𝜃̂𝑇 𝜉6 𝛾𝑑̈ (𝜃̂𝑇 𝜉5 𝑘𝛼̃ 𝛼̃̇ ) 𝜃̂3𝑇 𝜉4 𝛼 𝑈43(14)𝛽2 𝐶2 𝛼 𝑈2 𝛽2 ,where 𝐶2 and 𝛽2 are similar terms containing the uncertainparameters. They can also be presented by the vectors of theuncertain parameters and the regressors𝐶2 𝜃3𝑇 𝜉4 ,̃̇ 𝛾𝑑̈ ,𝛽2 𝜃3𝑇 𝜉5 𝑘𝛼̃ 𝛼(15)𝜃4𝑇 𝜉6 , 2𝑀 2𝑀 𝐴𝛼0𝛼0; 𝐶𝑇𝛼 ; 𝐶𝑇,𝜙 ; 𝐶𝑇,𝜙 ; 𝐶𝑇,𝜙; 𝐶𝑇 ; 𝐶𝑇 𝑑 ; 𝐶𝑇0 ; 𝐶𝑀; 𝐶𝑀],𝜃3 [𝐶𝑇,𝜙𝛿 𝑘𝑍 𝑍 (𝜃̂3𝑇 𝜉4 1) 𝛼̃,̂𝑈𝛿[ 𝑒 ] 𝐵 1 [ ̂1 ] .𝛿𝑐𝑈2𝛿𝜃4 [𝐶𝑀𝑒 ; 𝐶𝑀𝑐 ] ,𝜉4 𝑞[𝑧 𝜙; 𝑧 ; 𝑧 𝑀 2 𝜙; 0; 0; 0; 0; 0; 𝑆𝑐; 0] ,𝐼𝑦𝑦 𝑇 𝑇 𝑇 𝜉5 𝑞 2 2[0; 0; 0; 𝑧𝑇 𝑀 𝜙; 𝑧𝑇 𝜙; 𝑧𝑇 𝑀 ; 𝑧𝑇 𝐴 𝑑 ; 𝑧𝑇 ; 0; 𝑆𝑐] ,𝐼𝑦𝑦𝜉6 𝑞𝑆𝑐[𝛿 ; 𝛿 ] .𝐼𝑦𝑦 𝑒 𝑐(16)̇ℎ̃ 𝑘̃ℎ ̃ℎ 𝑉̃𝛾,𝛾̃̇ 𝑘𝛾̃𝛾̃ 𝑉̃ℎ 𝑦𝛼̃ 𝜃̃1𝑇 𝜉1 𝛼 𝜃̃1𝑇 𝜉2 𝜃̃2𝑇 𝜉3 ,where 𝑦𝛼̃ 𝜃̂1𝑇 𝜉1 𝛼̃, 𝑦𝛾̃ 𝜃̂3𝑇 𝜉4 𝛾̃.For ensuring the stability of the altitude loop, the newformulation (20) is divided into the altitude-flight-path anglesubsystem and the attack angle-pitch rate subsystem. Asillustrated in Figure 1, these two subsystems constitute astructure of interconnection. It is seen that 𝑦𝛼̃ and 𝑦𝛾̃ actas the input and output of the altitude-flight-path anglesubsystem and 𝑦𝛾̃, 𝑦𝛼̃ are the input and output of the attackangle-pitch rate subsystem, respectively.For the above interconnection subsystems, input-to-statestability will be analyzed via small gain theorem. Firstly, thedefinition of the asymptotic 𝐿 norm ‖ ‖𝑎 is given [19]‖𝜆‖𝑎 : lim sup 𝜆 .𝑡 (17)Due to the uncertainty of the aerodynamic parameters, 𝜃𝑖 ,𝑖 1, . . . , 4 will change with flight of hypersonic vehicles.Therefore it is necessary to estimate their values by theadaptive technique. Let 𝜃̂𝑖 , 𝜃̃𝑖 be the estimate vector and theestimate error vector of 𝜃𝑖 , where 𝜃̃𝑖 𝜃𝑖 𝜃̂𝑖 , 𝑖 1, . . . , 4.Assumption 1. The aerodynamic parameters 𝜃𝑖 , 𝑖 1, . . . , 4are bounded; they lie in a compact convex set.In order to guarantee tracking performance of hypersonic vehicles, the equivalent control inputs 𝑈1 and 𝑈2 are(20)𝑍̇ 𝑘𝑍 𝑍 𝛼̃ 𝑦𝛾̃ 𝜃̃3𝑇 𝜉4 𝛼 𝜃̃3𝑇 𝜉5 𝜃̃4𝑇 𝜉6 ,So (14) can be reformulated as̃̇ 𝛾𝑑̈ .𝑍̇ 𝜃3𝑇 𝜉4 𝛼 𝜃4𝑇 𝜉6 𝜃3𝑇 𝜉5 𝑘𝛼̃ 𝛼(19)Combining (18), the state error dynamics about thealtitude loop is transformed into the following equations:̃̇ 𝑍 𝑘𝛼̃ 𝛼̃,𝛼where 2𝛼𝑀 (18)where 𝑘𝛾̃ 0, 𝑘𝑍 0 are the design parameters for 𝛾̃ and 𝑍.Let 𝛿 [𝛿𝑒 , 𝛿𝑐 ]. There is U 𝐵𝛿 according to(7) and (14). 𝐵 is a coefficient matrix and is equal to[(𝑞𝑆/𝑚𝑉)𝜃̂2𝑇 ; (𝑞𝑆𝑐/𝐼𝑦𝑦 )𝜃̂4𝑇 ]. The real inputs of the altitude loopcan be obtained as follows:0𝑧𝑇 𝑇0 𝑞𝑆𝑐𝐶𝑀̃̇ 𝛾𝑑̈ 𝑘𝛼̃ 𝛼𝐼𝑦𝑦 𝑈2 ̂1 𝜃̂𝑇 𝜉3 𝜃̂𝑇 𝜉1 𝛼 (𝜃̂𝑇 𝜉2 𝑔 cos 𝛾 𝛾𝑑̇ )𝑈211𝑉 (𝜃̂1𝑇 𝜉1 𝑘𝛾̃) 𝛾̃ 𝑉̃ℎ,𝛿𝑐𝛼𝐶 𝑒 𝛿 𝐶𝑀𝛿𝑐 𝑧𝑇 𝑇 𝑞𝑆𝑐𝐶𝑀 𝑞𝑆𝑐 𝑀 𝑒 𝛼𝐼𝑦𝑦𝐼𝑦𝑦 designed, respectively, by replacing the uncertain parametervector 𝜃𝑖 with its estimate vector and estimate error vector(21)Then, define 𝜓1 ̃ℎ2 𝛾̃2 , and choose the Lyapunov function candidate of the altitude-flight-path angle subsystem as𝑊1 1 ̃211(ℎ 𝛾̃2 ) 𝜃̃1𝑇 𝜏1 1 𝜃̃1 𝜃̃2𝑇 𝜏2 1 𝜃̃2 .222(22)Its time derivative iṡ̇̇𝑊̇ 1 ̃ℎℎ̃ 𝛾̃𝛾̃̇ 𝜃̃1𝑇 𝜏1 1 𝜃̂1 𝜃̃2𝑇 𝜏2 1 𝜃̂2̇ 𝑘̃ℎ ̃ℎ2 𝑘𝛾̃𝛾̃2 𝛾̃𝑦𝛼̃ 𝜃̃1𝑇 𝜏1 1 {𝜏1 𝛾̃ (𝜉1 𝛼 𝜉2 ) 𝜃̂1 }̇ 𝜃̃2𝑇 𝜏2 1 (𝜏2 𝛾̃𝜉3 𝜃̂2 ) .(23)

Journal of Applied Mathematics5Substituting (29) in (28), we can acquirey𝛼̃ 2 𝑊̇ 2 min {𝑘𝛼̃ , 𝑘𝑍 } 𝜓2 𝜓2 𝑦𝛾̃ .Altitude-ﬂight-pathangle(30)When 𝜓2 𝑦𝛾̃ / min{𝑘𝛼̃ , 𝑘𝑍 }, 𝑊̇ 2 0. Similarly, ‖𝜓2 ‖𝑎 ‖𝑦𝛾̃‖𝑎 / min{𝑘𝛼̃ , 𝑘𝑍} can be obtained, and 𝑊2 is input-to-statestable as well. Because 𝑦 𝜃̂𝑇 𝜉 𝛼̃, there is𝛼̃Attack angle-pitchy𝛾̃𝜃̂1𝑇 𝜉1 𝑇 𝑇 𝑦 . (31) 𝑦𝛼̃ 𝑎 𝜃̂1 𝜉1 𝛼̃ 𝑎 𝜃̂1 𝜉1 𝜓2 𝑎 min {𝑘𝛼̃ , 𝑘𝑍 } 𝛾̃ 𝑎rateThe interconnection formulation (20) is input-to-state stableaccording to small-gain theorem if we choose proper designparameters to make the following equation holdsFigure 1: Interconnection subsystem structure.𝜃̂3𝑇 𝜉4𝜃̂1𝑇 𝜉1 1.min {𝑘̃ℎ , 𝑘𝛾̃} min {𝑘𝛼̃ , 𝑘𝑍 }The adaptive laws of 𝜃̂1 , 𝜃̂2 are designed aṡ𝜃̂1 𝜏1 𝛾̃ (𝜉1 𝛼 𝜉2 ) ,̇𝜃̂2 𝜏2 𝛾̃𝜉3 ,(24)where 𝜏1 and 𝜏2 are the adaptive parameters.By (24), (23) becomes 2 𝑊̇ 1 min {𝑘̃ℎ , 𝑘𝛾̃} 𝜓1 𝜓1 𝑦𝛼̃ .(25)As a consequence, it satisfies 𝑊1 which is negative definitewhen 𝜓1 𝑦𝛼̃ / min{𝑘̃ℎ , 𝑘𝛾̃}. Then 𝑊1 is a input-to-statestable Lyapunov function. According to the lemma in [19],we know that ‖𝜓1 ‖𝑎 ‖𝑦𝛼̃ ‖𝑎 / min{𝑘̃ℎ , 𝑘𝛾̃}. As 𝑦𝛾̃ 𝜃̂3𝑇 𝜉4 𝛾̃,the following formulation is obtained:𝜃̂3𝑇 𝜉4 𝑦𝛾̃ 𝜃̂3𝑇 𝜉4 𝛾̃ 𝜃̂3𝑇 𝜉4 𝜓1 𝑎 𝑦𝛼̃ 𝑎 . 𝑎 𝑎 min {𝑘̃ℎ , 𝑘𝛾̃}(26)For the attack angle-pitch rate subsystem, 𝜓2 𝛼̃2 𝑍2is defined, and the following Lyapunov function candidate ischosen:𝑊2 1 211(̃𝛼 𝑍2 ) 𝜃̃3𝑇 𝜏3 1 𝜃̃3 𝜃̃4𝑇 𝜏4 1 𝜃̃4 .2221 1(27) (32)Therefore the tracking errors and estimate errors of thealtitude loop can converge to a small neighborhood of origin.3.2. Velocity Controller. Since velocity is controlled by 𝜙directly, the adaptive dynamic inversion method is used. Let̃ 𝑉 𝑉ref ; the error dynamics of velocity is written as𝑉̃̇ 𝑇 cos 𝛼 𝐷 𝑔 sin 𝛾 𝑉reḟ𝑉𝑚 𝑞 (𝐶𝑇,𝜙 𝜙 𝐶𝑇 ) cos 𝛼 𝑞𝑆𝐶𝐷𝑚(33)̇ . 𝑔 sin 𝛾 𝑉refFor existence of uncertain parameters, the following vectorsand repressors are defined(𝛼 Δ𝜏1 )𝜃5 [𝐶𝐷𝛼𝛿(𝛼 Δ𝜏1 )2𝛿22𝛿𝛿; 𝐶𝐷𝛿𝑐; 𝐶𝐷𝑒 ; 𝐶𝐷𝑒 ; 𝐶𝐷; 𝐶𝐷𝑐 ;𝛼𝛿𝐴𝑀 20; 𝐶𝑇 𝑑 ; 𝐶𝑇𝛼 ; 𝐶𝑇 ; 𝐶𝑇0 ] ,𝐶𝐷 𝑒 ; 𝐶𝐷 𝑐 ; 𝐶𝐷𝛼𝑀 2𝑀 2𝛼0; 𝐶𝑇,𝜙 ; 𝐶𝑇,𝜙 ; 𝐶𝑇,𝜙],𝜃6 [𝐶𝑇,𝜙𝜉7 𝑞 [𝑆𝛼; 𝑆𝛼2(34); 𝑆𝛿𝑒2 ; 𝑆𝛿𝑒 ; 𝑆𝛿𝑐2 ; 𝑆𝛿𝑐 ; 𝑆𝛼𝛿𝑒 ; 𝑆𝛼𝛿𝑐 ; 𝑆; 2cos 𝛼; cos 𝛼] , 𝐴 𝑑 cos 𝛼; 𝛼 cos 𝛼; 𝑀 Its time derivative iṡ̇̃̇ 𝑍𝑍̇ 𝜃̃3𝑇 𝜏3 1 𝜃̂3 𝜃̃4𝑇 𝜏4 1 𝜃̂4𝑊̇ 2 𝛼̃𝛼 2 2; 𝑀 ; 1] .𝜉8 𝑞 cos 𝛼 [𝛼; 𝛼𝑀 ̇ 𝑘𝛼̃ 𝛼̃2 𝑘𝑍 𝑍2 𝑍𝑦𝛾̃ 𝜃̃3𝑇 𝜏3 1 {𝜏3 𝑍 (𝜉4 𝛼 𝜉5 ) 𝜃̂3 }̇ 𝜃̃4𝑇 𝜏4 1 (𝜏4 𝑍𝜉6 𝜃̂4 ) ,(28)̇𝜃̂4 𝜏4 𝑍𝜉6 .𝜃𝑇 𝜉 𝜙 𝜃5𝑇 𝜉7̃̇ 6 8̇ .𝑉 𝑔 sin 𝛾 𝑉ref𝑚(35)Then the control input 𝜙 is designed aswhere the adaptive laws of 𝜃̂3 , 𝜃̂4 are determined aṡ𝜃̂3 𝜏3 𝑍 (𝜉4 𝛼 𝜉5 ) ,Consequently,𝜙 (29)̃ 𝑚𝑉reḟ 𝑚𝑔 sin 𝛾 𝜃̂𝑇 𝜉7 𝑚𝑘̃V 𝑉5,𝜃̂𝑇 𝜉6 8where 𝑘̃V 0 is a design parameter.(36)

6Journal of Applied Mathematics 10479608.7579408.77920V (ft/s)h (ft)8.658.6790078808.55786078408.501020304050 60t (s)7080090 100hrefh1020304050 60t (s)708090 100VrefV(a) Altitude(b) Velocity 1080.023 370.02265𝛾 (rad)0.020.0194320.01810.01700.01601020304050 60t (s)7080 190 1000102030405060708090 100t (s)𝛾d𝛾𝛼d𝛼(c) Attack angle𝜂 (ft·slug )𝛼 (rad)0.021(d) Flight-path 0.10.50 0.10.480102030405060708090 1000t (s)𝜂1𝜂21020304050 60t (s)𝜂3(f) Fuel-to-air ratio(e) Flexible statesFigure 2: Continued.708090 100

Journal of Applied Mathematics70.11 0.050.1 0.06 0.07𝛿c (rad)𝛿e (rad)0.090.080.07 0.08 0.09 0.10.06 0.110.05 0.120.0401020304050 60t (s)708090 100(g) Elevator deflection 0.1301020304050 60t (s)708090 100(h) Canard deflectionFigure 2: Climbing maneuver with longitudinal acceleration for case one.Determine the Lyapunov function candidate as1 ̃2 1 ̃𝑇 1 ̃ 1 ̃𝑇 1 ̃ 𝜃5 𝜏5 𝜃5 𝜃6 𝜏6 𝜃6 .𝑊3 𝑚𝑉222(37)Its time derivative is̃ (𝜃𝑇 𝜉8 𝜙 𝜃𝑇 𝜉7 𝑚𝑔 sin 𝛾 𝑚𝑉reḟ )𝑊̇ 3 𝑉65̇̇ 𝜃̃5𝑇 𝜏5 1 𝜃̂5 𝜃̃6𝑇 𝜏6 1 𝜃̂6̃ 𝜃̃𝑇 𝜉7 ) 𝜃̃𝑇 𝜏 1 𝜃̂̇ 5 𝜃̃𝑇 𝜏 1 𝜃̂̇ 6̃ (𝜃̃𝑇 𝜉8 𝜙 𝑚𝑘̃V 𝑉 𝑉655 56 6̃2 𝜃̃𝑇 𝜏 1 (𝜏5 𝑉𝜉̃ 7 𝜃̂̇ 5 ) 𝜃̃𝑇 𝜏 1 (𝜏6 𝜙𝜉8 𝜃̂̇ 6 ) . 𝑚𝑘̃V 𝑉5 56 6(38)The adaptive laws of 𝜃̂5 , 𝜃̂6 are obtained in the following:̇̃ 7,𝜃̂5 𝜏5 𝑉𝜉̇𝜃̂6 𝜏6 𝜙𝜉8 .(39)̃2 0; that is, when 𝑡 Thus (38) becomes 𝑊̇ 3 𝑚𝑘̃V 𝑉 , the tracking errors and estimate errors of the velocitysubsystem can converge to zero finally.Therefore the accurate tracking performance and thestability of altitude and velocity can be guarranteed by theproposed method.4. Numerical SimulationThe feasibility of the proposed method is verified based ona flexible model (1)–(3). The initial trim conditions are ℎ 85000 ft, 𝑉 7846 ft/s, 𝛼 0.0174 rad, 𝛾 0 rad, 𝑞 0 rad/s, 𝜂1 0.4588 ft sulg, 𝜂2 0.08726 ft sulg, and 𝜂3 0.03671 ft sulg. Two cases are studied here.Case one is a climbing maneuver with longitudinal acceleration, and the expected equilibrium is 𝛼 0.0219 rad.The increments of altitude and velocity are 2000 ft, and100 ft/s respectively. Case two is a descending maneuver withvelocity reducing gradually, and its corresponding equilibrium is 𝛼 0.0158 rad. The decreasing of altitude is 1000 ftand that of velocity is 100 ft/s.For these two cases, The corresponding reference commands are generated by filtering step reference commandswith a second-order profiler with 𝜔 0.1 rad/s and 𝜉 0.9.The simulation results of case one are shown in Figure 2.The simulation results of case two are shown in Figure 3.From Figures 2(a), 2(b), 3(a), and 3(b), it is seen thatthe controller can provide stable and accurate tracking ofthe reference trajectories for the two cases, and the trackingerrors of altitude and velocity remain remarkably small.Figures 2(c), 2(d), 3(c), and 3(d) show that both the signalsof the flight-path angle and the attack angle can also followthe change of virtual control commands closely.For the flexible dynamics, its effect on aerodynamicmodel is neglected, and its motion is taken as externalperturbation during the control design. That means thatthe forebody turn angle Δ𝜏1 and the aftbody vertex angleΔ𝜏2 are equal to zero in model (3) when the controller isdesigned. Simultaneously, the second-order equation aboutflexible states 𝜂𝑖 and 𝜂𝑖̇ is not considered. From Figures 2(e)and 3(e), we can know that the flexible states can converge tostable states ultimately although the flexible dynamics is nottaken into account directly. This denotes that our controllerhas the strong robustness, and it is suitable to control theflexible hypersonic vehicle.Moreover, the variation ranges of the control inputs thatis, fuel-to-air ratio, elevator deflection, and canard deflectionare bounded according to Figures 2(f), 2(g), 2(h), 3(f), 3(g),and 3(h).In summary, the nonminimum phase behavior is suppressed successfully, and the excellent closed-loop behaviorof air-breathing hypersonic vehicle can be obtained by theproposed controller for the cases of maneuver of altitude andvelocity.

8Journal of Applied Mathematics8.5278608.578408.487820V (ft/s)h (ft) 0090 10010203040t (s)hrefh50 60t (s)708090 100VrefV(a) Altitude(b) Velocity 10 30.50.017500.017 0.5𝛾 (rad)0.016 1.5 2 2.5 30.0155 3.50.01501020304050 60t (s)7080 490 100010𝛼d𝛼304050 60t (s)708090 1008090 100(d) Flight-path angle0.80.580.70.560.60.540.50.520.4𝜙 0.50.30.480.20.10.4600.44 0.120𝛾d𝛾(c) Attack angle𝜂 (ft·slug )𝛼 (rad) 10.01650.420102030405060708090 1000𝜂1𝜂2102030405060t (s)t (s)𝜂3(f) Fuel-to-air ratio(e) Flexible statesFigure 3: Continued.70

90.13 0.030.12 0.040.11 0.050.1 0.06𝛿c (rad)𝛿e (rad)Journal of Applied Mathematics0.090.08 0.07 0.08 0.090.07 0.10.06 0.110.05 0.120.040102030405060708090 100 0.13t (s)(g) Elevator deflection01020304050 60t (s)708090 100(h) Canard deflectionFigure 3: Descending maneuver with velocity reducing gradually for case two.5. ConclusionFor the flexible air-breathing hypersonic vehicle with canardcontrol surface, the controller is designed based on thenonlinear adaptive equivalent control strategy under interconnected structure. The equivalent control inputs are introduced and designed to replace the terms about elevator andcanard in the flight-path angle dynamics and the pitchrate dynamics for eliminating the nonminimum phase. Theuncertain aerodynamic parameters are identified online bythe adaptive method. And input-to-state stability of theinterconnection subsystems is guaranteed by small-gain theorem. Similarly, the adaptive dynamic inversion approach isadopted in the velocity loop. With our approach, the stableand accurate tracking of the hypersonic vehicle model withnonminimum phase can be realized.AcknowledgmentsThis work is supported by National Natural Science Foundation of China under Grant 61074064 and Natural Science Foundation of Tianjin 12JCZDJC30300. The authorsare grateful to the anonymous reviewers for their helpfulcomments and constructive suggestions with regard to thispaper.[4][5][6][7][8][9]References[1] M. A. Bolender and D. B. Doman, “Nonlinear longitudinaldynamical model of an air-breathing hypersonic vehicle,” Journal of Spacecraft and Rockets, vol. 44, no. 2, pp. 374–387, 2007.[2] J. T. Parker, A. Serrani, S. Yurkovich, M. A. Bolender, and D. B.Doman, “Control-oriented modeling of an air-breathing hypersonic vehicle,” Journal of Guidance, Control, and Dynamics, vol.30, no. 3, pp. 856–869, 2007.[3] B. Fidan, M. Kuipers, P. A. Ioannou, and M. Mirmirani, “Longitudinal motion control of air-breathing hypersonic vehiclesbased on time-varying models,” in Proceedings of the 14thAIAA/AHI International Space Planes and Hypersonics Systems[10][11][12]Technologies Conference, pp. 1705–1717, AIAA, Canberra, Australia, November 2006.K. P. Groves, D. O. Sigthorsson, A. Serrani, S. Yurkovich, M.A. Bolender, and D. B. Doman, “Reference command trackingfor a linearized model of an air-breathing hypersonic vehicle,”in Proceedings of the AIAA Guidance, Navigation, and ControlConference, pp. 2901–2914, AIAA, San Francisco, Calif, USA,August 2005.D. O. Sigthorsson, P. Jankovsky, A. Serrani, S. Yurkovich,M. A. Bolender, and D. B. Doman, “Robust linear outputfeedback control of an airbreathing hypersonic vehicle,” Journalof Guidance, Control, and Dynamics, vol. 31, no. 4, pp. 1052–1066, 2008.L. G. Wu, X. B. Yang, and F. B. Li, “Nonfragile output trackingcontrol of hypersonic air-breathing vehicles with LPV model,”IEEE/ASME Transactions on Mechatronics, vol. 18, no. 4, pp.1280–1288, 2013.H. P. Lee, S. E. Reiman, C. H. Dillon, and H. M. Youssef, “Robustnonlinear dynamic inversion control for a hypersonic cruisevehicle,” in Proceedings of the AIAA Guidance, Navigation, andControl Conference and Exhibit, pp. 3380–3388, Hilton Head,SC, USA, August 2007.H. J. Xu, M. D. Mirmirani, and P. A. Ioannou, “Adaptive slidingmode control design for a hypersonic flight vehicle,” Journal ofGuidance, Control, and Dynamics, vol. 27, no. 5, pp. 829–838,2004.B. L. Tian, Q. Zong, J. Wang, and F. Wang, “Quasi-continuoushigh-order sliding mode controller design for reusable launchvehicles in reentry phase,” Aerospace Science and Technology,vol. 28, no. 1, pp. 198–207, 2013.B. Xu, F. Sun, H. Liu, and J. Ren, “Adaptive Kriging controllerdesign for hypersonic flight vehicle via back-stepping,” IETControl Theory and Applications, vol. 6, no. 4, pp. 487–497, 2012.T. E. Gibson, L. G. Crespo, and A. M. Annaswamy, “Adaptivecontrol of hypersonic vehicles in the presence of modelinguncertainties,” in Proceedings of the 2009 American ControlConference (ACC ’09), pp. 3178–3183, St. Louis, Mo, USA, June2009.H. Buschek and A. J. Calise, “Uncertainty modeling and fixedorder controller design for a hypersonic vehicle model,” Journal

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dynamic parameters. en the interconnection subsystems including the altitude- ight-path angle dynamics and the attack angle-pitch rate dynamics are constructed. Via the small-gain method, the system is proven to be input-to-state stable. In the velocity loop, the adaptive dynamic inversion controller is designed. Simulation results show the .

Research Article Nonlinear Adaptive Equivalent Control Based . - Hindawi

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