Output Feedback Dynamic Surface Controller Design for Airbreathing Hypersonic Flight Vehicle

Transcription

1 186 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. APRIL 015 Output Feedback Dynamic Surface Controller Design for Airbreathing Hypersonic Flight Vehicle Delong Hou Qing Wang and Chaoyang Dong Abstract This paper addresses issues related to nonlinear robust output feedback controller design for a nonlinear model of airbreathing hypersonic vehicle. The control objective is to realize robust tracking of velocity and altitude in the presence of immeasurable states uncertainties and varying flight conditions. A novel reduced order fuzzy observer is proposed to estimate the immeasurable states. Based on the information of observer and the measured states a new robust output feedback controller combining dynamic surface theory and fuzzy logic system is proposed for airbreathing hypersonic vehicle. The closedloop system is proved to be semi-globally uniformly ultimately bounded SUUB) and the tracking error can be made small enough by choosing proper gains of the controller filter and observer. Simulation results from the full nonlinear vehicle model illustrate the effectiveness and good performance of the proposed control scheme. Index Terms Hypersonic flight vehicle immeasurable states output feedback control model uncertainties fuzzy logic system dynamic surface control. I. INTRODUCTION AIRBREATHING hypersonic vehicles AHVs) have been studied broadly due to the prospects of high speed transportation and affordable space access for a long time. For the purpose of flight safety and high accuracy control system design has become a core problem for AHV. However the following characteristics of AHV make this problem full of challenges [1 5]. 1) There are strong couplings between propulsive and aerodynamic forces which are caused by the integrated configuration of the airframe and the scramjet engine. ) The uncertainties of aerodynamic parameters [1] are derived from large-scale variations of altitude and velocity. 3) The immeasurable states exist in hypersonic condition. In the past decades amounts of work has been done for developing control system of hypersonic flight vehicle. Mainly the control system design can be classified into two categories based on the linear and nonlinear models respectively. For the controller design of the linearized dynamic models of hypersonic vehicles several methods are adopted While considering the problems of different complexity such as decentralized control [6] linear quadratic regulator LQR) approach [7] gain Manuscript received October ; accepted April This work was supported by Natural National Science Foundation of China ). Recommended by Associate Editor Changyin Sun. Citation: Delong HouQing Wang Chaoyang Dong. Output feedback dynamic surface controller design for airbreathing hypersonic flight vehicle. IEEE/CAA Journal of Automatica Sinica 015 ): Delong Hou is with the School of Automation Science and Electrical Engineering Beihang University Beijing China and also with Beijing Institute of Electronic System Engineering Beijing China dlh8855@163.com). Qing Wang and Chaoyang Dong are with the School of Automation Science and Electrical Engineering Beihang University Beijing China wangqing@buaa.edu.cn; dongchaoyang@buaa.edu.cn). scheduling method [8] etc. On the assumption that the fight dynamics at a certain point can be denoted by the linear combination of the model on adjacent design points the linear parameter varying LPV) model of hypersonic vehicle is proposed. Various methods are developed for LPV system of hypersonic vehicle [9 10]. T-S fuzzy system based method is also adopted to synthesize controller for hypersonic vehicle based on linearized dynamical model [11]. Linearized dynamical model for hypersonic vehicle is obtained through Jacobian linearization under certain conditions which brings loss of dynamic characteristic to some degree. Actually the hypersonic flight vehicle is a nonlinear system and the controllers based on nonlinear model are more accurate than the linear. The nonlinear methods include feedback linearization [1 13] sliding mode [14 15] etc. Feedback linearization requires repetitive differentiations of system nonlinear terms which is difficult to achieve in real world. Sliding mode method is easy to cause chattering of the input. At present as a kind of convenient control design method for strict feedback nonlinear system backstepping has received a lot of research attention for applications in hypersonic vehicle control area [16 17]. However the methods mentioned above require full information of the states which limits their practical applications. In fact the angle of attack AOA) is difficult to be measured in hypersonic condition. Hypersonic aerodynamic heating decline the performance of the commonly used air-data sensors. Flush air-data sensor FADS) is another instrument for AOA measurement but when used in hypersonic condition modeling of aerodynamic heating process is also a hard task which is the fundamental problem of FADS []. Observer-based output feedback control is a feasible method for hypersonic vehicle [ 13 18] in the presence of immeasurable states. In [13] a sliding mode observer SMO) is used to construct the immeasurable states. However the observer is developed based on the exactly known parameters in the dynamic equations. When there exist parameters uncertainties the equilibrium point of this observer is not the origin which brings steady estimation errors. Besides the controller is designed based on feedback linearization and sliding mode control. However complex derivative signals of altitude and velocity are essential but unavailable. Reference [18] proposes a kind of robust output feedback scheme for linearized hypersonic vehicle model. The information used in the controller only includes velocity altitude pitch rate and normal acceleration while the required state signals in the nominal controller are replaced by the signals from the full state observer. Reference [] proposes an output feedback controller combining backstepping and sliding mode observer. The stability of the closed-loop system and the convergence of the output

2 HOU et al.: OUTPUT FEEDBACK DYNAMIC SURFACE CONTROLLER DESIGN FOR AIRBREATHING HYPERSONIC FLIGHT VEHICLE 187 tracking error are verified based on the small-gain theorem. However parameter uncertainties are not considered and the problem of explosion of complexity arising from the differentiation of the intermediate virtual control exists in []. Besides the small-gain theorem holds only when tracking errors are out of certain range so the stability proof is not complete. A number of difficulties still exist in output feedback controller for hypersonic vehicle system. 1) The system cannot be exactly expressed in strict feedback form. ) Forces and moment of hypersonic vehicle are the functions of AOA and its high order terms. 3) Altitude information cannot be used as unique information to estimate AOA because its precision is much lower than the measured values of pitch angle and pitch angle rate. 4) The aerodynamic coefficients are uncertain to some degree. These four difficulties lead to the failure of commonly used high-gain observer or K-filter. Reference [19] solves a class of nonlinear systems output feedback tracking problem combining the modified high-gain observer and the adaptive backstepping controller but the uncertainties are dominated by output-dependent functions and the immeasurable states are constructed using the first state information which is also the output variable. The virtual control coefficients in [19] are ones which is another advantage of output feedback controller design. Reference [0] proposes an output feedback control scheme for a class of stochastic nonlinear systems combining a kind of full-order observer and backstepping approach in which the virtual control gains are also ones. Output feedback control for stabilization problem of nonlinear system is summarized in [1]. The control schemes above are not suitable for the hypersonic vehicle model. In this paper the objective is to construct a nonlinear output feedback tracking controller for hypersonic vehicle. The main contributions of this paper are summarized as follows. 1) The local characteristic of thrust and pitching moment with respect to AOA in the argument range is first validated. The advantage of this characteristic is discussed in Remark 1. ) A reduced order observer is first proposed to estimate the value of immeasurable AOA based on the information of pitch angle and its rate. 3) The output feedback fuzzy dynamic surface technique [] is first adopted for hypersonic vehicle control which drives the trajectories of the velocity and altitude tracking errors into an arbitrarily small neighborhood of the origin. Fuzzy logic system method is used to compensate the effects of parameter uncertainties. Besides these contributions continuous hyperbolic tangent function is used in the virtual and actual control design to eliminate the effect of fuzzy estimation. Numerical simulations of various situations are presented. The maneuver is performed under several conditions to demonstrate that the control laws are valid for the entire flight envelope. It is shown that trajectory control is established for the closed-loop system even in the presence of uncertainties and immeasurable states in the vehicle model. The reminder of this paper is organized as follows. The hypersonic vehicle longitudinal motion model is described in Section II. In Section III the flight control system for hypersonic vehicles adopting the fuzzy reduced order observer and fuzzy dynamic surface technique is addressed. A comprehensive stability analysis is given for the closed-loop system. In Section IV numerical simulations on the longitudinal hypersonic vehicle model are carried out to validate the proposed controller. Finally brief concluding remarks end the paper in Section V. II. PLANT MODEL AND PROBLEM FORMULATION The hypersonic vehicle model considered in this study is the curve-fitted model CFM) given by Fiorentini [16] where the complex forces and moment are approximated in curvefitted form. The control inputs of this model are the elevator deflection angle and fuel-to-air ratio. The longitudinal dynamic equations are written as ḣ = V sin γ T cos α D V = m g sin γ L+T sin α γ = mv g V cos γ L+T sin α α = mv + Q + g V cos γ 1) θ = Q Q = M η i = ζ i ω i η i ωi η i + N i i = 1 3 where h V γ α θ Q and η i are altitude velocity flight path angle FPA) AOA pitch angle pitch rate and the ith generalized elastic coordinate of the vehicle respectively. T D L M and N i denote thrust drag lift pitching moment and the ith generalized force respectively. In the CFM by defining η = [η 1 η η 3 ] T these forces and moment are approximated as follows: T qs[c TΦ α)φ + C T α) + C η T η] D = qsc D α δ e η) ) L = qsc L α δ e η) M = z T T + qs cc M α δ e η) where q = 1/)ρV is the dynamic pressure C TΦ C T C L and C M are forces and moment coefficients. These coefficients are defined as follows: C TΦ α) = CT Φα3 α 3 + CT Φα α + CT Φαα + CΦ T + C TΦ C T α) = CT 3 α3 + CT α + CT 1 α + C0 T + C T C L α δ e ) = CL α α + Cδe L δ e + CL 0 + Cη L η + C L C D α δ e ) = CD α α + CD α α + Cδ e D δ e + C δe D δ e + CD 0 + C η D η + C D C M α δ e ) = CM α α + CM α α + Cδe M δ e + CM 0 + C η j = [Cη1 j η1 = [Ni N η i C η M η + C M 0 C η j 0 C η3 j 0] j = T M L D 0 N η i 0 N η3 i 0] i = 1 3. The rest of nomenclatures are revealed in Table I. The admissible ranges of states and inputs are displayed in Table II. Remark 1. The CFM is derived from the true model TM) in [3] by expressing T L D M in the curve-fitted form. This manner is similar to the method of Parker [3]. The CFM can depict the TM with sufficient accuracy. All the dominant features including the coupling between thrust and aerodynamic forces the effect of the thrust engine) on the 3)

3 188 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. APRIL 015 moment the flexibility effect and the nonlinearity of force and moment are all retained in this model. Due to the lack of data for the TM the CFM is taken as the simulation model to produce true states of AHV in this paper. The data of CFM can be found in [4]. TABLE I NOMENCLATURES OF HYPERSONIC VEHICLE as T α 1 Φ) T α Φ) σ T α 1 α Mα 1 δ e ) Mα δ e ) σ M α 1 α α 1 α Ω α where Ω α denotes the argument range for α which is listed in Table II. Symbol Meaning Moment of inertia kg m ρ Density of air kg m 3 S Reference area m c Mean aerodynamic chord m Φ Fuel-to-air ratio δ e Elevator deflection angle g Gravity acceleration m/s Ma Mach number z T Thrust to moment coupling coefficient TABLE II ADMISSIBLE RANGES OF STATES INPUTS DYNAMIC PRESSURE AND MACH NUMBER Item Min Value Max Value V m/s) hm) γdeg) 3 3 αdeg) 5 10 Φ ) δ e 5 5 qpa) Ma ) 7 1 From equations 1)-3) thrust T and moment M are expressed in nonlinear form. We calculate T for different Φ and α and M for different α while the elevator angle is the trim value and the relationships are depicted in Figs. 1 and. Fig. 1. T for different α and Φ. It is obtained that T and M satisfy the locally Lipschitz condition with respect to AOA in the argument range according to Figs. 1 and. Consequently for the value of Φ and δ e in the admissible range T α Φ) and Mα δ e ) satisfy the Lipschitz condition with some Lipschitz constants σ T and σ M Fig.. M for different α. According to 1) the velocity is mainly controlled by the throttle setting Φ while the change of the altitude is governed by the elevator deflection δ e. Therefore we separate the longitudinal dynamics into two subsystems and design the velocity controller and altitude controller separately. The velocity is considered separately from the rest of dynamics and the model is denoted as V = f v + g v Φ 4) where f v = qsc T α) cos α/m qsc D /m g sin γ g v x 1 x x 3 V ) = qsc TΦ α) cos α/m. The rest of the hypersonic vehicle dynamic model constitutes the altitude tracking loop. The selected states for altitude loop are defined as x = [x 1 x x 3 x 4 ] T = [h/h c γ θ q] T. The transformation of altitude is to keep all states in the same size scale. Due to the fact that cos γ 1 if γ is small enough the CFM of altitude loop can be transformed into ẋ 1 = g 1 V )sin x 5) ẋ = f x V ) + g V )x 3 + w 1 α α ref δ e η) 6) ẋ 3 = g 3 x 4 7) where ẋ 4 = f 4 x x 3 V ) + g 4 V )u + w α α ref δ e η) 8) g 1 V ) = V/h c g V ) = C α LqS + k T 1 )/mv ) f x V ) = f 0 x V ) g V ) γ f 0 x 1 V ) = C 0 LqS/mV ) g/v g 3 = 1 f 4 = [z T k T α + C c T ) + qs ck M α + C 0 M )]/ g 4 = qs cc δe M / u = δ e k T 1 = CT α3 α ref ) 3 + CT α α ref ) + CT α α ref ) + CT c k T = CT α3 α ref ) + CT α α ref + CT α k M = C α M α ref + C α M.

4 HOU et al.: OUTPUT FEEDBACK DYNAMIC SURFACE CONTROLLER DESIGN FOR AIRBREATHING HYPERSONIC FLIGHT VEHICLE 189 The goal of this study is to synthesize a controller using measurable state information realizing altitude and velocity tracking. Remark. It can be concluded that w 1 and w are mainly constituted by three parts namely the difference between α ref and α coefficient uncertainties and generalized elastic coordinates. From the locally Lipschitz characteristic of T and M we can obtain that the first one can also be seen as coefficient uncertainty. The following assumptions are made for developing the output feedback based control laws. Assumption 1. The generalized elastic coordinate is mainly affected by AOA and is much less affected by δ e. Assumption. The reference signal h d t) is a sufficiently smooth function of t and h d t) ḣ d t) ḧdt) are bounded within a known compact set Ω hd = {[h d ḣd ḧd] T : h d + ḣ d + ḧ d h 0 } R 3 where h 0 is a known positive constant. Remark 3. Assumption 1 is made according to the scale of Ni α and N δe i ; N δe i is much smaller than Ni α. Assumption 1 will be verified in the simulation. III. OUTPUT FEEDBACK DYNAMIC SURFACE CONTROLLER The strategy chosen here is an output feedback controller based on dynamic surface control DSC). First a fuzzy reduced order observer is designed for altitude loop to estimate AOA. Simultaneously the estimated value of FPA is obtained based on the value of pitch angle and estimated value of AOA. Then the dynamic surface control is applied using the estimated value of FPA pitch angle pitch angle as virtual control inputs and the estimation error of FPA and uncertainties are considered as disturbances whose behavior must be dominated. A. Fuzzy Logic Systems and Reduced Order Observer Design for Altitude Loop First we introduce the following useful lemma on fuzzy logic systems FLSs). Lemma 1 [5]. Let fx) be a continuous function of vector x defined on a compact set U. Then for any constant ε > 0 there exists the following FLSs: such that ˆfx ϑ) = ϑ T ϕx) 9) sup ˆfx ϑ) fx) ε 10) x U where ϑ is the optimal parameter defined as ϑ = arg min sup ˆfx ϑ) ) fx). ϑ Ω x U ϕx) = [ϕ 1 x) ϕ N x)] is the rector of fuzzy basis functions ϑ is the vector of weighting coefficients. ϕ l x) l = 1 N) are defined as n µ F l i x i ) i=1 ϕ l x) = N n i=1 µ F x i)) i l l=1 where µ F l i x i ) is the fuzzy membership function. In the model presented in 1) γ and α are two immeasurable states. Due to the relationship that α = θ γ we only need to estimate one of them. We select α as the variable to be estimated. Defining x b = [α α ref δ e η] T w i α α ref δ e η) i = 1 ) can be denoted as w 1 x b ) and w x b ). According to Lemma 1 each unknown nonlinear uncertain function ŵ j can be approximated by a FLS in the form of ŵ i x b ϑ i ) = ϑ T i ϕ i x b ). 11) Denoting x b = [ˆα α ref δ e η ] T where η is the trim value of η one has ŵ i x b ϑ i ) = ϑ T i ϕ i x b ). 1) Define the optimal parameter vectors ϑ i i = 1 ) as ϑ i = arg min ϑ i Ω i sup ŵ i x b ϑ 1 ) w i x b ) 13) x b U b x b U b where Ω i U b and U b are compact regions for ϑ i x b and x b respectively. The minimum estimation errors ε i and estimation errors ε i are defined as ε i = w i x b ) ŵ i x b ϑ i ) 14) ε i = w i x b ) ŵ i x b ϑ i ). 15) Assumption 3. With the boundedness of η there exist known constants ε 0 i > 0 and ε 0 i > 0 such that ε i ε 0 i and ε i ε 0 i i = 1 ). A reduced order observer for AOA dynamics is formulated as { ˆα = ξ + Ly = ξ + l 1 θ + l q ξ = Mξ + Nδ e + Ry + C = Mξ + Nδ e + r 1 θ + r q + C 16) where y = [θ q] T is the measurable states vector L = [l 1 l ] is the observer gain coefficient matrix and these gain coefficients are defined as M = qscα L + k T 1 l z T k T + qs ck M ) mv r 1 = l 1 = 0 r = qscα L + k T 1 mv N = qscδe L mv l qs cc δe M l z T k T + qs ck M ) )l + 1 C = qsc0 L mv + g V l z T C c T + qs cc0 M ) ŵ 1 x b ϑ 1 ) l ŵ x b ϑ ). Let e = α = α ˆα then the estimation error γ = e. The estimation error dynamics can be deduced as ė = α ˆα = qscα L α + C0 L ) + k T 1α mv + q + g V w 1 [Mξ + Nδ e + r 1 θ + r q + C + l 1 q + l z T k T α + C c T )+ qs ck M α + C δe M δ e + C 0 M )) + l w ] = Me ε 1 ε. 17)

5 190 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. APRIL 015 The stability of the estimation error is given by the following proposition. Proposition 1. By using the observer in 16) the estimation error boundedly converge to a small value and it can be made small enough by e ) = ε ε) 0 1 M+1). Proof. Consider the Lyapunov function candidate as V e = 1 e. 18) The derivative along the trajectory of 17) results in V e = eė = eme ε 1 ε ) Me + ε ε 0 ) e M + 1 ε 0 )e ε) 0 ε 0 M + 1)V e ε) 0. According to the differential inequality theory there exists the following inequality by induction V e t) V e 0)e M+1)t + t 0 e M+1)t τ) ε0 1 + ε 0 ) dτ when t V ) ε0 1 +ε0 ) M+1) that is to say e ) ε ε) 0 1 M+1). Thus we can obtain the conclusion in Proposition 1. The final error dynamics can be seen as a system disturbed by the fuzzy estimation error with the attenuation level of M + 1). B. Controller Design for Altitude Loop In this section an adaptive fuzzy controller and parameter adaptive laws are to be developed in the framework of the backstepping design and DSC technique so that all the signals in the closed-loop system are SUUB and the tracking errors of altitude are as small as desired. Step 1. Define the dynamic surface error tracking error) as z 1 := x 1 h d whose time derivative while considering 5) is ż 1 = g 1 V ) x ḣd. 19) Although the precision of altitude signal is not high enough to estimate AOA it can be taken into altitude tracking error calculation. The nominal command of virtual control variable x is designed as x ref0 = 1 g 1 V ) k 1z 1 + ḣd). 0) To avoid the explosion of complexity in calculating the derivative a first-order filter is introduced to obtain the differentiation of the virtual control input i.e. x ref = T x ref x ref0 ) x ref 0) = x ref0 0) 1) where T is the time constant of filter and should be as large as possible to promise the fast tracking. Step. Define the dynamic surface equation in Eq. as z := sin ˆx x ref whose dynamics can be calculated by 6) i.e. ż = cosˆx )[f x V ) + g V )x 3 + w 1 x b ) + ė] ẋ ref ) where x 3 is the virtual control input of this step the nominal value of x ref0 3 is chosen to satisfy the following equation: g V )x ref0 3 = 1 cosˆx ) [ k z g 1 z 1 ε ε 0 1+ε 0 )tanh z ) + x ref ] ε f 0 x V ) + g V ) ˆγ ŵ 1 x b ϑ 1 ). 3) Let the nominal virtual control variable value passing through the following first-order filter: ẋ ref 3 = T 3 x ref 3 x ref0 3 ) x ref 3 0) = x ref0 3 0) 4) where T 3 is the time constant of filter. Step 3. Define dynamic surface as z 3 dynamics can be calculated from 6) i.e. ż 3 = x 4 ẋ ref 3 := x 3 x ref 3 its where x 4 is the virtual control variable of this step and its nominal value is selected as x ref0 4 = k 3 z 3 + ẋ ref 3 z cosˆx )g V ). 5) This nominal virtual control variable value will also pass through the following first-order filter to get the derivative of the virtual control variable: ẋ ref 4 = T 4 x ref 4 x ref0 4 ) x ref 4 0) = x ref0 4 0). 6) where T 4 is the filter time constant. Step 4. Define dynamic surface as z 4 := x 4 x ref 4. From 8) the dynamics of z 4 is calculated as ż 4 = f 4 x x 3 V ) + g 4 V )u + w x b ) ẋ ref 4. 7) Select the control input to satisfy g 4 V )u = k 4 z 4 f 4 ˆx x 3 V ) + ẋ ref 4 z 3 ŵ x b ϑ ) ) ε 0 z4 tanh. 8) ε 4 C. Controller Design for Velocity Loop The controller for velocity loop adopts dynamic inversion method. Defining tracking error as z V = V V d then we can deduce the derivative of z V as ż V = ˆf v + ĝ v Φ + d v V d 9) where d v denotes the uncertainties of velocity loop which is induced by observer error of α. It can be estimated from the following extended state observer [6 7] : E 11 = Z 11 V Ż 11 = Z 1 + ˆf v β 11 E 11 + ĝ v Φ 30) Ż 1 = β 1 fale 11 λ v ε v ) where 0 < λ v < 1 ε v > 0 β 11 > 0 β 1 > 0 are design coefficients and the function fal is defined as { E 11 λv sgne 11 ) E 11 > ε v fale 11 λ v ε v ) = E 11 31) others. ε 1 λv v Then the controller can be designed as Φ = ĝ 1 v V ) k v z V ˆf v Z 1 + V d ). 3)

6 HOU et al.: OUTPUT FEEDBACK DYNAMIC SURFACE CONTROLLER DESIGN FOR AIRBREATHING HYPERSONIC FLIGHT VEHICLE 191 D. Stability and Performance Analysis In this section we will prove that by using the proposed output feedback dynamic surface control scheme the semiglobal stability of the closed-loop system can be guaranteed. Furthermore the tracking performance can be achieved through tuning control coefficients under an initialization error constraint condition. Define the error between the nominal and actual values of virtual control variable as δ = x ref x ref0 δ 3 = x ref 3 x ref0 3 δ 4 = x ref 4 x ref0 4. Then the actual state variable can be denoted as x = e + z + x ref0 + δ x 3 = z 3 + x ref0 3 + δ 3 x 4 = z 4 + x ref0 4 + δ 4. The closed-loop dynamics of four dynamic surfaces of altitude loop are deduced as ż 1 =g 1 V )x ḣd = g 1 V )e + z + δ + x ref0 ) ḣd = k 1 z 1 + g 1 V )e + z + δ ) 33) ż = k z + cosˆx )g V )z 3 + δ 3 + e) + cosˆx ) [w 1 x b ) ŵ 1 x b ϑ 1 )] + cosˆx )ė g 1 z 1 ε ε 0 1+ε 0 )tanh z ε ) 34) ż 3 = k 3 z 3 + z 4 + δ 4 z g V ) 35) ż 4 = k 4 z 4 + z T k T + qs ck M )e + w x b ) ŵ x b ϑ ) z 3 ε 0 tanh z4 ε 4 ). 36) The boundary layers δ i i = 3 4) and the three derivatives satisfy the following relations: δ i δi = δ i T i x ref0 i x ref i ) δ i ẋ ref0 i T i δi + δ i ψ i z 1 z z 3 z 4 δ 1 δ δ 3 h d ḣd ḧd) 37) where ψ i is a continuous function. By defining ϑ 1 = ϑ 1 ϑ 1 ϑ = ϑ ϑ then the FLS estimation error can be denoted as w i x b ) ŵ i x b ϑ i ) = [w i x b ) ŵ i x b ϑ i )] + [ŵ i x b ϑ i ) ŵ i x b ϑ i )] = ε i ϑ T i ϕ i x b ). 38) The following lemma will be used in stability analysis. Lemma [8]. For any constant ε > 0 χ R the following inequality holds: χ ) 0 χ χ tanh k p ε ε where k p is a constant satisfying k p = e kp+1). We conclude the stability of the closed-loop altitude subsystem in the following theorem. Theorem 1. Given motion equations 6) 8) for the altitude loop of hypersonic flight vehicle the proposed dynamic surface controller 8) together with the FLSs estimator 16) there exist proper positive numbers k i T j i = j = 3 4) and a negative number M such that all signals of the closed-loop altitude system are uniformly bounded and the altitude tracking error z 1 converges to a residual set that can be made arbitrarily small by properly choosing some design parameters. Proof. Define the Lyapunov function candidate as V al = 1 4 zi + 1 i=1 4 δj + 1 j= k=1 Γ 1 ϑk ϑ T k ϑ k + 1 e. 39) Along the trajectories 9)-3) the derivative of the Lyapunov function candidate can be calculated as V al = z 1 [ k 1 z 1 + g 1 V )e + z + δ )]+ z [ k z + cosˆx )g V )z 3 + δ 3 + e) + cosˆx ) [w 1 x b ) ŵ 1 x b ϑ 1 )] g 1 z 1 + cosˆx )ė ε ε 0 1+ε 0 ) tanh z ε )] + z 3 [ k 3 z 3 + z 4 + δ 4 z cosˆx )g V )]+ z 4 [ k 4 z 4 + z T k T + qs ck M )e + w x b ) ŵ x b ϑ ) z 3 ε 0 tanh z 4 ε 4 )] + k=1 Γ 1 ϑ T ϑk k ϑ k + eė. 4 δ j δj + j= By simplifying the equation the derivative of Lyapunov function is transformed into V al =z 1 [ k 1 z 1 + g 1 V )e + δ )]+ z [ k z + cosˆx )g V )z 3 + δ 3 + e) + cosˆx ) [w 1 x b ) ŵ 1 x b ϑ 1 )] + cosˆx )ė ε ε ε 0 ) ) z tanh ] + z 3 [ k 3 z 3 + δ 4 ] + z 4 [ k 4 z 4 + ε z T k T + qs ck M )e + w x b ) ŵ x I b ϑ ) yy ) ε 0 z4 4 tanh ] + δ j δj + Γ 1 ϑ T ϑk k ϑ k + eė = ε 4 j= k=1 k 1 z 1 + g 1 V )z 1 e + δ ) k z + g V ) cosˆx )z δ 3 + e) + z cosˆx )ε 1 ϑ T 1 ϕ 1 x b )) + z cosˆx ) ) Me ε 1 ε ) ε ε 0 1+ε 0 z )z tanh k 3 z3 + z 3 δ 4 k 4 z4 + k M ez 4 + z 4 ε ϑ T ϕ x b )) ε 0 z 4 tanh z 4 4 ) + δ j δj + Γ 1 ε ϑ T ϑk k ϑ k + eė 40) 4 j= k=1 where k M = z T k T + qs ck M )/. By using Young inequality we have the following relationships: g 1 V )z 1 e 1 g 1V )e + z1) g 1 V )z 1 δ 1 z 1 + g1v )δ) g V ) cosˆx )z δ 3 1 z + gv )δ3) g V ) cosˆx )z e 1 e + gv )z) 41) z cosˆx )Me 1 M z + e ) z 3 δ 4 1 z 3 + δ4) k M ez 4 1 k M z 4 + e ). ε

7 19 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. APRIL 015 According to 37) it can be deduced that V al k 1 z g 1V )e + z 1) + 1 z 1 + g 1V )δ ) k z + 1 z + g V )δ 3) + 1 e + g V )z ) + z ε 1+ z ϑt 1 ϕ 1 x b ) + 1 M z + e ) + z ε ε 0 ) ) ε ε 0 1+ε 0 z )z tanh k 3 z3 + 1 ε z 3 + δ4) k 4 z4 + 1 ) k M z4 + e ) + z 4 ε ε 0 z4 z 4 tanh + z 4 ϑ T ϕ x b ) + 4 δ j δj + j= k=1 ε 4 Γ 1 ϑ T ϑk k ϑ k + eė k 1 1)z g 1V )e + 1 g 1V )δ k 1 1 g V ))z + 1 g V )δ e + z ε ε ε 0 ) ε ε 0 1+ε 0 )z tanh z z ϑt 1 ϕ 1 x b ) Γ 1 ϑ T ϑ ϑ M z + e ) k 3 1 ) z3 + 1 δ 4 k 4 z4 + 1 k M z4 + e )+ z4 ε 0 ε 0 z 4 tanh z4 ε 4 ε ) + ) + z 4 ϑ T ϕ x b ) Γ 1 ϑ ϑt ϑ + 4 T i δi + δ i ψ i ) + eme ε 1 ε ). 4) j= Choose the adaptation functions ϑ 1 and ϑ as ϑ 1 = Γ ϑ1 z ϕ 1 x b ) + β 1 ϑ 1 ) 43) ϑ = Γ ϑ z 4 ϕ x b ) + β ϑ ). 44) Combining 39) and 40) and using the facts which are acquired through completion of squares we have β 1 ϑ T 1 ϑ 1 β 1 ϑ 1 β ϑ T ϑ β ϑ Then it can be concluded that + β 1 ϑ 1 45) + β ϑ. 46) V al k 1 1)z 1 k 1 1 g V ) 1 M )z k 3 1 )z 3 k 4 1 k M )z4 T 3 1 g V ))δ3+ M + 1 g 1V ) ε0 1 + ε 0 ) 4 ςe e + δ i ψ i j= T 1 g 1V ))δ T 4 1 )δ 4 β 1 ϑ 1 β ϑ + k p ς + ς e + β ϑ + β 1 ϑ 1. 47) Define the following sets: A := {h d ḣd ḧd) : h d + ḣ d + ḧ d h 0 } B :={z 1 z z 3 z 4 δ δ 3 δ 4 ) : z 1 + z + z 3 + z 4 + δ + δ 3 + δ 4 p} where p is a suitable positive number. Note that A and B are compact sets on R 3 and R 7 which makes A B a compact set on R 10 the continuous functions ψ j in 33) for j = 3 4 have maximums on A B i.e. M ψj. Moreover by using Young inequality we have δ j ψ j δ j M ψj δ j M ψj ς M + ς M ) j = ) where ς M is an arbitrary constant. In view of 44) 43) can be rewritten as V al k 1 1)z 1 k 1 1 g V ) 1 M )z k 3 1 )z 3 k 4 1 k M )z 4 T 3 1 g V ) M ψ3 )δ3 + M + 1 ς M g 1V )+ 3 + ε0 1 + ε 0 )e T 1 g 1V ) M ψ )δ ς M ς e T 4 1 M ψ4 )δ4 β 1 ϑ 1 β ϑ + 3ς M ς M + k p ς + ς e + β ϑ + β 1 ϑ 1. 49) Let the control gain filter gain and observer gain satisfy the following inequalities: k 1 > 1 k > g V ) + 1 M k 3 > 1 k 4 > 1 k M T > 1 g 1V ) + M ψ ς M T 3 > 1 g V ) + M ψ3 ς M T 4 > 1 + M ψ4 ς M M < 1 g 1V ) 3 ε0 1 +ε0 ς. e Then 45) can be transformed to where α 0 =min 50) V al α 0 V al + ε 0 51) k 1 1) k 1 1 g V ) 1 M ) k 3 1 ) k 4 k M ) T 4 1 M ψ4 ς M ) β 1 Γ ϑ1 T 3 1 g V ) M ψ3 ς M ) T 1 g 1V ) M ψ ς M ) M + g 1 V ) ε0 1 +ε0 ) β Γ ϑ ε 0 = 3ς M + k pς + ς e + β 1 ϑ 1 ς e + β ϑ. We can see that the value of α 0 can be adjusted by selecting the gains of controller filter and observer. If the gains satisfy α 0 > ε0 p then V al < 0 is on the boundary of compact set A B that is to say the boundary of compact set A B is an invariant set which proves the boundedness of the closedloop system. Moreover solving 47) yields V al t) V al 0)e α0t + t 0 e α0t τ) ε 0 dτ. 5)

8 HOU et al.: OUTPUT FEEDBACK DYNAMIC SURFACE CONTROLLER DESIGN FOR AIRBREATHING HYPERSONIC FLIGHT VEHICLE 193 When t V ) ε0 α 0 that is z 1 ) ε0 α 0. This inequality implies that any given tracking error limitation by properly choosing gains can be guaranteed. The extended state observer can estimate the disturbance d v in finite time and the estimation error d v = d v Z 1 can be extremely small [6 7] so the tracking error dynamics can be approximated by ż V = k v z V which guarantees the stability of velocity loop and convergence of velocity tracking error. The generalized elastic coordinate system can be treated as a disturbance to the rigid-body system [9]. Theorem 1 shows that the closed-loop altitude system has strong disturbance attenuation ability. Velocity loop also possesses high disturbance attenuation ability with the help of extended state observer. Thus the boundedness of generalized elastic coordinate can guarantee the stability of rigid system. With Assumption 1 the dynamics of generalized elastic coordinate is mainly affected by AOA so the boundedness of AOA can guarantee the stability of the structure elastic system from the system dynamics in 1). Thus the stability of the whole system can be guaranteed by the small-gain theorem [30]. Figs. 3 and 4 confirm that the controllers provide stable tracking of the reference trajectories of altitude and velocity and exact convergence to the reference command. The flightpath angle reference command as depicted in Fig. 5 is well tracked. The flight-path angle is smaller than 1.5 deg during the whole process which validates the small angle assumption used in Section II. IV. SIMULATIONS To show the performance of the controllers proposed in previous sections simulations have been performed on the fully nonlinear vehicle model described by 1). The vehicle is desired to track the velocity and altitude references initialized at V 0 = m s 1 h 0 = m γ 0 = 0 deg. The reference commands of velocity and altitude are generated by filtering step reference commands using two second-order prefilters with natural frequency ω f = 0.03 rad/s and damping factor ξ f = Reference h d t) is generated to let the vehicle climb 6.1 km in about 180 s in two steps whereas the velocity reference is one step signal with the increase of 8.6 m/s. The simulations are accomplished in three steps corresponding to the nominal model uncertain model with compensation and the comparisons with other schemes. The compensation-control involves fuzzy logic compensation and uncertainty estimation error compensation. The parameters used in the controllers of all simulations are shown in Table III. The first simulation is only for the nominal model indicating the uncertainties are ignored that is to say w 1 = w = 0. The reference signals of virtual control variables are chosen according to equations 0) 3) and 5) and the control input is computed from 8). In this study the constraints on states and inputs are dealt with indirectly by tuning the controller gains including the parameters of the pre-filter and dynamic surface gain k i i = 1 4). The selection of gain k i should make the inner loop respond quicker than the outer loop. The controller gains used in all simulations are shown in Table III. TABLE III CONTROLLER PARAMETERS Fig. 3. Fig. 4. The tracking curve of altitude command. The tracking curve of velocity command. Gain Value k k 1.5 k 3.8 T i i = 1 4) 15 l 5 k 4.8 Fig. 5. The tracking curve of FPA. Figs. 6 and 7 show the tracking performance of the virtual control variables of pitch angle and its rate. The virtual control

9 194 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. APRIL 015 commands are smooth within their bounds and are nicely tracked. The estimation curve of AOA is given in Fig. 9 which shows the good performance of the proposed estimator. The second simulation is performed for the system with uncertainties. The references of velocity altitude and flightpath angle are the same as those in the first simulation. It is assumed that the true value of some aerodynamic parameters including CL α Cα D Cα M are unknown and different from the nominal value within the tolerance of 5 %. Additionally the mass and moment of inertia vary by 10 % of their nominal values. The simulation is conducted considering the effects of the generalized elastic coordinates. Fig. 6. The tracking curve of pitch angle. According to the simulation results in nominal condition it is clear that even though the exact information on AOA is immeasurable the proposed adaptive fuzzy output feedback controller guarantees both the stability and good tracking performance of the closed-loop system. Fig. 7. Fig. 8. The tracking curve of pitch angle rate. The curve of elevator deflection. Fig. 9. The estimate in curve of AOA. The fuzzy weighting values are computed with update laws 39) and 40). The gains of the update laws are selected as T ϑ1 = 80 β 1 = 0.05 T ϑ = 140 β = Gaussian membership function is used and the variances should be in the suitable range corresponding to the partitioning points selected. From Remark 1 the uncertainty from the difference between α ref and α can also be seen as coefficient uncertainty so w 1 and w can be predigested as the functions of α δ e and η. In the FLSs we use ˆα δ e and η as inputs. The partitioning points are chosen as deg and deg for ˆα and δ e respectively. The partitioning points for η are selected according to the scale of each generalized elastic coordinate which can be estimated by the trim value. In this simulation we select 1.5 to 1.5 with the interval of 0.3 for η to 0.6 with the interval of 0.06 for η and 0.1 to 0.1 with the interval of 0.03 for η 3. The fuzzy membership functions for ˆα are given as follows and the fuzzy membership functions for δ e and η are similar to ˆα: µ F 1 1 ˆα) = 1 1+e 80 / ˆα+4/57.3) µ F1 µ F ˆα) = e 41 µ F 3 1 ˆα) = e ˆα ) 1 µ F 5 1 ˆα) =. 80/ ˆα 8 1+e 57.3 ) ˆα ˆα) = e 0.04 ) ˆα ) It can be seen that the tracking error of altitude remains very well behaved in the presence of model uncertainties. The tracking errors remain remarkably small during the entire maneuver and vanish asymptotically. The good performance of the fuzzy compensators are demonstrated. In fact extensive studies are performed on changing the fuzzy membership functions and weighing update gains for exact estimation of model uncertainties. The effect of δ e on the generalized elastic coordinate is also analyzed in this simulation. It can be seen from Fig. 15 that

10 HOU et al.: OUTPUT FEEDBACK DYNAMIC SURFACE CONTROLLER DESIGN FOR AIRBREATHING HYPERSONIC FLIGHT VEHICLE 195 the effect of δ e is much smaller than that of other parts mainly the angle of attack. proposed observer. The design parameters of SMO are selected according to the principle given in [13]. The simulation is conducted in the same uncertainty condition as in the second simulation. Fig. 10. The tracking curve of altitude command with uncertainty. Fig. 13. The tracking error of altitude. Fig. 11. The tracking error of altitude. Fig. 14. The estimation curve of AOA. Fig. 1. The tracking curve of altitude command with uncertainty. Reference [13] proposes a kind of sliding mode observer to estimate FPA and AOA. The observer and controller are designed separately. In this paper we will give the off-line estimation results of SMO and make a comparison with our Fig. 15. The curve of elevator deflection.

11 196 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. APRIL 015 The simulation results are given in Figs From Fig. 17 it can be seen that the estimation error of pitch rate converges to zero quickly but this do not ensure the convergence of AOA estimation error which can be seen from Fig. 19. The existence of uncertainty changes the origin of the observer which make estimation value of AOA cannot converge to its true value. The estimation error of AOA affects the convergence of altitude and FPA which can be seen from Figs. 16 and 18. In the proposed method of this paper the observer and controller are designed in associated manner. The uncertainty is estimated by FLSs whose parameters are adaptively tuned in the controller. Thus the uncertainty is well compensated in the estimator. Besides the stability analysis of observer can be proved in theory. V. CONCLUSION The usual air-data measurement system cannot work well due to hypersonic aerodynamic heating. Thus output feedback controller is an essential issue for generic airbreathing hypersonic flight vehicle. A novel output feedback control scheme is developed in this paper. Estimated AOA value is acquired through a reduced order observer only using pitch angle and its rate information. The control system is constructed based on the AOA signal from observer. Dynamic surface technique is used in controller design for avoiding complexity explosion of traditional backstepping control. Fuzzy logic based compensator is used in the controller to dispose the inaccuracy and uncertainty of the model. Different from the existing output feedback controller in the literature this scheme not only uses limited information but also takes model uncertainty into consideration which greatly strengthen the robustness of control system. At last simulation results demonstrate the good performance of the proposed method. Fig. 16. Altitude estimation error with SMO. Fig. 18. FPA estimation curve with SMO. Fig. 17. Pitch estimation error with SMO. Fig. 19. AOA estimation curve with SMO. Remark 4. Comparing to state feedback based backstepping control scheme [16 17] for the hypersonic vehicle the proposed scheme in this paper is constructed using limited state information but achieves relatively good tracking performance of velocity and altitude. Additionally dynamic surface technique used in this paper avoids the complexity explosion of backstepping control. REFERENCES [1] Chavez F R Schmidt D K. Uncertainty modeling for multivariablecontrol robustness analysis of elastic high-speed vehicles. Journal of Guidance Control and Dynamics ): [] Zong Q Ji Y H Zeng F L Liu H L. Output feedback backstepping control for a generic hypersonic vehicle via small gain theorem. Aerospace Science and Technology 01 31): [3] Bolender M A Doman D B. Nonlinear longitudinal dynamical model of an air-breathing hypersonic vehicle. Journal of Spacecraft and Rockets ):

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Decentralized adaptive outputfeedback controller design for stochastic nonlinear interconnected systems. Automatica ): [1] Jiang Zhong-Ping Huang Jie. Stabilization and output regulation by nonlinear feedback: a brief overview. Acta Automatica Sinica ): in Chinese) [] Tong S C Li Y M Feng G Li T S. Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems. IEEE Transactions on Systems Man and Cybernetics-Part B: Cybernetics ): [3] Parker J T Serrani A Yurkovich S Bolender M A Doman D B. Controloriented modeling of an air-breathing hypersonic vehicle. Journal of Guidance Control and Dynamics ): [4] Fiorentini L. Nonlinear Adaptive Controller Design for Air-Breathing Hypersonic Vehicles [Ph. D. dissertation] The Ohio State University USA 010. [5] Tong S C Li Y M Feng G Li T S. Observer-based adaptive fuzzy backstepping dynamic surface control for a class of non-linear systems with unknown time delays. IET Control Theory and Applications ): [6] Han J Q. From PID to active disturbance rejection control. IEEE Transactions on Industrial Electronics ): [7] Xia Y Q Zhu Z Fu M Y Wang S. Attitude tracking of rigid spacecraft with bounded disturbances. IEEE Transactions on Industrial Electronics ): [8] Chen M Ge S S Ren B B. Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints. Automatica ): [9] Sun H F Yang Z L Zeng J P. New tracking-control strategy for airbreathing hypersonic vehicles. Journal of Guidance Control and Dynamics ): [30] Jiang Z P Teel A R Praly L. Small-gain theorem for ISS systems and application. Mathematics of Control Signals and Systems ): Delong Hou received his bachelor degree in flight vehicle design and engineering from Northwestern Polytechnical University in 009. He is currently a Ph. D. candidate at the School of Automation Science and Electrical Engineering Beihang University. His research interest covers nonlinear control theory and its application on flight vehicle. Corresponding of this paper. Qing Wang received her bachelor master and Ph. D. degrees from Northwestern Polytechnical University in and 1996 respectively. She is currently a professor at the School of Automation Science and Electrical Engineering Beihang University. Her research interest covers guidance and control of aerospace flight vehicles fault diagnosis and tolerant control switched control and its application on flight control system. Chaoyang Dong received his bachelor degree from Beihang University in 1989 master degree in flight dynamics from Northwestern Polytechnical University in 199 and Ph. D. degree in guidance navigation and control from Beihang University in 007. He is currently a professor of at the School of Aeronautic Science and Engineering Beihang University. His research interest covers vehicle modeling and control simulation of aerospace vehicles and synthesis of aerospace electrical system.