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, 2, 3, 4, 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.

2) 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 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^{[12, 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^[2].

Observer-based output feedback control is a feasible method for hypersonic vehicle^{[2, 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 ^[2] proposes an output feedback controller combining backstepping and sliding mode observer. The stability of the closed-loop system and the convergence of the output 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 ^[2]. 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.

2) 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 ^[20] 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 ^[21]. 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.

2) 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^[22] 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 curve-fitted form. The control inputs of this model are the elevator deflection angle and fuel-to-air ratio. The longitudinal dynamic equations are written as

$\begin{align} \begin{cases} \dot h = V\sin \gamma,\\ \dot V = \frac{{T\cos \alpha - D}}{m} - g\sin \gamma,\\ \dot \gamma = \frac{{L + T\sin \alpha }}{{mV}} - \frac{g}{V}\cos \gamma,\\ \dot \alpha = - \frac{{L + T\sin \alpha }}{{mV}} + Q + \frac{g}{V}\cos \gamma,\\ \dot \theta = Q,\\ \dot Q = \frac{M}{{{I_{yy}}}},\\ {\ddot \eta _i} = - 2{\zeta _i}{\omega _i}{\dot \eta _i} - \omega _i^2{\eta _i} + {N_i},\quad i = 1,2,3, \end{cases} \end{align}$

(1)

where $h$,$V$,$\gamma $,$\alpha $,$\theta $,$Q$ and ${\eta _i}$ are altitude,velocity,flight path angle (FPA),AOA,pitch angle,pitch rate and the $i$th generalized elastic coordinate of the vehicle,respectively. $T$,$D$,$L$,$M$ and ${N_i}$ denote thrust,drag,lift,pitching moment and the $i$th generalized force,respectively. In the CFM,by defining $\boldsymbol \eta = {[{\eta _1}~~{\eta _2}~~{\eta _3}]^{\rm T}}$,these forces and moment are approximated as follows:

$\begin{align} \begin{cases} T \approx \bar qS[{C_{T,\Phi }}(\alpha )\Phi + {C_T}(\alpha )+\boldsymbol C_T^\eta \boldsymbol \eta],\\ D = \bar qS{C_D}(\alpha ,{\delta _e},\boldsymbol\eta),\\ L = \bar qS{C_L}(\alpha ,{\delta _e},\boldsymbol\eta),\\ M = {z_T}T + \bar qS\bar c{C_M}(\alpha ,{\delta _e},\boldsymbol\eta), \end{cases} \end{align}$

(2)

where $\bar q = (1/2)\rho {V^2}$ is the dynamic pressure, ${C_{T,\Phi }}$,${C_T}$,${C_L}$ and ${C_M}$ are forces and moment coefficients. These coefficients are defined as follows:

$\begin{align} \left\{\begin{array}{*{20}lll}{C_{T,\Phi }}(\alpha ) = C_T^{\Phi {\alpha ^3}}{\alpha ^3} + C_T^{\Phi {\alpha ^2}}{\alpha ^2} + C_T^{\Phi \alpha }\alpha + C_T^\Phi+ \Delta {C_{T,\Phi }},\\ {C_T}(\alpha ) = C_T^3{\alpha ^3} + C_T^2{\alpha ^2} + C_T^1\alpha + C_T^0 + \Delta {C_T},\\ {C_L}(\alpha ,{\delta _e}) = C_L^\alpha \alpha + C_L^{{\delta _e}}{\delta _e} + C_L^0 + \boldsymbol C_L^\eta \boldsymbol\eta + \Delta {C_L},\\ {C_D}(\alpha ,{\delta _e}) = C_D^{{\alpha ^2}}{\alpha ^2} + C_D^\alpha \alpha + C_D^{\delta _e^2}\delta _e^2 + C_D^{{\delta _e}}{\delta _e} + C_D^0 +\\ \quad~ ~\qquad\qquad \boldsymbol C_D^\eta \boldsymbol\eta + \Delta {C_D},\\ {C_M}(\alpha ,{\delta _e}) = C_M^{{\alpha ^2}}{\alpha ^2} + C_M^\alpha \alpha + C_M^{{\delta _e}}{\delta _e} + C_M^0 +\\ \quad~~ \qquad\qquad \boldsymbol C_M^\eta \boldsymbol\eta + \Delta {C_M},\\ \boldsymbol {C}_j^\eta = [C_j^{{\eta _1}}~~0~~C_j^{{\eta _2}}~~0~~C_j^{{\eta _3}}~~0],~~j = T,M,L,D,\\ \boldsymbol N_i^\eta = [N_i^{{\eta _1}}~~0~~N_i^{{\eta _2}}~~0~~N_i^{{\eta _3}}~~0],\quad i = 1,2,3. \end{array}\right. \end{align}$

(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^[23]. 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 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 ^[24].

From equations (1)-(3),thrust $T$ and moment $M$ are expressed in nonlinear form. We calculate $T$ for different $\Phi $ and $\alpha $,and $M$ for different $\alpha$ while the elevator angle is the trim value,and the relationships are depicted in Figs. 1 and 2.

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 2. Consequently,for the value of $\Phi $ and ${\delta _e}$ in the admissible range,$T(\alpha ,\Phi )$ and $M(\alpha ,{\delta _e})$ satisfy the Lipschitz condition with some Lipschitz constants ${\sigma _T}$ and ${\sigma _M}$ as

$\begin{align*} &\left| {T({\alpha _1},\Phi ) - T({\alpha _2},\Phi )} \right| \le {\sigma _T}\left| {{\alpha _1} - {\alpha _2}} \right|,\\ &\left| {M({\alpha _1},{\delta _e}) - M({\alpha _2},{\delta _e})} \right| \le {\sigma _M}\left| {{\alpha _1} - {\alpha _2}} \right|, {\alpha _1},\quad {\alpha _2} \in {\Omega _\alpha }, \end{align*}$

where ${\Omega _\alpha }$ denotes the argument range for $\alpha$ which is listed in Table II.

According to (1),the velocity is mainly controlled by the throttle setting $\Phi$,while the change of the altitude is governed by the elevator deflection ${\delta _e}$. 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

$\begin{align} \dot V = {f_v} + {g_v}\Phi, \end{align}$

(4)

where ${f_v} = \bar qS{C_T}(\alpha )\cos \alpha /m - \bar qS{C_D}/m - g\sin \gamma $,${g_v}({x_1},{x_2},{x_3},V) = \bar qS{C_{T,\Phi }}(\alpha )\cos \alpha /m$.

The rest of the hypersonic vehicle dynamic model constitutes the altitude tracking loop. The selected states for altitude loop are defined as

$\begin{eqnarray*} \boldsymbol x = {[{x_1},{x_2},{x_3},{x_4}]^{\rm T}} = {[h/{h_c},\gamma ,\theta ,q]^{\rm T}}. \end{eqnarray*}$

The transformation of altitude is to keep all states in the same size scale. Due to the fact that $\cos \gamma \approx 1$ if $\gamma $ is small enough,the CFM of altitude loop can be transformed into

$\begin{array}{*{20}{l}} {{{\dot x}_1}}&{ = {g_1}(V)\sin {x_2},} \\ {{{\dot x}_2}}&{ = {f_2}({x_2},V) + {g_2}(V){x_3} + {w_1}(\alpha ,{\alpha ^{ref}},{\delta _e},{\mathbf{\eta }}),} \\ {{{\dot x}_3}}&{ = {g_3}{x_4},} \\ {{{\dot x}_4}}&{ = {f_4}({x_2},{x_3},V) + {g_4}(V)u + {w_2}(\alpha ,{\alpha ^{ref}},{\delta _e},,{\mathbf{\eta }}),} \end{array}$

(5-8)

where

$\begin{align*} &{g_1}(V) = V/{h_c},~~~{g_2}(V) = {{(C_L^\alpha qS + {k_{T1}})}/{(mV)}},\\ & {f_2}({x_2},V) = {f_{20}}({x_2},V) - {g_2}(V) \times \gamma ,\\ & {f_{20}}({x_1},V) = {{C_L^0qS}/{(mV)}} - {g/V},~~~{g_3} = 1,\\ &{f_4} = {{[{z_T}({k_{T2}}\alpha + C_T^c) + \bar qS\bar c({k_M}\alpha + C_M^0)]}/{{I_{yy}}}},\\ & {g_4} = {{qS\bar cC_M^{{\delta _e}}}/{{I_{yy}}}},~~~u = {\delta _e},\\ &{k_{T1}} = C_T^{{\alpha ^3}}{({\alpha ^{ref}})^3} + C_T^{{\alpha ^2}}{({\alpha ^{ref}})^2} + C_T^\alpha ({\alpha ^{ref}}) + C_T^c,\\ &{k_{T2}} = C_T^{{\alpha ^3}}{({\alpha ^{ref}})^2} + C_T^{{\alpha ^2}}{\alpha ^{ref}} + C_T^\alpha ,\\ &{k_M} = C_M^{{\alpha ^2}}{\alpha ^{ref}} + C_M^\alpha . \end{align*}$

The goal of this study is to synthesize a controller using measurable state information realizing altitude and velocity tracking.

Remark 2. It can be concluded that ${w_1}$ and ${w_2}$ are mainly constituted by three parts,namely,the difference between ${\alpha ^{ref}}$ and $\alpha $,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 ${\delta _e}$.

Assumption 2. The reference signal ${h_d}(t)$ is a sufficiently smooth function of $t$,and ${h_d}(t)$,${\dot h_d}(t)$,${\ddot h_d}(t)$ are bounded within a known compact set

$\begin{align*} {\Omega _{hd}} = \{ {[{h_d},{\dot h_d},{\ddot h_d}]^{\rm T}}:h_d^2 + \dot h_d^2 + \ddot h_d^2 \le {h_0}\} \in {{\bf R}^3}, \end{align*}$

where ${h_0}$ is a known positive constant.

Remark 3. Assumption 1 is made according to the scale of $N_i^\alpha $ and $N_i^{{\delta _e}}$; $N_i^{{\delta _e}}$ is much smaller than $N_i^\alpha $. 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^[25] Let $f(\boldsymbol x)$ be a continuous function of vector $\boldsymbol x$ defined on a compact set $U$. Then for any constant $\varepsilon > 0$,there exists the following FLSs:

$\begin{align} \hat f(\boldsymbol x|\boldsymbol\vartheta ) = {\boldsymbol\vartheta ^{*{\rm T}}}\boldsymbol\varphi (\boldsymbol x), \end{align}$

(9)

such that

$\begin{align} \mathop {\sup }\limits_{\boldsymbol x \in U} \left| {\hat f(\boldsymbol x|\boldsymbol\vartheta ) - f(\boldsymbol x)} \right| \le \varepsilon, \end{align}$

(10)

where ${\boldsymbol\vartheta ^*}$ is the optimal parameter defined as

$\begin{align*} {\boldsymbol \vartheta ^*} = \arg \mathop {\min }\limits_{\boldsymbol\vartheta \in \Omega } \left( {\mathop {\sup }\limits_{\boldsymbol x \in U} \left| {\hat f(\boldsymbol x|\boldsymbol\vartheta ) - f(\boldsymbol x)} \right|} \right). \end{align*}$

$\boldsymbol\varphi (x) = [{\varphi _1}(x),\cdots ,{\varphi _N}(x)]$ is the rector of fuzzy basis functions, $\boldsymbol\vartheta$ is the vector of weighting coefficients. ${\varphi _l}(x)$ $(l=1,\cdots ,N)$ are defined as

$\begin{align*} {\varphi _l}(x) = \frac{{\prod\limits_{i = 1}^n {{\mu _{F_i^l}}({x_i})} }}{{\sum\limits_{l = 1}^N {(\prod\nolimits_{i = 1}^n {{\mu _{F_i^l}}({x_i})} )} }}, \end{align*}$

where ${\mu _{F_i^l}}({x_i})$ is the fuzzy membership function.

In the model presented in (1),$\gamma $ and $\alpha $ are two immeasurable states. Due to the relationship that $\alpha = \theta - \gamma $,we only need to estimate one of them. We select $\alpha $ as the variable to be estimated. Defining ${\boldsymbol x_b} = {[\alpha ,{\alpha ^{ref}},{\delta _e},\boldsymbol \eta ]^{\rm T}}$,${w_i}(\alpha ,{\alpha ^{ref}},{\delta _e},\boldsymbol \eta)$ ($i = 1,2$) can be denoted as ${w_1}({\boldsymbol x_b})$ and ${w_2}({\boldsymbol x_b})$. According to Lemma 1,each unknown nonlinear uncertain function ${\hat w_j}$ can be approximated by a FLS in the form of

$\begin{align} {\hat w_i}({\boldsymbol x_b}|{\boldsymbol\vartheta _i}) =\boldsymbol \vartheta _i^{\rm T}{\boldsymbol\varphi _i}({\boldsymbol x_b}). \end{align}$

(11)

Denoting ${\underline{\boldsymbol x}_b} = {[\hat \alpha ,{\alpha ^{ref}},{\delta _e},{\eta ^ * }]^{\rm T}}$,where ${\boldsymbol\eta ^ * }$ is the trim value of ${\boldsymbol\eta}$,one has

$\begin{align} {\hat w_i}({\underline{\boldsymbol x}_b}|{\boldsymbol\vartheta _i}) =\boldsymbol \vartheta _i^{\rm T}{\boldsymbol\varphi _i}({\underline{\boldsymbol x}_b}). \end{align}$

(12)

Define the optimal parameter vectors $\boldsymbol\vartheta _i^*$ ($i = 1,2$) as

$\vartheta _{i}^{*}=\arg \underset{{{\vartheta }_{i}}\in {{\Omega }_{i}}}{\mathop{\min }}\,\left( \underset{\begin{matrix} {{\mathbf{x}}_{b}}\in {{U}_{b}} \\ {{\underline{\mathbf{x}}}_{b}}\in {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{U}}}_{b}}\text{ } \\ \end{matrix}}{\mathop{\sup }}\,\left| {{{\hat{w}}}_{i}}({{\mathbf{x}}_{b}}|{{\vartheta }_{1}})-{{w}_{i}}({{\mathbf{x}}_{b}}) \right| \right),$

(13)

where ${\Omega _i}$,${U_b}$ and ${\underline{U}_b}$ are compact regions for ${\boldsymbol\vartheta _i}$,${{\boldsymbol x}_{b}}$ and ${\underline{\boldsymbol x}_b}$,respectively. The minimum estimation errors $\varepsilon _i^ * $ and estimation errors ${{\varepsilon }_{i}}$ are defined as

$\begin{align} &\varepsilon _i^* = {w_i}({\boldsymbol x_b}) - {\hat w_i}({\underline{\boldsymbol x}_b}|\boldsymbol\vartheta _i^*),\\ &{\varepsilon _i} = {w_i}({\boldsymbol x_b}) - {\hat w_i}({\underline{\boldsymbol x}_b}|{\boldsymbol\vartheta _i}). \end{align}$

(14-15)

Assumption 3. With the boundedness of ${\boldsymbol \eta}$,there exist known constants $\varepsilon _i^{*0} > 0$ and $\varepsilon _i^0 > 0$,such that $\left| {\varepsilon _i^*} \right| \le \varepsilon _i^{*0}$ and $\left| {{\varepsilon _i}} \right| \le \varepsilon _i^0$ $(i = 1,2)$.

A reduced order observer for AOA dynamics is formulated as

$\begin{align} \begin{cases} \hat \alpha = \xi + \boldsymbol L\boldsymbol y = \xi + {l_1}\theta + {l_2}q,\\ \dot \xi = M\xi + N{\delta _e} + Ry + C = M\xi + N{\delta _e} + {r_1}\theta + {r_2}q + C, \end{cases} \end{align}$

(16)

where $\boldsymbol y = {[\theta ,q]^{\rm T}}$ is the measurable states vector,$\boldsymbol L = [{l_1},{l_2}]$ is the observer gain coefficient matrix,and these gain coefficients are defined as

$\begin{align*} & M = - \frac{{\bar qSC_L^\alpha + {k_{T1}}}}{{mV}} - \frac{{{l_2}({z_T}{k_{T2}} + \bar qS\bar c{k_M})}}{{{I_{yy}}}},\\ & {r_1} = {l_1} = 0,\nonumber \\ & {r_2} = ( - \frac{{\bar qSC_L^\alpha + {k_{T1}}}}{{mV}} - \frac{{{l_2}({z_T}{k_{T2}} + \bar qS\bar c{k_M})}}{{{I_{yy}}}}){l_2} + 1 ,\\ & N = - \frac{{\bar qSC_L^{{\delta _e}}}}{{mV}} - \frac{{{l_2}\bar qS\bar cC_M^{{\delta _e}}}}{{{I_{yy}}}},\\ & C = - \frac{{\bar qSC_L^0}}{{mV}} + \frac{g}{V} - \frac{{{l_2}({z_T}C_T^c + \bar qS\bar cC_M^0)}}{{{I_{yy}}}} - {{\hat w}_1}({\underline{\boldsymbol x}_b}|{\boldsymbol\vartheta _1}) -\\ &\qquad {\rm{ }}{l_2}{{\hat w}_2}({{\underline{\boldsymbol x}_b}}|{\boldsymbol\vartheta _2}). \end{align*}$

Let $e = \tilde \alpha = \alpha - \hat \alpha $,then the estimation error $\tilde \gamma = - e$. The estimation error dynamics can be deduced as

$ \begin{align} \dot e &= \dot \alpha - \dot {\hat \alpha} =\notag\\ & - \frac{{\bar qS(C_L^\alpha \alpha + C_L^0) + {k_{T1}}\alpha }}{{mV}} + q + \frac{g}{V} - {w_1} - \notag\\ &[M\xi + N{\delta _e} + {r_1}\theta + {r_2}q + C + {l_1}q + \frac{{{l_2}}}{{{I_{yy}}}}({z_T}({k_{T2}}\alpha + C_T^c) + \notag\\ &\bar qS\bar c({k_M}\alpha + C_M^{{\delta _e}}{\delta _e} + C_M^0)) + {l_2}{w_2}] =\notag\\ & Me - {\varepsilon _1} - {\varepsilon _2}. \end{align}$

(17)

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 $\left| {e(\infty )} \right| = (\varepsilon _1^0 + \varepsilon _2^0)\sqrt {\frac{1}{{ - (2M + 1)}}} $.

Proof. Consider the Lyapunov function candidate as

$\begin{align} {V_e} = \frac{1}{2}{e^2}. \end{align}$

(18)

The derivative along the trajectory of (17) results in

$\begin{align*} {{\dot V}_e} =~& e\dot e = e(Me - {\varepsilon _1} - {\varepsilon _2}) \le\\ & M{e^2} + (\varepsilon _1^0 + \varepsilon _2^0)\left| e \right| \le\\ & (M + \frac{1}{2}){e^2} + {\frac{\left(\varepsilon _1^0 + \varepsilon _2^0\right)^2}{2}} \le\\ & (2M + 1){V_e} + {\frac{\left(\varepsilon _1^0 + \varepsilon _2^0\right)^2}{2}}. \end{align*}$

According to the differential inequality theory,there exists the following inequality by induction

$\begin{align*} {V_e}(t) \le {V_e}(0){{\rm e}^{(2M + 1)t}} + \int_0^t {{{\rm e}^{(2M + 1)(t - \tau )}}\frac{{{{(\varepsilon _1^0 + \varepsilon _2^0)}^2}}}{2}{\rm d}\tau }, \end{align*}$

when $t \to \infty $,$V(\infty ) \le \frac{{{{(\varepsilon _1^0 + \varepsilon _2^0)}^2}}}{{ -2 (2M + 1)}}$,that is to say $\left| {e(\infty )} \right| \le (\varepsilon _1^0 + \varepsilon _2^0)\sqrt {\frac{1}{{ - (2M + 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 $-(2M + 1)$.

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

$\begin{align} {\dot z_1} = {g_1}(V){\bar x_2} - {\dot h_d}. \end{align}$

(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 ${\bar x_2}$ is designed as

$\begin{align} \bar x_2^{ref0} = \frac{1}{{{g_1}(V)}}( - {k_1}{z_1} + {\dot h_d}). \end{align}$

(20)

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.,

$\begin{align} \dot {\bar x}_2^{ref} = - {T_2}({\bar x}_2^{ref} - {\bar x}_2^{ref0}),\quad {\bar x}_2^{ref}(0) = {\bar x}_2^{ref0}(0), \end{align}$

(21)

where ${T_2}$ is the time constant of filter and should be as large as possible to promise the fast tracking.

Step 2. Define the dynamic surface equation in Eq. as ${z_2}: = \sin{\hat x_2} - {\bar x}_2^{ref}$,whose dynamics can be calculated by (6),i.e.,

$\begin{align} {\dot z_2} = \cos({\hat x_2})[{f_2}({x_2},V) + {g_2}(V){x_3} + {w_1}({\boldsymbol x_b}) + \dot e]- \dot x_2^{ref}, \end{align}$

(22)

where ${x_3}$ is the virtual control input of this step,the nominal value of $x_3^{ref0}$ is chosen to satisfy the following equation:

$\begin{align} {g_2}(V)x_3^{ref0}= &\frac{1}{{\cos ({{\hat x}_2})}}[- {k_2}{z_2} - {g_1}{z_1}-\notag\\ &(\varepsilon _1^{*0} + \varepsilon _1^0{\rm{ + }}\varepsilon _2^0{\rm{)tanh}}(\frac{z_2}{\varepsilon _2})+ \dot {\bar x}_2^{ref}]-\notag\\ &{f_{20}}({x_2},V) + {g_2}(V) \times \hat \gamma - {{\hat w}_1}({\underline{\boldsymbol x}_b} |{\boldsymbol\vartheta _1}). \end{align}$

(23)

Let the nominal virtual control variable value passing through the following first-order filter:

$\begin{align} \dot x_3^{ref} = - {T_3}(x_3^{ref} - x_3^{ref0}),\quad x_3^{ref}(0) = x_3^{ref0}(0), \end{align}$

(24)

where ${T_3}$ is the time constant of filter.

Step 3. Define dynamic surface as ${z_3}: = {x_3} - x_3^{ref}$,its dynamics can be calculated from (6),i.e.,

$\begin{eqnarray*} {\dot z_3} = {x_4} - \dot x_3^{ref}, \end{eqnarray*}$

where ${x_4}$ is the virtual control variable of this step,and its nominal value is selected as

$\begin{align} x_4^{ref0} = - {k_3}{z_3} + \dot x_3^{ref} - {z_2}\cos({\hat x_2}){g_2}(V). \end{align}$

(25)

This nominal virtual control variable value will also pass through the following first-order filter to get the derivative of the virtual control variable:

$\begin{align} \dot x_4^{ref} = - {T_4}(x_4^{ref} - x_4^{ref0}),\quad x_4^{ref}(0) = x_4^{ref0}(0). \end{align}$

(26)

where ${T_4}$ is the filter time constant.

Step 4. Define dynamic surface as ${z_4}: = {x_4} - x_4^{ref}$. From (8),the dynamics of ${z_4}$ is calculated as

$\begin{align} {\dot z_4} = {f_4}({x_2},{x_3},V) + {g_4}(V)u + {w_2}({\boldsymbol x_b}) - \dot x_4^{ref}. \end{align}$

(27)

Select the control input to satisfy

$\begin{align} {g_4}(V)u=& - {k_4}{z_4} - {f_4}({{\hat x}_2},{x_3},V) + \dot x_4^{ref} - {z_3} - {{\hat w}_2}({\underline{\boldsymbol x}_b}|{\boldsymbol\vartheta _2}) -\notag\\ &~\varepsilon _2^{*0}{\rm{tanh}}\left(\frac{z_4}{\varepsilon _4}\right). \end{align}$

(28)

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

$\begin{align} {\dot z_V} = {\hat f_v} + {\hat g_v}\Phi + {d_v} - {\dot V_d}, \end{align}$

(29)

where ${d_v}$ denotes the uncertainties of velocity loop,which is induced by observer error of $\alpha$. It can be estimated from the following extended state observer^{[26, 27]}:

$\begin{align} \begin{cases} {E_{11}} = {Z_{11}} - V,\\ {{\dot Z}_{11}} = {Z_{12}} + {{\hat f}_v} - {\beta _{11}}{E_{11}} + {{\hat g}_v}\Phi,\\ {{\dot Z}_{12}} = - {\beta _{12}}fal({E_{11}},{\lambda _v},{\varepsilon _v}), \end{cases} \end{align}$

(30)

where $0 < \lambda _v < 1$,$\varepsilon _v>0$,$\beta _{11}>0$, $\beta _{12}>0$ are design coefficients,and the function $fal$ is defined as

$\begin{align} \!\!\!\!\!\!\! fa{l}({E_{11}},{\lambda _v},{\varepsilon _v}) = \left\{ \begin{array}{llll} {\left| {{E_{11}}} \right|^{{\lambda _v}}}{\mathop{\rm sgn}} ({E_{11}}),& \left| {{E_{11}}} \right| > {\varepsilon _v},\\ \frac{E_{11}}{\varepsilon _v^{1 - {\lambda _v}}},&\mbox{others}. \end{array} \right. \end{align}$

(31)

Then the controller can be designed as

$\begin{align} \Phi = \hat g_v^{ - 1}(V)( - {k_v}{z_V} - {\hat f_v} - {Z_{12}} + {\dot V_d}). \end{align}$

(32)

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

$\begin{eqnarray*} {\delta _2} = x_2^{ref} - x_2^{ref0},{\delta _3} = x_3^{ref} - x_3^{ref0},{\delta _4} = x_4^{ref} - x_4^{ref0}. \end{eqnarray*}$

Then the actual state variable can be denoted as ${x_2} = e + {z_2} + x_2^{ref0} + {\delta _2}$,${x_3} = {z_3} + x_3^{ref0} + {\delta _3}$,${x_4} = {z_4} + x_4^{ref0} + {\delta _4}$. The closed-loop dynamics of four dynamic surfaces of altitude loop are deduced as

$\begin{align} {{\dot z}_1} =& {g_1}(V){x_2} - {{\dot h}_d} = {g_1}(V)(e + {z_2} + {\delta _2} + x_2^{ref0}) - {{\dot h}_d} =\nonumber\\ & - {k_1}{z_1} + {g_1}(V)(e + {z_2} + {\delta _2}),\\ {{\dot z}_2} =& - {k_2}{z_2} + \cos ({{\hat x}_2}){g_2}(V)({z_3} + {\delta _3} + e) +\cos ({{\hat x}_2})\times\nonumber\\ &[{w_1}({\boldsymbol x_b}) - {{\hat w}_1}({\underline{\boldsymbol x}_b}|{\boldsymbol\vartheta _1})] + \cos ({{\hat x}_2}) \dot e - {g_1}{z_1} -\nonumber\\ &(\varepsilon _1^{*0} + \varepsilon _1^0{\rm{ + }}\varepsilon _2^0{\rm{)tanh}}\left(\frac{z_2}{\varepsilon _2}\right),\\ {\dot z_3} =& - {k_3}{z_3} + {z_4} + {\delta _4} - {z_2}{g_2}(V),\\ {{\dot z}_4} = & - {k_4}{z_4} + \frac{({z_T}{k_{T2}} + \bar qS\bar c{k_M})e}{I_{yy}} + {w_2}({\boldsymbol x_b}) - \nonumber\\ &{{\hat w}_2}({{\underline{\boldsymbol x}_b}}|{\boldsymbol\vartheta _2}) - {z_3} - \varepsilon _2^{*0}{\rm{tanh}}\left(\frac{z_4}{\varepsilon _4}\right). \end{align}$

(33-36)

The boundary layers ${\delta _i}$ ($i = 2,3,4$) and the three derivatives satisfy the following relations:

$\begin{align} {\delta _i}{{\dot \delta }_i} &= {\delta _i}{T_i}(x_i^{ref0} - x_i^{ref}) - {\delta _i}\dot x_i^{ref0} \le\notag\\ &- {T_i}\delta _i^2 + \left| {{\delta _i}} \right|{\psi _i}({z_1},{z_2},{z_3},{z_4},{\delta _1},{\delta _2},{\delta _3},{h_d},{{\dot h}_d},{{\ddot h}_d}), \end{align}$

(37)

where ${\psi _i}$ is a continuous function.

By defining ${\tilde {\boldsymbol \vartheta }_1} = \boldsymbol\vartheta _1 - {\boldsymbol\vartheta _1}^*$,${\tilde {\boldsymbol\vartheta }_2} =\boldsymbol \vartheta _2 - {\boldsymbol\vartheta _2}^*$,then the FLS estimation error can be denoted as

$\begin{align} &{w_i}({\boldsymbol x_b}) - {{\hat w}_i}({{\underline{\boldsymbol x}_b}}|{\boldsymbol \vartheta _i})= \notag\\ &\quad [{w_i}({\boldsymbol x_b}) - {{\hat w}_i}({{\underline{\boldsymbol x}_b}}|\boldsymbol\vartheta _i^*)] + [{{\hat w}_i}({{\underline{\boldsymbol x}_b}}|\boldsymbol\vartheta _i^*) - {{\hat w}_i} ({{\underline{\boldsymbol x}_b}}|{\boldsymbol\vartheta _i})]= \notag\\ &\quad \varepsilon _i^* - \boldsymbol \vartheta _i^{\rm T}{\boldsymbol\varphi _i}({\underline{\boldsymbol x}_b}). \end{align}$

(38)

The following lemma will be used in stability analysis.

Lemma 2^[28] For any constant $\varepsilon > 0$,$\chi \in \bf{R}$,the following inequality holds:

$\begin{eqnarray*} 0 \le \left| \chi \right| - \chi \tanh \left(\frac{\chi }{\varepsilon }\right) \le {k_p}\varepsilon, \end{eqnarray*}$

where ${k_p}$ is a constant satisfying ${k_p} = {{\rm e}^{ - ({k_p} + 1)}}$.

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 (28),together with the FLSs estimator (16),there exist proper positive numbers ${k_i}$,${T_j}$ ($i = 1,2,3,4$,$j = 2,3,4$) and a negative number $M$,such that all signals of the closed-loop altitude system are uniformly bounded, and the altitude tracking error ${{\text{z}}_{\text{1}}}$ converges to a residual set that can be made arbitrarily small by properly choosing some design parameters.

Proof. Define the Lyapunov function candidate as

$\begin{align} {V_{{\rm{al}}}} = \frac{1}{2}\sum\limits_{i = 1}^4 {z_i^2} + \frac{1}{2}\sum\limits_{j = 2}^4 {\delta _j^2} + \frac{1}{2}\sum\limits_{k = 1}^2 {\Gamma _{\vartheta k}^{ - 1}\tilde {\boldsymbol \vartheta} _k^{\rm {T}}\tilde {\boldsymbol \vartheta} _k} + \frac{1}{2}{e^2}. \end{align}$

(39)

Along the trajectories (29)-(32),the derivative of the Lyapunov function candidate can be calculated as

$\begin{align} &{{\dot V}_{{\rm{al}}}} = {z_1}[- {k_1}{z_1} + {g_1}(V)(e + {z_2} + {\delta _2})] + \notag\\ &\quad{z_2}[- {k_2}{z_2} + \cos ({{\hat x}_2}){g_2}(V)({z_3} + {\delta _3} + e) + \cos ({{\hat x}_2})\times\notag\\ &\quad [{w_1}({\boldsymbol x_b})-{{\hat w}_1}({\underline{\boldsymbol x}_b}|{\boldsymbol\vartheta _1})]- {g_1}{z_1} + \cos ({{\hat x}_2}) \dot e - (\varepsilon _1^{*0} + \varepsilon _1^0{\rm{ + }}\varepsilon _2^0)\times\notag\\ &\quad {\rm tanh}(\frac{z_2}{\varepsilon _2})] + {z_3}[- {k_3}{z_3} + {z_4} + {\delta _4} - {z_2}\cos({{\hat x}_2}) {g_2}(V)] + \notag\\ &\quad {z_4}[- {k_4}{z_4} + \frac{({z_T}{k_{T2}} + \bar qS\bar c{k_M})e}{I_{yy}} + {w_2}({\boldsymbol x_b}) - \notag\\ &\quad {{\hat w}_2}({\underline{\boldsymbol x}_b}|{\boldsymbol\vartheta _2}) - {z_3} - \varepsilon _2^{*0}{\rm{tanh}}(\frac{z_4}{\varepsilon _4})] + \sum\limits_{j = 2}^4 {{\delta _j}{{\dot \delta }_j}} + \notag\\ &\quad \sum\limits_{k = 1}^2 {\Gamma _{\vartheta k}^{ - 1}{{\tilde {\boldsymbol \vartheta} }^{\rm {T}}_k}{{\dot {\tilde {\boldsymbol \vartheta} }}_k}} + e\dot e.\notag \end{align}$

By simplifying the equation,the derivative of Lyapunov function is transformed into

$\begin{align} {{\dot V}_{{\rm{al}}}} =& {z_1}[- {k_1}{z_1} + {g_1}(V)(e + {\delta _2})] + \notag\\ & {z_2}[- {k_2}{z_2} +\cos ({{\hat x}_2}){g_2}(V)(z_3+{\delta _3} + e) +\cos ({{\hat x}_2})\times\notag\\ &[{w_1}({\boldsymbol x_b}) - {{\hat w}_1}({\underline{\boldsymbol x}_b}|{\boldsymbol\vartheta _1})] + \cos ({{\hat x}_2})\dot e - (\varepsilon _1^{*0} + \varepsilon _1^0+\varepsilon _2^0)\times\notag \\ &{\rm tanh}\left(\frac{z_2}{\varepsilon _2}\right)] + {z_3}[- {k_3}{z_3} + {\delta _4}] + {z_4}[- {k_4}{z_4} + \notag\\ &\frac{({z_T}{k_{T2}} + \bar qS\bar c{k_M})e}{I_{yy}} + {w_2}({\boldsymbol x_b}) - {{\hat w}_2} ({\underline{\boldsymbol x}_b}|{\boldsymbol\vartheta _2}) - \notag\\ &\varepsilon _2^{*0}{\rm{tanh}}\left(\frac{z_4}{\varepsilon _4}\right)] + \sum\limits_{j = 2}^4 {{\delta _j} {{\dot \delta }_j}} + \sum\limits_{k = 1}^2 {\Gamma _{\vartheta k}^{ - 1}{{\tilde {\boldsymbol \vartheta} }^{\rm {T}}_k}{{\dot {\tilde {\boldsymbol \vartheta} }}_k}} + e\dot e {\rm{ }}= \notag\\ & - {k_1}z_1^2 + {g_1}(V){z_1}(e + {\delta _2}) - {k_2}z_2^2 + {g_2}(V) \cos ({{\hat x}_2}){z_2}\times\notag\\ &({\delta _3} + e) + {z_2} \cos ({{\hat x}_2})(\varepsilon _1^* - \tilde {\boldsymbol\vartheta}_1^{\rm {T}}{\boldsymbol\varphi _1}({\underline{\boldsymbol x}_b})) + {z_2} \cos ({{\hat x}_2})\times\notag\\ &(Me - {\varepsilon _1} - {\varepsilon _2}) - (\varepsilon _1^{*0} + \varepsilon _1^0{\rm{ + }}\varepsilon _2^0{\rm{)}}{z_2}{\rm{tanh}}\left(\frac{z_2}{\varepsilon _2}\right) - \notag\\ &{k_3}z_3^2 + {z_3}{\delta _4} - {k_4}z_4^2 + {{\bar k}_M}e{z_4} + {z_4}(\varepsilon _2^* - \tilde {\boldsymbol \vartheta} _2^{\rm {T}}{\boldsymbol\varphi _2} ({\underline{\boldsymbol x}_b})) - \notag\\ &\varepsilon _2^{*0}{z_4}{\rm{tanh}}(\frac{z_4}{\varepsilon _4}) + \sum\limits_{j = 2}^4 {{\delta _j}{{\dot \delta }_j}} + \sum\limits_{k = 1}^2 {\Gamma _{\vartheta k}^{ - 1}{{\tilde {\boldsymbol \vartheta} }^{\rm{T}}_k}{{\dot {\tilde {\boldsymbol \vartheta}} }_k}} + e\dot e, \end{align}$

where ${\bar k_M} = ({z_T}{k_{T2}} + \bar qS\bar c{k_M})/{I_{yy}}$. By using Young inequality,we have the following relationships:

$\begin{align} \begin{cases} {g_1}(V){z_1}e \le \frac{1}{2}(g_1^2(V){e^2} + z_1^2),\\ {g_1}(V){z_1}{\delta _2} \le \frac{1}{2}(z_1^2 + g_1^2(V)\delta _2^2),\\ {g_2}(V) \cos ({{\hat x}_2}){z_2}{\delta _3} \le \frac{1}{2}(z_2^2 + g_2^2(V)\delta _3^2),\\ {g_2}(V) \cos ({{\hat x}_2}){z_2}e \le \frac{1}{2}({e^2} + g_2^2(V)z_2^2),\\ {z_2} \cos ({{\hat x}_2})Me \le \frac{1}{2}({M^2}z_2^2 + {e^2}),\\ {z_3}{\delta _4} \le \frac{1}{2} (z_3^2 + \delta _4^2),\\ {\bar k_M}e{z_4} \le \frac{1}{2}(\bar k_M^2z_4^2 + {e^2}). \end{cases} \end{align}$

(41)

According to (37),it can be deduced that

$\begin{gathered} {{\dot V}_{{\text{al}}}} \leqslant - {k_1}z_1^2 + \frac{1}{2}(g_1^2(V){e^2} + z_1^2) + \frac{1}{2}(z_1^2 + g_1^2(V)\delta _2^2) \hfill \\ {k_2}z_2^2 + \frac{1}{2}(z_2^2 + g_2^2(V)\delta _3^2) + \frac{1}{2}({e^2} + g_2^2(V)z_2^2) + {z_2}\varepsilon _1^* \hfill \\ + {z_2}\widetilde \vartheta _1^{\text{T}}{{\mathbf{\varphi }}_1}({\underline {\mathbf{x}} _b}) + \frac{1}{2}{z_2}\widetilde \vartheta _1^{\text{T}}{{\mathbf{\varphi }}_1}({{\mathbf{x}}_b})2({M^2}z_2^2 + {e^2}) + \left| {{z_2}(\varepsilon _1^0 + \varepsilon _2^0)} \right| - \hfill \\ \end{gathered}$

$\begin{gathered} (\varepsilon _1^{*0} + \varepsilon _1^0{\text{ + }}\varepsilon _2^0){z_2}{\text{tanh}}\left( {\frac{{{z_2}}}{{{\varepsilon _2}}}} \right) - {k_3}z_3^2 + \frac{1}{2}(z_3^2 + \delta _4^2) - \hfill \\ {k_4}z_4^2 + \frac{1}{2}(\bar k_M^2z_4^2 + {e^2}) + {z_4}\varepsilon _2^* - \varepsilon _2^{*0}{z_4}{\text{tanh}}\left( {\frac{{{z_4}}}{{{\varepsilon _4}}}} \right) + \hfill \\ {z_4}\vartheta _2^{\text{T}}{{\mathbf{\varphi }}_2}({\underline {\mathbf{x}} _b}) + \sum\limits_{j = 2}^4 {{\delta _j}{{\dot \delta }_j}} + \sum\limits_{k = 1}^2 {\Gamma _{\vartheta k}^{ - 1}\widetilde \vartheta _k^{\text{T}}{{\mathop {\widetilde \vartheta }\limits^. }_k}} + e\dot e \leqslant \hfill \\ \end{gathered}$

$\begin{gathered} - ({k_1} - 1)z_1^2 + \frac{1}{2}g_1^2(V){e^2} + \frac{1}{2}g_1^2(V)\delta _2^2 - \hfill \\ ({k_2} - \frac{1}{2} - \frac{1}{2}g_2^2(V))z_2^2 + \frac{1}{2}g_2^2(V)\delta _3^2 + \frac{1}{2}{e^2} + \hfill \\ \left| {{z_2}(\varepsilon _1^{*0} + \varepsilon _1^0 + \varepsilon _2^0)} \right| - (\varepsilon _1^{*0} + \varepsilon _1^0{\text{ + }}\varepsilon _2^0){z_2}{\text{tanh}}\left( {\frac{{{z_2}}}{{{\varepsilon _2}}}} \right) + \hfill \\ {z_2}\widetilde \vartheta _1^{\text{T}}{{\mathbf{\varphi }}_1}({\underline {\mathbf{x}} _b}) - \Gamma _{\vartheta 1}^{ - 1}\widetilde \vartheta _1^{\text{T}}{\mathop \vartheta \limits^. _1} + \frac{1}{2}({M^2}z_2^2 + {e^2}) - \hfill \\ \end{gathered}$

$\begin{gathered} \left( {{k_3} - \frac{1}{2}} \right)z_3^2 + \frac{1}{2}\delta _4^2 - {k_4}z_4^2 + \frac{1}{2}(\bar k_M^2z_4^2 + {e^2}) + \hfill \\ \left| {{z_4}\varepsilon _2^{*0}} \right| - \varepsilon _2^{*0}{z_4}{\text{tanh}}\left( {\frac{{{z_4}}}{{{\varepsilon _4}}}} \right) + {z_4}\vartheta _2^{\text{T}}{\varphi _2}({\underline {\mathbf{x}} _b}) - \Gamma _{\vartheta 2}^{ - 1}\vartheta _2^{\text{T}}{\mathop \vartheta \limits^. _2} + \hfill \\ \sum\limits_{j = 2}^4 {( - {T_i}\delta _i^2 + \left| {{\delta _i}} \right|{\psi _i})} + e(Me - {\varepsilon _1} - {\varepsilon _2}). \hfill \\ \end{gathered} $

(42)

Choose the adaptation functions ${\boldsymbol\vartheta _1}$ and ${\boldsymbol\vartheta _2}$ as

$\begin{align} {\dot {\boldsymbol\vartheta _1}} = {\Gamma _{\boldsymbol\vartheta 1}}(-{z_2}{{\boldsymbol \varphi} _1}({\boldsymbol x_b}) + {\beta _1}{\boldsymbol\vartheta _1}), \end{align}$

(43)

$\begin{align} {\dot{\boldsymbol \vartheta _2}} = {\Gamma _{\boldsymbol\vartheta 2}}(-{z_4}{{\boldsymbol \varphi} _2}({\boldsymbol x_b}) + {\beta _2}{{\boldsymbol\vartheta} _2}). \end{align}$

(44)

Combining (39) and (40),and using the facts which are acquired through completion of squares,we have

$\begin{align} - {\beta _1}\boldsymbol \vartheta _1^{\rm T}{\boldsymbol \vartheta _1} \le - \frac{{{\beta _1}{{\left\| {{{\boldsymbol \vartheta }_1}} \right\|}^2}}}{2} + \frac{{{\beta _1}{{\left\| {\boldsymbol \vartheta _1^*} \right\|}^2}}}{2}, \end{align}$

(45)

$\begin{align} - {\beta _2}\boldsymbol \vartheta _2^{\rm T}{\boldsymbol \vartheta _2} \le - \frac{{{\beta _2}{{\left\| {{{\boldsymbol \vartheta }_2}} \right\|}^2}}}{2} + \frac{{{\beta _2}{{\left\| {\boldsymbol\vartheta _2^*} \right\|}^2}}}{2}. \end{align}$

(46)

Then it can be concluded that

$\begin{align} {{\dot V}_{{\rm{al}}}} \le & - ({k_1} - 1)z_1^2 - ({k_2} - {\frac{1}{2}} - \frac{1}{2}g_2^2(V) - \frac{1}{2}{M^2}) z_2^2 - \notag \\ &({k_3} - {\frac{1}{2}})z_3^2 - ({k_4} - \frac{1}{2}\bar k_M^2)z_4^2 - ({T_3} - \frac{1}{2}g_2^2(V)) \delta _3^2 + \notag\\ &\left(M + \frac{1}{2}g_1^2(V) + \frac{3}{2} + \frac{{\varepsilon _1^0 + \varepsilon _2^0}}{{2\varsigma _e^2}}\right) {e^2} + \sum\limits_{j = 2}^4 {\left| {{\delta _i}} \right|{\psi _i}}-\notag\\ &({T_2} - \frac{1}{2}g_1^2(V))\delta _2^2 - ({T_4} - {\frac{1}{2}}) \delta _4^2 - \frac{{{\beta _1}{{\left\| {{{\boldsymbol \vartheta }_1}} \right\|}^2}}}{2}- \notag\\ &\frac{{{\beta _2}{{\left\| {{{\boldsymbol \vartheta }_2}} \right\|}^2}}}{2} + 2{k_p}\varsigma + \frac{{\varsigma _e^2}}{2} + \frac{{{\beta _2}{{\left\| {\boldsymbol\vartheta _2^* } \right\|}^2}}}{2} + \frac{{{\beta _1}{{\left\| {\boldsymbol\vartheta _1^*} \right\|}^2}}}{2}. \end{align}$

(47)

Define the following sets:

$\begin{eqnarray*} A: = \{ ({h_d},{\dot h_d},{\ddot h_d}):h_d^2 + \dot h_d^2 + \ddot h_d^2 \le {h_0}\}, \end{eqnarray*}$

$\begin{align*} B: = &\{(z_1,z_2,z_3,z_4,\delta_2,\delta_3,\delta_4): \\ &z_1^2 + z_2^2 + z_3^2 + z_4^2 + \delta _2^2 + \delta _3^2 + \delta _4^2 \le p\}, \end{align*}$

where $p$ is a suitable positive number. Note that $A$ and $B$ are compact sets on ${\bf{R}}^3$ and ${\bf {R}}^7$,which makes $A \times B$ a compact set on ${\bf{R}}^{10}$,the continuous functions ${\psi _j}$ in (33) for $j = 2,3,4$,have maximums on $A \times B$, i.e.,${M_{\psi j}}$. Moreover,by using Young inequality,we have

$\begin{align} \left| {{\delta _j}} \right|{\psi _j} \le \left| {{\delta _j}} \right|{M_{\psi j}} \le (\frac{\delta _j^2M_{\psi j}^2}{2{\varsigma _M}} + \frac{\varsigma _M}{2}),\quad j = 2,3,4, \end{align}$

(48)

where $\varsigma _M$ is an arbitrary constant. In view of (44), (43) can be rewritten as

$\begin{align} {{\dot V}_{{\rm{al}}}} \le& - ({k_1} - 1)z_1^2 - ({k_2} - {\frac{1}{2}} - \frac{1}{2}g_2^2(V) - \notag\\ &\frac{1}{2}{M^2})z_2^2 - ({k_3} - {\frac{1}{2}})z_3^2 - ({k_4} - \frac{1}{2}\bar k_M^2)z_4^2 - \notag\\ &({T_3} - \frac{1}{2}g_2^2(V) - \frac{{M_{\psi 3}^2}}{{2{\varsigma _M}}})\delta _3^2 + (M + \frac{1}{2} g_1^2(V) + \notag\\ &\frac{3}{2} + \frac{{\varepsilon _1^0 + \varepsilon _2^0}}{{2\varsigma _e^2}}){e^2} - ({T_2} - \frac{1}{2}g_1^2(V) - \frac{{M_{\psi 2}^2}}{{2{\varsigma _M}}})\delta _2^2 - \notag \\ &({T_4} - {\frac{1}{2}} - \frac{{M_{\psi 4}^2}}{{2{\varsigma _M}}})\delta _4^2 - \frac{{{\beta _1} {{\left\| {{{\boldsymbol \vartheta }_1}} \right\|}^2}}}{2} - \frac{{{\beta _2}{{\left\| {{{\boldsymbol \vartheta }_2}} \right\|}^2}}}{2} + \frac{{3{\varsigma _M}}}{2} + \notag \\ & 2{k_p}\varsigma + \frac{{\varsigma _e^2}}{2} + \frac{{{\beta _2}{{\left\| {\boldsymbol\vartheta _2^*} \right\|}^2}}}{2} + \frac{{{\beta _1}{{\left\| {\boldsymbol\vartheta _1^*} \right\|}^2}}}{2}. \end{align}$

(49)

Let the control gain,filter gain and observer gain satisfy the following inequalities:

$\begin{align} \left\{\begin{array}{*{20}lll}{k_1} > 1,{k_2} > {\frac{1}{2}} + \frac{1}{2}g_2^2(V) + \frac{1}{2}{M^2},\\ {k_3} > {\frac{1}{2}},~{k_4} > \frac{1}{2}\bar k_M^2,~{T_2} > \frac{1}{2}g_1^2(V) + \frac{{M_{\psi 2}^2}}{{2{\varsigma _M}}},\\ {T_3} > \frac{1}{2}g_2^2(V) + \frac{{M_{\psi 3}^2}}{{2{\varsigma _M}}},~{T_4} > {\frac{1}{2}} + \frac{{M_{\psi 4}^2}}{{2{\varsigma _M}}},\\ M < - \frac{1}{2}g_1^2(V) - \frac{3}{2} - \frac{{\varepsilon _1^0 + \varepsilon _2^0}}{{2\varsigma _e^2}}.\end{array}\right. \end{align}$

(50)

Then,(45) can be transformed to

$\begin{align} {\dot V_{{\rm{al}}}} \le - 2{\alpha _0}{V_{{\rm{al}}}} + {\varepsilon _0}, \end{align}$

(51)

where

${\alpha _0}\! =\! \min \left\{ \begin{array}{l}\! ({k_1} - 1),({k_2} - {\frac{1}{2}} - \frac{1}{2} g_2^2(V) - \frac{1}{2}{M^2}),\\ ({k_3} - \frac{1}{2}),({k_4} - \frac{{\bar k_M^2}}{2}),({T_4} - \frac{1}{2} - \frac{{M_{\psi 4}^2}}{{2{\varsigma _M}}}),{\beta _1}{\Gamma _{\vartheta 1}},\\ ({T_3} - \frac{1}{2}g_2^2(V) - \frac{{M_{\psi 3}^2}}{{2{\varsigma _M}}}),({T_2} - \frac{1}{2}g_1^2(V) - \\ \frac{{M_{\psi 2}^2}}{{2{\varsigma _M}}}),- (M + \frac{{g_1^2(V)}}{2} + \frac{3}{2} + \frac{{\varepsilon _1^0 + \varepsilon _2^0}}{{2\varsigma _e^2}}),{\beta _2}{\Gamma _{\vartheta 2}} \\\! \end{array} \right\},\\ {\varepsilon _0} = \frac{{3{\varsigma _M}}}{2} + 2{k_p}\varsigma + \frac{{\varsigma _e^2}}{2} + \frac{{{\beta _1}{{\left\| {\boldsymbol\vartheta _1^*} \right\|}^2}}}{2} + \frac{{{\beta _2}{{\left\| {\boldsymbol\vartheta _2^*} \right\|}^2}}}{2}.$

We can see that the value of ${\alpha _0}$ can be adjusted by selecting the gains of controller,filter and observer. If the gains satisfy ${\alpha _0} > \frac{{{\varepsilon _0}}}{{2p}}$, then ${\dot V_{{\rm{al}}}} < 0$ is on the boundary of compact set $A \times B$,that is to say,the boundary of compact set $A \times B$ is an invariant set,which proves the boundedness of the closed-loop system. Moreover,solving (47) yields

$\begin{align} {V_{{\rm{al}}}}(t) \le {V_{{\rm{al}}}}(0){{\rm e}^{ - 2{\alpha _0}t}} + \int_0^t {{{\rm e}^{ - 2{\alpha _0}(t - \tau )}}{\varepsilon _0}{\rm d}\tau.} \end{align}$

(52)

When $t \to \infty $,$V(\infty ) \le \frac{{{\varepsilon _0}}}{{2{\alpha _0}}}$,that is,$\left| {{z_1}(\infty )} \right| \le \sqrt {\frac{{{\varepsilon _0}}}{{{\alpha _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 ${\tilde d_v} = {d_v} - {Z_{12}}$ can be extremely small^{[26, 27]},so the tracking error dynamics can be approximated by ${\dot z_V} = - {k_v}{z_V}$,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^[29]. 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].

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 V₀ = 2 347.6 m $ \cdot {{\rm s}^{ - 1}}$,h₀ = 25 908 m, ${\gamma _0}$ = 0 deg. The reference commands of velocity and altitude are generated by filtering step reference commands using two second-order pre-filters with natural frequency ${\omega _f} = 0.03$ rad/s and damping factor ${\xi _f} = 0.95$. 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 228.6 ${\rm m}/{\rm s}$.

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,${w_1} = {w_2} = 0$. The reference signals of virtual control variables are chosen according to equations (20),(23) and (25),and the control input is computed from (28). 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 ${k_i}$ $(i = 1,\cdots ,4)$. 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.

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 flight-path 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.

Figs. 6 and 7 show the tracking performance of the virtual control variables of pitch angle and its rate. The virtual control 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.

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.

The second simulation is performed for the system with uncertainties. The references of velocity,altitude,and flight-path angle are the same as those in the first simulation. It is assumed that the true value of some aerodynamic parameters including $C_L^\alpha $,$C_D^{{\alpha ^2}}$,$C_M^\alpha $ 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.

The fuzzy weighting values are computed with update laws (39) and (40). The gains of the update laws are selected as ${T_{\vartheta 1}} = 80$,${\beta _1} = 0.05$,${T_{\vartheta 2}} = 140$,${\beta _2} = 0.05$. 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 ${\alpha ^{ref}}$ and $\alpha $ can also be seen as coefficient uncertainty,so ${w_1}$ and ${w_2}$ can be predigested as the functions of $\alpha $,${\delta _e}$ and $\eta $. In the FLSs,we use $\hat \alpha $,${\delta _e}$ and ${\eta ^ * }$ as inputs. The partitioning points are chosen as -4,-1,2, 5,8 deg,and -20,-15,-10,-5,0,5,10,15,20 deg for $\hat \alpha $ and ${\delta _e}$,respectively. The partitioning points for $\eta $ 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 ${\eta _1}$,$-$0.6 to 0.6 with the interval of 0.06 for ${\eta _2}$ and $-$0.12 to 0.12 with the interval of 0.03 for ${\eta _3}$. The fuzzy membership functions for $\hat \alpha $ are given as follows,and the fuzzy membership functions for ${\delta _e}$ and $\eta $ are similar to $\hat \alpha $:

$\begin{gathered} {\mu _{F_1^1}}(\hat \alpha ) = \frac{1}{{1 + {{\text{e}}^{80/(\hat \alpha + 4/57.3)}}}}, \hfill \\ {\mu _{F_1^2}}(\hat \alpha ) = {{\text{e}}^{ - {{(\frac{{\hat \alpha + \frac{1}{{57.3}}}}{{0.04}})}^2}}}, \hfill \\ {\mu _{F_1^3}}(\hat \alpha ) = {{\text{e}}^{ - {{(\frac{{\hat \alpha - \frac{2}{{57.3}}}}{{0.04}})}^2}}}, \hfill \\ {\mu _{F_1^4}}(\hat \alpha ) = {{\text{e}}^{ - {{(\frac{{\hat \alpha - \frac{5}{{57.3}}}}{{0.04}})}^2}}}, \hfill \\ {\mu _{F_1^5}}(\hat \alpha ) = \frac{1}{{1 + {{\text{e}}^{ - 80/(\hat \alpha - \frac{8}{{57.3}})}}}}. \hfill \\ \end{gathered} $

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 $\delta _e$ on the generalized elastic coordinate is also analyzed in this simulation. It can be seen from Fig. 15 that the effect of $\delta _e$ is much smaller than that of other parts,mainly the angle of attack.

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 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.

The simulation results are given in Figs. 16~19. 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.

Remark 4. Comparing to state feedback based back-stepping 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.

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.