Extended state observer-based back-stepping control for hypersonic reentry vehicle with input constraints

Extended state observer-based back-stepping control for hypersonic reentry vehicle with input... Abstract This paper investigates the attitude-tracking control problem of hypersonic reentry vehicle in cases of multiple uncertainties, external disturbances and input constraints. The controller design is based on synthesizing the extended state observer (ESO) into a back-stepping control technique. This control-oriented model is formulated with mismatched and matched uncertainties. They reflect the total disturbances that group different types of aerodynamic uncertainties and external moment disturbances. In order to improve the system robustness, a sigmoid function-based ESO is first proposed. This will estimate the total disturbance and is equipped with a controller. The sigmoid smooth function is also introduced for the purpose of handling the input constraints. This will approximate saturation and guarantee that the control input is bounded. Error states between the actual control input and the desired control input are integrated to compensate for the saturation effect. Following this, the stability of the closed-loop system is proved within the Lyapunov theory framework. Several simulations are then investigated to illustrate the effectiveness of the proposed constrained attitude control scheme. 1. Introduction Considerable control design research related to hypersonic reentry vehicle (HRV) stabilization strategies has been presented in recent decades. This growing focus is due to increasing potential applications for the technology and the arising theoretical challenges in attitude control (Geng et al., 2013; Wang et al., 2001; Corban et al., 2001). HRV systems present unique control challenges, especially in the reentry stage, under the condition of dramatically changing flight environments (Jiang et al., 2013), model uncertainties, external moment disturbances and unknown aerodynamic characteristics as well as actuator saturation (Bao, 2013; Zhang et al., 2014). The disturbance rejection ability of control systems must be improved to meet the requirement for high-control performance and flight reliability, particularly in the presence of input constraints. Note that the HRV dynamic model is a second-order system that mainly suffers from three types of uncertainties: aerodynamic uncertainties, dynamic model uncertainties and external moment disturbances. From a control point of view these uncertainties generally fall into the category of either matched or mismatched uncertainties. It is well known that controller design based on multiple time scaling (Naidu & Calise, 2001) is an effective methodology to tackle mismatched uncertainties. The design process can be simplified sufficiently by using time-scale collocation architecture (Yu & Li, 2015). Back-stepping methodology (Krstic et al., 1995) is also an efficient tool to solve the control problem for non-linear system with mismatched uncertainties. It is easily incorporated with other control techniques to deal with unknown disturbances as an alternative approach. In addition, more rigourous mathematical proof can be given based on the Lyapunov stability theorem (Lee & Kim, 2001) without the time-scale assumption. A composite anti-disturbance controller (Sun & Guo, 2014) is synthesized by introducing disturbance estimations into the design of virtual control laws to compensate for the mismatched disturbances. A drawback with the implementation of an adaptive back-stepping controller is the need to compute virtual derivatives at each step. This is called the ‘explosion of terms’ problem and dynamic surface control (Chen & Yu, 2014) or filter technique (Swaroop et al., 2000; Wang & Huang, 2005; Dong et al., 2014) can be employed to counter this. Filter errors are seldom considered in the stability analysis of the closed-loop system, however. Another difficulty worth studying is actuator saturation (Gou et al., 2017), which can severely limit system performance and even lead to instability. Actuator saturation is unavoidable in a practical system so it is necessary to improve the control system capability to deal with this. This is particularly important for the attitude control system of HRV that suffers more occasions of actuator saturation during entry phase because of large attitude manoeuvres. It must be equipped with an appropriate controller to handle the harmful effects of saturation and improve safety and reliability. This can be done by employing a design-bounded controller to bound the control input so that it will not violate the limitation as shown in the study by Boškovic et al. (2012). However, the stability can be obtained only under some restrictive assumptions (Huang et al., 2001) and the controller is conservative. Alternatively, an auxiliary system driven by the error between desired control input and actual control input could be employed to reduce the effects of saturation and guarantee system stability (Chen et al., 2001). However, the saturation is a non-smooth and non-differentiable function, which prevents the back-stepping technique from being utilized directly (Wen et al., 2017). While previous research results related to actuator saturations have been implemented successfully, robustness against uncertainties and disturbances as well as input constraints is still worth comprehensive study. Aerodynamic uncertainties, dynamic model uncertainties and external disturbances are seldom considered simultaneously in designing constrained controllers for HRV. A controller with an adaptive law driven by tracking error has drawbacks when the system faces control constraints (Sonneveldt et al., 2017). During the saturation period, tracking errors increase not only because of external disturbances but also due to actuator limitations. The updated law seeks the desired performance only according to tracking errors and this leads to a larger and more aggressive control input. This problem also increases saturation and extends its time period. The extended state observer (ESO)-based back-stepping method explored in this paper seeks to avoid this problem by applying a novel sigmoid function-based ESO (SESO) to compensate the disturbances instead of using adaptive laws to estimate the bounds of disturbances. An ESO can actively compensate for the total disturbance in the dynamic model (Huang & Xue, 2014). In the past decades, ESO has been developed and widely used in the control field. For example, the finite time ESO (Deefort et al., 2011) developed recently can increase the speed and efficacy of the observer (Zhao & Yang, 2016). Another finite time ESO was proposed in the study by (Xiong et al., 2006) with complex structure that makes convergence proof overcomplicated and it is also difficult to obtain the ESO parameters tuning guideline. In order to realize the finite-time convergence, non-linear sign function is normally introduced to construct the observer. This will inevitably lead to the chattering of estimation and makes it difficult for practical application. In addition, sign function is more sensitive to noise because it is not smooth around zero, this should be avoided for practical application. The sign function also can be found in traditional ESO (Han, 2009). This makes it quite difficult to give rigourous proof (Guo & Zhao, 2011). These undesirable features need to be removed by replacing the sign functions by novel smooth functions. In contrast with sign function, from Shao’s recent work we can know that sigmoid function has the noise-attenuation ability which can be obtained by adjusting the function parameters. Moreover, sigmoid function (Shao & Wang, 2016) is characterized by integrating the non-linear and linear terms naturally (Shao et al., 2017), which is also a real-valued and continuous differentiable function with no singularities. In light of this, a convergent system can be constructed based on sigmoid function and then an ESO can be designed. The stability and convergence of the proposed ESO can be proven based on the proposed convergent system. This paper will explore this method and attempt to develop a novel smooth ESO. A new control strategy for HRV based on the back-stepping technique is proposed, which does not introduce a conventional time-scale assumption. This takes into account aerodynamic and dynamic model uncertainties, external moment disturbances and the effects of actuator saturation as discussed above. To facilitate the controller design a new control-oriented model was formulated by lumping the multiple uncertainties into total disturbance. In this process, mismatched disturbance are considered in the two-order HRV system. To improve system robustness a novel SESO was also applied to reconstruct and compensate the total disturbance in the controller. The proposed SESO can estimate disturbance more smoothly compared to a sign function-based ESO. To deal with control input constraints, an auxiliary system and variable were introduced to reduce the harmful effects of saturation. This is done in place of approximating the saturation with smooth function which will limit control inputs. Saturation time can be reduced significantly in this way. The stability of this closed-loop system has been rigourously proven using the Lyapunov theory. It takes estimation error, auxiliary systems and input constraints into consideration. In comparison to existing works, the main contributions of this paper are summarized as follows: 1. A new control-oriented model is developed in the presence of aerodynamic uncertainties, dynamic model uncertainties and external moment disturbance as well as actuator saturation. 2. A novel SESO is developed based on Lyapunov methodology to observe and compensate the total disturbance. Compared with the existing ESO, the SESO shows smooth estimation with simple structure. The fundamental tuning rules of the observer parameters are provided. 3. Two ways are combined to deal with input constraints. First, the smooth bounded function is applied to approximate saturation, making the control signals limited. Secondly, in case of the dramatically increasing inputs caused by perturbations, the auxiliary variables are applied to reduce saturation effects. The rest of this paper is organized as follows. Section 2 will formulate the control-oriented model of the HRV. The proposed SESO design method will be presented in Section 3. In Section 4 the back-stepping controller design is presented based on the novel SESO and stability analysis of the closed-loop system is given. Section 5 will present the simulation results to demonstrate the effectiveness of this proposed approach and conclusions are provided in Section 6. 2. Preliminaries and system description 2.1 Preliminaries The following notations and lemma are introduced and used in the analysis and design of the SESO. For any variable $$x_{i}$$ and any constant $$a_{i}>0,\, b_{i} >0$$, i = 1, …n, let   \begin{align} \begin{aligned} \textrm{sig}(x_{i} )&=\textrm{sig}(x_{i} ;a_{i},b_{i} )=a_{i} \left[(1+e^{-b_{i} x_{i} } )^{-1} -0.5\right] \\[-2pt] \mathbf{sig}(\boldsymbol{x})&=\left[\textrm{sig}(x_{1} ;a_{1},b_{1} )\, \ldots \, \textrm{sig}(x_{n} ;a_{n},b_{n} )\right]^\textrm{T}, \end{aligned} \end{align} (1)where sig(•) denotes the sigmoid function. 2.2 System description In this section, dynamic equations of rotational motion referenced to a vehicle-body-fixed coordinate system are presented (Van Soest, 2006). The equations are described as   \begin{align} \begin{cases} \dot{\alpha }=\omega_{z} -\omega_{x} \cos \alpha \tan \beta +\omega_{y} \sin \alpha \tan \beta -\frac{1}{mV\cos \beta } (L+mg\cos \theta \cos \mu )+d_{\alpha } \\[2pt] \dot{\beta }=\omega_{x} \sin \alpha +\omega_{y} \cos \alpha + \frac{1}{mV} \left(C-mg\cos \theta \sin \mu \right)+d_{\beta } \\[2pt] \dot{\mu }=\omega_{x} \frac{\cos \alpha }{\cos \beta } -\omega_{y} \frac{\sin \alpha }{\cos \beta } +\frac{1}{mV} \big[L(\sin \theta \sin \mu +\tan \beta ) + C\sin \theta \cos \mu +mg\cos \theta \cos \mu \tan \beta \big]+d_{\mu }, \end{cases} \end{align} (2)where $$\alpha ,\beta $$ and $$\mu $$ denote the angle of attack (AOA), sideslip angle and bank angle, respectively. $$\omega _{x}$$, $$\omega _{y}$$ and $$\omega _{z}$$ denote the roll, yaw, and pitch rate in that order. $$d_{\alpha }, d_{\beta }$$ and $$d_{\mu }$$ denote the model uncertainties. The lift force is L and C is the side force. Mass is m, g is the acceleration of gravity and V and $$\theta $$ are the velocity and flight path angle, respectively. The rotational dynamic model is denoted as follows:   \begin{align} \begin{cases} I_{x} \dot{\omega }_{x} =(I_{y} -I_{z} )\omega_{y} \omega_{z} +\left(C_{l} +\Delta C_{l} +C_{l,\omega_{x} } \frac{\omega_{x} b_{ref} }{2V} + C_{l,\omega_{y} } \frac{\omega_{y} b_{ref} }{2V} \right)qb_{ref} S_{ref} +d_{l} \\[5pt] I_{y} \dot{\omega }_{y} =(I_{z} -I_{x} )\omega_{x} \omega_{z} +\left(C_{m} +\Delta C_{m} +C_{m,\omega_{x}} \frac{\omega_{x} b_{ref} }{2V} + C_{m,\omega_{y} } \frac{\omega_{y} b_{ref} }{2V} \right)qb_{ref} S_{ref} +X_{cg} C+d_{m} \\[5pt] I_{z} \dot{\omega }_{z} =(I_{x} -I_{y} )\omega_{y} \omega_{x} +\left(C_{n} +\Delta C_{n} +C_{n,\omega_{x} } \frac{\omega_{z} c_{ref} }{2V} \right)qc_{ref} S_{ref} + X_{cg} (D\sin \alpha +L\cos \alpha )+d_{n}, \end{cases} \end{align} (3)where $$X_{cg}$$ represents the longitude distance from the moment reference centre to the vehicle centre of gravity, $$q={\rho V^{2} \mathord{\left / {\vphantom{\rho V^{2} 2}} \right . } 2}$$ is the dynamic pressure and $$\rho $$ is atmospheric density. The reference length is $$b_{ref}$$, mean aerodynamic chord is $$c_{ref}$$ and $$S_{ref}$$ is the aerodynamic reference area. Aerodynamic moment uncertainty coefficients are $$\Delta C_{l},\Delta C_{m}$$ and $$\Delta C_{n}$$. $$d_{l},d_{m}$$ and $$d_{n}$$ denote the moment disturbances. For the purpose of reflecting an accurate reentry flight dynamic and developing a realistic control-oriented model, $$d_{\omega _{x}}, d_{\omega _{y}}$$ and $$d_{\omega _{z}}$$ represent the compound moment disturbances of the three channels, respectively. In this way (3) can be described as   \begin{align} \begin{cases} I_{x} \dot{\omega }_{x} =(I_{y} -I_{z} )\omega_{y} \omega_{z} +C_{l} qb_{ref} S_{ref} +d_{\omega_{x} } \\[2pt] I_{y} \dot{\omega }_{y} =(I_{z} -I_{x} )\omega_{x} \omega_{z} +C_{m} qb_{ref} S_{ref} +d_{\omega_{y} } \\[2pt] I_{z} \dot{\omega }_{z} =(I_{x} -I_{y} )\omega_{y} \omega_{x} +C_{n} qc_{ref} S_{ref} +d_{\omega_{z} }, \end{cases} \end{align} (4)where   \begin{align} \begin{cases} d_{\omega_{x} } =(\Delta C_{l} +C_{l,\Delta } )qb_{ref} S_{ref} +d_{l} \\ d_{\omega_{y} } =(\Delta C_{m} +C_{m,\Delta } )qb_{ref} S_{ref} +X_{cg} C+d_{m} \\ d_{\omega_{z} } =(\Delta C_{n} +C_{n,\Delta } )qc_{ref} S_{ref} +X_{cg} (D\sin \alpha +L\cos \alpha )+d_{n}, \end{cases} \end{align} (5)where D is the drag force and $$C_{l,\Delta }$$, $$C_{m,\Delta }$$ and $$C_{n,\Delta }$$ are the aerodynamic moment coefficients for angle rates. These are regarded as the uncertainties and are described as follows:   \begin{align} \begin{cases} C_{l,\Delta } =C_{l,\omega_{x} } \frac{\omega_{x} b_{ref} }{2V} +C_{l,\omega_{y} } \frac{\omega_{y} b_{ref} }{2V} \\[2pt] C_{m,\Delta } =C_{m,\omega_{x} } \frac{\omega_{x} b_{ref} }{2V} +C_{m,\omega_{y} } \frac{\omega_{y} b_{ref} }{2V} \\[2pt] C_{n,\Delta} = C_{n,\omega_{x} } \frac{\omega_{z} c_{ref} }{2V}. \end{cases} \end{align} (6)In equation (6), $$C_{l},C_{m}$$ and $$C_{n}$$ denote the normal rolling, yawing and pitching moment coefficients correspondingly. These are expressed as   \begin{align} \begin{cases} C_{l} = C_{l,\beta } \beta +C_{l,\delta e} \delta_{e} +C_{l,\delta a} \delta_{a} +C_{l,\delta r} \delta_{r} \\ C_{m} = C_{m,\beta } \beta +C_{m,\delta e} \delta_{e} +C_{m,\delta a} \delta_{a} +C_{m,\delta r} \delta_{r} \\ C_{n} = C_{n,\alpha } +C_{n,\delta e} \delta_{e} +C_{n,\delta a} \delta_{a} +C_{n,\delta r} \delta_{r}, \end{cases} \end{align} (7)where $$\delta _{e},\delta _{a}$$ and $$\delta _{r}$$ are the left elevon, right elevon and the rudder fin deflections. The nominal aerodynamic drag, lift and side force are denoted as   \begin{align} \begin{cases} D = C_{D} qS_{ref} = qS_{ref} (C_{D,\alpha } +C_{D,\delta_{e} } +C_{D,\delta_{a} } +C_{D,\delta_{r} } ) \\ L = C_{L} qS_{ref} = qS_{ref} (C_{L,\alpha } +C_{L,\delta_{e} } +C_{L,\delta_{a} } ) \\ C = C_{C} qS_{ref} = qS_{ref} (C_{C,\beta } \beta +C_{C,\delta_{e} } +C_{C,\delta_{a} } +C_{C,\delta_{r} } ), \end{cases} \end{align} (8)where $$C_{D},C_{L}$$ and $$C_{C}$$ represent the aerodynamic force coefficients. We can see from (7) and (8) that aerodynamic coefficients are the function of AOA, slide angle and Mach number as well as deflection angles. Following this, model equations (2)–(3) can be expressed in matrix form   \begin{align} \dot{\boldsymbol{\varOmega} }=\boldsymbol{f}_{1} (\boldsymbol{\varOmega},\boldsymbol{\omega} )+\boldsymbol{g}_{1} (\boldsymbol{\varOmega} )\boldsymbol{\omega} +\boldsymbol{g}_{\Delta } \boldsymbol{u}+\boldsymbol{d}_{\varOmega } \end{align} (9)  \begin{align} \dot{\boldsymbol{\omega} }=\boldsymbol{f}_{2} (\boldsymbol{\varOmega},\boldsymbol{\omega} )+\boldsymbol{g}_{2} (\boldsymbol{\omega} )\boldsymbol{u}+\boldsymbol{d}_{\omega }, \end{align} (10)where $$\boldsymbol{\varOmega } =[\alpha \; \beta \; \mu ]^{\textrm{T}}$$ is the attitude angle vector, $$\boldsymbol{\omega } =[\omega _{x}\, \omega _{y}\, \omega _{z} ]^{\textrm{T}} $$is the angular rate vector and $$\boldsymbol{u}=[\delta _{e}\, \delta _{a}\, \delta _{r} ]^{\textrm{T}}$$ is the control input vector. The unknown disturbance induced by model uncertainties is $$\boldsymbol{d}_{\varOmega } =[d_{\alpha }\, d_{\beta }\, d_{\mu } ]^{\textrm{T}}$$ and $$\boldsymbol{d}_{\omega } = [d_{\omega _{x} }\, d_{\omega _{y} }\, d_{\omega _{z}} ]^{\textrm{T}}$$ is the compound moment disturbance vector. The vectors $$\boldsymbol{f}_{1} =[\,f_{\alpha } \; f_{\beta } \; f_{\mu } ]^{\textrm{T}}$$ and $$\boldsymbol{f}_{2} =[\,f_{\omega _{x} }\ f_{\omega _{y} }\ f_{\omega _{z} } ]^{\textrm{T}}$$ as well as the matrices $$\boldsymbol{g}_{1},\boldsymbol{g}_{\Delta }$$ and $$\boldsymbol{g}_{2}$$ are as follows:   \begin{align} \begin{cases} f_{\alpha } =\frac{1}{MV\cos \beta } [-qSC_{Y,\alpha } -Mg\cos \theta \cos \mu ] \\ f_{\beta } =\frac{1}{MV} (-qSC_{Z,\beta } \beta -Mg\cos \theta \sin \mu ) \\ f_{\mu } =\frac{g}{V} \cos \theta \cos \mu \tan \beta +\frac{1}{MV} [qSC_{Y,\alpha } (\sin \theta \sin \mu + \tan \beta )+qSC_{Z,\beta } \beta \sin \theta \cos \mu ] \end{cases} \end{align} (11)  \begin{align} \begin{cases} f_{\omega_{x} } =\frac{(I_{y} -I_{z} )}{I_{x} } \omega_{y} \omega_{z} +\frac{1}{I_{x} } C_{l,\beta } \beta qb_{ref} S_{ref} \\[2pt] f_{\omega_{y} } =\frac{(I_{z} -I_{x} )}{I_{y} } \omega_{x} \omega_{z} +\frac{1}{I_{y} } C_{m\mathrm{,}a} qb_{ref} S_{ref} \\[2pt] f_{\omega_{z} } = \frac{(I_{x} -I_{y} )}{I_{z} } \omega_{x} \omega_{z} +\frac{1}{I_{z} } C_{n,\alpha } qc_{ref} S_{ref} \end{cases} \end{align} (12)  \begin{align} \boldsymbol{g}_{1} = \left[\begin{array}{@{}ccc@{}} {-\cos \alpha \tan \beta } & {\sin \alpha \tan \beta } & {1} \\{\sin \alpha } & {\cos \alpha } & {0} \\{\sec \beta \cos \alpha } & {-\sec \beta \sin \alpha } & {0} \end{array}\right] \end{align} (13)  \begin{align} \boldsymbol{g}_{\Delta } = \left[\begin{array}{@{}ccc@{}} {g_{\alpha,\delta_{e} } } & {g_{\alpha,\delta_{a} } } & {0} \\{g_{\beta,\delta_{e} } } & {g_{\beta,\delta_{a} } } & {g_{\beta,\delta_{r} } } \\{g_{\mu,\delta_{e} } } & {g_{\mu,\delta_{a} } } & {g_{\mu,\delta_{r} } } \end{array}\right] \end{align} (14)  \begin{align} \boldsymbol{g}_{2} = \left[\begin{array}{@{}ccc@{}} {g_{\omega_{x},\delta_{e} } } & {g_{\omega_{x},\delta_{a} } } & {g_{\omega_{x},\delta_{r} } } \\{g_{\omega_{y},\delta_{e} } } & {g_{\omega_{y},\delta_{a} } } & {g_{\omega_{y},\delta_{r} } } \\{g_{\omega_{z},\delta_{e} } } & {g_{\omega_{z},\delta_{a} } } & {g_{\omega_{z},\delta_{r} } } \end{array}\right]\!, \end{align} (15)where $$g_{i,j}$$ denote the aerodynamic coefficients with $$i=\alpha ,\beta ,\mu , \omega _{x},\omega _{y},\omega _{z},\, j=\delta _{e},\delta _{a},\delta _{r}$$. The term $$\boldsymbol{g}_{\Delta } \boldsymbol{\delta }$$ is the deflection coupling effect for angular subsystems which could be regarded as disturbance. Therefore, for dynamic model (9), it can be deduced   \begin{align} \dot{\boldsymbol{\varOmega} }=\boldsymbol{f}_{1} (\boldsymbol{\varOmega},\boldsymbol{\omega} )+\boldsymbol{g}_{1} (\boldsymbol{\varOmega} )\boldsymbol{\omega} +\boldsymbol{\varphi}_{1}, \end{align} (16)where $$\boldsymbol{\varphi }_{1} =\boldsymbol{g}_{\Delta } \boldsymbol{u}+\boldsymbol{d}_{\varOmega }$$ denotes the ‘total disturbance’ of angular subsystem which includes coupling with aerodynamic force generated by deflection angles and model uncertainties. A further issue is uncertainties that exist in aerodynamic moment coefficients making it difficult to acquire the accurate $$\boldsymbol{g}_{2}$$ in advance and keep $$\boldsymbol{g}_{2}$$ invertible during the whole flight envelope. To solve this problem $$\boldsymbol{g}_{20}$$ is instead used for the controller design, which is the nominal data obtained based on the ground test for typical flight conditions. Therefore, dynamic model (10) can be rewritten as   \begin{align} \dot{\boldsymbol{\omega} }=\boldsymbol{f}_{2} (\boldsymbol{\varOmega}, \boldsymbol{\omega} )+\boldsymbol{g}_{20} \boldsymbol{u}+\boldsymbol{\varphi}_{2}, \end{align} (17)where $$\boldsymbol{g}_{20} =\boldsymbol{g}_{2} |_{\alpha =\textrm{const,Ma=const}},\boldsymbol{\varphi }_{2} =(\boldsymbol{g}_{2} -\boldsymbol{g}_{20} )\boldsymbol{u}+\boldsymbol{d}_{\omega }$$ denotes the ‘total disturbance’ of the angular rate subsystem including compound moment disturbances. From another point of view, $$(\boldsymbol{g}_{2} -\boldsymbol{g}_{20} )\boldsymbol{u}$$ is also representative of coefficient uncertainties included in the term $$\boldsymbol{\varphi }_{2}$$. The final problem concerns input constraints. The vector of actual control deflections generated by actuators is defined as u = sat(v). In this case $$\mathbf{sat}(\boldsymbol{v})= [\textrm{sat}(v_{1} ),\, \textrm{sat}(v_{2} ),\, \textrm{sat}(v_{3})]^{\textrm{T}}$$ denotes the non-linear saturation characteristic, v is the desired control signal to be designed and $$\textrm{sat}(v_{i})$$ is defined as   \begin{align} \textrm{sat}(v_{i})=\left\{\begin{array}{@{}ll@{}} \begin{array}{l} {u_{i\max } \textrm{sign}(v_{i} )} \\{v_{i} (t)} \end{array} \begin{array}{l} \left|v_{i} \right|> u_{i\max } \\{ \left|v_{i} \right| < u_{i\max } } \end{array} \end{array}\right. (i=1,2,3), \end{align} (18)where $$u_{i\max }$$ is the known bound of the ith actuator. In this paper the saturation is approximated by the following smooth sigmoid function:   \begin{align} h(v_{i} )=\textrm{sig}(v_{i} ;2u_{i\max },b_{i} )=2u_{i\max } \left[(1+e^{-b_{i} v_{i} } )^{-1} -0.5\right]. \end{align} (19) According to the property of sigmoid function it is known that $$h(v_{i} )$$ is smooth and bounded, meaning: $$\left |h(v_{i} )\right |<u_{i\max }$$. It is clear that there exists a different j(v) between sat(v) and h(v). Therefore,   \begin{align} \mathbf{sat}(\boldsymbol{v})=\boldsymbol{h}(\boldsymbol{v})+\boldsymbol{j}(\boldsymbol{v}) \end{align} (20)following this, (17) can be rewritten as   \begin{align} \dot{\boldsymbol{\omega} }=\boldsymbol{f}_{2} (\boldsymbol{\varOmega},\; \boldsymbol{\omega} )+\boldsymbol{g}_{20} \boldsymbol{h}(\boldsymbol{v})+\boldsymbol{g}_{20}\ \boldsymbol{j}(\boldsymbol{v})+\boldsymbol{\varphi}_{2}. \end{align} (21) Remark 1 Sigmoid function is used as an alternative to the hyperbolic tangent function to approximate saturation Wen et al. (2011) in this paper. In this case, parameter b can be adjusted according to the actuator response requirement. It can be seen from Fig. 1 that the larger b, the more sensitive the actuator is to the desired control input. The actuator will become relatively insensitive when the input signal is close to violating limitations. To an extent this will provide more freedom for enhanced performance of the controller. Note that j(v) is not differentiable at the sharp corner so it cannot be lumped into the ‘total disturbance’ as a part of $$\boldsymbol{\varphi }_{2}$$. Fig. 1. View largeDownload slide Saturation function h(v) and sat(v). Fig. 1. View largeDownload slide Saturation function h(v) and sat(v). So far the control-oriented model, (16) and (21) has been formulated successfully for controller design. Aerodynamic coefficient uncertainties, coupling effects, unknown external moment disturbances as well as input constraints are reflected in this model. The feature of dynamic systems (16) and (21) is that mismatched uncertainties $$\boldsymbol{\varphi }_{1}$$ and matched uncertainties $$\boldsymbol{\varphi }_{2}$$ are both considered in formulating the model. As an alternative way to deal with this kind of uncertainty problem, adaptive methods can be combined in controller design. Adaptive law driven by tracking errors will make actuator saturation periods longer. In light of this, ESO-based control methods to handling the mismatch problem in consideration of input saturations will be explored in this paper. The divide control design process based on back-stepping technique can be divided into two steps. First the virtual control input must be designed to guarantee angular subsystem (16) is stable. Then the actual control input for angular rate subsystem (21) can be modified. In order to enhance the robustness of control law at each step SESO must be integrated to tackle unknown disturbances. Following this the actual control input can be designed. 3. Novel ESO design based on sigmoid function In this section a general method to construct ESO is provided. A novel convergent system is first proposed with smooth sigmoid function and based on this the SESO can be developed. Consider the following general nonlinear system with uncertainty   \begin{align} \dot{x}=f(x)+g(x)u+w, \end{align} (22)where u ∈ R is control input and x is the measured output. System functions are f(x) and g(x) while w is uncertainty. In order to estimate the generalized disturbance w based on the principle of the ESO design, extended state variable $$x_{2}=w$$ is introduced. This is defined with $$\dot{x}_{2}=p$$. The following assumption is outlined below. Assumption 1 It is assumed p is unknown but bounded, that is to say $$p\le \varpi $$, where $$\varpi $$ is a positive constant. Afterwards, the system (22) can be extended as   \begin{align} \begin{cases} {\dot{x}_{1} =f(x_{1} )+g(x_{1} )u+x_{2} } \\{\dot{x}_{2} =p}. \end{cases} \end{align} (23)According to the ESO technique the following standard ESO can be constructed:   \begin{align} \begin{cases} \dot{\hat{x}}_{1} =f(x_{1} )+g(x_{1} )u+\hat{x}_{2} -\beta_{1} h_{1} (x_{1} -\hat{x}_{1} ) \\ \dot{\hat{x}}_{2} =-\beta_{2} h_{2} (x_{1} -\hat{x}_{1} )\!. \end{cases} \end{align} (24) The challenge is to design functions h(•) and parameters $$\beta _{1},\beta _{2}$$, such that $$\hat{x}_{1}$$ and $$\hat{x}_{2}$$ tend to be the estimations of the state $$x_{1}$$ and the uncertainty w, respectively. Combining (23) and (24) provides the following error state system   \begin{align} \begin{cases} {\dot{e}_{1} =-\beta_{1} h_{1} (e_{1} )+e_{2} } \\{\dot{e}_{2} =-\beta_{2} h_{2} (e_{1} )-p}, \end{cases} \end{align} (25)where $$e_{1} =\hat{x}_{1} -x_{1},\, e_{2} =\hat{x}_{2} -x_{2}$$ are estimation errors. If h(•) function can be found which can make the error system (25) convergent, then the ESO can be designed as (24). According to the Super-Twisting Algorithm (STA) (Pisano & Usai, 2007), $$h_{1} (e_{1} )= \left |e_{1} \right |^{{1\mathord{\left / {\vphantom{1 2}} \right . } 2} } \textrm{sign}(e_{1} ),\, h_{2} (e_{1} )=\textrm{sign}(e_{1})$$ can be selected. This means   \begin{align} \begin{cases} \dot{e}_{1} =-\beta_{1} \left|e_{1} \right|{}^{1/2}\textrm{sign}(e_{1} )+e_{2} \\ \dot{e}_{2} =-\beta_{2} \textrm{sign}(e_{1})-p. \end{cases} \end{align} (26)According to Moreno & Osorio (2015) the convergence can be proven by constructing a Lyapunov function. It is worth noting that the existence of function sign(•) will lead to a chattering result of estimations that will have a negative impact on the actuators. A new convergent system is explored by changing the function of h(•) which is crucial to the ESO performance with properly chosen $$\beta _{1},\beta _{2} $$ in (24). Inspired by Shao’s recent work Shao & Wang (2016), sigmoid function has been selected as h(•). This SESO is given as   \begin{align} \begin{cases} {\dot{\hat{x}}_{1} =f(x_{1} )+g(x_{1} )u+\hat{x}_{2} -\beta_{1}\, \textrm{sig}(e_{1} ;a,b)} \\{\dot{\hat{x}}_{2} =-\beta_{2}\, \textrm{sig}(e_{1} ;a,b)}, \end{cases} \end{align} (27)where $$a,b,\beta _{1}>0,\beta _{2} >0$$ are the observed parameters which will be chosen accounting to desired estimation precision. Theorem 1 This concerns the non-linear system described by plant (22) in combination with SESO (27). If Assumption 1 is satisfied, appropriate observer parameters $$a>2\psi $$, $$b={\beta _{2} \mathord{\left / {\vphantom{\beta _{2} \varepsilon }} \right . } \varepsilon }$$, $$\beta _{1} <\sqrt{8\sqrt{2} +12}$$ will exist where $$\varepsilon>0,\beta _{2} >0$$ such that   \begin{align} \left\| \zeta \right\|_{2} \le \Pi \buildrel\Delta\over= \left\{\zeta \left|\left\| \zeta \right\| \le \frac{4\varpi l}{\beta_{2} \sigma_{\min } (\boldsymbol{Q})} \right. \right\} \end{align} (28)is satisfied. Where $$\boldsymbol{\zeta } =\left[\textrm{sig}(e_{1} ;a,b)\ e_{2} \right]^{\textrm{T}}$$. The observer outputs $$\hat{x}_{1}$$ and $$\hat{x}_{2}$$ will converge into a residual region of actual states $$x_{1}$$ and extended state $$x_{2}$$, respectively. Proof. In order to examine stability, error dynamics must be developed. Combining (23) and (27), can provide the following error state system:   \begin{align} \begin{cases} {\dot{e}_{1} =e_{2} -\beta_{1}\, \textrm{sig}(e_{1} ;a,b)} \\{\dot{e}_{2} =-\beta_{2}\, \textrm{sig}(e_{1} ;a,b)-p}. \end{cases} \end{align} (29)Step1. Consider the following Lyapunov candidate   \begin{align} \begin{aligned} L_{1} &=\textrm{sig}^{2} (e_{1} ) + \left(\frac{\beta_{1}}{2} \textrm{sig}(e_{1})-e_{2} \right)^{2} \\ &=\left(\frac{{\beta_{1}^{2}} }{4} +1\right)\textrm{sig}^{2} (e_{1} )+{e_{2}^{2}} -\beta_{1}\, \textrm{sig}(e_{1} )e_{2}. \end{aligned} \end{align} (30)Letting   \begin{align*} \boldsymbol{\zeta} &=\big[\textrm{sig}(e_{1} ;a_{1},\beta_{1} e_{1}\, \textrm{sig}(e_{1} )_{1} )\quad e_{2} \big]^{\textrm{T}} \\ \boldsymbol{P}&=\frac{1}{2} \left[\begin{array}{@{}cc@{}} {\frac{{\beta_{1}^{2}} }{2} +2} & {-\beta_{1} } \\{-\beta_{1} } & {2} \end{array}\right]. \end{align*}It can be expressed as   \begin{align} L_{1} =\boldsymbol{\zeta}^{\textrm{T}} \boldsymbol{P}\boldsymbol{\zeta}. \end{align} (31)Noting that sig(•) is differentiable, raising the fact that   $$ \frac{d[\textrm{sig}(e_{1} )]}{dt} = \left[\frac{b}{a} (\frac{1}{4} a^{2} -\textrm{sig}^{2} (e_{1} ))\right]\dot{e}_{1}. $$In differentiating $$L_{1}$$ with respect to time gives   \begin{align} \begin{aligned} \dot{L}_{1} &=\left(\frac{{\beta_{1}^{2}} }{4} +1\right)2\,\textrm{sig}(e_{1} )\frac{b}{a} \left(\frac{1}{4} a^{2} -\textrm{sig}^{2} (e_{1} )\right)\dot{e}_{1} +2e_{2} \dot{e}_{2} \\ &\quad -\beta_{1} \frac{b}{a} \left(\frac{1}{4} a^{2} -\textrm{sig}^{2} (e_{1} )\right)\dot{e}_{1} e_{2} -\beta_{1} \textrm{sig}(e_{1} )\dot{e}_{2}. \end{aligned} \end{align} (32)Substituting error sate (29) into (32) yields   \begin{align} \begin{aligned} \dot{L}_{1} &=\left(2\chi \frac{b}{a} \beta_{1} \gamma -\chi \frac{ab}{2} \beta_{1} +\beta_{1} \beta_{2} \right)\textrm{sig}^{2} (e_{1} )+\left[\frac{ab}{4} {\beta_{1}^{2}} -\frac{b}{a} {\beta_{1}^{2}} \gamma -2\beta_{2} +\chi \frac{ab}{2} -2\chi \frac{b}{a} \gamma \right]\textrm{sig}(e_{1} )e_{2} \\ &\quad -\beta_{1} \left(\frac{ab}{4} -\frac{b}{a} \gamma \right){e_{2}^{2}} -2e_{2} \varpi +\beta_{1}\, \textrm{sig}(e_{1} )\varpi, \end{aligned} \end{align} (33)where $$\chi =({{\beta _{1}^{2}} \mathord{\left / {\vphantom{{\beta _{1}^{2}} 4}} \right . } 4} +1),\gamma =\textrm{sig}^{2} (e_{1})$$, giving   \begin{align} \begin{aligned} \dot{L}_{1} &=\left((\frac{4\gamma -a^{2} }{8a} )b{\beta_{1}^{3}} + (\frac{4\gamma -a^{2} }{2a} )b\beta_{1} +\beta_{1} \beta_{2} \right)\textrm{sig}^{2} (e_{1} ) \\ &\quad +\left[\frac{a^{2} -4\gamma }{4a} b{\beta_{1}^{2}} -2\beta_{2} +(\frac{a^{2} -4\gamma }{8a} ){\beta_{1}^{2}} b +\frac{a^{2} -4\gamma }{2a} b\right]\textrm{sig}(e_{1} )e_{2}\\ &\quad -\beta_{1} \left(\frac{a^{2} -4\gamma }{4a} \right)b{e_{2}^{2}}-2e_{2} \varpi +\beta_{1}\, \textrm{sig}(e_{1} )\varpi. \end{aligned} \end{align} (34)A simple calculation provides the formula   \begin{align} \begin{aligned} \dot{L}_{1} &=\left(-\frac{\varepsilon }{4} b{\beta_{1}^{3}} -\varepsilon b\beta_{1} +\beta_{1} \beta_{2} \right)\textrm{sig}^{2} (e_{1} )+\left(\frac{\varepsilon }{2} b{\beta_{1}^{2}} -2\beta_{2} +\frac{\varepsilon }{4} {\beta_{1}^{2}} b+\varepsilon b\right)\textrm{sig}(e_{1} )e_{2} \\ &\quad -\beta_{1} \frac{\varepsilon }{2} b{e_{2}^{2}} -2e_{2} \varpi +\beta_{1}\, \textrm{sig}(e_{1} )\varpi, \end{aligned} \end{align} (35)where $${(4\gamma -a^{2} )\mathord{\left / {\vphantom{(4\gamma -a^{2} ) 2a}} \right . } 2a} =-\varepsilon ,\varepsilon>0$$, let $$b={\beta _{2} \mathord{\left / {\vphantom{\beta _{2} \varepsilon }} \right . } \varepsilon }$$ giving   \begin{align} \begin{aligned} \dot{L}_{1} &=-\frac{1}{4} \beta_{2} \left[{\beta_{1}^{3}}\, \textrm{sig}^{2} (e_{1} )+(4-3{\beta_{1}^{2}} )\textrm{sig}(e_{1} )e_{2} + 2\beta_{1} {e_{2}^{2}} \right]-2e_{2} \varpi +\beta_{1}\, \textrm{sig}(e_{1} )\varpi \\ & =-\frac{1}{4} \beta_{2} \zeta^{T} Q\zeta +\varpi \bar{B}\zeta, \end{aligned} \end{align} (36)where   $$ Q=\frac{1}{2} \left[\begin{array}{@{}cc@{}} {2{\beta_{1}^{3}} } & {4-3{\beta_{1}^{2}} } \\{4-3{\beta_{1}^{2}} } & {4\beta_{1} } \end{array}\right],\quad\bar{B}=\left[\begin{array}{@{}c@{}} {\beta_{1} } \\{-2} \end{array}\right]^{\textrm{T}}. $$It is clear that Q > 0, if $$-{\beta _{1}^{4}} -16+24{\beta _{1}^{2}}>0$$, meaning $$0<\beta _{1} <\sqrt{8\sqrt{2} +12}$$, let $$\| \bar{B} \|_{2} =\sqrt{4+{\beta _{1}^{2}} } =l$$ yields   \begin{align} \begin{aligned} \dot{L}_{1} &=-\frac{1}{4} \beta_{2} \zeta^{T} Q\zeta +\varpi l\zeta \\ & \le -\frac{1}{4} \beta_{2} \sigma_{\min } (Q)\left\| \zeta \right\|_{2}^{2} +\varpi l\left\| \zeta \right\|_{2} \\ & =-\left(\frac{1}{4} \beta_{2} \sigma_{\min } (Q)\left\| \zeta \right\|_{2} -\varpi l\right)\left\| \zeta \right\|_{2}. \end{aligned} \end{align} (37)From this the conclusion that $$\dot{L}_{1} < 0$$ if $${\beta _{2} \sigma _{\min } (Q)\left \| \zeta \right \|_{2} \mathord{\left / {\vphantom{\beta _{2} \sigma _{\min } (Q)\left \| \zeta \right \|_{2} 4}} \right . } 4}>\varpi l$$ can be obtained so that error system (29) is asymptotically stable and the error vector will converge into the following region:   \begin{align} \Pi\, \buildrel\Delta\over= \left\{\boldsymbol{\zeta} \left|\left\| \boldsymbol{\zeta} \right\| \le \frac{4\varpi l}{\beta_{2} \sigma_{\min } (\boldsymbol{Q})} \right. \right\}\!. \end{align} (38)From (38) it can be understood that the larger $$\beta _{2}$$ the smaller the region $$\Pi $$. Additionally, according to (31) the following inequality is present:   \begin{align} \sigma_{\min } (\boldsymbol{P})\left\| \boldsymbol{\zeta} \right\|_{2}^{2} \le L(\boldsymbol{\zeta} )\le \sigma_{\max } (\boldsymbol{P})\left\| \boldsymbol{\zeta} \right\|_{2}^{2}, \end{align} (39)where $$\sigma _{\min } (\bullet )$$ and $$\sigma _{\max } (\bullet )$$ represent the minimum and maximum eigenvalue of the matrix. $$\left \| \boldsymbol{\zeta } \right \|_{2}^{2} =\textrm{sig}^{2} (e_{1} )+{e_{2}^{2}} $$, let $$\gamma =\textrm{sig}^{2} (e_{1} )\le{a^{2} \mathord{\left / {\vphantom{a^{2} 4}} \right . } 4}$$. Next   \begin{align} \left|e_{2} \right|=\sqrt{\left\| \boldsymbol{\zeta} \right\|_{2}^{2} -\gamma } \le \sqrt{\left(\frac{4\varpi l}{\beta_{2} \sigma_{\min } (\boldsymbol{Q})} \right)^{2} -\gamma } =\psi, \end{align} (40)where $$\psi>0$$. Step2. Consider the following Lyapunov candidate:   \begin{equation} L_{2} =\frac{1}{2} {e_{1}^{2}} \end{equation} (41)  \begin{align} \begin{aligned} \dot{L}_{2} &=e_{1} \dot{e}_{1} =-\beta_{1} e_{1}\, \textrm{sig}(e_{1} )+e_{1} e_{2} \\ & \le -\beta_{1} e_{1}\, \textrm{sig}(e_{1} )+\left|e_{1} \right|\psi. \end{aligned} \end{align} (42)Noting that $$\textrm{sig}(e_{1} )\in (-0.5a,0.5a)$$, if $$0.5a>\psi $$, b is large enough, we can select $$\beta _{1} $$ to make $$\left |\beta _{1}\, \textrm{sig}(e_{1} )\right |>\psi $$, there is   \begin{align} \dot{L}_{2} \le -\left|e_{1} \right|(\left|\beta_{1}\, \textrm{sig}(e_{1} )\right|-\psi )\le 0. \end{align} (43)From (43) we can know that $$\dot{L}_{2} $$ will be negative outside the set $$\{ |e_{1} |\le \textrm{sig}^{-1} ({\psi \mathord{ / {\vphantom{\psi \beta _{1} }} . } \beta _{1} } ) \}$$. Thus, $$\left |e_{1} \right |$$ is uniformly ultimately bounded. Therefore, the trajectories of the error state system are convergent, demonstrating the consistently bounded stability of $$\boldsymbol{\zeta } $$. This means that if observer parameters are chosen properly, the error system converges to the small region of zero and the observer outputs meet in a residual region of actual states $$x_{1}$$ and extended state $$x_{2} $$, respectively. This completes the proof of Theorem 1. Remark 2 Looking at (40) and (43), it is clear that estimation errors $$e_{1}$$ and $$e_{2}$$ are determined by parameters $$\beta _{1},\beta _{2},a$$ and b. In tuning these parameters appropriately, estimation errors can be minimized to a level enabling extended state $$x_{2}$$ to be observed effectively by the SESO. The fundamental selection of the parameters can be chosen as $$\beta _{1} < \sqrt{8\sqrt{2} +12}, \beta _{2}>0, \ a>2\psi $$, $$b={\beta _{2} \mathord{\left / {\vphantom{\beta _{2} \varepsilon }} \right . } \varepsilon }>0$$. An appropriately large $$\beta _{2} $$ can also be selected such that $$\left \| \boldsymbol{\zeta } \right \| $$ is small enough, although $$\varpi $$ is unknown. A relatively larger a can be chosen to make $$\left |e_{1} \right |$$ as small as required. Remark 3 In order to apply this general method to SESO design, the derivative of perturbation w should exist and be bounded. This will satisfy Assumption 2. In practical systems this ESO cannot be used to observe the non-differentiable disturbances. The advantage to this however is that there is no need to know the disturbance boundary as is necessary with the sliding mode observer in the study by Yan & Edwards (2007). At the end of this subsection, a comparative simulation example is conducted to demonstrate the superiority of the proposed SESO in observing performance. ESO1 represents the ESO design based on the STA convergent system (Moreno & Osorio, 2015) and ESO2 denotes the proposed ESO in this paper. In the simulation let disturbance $$w=\textrm{sign}(\sin (t))$$ which is a harsh simulation condition for the observers. The two ESOs are designed as follows: ESO 1:  \begin{align} \begin{cases} {\dot{\hat{x}}_{1} =f(x_{1} )+g(x_{1} )u+\hat{x}_{2} -d_{1} \left|x_{1} -\hat{x}_{1} \right|{}^{1/2} \textrm{sign}(x_{1} -\hat{x}_{1} )} \\ {\dot{\hat{x}}_{2} =-d_{2}\, \textrm{sign}(x_{1} -\hat{x}_{1} )}, \end{cases} \end{align} (44)where observer gains are selected as $$d_{1} =5,d_{2} =20$$. ESO 2:  \begin{align} \begin{cases} {\dot{\hat{x}}_{1} =f(x_{1} )+g(x_{1} )u+\hat{x}_{2} -\beta_{1} \textrm{sig}(x_{1} -\hat{x}_{1} )} \\{\dot{\hat{x}}_{2} =-\beta_{2}\, \textrm{sig}(x_{1} -\hat{x}_{1} )}, \end{cases} \end{align} (45)where observer parameters are selected as $$\beta _{1} =5,\beta _{2} =20$$, a = 2, b = 20. The same observer gains are selected in the simulation. Estimation performances are diverse because the convergent systems are constructed using different h(•) functions. Figure 2 illustrates the ESO 2 displays superior observing performance without overshooting and chattering even in the presence of square wave disturbance which has instantaneous transitions. Therefore, SESO is more suitable for the application of practical controller. Fig. 2. View largeDownload slide Observer results of ESO1 and ESO2 Fig. 2. View largeDownload slide Observer results of ESO1 and ESO2 4. Controller design based on SESO The objective of the control design is to force the output $$\boldsymbol{\varOmega }$$ to track the guidance command $$\boldsymbol{\varOmega }_{d}$$ by specifying the fin deflections $$\delta _{e}$$, $$\delta _{a}$$ and $$\delta _{r}$$, for control-oriented models (16) and (21). The proposed SESO will be applied to reconstruct the total disturbance in the design. First-order filter and auxiliary variables will be integrated into the back-stepping design to derive the real control input and reduce saturation effects, respectively. The procedure is presented in the following subsections. Before designing the controller, the following assumption must be raised: Assumption 2 The gain matrices $$\boldsymbol{g}_{1}$$ and $$\boldsymbol{g}_{20}$$ are invertible. This means $$\textrm{det}(\boldsymbol{g}_{1})=-1/\cos \beta $$. The singularity of $$\boldsymbol{g}_{1}^{-1}$$ occurs if the sideslip angle $$\beta =\pm 90^{\circ }$$. This situation will not occur as the controller attempts to keep $$\beta =0^{\circ }$$. The nominal gain matrix can be selected to make $$\boldsymbol{g}_{20}$$ invertible so this assumption is reasonable. 4.1 Structure of the robust back-stepping control scheme The structure of the proposed control scheme for HRV is shown in Fig. 3 and is based on back-stepping technology. The SESOs are adopted to compensate total disturbances existing in the two subsystems. Once the disturbances $$\boldsymbol{\varphi }_{1}$$ and $$\boldsymbol{\varphi }_{2}$$ existing in (16) and (21) are estimated, the actual control input can be modified based on $$\hat{\boldsymbol{\varphi }}_{i},\ i=1,2$$. In addition, h(v) is introduced as the input of the actuator to deal with saturation. In this way the control inputs will be limited and will not conflict with the admissible maximum value. To reduce saturation effects an auxiliary system driven by $$\varDelta \boldsymbol{v}$$ is applied to motivate the control input to decrease appropriately when saturation occurs. These are the key features of the proposed control method. Fig. 3. View largeDownload slide The structure of the SESO based back-stepping control for HRV. Fig. 3. View largeDownload slide The structure of the SESO based back-stepping control for HRV. 4.2 Back-stepping procedure Back-stepping control is proposed for the HRV non-linear system in this subsection. The recursive design procedure contains two steps. First, virtual control law is designed to guarantee the stability of the angular subsystem (16). In step 2, an overall control law v is constructed for the angular rate subsystem (21). Step 1. The following error state can be defined as   \begin{align} \boldsymbol{z}_{1} =\boldsymbol{\varOmega}-\boldsymbol{\varOmega}_{d}. \end{align} (46)Considering (16) and taking the derivation of (46),   \begin{align} \dot{\boldsymbol{z}}_{1} =\boldsymbol{f}_{1} +\boldsymbol{g}_{1} \boldsymbol{\omega}+\boldsymbol{\varphi }_{1} -\dot{\boldsymbol{\varOmega }}_{d} \end{align} (47)the virtual control law is then designed as follows:   \begin{align} \boldsymbol{\omega}_{c} =-\boldsymbol{g}_{1}^{-1} (\boldsymbol{k}_{1} \boldsymbol{z}_{1} +\boldsymbol{f}_{1} -\dot{\boldsymbol{\varOmega }}_{d} +\hat{\boldsymbol{\varphi }}_{1} ), \end{align} (48)where $$\boldsymbol{k}_{1}=\textrm{diag}[k_{11},k_{12},k_{13}]$$, $$k_{1i}>0, \ i=1,2,3$$ are parameters to be designed and $$\hat{\boldsymbol{\varphi }}_{1}$$ is the estimation of $$\boldsymbol{\varphi }_{1}$$ to be obtained by the SESO proposed later. A new state variable $$\boldsymbol{\omega }_{d}$$ is introduced here so $$\dot{\boldsymbol{\omega }}_{c}$$ can be analytically expressed as $$\dot{\boldsymbol{\omega }}_{d}$$. This can be obtained by the following first-order filter   \begin{align} \dot{\boldsymbol{\omega }}_{d} =(\boldsymbol{\omega }_{c} -\boldsymbol{\omega }_{d} -\tau \boldsymbol{g}_{1} \boldsymbol{z}_{1} ){/}\tau ,\quad \boldsymbol{\omega }_{d} \left(0\right)=\boldsymbol{\omega }_{c} \left(0\right), \end{align} (49)where $$\tau $$ is the positive filter parameter to be designed. The following auxiliary variables are defined:   \begin{align} \boldsymbol{\gamma }=\boldsymbol{\omega} -\boldsymbol{\omega }_{d} \end{align} (50)  \begin{align} \boldsymbol{\xi} =\boldsymbol{\omega }_{c} -\boldsymbol{\omega }_{d}, \end{align} (51)where $$\boldsymbol{\gamma }$$ is a new error state and $$\boldsymbol{\xi }$$ is called ‘filter error’. Next is $$\boldsymbol{\omega }=\boldsymbol{\gamma }+\boldsymbol{\omega }_{c}-\boldsymbol{\xi }$$, thus the error dynamics $$\dot{\boldsymbol{z}}_{1}$$ can be calculated as   \begin{align} \begin{aligned} \dot{\boldsymbol{z}}_{1} &=\boldsymbol{f}_{1} +\boldsymbol{g}_{1} \left(\boldsymbol{\gamma }+\boldsymbol{\omega }_{c} -\boldsymbol{\xi }\right)+\boldsymbol{\varphi }_{1} -\dot{\boldsymbol{\varOmega }}_{d} \\ &=\boldsymbol{f}_{1} +\boldsymbol{g}_{1} \boldsymbol{\gamma}-\boldsymbol{g}_{1} \boldsymbol{\xi}-\dot{\boldsymbol{\varOmega}}_{d} +\boldsymbol{g}_{1} \boldsymbol{\omega }_{c} +\boldsymbol{\varphi }_{1} \\ & =\boldsymbol{A}_{1} +\boldsymbol{B}_{1} \boldsymbol{\omega }_{c} +\boldsymbol{\varphi }_{1}, \end{aligned} \end{align} (52)where $$\boldsymbol{A}_{1}=\boldsymbol{f}_{1}+\boldsymbol{g}_{1}\boldsymbol{\gamma }-\boldsymbol{g}_{1}\boldsymbol{\xi }-\dot{\boldsymbol{\varOmega }}_{d}$$, $$\boldsymbol{B}_{1}=\boldsymbol{g}_{1}$$. Based on the ESO design principle, the extended state variable can be introduced as $$\boldsymbol{z}_{2}=\boldsymbol{\varphi }_{1}$$, which is defined with $$\dot{\boldsymbol{z}}_{2}=\boldsymbol{p}_{1}(t)$$. The following assumption is given: Assumption 3 It is assumed $$\boldsymbol{P}_{1}(t)$$ is unknown but bounded, that is to say $$p_{1i}(t)\leq \varpi _{1}, i=1,2,3$$ where $$\varpi _{1}>0$$. Following this, the system (52) can be extended as   \begin{align} \begin{cases} {\dot{\boldsymbol{z}}_{1} =\boldsymbol{A}_{1} +\boldsymbol{B}_{1} \boldsymbol{ \omega }_{c} +\boldsymbol{z}_{2} } \\{\dot{\boldsymbol{z}}_{2} =\boldsymbol{p}_{1} \left(t\right)}. \end{cases} \end{align} (53)The SESO can then be described as follows:   \begin{equation} \left\{\begin{array}{l} \boldsymbol{E}_{11} =\boldsymbol{Z}_{11} -\boldsymbol{z}_{1} \\ \dot{\boldsymbol{Z}}_{1} =\boldsymbol{A}_{1} +\boldsymbol{B}_{1} \boldsymbol{\omega }_{c} +\boldsymbol{ Z}_{2} -\boldsymbol{\beta }_{z1} \mathbf{sig}\left(\boldsymbol{E}_{11} \right) \\ \dot{\boldsymbol{ Z}}_{2} =-\boldsymbol{\beta }_{z2} \mathbf{sig}\left(\boldsymbol{E}_{11} \right), \end{array}\right. \end{equation} (54)where $$\boldsymbol{Z}_{1}$$ and $$\boldsymbol{Z}_{2}$$ denote the output of the observer which will approach $$\boldsymbol{z}_{1}$$ and extended state $$\boldsymbol{z}_{2}$$, respectively. The estimation error is $$\boldsymbol{E}_{11}$$, $$\boldsymbol{\beta }_{zi}=\textrm{diag}(\beta _{zi1},\beta _{zi2},\beta _{zi3})$$ is the observer gain with $$\beta _{zij}>0$$, i = 1, 2, j = 1, 2, 3, which will be defined according to the desired estimation precision. In order to examine stability error dynamics must be developed. Combining (53) and (54) gives   \begin{align} \begin{cases} {\dot{\boldsymbol{E}}_{11} =-\boldsymbol{\beta }_{z1} \mathbf{sig}\left(\boldsymbol{ E}_{11} \right)+\boldsymbol{ E}_{12} } \\{\dot{\boldsymbol{E}}_{12} =-\boldsymbol{\beta }_{z2} \mathbf{sig}\left(\boldsymbol{E}_{11} \right)-\boldsymbol{p}_{1} \left(t\right)}, \end{cases} \end{align} (55)where $$\boldsymbol{E}_{12}=\boldsymbol{Z}_{2},\boldsymbol{z}_{2}$$ is the estimation error. Next, substituting (48) into (52) yields   \begin{align} \dot{\boldsymbol{z}}_{1} =-\boldsymbol{k}_{1} \boldsymbol{z}_{1} +\boldsymbol{g}_{1} \boldsymbol{\gamma }-\boldsymbol{g}_{1} \boldsymbol{\xi }+\boldsymbol{E}_{12}. \end{align} (56)Considering the ‘filter error’ $$\boldsymbol{\xi }$$, the following Lyapunov function candidate is written as   \begin{align} V_{1} =\frac{1}{2} \boldsymbol{z}_{1}^{{\textrm T}} \boldsymbol{z}_{1} +\frac{1}{2} \boldsymbol{\xi }^{{\textrm T}} \boldsymbol{\xi }. \end{align} (57)This definition of $$\boldsymbol{\xi }$$ gives the equation $$\dot{\boldsymbol{\zeta }}=-\boldsymbol{\xi }/\tau +\boldsymbol{g}_{1}\boldsymbol{z}_{1}+\dot{\boldsymbol{\omega }}_{c}$$. The time derivation of $$V_{1}$$ is given by   \begin{equation} \dot{V}_{1} =-\boldsymbol{z}_{1}^{\textrm{T}} \boldsymbol{k}_{1} \boldsymbol{z}_{1} +\boldsymbol{z}_{1}^{\textrm{T}} \boldsymbol{g}_{1} \boldsymbol{\gamma}-\boldsymbol{z}_{1}^{\textrm{T}} \boldsymbol{g}_{1} \boldsymbol{\xi}+\boldsymbol{z}_{1}^{\textrm{T}} \boldsymbol{E}_{12} -\left\| \boldsymbol{\xi}\right\| ^{2} {/} \tau +\boldsymbol{\xi }^{\textrm{T}} \boldsymbol{g}_{1} \boldsymbol{z}_{1} +\boldsymbol{\xi}^{\textrm{T}} \dot{\boldsymbol{\omega}}_{c} \end{equation} (58)considering the following assumption: Assumption 4 The derivation of virtual control is bounded, $$\Vert \dot{\boldsymbol{\omega }}_{c}\Vert \leq \vartheta $$, where $$\vartheta $$ is a positive constant, and if $$\frac{1-\tau }{\tau }\geq \tau _{0}>0$$, we can have   \begin{align} \begin{aligned} -\frac{1}{\tau } \left\| \boldsymbol{\xi }\right\| ^{2} +\boldsymbol{\xi }^{{\textrm{T}}} \vartheta &\le -\left(\frac{1-\tau }{\tau } \right)\left\| \boldsymbol{ \xi }\right\| ^{2} -\left(\left\| \boldsymbol{ \xi }\right\| -\frac{\vartheta }{2} \right)^{2} +\frac{\vartheta ^{2} }{4} \\ &\le -\tau _{0} \left\| \boldsymbol{ \xi }\right\| ^{2} +\frac{\vartheta ^{2} }{4} \end{aligned} \end{align} (59)the derivative of $$V_{1}$$ is then calculated as   \begin{align} \dot{V}_{1} \le -\boldsymbol{ z}_{1}^{\textrm{T}} \boldsymbol{ k}_{1} \boldsymbol{ z}_{1} -\tau _{0} \left\| \boldsymbol{ \xi }\right\| ^{2} +\boldsymbol{ z}_{1}^{\textrm{T}} \boldsymbol{ g}_{1} \boldsymbol{ \gamma }+\vartheta ^{2} / 4 +\boldsymbol{ z}_{1}^{\textrm{T}} \boldsymbol{ E}_{12}, \end{align} (60)the term $$\boldsymbol{z}_{1}^{\textrm{T}}\boldsymbol{g}_{1}\boldsymbol{\gamma }$$ in (60) will be cancelled during the next control design procedure. The other terms will be considered in the stability analysis of the closed-loop system. Step 2. The following auxiliary system can be constructed to compensate the constraint effect caused by saturation   \begin{align} \dot{\boldsymbol{ \lambda }}=-k_{\lambda } \boldsymbol{ \lambda }+\boldsymbol{ g}_{20} \boldsymbol{ h}\left(\boldsymbol{ v}\right)-\boldsymbol{ g}_{20} \boldsymbol{ v}, \end{align} (61)where $$k_{\lambda }>0$$. The new error state vector is defined as   \begin{align} \boldsymbol{ s}_{1} =\boldsymbol{ \gamma }-\boldsymbol{ \lambda }. \end{align} (62)Taking the derivative of $$\boldsymbol{s}_{1}$$ with respect to time provides   \begin{align} \begin{aligned} \dot{\boldsymbol{ s}}_{1} &=\boldsymbol{ f}_{2} +\boldsymbol{ g}_{20} \boldsymbol{ h}\left(\boldsymbol{ v}\right)+\boldsymbol{ g}_{20}\, \boldsymbol{ j}\left(\boldsymbol{ v}\right)+\boldsymbol{ \varphi }_{2} -\dot{\boldsymbol{ \omega }}_{d} -\dot{\boldsymbol{ \lambda }} \\ & =\boldsymbol{ f}_{2} +\boldsymbol{ g}_{20} \boldsymbol{ j}\left(\boldsymbol{ v}\right)-\dot{\boldsymbol{ \omega }}_{d} +k_{\lambda } \boldsymbol{ \lambda }+\boldsymbol{ g}_{20} \boldsymbol{ v}+\boldsymbol{ \varphi }_{2} \\ & =\boldsymbol{ A}_{2} +\boldsymbol{ B}_{2} \boldsymbol{ v}+\boldsymbol{ \varphi }_{2}, \end{aligned} \end{align} (63)where $$\boldsymbol{A}_{2}=\boldsymbol{f}_{2}+\boldsymbol{g}_{20}\,\boldsymbol{j}(\boldsymbol{v})-\dot{\boldsymbol{\omega }}_{d}+k_{\lambda }\boldsymbol{\lambda }$$, $$\boldsymbol{B}_{2}=\boldsymbol{g}_{20}$$, then the actual control input is proposed as   \begin{align} \begin{aligned} \boldsymbol{ v}&=\boldsymbol{ B}_{2} ^{-1} \left(-\boldsymbol{ k}_{2} \boldsymbol{ s}_{1} -\boldsymbol{ A}_{2} -\hat{\boldsymbol{ \varphi }}_{2} +\boldsymbol{ v}_{a} \right) \\ \boldsymbol{ v}_{a} &=\boldsymbol{ z}_{a} -\frac{\boldsymbol{ s}_{1} h_{a} }{\eta ^{2} +\left\| \boldsymbol{ s}_{1} \right\| ^{2} } \\ \dot{\boldsymbol{ z}}_{a} &=-k_{\boldsymbol{ z}_{a} } \boldsymbol{ z}_{a} -\frac{\boldsymbol{ z}_{a} f_{a} }{\left\| \boldsymbol{ z}_{a} \right\| ^{2} } \\ \dot{\eta }&=-k_{\eta } \eta -\frac{\eta h_{a} }{\eta ^{2} +\left\| \boldsymbol{ s}_{1} \right\| ^{2} }, \end{aligned} \end{align} (64)where $$k_{z_{a}}>0$$, $$h_{a}=\Vert \boldsymbol{s}_{1}\Vert ^{2}/2$$, $$f_{a}=\boldsymbol{\lambda }^{\textrm{T}}\boldsymbol{g}_{1}\boldsymbol{z}_{1}-\Vert \boldsymbol{g}_{20}\varDelta \boldsymbol{v}\Vert ^{2}/4$$. $$\varDelta \boldsymbol{v}=\boldsymbol{h}(\boldsymbol{v})-\boldsymbol{v}$$ is the error state generated by desired controller output and sigmoid function output. Note that $$\varDelta \boldsymbol{v}$$ is smooth and it will shrink the control signals when the desired control inputs exceed the maximum authority. This means the saturation time could be shortened. The estimation of $$\boldsymbol{\varphi }_{2}$$ is $$\hat{\boldsymbol{\varphi }}_{2}$$, which will be obtained by the following SESO. Based on the principle of the SESO design, extended state variable $$\boldsymbol{s}_{2}=\boldsymbol{\varphi }_{2}$$ are now introduced which are defined with $$\dot{\boldsymbol{s}}_{2}=\boldsymbol{p}_{2}(t)$$. The following assumption is given: Assumption 5 It is assumed $$\boldsymbol{p}_{2}(t)$$ is unknown but bounded, that is to say $$p_{2i}(t)\leq \varpi _{2},i=1,2,3$$, where $$\varpi _{2}>0$$. The system (63) can then be extended as   \begin{align} \begin{cases} {\boldsymbol{ s}_{1} =\boldsymbol{ A}_{2} +\boldsymbol{ B}_{2} \boldsymbol{ v}+\boldsymbol{ s}_{2} } \\{\dot{\boldsymbol{ s}}_{2} =\boldsymbol{ p}_{2} \left(t\right)}, \end{cases} \end{align} (65)the SESO can be described as follows:   \begin{align} \begin{cases} {\boldsymbol{ E}_{21} =\boldsymbol{ S}_{1} -\boldsymbol{ s}_{1} } \\{\dot{\boldsymbol{ S}}_{1} =\boldsymbol{ A}_{2} +\boldsymbol{ B}_{2} \boldsymbol{ v}+\boldsymbol{ S}_{2} -\boldsymbol{ \beta }_{s1} \mathbf{sig}\left({\textbf{E}}_{21} \right)} \\{\dot{\boldsymbol{ S}}_{2} =-\boldsymbol{\beta }_{s2} \mathbf{sig}\left({\textbf{E}}_{21} \right)}, \end{cases} \end{align} (66)where $$\boldsymbol{S}_{1}$$ and $$\boldsymbol{S}_{2}$$ are the observer outputs. They will approach $$\boldsymbol{s}_{1}$$ and extended state $$\boldsymbol{s}_{2}$$, respectively. The estimation error is $$\boldsymbol{E}_{22}$$, $$\boldsymbol{\beta }_{si}=\textrm{diag}(\beta _{si1},\beta _{si2},\beta _{si3})$$ is the observer gain with $$\beta _{sij}>0$$, i = 1, 2, j = 1, 2, 3, which will be defined according to the desired estimation precision. Error dynamics must be developed to examine stability. Combining (65) and (66) provides   \begin{align} \begin{cases} {\dot{\boldsymbol{ E}}_{21} =-\boldsymbol{ \beta }_{s1} \mathbf{sig}\left(\boldsymbol{ E}_{21} \right)+\boldsymbol{ E}_{22} } \\{\dot{\boldsymbol{ E}}_{22} =-\boldsymbol{ \beta }_{s2} \mathbf{sig}\left(\boldsymbol{ E}_{21} \right)-\boldsymbol{ p}_{1} \left(t\right)}, \end{cases} \end{align} (67)where $$\boldsymbol{E}_{22}=\boldsymbol{S}_{2}-\boldsymbol{s}_{2}$$ is the estimation error state. So far the controller and SESO have been established successfully. Stability analysis will be given in the following section. Note that compared with the adaptive method to handle unknown total disturbance, tracking error states are not used for constructing the SESO. This means saturation will not become worse under the action of disturbance estimation. It can be concluded from this that a controller design based on SESO will be more robust against actuator saturation to a certain extent. Remark 1 In the controller design process of an angular rate subsystem, the inversion of $$\boldsymbol{g}_{2}$$ may not exist. The normal value $$\boldsymbol{g}_{20}$$ can therefore be selected to guarantee its non-singularity. In this way, control inputs can be kept smooth without singularity while aerodynamic data acquired by ground testing and computing loads can be reduced.The back-stepping controller based on SESO has been designed successfully. Tracking performance depends crucially on SESO estimation accuracy so observer gains parameters must be correctly selected before tuning the controller gains. 4.3 Stability analysis of closed-loop dynamics In this subsection, the stability of the closed-loop system can be established by the following theorem: Theorem 2 Consider the HRV attitude systems (16) and (21). By applying control laws (48) and (64) in combination with SESOs (54) and (66), the closed-loop system is guaranteed to be uniformly bounded stable and tracking errors will converge to a small set around zero. Proof. In order to examine the stability of the closed-loop system, firstly, consider the Lyapunov function candidate   \begin{align} V_{2} =\frac{1}{2} \boldsymbol{ s}_{1}^{{\textrm T}} \boldsymbol{ s}_{1} +\frac{1}{2} \boldsymbol{ \lambda }^{{\textrm T}} \boldsymbol{ \lambda }+\frac{1}{2} \boldsymbol{ z}_{a}^{{\textrm T}} \boldsymbol{ z}_{a} +\frac{1}{2} \eta ^{2}, \end{align} (68)its time derivation is   \begin{align} \dot{V}_{2} =\boldsymbol{ s}_{1}^{{\textrm T}} \dot{\boldsymbol{ s}}_{1}^{{\textrm T}} +\boldsymbol{ \lambda }^{{\textrm T}} \dot{\boldsymbol{ \lambda }}+\boldsymbol{ z}_{a}^{{\textrm T}} \dot{\boldsymbol{ z}}_{a} +\eta \dot{\eta }. \end{align} (69)Substituting control law (64) into (69) gives   \begin{align} \dot{V}_{2} =-\boldsymbol{ s}_{{1}}^{{\textrm T}} \boldsymbol{ k}_{2} \dot{\boldsymbol{ s}}_{{1}}^{{\textrm T}} -\boldsymbol{ s}_{{1}}^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{ z}_{1} +\boldsymbol{ s}_{{ 1}}^{{\textrm T}} \boldsymbol{ E}_{22} +W, \end{align} (70)where $$W=\boldsymbol{s}_{1}^{\textrm T}\boldsymbol{v}_{a}+\boldsymbol{z}_{z}^{\textrm T}\dot{\boldsymbol{z}}_{a}+\eta \dot{\eta }+\boldsymbol{\lambda }^{\textrm T}\dot{\boldsymbol{\lambda }}$$. Considering control law (64) yields   \begin{align} \begin{aligned} W&\le \left\| \boldsymbol{ z}_{a} \boldsymbol{ s}_{1} \right\| -k_{\boldsymbol{ z}_{a} } \left\| \boldsymbol{ z}_{a} \right\| ^{2} -\frac{\left\| s_{1} \right\| ^{2} }{2} -k_{\eta } \eta ^{2} -k_{\lambda } \left\| \boldsymbol{ \lambda }\right\| ^{2} +\left\| \boldsymbol{ \lambda }\right\| \left\| \boldsymbol{ g}_{20} {\varDelta} \boldsymbol{v}\right\| -\frac{1}{4} \left\| \boldsymbol{ g}_{20} \varDelta \boldsymbol{v}\right\|^{2} -\boldsymbol{\lambda }^{{\textrm T}} \boldsymbol{g}_{1} \boldsymbol{z}_{1} \\ &=-\left(k_{\boldsymbol{z}_{a} } \frac{1}{2} \right)\left\| \boldsymbol{z}_{a} \right\| ^{2} -\frac{1}{2} \left(\left\| \boldsymbol{z}_{a} \right\| -\left\| \boldsymbol{s}_{1} \right\| \right)^{2} k_{\eta } \eta ^{2} -\left(k_{\lambda } -1\right)\left\| \boldsymbol{\lambda }\right\| ^{2} -\left(\left\| \boldsymbol{\lambda }\right\| -\frac{1}{2} \left\| \boldsymbol{g}_{20} \varDelta \boldsymbol{v}\right\| \right)^{2} -\boldsymbol{\lambda }^{{\textrm T}} \boldsymbol{g}_{1} \boldsymbol{z}_{1} \\ &\le -\bar{k}_{\boldsymbol{ z}_{a} } \left\| \boldsymbol{z}_{a} \right\| ^{2} -k_{\eta } \eta ^{2} -\bar{k}_{\lambda } \left\| \boldsymbol{\lambda }\right\| ^{2} -\boldsymbol{\lambda }^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{z}_{1}, \end{aligned} \end{align} (71)where $$\bar{k}_{z_{a}}=k_{z_{a}}-\frac{1}{2}>0$$, $$\bar{k}_{\lambda }=k_{\lambda }-1>0$$. Then one has   \begin{equation} \dot{V}_{2} \le -\boldsymbol{s}_{{1}}^{{\textrm T}} \boldsymbol{ k}_{2} \dot{\boldsymbol{ s}}_{{1}}^{{\textrm T}} -\boldsymbol{ s}_{{1}}^{\boldsymbol{T}} \boldsymbol{ g}_{1} \boldsymbol{z}_{1} +\boldsymbol{ s}_{1}^{{\textrm T}} \boldsymbol{ E}_{22} -\bar{k}_{z_{a} } \left\| \boldsymbol{ z}_{a} \right\| ^{2} -k_{\eta } \eta ^{2} -\bar{k}_{\lambda } \left\| \boldsymbol{\lambda }\right\| ^{2} -\boldsymbol{\lambda }^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{z}_{1}. \end{equation} (72)Following this, the augmented Lyapunov function is selected   \begin{align} V=V_{1} +V_{2}. \end{align} (73)Its derivative with respect to time is   \begin{align} \begin{aligned} \dot{V}&\le -\boldsymbol{z}_{1}^{{\textrm T}} \boldsymbol{k}_{1} \boldsymbol{z}_{1} -\tau _{0} \left\| \boldsymbol{\xi }\right\| ^{2} +\boldsymbol{ z}_{1}^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{\gamma }_{1} +\vartheta ^{2} /4+\boldsymbol{ z}_{1}^{{\textrm T}} \boldsymbol{E}_{12}-\boldsymbol{ s}_{1}^{{\textrm T}} \boldsymbol{k}_{2} \dot{\boldsymbol{ s}}_{1}^{T} -\boldsymbol{ s}_{1}^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{z}_{1} -\bar{k}_{z_{a} } \left\| \boldsymbol{z}_{a} \right\| ^{2} \\ &\quad+\boldsymbol{s}_{1}^{{\textrm T}} \boldsymbol{ E}_{22} -k_{\eta } \eta ^{2} -\bar{k}_{\lambda } \left\| \boldsymbol{\lambda }\right\| ^{2} -\boldsymbol{ \lambda }^{{\textrm T}} \boldsymbol{g}_{1} \boldsymbol{ z}_{1}. \end{aligned} \end{align} (74)Theorem 1 shows that observer errors converge into residual set of zero. This means $$\Vert \boldsymbol{E}_{12}\Vert <\zeta _{1}$$ and $$\Vert \boldsymbol{E}_{22}\Vert <\zeta _{2}$$ can be satisfied, where $$\psi _{1}$$ and $$\psi _{2}$$ are positive but unknown constants. This is seen as   \begin{equation} \dot{V}\le -k_{1\min } \left\| \boldsymbol{z}_{1} \right\| ^{2} -k_{2\min } \left\| \boldsymbol{s}_{1} \right\| ^{2} +\varsigma _{1} \left\| \boldsymbol{z}_{1} \right\| +\varsigma _{2} \left\| \boldsymbol{ s}_{1} \right\| -\tau _{0} \left\| \boldsymbol{\xi }\right\| ^{2} -\bar{k}_{\boldsymbol{z}_{a} } \left\| \boldsymbol{z}_{a} \right\| ^{2} -k_{\eta } \eta ^{2} -\bar{k}_{\lambda } \left\| \boldsymbol{\lambda }\right\| ^{2} +\vartheta ^{2} /4,\quad \end{equation} (75)where $$k_{1\min }=\min \{k_{1i}\}$$, $$k_{2\min }=\{k_{2i}\}$$. Consider the following two inequalities   \begin{align} \begin{aligned} -k_{1\min } \left\| \boldsymbol{ z}_{1} \right\| ^{2} +\varsigma _{1} \left\| \boldsymbol{ z}_{1} \right\| &= -\left(k_{1\min } -1\right)\left\| \boldsymbol{z}_{1} \right\| ^{2} -\left(\left\| \boldsymbol{ z}_{1} \right\| -\frac{\varsigma _{1} }{2} \right)^{2} +\frac{\varsigma _{1} ^{2} }{4} \le -c_{1} \left\| \boldsymbol{ z}_{1} \right\| ^{2} +\frac{\varsigma _{1} ^{2} }{4} \\ -k_{2\min } \left\| \boldsymbol{ s}_{1} \right\| ^{2} +\varsigma _{1} \left\| \boldsymbol{ s}_{1} \right\| &= -\left(k_{2\min } -1\right)\left\| \boldsymbol{s}_{1} \right\| ^{2} -\left(\left\| \boldsymbol{ s}_{1} \right\| -\frac{\varsigma _{2} }{2} \right)^{2} +\frac{\varsigma _{2} ^{2} }{4} \le -c_{2} \left\| \boldsymbol{s}_{1} \right\| ^{2} +\frac{\varsigma _{2} ^{2} }{4}, \end{aligned} \end{align} (76)where $$c_{1}=k_{1\min }-1>0, c_{2}=k_{2\min }-1>0$$. Then we have   \begin{align} \begin{aligned} \dot{V}&\le 2c\frac{1}{2} \left\| \boldsymbol{z}_{1} \right\| ^{2} -2c_{2} \frac{1}{2} \left\| \boldsymbol{s}_{1} \right\| ^{2} -2\tau _{0} \frac{1}{2} \| \boldsymbol{\xi }_{1} \| ^{2} -2\overline{k}_{\boldsymbol{ z}_{a} } \frac{1}{2} \left\| \boldsymbol{ z}_{a} \right\| ^{2} \\ &\quad-2k_{\eta } \frac{1}{2} \eta ^{2} -2\overline{k}_{\lambda } \frac{1}{2} \left\| \boldsymbol{ \lambda }\right\| ^{2} +\frac{\varsigma _{1}^{2} }{4} +\frac{\varsigma _{2}^{2} }{4} +\frac{\vartheta ^{2} }{4} \\ &\le -\boldsymbol{ \xi }V+\vartheta, \end{aligned} \end{align} (77)where   \begin{align*} \xi &=\min \left\{2_{c_{1} } ,2_{c_{2} } ,2\tau _{0} ,2\overline{k}_{\boldsymbol{z}_{a} } ,2k_{\eta } ,2\overline{k}_{\lambda } \right\} \\ \vartheta &=\frac{\varsigma _{1}^{2} }{4} +\frac{\varsigma _{2}^{2} }{4} +\frac{\vartheta ^{2} }{4}. \end{align*}Integration of (77) yields   \begin{align} 0\le V\left(t\right)\le \frac{\vartheta }{\xi } +\left(V\left(0\right)-\frac{\vartheta }{\xi } \right)\exp (-\xi t)\; \; \quad\forall\, t>0. \end{align} (78) It follows that the Lyapunov function candidate V is uniformly ultimately bounded. From (57) and (73) we have $$\Vert \boldsymbol{z}_{1}\Vert ^{2}\leq 2V(t)$$, it is obtained that the tracking error state $$\boldsymbol{z}_{1}$$ asymptotically converges to a compact set $$\varTheta $$ defined by   \begin{align} {\varTheta }:=\left\{\boldsymbol{ z}_{1} \bigg|\left\| \boldsymbol{ z}_{1} \right\| \le \sqrt{2\left(V(0)+\frac{\vartheta }{\xi } \right)} \right\}. \end{align} (79)This completes the proof of Theorem 2. A composite controller has been designed successfully based on back-stepping method and disturbance observer technique. The main difference between the approach presented and the conventional back-stepping method is that disturbance estimation is integrated to the virtual control law and employed to handle mismatched disturbances. Input constraints are handled by combining approximation function and auxiliary variables. 5. Simulation and analysis Several simulations are conducted in this section to demonstrate the performance of the proposed SESO-based back-stepping controller. Consider HRV system models (2) and (3) with the following initial parameters: The guidance command signals are given as   \begin{align*} \boldsymbol{\varOmega}_{d}&=\left[\begin{array}{@{}ccc@{}}\alpha_{d},&\beta_{d},&\mu_{d}\end{array}\right]^{\textrm T}\\ & = \left[\begin{array}{@{}ccc@{}}\textrm{sign}(\textrm{sin}(0.11t))+5,&0,&5\textrm{sin}(0.5t)+5\textrm{sin}(0.15t)\end{array}\right]^{\textrm T}\deg \end{align*}In addition, external disturbances and uncertainties are added to the system. These are described as   \begin{align*} \left[\begin{array}{@{}c@{}}d_{\alpha}\\ d_{\beta}\\ d_{\mu}\end{array}\right]&=\left[\begin{array}{@{}c@{}}\textrm{sin}(0.1t)+\textrm{sin}(0.2t)\\ \textrm{sin}(0.1t)+\textrm{sin}(0.2t)\\ \textrm{sin}(0.1t)+\textrm{sin}(0.2t)\end{array}\right]\times 0.01\textrm{rad}/s\\ \left[\begin{array}{@{}c@{}}d_{l}\\ d_{m}\\ d_{n}\end{array}\right]&= \left[\begin{array}{@{}ccc@{}}I_{x}& &\\ & I_{y}&\\ &&I_{z}\end{array}\right] \left[\begin{array}{@{}c@{}}\textrm{sin}(0.1t)+\textrm{sin}(0.2t)\\ 0.1\textrm{sin}(0.1t)+0.1\textrm{sin}(0.2t)\\ \textrm{sin}(0.1)+\textrm{sin}(0.2t)\end{array}\right]\textrm{N}\cdot\textrm{m}\\ \left[\begin{array}{@{}c@{}}\varDelta C_{l}\\ \varDelta C_{m}\\ \varDelta C_{n}\end{array}\right]&=\left[\begin{array}{@{}c@{}}-0.3C_{l}+0.001\\ -0.3C_{m}+0.001\\ -0.3C_{n}+0.001\end{array}\right]. \end{align*} Here, coefficients of aerodynamic moments are reduced by approximately 30%. Moment disturbances and model uncertainties are also considered. These multiple uncertainties and disturbances are additional aggressive scenarios for the simulations. For the purpose of comparison, three cases analyses are presented. System initial states, uncertain aerodynamic parameters, external disturbances and input constraints are the same for the following numerical simulations: Case 1. The conventional back-stepping controller is applied in this case. It represents the controller without auxiliary system (61), approximate function (19) and SESOs (54) and (66), and is designed as follows:   \begin{align} \begin{aligned} \boldsymbol{\omega}_{c,\textrm{tra}} &=\boldsymbol{g}_{1}^{-1} (-\boldsymbol{k}_{1} \boldsymbol{z}_{1}-\boldsymbol{f}_{1} +\dot{\boldsymbol{\varOmega}}_{d} ) \\ \boldsymbol{v}_{\textrm{tra}} &=\boldsymbol{g}_{20}^{-1} (-\boldsymbol{k}_{2} \boldsymbol{\gamma}-\boldsymbol{f}_{2} +\dot{\boldsymbol{\omega}}_{d} ). \end{aligned} \end{align} (80) Case 2. The SESO-based back-stepping (SESOB) strategy proposed in this paper is applied here. Case 3. The adaptive back-stepping controller designed in the study by Wang et al., (2015) is used here to demonstrate the advantages of the SESOB in dealing with saturation issues. Only the attitude angles and input responses are given in this case due to page limitation. Control parameters selected can be found in Table 1. Table 1. Initial values for HRV Notation  Value  Unit  $$\boldsymbol{\varOmega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}}$$  $$\deg $$  $$\boldsymbol{\omega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}} $$  $$\deg /s$$  $$\boldsymbol{I}_{i} (i=x,y,z)$$  $$\left [\begin{matrix} {9.18}\ \, {90.3}\ \,{90.3} \end{matrix}\right ]\times e^{5} $$  $$kg\cdot m^{2} $$  $$\delta _{e},\delta _{a},\delta _{r} $$  [−30, 30]  $$\deg $$  Notation  Value  Unit  $$\boldsymbol{\varOmega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}}$$  $$\deg $$  $$\boldsymbol{\omega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}} $$  $$\deg /s$$  $$\boldsymbol{I}_{i} (i=x,y,z)$$  $$\left [\begin{matrix} {9.18}\ \, {90.3}\ \,{90.3} \end{matrix}\right ]\times e^{5} $$  $$kg\cdot m^{2} $$  $$\delta _{e},\delta _{a},\delta _{r} $$  [−30, 30]  $$\deg $$  View Large Table 1. Initial values for HRV Notation  Value  Unit  $$\boldsymbol{\varOmega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}}$$  $$\deg $$  $$\boldsymbol{\omega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}} $$  $$\deg /s$$  $$\boldsymbol{I}_{i} (i=x,y,z)$$  $$\left [\begin{matrix} {9.18}\ \, {90.3}\ \,{90.3} \end{matrix}\right ]\times e^{5} $$  $$kg\cdot m^{2} $$  $$\delta _{e},\delta _{a},\delta _{r} $$  [−30, 30]  $$\deg $$  Notation  Value  Unit  $$\boldsymbol{\varOmega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}}$$  $$\deg $$  $$\boldsymbol{\omega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}} $$  $$\deg /s$$  $$\boldsymbol{I}_{i} (i=x,y,z)$$  $$\left [\begin{matrix} {9.18}\ \, {90.3}\ \,{90.3} \end{matrix}\right ]\times e^{5} $$  $$kg\cdot m^{2} $$  $$\delta _{e},\delta _{a},\delta _{r} $$  [−30, 30]  $$\deg $$  View Large Table 2. Controller parameters Parameter  Value  Parameter  Value  $$\boldsymbol{k}_{1} $$  diag(3, 3, 3)  $$\boldsymbol{k}_{2} $$  diag(6, 6, 6)  $$\boldsymbol{\beta }_{s1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{s2} $$  diag(25, 25, 25)  $$\boldsymbol{\beta }_{f1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{f2} $$  diag(22, 22, 22)  a  2  b  35  Parameter  Value  Parameter  Value  $$\boldsymbol{k}_{1} $$  diag(3, 3, 3)  $$\boldsymbol{k}_{2} $$  diag(6, 6, 6)  $$\boldsymbol{\beta }_{s1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{s2} $$  diag(25, 25, 25)  $$\boldsymbol{\beta }_{f1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{f2} $$  diag(22, 22, 22)  a  2  b  35  View Large Table 2. Controller parameters Parameter  Value  Parameter  Value  $$\boldsymbol{k}_{1} $$  diag(3, 3, 3)  $$\boldsymbol{k}_{2} $$  diag(6, 6, 6)  $$\boldsymbol{\beta }_{s1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{s2} $$  diag(25, 25, 25)  $$\boldsymbol{\beta }_{f1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{f2} $$  diag(22, 22, 22)  a  2  b  35  Parameter  Value  Parameter  Value  $$\boldsymbol{k}_{1} $$  diag(3, 3, 3)  $$\boldsymbol{k}_{2} $$  diag(6, 6, 6)  $$\boldsymbol{\beta }_{s1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{s2} $$  diag(25, 25, 25)  $$\boldsymbol{\beta }_{f1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{f2} $$  diag(22, 22, 22)  a  2  b  35  View Large Simulation results of Case 1 are depicted in Fig. 4. It can be seen that the AOA, sideslip angle and bank angle track their respective guidance commands with increased tracking errors although the closed-system tends to be stable. Satisfactory tracking performances cannot be obtained without the compensation of SESOs. As seen in Fig. 4(c), the control signals violate limitations as sharp corners have a negative effect on the actuators. Fig. 4. View largeDownload slide Response curves of Case 1. Fig. 4. View largeDownload slide Response curves of Case 1. View largeDownload slide View largeDownload slide Tracking results of Case 2 are depicted in Fig. 5. It is clear here that better altitude tracking specification is achieved with smaller tracking errors. This is because disturbances are estimated and compensated directly in the controller via SESOs. As illustrated in Fig. 5(c), control signals will increase to some extent but will never violate limitations based on the property of sigmoid function. This means saturated inputs will not occur and inputs are kept smooth during saturation periods. These are the features of the proposed anti-windup method. Compared with the results of Case 1, tracking performance is improved significantly. View largeDownload slide View largeDownload slide Fig. 5. View largeDownload slide Response curves of Case 2. Fig. 5. View largeDownload slide Response curves of Case 2. Performance estimations under Case 2 can be seen in Fig. 5(d) and (e). It is verified that the SESO can reconstruct the total disturbance efficiently and smoothly. Fig. 6. View largeDownload slide Response curves of Case 3. Fig. 6. View largeDownload slide Response curves of Case 3. Simulation results of Case 3 are depicted in Fig. 6. Although the adaptive back-stepping controller shows superior control precision compared with the traditional back-stepping controller in Case 1, the tracking performance is not comparable to that of the SESOB in Case 2. An increased saturation occurrence is illustrated by the input response in Fig. 6(b). Once the inputs violate the limitation, the adaptive law will be motivated by tracking errors to generate larger control signals to track the reference. This makes the controller aggressive and leads to longer saturation times and more of them. This simulation verifies previous results. In terms of input constraint and multiple uncertainties, the proposed control strategy based on SESO can achieve enhanced control performance in comparison with the adaptive method. Table 3. Tracking errors in different cases of controllers   Case 1  Case 2  Case 3  $$z_{11} (\deg )$$  6  0.5  1  $$z_{12} (\deg )$$  0.7  0.1  0.2  $$z_{13} (\deg )$$  2  0.1  1    Case 1  Case 2  Case 3  $$z_{11} (\deg )$$  6  0.5  1  $$z_{12} (\deg )$$  0.7  0.1  0.2  $$z_{13} (\deg )$$  2  0.1  1  Table 3. Tracking errors in different cases of controllers   Case 1  Case 2  Case 3  $$z_{11} (\deg )$$  6  0.5  1  $$z_{12} (\deg )$$  0.7  0.1  0.2  $$z_{13} (\deg )$$  2  0.1  1    Case 1  Case 2  Case 3  $$z_{11} (\deg )$$  6  0.5  1  $$z_{12} (\deg )$$  0.7  0.1  0.2  $$z_{13} (\deg )$$  2  0.1  1  The tracking errors in different cases of controllers can be found in Table 3. 6. Conclusion This paper has focused on the ESO-based back-stepping controller design for HRV. The new control-oriented model has been subject to multiple uncertainties, disturbances and actuator saturations. A novel SESO has been presented and combined successfully with the back-stepping method to develop a controller to cope with mismatched and matched uncertainties. The stability of this SESO has been proven through constructing a Lyapunov function. Two methods have been combined successfully to handle to effects of input constraint. First, Sigmoid function was adopted to approximate saturation so the limited inputs are guaranteed. The error state between limited control input and the desired control input was also applied to integrate the controller. This resulted in saturation time being reduced significantly. The proposed control method was compared with conventional back-stepping and adaptive back-stepping control from existing literature. Results demonstrate that the SESO-based strategy shows superior robustness against aerodynamic uncertainties, external disturbances and input constraints. The proposed composite control method combines SESO and back-stepping techniques, providing a general solution for high-order systems with mismatched disturbance and saturation problems. Further research will focus on the ESO-based control strategy for general high-order system in the presence of actuator faults such as loss of effectiveness. Acknowledgements The author CC would like to thank the China Scholarship Council (CSC) for the financial support during his visit at ETH (Swiss Federal Institute of Technology Zurich). The authors would like to thank Mina from ASL (Autonomous Systems Lab), ETH, Switzerland for his writing and proofreading assistance. Funding National Natural Science Foundation of China (61403103, 61673135, 61603114). References Bao, W. ( 2013) Present situation and development tendency of aerospace control techniques. Acta Automatica Sinica , 39, 697– 702. Google Scholar CrossRef Search ADS   Boškovic, J. D., Li, S.-M. & Mehra, R. K. ( 2001) Robust adaptive variable structure control of spacecraft under control input saturation. J. Guid. Control Dyn. , 24, 14– 22. Google Scholar CrossRef Search ADS   Chen, M., Ge, S. S. & Ren, B. ( 2011) Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints. Automatica , 47, 452– 465. 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Extended state observer-based back-stepping control for hypersonic reentry vehicle with input constraints

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© The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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Abstract

Abstract This paper investigates the attitude-tracking control problem of hypersonic reentry vehicle in cases of multiple uncertainties, external disturbances and input constraints. The controller design is based on synthesizing the extended state observer (ESO) into a back-stepping control technique. This control-oriented model is formulated with mismatched and matched uncertainties. They reflect the total disturbances that group different types of aerodynamic uncertainties and external moment disturbances. In order to improve the system robustness, a sigmoid function-based ESO is first proposed. This will estimate the total disturbance and is equipped with a controller. The sigmoid smooth function is also introduced for the purpose of handling the input constraints. This will approximate saturation and guarantee that the control input is bounded. Error states between the actual control input and the desired control input are integrated to compensate for the saturation effect. Following this, the stability of the closed-loop system is proved within the Lyapunov theory framework. Several simulations are then investigated to illustrate the effectiveness of the proposed constrained attitude control scheme. 1. Introduction Considerable control design research related to hypersonic reentry vehicle (HRV) stabilization strategies has been presented in recent decades. This growing focus is due to increasing potential applications for the technology and the arising theoretical challenges in attitude control (Geng et al., 2013; Wang et al., 2001; Corban et al., 2001). HRV systems present unique control challenges, especially in the reentry stage, under the condition of dramatically changing flight environments (Jiang et al., 2013), model uncertainties, external moment disturbances and unknown aerodynamic characteristics as well as actuator saturation (Bao, 2013; Zhang et al., 2014). The disturbance rejection ability of control systems must be improved to meet the requirement for high-control performance and flight reliability, particularly in the presence of input constraints. Note that the HRV dynamic model is a second-order system that mainly suffers from three types of uncertainties: aerodynamic uncertainties, dynamic model uncertainties and external moment disturbances. From a control point of view these uncertainties generally fall into the category of either matched or mismatched uncertainties. It is well known that controller design based on multiple time scaling (Naidu & Calise, 2001) is an effective methodology to tackle mismatched uncertainties. The design process can be simplified sufficiently by using time-scale collocation architecture (Yu & Li, 2015). Back-stepping methodology (Krstic et al., 1995) is also an efficient tool to solve the control problem for non-linear system with mismatched uncertainties. It is easily incorporated with other control techniques to deal with unknown disturbances as an alternative approach. In addition, more rigourous mathematical proof can be given based on the Lyapunov stability theorem (Lee & Kim, 2001) without the time-scale assumption. A composite anti-disturbance controller (Sun & Guo, 2014) is synthesized by introducing disturbance estimations into the design of virtual control laws to compensate for the mismatched disturbances. A drawback with the implementation of an adaptive back-stepping controller is the need to compute virtual derivatives at each step. This is called the ‘explosion of terms’ problem and dynamic surface control (Chen & Yu, 2014) or filter technique (Swaroop et al., 2000; Wang & Huang, 2005; Dong et al., 2014) can be employed to counter this. Filter errors are seldom considered in the stability analysis of the closed-loop system, however. Another difficulty worth studying is actuator saturation (Gou et al., 2017), which can severely limit system performance and even lead to instability. Actuator saturation is unavoidable in a practical system so it is necessary to improve the control system capability to deal with this. This is particularly important for the attitude control system of HRV that suffers more occasions of actuator saturation during entry phase because of large attitude manoeuvres. It must be equipped with an appropriate controller to handle the harmful effects of saturation and improve safety and reliability. This can be done by employing a design-bounded controller to bound the control input so that it will not violate the limitation as shown in the study by Boškovic et al. (2012). However, the stability can be obtained only under some restrictive assumptions (Huang et al., 2001) and the controller is conservative. Alternatively, an auxiliary system driven by the error between desired control input and actual control input could be employed to reduce the effects of saturation and guarantee system stability (Chen et al., 2001). However, the saturation is a non-smooth and non-differentiable function, which prevents the back-stepping technique from being utilized directly (Wen et al., 2017). While previous research results related to actuator saturations have been implemented successfully, robustness against uncertainties and disturbances as well as input constraints is still worth comprehensive study. Aerodynamic uncertainties, dynamic model uncertainties and external disturbances are seldom considered simultaneously in designing constrained controllers for HRV. A controller with an adaptive law driven by tracking error has drawbacks when the system faces control constraints (Sonneveldt et al., 2017). During the saturation period, tracking errors increase not only because of external disturbances but also due to actuator limitations. The updated law seeks the desired performance only according to tracking errors and this leads to a larger and more aggressive control input. This problem also increases saturation and extends its time period. The extended state observer (ESO)-based back-stepping method explored in this paper seeks to avoid this problem by applying a novel sigmoid function-based ESO (SESO) to compensate the disturbances instead of using adaptive laws to estimate the bounds of disturbances. An ESO can actively compensate for the total disturbance in the dynamic model (Huang & Xue, 2014). In the past decades, ESO has been developed and widely used in the control field. For example, the finite time ESO (Deefort et al., 2011) developed recently can increase the speed and efficacy of the observer (Zhao & Yang, 2016). Another finite time ESO was proposed in the study by (Xiong et al., 2006) with complex structure that makes convergence proof overcomplicated and it is also difficult to obtain the ESO parameters tuning guideline. In order to realize the finite-time convergence, non-linear sign function is normally introduced to construct the observer. This will inevitably lead to the chattering of estimation and makes it difficult for practical application. In addition, sign function is more sensitive to noise because it is not smooth around zero, this should be avoided for practical application. The sign function also can be found in traditional ESO (Han, 2009). This makes it quite difficult to give rigourous proof (Guo & Zhao, 2011). These undesirable features need to be removed by replacing the sign functions by novel smooth functions. In contrast with sign function, from Shao’s recent work we can know that sigmoid function has the noise-attenuation ability which can be obtained by adjusting the function parameters. Moreover, sigmoid function (Shao & Wang, 2016) is characterized by integrating the non-linear and linear terms naturally (Shao et al., 2017), which is also a real-valued and continuous differentiable function with no singularities. In light of this, a convergent system can be constructed based on sigmoid function and then an ESO can be designed. The stability and convergence of the proposed ESO can be proven based on the proposed convergent system. This paper will explore this method and attempt to develop a novel smooth ESO. A new control strategy for HRV based on the back-stepping technique is proposed, which does not introduce a conventional time-scale assumption. This takes into account aerodynamic and dynamic model uncertainties, external moment disturbances and the effects of actuator saturation as discussed above. To facilitate the controller design a new control-oriented model was formulated by lumping the multiple uncertainties into total disturbance. In this process, mismatched disturbance are considered in the two-order HRV system. To improve system robustness a novel SESO was also applied to reconstruct and compensate the total disturbance in the controller. The proposed SESO can estimate disturbance more smoothly compared to a sign function-based ESO. To deal with control input constraints, an auxiliary system and variable were introduced to reduce the harmful effects of saturation. This is done in place of approximating the saturation with smooth function which will limit control inputs. Saturation time can be reduced significantly in this way. The stability of this closed-loop system has been rigourously proven using the Lyapunov theory. It takes estimation error, auxiliary systems and input constraints into consideration. In comparison to existing works, the main contributions of this paper are summarized as follows: 1. A new control-oriented model is developed in the presence of aerodynamic uncertainties, dynamic model uncertainties and external moment disturbance as well as actuator saturation. 2. A novel SESO is developed based on Lyapunov methodology to observe and compensate the total disturbance. Compared with the existing ESO, the SESO shows smooth estimation with simple structure. The fundamental tuning rules of the observer parameters are provided. 3. Two ways are combined to deal with input constraints. First, the smooth bounded function is applied to approximate saturation, making the control signals limited. Secondly, in case of the dramatically increasing inputs caused by perturbations, the auxiliary variables are applied to reduce saturation effects. The rest of this paper is organized as follows. Section 2 will formulate the control-oriented model of the HRV. The proposed SESO design method will be presented in Section 3. In Section 4 the back-stepping controller design is presented based on the novel SESO and stability analysis of the closed-loop system is given. Section 5 will present the simulation results to demonstrate the effectiveness of this proposed approach and conclusions are provided in Section 6. 2. Preliminaries and system description 2.1 Preliminaries The following notations and lemma are introduced and used in the analysis and design of the SESO. For any variable $$x_{i}$$ and any constant $$a_{i}>0,\, b_{i} >0$$, i = 1, …n, let   \begin{align} \begin{aligned} \textrm{sig}(x_{i} )&=\textrm{sig}(x_{i} ;a_{i},b_{i} )=a_{i} \left[(1+e^{-b_{i} x_{i} } )^{-1} -0.5\right] \\[-2pt] \mathbf{sig}(\boldsymbol{x})&=\left[\textrm{sig}(x_{1} ;a_{1},b_{1} )\, \ldots \, \textrm{sig}(x_{n} ;a_{n},b_{n} )\right]^\textrm{T}, \end{aligned} \end{align} (1)where sig(•) denotes the sigmoid function. 2.2 System description In this section, dynamic equations of rotational motion referenced to a vehicle-body-fixed coordinate system are presented (Van Soest, 2006). The equations are described as   \begin{align} \begin{cases} \dot{\alpha }=\omega_{z} -\omega_{x} \cos \alpha \tan \beta +\omega_{y} \sin \alpha \tan \beta -\frac{1}{mV\cos \beta } (L+mg\cos \theta \cos \mu )+d_{\alpha } \\[2pt] \dot{\beta }=\omega_{x} \sin \alpha +\omega_{y} \cos \alpha + \frac{1}{mV} \left(C-mg\cos \theta \sin \mu \right)+d_{\beta } \\[2pt] \dot{\mu }=\omega_{x} \frac{\cos \alpha }{\cos \beta } -\omega_{y} \frac{\sin \alpha }{\cos \beta } +\frac{1}{mV} \big[L(\sin \theta \sin \mu +\tan \beta ) + C\sin \theta \cos \mu +mg\cos \theta \cos \mu \tan \beta \big]+d_{\mu }, \end{cases} \end{align} (2)where $$\alpha ,\beta $$ and $$\mu $$ denote the angle of attack (AOA), sideslip angle and bank angle, respectively. $$\omega _{x}$$, $$\omega _{y}$$ and $$\omega _{z}$$ denote the roll, yaw, and pitch rate in that order. $$d_{\alpha }, d_{\beta }$$ and $$d_{\mu }$$ denote the model uncertainties. The lift force is L and C is the side force. Mass is m, g is the acceleration of gravity and V and $$\theta $$ are the velocity and flight path angle, respectively. The rotational dynamic model is denoted as follows:   \begin{align} \begin{cases} I_{x} \dot{\omega }_{x} =(I_{y} -I_{z} )\omega_{y} \omega_{z} +\left(C_{l} +\Delta C_{l} +C_{l,\omega_{x} } \frac{\omega_{x} b_{ref} }{2V} + C_{l,\omega_{y} } \frac{\omega_{y} b_{ref} }{2V} \right)qb_{ref} S_{ref} +d_{l} \\[5pt] I_{y} \dot{\omega }_{y} =(I_{z} -I_{x} )\omega_{x} \omega_{z} +\left(C_{m} +\Delta C_{m} +C_{m,\omega_{x}} \frac{\omega_{x} b_{ref} }{2V} + C_{m,\omega_{y} } \frac{\omega_{y} b_{ref} }{2V} \right)qb_{ref} S_{ref} +X_{cg} C+d_{m} \\[5pt] I_{z} \dot{\omega }_{z} =(I_{x} -I_{y} )\omega_{y} \omega_{x} +\left(C_{n} +\Delta C_{n} +C_{n,\omega_{x} } \frac{\omega_{z} c_{ref} }{2V} \right)qc_{ref} S_{ref} + X_{cg} (D\sin \alpha +L\cos \alpha )+d_{n}, \end{cases} \end{align} (3)where $$X_{cg}$$ represents the longitude distance from the moment reference centre to the vehicle centre of gravity, $$q={\rho V^{2} \mathord{\left / {\vphantom{\rho V^{2} 2}} \right . } 2}$$ is the dynamic pressure and $$\rho $$ is atmospheric density. The reference length is $$b_{ref}$$, mean aerodynamic chord is $$c_{ref}$$ and $$S_{ref}$$ is the aerodynamic reference area. Aerodynamic moment uncertainty coefficients are $$\Delta C_{l},\Delta C_{m}$$ and $$\Delta C_{n}$$. $$d_{l},d_{m}$$ and $$d_{n}$$ denote the moment disturbances. For the purpose of reflecting an accurate reentry flight dynamic and developing a realistic control-oriented model, $$d_{\omega _{x}}, d_{\omega _{y}}$$ and $$d_{\omega _{z}}$$ represent the compound moment disturbances of the three channels, respectively. In this way (3) can be described as   \begin{align} \begin{cases} I_{x} \dot{\omega }_{x} =(I_{y} -I_{z} )\omega_{y} \omega_{z} +C_{l} qb_{ref} S_{ref} +d_{\omega_{x} } \\[2pt] I_{y} \dot{\omega }_{y} =(I_{z} -I_{x} )\omega_{x} \omega_{z} +C_{m} qb_{ref} S_{ref} +d_{\omega_{y} } \\[2pt] I_{z} \dot{\omega }_{z} =(I_{x} -I_{y} )\omega_{y} \omega_{x} +C_{n} qc_{ref} S_{ref} +d_{\omega_{z} }, \end{cases} \end{align} (4)where   \begin{align} \begin{cases} d_{\omega_{x} } =(\Delta C_{l} +C_{l,\Delta } )qb_{ref} S_{ref} +d_{l} \\ d_{\omega_{y} } =(\Delta C_{m} +C_{m,\Delta } )qb_{ref} S_{ref} +X_{cg} C+d_{m} \\ d_{\omega_{z} } =(\Delta C_{n} +C_{n,\Delta } )qc_{ref} S_{ref} +X_{cg} (D\sin \alpha +L\cos \alpha )+d_{n}, \end{cases} \end{align} (5)where D is the drag force and $$C_{l,\Delta }$$, $$C_{m,\Delta }$$ and $$C_{n,\Delta }$$ are the aerodynamic moment coefficients for angle rates. These are regarded as the uncertainties and are described as follows:   \begin{align} \begin{cases} C_{l,\Delta } =C_{l,\omega_{x} } \frac{\omega_{x} b_{ref} }{2V} +C_{l,\omega_{y} } \frac{\omega_{y} b_{ref} }{2V} \\[2pt] C_{m,\Delta } =C_{m,\omega_{x} } \frac{\omega_{x} b_{ref} }{2V} +C_{m,\omega_{y} } \frac{\omega_{y} b_{ref} }{2V} \\[2pt] C_{n,\Delta} = C_{n,\omega_{x} } \frac{\omega_{z} c_{ref} }{2V}. \end{cases} \end{align} (6)In equation (6), $$C_{l},C_{m}$$ and $$C_{n}$$ denote the normal rolling, yawing and pitching moment coefficients correspondingly. These are expressed as   \begin{align} \begin{cases} C_{l} = C_{l,\beta } \beta +C_{l,\delta e} \delta_{e} +C_{l,\delta a} \delta_{a} +C_{l,\delta r} \delta_{r} \\ C_{m} = C_{m,\beta } \beta +C_{m,\delta e} \delta_{e} +C_{m,\delta a} \delta_{a} +C_{m,\delta r} \delta_{r} \\ C_{n} = C_{n,\alpha } +C_{n,\delta e} \delta_{e} +C_{n,\delta a} \delta_{a} +C_{n,\delta r} \delta_{r}, \end{cases} \end{align} (7)where $$\delta _{e},\delta _{a}$$ and $$\delta _{r}$$ are the left elevon, right elevon and the rudder fin deflections. The nominal aerodynamic drag, lift and side force are denoted as   \begin{align} \begin{cases} D = C_{D} qS_{ref} = qS_{ref} (C_{D,\alpha } +C_{D,\delta_{e} } +C_{D,\delta_{a} } +C_{D,\delta_{r} } ) \\ L = C_{L} qS_{ref} = qS_{ref} (C_{L,\alpha } +C_{L,\delta_{e} } +C_{L,\delta_{a} } ) \\ C = C_{C} qS_{ref} = qS_{ref} (C_{C,\beta } \beta +C_{C,\delta_{e} } +C_{C,\delta_{a} } +C_{C,\delta_{r} } ), \end{cases} \end{align} (8)where $$C_{D},C_{L}$$ and $$C_{C}$$ represent the aerodynamic force coefficients. We can see from (7) and (8) that aerodynamic coefficients are the function of AOA, slide angle and Mach number as well as deflection angles. Following this, model equations (2)–(3) can be expressed in matrix form   \begin{align} \dot{\boldsymbol{\varOmega} }=\boldsymbol{f}_{1} (\boldsymbol{\varOmega},\boldsymbol{\omega} )+\boldsymbol{g}_{1} (\boldsymbol{\varOmega} )\boldsymbol{\omega} +\boldsymbol{g}_{\Delta } \boldsymbol{u}+\boldsymbol{d}_{\varOmega } \end{align} (9)  \begin{align} \dot{\boldsymbol{\omega} }=\boldsymbol{f}_{2} (\boldsymbol{\varOmega},\boldsymbol{\omega} )+\boldsymbol{g}_{2} (\boldsymbol{\omega} )\boldsymbol{u}+\boldsymbol{d}_{\omega }, \end{align} (10)where $$\boldsymbol{\varOmega } =[\alpha \; \beta \; \mu ]^{\textrm{T}}$$ is the attitude angle vector, $$\boldsymbol{\omega } =[\omega _{x}\, \omega _{y}\, \omega _{z} ]^{\textrm{T}} $$is the angular rate vector and $$\boldsymbol{u}=[\delta _{e}\, \delta _{a}\, \delta _{r} ]^{\textrm{T}}$$ is the control input vector. The unknown disturbance induced by model uncertainties is $$\boldsymbol{d}_{\varOmega } =[d_{\alpha }\, d_{\beta }\, d_{\mu } ]^{\textrm{T}}$$ and $$\boldsymbol{d}_{\omega } = [d_{\omega _{x} }\, d_{\omega _{y} }\, d_{\omega _{z}} ]^{\textrm{T}}$$ is the compound moment disturbance vector. The vectors $$\boldsymbol{f}_{1} =[\,f_{\alpha } \; f_{\beta } \; f_{\mu } ]^{\textrm{T}}$$ and $$\boldsymbol{f}_{2} =[\,f_{\omega _{x} }\ f_{\omega _{y} }\ f_{\omega _{z} } ]^{\textrm{T}}$$ as well as the matrices $$\boldsymbol{g}_{1},\boldsymbol{g}_{\Delta }$$ and $$\boldsymbol{g}_{2}$$ are as follows:   \begin{align} \begin{cases} f_{\alpha } =\frac{1}{MV\cos \beta } [-qSC_{Y,\alpha } -Mg\cos \theta \cos \mu ] \\ f_{\beta } =\frac{1}{MV} (-qSC_{Z,\beta } \beta -Mg\cos \theta \sin \mu ) \\ f_{\mu } =\frac{g}{V} \cos \theta \cos \mu \tan \beta +\frac{1}{MV} [qSC_{Y,\alpha } (\sin \theta \sin \mu + \tan \beta )+qSC_{Z,\beta } \beta \sin \theta \cos \mu ] \end{cases} \end{align} (11)  \begin{align} \begin{cases} f_{\omega_{x} } =\frac{(I_{y} -I_{z} )}{I_{x} } \omega_{y} \omega_{z} +\frac{1}{I_{x} } C_{l,\beta } \beta qb_{ref} S_{ref} \\[2pt] f_{\omega_{y} } =\frac{(I_{z} -I_{x} )}{I_{y} } \omega_{x} \omega_{z} +\frac{1}{I_{y} } C_{m\mathrm{,}a} qb_{ref} S_{ref} \\[2pt] f_{\omega_{z} } = \frac{(I_{x} -I_{y} )}{I_{z} } \omega_{x} \omega_{z} +\frac{1}{I_{z} } C_{n,\alpha } qc_{ref} S_{ref} \end{cases} \end{align} (12)  \begin{align} \boldsymbol{g}_{1} = \left[\begin{array}{@{}ccc@{}} {-\cos \alpha \tan \beta } & {\sin \alpha \tan \beta } & {1} \\{\sin \alpha } & {\cos \alpha } & {0} \\{\sec \beta \cos \alpha } & {-\sec \beta \sin \alpha } & {0} \end{array}\right] \end{align} (13)  \begin{align} \boldsymbol{g}_{\Delta } = \left[\begin{array}{@{}ccc@{}} {g_{\alpha,\delta_{e} } } & {g_{\alpha,\delta_{a} } } & {0} \\{g_{\beta,\delta_{e} } } & {g_{\beta,\delta_{a} } } & {g_{\beta,\delta_{r} } } \\{g_{\mu,\delta_{e} } } & {g_{\mu,\delta_{a} } } & {g_{\mu,\delta_{r} } } \end{array}\right] \end{align} (14)  \begin{align} \boldsymbol{g}_{2} = \left[\begin{array}{@{}ccc@{}} {g_{\omega_{x},\delta_{e} } } & {g_{\omega_{x},\delta_{a} } } & {g_{\omega_{x},\delta_{r} } } \\{g_{\omega_{y},\delta_{e} } } & {g_{\omega_{y},\delta_{a} } } & {g_{\omega_{y},\delta_{r} } } \\{g_{\omega_{z},\delta_{e} } } & {g_{\omega_{z},\delta_{a} } } & {g_{\omega_{z},\delta_{r} } } \end{array}\right]\!, \end{align} (15)where $$g_{i,j}$$ denote the aerodynamic coefficients with $$i=\alpha ,\beta ,\mu , \omega _{x},\omega _{y},\omega _{z},\, j=\delta _{e},\delta _{a},\delta _{r}$$. The term $$\boldsymbol{g}_{\Delta } \boldsymbol{\delta }$$ is the deflection coupling effect for angular subsystems which could be regarded as disturbance. Therefore, for dynamic model (9), it can be deduced   \begin{align} \dot{\boldsymbol{\varOmega} }=\boldsymbol{f}_{1} (\boldsymbol{\varOmega},\boldsymbol{\omega} )+\boldsymbol{g}_{1} (\boldsymbol{\varOmega} )\boldsymbol{\omega} +\boldsymbol{\varphi}_{1}, \end{align} (16)where $$\boldsymbol{\varphi }_{1} =\boldsymbol{g}_{\Delta } \boldsymbol{u}+\boldsymbol{d}_{\varOmega }$$ denotes the ‘total disturbance’ of angular subsystem which includes coupling with aerodynamic force generated by deflection angles and model uncertainties. A further issue is uncertainties that exist in aerodynamic moment coefficients making it difficult to acquire the accurate $$\boldsymbol{g}_{2}$$ in advance and keep $$\boldsymbol{g}_{2}$$ invertible during the whole flight envelope. To solve this problem $$\boldsymbol{g}_{20}$$ is instead used for the controller design, which is the nominal data obtained based on the ground test for typical flight conditions. Therefore, dynamic model (10) can be rewritten as   \begin{align} \dot{\boldsymbol{\omega} }=\boldsymbol{f}_{2} (\boldsymbol{\varOmega}, \boldsymbol{\omega} )+\boldsymbol{g}_{20} \boldsymbol{u}+\boldsymbol{\varphi}_{2}, \end{align} (17)where $$\boldsymbol{g}_{20} =\boldsymbol{g}_{2} |_{\alpha =\textrm{const,Ma=const}},\boldsymbol{\varphi }_{2} =(\boldsymbol{g}_{2} -\boldsymbol{g}_{20} )\boldsymbol{u}+\boldsymbol{d}_{\omega }$$ denotes the ‘total disturbance’ of the angular rate subsystem including compound moment disturbances. From another point of view, $$(\boldsymbol{g}_{2} -\boldsymbol{g}_{20} )\boldsymbol{u}$$ is also representative of coefficient uncertainties included in the term $$\boldsymbol{\varphi }_{2}$$. The final problem concerns input constraints. The vector of actual control deflections generated by actuators is defined as u = sat(v). In this case $$\mathbf{sat}(\boldsymbol{v})= [\textrm{sat}(v_{1} ),\, \textrm{sat}(v_{2} ),\, \textrm{sat}(v_{3})]^{\textrm{T}}$$ denotes the non-linear saturation characteristic, v is the desired control signal to be designed and $$\textrm{sat}(v_{i})$$ is defined as   \begin{align} \textrm{sat}(v_{i})=\left\{\begin{array}{@{}ll@{}} \begin{array}{l} {u_{i\max } \textrm{sign}(v_{i} )} \\{v_{i} (t)} \end{array} \begin{array}{l} \left|v_{i} \right|> u_{i\max } \\{ \left|v_{i} \right| < u_{i\max } } \end{array} \end{array}\right. (i=1,2,3), \end{align} (18)where $$u_{i\max }$$ is the known bound of the ith actuator. In this paper the saturation is approximated by the following smooth sigmoid function:   \begin{align} h(v_{i} )=\textrm{sig}(v_{i} ;2u_{i\max },b_{i} )=2u_{i\max } \left[(1+e^{-b_{i} v_{i} } )^{-1} -0.5\right]. \end{align} (19) According to the property of sigmoid function it is known that $$h(v_{i} )$$ is smooth and bounded, meaning: $$\left |h(v_{i} )\right |<u_{i\max }$$. It is clear that there exists a different j(v) between sat(v) and h(v). Therefore,   \begin{align} \mathbf{sat}(\boldsymbol{v})=\boldsymbol{h}(\boldsymbol{v})+\boldsymbol{j}(\boldsymbol{v}) \end{align} (20)following this, (17) can be rewritten as   \begin{align} \dot{\boldsymbol{\omega} }=\boldsymbol{f}_{2} (\boldsymbol{\varOmega},\; \boldsymbol{\omega} )+\boldsymbol{g}_{20} \boldsymbol{h}(\boldsymbol{v})+\boldsymbol{g}_{20}\ \boldsymbol{j}(\boldsymbol{v})+\boldsymbol{\varphi}_{2}. \end{align} (21) Remark 1 Sigmoid function is used as an alternative to the hyperbolic tangent function to approximate saturation Wen et al. (2011) in this paper. In this case, parameter b can be adjusted according to the actuator response requirement. It can be seen from Fig. 1 that the larger b, the more sensitive the actuator is to the desired control input. The actuator will become relatively insensitive when the input signal is close to violating limitations. To an extent this will provide more freedom for enhanced performance of the controller. Note that j(v) is not differentiable at the sharp corner so it cannot be lumped into the ‘total disturbance’ as a part of $$\boldsymbol{\varphi }_{2}$$. Fig. 1. View largeDownload slide Saturation function h(v) and sat(v). Fig. 1. View largeDownload slide Saturation function h(v) and sat(v). So far the control-oriented model, (16) and (21) has been formulated successfully for controller design. Aerodynamic coefficient uncertainties, coupling effects, unknown external moment disturbances as well as input constraints are reflected in this model. The feature of dynamic systems (16) and (21) is that mismatched uncertainties $$\boldsymbol{\varphi }_{1}$$ and matched uncertainties $$\boldsymbol{\varphi }_{2}$$ are both considered in formulating the model. As an alternative way to deal with this kind of uncertainty problem, adaptive methods can be combined in controller design. Adaptive law driven by tracking errors will make actuator saturation periods longer. In light of this, ESO-based control methods to handling the mismatch problem in consideration of input saturations will be explored in this paper. The divide control design process based on back-stepping technique can be divided into two steps. First the virtual control input must be designed to guarantee angular subsystem (16) is stable. Then the actual control input for angular rate subsystem (21) can be modified. In order to enhance the robustness of control law at each step SESO must be integrated to tackle unknown disturbances. Following this the actual control input can be designed. 3. Novel ESO design based on sigmoid function In this section a general method to construct ESO is provided. A novel convergent system is first proposed with smooth sigmoid function and based on this the SESO can be developed. Consider the following general nonlinear system with uncertainty   \begin{align} \dot{x}=f(x)+g(x)u+w, \end{align} (22)where u ∈ R is control input and x is the measured output. System functions are f(x) and g(x) while w is uncertainty. In order to estimate the generalized disturbance w based on the principle of the ESO design, extended state variable $$x_{2}=w$$ is introduced. This is defined with $$\dot{x}_{2}=p$$. The following assumption is outlined below. Assumption 1 It is assumed p is unknown but bounded, that is to say $$p\le \varpi $$, where $$\varpi $$ is a positive constant. Afterwards, the system (22) can be extended as   \begin{align} \begin{cases} {\dot{x}_{1} =f(x_{1} )+g(x_{1} )u+x_{2} } \\{\dot{x}_{2} =p}. \end{cases} \end{align} (23)According to the ESO technique the following standard ESO can be constructed:   \begin{align} \begin{cases} \dot{\hat{x}}_{1} =f(x_{1} )+g(x_{1} )u+\hat{x}_{2} -\beta_{1} h_{1} (x_{1} -\hat{x}_{1} ) \\ \dot{\hat{x}}_{2} =-\beta_{2} h_{2} (x_{1} -\hat{x}_{1} )\!. \end{cases} \end{align} (24) The challenge is to design functions h(•) and parameters $$\beta _{1},\beta _{2}$$, such that $$\hat{x}_{1}$$ and $$\hat{x}_{2}$$ tend to be the estimations of the state $$x_{1}$$ and the uncertainty w, respectively. Combining (23) and (24) provides the following error state system   \begin{align} \begin{cases} {\dot{e}_{1} =-\beta_{1} h_{1} (e_{1} )+e_{2} } \\{\dot{e}_{2} =-\beta_{2} h_{2} (e_{1} )-p}, \end{cases} \end{align} (25)where $$e_{1} =\hat{x}_{1} -x_{1},\, e_{2} =\hat{x}_{2} -x_{2}$$ are estimation errors. If h(•) function can be found which can make the error system (25) convergent, then the ESO can be designed as (24). According to the Super-Twisting Algorithm (STA) (Pisano & Usai, 2007), $$h_{1} (e_{1} )= \left |e_{1} \right |^{{1\mathord{\left / {\vphantom{1 2}} \right . } 2} } \textrm{sign}(e_{1} ),\, h_{2} (e_{1} )=\textrm{sign}(e_{1})$$ can be selected. This means   \begin{align} \begin{cases} \dot{e}_{1} =-\beta_{1} \left|e_{1} \right|{}^{1/2}\textrm{sign}(e_{1} )+e_{2} \\ \dot{e}_{2} =-\beta_{2} \textrm{sign}(e_{1})-p. \end{cases} \end{align} (26)According to Moreno & Osorio (2015) the convergence can be proven by constructing a Lyapunov function. It is worth noting that the existence of function sign(•) will lead to a chattering result of estimations that will have a negative impact on the actuators. A new convergent system is explored by changing the function of h(•) which is crucial to the ESO performance with properly chosen $$\beta _{1},\beta _{2} $$ in (24). Inspired by Shao’s recent work Shao & Wang (2016), sigmoid function has been selected as h(•). This SESO is given as   \begin{align} \begin{cases} {\dot{\hat{x}}_{1} =f(x_{1} )+g(x_{1} )u+\hat{x}_{2} -\beta_{1}\, \textrm{sig}(e_{1} ;a,b)} \\{\dot{\hat{x}}_{2} =-\beta_{2}\, \textrm{sig}(e_{1} ;a,b)}, \end{cases} \end{align} (27)where $$a,b,\beta _{1}>0,\beta _{2} >0$$ are the observed parameters which will be chosen accounting to desired estimation precision. Theorem 1 This concerns the non-linear system described by plant (22) in combination with SESO (27). If Assumption 1 is satisfied, appropriate observer parameters $$a>2\psi $$, $$b={\beta _{2} \mathord{\left / {\vphantom{\beta _{2} \varepsilon }} \right . } \varepsilon }$$, $$\beta _{1} <\sqrt{8\sqrt{2} +12}$$ will exist where $$\varepsilon>0,\beta _{2} >0$$ such that   \begin{align} \left\| \zeta \right\|_{2} \le \Pi \buildrel\Delta\over= \left\{\zeta \left|\left\| \zeta \right\| \le \frac{4\varpi l}{\beta_{2} \sigma_{\min } (\boldsymbol{Q})} \right. \right\} \end{align} (28)is satisfied. Where $$\boldsymbol{\zeta } =\left[\textrm{sig}(e_{1} ;a,b)\ e_{2} \right]^{\textrm{T}}$$. The observer outputs $$\hat{x}_{1}$$ and $$\hat{x}_{2}$$ will converge into a residual region of actual states $$x_{1}$$ and extended state $$x_{2}$$, respectively. Proof. In order to examine stability, error dynamics must be developed. Combining (23) and (27), can provide the following error state system:   \begin{align} \begin{cases} {\dot{e}_{1} =e_{2} -\beta_{1}\, \textrm{sig}(e_{1} ;a,b)} \\{\dot{e}_{2} =-\beta_{2}\, \textrm{sig}(e_{1} ;a,b)-p}. \end{cases} \end{align} (29)Step1. Consider the following Lyapunov candidate   \begin{align} \begin{aligned} L_{1} &=\textrm{sig}^{2} (e_{1} ) + \left(\frac{\beta_{1}}{2} \textrm{sig}(e_{1})-e_{2} \right)^{2} \\ &=\left(\frac{{\beta_{1}^{2}} }{4} +1\right)\textrm{sig}^{2} (e_{1} )+{e_{2}^{2}} -\beta_{1}\, \textrm{sig}(e_{1} )e_{2}. \end{aligned} \end{align} (30)Letting   \begin{align*} \boldsymbol{\zeta} &=\big[\textrm{sig}(e_{1} ;a_{1},\beta_{1} e_{1}\, \textrm{sig}(e_{1} )_{1} )\quad e_{2} \big]^{\textrm{T}} \\ \boldsymbol{P}&=\frac{1}{2} \left[\begin{array}{@{}cc@{}} {\frac{{\beta_{1}^{2}} }{2} +2} & {-\beta_{1} } \\{-\beta_{1} } & {2} \end{array}\right]. \end{align*}It can be expressed as   \begin{align} L_{1} =\boldsymbol{\zeta}^{\textrm{T}} \boldsymbol{P}\boldsymbol{\zeta}. \end{align} (31)Noting that sig(•) is differentiable, raising the fact that   $$ \frac{d[\textrm{sig}(e_{1} )]}{dt} = \left[\frac{b}{a} (\frac{1}{4} a^{2} -\textrm{sig}^{2} (e_{1} ))\right]\dot{e}_{1}. $$In differentiating $$L_{1}$$ with respect to time gives   \begin{align} \begin{aligned} \dot{L}_{1} &=\left(\frac{{\beta_{1}^{2}} }{4} +1\right)2\,\textrm{sig}(e_{1} )\frac{b}{a} \left(\frac{1}{4} a^{2} -\textrm{sig}^{2} (e_{1} )\right)\dot{e}_{1} +2e_{2} \dot{e}_{2} \\ &\quad -\beta_{1} \frac{b}{a} \left(\frac{1}{4} a^{2} -\textrm{sig}^{2} (e_{1} )\right)\dot{e}_{1} e_{2} -\beta_{1} \textrm{sig}(e_{1} )\dot{e}_{2}. \end{aligned} \end{align} (32)Substituting error sate (29) into (32) yields   \begin{align} \begin{aligned} \dot{L}_{1} &=\left(2\chi \frac{b}{a} \beta_{1} \gamma -\chi \frac{ab}{2} \beta_{1} +\beta_{1} \beta_{2} \right)\textrm{sig}^{2} (e_{1} )+\left[\frac{ab}{4} {\beta_{1}^{2}} -\frac{b}{a} {\beta_{1}^{2}} \gamma -2\beta_{2} +\chi \frac{ab}{2} -2\chi \frac{b}{a} \gamma \right]\textrm{sig}(e_{1} )e_{2} \\ &\quad -\beta_{1} \left(\frac{ab}{4} -\frac{b}{a} \gamma \right){e_{2}^{2}} -2e_{2} \varpi +\beta_{1}\, \textrm{sig}(e_{1} )\varpi, \end{aligned} \end{align} (33)where $$\chi =({{\beta _{1}^{2}} \mathord{\left / {\vphantom{{\beta _{1}^{2}} 4}} \right . } 4} +1),\gamma =\textrm{sig}^{2} (e_{1})$$, giving   \begin{align} \begin{aligned} \dot{L}_{1} &=\left((\frac{4\gamma -a^{2} }{8a} )b{\beta_{1}^{3}} + (\frac{4\gamma -a^{2} }{2a} )b\beta_{1} +\beta_{1} \beta_{2} \right)\textrm{sig}^{2} (e_{1} ) \\ &\quad +\left[\frac{a^{2} -4\gamma }{4a} b{\beta_{1}^{2}} -2\beta_{2} +(\frac{a^{2} -4\gamma }{8a} ){\beta_{1}^{2}} b +\frac{a^{2} -4\gamma }{2a} b\right]\textrm{sig}(e_{1} )e_{2}\\ &\quad -\beta_{1} \left(\frac{a^{2} -4\gamma }{4a} \right)b{e_{2}^{2}}-2e_{2} \varpi +\beta_{1}\, \textrm{sig}(e_{1} )\varpi. \end{aligned} \end{align} (34)A simple calculation provides the formula   \begin{align} \begin{aligned} \dot{L}_{1} &=\left(-\frac{\varepsilon }{4} b{\beta_{1}^{3}} -\varepsilon b\beta_{1} +\beta_{1} \beta_{2} \right)\textrm{sig}^{2} (e_{1} )+\left(\frac{\varepsilon }{2} b{\beta_{1}^{2}} -2\beta_{2} +\frac{\varepsilon }{4} {\beta_{1}^{2}} b+\varepsilon b\right)\textrm{sig}(e_{1} )e_{2} \\ &\quad -\beta_{1} \frac{\varepsilon }{2} b{e_{2}^{2}} -2e_{2} \varpi +\beta_{1}\, \textrm{sig}(e_{1} )\varpi, \end{aligned} \end{align} (35)where $${(4\gamma -a^{2} )\mathord{\left / {\vphantom{(4\gamma -a^{2} ) 2a}} \right . } 2a} =-\varepsilon ,\varepsilon>0$$, let $$b={\beta _{2} \mathord{\left / {\vphantom{\beta _{2} \varepsilon }} \right . } \varepsilon }$$ giving   \begin{align} \begin{aligned} \dot{L}_{1} &=-\frac{1}{4} \beta_{2} \left[{\beta_{1}^{3}}\, \textrm{sig}^{2} (e_{1} )+(4-3{\beta_{1}^{2}} )\textrm{sig}(e_{1} )e_{2} + 2\beta_{1} {e_{2}^{2}} \right]-2e_{2} \varpi +\beta_{1}\, \textrm{sig}(e_{1} )\varpi \\ & =-\frac{1}{4} \beta_{2} \zeta^{T} Q\zeta +\varpi \bar{B}\zeta, \end{aligned} \end{align} (36)where   $$ Q=\frac{1}{2} \left[\begin{array}{@{}cc@{}} {2{\beta_{1}^{3}} } & {4-3{\beta_{1}^{2}} } \\{4-3{\beta_{1}^{2}} } & {4\beta_{1} } \end{array}\right],\quad\bar{B}=\left[\begin{array}{@{}c@{}} {\beta_{1} } \\{-2} \end{array}\right]^{\textrm{T}}. $$It is clear that Q > 0, if $$-{\beta _{1}^{4}} -16+24{\beta _{1}^{2}}>0$$, meaning $$0<\beta _{1} <\sqrt{8\sqrt{2} +12}$$, let $$\| \bar{B} \|_{2} =\sqrt{4+{\beta _{1}^{2}} } =l$$ yields   \begin{align} \begin{aligned} \dot{L}_{1} &=-\frac{1}{4} \beta_{2} \zeta^{T} Q\zeta +\varpi l\zeta \\ & \le -\frac{1}{4} \beta_{2} \sigma_{\min } (Q)\left\| \zeta \right\|_{2}^{2} +\varpi l\left\| \zeta \right\|_{2} \\ & =-\left(\frac{1}{4} \beta_{2} \sigma_{\min } (Q)\left\| \zeta \right\|_{2} -\varpi l\right)\left\| \zeta \right\|_{2}. \end{aligned} \end{align} (37)From this the conclusion that $$\dot{L}_{1} < 0$$ if $${\beta _{2} \sigma _{\min } (Q)\left \| \zeta \right \|_{2} \mathord{\left / {\vphantom{\beta _{2} \sigma _{\min } (Q)\left \| \zeta \right \|_{2} 4}} \right . } 4}>\varpi l$$ can be obtained so that error system (29) is asymptotically stable and the error vector will converge into the following region:   \begin{align} \Pi\, \buildrel\Delta\over= \left\{\boldsymbol{\zeta} \left|\left\| \boldsymbol{\zeta} \right\| \le \frac{4\varpi l}{\beta_{2} \sigma_{\min } (\boldsymbol{Q})} \right. \right\}\!. \end{align} (38)From (38) it can be understood that the larger $$\beta _{2}$$ the smaller the region $$\Pi $$. Additionally, according to (31) the following inequality is present:   \begin{align} \sigma_{\min } (\boldsymbol{P})\left\| \boldsymbol{\zeta} \right\|_{2}^{2} \le L(\boldsymbol{\zeta} )\le \sigma_{\max } (\boldsymbol{P})\left\| \boldsymbol{\zeta} \right\|_{2}^{2}, \end{align} (39)where $$\sigma _{\min } (\bullet )$$ and $$\sigma _{\max } (\bullet )$$ represent the minimum and maximum eigenvalue of the matrix. $$\left \| \boldsymbol{\zeta } \right \|_{2}^{2} =\textrm{sig}^{2} (e_{1} )+{e_{2}^{2}} $$, let $$\gamma =\textrm{sig}^{2} (e_{1} )\le{a^{2} \mathord{\left / {\vphantom{a^{2} 4}} \right . } 4}$$. Next   \begin{align} \left|e_{2} \right|=\sqrt{\left\| \boldsymbol{\zeta} \right\|_{2}^{2} -\gamma } \le \sqrt{\left(\frac{4\varpi l}{\beta_{2} \sigma_{\min } (\boldsymbol{Q})} \right)^{2} -\gamma } =\psi, \end{align} (40)where $$\psi>0$$. Step2. Consider the following Lyapunov candidate:   \begin{equation} L_{2} =\frac{1}{2} {e_{1}^{2}} \end{equation} (41)  \begin{align} \begin{aligned} \dot{L}_{2} &=e_{1} \dot{e}_{1} =-\beta_{1} e_{1}\, \textrm{sig}(e_{1} )+e_{1} e_{2} \\ & \le -\beta_{1} e_{1}\, \textrm{sig}(e_{1} )+\left|e_{1} \right|\psi. \end{aligned} \end{align} (42)Noting that $$\textrm{sig}(e_{1} )\in (-0.5a,0.5a)$$, if $$0.5a>\psi $$, b is large enough, we can select $$\beta _{1} $$ to make $$\left |\beta _{1}\, \textrm{sig}(e_{1} )\right |>\psi $$, there is   \begin{align} \dot{L}_{2} \le -\left|e_{1} \right|(\left|\beta_{1}\, \textrm{sig}(e_{1} )\right|-\psi )\le 0. \end{align} (43)From (43) we can know that $$\dot{L}_{2} $$ will be negative outside the set $$\{ |e_{1} |\le \textrm{sig}^{-1} ({\psi \mathord{ / {\vphantom{\psi \beta _{1} }} . } \beta _{1} } ) \}$$. Thus, $$\left |e_{1} \right |$$ is uniformly ultimately bounded. Therefore, the trajectories of the error state system are convergent, demonstrating the consistently bounded stability of $$\boldsymbol{\zeta } $$. This means that if observer parameters are chosen properly, the error system converges to the small region of zero and the observer outputs meet in a residual region of actual states $$x_{1}$$ and extended state $$x_{2} $$, respectively. This completes the proof of Theorem 1. Remark 2 Looking at (40) and (43), it is clear that estimation errors $$e_{1}$$ and $$e_{2}$$ are determined by parameters $$\beta _{1},\beta _{2},a$$ and b. In tuning these parameters appropriately, estimation errors can be minimized to a level enabling extended state $$x_{2}$$ to be observed effectively by the SESO. The fundamental selection of the parameters can be chosen as $$\beta _{1} < \sqrt{8\sqrt{2} +12}, \beta _{2}>0, \ a>2\psi $$, $$b={\beta _{2} \mathord{\left / {\vphantom{\beta _{2} \varepsilon }} \right . } \varepsilon }>0$$. An appropriately large $$\beta _{2} $$ can also be selected such that $$\left \| \boldsymbol{\zeta } \right \| $$ is small enough, although $$\varpi $$ is unknown. A relatively larger a can be chosen to make $$\left |e_{1} \right |$$ as small as required. Remark 3 In order to apply this general method to SESO design, the derivative of perturbation w should exist and be bounded. This will satisfy Assumption 2. In practical systems this ESO cannot be used to observe the non-differentiable disturbances. The advantage to this however is that there is no need to know the disturbance boundary as is necessary with the sliding mode observer in the study by Yan & Edwards (2007). At the end of this subsection, a comparative simulation example is conducted to demonstrate the superiority of the proposed SESO in observing performance. ESO1 represents the ESO design based on the STA convergent system (Moreno & Osorio, 2015) and ESO2 denotes the proposed ESO in this paper. In the simulation let disturbance $$w=\textrm{sign}(\sin (t))$$ which is a harsh simulation condition for the observers. The two ESOs are designed as follows: ESO 1:  \begin{align} \begin{cases} {\dot{\hat{x}}_{1} =f(x_{1} )+g(x_{1} )u+\hat{x}_{2} -d_{1} \left|x_{1} -\hat{x}_{1} \right|{}^{1/2} \textrm{sign}(x_{1} -\hat{x}_{1} )} \\ {\dot{\hat{x}}_{2} =-d_{2}\, \textrm{sign}(x_{1} -\hat{x}_{1} )}, \end{cases} \end{align} (44)where observer gains are selected as $$d_{1} =5,d_{2} =20$$. ESO 2:  \begin{align} \begin{cases} {\dot{\hat{x}}_{1} =f(x_{1} )+g(x_{1} )u+\hat{x}_{2} -\beta_{1} \textrm{sig}(x_{1} -\hat{x}_{1} )} \\{\dot{\hat{x}}_{2} =-\beta_{2}\, \textrm{sig}(x_{1} -\hat{x}_{1} )}, \end{cases} \end{align} (45)where observer parameters are selected as $$\beta _{1} =5,\beta _{2} =20$$, a = 2, b = 20. The same observer gains are selected in the simulation. Estimation performances are diverse because the convergent systems are constructed using different h(•) functions. Figure 2 illustrates the ESO 2 displays superior observing performance without overshooting and chattering even in the presence of square wave disturbance which has instantaneous transitions. Therefore, SESO is more suitable for the application of practical controller. Fig. 2. View largeDownload slide Observer results of ESO1 and ESO2 Fig. 2. View largeDownload slide Observer results of ESO1 and ESO2 4. Controller design based on SESO The objective of the control design is to force the output $$\boldsymbol{\varOmega }$$ to track the guidance command $$\boldsymbol{\varOmega }_{d}$$ by specifying the fin deflections $$\delta _{e}$$, $$\delta _{a}$$ and $$\delta _{r}$$, for control-oriented models (16) and (21). The proposed SESO will be applied to reconstruct the total disturbance in the design. First-order filter and auxiliary variables will be integrated into the back-stepping design to derive the real control input and reduce saturation effects, respectively. The procedure is presented in the following subsections. Before designing the controller, the following assumption must be raised: Assumption 2 The gain matrices $$\boldsymbol{g}_{1}$$ and $$\boldsymbol{g}_{20}$$ are invertible. This means $$\textrm{det}(\boldsymbol{g}_{1})=-1/\cos \beta $$. The singularity of $$\boldsymbol{g}_{1}^{-1}$$ occurs if the sideslip angle $$\beta =\pm 90^{\circ }$$. This situation will not occur as the controller attempts to keep $$\beta =0^{\circ }$$. The nominal gain matrix can be selected to make $$\boldsymbol{g}_{20}$$ invertible so this assumption is reasonable. 4.1 Structure of the robust back-stepping control scheme The structure of the proposed control scheme for HRV is shown in Fig. 3 and is based on back-stepping technology. The SESOs are adopted to compensate total disturbances existing in the two subsystems. Once the disturbances $$\boldsymbol{\varphi }_{1}$$ and $$\boldsymbol{\varphi }_{2}$$ existing in (16) and (21) are estimated, the actual control input can be modified based on $$\hat{\boldsymbol{\varphi }}_{i},\ i=1,2$$. In addition, h(v) is introduced as the input of the actuator to deal with saturation. In this way the control inputs will be limited and will not conflict with the admissible maximum value. To reduce saturation effects an auxiliary system driven by $$\varDelta \boldsymbol{v}$$ is applied to motivate the control input to decrease appropriately when saturation occurs. These are the key features of the proposed control method. Fig. 3. View largeDownload slide The structure of the SESO based back-stepping control for HRV. Fig. 3. View largeDownload slide The structure of the SESO based back-stepping control for HRV. 4.2 Back-stepping procedure Back-stepping control is proposed for the HRV non-linear system in this subsection. The recursive design procedure contains two steps. First, virtual control law is designed to guarantee the stability of the angular subsystem (16). In step 2, an overall control law v is constructed for the angular rate subsystem (21). Step 1. The following error state can be defined as   \begin{align} \boldsymbol{z}_{1} =\boldsymbol{\varOmega}-\boldsymbol{\varOmega}_{d}. \end{align} (46)Considering (16) and taking the derivation of (46),   \begin{align} \dot{\boldsymbol{z}}_{1} =\boldsymbol{f}_{1} +\boldsymbol{g}_{1} \boldsymbol{\omega}+\boldsymbol{\varphi }_{1} -\dot{\boldsymbol{\varOmega }}_{d} \end{align} (47)the virtual control law is then designed as follows:   \begin{align} \boldsymbol{\omega}_{c} =-\boldsymbol{g}_{1}^{-1} (\boldsymbol{k}_{1} \boldsymbol{z}_{1} +\boldsymbol{f}_{1} -\dot{\boldsymbol{\varOmega }}_{d} +\hat{\boldsymbol{\varphi }}_{1} ), \end{align} (48)where $$\boldsymbol{k}_{1}=\textrm{diag}[k_{11},k_{12},k_{13}]$$, $$k_{1i}>0, \ i=1,2,3$$ are parameters to be designed and $$\hat{\boldsymbol{\varphi }}_{1}$$ is the estimation of $$\boldsymbol{\varphi }_{1}$$ to be obtained by the SESO proposed later. A new state variable $$\boldsymbol{\omega }_{d}$$ is introduced here so $$\dot{\boldsymbol{\omega }}_{c}$$ can be analytically expressed as $$\dot{\boldsymbol{\omega }}_{d}$$. This can be obtained by the following first-order filter   \begin{align} \dot{\boldsymbol{\omega }}_{d} =(\boldsymbol{\omega }_{c} -\boldsymbol{\omega }_{d} -\tau \boldsymbol{g}_{1} \boldsymbol{z}_{1} ){/}\tau ,\quad \boldsymbol{\omega }_{d} \left(0\right)=\boldsymbol{\omega }_{c} \left(0\right), \end{align} (49)where $$\tau $$ is the positive filter parameter to be designed. The following auxiliary variables are defined:   \begin{align} \boldsymbol{\gamma }=\boldsymbol{\omega} -\boldsymbol{\omega }_{d} \end{align} (50)  \begin{align} \boldsymbol{\xi} =\boldsymbol{\omega }_{c} -\boldsymbol{\omega }_{d}, \end{align} (51)where $$\boldsymbol{\gamma }$$ is a new error state and $$\boldsymbol{\xi }$$ is called ‘filter error’. Next is $$\boldsymbol{\omega }=\boldsymbol{\gamma }+\boldsymbol{\omega }_{c}-\boldsymbol{\xi }$$, thus the error dynamics $$\dot{\boldsymbol{z}}_{1}$$ can be calculated as   \begin{align} \begin{aligned} \dot{\boldsymbol{z}}_{1} &=\boldsymbol{f}_{1} +\boldsymbol{g}_{1} \left(\boldsymbol{\gamma }+\boldsymbol{\omega }_{c} -\boldsymbol{\xi }\right)+\boldsymbol{\varphi }_{1} -\dot{\boldsymbol{\varOmega }}_{d} \\ &=\boldsymbol{f}_{1} +\boldsymbol{g}_{1} \boldsymbol{\gamma}-\boldsymbol{g}_{1} \boldsymbol{\xi}-\dot{\boldsymbol{\varOmega}}_{d} +\boldsymbol{g}_{1} \boldsymbol{\omega }_{c} +\boldsymbol{\varphi }_{1} \\ & =\boldsymbol{A}_{1} +\boldsymbol{B}_{1} \boldsymbol{\omega }_{c} +\boldsymbol{\varphi }_{1}, \end{aligned} \end{align} (52)where $$\boldsymbol{A}_{1}=\boldsymbol{f}_{1}+\boldsymbol{g}_{1}\boldsymbol{\gamma }-\boldsymbol{g}_{1}\boldsymbol{\xi }-\dot{\boldsymbol{\varOmega }}_{d}$$, $$\boldsymbol{B}_{1}=\boldsymbol{g}_{1}$$. Based on the ESO design principle, the extended state variable can be introduced as $$\boldsymbol{z}_{2}=\boldsymbol{\varphi }_{1}$$, which is defined with $$\dot{\boldsymbol{z}}_{2}=\boldsymbol{p}_{1}(t)$$. The following assumption is given: Assumption 3 It is assumed $$\boldsymbol{P}_{1}(t)$$ is unknown but bounded, that is to say $$p_{1i}(t)\leq \varpi _{1}, i=1,2,3$$ where $$\varpi _{1}>0$$. Following this, the system (52) can be extended as   \begin{align} \begin{cases} {\dot{\boldsymbol{z}}_{1} =\boldsymbol{A}_{1} +\boldsymbol{B}_{1} \boldsymbol{ \omega }_{c} +\boldsymbol{z}_{2} } \\{\dot{\boldsymbol{z}}_{2} =\boldsymbol{p}_{1} \left(t\right)}. \end{cases} \end{align} (53)The SESO can then be described as follows:   \begin{equation} \left\{\begin{array}{l} \boldsymbol{E}_{11} =\boldsymbol{Z}_{11} -\boldsymbol{z}_{1} \\ \dot{\boldsymbol{Z}}_{1} =\boldsymbol{A}_{1} +\boldsymbol{B}_{1} \boldsymbol{\omega }_{c} +\boldsymbol{ Z}_{2} -\boldsymbol{\beta }_{z1} \mathbf{sig}\left(\boldsymbol{E}_{11} \right) \\ \dot{\boldsymbol{ Z}}_{2} =-\boldsymbol{\beta }_{z2} \mathbf{sig}\left(\boldsymbol{E}_{11} \right), \end{array}\right. \end{equation} (54)where $$\boldsymbol{Z}_{1}$$ and $$\boldsymbol{Z}_{2}$$ denote the output of the observer which will approach $$\boldsymbol{z}_{1}$$ and extended state $$\boldsymbol{z}_{2}$$, respectively. The estimation error is $$\boldsymbol{E}_{11}$$, $$\boldsymbol{\beta }_{zi}=\textrm{diag}(\beta _{zi1},\beta _{zi2},\beta _{zi3})$$ is the observer gain with $$\beta _{zij}>0$$, i = 1, 2, j = 1, 2, 3, which will be defined according to the desired estimation precision. In order to examine stability error dynamics must be developed. Combining (53) and (54) gives   \begin{align} \begin{cases} {\dot{\boldsymbol{E}}_{11} =-\boldsymbol{\beta }_{z1} \mathbf{sig}\left(\boldsymbol{ E}_{11} \right)+\boldsymbol{ E}_{12} } \\{\dot{\boldsymbol{E}}_{12} =-\boldsymbol{\beta }_{z2} \mathbf{sig}\left(\boldsymbol{E}_{11} \right)-\boldsymbol{p}_{1} \left(t\right)}, \end{cases} \end{align} (55)where $$\boldsymbol{E}_{12}=\boldsymbol{Z}_{2},\boldsymbol{z}_{2}$$ is the estimation error. Next, substituting (48) into (52) yields   \begin{align} \dot{\boldsymbol{z}}_{1} =-\boldsymbol{k}_{1} \boldsymbol{z}_{1} +\boldsymbol{g}_{1} \boldsymbol{\gamma }-\boldsymbol{g}_{1} \boldsymbol{\xi }+\boldsymbol{E}_{12}. \end{align} (56)Considering the ‘filter error’ $$\boldsymbol{\xi }$$, the following Lyapunov function candidate is written as   \begin{align} V_{1} =\frac{1}{2} \boldsymbol{z}_{1}^{{\textrm T}} \boldsymbol{z}_{1} +\frac{1}{2} \boldsymbol{\xi }^{{\textrm T}} \boldsymbol{\xi }. \end{align} (57)This definition of $$\boldsymbol{\xi }$$ gives the equation $$\dot{\boldsymbol{\zeta }}=-\boldsymbol{\xi }/\tau +\boldsymbol{g}_{1}\boldsymbol{z}_{1}+\dot{\boldsymbol{\omega }}_{c}$$. The time derivation of $$V_{1}$$ is given by   \begin{equation} \dot{V}_{1} =-\boldsymbol{z}_{1}^{\textrm{T}} \boldsymbol{k}_{1} \boldsymbol{z}_{1} +\boldsymbol{z}_{1}^{\textrm{T}} \boldsymbol{g}_{1} \boldsymbol{\gamma}-\boldsymbol{z}_{1}^{\textrm{T}} \boldsymbol{g}_{1} \boldsymbol{\xi}+\boldsymbol{z}_{1}^{\textrm{T}} \boldsymbol{E}_{12} -\left\| \boldsymbol{\xi}\right\| ^{2} {/} \tau +\boldsymbol{\xi }^{\textrm{T}} \boldsymbol{g}_{1} \boldsymbol{z}_{1} +\boldsymbol{\xi}^{\textrm{T}} \dot{\boldsymbol{\omega}}_{c} \end{equation} (58)considering the following assumption: Assumption 4 The derivation of virtual control is bounded, $$\Vert \dot{\boldsymbol{\omega }}_{c}\Vert \leq \vartheta $$, where $$\vartheta $$ is a positive constant, and if $$\frac{1-\tau }{\tau }\geq \tau _{0}>0$$, we can have   \begin{align} \begin{aligned} -\frac{1}{\tau } \left\| \boldsymbol{\xi }\right\| ^{2} +\boldsymbol{\xi }^{{\textrm{T}}} \vartheta &\le -\left(\frac{1-\tau }{\tau } \right)\left\| \boldsymbol{ \xi }\right\| ^{2} -\left(\left\| \boldsymbol{ \xi }\right\| -\frac{\vartheta }{2} \right)^{2} +\frac{\vartheta ^{2} }{4} \\ &\le -\tau _{0} \left\| \boldsymbol{ \xi }\right\| ^{2} +\frac{\vartheta ^{2} }{4} \end{aligned} \end{align} (59)the derivative of $$V_{1}$$ is then calculated as   \begin{align} \dot{V}_{1} \le -\boldsymbol{ z}_{1}^{\textrm{T}} \boldsymbol{ k}_{1} \boldsymbol{ z}_{1} -\tau _{0} \left\| \boldsymbol{ \xi }\right\| ^{2} +\boldsymbol{ z}_{1}^{\textrm{T}} \boldsymbol{ g}_{1} \boldsymbol{ \gamma }+\vartheta ^{2} / 4 +\boldsymbol{ z}_{1}^{\textrm{T}} \boldsymbol{ E}_{12}, \end{align} (60)the term $$\boldsymbol{z}_{1}^{\textrm{T}}\boldsymbol{g}_{1}\boldsymbol{\gamma }$$ in (60) will be cancelled during the next control design procedure. The other terms will be considered in the stability analysis of the closed-loop system. Step 2. The following auxiliary system can be constructed to compensate the constraint effect caused by saturation   \begin{align} \dot{\boldsymbol{ \lambda }}=-k_{\lambda } \boldsymbol{ \lambda }+\boldsymbol{ g}_{20} \boldsymbol{ h}\left(\boldsymbol{ v}\right)-\boldsymbol{ g}_{20} \boldsymbol{ v}, \end{align} (61)where $$k_{\lambda }>0$$. The new error state vector is defined as   \begin{align} \boldsymbol{ s}_{1} =\boldsymbol{ \gamma }-\boldsymbol{ \lambda }. \end{align} (62)Taking the derivative of $$\boldsymbol{s}_{1}$$ with respect to time provides   \begin{align} \begin{aligned} \dot{\boldsymbol{ s}}_{1} &=\boldsymbol{ f}_{2} +\boldsymbol{ g}_{20} \boldsymbol{ h}\left(\boldsymbol{ v}\right)+\boldsymbol{ g}_{20}\, \boldsymbol{ j}\left(\boldsymbol{ v}\right)+\boldsymbol{ \varphi }_{2} -\dot{\boldsymbol{ \omega }}_{d} -\dot{\boldsymbol{ \lambda }} \\ & =\boldsymbol{ f}_{2} +\boldsymbol{ g}_{20} \boldsymbol{ j}\left(\boldsymbol{ v}\right)-\dot{\boldsymbol{ \omega }}_{d} +k_{\lambda } \boldsymbol{ \lambda }+\boldsymbol{ g}_{20} \boldsymbol{ v}+\boldsymbol{ \varphi }_{2} \\ & =\boldsymbol{ A}_{2} +\boldsymbol{ B}_{2} \boldsymbol{ v}+\boldsymbol{ \varphi }_{2}, \end{aligned} \end{align} (63)where $$\boldsymbol{A}_{2}=\boldsymbol{f}_{2}+\boldsymbol{g}_{20}\,\boldsymbol{j}(\boldsymbol{v})-\dot{\boldsymbol{\omega }}_{d}+k_{\lambda }\boldsymbol{\lambda }$$, $$\boldsymbol{B}_{2}=\boldsymbol{g}_{20}$$, then the actual control input is proposed as   \begin{align} \begin{aligned} \boldsymbol{ v}&=\boldsymbol{ B}_{2} ^{-1} \left(-\boldsymbol{ k}_{2} \boldsymbol{ s}_{1} -\boldsymbol{ A}_{2} -\hat{\boldsymbol{ \varphi }}_{2} +\boldsymbol{ v}_{a} \right) \\ \boldsymbol{ v}_{a} &=\boldsymbol{ z}_{a} -\frac{\boldsymbol{ s}_{1} h_{a} }{\eta ^{2} +\left\| \boldsymbol{ s}_{1} \right\| ^{2} } \\ \dot{\boldsymbol{ z}}_{a} &=-k_{\boldsymbol{ z}_{a} } \boldsymbol{ z}_{a} -\frac{\boldsymbol{ z}_{a} f_{a} }{\left\| \boldsymbol{ z}_{a} \right\| ^{2} } \\ \dot{\eta }&=-k_{\eta } \eta -\frac{\eta h_{a} }{\eta ^{2} +\left\| \boldsymbol{ s}_{1} \right\| ^{2} }, \end{aligned} \end{align} (64)where $$k_{z_{a}}>0$$, $$h_{a}=\Vert \boldsymbol{s}_{1}\Vert ^{2}/2$$, $$f_{a}=\boldsymbol{\lambda }^{\textrm{T}}\boldsymbol{g}_{1}\boldsymbol{z}_{1}-\Vert \boldsymbol{g}_{20}\varDelta \boldsymbol{v}\Vert ^{2}/4$$. $$\varDelta \boldsymbol{v}=\boldsymbol{h}(\boldsymbol{v})-\boldsymbol{v}$$ is the error state generated by desired controller output and sigmoid function output. Note that $$\varDelta \boldsymbol{v}$$ is smooth and it will shrink the control signals when the desired control inputs exceed the maximum authority. This means the saturation time could be shortened. The estimation of $$\boldsymbol{\varphi }_{2}$$ is $$\hat{\boldsymbol{\varphi }}_{2}$$, which will be obtained by the following SESO. Based on the principle of the SESO design, extended state variable $$\boldsymbol{s}_{2}=\boldsymbol{\varphi }_{2}$$ are now introduced which are defined with $$\dot{\boldsymbol{s}}_{2}=\boldsymbol{p}_{2}(t)$$. The following assumption is given: Assumption 5 It is assumed $$\boldsymbol{p}_{2}(t)$$ is unknown but bounded, that is to say $$p_{2i}(t)\leq \varpi _{2},i=1,2,3$$, where $$\varpi _{2}>0$$. The system (63) can then be extended as   \begin{align} \begin{cases} {\boldsymbol{ s}_{1} =\boldsymbol{ A}_{2} +\boldsymbol{ B}_{2} \boldsymbol{ v}+\boldsymbol{ s}_{2} } \\{\dot{\boldsymbol{ s}}_{2} =\boldsymbol{ p}_{2} \left(t\right)}, \end{cases} \end{align} (65)the SESO can be described as follows:   \begin{align} \begin{cases} {\boldsymbol{ E}_{21} =\boldsymbol{ S}_{1} -\boldsymbol{ s}_{1} } \\{\dot{\boldsymbol{ S}}_{1} =\boldsymbol{ A}_{2} +\boldsymbol{ B}_{2} \boldsymbol{ v}+\boldsymbol{ S}_{2} -\boldsymbol{ \beta }_{s1} \mathbf{sig}\left({\textbf{E}}_{21} \right)} \\{\dot{\boldsymbol{ S}}_{2} =-\boldsymbol{\beta }_{s2} \mathbf{sig}\left({\textbf{E}}_{21} \right)}, \end{cases} \end{align} (66)where $$\boldsymbol{S}_{1}$$ and $$\boldsymbol{S}_{2}$$ are the observer outputs. They will approach $$\boldsymbol{s}_{1}$$ and extended state $$\boldsymbol{s}_{2}$$, respectively. The estimation error is $$\boldsymbol{E}_{22}$$, $$\boldsymbol{\beta }_{si}=\textrm{diag}(\beta _{si1},\beta _{si2},\beta _{si3})$$ is the observer gain with $$\beta _{sij}>0$$, i = 1, 2, j = 1, 2, 3, which will be defined according to the desired estimation precision. Error dynamics must be developed to examine stability. Combining (65) and (66) provides   \begin{align} \begin{cases} {\dot{\boldsymbol{ E}}_{21} =-\boldsymbol{ \beta }_{s1} \mathbf{sig}\left(\boldsymbol{ E}_{21} \right)+\boldsymbol{ E}_{22} } \\{\dot{\boldsymbol{ E}}_{22} =-\boldsymbol{ \beta }_{s2} \mathbf{sig}\left(\boldsymbol{ E}_{21} \right)-\boldsymbol{ p}_{1} \left(t\right)}, \end{cases} \end{align} (67)where $$\boldsymbol{E}_{22}=\boldsymbol{S}_{2}-\boldsymbol{s}_{2}$$ is the estimation error state. So far the controller and SESO have been established successfully. Stability analysis will be given in the following section. Note that compared with the adaptive method to handle unknown total disturbance, tracking error states are not used for constructing the SESO. This means saturation will not become worse under the action of disturbance estimation. It can be concluded from this that a controller design based on SESO will be more robust against actuator saturation to a certain extent. Remark 1 In the controller design process of an angular rate subsystem, the inversion of $$\boldsymbol{g}_{2}$$ may not exist. The normal value $$\boldsymbol{g}_{20}$$ can therefore be selected to guarantee its non-singularity. In this way, control inputs can be kept smooth without singularity while aerodynamic data acquired by ground testing and computing loads can be reduced.The back-stepping controller based on SESO has been designed successfully. Tracking performance depends crucially on SESO estimation accuracy so observer gains parameters must be correctly selected before tuning the controller gains. 4.3 Stability analysis of closed-loop dynamics In this subsection, the stability of the closed-loop system can be established by the following theorem: Theorem 2 Consider the HRV attitude systems (16) and (21). By applying control laws (48) and (64) in combination with SESOs (54) and (66), the closed-loop system is guaranteed to be uniformly bounded stable and tracking errors will converge to a small set around zero. Proof. In order to examine the stability of the closed-loop system, firstly, consider the Lyapunov function candidate   \begin{align} V_{2} =\frac{1}{2} \boldsymbol{ s}_{1}^{{\textrm T}} \boldsymbol{ s}_{1} +\frac{1}{2} \boldsymbol{ \lambda }^{{\textrm T}} \boldsymbol{ \lambda }+\frac{1}{2} \boldsymbol{ z}_{a}^{{\textrm T}} \boldsymbol{ z}_{a} +\frac{1}{2} \eta ^{2}, \end{align} (68)its time derivation is   \begin{align} \dot{V}_{2} =\boldsymbol{ s}_{1}^{{\textrm T}} \dot{\boldsymbol{ s}}_{1}^{{\textrm T}} +\boldsymbol{ \lambda }^{{\textrm T}} \dot{\boldsymbol{ \lambda }}+\boldsymbol{ z}_{a}^{{\textrm T}} \dot{\boldsymbol{ z}}_{a} +\eta \dot{\eta }. \end{align} (69)Substituting control law (64) into (69) gives   \begin{align} \dot{V}_{2} =-\boldsymbol{ s}_{{1}}^{{\textrm T}} \boldsymbol{ k}_{2} \dot{\boldsymbol{ s}}_{{1}}^{{\textrm T}} -\boldsymbol{ s}_{{1}}^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{ z}_{1} +\boldsymbol{ s}_{{ 1}}^{{\textrm T}} \boldsymbol{ E}_{22} +W, \end{align} (70)where $$W=\boldsymbol{s}_{1}^{\textrm T}\boldsymbol{v}_{a}+\boldsymbol{z}_{z}^{\textrm T}\dot{\boldsymbol{z}}_{a}+\eta \dot{\eta }+\boldsymbol{\lambda }^{\textrm T}\dot{\boldsymbol{\lambda }}$$. Considering control law (64) yields   \begin{align} \begin{aligned} W&\le \left\| \boldsymbol{ z}_{a} \boldsymbol{ s}_{1} \right\| -k_{\boldsymbol{ z}_{a} } \left\| \boldsymbol{ z}_{a} \right\| ^{2} -\frac{\left\| s_{1} \right\| ^{2} }{2} -k_{\eta } \eta ^{2} -k_{\lambda } \left\| \boldsymbol{ \lambda }\right\| ^{2} +\left\| \boldsymbol{ \lambda }\right\| \left\| \boldsymbol{ g}_{20} {\varDelta} \boldsymbol{v}\right\| -\frac{1}{4} \left\| \boldsymbol{ g}_{20} \varDelta \boldsymbol{v}\right\|^{2} -\boldsymbol{\lambda }^{{\textrm T}} \boldsymbol{g}_{1} \boldsymbol{z}_{1} \\ &=-\left(k_{\boldsymbol{z}_{a} } \frac{1}{2} \right)\left\| \boldsymbol{z}_{a} \right\| ^{2} -\frac{1}{2} \left(\left\| \boldsymbol{z}_{a} \right\| -\left\| \boldsymbol{s}_{1} \right\| \right)^{2} k_{\eta } \eta ^{2} -\left(k_{\lambda } -1\right)\left\| \boldsymbol{\lambda }\right\| ^{2} -\left(\left\| \boldsymbol{\lambda }\right\| -\frac{1}{2} \left\| \boldsymbol{g}_{20} \varDelta \boldsymbol{v}\right\| \right)^{2} -\boldsymbol{\lambda }^{{\textrm T}} \boldsymbol{g}_{1} \boldsymbol{z}_{1} \\ &\le -\bar{k}_{\boldsymbol{ z}_{a} } \left\| \boldsymbol{z}_{a} \right\| ^{2} -k_{\eta } \eta ^{2} -\bar{k}_{\lambda } \left\| \boldsymbol{\lambda }\right\| ^{2} -\boldsymbol{\lambda }^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{z}_{1}, \end{aligned} \end{align} (71)where $$\bar{k}_{z_{a}}=k_{z_{a}}-\frac{1}{2}>0$$, $$\bar{k}_{\lambda }=k_{\lambda }-1>0$$. Then one has   \begin{equation} \dot{V}_{2} \le -\boldsymbol{s}_{{1}}^{{\textrm T}} \boldsymbol{ k}_{2} \dot{\boldsymbol{ s}}_{{1}}^{{\textrm T}} -\boldsymbol{ s}_{{1}}^{\boldsymbol{T}} \boldsymbol{ g}_{1} \boldsymbol{z}_{1} +\boldsymbol{ s}_{1}^{{\textrm T}} \boldsymbol{ E}_{22} -\bar{k}_{z_{a} } \left\| \boldsymbol{ z}_{a} \right\| ^{2} -k_{\eta } \eta ^{2} -\bar{k}_{\lambda } \left\| \boldsymbol{\lambda }\right\| ^{2} -\boldsymbol{\lambda }^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{z}_{1}. \end{equation} (72)Following this, the augmented Lyapunov function is selected   \begin{align} V=V_{1} +V_{2}. \end{align} (73)Its derivative with respect to time is   \begin{align} \begin{aligned} \dot{V}&\le -\boldsymbol{z}_{1}^{{\textrm T}} \boldsymbol{k}_{1} \boldsymbol{z}_{1} -\tau _{0} \left\| \boldsymbol{\xi }\right\| ^{2} +\boldsymbol{ z}_{1}^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{\gamma }_{1} +\vartheta ^{2} /4+\boldsymbol{ z}_{1}^{{\textrm T}} \boldsymbol{E}_{12}-\boldsymbol{ s}_{1}^{{\textrm T}} \boldsymbol{k}_{2} \dot{\boldsymbol{ s}}_{1}^{T} -\boldsymbol{ s}_{1}^{{\textrm T}} \boldsymbol{ g}_{1} \boldsymbol{z}_{1} -\bar{k}_{z_{a} } \left\| \boldsymbol{z}_{a} \right\| ^{2} \\ &\quad+\boldsymbol{s}_{1}^{{\textrm T}} \boldsymbol{ E}_{22} -k_{\eta } \eta ^{2} -\bar{k}_{\lambda } \left\| \boldsymbol{\lambda }\right\| ^{2} -\boldsymbol{ \lambda }^{{\textrm T}} \boldsymbol{g}_{1} \boldsymbol{ z}_{1}. \end{aligned} \end{align} (74)Theorem 1 shows that observer errors converge into residual set of zero. This means $$\Vert \boldsymbol{E}_{12}\Vert <\zeta _{1}$$ and $$\Vert \boldsymbol{E}_{22}\Vert <\zeta _{2}$$ can be satisfied, where $$\psi _{1}$$ and $$\psi _{2}$$ are positive but unknown constants. This is seen as   \begin{equation} \dot{V}\le -k_{1\min } \left\| \boldsymbol{z}_{1} \right\| ^{2} -k_{2\min } \left\| \boldsymbol{s}_{1} \right\| ^{2} +\varsigma _{1} \left\| \boldsymbol{z}_{1} \right\| +\varsigma _{2} \left\| \boldsymbol{ s}_{1} \right\| -\tau _{0} \left\| \boldsymbol{\xi }\right\| ^{2} -\bar{k}_{\boldsymbol{z}_{a} } \left\| \boldsymbol{z}_{a} \right\| ^{2} -k_{\eta } \eta ^{2} -\bar{k}_{\lambda } \left\| \boldsymbol{\lambda }\right\| ^{2} +\vartheta ^{2} /4,\quad \end{equation} (75)where $$k_{1\min }=\min \{k_{1i}\}$$, $$k_{2\min }=\{k_{2i}\}$$. Consider the following two inequalities   \begin{align} \begin{aligned} -k_{1\min } \left\| \boldsymbol{ z}_{1} \right\| ^{2} +\varsigma _{1} \left\| \boldsymbol{ z}_{1} \right\| &= -\left(k_{1\min } -1\right)\left\| \boldsymbol{z}_{1} \right\| ^{2} -\left(\left\| \boldsymbol{ z}_{1} \right\| -\frac{\varsigma _{1} }{2} \right)^{2} +\frac{\varsigma _{1} ^{2} }{4} \le -c_{1} \left\| \boldsymbol{ z}_{1} \right\| ^{2} +\frac{\varsigma _{1} ^{2} }{4} \\ -k_{2\min } \left\| \boldsymbol{ s}_{1} \right\| ^{2} +\varsigma _{1} \left\| \boldsymbol{ s}_{1} \right\| &= -\left(k_{2\min } -1\right)\left\| \boldsymbol{s}_{1} \right\| ^{2} -\left(\left\| \boldsymbol{ s}_{1} \right\| -\frac{\varsigma _{2} }{2} \right)^{2} +\frac{\varsigma _{2} ^{2} }{4} \le -c_{2} \left\| \boldsymbol{s}_{1} \right\| ^{2} +\frac{\varsigma _{2} ^{2} }{4}, \end{aligned} \end{align} (76)where $$c_{1}=k_{1\min }-1>0, c_{2}=k_{2\min }-1>0$$. Then we have   \begin{align} \begin{aligned} \dot{V}&\le 2c\frac{1}{2} \left\| \boldsymbol{z}_{1} \right\| ^{2} -2c_{2} \frac{1}{2} \left\| \boldsymbol{s}_{1} \right\| ^{2} -2\tau _{0} \frac{1}{2} \| \boldsymbol{\xi }_{1} \| ^{2} -2\overline{k}_{\boldsymbol{ z}_{a} } \frac{1}{2} \left\| \boldsymbol{ z}_{a} \right\| ^{2} \\ &\quad-2k_{\eta } \frac{1}{2} \eta ^{2} -2\overline{k}_{\lambda } \frac{1}{2} \left\| \boldsymbol{ \lambda }\right\| ^{2} +\frac{\varsigma _{1}^{2} }{4} +\frac{\varsigma _{2}^{2} }{4} +\frac{\vartheta ^{2} }{4} \\ &\le -\boldsymbol{ \xi }V+\vartheta, \end{aligned} \end{align} (77)where   \begin{align*} \xi &=\min \left\{2_{c_{1} } ,2_{c_{2} } ,2\tau _{0} ,2\overline{k}_{\boldsymbol{z}_{a} } ,2k_{\eta } ,2\overline{k}_{\lambda } \right\} \\ \vartheta &=\frac{\varsigma _{1}^{2} }{4} +\frac{\varsigma _{2}^{2} }{4} +\frac{\vartheta ^{2} }{4}. \end{align*}Integration of (77) yields   \begin{align} 0\le V\left(t\right)\le \frac{\vartheta }{\xi } +\left(V\left(0\right)-\frac{\vartheta }{\xi } \right)\exp (-\xi t)\; \; \quad\forall\, t>0. \end{align} (78) It follows that the Lyapunov function candidate V is uniformly ultimately bounded. From (57) and (73) we have $$\Vert \boldsymbol{z}_{1}\Vert ^{2}\leq 2V(t)$$, it is obtained that the tracking error state $$\boldsymbol{z}_{1}$$ asymptotically converges to a compact set $$\varTheta $$ defined by   \begin{align} {\varTheta }:=\left\{\boldsymbol{ z}_{1} \bigg|\left\| \boldsymbol{ z}_{1} \right\| \le \sqrt{2\left(V(0)+\frac{\vartheta }{\xi } \right)} \right\}. \end{align} (79)This completes the proof of Theorem 2. A composite controller has been designed successfully based on back-stepping method and disturbance observer technique. The main difference between the approach presented and the conventional back-stepping method is that disturbance estimation is integrated to the virtual control law and employed to handle mismatched disturbances. Input constraints are handled by combining approximation function and auxiliary variables. 5. Simulation and analysis Several simulations are conducted in this section to demonstrate the performance of the proposed SESO-based back-stepping controller. Consider HRV system models (2) and (3) with the following initial parameters: The guidance command signals are given as   \begin{align*} \boldsymbol{\varOmega}_{d}&=\left[\begin{array}{@{}ccc@{}}\alpha_{d},&\beta_{d},&\mu_{d}\end{array}\right]^{\textrm T}\\ & = \left[\begin{array}{@{}ccc@{}}\textrm{sign}(\textrm{sin}(0.11t))+5,&0,&5\textrm{sin}(0.5t)+5\textrm{sin}(0.15t)\end{array}\right]^{\textrm T}\deg \end{align*}In addition, external disturbances and uncertainties are added to the system. These are described as   \begin{align*} \left[\begin{array}{@{}c@{}}d_{\alpha}\\ d_{\beta}\\ d_{\mu}\end{array}\right]&=\left[\begin{array}{@{}c@{}}\textrm{sin}(0.1t)+\textrm{sin}(0.2t)\\ \textrm{sin}(0.1t)+\textrm{sin}(0.2t)\\ \textrm{sin}(0.1t)+\textrm{sin}(0.2t)\end{array}\right]\times 0.01\textrm{rad}/s\\ \left[\begin{array}{@{}c@{}}d_{l}\\ d_{m}\\ d_{n}\end{array}\right]&= \left[\begin{array}{@{}ccc@{}}I_{x}& &\\ & I_{y}&\\ &&I_{z}\end{array}\right] \left[\begin{array}{@{}c@{}}\textrm{sin}(0.1t)+\textrm{sin}(0.2t)\\ 0.1\textrm{sin}(0.1t)+0.1\textrm{sin}(0.2t)\\ \textrm{sin}(0.1)+\textrm{sin}(0.2t)\end{array}\right]\textrm{N}\cdot\textrm{m}\\ \left[\begin{array}{@{}c@{}}\varDelta C_{l}\\ \varDelta C_{m}\\ \varDelta C_{n}\end{array}\right]&=\left[\begin{array}{@{}c@{}}-0.3C_{l}+0.001\\ -0.3C_{m}+0.001\\ -0.3C_{n}+0.001\end{array}\right]. \end{align*} Here, coefficients of aerodynamic moments are reduced by approximately 30%. Moment disturbances and model uncertainties are also considered. These multiple uncertainties and disturbances are additional aggressive scenarios for the simulations. For the purpose of comparison, three cases analyses are presented. System initial states, uncertain aerodynamic parameters, external disturbances and input constraints are the same for the following numerical simulations: Case 1. The conventional back-stepping controller is applied in this case. It represents the controller without auxiliary system (61), approximate function (19) and SESOs (54) and (66), and is designed as follows:   \begin{align} \begin{aligned} \boldsymbol{\omega}_{c,\textrm{tra}} &=\boldsymbol{g}_{1}^{-1} (-\boldsymbol{k}_{1} \boldsymbol{z}_{1}-\boldsymbol{f}_{1} +\dot{\boldsymbol{\varOmega}}_{d} ) \\ \boldsymbol{v}_{\textrm{tra}} &=\boldsymbol{g}_{20}^{-1} (-\boldsymbol{k}_{2} \boldsymbol{\gamma}-\boldsymbol{f}_{2} +\dot{\boldsymbol{\omega}}_{d} ). \end{aligned} \end{align} (80) Case 2. The SESO-based back-stepping (SESOB) strategy proposed in this paper is applied here. Case 3. The adaptive back-stepping controller designed in the study by Wang et al., (2015) is used here to demonstrate the advantages of the SESOB in dealing with saturation issues. Only the attitude angles and input responses are given in this case due to page limitation. Control parameters selected can be found in Table 1. Table 1. Initial values for HRV Notation  Value  Unit  $$\boldsymbol{\varOmega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}}$$  $$\deg $$  $$\boldsymbol{\omega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}} $$  $$\deg /s$$  $$\boldsymbol{I}_{i} (i=x,y,z)$$  $$\left [\begin{matrix} {9.18}\ \, {90.3}\ \,{90.3} \end{matrix}\right ]\times e^{5} $$  $$kg\cdot m^{2} $$  $$\delta _{e},\delta _{a},\delta _{r} $$  [−30, 30]  $$\deg $$  Notation  Value  Unit  $$\boldsymbol{\varOmega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}}$$  $$\deg $$  $$\boldsymbol{\omega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}} $$  $$\deg /s$$  $$\boldsymbol{I}_{i} (i=x,y,z)$$  $$\left [\begin{matrix} {9.18}\ \, {90.3}\ \,{90.3} \end{matrix}\right ]\times e^{5} $$  $$kg\cdot m^{2} $$  $$\delta _{e},\delta _{a},\delta _{r} $$  [−30, 30]  $$\deg $$  View Large Table 1. Initial values for HRV Notation  Value  Unit  $$\boldsymbol{\varOmega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}}$$  $$\deg $$  $$\boldsymbol{\omega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}} $$  $$\deg /s$$  $$\boldsymbol{I}_{i} (i=x,y,z)$$  $$\left [\begin{matrix} {9.18}\ \, {90.3}\ \,{90.3} \end{matrix}\right ]\times e^{5} $$  $$kg\cdot m^{2} $$  $$\delta _{e},\delta _{a},\delta _{r} $$  [−30, 30]  $$\deg $$  Notation  Value  Unit  $$\boldsymbol{\varOmega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}}$$  $$\deg $$  $$\boldsymbol{\omega } (0)$$  $$\left [\begin{matrix} {0}\ {0}\ {0} \end{matrix}\right ]^{\textrm{T}} $$  $$\deg /s$$  $$\boldsymbol{I}_{i} (i=x,y,z)$$  $$\left [\begin{matrix} {9.18}\ \, {90.3}\ \,{90.3} \end{matrix}\right ]\times e^{5} $$  $$kg\cdot m^{2} $$  $$\delta _{e},\delta _{a},\delta _{r} $$  [−30, 30]  $$\deg $$  View Large Table 2. Controller parameters Parameter  Value  Parameter  Value  $$\boldsymbol{k}_{1} $$  diag(3, 3, 3)  $$\boldsymbol{k}_{2} $$  diag(6, 6, 6)  $$\boldsymbol{\beta }_{s1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{s2} $$  diag(25, 25, 25)  $$\boldsymbol{\beta }_{f1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{f2} $$  diag(22, 22, 22)  a  2  b  35  Parameter  Value  Parameter  Value  $$\boldsymbol{k}_{1} $$  diag(3, 3, 3)  $$\boldsymbol{k}_{2} $$  diag(6, 6, 6)  $$\boldsymbol{\beta }_{s1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{s2} $$  diag(25, 25, 25)  $$\boldsymbol{\beta }_{f1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{f2} $$  diag(22, 22, 22)  a  2  b  35  View Large Table 2. Controller parameters Parameter  Value  Parameter  Value  $$\boldsymbol{k}_{1} $$  diag(3, 3, 3)  $$\boldsymbol{k}_{2} $$  diag(6, 6, 6)  $$\boldsymbol{\beta }_{s1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{s2} $$  diag(25, 25, 25)  $$\boldsymbol{\beta }_{f1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{f2} $$  diag(22, 22, 22)  a  2  b  35  Parameter  Value  Parameter  Value  $$\boldsymbol{k}_{1} $$  diag(3, 3, 3)  $$\boldsymbol{k}_{2} $$  diag(6, 6, 6)  $$\boldsymbol{\beta }_{s1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{s2} $$  diag(25, 25, 25)  $$\boldsymbol{\beta }_{f1} $$  diag(2, 2, 2)  $$\boldsymbol{\beta }_{f2} $$  diag(22, 22, 22)  a  2  b  35  View Large Simulation results of Case 1 are depicted in Fig. 4. It can be seen that the AOA, sideslip angle and bank angle track their respective guidance commands with increased tracking errors although the closed-system tends to be stable. Satisfactory tracking performances cannot be obtained without the compensation of SESOs. As seen in Fig. 4(c), the control signals violate limitations as sharp corners have a negative effect on the actuators. Fig. 4. View largeDownload slide Response curves of Case 1. Fig. 4. View largeDownload slide Response curves of Case 1. View largeDownload slide View largeDownload slide Tracking results of Case 2 are depicted in Fig. 5. It is clear here that better altitude tracking specification is achieved with smaller tracking errors. This is because disturbances are estimated and compensated directly in the controller via SESOs. As illustrated in Fig. 5(c), control signals will increase to some extent but will never violate limitations based on the property of sigmoid function. This means saturated inputs will not occur and inputs are kept smooth during saturation periods. These are the features of the proposed anti-windup method. Compared with the results of Case 1, tracking performance is improved significantly. View largeDownload slide View largeDownload slide Fig. 5. View largeDownload slide Response curves of Case 2. Fig. 5. View largeDownload slide Response curves of Case 2. Performance estimations under Case 2 can be seen in Fig. 5(d) and (e). It is verified that the SESO can reconstruct the total disturbance efficiently and smoothly. Fig. 6. View largeDownload slide Response curves of Case 3. Fig. 6. View largeDownload slide Response curves of Case 3. Simulation results of Case 3 are depicted in Fig. 6. Although the adaptive back-stepping controller shows superior control precision compared with the traditional back-stepping controller in Case 1, the tracking performance is not comparable to that of the SESOB in Case 2. An increased saturation occurrence is illustrated by the input response in Fig. 6(b). Once the inputs violate the limitation, the adaptive law will be motivated by tracking errors to generate larger control signals to track the reference. This makes the controller aggressive and leads to longer saturation times and more of them. This simulation verifies previous results. In terms of input constraint and multiple uncertainties, the proposed control strategy based on SESO can achieve enhanced control performance in comparison with the adaptive method. Table 3. Tracking errors in different cases of controllers   Case 1  Case 2  Case 3  $$z_{11} (\deg )$$  6  0.5  1  $$z_{12} (\deg )$$  0.7  0.1  0.2  $$z_{13} (\deg )$$  2  0.1  1    Case 1  Case 2  Case 3  $$z_{11} (\deg )$$  6  0.5  1  $$z_{12} (\deg )$$  0.7  0.1  0.2  $$z_{13} (\deg )$$  2  0.1  1  Table 3. Tracking errors in different cases of controllers   Case 1  Case 2  Case 3  $$z_{11} (\deg )$$  6  0.5  1  $$z_{12} (\deg )$$  0.7  0.1  0.2  $$z_{13} (\deg )$$  2  0.1  1    Case 1  Case 2  Case 3  $$z_{11} (\deg )$$  6  0.5  1  $$z_{12} (\deg )$$  0.7  0.1  0.2  $$z_{13} (\deg )$$  2  0.1  1  The tracking errors in different cases of controllers can be found in Table 3. 6. Conclusion This paper has focused on the ESO-based back-stepping controller design for HRV. The new control-oriented model has been subject to multiple uncertainties, disturbances and actuator saturations. A novel SESO has been presented and combined successfully with the back-stepping method to develop a controller to cope with mismatched and matched uncertainties. The stability of this SESO has been proven through constructing a Lyapunov function. Two methods have been combined successfully to handle to effects of input constraint. First, Sigmoid function was adopted to approximate saturation so the limited inputs are guaranteed. The error state between limited control input and the desired control input was also applied to integrate the controller. This resulted in saturation time being reduced significantly. The proposed control method was compared with conventional back-stepping and adaptive back-stepping control from existing literature. Results demonstrate that the SESO-based strategy shows superior robustness against aerodynamic uncertainties, external disturbances and input constraints. The proposed composite control method combines SESO and back-stepping techniques, providing a general solution for high-order systems with mismatched disturbance and saturation problems. Further research will focus on the ESO-based control strategy for general high-order system in the presence of actuator faults such as loss of effectiveness. Acknowledgements The author CC would like to thank the China Scholarship Council (CSC) for the financial support during his visit at ETH (Swiss Federal Institute of Technology Zurich). The authors would like to thank Mina from ASL (Autonomous Systems Lab), ETH, Switzerland for his writing and proofreading assistance. Funding National Natural Science Foundation of China (61403103, 61673135, 61603114). References Bao, W. ( 2013) Present situation and development tendency of aerospace control techniques. Acta Automatica Sinica , 39, 697– 702. Google Scholar CrossRef Search ADS   Boškovic, J. D., Li, S.-M. & Mehra, R. K. ( 2001) Robust adaptive variable structure control of spacecraft under control input saturation. J. Guid. Control Dyn. , 24, 14– 22. Google Scholar CrossRef Search ADS   Chen, M., Ge, S. S. & Ren, B. ( 2011) Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints. Automatica , 47, 452– 465. 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Published: Apr 6, 2018

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