Robust adaptive constrained backstepping flight controller design for re-entry reusable launch vehicle under input constraint

Abstract

A nonlinear constrained controller is designed for a reusable launch vehicle during re-entry phase in the presence of model uncertainty, external disturbance, and input constraint, via combining sliding mode control and adaptive backstepping control. Since the complex coupling between the translational and rotational dynamics of reusable launch vehicle, a control-oriented model derived from rotational dynamic is used for controller design. During the virtual control input design procedure, a dynamic robust term is utilized to compensate for the uncertainty. In addition, a filter is applied to handle “explosion of terms” problem during the actual control input design. To reduce the computational burden, adaptive law is used to evaluate the unknown norm bound of the lumped uncertainty. An auxiliary system is constructed to compensate for the input constraint effect. The stability of the closed-loop system is analyzed based on Lyapunov theory. Simulation results demonstrate the validity of the developed controller in providing stable tracking of the guidance command by numerical simulation on the 6-degree-of-freedom model of reusable launch vehicle.

Introduction

Reusable launch vehicle (RLV) is designed to dramatically reduce the cost of accessing space by recovering and reusing after each mission. A major challenge posed in such mission is atmospheric re-entry. In this phase, a number of stringent constraints come into action that include constraints on heat flux, structural load, and so on, which restrict the vehicle to fly within a narrow flight corridor. ¹ Since advanced guidance and control technologies are critical for achieving safety, reliability, and cost requirements of RLV, these methods have got more and more attention over the past few years. The major mission of re-entry guidance is the on-board correction of the flight path to make up for model uncertainties and external disturbance. The impactful guidance law has been developed for calculating the referenced angle of attack (AOA) and bank angle. Following, the focus is to design an attitude control system to track the guidance command effectively, which is the key point of this article. Several approaches have been probed in the past, such as gain scheduling, ² dynamic inversion technique, ³ trajectory linearization control, ⁴ sliding mode control, ⁵ and state-dependent Riccati equation strategy. ⁶ All these methods provided good performance. But input constraint was not taken into account.

Control input saturation is often encountered in practical applications due to the fact that it is usually impossible to implement unlimited control signals. ⁷ Saturation is a potential problem for actuators of control systems. It is frequently one of the main sources of instability, degradation of system performance, and parasitic equilibrium points of a control system. ⁷ Physical input saturation on hardware dictates that the magnitude of the control signal is always constrained. ⁸ Moreover, control input saturation often severely limits system performance, giving rise to undesirable inaccuracy or leading instability. ⁹ In flight control system, the vehicle body may change seriously when input saturation occurs, and it may lead the vehicle to disintegrate. Thus, it is important to study the problem of flight control system design subjects to input constraint. In addition, the consideration of uncertainty and disturbance make the controller design more difficult.

Some methods are proposed to handle input constraint for flight control system. Anti-windup control was designed to handle input constraint of hypersonic vehicles (HSVs) while the uncertainty was not considered. ¹⁰ Model predictive control has been employed extensively because of its inherent capability to implement input constraint directly during the controller design procedure. ¹¹ However, it is dependent on the real-time receding horizon optimization and online optimization, and the determination of time-domain step size becomes the main barrier that restricts its application in HSV. ¹² H _∞ approach was proposed for a linearized flexible airbreathing hypersonic vehicle (FAHV) model, uncertain parameters, and input constraints, where the linearized model was obtained by the feedback linearization approach. ¹³ But the high-order derivatives of outputs need to be computed. Using the differential geometry principle and the total energy theory, advanced flight control laws were designed for HSV in the presence of actuator limitations. ¹⁴ Three adaptive fault control schemes were proposed for airbreathing hypersonic vehicle (AHV) with external disturbances, actuator faults, and input saturation. ¹⁵ For the latter two control approaches, it did not need to know the upper bound of the external disturbances and the real minimum value of actuator efficiency factor in advance. An adaptive backstepping attitude control scheme was developed for RLV in re-entry phase subjects to external disturbance and input constraint. ¹⁶ An adaptive dynamic surface control strategy was designed for a HSV with actuator signals’ magnitude, rate, and bandwidth constraints. ¹⁷ Then, to improve the tracking performance of designed controller and avoid a large initial control signal, an integral term was used at the level of dynamic surface control design. ¹⁸ Moreover, a robust adaptive dynamic surface control strategy was investigated for a HSV with parametric model uncertainty and input saturation. When the input saturations occurred, a compensation design was used. ¹⁹ A radial basis function neural network (RBFNN)-based adaptive dynamic surface control scheme was proposed for a HSV with the consideration of the magnitude, rate, and bandwidth constraints on actuator. ²⁰ Furthermore, sliding mode control combining nonlinear disturbance observer and RBFNN was designed for a near space vehicle, ²¹ where RBFNN was constructed as a compensator to avoid the saturation nonlinearity of rudders. Two continuous time-varying sliding mode-based attitude controllers were designed to achieve the robust tracking of the attitude commands of RLV with and without a priori knowledge of upper bound on the lumped uncertainty. ²² A finite-time robust flight controller, targeting for a re-entry vehicle with blended aerodynamic surfaces and a reaction control system, ²³ was presented for RLV. Based on multivariable smooth second-order sliding mode controller and disturbance observer, a re-entry attitude control strategy was proposed to guarantee that the guidance commands generated from the guidance system can be tracked in finite time. ²⁴ Multi-time scale smooth second-order sliding mode controller with disturbance observer was presented to ensure the finite-time re-entry attitude tracking despite the model parameter uncertainties and unknown external disturbances. ²⁵ An adaptive multivariable disturbance compensation scheme was proposed to provide the estimation for external disturbances where the bounds of the perturbations were not known. Based on the estimation, a continuous multivariable homogeneity second-order sliding mode controller was designed to ensure that the attitude tracking was achieved in finite time. ²⁶

The motivation of this article is to propose an attitude controller to achieve that a RLV in re-entry phase will be driven to track the guidance command under input constraint, model uncertainty, and external disturbance. The control-oriented model (COM) is first established, and it is a strict-feedback system with uncertainties. Then, a robust adaptive constrained backstepping control scheme is designed. A second-order filter is utilized to avoid repeated computation of the time derivative of virtual control input. An auxiliary system is applied to controller design for tackling input constraint. Then through Lyapunov technique, the stability of the closed-loop system is proved. To evaluate the effectiveness of the designed controller, 6-degree-of-freedom (6-DOF) simulation and analysis are carried out.

RLV model

In this section, the rotational equations of motion of RLV during re-entry phase are described.

Attitude model

The motion of 6-DOF unpowered rigid flight vehicle is divided into translational motion and rotational (attitude) motion. Translational motion is referenced to a flight-path coordinate system, and it is caused by the forces that act on the vehicle. It is used for generating trajectory and designing guidance law. Most applications assume steady, coordinated turns such that the sideslip angle is zero. The equations of translational motion are given by Groves et al. ²⁷

Since the primary task of the guidance and control system is to guide the RLV along a predetermined re-entry reference trajectory corridor. Because we focus on the control part of the guidance and control system, the translational equations of motion are not considered herein. The re-entry attitude controller design is mainly based on the rotational equations of motion. The attitude equations of motion, which govern the rigid body attitude dynamic of the vehicle during the re-entry flight, are given as follows^28,29

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\alpha }=-p\cos (\alpha )\tan (\beta )+q-r\sin (\alpha )\tan (\beta )}\\ \multicolumn{1}{c}{-\frac{\cos (\sigma )}{\cos (\beta )}·[\stackrel{·}{\gamma }-\stackrel{·}{\varphi }\cos (\chi )-(\stackrel{·}{\theta }+{\Omega }_E)\cos (\varphi )\sin (\chi )]}\\ \multicolumn{1}{c}{+\frac{\sin (\sigma )}{\cos (\beta )}[\stackrel{·}{\chi }\cos (\gamma )-\stackrel{·}{\varphi }\sin (\chi )\sin (\gamma )]}\\ \multicolumn{1}{c}{+\frac{\sin (\sigma )}{\cos (\beta )}[\varphi +(\stackrel{·}{\theta }+{\Omega }_E)·(\cos (\varphi )\cos (\chi )\sin (\gamma ))}\\ \multicolumn{1}{c}{-\sin (\varphi )\cos (\gamma )]}\end{array}$$ (1)

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\beta }=p\sin (\alpha )-r\cos (\alpha )}\\ \multicolumn{1}{c}{+\sin (\sigma )·[\stackrel{·}{\gamma }-\stackrel{·}{\varphi }\cos (\chi )+(\stackrel{·}{\theta }+{\Omega }_E)\cos (\varphi )\sin (\chi )]}\\ \multicolumn{1}{c}{+\cos (\sigma )·[\stackrel{·}{\chi }\cos (\gamma )-\stackrel{·}{\varphi }\sin (\chi )\sin (\gamma )]}\\ \multicolumn{1}{c}{-\cos (\sigma )(\stackrel{·}{\theta }+{\Omega }_E)·(\cos (\varphi )\cos (\chi )\sin (\gamma )}\\ \multicolumn{1}{c}{-\sin (\varphi )\cos (\gamma ))}\end{array}$$ (2)

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\sigma }=-p\cos (\alpha )\cos (\beta )-q\sin (\beta )-r\sin (\alpha )\cos (\beta )}\\ \multicolumn{1}{c}{+\stackrel{·}{\alpha }\sin (\beta )-\stackrel{·}{\chi }\sin (\gamma )-\stackrel{·}{\varphi }\sin (\chi )\cos (\gamma )}\\ \multicolumn{1}{c}{+(\stackrel{·}{\theta }+{\Omega }_E)[\cos (\varphi )\cos (\chi )\cos (\gamma )+\sin (\varphi )\sin (\gamma )]}\end{array}$$ (3)

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{p}=\frac{I_{\mathrm{zz}}{M}_x}{I_{\mathrm{xx}}{I}_{\mathrm{zz}}-I_{\mathrm{xz}}^2}+\frac{I_{\mathrm{xz}}{M}_z}{I_{\mathrm{xx}}{I}_{\mathrm{zz}}-I_{\mathrm{xz}}^2}+\frac{(I_{\mathrm{xx}}-I_{\mathrm{yy}}+I_{\mathrm{zz}})I_{\mathrm{xz}}}{I_{\mathrm{xx}}{I}_{\mathrm{zz}}-I_{\mathrm{xz}}^2}\mathrm{pq}}\\ \multicolumn{1}{c}{+\frac{(I_{\mathrm{yy}}-I_{\mathrm{zz}})I_{\mathrm{zz}}-I_{\mathrm{xz}}^2}{I_{\mathrm{xx}}{I}_{\mathrm{zz}}-I_{\mathrm{xz}}^2}\mathrm{qr}}\end{array}$$ (4)

$$\stackrel{·}{q}=\frac{M_y}{I_{\mathrm{yy}}}+\frac{I_{\mathrm{xz}}}{I_{\mathrm{yy}}}(r^2-p^2)+\frac{I_{\mathrm{zz}}-I_{\mathrm{xx}}}{I_{\mathrm{yy}}}\mathrm{pr}$$ (5)

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{r}=\frac{I_{\mathrm{xz}}{M}_x}{I_{\mathrm{xx}}{I}_{\mathrm{zz}}-I_{\mathrm{xz}}^2}+\frac{I_{\mathrm{xx}}{M}_z}{I_{\mathrm{xx}}{I}_{\mathrm{zz}}-I_{\mathrm{xz}}^2}+\frac{(I_{\mathrm{xx}}-I_{\mathrm{yy}})I_{\mathrm{xx}}+I_{\mathrm{xz}}^2}{I_{\mathrm{xx}}{I}_{\mathrm{zz}}-I_{\mathrm{xz}}^2}\mathrm{pq}}\\ \multicolumn{1}{c}{+\frac{(I_{\mathrm{yy}}-I_{\mathrm{xx}}-I_{\mathrm{zz}})I_{\mathrm{xz}}}{I_{\mathrm{xx}}{I}_{\mathrm{zz}}-I_{\mathrm{xz}}^2}\mathrm{qr}}\end{array}$$ (6)

where α, β, and σ, respectively, denote AOA, sideslip angle, and bank angle. p, q, and r, respectively, denote roll rate, pitch rate, and yaw rate. M_x , M_y , and M_z , respectively, denote rolling moment, pitching moment, and yawing moment. I_ij (i = x, y, z; j = x, y, z) are inertia. h, v, θ, ϕ, γ, and χ, respectively, denote altitude, velocity, longitude, latitude, flight-path angle, and heading angle. Ω _E is the Earth angular speed.

Robust adaptive constrained backstepping controller design

In this section, the COM for the original model (1)–(6) is first developed. Attitude controller is developed under input constraint, model uncertainties, and external disturbance.

COM

Since the states of equations (1)–(6) are coupled with trajectory states, the COM is proposed to design controller succinctly.

Since rotational motion of RLV is much faster than translational motion and the motion of the Earth, the time derivatives of both position and direction of the velocity, and the Earth’s angular velocity are considered to be negligible with respect to the rotational motion. Thus, the Earth’s angular velocity and the translational terms in the rotational equations can be set to zero, that is, Ω _E  = 0, and $\stackrel{·}{\gamma }=\stackrel{·}{\chi }=\stackrel{·}{\varphi }=\stackrel{·}{\theta }=0$ .

Then, the rotational equations (1)–(3) resulting in the following equations

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\alpha }=-p\cos (\alpha )\tan (\beta )+q-r\sin (\alpha )\tan (\beta )}\\ \multicolumn{1}{c}{\stackrel{·}{\beta }=p\sin (\alpha )-r\cos (\alpha )}\\ \multicolumn{1}{c}{\stackrel{·}{\sigma }=-p\cos (\alpha )\cos (\beta )-q\sin (\beta )}\\ \multicolumn{1}{c}{-r\sin (\alpha )\cos (\beta )+\stackrel{·}{\alpha }\sin (\beta )}\end{array}$$ (6)

Considering the uncertainties induced by model simplification, the above equations can be written as

$$\stackrel{·}{\alpha }=-p\cos \alpha \tan \beta +q-r\sin \alpha \tan \beta +\Delta f_{11}$$ (7)

$$\stackrel{·}{\beta }=p\sin \alpha -r\cos \alpha +\Delta f_{12}$$ (8)

$$\stackrel{·}{\sigma }=-p\cos \alpha \cos \beta -q\sin \beta -r\sin \alpha \cos \beta +\Delta f_{13}$$ (9)

where $\Delta f_{1i}(i=1,2,3)$ represent uncertainties result in model simplification.

Taking the model uncertainty $\Delta \mathbf{I}$ and external disturbance $\Delta {\mathbf{D}}_1=[\begin{array}{c}\hfill d_{21}{d}_{22}{d}_{23}\hfill \end{array}{]}^T$ into account, equations (4)–(6) can be written as

$$(\mathbf{I}+\Delta \mathbf{I})\stackrel{·}{\mathbf{\omega }}=-\mathbf{\omega }(\mathbf{I}+\Delta \mathbf{I})\mathbf{\omega }+\mathbf{u}+\Delta {\mathbf{D}}_1$$ (10)

Based on equations (7)–(10), we obtain COM that is used for controller design, and it can be denoted as the following matrix form

$$\stackrel{·}{\mathbf{\omega }}=\mathbf{R}\mathbf{\omega }+\Delta {\mathbf{f}}_1$$ (11)

$$\stackrel{·}{\mathbf{\omega }}=-{\mathbf{I}}^{-1}{\mathbf{\Phi }\mathbf{I}\mathbf{\omega }\mathbf{+}\mathbf{I}}^{-1}\mathbf{u}\mathbf{+}\mathbf{\Delta }\mathbf{D}$$ (12)

Here, ${\mathbf{\Delta }\mathbf{D}\mathbf{=}\mathbf{I}}^{-1}{\mathbf{(}\mathbf{D}}_1-\mathbf{\Delta }\mathbf{I}\stackrel{·}{\mathbf{\omega }}-\mathbf{\Phi }\mathbf{\Delta }\mathbf{I}\mathbf{\omega }\mathbf{)},\mathbf{\omega }=[\begin{array}{c}\hfill pqr\hfill \end{array}{]}^T$ is the attitude angular rate vector. $\mathbf{\omega }=[\begin{array}{c}\hfill \alpha \beta \sigma \hfill \end{array}{]}^T$ is the attitude angle vector. ${\mathbf{\Delta }\mathbf{f}}_1={\left[\begin{array}{c}\hfill \Delta f_{11}\Delta f_{12}\Delta f_{13}\hfill \end{array}\right]}^T$ is the uncertain vector. $\mathbf{u}=[\begin{array}{c}\hfill M_x{M}_y{M}_z\hfill \end{array}{]}^T$ is the control input vector. The matrixes $\mathbf{I}$ , $\mathbf{\Phi }$ , and $\mathbf{R}$ are as follows

$$\mathbf{I}=\left[\begin{array}{c}\hfill I_{\mathrm{xx}}-I_{\mathrm{xy}}-I_{\mathrm{xz}}\hfill \\ \multicolumn{1}{c}{-I_{\mathrm{xy}}{I}_{\mathrm{yy}}-I_{\mathrm{yz}}}\\ \multicolumn{1}{c}{-I_{\mathrm{xz}}-I_{\mathrm{yz}}{I}_{\mathrm{zz}}}\end{array}\right],\mathbf{\Phi }=\left[\begin{array}{c}\hfill 0-rq\hfill \\ \multicolumn{1}{c}{r0-p}\\ \multicolumn{1}{c}{-qp0}\end{array}\right]\mathbf{R}=\left[\begin{array}{c}\hfill -\cos \alpha \tan \beta 1-\sin \alpha \tan \beta \hfill \\ \multicolumn{1}{c}{\sin \alpha 0-\cos \alpha }\\ \multicolumn{1}{c}{-\cos \alpha \cos \beta -\sin \beta -\sin \alpha \cos \beta }\end{array}\right]$$

Dynamics (11) and (12) have a strict-feedback form since the uncertain terms $\Delta {\mathit{f}}_1$ and $\Delta \mathit{D}$ do not satisfy linear parameterization, and backstepping technique is applied to design controller. Backstepping control is a powerful and systematic technique that recursively interlaces the choice of a Lyapunov function with the feedback control design. The main advantage of this method is the systematic construction of a Lyapunov function for the nonlinear systems, and the control goal can be achieved with reduced control effort. However, the repeated analytic computation of time derivative of virtual control input causes the “explosion of terms” problem, and a second-order filter is employed to avoid it. Moreover, the bound of uncertain term is estimated by designing an adaptive law.

For HSV control system, it is inevitable that actuator is limited especially the magnitude constraints of actuator inputs. Thus, the control input is constrained and defined as follows

$$u_i=\{\begin{array}{c}\hfill u_{imax},\begin{array}{c}\hfill u_{0i}\ge u_{imax}\hfill \end{array}\hfill \\ \multicolumn{1}{c}{u_{0i},\begin{array}{c}\hfill -u_{imax}<u_{0i}<u_{imax}\hfill \end{array}}\\ \multicolumn{1}{c}{-u_{imax},\begin{array}{c}\hfill u_{0i}\le -u_{imax}\hfill \end{array}}\end{array},i=1,2,3.$$ (13)

where $\mathbf{u}=[\begin{array}{c}\hfill u_1{u}_2{u}_3\hfill \end{array}{]}^T=[\begin{array}{c}\hfill M_x{M}_y{M}_z\hfill \end{array}{]}^T$ , ${\mathbf{u}}_0=[\begin{array}{c}\hfill u_{01}{u}_{02}{u}_{03}\hfill \end{array}{]}^T$ is the control input to be designed in the following subsection. The maximum value of three control inputs u_i is defined as $\overline{u}=u_{imax}$ .

Controller design

To proceed, the following assumption is made for control inputs u_i for the input constraints.

Assumption 1

The unknown external disturbances are assumed to be bounded by $\left|d_{2i}\right|\le {\lambda }_i$ and satisfy the following inequality

$$\Vert {\mathbf{D}}_1\Vert \le {\overline{D}}_1=\sqrt{\sum _{i=1}^3{\lambda }_i^2}$$ (14)

where λ_i are known positive constants.

Assumption 2

The relationship between $\overline{u}$ and ${\overline{D}}_1$ is addressed as follows: $\overline{u}>{\overline{D}}_1$ . ³⁰

The control objective is that a controller is proposed for COM (11) and (12) to assure that the attitude angle $\mathbf{\omega }$ tracks the guidance command ${\mathbf{\omega }}_d$ under input constraint, model uncertainties, and external disturbance, under assumption that elements of ${\mathbf{\omega }}_d$ and their first-order time derivatives are continuous and bounded.

Due to the backstepping procedure, the virtual control input is first designed based on equation (11), and then the actual control input is designed based on equation (12) and input constraint (13). The design procedure is as follows. The attitude angle subsystem and the attitude angular rate subsystem are outer loop and inner loop, respectively. The outer loop is employed to develop virtual control input, which is used as the reference signal for the angular rate subsystem. Combining it with the auxiliary system, the actual control input is developed.

Virtual control input design for attitude angle subsystem

In this subsection, attitude angular rate is regarded as the virtual control input of attitude angle subsystem (11). Tracking error for the attitude angle and error signal of attitude angular rate are as follows

$${\mathbf{z}}_1=\mathbf{\omega }-{\mathbf{\omega }}_d$$ (15)

$${\mathbf{z}}_2=\mathbf{\omega }-{\mathbf{\omega }}_d$$ (16)

where ${\mathbf{\omega }}_d=[\begin{array}{c}\hfill {\alpha }_d{\beta }_d{\sigma }_d\hfill \end{array}{]}^T$ is the guidance command. ${\mathbf{\omega }}_d$ is the reference command of the attitude rate subsystem and it will be designed in the following section.

The Lyapunov candidate function is chosen as

$$V_1=0.5{\mathbf{z}}_1^T{\mathbf{z}}_1$$ (17)

Based on equations (11) and (15), the time derivative of (17) is given by

$${\stackrel{·}{V}}_1={\mathbf{z}}_1^T{\stackrel{·}{\mathbf{z}}}_1={\mathbf{z}}_1^T\mathbf{R}{\omega }_d+{\mathbf{z}}_1^T\Delta {\mathbf{f}}_1+{\mathbf{z}}_1^T{\mathbf{Rz}}_2-{\mathbf{z}}_1^T{\stackrel{·}{\mathbf{\omega }}}_d$$ (18)

In above equation, the term ${\mathbf{Rz}}_2$ needs to be determined in the next controller design step, and it is about the states of RLV. According to backstepping control design procedure, the error ${\mathbf{z}}_2$ can converge to a random small value through the next step controller design, so it is reasonable to treat the term ${\mathbf{Rz}}_2$ as an uncertain term. The terms $\Delta {\mathbf{f}}_1$ and ${\mathbf{Rz}}_2$ are bounded in view of their physical backgrounds, and the term $\Delta {\mathbf{f}}_2={\mathbf{Rz}}_2+\Delta {\mathbf{f}}_1$ is in the certain bound. A three-order robust term is utilized to compensate the influence of uncertainty $\Delta {\mathbf{f}}_2$ . The dynamic of the robust term is defined as

$${\stackrel{·}{\mathbf{u}}}_s={\mathbf{\xi }}_1\mathbf{Sgn}(z_1)$$ (19)

with ${\mathbf{\xi }}_1=\mathrm{diag}({\xi }_{11}{\xi }_{12}{\xi }_{13})>0$ and $\mathbf{Sgn}({\mathbf{z}}_1)=[\mathrm{sign}(z_{11})\mathrm{sign}(z_{12})\mathrm{sign}(z_{13}{)]}^T$ .

Based on equations (18) and (19), the virtual control input is designed as

$${\mathbf{\omega }}_d{=\mathbf{R}}^{-1}(-{\mathbf{k}}_1{\mathbf{z}}_1-{\mathbf{u}}_s+{\stackrel{·}{\mathbf{\omega }}}_d)$$ (20)

Remark 1

From equation (19), though there is sign function in the dynamic equation of robust term, the robust term ${\mathbf{u}}_s$ is included in control input (20) after integrating. It does not have sign function and the chattering problem is thus avoided.

The estimation error of $\Delta {\mathbf{f}}_2$ is defined as $\tilde{\mathbf{E}}={\mathbf{u}}_s-\Delta {\mathbf{f}}_2$ and $\Delta {\mathbf{f}}_2=[f_{21}{f}_{22}{f}_{23}{]}^T$ . Using derivative with respect to time, then

$$\stackrel{·}{\stackrel{~}{\mathbf{E}}}={\stackrel{·}{\mathbf{u}}}_s-\Delta {\stackrel{·}{\mathbf{f}}}_2$$ (21)

Let $\mathbf{F}\mathbf{(}t)=\Delta {\stackrel{·}{\mathbf{f}}}_2=[\begin{array}{c}\hfill {\stackrel{·}{f}}_{21}(t){\stackrel{·}{f}}_{22}(t){\stackrel{·}{f}}_{23}(t)\hfill \end{array}{]}^T$ . From equations (11), (15), (19)–(21), the following system is obtained

$$\left(\begin{array}{c}\hfill \stackrel{·}{\stackrel{~}{\mathbf{E}}}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathbf{z}}}_1}\end{array}\right)=\left(\begin{array}{c}\hfill 00\hfill \\ \multicolumn{1}{c}{-{\mathbf{E}}_0-{\mathbf{k}}_1}\end{array}\right)\left(\begin{array}{c}\hfill \tilde{\mathbf{E}}\hfill \\ \multicolumn{1}{c}{{\mathbf{z}}_1}\end{array}\right)+\left(\begin{array}{c}\hfill {\mathbf{\xi }}_1\mathbf{Sgn}({\mathbf{z}}_1)-\mathbf{F}(\mathbf{t})\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)$$ (22)

where ${\mathbf{E}}_0$ is an identity matrix.

Based on above equation, we have ${\stackrel{·}{\mathbf{z}}}_1=-\tilde{\mathbf{E}}-{\mathbf{k}}_1{\mathbf{z}}_1$ , differentiating it with respect to time, then ${\stackrel{··}{\mathbf{z}}}_1=-\stackrel{·}{\stackrel{~}{\mathbf{E}}}-{\mathbf{k}}_1{\stackrel{·}{\mathbf{z}}}_1$ . Substituting $\stackrel{·}{\stackrel{~}{\mathbf{E}}}$ into ${\stackrel{··}{\mathbf{z}}}_1$ , we have ${\stackrel{··}{\mathbf{z}}}_1+{\mathbf{k}}_1{\stackrel{·}{\mathbf{z}}}_1=-{\mathbf{\xi }}_1\mathbf{Sgn}({\mathbf{z}}_1)+\mathbf{F}(t)$ . On the basis of the definitions of ${\mathbf{z}}_1,{\stackrel{·}{\mathbf{u}}}_s,\mathbf{\xi },\mathbf{Sgn}({\mathbf{z}}_1)$ and $\mathbf{F}(t)$ , the following equation holds

$$\{\begin{array}{c}\hfill {\stackrel{··}{z}}_{11}+k_{11}{\stackrel{·}{z}}_{11}=-{\xi }_{11}\mathrm{sign}(z_{11})+{\stackrel{·}{f}}_{21}(t)\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{z}}_{12}+k_{12}{\stackrel{·}{z}}_{12}=-{\xi }_{12}\mathrm{sign}(z_{12})+{\stackrel{·}{f}}_{22}(t)}\\ \multicolumn{1}{c}{{\stackrel{··}{z}}_{13}+k_{13}{\stackrel{·}{z}}_{13}=-{\xi }_{13}\mathrm{sign}(z_{13})+{\stackrel{·}{f}}_{23}(t)}\end{array}$$ (23)

The right-hand side of (23) is discrete, according to Peng et al., ³¹ it should satisfy

$$\{\begin{array}{c}\hfill {\stackrel{··}{z}}_{1i}+k_{1i}{\stackrel{·}{z}}_{1i}=-{\mu }_{1i}^{-},z_{1i}>0,i=1,2,3\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{z}}_{1i}+k_{1i}{\stackrel{·}{z}}_{1i}={\mu }_{1i}^{-},z_{1i}\le 0,i=1,2,3}\end{array}$$ (24)

$$\{\begin{array}{c}\hfill {\stackrel{··}{z}}_{1i}+k_{1i}{\stackrel{·}{z}}_{1i}=-{\mu }_{1i}^{+},z_{1i}>0,i=1,2,3\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{z}}_{1i}+k_{1i}{\stackrel{·}{z}}_{1i}={\mu }_{1i}^{+},z_{1i}\le 0,i=1,2,3}\end{array}$$ (25)

where ${\mu }_{1i}^{-}={\xi }_{1i}-\left|{\stackrel{·}{f}}_{2i}\right|$ and ${\mu }_{1i}^{+}={\xi }_{1i}+\left|{\stackrel{·}{f}}_{2i}\right|$ .

Based on equations (24) and (25), the following equations are obtained

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{··}{z}}_{1i}+k_{1i}{\stackrel{·}{z}}_{1i}=-{\mu }_{1i}\mathrm{sign}(z_{1i}),}\\ \multicolumn{1}{c}{{\mu }_{1i}\in ({\mu }_{1i}^{-},{\mu }_{1i}^{+}),i=1,2,3}\end{array}$$ (26)

If we define ${\mathbf{U}}_1=\mathrm{diag}({\mu }_{11}{\mu }_{12}{\mu }_{13}),{\mathbf{U}}_1^{-}=\mathrm{diag}({\mu }_{11}^{-}{\mu }_{12}^{-}{\mu }_{13}^{-}),{\mathbf{U}}_1^{+}=\mathrm{diag}({\mu }_{11}^{+}{\mu }_{12}^{+}{\mu }_{13}^{+})$ , then equation (26) can be written as follows

$${\stackrel{··}{\mathbf{z}}}_1+{\mathbf{k}}_1{\stackrel{·}{\mathbf{z}}}_1=-{\mathbf{U}}_1\mathbf{Sgn}({\mathbf{z}}_1)$$ (27)

It is noted that equation (27) is a switched system, which is a hybrid system that is composed of a family of continuous-time and discrete-time subsystems, and a rule orchestrating the switching between these subsystems. It has been applied in engineering area (see Zhang et al. ³² and Wu et al. ³³ ). The stability analysis of the system (22) is transformed into stability analysis for the switched system (27), which is carried out in Lyapunov framework.

Theorem 1

Considering the switched system (27), with the given gain matrixes ${\mathbf{k}}_1,\mathbf{\xi }>0$ , if there are proper values of parameters $a_{\mathrm{ji}},j=1,\dots ,4;i=1,2,3$ , the following relationships can be satisfied

$$a_{\mathrm{ji}}>0,a_{\mathrm{ji}}=k_{2i}{a}_{2i},a_{4i}=a_{3i},k_{2i}{a}_{3i}>a_{2i},a_{1i}{a}_{3i}>a_{2i}^2$$ (28)

Then the switched system (27) is stable.

Proof

The Lyapunov candidate function is constructed as

$$V_1=\sum _{i=1}^3(0.5{\mathbf{P}}_i^T{\mathbf{Q}}_i{\mathbf{P}}_i+a_{4i}{\mu }_{1i}\left|z_{1i}\right|)$$ (29)

where ${\mathbf{P}}_i={(\begin{array}{c}\hfill z_{1i}{\stackrel{·}{z}}_{1i}\hfill \end{array})}^T$ and ${\mathbf{Q}}_i=\left(\begin{array}{c}\hfill a_{1i}{a}_{2i}\hfill \\ \multicolumn{1}{c}{a_{2i}{a}_{3i}}\end{array}\right),i=1,2,3$ are positive definite matrixes, which determine that

$$a_{\mathrm{ji}}>0,a_{1i}{a}_{3i}>a_{2i}^2,j=1,\dots ,4$$

The time derivative of V ₁ is given by

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=\sum _{i=1}^3[a_{3i}{\stackrel{·}{z}}_{1i}(-k_{1i}{\stackrel{·}{z}}_{1i}-{\mu }_{1i}\mathrm{sign}(z_{1i}))]}\\ \multicolumn{1}{c}{+\sum _{i=1}^3[a_{2i}{z}_{1i}(-k_{1i}{\stackrel{·}{z}}_{1i}-{\mu }_{1i}\mathrm{sign}(z_{1i}))]}\\ \multicolumn{1}{c}{+\sum _{i=1}^3[a_{1i}{z}_{1i}{\stackrel{·}{z}}_{1i}+a_{2i}{\stackrel{·}{z}}_{1i}^2+{\mu }_{1i}{a}_{4i}{\stackrel{·}{z}}_{1i}\mathrm{sign}(z_{1i})]}\end{array}$$ (30)

Based on what was mentioned, equation (30) becomes

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1}\end{array}\le \sum _{i=1}^3[(a_{1i}-k_{1i}{a}_{2i})z_{1i}{\stackrel{·}{z}}_{1i}-(k_{1i}{a}_{3i}-a_{2i}){\stackrel{·}{z}}_{1i}^2]+\sum _{i=1}^3[{\mu }_{1i}(a_{4i}-a_{3i}){\stackrel{·}{z}}_{1i}\mathrm{sign}(z_{1i})-{\mu }_{1i}{a}_{2i}\left|z_{1i}\right|]$$ (31)

If $a_{\mathrm{ji}},a_{1i}-k_{1i}{a}_{2i}=0,a_{4i}-a_{3i}=0$ and $k_{1i}{a}_{3i}-a_{2i}>0$ , then

$${\stackrel{·}{V}}_1\le -\sum _{i=1}^3[(k_{1i}{a}_{3i}-a_{2i}){\stackrel{·}{z}}_{1i}^2+{\mu }_{1i}{a}_{2i}\left|z_{1i}\right|]\le 0$$ (32)

Therefore, the switched system is stable. The stability of system (27) and attitude angle subsystem (11) is assured.

Actual control input design for attitude angular rate subsystem

To proceed, on the basis of attitude angular rate dynamic (12) and virtual control input (20), the actual control input for RLV is designed.

The time derivative of (16) is

$${\stackrel{·}{\mathbf{z}}}_2=-{\mathbf{I}}^{-1}\mathbf{\Phi }\mathbf{I}\mathbf{\omega }+{\mathbf{I}}^{-1}\mathbf{u}+\Delta \mathbf{D}-{\stackrel{·}{\mathbf{\omega }}}_d$$ (33)

From above equation, the time derivative of ${\mathbf{\omega }}_d$ needs to be computed, due to the nonlinearity and uncertainty of the COM (17) and (18), it is difficult to obtain ${\stackrel{·}{\mathbf{\omega }}}_d$ , so a second-order filter is used to avoid the analytic computation to cope with “explosion of terms” problem.

The second-order filter is depicted as follows ³⁴

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathbf{\rho }}}_1=-\frac{{\mathbf{\rho }}_1-{\mathbf{\omega }}_d}{{\tau }_1}-\frac{{\psi }_1({\mathbf{\rho }}_1-{\mathbf{\omega }}_d)}{\Vert {\mathbf{\rho }}_1-{\mathbf{\omega }}_d\Vert +{\epsilon }_1}}\\ \multicolumn{1}{c}{{\stackrel{·}{\mathbf{\rho }}}_2=-\frac{{\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1}{{\tau }_2}-\frac{{\psi }_2({\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1)}{\Vert {\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1\Vert +{\epsilon }_2}}\end{array}$$ (34)

where ${\tau }_l$ is the filter time constant, whereas ${\psi }_l$ and ${\epsilon }_l$ are positive constants, $l=1,2$ .

The following assumption holds due to the physical backgrounds of RLV.

Assumption 3

The uncertain vector $\Delta \mathbf{D}$ is assumed to be bounded by an unknown bound constant δ > 0, that is, $\Vert \Delta \mathbf{D}\Vert \le \delta$ .

Since the bound δ is unknown, an adaptive law will be designed to estimate δ online, and $\hat{\delta }$ will be employed to denote the estimation.

The Lyapunov function is chosen as $V_{20}=0.5{\mathbf{z}}_2^T{\mathbf{z}}_2$ . Based on equations (12) and (16), the time derivative of it is

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{20}=-{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\mathbf{\Phi }\mathbf{I}\mathbf{\omega }+{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\mathbf{u}+{\mathbf{z}}_2^T\Delta \mathbf{D}-{\mathbf{z}}_2^T{\mathbf{\rho }}_2}\\ \multicolumn{1}{c}{\le -{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\mathbf{\Phi }\mathbf{I}\mathbf{\omega }+{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\mathbf{u}+\Vert {\mathbf{z}}_2\Vert \delta -{\mathbf{z}}_2^T{\mathbf{\rho }}_2}\end{array}$$ (35)

For input constraint (13), an auxiliary system is employed to analyze the effect, the states of which are used in controller design and stability analysis. The formulation of this system is ³⁵

$${\stackrel{·}{\mathbf{e}}}_{\mathbf{u}}=\{\begin{array}{c}\hfill -{\mathbf{k}}_3{\mathbf{e}}_{\mathbf{u}}-{\mathbf{I}}^{-1}\Delta \mathbf{u}-\frac{\left|{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\Delta \mathbf{u}\right|+\frac{1}{2}\Vert \Delta \mathbf{u}{\Vert }^2}{\Vert {\mathbf{e}}_{\mathbf{u}}{\Vert }^2}{\mathbf{e}}_{\mathbf{u}},\Vert {\mathbf{e}}_{\mathbf{u}}\Vert \ge {\eta }_1\hfill \\ \multicolumn{1}{c}{0,\Vert {\mathbf{e}}_{\mathbf{u}}\Vert <{\eta }_1}\end{array}$$ (36)

$\Delta \mathbf{u}=\mathbf{u}-{\mathbf{u}}_0$ , ${\mathbf{k}}_3>0$ is a diagonal matrix, and ${\eta }_1>0$ .

The control input can be designed as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{u}}_0=\mathbf{I}(-{\mathbf{k}}_2({\mathbf{z}}_2-{\mathbf{e}}_{\mathbf{u}})-\frac{a\hat{\delta }{\mathbf{z}}_2}{\Vert {\mathbf{z}}_2\Vert +\epsilon }}\\ \multicolumn{1}{c}{-{\mathbf{R}}^T{\mathbf{z}}_1+{\mathbf{\rho }}_2+{\mathbf{I}}^{-1}\mathbf{\Phi }\mathbf{I}\mathbf{\omega })}\end{array}$$ (37)

where ${\mathbf{k}}_2>0$ .

The adaptive law for $\hat{\delta }$ is constructed as

$$\stackrel{·}{\hat{\delta }}=\frac{\mathrm{ac}\Vert {\mathbf{z}}_2{\Vert }^2}{\Vert {\mathbf{z}}_2\Vert +\epsilon }$$ (38)

where $a>1,c,\epsilon >0$ .

Remark 2

During the control input design procedure, ${\stackrel{·}{\mathbf{\omega }}}_d$ is instead by simpler algebraic operations ${\rho }_2$ defined by the filter (34) to avoid computing the analytic derivative of the ${\mathbf{\omega }}_d$ .

In the following section, with the controller (37) and input constraint (13), the stability of the closed-loop system is analyzed via Lyapunov theory.

Theorem 2

Considering the attitude systems (11) and (12), under assumptions 1–3, the designed controller (37), integrating with adaptive law (38), assures the control system stable.

Proof

Considering the estimation errors, tracking errors, and the states of the auxiliary system, the composite Lyapunov function can be chosen as

$$V_2=V_{20}+\frac{1}{2c}{\tilde{\delta }}^2+\frac{1}{2}{\mathbf{e}}_{\mathbf{u}}^T{\mathbf{e}}_{\mathbf{u}}$$ (39)

where $\tilde{\delta }=\hat{\delta }-\delta$ is the estimation error of δ.

Taking the time derivative of (39), then

$${\stackrel{·}{V}}_2={\stackrel{·}{V}}_{20}+\frac{1}{c^{*}{\tilde{\delta }}^{*}\stackrel{·}{\hat{\delta }}+{\mathbf{e}}_{\mathbf{u}}^T{}^{*}{\stackrel{·}{\mathbf{e}}}_{\mathbf{u}}}$$ (40)

From equations (20) and (35)–(38), we have

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{20}+\frac{1}{c^{*}{\tilde{\delta }}^{*}\stackrel{·}{\hat{\delta }}}\le {\mathbf{z}}_1^T\mathbf{R}({\mathbf{z}}_2+{\mathbf{\omega }}_d)+\Vert {\mathbf{z}}_2\Vert \delta }\\ \multicolumn{1}{c}{+\frac{a\tilde{\delta }\Vert {\mathbf{z}}_2{\Vert }^2}{\Vert {\mathbf{z}}_2\Vert +\epsilon }-{\mathbf{z}}_1^T{\stackrel{·}{\mathbf{\omega }}}_d+{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\mathbf{\Phi }\mathbf{I}\mathbf{\omega }}\\ \multicolumn{1}{c}{+{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\mathbf{u}-{\mathbf{z}}_2^T{\mathbf{\rho }}_2-{\mathbf{k}}_3\Vert {\mathbf{e}}_{\mathbf{u}}{\Vert }^2-\left|{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\Delta \mathbf{u}\right|}\\ \multicolumn{1}{c}{-{\mathbf{e}}_{\mathbf{u}}^T\Delta \mathbf{u}-\frac{1}{2}\Vert \Delta \mathbf{u}{\Vert }^2}\\ \multicolumn{1}{c}{\le -k_{2,min}\Vert {\mathbf{z}}_2{\Vert }^2+(\Vert {\mathbf{z}}_2\Vert \delta -\frac{a\delta \Vert {\mathbf{z}}_2{\Vert }^2}{\Vert {\mathbf{z}}_2\Vert +\epsilon })}\\ \multicolumn{1}{c}{+{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\Delta \mathbf{u}+{\mathbf{z}}_2^T{\mathbf{k}}_2{\mathbf{e}}_{\mathbf{u}}-{\mathbf{k}}_3{\mathbf{e}}_{\mathbf{u}}^T{\mathbf{e}}_{\mathbf{u}}-\left|{\mathbf{z}}_2^T{\mathbf{I}}^{-1}\Delta \mathbf{u}\right|}\\ \multicolumn{1}{c}{-{\mathbf{e}}_{\mathbf{u}}^T{\mathbf{I}}^{-1}\Delta \mathbf{u}-\frac{1}{2}\Vert \Delta \mathbf{u}{\Vert }^2}\end{array}$$ (41)

Since

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{z}}_2^T{\mathbf{k}}_2{\mathbf{e}}_{\mathbf{u}}-k_{3,\min }{\mathbf{e}}_{\mathbf{u}}^T{\mathbf{e}}_{\mathbf{u}}-\frac{1}{2}{\tau }^2\Delta {\mathbf{u}}^T\Delta \mathbf{u}}\\ \multicolumn{1}{c}{-{\mathbf{e}}_{\mathbf{u}}^T{\mathbf{I}}^{-1}\Delta \mathbf{u}\le \frac{1}{2}\Vert {\mathbf{k}}_2{\Vert }^2\Vert {\mathbf{z}}_2{\Vert }^2+\frac{1}{2}{\mathbf{e}}_{\mathbf{u}}^T{\mathbf{e}}_{\mathbf{u}}-k_{3,\min }{\mathbf{e}}_{\mathbf{u}}^T{\mathbf{e}}_{\mathbf{u}}}\\ \multicolumn{1}{c}{-\frac{1}{2}{\tau }^2\Delta {\mathbf{u}}^T\Delta \mathbf{u}+\frac{1}{2}{\mathbf{e}}_{\mathbf{u}}^T{\mathbf{e}}_{\mathbf{u}}}\\ \multicolumn{1}{c}{+\frac{1}{2}{\tau }^2\Delta {\mathbf{u}}^T\Delta \mathbf{u}\le \frac{1}{2}\Vert {\mathbf{k}}_2{\Vert }^2\Vert {\mathbf{z}}_2{\Vert }^2-(k_{3,\min }-1){\mathbf{e}}_{\mathbf{u}}^T{\mathbf{e}}_{\mathbf{u}}}\end{array}$$

and if $\Vert {\mathbf{z}}_2\Vert \ge \epsilon /(a-1),(a>1)$ , the following inequality holds

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2\le -(k_{2,\min }-\frac{1}{2}\Vert {\mathbf{k}}_2{\Vert }^2)\Vert {\mathbf{z}}_2{\Vert }^2-(k_{3,\min }-1){\mathbf{e}}_{\mathbf{u}}^T{\mathbf{e}}_{\mathbf{u}}}\\ \multicolumn{1}{c}{\le 0}\end{array}$$ (42)

where $k_{2,min}$ and $k_{3,min}$ are the minimum element of matrixes ${\mathbf{k}}_2$ and ${\mathbf{k}}_3$ , respectively.

Based on what was discussed above, if $\Vert {\mathbf{z}}_2\Vert \ge \epsilon /(a-1),k_{2,min}>\Vert {\mathbf{k}}_2{\Vert }^2/2,k_{3,min}>1,{\stackrel{·}{V}}_2\le 0$ . So far, the control system is stable.

Theorem 3

Considering the closed-loop system described as (11) and (12), with assumptions 1–3, under the controller (38), adaptive law (38), the second-order filter (34) and the designed parameters, $k_{1i},\xi >0,i=1,\dots 3,k_{2,min}>\Vert {\mathbf{k}}_2{\Vert }^2/2,k_{3,min}>1,a>1,\epsilon >0,{\epsilon }_l>0,b_l,{\eta }_l>0,{\tau }_l>0,{\psi }_l>0,l=1,2$ . If there are proper values of parameters $a_{\mathrm{ji}},{\overline{\psi }}_l>1$ , then they satisfy the relationships, $\Vert {\mathbf{z}}_2\Vert >\epsilon /(a-1),$ $\Vert {\mathbf{\rho }}_1-{\mathbf{\omega }}_d\Vert >{\epsilon }_1/({\overline{\psi }}_1-1)$ , $\Vert {\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1\Vert >{\epsilon }_2/({\overline{\psi }}_2-1)$ , the control system is stable.

Proof

The Lyapunov candidate function is constructed as

$$V=V_1+V_2+\sum _{l=1}^{2}0.5{\mathbf{e}}_{\mathrm{fl}}^T{\mathbf{e}}_{\mathrm{fl}}$$ (43)

The time derivative is

$$\stackrel{·}{V}={\stackrel{·}{V}}_1+{\stackrel{·}{V}}_2+\sum _{l=1}^2{\mathbf{e}}_{\mathrm{fl}}^T{\stackrel{·}{\mathbf{e}}}_{\mathrm{fl}}$$ (44)

For the last term, it satisfies the following inequality

$$\begin{array}{c}\multicolumn{1}{c}{\sum _{l=1}^2{\mathbf{e}}_{\mathrm{fl}}^T{\stackrel{·}{\mathbf{e}}}_{\mathrm{fl}}\le -\Vert {({\mathbf{\rho }}_1-{\mathbf{\omega }}_d)}^T\Vert c_1(\frac{{\overline{\psi }}_1\Vert {\mathbf{\rho }}_1-{\mathbf{\omega }}_d\Vert }{\Vert {\mathbf{\rho }}_1-{\mathbf{\omega }}_d\Vert +{\epsilon }_1}-1)}\\ \multicolumn{1}{c}{-\Vert {({\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1)}^T\Vert c_2(\frac{{\overline{\psi }}_2\Vert {\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1\Vert }{\Vert {\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1\Vert +{\epsilon }_2}-1)}\end{array}$$ (45)

with the following assumptions: (1) the time derivative of ${\mathbf{\omega }}_d$ is bounded with a known positive constant, $c_1$ , which satisfies $\Vert {\stackrel{·}{\mathbf{\omega }}}_d\Vert \le \Vert {\stackrel{·}{\mathbf{\omega }}}_d{\Vert }_{max}\le c_1$ and ${\psi }_1={\overline{\psi }}_1{c}_1,{\overline{\psi }}_1>1$ . (2) ${\stackrel{·}{\mathbf{\rho }}}_1$ is bounded with a constant, $c_2>0$ , which satisfies $\Vert {\stackrel{·}{\mathbf{\rho }}}_1\Vert \le \Vert {\stackrel{·}{\mathbf{\rho }}}_1{\Vert }_{max}\le c_2$ and ${\psi }_2={\overline{\psi }}_2{c}_2,{\overline{\psi }}_2>1$ .

If $\Vert {\mathbf{\rho }}_1-{\mathbf{\omega }}_d\Vert >{\epsilon }_1/({\overline{\psi }}_1-1)$ and $\Vert {\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1\Vert >{\epsilon }_2/({\overline{\psi }}_2-1)$ , $\sum _{l=1}^2{\mathbf{e}}_{\mathrm{fl}}^T{\stackrel{·}{\mathbf{e}}}_{\mathrm{fl}}\le 0$ . As shown in Bollino, ²⁹ the convergence of the filter is assured, and the estimation error can be guaranteed within the compact set determined in the following form: $\Vert {\mathbf{\rho }}_1-{\mathbf{\omega }}_d\Vert \le {\epsilon }_1/({\overline{\psi }}_1-1)$ , $\Vert {\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1\Vert \le {\epsilon }_2/({\overline{\psi }}_2-1)$ . The estimation error of the filter can be adjusted sufficiently small by choosing appropriate values of ${\epsilon }_1$ and ${\epsilon }_2$ .

Therefore, combining with inequalities (32), (42), and (45), as long as $a_{\mathrm{ji}}>0,a_{1i}=k_{1i}{a}_{2i},k_{1i}{a}_{3i}>a_{2i}$ , $a_{1i}{a}_{3i}>a_{2i}^2,a_{4i}=a_{3i},\Vert {\mathbf{\rho }}_1-{\mathbf{\omega }}_d\Vert \le {\epsilon }_1/({\overline{\psi }}_1-1),\Vert {\mathbf{\rho }}_2-{\stackrel{·}{\mathbf{\rho }}}_1\Vert \ge {\epsilon }_2/({\overline{\psi }}_2-1),\stackrel{·}{V}\le 0$ holds. So far, $\stackrel{·}{V}$ is proved to be negative semi-definite, and it assures the closed-loop system’s Lyapunov stability.

Simulation results

In order to show that the proposed control strategy can work effectively, numerical simulations are performed and shown in this section. A detailed description of the guidance command is provided in Tian and Zong. ³⁶ The normal inertial matrix and the external disturbance vector are given as

$$\mathbf{I}=\left[\begin{array}{c}\hfill 4342700-17880\hfill \\ \multicolumn{1}{c}{09612000}\\ \multicolumn{1}{c}{-1788001131541}\end{array}\right],\Delta {\mathbf{D}}_1=\left[\begin{array}{c}\hfill 1+\sin (\pi t/125)+\sin (\pi t/250)\hfill \\ \multicolumn{1}{c}{1+\sin (\pi t/125)+\sin (\pi t/250)}\\ \multicolumn{1}{c}{1+\sin (\pi t/125)+\sin (\pi t/250)}\end{array}\right]\times {10}^4,$$

It is noted that the external disturbance is bounded by ${\overline{D}}_1=3\sqrt{3}\times {10}^4$ , and the control authority is assumed to be $\overline{u}=6\times {10}^4$ . Therefore, the control limits used for exploring the capability of the designed control strategy in adhering to the limit are ${\mathbf{u}}_{max}={(\begin{array}{c}\hfill 6\times {10}^{4}6\times {10}^{4}6\times {10}^4\hfill \end{array})}^T$ , ${\mathbf{u}}_{min}=-{\mathbf{u}}_{max}$ . The model uncertainty (inertia matrix uncertainty) is $\Delta \mathbf{I}=0.1\mathbf{I}$ . The initial flight conditions and the controller parameters are given, respectively, in Tables 1 and 2.

Table 1.

Initial flight condition of RLV.

State	Value	State	Value	State	Value
Altitude h 0 (ft)	2,60,000	FPA γ 0 (°)	−1.064	Bank angle σ0 (deg)	−74.48
Velocity v 0 (ft/s)	24,061	Heading angle χ 0 (°)	0	Roll rate p (°/s)	0
Longitude ϕ 0 (°)	0	AOA α 0 (°)	17.41	Pitch rate q (°)	0
Latitude θ 0 (°)	0	Sideslip angle β 0 (°)	0	Yaw rate r (°)	0

RLV: reusable launch vehicle, AOA: angle of attack, FPA: flight path angle.

Table 2.

Controller parameters.

Parameter	Value	Parameter	Value
k 1	$0.8\mathrm{diag}(\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array})$	η 1	0.01
k 2	$3\mathrm{diag}(\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array})$	η 2	0.001
k 3	$3\mathrm{diag}(\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array})$	τ 1, τ2	0.1
k 4	2	ψ 1, ψ 2	0.001
a	1.2	ε 1, ε2	0.001
c	0.001	ε	0.001

As shown in adaptive law (38), $\hat{\delta }$ is monotonically increasing. If it over-increases, the control input may increase too large. Thus, the adaptive law is revised as follows to suppress the over-increase of $\hat{\delta }$

$$\stackrel{·}{\hat{\delta }}=\{\begin{array}{c}\hfill \frac{\mathrm{ac}\Vert {\mathbf{z}}_2{\Vert }^2}{\Vert {\mathbf{z}}_2\Vert +\epsilon },\Vert {\mathbf{z}}_2\Vert \ge \frac{\epsilon }{a-1}\hfill \\ \multicolumn{1}{c}{0,\Vert {\mathbf{z}}_2\Vert <\frac{\epsilon }{a-1}}\end{array}$$ (46)

It is noted that the estimation change rate remains at zero after the error signal ${\mathbf{z}}_2$ achieving stability region. It induces that $\hat{\delta }$ will not change, so the amplitude of the controller will not be too large.

Time history of attitude angle tracking, tracking error and control inputs under input constraint, uncertain inertia matrix, and external disturbance are demonstrated in Figures 1–3. The local time histories of figures are also given in Figures 1–3 to show the dynamic process. Time history of tracking performance is given in Figures 1 and 2. They demonstrate that AOA, sideslip angle, and bank angle achieve the stable tracking of their respective guidance command despite input constraint, model uncertainty, and external disturbance. But from Figure 3, where the dotted line denotes the maximum control input and the solid line denotes the minimum control input, the proposed controller can still guarantee the stable tracking of attitude angle while the controller handles the input constraint effectively. Besides, the saturation of control inputs occurs only during the initial transient phase as shown in Figure 3, the saturation time is shorter than the transient time, after this period of time, input saturation no longer occurs.

Figure 1.

Time history of attitude angle tracking.

Figure 2.

Time history of attitude angle tracking error.

Figure 3.

Time history of control input.

It is shown that the designed control scheme achieves desired tracking performance with no obvious steady state error and the tracking errors remain small (from Figure 2) during the whole re-entry flight. Moreover, they suggest that it is necessary to consider input constraint (19), and the proposed control scheme is able to implement on input constraints.

To present the control performance of the designed control scheme (RACBC) in this article and the designed ABFTSMC scheme proposed in Wang et al., ³⁷ the simulation results of these two control schemes are given in Figures 4–6 for comparison.

Figure 4.

Time history of attitude angle tracking of RACBC and ABFTSMC.

Figure 5.

Time history of attitude angle tracking error of RACBC and ABFTSMC.

Figure 6.

Time history of control input of RACBC and ABFTSMC.

Both these control strategies (RACBC [robust adaptive constrained backstepping control] and ABFTSMC [adaptive backstepping finite time siliding mode control]) can handle input constraint effectively. It is worth pointing out that under the two controllers, saturation occurs during the transient state phase. After this phase, control inputs are within their limits. Compared to the ABFTSMC scheme, the proposed RACBC scheme at least has two advantages. First, the saturation time of control input is shorter than that of ABFTSMC, which implies that the designed controller has a higher ability of handling input constraint under model uncertainty and external disturbances. Second, although the settling time of attitude angle is a little longer than that of ABFTSMC, it has the better dynamic process of attitude angle tracking.

Conclusion

The attitude controller is designed for RLV in considering input constraint, model uncertainties, and external disturbance. The stable tracking of the attitude angles is assured by the proposed robust adaptive constrained backstepping controller; simultaneously, the input constraint is tackled effectively. The auxiliary system is introduced for coping with input constraint. Besides, the uncertainty generated by the model simplification is estimated and compensated via the high-order robust term. The unknown bound of lumped uncertainty, including the external disturbance and the uncertainty, is estimated by the robust adaptive law. The “explosion of terms” problem is eliminated through the second-order filter. Based on the Lyapunov theory, the stability of the closed-loop system is analyzed. Simulation results show that the presented controller has reliability of control for RLV. Future research plans will focus on that the control torque is transformed to control rudders. In this article, the effectiveness of the designed control schemes is evaluated in MATLAB/Simulink environment. In further research, using a real-time interface, the designed control schemes will be carried out under MATLAB/Simulink and runs on the DSPACE system, which is equipped by a power PC processor.