Output Feedback Stabilization of Spacecraft Autonomous Rendezvous Subject to Actuator Saturation

Qian Wang Guang-Ren Duan

Qian Wang and Guang-Ren Duan. Output Feedback Stabilization of Spacecraft Autonomous Rendezvous Subject to Actuator Saturation. International Journal of Automation and Computing, vol. 13, no. 5, pp. 428-437, 2016 doi:  10.1007/s11633-016-0952-z
Citation: Qian Wang and Guang-Ren Duan. Output Feedback Stabilization of Spacecraft Autonomous Rendezvous Subject to Actuator Saturation. International Journal of Automation and Computing, vol. 13, no. 5, pp. 428-437, 2016 doi:  10.1007/s11633-016-0952-z

doi: 10.1007/s11633-016-0952-z
基金项目: 

National Basic Research Program (973) of China 2012CB821205

Innovative Team Program of National Natural Science Foundation of China 61321062

Output Feedback Stabilization of Spacecraft Autonomous Rendezvous Subject to Actuator Saturation

Funds: 

National Basic Research Program (973) of China 2012CB821205

Innovative Team Program of National Natural Science Foundation of China 61321062

More Information
    Author Bio:

    Guang-Ren Duan received the B. Sc. degree in applied mathematics, and both the M. Sc. and Ph. D. degrees in control systems theory. From 1989 to 1991, he was a post-doctoral researcher at Harbin Institute of Technology, where he became a professor of control systems theory in 1991. Since August 2000, he has been elected specially employed professor at Harbin Institute of Technology sponsored by the Cheung Kong Scholars Program of the Chinese government. He is currently the director of the center for control theory and guidance technology at Harbin Institute of Technology. He is a chartered engineer in the UK, a senior member of IEEE and a fellow of IEE. His research interests include robust control, eigenstructure assignment, descriptor systems, missile autopilot control and magnetic bearing control. E-mail: g.r.duan@hit.edu.cn

    Corresponding author: Qian Wang received her B. Sc. degree in automation in 2008 and received M. Sc. degree in control theory and control engineering in 2011. She is currently the Ph. D. candidate of Center for Control Theory and Guidance Technology at Harbin Institute of Technology, China. Her research interests include gain scheduling control, nonlinear system, and spacecraft rendezvous. E-mail: wq@hit.edu.cn (Corresponding author)
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出版历程
  • 收稿日期:  2014-05-03
  • 录用日期:  2014-12-04
  • 网络出版日期:  2016-03-11
  • 刊出日期:  2016-10-01

Output Feedback Stabilization of Spacecraft Autonomous Rendezvous Subject to Actuator Saturation

doi: 10.1007/s11633-016-0952-z
    基金项目:

    National Basic Research Program (973) of China 2012CB821205

    Innovative Team Program of National Natural Science Foundation of China 61321062

    作者简介:

    Guang-Ren Duan received the B. Sc. degree in applied mathematics, and both the M. Sc. and Ph. D. degrees in control systems theory. From 1989 to 1991, he was a post-doctoral researcher at Harbin Institute of Technology, where he became a professor of control systems theory in 1991. Since August 2000, he has been elected specially employed professor at Harbin Institute of Technology sponsored by the Cheung Kong Scholars Program of the Chinese government. He is currently the director of the center for control theory and guidance technology at Harbin Institute of Technology. He is a chartered engineer in the UK, a senior member of IEEE and a fellow of IEE. His research interests include robust control, eigenstructure assignment, descriptor systems, missile autopilot control and magnetic bearing control. E-mail: g.r.duan@hit.edu.cn

    通讯作者: Qian Wang received her B. Sc. degree in automation in 2008 and received M. Sc. degree in control theory and control engineering in 2011. She is currently the Ph. D. candidate of Center for Control Theory and Guidance Technology at Harbin Institute of Technology, China. Her research interests include gain scheduling control, nonlinear system, and spacecraft rendezvous. E-mail: wq@hit.edu.cn (Corresponding author)

English Abstract

Qian Wang and Guang-Ren Duan. Output Feedback Stabilization of Spacecraft Autonomous Rendezvous Subject to Actuator Saturation. International Journal of Automation and Computing, vol. 13, no. 5, pp. 428-437, 2016 doi:  10.1007/s11633-016-0952-z
Citation: Qian Wang and Guang-Ren Duan. Output Feedback Stabilization of Spacecraft Autonomous Rendezvous Subject to Actuator Saturation. International Journal of Automation and Computing, vol. 13, no. 5, pp. 428-437, 2016 doi:  10.1007/s11633-016-0952-z
    • Spacecraft rendezvous is an important technology for the present and the future space missions[1]. There have been over 100 times of rendezvous operations since 1960s. From the technical point of view, the spacecraft rendezvous will become more and more autonomous in the future[1-3]. Firstly, some manned rendezvous missions were performed with great difficulty, such as high cost and high-risk. In order to save costs effectively and reduce risk, autonomous rendezvous is a better choice. Secondly, different from the conventional manned rendezvous mission, the autonomous rendezvous can reduce the working pressure on astronauts and improve the reliability of the mission.

      Considering a target spacecraft in a circular orbit and another chaser spacecraft in its neighborhood, the relative motion between two neighboring spacecrafts can be described by autonomous nonlinear differential equations. If the distance between them is much smaller than the orbit radius, the model is given by C-W equation, derived by Clohessey and Wiltshire in 1960[4]. Many advanced methods have been used to solve the rendezvous control problem. For example, model predictive control approach was developed for the spacecraft rendezvous in [5, 6], an adaptive backstepping control law is designed for spacecraft in proximity operation missions in [7], and the problem of rendezvous was cast into a stabilization problem analyzed by Lyapunov theory in [8].

      For astronautic missions, the control input constraint is an important issue that we must pay attention to. In practical engineering, the orbital control input force is limited due to the constraints of the thrust equipment, the limited quantity of the fuel and the power the engine provided and so on. Among the constraints, the constraint of thrust due to constrained thrust equipment is very important. So far, there are some results on these problems[9-13].

      In some situations, the chaser spacecraft is unable to fully obtain the real-time orbital information of the target spacecraft. For example, the trajectories of the two spacecraft are beyond the monitoring range of ground facilities which are unable to transmit real-time orbit information; the spacecraft autonomous rendezvous is performed near planets or other celestial bodies; or no communication between two spacecraft caused by the failure of target spacecraft. Under these situations, output feedback control is more applicable. Since the ability of static output feedback is generally limited, it is more realistic to use an observer-based output feedback controller, which is a dynamic output feedback controller that estimates the system states online. Therefore, from the practical point of view, observer-based output feedback design is important.

      The gain-scheduling approach is perhaps one of the most popular nonlinear control design approaches which has been widely applied in fields ranging from aerospace to process control[14, 15]. There are basically three kinds of scheduling approaches, namely, the continuous static scheduling approach[16-19], the continuous dynamic scheduling approach[20-22], and the discrete scheduling approach[23-25].

      Motivated by the above discussions, in this paper, we intend to develop an effective observer-based output feedback continuous dynamic gain scheduled controller based on the parametric Lyapunov equation approach[26] and gain scheduling technique. The proposed method is a solution of the semi-global stabilization problem for the circular orbit rendezvous system subject to actuator saturation by recognizing that the C-W equation is asymptotically null controllable with bounded controls (ANCBC). The main advantage of the proposed gain scheduling approach is that the time required for accomplishing the rendezvous mission can be significantly reduced.

      The rest of this paper is organized as follows. Section 2 presents the dynamic model of spacecraft autonomous rendezvous, and the stabilization control problem is formulated. In Section 3, the output feedback continuous dynamic gain scheduling controller is proposed to solve the semi-global stabilization control problem for spacecrafts autonomous rendezvous. Then, the numerical simulation is given to illustrate the effectiveness of the presented approach in Section 4. Finally, Section 5 concludes the paper.

      Notations. Throughout the paper, the notation used is fairly standard. We use AT, tr(A), λmin(A) and λmax(A) to denote the transpose, the trace, the minimal eigenvalue of matrix A and the maximal eigenvalue of matrix A, respectively. diag{·} stands for a block-diagonal matrix. In denotes the n×n identity matrix. I[p, q] denotes the sets of integers [p, p+1, …, q]. For a real symmetric matrix P, the notation P > 0 is used to denote its positive definiteness. The function sgn is defined as sgn(y)=1 if y≥0 and sgn(y)=-1 if y < 0. The function sat:RmRm is a vector-valued standard saturation function, i.e.

      $\text{sat}(u)={{\left[\begin{matrix} \text{sat}({{u}_{1}}) & \text{sat}({{u}_{2}}) & \cdots & \text{sat}({{u}_{m}}) \\ \end{matrix} \right]}^{\text{T}}}$

      and $\text{sat}({{u}_{i}})=\text{sgn}({{u}_{i}})\min \left\{ 1,\left| {{u}_{i}} \right| \right\},i\in \text{I}\left[1,m \right].$

    • In this section, with the independent continuous control accelerations being used as the control signals to C-W equations[4], we establish the relative motion equation with input saturation. Then, the control problems to be concerned in this paper are presented.

      The spacecraft rendezvous system is illustrated in Fig. 1. We assume that the two spacecraft (the target and chaser) are adjacent, and that the target spacecraft is in a circular orbit whose radius is R. Assume that the vector from the target spacecraft to the chaser spacecraft is denoted by r. The right-handed coordinate system (rotating coordinate system) O-XYZ is fixed at the center of mass of the target with X axis along the radial direction, Y axis along the flight direction of the target, and Z axis out of the orbit plane, respectively. Denote the gravitational parameter μ=GM where M is the mass of the center of the planet and G is the gravitational constant. Then, the orbit rate is given by $n=\frac{{{\mu }^{\frac{1}{2}}}}{{{R}^{\frac{3}{2}}}}$.

      Figure 1.  Circular orbit and coordinate system

      The relative motion between the target and chaser can be governed by Newton's equations[27]

      $\left\{ \begin{array}{*{35}{l}} \ddot{x}=2n\dot{y}+{{n}^{2}}(R+x)-\sigma \mu (R+x)+\text{sa}{{\text{t}}_{{{\alpha }_{X}}}}({{a}_{x}}) \\ \ddot{y}=-2n\dot{x}+{{n}^{2}}y-\sigma \mu y+\text{sa}{{\text{t}}_{{{\alpha }_{Y}}}}({{a}_{y}}) \\ \ddot{z}=-\sigma \mu z+\text{sa}{{\text{t}}_{{{\alpha }_{Z}}}}({{a}_{z}}) \\ \end{array} \right.$

      (1)

      where $\sigma ={{({{(R+x)}^{2}}+{{y}^{2}}+{{z}^{2}})}^{-\frac{3}{2}}}$; ax, ay and az are the accelerations supplied by the thrusts in the three directions; x, y, and z denote respectively the relative positions information of the spacecraft rendezvous system in the three directions; αX, αY and αZ are respectively the maximal accelerations that the thrusts can generate in the three directions, and $\text{sa}{{\text{t}}_{\alpha }}(u)=\alpha \text{sat}(\frac{u}{%\alpha })$ with α>0 being the saturation level and $\mathrm{sat}% _{1}(u)$ being denoted by $\mathrm{sat}(u)$ in short. By the first order Taylor expansion of $\sigma$ at the origin, the obtained linearized equation of (1) is

      $\left\{ \begin{array}{*{35}{l}} \ddot{x}=2n\dot{y}+3{{n}^{2}}x+\text{sa}{{\text{t}}_{{{\alpha }_{X}}}}({{a}_{x}}) \\ \ddot{y}=-2n\dot{x}+\text{sa}{{\text{t}}_{{{\alpha }_{Y}}}}({{a}_{y}}) \\ \ddot{z}=-{{n}^{2}}z+\text{sa}{{\text{t}}_{{{\alpha }_{Z}}}}({{a}_{z}}) \\ \end{array} \right.$

      (2)

      which is known as the Hill's equation or Clohessy-Wiltshire equation[4] without saturation nonlinearity. Notice that, by denoting $D=\mathrm{diag}\{ \alpha _{X},\alpha _{Y},\alpha _{Z}\}$ and $a={{[{{a}_{x}},{{a}_{y}},{{a}_{z}}]}^{\text{T}}}$, we have

      $u\triangleq {{[\text{sa}{{\text{t}}_{{{\alpha }_{X}}}}({{a}_{x}}),\text{sa}{{\text{t}}_{{{\alpha }_{Y}}}}({{a}_{y}}),\text{sa}{{\text{t}}_{{{\alpha }_{Z}}}}({{a}_{z}})]}^{\text{T}}}=D\text{sat}({{D}^{-1}}a).$

      (3)

      By choosing the state vector and control vector

      ${{x}_{p}}={{\left[\begin{matrix} x & y & z & {\dot{x}} & {\dot{y}} & {\dot{z}} \\ \end{matrix} \right]}^{\text{T}}},U={{D}^{-1}}a$

      (4)

      the relative motion equation (2) can be rewritten as

      ${{\dot{x}}_{p}}={{A}_{p}}{{x}_{p}}+{{B}_{p}}\text{sat}(U)$

      (5)

      in which

      $\begin{align} & {{A}_{p}}=\left[\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 3{{n}^{2}} & 0 & 0 & 0 & 2n & 0 \\ 0 & 0 & 0 & -2n & 0 & 0 \\ 0 & 0 & -{{n}^{2}} & 0 & 0 & 0 \\ \end{matrix} \right] \\ & {{B}_{p}}=\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]D. \\ \end{align}$

      (6)

      We assume that only the relative position is available for feedback, namely, the output

      $y={{C}_{p}}{{x}_{p}}$

      (7)

      where

      ${{C}_{p}}=\left[\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ \end{matrix} \right]$

      (8)

      is available for the controller design. In this paper, an observer-based output feedback gain scheduled controller will be designed based on the relative position signal (7).

      The linearized version of the spacecraft rendezvous system, namely, the linear system in (5) and (7), possesses the following property.

      Proposition 1. The matrix pair $(A_{p},B_{p})$ is controllable, $(A_{p},C_{p})$ is observable and all the eigenvalues of $A_{p}$ are on the imaginary axis, namely, $(A_{p},B_{p})$ is ANCBC.

      For linear systems, it is well known that the separation principle[28] holds: the state feedback and the observer gain can be designed separately to guarantee the stability of the augmented closed-loop system. The separation principle is also available for the design of control systems with input saturation[29].

      We now consider these problems and construct an observer-based gain scheduling output feedback controller. With this aim, we consider a full-order Luenberger state observer[28]

      ${{\dot{\hat{x}}}_{p}}={{A}_{p}}{{\hat{x}}_{p}}+{{B}_{p}}\text{sat}\left( U \right)-L{{C}_{p}}\left( {{x}_{p}}-{{{\hat{x}}}_{p}} \right)$

      (9)

      where $\widehat{x}_{p}\in \mathbf{R}^{6}$ is the estimate of the state and $L \in \mathbf{R}^{6\times 3}$ defines the estimation dynamics. The controller is given as

      $U={{K}_{p}}{{\hat{x}}_{p}}.$

      (10)

      Let $e=x_{p}-\widehat{x}_{p}$ be the observer error. By choosing

      $X=\left[\begin{matrix} {{x}_{p}} \\ e \\ \end{matrix} \right]=\left[\begin{matrix} {{I}_{6}} & 0 \\ {{I}_{6}} & -{{I}_{6}} \\ \end{matrix} \right]\left[\begin{matrix} {{x}_{p}} \\ {{{\hat{x}}}_{p}} \\ \end{matrix} \right]$

      the augmented closed-loop system is written as

      $\left\{ \begin{array}{*{35}{l}} \dot{X}=AX+B\text{sat}(U) \\ y=CX \\ \end{array} \right.$

      (11)

      where

      $A=\left[\begin{matrix} {{A}_{p}} & 0 \\ 0 & {{A}_{p}}+L{{C}_{p}} \\ \end{matrix} \right],B=\left[\begin{matrix} {{B}_{p}} \\ 0 \\ \end{matrix} \right],C=\left[\begin{matrix} {{C}_{p}} & 0 \\ \end{matrix} \right]$

      (12)

      $U=KX=\left[\begin{matrix} {{K}_{p}} & -{{K}_{p}} \\ \end{matrix} \right]\left[\begin{matrix} {{x}_{p}} \\ e \\ \end{matrix} \right].$

      (13)

      The spacecraft rendezvous process can be described by the transformation of the state vector $x_{p}$ from nonzero initial states $x_{p}(t_{0})\neq 0$ to the terminal state $x_{p}(t_{f})=0$, where $t_{f}$ is the rendezvous time. In this paper, the proposed gain scheduled control law will be utilized to accomplish such a mission.

      The problem we are to solve can then be stated as follows.

      Problem 1. Design an output feedback continuous dynamic gain scheduled controller $U$ for system composed by (5) and (7) such that the closed-loop system is asymptotically stable with an arbitrary large yet bounded domain of attraction.

      The following lemma will be used in this paper.

      Lemma 1[30]. Let $L,E$ and $N$ are real matrices of appropriate dimensions with $N^{\mathrm{T}}N\leq I_{n}$. Then, for any scalar $% \delta >0$, there holds $LNE+E^{\mathrm{T}}N^{\mathrm{T}}L^{\mathrm{T}}\leq \delta ^{-1}LL^{\mathrm{T}}+\delta E^{\mathrm{T}}E.$

    • In this section, we will design an observer-based output feedback gain scheduled controller to solve Problem 1. Our obtained solution is mainly based on a class of parametric Lyapunov equation.

      Lemma 2[26]. Let $\gamma>0$ be a given scalar. Consider the following parametric Lyapunov equation

      $P{{A}_{p}}+A_{p}^{\text{T}}P-P{{B}_{p}}B_{p}^{\text{T}}P+\gamma P=0$

      (14)

      and the associated feedback gain $K_{p}=-B_{p}^{\mathrm{T}}P$.

      1) There exists a unique matrix $P(\gamma)>0$ which solves the parametric Lyapunov equation (14), $P(\gamma)=W^{-1}(\gamma)$, where $W(\gamma)$ is the unique positive-definite solution to the following Lyapunov matrix equation

      $W{{\left( {{A}_{p}}+\frac{\gamma }{2}I \right)}^{\text{T}}}+\left( {{A}_{p}}+\frac{\gamma }{2}I \right)W={{B}_{p}}B_{p}^{\text{T}}.$

      (15)

      2) There holds $\lambda _{i}\left( A_{p}+B_{p}K_{p}\right) = -\frac{\gamma }{2}% ,i\in \mathrm{I}\left[1,6\right] .$ Hence, if $\gamma \geq 0,$ then the state of the closed-loop system $\dot{x}_{p}\left( t\right) =\left( A_{p}+B_{p}K_{p}\right) x_{p}\left( t\right) $ converges to the origin no slower than $\exp \left( -\frac{\gamma }{2}t \right).$

      3) For any $\gamma \in \mathbf{R}$, the unique positive definite solution $P(\gamma )$ is continuously differentiable and strictly increasing with respect to $\gamma $, i.e. $\frac{{\rm d}P( \gamma )}{{\mathrm{d}}\gamma} >0$.

      4) $\underset{r\rightarrow 0^{+}}{\lim }P(\gamma)=0$.

      5) $\mathrm{tr}\left( B_{p}^\mathrm{T}P\left( \gamma \right) B_{p}\right) =6\gamma.$

      Let

      $P(\gamma )=\left[\begin{matrix} 12\gamma P(\gamma ) & 0 \\ 0 & \delta \left( \gamma \right){{P}_{e}} \\ \end{matrix} \right]$

      (16)

      where $P_{e}$ is the unique positive definite solution to the following Lyapunov equation

      ${{({{A}_{p}}+L{{C}_{p}})}^{\text{T}}}{{P}_{e}}+{{P}_{e}}({{A}_{p}}+L{{C}_{p}})=-{{I}_{6}}$

      (17)

      and

      $\delta \left( \gamma \right)=\frac{12\gamma \text{tr}(P)}{%{{\lambda }_{\min }}\left( {{P}_{e}} \right)}.$

      (18)

      Assume that the initial condition for system (11) comes from a prescribed bounded set $\Omega \in \mathbf{R}^{12}$. Then, we define $\gamma _{0}$ as

      ${{\gamma }_{0}}={{\gamma }_{0}}\left( \Omega \right)=\underset{X\in \Omega }{\mathop{\min }}\,\left\{ \gamma :{{X}^{\text{T}}}P\left( \gamma \right)X=1 \right\}.$

      (19)

      The existence of $\gamma_{0}$ is due to the facts that $\lim_{\gamma \rightarrow 0^{+}}P\left( \gamma\right) =0$ and that $\frac{{\rm d}{P}\left( \gamma\right)}{{\rm d}\gamma }>0$.

      Now, we define the ellipsoids associate with the quadratic function ${{X}^{\text{T}}}P(\gamma )X$ as

      $E(P(\gamma ))=\left\{ X\in {{\mathbf{R}}^{12}}:{{X}^{\text{T}}}P(\gamma )X\le 1 \right\}.$

      (20)

      We also consider the following sets

      $L=\left\{ X\in {{\mathbf{R}}^{12}}:\left\| KX \right\|\le 1 \right\}$

      (21)

      which is the area in the state space where the actuators with the control $U=KX,$ are not saturated. In view of (20), we have, for any $X\in{E}({P}(\gamma))$.

      $\begin{align} & {{\left\| KX \right\|}^{2}}={{X}^{\text{T}}}{{K}^{\text{T}}}KX= \\ & \left[ \begin{matrix} x_{p}^{\text{T}} & {{e}^{\text{T}}} \\ \end{matrix} \right]\left[ \begin{matrix} -P{{B}_{p}} \\ P{{B}_{p}} \\ \end{matrix} \right]\times \\ & \left[ \begin{matrix} -B_{p}^{\text{T}}P & B_{p}^{\text{T}}P \\ \end{matrix} \right]\left[ \begin{matrix} {{x}_{p}} \\ e \\ \end{matrix} \right]= \\ & {{X}^{\text{T}}}\left[ \begin{matrix} P{{B}_{p}}B_{p}^{\text{T}}P & -P{{B}_{p}}B_{p}^{\text{T}}P \\ -P{{B}_{p}}B_{p}^{\text{T}}P & P{{B}_{p}}B_{p}^{\text{T}}P \\ \end{matrix} \right]X= \\ & {{X}^{\text{T}}}\left[ \begin{matrix} {{P}^{\frac{1}{2}}} & 0 \\ 0 & {{P}^{\frac{1}{2}}} \\ \end{matrix} \right]\times \\ & \left[ \begin{matrix} {{P}^{\frac{1}{2}}}{{B}_{p}}B_{p}^{\text{T}}{{P}^{\frac{1}{2}}} & -{{P}^{\frac{1}{2}}}{{B}_{p}}B_{p}^{\text{T}}{{P}^{\frac{1}{2}}} \\ -{{P}^{\frac{1}{2}}}{{B}_{p}}B_{p}^{\text{T}}{{P}^{\frac{1}{2}}} & {{P}^{\frac{1}{2}}}{{B}_{p}}B_{p}^{\text{T}}{{P}^{\frac{1}{2}}} \\ \end{matrix} \right]\times \\ & \left[ \begin{matrix} {{P}^{\frac{1}{2}}} & 0 \\ 0 & {{P}^{\frac{1}{2}}} \\ \end{matrix} \right]X\le \\ & \left( \text{tr}\left( {{P}^{\frac{1}{2}}}{{B}_{p}}B_{p}^{\text{T}}{{P}^{\frac{1}{2}}} \right)+\text{tr}\left( {{P}^{\frac{1}{2}}}{{B}_{p}}B_{p}^{\text{T}}{{P}^{\frac{1}{2}}} \right) \right){{X}^{\text{T}}}\times \\ & \left[ \begin{matrix} {{P}^{\frac{1}{2}}} & 0 \\ 0 & {{P}^{\frac{1}{2}}} \\ \end{matrix} \right]\left[ \begin{matrix} {{P}^{\frac{1}{2}}} & 0 \\ 0 & {{P}^{\frac{1}{2}}} \\ \end{matrix} \right]X= \\ & \left( \text{tr}\left( B_{p}^{\text{T}}P{{B}_{p}} \right)+\text{tr}\left( B_{p}^{\text{T}}P{{B}_{p}} \right) \right)\times \\ & {{X}^{\text{T}}}\left[ \begin{matrix} P & 0 \\ 0 & P \\ \end{matrix} \right]X= \\ & 12\gamma {{X}^{\text{T}}}\left[ \begin{matrix} P & 0 \\ 0 & P \\ \end{matrix} \right]X= \\ & {{X}^{\text{T}}}\left[ \begin{matrix} 12\gamma P & 0 \\ 0 & 12\gamma P \\ \end{matrix} \right]X= \\ & {{X}^{\text{T}}}\tilde{P}(\gamma )X. \\ \end{align}$

      Then, it follows from (18) and the fact ${{\lambda }_{\min }}\left( Q \right)I<Q<{{\lambda }_{\max }}\left( Q \right)I$ for any symmetric matrix $Q>0$ that $\widetilde{P}(\gamma) \leq {P}(\gamma)$. Then, we can conclude that

      $E(P(\gamma ))\subseteq L.$

      (22)

      Hence, the actuators are not saturated for any $X\in {E}({P}(\gamma))$ and $\mathrm{sat}\left( {K}X\right) $ can be simplified as ${K}X,\forall X\in {E}% ({P}(\gamma)),$ namely

      $X\in E(P(\gamma ))\Rightarrow \text{sat}\left( KX \right)=KX.$

      (23)

      Now, we present the main result of this section.

      Theorem 1. Let $P(\gamma )$ be the unique positive definite solution to the parametric Lyapunov equation in (14). Then, the continuous dynamic gain scheduled output feedback controller

      $U=K\left( \gamma \right)X$

      (24)

      and the following scheduled function

      $\dot{\gamma }=\frac{{{\gamma }^{2}}(1-\text{sat}(\frac{\gamma }{{{\gamma }^{*}}}))x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}}{\rho +\tau \Psi }$

      (25)

      solve Problem 1 with the set

      $E(P({{\gamma }_{0}}))=\left\{ X\in {{\mathbf{R}}^{12}}:{{X}^{\text{T}}}P({{\gamma }_{0}})X\le 1 \right\}$

      within the domain of attraction of the closed-loop system, where

      $K\left( \gamma \right)=\left[\begin{matrix} -B_{p}^{\text{T}}P\left( \gamma \right) & B_{p}^{\text{T}}P\left( \gamma \right) \\ \end{matrix} \right]$

      (26)

      ${{\gamma }^{*}}=\frac{\tau -\varsigma }{12\tau {{\lambda }_{\min }}\left( {{P}_{e}} \right)+\eta }$

      (27)

      and

      $\begin{align} & \Psi =x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+\gamma x_{p}^{\text{T}}\frac{\text{d}P\left( \gamma \right)}{\text{d}\gamma }{{x}_{p}}+ \\ & \frac{{{\lambda }_{\max }}\left( {{P}_{e}} \right)x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}\left( \text{tr}\left( P \right)+\gamma \text{tr}\left( \frac{\text{d}P\left( \gamma \right)}{\text{d}\gamma } \right) \right)}{\eta \text{tr}\left( P \right)} \\ \end{align}$

      (28)

      here $\rho>0$, $\tau>1$ and $\eta>0$ are given scalars, $\varsigma>0$ is a sufficiently small positive number and $\gamma \left( t\right) $ is uniformly ultimately bounded, namely

      $\gamma (t)\le {{\gamma }^{*}}.$

      (29)

      Proof. For simplicity, we denote $P(\gamma(t))$ as $P(\gamma)$ and $\gamma(t)$ as $\gamma$. It follows from (29) and (27) that

      $\dot{\gamma }=\left\{ \begin{matrix} \frac{{{\gamma }^{2}}\delta x_{p}^{\text{T}}P{{x}_{p}}}{\rho +\tau \Psi },0<\frac{\gamma }{{{\gamma }^{*}}}<1 \\ 0,\frac{\gamma }{{{\gamma }^{*}}}\ge 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.$

      (30)

      where $\delta=1-\frac{\gamma }{\gamma ^{\ast }}$ which satisfies $0<\delta<1$. Hence,

      $\dot{\gamma }\le \frac{{{\gamma }^{2}}x_{p}^{\text{T}}P{{x}_{p}}}{\rho +\tau \Psi }.$

      Notice that, as $\frac{\mathrm{d}}{\mathrm{d} \gamma }P\left( \gamma \right) >0,\forall \gamma>0,$ the right-hand-side of (26) is well-defined for all $t\geq0.$ We write the closed-loop system as

      $\dot{X}=AX+B\text{sat}\left( KX \right).$

      (31)

      Consider the following Lyapunov function

      $V\left( X \right)=V\left( {{x}_{p}} \right)+V\left( e \right)$

      (32)

      where

      $V\left( {{x}_{p}} \right)=12\gamma x_{p}^{\text{T}}P(\gamma ){{x}_{p}}$

      (33)

      and

      $V\left( e \right)=\delta (\gamma ){{e}^{\text{T}}}{{P}_{e}}e.$

      (34)

      Then, the time-derivative of $% V\left( X\right) $ along the trajectories of system (31) can be evaluated as

      $\dot{V}\left( X \right)=\dot{V}\left( {{x}_{p}} \right)+\dot{V}\left( e \right).$

      (35)

      According to (25), we have

      $\begin{align} & \dot{V}\left( {{x}_{p}} \right)=24\gamma \dot{x}_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+12\dot{\gamma }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+ \\ & 12\gamma \dot{\gamma }x_{p}^{\text{T}}\frac{\text{d}P\left( \gamma \right)}{\text{d}\gamma }{{x}_{p}}= \\ & 24\gamma {{\left( {{A}_{p}}{{x}_{p}}+{{B}_{p}}\text{sat}\left( K(\gamma )X \right) \right)}^{\text{T}}}P\left( \gamma \right){{x}_{p}}+ \\ & 12\dot{\gamma }\left( x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+\gamma x_{p}^{\text{T}}\frac{\text{d}P\left( \gamma \right)}{\text{d}\gamma }{{x}_{p}} \right)\le \\ & 24\gamma {{\left( {{A}_{p}}{{x}_{p}}+{{B}_{p}}\text{sat}\left( K(\gamma )X \right) \right)}^{\text{T}}}P\left( \gamma \right){{x}_{p}}+ \\ & \frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}} \\ \end{align}$

      (36)

      which, in view of (23), can be continued as

      $\begin{align} & \dot{V}({{x}_{p}})\le 12\gamma x_{p}^{\text{T}}\left( P\left( \gamma \right){{A}_{p}}+A_{p}^{\text{T}}P\left( \gamma \right) \right){{x}_{p}}+ \\ & 24\gamma {{X}^{\text{T}}}{{K}^{\text{T}}}(\gamma )B_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+\frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}= \\ & 12\gamma x_{p}^{\text{T}}\left( -\gamma P\left( \gamma \right)+P\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}P\left( \gamma \right) \right){{x}_{p}}+ \\ & 24\gamma \left[\begin{matrix} x_{p}^{\text{T}} & {{e}^{\text{T}}} \\ \end{matrix} \right]\left[\begin{matrix} -P\left( \gamma \right){{B}_{p}} \\ P\left( \gamma \right){{B}_{p}} \\ \end{matrix} \right]B_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+ \\ & \frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}= \\ & -12{{\gamma }^{2}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+12\gamma x_{p}^{\text{T}}P\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}- \\ & 24\gamma x_{p}^{\text{T}}P\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+\frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+ \\ & 24\gamma {{e}^{\text{T}}}P\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}. \\ \end{align}$

      (37)

      Then, in view of Lemma 1, (37) can be further continued as

      $\begin{align} & \dot{V}\left( {{x}_{p}} \right)\le -12{{\gamma }^{2}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+12\gamma x_{p}^{\text{T}}P\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}- \\ & 24\gamma x_{p}^{\text{T}}P\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+ \\ & 12\gamma x_{p}^{\text{T}}P\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+ \\ & 12\gamma {{e}^{\text{T}}}P\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}P\left( \gamma \right)e+\frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}= \\ & -12{{\gamma }^{2}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+12\gamma {{e}^{\text{T}}}P\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}P\left( \gamma \right)e+ \\ & \frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}= \\ & -12{{\gamma }^{2}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+6\gamma {{e}^{\text{T}}}\left[ \begin{matrix} {{P}^{\frac{1}{2}}}\left( \gamma \right) & {{P}^{\frac{1}{2}}}\left( \gamma \right) \\ \end{matrix} \right]\times \\ & \left[ \begin{matrix} {{P}^{\frac{1}{2}}}\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}{{P}^{\frac{1}{2}}}\left( \gamma \right) & 0 \\ 0 & {{P}^{\frac{1}{2}}}\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}{{P}^{\frac{1}{2}}}\left( \gamma \right) \\ \end{matrix} \right]\times \\ & [\begin{matrix} {{P}^{\frac{1}{2}}}\left( \gamma \right) \\ {{P}^{\frac{1}{2}}}\left( \gamma \right) \\ \end{matrix}]e+\frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}\le \\ & -12{{\gamma }^{2}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+ \\ & 12\gamma \left( 2\text{tr}\left( {{P}^{\frac{1}{2}}}\left( \gamma \right){{B}_{p}}B_{p}^{\text{T}}{{P}^{\frac{1}{2}}}\left( \gamma \right) \right) \right){{e}^{\text{T}}}P\left( \gamma \right)e+ \\ & \frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}= \\ & -12{{\gamma }^{2}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+ \\ & 12\gamma \left( \text{tr}\left( B_{p}^{\text{T}}P\left( \gamma \right){{B}_{p}} \right)+\text{tr}\left( B_{p}^{\text{T}}P\left( \gamma \right){{B}_{p}} \right) \right)\times \\ & {{e}^{\text{T}}}P\left( \gamma \right)e+\frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}= \\ & -12{{\gamma }^{2}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+144{{\gamma }^{2}}{{e}^{\text{T}}}P\left( \gamma \right)e+ \\ & \frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}\le \\ & -12{{\gamma }^{2}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+144{{\gamma }^{2}}\text{tr}\left( P\left( \gamma \right) \right){{e}^{\text{T}}}e+ \\ & \frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}. \\ \end{align}$

      The derivative of $\delta$ with respect to $t$ is

      $\begin{align} & \dot{\delta }\left( \gamma \right)=\frac{12\dot{\gamma }\text{tr}\left( P \right)}{{{\lambda }_{\min }}\left( {{P}_{e}} \right)}+\frac{12\gamma \dot{\gamma }\text{tr}\left( \frac{\text{d}P}{\text{d}\gamma } \right)}{{{\lambda }_{\min }}\left( {{P}_{e}} \right)}= \\ & \frac{12\dot{\gamma }}{{{\lambda }_{\min }}\left( {{P}_{e}} \right)}\left( \text{tr}\left( P \right)+\gamma \text{tr}\left( \frac{\text{d}P}{\text{d}\gamma } \right) \right). \\ \end{align}$

      (38)

      On the other hand, it follows from (17), (25) and (38) that

      $\begin{align} & \dot{V}\left( e \right)=2\delta \left( \gamma \right){{{\dot{e}}}^{\text{T}}}{{P}_{e}}e+\dot{\delta }(\gamma ){{e}^{\text{T}}}{{P}_{e}}e\le \\ & 2\delta \left( \gamma \right){{\left( \left( {{A}_{p}}+L{{C}_{p}} \right)e \right)}^{\text{T}}}{{P}_{e}}e+ \\ & \frac{12\eta {{\gamma }^{2}}\text{tr}(P)}{\tau {{\lambda }_{\min }}\left( {{P}_{e}} \right){{\lambda }_{\max }}\left( {{P}_{e}} \right)}{{e}^{\text{T}}}{{P}_{e}}e\le \\ & \delta \left( \gamma \right){{e}^{\text{T}}}\left( {{P}_{e}}\left( {{A}_{p}}+L{{C}_{p}} \right)+{{\left( {{A}_{p}}+L{{C}_{p}} \right)}^{\text{T}}}{{P}_{e}} \right)e+ \\ & \frac{12\eta {{\gamma }^{2}}\text{tr}(P)}{\tau {{\lambda }_{\min }}\left( {{P}_{e}} \right)}{{e}^{\text{T}}}e=-\delta \left( \gamma \right){{e}^{\text{T}}}e+\frac{12\eta {{\gamma }^{2}}\text{tr}(P)}{\tau {{\lambda }_{\min }}\left( {{P}_{e}} \right)}{{e}^{\text{T}}}e. \\ \end{align}$

      Thus, we obtain

      $\begin{align} & \dot{V}\left( X \right)\le -12{{\gamma }^{2}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}+144{{\gamma }^{2}}\text{tr}\left( P\left( \gamma \right) \right){{e}^{\text{T}}}e+ \\ & \frac{12{{\gamma }^{2}}}{\tau }x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}-\delta \left( \gamma \right){{e}^{\text{T}}}e+\frac{12\eta {{\gamma }^{2}}\text{tr}(P)}{\tau {{\lambda }_{\min }}\left( {{P}_{e}} \right)}{{e}^{\text{T}}}e. \\ \end{align}$

      (39)

      Then, it follows from (18), (27) and (29) that

      $\dot{V}\left( X \right)\le -{{\varepsilon }_{1}}x_{p}^{\text{T}}P\left( \gamma \right){{x}_{p}}-{{\varepsilon }_{2}}{{e}^{\text{T}}}e<0$

      (40)

      where

      ${{\varepsilon }_{1}}=12{{\gamma }^{2}}\left( 1-\frac{1}{\tau } \right)>0$

      and

      $\begin{align} & {{\varepsilon }_{2}}=\delta \left( \gamma \right)-144{{\gamma }^{2}}\text{tr}\left( P \right)-\frac{12\eta {{\gamma }^{2}}\text{tr}\left( P \right)}{\tau {{\lambda }_{\min }}\left( {{P}_{e}} \right)}= \\ & \frac{12\gamma \text{tr}\left( P \right)}{\tau {{\lambda }_{\min }}\left( {{P}_{e}} \right)}\left( \tau -12\gamma \tau {{\lambda }_{\min }}\left( {{P}_{e}} \right)-\eta \gamma \right)>0. \\ \end{align}$

      Hence, the closed-loop system is asymptotically stable with an arbitrary large yet bounded ellipsoid set ${E}\left( {P}(\gamma _{0})\right) $ contained in the domain of attraction. $\square$

      A couple of remarks on Theorem 1 are given in order.

      Remark 1. Since $(A_{p},C_{p})$ is observable, the poles of $A_{p}+LC_{p}$ can be placed arbitrarily by choosing a suitable gain matrix $L$ and thus $\lambda _{\min }\left( P_{e}\right) $ can be made arbitrarily small by making the real parts of the eigenvalues of $A_{p}+LC_{p}$ negative enough. Hence, it follows from (29) and (27) that the design parameter $\gamma(t)$ in Theorem 1 can be made big enough to improve the convergence rate of the closed-loop system. We know that a small $\lambda _{\min }\left( P_{e}\right) $ means a fast convergence rate of the error system[31]

      $\dot{e}=\left( {{A}_{p}}+L{{C}_{p}} \right)e$

      (41)

      which further implies that the observer based output feedback controller recovers almost the state feedback controller, that is, $K_{p} \widehat{x}_{p}\approx K_{p} x_{p}$.

      Remark 2. For any initial state $X(t_{0})$, $\gamma _{0}$ can be chosen as the solution to the nonlinear equation

      ${{X}^{\text{T}}}P\left( {{\gamma }_{0}} \right)X=1.$

      (42)

      The unique solution of (42) can be obtained by the bisection method in view of the monotonicity of ${P}(\gamma)$ with respect to $\gamma $[10].

      Remark 3. It follows from Theorem 1 that $\gamma$ becomes larger and larger during the convergence of the state. According to $\frac{\mathrm{d}P(\gamma)}{\mathrm{d}\gamma}>0$ that a larger $\gamma$ means a larger $P(\gamma)$, thus, a larger norm of the feedback gain $K=\left[ \begin{matrix} -B_{p}^{\text{T}}P\left( \gamma \right) & B_{p}^{\text{T}}P\left( \gamma \right) \\ \end{matrix} \right]$, which increases the control force and improves the convergence rate of the closed-loop system. Hence, the proposed gain scheduled controller (24)-(27) can improve the dynamic performance of the closed-loop system, which will be illustrated by numerical simulation in Section 4.

    • In this section, we will carry out numerical simulations to demonstrate the effectiveness of the proposed approach to the control of circular orbital rendezvous system. It follows from Proposition 1 that we can use the gain scheduled controller in (24)-(27) to solve the stabilization problem for the system composed by (5) and (7). Different from the controller design which is based on the linearized model (2), our simulation will be carried out directly on the nonlinear model described by (1).

      Suppose that the target spacecraft is on a geosynchronous orbit of radius $R=42 241 \mathrm{km}$ with an orbital period of 24 hours. Thus the orbit rate can be computed as $n=7.272 2\times10^{-5} \mathrm{rad/s}$. For simulation purpose, choose the initial condition in the target orbital coordinate system as

      ${{x}_{p}}(0)={{\left[\begin{matrix} 10000 & 10000 & 10000 & 5 & 3 & -1 \\ \end{matrix} \right]}^{\text{T}}}.$

      Namely, all the distances between the target and the chaser spacecraft in the three directions, X, Y and Z, are $10 000 \mathrm{m}$, respectively. The relative velocity are respectively $5 \mathrm{m/s}$, $3 \mathrm{m/s}$ and $-1 \mathrm{m/s}$ in the three directions. Assume that the accelerations supplied by the thrusts in the three directions satisfy, respectively, $\left \vert a_{x}\right \vert \leq0.1=2\alpha$, $\left \vert a_{y}\right \vert \leq0.1=2\alpha$, and $\left \vert a_{z}\right \vert \leq0.05=\alpha$, where $\alpha=0.05$. The initial condition for system (41) is chosen as

      $e\left( 0 \right)={{\left[\begin{matrix} 10 & 10 & 10 & 1 & 1 & 1 \\ \end{matrix} \right]}^{\text{T}}}.$

      We choose the expected poles of $A_{p}+LC_{p}$ as $\left\{ -\lambda ,-\lambda ,-\lambda +n\mathrm{j},-\lambda -n\mathrm{j},-\lambda +n\mathrm{j},-\lambda -n\mathrm{j}\right\}$, where $-\lambda$ is the real part of the expected closed-loop poles and $\mathrm{j}=\sqrt{-1}$. With the expected poles, we can compute the gain matrix $L$. In accordance to (17), we can compute the matrix $P_{e}$, $\lambda _{\max }\left( P_{e}\right)$ and $\lambda _{\min }\left( P_{e}\right)$, respectively. Set $\lambda$ as 0.1, 0.5, 1, 5, 10, 15, 20, 25, 30 and 35, respectively, the relationship between $\lambda _{\min }\left( P_{e}\right)$ and $\lambda$ is recorded in Fig. 2, which demonstrates that $\lambda _{\min }\left( P_{e}\right) $ can be small enough by choosing reasonable $\lambda$.

      Figure 2.  The curve of the relation between λ and λmin (Pe)

      Let $\lambda=1$, $\tau=1.000 1$, $\rho=0.01$, $\eta=10$ and $\varsigma=1\times10^{-10}$ for simulation. With these parameters, we get the observer gain matrix

      $L=\left[\begin{matrix} -2 & -0.00019731 & 6.7067\times {{10}^{-6}} \\ 0.0002517 & -2 & 3.2391\times {{10}^{-6}} \\ -2.8637\times {{10}^{-5}} & -3.0144\times {{10}^{-5}} & -2 \\ -1 & -0.00034275 & 6.7075\times {{10}^{-6}} \\ 0.00039715 & -0.99998 & 3.2373\times {{10}^{-6}} \\ -2.8634\times {{10}^{-5}} & -3.0145\times {{10}^{-5}} & -1 \\ \end{matrix} \right].$

      In view of Remark 2 and the prescribed initial conditions, we get $\gamma_{0}=0.000 305$. We can compute the unique positive definite solution to the parametric Lyapunov equation (14) as follows

      $P=\frac{1}{{{\alpha }^{2}}}\left[\begin{matrix} {{P}_{11}} & 0 & {{P}_{13}} & 0 \\ 0 & 2\gamma {{n}^{2}}+{{\gamma }^{3}} & 0 & {{\gamma }^{2}} \\ P_{13}^{\text{T}} & 0 & {{P}_{22}} & 0 \\ 0 & {{\gamma }^{2}} & 0 & 2\gamma \\ \end{matrix} \right]$

      (43)

      in which $P_{11},P_{13}$ and $P_{33}$ are respectively given by (44), (45) and (46). The matrix $\frac{\mathrm{d}P}{\mathrm{d}\gamma}$ can then be computed accordingly. The resulting observer-based output feedback gain scheduling controller can be computed according to (24)-(27). And we calculate $\gamma^{\ast}=0.073 999$. For comparison purpose, the closed-loop system will also be simulated for $\frac{\mathrm{d}\gamma }{ \mathrm{d}t}=0$ which corresponds to a normal time-invariant controller.

      The state trajectories of the closed-loop system are plotted in Figs. 3-5, observer errors are recorded in Figs. 6 and 7 and the control accelerations for the closed-loop system are recorded in Fig. 8, respectively. From Figs. 3-7, we can see that the rendezvous mission is accomplished by using the observer-based output feedback continuous dynamic gain scheduled controller (24)-(27) at about $t_{f}=14 000 \mathrm{s}$ which saves more than $6 000 \mathrm{s}$ compared with the normal time-invariant controller corresponding to $\frac{\mathrm{d}\gamma}{\mathrm{d}t}=0$, which shows the effectiveness of the proposed approach. Fig. 8 shows that the proposed controller not only makes full use of the actuator capacity, but also guarantees that the control inputs do not exceed the maximal control inputs during the whole rendezvous process. From Figs. 9 and 10, we can see that the parameter $\gamma(t)$ increases monotonically and is uniformly ultimately bounded. The ultimate value of $\gamma(t)$ is about 0.007 38 which satisfies (29) and (27), that is, $0.007 38<0.073 999$. Fig. 10 shows the derivative of the design parameter $\gamma(t)$ becomes zero when $\gamma(t)$ increases to the ultimate value.

      ${{P}_{11}}=\left[\begin{align} & \frac{\left( 2610{{n}^{10}}+3073{{\gamma }^{2}}{{n}^{8}}+1060{{\gamma }^{4}}{{n}^{6}}+174{{\gamma }^{6}}{{n}^{4}}+10{{n}^{2}}{{\gamma }^{8}}+{{\gamma }^{10}} \right)\gamma }{4\left( {{n}^{2}}+{{\gamma }^{2}} \right)\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)}-\frac{3{{n}^{3}}\left( 25{{n}^{4}}+18{{\gamma }^{2}}{{n}^{2}}+5{{\gamma }^{4}} \right){{\gamma }^{2}}}{2\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{3{{n}^{3}}\left( 25{{n}^{4}}+18{{\gamma }^{2}}{{n}^{2}}+5{{\gamma }^{4}} \right){{\gamma }^{2}}}{2\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\left( 25{{n}^{4}}+14{{\gamma }^{2}}{{n}^{2}}+{{\gamma }^{4}} \right)\left( {{n}^{2}}+{{\gamma }^{2}} \right){{\gamma }^{3}}}{4\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)} \\ \end{align} \right]$

      (44)

      ${{P}_{13}}=\left[\begin{align} & \frac{\left( {{\gamma }^{2}}+3{{n}^{2}} \right)\left( {{\gamma }^{6}}+9{{\gamma }^{4}}{{n}^{2}}+159{{\gamma }^{2}}{{n}^{4}}+367{{n}^{6}} \right){{\gamma }^{2}}}{4\left( {{n}^{2}}+{{\gamma }^{2}} \right)\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)}\frac{n\left( 720{{n}^{8}}+613{{n}^{6}}{{\gamma }^{2}}+111{{n}^{4}}{{\gamma }^{4}}+3{{n}^{2}}{{\gamma }^{6}}+{{\gamma }^{8}} \right)\gamma }{2\left( {{n}^{2}}+{{\gamma }^{2}} \right)\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{n\left( 55{{n}^{4}}+20{{\gamma }^{2}}{{n}^{2}}+{{\gamma }^{4}} \right){{\gamma }^{3}}}{2\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{\left( 75{{n}^{6}}-7{{\gamma }^{2}}{{n}^{4}}-11{{\gamma }^{4}}{{n}^{2}}-{{\gamma }^{6}} \right){{\gamma }^{2}}}{4\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)} \\ \end{align} \right]$

      (45)

      ${{P}_{33}}=\left[ \begin{matrix} \frac{\left( 45{{n}^{8}}+388{{n}^{6}}{{\gamma }^{2}}+138{{n}^{4}}{{\gamma }^{4}}+12{{n}^{2}}{{\gamma }^{6}}+{{\gamma }^{8}} \right)\gamma }{2\left( {{n}^{2}}+{{\gamma }^{2}} \right)\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)} & \frac{3{{n}^{3}}\left( 47{{n}^{4}}+10{{\gamma }^{2}}{{n}^{2}}-{{\gamma }^{4}} \right){{\gamma }^{2}}}{\left( {{n}^{2}}+{{\gamma }^{2}} \right)\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)} \\ \frac{3{{n}^{3}}\left( 47{{n}^{4}}+10{{\gamma }^{2}}{{n}^{2}}-{{\gamma }^{4}} \right){{\gamma }^{2}}}{\left( {{n}^{2}}+{{\gamma }^{2}} \right)\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)} & \frac{\left( 405{{n}^{8}}+244{{n}^{6}}{{\gamma }^{2}}+66{{n}^{4}}{{\gamma }^{4}}+12{{n}^{2}}{{\gamma }^{6}}+{{\gamma }^{8}} \right)\gamma }{2\left( {{n}^{2}}+{{\gamma }^{2}} \right)\left( 225{{n}^{6}}+91{{\gamma }^{2}}{{n}^{4}}+11{{\gamma }^{4}}{{n}^{2}}+{{\gamma }^{6}} \right)} \\ \end{matrix} \right].$

      (46)

      Figure 3.  Relative positions in the X-axis, Y-axis and Z-axis

      Figure 4.  Relative velocities in the X-axis, Y-axis and Z-axis

      Figure 5.  Relative distances and velocities, where $d=\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}$ and $\upsilon =\sqrt{{{{\dot{x}}}^{2}}+{{{\dot{y}}}^{2}}+{{{\dot{z}}}^{2}}}$

      Figure 6.  Observer position errors in the three directions

      Figure 7.  Observer velocity errors in the three directions

      Figure 8.  Control accelerations in the X-axis, Y-axis and Z-axis

      Figure 9.  The curve of design parameter γ(t)

      Figure 10.  The curve of $\frac{\text{d}\gamma \left( t \right)}{\text{d}t}$

    • This paper has proposed an observer-based output feedback continuous dynamic gain scheduled controller for a spacecraft rendezvous system subject to actuator saturation. The main advantage of the proposed method is that the convergence rate of the closed-loop system can be improved by the design parameter online. Simulation results show that the proposed approach finished the rendezvous mission successfully.

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