Nomenclature

$\Theta$

Attitude angle of the rigid platform

$\omega$

Angular velocity of the rigid platform

$J_0$

Inertia matrix of the rigid platform

$r_0$

Radius of the rigid platform

$w_L$

Vibration vector of the coilable mast

$\Phi$

Modal functions of the coilable mast

$\eta$

Modal coordinates of the coilable mast

$M, K, F$

Coupled matrices

$\Pi_1, \Pi_2$

Coupled matrices

$\sigma$

MRP, Modified Rodriguez Parameters

SOS

Sum of squares

LMI

Linear matrix inequality

LTI

Linear time invariant

2.0 Dynamics modeling

The diagram of the flexible satellite is illustrated in Fig. 1, featuring a flexible coilable mast exhibiting torsional and bending modes. The inertial frame and the satellite body-fixed frame are denoted by ${\mathcal{F}_i}\left( {{o_i}{x_i}{y_i}{z_i}} \right)$ and ${\mathcal{F}_b}\left( {{o_b}{x_b}{y_b}{z_b}} \right)$ . A floating frame, ${\mathcal{F}_L}\left( {{o_L}{x_L}{y_L}{z_L}} \right)$ , is employed to fix onto the flexible attachment and represent the vibration vector. For simplicity in describing the control problem, we disregard the motion of the satellite in orbit.

Figure 1. The diagram of the flexible satellite.

Regarding the rigid platform, ${\rm{\Theta }}\left( {\rm{t}} \right) = {\left[ {\begin{array}{c@{\quad}c@{\quad}c}\varphi & {}\phi & {}\theta \end{array}} \right]^T} \in {\mathbb{R}^3}$ represents the attitude angle, and $\omega \left( t \right) = {\left[ {\begin{array}{c@{\quad}c@{\quad}c}{{\omega _x}}& {}{{\omega _y}}& {}{{\omega _z}}\end{array}} \right]^T} \in {\mathbb{R}^3}$ represents the angular velocity. Concerning the flexible attachment, ${w_L}\left( {x,t} \right) = {\left[ {\begin{array}{c@{\quad}c@{\quad}c}{{w_x}} & {}{{w_y}} {}&{{w_z}}\end{array}} \right]^T} \in {\mathbb{R}^3}$ denotes its vibration vector, with ${w_x}\left( {x,t} \right)$ representing torsional deformation and ${w_i}\left( {x,t} \right)\left( {i = y,z} \right)$ representing bending deformation. The vibration vector is represented by ${r_P}\left( {x,t} \right) = {[\begin{array}{c@{\quad}c@{\quad}c}{{r_0} + x}& {}{{w_y}} {}&{{w_z}}\end{array}]^T}$ in the floating frame and ${r_L}\left( {x,t} \right) = R\left( {\rm{\Theta }} \right){r_P}\left( {x,t} \right)$ in the inertial frame, where $R\left( {\rm{\Theta }} \right)$ is the rotation matrix between the floating frame and the inertial frame. The kinetic energy of the rigid platform, torsional kinetic energy of the flexible attachment and bending kinetic energy of the flexible attachment are denoted by ${E_{k0}}$ , ${E_{k1}}$ and ${E_{k2}}$ , respectively. Then the kinetic energy of the entire satellite can be expressed as follows:

(1) \begin{align}{E_k} &= {E_{k0}} + {E_{k1}} + {E_{k2}}\nonumber \\[5pt] &= \frac{1}{2}{{\dot \Theta }^T}{J_0}\dot \Theta + \frac{1}{2}\overline {{J_x}} \int_0^L {{{\left( {\dot \varphi + {{\dot w}_x}} \right)}^2}dx} + \frac{1}{2}\overline {\rho A} \int_0^L {\dot r_L^T} {{\dot r}_L}dx.\end{align}

Where, ${J_0}$ denotes the inertia matrix of the rigid platform. The rotational inertia $\overline {{J_x}} $ and mass $\overline {\rho A} $ of the coilable mast, along with other equivalent parameters, can be calculated using the equivalent modeling method [Reference Liu, Cao, Zhang, Wei and Zhu35, Reference Liu, Cao and Zhu36]. The corresponding expressions are provided in the appendix.

The potential energy stored in the flexible attachment is presented as follows:

(2) \begin{align}{E_p} = {1 \over 2}\mathop \int \nolimits_0^L {\left( {w_L^{\prime\prime}} \right)^T}{D_L}w_L^{\prime\prime}dx.\end{align}

The coupling matrix ${D_L}$ is defined as follows:

\begin{align*}{D_L} = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{\overline {GJ} }&{{\kappa _1}}&{{\kappa _2}}\\[5pt]{{\kappa _1}}&{\overline {E{I_y}} }&{{\kappa _3}}\\[5pt]{{\kappa _2}}&{{\kappa _3}}&{\overline {E{I_z}} }\end{array}} \right].\end{align*}

The expressions of its coefficients are also provided in the appendix. The variation of the kinetic energy can be expressed as follows:

(3) \begin{align}\delta {E_{k0}} = - \delta {{\rm{\Theta }}^T}\left( {{J_0}\dot \omega + \omega \times {J_0}\omega } \right),\end{align}

(4) \begin{align}\delta {E_{k1}} &= - \overline {{J_x}} \int_0^L {\delta (\varphi + {w_x})\left( {\ddot \varphi + {{\ddot w}_x}} \right)dx} \nonumber\\[3pt] &= - \int_0^L {\left( {\delta {\Theta ^T} + \delta w_L^T} \right)\left( {\overline {{J_x}} {\Delta _1}} \right)\left( {\dot \omega + {{\ddot w}_L}} \right)dx},\end{align}

(5) \begin{align}\delta {E_{k2}} &= - \overline {\rho A} \int_0^L {\delta r_L^T{{\ddot r}_L}} dx\nonumber\\[3pt] &= - \int_0^L {\delta {{\left( {{{\dot r}_P} + \omega \times {r_P}} \right)}^T}\overline {\rho A} \left( {{{\ddot r}_p} + 2\omega \times {{\dot r}_p} + \dot \omega \times {r_p}} \right)} dx\nonumber\\[3pt] &= - \int_0^L {\left( {\delta w_L^T{\Delta _2} + \delta {\Theta ^T}{r_P} \times } \right)\overline {\rho A} \left( {{{\ddot r}_p} + 2\omega \times {{\dot r}_p} + \dot \omega \times {r_p} + {{\left( {\omega \times } \right)}^2}{r_p}} \right)} dx\nonumber\\[3pt] &\approx - \int_0^L {\left( {\delta w_L^T{\Delta _2} + \delta {\Theta ^T}{r_P} \times } \right)\overline {\rho A} \left( {{{\ddot r}_p} + 2\omega \times {{\dot r}_p} + \dot \omega \times {r_p} + {{\left( {\omega \times } \right)}^2}{r_p}} \right)} dx \end{align}

Where ${{\rm{\Delta }}_1} = diag\left\{ {1,0,0} \right\}$ and ${{\rm{\Delta }}_2} = diag\left\{ {0,1,1} \right\}$ . Due to the microsatellite’s generally small angular velocity, the quadratic coupling term ${\left( {\omega \times } \right)^2}$ in Equation (5) is neglected. Let $\omega \times \in {\mathbb{R}^{3 \times 3}}$ denote the skew-symmetric matrix of vector $\omega $ . The variation of the potential energy is then presented as follows:

(6) \begin{align}\delta {E_U} = \mathop \int \nolimits_0^L \delta w_L^T{D_L}w_L^{\left( 4 \right)}dx.\end{align}

The general form of Hamilton’s principle of variation principle is expressed as follows [Reference Meng, He, Yang, Liu and You37]:

(7) \begin{align}\mathop \int \nolimits_{{t_1}}^{{t_2}} \left( {\delta {E_K} - \delta {E_U} + \delta W} \right)dt = 0.\end{align}

Where, $\delta W = \delta {{\rm{\Theta }}^T}\left( {{T_c} + {T_d}} \right)$ denotes the variation of the work done by the external torque, while ${T_c} \in {\mathbb{R}^3}$ and ${T_d} \in {\mathbb{R}^3}$ denote the control torque provided by the actuators and the disturbances, respectively. By substituting Equations (1)–(6) into Equation (7), we obtain the dynamics model represented as a set of partial differential equations (PDEs) as Equation (8):

(8) \begin{equation}\left\{\begin{aligned}& {J_0}\dot \omega + \omega \times {J_0}\omega + \overline {{J_x}} {\Delta _1}\int_0^L {(\dot \omega + {{\ddot w}_L})dx} + \overline {\rho A} \int_0^L {{r_P} \times \left( {{{\ddot r}_p} + 2\omega \times {{\dot r}_p} + \dot \omega \times {r_p}} \right)dx} = {T_c} + {T_d} \\[5pt]& {D_L}w_L^{(4)} + {\Delta _1}\overline {{J_x}} \left( {\dot \omega + {{\ddot w}_L}} \right) + {\Delta _2}\overline {\rho A} \left( {{{\ddot r}_p} + 2\omega \times {{\dot r}_p} + \dot \omega \times {r_p}} \right) = 0.\end{aligned}\right.\end{equation}

The boundary conditions of flexible attachment are given as below [Reference Liu, Cao and Wei38]:

(9) \begin{align}\left\{ {\begin{array}{*{20}{l}}{} {}{{w_x}\left( {0,t} \right) = {w_y}\left( {0,t} \right) = {w_z}\left( {0,t} \right) = 0}\\[5pt] {} {}{{w_x}^{\prime}\left( {L,t} \right) = {w_y}^{\prime}\left( {0,t} \right) = {w_z}^{\prime}\left( {0,t} \right) = 0}\\[5pt] {} {}{{w_y}^{\prime\prime}\left( {L,t} \right) = {w_z}^{\prime\prime}\left( {L,t} \right) = 0}\\[5pt] {} {}{{w_y}^{\prime\prime\prime}\left( {L,t} \right) = {w_z}^{\prime\prime\prime}\left( {L,t} \right) = 0}\end{array}} \right. .\end{align}

For the flexible attachment, the discrete form of the continuous vibration vector can be expressed as:

(10) \begin{align}{w_i}\left( {x,t} \right) = {{\rm{\Phi }}_i}\left( x \right){\eta _i}\left( t \right) = \mathop \sum \limits_{j = 1}^n {\phi _{ij}}\left( x \right){\eta _{ij}}\left( t \right),i = x,y,z.\end{align}

Let ${{\rm{\Phi }}_i} = \left[ {{\phi _{i1}}, \cdots, {\phi _{in}}} \right] \in {\mathbb{R}^{1 \times n}}$ represents the modal functions, and ${\eta _i} = {\left[ {{\eta _{i1}}, \cdots, {\eta _{in}}} \right]^T} \in {\mathbb{R}^{n \times 1}}$ represents the modal coordinates. By defining ${{\rm{\Phi }}_L} = diag\left\{ {{{\rm{\Phi }}_x},{{\rm{\Phi }}_y},{{\rm{\Phi }}_z}} \right\}$ and ${\eta _L} = {\left[ {\begin{array}{*{20}{l}}{\eta _x^T} {}{\eta _y^T} {}{\eta _z^T}\end{array}} \right]^T}$ , we can obtain the discretised dynamic model as follows:

(11) \begin{align}\left\{ {\begin{array}{*{20}{l}}{} {}{J\dot \omega + \omega \times {J_0}\omega + F{{\ddot \eta }_L} + {{\rm{\Pi }}_1}\left( {\omega, {\eta _L}} \right) = {T_c} + {T_d}}\\[5pt] {} {}{M{{\ddot \eta }_L} + K{\eta _L} + {F^T}\dot \omega + {{\rm{\Pi }}_2}\left( {\omega, {\eta _L}} \right) = 0}.\end{array}} \right.\end{align}

Where, $J = {J_0} + {{\rm{\Delta }}_1}\overline {{J_x}} L$ , the matrices $M$ , $K$ and $F$ are related to the mode function of the flexible beam, which can bedefined as follows:

(12) \begin{align}M &= \int_0^L {\Phi _L^T\left( {\overline {{J_x}} {\Delta _1} + \overline {\rho A} {\Delta _2}} \right){\Phi _L}dx} \nonumber\\[5pt] K &= \int_0^L {{{\left( {{\Phi _L}^{\prime \prime }} \right)}^T}{D_L}{\Phi _L}^{\prime \prime }dx}. \nonumber\\[5pt] F &= \overline {{J_x}} {\Delta _1}\int_0^L {{\Phi _L}dx} \end{align}

Where ${{\rm{\Phi }}_L}$ represent the torsional mode function of the flexible beam on the x axis and the bending mode function on the y and z axes. The mode function needs to satisfy the boundary conditions of flexible attachment. The coupling matrices ${{\rm{\Pi }}_1}\left( {\omega, {\eta _L}} \right)$ and ${{\rm{\Pi }}_2}\left( {\omega, {\eta _L}} \right)$ are related to $\omega $ , ${\eta _L}$ and their derivatives, which can be derived as follows:

(13) \begin{align}{\Pi _1}\left( {\omega, {\eta _L}} \right) &= \overline {\rho A} \int_0^L {{r_P} \times \left( {{{\ddot r}_p} + 2\omega \times {{\dot r}_p} + \dot \omega \times {r_p}} \right)dx} \nonumber\\[5pt]{\Pi _2}\left( {\omega, {\eta _L}} \right) &= {\Delta _2}\overline {\rho A} \int_0^L {\left( {2\omega \times {{\dot r}_p} + \dot \omega \times {r_p}} \right)dx}.\end{align}

For the large-angle rest-to-rest attitude manoeuver, the modified Rodriguez parameters (MRP) are used to describe the attitude [Reference Ren39]. MRP are expressed as $\sigma = {\left[ {\begin{array}{*{20}{l}}{{\sigma _1}} {}{{\sigma _2}} {}{{\sigma _3}}\end{array}} \right]^T} = \vec k{\rm{tan}}\frac{{\rm{\Phi }}}{4} \in {\mathbb{R}^3}$ , where $\vec k$ represents the rotation spindle and ${\rm{\Phi }}$ represents the rotation angle. The kinematic model of the satellite described by MRP is as follows:

(14) \begin{align}\dot \sigma = A\left( \sigma \right)\omega = \frac{1}{4}\left[ {\begin{array}{c@{\quad}c@{\quad}c}{1 - {\sigma ^2} + 2\sigma _1^2}&{2\left( {{\sigma _1}{\sigma _2} - {\sigma _3}} \right)}&{2\left( {{\sigma _1}{\sigma _3} + {\sigma _2}} \right)}\\[5pt]{2\left( {{\sigma _2}{\sigma _1} + {\sigma _3}} \right)}&{1 - {\sigma ^2} + 2\sigma _2^2}&{2\left( {{\sigma _2}{\sigma _3} - {\sigma _1}} \right)}\\[5pt]{2\left( {{\sigma _1}{\sigma _3} - {\sigma _2}} \right)}&{2\left( {{\sigma _2}{\sigma _3} + {\sigma _1}} \right)}&{1 - {\sigma ^2} + 2\sigma _3^2}\end{array}} \right]\omega. \end{align}

Where, ${\sigma ^2} = \sigma _1^2 + \sigma _2^2 + \sigma _3^2$ , and $A\left( \sigma \right)$ is then expresses by $A\left( \sigma \right) = \frac{1}{4}\left( {\left( {1 - {\sigma ^2}} \right)I + 2\sigma \times + 2\sigma {\sigma ^T}} \right)$ . Let $\left\{ {\begin{array}{*{20}{l}}{{\sigma _e} = \sigma - {\sigma _r}}\\[5pt] {{\omega _e} = \omega - {\omega _r}}\end{array}} \right.$ represent the attitude tracking error, where ${\sigma _r}$ and ${\omega _r}$ are the desired statuses provided by the attitude control unit (ACU). The attitude manoeuver tracking model is then presented as follows:

(15) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{{\dot \sigma }_e}} {}{ = A\left( {{\sigma _e} + {\sigma _r}} \right)\left( {{\omega _e} + {\omega _r}} \right) - {{\dot \sigma }_r}}\\[5pt] {{{\dot \omega }_e}} {}{ = - {J^{ - 1}}\left( {{\omega _e} + {\omega _r}} \right) \times {J_0}\left( {{\omega _e} + {\omega _r}} \right) + {J^{ - 1}}{T_c} + {J^{ - 1}}\left( {{T_d} - {{\rm{\Pi }}_1}\left( {\omega, {\eta _L}} \right)} \right) - {{\dot \omega }_r}}\end{array}} \right.\!\!.\end{align}

3.0 Problem statement

The primary objective of this paper is to achieve precise large angle attitude manoeuver tracking control for the flexible spacecraft under input constraints while mitigating the external disturbances. Unlike conventional attitude attitude manoeuver problems, special attention must be given to maintaining real-time tracking of the desired path to prevent vibration of the coilable mast. Since the microsatellite lacks a modal measurement sensor, the nonlinear observer is employed to assess the influence of the flexible attachment on the rigid platform. In summary, the control problem can be formulated as follows:

1. 1. Ensure that the tracking errors for the attitude manoeuver, denoted as ${\sigma _e}$ and ${\omega _e}$ , converge asymptotically to the zero.

2. 2. Limit the ${L_2}$ -gain form the input disturbance to the controlled output to be less than a given constant under the zero initial condition [Reference Hu40].

3. 3. Ensure that the controller output remains within the actuator boundaries, denoted as $\left\| {{T_{ci}}} \right\| \le \left\| {{T_{\max i}}} \right\|\left( {i = x,y,z} \right)$ .

4.0 Manoeuver tracking controller

The manoeuver tracking model in Equation (15) is reformulated as a state-space equation with the following structure:

(16) \begin{align}\dot x = A\left( {x + {x_r}} \right)\left( {x + {x_r}} \right) - {\dot x_r} + Bu + Bw,\end{align}

\begin{align*}A\left( {x + {x_r}} \right) = \left[ {\begin{array}{c@{\quad}c}0& {}{A\left( {{\sigma _e} + {\sigma _r}} \right)} {}{}\\[5pt] 0 & {}{ - {J^{ - 1}}\left( {{\omega _e} + {\omega _r}} \right) \times {J_0}} {}{}\end{array}} \right]{\rm{\;\;\;\;}}B = \left[ {\begin{array}{c}0\\[5pt] {{J^{ - 1}}}\end{array}} \right].\end{align*}

Where, $x = {\left[ {\begin{array}{c@{\quad}c}{\sigma _e^T} & {}{\omega _e^T}\end{array}} \right]^T} \in {\mathbb{R}^6}$ denotes the state variable, $u = {T_c}$ denotes the control torque, and $w = {T_d} - {{\rm{\Pi }}_1}\left( {\omega, {\eta _L}} \right)$ denotes the cluster disturbances composed of environment and flexible vibration. The controller proposed in this paper includes: (1) a compensator based on NDO; (2) a feedforward controller; and (3) an output feedback controller. The controller can be represented as $u = {u_c} + {u_r} + {u_n}$ , and the block diagram of the proposed control system is depicted in Fig. 2.

Figure 2. The block diagram of proposed control system.

4.1 NDO-based sompensator

As there is no modal measurement sensor installed on the microsatellite for the flexible attachment, some papers employ a modal observer to estimate the modal vibration [Reference Bai, Zhou, Sun and Zeng41]. However, the high order of the observer is often challenging to implement in engineering. To address this, we consider the high-order dynamic states and environment disturbances as cluster disturbances and construct a novel NDO as follows:

(17) \begin{align}\left\{ {\begin{array}{*{20}{l}}{\dot \kappa } {}{ = - {k_g}{J^{ - 1}}\left( { - \left( {{\omega _e} + {\omega _r}} \right) \times {J_0}\left( {{\omega _e} + {\omega _r}} \right) - {{\dot \omega }_r} + {T_c} + \hat w} \right)}\\[5pt] {\hat w} {}{ = \kappa + g\left( {{\omega _e}} \right)}\end{array}} \right..\end{align}

Where, $\kappa \left( t \right) \in {\mathbb{R}^3}$ denotes an auxiliary variable in NDO; $\hat w$ denotes an observation of the cluster disturbances; $g\left( {{\omega _e}} \right)$ denotes a nonlinear function matrix related to ${w_e}$ ; and ${k_g} = \frac{{\partial g\left( {{\omega _e}} \right)}}{{\partial {\omega _e}}} \in {\mathbb{R}^{3 \times 3}}$ denotes the gain of NDO. The compensator is then expressed as follows: ${u_c} = - \hat w = - \kappa - g\left( {{\omega _e}} \right)$ .

Theorem 1. The observation error of the compensator based on NDO for cluster disturbances is expressed as: ${e_w} = w - \hat w$ . When ${k_g} \gt 0$ is satisfied, ${e_w}$ converges the exponent to zero.

Proof. When the environmental disturbance torque $w$ changes slowly, it can be considered that its derivative is zero. The differential of ${e_w}\left( t \right)$ is presented as follows:

(18) \begin{align} {{\dot e}_w} &= \dot w - {\dot {\hat w}} \approx - \dot \kappa - {k_g}\dot x\nonumber\\[5pt] &= {k_g}{J^{ - 1}}\left( { - \omega \times {J_0}\omega - {{\dot \omega }_r} + {T_c} + \hat w} \right) - {k_g}{{\dot \omega }_e}.\nonumber\\[5pt] &= - {k_g}{J^{ - 1}}{e_w}\end{align}

Further, ${\dot e_w} + {k_g}{J^{ - 1}}{e_w} = 0$ is obtained and ${e_w}$ will converge exponentially to zero for ${k_g}{J^{ - 1}} \gt 0$ .

4.2 Feedforward controller

The feedforward controller is employed to manoeuver the satellite along the desired path without the effects of disturbances and model parameter uncertainly. It is expressed as: ${u_r} = \left( {{\omega _e} + {\omega _r}} \right) \times {J_0}{\omega _r} + J{\dot \omega _r}$ . In this paper, the rotation angle in MRP is planned using the sine profile to reduce excessive vibration. The desired path is represented as follows:

(19) \begin{align}{{{\dot \Phi }}_r}\left( t \right) = \left\{ {\begin{array}{l@{\quad}l}{} {}{ - \dfrac{1}{\alpha }{A_s}{\rm{cos}}\left( {\alpha t} \right) + \beta } {}{} & {}{0 \lt t \lt T}\\[5pt] {} {}0 & {}{} {}{t \geqslant T}\end{array}} \right..\end{align}

Equation (19) shows that the differential of the desired rotation angle changes sinusoidally, and the rotation of the rotation angle itself can be obtained by integral operation as follows:

(20) \begin{align}{{\rm{\Phi }}_r}\left( t \right) = \left\{ {\begin{array}{l@{\quad}l}{} {}{ - \dfrac{1}{{{\alpha ^2}}}{A_s}{\rm{sin}}\!\left( {\alpha t} \right) + \beta t + {{\rm{\Phi }}_0}} {}{}& {}{0 \lt t \lt T}\\[8pt] {} {}{{{\rm{\Phi }}_d}} {}{}& {}{t \geqslant T}\end{array}} \right..\end{align}

Where, ${{\rm{\Phi }}_0}$ and ${{\rm{\Phi }}_d}$ are the initial and desired states in rest-to-rest manoeuver. By substituting ${\sigma _r}\left( t \right)$ and ${\dot \sigma _r}\left( t \right)$ into Equation (14), the desired angular velocity ${\omega _r}\left( t \right)$ is obtained. The desired path ${\dot \omega _r}\left( t \right)$ can be computed by difference operation. The specified parameters $\alpha $ , $\beta $ , and ${A_s}$ in the sine profile path meet the following constraints:

(21) \begin{align}\left\{ {\begin{array}{*{20}{l}}{} {}{ - \dfrac{1}{{{\alpha ^2}}}{A_s}{\rm{sin}}\!\left( {\alpha T} \right) + \beta T = {{\rm{\Phi }}_d} - {{\rm{\Phi }}_0}}\\[12pt] {} {}{ - \dfrac{1}{\alpha }{A_s}{\rm{cos}}\left( {\alpha \frac{T}{2}} \right) + \beta = 2{{{{\dot \Phi }}}_{{\rm{max}}}}}\end{array}} \right..\end{align}

4.3 Output feedback controller

Based on Equation (16), the objects of the feedback controller are as follow:

(22) \begin{align}\dot x = A\left( {x + {x_r}} \right)x + B{e_w} + B{u_n}.\end{align}

The role of the output feedback controller ${u_n}$ is to ensure that the ${L_2}$ -gain from the observation error ${e_w}$ to the controlled output $z = Cx + D{e_w}$ does not exceed the given constant $0 \lt \gamma \lt 1$ . It always the case that $\int_0^T {\left( {{{\left\| {z\left( t \right)} \right\|}^2} - {\gamma ^2}{{\left\| {{e_w}\left( t \right)} \right\|}^2}} \right)dt} \le 0$ exists for $T \gt 0$ when $x\left( 0 \right) = 0$ and ${e_w} \in {L_2}\left[ {0,\infty } \right)$ .

Lemma 1. [Reference Prajna, Papachristodoulou and Wu33] For a polynomial $p\left( {{x_1},{x_2}, \cdots, {x_n}} \right)\Delta = p\left( x \right)$ , if there are polynomials ${f_1}\left( x \right), \cdots, {f_m}\left( x \right)$ so that $p\left( x \right)$ can be written as a sum of squares (SOS), such as:

(23) \begin{align}p\left( x \right) = \mathop \sum \limits_{i = 1}^m f_i^2\left( x \right).\end{align}

Then such polynomials as $p\left( x \right)$ are called SOS polynomials, and obviously, all SOS polynomials are non-negative. The set of SOS polynomials is expressed as ${{\rm{\Sigma }}_{SOS}}$ , e.g. $p\left( x \right) \in {{\rm{\Sigma }}_{SOS}}$ .

Lemma 2. (The S-procedure lemma). [Reference Turki, Gritli and Belghith42] Let ${F_0}, \cdots, {F_N} \in {\mathbb{R}^{n \times n}}$ by symmetric matrices. If there exist scalar variables ${\varepsilon _1}, \cdots, {\varepsilon _N} \gt 0$ such that ${F_0} - \mathop \sum \limits_{i = 1}^N {\varepsilon _i}{F_i} \gt 0$ , the condition holds: ${v^T}{F_0}v \gt 0$ for all $v \ne 0$ such that ${v^T}{F_i}v \geqslant 0\left( {i = 1, \cdots, N} \right)$ .

Lemma 3. [Reference Prajna, Papachristodoulou and Wu33] Let $F\left( x \right)$ be an $N \times N$ symmetric polynomial matrix of variable $x$ and $z\left( x \right)$ be a column monomials of $x$ . The degree of $F\left( x \right)$ is $2d$ and the degree of $z\left( x \right)$ is no greater than $d$ . For the following three conditions:

1. 1. $F\left( x \right) \geqslant 0$ for all $x \in {\mathbb{R}^n}$ .

2. 2. ${v^T}F\left( x \right)v \in {\Sigma _{SOS}}$ , where $v \in {\mathbb{R}^N}$ .

3. 3. There exists a positive semidefinite matrix $Q$ such that ${v^T}F\left( x \right)v = {\left( {v \otimes \left( x \right)} \right)^T}Q\left( {v \otimes z\left( x \right)} \right)$ , where $otimes$ denotes the Kronecker product.

Then (ii) $ \Rightarrow $ (i) and (ii) $ \Leftrightarrow $ (iii).

Lemma 4 (Schur complement lemma). [Reference Turki, Gritli and Belghith42, Reference Boyd, El Ghaoui, Feron and Balakrishnan43] Let $X$ be a symmetric matrix of real numbers given by $X = \left[ {\begin{array}{c@{\quad}c}A & {}B\\[5pt] {{B^T}}& {}C\end{array}} \right]$ . Then there are three equivalent conditions as follows:

1. 1. ${\rm{X}} \gt 0$ .

2. 2. ${\rm{A}} \gt 0$ and ${\rm{C}} - {{\rm{B}}^{\rm{T}}}{{\rm{A}}^{ - 1}}{\rm{B}} \gt 0$ .

3. 3. ${\rm{C}} \gt 0$ and ${\rm{A}} - {\rm{B}}{{\rm{C}}^{ - 1}}{{\rm{B}}^{\rm{T}}} \gt 0$ .

Theorem 2. For the Equation (22), the system is asymptotically stable and $\int_0^T {\left( {{{\left\| {z\left( t \right)} \right\|}^2} - {\gamma ^2}{{\left\| {{e_w}\left( t \right)} \right\|}^2}} \right)dt} \le 0$ exists for any $T \gt 0$ , when the polynomial matrix $K\left( {x + {x_r}} \right)$ and the symmetric positive definite constant matrix $P$ exit and satisfy the conditions as follows:

(24) \begin{align} \left[ {\begin{array}{c@{\quad}c@{\quad}c}{\Xi \left( {P{A_c}\left( {x + {x_r}} \right)} \right)}&{PB}&{{C^T}}\\[5pt]*&{ - \gamma I}&{{D^T}}\\[5pt]*&*&{ - \gamma I}\end{array}} \right] < 0.\end{align}

Where, ${\rm{*}}$ denotes the same element in the symmetric matrix, ${A_c}\left( {x + {x_r}} \right) = A\left( {x + {x_r}} \right) + BK\left( {x + {x_r}} \right)P$ , and ${\rm{\Xi }}\left( {P{A_c}\left( {x + {x_r}} \right)} \right) = P{A_c}\left( {x + {x_r}} \right) + A_c^T\left( {x + {x_r}} \right)P$ .

The controller is expressed as: ${u_n} = K\left( {x + {x_r}} \right)Px$ .

Proof. By multiplying both sides of the inequality 24 by the matrix $diag\left\{ {{\gamma ^{0.5}}I,{\gamma ^{0.5}}I,{\gamma ^{ - 0.5}}I} \right\}$ , and setting $G = \gamma P$ , the inequality can be rewritten as follows:

(25) \begin{align}\left[ {\begin{array}{c@{\quad}c@{\quad}c}{\Xi \left( {G{A_c}\left( {x + {x_r}} \right)} \right)}&{GB}&{{C^T}}\\[5pt]*&{ - {\gamma ^2}I}&{{D^T}}\\[5pt]*&*&{ - I}\end{array}} \right] < 0.\end{align}

Assume $V\left( x \right) = {x^T}\left( t \right)Gx\left( t \right)$ is the Lyapunov function. As $G \gt 0$ and ${\rm{\Xi }}\left( {G{A_c}\left( {x + {x_r}} \right)} \right) \lt 0$ , the system is asymptotically stable. According to 4, inequality 25 can be rewritten as:

(26) \begin{align}\left[ {\begin{array}{c@{\quad}c}{{\rm{\Xi }}\left( {G{A_c}\left( {x + {x_r}} \right)} \right)} {}& {GB}\\[5pt] {}\ast & {}{ - {\gamma ^2}I}\end{array}} \right] + \left[ {\begin{array}{*{20}{l}}{{C^T}}\\[5pt] {{D^T}}\end{array}} \right]\left[ {\begin{array}{c@{\quad}c}C & {}D\end{array}} \right] \lt 0\end{align}

Therefore, for any $t \gt 0$ , there

(27) \begin{align}&{\left\| {z(t)} \right\|^2} - {\gamma ^2}{\left\| {{e_w}(t)} \right\|^2} + \dot V(x)\nonumber\\[5pt] &\quad = {z^T}(t)z(t) - {\gamma ^2}e_w^T(t){e_w}(t) + \dot V(x)\nonumber\\[5pt] &\quad = {z^T}(t)z(t) - {\gamma ^2}e_w^T(t)w(t)\nonumber\\[5pt] &\qquad + 2{x^T}(t)G\left( {{A_c}\left( {x + {x_r}} \right)x(t) + B{e_w}(t)} \right)\nonumber\\[5pt] &\quad = {\left[ {\begin{array}{c} {x(t)}\\[5pt] {{e_w}(t)}\end{array}} \right]^T}\left( {\left[ {\begin{array}{c@{\quad}c}{\Xi \left( {G{A_c}\left( {x + {x_r}} \right)} \right)}&{GB}\\[5pt] \ast &{ - {\gamma ^2}I}\end{array}} \right] + \left[ {\begin{array}{c} {{C^T}}\\[5pt] {{D^T}}\end{array}} \right]\left[ {\begin{array}{c@{\quad}c} C&D \end{array}} \right]} \right)\left[ {\begin{array}{c} {x(t)}\\[5pt] {{e_w}(t)}\end{array}} \right] < 0.\end{align}

The integration of the inequality 27 from $t = 0$ to $t = T$ is shown below:

(28) \begin{align} \int_0^T {\left( {{{\left\| {z(t)} \right\|}^2} - {\gamma ^2}{{\left\| {{e_w}(t)} \right\|}^2}} \right)dt} + V\left( {x(T)} \right) - V\left( {x(0)} \right) < 0 \end{align}

Since the system is asymptotically stable, it can be shown that $\int_0^T {\left( {{{\left\| {z(t)} \right\|}^2} - {\gamma ^2}{{\left\| {{e_w}(t)} \right\|}^2}} \right)dt} < 0$ exists.

When $C = I$ and $D = 0$ , the inequality 24 can be reduced to:

(29) \begin{align}\left[ {\begin{array}{c@{\quad}c@{\quad}c}{\Xi \left( {P{A_c}\left( {x + {x_r}} \right)} \right)}&{PB}&I\\[5pt]*&{ - \gamma I}&0\\[5pt]*&*&{ - \gamma I}\end{array}} \right] < 0.\end{align}

The inequality 29 contains terms with unknown variables $P$ and $K\left( x \right)$ , making it more challenging to solve. By multiplying both sides of the inequality by $Q = {P^{ - 1}}$ , we can rewrite it as follows:

(30) \begin{align}\begin{array}{c}\Xi \left( {A(x + {x_r})Q + BK(x + {x_r})} \right) < 0\\[7pt]\left[ {\begin{array}{c@{\quad}c@{\quad}c}{\Xi \left( {A(x + {x_r})Q + BK(x + {x_r})} \right)}&B&Q\\[5pt]*&{ - \gamma I}&0\\[5pt]*&*&{ - \gamma I}\end{array}} \right] < 0.\end{array}\end{align}

Solving the constraints 30 is equivalent to solving a state-dependent LMI problem in infinite dimensions, which is computationally difficult. Based on 1, the LMI constraints (30) can be rewritten as the SOS constraints as shown below:

(31) \begin{align}v_1^T\left( {Q - {\varepsilon _1}I} \right){v_1} \in {{\rm{\Sigma }}_{SOS}},\end{align}

(32) \begin{align} - v_2^T\left( {{\rm{\Xi }}\left( {A\left( {x + {x_r}} \right)Q + BK\left( {x + {x_r}} \right)} \right) + {\varepsilon _2}I} \right){v_2} \in {{\rm{\Sigma }}_{SOS}},\end{align}

(33) \begin{align} - v_3^T\left( {\left[ {\begin{array}{c@{\quad}c@{\quad}c}{\Xi \left( {A(x + {x_r})Q + BK(x + {x_r})} \right)}&B&Q\\[5pt]*&{ - \gamma I}&0\\[5pt]*&*&{ - \gamma I}\end{array}} \right] + {\varepsilon _3}I} \right){v_2} \in {\Sigma _{SOS}}.\end{align}

Where, ${\varepsilon _1},{\varepsilon _2},{\varepsilon _3} \gt 0$ and ${v_1},{v_2},{v_3} \in {\mathbb{R}^6}$ are given parameters. For this SOS optimisation problem, it can be solved by SOSTOOLS. As a simplifying case, when $A$ is constant matrix, it transforms into the LMI problem for linear time-invariant (LTI) systems.

Considering the controller’s form, when a large rotation angle is required, a correspondingly large control torque is also needed. However, the control torque provided by actuators is limited. Therefore, a saturation constraint is incorporated into the control torque as shown below:

(34) \begin{align}\bar u = Sa{t_u}\left( {u\left( t \right)} \right) = {T_{{\rm{max}}}}{\rm{tanh}}\left( {\frac{{u\left( t \right)}}{{{T_{{\rm{max}}}}}}} \right).\end{align}

5.0 Case study

This paper focuses on the microsatellite SSS-1 (Fig. 3), which was launched on October 14, 2021. The satellite is equipped with a coilable mast that exhibits bending and torsional vibrations in multi-direction, causing disturbances to the attitude stability of the platform during ground imaging as shown in Fig. 4. After deployment in orbit, it can be used to stabilise the satellite’s orientation and extend some sensors away from the satellite’s body to minimise electromagnetic interference from the platform. Throughout the satellite’s lifecycle, the main control objectives of the ADCS include:

Figure 3. SSS-1 satellite.

Figure 4. Ground imaging by SSS-1.

1. 1. Utilising the magnetic coils to damp the satellite’s motion after separation from the rocket;

2. 2. Finding the sun’s position through the sun sensor and cruising to face it, ensuring adequate energy supply;

3. 3. Maintaining a stable three-axis orientation towards the sun or the Earth by the command from the ground, in order to complete payload tasks such as deployment of the coilable mast and remote sensing camera imaging.

The main performance of the ADCS is shown in Table 1.

Table 1. Performance of ADCS

In this section, the control performance of the satellite in multiple manoeuver situations is studied through two cases. In Case 1, the satellite is required to perform a large-angle rest-to-rest attitude manoeuver in the presence of external disturbances ${T_{d1}}$ . In Case 2, the satellite needs to complete three large-angle attitude manoeuvers within a specified time while facing external disturbances ${T_{d2}}$ . The model parameters of the simulation are provided as follows:

• • Inertia matrix of the rigid platform, ${J_0} = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{3.6} & {}{0.3}& {}{0.2}\\[5pt] {0.3}& {}{3.4}& {}0\\[5pt] {0.2}& {}0& {}{1.2}\end{array}} \right]{\rm{\;\;kg}} \cdot {{\rm{m}}^2}$

• • Radius of the rigid platform, ${r_0} = 0.15{\rm{\;\;m}}$

• • Length of the coilable mast, $L = 4{\rm{\;\;m}}$

• • Maximum control torque, ${T_{{\rm{max}}}} = 5.0 \times {10^{ - 3}}{\rm{\;\;N}} \cdot {\rm{m}}$

• • Attitude manoeuver velocity, ${{{\dot \Phi }}_{{\rm{max}}}} = 0.5^{\circ}/{\rm{s}}$

• • Rotation spindle vector, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}} \over k} = {\left[ {\begin{array}{c@{\quad}c@{\quad}c}{0.248} {}&{ - 0.465} {}&{0.850}\end{array}} \right]^T}$

• • NDO gain, ${k_g} = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{20}& {}{10}& {}{12}\\[5pt] {10} {}&{28}& {}8\\[5pt] {12} {}&8& {}{16}\end{array}} \right]$

• • External disturbance 1, ${T_{d1}} = \left[ {\begin{array}{*{20}{l}}{{\rm{sin}}\!\left( {0.05\pi t - 0.4\pi } \right) + {\rm{sin}}\!\left( {0.04\pi t} \right)}\\[5pt] {{\rm{sin}}\!\left( {0.05\pi t + 0.5\pi } \right) + {\rm{sin}}\!\left( {0.04\pi t} \right)}\\[5pt] {{\rm{sin}}\!\left( {0.05\pi t + 0.6\pi } \right) + {\rm{sin}}\!\left( {0.04\pi t} \right)}\end{array}} \right] \times {10^{ - 3}}{\rm{\;\;N}} \cdot {\rm{m}}$

• • External disturbance 2, ${T_{d2}} = \left[ {\begin{array}{*{20}{l}}{3\left( {{\rm{sin}}\!\left( {0.2\pi t - 0.4\pi } \right) + {\rm{sin}}\!\left( {0.4\pi t} \right)} \right)}\\[5pt] {3\left( {{\rm{sin}}\!\left( {0.2\pi t + 0.5\pi } \right) + {\rm{sin}}\!\left( {0.4\pi t} \right)} \right)}\\[5pt] {3\left( {{\rm{sin}}\!\left( {0.2\pi t + 0.6\pi } \right) + {\rm{sin}}\!\left( {0.4\pi t} \right)} \right)}\end{array}} \right] \times {10^{ - 3}}{\rm{\;\;N}} \cdot {\rm{m}}$

The parameters in the SOS constraints are chosen as ${\varepsilon _1} = 0.01$ , ${\varepsilon _2} = 0.001$ , ${\varepsilon _3} = 0.001$ and $\gamma = 0.4$ . The output feedback controller is then obtained based on SOSTOOLS as follows:

\begin{equation*}\begin{aligned}{u_{n1}}\left( t \right) &= - \left( {3700 + 3300{\Upsilon ^2}} \right)\left( {{\sigma _{ex}} + {\omega _{ex}}} \right) - \left( {200 + 180{\Upsilon ^2}} \right)\left( {{\sigma _{ey}} + {\omega _{ey}}} \right) \\[5pt]& \qquad - \left( {140 + 140{\Upsilon ^2}} \right)\left( {{\sigma _{ez}} + {\omega _{ez}}} \right)\\[5pt]{u_{n2}}\left( t \right) &= - \left( {200 + 190{\Upsilon ^2}} \right)\left( {{\sigma _{ex}} + {\omega _{ex}}} \right) - \left( {3500 + 3200{\Upsilon ^2}} \right){\omega _{ey}}\\[5pt]{u_{n3}}\left( t \right) &= - \left( {160 + 160{\Upsilon ^2}} \right)\left( {{\sigma _{ex}} + {\omega _{ex}}} \right) - \left( {1500 + 1200{\Upsilon ^2}} \right){\omega _{ez}}\end{aligned}\end{equation*}

Where ${{\rm{\Upsilon }}^2} = \sigma _1^2 + \sigma _2^2 + \sigma _3^2 + \omega _x^2 + \omega _y^2 + \omega _z^2$ . The polynomial ${u_n}\left( {x,{x_e}} \right)$ involves ignoring monomials with tiny coefficients, leading to a more concise controller expression.

Different from the general flexible solar panel accessories, the coupling relationship between the coilable mast and the rigid platform is reflected in that when the rigid platform performs attitude manoeuver, the flexible mast will simultaneously undergo torsional deformation along the X-axis and bending deformation along the Y-axis and Z-axis. In order to reveal more dynamic behaviours caused by the vibration of the flexible mast and simulate the actual spacecraft attitude manoeuver situation. Here, by setting the driving torque of the controller and setting the initial angular velocity of the spacecraft to zero, the attitude motion and flexible vibration of the spacecraft are observed. The control torque of the actuator is shown as follows:

\begin{equation*}{T_c} = \left\{ {\begin{array}{l@{\quad}l}{{{\left[ {\begin{array}{c@{\quad}c@{\quad}c}{0.2}&{0.2}&{0.2}\end{array}} \right]}^T}\;{\rm{N}} \cdot {\rm{m}}}&{0 \le t < 2\;{\rm{s}}}\\[5pt]{{{\left[ {\begin{array}{c@{\quad}c@{\quad}c}0&0&0\end{array}} \right]}^T}}&{2\;{\rm{s}} \le t < 20\;{\rm{s}}}\\[5pt]{ - {{\left[ {\begin{array}{c@{\quad}c@{\quad}c}{0.2}&{0.2}&{0.2}\end{array}} \right]}^T}\;{\rm{N}} \cdot {\rm{m}}}&{20\;{\rm{s}} \le t < 22\;{\rm{s}}}\end{array}} \right.\end{equation*}

The Fig. 5 depicts the attitude manoeuvering angular velocity of the spacecraft under the influence of vibration interference from the coiled mast. As a result, the spacecraft’s attitude stability diminishes, subsequently impacting the accuracy of its attitude manoeuvering. In the course of the spacecraft platform’s attitude manoeuver, the coilable mast experiences the influence of the rigid-flexible coupling effect, manifesting in torsional and bending vibration responses, as delineated in Fig. 6.

Figure 5. Attitude manoeuvering angular velocity.

Figure 6. Vibration at the tip of the flexible mast.

Case 1 To empirically validate the efficacy of the introduced robust control strategy, this study conducts a comparative analysis against the conventional PD controller employed in SSS-1 satellite. The evaluation encompasses a comprehensive examination of the manoeuvering performance and vibrational characteristics exhibited by the flexible appendages in both control paradigms. The specific formulation of the PD controller is articulated as follows:

(35) \begin{align}{T_{PD}} = {K_P}{\sigma _e} + {K_D}{\omega _e},{\rm{\;\;\;\;\;\;}}\;\;{\rm{\;\;}}{K_P},{K_D} \in {\mathbb{R}^{3 \times 3}}.\end{align}

Where, ${K_P}$ , ${K_D}$ denote the parameters of the PD controller. The satellite’s attitude undergoes an adjustment from $0^{\circ}$ to $40^{\circ}$ , with initial and desired angular velocities both set to $0^{\circ}/{\rm{s}}$ . Figures 7 and 8 present the performance of attitude manoeuver tracking, illustrating the planned S-shaped curve for the desired path. Throughout the manoeuver phase, the satellite effectively follows the desired path of MRP and angular velocity, successfully reaching the predetermined position at about 70 seconds.

Figure 7. MRP and angular velocity in Case 1 (proposed controller).

Figure 8. Tracking errors in Case 1 (proposed Controller).

Analysis of Fig. 8 reveals that the errors in MRP and angular velocity during the manoeuver phase dose not exceed $6.00 \times {10^{ - 7}}$ and $2.10 \times {10^{ - 5}}{}^{\circ}/{\rm{s}}$ , respectively. Upon reaching the desired position, the error in MRP and angular velocity remains below $5.40 \times {10^{ - 8}}$ and $2.54 \times {10^{ - 6}}{}^{\circ}/{\rm{s}}$ . In contrast, the angular velocity tracking error of the PD controller surpasses $0.011^{\circ}/{\rm{s}}$ during manoeuvering, accompanied by a notable deficiency in post-manoeuver attitude control stability, with the attitude angle stability exceeding $0.007^{\circ}/{\rm{s}}$ in Fig. 9. It is evident from these observations that the PD controller falls short of meeting the stringent demands for high-precision trajectory tracking, particularly in the context of extensive spacecraft attitude manoeuvers. This inadequacy holds the potential to compromise the efficacy of vibration suppression in the coilable mast.

Figure 9. Tracking errors in Case 1 (PD controller).

Figures 10 and 11 illustrate the torsional and bending deformations at the tip of the flexible beam model by applying proposed controller and PD controller. Initially, the coilable mast is in equilibrium, subject to vibrations induced by attitude manoeuvering. The proposed controller effectively mitigates the vibrational amplitude of the flexible attachment.

Figure 10. Vibration at the tip in Case 1 (proposed controller).

Figure 11. Vibration at the tip in Case 1 (PD controller).

Figures 12 and 13 illustrate the performance of attitude manoeuver tracking in Case 2, indicating that the satellite maintains a high tracking and control capability during continuous manoeuver tasks. However, the control accuracy has slightly decreased due to excessive external disturbances. Figure 13 depicts the error in MRP and angular velocity, which remains within the acceptable limits for the requirements (less than $3.98 \times {10^{ - 7}}$ and $3.48 \times {10^{ - 4}}{}^{\circ}/{\rm{s}}$ , respectively).

Figure 12. MRP and angular velocity in Case 2 (proposed controller).

Figure 13. Tracking errors in Case 2 (proposed controller).

Moreover, the simulation results further validate the performance of NDO. The estimations for cluster disturbances are shown as Fig. 14. The observation errors are shown as Fig. 15. The observation error is less than $4.32 \times {10^{ - 3}}{\rm{\;\;N}} \cdot {\rm{m}}$ in Case 1 and remains below $5.10 \times {10^{ - 3}}{\rm{\;\;N}} \cdot {\rm{m}}$ even under external disturbances with higher frequencies and larger magnitudes in Case 2. The ${L_2}$ -gain robust controller effectively suppresses observation errors.

Figure 14. Observation by NDO.

Figure 15. Observation errors by NDO.

Finally, Fig. 16 illustrates the control torque in Case 1 and Case 2. Despite saturation of control torque in Case 2 due to excessive disturbances, the control accuracy remains uncompromised.

Figure 16. Control torque provided by the actuators.