logo

SCIENCE CHINA Information Sciences, Volume 63 , Issue 7 : 172002(2020) https://doi.org/10.1007/s11432-020-2916-8

Observer-based multi-objective parametric design for spacecraft with super flexible netted antennas

More info
  • ReceivedMar 9, 2020
  • AcceptedApr 28, 2020
  • PublishedJun 10, 2020

Abstract

A multi-objective parametric design method that based on therobust observer is proposed for the attitude control of satellites withsuper flexible netted antennas. First, a parametric observer-based controller is obtained based on the eigen-structure assignment theory. The closed-loop poles are assigned to desiredpositions or regions, and full degrees of freedom of the design, which are characterized by a set of parameters, are preserved under the proposed control law. Second, the obtainedparameters are comprehensively optimized to make the closed-loop system havelower eigenvalue sensitivity, a smaller control gain, and stronger toleranceto high-order unmodeled dynamics and external disturbances. Finally,comparative simulations are carried out based on practical engineeringparameters of a satellite in order to verify the effect of the proposed method, andalso to show their superiority over the traditional proportional-integral-derivative (PID) controller with filters and thetraditional dynamic compensators.


Acknowledgment

This work was supported by Major Program of National Natural Science Foundation of China (Grant Nos. 61690210, 61690212), Self-Planned Task of State Key Laboratory of Robotics and System (HIT) (Grant No. SKLRS201716A), and National Natural Science Foundation of China (Grant No. 61333003). The authors are very grateful to the anonymous reviewers for their meaningful suggestions and comments.


Supplement

Appendix

Proof of Theorem protect 3.2

To prove this theorem, the following lemma which was presented in [22] is needed.

Lemma 3. Let the system (3)–(6) satisfy condition (ref co. Then the right coprime polynomial matrices $N\left(~s\right)~$ and $ D\left(~s\right)~$ satisfying the following right coprime factorization (RCF): \begin{equation}\left( sI-A\right) ^{-1}B=N\left( s\right) D^{-1}\left( s\right) \tag{78}\end{equation} are given by \begin{eqnarray}N\left( s\right) &=&\left[ { \begin{array}{c} {\gamma {s^{2}}+{a_{2}}s+{a_{1}}} \\ {-b_{y}\gamma {s^{2}}} \\ {\gamma {s^{3}}+{a_{2}}{s^{2}}+{a_{1}}s} \\ {-b_{y}\gamma {s^{3}}} \end{array} }\right] , \tag{79} \\ D\left( s\right) &=&-{s^{4}}+{I_{y}}{a_{2}}{s^{3}}+{I_{y}}{a_{1}}{s^{2}.} \tag{80} \end{eqnarray}

According to the eigenstructure assignment result in [23], when the system (3)–(6) is controllable, that is, when the condition ( 14) holds, complete parametric forms of the gain matrix $K$ and a corresponding nonsingular matrix $V$ satisfying \begin{equation}\left( A+BK\right) V=V\Lambda _{c}, \tag{81}\end{equation} where $\Lambda~_{c}$ is shown in (68), can be given by \begin{equation}\left \{ \begin{array}{l} K=WV^{-1}, \\ V=[ \begin{array}{cccc} \hat{v}_{1} & \hat{v}_{2} & \hat{v}_{3} & \hat{v}_{4} \end{array} ] , \\ W=[ \begin{array}{cccc} \hat{w}_{1} & \hat{w}_{2} & \hat{w}_{3} & \hat{w}_{4} \end{array} ] , \end{array} \right. \tag{82}\end{equation} with \begin{equation*}\left \{ \begin{array}{l} \hat{v}_{1}=N\left( \alpha _{1}+\alpha _{2} \mathrm{i}\right) \left( f_{1}+f_{2} \mathrm{i}\right) , \\ \hat{v}_{2}=N\left( \alpha _{1}-\alpha _{2} \mathrm{i}\right) \left( f_{1}-f_{2} \mathrm{i}\right) , \\ \hat{v}_{3}=N\left( \alpha _{3}\right) f_{3}, \hat{v}_{4}=N\left( \alpha _{4}\right) f_{4}, \end{array} \right.\end{equation*} and \begin{equation*}\left \{ \begin{array}{l} \hat{w}_{1}=D\left( \alpha _{1}+\alpha _{2} \mathrm{i}\right) \left( f_{1}+f_{2} \mathrm{i}\right) , \\ \hat{w}_{2}=D\left( \alpha _{1}-\alpha _{2} \mathrm{i}\right) \left( f_{1}-f_{2} \mathrm{i}\right) , \\ \hat{w}_{3}=D\left( \alpha _{3}\right) f_{3}, \hat{w}_{4}=D\left( \alpha _{4}\right) f_{4}, \end{array} \right.\end{equation*} where $N\left(~s\right)~\in~\mathbb{R}^{4\times~1}[s]$ and $D\left( s\right)~\in~\mathbb{R}[s]$ are a pair of polynomial matrices satisfying the RCF (78), and $f_{i},\alpha~_{i},i=1,2,3,4,$ are parameters satisfying the following constraint: \begin{equation}\det \left( V\right) =\Delta _{c}\neq 0. \tag{83}\end{equation}

It is known from Lemma 3 that such $N\left(~s\right)~$ and $ D\left(~s\right)~$ can be given by (79) and (80), respectively. Then, through simple deductions, we can obtain the expression of $\Delta~_{c} $ as shown in Constraint C1.

It is easy to see that $\hat{v}_{1}$ and $\hat{v}_{2}$ are complex conjugates to each other, so do $\hat{w}_{1}$ and $\hat{w}_{2}.$ Therefore, assume that \begin{equation}\hat{v}_{1}=\vartheta _{vR}+\vartheta _{vI} \mathrm{i}, \hat{v} _{2}=\vartheta _{vR}-\vartheta _{vI} \mathrm{i}, \tag{84}\end{equation} \begin{equation}\hat{w}_{1}=\vartheta _{wR}+\vartheta _{wI} \mathrm{i}, \hat{w} _{2}=\vartheta _{wR}-\vartheta _{wI} \mathrm{i}. \tag{85}\end{equation} In view of the first formula in (82), we have \begin{equation}\hat{w}_{i}=K\hat{v}_{i}, i=1,2,3,4. \tag{86}\end{equation} Substituting (84) and (85) into (86), we obtain the following linear equation: \begin{equation*}W_{0}=KV_{0},\end{equation*} where \begin{equation}W_{0}=\left[ \begin{array}{cccc} \vartheta _{wR} & \vartheta _{wI} & \hat{w}_{3} & \hat{w}_{4} \end{array} \right] =\left[ \begin{array}{cccc} w_{1} & w_{2} & w_{3} & w_{4} \end{array} \right] , \tag{87}\end{equation} \begin{equation}V_{0}=\left[ \begin{array}{cccc} \vartheta _{vR} & \vartheta _{vI} & \hat{v}_{3} & \hat{v}_{4} \end{array} \right] =\left[ \begin{array}{cccc} v_{1} & v_{2} & v_{3} & v_{4} \end{array} \right] , \tag{88}\end{equation} with $v_{i},~w_{i},~i=1,2,3,4$ being given by (24)–(27). Obviously, when Eq. (83) holds, $V_{0}$ is also nonsingular. Thus the matrix $K$ can be given by (22). Combining (82), (84) and (88), gives the expression of $V$ shown in (23). Then the proof is completed.

Proof of Theorem protect 3.3

Similarly, to prove Theorem 3.3, the following result obtained in [22] is needed.

Lemma 4. Let the system (3)–(6) satisfy condition ( 14), and then the right coprime polynomial matrices $H\left(~s\right)~$ and $L\left(~s\right)~$ satisfying the following RCF: \begin{equation}\left( sI-A^{\mathrm{T}}\right) ^{-1}C^{\mathrm{T}}=H\left( s\right) L^{-1}\left( s\right) \tag{89}\end{equation} are given by \begin{eqnarray}H\left( s\right) &=&\left[ { \begin{array}{cc} 1 & 0 \\ 0 & {{a_{1}}b_{y}s} \\ 0 & {-{s^{2}}+{I_{y}}{a_{2}}s+{I_{y}}{a_{1}}} \\ 0 & {{a_{2}}b_{y}s+{a_{1}}b_{y}} \end{array} }\right] , \tag{90} \\ L\left( s\right) &=&\left[ { \begin{array}{cc} s & 0 \\ {-1} & {-{s^{3}}+{I_{y}}{a_{2}}{s^{2}}+{I_{y}}{a_{1}}s} \end{array} }\right] . \tag{91} \end{eqnarray}

Based on the eigenstructure assignment theory shown in [23], when $ \left(~A^{\mathrm{T}},~C^{\mathrm{T}}\right)~$ is controllable, that is, when the condition (14) holds, complete parametric forms of the gain matrix $L$ and a corresponding nonsingular matrix $T$ satisfying \begin{equation}T^{\mathrm{T}}\left( A+LC\right) =\Lambda _{o}T^{\mathrm{T}}, \tag{92}\end{equation} where $\Lambda~_{o}$ is shown in (68), can be given by \begin{equation}\left \{ \begin{array}{l} L=T^{-\mathrm{T}}Z^{\mathrm{T}}, \\ T=\left[ \begin{array}{cccc} \hat{t}_{1} & \hat{t}_{2} & \hat{t}_{3} & \hat{t}_{4} \end{array} \right] , \\ Z=\left[ \begin{array}{cccc} \hat{z}_{1} & \hat{z}_{2} & \hat{z}_{3} & \hat{z}_{4} \end{array} \right] , \end{array} \right. \tag{93}\end{equation} with \begin{equation*}\left \{ \begin{array}{ll} \hat{t}_{1}=H\left( \tilde{\alpha}_{1}+\tilde{\alpha}_{2} \mathrm{i}\right) \left( g_{1}+g_{2} \mathrm{i}\right) , \\ \hat{t}_{2}=H\left( \tilde{\alpha}_{1}-\tilde{\alpha}_{2} \mathrm{i}\right) \left( g_{1}-g_{2} \mathrm{i}\right) , \\ \hat{t}_{3}=H\left( \tilde{\alpha}_{3}\right) g_{3}, \hat{t}_{4}=H\left( \tilde{\alpha}_{4}\right) g_{4}, \end{array} \right.\end{equation*} and \begin{equation*}\left \{ \begin{array}{l} \hat{z}_{1}=L\left( \tilde{\alpha}_{1}+\tilde{\alpha}_{2} \mathrm{i}\right) \left( g_{1}+g_{2} \mathrm{i}\right) , \\ \hat{z}_{2}=L\left( \tilde{\alpha}_{1}-\tilde{\alpha}_{2} \mathrm{i}\right) \left( g_{1}-g_{2} \mathrm{i}\right) , \\ \hat{z}_{3}=L\left( \tilde{\alpha}_{3}\right) g_{3}, \hat{z}_{4}=L\left( \tilde{\alpha}_{4}\right) g_{4}, \end{array} \right.\end{equation*} where $H\left(~s\right)~\in~\mathbb{R}^{4\times~2}[s]$ and $L\left( s\right)~\in~\mathbb{R}^{2\times~2}[s]$ are a pair of polynomial matrices satisfying the RCF (89), and $\tilde{\alpha}_{i}$, $g_{i}=[{\tiny \begin{array}{c} g_{i1}~\\ g_{i2} \end{array}} ]~,~g_{i1},g_{i2}\in~\mathbb{R},~i=1,2,3,4$ are parameters satisfying the following constraint: \begin{equation}\det \left( T\right) =\Delta _{o}\neq 0. \tag{94}\end{equation} It is known from Lemma 4 that such $H\left(~s\right)~$ and $ L\left(~s\right)~$ can be given by (90) and (91), respectively. Then, substituting (90), (91) and (93) into (94), gives the expression of $\Delta~_{o}$ as shown in Constraint C2.

It is easy to see that $\hat{t}_{1}$ and $\hat{t}_{2}$ are complex conjugates to each other, so do $\hat{z}_{1}$ and $\hat{z}_{2}.$ Therefore, assume that \begin{equation}\hat{t}_{1}=\xi _{tR}+\xi _{tI} \mathrm{i}, \hat{t}_{2}=\xi _{tR}-\xi _{tI} \mathrm{i}, \tag{95}\end{equation} \begin{equation}\hat{z}_{1}=\xi _{zR}+\xi _{zI} \mathrm{i}, \hat{z}_{2}=\xi _{zR}-\xi _{zI} \mathrm{i}. \tag{96}\end{equation} Considering the first formula in (93), we have \begin{equation}\hat{z}_{i}=L^{\mathrm{T}}\hat{t}_{i}, i=1,2,3,4. \tag{97}\end{equation} Substituting (95) and (96) into (97), we obtain the following linear equation: \begin{equation*}Z_{0}=L^{\mathrm{T}}T_{0},\end{equation*} where \begin{equation}Z_{0}=\left[ \begin{array}{cccc} \xi _{zR} & \xi _{zI} & \hat{z}_{3} & \hat{z}_{4} \end{array} \right] =\left[ \begin{array}{cccc} z_{1} & z_{2} & z_{3} & z_{4} \end{array} \right] , \tag{98}\end{equation} \begin{equation}T_{0}=\left[ \begin{array}{cccc} \xi _{tR} & \xi _{tI} & \hat{t}_{3} & \hat{t}_{4} \end{array} \right] =\left[ \begin{array}{cccc} t_{1} & t_{2} & t_{3} & t_{4} \end{array} \right] , \tag{99}\end{equation} with $t_{i},~z_{i},~i=1,2,3,4$ being given by (36)–(39). Obviously, when Eq. (94) holds, $T_{0}$ is also nonsingular. Thus the matrix $L$ can be given by (34). Combining (93), (95) and (99), gives the expression of $T$ shown in (35). Then the proof is completed.

Proof of Theorem protect 4.2

Let \begin{equation*}V^{-\mathrm{T}}=\left[ \begin{array}{cccc} \tilde{v}_{1} & \tilde{v}_{2} & \tilde{v}_{3} & \tilde{v}_{4} \end{array} \right] , T^{-\mathrm{T}}=\left[ \begin{array}{cccc} \tilde{t}_{1} & \tilde{t}_{2} & \tilde{t}_{3} & \tilde{t}_{4} \end{array} \right].\end{equation*} Then, according to (56), the following relations hold: \begin{equation*}\tilde{v}_{i}=\frac{1}{\Delta _{c}}v_{i}^{\ast }, \tilde{t}_{i}=\frac{1}{ \Delta _{o}}t_{i}^{\ast }, i=1,2,3,4.\end{equation*}

It can be seen that $V$ and $V^{-1}$ are, respectively, the right and the left eigenvector matrices of $A_{c}$. Thus, according to Lemma 1 in Para4 we have \begin{eqnarray*}\frac{\partial \lambda _{i}\left( A_{c}\right) }{\partial \Delta a_{j}} &=& \tilde{v}_{i}^{\mathrm{T}}\frac{\partial A_{c}}{\partial \Delta a_{j}}v_{i}= \tilde{v}_{i}^{\mathrm{T}}A_{j}v_{i}=\frac{1}{\Delta _{c}}\left( v_{i}^{\ast }\right) ^{\mathrm{T}}A_{j}v_{i}, \\ i &=&1,2,3,4, j=1,2. \end{eqnarray*} Similarly, considering that $T^{\mathrm{T}}$ and $T^{-\mathrm{T}}$ are, respectively, the left and the right eigenvector matrices of $A_{o}$, it can be known from Lemma 1 in [26] that \begin{eqnarray*}\frac{\partial \lambda _{i}\left( A_{o}\right) }{\partial \Delta a_{j}} &=&t_{i}^{\mathrm{T}}\frac{\partial A_{o}}{\partial \Delta a_{j}}\tilde{t} _{i}=t_{i}^{\mathrm{T}}A_{j}\tilde{t}_{i}=\frac{1}{\Delta _{o}}t_{i}^{ \mathrm{T}}A_{j}t_{i}^{\ast }, \\ i &=&1,2,3,4, j=1,2, \end{eqnarray*} holds. Then, the proof is completed.

Proof of Theorem protect 4.3

At first, let us discuss the eigenstructure of $A_{z}$ as a preliminary. Let \begin{equation}T_{z}^{\mathrm{T}}=\left[ \begin{array}{cc} V^{-1}-Q_{\ast }T^{\mathrm{T}} & Q_{\ast }T^{\mathrm{T}} \\ -T^{\mathrm{T}} & T^{\mathrm{T}} \end{array} \right] , \tag{100}\end{equation} and \begin{equation}V_{z}=\left[ \begin{array}{cc} V & -VQ_{\ast } \\ V & T^{-\mathrm{T}}-VQ_{\ast } \end{array} \right] , \tag{101}\end{equation} where $T$ and $V$ are given by (23) and (35), respectively. It is known from Theorems 3.2 and 3.3 that, when $K$ and $L$ are taken as (22) and (34), respectively, the relations (ref as1 and (92) hold. Thus, in view of (100) and (101), we can verify that \begin{equation}T_{z}^{\mathrm{T}}V_{z}=I, \tag{102}\end{equation} and \begin{equation}T_{z}^{\mathrm{T}}A_{z}V_{z}=\left[ \begin{array}{cc} \Lambda _{c} & -\Lambda _{c}Q_{\ast }+Q_{\ast }\Lambda _{o}+V^{-1}BKT^{- \mathrm{T}} \\ 0 & \Lambda _{o} \end{array} \right] . \tag{103}\end{equation} According to the matrix equation theory, there exists a unique solution to the following linear matrix equation with respect to $Q$: \begin{equation}\Lambda _{c}Q-Q\Lambda _{o}=V^{-1}BKT^{-\mathrm{T}}. \tag{104}\end{equation} With the help of matrix vectorization operations, it can be easily verified that $Q_{\ast~}$ given by (66) is the unique solution of the matrix equation (104). Thus, Eq. (103) can be simplified as \begin{equation}T_{z}^{\mathrm{T}}A_{z}V_{z}=\Lambda _{z}, \tag{105}\end{equation} where $\Lambda~_{z}$ is given by (68).

Then, let us discuss the explicit expression of $\left~\Vert~G_{c}\left( s\right)~\right~\Vert~_{2}.$ Considering (105), the function $\left~\Vert G_{c}\left(~s\right)~\right~\Vert~_{2}$ can be transformed into \begin{equation*}\left \Vert G_{c}\left( s\right) \right \Vert _{2}=\left \Vert C_{z}V_{z}\left( sI-\Lambda _{z}\right) ^{-1}T_{z}^{\mathrm{T}}D_{z}\right \Vert _{2}.\end{equation*} According to Theorems 3.2 and 3.3, when $K$ and $L$ are taken as (22) and (34), respectively, both $A_{c}$ and $A_{o} $ are stable. Then, from (105), we know that $A_{z}$ is also stable. Therefore, it is known from Lemma 4.1 of the previous study 1) that there exist unique symmetric positive definite solutions $P_{1}$ and $P_{2}$ to the following Lyapunov matrix equations: \begin{equation}\Lambda _{z}P_{1}+P_{1}\Lambda _{z}=-T_{z}^{\mathrm{T}}D_{z}D_{z}^{\mathrm{T} }T_{z}, \tag{106}\end{equation} and \begin{equation}\Lambda _{z}P_{2}+P_{2}\Lambda _{z}=-V_{z}^{\mathrm{T}}C_{z}^{\mathrm{T} }C_{z}V_{z}, \tag{107}\end{equation} and $\left~\Vert~G_{c}\left(~s\right)~\right~\Vert~_{2}$ can be given by \begin{eqnarray*}\left \Vert G_{dy_{p}}\left( s\right) \right \Vert _{2} =\left( \mathrm{trace }\left( C_{z}V_{z}P_{1}V_{z}^{\mathrm{T}}C_{z}^{\mathrm{T}}\right) \right) ^{ \frac{1}{2}} =\left( \mathrm{trace}\left( D_{z}^{\mathrm{T}}T_{z}P_{2}T_{z}^{\mathrm{T} }D_{z}\right) \right) ^{\frac{1}{2}}. \end{eqnarray*} In view of (100) and (101), with the help of matrix vectorization operations, it can be verified that $P_{1}^{\ast~}$ and $P_{2}^{\ast~}$ which are given by (66) are the unique solutions of the matrix equations (106) and (107), respectively. Thus the result (ref rrr can be obtained. Then the proof is completed.

Duan G-R, Liu G P, Thompson S. Disturbance attenuation in Luenberger function observer designs—a parametric approach. IFAC Proc Vol, 2000, 33: 41–46.


References

[1] Kida T, Yamaguchi I, Chida Y. On-Orbit Robust Control Experiment of Flexible Spacecraft ETS-VI. J Guidance Control Dyn, 1997, 20: 865-872 CrossRef ADS Google Scholar

[2] Xiao B, Hu Q, Zhang Y. Adaptive Sliding Mode Fault Tolerant Attitude Tracking Control for Flexible Spacecraft Under Actuator Saturation. IEEE Trans Contr Syst Technol, 2012, 20: 1605-1612 CrossRef Google Scholar

[3] Hu Q, Ma G. Variable structure control and active vibration suppression of flexible spacecraft during attitude maneuver. Aerospace Sci Tech, 2005, 9: 307-317 CrossRef Google Scholar

[4] Hu Q, Ma G, Xie L. Robust and adaptive variable structure output feedback control of uncertain systems with input nonlinearity. Automatica, 2008, 44: 552-559 CrossRef Google Scholar

[5] Jiang Y, Hu Q, Ma G. Adaptive backstepping fault-tolerant control for flexible spacecraft with unknown bounded disturbances and actuator failures.. ISA Trans, 2010, 49: 57-69 CrossRef PubMed Google Scholar

[6] Wu A G, Dong R Q, Zhang Y. Adaptive Sliding Mode Control Laws for Attitude Stabilization of Flexible Spacecraft With Inertia Uncertainty. IEEE Access, 2019, 7: 7159-7175 CrossRef Google Scholar

[7] Xu S, Cui N, Fan Y. Flexible Satellite Attitude Maneuver via Adaptive Sliding Mode Control and Active Vibration Suppression. AIAA J, 2018, 56: 4205-4212 CrossRef ADS Google Scholar

[8] Huo J, Meng T, Song R. Adaptive prediction backstepping attitude control for liquid-filled micro-satellite with flexible appendages. Acta Astronaut, 2018, 152: 557-566 CrossRef ADS Google Scholar

[9] Wu S, Chu W, Ma X. Multi-objective integrated robust H control for attitude tracking of a flexible spacecraft. Acta Astronaut, 2018, 151: 80-87 CrossRef ADS Google Scholar

[10] Liu C, Sun Z, Shi K. Robust dynamic output feedback control for attitude stabilization of spacecraft with nonlinear perturbations. Aerospace Sci Tech, 2017, 64: 102-121 CrossRef Google Scholar

[11] Bai H, Huang C, Zeng J. Trans Institute Measurement Control, 2019, 41: 2026-2038 CrossRef Google Scholar

[12] Liu C, Shi K, Sun Z. Robust H controller design for attitude stabilization of flexible spacecraft with input constraints. Adv Space Res, 2019, 63: 1498-1522 CrossRef ADS Google Scholar

[13] Liu L, Cao D, Wei J. Rigid-Flexible Coupling Dynamic Modeling and Vibration Control for a Three-Axis Stabilized Spacecraft. J Vib Acoustics, 2017, 139: 041006 CrossRef Google Scholar

[14] Yan R, Wu Z. Super-twisting disturbance observer-based finite- time attitude stabilization of flexible spacecraft subject to complex disturbances. J Vib Control, 2019, 25: 1008-1018 CrossRef Google Scholar

[15] Yadegari H, Khouane B, Yukai2b Z, et al. Disturbance observer based anti-disturbance fault tolerant control for flexible satellites. ADVANCES IN AIRCRAFT AND SPACECRAFT SCIENCE, 2018, 5: 459-475 DOI: 10.12989/aas.2018.5.4.459. Google Scholar

[16] Zhong C, Wu L, Guo J. Robust adaptive attitude manoeuvre control with finite-time convergence for a flexible spacecraft. Trans Institute Measurement Control, 2018, 40: 425-435 CrossRef Google Scholar

[17] Hu Q. Robust adaptive sliding mode attitude control and vibration damping of flexible spacecraft subject to unknown disturbance and uncertainty. Trans Institute Measurement Control, 2012, 34: 436-447 CrossRef Google Scholar

[18] Smaeilzadeh S M, Golestani M. A finite-time adaptive robust control for a spacecraft attitude control considering actuator fault and saturation with reduced steady-state error. Trans Institute Measurement Control, 2019, 41: 1002-1009 CrossRef Google Scholar

[19] Li L, Liu J. Neural-network-based adaptive fault-tolerant vibration control of single-link flexible manipulator. Trans Institute Measurement Control, 2020, 42: 430-438 CrossRef Google Scholar

[20] Fu Y, Liu Y, Huang D. Adaptive boundary control and vibration suppression of a flexible satellite system with input saturation. Trans Institute Measurement Control, 2019, 41: 2666-2677 CrossRef Google Scholar

[21] Zhong C, Guo Y, Yu Z. Finite-time attitude control for flexible spacecraft with unknown bounded disturbance. Trans Institute Measurement Control, 2016, 38: 240-249 CrossRef Google Scholar

[22] Wu Y, Lin B, Zeng H. Parametric Multi-Objective Design for Spacecrafts with Super Flexible Netted Antennas (in Chinese). Control Theory Appl, 2019, 36: 766--773. Google Scholar

[23] Guang-Ren Duan . Solutions of the equation AV+BW=VF and their application to eigenstructure assignment in linear systems. IEEE Trans Automat Contr, 1993, 38: 276-280 CrossRef Google Scholar

[24] Duan G R. Solution to matrix equation AV + BW = EVF and eigenstructure assignment for descriptor systems. Automatica, 1992, 28: 639-642 CrossRef Google Scholar

[25] Irwin G W, Liu G P, Duan G R. Disturbance attenuation in linear systems via dynamical compensators: A parametric eigenstructure assignment approach. IEE Proc - Control Theor Appl, 2000, 147: 129-136 CrossRef Google Scholar

[26] Duan G R. Robust eigenstructure assignment via dynamical compensators. Automatica, 1993, 29: 469-474 CrossRef Google Scholar

[27] Duan G R, Liu G P, Thompson S. Disturbance decoupling in descriptor systems via output feedback-a parametric eigenstructure assignment approach. In: Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, 2000. 3660--3665. Google Scholar

[28] Wu W J, Duan G R. Gain scheduled control of linear systems with unsymmetrical saturation actuators. Int J Syst Sci, 2016, 47: 3711-3719 CrossRef ADS Google Scholar

[29] Wang Q, Zhou B, Duan G R. Robust gain scheduled control of spacecraft rendezvous system subject to input saturation. Aerospace Sci Tech, 2015, 42: 442-450 CrossRef Google Scholar

[30] Duan G R. Simple algorithm for robust pole assignment in linear output feedback. IEE Proc D Control Theor Appl UK, 1992, 139: 465 CrossRef Google Scholar

[31] Duan G R, Thompson S, Liu G P. Separation principle for robust pole assignment-an advantage of full-order state observers. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, 1999. 76--78. Google Scholar

[32] Duan G R, Liu G P, Thompson S. Disturbance Attenuation in Luenberger Function Observer Designs - A Parametric Approach. IFAC Proc Volumes, 2000, 33: 41-46 CrossRef Google Scholar

  • Figure 1

    Structure of classic PID controller.

  • Figure 2

    (Color online) The index values $J_{\mathrm{sen}}$ under the three control methods.

  • Figure 3

    (Color online) (a) Pitch angle; (b) pitch angular velocity; (c) control torque.

  • Table 1  

    Table 1Symbols

    Symbol Meaning
    $\mathrm{diag}\left(~s_{1},s_{2},\ldots,s_{n}\right)$$\text{Diagonal~matrix~with~}s_{1},s_{2},\ldots,s_{n}\text{~as~diagonal~elements}$
    $\lambda~_{i}\left(~M\right)$textThe $i$text-th eigenvalue of a matrix $M$
    $\mathrm{trace}\left(~M\right)$textSum of diagonal elements of a matrix $M$
    $\mathrm{Blockdiag}\left(~M_{1},M_{2},\ldots,M_{n}\right)$textBlock diagonal matrix with $M_{1},M_{2},\ldots,M_{n}$text as diagonal elements
    $\mathrm{vec}(~[ \begin{array}{cccc} \eta~_{1}~&~\eta~_{2}~&~\cdots~&~\eta~_{n} \end{array} ]~)$$[ \begin{array}{cccc} \eta~_{1}^{\mathrm{T}}~&~\eta~_{2}^{\mathrm{T}}~&~\cdots~&~\eta~_{n}^{\mathrm{T} } \end{array} ]~^{\mathrm{T}}$
    $\mathrm{unvec}(~[ \begin{array}{cccc} \eta~_{1}^{\mathrm{T}}~&~\eta~_{2}^{\mathrm{T}}~&~\cdots~&~\eta~_{n}^{\mathrm{T} } \end{array} ]~^{\mathrm{T}})$$[ \begin{array}{cccc} \eta~_{1}~&~\eta~_{2}~&~\cdots~&~\eta~_{n} \end{array} ] $
    $A\otimes~B$ $\text{Kronecker~product~of~}A\text{~and~}B$
  • Table 2  

    Table 2Nominal values of parameters

    Parameter Value Unit
    $I_{y}$ 20667.25 $\mathrm{ kg\cdot~m}^{2}$
    $b$ $-108.88$ $\sqrt{\mathrm{kg}}\mathrm{\cdot~m}$
    $\xi$ $0.005$
    $\Lambda~_{y}$ $2\pi~\times~0.151$

Copyright 2020  CHINA SCIENCE PUBLISHING & MEDIA LTD.  中国科技出版传媒股份有限公司  版权所有

京ICP备14028887号-23       京公网安备11010102003388号