#  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
Share
Rating

### Abstract ### 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  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 , 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  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 , 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  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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

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

 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

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

 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

 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

 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

 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

 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

 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 $\mathrm{ kg\cdot~m}^{2}$ $b$ $-108.88$ $\sqrt{\mathrm{kg}}\mathrm{\cdot~m}$ $\xi$ $0.005$ – $\Lambda~_{y}$ $2\pi~\times~0.151$ –