Manoeuvring and vibration reduction of a flexible spacecraft integrating optimal sliding mode controller and distributed piezoelectric sensors/actuators

After publication, it was determined that this paper contained an error where Equation ( 36) cannot be solved because it is constrained by the inequality of Equation (30). To overcome this, we here change the mistake into a static output feedback problem. Equation (29) can be rewritten in a first‐order state‐space form 31\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$$ \dot{X}\,{=}\,\tilde{A}X+\tilde{B}\overline{G}\tilde{C}X$$\end{document}where\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*} X\,{=}\,[{\eta^{\rm{T}},\dot{\eta}{}^{\rm{T}},\xi^{\rm{T}},\dot{\xi}{}^{\rm{T}}}]^{\rm{T}}, \quad \tilde{A}\,{=}\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & I & 0 & 0\\ \noalign{\vspace*{6pt}} {-\Lambda} & {-C_d} & 0 & 0\\ \noalign{\vspace*{6pt}} 0 & 0 & 0 & I\\ \noalign{\vspace*{6pt}} {\Lambda _f \overline{C}_s} & 0 & {-\Lambda _f} & {-C_f} \end{array}\right],\quad \tilde{B}\,{=}\left[\begin{array}{c} 0\\ \noalign{\vspace*{6pt}} {\overline{B}_a}\\ \noalign{\vspace*{6pt}} 0\\ \noalign{\vspace*{6pt}} 0 \end{array}\right] \end{eqnarray*}\end{document}and C̃ = [0 0 Λ 0]. The following cost function is to be minimized by the feedback gains Ḡ: 32\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$$J\,{=}\,{\frac{1}{2}}\int\nolimits_0^\infty {(X^{\rm{T}}Q_s X+v_a^{\rm{T}} Q_v v_a )\,\rm{d}t}$$\end{document}From the above Equation ( 31), the cost function designed in Equation ( 32) becomes 33\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$$J\,{=}\,{\frac{1}{2}}X(0)^{\rm{T}}[P_s+P_u ]X(0)$$\end{document}where 34\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$$P_s\,{=}\,\int\nolimits_0^\infty {{\rm{e}}^{(\tilde{A}+\tilde{B}\overline{G}\tilde{C})^{\rm{T}}t}} Q_s {\rm{e}}^{(\tilde{A}+\tilde{B}\overline{G}\tilde{C})t}\,\rm{d}t \quad\mbox{and}\quad P_u\,{=}\,\int\nolimits_0^\infty {{\rm{e}}^{(\tilde{A}+\tilde{B}\overline{G}\tilde{C})^{\rm{T}}t}} G^{\rm{T}}Q_v Ge^{(\tilde {A}+\tilde{B}\overline{G}\tilde{C})t}\,\rm{d}t$$\end{document}The optimization problem can be restated as follows: 35\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$$J_{\min}\,{=}\,X(0)^{\rm{T}}PX(0) $$\end{document}subject to 36\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$$(\tilde{A}+\tilde{B}\overline{G}\tilde{C})^{\rm{T}}P+P(\tilde{A}+\tilde{B}\bar {G}\tilde{C})+Q_s \le 0$$\end{document}where 37\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$$\overline{P}\,{=}\,P_s+P_u\quad\mbox{and}\quad \overline{Q}_s\,{=}\, Q_s+[0\enspace 0\enspace 1\enspace 0]^{\rm{T}}\Lambda \overline{G}^{\rm{T}}Q_v \overline{G}\Lambda [0\enspace 0 \enspace 1 \enspace 0]$$\end{document}Note that this kind of constraint optimization problem is a standard static output feedback problem, which may be solved by using the efficient software package LMI Control Toolbox for MATLAB. For space limitation, the details of the derivative process are omitted here.