Robust $H_\infty$ asynchronous fault detection for uncertain singular hybrid systems based on Hmm strategy

Abstract

This paper researches robust $H_\infty$ asynchron-ous fault detection for uncertain singular Markov jump systems with time-varying delays based on hidden Marko-v model strategy. The aim is to implement asynchronous fault detection for uncertain singular Markov jump system and realize stochastic admissibility with $H_\infty$ performance level for augmented uncertain singular Markov jump fault detection system. By applying singular value decomposition method and free weighting matrix technique, modified admissibility conditions are addressed based on Lyapunov stability theory. Robust fault detection problem is translated into $H_\infty$ filter design in this work. A hidden Markov model is used to describe a kind of asynchronous phenomenon produced by original system’s modes and fault detection filter’s modes, and the desired filter gains are obtained by solving linear matrix inequalities. Finally, a numerical example and a direct current motor system are used to verify the effectiveness of this approach.

Appendix

Conditions of theorem 2

$$\begin{aligned} \begin{aligned} \breve{X}&=\left[ \begin{array}{cccccccccc} X^{1}_{11} &{} 0&{}0&{}X^{1}_{12}&{}0&{}0\\ *&{} X^{2}_{11}&{}0&{}0&{}X^{2}_{12}&{}0\\ *&{}*&{}X^{3}_{11}&{}0&{}0&{}X^{3}_{12}\\ *&{}*&{}*&{}X^{1}_{22}&{}0&{}0\\ *&{}*&{}*&{}*&{}X^{2}_{22}&{}0\\ *&{}*&{}*&{}*&{}*&{}X^{3}_{22} \end{array} \right] ,\\ \Sigma _{1}&=\left[ \begin{array}{cccccc} X^{1}_{11}&{}0&{}0\\ *&{}X^{2}_{11}&{}0\\ *&{}*&{}X^{3}_{11} \end{array} \right] ,~~~~~~~~ \Sigma _{2}=\left[ \begin{array}{cccccc} X^{1}_{12}&{}0&{}0\\ *&{}X^{2}_{12}&{}0\\ *&{}*&{}X^{3}_{12} \end{array} \right] ,\\ \Sigma _{3}&=\left[ \begin{array}{cccccc} N_{11}&{}N_{12}&{}N_{13}\\ *&{}N_{22}&{}N_{23}\\ *&{}*&{}N_{33} \end{array} \right] ,~~~~~~~~ \Sigma _{4}=\left[ \begin{array}{cccccc} X^{1}_{22}&{}0&{}0\\ *&{}X^{2}_{22}&{}0\\ *&{}*&{}X^{3}_{22} \end{array} \right] ,\\ \Sigma _{5}&=\left[ \begin{array}{cccccc} \hat{N}_{11}&{}\hat{N}_{12}&{}\hat{N}_{13}\\ *&{}\hat{N}_{22}&{}\hat{N}_{23}\\ *&{}*&{}\hat{N}_{33} \end{array} \right] ,~~~~~~~~~~~~~ \Upsilon _{1i\theta }=\left[ \begin{array}{cccccccc} \tilde{\gamma }_{1i\theta }&{}\tilde{\gamma }_{2i\theta }\\ *&{}\tilde{\gamma }_{3i\theta } \end{array} \right] , \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Phi _{i\theta 2}&=\left[ \begin{array}{cccccc} \Phi _{14i\theta }&{}\Phi _{15i\theta }&{}\Phi _{16i\theta }\\ \Phi _{24i\theta }&{}\Phi _{25i\theta }&{}\Phi _{26i\theta }\\ \Phi _{34i}&{}\Phi _{35i}&{}\Phi _{36i} \end{array} \right] ,~~~~~~~~ \Upsilon _{2i\theta }=\left[ \begin{array}{cccccccc} \tilde{\gamma }_{4i\theta }&{}\tilde{\gamma }_{5i\theta }\\ \tilde{\gamma }_{6i\theta }&{}\tilde{\gamma }_{7i\theta } \end{array} \right] ,\\ \Phi _{i\theta 1}&=\left[ \begin{array}{cccccc} \Phi _{11i\theta }&{}\Phi _{12i\theta }&{}\Phi _{13i\theta }\\ *&{}\Phi _{22i\theta }&{}\Phi _{23i\theta }\\ *&{}*&{}\Phi _{33i} \end{array} \right] , \Phi _{i\theta 5}=\left[ \begin{array}{cccccc} \Phi _{44i}&{}\Phi _{45i}&{}\Phi _{46i}\\ *&{}\Phi _{55}&{}\Phi _{56i}\\ *&{}*&{}\Phi _{66i} \end{array} \right] ,\\ \Sigma _{6}&=\left[ \begin{array}{cccccc} E^{T}R_{1}E&{}0&{}0\\ *&{}E_{f}^{T}R_{2}E_{f}&{}0\\ *&{}*&{}R_{3} \end{array} \right] ,~~~~~~ \tilde{\gamma }_{1}^{1}=\left[ \begin{array}{cccccccc} \Upsilon _{88}&{}\Upsilon _{89}\\ *&{}\Upsilon _{99}\\ \end{array} \right] , \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \tilde{\gamma }_{2i\theta }&=\left[ \begin{array}{cccccc} \Upsilon _{15i\theta }&{}\Upsilon _{16i\theta }&{}\Upsilon _{17i\theta }\\ \Upsilon _{25i\theta }&{}\Upsilon _{26i\theta }&{}\Upsilon _{27i\theta }\\ \Upsilon _{35i}&{}\Upsilon _{36i}&{}\Upsilon _{37i}\\ \Upsilon _{45i}&{}\Upsilon _{46i}&{}\Upsilon _{47i}\\ \end{array} \right] , \tilde{\gamma }_{3i\theta }=\left[ \begin{array}{cccccc} \Upsilon _{55}&{}\Upsilon _{56i}&{}\Upsilon _{57i\theta }\\ *&{}\Upsilon _{66i}&{}\Upsilon _{67i\theta }\\ *&{}*&{}\Upsilon _{77} \end{array} \right] ,\\ \tilde{\gamma }_{1i\theta }&=\left[ \begin{array}{cccccc} \Upsilon _{11i\theta }&{}\Upsilon _{12i\theta }&{}\Upsilon _{13i\theta }&{}\Upsilon _{14i\theta }\\ *&{}\Upsilon _{22i\theta }&{}\Upsilon _{23i\theta }&{}\Upsilon _{24i\theta }\\ *&{}*&{}\Upsilon _{33i}&{}\Upsilon _{34i}\\ *&{}*&{}*&{}\Upsilon _{44i} \end{array} \right] ,~~~ \Upsilon _{5i}=\left[ \begin{array}{cccccccc} 0&{}0\\ \tilde{\Upsilon }_{3i}^{3}&{}\tilde{\Upsilon }_{4i}^{4} \end{array} \right] ,\\ \tilde{\gamma }_{4i\theta }&\!=\!\left[ \begin{array}{cccccc} \Upsilon _{18i\theta }&{}\Upsilon _{19}&{}\Upsilon ^{1}_{11i\theta }\\ \Upsilon _{28i\theta }&{}\Upsilon _{29}&{}\Upsilon ^{1}_{21i\theta }\\ \Upsilon _{38i\theta }&{}\Upsilon _{39}&{}\Upsilon ^{1}_{31i\theta } \end{array} \right] \!,\! \tilde{\gamma }_{5i\theta }\!=\!\left[ \begin{array}{cccccc} \Upsilon ^{1}_{12i\theta }&{}\Upsilon ^{1}_{13i\theta }&{}\Upsilon ^{1}_{14i}&{}\Upsilon ^{1}_{15i\theta }\\ \Upsilon ^{1}_{22i\theta }&{}\Upsilon ^{1}_{23i\theta }&{}\Upsilon ^{1}_{24i}&{}\Upsilon ^{1}_{25i\theta }\\ \Upsilon ^{1}_{32i\theta }&{}\Upsilon ^{1}_{33i\theta }&{}\Upsilon ^{1}_{34}&{}\Upsilon ^{1}_{35i\theta }\\ \end{array} \right] \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \tilde{\gamma }_{6i\theta }&=\left[ \begin{array}{cccccc} 0&{}0&{}\Upsilon ^{1}_{41i\theta }\\ 0&{}0&{}\Upsilon ^{1}_{51i\theta }\\ 0&{}0&{}\Upsilon ^{1}_{61i\theta }\\ \Upsilon _{78}&{}\Upsilon _{79}&{}0 \end{array} \right] ,\! \tilde{\gamma }_{7i\theta }=\left[ \begin{array}{cccccc} \Upsilon ^{1}_{42i\theta }&{}\Upsilon ^{1}_{43i\theta }&{}0&{}0\\ \Upsilon ^{1}_{52i\theta }&{}\Upsilon ^{1}_{53i\theta }&{}0&{}0\\ \Upsilon ^{1}_{62i\theta }&{}\Upsilon ^{1}_{63i\theta }&{}0&{}0\\ 0&{}0&{}\Upsilon ^{1}_{74i}&{}0 \end{array} \right] ,\\ \Upsilon _{3i\theta }&=\left[ \begin{array}{cccccccc} \tilde{\mathfrak {N}}_{1i}&{}0&{}\tilde{\gamma }_{8i\theta }&{}0\\ 0&{}\tilde{\gamma }_{9i\theta }&{}0&{}0\\ \tilde{\mathfrak {N}}_{2i}&{}0&{}0&{}\tilde{\mathfrak {N}}_{3i} \end{array} \right] , \tilde{\mathfrak {N}}_{1i}=\left[ \begin{array}{cccccc} \mathfrak {N}^{T}_{1i}&{}\mathfrak {N}^{T}_{1i}\\ 0&{}0\\ 0&{}0 \end{array} \right] ,\\ \tilde{\gamma }_{9i\theta }&=\left[ \begin{array}{cccccc} \Upsilon ^{2}_{42i\theta }&{}{0}&{}{0}\\ 0&{}0&{}\Upsilon ^{2}_{52i\theta }\\ 0&{}0&{}\Upsilon ^{2}_{62i\theta } \end{array} \right] ,~~~~~~~ \Upsilon _{4i}=\left[ \begin{array}{cccccccc} \tilde{\gamma }_{1}^{1}&{}0&{}0&{}0\\ *&{}\tilde{\varpi }&{}\tilde{\gamma }_{2i}^{2}&{}0\\ *&{}*&{}-I&{}0\\ *&{}*&{}*&{}-\varepsilon _{1}I \end{array} \right] , \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \tilde{\Upsilon }_{3i}^{3}&=\left[ \begin{array}{cccccccc} 0&{}0&{}\Upsilon ^{3}_{4i}\\ 0&{}0&{}0 \end{array} \right] , \tilde{\Upsilon }_{4i}^{4}=\left[ \begin{array}{cccccccc} 0&{}0&{}0&{}\Upsilon ^{3}_{5i}\\ 0&{}0&{}0&{}0 \end{array} \right] , \tilde{\mathfrak {N}}^{T}_{2i}=\left[ \begin{array}{cccccc} \mathfrak {N}_{2i}\\ 0 \end{array} \right] ,\\ \tilde{\mathfrak {N}}^{T}_{3i}&=\left[ \begin{array}{cccccc} \mathfrak {N}_{2i}\\ \mathfrak {N}_{2i}\\ 0 \end{array} \right] ,~~ \tilde{\gamma }_{8i\theta }=\left[ \begin{array}{cccccc} \Upsilon ^{2}_{13i\theta }\\ \Upsilon ^{2}_{23i\theta }\\ \Upsilon ^{2}_{33i\theta } \end{array} \right] ,~~~ \Phi _{i\theta 4}=\left[ \begin{array}{cccccc} N^{T}_{1}\\ 0\\ 0 \end{array} \right] ,\\ \tilde{\gamma }_{2i}^{2}&=\left[ \begin{array}{cccccccc} \Upsilon ^{3}_{1i}\\ \Upsilon ^{3}_{2i}\\ \Upsilon ^{3}_{3i} \end{array} \right] ,~~ \Phi _{i\theta 3}=\left[ \begin{array}{cccccc} \Phi _{17i\theta }\\ \Phi _{27i\theta }\\ \Phi _{37i} \end{array} \right] ,~~~ \Phi _{i\theta 6}=\left[ \begin{array}{cccccc} \Phi _{47i\theta }\\ \Phi _{57i\theta }\\ \Phi _{67i} \end{array} \right] ,\\ \tilde{\varpi }&={ -diag\left\{ \varpi I,~\varpi I,~\varpi I\right\} ,}\\ \Upsilon _{6}&=-diag\left\{ \varepsilon ^{-1}_{1}I,~ \varepsilon _{2}I, ~ \varepsilon ^{-1}_{2}I,~ \varepsilon _{3}I,~ \varepsilon ^{-1}_{3}I, ~ \varepsilon _{4}I, ~\varepsilon ^{-1}_{4}I\right\} \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \Phi _{11i\theta }&=sym\{T_{1i}A_{i}+\mathcal {B}_{fi\theta }C_{i}\}+\sum \limits _{j=1}^{N}\pi _{ij}E^{T}\tilde{P}_{j1}E,\\ \Phi _{12i\theta }&=\mathcal {A}_{fi\theta }+A^{T}_{i}T^{T}_{2i}+C^{T}_{i}\mathcal {B}^{T}_{fi\theta }+\sum \limits _{j=1}^{N}\pi _{ij}E^{T}\tilde{P}_{j2}E_{f},\\ \Phi _{13i\theta }&=T_{4i}A_{w}+A^{T}_{i}T^{T}_{3i}+C^{T}_{i}\mathcal {B}^{T}_{fi\theta }+\sum \limits _{j=1}^{N}\pi _{ij}E^{T}\tilde{P}_{j3}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Phi _{22i\theta }&=sym\{\mathcal {A}_{fi\theta }\}+\sum \limits _{j=1}^{N}\pi _{ij}E_{f}^{T}\tilde{P}_{j4}E_{f},\\ \Phi _{23i\theta }&=T_{5i}A_{w}+\mathcal {A}^{T}_{fi\theta }+\sum \limits _{j=1}^{N}\pi _{ij}E_{f}^{T}\tilde{P}_{j5},\\ \Phi _{33i}&=sym\{T_{6i}A_{w}\}+\sum \limits _{j=1}^{N}\pi _{ij}\tilde{P}_{j6}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Phi _{14i\theta }&=E^{T}\tilde{P}^{T}_{i1}+L_{1i}J^{T}_{1}-T_{1i}+A^{T}_{i}T^{T}_{11i}+C^{T}_{i}\mathcal {B}^{T}_{fi\theta },\\ \Phi _{15i\theta }&=E^{T}\tilde{P}_{i2}+L_{2i}J^{T}_{2}-T+A^{T}_{i}T^{T}_{21i}+C^{T}_{i}\mathcal {B}^{T}_{fi\theta },\\ \Phi _{16i\theta }&=E^{T}\tilde{P}_{i3}+L_{3i}J^{T}_{3}-T_{4i}+A^{T}_{i}T^{T}_{31i}+C^{T}_{i}\mathcal {B}^{T}_{fi\theta },\\ \Phi _{24i\theta }&=E^{T}_{f}\tilde{P}^{T}_{i2}+L_{4i}J^{T}_{1}-T_{2i}+A^{T}_{fi\theta },\\ \Phi _{25i\theta }&=E^{T}_{f}\tilde{P}^{T}_{i4}+L_{5i}J^{T}_{2}-T+A^{T}_{fi\theta },\\ \Phi _{26i\theta }&=E^{T}_{f}\tilde{P}_{i5}+L_{6i}J^{T}_{3}-T_{5i}+A^{T}_{fi\theta },\\ \Phi _{34i}&=\tilde{P}^{T}_{i3}+L_{7i}J^{T}_{1}-T_{3i}+A^{T}_{w}T^{T}_{41i},\\ \Phi _{35i}&=\tilde{P}^{T}_{i5}+L_{8i}J^{T}_{2}-T+A^{T}_{w}T^{T}_{51i},\\ \Phi _{36i}&=\tilde{P}^{T}_{i6}+L_{9i}J^{T}_{3}-T_{6i}+A^{T}_{w}T^{T}_{61i}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Phi _{17i\theta }&=T_{1i}M_{1i}+\mathcal {B}_{fi\theta }M_{2i},\\ \Phi _{37i\theta }&=T_{3i}M_{1i}+\mathcal {B}_{fi\theta }M_{2i},\\ \Phi _{47i\theta }&=T_{11i}M_{1i}+\mathcal {B}_{fi\theta }M_{2i},\\ \Phi _{57i\theta }&=T_{21i}M_{1i}+\mathcal {B}_{fi\theta }M_{2i},\\ \Phi _{67i\theta }&=T_{31i}M_{1i}+\mathcal {B}_{fi\theta }M_{2i},\\ \Phi _{44i}&=-sym\{T_{11i}\},~~~~~~~~~ \Phi _{45i}=-sym\{T^{T}_{21i}\}-T,\\ \\ \Phi _{27i\theta }&=T_{2i}M_{1i}+\mathcal {B}_{fi\theta }M_{2i},\\ \Phi _{46i}&=-sym\{T^{T}_{31i}\}-T_{41i},~~ \Phi _{55}=-sym\{T\},\\ \Phi _{56i}&=-T^{T}-T_{51i},~~~~~~~~~~~ \Phi _{66i}=-sym\{T_{61i}\}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon _{11i\theta }&=sym\{\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }C_{i}+T_{1i}A_{i}\}+\sum \limits _{j=1}^{N}\pi _{ij}E^{T}\tilde{P}_{j1}E+\\ {}&Q_{1}+sym\{N_{11}\}+\tau X^{1}_{11}, \Upsilon _{44i}=-sym\{T_{11i}\}+\tau R_{1},\\ \Upsilon _{12i\theta }&=\sum \limits _{\theta =1}^{M}\lambda _{i\theta }(\mathcal {A}_{fi\theta }+C^{T}_{i}\mathcal {B}^{T}_{fi\theta })+A^{T}_{i}T^{T}_{2i}+N_{12}+N_{12}\\ {}&+\sum \limits _{j=1}^{N}\pi _{ij}E^{T}\tilde{P}_{j2}E_{f},~~~ \Upsilon _{18i\theta }=-N_{12}+\hat{N}_{12},\\ \Upsilon _{13i\theta }&=T_{4i}A_{w}+A^{T}_{i}T^{T}_{3i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }C^{T}_{i}\mathcal {B}^{T}_{fi\theta }\\ {}&+\sum \limits _{j=1}^{N}\pi _{ij}E^{T}\tilde{P}_{j3}+N_{13}+N_{13},~ ~\Upsilon _{45i}=-T-T^{T}_{21i}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon _{22i\theta }&=sym\{\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {A}_{fi\theta }\}+\sum \limits _{j=1}^{N}\pi _{ij}E_{f}^{T}\tilde{P}_{j4}E_{f}+Q_{2}\\ {}&+sym\{N_{22}\}+ \tau X^{2}_{11},~~~~~~ \Upsilon _{46i}=-T_{41i}-T^{T}_{31i},\\ \Upsilon _{23i\theta }&\!=\!T_{5i}A_{w}\!+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {A}^{T}_{fi\theta }\!+\sum \limits _{j=1}^{N}\pi _{ij}E_{f}^{T}\tilde{P}_{j5}\!+N_{23}+N_{23},\\ \Upsilon _{55}&=-sym\{T\}+\tau R_{2},~~~~~~~ \Upsilon _{56i}=-T_{51i}-T^{T},\\ \Upsilon _{33i}&=sym\{T_{6i}A_{w}+N_{33}\}+\sum \limits _{j=1}^{N}\pi _{ij}\tilde{P}_{j6}+Q_{3}+X^{3}_{11},\\ \Upsilon _{14i\theta }&=E^{T}\tilde{P}^{T}_{i1}+L_{1i}J^{T}_{1}\!-T_{1i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }C^{T}_{i}\mathcal {B}^{T}_{fi\theta }+A^{T}_{i}T^{T}_{11i}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon _{19}&=-N_{13}+\hat{N}_{13},~~~~ \Upsilon _{29}=-N_{23}+\hat{N}_{23},\\ \Upsilon _{15i\theta }&=E^{T}\tilde{P}_{i2}+L_{2i}J^{T}_{2}-T+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }C^{T}_{i}\mathcal {B}^{T}_{fi\theta }+A^{T}_{i}T^{T}_{21i},\\ \Upsilon _{16i\theta }&=E^{T}\tilde{P}_{i3}+L_{3i}J^{T}_{3}-T_{4i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }C^{T}_{i}\mathcal {B}^{T}_{fi\theta }+A^{T}_{i}T^{T}_{31i},\\ \Upsilon _{66i}&=-sym\{T^{T}_{61i}\}+\tau R_{3}, \Upsilon _{28i\theta }=-N_{22}+\hat{N}^{T}_{22}+\tau X^{2}_{12}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon _{24i\theta }&=E_{f}^{T}\tilde{P}^{T}_{i2}+L_{4i}J^{T}_{1}-T_{2i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {A}^{T}_{fi\theta },\\ \Upsilon _{25i\theta }&=E_{f}^{T}\tilde{P}^{T}_{i4}+L_{5i}J^{T}_{2}-T+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {A}^{T}_{fi\theta },\\ \Upsilon _{26i\theta }&=E_{f}^{T}\tilde{P}_{i5}+L_{6i}J^{T}_{3}-T_{5i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {A}^{T}_{fi\theta },\\ \Upsilon _{34i}&=\tilde{P}^{T}_{i3}+L_{7i}J^{T}_{1}-T_{3i}+A^{T}_{w}T^{T}_{41i},\\ \Upsilon _{35i}&=\tilde{P}^{T}_{i5}+L_{8i}J^{T}_{2}-T+A^{T}_{w}T^{T}_{51i},\\ \Upsilon _{36i}&=\tilde{P}_{i6}+L_{9i}J^{T}_{3}-T_{6i}+A^{T}_{w}T^{T}_{61i}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon _{17i\theta }&=T_{1i}A_{di}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }C_{di}-N_{11}+\hat{N}^{T}_{11}+\tau X^{1}_{12},\\ \Upsilon _{27i\theta }&=T_{2i}A_{di}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }C_{di}-N_{12}+\hat{N}^{T}_{12},\\ \Upsilon _{99}&=-(1-\mu )Q_{3}-sym\{\hat{N}_{33}\}+\tau X^{3}_{22},\\ \Upsilon _{37i\theta }&=T_{3i}A_{di}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }C_{di}-N^{T}_{13}+\hat{N}^{T}_{13},\\ \\ \Upsilon _{38i\theta }&=-N^{T}_{23}+\hat{N}^{T}_{23},~~ \Upsilon _{39}=-N_{33}+\hat{N}^{T}_{33}+\tau X^{3}_{12}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon _{47i\theta }&=\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }C_{di}+T_{11i}A_{di},\\ \Upsilon _{57i\theta }&=\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }C_{di}+T_{21i}A_{di},\\ \Upsilon _{67i\theta }&=\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }C_{di}+T_{31i}A_{di},\\ \Upsilon _{77}&=-(1-\mu )Q_{1}-sym\{\hat{N}_{11}\}+\tau X^{1}_{22},\\ \Upsilon _{78}&=-\hat{N}_{12}-\hat{N}_{12},~~ \Upsilon _{79}=-\hat{N}_{13}-\hat{N}_{13},\\ \Upsilon _{88}&=-(1-\mu )Q_{2}-sym\{\hat{N}_{22}\}+\tau X^{2}_{22},\\ \Upsilon _{89}&=-\hat{N}_{23}-\hat{N}_{23}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon ^{1}_{11i\theta }&=T_{1i}B_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }G_{i},\\ \Upsilon ^{1}_{12i\theta }&=T_{1i}D_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }H_{i},\\ \Upsilon ^{1}_{13i\theta }&=T_{1i}F_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }V_{i}+T_{4i}B_{w},\\ \Upsilon ^{1}_{21i\theta }&=T_{2i}B_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }G_{i},\\ \Upsilon ^{1}_{22i\theta }&=T_{2i}D_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }H_{i}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon ^{1}_{23i\theta }&=T_{2i}F_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }V_{i}+T_{5i}B_{w},\\ \Upsilon ^{1}_{31i\theta }&=T_{3i}B_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }G_{i},\\ \Upsilon ^{1}_{32i\theta }&=T_{3i}D_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }H_{i},\\ \Upsilon ^{1}_{33i\theta }&=T_{3i}F_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }V_{i}+T_{6i}B_{w},\\ \Upsilon ^{1}_{41i\theta }&=T_{11i}B_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }G_{i}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon ^{1}_{42i\theta }&=T_{11i}D_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }H_{i},\\ \Upsilon ^{1}_{43i\theta }&=T_{11i}F_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }V_{i}+T_{41i}B_{w},\\ \Upsilon ^{1}_{51i\theta }&=T_{21i}B_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }G_{i},\\ \Upsilon ^{1}_{52i\theta }&=T_{21i}D_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }H_{i},\\ \Upsilon ^{1}_{53i\theta }&=T_{21i}F_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }V_{i}+T_{51i}B_{w},\\ \Upsilon ^{1}_{61i\theta }&=T_{31i}B_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }G_{i},\\ \Upsilon ^{1}_{62i\theta }&=T_{31i}D_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }H_{i}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon ^{1}_{63i\theta }&=T_{31i}F_{i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }V_{i}+T_{61i}B_{w},\\ \Upsilon ^{1}_{15i\theta }&=T_{1i}M_{1i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }M_{2i},\\ \Upsilon ^{1}_{25i\theta }&=T_{2i}M_{1i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }M_{2i},\\ \Upsilon ^{1}_{35i\theta }&=T_{3i}M_{1i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }M_{2i},\\ \Upsilon ^{2}_{42i\theta }&=\Upsilon ^{2}_{13i\theta }=T_{11i}M_{1i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }M_{2i},\\ \Upsilon ^{2}_{52i\theta }&=\Upsilon ^{2}_{23i\theta }=T_{21i}M_{1i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }M_{2i}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon ^{1}_{14i}&=[\sqrt{\lambda _{i1}}C^{T}_{i}\mathcal {D}^{T}_{fi1}~\sqrt{\lambda _{i2}}C^{T}_{i}\mathcal {D}^{T}_{fi2}\cdots ~\sqrt{\lambda _{iM}}C^{T}_{i}\mathcal {D}^{T}_{fiM}],\\ \Upsilon ^{1}_{24i}&=[\sqrt{\lambda _{i1}}\mathcal {C}^{T}_{fi1}~\sqrt{\lambda _{i2}}\mathcal {C}^{T}_{fi2}\cdots ~\sqrt{\lambda _{iM}}\mathcal {C}^{T}_{fiM}],\\ \Upsilon ^{1}_{74i}&=[\sqrt{\lambda _{i1}}C^{T}_{di}\mathcal {D}^{T}_{fi1}~\sqrt{\lambda _{i2}}C^{T}_{di}\mathcal {D}^{T}_{fi2}\cdots ~\sqrt{\lambda _{iM}}C^{T}_{di}\mathcal {D}^{T}_{fiM}],\\ \Upsilon ^{1}_{34i}&=-[\sqrt{\lambda _{i1}}C^{T}_{w}~\sqrt{\lambda _{i2}}C^{T}_{w}\cdots ~\sqrt{\lambda _{iM}}C^{T}_{w}],\\ \Upsilon ^{3}_{1i}&=[\sqrt{\lambda _{i1}}G^{T}_{i}\mathcal {D}_{fi1}~\sqrt{\lambda _{i2}}G^{T}_{i}\mathcal {D}_{fi2}\cdots ~\sqrt{\lambda _{iM}}G^{T}_{i}\mathcal {D}_{fiM}], \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Upsilon ^{2}_{62i\theta }&=\Upsilon ^{2}_{33i\theta }=T_{31i}M_{1i}+\sum \limits _{\theta =1}^{M}\lambda _{i\theta }\mathcal {B}_{fi\theta }M_{2i},\\ \Upsilon ^{3}_{2i}&=[\sqrt{\lambda _{i1}}H^{T}_{i}\mathcal {D}_{fi1}~\sqrt{\lambda _{i2}}H^{T}_{i}\mathcal {D}_{fi2}\cdots ~\sqrt{\lambda _{iM}}H^{T}_{i}\mathcal {D}_{fiM}],\\ \Upsilon ^{3}_{3i}&=\left[ \begin{array}{cccccc} \sqrt{\lambda _{i1}}V^{T}_{i}\mathcal {D}_{fi1}-D_w&\sqrt{\lambda _{i2}}V^{T}_{i}\mathcal {D}_{fi2}-D_w&\end{array}\right. \\&\left. \begin{array}{ccc} \cdots&\sqrt{\lambda _{iM}}V^{T}_{i}\mathcal {D}_{fiM}-D_w \end{array} \right] ,\\ \Upsilon ^{3}_{4i}&=\Upsilon ^{3}_{5i}=\left[ \begin{array}{cccccc} \sqrt{\lambda _{i1}}M^{T}_{2i}\mathcal {D}^{T}_{fi1}&\sqrt{\lambda _{i2}}M^{T}_{2i}\mathcal {D}^{T}_{fi2}&\end{array}\right. \\&\left. \begin{array}{ccc} \cdots&\sqrt{\lambda _{iM}}M^{T}_{2i}\mathcal {D}^{T}_{fiM} \end{array} \right] ^{T}.\\ \end{aligned} \end{aligned}$$