Tao Jiang 2024a - Revision history

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@@ Line 1,213: / Line 1,213: @@
 <span id='OLE_LINK12'></span><span id='OLE_LINK13'></span>In this paper, the problem of a general Lipschitz nonlinear system with actuator fault and unmatched disturbance is investigated. Specifically, a disturbance observer is designed to estimate the mismatched disturbance first. Then, an observer-based integral sliding mode fault tolerant control scheme is proposed. In order to guarantee the stability of the system, three adaptive control laws are constructed because of the unknown nonlinear function parameters and the unmodeled uncertainty. Finally, two examples are given to illustrate the effectiveness of the proposed method. In our future work, we would like to focus on the fault-tolerant control methods for multiple faults and disturbances and their applications.
-==Reference==
+==References==
 <div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">

@@ Line 1,253: / Line 1,253: @@
 [18]  Kempf C.J.,  Kobayashi S. Disturbance observer and feed forward design for a high-speed direct-drive position table, IEEE Trans. Control Syst. Technol.,  7(5):513–526, 1999.
-[19] X.S. Chen, J. Yang, S.H. Li, and Q. Li ,‘Disturbance observer based multi-variable control of ball mill grinding circuits’, J. Process Control, 19, 7,  1205–1213, Jul.2009.
+[19]  Chen X.S.,  Yang J.,  Li S.H.,   Li Q. Disturbance observer based multi-variable control of ball mill grinding circuits, J. Process Control, 19(7):1205–1213, 2009.
-[20] W.H. Chen, Disturbance observer based control for nonlinear systems, IEEE/ASME Trans. Mechatronics, 9, 4,  706–710, Dec. 2004.
+[20]  Chen W.H. Disturbance observer based control for nonlinear systems, IEEE/ASME Trans. Mechatronics, 9(4):706–710, 2004.
-[21] X. Yao and L. Guo, Composite anti-disturbance control for Markovian jump nonlinear systems via disturbance observer, Automatica,  49,  8,  2538–2545, Aug. 2013.
+[21]  Yao X.,   Guo L. Composite anti-disturbance control for Markovian jump nonlinear systems via disturbance observer, Automatica,  49(8):2538–2545,  2013.
-[22] H.N. Wu, ZY. Liu, and L. Guo, Robust  <math>L_{\infty }</math> -gain fuzzy disturbance observer-based control design with adaptive bounding for a hypersonic vehicle, IEEE Trans. Fuzzy Syst.,  22,  6,  1401–1412, Dec. 2014.
+[22]  Wu H.N.,  Liu Z.Y.,  Guo  L. Robust  <math>L_{\infty }</math>-gain fuzzy disturbance observer-based control design with adaptive bounding for a hypersonic vehicle, IEEE Trans. Fuzzy Syst.,  22(6):1401–1412,  2014.
-[23] H. Zhang, J .Han, C. Luo, and Y. Wang, Fault-Tolerant Control of a Nonlinear System Based on Generalized Fuzzy Hyperbolic Model and Adaptive Disturbance Observer, IEEE Trans. Syst, Man, Cybern, Syst., 47 (8) :2289-2300, Aug.2017.
+[23]  Zhang H.,  Han J.,  Luo C., Wang Y. Fault-tolerant control of a nonlinear system based on generalized fuzzy hyperbolic model and adaptive disturbance observer, IEEE Trans. Syst, Man, Cybern, Syst., 47(8):2289-2300, 2017.
-[24] X.J. Wei, and L. Guo, Composite disturbance-observer-based control and H-infinity control for complex continuous models, Int. J. Robust Nonlinear Control, 20, (1),  106–118, Apr.2009.
+[24]  Wei X.J.,  Guo L. Composite disturbance-observer-based control and H-infinity control for complex continuous models, Int. J. Robust Nonlinear Control, 20(1):106–118, 2009.
-[25] J. Yang, S. Li, and X. Yu, Sliding-Mode Control for Systems With Mismatched Uncertainties via a Disturbance Observer, IEEE Trans. Ind. Electron., 60 (1) :160-169, Jan.2013.
+[25]  Yang J.,  Li S.,  Yu X. Sliding-mode control for systems with mismatched uncertainties via a disturbance observer, IEEE Trans. Ind. Electron., 60 (1):160-169, 2013.
-[26] H. S. Wu, Adaptive robust tracking and model following of uncertain dynamical systems with multiple time delays, IEEE Trans. Autom. Control,  49,  4,  611–616, Apr. 2004.
+[26]  Wu H.S. Adaptive robust tracking and model following of uncertain dynamical systems with multiple time delays, IEEE Trans. Autom. Control,  49(4):611–616, 2004.
-[27] J.W. Zhu, and G.H Yang, Fault-tolerant control for linear systems with multiple faults and disturbances based on augmented intermediate estimator, IET Control Theory & Applications, 11, 2, 164-172, Jan.2017.
+[27]  Zhu J.W., Yang G.H. Fault-tolerant control for linear systems with multiple faults and disturbances based on augmented intermediate estimator, IET Control Theory & Applications, 11(2):164-172, 2017.

@@ Line 1,223: / Line 1,223: @@
 [3]  Lin F.J.,  Hung Y.C.,  Tsai M.T. Fault-tolerant control for six-phase PMSM  drive system via intelligent complementary sliding mode control using TSKFNN-AMF, IEEE Trans. Ind. Electron., 60(12):5747–5762, 2013.
-[4] J. Shao, Z. Deng, and Y. Gu, Fault-Tolerant Control of Position Signals for Switched Reluctance Motor Drives, IEEE Trans Ind. Appli., 53 3, 2959-2966, May.2017.
+[4]  Shao J.,  Deng Z.,   Gu Y. Fault-tolerant control of position signals for switched reluctance motor drives, IEEE Trans Ind. Appli., 53(3):2959-2966, 2017.
-[5] Y. Wang, L. Xie, and C. E. de Souza, Robust control of a class of uncertain nonlinear systems, Syst. Control Lett.,  19,  2,  139–149, Aug.1992.
+[5] Wang  Y.,  Xie L.,   de Souza C.E. Robust control of a class of uncertain nonlinear systems, Syst. Control Lett.,  19(2):139–149, 1992.
-[6] P. Shi, M. Liu, and L. Zhang, Fault-tolerant sliding mode observer synthesis of Markovian jump systems using quantized measurements, IEEE Trans. Ind. Electron.,  62,  9,  5910–5918, Sep. 2015.
+[6]  Shi P.,  Liu M.,  Zhang L. Fault-tolerant sliding mode observer synthesis of Markovian jump systems using quantized measurements, IEEE Trans. Ind. Electron.,  62(9):5910–5918,  2015.
-[7] Q. Hu and B. Xiao, Fault-tolerant sliding mode attitude control for flexible spacecraft under loss of actuator effectiveness, Nonlin. Dyn.,  64,  1,  13–23, Jun.2011.
+[7]  Hu Q.,  Xiao B. Fault-tolerant sliding mode attitude control for flexible spacecraft under loss of actuator effectiveness, Nonlin. Dyn.,  64(1):13–23, 2011.
-[8] S.K. Kommuri, M. Defoort, H.R. Karimi，and K.C. Veluvolu, A Robust Observer-Based Sensor Fault-Tolerant Control for PMSM in Electric Vehicles, IEEE Trans. Ind. Electron.,  63,  12,  7671–7681, Dec. 2016.
+[8]  Kommuri S.K.,  Defoort M.,  Karimi H.R.,   Veluvolu K.C. A robust observer-based sensor fault-tolerant control for PMSM in electric vehicles, IEEE Trans. Ind. Electron.,  63(12):7671–7681, 2016.
-[9] K. Patan, Neural network-based model predictive control: Fault tolerance and stability, IEEE Trans. Control Syst. Technol.,  23,  3,  1147–1155, May 2015.
+[9]  Patan K. Neural network-based model predictive control: Fault tolerance and stability, IEEE Trans. Control Syst. Technol.,  23(3):1147–1155, 2015.
-[10] G.H. Yang and D. Ye, Reliable   control of linear systems with adaptive mechanism, IEEE Trans. Autom. Control,  55,  1,  242–247, Jan. 2010.
+[10]  Yang G.H.,  Ye  D. Reliable   control of linear systems with adaptive mechanism, IEEE Trans. Autom. Control,  55(1):242–247, 2010.
-[11] W. Wang and C. Y. Wen, Adaptive actuator failure compensation control of uncertain nonlinear systems with guaranteed transient performance, Automatica,  46,  12,  2082–2091, Dec.2010.
+[11]  Wang W.,   Wen C.Y. Adaptive actuator failure compensation control of uncertain nonlinear systems with guaranteed transient performance, Automatica,  46(12):2082–2091, 2010.
-[12] D. Ye, and G.H.Yang, Adaptive fault-tolerant tracking control against actuator faults with application to flight control, IEEE Trans. Control Syst. Technol., 14,  1088–1096, Nov.2006.
+[12]  Ye D.,  Yang G.H. Adaptive fault-tolerant tracking control against actuator faults with application to flight control, IEEE Trans. Control Syst. Technol., 14:1088–1096, 2006.
-[13] Z. Zuo, D.W.C. Ho, and Y.Wang, Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation, Automatica,  46, 3,  569–576, Mar.2010.
+[13]  Zuo Z.,  Ho D.W.C.,  Wang Y. Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation, Automatica,  46(3):569–576, 2010.
-[14] S. C. Tong, B. Huo, and Y. M. Li, Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failures, IEEE Trans. Fuzzy Syst.,  22,  1,  1–15, Feb. 2014.
+[14]  Tong S.C.,  Huo B.,  Li Y.M. Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failures, IEEE Trans. Fuzzy Syst.,  22(1):1–15, 2014.
-[15] Y.X. Li, and G.H. Yang, Robust Adaptive Fault-Tolerant Control for a Class of Uncertain Nonlinear Time Delay Systems, IEEE Trans. Syst, Man, Cybern, Syst., 47, 7,  1554-1563, Jul.2017.
+[15]  Li Y.X.,  Yang G.H. Robust adaptive fault-tolerant control for a class of uncertain nonlinear time delay systems, IEEE Trans. Syst, Man, Cybern, Syst., 47(7):1554-1563, 2017.
-[16] M.Van, S.S. Ge, and H .Ren, Robust Fault-Tolerant Control for a Class of Second-Order Nonlinear Systems Using an Adaptive Third-Order Sliding Mode Control, IEEE Trans. Syst, Man, Cybern, Syst., 47, 2,  221-228, Feb. 2017.
+[16] Van M.,  Ge S.S., Ren H. Robust fault-tolerant control for a class of second-order nonlinear systems using an adaptive third-order sliding mode control, IEEE Trans. Syst, Man, Cybern, Syst., 47(2):221-228, 2017.
-[17] Y.W. Liang, S.D. Xu, and C.L. Tsai, Study of VSC reliable designs with application to spacecraft attitude stabilization, IEEE Trans. Control Syst. Technol.,  15,  2,  332–338, Mar. 2007.
+[17]  Liang Y.W.,  Xu S.D.,  Tsai C.L. Study of VSC reliable designs with application to spacecraft attitude stabilization, IEEE Trans. Control Syst. Technol.,  15(2):332–338, 2007.
-[18] C.J. Kempf, and S. Kobayashi. Disturbance observer and feed forward design for a high-speed direct-drive position table, IEEE Trans. Control Syst. Technol.,  7, 5, 513–526, Sep.1999.
+[18]  Kempf C.J.,  Kobayashi S. Disturbance observer and feed forward design for a high-speed direct-drive position table, IEEE Trans. Control Syst. Technol.,  7(5):513–526, 1999.
 [19] X.S. Chen, J. Yang, S.H. Li, and Q. Li ,‘Disturbance observer based multi-variable control of ball mill grinding circuits’, J. Process Control, 19, 7,  1205–1213, Jul.2009.

← Older revision		Revision as of 15:37, 18 March 2024
Line 947:		Line 947:


−	According to Barbalat Lemma, we have <math>\underset{t\rightarrow \infty}{lim}{\Vert x\left(t\right)\Vert }^2=</math><math>0</math>. This completes the proof.	+	According to Barbalat Lemma, we have <math>\underset{t\rightarrow \infty}{\lim}{\Vert x\left(t\right)\Vert }^2=</math><math>0</math>. This completes the proof.

	''Remark 4''. Compared with the results in [15], where the nonlinear function is matched, i.e., <math>\Delta f\left(x,t\right)</math> is in the control channel. In this paper, <math>\Delta f\left(x,t\right)</math> exists in the different channel from the control input, i.e., which means that <math>\Delta f\left(x,t\right)</math> is more general.		''Remark 4''. Compared with the results in [15], where the nonlinear function is matched, i.e., <math>\Delta f\left(x,t\right)</math> is in the control channel. In this paper, <math>\Delta f\left(x,t\right)</math> exists in the different channel from the control input, i.e., which means that <math>\Delta f\left(x,t\right)</math> is more general.

@@ Line 549: / Line 549: @@
 d_{\theta 1} & d_{\theta 2} & ... & d_{\theta m_1}
 \end{array}\right]}^T\in R^{m_2\times 1},\\
-m_1+m_2=\nu \end{array}</math>.
+m_1+m_2=\nu \end{array}</math>
 |}
 |}

@@ Line 539: / Line 539: @@
 {\overline{B}}_{d_1}=\left[\begin{array}{cccc}
 {\overline{B}}_{d_{\kappa 1}} & {\overline{B}}_{d_{\kappa 2}} & ... & {\overline{B}}_{d_{\kappa m1}}
-\end{array}\right]\in R^{n\times m_1},
+\end{array}\right]\in R^{n\times m_1},  \\
 {\overline{B}}_{d_2}=\left[\begin{array}{cccc}
 {\overline{B}}_{d_{\theta 1}} & {\overline{B}}_{d_{\theta 2}} & ... & {\overline{B}}_{d_{\theta m2}}
@@ Line 546: / Line 546: @@
 d_{\kappa 1} & d_{\kappa 2} & ... & d_{\kappa m_1}
 \end{array}\right]}^T\in R^{m_1\times 1}, \\
-& d_2={\left[\begin{array}{cccc}
+ d_2={\left[\begin{array}{cccc}
 d_{\theta 1} & d_{\theta 2} & ... & d_{\theta m_1}
 \end{array}\right]}^T\in R^{m_2\times 1},\\

@@ Line 536: / Line 536: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" |<math>\begin{array}{ll}
+| style="text-align: center;" |<math>\begin{array}{l}
 {\overline{B}}_{d_1}=\left[\begin{array}{cccc}
 {\overline{B}}_{d_{\kappa 1}} & {\overline{B}}_{d_{\kappa 2}} & ... & {\overline{B}}_{d_{\kappa m1}}
-\end{array}\right]\in R^{n\times m_1},  \quad & {\overline{B}}_{d_2}=\left[\begin{array}{cccc}
+\end{array}\right]\in R^{n\times m_1},
 {\overline{B}}_{d_{\theta 1}} & {\overline{B}}_{d_{\theta 2}} & ... & {\overline{B}}_{d_{\theta m2}}
 \end{array}\right]\in R^{n\times m_2},\\
 d_1={\left[\begin{array}{cccc}
 d_{\kappa 1} & d_{\kappa 2} & ... & d_{\kappa m_1}
-\end{array}\right]}^T\in R^{m_1\times 1}, \quad & d_2={\left[\begin{array}{cccc}
+\end{array}\right]}^T\in R^{m_1\times 1}, \\
 d_{\theta 1} & d_{\theta 2} & ... & d_{\theta m_1}
-\end{array}\right]}^T\in R^{m_2\times 1},\quad  \mbox{and}\quad  m_1+m_2=\nu \end{array}</math>.
+\end{array}\right]}^T\in R^{m_2\times 1},\\
 |}
 |}

@@ Line 924: / Line 924: @@
 |-
 | <math>\begin{array}{c}
-V\left(t\right)\leq V\left(t_0\right)-{\int }_{t_0}^t{\lambda }_{min}\left(Q_1\right){\Vert x\left(s\right)\Vert }^2ds+{\int }_{t_0}^t{\zeta }_{\kappa }\sigma \left(s\right)ds\\
+V\left(t\right)\leq V\left(t_0\right)-{\int }_{t_0}^t{\lambda }_{\min}\left(Q_1\right){\Vert x\left(s\right)\Vert }^2ds+{\int }_{t_0}^t{\zeta }_{\kappa }\sigma \left(s\right)ds\\
-\leq V\left(t_0\right)-{\int }_{t_0}^t{\lambda }_{min}\left(Q_1\right){\Vert x\left(s\right)\Vert }^2ds+{\zeta }_{\kappa }{\sigma }^{{_\ast}}
+\leq V\left(t_0\right)-{\int }_{t_0}^t{\lambda }_{\min}\left(Q_1\right){\Vert x\left(s\right)\Vert }^2ds+{\zeta }_{\kappa }{\sigma }^{{_\ast}}
 \end{array}</math>
 |}
@@ Line 931: / Line 931: @@
 |}
-which means the system described in Eq.(33) is bounded.  <math>{\lambda }_{min}\left(Q{}_{}^1\right)</math> denotes the minimum eigenvalue of  <math>Q_1</math>, and  <math>-Q_1=A^TP+PA+P_1{}^TP_1+Q</math>. Eq.(68) also implies
+which means the system described in Eq.(33) is bounded.  <math>{\lambda }_{\min}\left(Q{}_{}^1\right)</math> denotes the minimum eigenvalue of  <math>Q_1</math>, and  <math>-Q_1=A^TP+PA+P_1{}^TP_1+Q</math>. Eq.(68) also implies
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 938: / Line 938: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>{\int }_{t_0}^t{\lambda }_{min}\left(Q_1\right){\Vert x\left(s\right)\Vert }^2ds\leq V\left(t_0\right)+</math><math>{\zeta }_{\kappa }{\sigma }^{{_\ast}}</math>
+| <math>{\int }_{t_0}^t{\lambda }_{\min}\left(Q_1\right){\Vert x\left(s\right)\Vert }^2ds\leq V\left(t_0\right)+</math><math>{\zeta }_{\kappa }{\sigma }^{{_\ast}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (63)

@@ Line 1,067: / Line 1,067: @@
 |-
 | style="text-align: center;" | <math>A=\left[\begin{array}{ccccc}
--\frac{1}{\tau } & 0 & 0 & 0 & 0\\
+-\displaystyle\frac{1}{\tau } & 0 & 0 & 0 & 0\\
-& -\frac{b_0}{a} & 0 & -\frac{c_0}{a} & \displaystyle\frac{c_0}{a}\\
+& -\displaystyle\frac{b_0}{a} & 0 & -\displaystyle\frac{c_0}{a} & \displaystyle\frac{c_0}{a}\\
 \displaystyle\frac{1}{a} & 0 & 0 & \displaystyle\frac{c}{a} & -\displaystyle\frac{c_0}{a}\\
 & 1 & 0 & 0 & 0\\