# American Institute of Mathematical Sciences

September  2020, 10(3): 367-379. doi: 10.3934/naco.2020008

## Fault estimation and optimization for uncertain disturbed singularly perturbed systems with time-delay

 Beijing Key Laboratory of Fieldbus Technology and Automation, North China University of Technology, Beijing 100144, China

* Corresponding author: Lei Liu, liulei_sophia@163.com

Received  May 2019 Revised  September 2019 Published  February 2020

Fund Project: The first author is supported by Youth Foundation of Beijing Nature Science Grant (No.4154068), the Youth Talent Cultivation Program of Beijing, the National Natural Science Foundation of China (No.61473002, No.61573024), North China University of Technology Yuyou Talent Support Program and the Fundamental Research Funds for Beijing Universities (No.110052971921/030)

This paper presents a observer-based fault estimation method for a class of singularly perturbed systems subjected to parameter uncertainties and time-delay in state and disturbance signal with finite energy. To solve the estimation problem involving actuator fault and sensor fault for the uncertain disturbed singularly perturbed systems with time-delay, the problem we studied is firstly transformed into a standard $H_\infty$ control problem, in which the performance index $\gamma$ represents the attenuation of finite energy disturbance. By adopting Lyapunov function with the $\varepsilon$-dependence, a sufficient condition can be derived which enables the designed observer to estimate different kinds of fault signals stably and accurately, and the result obtained by dealing with small perturbation parameter in this way is less conservative. A novel multi-objective optimization scheme is then proposed to optimal disturbance attenuation index $\gamma$ and system stable upper bound $\varepsilon^*$, in this case, the designed observer can estimate the fault signals better in the presence of interference when the systems guarantee maximum stability bound. In the end, the validity and correctness of proposed scheme is verified by comparing the error between the estimated faults and the actual faults.

Citation: Lei Liu, Shaoying Lu, Cunwu Han, Chao Li, Zejin Feng. Fault estimation and optimization for uncertain disturbed singularly perturbed systems with time-delay. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 367-379. doi: 10.3934/naco.2020008
##### References:
 [1] Z. Bougatef, N. Abdelkrim, A. Tellili, et al., Fault diagnosis and accommodation for singularly perturbed time-delayed systems: descriptor approach, in 18th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering, (2017), 86–92. Google Scholar [2] G. R. Duan, H. H. Yu, A. G. Wu, et al., Analysis and Design of Descriptor Linear Systems, 1st edition, Beijing: Science Press, 2012. doi: 10.1007/978-1-4419-6397-0.  Google Scholar [3] S. H. Jiang, Stability analysis of time-varying time-delay uncertain singular systems, Journal of Tonghua Normal University, 38 (2017), 30-32.   Google Scholar [4] H. Y. Li, Y. Y. Wang, D. Y. Yao, et al., A sliding mode approach to stabilization of nonlinear Markovian jump singularly perturbed systems, Automatica, 97 (2018), 404-413. doi: 10.1016/j.automatica.2018.03.066.  Google Scholar [5] D. Liu, Y. Yang and Y. Zhang, Robust fault estimation for singularly perturbed systems with Lipschitz nonlinearity, Journal of The Franklin Institute, 353 (2016), 876-890.  doi: 10.1016/j.jfranklin.2016.01.009.  Google Scholar [6] H. S. Liu and Y. Huang, Robust adaptive output feedback tracking control for flexible-joint robot manipulators based on singularly perturbed decoupling, Robotica, 36 (2018), 822-838.   Google Scholar [7] L. Liu, Y. Yang and W. Liu, Unified optimization of $H_\infty$ index and upper stability bound for singularly perturbed systems, Optimization Letters, 8 (2014), 1889-1904.  doi: 10.1007/s11590-013-0686-6.  Google Scholar [8] L. Liu, S. Y. Lu, C. W. Han, et al., Robust $H_\infty$ control for uncertain singularly perturbed systems with time-delay, in China Control Conference, (2017), 3147–3152. Google Scholar [9] L. Liu , X. F. Yan, C. W. Han, et al., Fault diagnosis and optimal fault-tolerant control of singularly perturbed systems based on PI observer, Control and Decision, 31 (2016), 1867-1872. Google Scholar [10] W. Q. Liu, M. Paskota, V. Sreeram, et al., Improvement on stability bounds for singularly perturbed systems via state feedback, International Journal of Systems Science, 28 (1997), 571-578. Google Scholar [11] P. Mei and Y. Zou, Study on robust stability of uncertain singular perturbation systems with time delay, Control and Decision, 23 (2008), 392-396.   Google Scholar [12] M. Ndiaye, W. Liu and Z. M. Wang, Robust ISS stabilization on disturbance for uncertain singularly perturbed systems, IMA Journal of Mathematical Control and Information, 35 (2018), 1115-1127.  doi: 10.1093/imamci/dnx017.  Google Scholar [13] H. Shen, F. Li, Z. Wu, J. H. Park, et al., Fuzzy-model-based nonfragile control for nonlinear singularly perturbed systems with semi-Markov jump parameters, IEEE Transactions on Fuzzy Systems, 26 (2018), 3428-3439. Google Scholar [14] H. Shen, Y. Men, Z. G. Wu, et al., Nonfragile $H_\infty$ control for fuzzy markovian jump systems under fast sampling singular perturbation, IEEE Transactions on Systems, Man and Cybernetics: Systems, (2017), 1–12. Google Scholar [15] H. Shen, Y. Men, Z. Wu, J. Cao, et al., Network-based quantized control for fuzzy singularly perturbed semi-Markov jump systems and its application, IEEE Transactions on Circuits and Systems I: Regular Papers, 66 (2019), 1130-1140.  Google Scholar [16] F. Q. Sun, Guaranteed performance control of a time-varying time-delay uncertain singular perturbation system, Journal of Jilin University, 6 (2015), 637-643.   Google Scholar [17] A. Tellili, M. N. Abdelkrim and M. Benrejeb, Model-based fault diagnosis of two-time scales singularly perturbed systems, in International Symposium on Control, (2004), 819–822. Google Scholar [18] A. Tellili and M. N. Abdelkrim, Fault diagnosis and reconfigurable control of singularly perturbed systems using GIMC structure, International Journal of Computer Applications, 44 (2012), 31-35.   Google Scholar [19] A. Tellili and M. N. Abdelkrim, Realiable $H_\infty$ controller design for singularly perturbed systems with sensor failure, in IEEE International Conference on Industrial Technology, (2004), 1636–1641. Google Scholar [20] A. Tellili, M. N. Abdelkrim and M. Benrejeb, Reliable $H_\infty$ control of multiple time scales singularly perturbed systems with sensor failure, International Journal of Control, 80 (2007), 659-665.  doi: 10.1080/00207170601009634.  Google Scholar [21] G. X. Wang, J. Wu, B. F. Zeng, et. al., A nonlinear adaptive sliding mode control strategy for modular high-temperature gas-cooled reactors, Progress in Nuclear Energy, 113 (2019), 53-61. Google Scholar [22] Y. Y. Wang, W. Liu and Z. M. Wang, Robust $H_\infty$ control of uncertain singular perturbation system with time delay, Journal of Beijing University of Technology, 42 (2016), 217-222.   Google Scholar [23] J. Xu, C. Cai and Y. Zou, A novel method for fault detection in singularly perturbed systems via the finite frequency strategy, Journal of The Franklin Institute, 352 (2015), 5061-5084. doi: 10.1016/j.jfranklin.2015.08.001.  Google Scholar [24] J. Xu and Y. G. Niu, A finite frequency approach for fault detection of fuzzy singularly perturbed systems with regional pole assignment, Neurocomputing, 325 (2019), 200-201.   Google Scholar [25] J. Xu, Y. G. Niu, E. Fridman and et. al, Finite frequency $H_\infty$ control of singularly perturbed Euler-Lagrange systems: An artificial delay approach, Inernational Journal of Robust and Nonlinear Control, 29 (2019), 353-374.  doi: 10.1002/rnc.4383.  Google Scholar [26] K. K. Xu, Singular Perturbation in the Control Systems, Beijing: Science Press, 1986.   Google Scholar [27] C. Yang, L. Ma, X. Ma, et al., Stability analysis of singularly perturbed control systems with actuator saturation, Journal of The Franklin Institute, 353 (2016), 1284-1296. doi: 10.1016/j.jfranklin.2015.12.013.  Google Scholar [28] C. Yang, Z. Che, J. Fu, et al., Passivity-based integral sliding mode control and " - bound estimation for uncertain singularly perturbed systems with disturbances, IEEE Transactions on Circuits and Systems II: Express Briefs, 66 (2019), 452-456. doi: 10.1155/2015/926762.  Google Scholar [29] C. Yang, Z. Che and L. Zhou, Integral sliding mode control for singularly perturbed systems with mismatched disturbances, Circuits, Systems, and Signal Processing, 38 (2019), 1561-1582.  doi: 10.1007/s00034-018-0925-2.  Google Scholar [30] C. Yang and Q. Zhang, Multiobjective control for T-S fuzzy singularly perturbed systems, IEEE Transactions Fuzzy Systems, 17 (2009), 104-115.   Google Scholar [31] D. M. Yang, Q. L. Zhang, B. Yao, et al., Singular Systems, 1st edition, Beijing: Science Press, 2004. Google Scholar [32] M. R. Zhou, W. Z. Lin, M. K. Ni, et al., Introduction to Singular Perturbation, Beijing: Science Press, 2014.  Google Scholar

show all references

##### References:
 [1] Z. Bougatef, N. Abdelkrim, A. Tellili, et al., Fault diagnosis and accommodation for singularly perturbed time-delayed systems: descriptor approach, in 18th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering, (2017), 86–92. Google Scholar [2] G. R. Duan, H. H. Yu, A. G. Wu, et al., Analysis and Design of Descriptor Linear Systems, 1st edition, Beijing: Science Press, 2012. doi: 10.1007/978-1-4419-6397-0.  Google Scholar [3] S. H. Jiang, Stability analysis of time-varying time-delay uncertain singular systems, Journal of Tonghua Normal University, 38 (2017), 30-32.   Google Scholar [4] H. Y. Li, Y. Y. Wang, D. Y. Yao, et al., A sliding mode approach to stabilization of nonlinear Markovian jump singularly perturbed systems, Automatica, 97 (2018), 404-413. doi: 10.1016/j.automatica.2018.03.066.  Google Scholar [5] D. Liu, Y. Yang and Y. Zhang, Robust fault estimation for singularly perturbed systems with Lipschitz nonlinearity, Journal of The Franklin Institute, 353 (2016), 876-890.  doi: 10.1016/j.jfranklin.2016.01.009.  Google Scholar [6] H. S. Liu and Y. Huang, Robust adaptive output feedback tracking control for flexible-joint robot manipulators based on singularly perturbed decoupling, Robotica, 36 (2018), 822-838.   Google Scholar [7] L. Liu, Y. Yang and W. Liu, Unified optimization of $H_\infty$ index and upper stability bound for singularly perturbed systems, Optimization Letters, 8 (2014), 1889-1904.  doi: 10.1007/s11590-013-0686-6.  Google Scholar [8] L. Liu, S. Y. Lu, C. W. Han, et al., Robust $H_\infty$ control for uncertain singularly perturbed systems with time-delay, in China Control Conference, (2017), 3147–3152. Google Scholar [9] L. Liu , X. F. Yan, C. W. Han, et al., Fault diagnosis and optimal fault-tolerant control of singularly perturbed systems based on PI observer, Control and Decision, 31 (2016), 1867-1872. Google Scholar [10] W. Q. Liu, M. Paskota, V. Sreeram, et al., Improvement on stability bounds for singularly perturbed systems via state feedback, International Journal of Systems Science, 28 (1997), 571-578. Google Scholar [11] P. Mei and Y. Zou, Study on robust stability of uncertain singular perturbation systems with time delay, Control and Decision, 23 (2008), 392-396.   Google Scholar [12] M. Ndiaye, W. Liu and Z. M. Wang, Robust ISS stabilization on disturbance for uncertain singularly perturbed systems, IMA Journal of Mathematical Control and Information, 35 (2018), 1115-1127.  doi: 10.1093/imamci/dnx017.  Google Scholar [13] H. Shen, F. Li, Z. Wu, J. H. Park, et al., Fuzzy-model-based nonfragile control for nonlinear singularly perturbed systems with semi-Markov jump parameters, IEEE Transactions on Fuzzy Systems, 26 (2018), 3428-3439. Google Scholar [14] H. Shen, Y. Men, Z. G. Wu, et al., Nonfragile $H_\infty$ control for fuzzy markovian jump systems under fast sampling singular perturbation, IEEE Transactions on Systems, Man and Cybernetics: Systems, (2017), 1–12. Google Scholar [15] H. Shen, Y. Men, Z. Wu, J. Cao, et al., Network-based quantized control for fuzzy singularly perturbed semi-Markov jump systems and its application, IEEE Transactions on Circuits and Systems I: Regular Papers, 66 (2019), 1130-1140.  Google Scholar [16] F. Q. Sun, Guaranteed performance control of a time-varying time-delay uncertain singular perturbation system, Journal of Jilin University, 6 (2015), 637-643.   Google Scholar [17] A. Tellili, M. N. Abdelkrim and M. Benrejeb, Model-based fault diagnosis of two-time scales singularly perturbed systems, in International Symposium on Control, (2004), 819–822. Google Scholar [18] A. Tellili and M. N. Abdelkrim, Fault diagnosis and reconfigurable control of singularly perturbed systems using GIMC structure, International Journal of Computer Applications, 44 (2012), 31-35.   Google Scholar [19] A. Tellili and M. N. Abdelkrim, Realiable $H_\infty$ controller design for singularly perturbed systems with sensor failure, in IEEE International Conference on Industrial Technology, (2004), 1636–1641. Google Scholar [20] A. Tellili, M. N. Abdelkrim and M. Benrejeb, Reliable $H_\infty$ control of multiple time scales singularly perturbed systems with sensor failure, International Journal of Control, 80 (2007), 659-665.  doi: 10.1080/00207170601009634.  Google Scholar [21] G. X. Wang, J. Wu, B. F. Zeng, et. al., A nonlinear adaptive sliding mode control strategy for modular high-temperature gas-cooled reactors, Progress in Nuclear Energy, 113 (2019), 53-61. Google Scholar [22] Y. Y. Wang, W. Liu and Z. M. Wang, Robust $H_\infty$ control of uncertain singular perturbation system with time delay, Journal of Beijing University of Technology, 42 (2016), 217-222.   Google Scholar [23] J. Xu, C. Cai and Y. Zou, A novel method for fault detection in singularly perturbed systems via the finite frequency strategy, Journal of The Franklin Institute, 352 (2015), 5061-5084. doi: 10.1016/j.jfranklin.2015.08.001.  Google Scholar [24] J. Xu and Y. G. Niu, A finite frequency approach for fault detection of fuzzy singularly perturbed systems with regional pole assignment, Neurocomputing, 325 (2019), 200-201.   Google Scholar [25] J. Xu, Y. G. Niu, E. Fridman and et. al, Finite frequency $H_\infty$ control of singularly perturbed Euler-Lagrange systems: An artificial delay approach, Inernational Journal of Robust and Nonlinear Control, 29 (2019), 353-374.  doi: 10.1002/rnc.4383.  Google Scholar [26] K. K. Xu, Singular Perturbation in the Control Systems, Beijing: Science Press, 1986.   Google Scholar [27] C. Yang, L. Ma, X. Ma, et al., Stability analysis of singularly perturbed control systems with actuator saturation, Journal of The Franklin Institute, 353 (2016), 1284-1296. doi: 10.1016/j.jfranklin.2015.12.013.  Google Scholar [28] C. Yang, Z. Che, J. Fu, et al., Passivity-based integral sliding mode control and " - bound estimation for uncertain singularly perturbed systems with disturbances, IEEE Transactions on Circuits and Systems II: Express Briefs, 66 (2019), 452-456. doi: 10.1155/2015/926762.  Google Scholar [29] C. Yang, Z. Che and L. Zhou, Integral sliding mode control for singularly perturbed systems with mismatched disturbances, Circuits, Systems, and Signal Processing, 38 (2019), 1561-1582.  doi: 10.1007/s00034-018-0925-2.  Google Scholar [30] C. Yang and Q. Zhang, Multiobjective control for T-S fuzzy singularly perturbed systems, IEEE Transactions Fuzzy Systems, 17 (2009), 104-115.   Google Scholar [31] D. M. Yang, Q. L. Zhang, B. Yao, et al., Singular Systems, 1st edition, Beijing: Science Press, 2004. Google Scholar [32] M. R. Zhou, W. Z. Lin, M. K. Ni, et al., Introduction to Singular Perturbation, Beijing: Science Press, 2014.  Google Scholar
Curves of estimated actuator fault
Curves of estimated actuator fault
Curves of the actuator fault estimation error
Curves of the actuator fault estimation error
Curves of estimated sensor fault
Curves of estimated sensor fault
Curves of the sensor fault estimation error
Curves of the sensor fault estimation error
Estimation error curves of fault state vector
Estimation error curves of original system state vector
 [1] Junlin Xiong, Wenjie Liu. $H_{\infty}$ observer-based control for large-scale systems with sparse observer communication network. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 331-343. doi: 10.3934/naco.2020005 [2] Liqiang Jin, Yanyan Yin, Kok Lay Teo, Fei Liu. Event-triggered mixed $H_\infty$ and passive control for Markov jump systems with bounded inputs. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020024 [3] Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $\mathcal{H}_{\infty}$ control for fuzzy markovian jump systems based on sampled-data control method. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020368 [4] Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $\bf{M/G/1}$ fault-tolerant machining system with imperfection. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019096 [5] Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020011 [6] M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365 [7] Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035 [8] Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $-\Delta u = \lambda f(u)$ as $\lambda\to-\infty$. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020293 [9] Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $D$-$BMAP/G/1$ queueing system under $N$-policy and its cost optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020135 [10] Min Zhao, Changzheng Qu. The two-component Novikov-type systems with peaked solutions and $H^1$-conservation law. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020245 [11] Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303 [12] Li-Min Wang, Jing-Xian Yu, Jia Shi, Fu-Rong Gao. Delay-range dependent $H_\infty$ control for uncertain 2D-delayed systems. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 11-23. doi: 10.3934/naco.2015.5.11 [13] Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $L^1$-control in coefficients for quasi-linear Dirichlet boundary value problems with $BMO$-anisotropic $p$-Laplacian. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020021 [14] Hongru Ren, Shubo Li, Changxin Lu. Event-triggered adaptive fault-tolerant control for multi-agent systems with unknown disturbances. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020379 [15] Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $H = H_1(x)+H_2(y)$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004 [16] Weiwei Ao, Chao Liu. Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in $\mathbb{R}^2$ when exponent approaches $+\infty$. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 5047-5077. doi: 10.3934/dcds.2020211 [17] Dean Crnković, Nina Mostarac, Bernardo G. Rodrigues, Leo Storme. $s$-PD-sets for codes from projective planes $\mathrm{PG}(2,2^h)$, $5 \leq h\leq 9$. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020075 [18] Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, Jawad Salhi. Null controllability for singular cascade systems of $n$-coupled degenerate parabolic equations by one control force. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020080 [19] Sanjiban Santra. On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1441-1460. doi: 10.3934/dcds.2018059 [20] Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094

Impact Factor: