Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays

Previous Article
Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion
JIMO Home
This Issue
Next Article
Time-inconsistent consumption-investment problem for a member in a defined contribution pension plan

October 2016, 12(4): 1535-1556. doi: 10.3934/jimo.2016.12.1535

Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays

Hongbiao Fan ^1, , Jun-E Feng ^2, and Min Meng ^2,

1.	School of Mathematics Sicences, Dezhou University, Dezhou 253600, China
2.	School of Mathematics, Shandong University, Jinan 250100, China, China

Received December 2014 Revised October 2015 Published January 2016

Abstract
Related Papers

This paper investigates piecewise observer design for rectangular discrete fuzzy descriptor systems with multiple time-varying delays. Via a series of simple transformations, the considered rectangular descriptor plants are converted into standard ones with multiple time-varying delays. Then, two sufficient delay-dependent conditions for existence of piecewise fuzzy observers are derived based on piecewise Lyapunov functions. Finally, two numerical examples are presented to show the effectiveness of the theoretical results.

Keywords: time-varying delays., piecewise observer, Fuzzy descriptor system, piecewise Lyapunov function.

Mathematics Subject Classification: Primary: 93C10, 93C55; Secondary: 93B0.

Citation: Hongbiao Fan, Jun-E Feng, Min Meng. Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1535-1556. doi: 10.3934/jimo.2016.12.1535

References:

[1]	S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Society for Industrial and Applied Mathematics, (1994). doi: 10.1137/1.9781611970777.fm.
[2]	S. Cao, N. W. Rees and G. Feng, Analysis and design of fuzzy control systems using dynamic fuzzy-state space models,, IEEE Transactions on Fuzzy Systems, 7 (1999), 192. doi: 10.1109/91.755400.
[3]	Y. Y. Cao and P. M. Frank, Robust $H_{\infty}$ disturbance attenuation for a class of uncertain discrete-time fuzzy systems,, IEEE Transactions on Fuzzy Systems, 8 (2000), 406. doi: 10.1109/91.868947.
[4]	Q. Chai, L. Ryan, K. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems,, Journal of Industrial and Management Optimization, 9 (2013), 471. doi: 10.3934/jimo.2013.9.471.
[5]	B. S. Chen, C. H. Tseng and H. J. Uang, Mixed $H_{2}/H_{\infty}$ fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach,, IEEE Transactions on Fuzzy Systems, 8 (2000), 249. doi: 10.1109/91.855915.
[6]	M. Darouach, M. Zasadzinski and M. Hayar, Reduced-order observer design for descriptor systems with unknown inputs,, IEEE Transactions on Automatic Control, 41 (1996), 1068. doi: 10.1109/9.508918.
[7]	D. Essawy, Adaptive control of nonlinear systems using fuzzy systems,, Journal of Industrial and Management Optimization, 6 (2010), 861. doi: 10.3934/jimo.2010.6.861.
[8]	G. Feng, Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions,, IEEE Transactions on Fuzzy Systems, 12 (2004), 22. doi: 10.1109/TFUZZ.2003.819833.
[9]	H. Gao and T. Chen, New results on stability of discrete-time systems with time-varying state delay,, IEEE Transactions on Automatic Control, 52 (2007), 328. doi: 10.1109/TAC.2006.890320.
[10]	H. Gao, J. Lam, C. Wang and Y. Wang, Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay,, IEE Proceedings-Control Theory and Applications, 151 (2004), 691. doi: 10.1049/ip-cta:20040822.
[11]	Z. Gao, X. Shi and S. Ding, Fuzzy state/disturbance observer design for T-S fuzzy systems with application to sensor fault estimation,, IEEE Transactions on Systems, 38 (2008), 875. doi: 10.1109/TSMCB.2008.917185.
[12]	T. M. Guerra and L. Vermeiren, LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form,, Automatica, 40 (2004), 823. doi: 10.1016/j.automatica.2003.12.014.
[13]	A. Hmamed, Constrained regulation of linear discrete-time systems with time delay: Delay-dependent and delay-independent conditions,, International Journal of Systems Science, 31 (2000), 529. doi: 10.1080/002077200291109.
[14]	Y. Hosoe and T. Hagiwara, Robust stability analysis based on finite impulse response scaling for discrete-time linear time-invariant systems,, IET Control Theory and Applications, 7 (2013), 1463. doi: 10.1049/iet-cta.2013.0053.
[15]	C. Jiang, K. Teo, R. Loxton and G. Duan, A neighboring extremal solution for an optimal switched impulsive control problem,, Journal of Industrial and Management Optimization, 8 (2012), 591. doi: 10.3934/jimo.2012.8.591.
[16]	M. Johansson, A. Rantzer and K.-E. Årzén, Piecewise quadratic stability of fuzzy systems,, IEEE Transactions on Fuzzy Systems, 7 (1999), 713. doi: 10.1109/91.811241.
[17]	D. Koenig, Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation,, IEEE Transactions on Automatic Control, 50 (2005), 212. doi: 10.1109/TAC.2004.841889.
[18]	A. Kumar and P. Daoutidis, Control of Nonlinear Differential Algebraic Equation Systems with Applications to Chemical Processes,, Chapman & Hall/CRC, (1999). doi: 10.1007/978-94-017-3594-0_4.
[19]	F. Li, P. Shi, L. Wu and X. Zhang, Fuzzy-model-based D-stability and non-fragile control for discrete-time descriptor systems with multiple delays,, IEEE Transactions on Fuzzy Systems, 22 (2013), 1019. doi: 10.1109/TFUZZ.2013.2272647.
[20]	X. Liu and Q. Zhang, New approaches to $H_{\infty}$ controller designs based on fuzzy observers for T-S fuzzy systems via LMI,, Automatica, 39 (2003), 1571. doi: 10.1016/S0005-1098(03)00172-9.
[21]	S. Ma and Z. Cheng, Observer design for discrete time-delay singular systems with unknown inputs,, American Control Conference, 6 (2005), 4215. doi: 10.1109/ACC.2005.1470640.
[22]	Y. Ma and G. Yang, Stability analysis for linear discrete-time systems subject to actuator saturation,, Control Theory and Technology, 8 (2010), 245. doi: 10.1007/s11768-010-7261-9.
[23]	S. K. Nguang and P. Shi, $H_{\infty}$ fuzzy output feedback control design for nonlinear systems: An LMI approach,, IEEE Transactions on Fuzzy Systems, 11 (2003), 331. doi: 10.1109/TFUZZ.2003.812691.
[24]	R. Palm and P. Bergsten, Sliding mode observer for a Takagi-Sugeno fuzzy system,, The Ninth IEEE International Conference on Fuzzy Systems, 2 (2000), 665. doi: 10.1109/FUZZY.2000.839072.
[25]	J. Qiu, G. Feng and H. Gao, Static-Output-Feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions,, IEEE Transactions on Fuzzy Systems, 21 (2013), 245. doi: 10.1109/TFUZZ.2012.2210555.
[26]	J. Qiu, G. Feng and H. Gao, Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements,, IEEE Transactions on Fuzzy Systems, 20 (2012), 1046. doi: 10.1109/TFUZZ.2012.2191790.
[27]	J. Qiu, G. Feng and H. Gao, Fuzzy-model-based piecewise $H_{\infty}$ static-output-feedback controller design for networked nonlinear systems,, IEEE Transactions on Fuzzy Systems, 18 (2010), 919. doi: 10.1109/TFUZZ.2010.2052259.
[28]	R. Riaza, Differential-Algebraic Systems: Analytical Aspects And Circuit Applications,, World Scientific, (2008). doi: 10.1016/0098-1354(88)85052-X.
[29]	H. Shi, G. Xie and W. Luo, Controllability analysis of linear discrete time systems with time delay in state,, Abstract and Applied Analysis, (2012). doi: 10.1155/2012/490903.
[30]	T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control,, IEEE Transactions on Systems, 15 (1985), 116.
[31]	T. Taniguchi, K. Tanaka, K. Yamafuji and H. Wang, Fuzzy descriptor systems:stability analysis and design via LMIs,, American Control Conference, 3 (1999), 1827. doi: 10.1109/ACC.1999.786165.
[32]	Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold,, Journal of Industrial and Management Optimization, 5 (2009), 467. doi: 10.3934/jimo.2009.5.467.
[33]	Z. Wang, Y. Shen, X. Zhang and Q. Wang, Observer design for discrete-time descriptor systems: An LMI approach,, Systems & Control Letters, 61 (2012), 683. doi: 10.1016/j.sysconle.2012.03.006.
[34]	J. Xiong and J. Lam, Stabilization of linear systems over networks with bounded packet loss,, Automatica, 43 (2007), 80. doi: 10.1016/j.automatica.2006.07.017.
[35]	S. Xu and J. Lam, Robust $H_{\infty}$ control for uncertain discrete-time-delay fuzzy systems via output feedback controllers,, IEEE Transactions on Fuzzy Systems, 13 (2005), 82. doi: 10.1109/TFUZZ.2004.839661.
[36]	Q. Zhang, C. Liu and X. Zhang, Complexity, Analysis and Control of Singular Biological Systems,, Springer, (2012). doi: 10.1007/978-1-4471-2303-3.
[37]	B. Zhu, Q. Zhang and C. Chang, Delay-dependent disspative control for a class of non-linear system via Takagi-Sugeno fuzzy descriptor model with time delay,, IET Control Theory and Applications, 8 (2014), 451. doi: 10.1049/iet-cta.2013.0438.

show all references

References:

[1]	S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Society for Industrial and Applied Mathematics, (1994). doi: 10.1137/1.9781611970777.fm.
[2]	S. Cao, N. W. Rees and G. Feng, Analysis and design of fuzzy control systems using dynamic fuzzy-state space models,, IEEE Transactions on Fuzzy Systems, 7 (1999), 192. doi: 10.1109/91.755400.
[3]	Y. Y. Cao and P. M. Frank, Robust $H_{\infty}$ disturbance attenuation for a class of uncertain discrete-time fuzzy systems,, IEEE Transactions on Fuzzy Systems, 8 (2000), 406. doi: 10.1109/91.868947.
[4]	Q. Chai, L. Ryan, K. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems,, Journal of Industrial and Management Optimization, 9 (2013), 471. doi: 10.3934/jimo.2013.9.471.
[5]	B. S. Chen, C. H. Tseng and H. J. Uang, Mixed $H_{2}/H_{\infty}$ fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach,, IEEE Transactions on Fuzzy Systems, 8 (2000), 249. doi: 10.1109/91.855915.
[6]	M. Darouach, M. Zasadzinski and M. Hayar, Reduced-order observer design for descriptor systems with unknown inputs,, IEEE Transactions on Automatic Control, 41 (1996), 1068. doi: 10.1109/9.508918.
[7]	D. Essawy, Adaptive control of nonlinear systems using fuzzy systems,, Journal of Industrial and Management Optimization, 6 (2010), 861. doi: 10.3934/jimo.2010.6.861.
[8]	G. Feng, Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions,, IEEE Transactions on Fuzzy Systems, 12 (2004), 22. doi: 10.1109/TFUZZ.2003.819833.
[9]	H. Gao and T. Chen, New results on stability of discrete-time systems with time-varying state delay,, IEEE Transactions on Automatic Control, 52 (2007), 328. doi: 10.1109/TAC.2006.890320.
[10]	H. Gao, J. Lam, C. Wang and Y. Wang, Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay,, IEE Proceedings-Control Theory and Applications, 151 (2004), 691. doi: 10.1049/ip-cta:20040822.
[11]	Z. Gao, X. Shi and S. Ding, Fuzzy state/disturbance observer design for T-S fuzzy systems with application to sensor fault estimation,, IEEE Transactions on Systems, 38 (2008), 875. doi: 10.1109/TSMCB.2008.917185.
[12]	T. M. Guerra and L. Vermeiren, LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form,, Automatica, 40 (2004), 823. doi: 10.1016/j.automatica.2003.12.014.
[13]	A. Hmamed, Constrained regulation of linear discrete-time systems with time delay: Delay-dependent and delay-independent conditions,, International Journal of Systems Science, 31 (2000), 529. doi: 10.1080/002077200291109.
[14]	Y. Hosoe and T. Hagiwara, Robust stability analysis based on finite impulse response scaling for discrete-time linear time-invariant systems,, IET Control Theory and Applications, 7 (2013), 1463. doi: 10.1049/iet-cta.2013.0053.
[15]	C. Jiang, K. Teo, R. Loxton and G. Duan, A neighboring extremal solution for an optimal switched impulsive control problem,, Journal of Industrial and Management Optimization, 8 (2012), 591. doi: 10.3934/jimo.2012.8.591.
[16]	M. Johansson, A. Rantzer and K.-E. Årzén, Piecewise quadratic stability of fuzzy systems,, IEEE Transactions on Fuzzy Systems, 7 (1999), 713. doi: 10.1109/91.811241.
[17]	D. Koenig, Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation,, IEEE Transactions on Automatic Control, 50 (2005), 212. doi: 10.1109/TAC.2004.841889.
[18]	A. Kumar and P. Daoutidis, Control of Nonlinear Differential Algebraic Equation Systems with Applications to Chemical Processes,, Chapman & Hall/CRC, (1999). doi: 10.1007/978-94-017-3594-0_4.
[19]	F. Li, P. Shi, L. Wu and X. Zhang, Fuzzy-model-based D-stability and non-fragile control for discrete-time descriptor systems with multiple delays,, IEEE Transactions on Fuzzy Systems, 22 (2013), 1019. doi: 10.1109/TFUZZ.2013.2272647.
[20]	X. Liu and Q. Zhang, New approaches to $H_{\infty}$ controller designs based on fuzzy observers for T-S fuzzy systems via LMI,, Automatica, 39 (2003), 1571. doi: 10.1016/S0005-1098(03)00172-9.
[21]	S. Ma and Z. Cheng, Observer design for discrete time-delay singular systems with unknown inputs,, American Control Conference, 6 (2005), 4215. doi: 10.1109/ACC.2005.1470640.
[22]	Y. Ma and G. Yang, Stability analysis for linear discrete-time systems subject to actuator saturation,, Control Theory and Technology, 8 (2010), 245. doi: 10.1007/s11768-010-7261-9.
[23]	S. K. Nguang and P. Shi, $H_{\infty}$ fuzzy output feedback control design for nonlinear systems: An LMI approach,, IEEE Transactions on Fuzzy Systems, 11 (2003), 331. doi: 10.1109/TFUZZ.2003.812691.
[24]	R. Palm and P. Bergsten, Sliding mode observer for a Takagi-Sugeno fuzzy system,, The Ninth IEEE International Conference on Fuzzy Systems, 2 (2000), 665. doi: 10.1109/FUZZY.2000.839072.
[25]	J. Qiu, G. Feng and H. Gao, Static-Output-Feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions,, IEEE Transactions on Fuzzy Systems, 21 (2013), 245. doi: 10.1109/TFUZZ.2012.2210555.
[26]	J. Qiu, G. Feng and H. Gao, Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements,, IEEE Transactions on Fuzzy Systems, 20 (2012), 1046. doi: 10.1109/TFUZZ.2012.2191790.
[27]	J. Qiu, G. Feng and H. Gao, Fuzzy-model-based piecewise $H_{\infty}$ static-output-feedback controller design for networked nonlinear systems,, IEEE Transactions on Fuzzy Systems, 18 (2010), 919. doi: 10.1109/TFUZZ.2010.2052259.
[28]	R. Riaza, Differential-Algebraic Systems: Analytical Aspects And Circuit Applications,, World Scientific, (2008). doi: 10.1016/0098-1354(88)85052-X.
[29]	H. Shi, G. Xie and W. Luo, Controllability analysis of linear discrete time systems with time delay in state,, Abstract and Applied Analysis, (2012). doi: 10.1155/2012/490903.
[30]	T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control,, IEEE Transactions on Systems, 15 (1985), 116.
[31]	T. Taniguchi, K. Tanaka, K. Yamafuji and H. Wang, Fuzzy descriptor systems:stability analysis and design via LMIs,, American Control Conference, 3 (1999), 1827. doi: 10.1109/ACC.1999.786165.
[32]	Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold,, Journal of Industrial and Management Optimization, 5 (2009), 467. doi: 10.3934/jimo.2009.5.467.
[33]	Z. Wang, Y. Shen, X. Zhang and Q. Wang, Observer design for discrete-time descriptor systems: An LMI approach,, Systems & Control Letters, 61 (2012), 683. doi: 10.1016/j.sysconle.2012.03.006.
[34]	J. Xiong and J. Lam, Stabilization of linear systems over networks with bounded packet loss,, Automatica, 43 (2007), 80. doi: 10.1016/j.automatica.2006.07.017.
[35]	S. Xu and J. Lam, Robust $H_{\infty}$ control for uncertain discrete-time-delay fuzzy systems via output feedback controllers,, IEEE Transactions on Fuzzy Systems, 13 (2005), 82. doi: 10.1109/TFUZZ.2004.839661.
[36]	Q. Zhang, C. Liu and X. Zhang, Complexity, Analysis and Control of Singular Biological Systems,, Springer, (2012). doi: 10.1007/978-1-4471-2303-3.
[37]	B. Zhu, Q. Zhang and C. Chang, Delay-dependent disspative control for a class of non-linear system via Takagi-Sugeno fuzzy descriptor model with time delay,, IET Control Theory and Applications, 8 (2014), 451. doi: 10.1049/iet-cta.2013.0438.

[1]	Dinh Cong Huong, Mai Viet Thuan. State transformations of time-varying delay systems and their applications to state observer design. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 413-444. doi: 10.3934/dcdss.2017020
[2]	Mohammad-Sahadet Hossain. Projection-based model reduction for time-varying descriptor systems: New results. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 73-90. doi: 10.3934/naco.2016.6.73
[3]	Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559
[4]	Tingwen Huang, Guanrong Chen, Juergen Kurths. Synchronization of chaotic systems with time-varying coupling delays. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1071-1082. doi: 10.3934/dcdsb.2011.16.1071
[5]	Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150
[6]	Yangzi Hu, Fuke Wu. The improved results on the stochastic Kolmogorov system with time-varying delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1481-1497. doi: 10.3934/dcdsb.2015.20.1481
[7]	Ruoxia Li, Huaiqin Wu, Xiaowei Zhang, Rong Yao. Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation. Mathematical Control & Related Fields, 2015, 5 (4) : 827-844. doi: 10.3934/mcrf.2015.5.827
[8]	Wei Feng, Xin Lu. Global stability in a class of reaction-diffusion systems with time-varying delays. Conference Publications, 1998, 1998 (Special) : 253-261. doi: 10.3934/proc.1998.1998.253
[9]	Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263
[10]	Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure & Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667
[11]	Peter Giesl, Sigurdur Hafstein. Existence of piecewise linear Lyapunov functions in arbitrary dimensions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3539-3565. doi: 10.3934/dcds.2012.32.3539
[12]	Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839
[13]	Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
[14]	Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653
[15]	Roberta Fabbri, Russell Johnson, Sylvia Novo, Carmen Núñez. On linear-quadratic dissipative control processes with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 193-210. doi: 10.3934/dcds.2013.33.193
[16]	Robert G. McLeod, John F. Brewster, Abba B. Gumel, Dean A. Slonowsky. Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs. Mathematical Biosciences & Engineering, 2006, 3 (3) : 527-544. doi: 10.3934/mbe.2006.3.527
[17]	Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693
[18]	Marat Akhmet, Duygu Aruğaslan. Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 457-466. doi: 10.3934/dcds.2009.25.457
[19]	Sigurdur F. Hafstein, Christopher M. Kellett, Huijuan Li. Computing continuous and piecewise affine lyapunov functions for nonlinear systems. Journal of Computational Dynamics, 2015, 2 (2) : 227-246. doi: 10.3934/jcd.2015004
[20]	Anass Belcaid, Mohammed Douimi, Abdelkader Fassi Fihri. Recursive reconstruction of piecewise constant signals by minimization of an energy function. Inverse Problems & Imaging, 2018, 12 (4) : 903-920. doi: 10.3934/ipi.2018038

2018 Impact Factor: 1.025

American Institute of Mathematical Sciences