# American Institute of Mathematical Sciences

August  2019, 2(3): 251-266. doi: 10.3934/mfc.2019016

## Eigenstructure assignment for polynomial matrix systems ensuring normalization and impulse elimination

 1 School of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China 2 School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China

* Corresponding author: Peizhao Yu

Received  May 2019 Revised  July 2019 Published  September 2019

Fund Project: The first author is supported by the National Natural Science Foundation of China grant 61903342 and the Doctor fund project of Zhengzhou University of Light Industry grant 2017BSJJ009; The second author is supported by the National Natural Science Foundation of China grant 61473202

In this paper, eigenstructure assignment problems for polynomial matrix systems ensuring normalization and impulse elimination are considered. By using linearization method, a polynomial matrix system is transformed into a descriptor linear system without changing the eigenstructure of original system. By analyzing the characteristic polynomial of the desired system, the normalizable condition under feedback is given, and moreover, the parametric expressions of controller gains for eigenstructure ensuring normalization are derived by singular value decomposition. Impulse elimination in polynomial matrix systems is investigated when the normalizable condition is not satisfied. The parametric expressions of controller gains for impulse elimination ensuring finite eigenstructure assignment are formulated. The solving algorithms of corresponding controller gains for eigenstructure assignment ensuring normalization and impulse elimination are also presented. Numerical examples show the effectiveness of proposed method.

Citation: Peizhao Yu, Guoshan Zhang. Eigenstructure assignment for polynomial matrix systems ensuring normalization and impulse elimination. Mathematical Foundations of Computing, 2019, 2 (3) : 251-266. doi: 10.3934/mfc.2019016
##### References:
 [1] T. H. S. Abdelaziz, Robust pole placement for second-order linear systems using velocity-plus-acceleration feedback, IET Control Theory and Applications, 7 (2013), 1843-1856. doi: 10.1049/iet-cta.2013.0039. Google Scholar [2] T. H. S. Abdelaziz, Parametric approach for eigenstructure assignment in descriptor second-order systems via velocity-plus-acceleration feedback, Journal Dynamic Systems, Measurement, and Control, 136 (2014), 044505. doi: 10.1115/1.4026876. Google Scholar [3] E. N. Antoniou and S. Vologiannidis, A new family of companion forms of polynomial matrices, Electronic Journal of Linear Algebra, 11 (2004), 78-87. doi: 10.13001/1081-3810.1124. Google Scholar [4] Z. J. Bai and Q. Y. Wan, Partial quadratic eigenvalue assignment in vibrating structures using receptances and system matrices, Mechanical Systems and Signal Processing, 88 (2017), 290-301. doi: 10.1016/j.ymssp.2016.11.020. Google Scholar [5] M. Chu and N. D. Buono, Total decoupling of general quadratic pencils, Part Ⅱ: Structure preserving isospectral flows, Journal of Sound and Vibration, 309 (2008), 112-128. doi: 10.1016/j.jsv.2007.05.052. Google Scholar [6] G. R. Duan, Analysis and Design of Descriptor Linear Systems, New York: Springer; 2010. doi: 10.1007/978-1-4419-6397-0. Google Scholar [7] G. R. Duan, Parametric eigenstructure assignment in second-order descriptor linear systems, IEEE Transactions on Automatic Control, 49 (2004), 1789-1794. doi: 10.1109/TAC.2004.835580. Google Scholar [8] G. R. Duan and Y. L. Wu, Robust pole assignment in matrix descriptor second-order linear systems, Transactions of the Institute of Measurement and Control, 27 (2005), 279-295. doi: 10.1191/0142331205tm149oa. Google Scholar [9] G. R. Duan, Two parametric approaches for eigenstructure assignment in high-order linear systems, J. Control Theory Appl., 1 (2003), 59-64. doi: 10.1007/s11768-003-0009-z. Google Scholar [10] G. R. Duan, Parametric approaches for eigenstructure assignment in high-order descriptor linear systems, IEEE Conference on Decision and Control, 49 (2004), 1789-1795. doi: 10.1109/TAC.2004.835580. Google Scholar [11] G. R. Duan and H. H. Yu, Parametric approaches for eigenstructure assignment in high-order descriptor linear systems, Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 1399-1404. Google Scholar [12] G. R. Duan and H. H. Yu, Complete eigenstructure assignment in high-order descriptor linear systems via proportional plus derivative state feedback, Proceedings of the 6th World Congress on Intelligent Control and Automation, (2006), 500-505. Google Scholar [13] R. Galindo, Input/output decoupling of square linear systems by dynamic two-parameter stabilizing control, Asian Journal of Control, 18 (2016), 2310-2316. doi: 10.1002/asjc.1285. Google Scholar [14] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, New York: Academic Press; 1982. Google Scholar [15] D. Henrion and J. C. Zúñiga, Detecting infinite zeros in polynomial matrices, IEEE Transactions on Circuits and systems Ⅱ-Express Briefs, 52 (2005), 744-745. doi: 10.1109/TCSII.2005.852931. Google Scholar [16] M. Hou, Controllability and elimination of impulsive modes in descriptor systems, IEEE Transactions on Automatic Control, 49 (2004), 1723-1727. doi: 10.1109/TAC.2004.835392. Google Scholar [17] Y. Ilyashenko and O. Romaskevich, Sternberg linearization Theorem for skew products, Journal of Dynamical and Control Systems, 22 (2016), 595-614. doi: 10.1007/s10883-016-9319-6. Google Scholar [18] S. Johansson, B. Kågström and P. V. Dooren, Stratification of full rank polynomial matrices, Linear Algebra and its Applications, 439 (2013), 1062-1090. doi: 10.1016/j.laa.2012.12.013. Google Scholar [19] D. T. Kawano, M. Morzfeld and F. Ma, The decoupling of second-order linear systems with a singular mass matrix, Journal of Sound and Vibration, 332 (2013), 6829-6846. doi: 10.1016/j.jsv.2013.08.005. Google Scholar [20] Y. Kim, H. S. Kim and J. L. Junkins, Eigenstructure assignment algorithm for mechanical second-order systems, AIAA Journal Guidance, Control and Dynamics, 22 (1999), 729-731. doi: 10.2514/2.4444. Google Scholar [21] J. Li, Y. F. Teng and Q. L. Zhang, et al, Eliminating impulse for descriptor system by derivative output feedback, Journal of Applied Mathematics, 2014 (2014), Art. ID 265601, 15 pp. doi: 10.1155/2014/265601. Google Scholar [22] H. Liu, B. X. He and X. P. Chen, Minimum norm partial quadratic eigenvalue assignment for vibrating structures using receptance method, Mechanical Systems and Signal Processing, 123 (2019), 131-142. doi: 10.1016/j.ymssp.2019.01.006. Google Scholar [23] P. Losse and V. Mehrmann, Controllability and observability of second order descriptor systems, SIAM Journal on Control and Optimization, 47 (2008), 1351-1379. doi: 10.1137/060673977. Google Scholar [24] D. S. Mackey, N. Mackey, C. Mehl and et al, Vector spaces of linearizations for matrix polynomials, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 971-1004. doi: 10.1137/050628350. Google Scholar [25] M. Morzfeld and F. Ma, The decoupling of damped linear systems in configuration and state spaces, Journal of Sound and Vibration, 330 (2011), 155-161. doi: 10.1016/j.jsv.2010.09.005. Google Scholar [26] I. M. Stamova, Parametric stability of impulsive functional differential equations, Journal of Dynamical and Control Systems, 14 (2008), 235-250. doi: 10.1007/s10883-008-9037-9. Google Scholar [27] F. D. Terán, F. M. Dopico and D. S. Mackey, Fiedler companion linearizations and the recovery of minimal indices, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2181-2204. doi: 10.1137/090772927. Google Scholar [28] F. D. Terán, F. M. Dopico and D. S. Mackey, Linearizations of singular matrix polynomials and the recovery of minimal indices, Electronic Journal of Linear Algebra, 18 (2009), 371-402. doi: 10.13001/1081-3810.1320. Google Scholar [29] A. I. Vardulakis, Linear Multivariable Control: Algebraic Analysis and Synthesis Methods, Chichester: Wiley, 1991. Google Scholar [30] H. Wang, S. Duan and C. Li, et al, Stability criterion of linear delayed impulsive differential systems with impulse time windows, International Journal of Control, Automation and Systems, 24 (2016), 174–180.Google Scholar [31] S. Xu and J. Qian, Orthogonal basis selection method for robust partial eigenvalue assignment problem in second-order control systems, Journal of Sound and Vibration, 317 (2008), 1-19. doi: 10.1016/j.jsv.2008.03.002. Google Scholar [32] C. L. Yang, J. Z. Liu and Y. Liu, Solutions of the generalized Sylvester matrix equation and the application in eigenstructure assignment, Asian Journal of Control, 14 (2012), 1669-1675. doi: 10.1002/asjc.448. Google Scholar [33] H. H. Yu and G. R. Duan, ESA in high-order linear systems via output feedback, Asian Journal of Control, 11 (2009), 336-343. doi: 10.1002/asjc.111. Google Scholar [34] P. Z. Yu and G. S. Zhang, Infinite zero structure of polynomial matrix and impulsive modes of polynomial matrix systems, Proceedings of the 34th Chinese Control Conference (CCC), (2015), 278-282. Google Scholar [35] P. Z. Yu and G. S. Zhang, Eigenstructure assignment and impulse elimination for singular second-order system via feedback control, IET Control Theory and Applications, 10 (2016), 869-876. doi: 10.1049/iet-cta.2015.1189. Google Scholar [36] G. S. Zhang and W. Q. Liu, Impulsive mode elimination for descriptor systems by a structured P-D feedback, IEEE Transactions on Automatic Control, 56 (2011), 2968-2973. doi: 10.1109/TAC.2011.2160597. Google Scholar [37] L. Zhang, F. Yu and X. T. Wang, An algorithm of partial eigenstructure assignment for high-order systems, Mathematical Methods in the Applied Sciences, 41 (2018), 6070-6079. doi: 10.1002/mma.5118. Google Scholar [38] B. Zhang, Eigenstructure assignment for linear descriptor systems via output feedback, Asian Journal of Control, 21 (2019), 759-769. doi: 10.1002/asjc.1763. Google Scholar [39] Z. Zhang and N. Wong, Canonical projector techniques for analyzing descriptor systems, International Journal of Control, Automation and Systems, 12 (2014), 71-83. doi: 10.1007/s12555-012-0234-7. Google Scholar [40] J. Zhang, H. Ouyang and J. Yang, Partial eigenstructure assignment for undamped vibration systems using acceleration and displacement feedback, Journal of Sound and Vibration, 333 (2014), 1-12. doi: 10.1016/j.jsv.2013.08.040. Google Scholar [41] D. Z. Zheng, Linear System Theory, 2$^{nd}$ edition, Tsinghua University Press, Beijing, 2002.Google Scholar

show all references

##### References:
 [1] T. H. S. Abdelaziz, Robust pole placement for second-order linear systems using velocity-plus-acceleration feedback, IET Control Theory and Applications, 7 (2013), 1843-1856. doi: 10.1049/iet-cta.2013.0039. Google Scholar [2] T. H. S. Abdelaziz, Parametric approach for eigenstructure assignment in descriptor second-order systems via velocity-plus-acceleration feedback, Journal Dynamic Systems, Measurement, and Control, 136 (2014), 044505. doi: 10.1115/1.4026876. Google Scholar [3] E. N. Antoniou and S. Vologiannidis, A new family of companion forms of polynomial matrices, Electronic Journal of Linear Algebra, 11 (2004), 78-87. doi: 10.13001/1081-3810.1124. Google Scholar [4] Z. J. Bai and Q. Y. Wan, Partial quadratic eigenvalue assignment in vibrating structures using receptances and system matrices, Mechanical Systems and Signal Processing, 88 (2017), 290-301. doi: 10.1016/j.ymssp.2016.11.020. Google Scholar [5] M. Chu and N. D. Buono, Total decoupling of general quadratic pencils, Part Ⅱ: Structure preserving isospectral flows, Journal of Sound and Vibration, 309 (2008), 112-128. doi: 10.1016/j.jsv.2007.05.052. Google Scholar [6] G. R. Duan, Analysis and Design of Descriptor Linear Systems, New York: Springer; 2010. doi: 10.1007/978-1-4419-6397-0. Google Scholar [7] G. R. Duan, Parametric eigenstructure assignment in second-order descriptor linear systems, IEEE Transactions on Automatic Control, 49 (2004), 1789-1794. doi: 10.1109/TAC.2004.835580. Google Scholar [8] G. R. Duan and Y. L. Wu, Robust pole assignment in matrix descriptor second-order linear systems, Transactions of the Institute of Measurement and Control, 27 (2005), 279-295. doi: 10.1191/0142331205tm149oa. Google Scholar [9] G. R. Duan, Two parametric approaches for eigenstructure assignment in high-order linear systems, J. Control Theory Appl., 1 (2003), 59-64. doi: 10.1007/s11768-003-0009-z. Google Scholar [10] G. R. Duan, Parametric approaches for eigenstructure assignment in high-order descriptor linear systems, IEEE Conference on Decision and Control, 49 (2004), 1789-1795. doi: 10.1109/TAC.2004.835580. Google Scholar [11] G. R. Duan and H. H. Yu, Parametric approaches for eigenstructure assignment in high-order descriptor linear systems, Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 1399-1404. Google Scholar [12] G. R. Duan and H. H. Yu, Complete eigenstructure assignment in high-order descriptor linear systems via proportional plus derivative state feedback, Proceedings of the 6th World Congress on Intelligent Control and Automation, (2006), 500-505. Google Scholar [13] R. Galindo, Input/output decoupling of square linear systems by dynamic two-parameter stabilizing control, Asian Journal of Control, 18 (2016), 2310-2316. doi: 10.1002/asjc.1285. Google Scholar [14] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, New York: Academic Press; 1982. Google Scholar [15] D. Henrion and J. C. Zúñiga, Detecting infinite zeros in polynomial matrices, IEEE Transactions on Circuits and systems Ⅱ-Express Briefs, 52 (2005), 744-745. doi: 10.1109/TCSII.2005.852931. Google Scholar [16] M. Hou, Controllability and elimination of impulsive modes in descriptor systems, IEEE Transactions on Automatic Control, 49 (2004), 1723-1727. doi: 10.1109/TAC.2004.835392. Google Scholar [17] Y. Ilyashenko and O. Romaskevich, Sternberg linearization Theorem for skew products, Journal of Dynamical and Control Systems, 22 (2016), 595-614. doi: 10.1007/s10883-016-9319-6. Google Scholar [18] S. Johansson, B. Kågström and P. V. Dooren, Stratification of full rank polynomial matrices, Linear Algebra and its Applications, 439 (2013), 1062-1090. doi: 10.1016/j.laa.2012.12.013. Google Scholar [19] D. T. Kawano, M. Morzfeld and F. Ma, The decoupling of second-order linear systems with a singular mass matrix, Journal of Sound and Vibration, 332 (2013), 6829-6846. doi: 10.1016/j.jsv.2013.08.005. Google Scholar [20] Y. Kim, H. S. Kim and J. L. Junkins, Eigenstructure assignment algorithm for mechanical second-order systems, AIAA Journal Guidance, Control and Dynamics, 22 (1999), 729-731. doi: 10.2514/2.4444. Google Scholar [21] J. Li, Y. F. Teng and Q. L. Zhang, et al, Eliminating impulse for descriptor system by derivative output feedback, Journal of Applied Mathematics, 2014 (2014), Art. ID 265601, 15 pp. doi: 10.1155/2014/265601. Google Scholar [22] H. Liu, B. X. He and X. P. Chen, Minimum norm partial quadratic eigenvalue assignment for vibrating structures using receptance method, Mechanical Systems and Signal Processing, 123 (2019), 131-142. doi: 10.1016/j.ymssp.2019.01.006. Google Scholar [23] P. Losse and V. Mehrmann, Controllability and observability of second order descriptor systems, SIAM Journal on Control and Optimization, 47 (2008), 1351-1379. doi: 10.1137/060673977. Google Scholar [24] D. S. Mackey, N. Mackey, C. Mehl and et al, Vector spaces of linearizations for matrix polynomials, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 971-1004. doi: 10.1137/050628350. Google Scholar [25] M. Morzfeld and F. Ma, The decoupling of damped linear systems in configuration and state spaces, Journal of Sound and Vibration, 330 (2011), 155-161. doi: 10.1016/j.jsv.2010.09.005. Google Scholar [26] I. M. Stamova, Parametric stability of impulsive functional differential equations, Journal of Dynamical and Control Systems, 14 (2008), 235-250. doi: 10.1007/s10883-008-9037-9. Google Scholar [27] F. D. Terán, F. M. Dopico and D. S. Mackey, Fiedler companion linearizations and the recovery of minimal indices, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2181-2204. doi: 10.1137/090772927. Google Scholar [28] F. D. Terán, F. M. Dopico and D. S. Mackey, Linearizations of singular matrix polynomials and the recovery of minimal indices, Electronic Journal of Linear Algebra, 18 (2009), 371-402. doi: 10.13001/1081-3810.1320. Google Scholar [29] A. I. Vardulakis, Linear Multivariable Control: Algebraic Analysis and Synthesis Methods, Chichester: Wiley, 1991. Google Scholar [30] H. Wang, S. Duan and C. Li, et al, Stability criterion of linear delayed impulsive differential systems with impulse time windows, International Journal of Control, Automation and Systems, 24 (2016), 174–180.Google Scholar [31] S. Xu and J. Qian, Orthogonal basis selection method for robust partial eigenvalue assignment problem in second-order control systems, Journal of Sound and Vibration, 317 (2008), 1-19. doi: 10.1016/j.jsv.2008.03.002. Google Scholar [32] C. L. Yang, J. Z. Liu and Y. Liu, Solutions of the generalized Sylvester matrix equation and the application in eigenstructure assignment, Asian Journal of Control, 14 (2012), 1669-1675. doi: 10.1002/asjc.448. Google Scholar [33] H. H. Yu and G. R. Duan, ESA in high-order linear systems via output feedback, Asian Journal of Control, 11 (2009), 336-343. doi: 10.1002/asjc.111. Google Scholar [34] P. Z. Yu and G. S. Zhang, Infinite zero structure of polynomial matrix and impulsive modes of polynomial matrix systems, Proceedings of the 34th Chinese Control Conference (CCC), (2015), 278-282. Google Scholar [35] P. Z. Yu and G. S. Zhang, Eigenstructure assignment and impulse elimination for singular second-order system via feedback control, IET Control Theory and Applications, 10 (2016), 869-876. doi: 10.1049/iet-cta.2015.1189. Google Scholar [36] G. S. Zhang and W. Q. Liu, Impulsive mode elimination for descriptor systems by a structured P-D feedback, IEEE Transactions on Automatic Control, 56 (2011), 2968-2973. doi: 10.1109/TAC.2011.2160597. Google Scholar [37] L. Zhang, F. Yu and X. T. Wang, An algorithm of partial eigenstructure assignment for high-order systems, Mathematical Methods in the Applied Sciences, 41 (2018), 6070-6079. doi: 10.1002/mma.5118. Google Scholar [38] B. Zhang, Eigenstructure assignment for linear descriptor systems via output feedback, Asian Journal of Control, 21 (2019), 759-769. doi: 10.1002/asjc.1763. Google Scholar [39] Z. Zhang and N. Wong, Canonical projector techniques for analyzing descriptor systems, International Journal of Control, Automation and Systems, 12 (2014), 71-83. doi: 10.1007/s12555-012-0234-7. Google Scholar [40] J. Zhang, H. Ouyang and J. Yang, Partial eigenstructure assignment for undamped vibration systems using acceleration and displacement feedback, Journal of Sound and Vibration, 333 (2014), 1-12. doi: 10.1016/j.jsv.2013.08.040. Google Scholar [41] D. Z. Zheng, Linear System Theory, 2$^{nd}$ edition, Tsinghua University Press, Beijing, 2002.Google Scholar
The responses of $x$ and $\dot{x}$ for normalization
The responses of $x$ and $\dot{x}$ for impulse elimination
The responses of ${x}$ and $\dot{x}$ for impulse elimination
The responses of $\ddot{x}$ and $x^{(3)}$ for impulse elimination
 [1] Vladimir Pozdyayev. 2D system analysis via dual problems and polynomial matrix inequalities. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 491-504. doi: 10.3934/naco.2016022 [2] Ayla Sayli, Ayse Oncu Sarihan. Statistical query-based rule derivation system by backward elimination algorithm. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1341-1356. doi: 10.3934/dcdss.2015.8.1341 [3] Fernando Hernando, Diego Ruano. New linear codes from matrix-product codes with polynomial units. Advances in Mathematics of Communications, 2010, 4 (3) : 363-367. doi: 10.3934/amc.2010.4.363 [4] Simone Vazzoler. A note on the normalization of generating functions. Journal of Geometric Mechanics, 2018, 10 (2) : 209-215. doi: 10.3934/jgm.2018008 [5] Omer Faruk Yilmaz, Mehmet Bulent Durmusoglu. A performance comparison and evaluation of metaheuristics for a batch scheduling problem in a multi-hybrid cell manufacturing system with skilled workforce assignment. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1219-1249. doi: 10.3934/jimo.2018007 [6] Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691 [7] Bettina Klaus, Frédéric Payot. Paths to stability in the assignment problem. Journal of Dynamics & Games, 2015, 2 (3&4) : 257-287. doi: 10.3934/jdg.2015004 [8] Marina Núñez, Carles Rafels. A survey on assignment markets. Journal of Dynamics & Games, 2015, 2 (3&4) : 227-256. doi: 10.3934/jdg.2015003 [9] Thierry Paul, David Sauzin. Normalization in Banach scale Lie algebras via mould calculus and applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4461-4487. doi: 10.3934/dcds.2017191 [10] Dequan Yue, Wuyi Yue. Block-partitioning matrix solution of M/M/R/N queueing system with balking, reneging and server breakdowns. Journal of Industrial & Management Optimization, 2009, 5 (3) : 417-430. doi: 10.3934/jimo.2009.5.417 [11] Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103 [12] Alain Bensoussan, John Liu, Jiguang Yuan. Singular control and impulse control: A common approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 27-57. doi: 10.3934/dcdsb.2010.13.27 [13] Brahim El Asri. The value of a minimax problem involving impulse control. Journal of Dynamics & Games, 2019, 6 (1) : 1-17. doi: 10.3934/jdg.2019001 [14] David Cantala, Juan Sebastián Pereyra. Endogenous budget constraints in the assignment game. Journal of Dynamics & Games, 2015, 2 (3&4) : 207-225. doi: 10.3934/jdg.2015002 [15] Xiaobin Mao, Hua Dai. Partial eigenvalue assignment with time delay robustness. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 207-221. doi: 10.3934/naco.2013.3.207 [16] Hao Yang, Hang Qiu, Leiting Chen. An optimized direction statistics for detecting and removing random-valued impulse noise. Journal of Industrial & Management Optimization, 2018, 14 (2) : 597-611. doi: 10.3934/jimo.2017062 [17] Yoon Mo Jung, Taeuk Jeong, Sangwoon Yun. Non-convex TV denoising corrupted by impulse noise. Inverse Problems & Imaging, 2017, 11 (4) : 689-702. doi: 10.3934/ipi.2017032 [18] Michael Basin, Mark A. Pinsky. Control of Kalman-like filters using impulse and continuous feedback design. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 169-184. doi: 10.3934/dcdsb.2002.2.169 [19] Jianbin Yang, Cong Wang. A wavelet frame approach for removal of mixed Gaussian and impulse noise on surfaces. Inverse Problems & Imaging, 2017, 11 (5) : 783-798. doi: 10.3934/ipi.2017037 [20] Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a mean-reverting inventory with quadratic costs. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1685-1700. doi: 10.3934/jimo.2018027

Impact Factor: