# American Institute of Mathematical Sciences

January  2009, 5(1): 153-159. doi: 10.3934/jimo.2009.5.153

## $H_\infty$ optimal stabilization of a class of uncertain impulsive systems: An LMI approach

 1 Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia 2 Department of Mathematics and Statistics, Curtin University, G.P.O. Box U1987, Perth, WA 6845

Received  March 2008 Revised  November 2008 Published  December 2008

This paper studies $H_\infty$ optimal control problems for a class of impulsive dynamical systems with norm-bounded time-varying uncertainty. By using a linear matrix inequality approach, some sufficient conditions are established to ensure both internally asymptotical stability and $H_\infty$ optimal performance of the impulsive closed-loop system. Moreover, based on the stability criteria, a linear time-invariant stabilizing control law is designed. Finally, a numerical example is presented to illustrate the effectiveness of our results.
Citation: Honglei Xu, Kok Lay Teo. $H_\infty$ optimal stabilization of a class of uncertain impulsive systems: An LMI approach. Journal of Industrial & Management Optimization, 2009, 5 (1) : 153-159. doi: 10.3934/jimo.2009.5.153
 [1] 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 [2] 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 [3] 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 [4] Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275 [5] Zhong-Qiang Wu, Xi-Bo Zhao. Frequency $H_{2}/H_{∞}$ optimizing control for isolated microgrid based on IPSO algorithm. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1565-1577. doi: 10.3934/jimo.2018021 [6] C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial & Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435 [7] Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1 [8] Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989 [9] Canghua Jiang, Kok Lay Teo, Ryan Loxton, Guang-Ren Duan. A neighboring extremal solution for an optimal switched impulsive control problem. Journal of Industrial & Management Optimization, 2012, 8 (3) : 591-609. doi: 10.3934/jimo.2012.8.591 [10] Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164 [11] Evgeny I. Veremey, Vladimir V. Eremeev. SISO H-Optimal synthesis with initially specified structure of control law. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 121-138. doi: 10.3934/naco.2017009 [12] Vadim Azhmyakov, Alex Poznyak, Omar Gonzalez. On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach. Journal of Industrial & Management Optimization, 2013, 9 (3) : 579-593. doi: 10.3934/jimo.2013.9.579 [13] T. Tachim Medjo, Louis Tcheugoue Tebou. Robust control problems in fluid flows. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 437-463. doi: 10.3934/dcds.2005.12.437 [14] Qiying Hu, Wuyi Yue. Optimal control for discrete event systems with arbitrary control pattern. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 535-558. doi: 10.3934/dcdsb.2006.6.535 [15] Bangyu Shen, Xiaojing Wang, Chongyang Liu. Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 369-380. doi: 10.3934/naco.2015.5.369 [16] Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367 [17] Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1 [18] Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial & Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63 [19] Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107 [20] Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149

2018 Impact Factor: 1.025