American Institute of Mathematical Sciences

Advanced
October  2009, 5(4): 737-747. doi: 10.3934/jimo.2009.5.737

Optimal control of piecewise affine systems with piecewise affine state feedback

Changzhi Wu 1, , Kok Lay Teo 2, and  Volker Rehbock 3,

1. 

Department of Mathematics, Chongqing Normal University, Shapingba, Chongqing, 400047, China
2. 

Department of Mathematics and Statistics, Curtin University, G.P.O. Box U1987, Perth, WA 6845
3. 

Department of Mathematics and Statistics, Curtin University of Technology, Perth, 6002, WA, Australia

Received  May 2008 Revised  May 2009 Published  August 2009

In this paper, we consider a class of optimal control problems involving piecewise affine (PWA) systems with piecewise affine state feedback. We first show that if the piecewise affine state feedback control is assumed to be continuous at the switching boundaries, then the number of switching amongst PWA systems is finite. On this basis, this optimal control problem is transformed into a discrete valued optimal control problem. For this discrete valued optimal control problem, we introduce the time scaling transform to convert it into an equivalent constrained optimal parameter selection problem, for which it can be solved by existing optimal control techniques for optimal parameter selection problems. A numerical example is solved so as to illustrate the proposed method.
Keywords: Piecewise affine system; State feedback; Optimal control..
Mathematics Subject Classification: 34H05, 49M3.
Citation: Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial and Management Optimization, 2009, 5 (4) : 737-747. doi: 10.3934/jimo.2009.5.737
[1]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control and Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185
[2]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control and Related Fields, 2021, 11 (3) : 555-578. doi: 10.3934/mcrf.2021012
[3]

Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021
[4]

Bangyu Shen, Xiaojing Wang, Chongyang Liu. Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 369-380. doi: 10.3934/naco.2015.5.369
[5]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[6]

Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607
[7]

Sanmei Zhu, Jun-e Feng, Jianli Zhao. State feedback for set stabilization of Markovian jump Boolean control networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1591-1605. doi: 10.3934/dcdss.2020413
[8]

Meng Zhang, Kaiyuan Liu, Lansun Chen, Zeyu Li. State feedback impulsive control of computer worm and virus with saturated incidence. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1465-1478. doi: 10.3934/mbe.2018067
[9]

Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241
[10]

Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753
[11]

Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control and Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007
[12]

Fulvia Confortola, Elisa Mastrogiacomo. Feedback optimal control for stochastic Volterra equations with completely monotone kernels. Mathematical Control and Related Fields, 2015, 5 (2) : 191-235. doi: 10.3934/mcrf.2015.5.191
[13]

Qi Lü, Tianxiao Wang, Xu Zhang. Characterization of optimal feedback for stochastic linear quadratic control problems. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 11-. doi: 10.1186/s41546-017-0022-7
[14]

Chongyang Liu, Wenjuan Sun, Xiaopeng Yi. Optimal control of a fractional smoking system. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022071
[15]

Vincent Renault, Michèle Thieullen, Emmanuel Trélat. Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics. Networks and Heterogeneous Media, 2017, 12 (3) : 417-459. doi: 10.3934/nhm.2017019
[16]

Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014
[17]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065
[18]

Qiying Hu, Chen Xu, Wuyi Yue. A unified model for state feedback of discrete event systems II: Control synthesis problems. Journal of Industrial and Management Optimization, 2008, 4 (4) : 713-726. doi: 10.3934/jimo.2008.4.713
[19]

Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096
[20]

Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy