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

Optimal control of piecewise affine systems with piecewise affine state feedback

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.
Citation: Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial & 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 & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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 & Heterogeneous Media, 2017, 12 (3) : 417-459. doi: 10.3934/nhm.2017019

[10]

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 & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014

[11]

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

[12]

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

[13]

Claude Stolz. On estimation of internal state by an optimal control approach for elastoplastic material. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 599-611. doi: 10.3934/dcdss.2016014

[14]

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

[15]

Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553

[16]

Christian Clason, Barbara Kaltenbacher. Avoiding degeneracy in the Westervelt equation by state constrained optimal control. Evolution Equations & Control Theory, 2013, 2 (2) : 281-300. doi: 10.3934/eect.2013.2.281

[17]

Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006

[18]

Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739

[19]

Tiantian Wu, Xiao-Song Yang. A new class of 3-dimensional piecewise affine systems with homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5119-5129. doi: 10.3934/dcds.2016022

[20]

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

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]