2011, 2011(Special): 312-321. doi: 10.3934/proc.2011.2011.312

Invariant feedback design for control systems with lie symmetries - A kinematic car example

1. 

Saarland University, Chair of Systems Theory and Control Engineering, Germany, Germany

2. 

TU Dresden, Institut für Regelungs- und Steuerungstheorie, Germany

Received  July 2010 Revised  February 2011 Published  October 2011

Invariant tracking control design for control systems in state representation with classical Lie point symmetry is considered. The relevance of the invariance aspect is motivated by an exemplary control design for the kinematic car. Introducing the perpendicular distance between the position of the rear axle midpoint and the reference trajectory in combination with the contouring error as tracking error, an invariant feedback w.r.t. the action of $SE$(2) is derived. Generally, the use of compatible tracking errors allows the application of well-known feedback design approaches yielding an error dynamics that is invariant w.r.t. the group action. A normalization procedure allowing the construction of invariant tracking errors is recalled for which a geometric interpretation is presented.
Citation: Carsten Collon, Joachim Rudolph, Frank Woittennek. Invariant feedback design for control systems with lie symmetries - A kinematic car example. Conference Publications, 2011, 2011 (Special) : 312-321. doi: 10.3934/proc.2011.2011.312
[1]

Michael Hochman. Smooth symmetries of $\times a$-invariant sets. Journal of Modern Dynamics, 2018, 13: 187-197. doi: 10.3934/jmd.2018017

[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 and Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303

[3]

Sie Long Kek, Mohd Ismail Abd Aziz. Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 275-288. doi: 10.3934/naco.2015.5.275

[4]

Yi Gao, Rui Li, Yingjing Shi, Li Xiao. Design of path planning and tracking control of quadrotor. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2221-2235. doi: 10.3934/jimo.2021063

[5]

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

[6]

Leonardo Colombo, Fernando Jiménez, David Martín de Diego. Variational integrators for mechanical control systems with symmetries. Journal of Computational Dynamics, 2015, 2 (2) : 193-225. doi: 10.3934/jcd.2015003

[7]

Irina Berezovik, Carlos García-Azpeitia, Wieslaw Krawcewicz. Symmetries of nonlinear vibrations in tetrahedral molecular configurations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2473-2491. doi: 10.3934/dcdsb.2018261

[8]

Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 127-151. doi: 10.3934/naco.2016004

[9]

Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control and Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019

[10]

Zhao-Han Sheng, Tingwen Huang, Jian-Guo Du, Qiang Mei, Hui Huang. Study on self-adaptive proportional control method for a class of output models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 459-477. doi: 10.3934/dcdsb.2009.11.459

[11]

James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445

[12]

Constantin Christof, Dominik Hafemeyer. On the nonuniqueness and instability of solutions of tracking-type optimal control problems. Mathematical Control and Related Fields, 2022, 12 (2) : 421-431. doi: 10.3934/mcrf.2021028

[13]

Christian Meyer, Stephan Walther. Optimal control of perfect plasticity part I: Stress tracking. Mathematical Control and Related Fields, 2022, 12 (2) : 275-301. doi: 10.3934/mcrf.2021022

[14]

Chao Chen, Shanlin Yi, Feng Wang, Chengxi Zhang, Qingmin Yu. Prescribed performance tracking control of multi-link robotic manipulator with uncertainties. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022012

[15]

E. Fossas-Colet, J.M. Olm-Miras. Asymptotic tracking in DC-to-DC nonlinear power converters. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 295-307. doi: 10.3934/dcdsb.2002.2.295

[16]

Jin-Zi Yang, Yuan-Xin Li, Ming Wei. Fuzzy adaptive asymptotic tracking of fractional order nonlinear systems with uncertain disturbances. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1615-1631. doi: 10.3934/dcdss.2021144

[17]

Chaudry Masood Khalique, Muhammad Usman, Maria Luz Gandarais. Nonlinear differential equations: Lie symmetries, conservation laws and other approaches of solving. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : i-ii. doi: 10.3934/dcdss.2020415

[18]

Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040

[19]

María Rosa, María de los Santos Bruzón, María de la Luz Gandarias. Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1331-1339. doi: 10.3934/dcdss.2015.8.1331

[20]

Ying Wu, Zhaohui Yuan, Yanpeng Wu. Optimal tracking control for networked control systems with random time delays and packet dropouts. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1343-1354. doi: 10.3934/jimo.2015.11.1343

 Impact Factor: 

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (0)

[Back to Top]