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2013, 2013(special): 393-406. doi: 10.3934/proc.2013.2013.393

The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems

Byungik Kahng 1, and  Miguel Mendes 2,

1. 

Department of Mathematics and Information Sciences, University of North Texas at Dallas, Dallas, TX 75241, United States
2. 

Departamento de Engenharia Civil, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias 4200 - 465 Porto, Portugal

Received  September 2012 Revised  March 2013 Published  November 2013

The main topic of this paper is the controllability/reachability problems of the maximal invariant sets of non-linear discrete-time multiple-valued iterative dynamical systems. We prove that the controllability/reachability problems of the maximal full-invariant sets of classical control dynamical systems are equivalent to those of the maximal quasi-invariant sets of disturbed control dynamical systems, when modeled by the iterative dynamics of multiple-valued self-maps. Also, we prove that the afore-mentioned maximal full-invariant sets and maximal quasi-invariant sets are countably infinite step controllable under some appropriate conditions. We take an abstract set theoretical approach, so that our main theorems remain valid regardless of the topological structure of the space or the analytical structure of the dynamics.
Keywords: controllability, Maximal invariance, orbit-chain..
Mathematics Subject Classification: Primary: 93C05, 93C25, 93C55; Secondary: 37E9.
Citation: Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393
References:
[1]

E. Akin, "The General Topology of Dynamical Systems," Graduate Studies in Mathematics, 1. American Mathematical Society, Providence, RI, 1993.

[2]

Z. Artstein and S. Rakovic, Feedback invariance under uncertainty via set-iterates, Automatica J. IFAC, 44 (2008), 520-525.

[3]

P. Ashwin, X. C. Fu, T. Nishikawa and K. Zyczkowski, Invariant sets for discontinuous parabolic area-preserving torus maps, Nonlinearity, 13 (2000), 819-835.

[4]

P. Ashwin, X. C. Fu and J. R. Terry, Riddling and invariance for discontinuous maps preserving lebesgue measure, Nonlinearity, 15 (2002), 633-645.

[5]

A. Bemporad, F. D. Torrisi and M. Morari, "Optimization-based verification and stability characterization of piecewise affine and hybrid systems," in Proc. 3rd International Workshop on Hybrid Systems: Computation and Control, N. A. Lynch and B. H. Krogh, eds., London, UK, 2000, Springer-Verlag.

[6]

D. Bertsekas, Infinite time reachability of state-space regions by using feedback control, IEEE Transactions on Automatic Control, AC-17 (1972), 604-613.

[7]

F. Blanchini and S. Miani, "Set-Theoretic Methods in Control," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2008.

[8]

T. Das, K. Lee and M. Lee, $c^1$-persistently continuum-wise expansive homoclinic classes and recurrent sets, Topology and its Applications, 160 (2013), 350-359.

[9]

X. Fu, F. Chen and X. Zhao, Dynamical properties of 2-torus parabolic maps, Nonlinear Dynamics, 50 (2007), 539-549.

[10]

X. Fu and J. Duan, Global attractors and invariant measures for non-invertible planar piecewise isometric maps, Phys. Lett. A, 371 (2007), 285-290.

[11]

______, On global attractors for a class of nonhyperbolic piecewise affine maps, Physica D, 237 (2008), 3369-3376.

[12]

A. A. Julius and A. J. van der Schaft, The maximal controlled invariant set of switched linear systems, Proc. 41st IEEE Conf. on Decision and Control, (2002), pp. 3174-3179.

[13]

B. Kahng, Chains of minimal image sets can attain arbitrary length until they reach maximal invariant sets,, preprint., (). 

[14]

_______, The invariant set theory of multiple valued iterative dynamical systems, in Recent Advances in System Science and Simulation in Engineering, 7 (2008), 19-24.

[15]

_______, Maximal invariant sets of multiple valued iterative dynamics in disturbed control systems, Int. J. Circ. Sys. and Signal Processing, 2 (2008), 113-120.

[16]

_______, Positive invariance of multiple valued iterative dynamical systems in disturbed control models, in "Proc. IEEE Med. Control Conf." Thessaloniki, Greece, 2009, pp. 663-668.

[17]

_______, Singularities of 2-dimensional invertible piecewise isometric dynamics, Chaos, 19 (2009), p. 023115.

[18]

_______, The approximate control problems of the maximal invariant sets of non-linear discrete-time dis-turbed control dynamical systems: an algorithmic approach, in Proc. Int. Conf. on Control and Auto. and Sys. Gyeonggi-do, Korea, 2010, pp. 1513-1518.

[19]

_______, Multiple valued iterative dynamics models of nonlinear discrete-time control dynamical systems with disturbance, J. Korean Math. Soc., 50 (2013), 17-39.

[20]

E. Kerrigan, J. Lygeros and J. M. Maciejowski, "A Geometric Approach To Reachability Computations For Constrained Discrete-Time Systems," in IFAC World Congress, Barcelona, Spain, 2002.

[21]

E. Kerrigan and J. M. Maciejowski, "Invariant Sets for Constrained Nonlinear Discrete-Time Systems with Application to Feasibility in Model Predictive Control," in Proc. 39th IEEE Conf. on Decision and Control, Sydney, Australia, 2000.

[22]

X. D. Koutsoukos and P. J. Antsaklis, Safety and reachability of piecewise-linear hybrid dynamical systems based on discrete abstractions, J. Discr. Event Dynam. Sys., 13 (2003), 203-243.

[23]

K. Lee and M. Lee, Hyperbolicity of $c^1$-stably expansive homoclinic classes, Discr. and Contin. Dynam. Sys., 27 (2010), 1133-1145.

[24]

_______, Stably inverse shadowable transitive sets and dominated splitting, Proc. Amer. Math. Soc., 140 (2012), 217-226.

[25]

K. Lee, K. Moriyasu and K. Sakai, $c^1$-stable shadowing diffeomorphisms, Discr. and Contin. Dynam. Sys., 22 (2008), 683-697.

[26]

J. H. Lowenstein, G. Poggiaspalla and F. Vivaldi, Sticky orbits in a kicked-oscillator model, Dynam. Sys., 20 (2005), 413-451.

[27]

D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert, Constrained model predictive control: Stability and optimality, Automatica, 36 (2000), 789-814.

[28]

M. Mendes, "Dynamics of Piecewise Isometric Systems with Particular Emphasis to the Goetz Map," Ph. D. Thesis, University of Surrey, 2001.

[29]

_______, Quasi-invariant attractors of piecewise isometric systems, Discr. Contin. Dynam. Sys., 9 (2003), 323-338.

[30]

C. J. Ong and E. G. Gilbert, Constrained linear systems with disturbances: Enlargement of their maximal invariant sets by nonlinear feedback, (2006), pp. 5246-5251.

[31]

S. V. Rakovic and M. Fiacchini, "Invariant Approximations of the Maximal Invariant Set or Encircling the Square," in IFAC World Congress, Seoul, Korea, July 2008.

[32]

S. V. Rakovic, E. Kerrigan and D. Q. Mayne, "Optimal Control of Constrained Piecewise Affine Systems with State-Dependent and Input-Dependent Distrubances," in Proc. 16th Int. Sympo. on Mathematical Theory of Networks and Systems, Katholieke Universiteit Leuven, Belgium, July 2004.

[33]

S. V. Rakovic, E. Kerrigan, D. Q. Mayne and J. Lygeros, Reachability analysis of discrete-time systems with disturbances, IEEE Transactions on Automatic Control, 51 (2006), 546-561.

[34]

C. Tomlin, I. Mitchell, A. Bayen and M. Oishi, Computational techniques for the verification and control of hybrid systems, Proc. IEEE, 91 (2003), 986-1001.

[35]

F. D. Torrisi and A. Bemporad, "Discrete-Time Hybrid Modeling and Verification," in Proc. 40th IEEE Conf. on Decision and Control, 2001.

[36]

R. Vidal, S. Schaffert, O. Shakernia, J. Lygeros and S. Sastry, "Decidable and Semi-Decidable Controller Synthesis for Classes of Discrete Time Hybrid Systems," in Proc. 40th IEEE Conf. on Decision and Control, 2001.

show all references

References:
[1]

E. Akin, "The General Topology of Dynamical Systems," Graduate Studies in Mathematics, 1. American Mathematical Society, Providence, RI, 1993.

[2]

Z. Artstein and S. Rakovic, Feedback invariance under uncertainty via set-iterates, Automatica J. IFAC, 44 (2008), 520-525.

[3]

P. Ashwin, X. C. Fu, T. Nishikawa and K. Zyczkowski, Invariant sets for discontinuous parabolic area-preserving torus maps, Nonlinearity, 13 (2000), 819-835.

[4]

P. Ashwin, X. C. Fu and J. R. Terry, Riddling and invariance for discontinuous maps preserving lebesgue measure, Nonlinearity, 15 (2002), 633-645.

[5]

A. Bemporad, F. D. Torrisi and M. Morari, "Optimization-based verification and stability characterization of piecewise affine and hybrid systems," in Proc. 3rd International Workshop on Hybrid Systems: Computation and Control, N. A. Lynch and B. H. Krogh, eds., London, UK, 2000, Springer-Verlag.

[6]

D. Bertsekas, Infinite time reachability of state-space regions by using feedback control, IEEE Transactions on Automatic Control, AC-17 (1972), 604-613.

[7]

F. Blanchini and S. Miani, "Set-Theoretic Methods in Control," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2008.

[8]

T. Das, K. Lee and M. Lee, $c^1$-persistently continuum-wise expansive homoclinic classes and recurrent sets, Topology and its Applications, 160 (2013), 350-359.

[9]

X. Fu, F. Chen and X. Zhao, Dynamical properties of 2-torus parabolic maps, Nonlinear Dynamics, 50 (2007), 539-549.

[10]

X. Fu and J. Duan, Global attractors and invariant measures for non-invertible planar piecewise isometric maps, Phys. Lett. A, 371 (2007), 285-290.

[11]

______, On global attractors for a class of nonhyperbolic piecewise affine maps, Physica D, 237 (2008), 3369-3376.

[12]

A. A. Julius and A. J. van der Schaft, The maximal controlled invariant set of switched linear systems, Proc. 41st IEEE Conf. on Decision and Control, (2002), pp. 3174-3179.

[13]

B. Kahng, Chains of minimal image sets can attain arbitrary length until they reach maximal invariant sets,, preprint., (). 

[14]

_______, The invariant set theory of multiple valued iterative dynamical systems, in Recent Advances in System Science and Simulation in Engineering, 7 (2008), 19-24.

[15]

_______, Maximal invariant sets of multiple valued iterative dynamics in disturbed control systems, Int. J. Circ. Sys. and Signal Processing, 2 (2008), 113-120.

[16]

_______, Positive invariance of multiple valued iterative dynamical systems in disturbed control models, in "Proc. IEEE Med. Control Conf." Thessaloniki, Greece, 2009, pp. 663-668.

[17]

_______, Singularities of 2-dimensional invertible piecewise isometric dynamics, Chaos, 19 (2009), p. 023115.

[18]

_______, The approximate control problems of the maximal invariant sets of non-linear discrete-time dis-turbed control dynamical systems: an algorithmic approach, in Proc. Int. Conf. on Control and Auto. and Sys. Gyeonggi-do, Korea, 2010, pp. 1513-1518.

[19]

_______, Multiple valued iterative dynamics models of nonlinear discrete-time control dynamical systems with disturbance, J. Korean Math. Soc., 50 (2013), 17-39.

[20]

E. Kerrigan, J. Lygeros and J. M. Maciejowski, "A Geometric Approach To Reachability Computations For Constrained Discrete-Time Systems," in IFAC World Congress, Barcelona, Spain, 2002.

[21]

E. Kerrigan and J. M. Maciejowski, "Invariant Sets for Constrained Nonlinear Discrete-Time Systems with Application to Feasibility in Model Predictive Control," in Proc. 39th IEEE Conf. on Decision and Control, Sydney, Australia, 2000.

[22]

X. D. Koutsoukos and P. J. Antsaklis, Safety and reachability of piecewise-linear hybrid dynamical systems based on discrete abstractions, J. Discr. Event Dynam. Sys., 13 (2003), 203-243.

[23]

K. Lee and M. Lee, Hyperbolicity of $c^1$-stably expansive homoclinic classes, Discr. and Contin. Dynam. Sys., 27 (2010), 1133-1145.

[24]

_______, Stably inverse shadowable transitive sets and dominated splitting, Proc. Amer. Math. Soc., 140 (2012), 217-226.

[25]

K. Lee, K. Moriyasu and K. Sakai, $c^1$-stable shadowing diffeomorphisms, Discr. and Contin. Dynam. Sys., 22 (2008), 683-697.

[26]

J. H. Lowenstein, G. Poggiaspalla and F. Vivaldi, Sticky orbits in a kicked-oscillator model, Dynam. Sys., 20 (2005), 413-451.

[27]

D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert, Constrained model predictive control: Stability and optimality, Automatica, 36 (2000), 789-814.

[28]

M. Mendes, "Dynamics of Piecewise Isometric Systems with Particular Emphasis to the Goetz Map," Ph. D. Thesis, University of Surrey, 2001.

[29]

_______, Quasi-invariant attractors of piecewise isometric systems, Discr. Contin. Dynam. Sys., 9 (2003), 323-338.

[30]

C. J. Ong and E. G. Gilbert, Constrained linear systems with disturbances: Enlargement of their maximal invariant sets by nonlinear feedback, (2006), pp. 5246-5251.

[31]

S. V. Rakovic and M. Fiacchini, "Invariant Approximations of the Maximal Invariant Set or Encircling the Square," in IFAC World Congress, Seoul, Korea, July 2008.

[32]

S. V. Rakovic, E. Kerrigan and D. Q. Mayne, "Optimal Control of Constrained Piecewise Affine Systems with State-Dependent and Input-Dependent Distrubances," in Proc. 16th Int. Sympo. on Mathematical Theory of Networks and Systems, Katholieke Universiteit Leuven, Belgium, July 2004.

[33]

S. V. Rakovic, E. Kerrigan, D. Q. Mayne and J. Lygeros, Reachability analysis of discrete-time systems with disturbances, IEEE Transactions on Automatic Control, 51 (2006), 546-561.

[34]

C. Tomlin, I. Mitchell, A. Bayen and M. Oishi, Computational techniques for the verification and control of hybrid systems, Proc. IEEE, 91 (2003), 986-1001.

[35]

F. D. Torrisi and A. Bemporad, "Discrete-Time Hybrid Modeling and Verification," in Proc. 40th IEEE Conf. on Decision and Control, 2001.

[36]

R. Vidal, S. Schaffert, O. Shakernia, J. Lygeros and S. Sastry, "Decidable and Semi-Decidable Controller Synthesis for Classes of Discrete Time Hybrid Systems," in Proc. 40th IEEE Conf. on Decision and Control, 2001.

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