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Pseudo-continuous multi-dimensional multi-mode systems

Behavior, structure and optimal control

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Abstract

The dynamics of multi-mode multi-dimensional (M 3 D) hybrid systems is described. Such systems have modes of different dimensions, for which the state space is defined as a fibre bundle. The implications of the behavior at the mode transitions is investigated in detail, for which pseudo-continuity is introduced. An M 3 D system is pseudo-continuous if instantaneous switching via higher dimensional modes does not have any effect. Canonical forms and parameterizations are derived for pseudo-continuous M 3 D systems. The system may be actively controlled (exo-M 3 D), or passively switched via a fixed switching surface (auto-M 3 D). M 3 D systems are of interest in the approximate and reduced order modeling for nonlinear systems and the remote control over one-way communication channels. The optimal timing (switching) control for such M 3 D systems is solved in the general case. Necessary conditions for a stationary solution are derived and shown to extend those of the equal dimension case (Egerstedt et al. 2003). We also give a specific solution for the linear quadratic problem, involving a generalization of the Riccati equation. This problem is of interest in deriving neighboring extremal solutions for the control under small perturbations of s nominal solution. Some suggestions towards determining the optimal mode sequence are given. We illustrate the problem with the optimal control for a spring assisted high jump (aka man on a trampoline).

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Notes

  1. The llcm of two polynomial matrices R i [ξ] ∈ ℝ, i = 1,2 is only defined if they have the same number of columns, q 1 = q 2 = q, p 1 + p 2 > q, and \([R_1(\xi)^\top,R_2(\xi)^\top]^\top\) has full column rank.

References

  • Aboufadel E (1996) A Mathematician catches a baseball. Am Math Mon 103(10):870–878

    Article  MathSciNet  MATH  Google Scholar 

  • Azhmyakov V, Boltyanski VG, Poznyak A (2008) Optimal control of impulsive hybrid systems. Nonlinear Analysis: Hybrid Systems 2:1089–1097

    Article  MathSciNet  MATH  Google Scholar 

  • Azhmyakov V, Galvan-Guerra R, Egerstedt M (2009) Hybrid LQ-optimization using dynamic programming. In: Proceedings of the 2009 American control conference. St. Louis, pp 3617–3623

  • Boccadero M, Wardi Y, Egerstedt M, Verriest EI (2005) Optimal control of switching surfaces in hybrid dynamical systems. Discrete Event Dyn Syst 15(4):433–448

    Article  MathSciNet  Google Scholar 

  • Branicky MS, Borkar VS, Mitter S (1998) A unified framework for hybrid control theory: model and optimal control theory. IEEE Trans Auto Control 43(1):31–45

    Article  MathSciNet  MATH  Google Scholar 

  • Brockett RW (1993) Hybrid models for motion description control systems. In: Trentelman HL, Willems JC (eds) Essays on control: perspectives in the theory and its applications. Birkhäuser

  • Bryson AE, Ho YC (1975) Applied optimal control. Hemisphere

  • Egerstedt M, Wardi Y, Delmotte F (2003) Optimal control of switching times in switched dynamical systems. In: Proc. 42nd conference on decision and control. Maui, HI, pp 2138–2143

  • Feld’baum AA (1960–1961) Dual control theory. I–IV. Automation Remote Control 21, 22:874–880, 1033–1039, 112, 109–121

    MathSciNet  Google Scholar 

  • Freda F, Oriolo G (2007) Vision-based interception of a moving target with a nonholonomic mobile robot. Robot Auton Syst 55:419–432

    Article  Google Scholar 

  • Ghose K, Horiuchi TK, Krishnaprasad PS, Moss CF (2006) Echolocating bats use a nearly time-optimal strategy to intercept prey. PLoS Biol 4(5):865–873, e108. doi:10.1371/journal.pbio.0040108

    Article  Google Scholar 

  • Gratzer GA (1971) Lattice theory, Freeman

  • Heggie D, Hut P (2003) The gravitational million-body problem. Cambridge University Press

  • Jacobson N (1985) Basic algebra I, 2nd edn. Freeman and Comp

  • Kelly HJ (1962) Guidance theory and extremal fields. IRE Transactions on Automatic Control 7(5):75–82

    Article  Google Scholar 

  • Pesch HJ (1989) Real-time computation of feedback controls for constrained optimal control problems. Part 1: neighboring extremals. Optim Control Appl Methods 10:129–145

    Article  MathSciNet  MATH  Google Scholar 

  • Petreczky P, van Schuppen JH (2010) Realization theory for hybrid systems. IEEE Trans Auto Control 55(10):2282–2297

    Article  Google Scholar 

  • Polderman JW, Willems JC (1998) Introducton to mathematical systems theory. A behavioral approach. Springer

  • Šiljak DD (1978) Large scale dynamic systems. Dover

  • Suluh A, Sugar T, McBeath M (2001) Spatial navigation principles: applications to mobile robotics. In: Proceedings of the 2001 IEEE int’l conf. on robotics and automation. Seoul, Korea, pp 1689–1694

  • Sussmann H (1999) A maximum principle for hybrid optimal control problems. In: Proc. 38th conference on decision and control. Phoenix, AZ, pp 425–430

  • van der Schaft A, Schumacher H (2000) An introduction to hybrid dynamical systems. Lecture Notes in Control and Information Sciences, vol 251. Springer

  • Verriest EI (1992) Logic, geometry, and algebra in modeling. In: Proceedings of the 2nd IFAC workshop on algebraic-geometric methods in system theory. Prague, CZ, pp FP.1–FP.4

  • Verriest EI (2003) Regularization method for optimally switched and impulsive systems with biomedical applications. In: Proceedings of the 42th IEEE conference on decision and control. Maui, HI, pp 2156–2161

  • Verriest EI (2005) Optimal control for switched distributed delay systems with refractory period. In: Proceedings of the 44th IEEE conference on decision and control. Sevilla, Spain, pp 374–379

  • Verriest EI (2006) Multi-mode multi-dimensional systems. In: Proceedings of the 17th international symposium on mathematical theory of networks and systems. Kyoto, Japan, pp 1268–1274

  • Verriest EI (2009a) Multi-mode multi-dimensional systems with application to switched systems with delay. In: Proceedings of the 48th conference on decision and control and 28th Chinese control conference. Shangai, People’s Republic of Chian, pp 3958–3963

  • Verriest EI (2009b) Multi-mode multi-dimensional systems with poissonian sequencing. The Brockett Legacy Issue of Communications in Information and Systems, vol 9(1), pp 77–102

  • Verriest EI (2010) Multi-mode, multi-dimensional systems: structure and optimimal control. In: Proceedings of the 49-th IEEE conference on decision and control, Atlanta, GA, pp 7021–7026

  • Verriest EI, Gray WS (2006) Geometry and topology of the state space via balancing. In: Proceedings of the 17th international symposium on mathematical theory of networks and systems. Kyoto, Japan, pp 840–848

  • Xu X, Antsaklis P (2002) Optimal Control of Switched Autonomous Systems. In: Proc. 41st conference on decision and control. Las Vegas, NV, pp 4401–4406

  • Yeung D, Verriest EI (2006) A stochastic approach to optimal switching between control and observation. In: Proceedings of the 45-th IEEE conference on decision and control. San Diego, CA, pp 2655–2660

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  1. School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0250, USA

    Erik I. Verriest

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  1. Erik I. Verriest
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Verriest, E.I. Pseudo-continuous multi-dimensional multi-mode systems. Discrete Event Dyn Syst 22, 27–59 (2012). https://doi.org/10.1007/s10626-011-0113-z

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  • DOI: https://doi.org/10.1007/s10626-011-0113-z

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