Discrete Event Dynamic Systems

, Volume 22, Issue 1, pp 27–59 | Cite as

Pseudo-continuous multi-dimensional multi-mode systems

Behavior, structure and optimal control
  • Erik I. Verriest


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).


Hybrid systems Switched systems Optimal control Structure Canonical form Parameterization 


  1. Aboufadel E (1996) A Mathematician catches a baseball. Am Math Mon 103(10):870–878MathSciNetMATHCrossRefGoogle Scholar
  2. Azhmyakov V, Boltyanski VG, Poznyak A (2008) Optimal control of impulsive hybrid systems. Nonlinear Analysis: Hybrid Systems 2:1089–1097MathSciNetMATHCrossRefGoogle Scholar
  3. 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 3617–3623Google Scholar
  4. Boccadero M, Wardi Y, Egerstedt M, Verriest EI (2005) Optimal control of switching surfaces in hybrid dynamical systems. Discrete Event Dyn Syst 15(4):433–448MathSciNetCrossRefGoogle Scholar
  5. 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):31–45MathSciNetMATHCrossRefGoogle Scholar
  6. 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. BirkhäuserGoogle Scholar
  7. Bryson AE, Ho YC (1975) Applied optimal control. HemisphereGoogle Scholar
  8. 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 2138–2143Google Scholar
  9. Feld’baum AA (1960–1961) Dual control theory. I–IV. Automation Remote Control 21, 22:874–880, 1033–1039, 112, 109–121MathSciNetGoogle Scholar
  10. Freda F, Oriolo G (2007) Vision-based interception of a moving target with a nonholonomic mobile robot. Robot Auton Syst 55:419–432CrossRefGoogle Scholar
  11. 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 CrossRefGoogle Scholar
  12. Gratzer GA (1971) Lattice theory, FreemanGoogle Scholar
  13. Heggie D, Hut P (2003) The gravitational million-body problem. Cambridge University PressGoogle Scholar
  14. Jacobson N (1985) Basic algebra I, 2nd edn. Freeman and CompGoogle Scholar
  15. Kelly HJ (1962) Guidance theory and extremal fields. IRE Transactions on Automatic Control 7(5):75–82CrossRefGoogle Scholar
  16. Pesch HJ (1989) Real-time computation of feedback controls for constrained optimal control problems. Part 1: neighboring extremals. Optim Control Appl Methods 10:129–145MathSciNetMATHCrossRefGoogle Scholar
  17. Petreczky P, van Schuppen JH (2010) Realization theory for hybrid systems. IEEE Trans Auto Control 55(10):2282–2297CrossRefGoogle Scholar
  18. Polderman JW, Willems JC (1998) Introducton to mathematical systems theory. A behavioral approach. SpringerGoogle Scholar
  19. Šiljak DD (1978) Large scale dynamic systems. DoverGoogle Scholar
  20. 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 1689–1694Google Scholar
  21. Sussmann H (1999) A maximum principle for hybrid optimal control problems. In: Proc. 38th conference on decision and control. Phoenix, AZ, pp 425–430Google Scholar
  22. van der Schaft A, Schumacher H (2000) An introduction to hybrid dynamical systems. Lecture Notes in Control and Information Sciences, vol 251. SpringerGoogle Scholar
  23. 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.4Google Scholar
  24. 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 2156–2161Google Scholar
  25. 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 374–379Google Scholar
  26. 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 1268–1274Google Scholar
  27. 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 3958–3963Google Scholar
  28. 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 77–102Google Scholar
  29. 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 7021–7026Google Scholar
  30. 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 840–848Google Scholar
  31. Xu X, Antsaklis P (2002) Optimal Control of Switched Autonomous Systems. In: Proc. 41st conference on decision and control. Las Vegas, NV, pp 4401–4406Google Scholar
  32. 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 2655–2660Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.School of Electrical and Computer EngineeringGeorgia Institute of TechnologyAtlantaUSA

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