×

Pseudo-continuous multi-dimensional multi-mode systems: behavior, structure and optimal control. (English) Zbl 1242.93062

Summary: 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 a 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).

MSC:

93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93C35 Multivariable systems, multidimensional control systems
93B10 Canonical structure
93B25 Algebraic methods
49K15 Optimality conditions for problems involving ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aboufadel E (1996) A Mathematician catches a baseball. Am Math Mon 103(10):870–878 · Zbl 0892.00011 · doi:10.2307/2974611
[2] Azhmyakov V, Boltyanski VG, Poznyak A (2008) Optimal control of impulsive hybrid systems. Nonlinear Analysis: Hybrid Systems 2:1089–1097 · Zbl 1163.49038 · doi:10.1016/j.nahs.2008.09.003
[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 3617–3623
[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):433–448 · Zbl 1101.93054 · doi:10.1007/s10626-005-4060-4
[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):31–45 · Zbl 0951.93002 · doi:10.1109/9.654885
[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. Birkhäuser
[7] Bryson AE, Ho YC (1975) Applied optimal control. Hemisphere
[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 2138–2143
[9] Feld’baum AA (1960–1961) Dual control theory. I–IV. Automation Remote Control 21, 22:874–880, 1033–1039, 112, 109–121
[10] Freda F, Oriolo G (2007) Vision-based interception of a moving target with a nonholonomic mobile robot. Robot Auton Syst 55:419–432 · Zbl 05185188 · doi:10.1016/j.robot.2007.02.001
[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 · doi:10.1371/journal.pbio.0040108
[12] Gratzer GA (1971) Lattice theory, Freeman
[13] Heggie D, Hut P (2003) The gravitational million-body problem. Cambridge University Press · Zbl 1074.85001
[14] Jacobson N (1985) Basic algebra I, 2nd edn. Freeman and Comp · Zbl 0557.16001
[15] Kelly HJ (1962) Guidance theory and extremal fields. IRE Transactions on Automatic Control 7(5):75–82 · doi:10.1109/TAC.1962.1105503
[16] Pesch HJ (1989) Real-time computation of feedback controls for constrained optimal control problems. Part 1: neighboring extremals. Optim Control Appl Methods 10:129–145 · Zbl 0675.49023 · doi:10.1002/oca.4660100205
[17] Petreczky P, van Schuppen JH (2010) Realization theory for hybrid systems. IEEE Trans Auto Control 55(10):2282–2297 · Zbl 1368.93109 · doi:10.1109/TAC.2010.2044258
[18] Polderman JW, Willems JC (1998) Introducton to mathematical systems theory. A behavioral approach. Springer
[19] Šiljak DD (1978) Large scale dynamic systems. Dover · Zbl 0384.93002
[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 1689–1694
[21] Sussmann H (1999) A maximum principle for hybrid optimal control problems. In: Proc. 38th conference on decision and control. Phoenix, AZ, pp 425–430
[22] van der Schaft A, Schumacher H (2000) An introduction to hybrid dynamical systems. Lecture Notes in Control and Information Sciences, vol 251. Springer · Zbl 0940.93004
[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.4
[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 2156–2161
[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 374–379
[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 1268–1274
[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 3958–3963
[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 77–102 · Zbl 1194.93099
[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 7021–7026
[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 840–848
[31] Xu X, Antsaklis P (2002) Optimal Control of Switched Autonomous Systems. In: Proc. 41st conference on decision and control. Las Vegas, NV, pp 4401–4406
[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 2655–2660
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.