Hostname: page-component-6bf8c574d5-xtvcr Total loading time: 0 Render date: 2025-02-21T12:24:59.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized solutions to hybrid dynamical systems

Published online by Cambridge University Press:  07 February 2008

Ricardo G. Sanfelice ,
Rafal Goebel  and
Andrew R. Teel
Ricardo G. Sanfelice
Affiliation:
Center for Control, Dynamical Systems, and Computation; Department of Electrical and Computer Engineering; University of California, Santa Barbara, CA 93106-9560, USA; [email protected]; [email protected]
Rafal Goebel
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195, USA; [email protected]
Andrew R. Teel
Affiliation:
Center for Control, Dynamical Systems, and Computation; Department of Electrical and Computer Engineering; University of California, Santa Barbara, CA 93106-9560, USA; [email protected]; [email protected]
Get access

Abstract

Several recent results in the area of robust asymptotic stability of hybrid systems show that the concept of a generalized solution to a hybrid system is suitable for the analysis and design of hybrid control systems. In this paper, we show that such generalized solutions are exactly the solutions that arise when measurement noise in the system is taken into account.

Keywords

Hybrid systemsgeneralized solutionsdifferential inclusionsdifference inclusionsrobust controlhybrid feedback
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 699 - 724
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.C. Arkin, Behavior Based Robotics. The MIT Press (1998).
J.-P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag (1984).
Aubin, J.-P., Lygeros, J., Quincampoix, M., Sastry, S.S. and Seube, N., Impulse differential inclusions: a viability approach to hybrid systems. IEEE Trans. Aut. Cont. 47 (2002) 2–20. CrossRef
A. Back, J. Guckenheimer and M. Myers, A dynamical simulation facility for hybrid systems, in Hybrid Systems, Lect. Notes Comput. Sci. 36 (1993) 255–267.
D.D. Bainov and P.S. Simeonov, Systems with Impulse Effect: Stability, Theory, and Applications. Ellis Horwood Limited (1989).
Beker, O., Hollot, C.V., Chait, Y. and Han, H., Fundamental properties of reset control systems. Automatica 40 (2004) 905–915. CrossRef
Boccadoro, M., Wardi, Y., Egerstedt, M. and Verriest, E., Optimal control of switching surfaces in hybrid dynamical systems. Discrete Event Dyn. Syst. 15 (2005) 433–448. CrossRef
M.S. Branicky, Studies in hybrid systems: Modeling, analysis, and control. Ph.D. dissertation, Dept. Elec. Eng. and Computer Sci., MIT (1995).
Branicky, M.S., Borkar, V.S. and Mitter, S.K., A unified framework for hybrid control: Model and optimal control theory. IEEE Trans. Aut. Cont. 43 (1998) 31–45. CrossRef
R.W. Brocket, Hybrid Models for Motion Control Systems, in Essays on Control: Perspectives in the Theory and its Applications, H.L. Trentelman and J.C. Willems Eds., Birkhäuser (1993) 29–53.
Broucke, M. and Arapostathis, A., Continuous selections of trajectories of hybrid systems. Systems Control Lett. 47 (2002) 149–157. CrossRef
C. Cai, A.R. Teel and R. Goebel, Converse Lyapunov theorems and robust asymptotic stability for hybrid systems, in Proc. 24th American Control Conference (2005) 12–17.
Ceragioli, F., Some remarks on stabilization by means of discontinuous feedbacks. Systems Control Lett. 45 (2002) 271–281. CrossRef
Chellaboina, V., Bhat, S.P. and Haddad, W.H., An invariance principle for nonlinear hybrid and impulsive dynamical systems. Nonlin. Anal. 53 (2003) 527–550. CrossRef
Clarke, F.H., Ledyaev, Y.S., Sontag, E.D. and Subbotin, A.I., Asymptotic controllability implies feedback stabilization. IEEE Trans. Aut. Cont. 42 (1997) 1394–1407. CrossRef
J.C. Clegg, A nonlinear integrator for servomechanisms. Transactions A.I.E.E. 77 (Part II) 41–42, 1958.
P. Collins, A trajectory-space approach to hybrid systems, in Proc. 16th MTNS (2004).
P. Collins and J. Lygeros, Computability of finite-time reachable sets for hybrid systems, in Proc. 44th IEEE Conference on Decision Control (2005) 4688–4693.
Coron, J.-M. and Rosier, L., A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Systems Estimation Control 4 (1994) 67–84.
M. Egerstedt, Behavior based robotics using hybrid automata, in Hybrid Systems: Computation and Control, Lect. Notes Comput. Sci. 1790 (2000) 103–116.
Filippov, A.F., Differential equations with discontinuous right-hand sides (English). Matemat. Sbornik. 151 (1960) 99–128.
Garavello, M. and Piccoli, B., Hybrid necessary principle. SIAM J. Control Optim. 43 (2005) 1867–1887. CrossRef
Goebel, R. and Teel, A.R., Solutions to hybrid inclusions via set and graphical convergence with stability theory applications. Automatica 42 (2006) 573–587. CrossRef
R. Goebel, J.P. Hespanha, A.R. Teel, C. Cai and R.G. Sanfelice, Hybrid systems: Generalized solutions and robust stability, in Proc. 6th IFAC Symposium in Nonlinear Control Systems (2004) 1–12.
Hàjek, O., Discontinuous differential equations I. J. Diff. Eqn. 32 (1979) 149–170. CrossRef
H. Hermes, Discontinuous vector fields and feedback control, in Differential Equations and Dynamical Systems, J.K. Hale and J.P. LaSalle Eds., Academic Press, New York (1967) 155–165.
Hespanha, J.P., Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle. IEEE Trans. Aut. Cont. 49 (2004) 470–482. CrossRef
Hespanha, J.P., A model for stochastic hybrid systems with application to communication networks. Nonlinear Anal. (Special Issue on Hybrid Systems) 62 (2005) 1353–1383.
Kellet, C.M. and Teel, A.R., Smooth Lyapunov functions and robustness of stability for differential inclusions. Systems Control Lett. 52 (2004) 395–405. CrossRef
N.N. Krasovskii, Game-Theoretic Problems of Capture. Nauka, Moscow (1970).
N.N. Krasovskii and A.I. Subbotin, Game-Theoretical Control Problems. Springer-Verlag (1988).
Krishnan, K.R. and Horowitz, I.M., Synthesis of a non-linear feedback system with significant plant-ignorance for prescribed system tolerances. Inter. J. Control 19 (1974) 689–706. CrossRef
J. Lygeros, K.H. Johansson, S.S. Sastry and M. Egerstedt, On the existence of executions of hybrid automata, in Proc. 41st Conference on Decision and Control (1999) 2249–2254.
Lygeros, J., Johansson, K.H., Simić, S.N., Zhang, J. and Sastry, S.S., Dynamical properties of hybrid automata. IEEE Trans. Aut. Cont. 48 (2003) 2–17. CrossRef
A.N. Michel, L. Wang and B. Hu, Qualitative Theory of Dynamical Systems. Dekker (2001).
D. Nesic, L. Zaccarian and A.R. Teel, Stability properties of reset systems, in Proc. 16th IFAC World Congress in Prague (2005).
Prieur, C., Asymptotic controllability and robust asymptotic stabilizability. SIAM J. Control Optim. 43 (2005) 1888–1912. CrossRef
C. Prieur, R. Goebel and A.R. Teel, Results on robust stabilization of asymptotically controllable systems by hybrid feedback, in Proc. 44th IEEE Conference on Decision and Control and European Control Conference (2005) 2598–2603.
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer (1998).
Roup, A.V., Bernstein, D.S., Nersesov, S.G., Haddad, W.M. and Chellaboina, V., Limit cycle analysis of the verge and foliot clock escapement using impulsive differential equations and Poincaré maps. Inter. J. Control 76 (2003) 1685–1698. CrossRef
R.G. Sanfelice, R. Goebel and A.R. Teel, Results on convergence in hybrid systems via detectability and an invariance principle, in Proc. 24th IEEE American Control Conference (2005) 551–556.
J. Sprinkle, A.D. Ames, A. Pinto, H. Zheng and S.S. Sastry, On the partitioning of syntax and semantics for hybrid systems tools, in Proc. 44th IEEE Conference on Decision and Control and European Control Conference (2005).
Tang, S.Y. and Cheke, R.A., State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. 50 (2005) 257–292. CrossRef
Tavernini, L., Differential automata and their discrete simulators. Nonlin. Anal. 11 (1987) 665–683. CrossRef
Tavernini, L., Generic properties of impulsive hybrid systems. Dynamic Systems & Applications 13 (2004) 533–551.
S.E. Tuna, R.G. Sanfelice, M.J. Messina and A.R. Teel, Hybrid MPC: Open-minded but not easily swayed, in Proc. International Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control (2005).
A. van der Schaft and H. Schumacher, An Introduction to Hybrid Dynamical Systems, Lecture Notes in Control and Information Sciences. Springer (2000).
Witsenhausen, H.S., A class of hybrid-state continuous-time dynamic systems. IEEE Trans. Aut. Cont. 11 (1966) 161–167. CrossRef
L. Zaccarian, D. Nesic and A.R. Teel, First order reset elements and the Clegg integrator revisited, in Proc. 24th American Control Conference (2005) 563–568.