Orbit (control theory)

Orbit (control theory) The notion of orbit of a control system used in mathematical control theory is a particular case of the notion of orbit in group theory.[1][2][3] Inhalt 1 Definition 2 Orbit theorem (Nagano–Sussmann) 3 Logische Folge (Rashevsky–Chow theorem) 4 Siehe auch 5 Verweise 6 Further reading Definition Let {Anzeigestil { }{Punkt {q}}= f(q,u)} sei ein {Anzeigestil {mathematisch {C}}^{unendlich }} control system, wo {Anzeigestil { q}} belongs to a finite-dimensional manifold {Anzeigestil M} und {Anzeigestil u} belongs to a control set {Anzeigestil U} . Consider the family {Anzeigestil {mathematisch {F}}={f(cdot ,u)mid uin U}} and assume that every vector field in {Anzeigestil {mathematisch {F}}} is complete. Für jeden {Displaystyle Ende {mathematisch {F}}} and every real {Anzeigestil t} , denote by {Anzeigestil e^{tf}} der Fluss von {Anzeigestil f} at time {Anzeigestil t} .

The orbit of the control system {Anzeigestil { }{Punkt {q}}= f(q,u)} through a point {Anzeigestil q_{0}in m} is the subset {Anzeigestil {mathematisch {Ö}}_{q_{0}}} von {Anzeigestil M} definiert von {Anzeigestil {mathematisch {Ö}}_{q_{0}}={e^{t_{k}f_{k}}circ e^{t_{k-1}f_{k-1}}circ cdots circ e^{t_{1}f_{1}}(q_{0})mid kin mathbb {N} , t_{1},Punkte ,t_{k}in mathbb {R} , f_{1},Punkte ,f_{k}in {mathematisch {F}}}.} Remarks The difference between orbits and attainable sets is that, whereas for attainable sets only forward-in-time motions are allowed, both forward and backward motions are permitted for orbits. Im Speziellen, if the family {Anzeigestil {mathematisch {F}}} is symmetric (d.h., {Displaystyle Ende {mathematisch {F}}} dann und nur dann, wenn {displaystyle -fin {mathematisch {F}}} ), then orbits and attainable sets coincide.

The hypothesis that every vector field of {Anzeigestil {mathematisch {F}}} is complete simplifies the notations but can be dropped. In this case one has to replace flows of vector fields by local versions of them.

Orbit theorem (Nagano–Sussmann) Each orbit {Anzeigestil {mathematisch {Ö}}_{q_{0}}} is an immersed submanifold of {Anzeigestil M} .

The tangent space to the orbit {Anzeigestil {mathematisch {Ö}}_{q_{0}}} at a point {Anzeigestil q} is the linear subspace of {Anzeigestil T_{q}M} spanned by the vectors {Anzeigestil P_{*}f(q)} wo {Anzeigestil P_{*}f} denotes the pushforward of {Anzeigestil f} durch {Anzeigestil P} , {Anzeigestil f} belongs to {Anzeigestil {mathematisch {F}}} und {Anzeigestil P} is a diffeomorphism of {Anzeigestil M} des Formulars {Anzeigestil e^{t_{k}f_{k}}circ cdots circ e^{t_{1}f_{1}}} mit {Anzeigestil kin mathbb {N} , t_{1},Punkte ,t_{k}in mathbb {R} } und {Anzeigestil f_{1},Punkte ,f_{k}in {mathematisch {F}}} .

If all the vector fields of the family {Anzeigestil {mathematisch {F}}} are analytic, dann {Anzeigestil T_{q}{mathematisch {Ö}}_{q_{0}}=mathrm {Lie} _{q},{mathematisch {F}}} wo {Anzeigestil mathrm {Lie} _{q},{mathematisch {F}}} is the evaluation at {Anzeigestil q} of the Lie algebra generated by {Anzeigestil {mathematisch {F}}} with respect to the Lie bracket of vector fields. Andernfalls, the inclusion {Anzeigestil mathrm {Lie} _{q},{mathematisch {F}}subset T_{q}{mathematisch {Ö}}_{q_{0}}} holds true.

Logische Folge (Rashevsky–Chow theorem) Main article: Chow–Rashevskii theorem If {Anzeigestil mathrm {Lie} _{q},{mathematisch {F}}=T_{q}M} für jeden {displaystyle qin M} und wenn {Anzeigestil M} Ist verbunden, then each orbit is equal to the whole manifold {Anzeigestil M} .

See also Frobenius theorem (differential topology) References ^ Jurdjevic, Velimir (1997). Geometric control theory. Cambridge University Press. pp. xviii+492. ISBN 0-521-49502-4.[permanent dead link] ^ Sussmann, Héctor J.; Jurdjevic, Velimir (1972). "Controllability of nonlinear systems". J. Differentialgleichung. 12 (1): 95–116. doi:10.1016/0022-0396(72)90007-1. ^ Sussmann, Héctor J. (1973). "Orbits of families of vector fields and integrability of distributions". Trans. Amer. Mathematik. Soc. Amerikanische Mathematische Gesellschaft. 180: 171–188. doi:10.2307/1996660. JSTOR 1996660. Further reading Agrachev, Andrei; Sachkov, Juri (2004). "The Orbit Theorem and its Applications". Control Theory from the Geometric Viewpoint. Berlin: Springer. pp. 63–80. ISBN 3-540-21019-9. Kategorien: Control theory