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Control and stabilization of nonholonomic dynamic systems. (English) Zbl 0778.93084
Summary: A theoretical framework is established for the control of nonholonomic dynamic systems, i.e., dynamic systems with nonintegrable constraints. In particular, we emphasize control properties for nonholonomic systems that have no counterpart in holonomic systems. A model for nonholonomic dynamic systems is first presented in terms of differential-algebraic equations defined on a phase space. A reduction procedure is carried out to obtain reduced-order state equations. Feedback is then used to obtain a nonlinear control system in a normal form. The assumptions guarantee that the resulting normal form equations necessarily contain a nontrivial drift vector field. Conditions for smooth $\left({C}^{\infty }\right)$ asymptotic stabilization to an $m$-dimensional equilibrium manifold are presented; we also demonstrate that a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback. However, any equilibrium is shown to be strongly accessible and small time locally controllable. Finally, an approach using geometric phases is developed as a basis for the control of Caplygin dynamical systems, i.e., nonholonomic systems with certain symmetry properties which can be expressed by the fact that the constraints are cyclic in certain variables. The theoretical developments is applied to physical examples of systems that we have studied in detail elsewhere: the control of a knife edge moving on a plane surface and the control of a wheel rolling without slipping on a plane surface. The results of the paper are also applied to the control of a planar multibody system using angular momentum preserving control inputs since the angular momentum may be viewed as a nonholonomic constraint which is an invariant of the motion.
##### MSC:
 93D05 Lyapunov and other classical stabilities of control systems 93B05 Controllability 93C15 Control systems governed by ODE