Ultimate boundedness and asymptotic stability of a class of uncertain dynamical systems via continuous and discontinuous feedback control.

*(English)*Zbl 0662.93059Based on concepts from both the theory of variable structure systems and the theory of G. Leitmann and the second author [IEEE Trans. Autom. Control AC-26, 1139-1144 (1981; Zbl 0473.93056)], and G. Leitmann and B. R. Barmish [ibid. AC-27, 153-158 (1982; Zbl 0469.93043)], continuous and discontinuous feedback controls are developed which guarantee uniform ultimate boundedness of all motions of a class of imperfectly known dynamical systems with bounded uncertainty.

For a subclass of uncertain systems, the zero state can be (a) rendered ‘practically’ stable (in the sense that, given any neighbourhood of the zero state, there exists a control, in the proposed class of continuous feedback controls, which guarantees global uniform ultimate boundedness with respect to that neighbourhood), or (b) rendered globally uniformly asymptotically stable (in the sense of Lyapunov) by the proposed discontinuous feedback control.

The approach is illustrated by application to a Maglev suspension control system.

For a subclass of uncertain systems, the zero state can be (a) rendered ‘practically’ stable (in the sense that, given any neighbourhood of the zero state, there exists a control, in the proposed class of continuous feedback controls, which guarantees global uniform ultimate boundedness with respect to that neighbourhood), or (b) rendered globally uniformly asymptotically stable (in the sense of Lyapunov) by the proposed discontinuous feedback control.

The approach is illustrated by application to a Maglev suspension control system.

##### MSC:

93D20 | Asymptotic stability in control theory |

34D40 | Ultimate boundedness (MSC2000) |

93C10 | Nonlinear systems in control theory |

49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |

93C15 | Control/observation systems governed by ordinary differential equations |