Bursting phenomena in continuous-time adaptive systems with a \(\sigma\)- modification.

*(English)*Zbl 0614.93042In order to improve robustness of the continuous-time adaptive systems the so called \(\sigma\)-modification has been proposed recently. The present paper shows that this modification can introduce some sudden intermittent output error ”bursts” followed by a long period of ”smooth” system behaviour thus contradicting some results appearing in the literature.

In the regulation problem the equilibrium points are quite simple to be determined and only two possibilities can arise: (1) the unique equilibrium point is the trivial one; (2) the system presents three equilibrium points, one being the trivial one. The analysis of the variational equations around each equilibrium shows that for sufficiently small \(\sigma\) it is possible to have plants in which all equilibrium points are unstable. On the other hand it is known that a compact residual set for all system trajectories exists and must contain the equilibrium set. Thus, when all equilibria are unstable, sustained oscillations will occur within this residual set, suggesting in this case, that bursting phenomena can occur.

Some simulations are carried out and confirm this phenomenum. Moreover, it was observed that under sufficiently rich and strong excitations the error was ultimately bounded by a value that decreased to zero with \(\sigma\).

In the regulation problem the equilibrium points are quite simple to be determined and only two possibilities can arise: (1) the unique equilibrium point is the trivial one; (2) the system presents three equilibrium points, one being the trivial one. The analysis of the variational equations around each equilibrium shows that for sufficiently small \(\sigma\) it is possible to have plants in which all equilibrium points are unstable. On the other hand it is known that a compact residual set for all system trajectories exists and must contain the equilibrium set. Thus, when all equilibria are unstable, sustained oscillations will occur within this residual set, suggesting in this case, that bursting phenomena can occur.

Some simulations are carried out and confirm this phenomenum. Moreover, it was observed that under sufficiently rich and strong excitations the error was ultimately bounded by a value that decreased to zero with \(\sigma\).

##### MSC:

93C40 | Adaptive control/observation systems |

93B35 | Sensitivity (robustness) |

93D99 | Stability of control systems |

37C75 | Stability theory for smooth dynamical systems |