Adaptive Control, Filtering, and Signal Processing by Karl J. ÅStröM (auth.), K. J. Åström, G. C. Goodwin, P. R. PDF

By Karl J. ÅStröM (auth.), K. J. Åström, G. C. Goodwin, P. R. Kumar (eds.)

The region of adaptive structures, which encompasses recursive identity, adaptive keep an eye on, filtering, and sign processing, has been probably the most energetic components of the earlier decade. given that adaptive controllers are essentially nonlinear controllers that are utilized to nominally linear, potentially stochastic and time-varying platforms, their theoretical research is generally very tough. however, over the last decade a lot primary development has been made on a few key questions referring to their balance, convergence, functionality, and robustness. in addition, adaptive controllers were effectively hired in different useful functions, and feature even entered the marketplace.

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If the system is unstable (0' > 0) and if the initial condition is too large in magnitude there will not be any limit cycles, because the relay feedback is unable to keep Ixlfrom growing. Since the system is of first order, we have automatically W O. This means that the limit cycle is locally stable. Since W 0, convergence to the limit cycle will occur after one switch in the linear range. If the initial condition is such that Ix(O)\ < 1/0' then the limit cycle will be obtained after one switch .

0 ~; 24 KARL J . ASTROM 6. Conclusions. Relay feedback has played a role in adaptive control for a long time. Lately it has been used for automatic tuning and for initialization of adaptive cont rollers, see [3] and [17]. In this paper we have analyzed limit cycles and their stability in simple cases. Particular em phasis has been given to symmetric limit cycles. It should be observed that linear systems with relay feedback can exhibit very com plicated behavior. , shown that a second order oscillatory system with time delay can have different stable limit cycles depending on th e initial conditions.

To mod el some per turbations or inaccuracies in these models it is ofte n reasonable to introduce white noise in th ese systems to obtain linear stochastic distributed paramete r systems . For controlled linear distributed parameter systems it is oft en nat ur al to consider that th e cont rol occurs on the boundary or at dis cret e points becaus e in many applica tions it is impract ical or not feasible to a pply th e control throughout th e domain . Sin ce in many cont rol situations th ere are unknown par amet ers in th ese linear , stochas ti c distributed param et er systems, it is necessary to solve a stochastic adaptive control pr oblem .

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