Abstract
This study firstly considers the exponential stability of unforced linear systems of slowly time-varying dynamics. Possible switchings of the system structure to unstable dynamics during certain finite time intervals are admitted. The maintenance of global exponential stability does not necessarily require at most a finite number of switchings in the dynamics while infinitely many switches can also lead to stability. The mechanism to achieve stability under infinitely many switches in the dynamics is to maintain the system in the stable region during time intervals of sufficient large length without switches provided that the system dynamics evolves at a sufficiently small rate with time. Special attention is paid to the robust tolerance for a class of state disturbances and to the case of time-varying matrix of dynamics that possess either piecewise constant or constant eigenvalues. The obtained results can be relevant for their use in stability issues for the cases of multimodel non- adaptive and adaptive control with improved transient performances.
Highlights
It is well known that unforced piecewise - constant linear systems, whose associated matrix of the dynamics takes values in a set of strictly Hurwitzian matrices are not guaranteed to be exponentially stable[1,2,3]
The problem of switching operations between configurations of piecewise continuous stable dynamics is of growing interest in multimodel design with improved transient performances
The related problem of time- varying dynamics of piecewise constant eigenvalues are of relevant interest in adaptive control
Summary
It is well known that unforced piecewise - constant linear systems, whose associated matrix of the dynamics takes values in a set of strictly Hurwitzian matrices are not guaranteed to be exponentially stable[1,2,3]. The system has proven to be robustly stable in terms of ultimate boundedness against a class of non- linear state dependent disturbances when the unforced dynamics have associated stable piecewise constant eigenvalues during certain stabilization time- intervals of sufficiently large lengths.
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