STABILITY OF SWITCHED SYSTEMS WITH AVERAGE

It is shown that switching among stable linear systems results in a stable system provided that switching is “slow-on-the-average.” In particular, it is proved that exponential stability is achieved when the number of switches in any finite interval grows linearly with the length of the interval, and the growth rate is sufficiently small. Moreover, the exponential stability is uniform over all switchings with the above property. For switched systems with inputs this guarantees that several input-to-state induced norms are bounded uniformly over all slow-on-the-average switchings. These results extend to classes of nonlinear switched systems that satisfy suitable uniformity assumptions. In this paper it is also shown that, in a supervisory control context, scale-independent hysteresis can produce switching that is slow-on-the-average and therefore the results mentioned above can be used to study the stability of hysteresis-based adaptive control systems.

[1]  D. Mayne,et al.  Design issues in adaptive control , 1988 .

[2]  A. Morse,et al.  Applications of hysteresis switching in parameter adaptive control , 1992 .

[3]  Edward J. Davison,et al.  Switching control of a family of plants , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[4]  Fuad Kassab,et al.  Parallel algorithms for adaptive control: Robust stability , 1996 .

[5]  A. Morse.  Supervisory control of families of linear set-point controllers Part I. Exact matching , 1996, IEEE Trans. Autom. Control..

[6]  P. Ramadge,et al.  Model and controller selection policies based on output prediction errors , 1996, IEEE Trans. Autom. Control..

[7]  Yuan Wang,et al.  Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability , 1996, Math. Control. Signals Syst..

[8]  A. Stephen Morse,et al.  Control Using Logic-Based Switching , 1997 .

[9]  A. Morse.  Supervisory control of families of linear set-point controllers. 2. Robustness , 1997, IEEE Trans. Autom. Control..

[10]  Kumpati S. Narendra,et al.  Adaptive control using multiple models , 1997, IEEE Trans. Autom. Control..

[11]  P. Hespanha,et al.  Supervisory field‐oriented control of induction motors with uncertain rotor resistance , 1998 .

[12]  J.P. Hespanha,et al.  Supervisory control of families of noise suppressing controllers , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[13]  Joao P. Hespanha,et al.  Logic-based switching algorithms in control , 1998 .

[14]  P. Ramadge,et al.  Controller switching based on output prediction errors , 1998, IEEE Trans. Autom. Control..

[15]  A. Morse,et al.  Certainty equivalence implies detectability , 1998 .

[16]  A. Morse,et al.  Stability of switched systems with average dwell-time , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[17]  J.P. Hespanha,et al.  Towards the supervisory control of uncertain nonholonomic systems , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).