TY - JOUR
T1 - Nice reachability for planar bilinear control systems with applications to planar linear switched systems
AU - Margaliot, Michael
AU - Branicky, Michael S.
PY - 2009
Y1 - 2009
N2 - We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a "nice" control, specifically, a control that is a concatenation of a bang arc with either 1) a bang-bang control that is periodic after the third switch; or 2) a piecewise constant control with no more than two discontinuities. Under the additional assumption that the bilinear system is positive (or invariant for any proper cone), we show that the reachable set is spanned by a concatenation of a bang arc with either 1) a bang-bang control with no more than two discontinuities; or 2) a piecewise constant control with no more than two discontinuities. In particular, any point in the reachable set can be reached using a piecewise-constant control with no more than three discontinuities. Several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result. We demonstrate this with an example.
AB - We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a "nice" control, specifically, a control that is a concatenation of a bang arc with either 1) a bang-bang control that is periodic after the third switch; or 2) a piecewise constant control with no more than two discontinuities. Under the additional assumption that the bilinear system is positive (or invariant for any proper cone), we show that the reachable set is spanned by a concatenation of a bang arc with either 1) a bang-bang control with no more than two discontinuities; or 2) a piecewise constant control with no more than two discontinuities. In particular, any point in the reachable set can be reached using a piecewise-constant control with no more than three discontinuities. Several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result. We demonstrate this with an example.
KW - Lie algebra
KW - Lie brackets
KW - Maximum principle
KW - Metzler matrices
KW - Optimal control
KW - Positive linear systems
KW - Stability under arbitrary switching
UR - http://www.scopus.com/inward/record.url?scp=67649607702&partnerID=8YFLogxK
U2 - 10.1109/TAC.2009.2022905
DO - 10.1109/TAC.2009.2022905
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AN - SCOPUS:67649607702
SN - 0018-9286
VL - 54
SP - 1430
EP - 1435
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 6
ER -