TY - JOUR
T1 - Novel Gramians for linear semistable systems
AU - Cheng, Xiaodong
AU - Scherpen, Jacquelien M.A.
PY - 2020/5
Y1 - 2020/5
N2 - In this paper, the notions of pseudo Gramians are introduced for linear time-invariant semistable systems, which allow multiple semisimple poles at the origin. The proposed Gramians are the generalizations of standard Gramian matrices defined for asymptotically stable systems, and they can be computed by a set of Lyapunov equations. Furthermore, it is shown that the controllability and observability of a semistable system are indicated by the ranks of the pseudo Gramians, and the controllability and observability energy functions are also characterized using the pseudo Gramians. Additionally, the H2-norm and H∞-norm of a semistable system are analyzed, and then the results are used for the model reduction of semistable systems. Finally, the effectiveness of the methods is illustrated by an example of a gene regulation network.
AB - In this paper, the notions of pseudo Gramians are introduced for linear time-invariant semistable systems, which allow multiple semisimple poles at the origin. The proposed Gramians are the generalizations of standard Gramian matrices defined for asymptotically stable systems, and they can be computed by a set of Lyapunov equations. Furthermore, it is shown that the controllability and observability of a semistable system are indicated by the ranks of the pseudo Gramians, and the controllability and observability energy functions are also characterized using the pseudo Gramians. Additionally, the H2-norm and H∞-norm of a semistable system are analyzed, and then the results are used for the model reduction of semistable systems. Finally, the effectiveness of the methods is illustrated by an example of a gene regulation network.
KW - Controllability and observability
KW - Gramian matrix
KW - Linear system
KW - Model reduction
KW - Semistability
UR - http://www.scopus.com/inward/record.url?scp=85080023884&partnerID=8YFLogxK
U2 - 10.1016/j.automatica.2020.108911
DO - 10.1016/j.automatica.2020.108911
M3 - Article
AN - SCOPUS:85080023884
SN - 0005-1098
VL - 115
JO - Automatica
JF - Automatica
M1 - 108911
ER -