Temporal Linear System Structure

Piecewise constant rank systems and the differential Kalman decomposition are introduced in this note. Together these enable the detection of temporal uncontrollability/unreconstructability of linear continuous-time systems. These temporal properties are not detected by any of the four conventional Kalman decompositions. Moreover piecewise constant rank systems admit the state dimension to be time-variable. As demonstrated in this note linear continuous-time systems with variable state dimensions enable the well rounded realization theory suggested already by Kalman. The differential Kalman decomposition introduced in this note is associated with differential controllability and differential reconstructability. The system structure obtained from the differential Kalman decomposition may be interpreted as the temporal linear system structure. This note reveals that the difference between controllability and reachability as well as reconstructability and observability is entirely due to changes of the temporal linear system structure. Also, this note reveals how the differential Kalman decomposition relates to the conventional ones.

[1]  Willem L. De Koning,et al.  Minimal and non-minimal optimal fixed-order compensators for time-varying discrete-time systems , 2002, Autom..

[2]  G. Hostetter,et al.  Mathematical description of linear systems [Book reviews] , 1980, IEEE Transactions on Automatic Control.

[3]  R. Kálmán.  Mathematical description of linear dynamical systems , 1963 .

[4]  R. E. Kalman,et al.  On the Computation of the Reachable/Observable Canonical Form , 1982 .

[5]  Long Wang,et al.  On Controllability of Switched Linear Systems , 2002, IEEE Transactions on Automatic Control.

[6]  M. A. Athans,et al.  The role and use of the stochastic linear-quadratic-Gaussian problem in control system design , 1971 .

[7]  Michael A. Arbib,et al.  Topics in Mathematical System Theory , 1969 .

[8]  W. L. Koning,et al.  A Kalman decomposition to detect temporal linear system srtucture , 2007 .

[9]  Daniel Boley,et al.  Computing the Kalman decomposition: An optimal method , 1984 .

[10]  L. G. van Willigenburg,et al.  First order controllability and the time optimal control problem for rigid articulated arm robots with friction , 1990 .

[11]  L. Silverman,et al.  Equivalent Realizations of Linear Systems , 1969 .

[12]  Zoltán Szabó,et al.  Necessary and sufficient condition for the controllability of switching linear hybrid systems , 2004, Autom..

[13]  M. A. Kaashoek,et al.  Minimality and realization of discrete time-varying systems , 1992 .

[14]  Henrik Sandberg,et al.  Balanced truncation of linear time-varying systems , 2004, IEEE Transactions on Automatic Control.

[15]  T. Kailath,et al.  On generalized balanced realizations , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[16]  R. E. Kalman,et al.  Contributions to linear system theory , 1965 .

[17]  M. Athans.  The Role and Use of the Stochastic Linear-Quadratic-Gaussian Problem in Control System , 1971 .