A Kalman decomposition to detect temporal linear system srtucture

Feedback controllers for non-linear systems are often based on a linearized dynamic model. Such a linearized model may be temporarily uncontrollable and/or unreconstructable. This paper introduces the so-called differential Kalman decomposition of time-varying linear systems. It is based on differential controllability and differential reconstructability in conjunction with a linear time-varying continuous-time system description that allows the system structure and dimensions to change at certain time-instants. We show how these together enable the detection of what will be called temporal system structure. This structure among other things reveals the temporal loss of controllability and/or reconstructability. Moreover this paper shows how time-varying state-dimensions enable a satisfactory realization theory for time-varying linear systems and how our Kalman decomposition is linked to the conventional ones.