n-observability for linear systems

The
n
-observability problem is considered for linear dynamical systems. The discussion of the problem is divided into two sections: the first covers linear nonstationary continuous time systems and the second covers discrete time systems. A system is called
n
-observable at time
t'
, if the state can be reconstructed from n samples of the output history and from the input over this time interval. The first result gives a necessary and sufficient condition for a linear nonstationary continuous system to be
n
-observable at time
t'
. When this condition is satisfied, a formula is given by which the state at
t'
can be calculated from a knowledge of the system's output at n points in its history and from the system's input over this period of history. Although the theorem does provide a test for
n
-observability at
t'
, this test is not easily applied. To remedy this, further results relate this test to a more easily applied test for observability. These theorems connect the necessary and sufficient condition for
n
- observability with Kalman's necessary and sufficient condition for observability. The corresponding results for linear stationary continuous systems are given. For discrete time systems, the notation is purposely kept the same as for continuous time systems. In this way, the similarity of the relations of discrete to continuous systems is more obvious. The
n
-observability problem for discrete nonstationary systems is more obvious. The
n
-observability problem for discrete non-stationary systems is discussed, and a necessary and sufficient condition for
n
-observability is derived. Again, the corresponding results for stationary systems are given. Finally, the use of state reconstruction in the synthesis of feedback controllers is discussed. This discussion is limited to linear stationary continuous time system. An example is included to clarify the details.