Definition and Properties

The controllability and observability properties of a linear time-invariant system can be heuristically described as follows. The system dynamics described by the state variable (x) is excited by the input (u) and measured by the output (y). However, the input may not be able to excite all states (or, equivalently, to move them in an arbitrary direction). In this case we cannot fully control the system. Also, not all states may be represented at the output (or, equivalently, the system states cannot be recovered from a record of the output measurements). In this case we cannot fully observe the system. However, if the input excites all states, the system is controllable, and if all the states are represented in the output, the system is observable. More precise definitions follow.

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