Pole placement using dynamic compensators

The problem of designing a compensator to obtain arbitrary pole placement in the system consisting of the plant and compensator in cascade is considered. The design uses only those state variables which can be measured. It is shown that for a controllable observable plant a compensator of order
\beta = \min(\nu_{c} - 1, \nu_{o} - 1)
is sufficient to achieve this result. Here
\nu_{c}(\nu_{o})
is the controllability (observability) index of the plant. This result is obtained by first showing that any multi-input multi-output linear time-invariant system may be made controllable (observable) from a single input (output) using only output feedback. The main result is then proved in a constructive manner which explicitly relates the compensator parameters to the coefficients of the desired characteristic polynomial.