Abstract
The Ackermann's formula of pole placement for controllable linear time invariant (LTI) systems is extended to multi input LTI systems by employing generalized inversion of the system's controllability matrix instead of square inversion in the procedure of deriving the formula. The nullspace of the controllability matrix is affinely and redundantly parameterized in the modified Ackermann's formula by means of the Greville formula for solutions of consistent over-determined linear systems of algebraic equations. The redundant nullspace parameterizing variables are constrained such that the closed loop system matrix satisfies its characteristic equation. The infinite number of solutions for the multi input gain matrix are obtained explicitly in terms of the constrained nullspace parameterizing variables. Two examples are provided to illustrate the procedure.
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