Static output feedback pole placement via parameter-dependent Ackermann and Greville matrix formulae

The paper generalizes Ackermann's pole placement methodology to static output feedback pole placement (SOFPP) for general MIMO continuous LTI systems. The main tool used is Moore-Penrose generalized inversion of Kalman controllability matrix, which replaces square inversion that is performed in the original Ackermann's formulation for single input controllable systems. Additionally, affine and partially explicit parameterization of the accompanying nonempty controllability matrix nullspace is utilized to characterize solution non uniqueness of the MIMO SOFPP control problem and to solve for SOFPP feedback control gains. The proposed nullspace parameterization is inspired by the vector Greville formula for general solutions of linear algebraic equations, and it extends the formula to solve parameter-dependent (PD) linear matrix equations. The redundant nullspace parameterizing variables in the PD matrix Greville formula are constrained such that the closed loop system matrix satisfies its characteristic equation. Necessary and sufficient conditions for SOFPP solution existence are derived for general (possibly uncontrollable) continuous LTI systems, regardless of dimensionalities and ranks of system triplet and controllability matrices, and regardless of spectral multiplicities of the open loop system matrix and the sought closed loop system matrix. If an SOFPP control problem is solvable then all its control matrix gain solutions are synthesized partially explicitly via affine parameterization in terms of the constrained nullspace parameterizing variables. Five examples are provided to illustrate the proposed SOFPP control system analysis and design methodology.