Exact Multi-Input Pole Placement by Linear-Quadratic Synthesis

A method is developed that computes the control gain matrix, u=-C x, of a controllable multi-input, linear aystem, x = A x + B u, required to place the poles (eigenvalues) at arbitrary locations. The derivation of the method is based upon defining a quadratic performance index that is identically zero when the closed loop system has the desired eigenvalues. The numerical computations involve solving a set of n(n-m) linear algebraic equations where n is the number of states and m is the number of controls. For m ≫ 1, the C matrix is not unique and alternative gain matrices yaielding the desired n eigenvalues can be computed. The method is illustrated by both simDle analyt4cal and higher order numerical eramples. A robust computational program called POLESYS is described and applied to several examples.