Pole-by-Pole Shifting via a Linear-Quadratic Regulation

The linear-quadratic regulator and pole placement techniques are considered for designing continuous-time multivariable control systems. The proposed method combines the two approaches in a particular manner. The weighting matrices for the linear-quadratic optimization are constructed corresponding to a set of prescribed eigenvalues. In fact, a single eigenvalue (or a pair of complex conjugate eigenvalues) can be shifted at a time, leaving the remaining eigenvalues at their original positions. The simultaneous knowledge of the weights and the associated closed-loop eigenvalues provides the designer with the opportunity of interaction in both directions. Thereby eigenvalues located in undesired positions can be shifted to more suitable ones. The area into which each eigenvalue can be shifted is described in detail. The allowable shifts result in a faster and dampening feedback.