PolePlacement

Pole placement, also known as eigenstructure assignment, is a method in control theory for assigning the eigenvalues (poles) of a closed-loop system by state feedback. By selecting where the closed-loop poles reside in the complex plane, designers can influence stability, rise time, damping, and overall transient performance.

For a continuous-time linear time-invariant system ẋ = Ax + Bu, y = Cx, applying state feedback u = −Kx yields the closed-loop dynamics ẋ = (A − BK)x. The eigenvalues of A − BK are the closed-loop poles. If the pair (A, B) is controllable, it is possible to place the poles at desired locations (within practical limits) through an appropriate choice of K.

A common practical method is Ackermann’s formula, which provides an explicit way to compute K when the system is controllable. This involves forming the controllability matrix Wc = [B AB A^2B … A^{n−1}B] and selecting a desired characteristic polynomial p_d(s) = s^n + α_{n−1}s^{n−1} + … + α_0. Then K is derived from Wc and the polynomial in A corresponding to p_d(s). In many applications, numerical routines implemented in software tools (for example, MATLAB’s place or acker functions) are used to perform the computation.

Limitations include sensitivity to modeling errors and parameter variations, which can shift pole locations and degrade performance. If full-state feedback is not feasible, observers or dynamic compensators can be designed to realise pole placement in an output-feedback framework. Pole placement is widely used in aerospace, robotics, automotive control, and process control to meet specified transient requirements.