Polstellungen

Polstellungen refers to the method of pole placement in control theory, the design task of placing the closed-loop poles of a linear time-invariant system at specified locations in the complex plane. The goal is to shape the system’s dynamic response—such as rise time, settling time and damping—by choosing a suitable feedback or compensation that alters the system’s eigenstructure.

In a typical state-space description x' = Ax + Bu, y = Cx + Du, applying state feedback u = -Kx yields the closed-loop dynamics x' = (A - BK)x. The poles of the closed-loop transfer function are the eigenvalues of A - BK. By selecting K appropriately, engineers aim to position these poles at desired locations, often specified by a target characteristic polynomial. This approach is especially powerful because it directly targets transient performance.

Feasibility of Polstellungen depends on controllability. A system is controllable if the controllability matrix [B, AB, ..., A^{n-1}B] has full rank. If controllable, it is possible in principle to place the poles at any set of locations (within physical and numerical limits). For single-input systems, explicit methods such as Ackermann’s formula provide a constructive way to compute K from the desired polynomial. In multi-input cases, generalizations and numerical techniques—often involving optimization or convex programming—are used.

Limitations include model uncertainty, actuator constraints, and robustness to disturbances. Polstellungen assumes an accurate plant model and may not guarantee performance under nonlinearity, time-variance, or significant delays. Nevertheless, pole placement remains a foundational tool for achieving targeted dynamic behavior in linear control systems and is widely used in engineering practice.