Pole Placement by State Feedback
State feedback u = -K x can move a controllable system's closed-loop poles to any chosen locations, setting its dynamics by design.
Choosing the Poles Directly
Given a state-space model x' = A*x + B*u, apply the linear state feedback u = -K*x. The closed-loop dynamics become x' = (A - B*K)*x, so the closed-loop poles are the eigenvalues of A - B*K. Pole placement is choosing the gain matrix K to put those eigenvalues exactly where desired.
When it is possible
If the system is controllable, then for any desired set of closed-loop pole locations there exists a gain K that achieves them. Controllability is precisely the condition for full freedom in pole assignment. If the system is only stabilizable, the uncontrollable poles are fixed but the rest can still be placed.
How K is computed
- For single-input systems, Ackermann's formula gives K in closed form from the desired characteristic polynomial.
- For multi-input systems, the extra freedom is used to shape eigenvectors as well, and numerical methods choose a well-conditioned K.
- The desired poles are chosen for a target damping ratio and natural frequency, translating time-domain specs into pole locations.
Design trade-offs
It is tempting to place poles far into the left half-plane for a very fast response, but aggressive placement demands large gains and large control effort, and it amplifies noise and can saturate actuators. Poles should be placed only as fast as the actuators and sensors comfortably support. Placement that ignores control effort is a common beginner mistake.
Relation to optimal control
Pole placement sets closed-loop dynamics by hand. The linear-quadratic regulator instead chooses K to minimize a cost balancing state error against control effort, letting the pole locations emerge from that trade-off. LQR can be seen as a principled way to decide where to place the poles when direct choice is hard to justify.
Because it needs the full state, pole placement is usually paired with an observer that estimates the state from outputs, the two combined by the separation principle into a complete output-feedback controller.