State feedback and pole placement

For a real continuous-time LTI pair (A,B), controllability is equivalent to the ability to assign any real monic closed-loop characteristic polynomial by a real state-feedback gain. Desired nonreal poles must appear in conjugate pairs. This statement concerns the characteristic polynomial, not an arbitrary prescribed eigenvector basis or Jordan structure. In a controllable canonical form, the gain directly changes polynomial coefficients, which explains the theorem’s algebraic mechanism. Multi-input systems can offer several gains with the same poles, allowing additional numerical or effort considerations.

For a pair that is not controllable, a Kalman decomposition isolates an uncontrollable block whose eigenvalues feedback cannot move. Stabilizability means all those fixed modes are already Hurwitz, so there exists some gain making the whole closed loop Hurwitz. The continuous-time PBH test checks full row rank of [λI−A,B] for every λ with nonnegative real part. Thus stabilizability is enough for stabilization but not arbitrary full pole assignment. An uncontrollable eigenvalue at zero is not acceptable: remaining bounded is weaker than the strict decay required here.

When only y=Cx is measured and the input is known, a Luenberger observer uses the model plus an output correction: its state derivative is A x̂+Bu+L(y−C x̂). With e=x−x̂, the error satisfies ė=(A−LC)e in the exact noiseless model. Detectability is enough to select a stable error dynamics; observability permits arbitrary observer pole assignment. The feedback law then uses u=−K x̂ rather than an unavailable x. This construction does not assume sensors directly measure every state coordinate.

In coordinates (x,e), the combined dynamics is block triangular: the diagonal blocks are A−BK and A−LC. Consequently choosing both Hurwitz makes the exact observer-based closed loop Hurwitz. This is the linear separation principle. It does not say the plant is insensitive to model errors, sensor noise or saturation. Very fast observer poles can amplify measurement noise, while very fast control poles can demand excessive input. Those implementation effects require quantitative analysis beyond the eigenvalue union furnished by the ideal model.

Moving eigenvalues farther left suggests faster asymptotic decay, but large gains can make the command exceed actuator bounds. Once saturation occurs, the actual closed-loop vector field is no longer A−BK, so its linear proof is no longer global. Nonnormal closed-loop matrices can also show large transient growth despite stable poles. An eigenvalue computation therefore should be paired with state and input trajectories, conditioning of the eigenvectors, and a suitable energy or norm certificate. The next lesson develops one such certificate through Lyapunov equations.

Regulation and tracking are distinct tasks. The law u=−Kx regulates the origin when that origin is the intended equilibrium. A nonzero reference may require a feedforward equilibrium command, a reference prefilter or integral action; simply replacing x by x−r is not meaningful when the reference has different dimensions or is not a feasible equilibrium. Integral action adds states and changes the controllability and saturation questions. Good pole placement begins by specifying what must decay, which signals are measured and what commands the plant can physically accept.