State Feedback & Pole Placement

State feedback uses the current internal state, not only the scalar tracking error, to shape dynamics. It can move controllable closed-loop modes directly, which makes the relationship between model structure and transient behavior unusually transparent. The same transparency exposes the limits: unavailable state, weak actuation, uncertainty, and saturation do not disappear when the pole equation has a solution.

Feedback changes the state matrix #

For

$$\dot x=Ax+Bu,$$

choose

$$u=-Kx+Nr.$$

Then

$$\dot x=(A-BK)x+BNr.$$

The regulator dynamics are governed by $A-BK$. If $(A,B)$ is controllable, its $n$ eigenvalues can be assigned to any self-conjugate set using real $K$. If the pair is only stabilizable, the controllable modes may be assigned while the uncontrollable modes remain fixed and must already be stable.

Worked double-integrator placement #

For position $q$ and velocity $v$,

$$A=\begin{bmatrix}0&1\\0&0\end{bmatrix},\qquad B=\begin{bmatrix}0\\1\end{bmatrix},\qquad K=\begin{bmatrix}k_1&k_2\end{bmatrix}.$$

The closed-loop matrix and characteristic polynomial are

$$A-BK=\begin{bmatrix}0&1\\-k_1&-k_2\end{bmatrix},$$

$$\det(sI-(A-BK))=s^2+k_2s+k_1.$$

Desired poles at $-2$ and $-3$ give

$$(s+2)(s+3)=s^2+5s+6,$$

so $K=[6\;\;5]$ and $u=-6q-5v$. This coefficient match is exact for the nominal model. Whether the acceleration limit can deliver the demanded $u$ from the permitted initial conditions is a separate feasibility question.

Pole location encodes a transient, not a complete specification #

A continuous complex pair

$$s=-\zeta\omega_n\pm j\omega_n\sqrt{1-\zeta^2}$$

connects damping ratio $\zeta$ and natural frequency $\omega_n$ to a second-order approximation. Extra real poles are often placed faster so the pair dominates, but “five times faster” is only a heuristic. A fast pole may require high gain, amplify sensor noise through the eventual observer/controller, or excite a neglected actuator or structural mode.

Pole selection must reconcile:

• settling, overshoot, and damping requirements;
• peak and sustained control effort;
• actuator magnitude, rate, and bandwidth;
• sensor noise and state-estimation bandwidth;
• time delay and sampling;
• uncertain and omitted dynamics;
• eigenvector geometry, not only eigenvalues.

For MIMO systems, a chosen pole set generally admits many gains $K$. Those gains can have very different eigenvectors, control allocation, sensitivity, and robustness. Eigenvalue placement alone does not choose among them.

Ackermann's formula explains existence but is not the default algorithm #

For a controllable SISO pair with controllability matrix $\mathcal C$, desired monic polynomial $p_d(s)$, and $e_n^T=[0\;\cdots\;0\;1]$,

$$K=e_n^T\mathcal C^{-1}p_d(A)$$

under the matching companion-form convention. This is Ackermann's formula. It demonstrates that feedback can replace the characteristic polynomial when the controllability matrix is invertible.

Explicitly forming $\mathcal C^{-1}$ and high powers of $A$ is numerically fragile. Use a robust pole-assignment implementation based on orthogonal transformations, and verify the achieved eigenvalues, gain magnitude, and sensitivity rather than treating a returned $K$ as proof.

Regulation is not reference tracking #

The law $u=-Kx$ drives state toward the origin. Adding a reference directly as though it were a state generally gives the wrong steady-state gain. For a stable SISO closed loop with $D=0$, a constant prefilter can be chosen as

$$N=-\left[C(A-BK)^{-1}B\right]^{-1}$$

when the inverse exists, yielding unit nominal DC gain from $r$ to $y$. This is feedforward calibration: plant mismatch or a constant disturbance reintroduces error.

Robust zero error for a declared class of persistent references or disturbances requires an internal model, often an added integral state; see Tracking, Disturbance Rejection & the Internal Model Principle.

Discrete-time placement #

For

$$x_{k+1}=A_dx_k+B_du_k,$$

state feedback gives $A_d-B_dK$. Closed-loop poles must lie strictly inside the unit circle. A continuous pole $s_i$ maps under exact sampling to

$$z_i=e^{s_iT_s}.$$

Designing directly from a sampled model accounts for the zero-order hold, but not for unmodeled computation delay or jitter. Very fast desired continuous poles can collapse near the origin in the $z$-plane and demand control action the sampled actuator cannot supply.

Verification closes the gap from algebra to control #

After computing $K$:

1. confirm controllability or stabilizability with scaled numerical tests;
2. compute achieved poles and eigenvalue condition numbers;
3. simulate the full nonlinear model with saturation and rate limits;
4. inspect state peaks, input peaks, and recovery from representative initial conditions and disturbances;
5. perturb model parameters and include delays and neglected modes;
6. repeat with the actual observer, sampler, hold, and implementation timing;
7. verify startup and state initialization on hardware.

Pole placement is a synthesis primitive. A working controller is the resulting closed-loop mechanism under physical constraints and uncertainty.