Controllability, Observability & Structural Modes

Before asking for a controller gain or observer gain, ask whether the actuator and sensor geometry make the request possible. Controllability and observability are properties of a model together with chosen input and output channels, not of the state matrix $A$ alone.

Controllability: can the input reach the state? #

For

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

the forced response at time $T$ is

$$x(T)=e^{AT}x(0)+\int_0^T e^{A(T-\tau)}B u(\tau)\,d\tau.$$

The pair $(A,B)$ is controllable when a finite input can transfer any initial state to any final state in finite time. For an $n$-state continuous LTI system, the Kalman matrix test is

$$\mathcal C=[B\;AB\;A^2B\;\cdots\;A^{n-1}B],\qquad \mathop{\mathrm{rank}}\mathcal C=n.$$

The columns are input directions after repeated action by the dynamics. If their span omits a state direction, no input history through $B$ can independently set that direction.

Observability: can output history distinguish state? #

For the autonomous response $y(t)=Ce^{At}x(0)$, the pair $(C,A)$ is observable when different initial states always produce different output histories over a finite interval. Its Kalman matrix is

$$\mathcal O= \begin{bmatrix} C\\CA\\CA^2\\\vdots\\CA^{n-1} \end{bmatrix},\qquad \mathop{\mathrm{rank}}\mathcal O=n.$$

If $v\ne0$ lies in the nullspace of $\mathcal O$, adding $v$ to the initial state changes no measured output trajectory. An observer cannot reconstruct information the sensor channel never contains.

Controllability and observability are dual: the controllability test for $(A^T,C^T)$ is the observability test for $(C,A)$. The algebra is transposed, but the physical remedies differ: improve $B$ by changing actuator placement or authority; improve $C$ by changing sensor placement or measured variables.

PBH tests expose individual modes #

The Popov–Belevitch–Hautus tests ask whether an eigenmode is blocked:

$$\mathop{\mathrm{rank}}[\lambda I-A\;B]=n$$

for every eigenvalue $\lambda$ of $A$ gives controllability, while

$$\mathop{\mathrm{rank}} \begin{bmatrix} \lambda I-A\\C \end{bmatrix}=n$$

gives observability. Equivalently, no left eigenvector $w^T A=\lambda w^T$ may satisfy $w^TB=0$, and no right eigenvector $Av=\lambda v$ may satisfy $Cv=0$.

This mode-wise view answers the design question more directly: which pole cannot the actuator move, or which pole cannot the sensor see?

A hidden unstable mode #

Consider

$$A=\begin{bmatrix}-1&0\\0&1\end{bmatrix},\quad B=\begin{bmatrix}1\\0\end{bmatrix},\quad C=\begin{bmatrix}1&0\end{bmatrix}.$$

The first state is stable, actuated, and measured. The second state grows as $e^t$ but is neither actuated nor measured:

$$\mathcal C=\begin{bmatrix}1&-1\\0&0\end{bmatrix},\qquad \mathcal O=\begin{bmatrix}1&0\\-1&0\end{bmatrix}.$$

Both have rank one. The input/output transfer behavior can appear stable because the unstable mode is hidden, yet internal state diverges. Pole-zero cancellation at an interface is therefore not proof of internal stability.

Stabilizability and detectability are the practical thresholds #

Full controllability is stronger than needed for stabilization. A pair $(A,B)$ is stabilizable when every uncontrollable mode is already stable. State feedback can then stabilize all modes that need moving. Likewise, $(C,A)$ is detectable when every unobservable mode is stable; an observer error can then converge even if stable hidden modes are not reconstructed individually.

This distinction prevents needless rejection of models containing harmless stable decoupled states while still failing closed on hidden unstable dynamics.

Gramians add magnitude to the yes/no test #

For a stable continuous system, the infinite-horizon Gramians are

$$W_c=\int_0^\infty e^{At}BB^Te^{A^Tt}\,dt,$$

$$W_o=\int_0^\infty e^{A^Tt}C^TCe^{At}\,dt.$$

They solve Lyapunov equations

$$AW_c+W_cA^T+BB^T=0,$$

$$A^TW_o+W_oA+C^TC=0.$$

A small eigenvalue of $W_c$ identifies a direction requiring large control energy; a small eigenvalue of $W_o$ identifies a direction producing little output energy. Exact controllability or observability can coexist with an engineering-uselessly weak direction.

Rank is not a floating-point fact #

Powers of $A$ can make $\mathcal C$ and $\mathcal O$ badly scaled. Official numerical documentation warns that direct rank tests become unreliable for more than small systems. A matrix reported as full rank at one tolerance can be deficient at another.

Use a layered diagnosis:

1. scale states, inputs, and outputs in physical units;
2. inspect singular values rather than only an integer rank;
3. use PBH tests to identify the affected modes;
4. use controllability/observability staircase forms or Schur-based algorithms;
5. examine Gramians or finite-horizon energy when the system is stable;
6. perturb parameters within uncertainty and repeat the analysis.

Controllability and observability are design geometry. They tell where authority and information enter the model, and weak geometry warns about high gain, noise sensitivity, and fragile pole placement before synthesis begins.