Controllability / Observability Rank Test

Build the Kalman controllability and observability matrices for a 2nd- or 3rd-order LTI system and check their rank.

InputControllability: 𝒞 = [B AB A²B … Aⁿ⁻¹B] , observable iff rank 𝒞 = n , Observability: 𝒪 = [C ; CA ; CA² … CAⁿ⁻¹] , rank 𝒪 = n

System order nTestA[1,1]A[1,2]A[1,3] — optionalA[2,1]A[2,2]A[2,3] — optionalA[3,1] — optionalA[3,2] — optionalA[3,3] — optionalB/Cᵀ column [1]B/Cᵀ column [2]B/Cᵀ column [3] — optional

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The engineering

A state is controllable if some input can drive it anywhere, and observable if it leaves a fingerprint on the output. Kalman's rank test settles both: stack the powers of A acting on B (or C) and see whether the resulting n×n matrix spans the full state space. Full rank means no hidden, unreachable or invisible modes.

This card handles single-input / single-output systems up to 3rd order, taking A and either B (controllability) or Cᵀ (observability). Rank is found by Gaussian elimination with a small pivot tolerance, so a determinant that is 'nearly zero' still flags a rank deficiency — the common warning sign of a pole-zero cancellation or a symmetric mode that the actuator can't split.

Rule of thumb: if the determinant of the square Kalman matrix is essentially zero, you have an uncontrollable (or unobservable) subspace and no amount of gain tuning fixes it — you must change where the actuator or sensor sits.

Where this math comes from

Rudolf Kálmán introduced controllability and observability as formal, testable properties in his 1960 papers on the state-space viewpoint, the same body of work that produced the Kalman filter. Before that, control design lived almost entirely in the frequency domain, and the notion that a system could contain modes fundamentally beyond reach of any controller was not sharply defined.

Chi-Tsong Chen's textbook Linear System Theory and Design turned the rank tests into standard engineering practice for generations of students, laying out the controllability and observability matrices in exactly the stacked form this card computes.

1960Rudolf E. KálmánDefines controllability and observability and the rank criteria in the new state-space framework.
1963Kálmán, Ho & NarendraCanonical decomposition separating a system into its (un)controllable and (un)observable subspaces.
1984Chi-Tsong ChenLinear System Theory and Design codifies the rank tests as textbook practice.

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