ELEC4631 Chap.6 Controllability, Observability, Minimality and Pole-Zero Cancellation
Controllability, Observability, Minimality and Pole-Zero Cancellation
Controllability and observability are the reach and read tests of a state-space model. The controllability matrix [B AB ... A^(n-1)B] collects input directions propagated by the dynamics; rank n means every state direction is reachable and state-feedback poles may be assigned. The observability matrix stacks C, CA through CA^(n-1); rank n means no nonzero initial state produces an indistinguishable free-output record and observer poles may be assigned. The properties are dual because observability of (C,A) equals controllability of (A^T,C^T). They survive similarity transformations even though the test matrices change. A realisation that is both controllable and observable is minimal. Rank failure explains how an internal eigenvalue can be absent from a transfer function through pole-zero cancellation, making input-output stability weaker than a complete internal-stability claim.
What this chapter covers
- 01Controllability matrix and rank n
- 02Observability matrix and rank n
- 03State dimensions and matrix orientation
- 04Duality of controller and observer permissions
- 05Similarity invariance
- 06Minimal realisations
- 07Hidden modes and pole-zero cancellation
Test both permissions and minimality
- reach matrixAB=[1,-3]^T, so 𝒞=[B AB]=[[0,1],[1,-3]].
- reach conclusiondet(𝒞)=-1, hence rank(𝒞)=2=n and (A,B) is controllable.
- read matrixCA=[0,1], so 𝒪=[[1,0],[0,1]]=I.
- read conclusionrank(𝒪)=2=n and (C,A) is observable.
- design meaningBecause the pair is controllable, every two-state closed-loop pole can be assigned by state feedback; because it is observable, every observer-error pole can be assigned.
- minimalityControllable plus observable implies this two-state realisation is minimal.
Key terms
- Controllability
- Ability to transfer the state between arbitrary points in finite time; for LTI systems, the controllability matrix must have rank n.
- Observability
- Ability to reconstruct the initial state from input/output data over a finite interval; the observability matrix must have rank n.
- Duality
- Observability of (C,A) equals controllability of (A^T,C^T), enabling controller algorithms to be transposed into observer algorithms.
- Minimal realisation
- A controllable and observable realisation with the smallest possible state dimension for its transfer function.
- Hidden mode
- An uncontrollable or unobservable eigenmode that does not appear fully in the input-output map.
- Pole-zero cancellation
- An algebraic cancellation in the transfer function that can remove a state eigenvalue from visible poles without deleting the internal state dynamics.
Controllability, Observability, Minimality and Pole-Zero Cancellation FAQ
Why does the controllability matrix stop at A^(n-1)B?
Cayley–Hamilton expresses higher powers of A as combinations of I through A^(n-1), so later propagated input directions add no new span. With multiple inputs the matrix is wide, but the decisive comparison remains rank with n.
What does an observability null-space vector mean?
It is a nonzero initial state direction whose free output and the necessary output derivatives vanish, making it indistinguishable from zero through the available sensor. Observer correction cannot reconstruct that mode from y.
Do rank properties change under z=Tx?
No. The controllability matrix gains invertible transformations of its row/column coordinates, while the observability matrix changes correspondingly with T^-1. Their entries and bases change but rank and the underlying subspace dimensions do not.
Why are Gramians not the centre of this chapter?
The captured course materials and schedule support rank tests, duality, minimality and cancellation, then use them immediately for pole placement and observers. Gramians are valid broader theory, but giving them priority here would displace the course’s explicit LMI and tracking sequence.
What does a complete rank-test answer contain?
Write the state dimension n and matrix orientation before calculating. For controllability, form B, AB through A^(n-1)B as column blocks; for observability, stack C, CA through CA^(n-1) as row blocks. Reduce or use a determinant/minor appropriate to the shape, state the numerical rank and compare it explicitly with n. If full, name the design permission: all state-feedback or observer-error poles may be assigned. If deficient, give a nonzero null-space direction or identify the hidden eigenmode and discuss whether it is stable. End with minimality only after both properties have been tested. A transfer-function cancellation can support the explanation but does not replace the state-space ranks requested by the course.
How do reachability and visibility affect stability claims?
An uncontrollable unstable mode cannot be stabilised by state feedback because the input never reaches it. An unobservable unstable mode cannot be reconstructed by a standard observer because the output contains no information about it. Either can disappear from the transfer function, so a stable visible input-output map may coexist with unstable internal motion from certain initial conditions. By contrast, a hidden mode that is already stable may permit stabilisability or detectability even when full controllability or observability fails; later optimal-control theorems use those weaker conditions. For this chapter’s rank questions, state the full property asked, then use the mode-level interpretation to explain why a design is permitted or limited rather than treating rank as a detached matrix fact.
What variations should I practise beyond square SISO tests?
Use a three-state single-input example, a two-input plant whose controllability matrix is wide, and a two-output sensor whose observability matrix is tall. Rank—not determinant of the entire non-square matrix—is the invariant test. Include one case where C hides an eigenvector and one where B cannot excite a left eigenvector. Predict the hidden mode before row reduction, then confirm it through the matrix null space and transfer cancellation. An AI tutor can generate dimensions and check a rank, but require explicit A,B,C and solve the matrix yourself; unexplained software rank conflicts with the course’s analytical-working signal and teaches no design permission.
Exam move
Write 𝒞 and 𝒪 from dimensions before substituting numbers: B is n-by-m and C is p-by-n, so the reach matrix is n rows and the read matrix is n columns. Practise one full-rank and one deficient example, and for the deficient case identify an actual hidden mode or null-space direction rather than stopping at a number. Use duality to predict observer permission from the transposed controller problem. Pair every minimality decision with a transfer-function thought experiment: which eigenvalue could cancel, and would its internal stability still matter? Finish every rank calculation with the design consequence—arbitrary state-feedback poles, arbitrary observer poles or a limitation due to a hidden direction.
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