Questions: State-Space to Transfer Function Conversion

Question 1

A 5th-order state-space model is converted to a transfer function, then converted back to state-space using controllable canonical form. The resulting model has only 3 states. What does this tell you about the original system?

• A. The conversion algorithm has a numerical error — state dimension must be preserved
• B. The original model had 2 uncontrollable or unobservable modes that cancelled as pole-zero pairs in the transfer function and were permanently lost
• C. Controllable canonical form always reduces state dimension to match the transfer function order
• D. The system had 2 repeated eigenvalues that were merged during conversion

Answer: (B)

Converting state-space → transfer function → state-space always yields a minimal realization. If the round-trip produces a lower-order model, the original was non-minimal: it had hidden modes (uncontrollable or unobservable states) that appeared as pole-zero cancellations in the transfer function and were discarded. The canonical form reconstruction gives the McMillan degree of the transfer function, which equals the original state dimension only if the model was already minimal.

Question 2

A control engineer notices a pole-zero cancellation in a system's transfer function: a pole at s = +3 is cancelled by a zero at s = +3. Why is this dangerous?

• A. The cancellation makes the system's frequency response undefined at ω = 3 rad/s
• B. The cancelled right-half-plane pole represents an unstable internal mode that remains in the physical system — the state may diverge even though the output looks stable
• C. The cancellation reduces gain at all frequencies, degrading control performance
• D. Pole-zero cancellations are always benign — they simplify the transfer function without affecting system behavior

Answer: (B)

A pole-zero cancellation in the transfer function means the mode at s = +3 is either uncontrollable or unobservable — it disappears from the input-output description, but it is still physically present in the system. Because s = +3 is in the right half-plane, this hidden mode is unstable. The internal state corresponding to this mode will grow without bound over time, even while the output (which cannot 'see' the mode due to unobservability) appears well-behaved. This is one of the most dangerous failure modes in control design.

Question 3

Two state-space models with identical transfer functions represent the same physical dynamics and will behave identically in most operating conditions.

• T. True
• F. False

Answer: False

Two state-space models sharing a transfer function have the same input-output behavior, but may have completely different internal dynamics. Infinitely many state-space realizations correspond to any given transfer function, all related by invertible similarity transformations x̃ = Tx. More critically, a non-minimal realization has hidden modes (uncontrollable or unobservable states) that do not appear in the transfer function but are physically present — and those modes can be unstable, causing internal divergence while the outputs look fine. Identical transfer functions do not mean identical internal dynamics.

Question 4

A minimal realization of a transfer function is unique up to an invertible similarity transformation of the state vector.

• T. True
• F. False

Answer: True

All minimal realizations of a given transfer function are related by an invertible state transformation: if (A, B, C, D) and (Ã, B̃, C̃, D̃) are both minimal, then there exists an invertible matrix T such that à = TAT⁻¹, B̃ = TB, C̃ = CT⁻¹, D̃ = D. This means the eigenvalues of A (the poles), input-output behavior, and all input-output properties are the same across all minimal realizations — only the internal state coordinates differ. Minimality pins down the essential dynamics; the choice of state basis is free.

Question 5

Why is a pole-zero cancellation in a transfer function potentially dangerous in a real physical system, and what does minimality have to do with this?

Model answer: A pole-zero cancellation means a mode of the system is either uncontrollable (input cannot excite it) or unobservable (output cannot detect it). The mode is physically present in the system but hidden from the input-output description. If this hidden mode is unstable (right-half-plane pole), the corresponding state variable will grow without bound even though the output appears normal — a catastrophic failure invisible to any output-based measurement. Minimality is directly related: a minimal realization has no cancellations (it is both controllable and observable), so every pole in the transfer function corresponds to a physically active, observable, controllable mode. Non-minimal realizations have the dangerous hidden modes that minimality eliminates.

This is why control engineers should verify minimality before relying on transfer function analysis, and why the round-trip from state-space to transfer function and back is not a safe way to simplify a model — hidden modes are permanently discarded, not revealed.