Questions: Tests for Controllability and Observability

Question 1

A 4th-order system (n=4) has a controllability matrix Qc with rank 3. What can you conclude?

• A. The system is controllable — rank 3 out of 4 is close enough for practical purposes
• B. There is a 1-dimensional subspace of state space that the input can never reach, regardless of the control signal applied
• C. The system has one unstable pole that the controller cannot stabilize
• D. The system's transfer function has a pole-zero cancellation that reduces its effective order to 3

Answer: (B)

Controllability requires Qc to have full rank (rank = n = 4). A rank of 3 means the columns of Qc span only a 3-dimensional subspace of the 4-dimensional state space. There exists a direction in state space that no input can push the state toward — no matter how the control signal is chosen, the state can never reach that direction from the origin. Option A is wrong: rank deficiency is binary in its consequences, not a matter of degree. Option C is a different issue (stability without controllability). Option D is partially related (pole-zero cancellations can cause uncontrollability) but is not the direct conclusion from the rank test.

Question 2

A system is fully controllable but its observability matrix Qo has rank less than n. What is the consequence for observer and feedback design?

• A. No consequence — controllability is sufficient for full feedback design, observability only matters for open-loop systems
• B. You can design a state-feedback controller, but you cannot build an observer to estimate unmeasurable states — some state components are indistinguishable from the output
• C. The system will be unstable regardless of the feedback gain chosen
• D. The transfer function from input to output will be unstable

Answer: (B)

Controllability and observability are independent properties. A system can be controllable (all states reachable via input) without being observable (all states detectable from output). If Qo is rank-deficient, there exist distinct initial state vectors x₁(0) ≠ x₂(0) that produce identical output trajectories y(t). No measurement can distinguish them — the observer is blind to those state components. You can still design state feedback if you have full state access, but you cannot build a Luenberger observer (or Kalman filter) to reconstruct the unmeasurable states. Option A is wrong because modern control design (LQG, observer-based feedback) requires both properties.

Question 3

If a system's controllability matrix has full rank, the system is also very likely to be observable.

• T. True
• F. False

Answer: False

Controllability and observability are dual but entirely independent structural properties. A system can have any combination: controllable and observable, controllable but not observable, observable but not controllable, or neither. Controllability depends on the pair (A, B) — whether the input matrix B, through powers of A, can reach all of state space. Observability depends on the pair (A, C) — whether the output matrix C, through powers of A, can distinguish all initial states. Changing B (adding or removing actuators) affects controllability but not observability, and vice versa for changing C. There is no implication between the two rank conditions.

Question 4

An unstable hidden mode — a mode that is neither controllable nor observable — cannot be stabilized by any feedback controller that uses the system's existing inputs and outputs.

• T. True
• F. False

Answer: True

A hidden mode appears in neither the input-to-state reachable subspace nor the state-to-output observable subspace. This means no control signal can affect it (uncontrollable) and no measurement reveals its behavior (unobservable). A controller can only affect modes it can reach and observe; a hidden mode evolves freely under the autonomous dynamics ẋ = Ax. If this mode is unstable (the corresponding eigenvalue has positive real part), it will diverge without any possibility of correction. Kalman decomposition makes this explicit: the transfer function only reflects the controllable-and-observable subsystem, so an unstable hidden mode is completely invisible in the transfer function yet physically present and growing. The only remedies are hardware changes: adding actuators to make it controllable, or adding sensors to make it observable.

Question 5

What is a 'hidden mode' in a linear system, and why is an unstable hidden mode especially dangerous from a control engineering perspective?

Model answer: A hidden mode is an eigenmode of the system matrix A that is neither controllable (the input cannot excite it) nor observable (the output cannot reveal it). It corresponds to a subspace of state space that is decoupled from both the input and the output. In Kalman decomposition, the transfer function from input to output only captures modes that are both controllable and observable — hidden modes do not appear in the transfer function at all. An unstable hidden mode is especially dangerous because its divergence is invisible: the output looks well-behaved while the state is growing unboundedly. A controller acting only on the outputs sees no problem to correct, and even if it applies input, the input cannot reach the uncontrollable hidden mode. The system can catastrophically fail internally while appearing stable from the input-output perspective.

This is why rank tests are the first step in control design — not an optional mathematical formality. Discovering an unstable hidden mode after designing a controller means the controller is fundamentally flawed in a way that cannot be patched without redesigning the physical system (changing sensor or actuator placement).