Previous Year Question
Read the following statements:
Statement 1: A system is said to be completely controllable if it is possible to transfer the system state from an initial state X(t0) to any desired state X(t) in a specified finite time by a control vector u(t).
Statement 2: A system is said to be completely observable, if every state X(t0) can be completely identified by measurement of the outputs y(t) over the time interval 0 ≤ t ≤ ∞.
Choose the correct options from the following:
Correct answer: A
Explanation
Statement 1 gives the standard meaning of complete controllability: the input can drive the state to any desired state in finite time. Statement 2 gives the observability idea: the initial state can be reconstructed from output measurements. For an n-state LTI model, the algebraic checks are rank(𝒞)=n and rank(𝒪)=n. Hence option A.
Formula
Step-by-step solution
Correct Answer
The first statement describes reachability of the state by the input; the second describes recoverability of the state from output measurements.
What This Question Is Really Testing
State variables represent the internal condition of a dynamic system. Controllability asks whether the input can move that internal state where we want it. Observability asks whether output measurements contain enough information to determine the internal state.
Concept Foundation
For the LTI state model ẋ = Ax + Bu, y = Cx + Du, the two standard matrix tests are:
Step 1 — Evaluate Statement 1: Controllability
The statement says that a suitable control input can transfer the system from an initial state to any desired state in a specified finite time. That is the defining idea of complete controllability.
Step 2 — Evaluate Statement 2: Observability
The statement says that the state can be completely identified from measured outputs over the observation interval. That is the defining idea of complete observability.
Step 3 — Select the Statement Combination
Both definitions express the intended properties correctly, so the correct option is A.
Why This Method Works
The direction of information/action distinguishes the concepts: controllability is input → state; observability is state → measured output.
Independent Verification / Cross-Check
If A, B and C matrices were supplied, the rank tests above would provide an independent algebraic verification. Here the item asks only about the definitions.
Concept Extension
- A mode can be uncontrollable even if it appears in the state equations.
- A mode can be unobservable even though it affects internal state dynamics.
- Minimal realizations are both controllable and observable.
Exam Strategy
Use the mnemonic: Can I command the state? → controllability. Can I see/reconstruct the state? → observability.
Other options
B and C reject one correct definition; D rejects both. Only A accepts the two intended definitions together.
Common mistake
Interchanging the matrix tests: B generates the controllability matrix, while C generates the observability matrix.
Exam tip
Input → state = controllability; state → output = observability.
Quick method
Command it = controllability; see it = observability.