How it works
A system (A, B) is completely controllable if the controllability matrix Q_c = [B AB A²B ... Aⁿ⁻¹B] has rank n (full rank). A system (A, C) is completely observable if the observability matrix Q_o = [C; CA; CA²; ...; CAⁿ⁻¹] has rank n. Physical meaning of controllability: every state can be driven from any initial condition to any desired final state in finite time using an appropriate input u(t). Observability means every state can be determined from output measurements alone, even without direct access to the states.
Key points to remember
Duality principle: (A, B) is controllable if and only if (Aᵀ, Bᵀ) is observable. This means any observability test can be converted to a controllability test on the dual system. A system in phase variable (controllable canonical) form is always controllable. A system in observable canonical form is always observable. Pole placement by full-state feedback (Ackermann's formula: K = eₙᵀ·Q_c⁻¹·φ(A)) requires controllability. State observer (Luenberger observer) design requires observability. Minimum realisation of a transfer function retains only controllable and observable modes — pole-zero cancellations in G(s) indicate uncontrollable or unobservable modes.
Exam tip
The examiner always asks you to test controllability and observability by forming the respective matrices, computing their determinants, and concluding whether the system is controllable/observable — rank deficiency means det = 0, which you must state explicitly.