Consider a system described by

Which one of the following is correct?

Concept:

State space representation:

ẋ(t) = A(t)x(t) + B(t)u(t)

y(t) = C(t)x(t) + D(t)u(t)

y(t) is output

u(t) is input

x(t) is a state vector

A is a system matrix

This representation is continuous time-variant.

Controllability:

A system is said to be controllable if it is possible to transfer the system state from any initial state x(t0) to any desired state x(t) in a specified finite time interval by a control vector u(t)

Kalman’s test for controllability:

ẋ = Ax + Bu

Qc = {B AB A2B … An-1 B]

Qc = controllability matrix

If |Qc| = 0, system is not controllable

If |Qc|≠ 0, the system is controllable

Observability:

A system is said to be observable if every state x(t0) can be completely identified by measurement of output y(t) over a finite time interval.

Kalman’s test for observability:

Q0 = [CT ATCT (AT)2CT …. (AT)n-1 CT]

Q0 = observability testing matrix

If |Q0| = 0, system is not observable

If |Q0| ≠ 0, system is observable.

Calculation:

Controllability matrix Qc = [B    AB]

​​

Q_c = [B   AB]

|Q_c| = -1

|Qc|≠ 0

∴ System is controllable

Observability matrix Q0 = [CT    ATCT], i.e.

|Q0| = 1

|Q0| ≠ 0 = observable

∴ The system is controllable and observable

Important Points

Duality property of controllability and observability:

• If the pair (A, B) is controllable then the pair (AT, BT) is observable.
• If the pair of (A, C) is observable then the pair of (AT, CT) is controllable.

If there are no pole-zero cancellations in the transfer function then the system is completely controllable and observable.