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The transfer function G(S) = C(SI - A)-1b of the system
x' = Ax + bu
y = Cx + du
has no pole-zero cancellation. The system
  • a)
    is controllable and observable
  • b)
    is observable but uncontrollable
  • c)
    is controllable but unobservable
  • d)
    non of the above
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The transfer function G(S) = C(SI - A)-1b of the systemx = Ax + buy = ...
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.
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.
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The transfer function G(S) = C(SI - A)-1b of the systemx = Ax + buy = Cx + duhas no pole-zero cancellation. The systema)is controllable and observableb)is observable but uncontrollablec)is controllable but unobservabled)non of the aboveCorrect answer is option 'A'. Can you explain this answer?
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