Let \(X'=\begin{bmatrix}1 & 2 \\ 0 & 2\end{bmatrix}X+\begin{bmatrix}0\\1\end{bmatrix}U\)

U = [b, 0] X

Where b is an unknown constant. This system is

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, 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:

\(A = \left[ {\begin{array}{*{20}{c}} 1&2\\ 0&2 \end{array}} \right],\;B = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right],\;C = \left[ {\begin{array}{*{20}{c}} b&0 \end{array}} \right]\)

Observability:

\(CA = \left[ {\begin{array}{*{20}{c}} b&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1&2\\ 0&2 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} b&2b \end{array}} \right]\)

If |Q0| ≠ 0, system is observable.

Observable for all non-zero values of b.