If the system matrix of the control system is given as A = $[\begin{matrix} - 2 & 0 \\ 0 & - 2 \end{matrix}]$ and input matrix B = [0 1] and the output matrix as C = [1 3], then the state transition matrix is given by:

Question

Question

This question was previously asked in

UPPCL AE Electrical 2 April 2022 Official Paper (Shift 2)

Answer 1

Concept:

The state transition matrix is given by:

ϕ(t) = e^-At = L^-1 [{sI - A}^-1]

Calculation:

[sI - A] = $[\begin{matrix} s + 2 & 0 \\ 0 & s + 2 \end{matrix}]$

[sI - A]-1 = $\frac{1}{(s + 2)^{2}} [\begin{matrix} s + 2 & 0 \\ 0 & s + 2 \end{matrix}]$

[sI - A]-1 = $[\begin{matrix} \frac{1}{s + 2} & 0 \\ 0 & \frac{1}{s + 2} \end{matrix}]$

L-1 [{sI - A}-1] = $[\begin{array}{cc} e^{- 2 t} & 0 \\ 0 & e^{- 2 t} \end{array}]$

Additional Information Properties of state transition matrix:

1.) ϕ(0) = e-0t = 1

2.) $\frac{d ϕ (0)}{d t} = A$ , where A is the system matrix