An unforced linear time-invariant (LTI) system is represented by:
\(\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}}\\ {{{\dot x}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 1}&0\\ 0&{ - 2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}} \end{array}} \right]\)
If the initial condition are x1(0) = 1 and x2(0) = -1, the solution of the state equation is1
x1(t) = -1, x2(t) = 2
2
x1(t) = -e-t, x2(t) = 2e-t
3
x1(t) = e-t, x2(t) = -e-2t
4
x1(t) = -e-t, x2(t) = -2e-t