Obtain a state variable representation of the system governed by the differential equation: $${{{d^2}y} \over {d{t^2}}} + {{dy} \over {dt}} - 2y = u\left( t \right){e^{ - t}},\,\,\,$$ with the choice of state variables as $${x_1} = y,$$ $${x_2} = \left( {{{dy} \over {dt}} - y} \right){e^t}.$$ Aso find $${x_2}\left( t \right),$$ given that $$u(t)$$ is a unit step function and $${x_2}\left( 0 \right) = 0.$$

Consider the state equation $$\mathop X\limits^ \bullet \left( t \right) = Ax\left( t \right)$$
Given : $${e^{AT}} = \left[ {\matrix{ {{e^{ - t}} + t{e^{ - t}}} & {t{e^{ - t}}} \cr { - t{e^{ - t}}} & {{e^{ - t}} - t{e^{ - t}}} \cr } } \right]$$

(a) Find a set of states $${x_1}\left( 1 \right)$$ and $${x_2}\left( 1 \right)$$ such that $${x_1}\left( 2 \right) = 2.$$
(b) Show that $$\,{\left( {s{\rm I} - A} \right)^{ - t}} = \Phi \left( s \right) = {1 \over \Delta }\left[ {\matrix{ {s + 2} & 1 \cr { - 1} & s \cr } } \right];$$ $$\Delta = {\left( {s + 1} \right)^2}$$
(c) From $$\Phi \left( s \right),$$ find the matrix $$A$$.

The state-space representation of a system is given by $$\left[ {\matrix{ {\mathop {{X_1}}\limits^ \bullet } \cr {\mathop {{X_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 5} & 1 \cr { - 6} & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right].$$
Find the Laplace transform of the state transistion matrix. Find also the value of $${x_1}$$ at $$t=1$$ if $${x_1}\left( 0 \right) = 1$$ and $${x_2}\left( 0 \right) = 0.$$

Determine the transfer function of the system having the following state variable representation:
$$\eqalign{ & X = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr { - 40} & { - 44} & { - 14} \cr } } \right]x + \left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right]u \cr & y = \left[ {\matrix{ 0 & 1 & 0 \cr } } \right]x \cr} $$