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 $${x_2}\left( 0 \right) = 0.$$

The phase lead compensation is used to

The asymptotic magnitude Body plot of a system is given in Figure. Find the transfer function of the system analytically. It is known that the system is minimal phase system.

The number of roots on the equation $$2{s^4} + {s^3} + 3{s^2} + 5s + 7 = 0$$ that lie in the right half of $$S$$ plane is: