The asymptotic Bode plot of a transfer function is shown in the figure. the transfer function G(s) corresponding to this bode plot is

If the closed-loop transfer function of a control system is given as T(s)=$${{s - 5} \over {(s + 2)(s + 3)}},$$ then it is

The transfer function of a plant is $$$T\left(s\right)=\frac5{\left(s+5\right)\left(s^2+s+1\right)}$$$ The second-order approximation of T (s) using dominant pole concept is:

Consider a linear system whose state space Representation is $$\mathop x\limits^ \bullet \left( t \right) = AX\left( t \right).$$
If the initial state vector of the system is $$x\left( 0 \right) = \left[ {\matrix{ 1 \cr { - 2} \cr } } \right],$$
then the system response is $$x\left( t \right) = \left[ {\matrix{ {{e^{ - 2t}}} \cr { - 2{e^{ - 2t}}} \cr } } \right].$$
If the initial state vector of the system changes to $$x\left( 0 \right) = \left[ {\matrix{ 1 \cr { - 1} \cr } } \right],$$
then the system response becomes $$x\left( t \right) = \left[ {\matrix{ {{e^{ - t}}} \cr { - {e^{ - t}}} \cr } } \right].$$

The eigen value and eigen vector pairs $$\left( {{\lambda _{i,}}{V_i}} \right)$$ for the system are