Consider a transfer function $$G_p\left(s\right)\;=\;\frac{ps^2+3ps\;-2}{s^2+\left(3+p\right)s\;+\left(2-p\right)}$$ with 'p' a positive real parameter. The maximum value of 'p' until which G_p remains stable is ________.

The characteristic equation of a unity negative feedback system 1 + KG(s) = 0. The open loop transfer function G(s) has one pole at 0 and two poles at -1. The root locus of the system for varying K is shown in the figure.

The constant damping ratio line, for $$\xi$$ = 0.5 , intersects the root locus at point A. The distance from the origin to point A is given as 0.5. The value of K at point A is ________ .

In a Bode magnitude plot, which one of the following slopes would be exhibited at high frequencies by a 4^th order all-pole system?

The state transition matrix $$\phi \left( t \right)$$ of a system $$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ 0 & 1 \cr 0 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] is$$$