Can someone please explain WHY is it when the poles of F(s) have negative real parts, the system is stable. Why is it when the poles of F(s) have positive real parts the system is unstable? Why is it when the real parts of the poles of F(s) equal to 0 the system becomes metastable (oscillatory) Thanks
The reasons become clear when you do a partial fraction expansion of the output of the system and find its inverse Laplace transform. You'll find factors of exp^(p_i*t) in all the terms of the natural response of the system, where p_i is the corresponding pole of the expansion and t is the time. You can see what happens if the real part of the pole is positive or negative, exponential growth or decay. You can find a better runthrough here (go down to 'Poles and the Impulse Response'): http://www-control.eng.cam.ac.uk/gv/p6/Handout3.pdf