Hello Ritesh .. In the time BrownOut gets back to you,,
To clarify things, we must specify that when we say "right half plane" is unstable and "left half plane is stable", we're talking in s plane. If you want the z plane thing, feel free.
To get the left half plane "thingy", we must go back to constant coefficients differential equations and solving them with a characteristic equation. Second order, for example...
y'=dy/dt and y''=d2y/dt2 (I'm on a MacBook, don't hold typos on me)
Homogen equation (second part null)
a*y'' + b*y' + c*y = 0
We arrange a bit to make it look like : y'' + 2*Ksi*w*y' + w^2*y = 0 ..
Where Ksi is the damping coefficient, and w is omega, the natural pulsation of the system. (Hint : w^2= c/a... So Ksi is ?)
We look for a solution of the exponential form .. e^(s*t)
Characteristic equation of the differential equation above becomes : s^2 + 2*Ksi*w*s + w^2=0
Simple second degree. You find Delta' (Simplified Delta, we arranged the two so you don't get a 4, and you don't divide by 2)
Solutions will depend on the value of Ksi..
Ksi Superior to 1 (Delta Positive) : Two Simple (non double) Real solutions , e^(s1*t) and e^(s2*t)
Ksi = 1 (Delta Null) : The solution is double and real (A*t + B)*e^(s*t)
Ksi Inferior to 1 (Delta Negative) : Imaginary solution (two conjugates) ... s here is : s=u + i*v (i^2=-1)
First solution is (e^(u*t))*(Cos(v*t) + i*Sin(v*t)) (t being the time, v the imaginary part of s, and u the real part of s)
WHAT DOES ALL OF THIS MEAN ?
Notice that the Amplitude of these solutions DEPENDS on the REAL value of the solution.( The first and second cases where Delta is positive or null, s is real, so s=u.. No imaginary part)
They are exponential type. So, it's e^(u*t).
If u is positive, amplitude grows exponentially. And in the case of the imaginary solution, it's a sinusoide with an exponentially growing amplitude.. You don't want a system like that. It's a DIVERGENT system. Unstable.
If u is negative thought, amplitude gets "damped" with respect to time. It's a CONVERGENT system. Stable.