Stability

Theory

If you want to have a well-behaving dynamical system you want it to be stable. Stability means to have the output under control, such that the output does not blow up for a certain input, but that it just nicely does what you want it to do. If the output blows up it is called unstable, which is never desired in any system.

Stability can be evaluated by computing the poles of the transfer function. The poles are found by solving the polynomial in the denominator of the transfer function. There are in total three main locations for each pole to be in the complex plane. This plane has real values on the horizontal axis and imaginary values on the vertical one. This is because there holds $s = \sigma +j\omega$, which means that the poles can also be imaginary.

If the pole is in the left half plane then the pole is said to be stable. If it is in the right half plane the pole is unstable. The third case is that the pole is purely imaginary and therefore lies on the complex axis. This case is considered marginally stable.

A more clear visualization of this is shown in the following figure. The left half plane is arced more white and is the stable region. The right half plane is the unstable region.

From this figure, it gets also more clear that actually only the real part matters of the poles. Let's now evualte the different cases of stability with a few different transfer functions.

Stable Transfer Function

The different results in poles can easily be seen by means of the following example with transfer function $Y(s)$.

Now by solving the above transfer function for s in the denominator, there can be found that Y(s) is having a pole at the left half plane, namely $s = -\alpha$. As explained in the previous articles about the transfer function and impulse response, we can turn Y(s) back into the time domain by taking the inverse Laplace the transform of Y(s). This would take the form

By evaluating the output y(t) over time for the stable pole. There clearly can be seen that the output converges to zero over time in an exponential behavior, which visualizes the stability of this pole.

Unstable Transfer Function

If we now take the transfer function as

We obtain a pole for this system at $s = \alpha$, which is in the right half plane and therefore unstable. By computing the output in the time domain again, we get the following behavior.

The output y(t) is now clearly an exponential function. This means that the output will exponentially increase over time, so the output is 'blowing up' and therefore not desired. Its behavior is shown in the following figure, where indeed the exponential increase can be noticed, due to the unstable pole.

In this article, the importance of stability in systems is explained and shows that the pole determines a transfer function's stability. In the coming articles, there will be more explanations about Bode plots and Nyquist plots.