**Routh–Hurwitz stability criterion**

**08/15/17**

*“How can we analytically estimate the stability of a transfer function?”*

We know that transfer functions have a certain level of stability afforded to them. However, sometimes the functions can be quite complicated. So how can we analytically analyze their stability? Well, let’s think about it. First, let’s take our polynomial. Then, let’s take the coefficients for all of our functions. Then, let’s make a graph with the number of rows equal to the highest exponent and number of columns equal to half rounded up of that. Afterward, let’s place the coefficients on the first two columns such that the greatest coefficient goes on the top left, the next one goes beneath, the next one goes to the right of the first one, the next one goes below, and so on. Now let’s fill in the rest of the table by taking the difference between the coefficient in the first column two rows up multiplied by the coefficient up and right one, and then let’s subtract it from the value directly up two and right one minus the one at the first column one row up while dividing everything by the number directly up one row and put the result in place of the column . Let’s now repeat this pattern until we have the entire table filled out. If any coefficients in the first column are of a different sign, then the system will be unstable. This mathematical tool is known as the **Routh–Hurwitz stability criterion **and is used in designing all sorts of control systems.