Routh- Hurwitz Criterion

We will first examine the stable, unstable, and marginally stable systems before talking about the Routh-Hurwitz Criterion.

1. Stable System: The system is referred to as stable if all of the characteristic equation’s roots are located on the left half of the “S” plane.
2. Marginally Stable System: The system is referred to as marginally stable if all of its roots fall on the hypothetical axis of the “S” plane.
3. Unstable System: A system is consider unstable if all of its roots are located on the right side of the “S” plane.

Statement of Routh Hurwitz Criterion

According to the Routh Hurwitz criterion, a system can only be stable if and only if all of the roots in the first column have the same sign. If there are sign changes in the first column, the number of sign changes is equal to the number of roots in the characteristic equation in the right half of the s-plane, or the number of roots with positive real parts.

Necessary but not sufficient conditions for Stability

To make any system stable, we must adhere to a few requirements, or as we might say, there are a few prerequisites.

Consider a system with characteristic equation:

1. All the coefficients of the equation should have the same sign.
2. There should be no missing term.

We have no assurance that the system will be stable even if all the coefficients have the same sign and there are no missing terms. To do this, we use the Routh Hurwitz Criterion to assess the system’s stability. If the specified requirements are not met, The system is said to be unstable. The authors A. Hurwitz and E.J. Routh provide this criterion.

Advantages of Routh Hurwitz Criterion

1. Without solving the equation, we can determine whether the system is stable.
2. The system’s relative stability is simple to ascertain.
3. We can determine the range of K for stability using this method.
4. This technique also allows us to locate the intersection of the root locus and a hypothetical axis.

Limitations of Routh Hurwitz Criterion

1. This standard only applies to linear systems.
2. On the right and left sides of the S plane, the exact locations of the poles are not specified.
3. Using real coefficients is the only way to use the characteristic equation.

The Routh Hurwitz Criterion

The following polynomial property is worth thinking about.

When the coefficients a0, a1, ………………….an are all of the same sign, and none is zero.

Step 1: Arrange all the coefficients of the above equation in two rows:

Step 2: From these two rows we will form the third row:

And Step 3: Now, we shall form fourth row by using second and third row:

Step 4: We shall continue this procedure of forming a new rows: