Routh-Hurwitz Criteria: Simple Guide, Stable Systems!

System stability, a critical attribute in control engineering, often hinges on evaluating the roots of characteristic equations. The Routh-Hurwitz Criteria, a powerful technique, provides a clear method for ascertaining stability without explicitly solving for these roots. MIT, a leading institution in engineering research, has significantly contributed to the understanding and application of stability analysis techniques. This guide will simplify the Routh-Hurwitz Criteria, making it accessible to engineers and students alike and allowing them to apply it when using tools like MATLAB to assess system stability.

Routh-Hurwitz Criteria: A Straightforward Guide to System Stability

Understanding the stability of a system is crucial in various engineering disciplines, especially control systems. The routh hurwitz criteria provides a powerful tool for determining the stability of a linear time-invariant (LTI) system without explicitly solving for the roots of its characteristic equation. This guide breaks down the criteria in a simple, accessible manner.

What is System Stability?

Before diving into the routh hurwitz criteria, let’s define what we mean by system stability. In essence, a stable system is one where bounded inputs result in bounded outputs. Imagine pushing a swing; a stable swing will eventually settle back to equilibrium. An unstable swing, however, would exhibit oscillations that grow indefinitely.

• Stable System: Returns to equilibrium after a disturbance.
• Marginally Stable System: Oscillates indefinitely with constant amplitude.
• Unstable System: Exhibits oscillations that grow unbounded over time.

The Characteristic Equation

The routh hurwitz criteria operates on the characteristic equation of a system. This equation is typically derived from the system’s transfer function, which represents the relationship between the input and output in the Laplace domain. The general form of the characteristic equation is:

a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0 = 0

where:

• s is the Laplace variable (a complex number).
• a_n, a_{n-1}, ..., a_1, a_0 are real coefficients.

The roots of this equation, known as the system’s poles, determine its stability. A system is stable if and only if all the poles have negative real parts. The routh hurwitz criteria allows us to determine this without actually finding the roots.

Constructing the Routh Array

The heart of the routh hurwitz criteria is the Routh array. This array is a table constructed from the coefficients of the characteristic equation. Here’s how to build it:

1. First Two Rows: The first row consists of the coefficients of the even powers of s (starting with the highest). The second row consists of the coefficients of the odd powers of s (also starting with the highest).

   s^n	a_n	a_{n-2}	a_{n-4}	…
   s^{n-1}	a_{n-1}	a_{n-3}	a_{n-5}	…

2. Calculating Subsequent Rows: Each element in the subsequent rows is calculated based on the elements in the two rows directly above it. The general formula for calculating the element b_i in the third row is:

   b_i = - (1/a_{n-1}) * determinant([[a_n, a_{n-2i}], [a_{n-1}, a_{n-(2i+1)]])

   And for element c_i in the fourth row:

   c_i = - (1/b_1) * determinant([[a_{n-1}, a_{n-(2i+1)}], [b_1, b_{i+1}]])

   This pattern continues until you reach a row of all zeros or until you have a row corresponding to s^0.

   Important Note: If you encounter a zero as the first element in a row while constructing the array, replace it with a small positive number (epsilon, ε) and continue. This avoids division by zero and allows you to analyze the system’s stability. If the entire row is zero, you may need to take the derivative of an auxiliary polynomial. This situation indicates poles on the imaginary axis.

Applying the Routh-Hurwitz Criteria

Once the Routh array is constructed, the routh hurwitz criteria states:

• The number of sign changes in the first column of the Routh array is equal to the number of roots of the characteristic equation with positive real parts (i.e., unstable poles).

Therefore:

• For a stable system: All the elements in the first column of the Routh array must have the same sign (typically positive).
• For an unstable system: There will be at least one sign change in the first column. The number of sign changes indicates the number of poles in the right-half plane.

Example

Let’s consider the characteristic equation:

s^3 + 2s^2 + 3s + 4 = 0

The Routh array is constructed as follows:

s^3	1	3
s^2	2	4
s^1	-1	0
s^0	4	0

Calculating the elements:

• s^1 row: -(1/2) * determinant([[1, 3], [2, 4]]) = -(1/2) * (4 - 6) = 1. (Incorrect in original table – corrected here, should be -1)
• s^0 row: -(1/-1) * determinant([[2, 4], [-1, 0]]) = 1 * (0 + 4) = 4.

Notice there are two sign changes in the first column (from 2 to -1 and from -1 to 4). Therefore, the system is unstable and has two poles with positive real parts.

Common Scenarios and Special Cases

Row of Zeros

A row of zeros in the Routh array indicates the presence of roots on the imaginary axis (jω-axis) or roots that are symmetrically located about the origin in the s-plane. To analyze this further:

1. Form the Auxiliary Polynomial: Use the row above the row of zeros to form an auxiliary polynomial. For example, if the s^3 row is a b 0 0, and the s^2 row is a row of zeros, the auxiliary polynomial A(s) is formed from the s^3 row as A(s) = a*s^3 + b*s^1. (This example is wrong – corrected below). More correctly, if the s^3 row is a b, and the s^2 row is a row of zeros, the auxiliary polynomial A(s) is formed from the s^3 row as A(s) = a*s^2 + b*s^0. The power of s associated with the first term is always equal to the power represented by the row the polynomial is constructed from.
2. Differentiate the Auxiliary Polynomial: Differentiate A(s) with respect to s.
3. Replace the Row of Zeros: Replace the row of zeros with the coefficients of the derivative.
4. Continue the Routh Array: Proceed with the Routh array as before.

The roots of the auxiliary polynomial are also roots of the original characteristic equation. These roots are symmetrically located with respect to the origin.

Leading Coefficient is Zero

If the leading coefficient (a_n) of the characteristic equation is zero, it indicates that the equation is not of degree n. This often means there’s a root at infinity or that there’s a simpler system being represented. In these cases, it’s best to review the system modeling to ensure the characteristic equation is derived correctly. It is often possible to factor out s from the equation to reduce its order.

Repeated Roots

The Routh-Hurwitz criterion doesn’t directly indicate the presence of repeated roots. However, a row of zeros can often be a sign of the presence of repeated roots.

So there you have it! Hopefully, this makes the routh hurwitz criteria a little less daunting and a lot more useful in your design work. Now go build something stable!