Monday, August 23, 2010

Routh array, Routh Criterion and how to calculate the Routh Hurwitz array

The post is available also in PDF format here[mirror].

Trying to find a reasonable explanation of how to calculate Routh array, I found one here (while in book are usually given cryptic and tricky ways, even in Goodwin-Graebe).

Routh criterion

The Routh-Hurwitz stability criterion is a necessary (and frequently sufficient) method to establish the stability of a single-input, single-output (SISO), linear time invariant (LTI) control system.

One of the most popular algorithms to determine whether or not a polynomial is strictly Hurwitz, is Routh's algorithm. Consider a polynomial and its associated Routh array (see below). Then the number of roots with real part greater than zero is equal to the number of sign changes in the first column of the array.

Routh array

A tabular method of Routh criterion can be used to determine the stability when the roots of a higher order characteristic polynomial are difficult to obtain. Consider the polynomial

. The first two rows (two highest ones) of the Routh's array are formed using coefficients

in the increasing order:

Row	Power	First column	Second column	Third column
n				...
n-1	$s^{n-1}$			...

If there are no more coefficients then just add zeros in the table.

Then for each row after third row we need to calculate , , and so on. Now starting from row 3, each row contains the b's, c's, d's and so on as follows:

Row	Power	First column	Second column	Third column
n				...
n-1	$s^{n-1}$			...
n-2	$s^{n-2}$			...
n-3	$s^{n-3}$			...
n-4	$s^{n-4}$			...
&vellip#vdots;
0	$s^{0}$		0	0

Now we need to calculate the coefficients and so on. That can be done using the following determinants formulas:

and so on until . Remember to add zeros in the table when no more coefficients left - it will alleviate the calculations and make them more straightforward. The principle is the following: when one calculates , the first column in the determinant is always [column_1; row_1, row_2]"/> of the Routh table, e.g. .

Second column in the determinant contains , e.g. for calculations of .

The following illustration shows the principle:

Here dark-green rectangle depicts the first column of the determinant in the calculations of

, dark-green circle - denominator in the equation for

. Bright-green rectangle depicts the second column in the determinant for calculations of

. In the same fashion, dark-blue rectangle depicts the first column of the determinant in the calculations of

, dark-blue circle - denominator in the equation for

. Bright-blue rectangle depicts the second column in the determinant for calculations of

, and so on.

Example of calculation of Routh array

Consider the polynomial

. As above, the first two rows of the Routh's array are formed using coefficients

in the increasing order:

Row	Power	First column	Second column	Third column
n		1	3	1
n-1	$s^{3}$	1	2	0 ...

Now we need to calculate using mentioned above equations.

Thus our table will look like:

Row	Power	First column	Second column	Third column
n		1	3	1
n-1	$s^{3}$	1	2	0 ...
n-2	$s^{2}$	1	1	0 ...
n-3	$s^{1}$	1	0	0 ...
n-4	$s^{0}$	1	0	0 ...

From the array we note that there are no sign changes in the first column. According to Routh's criterion, this means that is a strictly Hurwitz polynomial.