Control Systems (CS)
Lecture-14
Routh-Herwitz Stability Criterion (Part A)
Dr. Dhafer Manea Hachim AL-HASNAWI
Assist Proof
Al-Furat Al-Awsat Technical University
Engineering Technical College / Najaf
email:coj.dfr@atu.edu.iq
1
Routh-Hurwitz Stability Criterion
• It is a method for determining continuous system
stability.
• The Routh-Hurwitz criterion states that “the number of
roots of the characteristic equation with positive real
parts is equal to the number of changes in sign of the
first column of the Routh array”.
Routh-Hurwitz Stability Criterion
This method yields stability information without the need to
solve for the closed-loop system poles.
Using this method, we can tell how many closed-loop system
poles are in the left half-plane, in the right half-plane, and on
the jw-axis. (Notice that we say how many, not where.)
The method requires two steps:
1. Generate a data table called a Routh table.
2. interpret the Routh table to tell how many closed-loop system
poles are in the LHP, the RHP, and on the jw-axis.
Routh-Hurwitz Stability Criterion
• The characteristic equation of the nth order continuous system can be written as:
• The stability criterion is applied using a Routh table which is defined as;
• Where a re coefficients of the characteristic equation.
Generating a Basic Routh Table
• First label the rows with powers of s from highest power of s down to lowest power
of s in a vertical column.
• Next form the first row of the Routh table, using the coefficients of the denominator
of the closed-loop transfer function (characteristic equation).
• Start with the coefficient of the highest power and skip every other power of s.
• Now form the second row with the coefficients of the denominator skipped in the
previous step.
• The table is continued horizontally and vertically until zeros are obtained.
• For convenience, any row can be multiplied or divide by a positive constant before
the next row is computed without changing the values of the rows below and
disturbing the properties of the Routh table.
Routh’s Stability Condition
• If the closed-loop transfer function has all poles in the left half of the s-plane,
the system is stable. Thus, a system is stable if there are no sign changes in
the first column of the Routh table.
• The Routh-Hurwitz criterion declares that the number of roots of the
polynomial that are lies in the right half-plane is equal to the number of sign
changes in the first column. Hence the system is unstable if the poles lies on
the right hand side of the s-plane.
Example: Generating a basic Routh Table.
• Only the first 2 rows of the array are obtained from the characteristic eq. the
remaining are calculated as follows;
Four Special Cases or Configurations in the First
Column Array of the Routh’s Table:
1. Case-I: No element in the first column is zero.
2. Case-II: A zero in the first column but some other elements of the row
containing the zero in the first column are nonzero.
3. Case-III: A zero in the first column and the other elements of the row
containing the zero are also zero.
4. Case-IV: As in the third case but with repeated roots on the jw -axis.
Case-I: No element in the first column is zero.
Second-Order System.
The characteristic polynomial of a second order system is given below
The Routh array is written as
Where
The requirement for a stable second order system is simply that all the
coefficient be positive or all the coefficient s be negative.
Third-Order System.
The characteristic polynomial of a third order system is given below
The Routh array is
Where
• The requirement for a stable third order system is that the coefficients be positive
and
• The condition when results in a marginally stability case (recognized as
Case-3 because there is a zero in the first column) and one pair of roots lies on
the imaginary axis in the s-plane.
Example-1: Find the stability of the continues system having the characteristic
equation of
The Routh table of the given system is computed and shown is the table below;
• Since there is no changes of the sign in the first column of the Routh table, it
means that all the roots of the characteristic equation have negative real parts
and hence this system is stable.
Example-2: Find the stability of the continues system having the characteristic
polynomial of a third order system is given below
• The Routh array is
• Because TWO changes in sign appear in the first column, we find that two roots
of the characteristic equation lie in the right hand side of the s-plane. Hence the
system is unstable.
Example-3: Determine a rang of values of a system parameter K for which the
system is stable.
• The Routh table of the given system is computed and shown is the table below;
• For system stability, it is necessary that the conditions 8 – k >0, and 1 + k > 0,
must be satisfied. Hence the rang of values of a system parameter k must be lies
between -1 and 8 (i.e., -1 < k < 8).
Example-4: Find the stability of the system shown below using Routh criterion.
The close loop transfer function is shown in the figure
The Routh table of the system is shown in the table
Because TWO changes in sign appear in the first column, we find that two roots of the
characteristic equation lie in the right hand side of the s-plane. Hence the system is
unstable.
