Routh
stability criterion
is a mathematical test to determine the
stability of a linear time invariant (LTI) control system.
This test requires the characteristic equation of the
control system under consideration. Characteristic
equation is nothing but equating the denominator of the closed loop
transfer function equal to zero.
Then arrange the characteristic equation terms in the decreasing
order of power of s from left to right as shown.
a0Sn+ a1Sn-1+
a2Sn-2+ a0Sn+ ... an-1S1+
anS0 = 0
Now, arrange the coefficients of the characteristic
equation into an array called Routh array as shown.
We get two rows. The first row consists of
coefficients a0 , a2 , a4 , a6 and so on. The second row consists of
coefficients a1 , a3 , a5 and so on.
Remember :- missing terms
are taken zero-coefficient.
All the remaining rows can be obtained from these two
rows as
and so on.
For the complete array obtained, Routh stability criterion
states that
"For a system to be stable, it is necessary and sufficient
that each term of the first column of the Routh array be positibe if a0 > 0 .
If this condition is not met, the system is unstable and the number of sign changes
of the terms of the first column of the Routh array = number of poles of the
given control system in the right half of the s-plane ".
Special
cases
when you will practice Routh stability criterion, you will find that in some problems, the
routh criterion breaks down. This may happen in two ways.
1. when the first terms
in any row is zero while the rest of the row has at least 1 non-zero term.
Because of this zero term, the terms in the next become infinite and the routh
test fails.
Ex :- S3-row
in
characteristic
equation
S5+ S4+ 2S3+ 2S2+
3S1+ 5 = 0
To overcome such situations, simply replace S by 1/Z
and apply Routh test for this newly obtained characteristic equation.
5Z5+ 3Z4+ 2Z3+ 2Z2+
Z1+ 1 = 0
2. when all the
elements in any row of the routh array are zero.
Ex :- S3-row
in
characteristic
equation
S6+2S5+ 8S4+ 12S3+
20S2+ 16S1+ 16 = 0
To solve such situations, make a polynominal from the
row just above the all zero row i.e S4-row
The polynomial will be S4+ 6S2+
8
, differentiate it w.r.t S,
we get 4S3+ 12S
Now, replace the all zero row with coefficients of the
above obtained polynomial i.e replace zeros with 4 and
12 .
Note :- Routh stability criterion only gives the number of roots in the right half of the s-plane. It gives no information about the nature and values of the roots.
Related concepts :-
(1) Block diagram reduction rules
(2) How to draw bode plot step-by-step
Related concepts :-
(1) Block diagram reduction rules
(2) How to draw bode plot step-by-step
