This document discusses stability analysis in the frequency domain using Routh's stability criterion. It defines absolute and relative stability and explains that Routh's criterion determines stability by analyzing the signs of coefficients in the characteristic equation's Routh array. The document provides examples of applying Routh's criterion to determine stability and calculating the range of a parameter value for stability. It also covers special cases and using the criterion to analyze relative stability by shifting the s-plane.

1. Classical
Control
Third Year
Second Semester
Dr. Amr A. Sharawi
T.A. Ghaidaa Eldeeb

2. STABILITY IN THE FREQUENCY
DOMAIN

3. Absolute Stability
• The most important characteristic of the dynamic
behavior of a control system is absolute stability.
• A control system is in equilibrium if, in the absence
of any disturbance or input, the output stays in the
same state.
• A linear time-invariant control system is stable if the
output eventually comes back to its equilibrium
state when the system is ONLY subjected to an
initial condition.

4. Absolute Stability
• A linear time-invariant (LTI) control system is
critically stable if oscillations of the output continue
forever.
• It is unstable if the output diverges without bound
from its equilibrium state when the system is
subjected to an initial condition.
• Thus, a LTI is stable, if it is a bounded-input-
bounded-output (BIBO) system.

5. Relative Stability
• An important system behavior (other than absolute
stability) to which we must give careful
consideration includes relative stability.
• Since a physical control system involves energy
storage, the output of the system, when subjected
to an input, cannot follow the input immediately
but exhibits a transient response before a steady
state can be reached.
• The duration of that transient response in the
practical sense is a measure of relative stability.

6. Time &
Frequency
Domain Stability
Analysis
Linear Time Invariant
System
(Frequency domain)
Complex Frequency
Domain
(Routh stability
analysis)
Real Frequency
Domain
(Nyquist stability
analysis)
Time-varying &
Nonlinear System
(Time domain)
Time Domain
Analysis
(Lyapunov stability
analysis)

7. Routh’s Stability Criterion
• The Routh’s stability criterion tells us whether or
not there are unstable roots in a polynomial
equation without actually solving for them.
• This stability criterion applies to polynomials with
only a finite number of terms.
• When the criterion is applied to a control system,
information about absolute stability can be
obtained directly from the coefficients of the
characteristic equation.

8. Routh’s Stability Analysis Procedure
• Write the polynomial in s in the following form:
where the coefficients are real quantities.
We assume that an an > 0; that is, any zero
root has been removed.
If any of the coefficients are zero or negative in the
presence of at least one positive coefficient,
a root or roots exist that are imaginary or that have
positive real parts.
Therefore, in such a case, the system is not stable.

9. Routh’s Stability Analysis Procedure
• If we are interested in only the absolute stability,
there is no need to follow the procedure further.
Note that all the coefficients must be positive.

10. Routh’s Stability Analysis Procedure
• If all coefficients are positive, arrange the coefficients of the
polynomial in rows and columns according to the following
pattern:

11. Routh’s Stability Analysis Procedure
• The process of forming rows continues until we run out of
elements. (The total number of rows is n+1). The
coefficients b1, b2, b3 , and so on, are evaluated as follows:

12. Routh’s Stability Analysis Procedure
• The evaluation of the b’s is continued until the remaining
ones are all zero.
• The same cross-multiplication pattern is followed in
evaluating the c’s, d’s, e’s, and so on. That is,

13. Routh’s Stability Analysis Procedure
This process is continued until the nth row has been
completed.
The complete array of coefficients is quasi-triangular.
In developing the array an entire row may be divided or
multiplied by a positive number in order to simplify the
subsequent numerical calculation without altering the
stability conclusion.

14. Implication of Routh’s Stability
Criterion
• Routh’s stability 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 coefficients of the first column of the array.
• The necessary and sufficient condition that all roots
of the characteristic equation lie in the left-half s-
plane is that all the coefficients of the characteristic
equation be positive and all terms in the first
column of the array have positive signs.

15. Example 5–11 – p. 214 – Ogata V
• Apply Routh’s stability criterion to the following
third-order polynomial:
• where all the coefficients are positive numbers.

16. Solution
• Routh’s array:
The condition that all roots have negative real parts is
given by

17. Example 5–12 – p. 214 – Ogata V
• Consider the following polynomial:
Solution
The first two rows can be obtained directly from the
given polynomial.
The second row can be divided by 2

18. Solution
• Having done so we get
𝑠3
1 2 0
We proceed with the remainder of the array as follows
The number of changes in sign of the coefficients in the
first column is 2.
This means that there are two roots with positive real
parts.

19. Special Cases
• If a first-column term in any row is zero, but the
remaining terms are not zero or there is no
remaining term, then the zero term is replaced by a
very small positive number ε and the rest of the
array is evaluated.

20. Example
• Consider the following equation:
The array of coefficients is
Now that the sign of the coefficient above the zero (ε) is
the same as that below it, it indicates that there are a pair
of imaginary roots (in that case at s = ±j).

21. Special Cases (Contd.)
• If, however, the sign of the coefficient above the
zero (ε) is opposite that below it, it indicates that
there are two sign changes of the coefficients in the
first column.
• In that case there are two coincident roots in the
right-half s-plane.

22. Relative Stability Analysis
• Routh’s stability criterion provides the answer to
• the question of absolute stability.
• This, in many practical cases, is not sufficient.
• We usually require information about the relative
stability of the system.
• A useful approach for examining relative stability is to
shift the s-plane axis and apply Routh’s stability
criterion.
• That is, we substitute
𝑠 = 𝑠−σ

23. Relative Stability Analysis
• into the characteristic equation of the system, write
the polynomial in terms of 𝑠 and apply Routh’s
stability criterion to the new polynomial in 𝑠
• The number of changes of sign in the first column of
the array developed for the polynomial in 𝑠 is equal
to the number of roots that are located to the right
of the vertical line s = –σ.
• Thus, this test reveals the number of roots that lie
to the right of the vertical line s = –σ.

24. Application of Routh’s Stability Criterion
to Control-System Analysis
• It is possible to determine the effects of changing
one or two parameters of a system by examining
the values that cause instability, particularly by
determining the stability range of one (or more)
parameter value.

25. Example
• Determine the range of K for stability for the system
shown.
Solution
The closed-loop transfer function is

26. Example (Contd.)
• The characteristic equation is
Routh’s array becomes
For stability, K must be positive, and all coefficients in
the first column must be positive. Therefore,