ECE 4150 - CONTROL SYSTEMS
CHAPTER 6
Chapter 6 - Relative Stability
• The Routh-Hurwitz criterion only verifies the absolute
stability of a system. i.e. it can ascertain whether the
system is stable, unstable, or marginally stable. It does
not provide an indication of relative stability, i.e. no
indication of how stable or unstable a system is.
• The relative stability of a system can be defined as a
property of the relative real parts of the roots, or in terms
of the damping coefficient of each pair of complex roots.
• If we consider that the farther the roots are from the jω-
axis, the more stable a system is
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ECE 4150 - CONTROL SYSTEMS
CHAPTER 6
If we shift the jω-axis we can utilize the Routh-Hurwitz
criterion to determine relative stability. For example if we
have two systems, System1 with the roots r2, r1, and ȓ1,
and System2 with roots r2, r3, and ȓ3, by moving the jωaxis to -σ1 and generating the Routh arrays we will see
that System1 will be marginally stable and System2
remains absolutely stable. This would indicate that
System2 is relatively more stable than System1.
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ECE 4150 - CONTROL SYSTEMS
CHAPTER 6
• Example: shift the axis to determine where the roots are
for
By inspection the function is stable, and is confirmed by
the Routh array
We can shift the jω-axis to -2 by substituting (sn-2) for s
this will place the jω-axis on the root s=-2
The resulting Routh array is
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ECE 4150 - CONTROL SYSTEMS
CHAPTER 6
If we further shift the jω-axis to -3 (sn=s+3 → s=sn–3), it
should place the root at s=-2 into the RHS of the
complex plane.
This is confirmed by the Routh array. If we did not know
the roots, by examining the Routh arrays for s and
s=sn–3 we could say that there is a root between s=0
and s=-3.
• Example: determine if all the roots of the following
characteristic equation have a real component less than
-1.
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ECE 4150 - CONTROL SYSTEMS
CHAPTER 6
which is telling us that the system is stable even when
we shift the axis by 1.
Since the auxiliary equation is a factor of the original
equation we can find the roots of the original equation,
and the third root via
Therefore the actual roots are: s=-1±j and s=-2.
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