8/18/2019 Absolute Stability

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  tability of

ystems

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Based on the above observations regarding location of

rOots

of the characteristic equation and

stability

of

the system, the s-plane can be divided into two regions. The left side

of

imaginary axis is

the stable region and

if

roots of characteristic equation occur in this

half

of s-plane, the system is

stable. The roots may be simple or may occur with any multiplicity. On the other hand, even

if

s-plane, the system is unstable.

If

simple roots occur on the

ja>-axis

which

is

the boundary for these two regions, the system

is

said to be marginally or limitedly stable.

If

roots

on a>-axis

are repeated, the system

is

unstable. The stable and unstable regions

of

jro

s-plane

Stable Unstable

Fig. 4.2 Demarcation of stable and unstable regions of s plane.

Hence, determination

of

stability of a linear time invariant system boils down to determining

whether any roots of the characteristic equation or poles of transfer function lie in the right half of

s-plane. One obvious method

of

determining stability

of

a system is to find all the roots

of

D s)

=

sn

+ a

l

sn

-

1

+ ..... +

an'

For a polynomial of degree

n

> 2, it is difficult to find the roots analytically. Numerical methods

for root determination for higher order polynomials is quite cumbersome. Hence, algebraic criteria

are developed to find out

if

any roots lie in the right half of s-plane. The actual location of the roots is

unimportant.

4.3 Necessary

Conditions for Stability

In section 4.2 we have seen that the system will be stable

if

the roots

of

the characteristic equation lie

in the left half of s-plane. The factors of the characteristic polynomial D s) can have terms like

s +

0 ,

s +

O k)2

+

rof

where

0 1

are positive and real. Thus

D s) = a

o

sn + a

l

sn - 1 + ..... +a

n

..... 4.4)

=

a

o

IT

s + 0 1)

IT

{ s +

O k)2

+

rof}

..... 4.5)

Since

0 1

and O k are all positive and real the product in eqn. 4.5) results in all positive and real

coefficients in the polynomial of

s.

Thus

if

the system

is

stable all the coefficients, ai must be

positive and real.

Further, since there are no negative terms involved in the product

of

eqn. 4.5), no cancellations

can occur and hence no coefficient can be zero. Thus none of the powers of

s

in between the highest

and lowest powers of

s

must be missing. But,

if

a root is present at the origin

an

is zero.

s-plane are

shown in Fig. 4.2.

simple

roots occur in the right half of