Automatic Control System

- 123 -

CHAPTER – 7
STABILITY OF LINEAR CONTROL SYSTEMS.

7.1. INTRODUCTION:
In preparation for the definition of stability, we define the two following types of responses for
linear time- invariant systems:
1. Zero-state response: The zero-state response is due to the input only; all the initial
conditions of the system are zero.
2. Zero-input response: The zero-input response is due to the initial conditions only; all
the inputs are zero.
From the principle of superposition, when a system is subject to both inputs and initial conditions, the
total response is written, Total response = Zero-state response + Zero-input response.
The definition just given applies to continuous-data as well as discrete-data system.
A system is said to be stable if its response cannot be made to increase indefinitely by the
application of a bounded input excitation. If the output approaches towards infinite value for sufficient
large time, the system is said to be unstable. A linear time invariant (LTI) system is stable if,
1. The system excited by a bounded input, the output is bounded (BIBO stability criteria).
2. In the absence of the input, the output tends towards zero (the equilibrium state of the
system). This is also termed asymptotic stable.
C ( s ) a m s m  a m1 s m1  .....  a1 s  a 0
Consider the transfer function,  …………...………. (7.1)
R(s) bn s n  bn 1 s n 1  .....  b1 s  b0

The output is given by, c(t )   g ( )r (t   )d ……...……………………………………….... (7.2)
0
1
Where, g ( )  L G ( s )  impulse response of the system. So, a system is said to be stable if the
impulse response approaches zero for sufficiently large time. If the impulse response approaches infinity
for sufficiently large time, the system is said to be unstable. If the impulse response approaches a
constant value for sufficiently large time, the system is said to be marginally stable.

7.2. BOUNDED-INPUT, BOUNDED-OUTPUT (BIBO) STABILITY-CONTINUOUS-DATA
SYSTEMS:
Let us consider a system that has input r(t), output c(t) and impulse response g(t). The impulse
response g(t) is inverse Laplace transform of G(s).
R(s) C(s)
G(s)
r(t) c(t)
Fig. 7.2.1
From figure 7.2.1 output is C(s) = G(s) R(s).

Applying convolution property of Laplace transform, c(t )   g ( )r (t   )d ,
0

If r(t) is bounded and has a magnitude M, then, r (t )  M   .
 
The magnitude of output can be expressed as c(t )   g ( )r (t   )d   g ( ) r (t   ) d
0 0

  M g ( ) d . ………………………………... (7.3)
0

, - 124 -

The output c(t ) is bounded when  g ( ) d   . [Note: Convolution in time domain is equal to
0
multiplication in frequency domain.]
We therefore conclude that the system is BIBO stable if and only if the impulse response is
absolutely integrable. Since G(s) is Laplace transform of g(t), we can also find the condition for stability
in frequency domain, which requires a revision of the topic poles and zeros.
[Poles: The poles of the system G(s) is defined as the roots of the denominator polynomial. That
is the values of s for which D(s) = 0.
Zeros: The zeros of the system G(s) is defined as the roots of the numerator polynomial. That is
the values of s for which N(s) = 0.]
7.3. RELATIONSHIP BETWEEN CHARACTERISTIC EQUATION ROOTS AND STABILITY:
The transfer function G(s), according to the Laplace transform defined as,

G ( s )  L[ g ( )]   g (t )e  st dt ……………………………………...…. (7.4)
0
Taking the absolute value on both sides of the last equation, we have
 
 st  st
G (s)   g (t )e dt   g (t ) e dt ………………..…………………. (7.5)
0 0

Since e  st  e t , where  is the real part of s. When s assumes a value of a pole of G(s), G(s) =  ,

equation (7.5) becomes    g (t ) e t dt …………………………………………………………. (7.6)
0

If one or more roots of the characteristic eq. are in the right-half s-plane or on the j -axis,   0 , then
e t  M  1 …………………………………………………………….. (7.7)
 
Equation (7.6) becomes    M g (t ) dt   g (t ) dt ………...…………………………………… (7.8)
0 0
This violates the BIBO stability requirement. Thus, for BIBO stability, the roots of the
characteristic equation or the poles of G(s), cannot be located in the right-half s-plane or on the j -axis,
in other words, they must all lie in the left-half s-plane. A system is said to be unstable if it is not BIBO
stable. When a system has roots on the j -axis, say, at s = j 0 and s = - j 0 , if the input is a sinusoid,
sin  0 t , then the output will be of the form of t sin  0 t , which is unbounded, and the system is unstable.

7.4. ZERO-INPUT AND ASYMPTOTIC STABILITY CONTINUOUS-DATA SYSTEMS:
Zero-input stability refers to the stability condition when the input is zero, and the system is
driven only by its initial conditions.
Let the input of an n th -order system be zero, and the output due to the initial conditions be c(t).
n 1
Then c(t) can be expressed as c(t )   g k (t )c k (t 0 ) ……………………………...…….. (7.9)
k 0

k d k c(t )
Where c (t 0 )  …………...………………………………………………. (7.10)
dt k
And g k (t ) denotes the zero-input response due to c k (t 0 ) . The zero-input stability is defined as
follows: If the zero-input response c(t), subject to the finite initial conditions, c k (t 0 ) , reaches zero as t
approaches infinity, the system is said to be zero-input stable, or stable; otherwise, the system is
unstable.