Lecture-Slides CHAPTER 06

ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali


Stability of Feedback Control Systems

Chapter 6


6.1: The Concept of Stability

6.2: The Routh-Hurwitz Stability Criterion

Outline


• Stability of a feedback control system

– Fundamentally important.

– An unstable closed-loop system is of no use in practice.

– Many physical systems are inherently open-loop unstable (e.g.

inverted pendulum, etc.).

– Among the performance specifications used in design, the

most important requirement is that the system must be stable.

– Classification:

• Relative stability

• Absolute stability



• Stable system:

– A dynamic system with a bounded response to a bounded

input.

– Bounded input bounded output (BIBO) stability

– A linear time-invariant control system is stable if the output

eventually comes back to its equilibrium state.

– It is marginally stable (neutral) if oscillations of the output

continue forever.

– It is unstable if the output diverges without bound from its

equilibrium state.



• The stability of a cone:



• Unit impulse response of a second order system for

various poles on s-plane



• Stability in the s-plane



• Mathematical description

– Consider the first-order differential equation:

– where 'a' is a constant

– Transfer function

– Unit impulse response:

– There are three cases:

i) a > 0, y(∞) = 0 → stable

ii) a = 0, y(∞) = 1 → neutral (marginally stable)

iii) a < 0, y(∞) = ∞ → unstable

– Stability is related to

• The location of poles of the system transfer function


assR

sY

1

0,

tetyat

trtay

dt

tdy




– A necessary and sufficient condition for a feedback system to

be stable is that ALL THE POLES of the system transfer

function have NEGATIVE REAL PARTS.



• Consider the transfer function of a feedback control

system:

– To check for stability, we have to find the poles of the

system.

– Poles of the system are the roots of the characteristic

equation

– The necessary conditions (but not sufficient) for the stability

are:

• All the coefficients have the same sign.

• Non of the coefficients are zero.


o

n

n

n

n

o

m

m

m

m

asasasa

bsbsbsbsT

1

1

1

1

1

1

01

1

1

o

n

n

n

nasasasasD


• RH stability method is used to assess system’s stability

– This method allow us to compute the number of poles in the

right half plane without computing the values of poles.

– The RH criterion is a necessary and sufficient criterion for

the stability of linear systems.



• The Routh-Hurwitz Criterion

– Take the characteristic equation:

– Make the Routh array/table:


1

321

321

531

42

3

2

1

h

ccc

bbb

aaa

aaa

s

s

s

s

s

nnn

nnn

o

n

n

n

n

01

1

1

o

n

n

n

nasasasasD


• Investigate the signs of the coefficients in the fist

column of Routh array.

– The number of roots of the characteristic equation with

positive real parts is equal to the number of changes in sign

of the first column of the Routh array



• Application of Routh's stability criterion in control

system analysis

– The R-H criterion can be used to determine the range of the

parameter values which maintain a stable system.

• Example:

– Consider closed-loop system shown below. Using the R-H

criterion, determine the range of K over which the system is

stable.



• Example continued …

– Transfer function:

– Routh’s table



• For stability, all the coefficients in the first column of

the Routh's table must be positive, i.e. we require:

4K/5 - 6 > 0 leading to K > 15/2

and K > 0

Therefore we choose a value for K > 15/2 for stability



• Determine the range of (K) and (a) for which the

system is stable.

Example 6.5

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Lecture-Slides CHAPTER 06