The stability of a control system is often extremely important and is generally a safety issue in the engineering of a system. An example to illustrate the importance of stability is the control of a nuclear reactor. An instability of this system could result in an unimaginable catastrophe.
DefinitionsThe stability of a system relates to its response to inputs or disturbances.
A system which remains in a constant state unless affected by an external action
and which returns to a constant state when the external action is removed can be considered
to be stable.
A systems stability can be defined in terms of its response to external impulse inputs..
Definition .a :
A system is stable if its impulse response approaches zero as time approaches
infinity..
The system stability can also be defined in terms of bounded (limited) inputs..
Definition .b:
A system is stable if every bounded input produces a bounded output.
Control analysis is concerned not only with the stability of a system but also the degree of stability of a system.. A typical system equation without considering the concept of integral action is of the form.
[ a _{2} D ^{2} + a _{1} D + a _{0} ].x = f(D) y
This is defined as be the highest order of D on the LHS as a equation of order 2.
The transient response, and as a result the stability, of such a system depends on
the coefficients a _{0}, a _{1} , a _{2}.
Assuming
a _{0} >0 then provided that a _{1} >0 and a _{2} >0 the complementary
function will not contain any positive time exponentials and the system will be stable.
If either a _{1} < 0 (negative damping) or a _{2} < 0 (negative mass) the transient response will contain
positive exponentials and the system will be unstable..
If a _{1} = 0 (As resulting from zero damping) then the complementary function
will oscillate indefinitely. This is not an unstable response but this marginally stable
response is not satisfactory.
Following are a number of plots to illustrate the types of stability responses resulting
from an input...
The notes below relate specifically to the Hurwitz stability criterion and is applied to time domain equations. The similar and
more generally used Routh stability criteria is described on a separate page with respect to Laplace transformed equations (using the complex variable s ) Routh Stability Criteria
Consider the generalised control equation
[a _{n}.D_{n} + a _{n1}.D_{n1} +..... a _{1}.D + a _{0} ]x = f(D) y
Assuming (or making) a _{0} is positive..
A determinant is created of the coefficients
a _{1}  a _{0}  0  0  0  0  0  .  .  . 
a _{3}  a _{2}  a _{1}  a _{0}  0  0  0  .  .  . 
a _{5}  a _{4}  a _{3}  a _{2}  a _{1}  a _{0}  0  .  .  . 
a _{7}  a _{6}  a _{5}  a _{4}  a _{3}  a _{2}  a _{1}  a _{0}  .  . 
a _{9}  a _{8}  a _{7}  a _{6}  a _{5}  a _{4}  a _{3}  a _{2}  a _{1}  a _{0} 
If an equation of order n is under consideration all factors of a_{n} and above are replaced by 0
For stability of an equation of degree 4 the necessary conditions are as follows
1)...
a _{1} > 0 , a _{2} > 0 , a _{3} > 0, a _{4} > 0
2)...Determinant

> 0 
That is (a _{1}. a _{2} — a _{0}.a _{3} ) > 0
3)...Determinant

> 0 
That is a _{1}(a _{2}.a _{3} — a _{1} a _{4} ) — a _{0} (a _{3}. a _{3}  a _{ 1} .0 ) > 0
Considering a control system which has an output loop variable x of the form
x = A cos ( ω_{n}. t ) as the complimentary function
(transient) part of the complete solution..
Treating this in exponential form x = Re.( A.e ^{jωnt} )
where ω is the natural frequency and A is the real amplitude..
considering the third order equation..
[ a _{3} D ^{3} + a _{2} D ^{2} + a _{1} D + a _{0} ]. A.e ^{jωnt} = 0
Now
D. A.e ^{jωnt} = jω_{n}.A.e ^{jωnt}
D^{2}. A.e ^{jωnt} = (jω_{n})^{2}.A.e ^{jωnt}
D^{3}. A.e ^{jωnt} = (jω_{n})^{3}.A.e ^{jωnt}
or in general terms
D^{r}. A.e ^{jωnt} = (jω_{n})^{r}.A.e ^{jωnt}
The equation above becomes.
(a _{3}. (jω_{n})^{3} + a _{2}. (jω_{n})^{2} + a _{1}.(jω_{n})^{1} + a _{0} ) .A.e ^{jωnt} = 0
As A.e ^{jω n t} is not zero the equation can be written..
(a _{3}. (jω_{n})^{3} + a _{2}. (jω_{n})^{2} + a _{1}.(jω_{n})^{1} + a _{0} ) = 0
Bring the real and imaginary terms together..
( a _{2}.ω_{n}^{2} + a _{0 } ) + ω_{n}.( a _{3}. ω_{n}^{2} + a _{1} ).j = 0
The real parts and the imaginary parts must each be equal to zero therefore..
ω_{n}^{2} = a_{0} / a_{2} = a_{1} / a_{3}
These conditions identify that the third order system is marginally stable and will oscillate continuously at a circular frequency ω_{n}...
Stability analysisTo know that the system is stable is not generally sufficient for the requirements of control
system design. There is a need for stability analysis to determine how close the system is
to instability and how much margin when disturbances are present and when the gain is adjusted..
The objectives of stability analysis is the determination of the following
The standard method of completing a system analysis includes the following steps..
A number of methods are available for determining the system characteristics including the following.