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.

The 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 _n-1.D_n-1 +..... a ₁.D + a ₀ ]x = f(D) y

Assuming (or making) a ₀ is positive..

A determinant is created of the coefficients

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

That is     (a ₁. a ₂ — a ₀.a ₃ )   >  0

3)...Determinant

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 ^jω_nt ) 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 ^jω_nt = 0

Now
D. A.e ^jω_nt = jω_n.A.e ^jω_nt
D². A.e ^jω_nt = (jω_n)².A.e ^jω_nt
D³. A.e ^jω_nt = (jω_n)³.A.e ^jω_nt
or in general terms
D^r. A.e ^jω_nt = (jω_n)^r.A.e ^jω_nt
The equation above becomes.

(a ₃. (jω_n)³ + a ₂. (jω_n)² + a ₁.(jω_n)¹ + a ₀ ) .A.e ^jω_nt = 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_n² + a ₀ ) + ω_n.(- a ₃. ω_n² + a ₁ ).j = 0

The real parts and the imaginary parts must each be equal to zero therefore..

ω_n² = a₀ / a₂ = a₁ / a₃

These conditions identify that the third order system is marginally stable and will oscillate continuously at a circular frequency ω_n...

To 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.