Definition for user with engineering background

Stability is the state or quality of being stable, which characterizes bodies or systems in a state of steadiness and equilibrium (Ferguson, 1992, p. 936).

 

Types of Stability:

 

Internal Stability: Also known as Asymptotic Stability

Internal Stability notion is related to the state space, implying that all the states will decay to zero. Internal Stability of a control system is strictly related to its transfer function G(s) = B(s)/A(s) and the function’s poles. The transfer function’s poles can either be real (different or repeated), or complex-conjugate poles. In determining internal stability, three states are possible:

• A system is asymptotically stable if all poles lie in the LHP (left hand plane)
• A system is marginally, or critically stable if no poles lie in the RHP (right hand plane), and some unrepeated roots lie on the imaginary axis.
• A system is asymptotically unstable if: (a) there exists at least one root in the RHP, or (b) repeated roots lie on the imaginary axis.

 

(Christian Schmid, 2005)

 

External Stability: Also known as BIBO (Bounded Input Bounded Output) Stability.

 

“It is a condition such that any bounded input yields a bounded output. This is to say that as long as we input a stable signal, we are guaranteed to have a stable output.”

For a continuous time system, Y(t) = H(t) * X(t)

The condition for BIBO stability is:  

 

 

For a discrete time system, Y(n) = H(n) * X(n)
The condition for BIBO stability is:  

 

(Wikipedia, 2006)

 

Internal Vs External Stability:

 

Internal stability implies external stability; however the opposite is not true. If a system is asymptotically stable, then it is also BIBO stable. However, if a system is BIBO stable, it does not essentially mean that the system should also be asymptotically stable.

 

 

References

Furguson, J.G. (1992). Webster's Disctionary. New York: Pamco Publishing Company, Inc. 

Christian Schimd (2005, September). Stability of Linear Control. Retrieved October 11, 2006, from    

     http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node38.html

 

Wikipedia (2006, August). BIBO Stability. Retrieved October 09, 2006, from

     http://en.wikipedia.org/wiki/BIBO_stability