Stable or Unstable System (Stability Property)

In the previous post we have seen linear and non linear systems. Here we will see how to determine whether the system is stable or unstable i.e. stability property. To define stability of a system we will use the term ‘BIBO’. It stands for Bounded Input Bounded Output. The meaning of word ‘bounded’ is some finite value. So bounded input means input signal is having some finite value. i.e. input signal is not infinite. Similarly bounded output means, the output signal attains some finite value i.e. the output is not reaching to infinite level.

Definition of stable system:

An infinite system is BIBO stable if and only if every bounded input produces bounded output.

Mathematical representation:

Let us consider some finite number Mx whose value is less than infinite. That means Mx < 8, so it’s a finite value. Then if input is bounded, we can write,

|x(n)| = Mx < 8

Similarly for C.T. system

|x(t)| = Mx < 8

Similarly consider some finite number My whose value is less than infinity. That means My < 8, so it’s a finite value. Then if output is bounded, we can write,

|y(n)| = My < 8

Similarly for continuous time system

|y(t)| = My < 8

Definition of Unstable system:

An initially system is said to be unstable if bounded input produces unbounded (infinite) output.

Significance:

  • Unstable system shows erratic and extreme behavior.
  • When unstable system is practically implemented then it causes overflow.

Solved problem on stability:

Determine whether the following discrete time functions are stable or not.

1)      y(n) = x(-n)

Solution: we have to check the stability of the system by applying bounded input. That means the value of x(-n) should be finite. So when input is bounded output will be bounded. Thus the given function is Stable system.

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