Stability

Definition

• The stability of a continuous or discrete-time system is determined by its response to inputs or disturbances.

• A continuous system is stable if its impulse response approaches zero as time approaches infinity.

• A continuous system is stable if every bounded input produces a bounded output.

Routh Stability Criterion

• The Routh criterion is a method for determining continuous system stability, for systems with an nth-order characteristic equation of the form:

• All the roots of the characteristic equation have negative real parts if and only if the elements of the first column of the Routh table have the same sign. Otherwise, the number of roots with positive real parts is equal to the number of changes of sign.

Example

Hurwitz Stability Criterion

• The Hurwitz criterion is another method for determining whether all the roots of the characteristic equation of a continuous system have negative real parts. This criterion is applied using determinants formed from the coefficients of the characteristic equation.

Example

• For n=3

Continued Fraction Stability Criterion

• This criterion is applied to the characteristic equation of a continuous system by forming a continued fraction from the odd and even portions of the equations, in the following manner.

• Form the fraction Q1/Q2, and then divide the denominator and invert the remainder to form a continued fraction:

• If h1, h2, … hn are all positive, then all the roots of Q(s)=0 have negative real parts.

Example

Stability Criteria for Discrete-Time Systems

• A stability criterion for discrete systems similar to the Routh criterion is called the Jury test. For this test, the coefficients of the characteristic equation are first arranged in the Jury array:

• Jury Test: Necessary and sufficient conditions for the roots of Q(z) = 0 to have magnitudes less than one are:

Example