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System Stability. DNT 354 - Control Principle. Date: 4 th September 2008 Prepared by: Megat Syahirul Amin bin Megat Ali Email: [email protected] Introduction System Stability Routh-Hurwitz Criterion Construction of Routh Table Determining System Stability. Contents.

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system stability
System Stability

DNT 354 - Control Principle

Date: 4th September 2008

Prepared by: MegatSyahirulAmin bin Megat Ali

Email: [email protected]



System Stability

Routh-Hurwitz Criterion

Construction of Routh Table

Determining System Stability


Stability is the most important system specification. If a system is unstable, the transient response and steady-state errors are in a moot point.

  • Definition of stability, for linear, time-invariant system by using natural response:
    • A system is stable if the natural response approaches zero as time approaches infinity.
    • A system is unstable if the natural response approaches infinity as time approaches infinity
    • A system is marginally stable if the natural response neither decays nor grows but remains constant or oscillates.

Definition of stability using the total response bounded-input, bounded-output (BIBO):

    • A system is stable if every bounded input yields a bounded output.
    • A system is unstable if any bounded input yields an unbounded output.
absolute relative stability

Absolute Stability:

    • The absolute stability indicates whether the system is stable or not.
    • This is indicated by the presence of one or more poles in RHP.
  • Relative Stability:
    • Relative stability refers to the degree of stability of a stable system described by above.
    • This depends on the transfer function of the system, which is represented by both the numerator (that yields the zeros) and denominator (that yields the poles).
    • This can then be referred to in the study of system response either in time or frequency domain.
Absolute & Relative Stability
system stability2

Unstable systems have closed-loop transfer functions with at least one pole in the right half-plane and/or poles of multiplicity greater than 1 on the imaginary axis.

System Stability
system stability3

Marginally stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity 1 and poles on the imaginary axis.

System Stability
determining system stability

To determine stability of a given system, we have to consider the manner in which the system is operating, whether open-loop or closed-loop.

    • If the system is operating in closed-loop, first find the closed loop transfer function.
    • Find the closed-loop poles.
    • If the order of the system is 2 or less, factorise the denominator of the transfer function. This will provide the roots of the polynomial, or the closed-loop poles of the system.
    • If the system order is higher than 2nd-order, use construct Routh table and apply Routh-Hurwitz Criterion.
    • Any poles that exist on the RHP will indicate that the system is unstable.
Determining System Stability
routh hurwitz criterion

Routh-Hurwitz Stability Criterion: The number of roots of the polynomial that are in the right half-plane is equal to the number of changes in the first column.

  • Systems with the transfer function having all poles in the LHP is stable.
  • Hence, we can conclude that a system is stable if there is no change of sign in the first column of its Routh table.
  • However, special cases exists when:
    • There exists zero only in the first column.
    • The entire row is zero.
Routh–Hurwitz Criterion
routh hurwitz criterion1

If a polynomial is given by:

  • The necessary conditions for stability are:
    • All the coefficients of the polynomial are of the same sign. If not, there are poles on the right hand side of the s-plane.
    • All the coefficient should exist accept for a0.
Routh–Hurwitz Criterion


an, an-1, …, a1, a0 are constants

n = 1, 2, 3,…, ∞

construction of routh table
Construction of Routh Table

Equivalent closed-loop transfer function

Initial layout for Routh table

Completed Routh table

further reading

Chapter 6

    • Nise N.S. (2004). Control System Engineering (4th Ed), John Wiley & Sons.
    • Dorf R.C., Bishop R.H. (2001). Modern Control Systems (9th Ed), Prentice Hall.
Further Reading…