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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]


Contents

Introduction

System Stability

Routh-Hurwitz Criterion

Construction of Routh Table

Determining System Stability

Contents


Introduction

  • 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.

Introduction


Introduction1

Introduction


Absolute relative stability

  • Absolute Stability: bounded-input, bounded-output (BIBO):

    • 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 stability1

Stable systems bounded-input, bounded-output (BIBO):have closed-loop transfer functions with poles only in the left half-plane.

System Stability


System stability2

Unstable systems bounded-input, bounded-output (BIBO):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: consider the manner in which the system is operating, whether open-loop or closed-loop.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: consider the manner in which the system is operating, whether open-loop or closed-loop.

  • 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

Where,

an, an-1, …, a1, a0 are constants

n = 1, 2, 3,…, ∞


Routh hurwitz criterion2

For the sufficient condition, we must form a Routh-array. consider the manner in which the system is operating, whether open-loop or closed-loop.

Routh-Hurwitz Criterion


Routh hurwitz criterion3

For the sufficient condition, we must formed a Routh-array. consider the manner in which the system is operating, whether open-loop or closed-loop.

Routh-Hurwitz Criterion


Construction of routh table
Construction of consider the manner in which the system is operating, whether open-loop or closed-loop.Routh Table

Equivalent closed-loop transfer function

Initial layout for Routh table

Completed Routh table


Determining system stability1

Example: consider the manner in which the system is operating, whether open-loop or closed-loop.How many roots exist on RHP?

Determining System Stability


Further reading

  • Chapter 6 consider the manner in which the system is operating, whether open-loop or closed-loop.

    • 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…



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