System stability
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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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DNT 354 - Control Principle

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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

  • 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.

Introduction


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 stability1

Stable systems have closed-loop transfer functions with poles only in the left half-plane.

System 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

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.

Routh-Hurwitz Criterion


Routh hurwitz criterion3

For the sufficient condition, we must formed a Routh-array.

Routh-Hurwitz Criterion


Construction of routh table

Construction of Routh Table

Equivalent closed-loop transfer function

Initial layout for Routh table

Completed Routh table


Determining system stability1

Example: How many roots exist on RHP?

Determining System Stability


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…


If we knew what we were doing it would not be called research would it

“If we knew what we were doing, it would not be called research, would it?…"

The End…


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