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System Stability. DNT 354 - Control Principle. Date: 4 th September 2008 Prepared by: Megat Syahirul Amin bin Megat Ali Email: megatsyahirul@unimap.edu.my. 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

## DNT 354 - Control Principle

Date: 4th September 2008

Prepared by: MegatSyahirulAmin bin Megat Ali

Email: megatsyahirul@unimap.edu.my

Introduction

System Stability

Routh-Hurwitz Criterion

Construction of Routh Table

Determining System Stability

### Contents

• 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

• 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 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

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

### System Stability

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

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

### 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 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

• 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,…, ∞

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

### Routh-Hurwitz Criterion

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

### Construction of Routh Table

Equivalent closed-loop transfer function

Initial layout for Routh table

Completed Routh table

Example: How many roots exist on RHP?

### Determining System Stability

• 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.