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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

- 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 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.

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

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.

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

- 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.

- 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.

- 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.

Where,

an, an-1, …, a1, a0 are constants

n = 1, 2, 3,…, ∞

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

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

Equivalent closed-loop transfer function

Initial layout for Routh table

Completed Routh table

Example: How many roots exist on RHP?

- 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.

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

The End…