lecture2 methods of determining stability-rhc

7/27/2019 Lecture2 Methods of Determining Stability-rhc

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Methods of Determining Stability

Eng R. L. Nkumbwa

Copperbelt University

2010


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Determining Stability

For all practical purposes, there is no need to

compute the complete system response todetermine stability.

When the system parameters are all know the

roots of the characteristic equation can be found

using various methods including the AutomaticControl Systems Software (ACSYS) in

conjunction with MATLAB Toolbox.

10/3/20132 Eng. R. L. Nkumbwa @ CBU2010


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Determining Stability

What are some of the methods for

determining stability in control engineering?



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Methods for Determining Stability

RouthHurwitz Criterion this is an algebraic method

that provides information on the absolute stability of a

system that has as characteristic equation with constant

coefficients. The criterion tests whether any of the roots

of the characteristic equation lie in the right half s-plane.

Nyquist Criterion this is a semi graphical method that

gives information on the difference between the numberof poles and zeros of the closed loop transfer function

that are in the right half s-plane by observing the

behavior of the Nyquist plot of the loop transfer function.



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Methods for Determining Stability

Bode Diagram this is the diagram plot ofthe magnitude of the loop transfer functionG(j) H (j) in dB and the phase of G (j)

H (j) in degrees, all verses frequency .

The stability of the closed loop system can bedetermined by observing the behavior ofthese plots.



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Tip Off

It will be evident throughout this course, that

most of the design and analysis techniqueson control systems will represent alternative

methods of solving the same problem.

The designer simply has to choose the bestanalytical tool, depending on the particular

situation.



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Tip Off

In the course of normal system design, it is not

merely necessary that the system be stable. It is essential that the system be sufficiently

stable that transient disturbances will decay

quickly enough to permit rapid recovery by the

controlled variable. In the next sections, however, the major

emphasis is on concepts and methods of

determining system stability.10/3/20137 Eng. R. L. Nkumbwa @ CBU2010


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Routh-Hurwitz Criterion

What is RHC Analysis?

The RouthHurwitz Criterion (RHC) representsa method of determining the location of zeros

of a polynomial with constant real coefficients

with respect to the left half and right half of the

s-plane, without actually solving for the zeros.



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Reminder



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Output Signals Characteristic

for Stability, Neutral & Unstable



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Stability Condition

A necessary and sufficient condition for a

feedback system to be stable is that allthe poles of the system transfer function

have negative real parts.



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Routh-Hurwitz Criterion Background

The stability analysis has occupied the interest ofmany engineers including Maxwell andVishnegradsky.

In the late 1800s, A. Hurwitz and E.J. Routhindependently published a method ofinvestigating the stability of a linear system.

The Routh-Hurwitz stability method provides ananswer to the question of stability by consideringthe characteristic equation of the system.



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Routh-Hurwitz Criterion

The Routh-Hurwitz criterion is based on

ordering the coefficients of the characteristicequation:



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Routhian Array

Into an array as follows:



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Routhian Rows

The rows of the schedule are then completed

as follows:



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Routhian Rows



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Hurwitz Determinants

Where,



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RHC Principle

The Routh-Hurwitz criterion states that thenumber of roots of D(s) with positive real parts isequal to the number of changes in sign in thefirst column of the Routh Array.

This criterion requires that there be no changesin sign in the first column for a stable system.

This requirement is both necessary andsufficient.



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Four Cases for First Column Array

No element in the first column is zero.

There is a zero in the first column, but otherelements of the row contain a zero in the first

column are non-zero.

There is a zero in the first column, the other

elements of the row contain a zero are also zero.

There is a zero in the first column, and the other

elements of the row contain the zero are also

zero, with repeated roots on the jw axis.10/3/201320 Eng. R. L. Nkumbwa @ CBU2010


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Example: Third-Order System

The characteristic polynomial of a third-

order system is:



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The polynomial satisfies all the necessary

conditions because all the coefficients existand are positive.

Utilizing the Routh array, we have:

10/3/201322Eng. R. L. Nkumbwa @ CBU

2010


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Because there are two changes in sign in the

first column, we find that two roots of D(s) liein the right- hand plane, and our prior

knowledge is confirmed.


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Exercise

To illustrate this method clearly, you are

required to do a research for each case.


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Routh-Hurwitz Criterion Limitations

Routh-Hurwitz Criterion is only valid, if the

characteristic equation is algebraic with real

coefficients. If any one of the coefficients is

complex or if the equation is not algebraic, for

example, containing experimental functions or

sinusoidal functions of s, RHC simply cannot beapplied.


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Routh-Hurwitz Criterion Limitations

The RHC cannot be applied to any other

stability boundaries in a complex plane, such

as the unit circle in the z-plane, which is the

stability boundary of discrete data systems.


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MATLAB & ACSYS LAB

Spend time in the lab learning MATLAB and

ACSYS Software.


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lecture2 methods of determining stability-rhc

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