FEEDCON - Lesson 10 - Classical Design in the s Domain

8/13/2019 FEEDCON - Lesson 10 - Classical Design in the s Domain

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Feedback ControlSystems Engineering

Lesson 10: Classical Design in the s-domain

Joshua NatividadFEEDCON

6 July 2007


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Objectives

Learn how to determine stability ofdynamic systems

Familiarize with Routh-Hurwitz stabilitycriterion


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References

R. S. Burns (2001), Advanced ControlEngineering , USA: Butterworth-Heinemann


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Stability of DynamicSystems

The response of a linear system to astimulus has two components:

steady-state terms which are directly related

to the input; and transient terms which are either exponential,

or oscillatory with an envelope of exponentialform


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Stability Defined

If the exponential terms decay as timeincreases, then the system is said to bestable.

If the exponential terms increase as timeincreases, then the system is said to beunstable.


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Stability Examples


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System Responses

Whichvariabledoes thestability of

the systemdependon?


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System Responses

Stability of adynamicsystemdepends uponthe sign of theexponential

index in thetime responsefunction,which involvesfinding realroots of the

characteristicequation.


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Stability and Roots of theCharacteristic Equation

The characteristic equation of asecond order system is given by

whose rootsare found from


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Response from Roots

The roots determine the response of thesystem,

Overdamping

Critical Damping

Underdamping


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Response from Roots

If the coefficient b were to be negative,then the roots would be


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Stable Response

Underdamping response, the timeresponse is given as


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Unstable Response


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Stability

If any of the roots of thecharacteristic equation

have positive real parts,then the system will be

unstable.--The statement is true even for higher ordered systems.


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Stability Criterion

Routh (1905) and Hurwitz (1875) gives amethod of indicating the presence andnumber of unstable roots, but not their

value.


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Routh-Hurwitz Criterion

If (a) is satisfied, then the necessary andsufficient condition for stability is either

b) All the Hurwitz determinants of the

polynomial are positive, or alternatively

c) All coefficients of the first column ofRouthsarray have the same sign. The

number of sign changes indicate thenumber of unstable roots.


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Hurwitz Determinants


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Routh Arrays


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Example

Check the stability of the system which hasthe following characteristic equation

Test 1: All coefficients are present andhave the same sign. Proceed to Test 2.


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Example

4th

order


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Example

3rd

order


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Example

-4

2


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Example

-4

-4


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Example

+16

--0


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Example

There are two sign changes in the columntherefore there are two roots with positivereal parts. Hence, the system is unstable.


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Stability of a Closed Loop System

The closed-loop transfer function wasdetermined to be

The zero or roots is determined from thedenominator and equated to zero to form

the characteristic equation


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Stability of a Closed-Loop System

Find the value of K1 such that the systemis unstable.

What is the transfer function?


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What is the transfer function?

The transfer function is


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The open loop gain constant is

such that the transfer function becomes


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Since the roots can be found from

the characteristic equation is therefore

AlternativeMethod


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Alternative Method

Start with the definition of the transferfunction

with H=1, the transfer function is writtenas


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Alternative Method

Multiply numerator and denominator with

to get

which simplifies into


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Alternative Method

Equate the denominator to zero in order toobtain the characteristic equation

BACK


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TheRouth Arrayis


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To produce a sign change in the firstcolumn,K 2(!).

SinceK= 8K1,

to make the system

just unstable,

K1

= 0.25.


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AtK= 2, the characteristic equation

becomes

Factorizing yields

{


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And the transient response is


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Special Cases of the Routh Array

Case 2: All elements in a row arezero.

If all the elements of a row is zero, replace

that row with derivatives of an auxiliarypolynomial, formed from the elements ofthe previous row.


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FEEDCON - Lesson 10 - Classical Design in the s Domain

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