# ELE 639 Lecture Notes - Lecture 2: Edward Routh, Adolf Hurwitz, Imaginary Number

ELE/BME639 Course Notes Winter 2011

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2 Stability

2.1 General Definition of Stability

Stability is an implicitly stated control objective. Intuitively, a closed loop system is stable if it does not

"blow up". For introduction, see Online Tutorials - sections on Basic Concepts and on Stability.

Recall from ELE532 that mathematically, stability is related to the location of the closed loop system

transfer function poles.

Definition: A system is stable in BIBO sense if, for every bounded input, the output remains bounded.

Consider now the transfer function of a basic closed loop system:

)()(1

)(

)( sHsG

sG

sGcl

)(

)(

)( sD

sN

sG

G

G

)(

)(

)( sD

sN

sH

H

H

)(

)(

)()()()(

)()(

)(

)(

)(

)(

1

)(

)(

)( sQ

sN

sNsNsDsD

sDsN

sD

sN

sD

sN

sD

sN

sG

HGHG

HG

H

H

G

G

G

G

cl

Equation 2-1

Characteristic equation of the closed loop system is:

0)(

sQ

Equation 2-2

tp

n

iinatural

forcednatural

forcednatural

jj

j

n

ii

i

cl

i

eKty

tytysYLty

sYsYsY

ps

K

ps

K

sU

sQ

sN

sY

sUsGsY

1

1

1

)(

)()()()(

)()()(

)(

)(

)(

)(

)()()(

Equation 2-3

Definition: Suppose that the closed loop system has a transfer function . The system is stable if, and

only if, the poles of shown in Equation 2-3 have strictly negative real parts: 0.

ELE/BME639 Course Notes Winter 2011

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To determine this condition analytically (as opposed to a numerical solution, such as provided by

MATLAB) a Criterion of Stability is defined - the Routh Test for Stability.

2.2 Locations in s-Plane vs. Time Response

Figure 2-1below shows three possible s-Plane locations for a pair of complex conjugate poles. Recall that

the pair of complex conjugate poles results in oscillatory time response, where the Real part of the pole

determines the decay rate of the response, while the Imaginary part of the pole determines the frequency

of oscillations.

Figure 2-1 Pole Locations vs. Time response

The first panel of Figure 2-1illustrates a stable response, the second panel illustrates an unstable response,

and the third panel illustrates a marginally stable response. The system respective behaviours will be

labeled as Stable, Unstable and Marginally Stable. Because we have the pair of complex conjugate poles,

the second, Ustable, case results in an oscillatory instability.

Note that these three behaviours can also be applied to a case where the system pole(s) are real. Real

poles result in transient(s) of exponential form. When 0, we will have a combination of

exponential decays, when 0 we will have a constant (step) response, and when 0, we will

have an exponentially increasing (but of single polarity, not oscillating) unstable response. Thus this case

is referred to as monotonic instability.

Refer to Online Tutorials for more on System Stability:

http://echo.ryerson.ca/flashcom/applications/controlsys/

ELE/BME639 Course Notes Winter 2011

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2.3 Stability in s-Domain: The Routh-Hurwitz Criterion of Stability

The original Criterion was formulated in a paper published in 1877 by Edward Routh, English

mathematician born in Upper Canada (now Quebec). In 1895 German mathematician Adolf Hurwitz

formulated the Criterion in its today's form, based on theory of polynomials. This is why the Criterion

bears both their names.

2.3.1 Necessary Condition for Stability

Definition: The Necessary Condition for Stability requires that all coefficients of the Characteristic

Equation polynomial are present and have the same sign. (In practice, it means that they should all be

positive, as negative signs would correspond to a negative Controller gain. A control system with a

negative gain is not practical as it would do exactly the opposite to the Command (Reference) input).

Where does the Necessary Condition come from? Consider the following. Once the Characteristic

Equation is factorized into a ZPK form, it will consist of two types of factors, 1st and 2nd order, as shown.

If the roots of these factors are in the LHP (i.e. the Stable Region of the s-plane), then the resulting

coefficients in these factors will be positive:

)(

)(

2kk

j

bsas

as

- stable factors

For example, 5

and 314

are factors corresponding to stable pole locations (-5 and -

1.5+j3.43, -1.5-j3.43, respectively), while 5

, 314

are factors corresponding to unstable pole

locations (+5 and +1.5+j3.43, +1.5-j3.43 respectively). Also note that one of the two poles corresponding

to the 314

factor is unstable (poles are: -5.53,+2.53).

0)(

)()()(

01

2

2

1

1

2

asasasasasQ

bsasassQ

n

n

n

n

kkk

jj

Equation 2-4

Conclusion 1: if only stable factors are present in Equation 2-4, after multiplication, the polynomial form of

the characteristic equation will have all powers of s terms present and all coefficients will be positive.

There is no possibility of having a negative sign or of a term cancellation resulting in a missing power of s,

since all factor signs were positive.

Conclusion 2: Any negative signs or any terms that are missing indicate presence of a factor or factors

describing unstable pole location(s).

Example:

013)( 3 sssQ roots are: 0.16 + j1.75, 0.16 - j1.75, -0.3222 (conjugate pair unstable)