MECH466 L6 Stability

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2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

University of British Columbia Lecture 6

Lecture 6

Stability

Routh

Routh--Hurwitz stability criterionHurwitz stability criterion

Course roadmap

Laplace transform Laplace transform Transfer function Transfer function

Models for systems

•

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical Linearization

Linearization Modeling

Modeling AnalysisAnalysis DesignDesign

Time response

•

•TransientTransient •

•Steady stateSteady state Frequency response

Frequency response

•

•Bode plotBode plot Stability

Stability

•

•RouthRouth--HurwitzHurwitz

•

•NyquistNyquist

Design specs Design specs Root locus Root locus Frequency domain Frequency domain

PID & Lead

PID & Lead--laglag Design examples

Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

Simple mechanical examples

We want mass to stay at x=0, but wind gave some We want mass to stay at x=0, but wind gave some initial speed (

initial speed (F(tF(t)=0). What will happen?)=0). What will happen?

How to characterize different behaviors with TF?How to characterize different behaviors with TF?

M

x(t

x(t)) f(t

f(t))

M

x(t

x(t))

f(t

f(t)) K

K

M

x(t

x(t)) f(t

f(t))

B

M

x(t

x(t))

f(t

f(t))

B B K K

Stability

Utmost important specification in control design!Utmost important specification in control design!

Unstable systems have to be stabilized by Unstable systems have to be stabilized by feedback.

feedback.

Unstable closedUnstable closed--loop systems are useless.loop systems are useless.

What happens if a system is unstable?What happens if a system is unstable?

•

• may hit mechanical/electrical may hit mechanical/electrical ““stopsstops””(saturation)(saturation)

•

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What happens if a system is unstable?

Tacoma Narrows Bridge (July 1

Tacoma Narrows Bridge (July 1--Nov.7, 1940)Nov.7, 1940)

2009

2009…… Wind

Wind--induced vibrationinduced vibration Collapsed!Collapsed!

Mathematical definitions of stability

BIBOBIBO(Bounded(Bounded--InputInput--BoundedBounded--Output) Output) stabilitystability: :

Any bounded input generates a bounded output.

Asymptotic stability Asymptotic stability ::

Any ICs generates

Any ICs generates y(ty(t) converging to zero.) converging to zero. BIBO stable

BIBO stable

system

u(t

u(t)) ICs=0ICs=0 y(ty(t))

Asymp

Asymp. stable . stable system

system

u(t

u(t)=0)=0

y(t

y(t))

ICs

Some terminologies

ZeroZero: roots of : roots of n(sn(s))

PolePole: roots of : roots of d(sd(s))

Characteristic polynomialCharacteristic polynomial: : d(sd(s))

Characteristic equationCharacteristic equation: : d(sd(s)=0)=0 Ex.

Ex.

Stability condition in s

-

domain

(Proof omitted, and not required)

For a system represented by a transfer For a system represented by a transfer function

function G(sG(s),),

system is BIBO stable system is BIBO stable

system is asymptotically stable system is asymptotically stable All the poles of

All the poles of G(sG(s) are in the open left ) are in the open left

half of the complex plane.

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Idea of stability condition

Asym

Asym. Stability: . Stability: (

(U(sU(s)=0))=0)

BIBO Stability: BIBO Stability:

(y(0)=0) (y(0)=0)

Example

Bounded if

Bounded if ReRe(α(α)>)>00

Remarks on stability

For a general system (nonlinear etc.), BIBO For a general system (nonlinear etc.), BIBO stability condition and asymptotic stability

stability condition and asymptotic stability

condition are different.

For For linear timelinear time--invariant (LTI) systemsinvariant (LTI) systems(to which (to which we can use Laplace transform and we can

we can use Laplace transform and we can

obtain a transfer function), the conditions

happen to be the same.

In this course, we are interested in only LTI In this course, we are interested in only LTI systems, we use simply

systems, we use simply ““stablestable””to mean both to mean both

BIBO and asymptotic stability.

Remarks on stability (cont

Remarks on stability (cont’

’d)

d)

Marginally stableMarginally stableifif

G(sG(s) has no pole in the open RHP (Right Half Plane), &) has no pole in the open RHP (Right Half Plane), &

G(sG(s) has at least one simple pole on ) has at least one simple pole on jwjw--axis, &axis, &

G(sG(s) has no multiple pole on ) has no multiple pole on jwjw--axis.axis.

UnstableUnstableif a system is neither stable nor if a system is neither stable nor marginally stable.

marginally stable. Marginally stable

Marginally stable NOT marginally stableNOT marginally stable

Stability summary

(BIBO, asymptotically) stable(BIBO, asymptotically) stableifif Re(s

Re(sii)<0 for all i.)<0 for all i.

marginally stablemarginally stableifif

Re(sRe(sii)<=0 for all i, and)<=0 for all i, and

simple root for simple root for Re(sRe(sii)=0)=0

unstableunstableifif

it is neither stable nor

marginally stable.

marginally stable. Let

Let ssiibe _bepoles_poles_{of G.}of G. Then, G is

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Mechanical examples: revisited

M

x(t

x(t)) f(t

f(t))

M

x(t

x(t))

f(t

f(t)) K

K

M

x(t

x(t)) f(t

f(t))

B

M

x(t

x(t))

f(t

f(t))

B B K K Poles= Poles= stable? stable? Poles= Poles= stable? stable? Poles= Poles= stable? stable? Poles= Poles= stable? stable?

Examples

Stable/marginally stable Stable/marginally stable /unstable /unstable ? ? ? ? ? ? ? ? ??? ???

Course roadmap

Laplace transform Laplace transform Transfer function Transfer function

Models for systems

•

•electricalelectrical

•

•mechanicalmechanical

•

•electromechanicalelectromechanical

Linearization

Linearization Modeling

Modeling AnalysisAnalysis DesignDesign

Time response

•

•TransientTransient

•

•Steady stateSteady state

Frequency response

•

•Bode plotBode plot

Stability

•

•RouthRouth--HurwitzHurwitz •

•NyquistNyquist

Design specs Design specs Root locus Root locus Frequency domain Frequency domain

PID & Lead

PID & Lead--laglag

Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

Routh

Routh-

-Hurwitz criterion

Hurwitz criterion

This is for LTI systems with a polynomialThis is for LTI systems with a polynomial

denominator (without sin,

denominator (without sin, coscos, exponential etc.), exponential etc.)

It determines if all the roots of a polynomial It determines if all the roots of a polynomial

lie in the open LHP (left halflie in the open LHP (left half--plane),plane),

or equivalently, have negative real parts.or equivalently, have negative real parts.

It also determines the number of roots of a It also determines the number of roots of a polynomial in the open RHP (right half

polynomial in the open RHP (right half--plane).plane).

It does It does NOTNOTexplicitly compute the roots.explicitly compute the roots.

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Polynomial and an assumption

Consider a polynomialConsider a polynomial

AssumeAssume

If this assumption does not hold, Q can be factored asIf this assumption does not hold, Q can be factored as

where

The following method applies to the polynomialThe following method applies to the polynomial

Routh

array

From the given

polynomial

Routh

array

(How to compute the third row)

Routh

array

(How to compute the fourth row)

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Routh

Routh-

-Hurwitz criterion

Hurwitz criterion

The number of roots The number of roots in the open right half in the open right half--plane plane

is equal to is equal to

the number of sign changes the number of sign changes in the

in the first columnfirst columnof Routhof Routharray.array.

Example 1

Routh

Routharrayarray

Two sign changes

in the first column

Two roots in RHP

Example 2

Routh

Routharrayarray

No sign changes

in the first column

No roots in RHP

Always same!

Example 3 (from slide 14)

Routh

Routharrayarray

No sign changes

in the first column

in the first column No roots in RHPNo roots in RHP

Always same!

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Simple & important criteria for stability

11stst_{order polynomial}_{order polynomial}

22ndnd_{order polynomial}_{order polynomial}

Higher order polynomialHigher order polynomial

Examples

All roots in open LHP? All roots in open LHP?

Yes / No Yes / No

Why no proof in textbooks?

“

“most undergraduate students are exposed to the most undergraduate students are exposed to the

Routh

Routh––Hurwitz criterion in their first introductory Hurwitz criterion in their first introductory controls course. This exposure, however, is at the

controls course. This exposure, however, is at the

purely algorithmic level in the sense that no attempt

is made whatsoever to explain why or how such an

algorithm works.

algorithm works.””

An Elementary Derivation of the Routh–Hurwitz Criterion

Ming-Tzu Ho, Aniruddha Datta, and S. P. Bhattacharyya IEEE Transactions on Automatic Control

vol. 43, no. 3, 1998, pp. 405-409.

Why no proof in textbooks? (cont

Why no proof in textbooks? (cont’

’d)

d)

“

“The principal reason for this is that the classical The principal reason for this is that the classical proof of the

proof of the RouthRouth--Hurwitz criterion relies on the Hurwitz criterion relies on the

notion of Cauchy indexes and Sturm

notion of Cauchy indexes and Sturm’’s theorem, s theorem, both of which are beyond the scope of

both of which are beyond the scope of

undergraduate students.

undergraduate students.””

“

“RouthRouth--Hurwitz criterion has become one of the few Hurwitz criterion has become one of the few results in control theory that most control engineers

results in control theory that most control engineers

are compelled to accept on faith.

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Summary and Exercises

Stability for LTI systemsStability for LTI systems

(BIBO, asymptotically) stable, marginally stable, unstable(BIBO, asymptotically) stable, marginally stable, unstable

Stability for Stability for G(sG(s) is determined by poles of G.) is determined by poles of G.

RouthRouth--Hurwitz stability criterionHurwitz stability criterionto determine stability to determine stability without explicitly computing the poles of a system

without explicitly computing the poles of a system

Next, examples of Next, examples of RouthRouth--Hurwitz criterionHurwitz criterion

ExercisesExercises

Read Sections 6.1 & 6.2.Read Sections 6.1 & 6.2.

Solve Problems 6.1, 6.9 & 6.14.Solve Problems 6.1, 6.9 & 6.14.