Describing Function analysis-v1 - Diee · Describing Function analysis of nonlinear systems –...

Describing Function analysisDescribing Function analysisof of

nonlinear systemsnonlinear systems

University of CagliariUniversity of CagliariDept of Electrical and Electronic Eng.Dept of Electrical and Electronic Eng.

Prof. Elio [email protected]

University of LeicesterDepartment of Engineering

Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008

System definition and problem statement

r: reference variablee: error signalu: control variablem: manipulated inputy: controlled variable

ym: measure of the outputz: feedback signald: disturbancen: measurement noise

Aactuator

Pprocess

Ccontroller

Ttrasmitter

+

++r(t) e(t) u(t) m(t)

d(t)

y(t)

ym(t)

_

Ffilter

z(t)

Controller

+

n(t)+

Plant



In many cases the system presents a nonlinear phenomenon which is fully characterised by its static characteristics, i.e., its dynamics can be neglected

Saturated actuators

Relay control

Gears backlash

Hysteresis in magnetic materials

Dead zone in electro-mechanical systems

u

m+M

-M




Saturated actuators

Relay control

Gears backlash



e

e+U

-U

+E

-E




Saturated actuators

Relay control

Gears backlash



u (driving gear)

+U-U

m (driven gear)




Saturated actuators

Relay control

Gears backlash



u

m

+M0

-M0




Saturated actuators

Relay control

Gears backlash



u+U

-U

m



Nonlinearities are not always a drawback, they can also a have a“stabilising” effect

A PC

H

+r(t) e(t) u(t) m(t) y(t)_

z(t)

( ) ( )24.111

ssssP

++=

( ) ( )ssH

1.015.0

+=

( ) 10=sC

( )( )2

1

±=

satsA

O

N



0 2 4 6 8 10 12 14 16 18 20-20

-15

-10

-5

0

5

10

15

20Step response

time [s]

Sys

tem

out

put

NOT saturated actuatorSaturated actuator

The saturated system oscillates but does not diverge



The NOT saturated system is not stable since its stability margin are negatives

• mg = -12 dB

• mϕ = - 47 deg

Bode Diagram of the difference transfer function

Frequency (rad/sec)

Pha

se (d

eg)

Mag

nitu

de (d

B)

10-1

100

101

102

-360

-270

-180

-90

System: sys Frequency (rad/sec): 1.68

Phase (deg): -227

System: sys Frequency (rad/sec): 0.941 Phase (deg): -180

-150

-100

-50

0

50 System: sys Frequency (rad/sec): 1.68 Magnitude (dB): -0.0708

System: sys Frequency (rad/sec): 0.94

Magnitude (dB): 12



Many systems can be reduced to a simplified form in which the all linear dynamics is concentrated in a unique block and the static non linear characteristics is represented by a separate block

f(u) P(jω)C(jω)

H(jω)

+r(t) e(t) u(t) m(t) y(t)_

z(t)

f(u) P(jω)H (jω)C (jω)C(jω)+r’(t) u(t) m(t) w(t)

_

r(t)



If constant (or very slowly varying) reference signals are considered, under some conditions it is possible to separate the low-frequency, almost static, behaviour defined by a working nominal condition and a high-frequency behaviour due to small variations around such a working point

f(u) P(jω)H (jω)C (jω)C(jω)+r’(t) u(t) m(t) w(t)

_

r(t)=R

f(u) G (jω)+r’(t)=kCR u(t) m(t) w(t)

_

( ) ( )( ) ( )( ) ( )

00

0

0

0

mkRku

twwtw

tmmtm

tuutu

GC −=

Δ+=

Δ+=

Δ+=



The non linear characteristics is translated so that the origin of the new reference Cartesian system is the point (u0, m0) in the original one

f’(u) G (jω)+Δr’(t)=0 Δu(t) Δm(t) Δw(t)

_

( )⎪⎩

⎪⎨⎧

−=

=

00

00

mkRku

ufm

GC

( )( ) ( ) 00

0

muufuf

ufm

−Δ+=Δ′

Δ′=Δ

Rkk

G

C

RkC

Gk1

−

u

m

Δu

Δm

u0

m0



The aim of the system analysis is to define which is the steady-state behaviour of the system defined by the variations around the nominal working point


_

If f’(u) is a passive sector function absolute stability tools allow for sufficientconditions for global asymptotic stability of the variation system, i.e, the steady state is characterised by constant values of the system variables

If the origin of the variation system is not stable does the varia-bles diverge to infinity or some periodic motion can appear?


Limit cycles

Limit cycle: a periodic oscillation around a constant working point

f(u) G (jω)+r’=5δ−1(τ) u(t) m(t) w(t)

_

( ) ( )( )

( ) ( )1116

116

2

,,1

14.1110

−=

+++=

satuf

ssssG

-6 -4 -2 0 2 4 6 8-1

-0.5

0

0.5

1

1.5

2

u

m

Δ u

Δ m

u0

m0

21

101

+−= um

115

115

0

0

=

=

m

u


Limit cycles

Limit cycle: a periodic oscillation around a constant working point

115

115

0

0

=

=

m

u

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

time [s]

outp

ut w

(t) w0


_

545.41150

0 ==w


Limit cycles

Limit cycle can be a drawback in control systems:

Instability of the equilibrium point

Wear and failure in mechanical systems

Loss of accuracy in regulation

Parameters of the limit cicle can be used to discriminate between acceptable and dangerous oscillations

oscillation frequency

oscillation magnitude

Electronic oscillators can be based on limit cycles


Describing Function - Assumptions

The describing Function approach to the analysis of steady-state oscillations in non linear systems is an approximate tool to estimate the limit cycle parameters.

It is based on the following assumptions

There is only one single nonlinear component

The nonlinear component is not dynamical and time invariant

The linear component has low-pass filter properties

The nonlinear characteristic is symmetric with respect to the origin



There is only one single nonlinear component

The system can be represented by a lumped parameters system with two main blocks:

•The linear part

•The nonlinear part

N.L. G (jω)+r’=const. u(t) m(t) w(t)

_



The nonlinear component is not dynamical and time invariant

The system is autonomous. All the system dynamics is concentrated in the linear part so that classical analysis tools such as Nyquist and Bode plots can be applied.

u

m+M

-M

ε+=

uuMm



The linear component has low-pass filter properties

This is the main assumption that allows for neglecting the higher frequency harmonics that can appear when a nonlinear system is driven by a harmonic signal

( ) ( ) K ,3 ,2=>> njnGjG ωω

The more the low-pass filter assumption is verified the more the estimation error affecting the limit cycle parameters is small



The nonlinear characteristic is symmetric with respect to the origin

This guarantees that the static term in the Fourier expansion of the output of the nonlinearity, subjected to an harmonic signal, can be neglected


_

Such an assumption is usually taken for the sake of simplicity, and it can be relaxed


Fourier expansion - Recall

Consider a periodic function

( ) constant real a is ),()(,)( TTtytytfty −==

( ) ( )( )∑∞

=

++=1

220 cossin2

)(k

TkTk tkbtkaaty ππ

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )tdtktfdttktfT

b

bk

tdtktfdttktfT

a

TTTk

TTTk

T

T

T

T

πππ

πππ

π

π

π

π

π

π

222

0

222

sin1sin20,2,1,0

cos1cos2

2

2

2

2

2

2

2

2

∫∫

∫∫

−−

−−

==

==

==

K


Describing Function – Harmonic balance


_

( )tUtu ωsin)( =Δ

( ) ( )( )∑∞

=

+=Δ1

sincos)(k

kk tkbtkatm ωω

( ) ( )( )( ) ( )ωϕω

ϕωϕω

jkGjkGG

tkbtkaGtw

kk

kkkkkk

∠==

+++=Δ ∑∞

=1sincos)(



tjUetu ω=Δ )(

k

kk

kkk

k

tjkjk

babaM

eeMtm k

arctan)(

22

1 =

+== ∑

∞

= ϑωϑ

Consider the polar representation of a complex number associatedwith the exponential form of harmonic signals

Taking into account the low-pass property of the linear part of the system

tjjj

k

tjkjk

jk eeMeGeeMeGtw kk ωϑϕωϑϕ 11

111

)( ≅=Δ ∑∞

=



A permanent oscillation in the loop appears if Δu(t)=-Δw(t)

tjjjtj eeMeGUe ωϑϕω 1111−=

N.L. G (jω)+Δr’(t)=0 Δu(t) Δm(t) Δw(t)

_

01 11 11 =+ ϑϕ jj e

UMeG

( ) ( ) 0,1 =+ ωω UNjG

( ) ( )111, jabU

UN +=ω is the Describing Function of the nonlinear term

Harmonic balance equation



The armonic balance equation is a necessary condition for the existence of limit cycles in the nonlinear system

The approximate analysis gives good estimates if the low-pass filter hypothesis is strongly verified. It is a good tools for engineers

The harmonic balance equation is similar to the characteristic polynomial function, i.e. it leads to the Nyquist condition for closed-loop stability

The Describing Function is a linear approximation of the static nonlinearity limited to the first harmonic

( ) ( )ωω

,1

UNjG −=

In most cases the Describing Function is not a function of the frequency and this simplifies the verification of the harmonic balance equation by means of the Nyquist plot of the transfer function



Nyquist Diagram

Real Axis

Imag

inar

y A

xis

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2-10

-8

-6

-4

-2

0

2

ω=0+

ω=+∞

G(jω)

U=0+∞← U (U0,ω0)

-\1/N(U)



Nyquist and DF Diagrams

Real Axis

Imag

inar

y A

xis

-8 -7 -6 -5 -4 -3 -2 -1 0 1-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

ω=100

ω=17.7828

ω=3.1623

ω=0.56234

ω=0.1

System: sys Real: -1.14 Imag: -2.02

Freq (rad/sec): 0.651

System: sys Real: -2.01

Imag: -0.021 Freq (rad/sec): 0.997

U

ω


Describing Function – Computation

The evaluation of coefficients a1 and b1 can be performed by means of both analytical calculation and numerical integration, depending on the type of nonlinearity involved

f’(u)Usin(ωt) Δm(t)

( ) ( )111, jabU

UN +=ω( ) ( )

( ) ( )dtttmT

b

dtttmT

a

T

T

T

T

ω

ω

sin2

cos2

2

2

2

2

1

1

∫

∫

−

−

Δ=

Δ=

The DF computation can be performed by means of its definition



( )π

ωU

MUN 4, =( ) ( ) ( )π

ϑϑπ

ωπ

MdMdtttmT

bT

T

4sin4sin2 22

2 01 ==Δ= ∫∫

−

Ideal relay

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

Δ u(t)

Δ m

(t)

+M

-M

0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time

Δ uΔ m

+M

-M

T

( ) ( ) 0cos2 2

2

1 =Δ= ∫−

dtttmT

aT

T

ω Because of the odd symmetry of the Δm(t) signal



Ideal relay

A limit cycle can exist if the relative degree of G(jω) is greater than two


Real Axis

Imag

inar

y A

xis

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

U

-1/N(U)

G(jω)

ω

(ωc;4M/π)

The negative reciprocal of the DF is the negative real axis in backward direction

The oscillation frequency is the critical frequency ωc of the linear system and the oscillation magnitude is proportional to the relay gain M



( )π

ωU

MjUN 4, −=( ) ( ) ( )

πϑϑ

πω

π

MdMdtttmT

aT

T

4cos2cos2 22

2 01 −=−=Δ= ∫∫

−

Pure hysteresis

( ) ( ) 0cos2 2

2

1 =Δ= ∫−

dtttmT

bT

T

ω Because of the even symmetry of the Δm(t) signal

0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time

Δ uΔ m

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Δ u

Δm

M

-M



A limit cycle can exist if the relative degree of G(jω) is greater than one and G(jω) is a type-0 system

The negative reciprocal of the DF is the negative imaginary axis in backward direction

The oscillation frequency is lower than the critical frequency ωc of the linear system and the oscillation’s magnitude is proportional to the relay gain M and to the modulus of the transfer function at phase -π/4


Real Axis

Imag

inar

y A

xis

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

G(jω)

ω

U

-1/N(U)

Pure hysteresis



( )

⎪⎪⎩

⎪⎪⎨

⎧

>−⎟⎠⎞

⎜⎝⎛−

≤−=

βπ

ββπ

βπ

ωU

UMj

UUM

UU

MjUN

2

2 414

4

,( ) ( )∫ ∫ =+−=γ

γ

ωωt

t

T

dttT

MdttT

Ma0

1

4

cos8cos8

Hysteretic relay

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Δ u

Δ m

+M

-M

β -β

If U≤β the hysteretic relay behaves as pure hysteresis

0 0.5 1 1.5 2 2.5 3-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Time

Δ uΔ m

tγ

( ) ( )2

01 14sin8sin8 4

⎟⎠⎞

⎜⎝⎛−=+−= ∫ ∫ U

MdttT

MdttT

Mbt

t

T

βπ

ωωγ

γ

The imaginary part of N(U) is proportional to the hysteresis area



Hysteretic relay

If β is larger than U the relay could not behave as a purehysteresis


Real Axis

Imag

inar

y A

xis

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

ω

G(jω )

U -1/N(U)

The oscillation frequency is lower than the critical frequency ωc of the linear system and the oscillation’s magnitude is proportional to the relay gain M

The negative reciprocal of the DF is the parallel to the negative real axis and with constant negative imaginary part



( )

⎪⎪⎩

⎪⎪⎨

⎧

>⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

≤

=

kMU

kUM

kUM

kUMk

kMUk

UN 2

1arcsin2π

Saturation

If U≤Μ the saturation behaves as pure gain

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=+= ∫ ∫

2

0

21 1arcsin2sin8sin8 4

kUM

kUM

kUMkUdtt

TMdttkU

Tb

t

t

T

πωω

γ

γ

0 0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Δ uΔ m

tγ

+M

-M

( ) ( ) 0cos2 2

2

1 =Δ= ∫−

dtttmT

aT

T


-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-4

-3

-2

-1

0

1

2

3

4

Δ u

Δ m

k

+M

-M



A limit cycle can exist if the relative degree of G(jω) is greater than two and gain k is sufficiently high

The negative reciprocal of the DF is part of the negative real axis in backward direction

The oscillation frequency is the critical frequency ωc of the linear system and the oscillation’s magnitude depends on the saturation parameters M and k

Saturation Nyquist and DF Diagrams

Real Axis

Imag

inar

y A

xis

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

-1/k

G(jω) ω

-1/N(U)

U



( )⎪⎩

⎪⎨

⎧

>⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−

≤

= ββββπ

β

UUUU

kk

U

UN2

1arcsin2

0

Dead zone

If U≤β the dead zone has no output

( )γωβ tU sin=

( ) ( ) 0cos2 2

2

1 =Δ= ∫−

dtttmT

aT

T


-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

Δ u

Δ m

+β

-β

k

0 0.5 1 1.5 2 2.5 3-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Time

Δ uΔ m

tγ

( )( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−=−= ∫

2

1 1arcsin2

2sinsin8 4

UUUkUdtttUk

Tb

T

t

βββππ

ωβωγ



A limit cycle can exist if the relative degree of G(jω) is greater than two and gain k is sufficiently high

The negative reciprocal of the DF is part of the negative real axis in forward direction

The oscillation frequency is the critical frequency ωc of the linear system and the oscillation magnitude depends on the dead zone parameters β and k

Dead zone Nyquist and DF Diagrams

Real Axis

Imag

inar

y A

xis

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-2.5

-2

-1.5

-1

-0.5

0

0.5

1

U=+∞

-1/N(U)

ω

G(jω)

-1/k



The nonlinear characteristics of the Dead Zone can be computed by subtracting the Saturation characteristics from a linear one

Dead zone

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

Δ u

Δ m

k

dead zone ψ(Δu)

saturation φ (Δu)

( ) ( )uku ΔΦ−=ΔΨ

( ) ( )UNkUN ΦΨ −=



The Describing function of a nonlinear characteristics can be computed as the combination of the Describing Functions of the elementary constituting nonlinear characteristics

( ) ( ) ( )UNUNUNt 21 +=

N2(U,ω)G (jω)

+Δr’(t)=0 Δu(t)Δm(t) Δw(t)

_

N1(U,ω)

Nt(U,ω)



A number of Describing Function can be computed by particularisation of the function

in which the parameter α defines a peculiar point of the nonlinear characteristics

( ) ( )[ ]21arcsin2 αααπ

α −+=Φ

m

u/U

+M

-M

k

+α

-α ( )( )⎪⎩

⎪⎨⎧

<Φ⋅

≥=

1

1

αα

αα

k

kN



( )( ) ( )⎪⎩

⎪⎨⎧

<Φ⋅−+

≥=

1

1

212

1

αα

αα

kkk

kN

m

u/U

+M

-M

k1

+α-α

k2

( )( )( )⎪⎩

⎪⎨⎧

<Φ−

≥=

11

10

αα

αα

kNu/U

+α-α

m

k



( ) ( )( )( ) ( )( )⎪

⎪⎩

⎪⎪⎨

⎧

<Φ−Φ⋅

≤<Φ−⋅

≥

=

1

11

10

βαβ

βαα

α

α

k

kN

m

u/U

+M

-M

k1

+β-α

k2

( )π

α MkN 4+=

+α-β

u/U

m

k+M

-M



m

u/U

+M

-M

( )π

α MN 4=

m

u/U

+M

-M

+α

−α( )

⎪⎩

⎪⎨⎧

<−

≥=

11410

2 ααπ

αα MN



( ) ( )[ ] ( )⎪⎩

⎪⎨⎧

<−−−Φ−

≥=

1141212

10

ααπαα

αα kjkNu/U

+α-α

m

k

( ) ( )[ ] ( )⎪⎩

⎪⎨⎧

<−+−Φ−

≥=

1141212

10

ααπαα

αα kjkNu/U

+α-α

m

k



( )⎪⎩

⎪⎨⎧

<+−

≥=

141410

, 2 απ

ααπ

αω

UMj

UMUN

u/U

m+M

-M

+α-α

( )⎪⎩

⎪⎨⎧

<−−

≥=

141410

, 2 απ

ααπ

αω

UMj

UMUNu/U

m+M

-M

+α-α



( )ωππ

ωUMj

UMUN 44, −=

( ) ( ) ( )ωj

UNUNUNt1

11 +=

1/jωG (jω)

+Δr’=0 Δu Δm Δw_

Nt(U,ω)

( ) ( )( )jM

UUN

++

−=− ω

ωωπ

ω 214,1

-8 -7 -6 -5 -4 -3 -2 -1 0 1-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

ω=0.1

ω=0.56234

ω=3.1623

ω=17.7828

ω=100

Real axis

Imag

inar

y ax

is

Negative inverse of the Describing Function

U

U

U

U


Stability of limit cycles

If the block N.L. is a pure constant k, the stability of the feedback system can be performed by means of the Nyquist criterion, which gives the number of roots with positive real roots of the Harmonic Balance Equation

N.L. G (jω)+Δr’(t)=0 Δu(t) Δm(t) Δw(t)

_

( ) 011 =+k

jG ω

The Nyquist criterion looks at the relative position of the transfer function G(jω) with respect to the point (-1/k, 0) in the complex plain.By extension the criterion can be applied to any point –α of the complex plain with reference to the Harmonic Balance Equation

( ) C∈=+ ααω ,01 jG



If the transfer function G(jω) represents a stable system, the reducedNyquist criterion can be applied, i.e. the closed loop stability can be stated if the reference point –α in the complex plain remains on the left-side when running along the Nyquist plot of G(jω) from ω=0+ to ω =+∞.

Nyquist plot of G(jω)

Real Axis

Imag

inar

y A

xis

-3 -2.5 -2 -1.5 -1 -0.5 0-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

ω=+∞

ω=0+

-1/k

-α

( )( )( )15.0

12 ++

=ωωω

ωjjj

jG

The closed loop is not stable with respect to the point (-1/k,0).

The closed loop is stable with respect to the point -α



Can the generalization of the Nyquist criterion be used to analyse the stability of a limit cycle?

If a perturbation of the magnitude of the periodic oscillation occurs, it tends to the original value as time passes.

What does it mean that a limit cycle is stable?

YES, it can

It is a marginally stable condition.How can a limit cycle be considered from the Nyquist criterion point of view?

By a point in the Describing Function plot.How can the magnitude of a limit cycle be represented?

Nyquist and DF plots

Real Axis

Imag

inar

y A

xis

-3 -2.5 -2 -1.5 -1 -0.5 0-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

G(jω) ω

U

A

BB’

B”

A’



Points B, B’, B”, A and A’represent possible magnitudes of the oscillation.

A and B re-present two possible limit cycles



Applying the reducedNyquist criterion with respect to:

point B’: oscillations tend to decrease (stable system)

point B”: oscillations tend to increase (unstable system)

point A’: oscillations tend to decrease (stable system)

B: Stable limit cycle A: Unstable limit cycle


Real Axis

Imag

inar

y Ax

is

-3 -2.5 -2 -1.5 -1 -0.5 0-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

G(jω ) ω

U

A

BB’ B”

A’

The linear system G(jω) is assumed to be stable in order to apply the reduced Nyquist criterion


DF Analysis – Example of application

Consider a DC motor with permanent magnets

0 V

+24 V

-24 V

M d.c.

The position of the motor shaft is measured by means of a rotational variable resistance.

The voltage on the rotational resistance drives the position of a relay that switches the motor supply voltage between +/- 24 V d.c.



The linear approximate model of a DC motor is the following

( ) ( ) ( ) ( )( ) ( ) ( )

( )

( ) ( )( ) ( )

lm

lm

rtem

e

em

rrrrr

BBB

JJJ

tiktC

tktedt

td

tBtCdt

tdJ

tedt

tdiLtiRtv

+=

+=

=

=

=

+=

++=

ω

ϑω

ωω

( )⎪⎩

⎪⎨⎧

≥−

<+=

d

d

r tvϑϑ

ϑϑ

24

24

Rr: rotor resistanceLr: rotor inductanceke: voltage feedback constantkt: torque constantJm: motor inertiaJl: load inertiaBm: motor friction coefficientBl: load friction coefficient

vr: rotor supply voltageir: rotor wound currentCem: electromagnetic torqueω: rotational speedθ: shatf angular position



The motor transfer functions are the following

( ) ( )( ) ( )( ) etaa

t

s kkBsJRsLk

sVssW

+++=

Ω=ω

( ) ( )( ) ( )( )( ) ( ) ( )

( )( ) ( )

( )( ) ( )[ ]2222

2

2222

-

JLkkBRBLJR

JLkkBRkj

JLkkBRBLJR

BLJRk

jjjkkBJjRLjj

kjVjjW

aetaaa

aetat

aetaaa

aat

etaa

t

r

ωωω

ω

ωω

ωωωωωω

ωωϑ

−+++

−+−

−+++

+=

ℑ+ℜ=+++

=Θ

=



Taking into account the parameters of the motor and of the load

R=0.4; % rotor resistance

L=0.001; % rotor inductance

ke=0.3; % voltage feedback constant

kt=0.3; % torque constant

Jm=0.01; % motor inertia

Jl=0.09 % load inertia

Bm=0.05; % motor friction coefficient

Bl=0.05; %load friction coefficient

J=Jm+Jl;

B=Bm+Bl;


Real Axis

Imag

inar

y A

xis

-2 -1.5 -1 -0.5 0 0.5 1-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

ω

U




( )( ) ( )( ) rad 0.17594M-Urad/s 056.360

=ℜ⋅==+

===ℑ crp

a

etacrjW

jWJL

kkBRp

ωπ

ωωω









J=Jm+Jl;

B=Bm+Bl;


Real Axis

Imag

inar

y A

xis

-20 -15 -10 -5 0

x 10-3

-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-4

System: Wp Real: -0.00568

Imag: -8.73e-007 Freq (rad/sec): 36.8

ω

U




( )( ) ( )( ) rad 0.17594M-Urad/s 056.360

=ℜ⋅==+

===ℑ crp

a

etacrjW

jWJL

kkBRp

ωπ

ωωω









J=Jm+Jl;

B=Bm+Bl;


Real Axis

Imag

inar

y A

xis

-20 -15 -10 -5 0

x 10-3

-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-4

System: Wp Real: -0.00568

Imag: -8.73e-007 Freq (rad/sec): 36.8

ω

U

STABLE limit cycle!!!!



The system presents a periodic steady-state oscillation

0 1 2 3 4 5 6 7 8 9 10-80

-60

-40

-20

0

20

40

60

80

Time

anglespeedcurrent




0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

Time

Angl

e




8.4 8.45 8.5 8.55 8.6 8.65 8.7 8.75 8.8

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

Time [s]

Angl

e[ra

d]

Tc=2π/ωc

2U=8M|G(jωc)|/π



What does it happen if the same control law is applied to the speed control problem?

Step Sign

Scope

24

Gain

V

Cr

ang

w

i

DC Motor

( ) ( )( ) etaa

t

kkBsJRsLksW

+++=ω

The linear plant is characterised by an all-pole transfer function with relative degree two, therefore there is no contact point between the Nyquist plot and the real negative axis, but the origin at ω=+∞ (corresponding to U=0 in the DF plot)




Real Axis

Imag

inar

y A

xis

-1 -0.5 0 0.5 1 1.5 2 2.5 3

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

U

ω

ω=0 ω=+∞U=0

The Nyquist plot of the linear system is tangentto the negative reciprocal of the DF at the origin, i.e., U=0 and ω=+∞

Maybe a sliding mode behaviour is established asymptotically!?



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

Time

anglespeed

Possibly a sort of dead-bit control, or finite time stabilisation seams to appear

Can it be a sliding mode behaviour?



By reducing the integration step of the simulation it is apparent that the stabilisation is asymptotic

Can it be a sliding mode behaviour?

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

1

2

3

4

5

6

7

Time [s]

Spe

ed [r

ad/s

]



The rotor current tends to a constant value, i.e. an asymptotic (2nd

order) sliding sliding mode appears

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-40

-30

-20

-10

0

10

20

30

40

50

60

Time [s]

Rot

or c

urre

nt [A

]

By reducing the integration step of the simulation it is apparent that the stabilisation is asymptotic



The DC motor is a second order system whose state variables are the rotor speed and current

( ) ( ) ( )( ) ( ) ( ) ( )

( )( )

( )

( ) ( ) ( ) ( )( )( )⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦⎤

⎢⎣⎡ +=++=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−=

+−−=

+=

ti

t

Jk

JBtti

Jk

tJBts

tvLti

t

LR

Lk

Jk

JB

tvL

tiLR

tLk

dttdi

tiJk

tJB

dttd

r

tr

t

r

rr

r

r

r

e

t

rr

rr

r

r

er

rt

ωωω

ω

ω

ωω

22

10

1

The system transfer function has a zero in –2, and if the system output is steared to zero in a finite time, than the system behaves as a first order system with time constant τ=0.5



A MATLAB-Simulink scheme is the following

r=2π

Sign

ScopeRelay

Manual Switch

2

Gain1

24

Gain

du/dt

Derivative

V

Cr

ang

w

i

DC Motor

Take care that because of the feedback, the new closed loop gain (assuming the rotor velocity as the output) will be 1/2



It is apparent that the shaft speed tends to the steady state value π as a first order system with τ = 0.5 s

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

Time [s]

Spe

ed [r

ad/s

]

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

Speed [rad/s]

Rot

or c

urre

nt [A

]Apart from a very short transient, the state trajectory tends to the steady-state values sliding on a linear manifold of the state space



The Nyquist plot of the linear system cross the negative reciprocal of the DF at the origin, i.e., U=0 and ω=+∞

A sliding mode behaviour is established in a finite time


Real Axis

Imag

inar

y A

xis

-2 -1 0 1 2 3 4 5 6 7 8-4

-3

-2

-1

0

1

2

ω

U


DF Analysis & Sliding Modes

Sliding modes are characterised by infinite frequency of proper ideal switching devices

Most sliding mode controllers use a sign function in the controller, i.e., an ideal relay, which can be approximately represented by its Describing Function

By the example it is apparent that a sliding mode behaviour is established in a finite time if the Nyquist plot of cross the inverse negative of the describing function at the point (U=0,ω=+∞)

It can also be derived that a sliding mode behaviour is established asymptotically if the Nyquist plot of is tangent to the inverse negative of the describing function at the point (U=0,ω=+∞)

TAKE CARE: the Describing function is an approximate tool



r=0

Sign

ScopeRelay

Manual Switch

2

Gain1

24

Gain

du/dt

Derivative

V

Cr

ang

w

i

DC Motor

Substitute the ideal relay with a hysteretic one with β=1

It is apparent that an ideal sliding mode cannot appear because of the switching delay



The limit cycle is apparent, with a periodTlc≅0.6 ms

The magnitude is very small because of the low-pass filter property of the motor transfer function

0.4 0.4001 0.4002 0.4003 0.4004 0.4005 0.4006 0.4007 0.4008 0.4009 0.401-1

-0.5

0

0.5

1x 10-5

Time [s]

Spe

ed [r

ad/s

]

0.4 0.4001 0.4002 0.4003 0.4004 0.4005 0.4006 0.4007 0.4008 0.4009 0.401

-20

-10

0

10

20

Sup

plu

volta

ge [V

]



The Nyquistplot of the linear system has no common point with the negative inverse of the Describing Function of the hysteretic relay

Describing function analysis can be useful in sliding mode systems when a common point is present


Real Axis

Imag

inar

y A

xis

-2 -1 0 1 2 3 4 5 6 7 8-4

-3

-2

-1

0

1

2

ω=+∞ ω=0

U=1 -1/N(U)

G(jω)



Most effective use of the Describing Function approach to sliding modes….

Analysis of the characteristics of a chattering behaviour due to unmodelleddynamics of sensors and/or actuators

What is chattering? It appears as oscillations of the system variables, whose magnitude is related to the influence of the neglected dynamics on the system

bandwidth

It is very close to a limit cycle



Consider the motor drive as a hysteretic switching device plus a time constant τa=0.1 s

r=2*pi

1

0.1s+1Transfer Fcn

SignScope3Scope2

ScopeRelay

Manual Switch

2

Gain1

24

Gain

du/dt

Derivative

V

Cr

ang

w

i

DC Motor


Real Axis

Imag

inar

y A

xis

-2 -1 0 1 2 3 4 5 6 7 8-4

-3

-2

-1

0

1

2

-1/N(U)

G(jω)



Nyquist plot parametersωlc=662 rad/s

Ulm=1.1833 rad/s2

Uls represents the magnitude of the oscillation of the sliding variable

-1/N(U)

G(j ω)

System: untitled1 Real: -0.0507 Imag: -0.0328 Freq (rad/sec): 662




Ulm=1.1833 rad/s2

Simulation resultsTls=9.7 ms

ωlc=646 rad/sUlm=1.8544 rad/s2

2.9 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Steady-state regime

Time [s]

Slid

ing

varia

ble

[rad/

s2

]

Differences are due to:• Numerical solution of

the simulation• Approximation of the

DF approach



The rotor velocity tends to its steady-state value as a first order system

with time constant τ=0.5s

Rotor wound current presents large variations

An approximate sliding mode is established

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

Time [s]

VelocityCurrent



⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦⎤

⎢⎣⎡ +=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

r

t

r

rr

r

r

r

e

t

r

iJk

JB

vLi

LR

Lk

Jk

JB

i

ωσ

ωω

2

10

&

&

s+−= ωω 2&

Full system dynamics

rr

tt

r

e

r

r

r

r vJLk

Jk

Lk

LR

JB

LR

JB

+⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛−+= ωσσ 222&Input-output dynamics

Internal reduced-order dynamics

ωσ ⎟⎠⎞

⎜⎝⎛ +−=

JBi

Jk

rt 2



σωω +−= 2&Internal reduced-order dynamics ( ) ( )ss

s Σ+

=Ω2

1

( ) ( )tUt lcωσσ sin0 +=

At steady-state( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

++

+=2

1argsin2

10

lclc

lc jt

jUt

ωω

ωωω

2.9 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 33.07

3.072

3.074

3.076

3.078

3.08

3.082

3.084

3.086

3.088

Time [s]

Spe

ed [r

ad/s

]


Uls=1.1833 rad/s2

Δωls=0.0018 rad/s

Simulation resultsTls=9.7 ms

ωlc=646 rad/sUls=1.8544 rad/s2

Δωls=0.0029 rad/s

0015.02

1≅

+lsjω



The Describing Function approach to the analysis of the chattering phenomenon in sliding mode control systems can be used to have an estimate of the chattering parameters, i.e., frequency and magnitude

The estimates are affected by an error which depends on the low-pass properties of the linear part of the plant

The sliding variable must be considered as the output of the nonlinear feedback system

The actual system output behaviour can be estimated by considering the reduced order dynamics

In the presence of a constant reference value, the nonlinear function can be not symmetric, therefore an equivalent gain of the nonlinearity has to be considered to estimate the constant mean steady-state value of the output

Describing Function analysis-v1 - Diee · Describing Function analysis of nonlinear systems –...

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Transcript of Describing Function analysis-v1 - Diee · Describing Function analysis of nonlinear systems –...