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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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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
e
e+U
-U
+E
-E
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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 (driving gear)
+U-U
m (driven gear)
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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
+M0
-M0
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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+U
-U
m
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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)
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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+=
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
System definition and problem statement
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
f’(u) G (jω)+Δr’(t)=0 Δu(t) Δm(t) Δw(t)
_
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?
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
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
f’(u) G (jω)+Δr’(t)=0 Δu(t) Δm(t) Δw(t)
_
545.41150
0 ==w
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
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 analysis of nonlinear systems – Prof Elio USAI – March 2008
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function - Assumptions
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)
_
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function - Assumptions
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function - Assumptions
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function - Assumptions
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
f’(u) G (jω)+Δr’(t)=0 Δu(t) Δm(t) Δw(t)
_
Such an assumption is usually taken for the sake of simplicity, and it can be relaxed
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
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 analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Harmonic balance
f’(u) G (jω)+Δr’(t)=0 Δu(t) Δm(t) Δw(t)
_
( )tUtu ωsin)( =Δ
( ) ( )( )∑∞
=
+=Δ1
sincos)(k
kk tkbtkatm ωω
( ) ( )( )( ) ( )ωϕω
ϕωϕω
jkGjkGG
tkbtkaGtw
kk
kkkkkk
∠==
+++=Δ ∑∞
=1sincos)(
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Harmonic balance
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 ∑∞
=
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Harmonic balance
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Harmonic balance
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Harmonic balance
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)
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Harmonic balance
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 analysis of nonlinear systems – Prof Elio USAI – March 2008
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( )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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
Ideal relay
A limit cycle can exist if the relative degree of G(jω) is greater than two
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
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( )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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
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
Nyquist and DF Diagrams
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( )
⎪⎪⎩
⎪⎪⎨
⎧
>−⎟⎠⎞
⎜⎝⎛−
≤−=
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
Hysteretic relay
If β is larger than U the relay could not behave as a purehysteresis
Nyquist and DF Diagrams
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( )
⎪⎪⎩
⎪⎪⎨
⎧
>⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛
≤
=
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
ω Because of the odd symmetry of the Δm(t) signal
-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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( )⎪⎩
⎪⎨
⎧
>⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛−
≤
= 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
ω Because of the odd symmetry of the Δm(t) signal
-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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
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 −=
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
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)
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( )( ) ( )⎪⎩
⎪⎨⎧
<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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( ) ( )( )( ) ( )( )⎪
⎪⎩
⎪⎪⎨
⎧
<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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
m
u/U
+M
-M
( )π
α MN 4=
m
u/U
+M
-M
+α
−α( )
⎪⎩
⎪⎨⎧
<−
≥=
11410
2 ααπ
αα MN
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( ) ( )[ ] ( )⎪⎩
⎪⎨⎧
<−−−Φ−
≥=
1141212
10
ααπαα
αα kjkNu/U
+α-α
m
k
( ) ( )[ ] ( )⎪⎩
⎪⎨⎧
<−+−Φ−
≥=
1141212
10
ααπαα
αα kjkNu/U
+α-α
m
k
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( )⎪⎩
⎪⎨⎧
<+−
≥=
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Describing Function – Computation
( )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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Stability of limit cycles
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Stability of limit cycles
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’
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Stability of limit cycles
Points B, B’, B”, A and A’represent possible magnitudes of the oscillation.
A and B re-present two possible limit cycles
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
Stability of 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
Nyquist and DF plots
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
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.
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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
=
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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;
Nyquist and DF plots
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
Taking into account the parameters of the motor and of the load
( )( ) ( )( ) rad 0.17594M-Urad/s 056.360
=ℜ⋅==+
===ℑ crp
a
etacrjW
jWJL
kkBRp
ωπ
ωωω
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;
Nyquist and DF plots
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
Taking into account the parameters of the motor and of the load
( )( ) ( )( ) rad 0.17594M-Urad/s 056.360
=ℜ⋅==+
===ℑ crp
a
etacrjW
jWJL
kkBRp
ωπ
ωωω
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;
Nyquist and DF plots
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!!!!
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
The system presents a periodic steady-state oscillation
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
Time
Angl
e
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
The system presents a periodic steady-state oscillation
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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)
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
Nyquist and DF plots
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!?
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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?
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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
]
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis – Example of application
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
Nyquist and DF plots
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
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
]
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
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
Nyquist and DF plots
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)
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
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
Nyquist and DF plots
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)
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
Nyquist plot parametersωlc=662 rad/s
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎥⎦⎤
⎢⎣⎡ +=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
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
]
Nyquist plot parametersωlc=662 rad/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
Describing Function analysis of nonlinear systems – Prof Elio USAI – March 2008
DF Analysis & Sliding Modes
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