Post on 18-Oct-2020
Adaptive Control – Landau, Lozano, M’Saad, Karimi1
Adaptive Control
Chapter 8: Robust digital control design
Adaptive Control – Landau, Lozano, M’Saad, Karimi2
Chapter 8:Robust digital control design
Adaptive Control – Landau, Lozano, M’Saad, Karimi3
r(t)
m
m
AB
TS1
ABq d−
R
u(t) y(t)
Controller
PlantModel
+
-
The R-S-T Digital Controller
Plant Model:)(
)(*)(
)()()(1
11
1
111
−
−−−
−
−−−− ===
qAqBq
qAqBqqHqG
dd
A
A
nn qaqaqA −−− +++= ...1)( 1
11 )(*...)( 111
11 −−−−− =++= qBqqbqbqB B
B
nn
R-S-T Controller: )()()1(*)()()( 111 tyqRdtyqTtuqS −−− −++=
Characteristic polynomial (closed loop poles):
)()()()()( 11111 −−−−−− += qRqBqqSqAqP d
Adaptive Control – Landau, Lozano, M’Saad, Karimi4
Digital control in the presence of disturbances and noise
Output sensitivity function(p y) )()()()(
)()()( 1111
111
−−−−
−−−
+=
zRzBzSzAzSzAzS yp
)()()()()()()( 1111
111
−−−−
−−−
+−
=zRzBzSzA
zRzAzSup
)()()()()()()( 1111
111
−−−−
−−−
+−
=zRzBzSzA
zRzBzS yb
)()()()()()()( 1111
111
−−−−
−−−
+=
zRzBzSzAzSzBzS yv
Input sensitivity function(p u)
Noise-output sensitivity function(b y)
Input disturbance-output sensitivity function(v y)
All four sensitivity functions should be stable !
T
R
1/S B/A
PLANT
r(t)u(t) y(t)+
-
p(t)
b(t)
++
++
(disturbance)
(measurement noise)
v(t)++
Adaptive Control – Landau, Lozano, M’Saad, Karimi5
Stability of closed loop discrete time systems
The Nyquist is used like in continuous time(can be displayed with WinReg ou Nyquist_OL.sci(.m))
)()()()()(
ωω
ωωω
jj
jjj
OL eSeAeReBeH
−−
−−− =
)()()()()()()(1)(
11
1111111
−−
−−−−−−− +
=+=zSzA
zRzBzSzAzHzS OLyp
Nyquist criterion (discrete time –O.L. is stable)
The Nyquist plot of the open loop transfer fct. HOL(e-jω) traversed in the sense of growingfrequencies (from 0 to 0.5fS) leaves the critical point[-1, j0] on the left
ω =0
HBO
(e )S = 1 +H
BO(e )
yp-1
Critical point
-1
Im H
Re Hω = π
-jω-jω
Adaptive Control – Landau, Lozano, M’Saad, Karimi6
The Nyquist plot of the open loop transfer fct. HOL(e-jω) traversed in the sense of growingfrequencies (from 0 et fS) leaves the critical point[-1, j0] on the left and the number of encirclements of the critical pointcounter clockwise should be equal to the number ofunstable poles in open loop.
Remarks:-The controller poles may becomeunstable if high performances arerequired without using an appropriatedesign method
-The Nyquist plot from 0.5fS to fS is the symmetric with respect to the real axisof the Nyquist plot from 0 to 0.5fS
Stability of closed loop discrete time systems
Nyquist criterion (discrete time –O.L. is unstable)
1 unstable pole in Open Loop
ω =0
ω = π
ω = 2π
-1
Im H
Re H
Stable Closed Loop (a)
ω = π
Stable Open Loop Unstable Closed Loop(b)
ω =0
Adaptive Control – Landau, Lozano, M’Saad, Karimi7
Marges de robustesse
The minimal distance with respect to the critical pointcharacterizes the robustness of the CL with respect touncertainties on the plant model parameters( or their variations)
-Gain margin ΔG-Phase margin Δφ-Delay margin Δτ-Modulus margin ΔM
-1
ΔΦ
Δ1
ΔΜ
1
Crossoverfrequency
Re H
Im H
G
|HOL|=1
ωCR
Adaptive Control – Landau, Lozano, M’Saad, Karimi8
( )fj
ypypOL
ezpourzRzBzSzA
zSzA
zSzSzHM
π21
1
max
1111
11
1
max
1
min
11
min
1
)()()()()()(
)()()(1
−−
−
−−−−
−−
−−−−−
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
===+=Δ
dBMdBMdBeS jyp )( 1
maxΔ−=Δ= −− ω
Modulus margin and sensitivity function
ω
dB
Syp
Syp
Syp
-1
min
-1
= - MΔ
= MΔ
Sypmax
0
= - Sypmax
Adaptive Control – Landau, Lozano, M’Saad, Karimi9
Robust stabilityTo assure stability in the presence of uncertainties (or variations)on the dynamic chatacteristics of the plant model
ωj
ypOLOLOL
ezzSzA
zPzSzA
zRzBzSzA
zSzHzHzH
−−
−−
−
−−
−−−−
−−−−−
==+
==+<−′
1
11
1
11
1111
11111
;)()(
)()()(
)()()()(
)()(1)()(Robust stabilitycondition
(sufficient cond.):
HOL – nominal F.T.; H’OL –Different from HOL (perturbed)
ω
dB
Syp
Syp
Syp
-1
min
-1
= - MΔ
= MΔ
Sypmax
0
= - Sypmax
Size of the tolerated uncertainity on HOL at each frequency (radius)
(*)
H
HOL
OL'
1 +
-1
Im H
Re H
HOL
Adaptive Control – Landau, Lozano, M’Saad, Karimi10
Frequency templates on the sur sensitivity functionsThe robust stability conditions allow to define frequency templateson the sensitivity functions which guarantee the delay margin andthe modulus margin;The templates are essential for designing a good controller
Frequency template on the noise-outputsensitivity function Syb for Δτ = TS
Frequency template on the output sensitivityfunction Syp for Δτ = TS and ΔΜ = 0.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-8
-7
-6
-5
-4
-3
-2
-1
0Noise-output Sensitivity Function Template
Mag
nitu
de (d
B)
Frequency (f/fs)
Template for Delay margin Δτ = Ts
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25
-20
-15
-10
-5
0
5
10Output Sensitivity Function Template
Mag
nitu
de (d
B)
Frequency (f/fs)
Modulus margin = 0.5Delay margin = Ts
Delay margin = Ts
Modulus margin templateDelay margin = Ts template
Output Sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi11
)(1
180ωjHG
OL
=Δ o180)( 180 −=∠ ωφ
1)( )(1800 =∠−=Δ crBOcr jHpour ωωφφ
ω
φτ
icr
ii
Δ=Δ min
( ) 1
maxmin
1
min)()()(1
−− ==+=Δ ωωω jSjSjHM ypypOL
Modulus margin
Gain margin
Phase margin
Delay margin
ωφτcr
Δ=Δ Several intersections points:
φφ iiΔ=Δ min If there are several intersections with the unit circle
pour
Robustness margins
Adaptive Control – Landau, Lozano, M’Saad, Karimi12
Robustness margins – typical values
Gain margin : ΔG ≥ 2 (6 dB) [min : 1,6 (4 dB)]
Phase margin : 30° ≤ Δφ ≤ 60°
Delay margin : fraction of system delay (10%) or of time response (10%) (often 1.TS)
Modulus margin : Δ M ≥ 0.5 (- 6 dB) [min : 0,4 (-8 dB)]
A modulus margin Δ M ≥ 0.5 implies ΔG ≥ 2 et Δφ > 29°Attention ! The converse is not generally true
The modulus margin defines also the tolerance with respectto nonlinearities
Adaptive Control – Landau, Lozano, M’Saad, Karimi13
Good gain and phase marginBad delay margin
Robustness margins
Good gain and phase marginBad modulus margin
.
Adaptive Control – Landau, Lozano, M’Saad, Karimi14
ω
dB
Syp
Syp
Syp
-1
min
-1
= - MΔ
= MΔ
Sypmax
0
= - Sypmax
( )fj
ypypOL
ezpourzRzBzSzA
zSzA
zSzSzHM
π21
1
max
1111
11
1
max
1
min
11
min
1
)()()()()()(
)()()(1
−−
−
−−−−
−−
−−−−−
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
===+=Δ
dBMdBMdBeS jyp )( 1
maxΔ−=Δ= −− ω
Modulus margin and sensitivity function
Minimum distance with respect to thecritical point
Critical regionfor design
Adaptive Control – Landau, Lozano, M’Saad, Karimi15
Correspondance Output Sensitivity Nyquist Plot
-1
ΔΦΔΜ
1
Crossoverfrequency
Re H
Im H
|HOL|=1
ωCR
ω
dB
Syp
Syp
Syp
-1
min
-1
= - MΔ
= MΔ
Sypmax
0
= - Sypmax
Adaptive Control – Landau, Lozano, M’Saad, Karimi16
– The open loop being stable, one has the property:
∫ =−SS
f.πf/fj
yp df)(eS50
0
2 0log
The sum of the areas between the curve of Syp and the axis 0dB taken withtheir sign is null
Disturbance attenuation in a frequency region implies amplificationof the disturbances in other frequency regions!
Properties of the output sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi17
Augmenting the attenuation or widening the attenuation zone
Reduction of the robustness(reduction of the modulus margin)
Higher amplification of disturbancesouside the attenuation zone
Properties of the output sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi18
Robust stabilityTo assure stability in the presence of uncertainties (or variations)on the dynamic chatacteristics of the plant model
ωj
ypOLOLOL
ezzSzA
zPzSzA
zRzBzSzA
zSzHzHzH
−−
−−
−
−−
−−−−
−−−−−
==+
==+<−′
1
11
1
11
1111
11111
;)()(
)()()(
)()()()(
)()(1)()(Robust stabilitycondition
(sufficient cond.):
HOL – nominal F.T.; H’OL –Different from HOL (perturbed)
ω
dB
Syp
Syp
Syp
-1
min
-1
= - MΔ
= MΔ
Sypmax
0
= - Sypmax
Size of the tolerated uncertainity on HOL at each frequency (radius)
(*)
H
HOL
OL'
1 +
-1
Im H
Re H
HOL
1111 )()()( −−−− −′< zHzHzS OLOLyp
Adaptive Control – Landau, Lozano, M’Saad, Karimi19
)()()()()()(
)()(
)()(
)()(
)()()()(
)()()()(
11
1111
1
1
1
1
1
1
11
11
11
11
−−
−−−−
−
−
−
−
−
−
−−
−−
−−
−− +<−
′′
⋅=−′′
zSzAzRzBzSzA
zAzB
zAzB
zSzR
zSzAzRzB
zSzAzRzB
)()()(
)()()(
)()()()()()(
)()( 11
11
1
11
1111
1
1
1
1−−
−−
−
−−
−−−−
−
−
−
−
==+
<−′′
zSzRzA
zPzRzA
zRzBzSzAzAzB
zAzB
up
Sup dB
Limitation of the actuator stress
00.5f e
Sup
-1
(Size of tolerated uncertainties)
Tolerance to plant additive uncertainty
H’OL HOL
(*)
From previous slide :
G’ G
G’ G
(**)
1111 )()(')(−−−− −< zGzGzSup
Adaptive Control – Landau, Lozano, M’Saad, Karimi20
Tolerance to plant normalized uncertainty(multiplicative uncertainty)
)()()(
)()()(
)()()()(
)()(
)()(
)()(
11
11
1
11
1111
1
1
1
1
1
1
−−
−−
−
−−
−−−−
−
−
−
−
−
−
==+
<−
′′
zSzRzB
zPzRzB
zRzBzSzA
zAzB
zAzB
zAzB
yb
From (**), previous slide:
The inverse of the modulus of the “complementary sensitivity function”gives at each frequency the tolerance with respect to “normalized(multiplicative) uncertainty”
Relation between additive and multiplicative uncertainty:
)'1()'('G
GGGGGGG −+=−+=
Adaptive Control – Landau, Lozano, M’Saad, Karimi21
Important message
Large values of the modulus of the sensitivity functions in a certain frequency region
Low tolerance to model uncertainty
Critical regions for control designNeed for a good model in these regions
Adaptive Control – Landau, Lozano, M’Saad, Karimi22
Small gain theorem
S1
S2
-u1
u2
y1
y2
11 <∞
S
12 ≤∞
S
S1: linear time invariant (state x)11 <
∞S
S2: 12 ≤∞
S
Then:
0)(lim;0)(lim;0)(lim 11 ===∞→∞→∞→
tytutxttt
It will be used to characterize “robust stability”
Adaptive Control – Landau, Lozano, M’Saad, Karimi23
Description of uncertainties in the frequency domain
Re H
Im H
Uncertainty disk(at a certain frequency)
1) It needs a description by a transfer function which may have any phase but a modulus < 12) The size of the radius will vary with the frequency and is characterized by a transfer function
Adaptive Control – Landau, Lozano, M’Saad, Karimi24
Additive uncertainty
)()()()(' 1111 −−−− += zWzzGzG aδ
)( 1−zδ any stable transfer function with 1)( 1 ≤∞
−zδ)( 1−zWa a stable transfer function
∞
−
∞
−−−− =−=− )()()(')()(' 111
max
11 zWzGzGzGzG a
ABzHSRK d /;/ −==
δ aW
K G-
+
+
aWupS−
δ-
1)()( 11 <∞
−− zWzS aupRobust stability condition:
Apply small gain theorem
Adaptive Control – Landau, Lozano, M’Saad, Karimi25
Multiplicative uncertainties
[ ])()(1)()(' 1111 −−−− += zWzzGzG mδ
)( 1−zδ any stable transfer function with 1)( 1 ≤∞
−zδ)( 1−zWm a stable transfer function
)()()( 111 −−− = zWzHzW ma
mWybS−
δ-
δ
K- +
+
G
mW
1)()( 11 <∞
−− zWzS mybRobust stability condition:
Adaptive Control – Landau, Lozano, M’Saad, Karimi26
Feedback uncertainties on the input
[ ])()(1)()('
11
11
−−
−−
+=
zWzzGzG
rδ)( 1−zδ any stable transfer function with 1)( 1 ≤
∞
−zδ
)( 1−zWr a stable transfer function
rWypS−
δ-
1)()( 11 <∞
−− zWzS rypRobust stability condition:
δ
K- +
-
G
mW
Adaptive Control – Landau, Lozano, M’Saad, Karimi27
Robust stability conditions
),(', δWHH Ρ∈ Family (set) of plant modelsRobust stability :The feedback system is asymptotically stable for all the plant models belonging to the family ),( δWΡ
• Additive uncertainties
1)()( 11 <∞
−− zWzS aup πωωω ≤≤< −−− 0)()( 1j
a
j
up eWeS
• Multiplicative uncertainties1)()( 11 <
∞
−− zWzS myb πωωω ≤≤< −−− 0)()( 1j
m
j
yb eWeS
• Feedback uncertainties on the input (or output)1)()( 11 <
∞
−− zWzS ryp πωωω ≤≤< −−− 0)()( 1j
r
j
yp eWeS
Adaptive Control – Landau, Lozano, M’Saad, Karimi28
Im G
Re Gω = π
G (e )-j ω
uncertaintydisk
δW
ω = 0
Robust Stability
Family of plant models:),,(' xyWGFG δ∈
G – nominal model; 1)( 1 ≤∞
−zδ
)( 1−zWxy - size of uncertainty
Robust stability condition:a related sensitivity
functiona type of uncertainty
1<∞xyxyWS
⇓1−
< xyxy WSdefines the size of thetolerated uncertainty
defines an upper templatefor the modulus of the
sensitivity function
There also lower templates (because of the relationship between various sensitivity fct.)
Adaptive Control – Landau, Lozano, M’Saad, Karimi29
Robust stability and templates for the sensitivity functions
Robust stability condition:
•The functions (the inverse of the size of the uncertainties) define an “upper” template for the sensitivity functions
• Conversely the frequency profile of can be interpreted interms of tolerated uncertainties
11 )( −−zW
πωωω ≤≤< −−− 0)()( 1j
z
j
xy eWeS
)( ωj
xy eS −
ω
dB
Syp
Syp
Syp
-1
min
-1
= - MΔ
= MΔ
Sypmax
0
= - Sypmax
Tolerated feedback uncertainty on the input
Sup dB
00.5f e
Sup
-1
Tolerated additive uncertainty
Adaptive Control – Landau, Lozano, M’Saad, Karimi30
( G ’ = G + δWa )
Sup dB
actuator effort
size of the tolerated additive uncertainty Wa
00.5f s
Sup-1
Templates for the Sensitivity Functions
Output SensitivityFunction
Input SensitivityFunction
Syp max= - MΔSyp dB
0.5fs0
delaymarginnominal
perform.
Dangerous zones.Need for good models inthese regions
Adaptive Control – Landau, Lozano, M’Saad, Karimi31
Templates for the output sensitivity functions Syp
Syp dB
0,5fe
Syp max= - MΔ
0
Performances
Robustness
Syp dB
0,5fe
Syp max= - MΔ
0
Attenuation zone
openingthe loop
Adaptive Control – Landau, Lozano, M’Saad, Karimi32
Shaping the sensitivity functions
1. Choice of the dominants et auxiliary poles of the closed loop2. Choice of the fixed part of the controller (HS and HR )3. Simultaneous choice of the fixed parts and the auxiliary poles
Procedure:
Basic shaping : use 1 and 2Fine shaping: use 3
There exist also tools for automatic sensitivity function shapingbased on convex optimization (Optreg from Adaptech)
Tools for sensitivity shaping: WinReg (Adaptech) and ppmaster.m
Adaptive Control – Landau, Lozano, M’Saad, Karimi33
Pole placement with sensitivity functions shaping
Performance specification for pole placement :• Desired dominant poles for the closed loop• The reference trajectory (tracking reference model)
Questions:• How to take into account the specifications in certain frequencyregions?
• How to guarantee the robustness of the controllers ?• How to take advantage from the degree of freedom forthe maximum number of poles which can be assigned ?
Answer:Shaping the sensitivity functions by:
- introducing auxiliary poles- introducing filters in the controllers
Adaptive Control – Landau, Lozano, M’Saad, Karimi34
Sensitivity functions - review
)()()()()()()( 1111
111
−−−−−
−−−
+=
qRqBqqSqAqSqAqS dyp
)()()()()()()( 1111
111
−−−−−
−−−
+−=
qRqBqqSqAqRqAqS dup
)()(')( 111 −−− = qHqRqR R )()(')( 111 −−− = qHqSqS S
)()()()()()()( 1111111 −−−−−−−− ==+ qPqPqPqRqBqqSqA FDd
Pre specified parts (filters)
Output sensitivity function:
Input sensitivity function:
Controller structure :
Dominant and auxiliary filters:
Study of the properties of the sensitivity functions in the frequency domain: q=z=ejω
Adaptive Control – Landau, Lozano, M’Saad, Karimi35
Properties of the output sensitivity function
P.1- The modulus of the output sensitivity function at a certainfrequency gives the amplification or attenuation factor of thedisturbance on the output
Syp(ω) < 1(0 dB) attenuation Syp(ω) > 1 amplification
Syp(ω) = 1 operation in open loop
P.2 ( ) 1max
)(−
=Δ ωjSM ypModulus margin
Adaptive Control – Landau, Lozano, M’Saad, Karimi36
P.3 – The open loop (KG) being stable one has the property:
∫ =−SS
f.πf/fj
yp df)(eS50
0
2 0log
The sum of the areas between the curve of Syp and the axis 0dB taken withtheir sign is null
Disturbance attenuation in a frequency region implies amplificationof the disturbances in other frequency regions!
Properties of the output sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi37
The asymptotically stable auxiliary poles (PF) lead in generalto the reduction of in the frequency regionscorresponding to the attenuation regions for 1/PF
Properties of the output sensitivity function
)( ωjS yp
FPnF qpqP )1()( 11 −− ′+= 05.05.0 −≤′≤− p
DF PPP nnn −≤
In many applications the introduction of damped high frequency auxiliarypoles is enough for assuring the required robustness margins
Adaptive Control – Landau, Lozano, M’Saad, Karimi38
22
11
22
11
1
1
11
)(
)(−−
−−
−
−
++
++=
qqqq
qP
qH
i
i
F
S
ααββ
Obtained by the discretization of :
200
2
200
2
22
)(ωωζωωζ
++
++=
ssss
sFden
num1
1
112
−
−
+−
=zz
Ts
e
with:
produce and attenuation (hole) at the normalized discretized frequency:
⎟⎠
⎞⎜⎝
⎛=
2arctan2 0 e
discTω
ω with attenuation: ⎟⎟⎠
⎞⎜⎜⎝
⎛=
den
numtM
ζζ
log20 dennum ζζ <( )
and has negligible effects at f << fdisc and at f >> fdisc
Properties of the output sensitivity function
Simultaneous introduction of a fixed part HSi and of a pair of auxiliary poles PFi of the form:
Adaptive Control – Landau, Lozano, M’Saad, Karimi39
For details see Landau: Commande des Systèmes, HermesEfective computation using: filter22.sci (.m)
Properties of the output sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi40
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25
-20
-15
-10
-5
0
5
Syp Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
ω = 0.4 rad/secω = 0.6 rad/secω = 1 rad/secTemplate for Modulus marginTemplate for Delay margin = Ts
Augmenting the attenuation or widening the attenuation zone
Higher amplification of disturbancesoutside the attenuation zone
Reduction of the robustness(reduction of the modulus margin)
Properties of the output sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi41
P.4 – Cancellation of the disturbance effect at a certain frequency:
sjj
Sjjj ffeSeHeAeSeA /20)()()()()( ; πωωωωωω ==′= −−−−−
{Zeros of Syp Allows introduction of zeros at desired frequencies
Properties of the output sensitivity function
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-40
-35
-30
-25
-20
-15
-10
-5
0
5
Syp Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
HS = 1 - q-1
HS = (1 + q-2)(1 - q-1)
Adaptive Control – Landau, Lozano, M’Saad, Karimi42
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25
-20
-15
-10
-5
0
5
Syp Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
HR = 1HR = 1 + q-2
P.5 - at frequencies where:)0(1)( dBjS yp =ω
sjj
Rjjj ffeReHeBeReB /20)()()()()( ** ; πωωωωωω ==′= −−−−−
Properties of the output sensitivity function
Allows introduction of zeros at desired frequencies
Adaptive Control – Landau, Lozano, M’Saad, Karimi43
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25
-20
-15
-10
-5
0
5
Syp Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
PF = 1PF = (1 - 0.375q-1)2
Template for Modulus marginTemplate for Delay margin = Ts
P.6 – Asymptotically stable auxiliary poles (PF) lead (in general) to the reduction of in the attenuationband of 1/PF
)( ωjS yp
FPnF qpqP )1()( 11 −− ′+= 05.05.0 −≤′≤− p
DF PPP nnn −≤
In many applications, introduction of high frequency auxiliary polesis enough for assuring the required robustness margins
Properties of the output sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi44
P.7 – Simultaneous introduction of a fixed part HSi and of a pairof auxiliary poles PFi having the form:
22
11
22
11
1
1
11
)(
)(−−
−−
−
−
++
++=
qqqq
qP
qH
i
i
F
S
ααββ
resulting from the dicretization of :
200
2
200
2
22
)(ωωζωωζ
++
++=
ssss
sFden
num1
1
112
−
−
+−
=zz
Ts
e
with:
introduces an attenuation at the normalized discretized frequency:
⎟⎠
⎞⎜⎝
⎛=
2arctan2 0 e
discTω
ω with the attenuation: ⎟⎟⎠
⎞⎜⎜⎝
⎛=
den
numtM
ζζ
log20 dennum ζζ <( )
and with negligible effect at f << fdisc and at f >> fdisc
Properties of the output sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi45
Effective computation with the function: filter22.sci (.m)
Properties of the output sensitivity function
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25
-20
-15
-10
-5
0
5
Syp Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
HS = 1, PF = 1HS = ( ω = 1.005, ζ = 0.21), PF = ( ω = 1.025, ζ = 0.34)
Adaptive Control – Landau, Lozano, M’Saad, Karimi46
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-40
-30
-20
-10
0
10
20Sup Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
HR = 1HR = 1 + 0.5q-1
HR = 1 + q-1
P.1 – Cancellation of the disturbance effect on the input at a certain frequency (Sup = 0):
sjj
Rj ffeReHeA /20)()()( ; πωωωω ==′ −−−
101)( 11 ≤<+= −− ββqqH R( active at 0.5fS)
Properties of the input sensitivity function
Allows introduction of zeros at desired frequencies
Rem: The system operate in open loop at this frequency
Adaptive Control – Landau, Lozano, M’Saad, Karimi47
P.2 – At frequencies where:
sjj
Sj ffeSeHeA /20)()()( ; πωωωω ==′ −−−
)()()( ω
ωω
j
jj
up eBeAeS
−
−− =0)( =ωjS yp
One has:
Consequence : strong attenuation of the disturbances should bedone only in the frequency regions where the system gainis enough large ( in order to preserve robustness and avoidtoo much stress on the actuator)
Inverse ofthe systemgain
Remember: gives the tolerance with respect to additive uncertainties on the model (high = weak robustness)
1)(
−ωjSup
)( ωjSup
Properties of the input sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi48
P.3 – Simultaneous introduction of a fixed part HRi and of a pairof auxiliary poles PFi having the form:
resulting from the dicretization of :
200
2
200
2
22
)(ωωζωωζ
++
++=
ssss
sFden
num1
1
112
−
−
+−
=zz
Ts
s
with:
introduces an attenuation at the normalized discretized frequency:
⎟⎠
⎞⎜⎝
⎛=
2arctan2 0 e
discTω
ω with the attenuation: ⎟⎟⎠
⎞⎜⎜⎝
⎛=
den
numtM
ζζ
log20 dennum ζζ <( )
and with negligible effect at f << fdisc and at f >> fdisc
22
11
22
11
1
1
11
)(
)(−−
−−
−
−
++
++=
qqqq
qP
qH
i
i
F
R
ααββ
Properties of the input sensitivity function
Adaptive Control – Landau, Lozano, M’Saad, Karimi49
Shaping the sensitivity functions - Example I
sTdqBqA e 1;2;3.0;7.01 11 ===−= −−Plant:
• Integrator• Dominant poles: discretization of a cont. time 2nd order system : ω0 = 1 rad/s, ζ = 0.9
Controller A :Attenuation band: 0 up to 0.058 Hz but ΔM < -6 dB and Δτ < Ts
Objective: same attenuation band but with ΔM > -6 dB and Δτ > Ts- insertion of auxiliary poles:
Specifications:
( )214.01 −−= qPF
Controller B : good margins but reduction of the attenuation band-insertion of pole-aero filter HS/PF centered at ω0 = 0.4 rad/s (0.064 Hz)
Controller C : good attenuation band but Syp > 6 dB - larger (slower) auxiliary poles (0.4 0.44)
Controller D : Correct
Adaptive Control – Landau, Lozano, M’Saad, Karimi50
Shaping the sensitivity functions - Example I
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25
-20
-15
-10
-5
0
5
10Syp Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
ABCDTemplate for Modulus marginTemplate for Delay margin = Ts
Adaptive Control – Landau, Lozano, M’Saad, Karimi51
Shaping the sensitivity functions - Example II
Plant (integrator): sTdqBqA s 1;2;5.0;1 11 ===−= −−
q-dBA
u(t) y(t)
Sinusoidal disturbance (0.25 Hz)
Low frequencies disturbances
+
+
+
Specifications:1. No attenuation of the sinusoidal disturbance at (0.25 Hz)2. Attenuation band at low frequencies : 0 à 0.03 Hz3. Disturbances amplification at 0.07 Hz: < 3dB 4. Modulus margin > -6 dB and Delay margin > T5. No integrator in the controller
Adaptive Control – Landau, Lozano, M’Saad, Karimi52
Shaping the sensitivity functions - Example II
- Fixed parts design : 1;1 2 =+= −SR HqH
Opening the loop at 0.25 Hz
-Dominant poles: discretization of a cont. time 2nd order system:ω0 = 0.628 rad/s, ζ = 0.9
Controller A : the specs. at 0.07 Hz are not fulfilled- insertion of a pole-zero filter HS/PF centered at ω0 = 0.44 rad/s
Controller B : Attenuation band smaller than that specified- dominant poles acceleration: ω0 = 0.9 rad/s
Controller C : Correct
Adaptive Control – Landau, Lozano, M’Saad, Karimi53
Shaping the sensitivity functions - Example II
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-15
-10
-5
0
5
10Syp Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
ABCTemplate for Modulus marginTemplate for Delay margin = Ts
Adaptive Control – Landau, Lozano, M’Saad, Karimi54
Robust Controller Design
Pole placement with sensitivity functions shaping
FD PPP =
RHRR '=
SHSS '=
Nominal performance: SRD HandHofpartandP
Allow to shape the sensitivity functions
-IterativeChoosing and using band stop filters(matlab toolbox « ppmaster » )
FjSjFiRi PHPH /,/FP
Several approaches to design :
-Convex optimization(see Langer, Landau, Automatica, June99, Optreg (Adaptech) )
Adaptive Control – Landau, Lozano, M’Saad, Karimi55
Position Control by means of a Flexible Transmission
Φm
LOAD
Φref
MOTORAXIS
AXISPOSITION
DAC
ADCCONTROLLER
DCMOTOR
R-S-TCONTROLLER
POSITIONTRANSDUCER
u(t) y(t)
For details see next slide and book
Adaptive Control – Landau, Lozano, M’Saad, Karimi56
Position Control by means of a Flexible Transmission
12 13 14 15 16 17 18-1.5
-1
-0.5
0
0.5
1
1.5Flexible Transmission: Output
Time (s) (Ts = 50 ms)A
mpl
itude
(Vol
t)
12 13 14 15 16 17 18
-0.2
0
0.2
0.4
Flexible Transmission: Control Signal
Time (s) (Ts = 50 ms)
Am
plitu
de (V
olt)
5 10 15 20 25-0.2
0
0.2
0.4
0.6
0.8
1
Flexible Transmission: Output
Time (s) (Ts = 50 ms)
Am
plitu
de (V
olt)
PositionReference
5 10 15 20 25
0
0.5
1
Flexible Transmission: Control Signal
Time (s) (Ts = 50 ms)
Am
plitu
de (V
olt)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-30
-25
-20
-15
-10
-5
0
5
10
15
20Flexible Transmission: Sup Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
HR = 1HR = 1, 4 aux. poles in α = 0.2HR = 1 + q-1, 4 aux. poles in α = 0.2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-20
-15
-10
-5
0
5
10Flexible Transmission: Syp Magnitude Frequency Responses
Frequency (f/fs)
Mag
nitu
de (d
B)
HR = 1HR = 1, 4 aux. poles in α = 0.2HR = 1 + q-1, 4 aux. poles in α = 0.2
Template for Modulus marginTemplate for Delay margin = Ts
RegulationTracking
Adaptive Control – Landau, Lozano, M’Saad, Karimi57
MIRROR
DETECTOR
RIGID FRAMES
LIGHTSOURCE
POT.
ENCODER
SERVO.
LOCALPOSITION
COMPUTERTACH.
ALUMINIUM
360° Flexible Arm
Adaptive Control – Landau, Lozano, M’Saad, Karimi58
Frequency characteristics Poles-Zeros
360° Flexible Arm
(Identified Model)
Adaptive Control – Landau, Lozano, M’Saad, Karimi59
1 2 3 4 5 6 7 8 9-30
-25
-20
-15
-10
-5
0
5
10
15
20
25
Frequence [Hz]
Mod
ule
[dB]
Syp - Sensibilité perturbation-sortie
A B
D C
gabarit
1 2 3 4 5 6 7 8 9-30
-20
-10
0
10
20
30
40
50
60
Frequence [Hz]M
odul
e [d
B]
Sup - Sensibilité perturbation-entrée
A
D
B
C
gabarit
A- without auxiliary polesB- with auxiliary polesC- with stop band filterD- with stop band filter
11 / FS PH22 / FR PH
Shaping the Sensitivity Functions
Output Sensitivity Function - Syp Input Sensitivity Function - Sup
template
template