Robust discrete time control
I.D. Landau
Emeritus Research Director at C.N.R.S
Laboratoire d’Automatique de Grenoble, (INPG/CNRS), France
April 2004,
Valencia
I.D. Landau A course on robust discrete time control, part1
1
Outline
Part I
Introduction, examples and basic concepts
Part II
Design of robust discrete time controllers
Part III
Special topics
I.D. Landau A course on robust discrete time control, part1
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Controller Design and Validation
Performance
specs.
Robustness
specs.
DESIGN
METHOD
MODEL(S)
IDENTIFICATION
Ref.
CONTROLLER
+
+
u
y
PLANT
1) Identification of the dynamic model
2) Performance and robustness specifications
3) Compatible robust controller design method
4) Controller implementation
5) Real-time controller validation
(and on site re-tuning)
6) Controller maintenance (same as 5)
(5) and (6) require
identification in closed-loop
I.D. Landau A course on robust discrete time control, part1
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Part I
Outline
• The R-S-T digital controller
• Robust control. What is it ?
• Examples (4 applications)
• Sensitivity functions
• Stability of closed loop discrete time systems
• Robustness margins
• Robust stability
• Small gain theorem/ Passivity theorem / Circle criterion
• Description of uncertainties and robust stability
• Templates for the sensitivity functions
I.D. Landau A course on robust discrete time control, part1
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The R-S-T Digital Controller
CLOCK
r(t)
Computer
(controller)
u(t)
D/A
+
ZOH
y(t)
PLANT
A/D
Discretized Plant
r(t)
Bm
Am
T
+
-
Controller
1
S
u(t)
R
q −d B
A
y(t)
Plant
Model
q −1 y (t ) = y (t − 1)
I.D. Landau A course on robust discrete time control, part1
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The R-S-T Digital Controller
r(t)
Bm
Am
+
T
-
1
S
R
Controller
u(t) q − d B
y(t)
A
Plant
Model
−d
−1
− d −1
−1
q
B
(
q
)
q
B
*
(
q
)
−1
−1
G
(
q
)
=
H
(
q
)
=
=
Plant Model:
A(q −1 )
A( q −1 )
A(q −1 ) = 1 + a1q −1 + ... + an q −n B(q −1 ) = b1q −1 + ... + bn q −n = q −1 B * ( q −1 )
A
A
R-S-T Controller:
B
B
S (q −1 )u (t ) = T (q −1 ) y * (t + d + 1) − R( q −1 ) y(t )
Characteristic polynomial (closed loop poles):
P(q −1 ) = A( q −1 ) S (q −1 ) + q −d B( q −1 ) R( q −1 )
I.D. Landau A course on robust discrete time control, part1
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Robust control
Robustness of what ? with respect to what ?
Robustness of an index of performance with respect to plant
model uncertainties.
Index of performance :
• closed loop stability
• time domain performance
• frequency domain performance
Plant model uncertainties:
• low quality design model
• uncertainties in a frequency range
• variations of the dynamic characteristics of the plant
I.D. Landau A course on robust discrete time control, part1
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Robust control
Robust controller : assures the desired index of performance of
the closed loop for a set of given uncertainties
Robust closed loop : the index of performance is guaranteed
for a set of given uncertainties
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Robustness of a control system
A control system is said to be robust for a set of given
uncertainties upon the nominal plant model if it guarantees
stability and performance for all plant models in this set.
•To characterize the robustness of a closed loop system a
frequency domain analysis is needed
• The sensitivity functions play a fundamental role in robustness
studies
• The study of closed loop stability in the frequency domain
gives valuable information for characterizing robustness
I.D. Landau A course on robust discrete time control, part1
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Robustness. Some questions
•How to characterize “model uncertainty”?
•How to deal with “model uncertainty” ?
•How to characterize “controller robustness” ?
Structured (parametric uncertainty)
Model uncertainty
Unstructured (specified in the frequency domain)
We will consider the “uncertainties” described in the frequency
domain (i.e. effect of parameter uncertainty should be converted
in the frequency domain).
Important concepts :
• nominal model, uncertainty model, family of plant models
• nominal stability, robust stability
• nominal performance, robust performance
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Nominal model, uncertainty model, family of plant models
Nominal model : the model used for design
Uncertainty model : the description of the uncertainties in the
frequency domain (magnitude, phase)
Family of plant models (P): characterized by the nominal plant model
and the uncertainty model
The different true “plant models” belong to the family P
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Nominal and robust stability and performance
Nominal stability : closed loop as. stable for the “nominal model”
Robust stability: closed loop as. Stable for all the plant models
belonging to the family (set) P
Nominal performance: performance for the “nominal model”
Robust performance: performance guaranteed for all plant
models belonging to the family (set) P
I.D. Landau A course on robust discrete time control, part1
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Robust stability and robust performance
Robust stability:
-1
-1
The closed loop poles remain inside the unit circle
(discrete time case) for all the models belonging to
the family P
1
r
c
1
Robust performance:
The closed loop poles remain inside the circle (c,r)
for all the models belonging to the family P
• robust performance condition can be translated in a
robust stability condition
• this is not the only way for solving robust performance design
I.D. Landau A course on robust discrete time control, part1
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Examples to illustrate robust control design
• Continuous steel casting
• Hot dip galvanizing
• Flexible transmission
• 360° flexible arm
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Continuous steel casting
Level to be controlled
Shaking
(to prevent adherence)
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Continuos steel casting
Plant (integrator):
A = 1 − q −1 ; B = 0.5q −1 ; d = 2 ; TS = 1s
Low frequency
disturbances
u(t)
-d
q B
A
+
Sinusoidal
disturbance (0.25Hz)
+
y(t)
+
Specifications:
1. No attenuation of the sinusoidal disturbance (0.25 Hz)
2. Attenuation band in low frequencies: 0 à 0.03 Hz
3. Disturbance amplification at 0.07 Hz: < 3dB
4. Modulus margin > -6 dB and Delay margin > TS
5. No integrator in the controller
I.D. Landau A course on robust discrete time control, part1
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Hot dip galvanizing. Control of the deposited zinc
Finished
product
Steel
strip
Measurement of
deposited mass
Air knife
Preheat oven
Zinc
bath
air
air
zinc
steel strip
I.D. Landau A course on robust discrete time control, part1
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Hot dip galvanizing. The control loops
Desired
deposited zinc mass per cm2
PRBS
(± ∆P)
m*
RST
controller
Measured
deposited zinc mass per cm2
P0
+
Pressure
controller
Pressurized
Gaz supply
Coating
process
+
Measurement
delay
m ± ∆m
• important time delay with respect to process dynamics
• time delay depends upon the steel strip speed
• sampling frequency tied to the steel strip speed
• constant integer delay in discrete time
• parameter variations of the process as a function
of the type of product
I.D. Landau A course on robust discrete time control, part1
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Hot dip galvanizing. Model and specifications
Plant model for a type of product :
q −7 (b q −1 )
1 + a1 q −1
1
Model : TS = 12 sec; b1 = 0.3; a1 = −0.2(− 0.3) )
Specifications (performance) :
•Modulus margin:
∆M ≥ 0.5
∆τ ≥ 2TS
•Delay margin:
•Integrator
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Hot dip galvanizing. Performance
I.D. Landau A course on robust discrete time control, part1
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Control of a Flexible Transmission
The flexible transmission
motor
load
axis
Φm
d.c.
motor
axis
position
Position
transducer
y(t)
Controller
D u(t)
A
C
I.D. Landau A course on robust discrete time control, part1
R-S-T
controller
A
D
C
Φref
21
Control of a Flexible Transmission
Sampling frequency : 20 Hz
A( q −1 ) = 1 − 1.609555q −1 + 1.87644q −2 − 1.49879q −3 + 0.88574q −4
B( q −1 ) = 0.3053q −1 + 0.3943q −2
d =2
Model
Vibration modes:
ω1 = 11.949 rad / sec, ζ 1 = 0.042 ; ω 2 = 31.462 rad / sec, ζ 2 = 0.023
Specifications:
ω 0 = 11 .94 rad / sec, ζ = 0.9
Dominant poles: ω 0 = 11 .94 rad / sec, ζ = 0.8
Tracking:
Robustness margins: ∆M ≥ 0.5 ∆τ ≥ 2TS
Zero steady state error (integrator)
Constraints on Sup :
Sup
max
≤ 10 dB for f ≥ 0.35 f S = 7 Hz
I.D. Landau A course on robust discrete time control, part1
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360° Flexible Arm
LIGHT
SOURCE
RIGID FRAMES
MIRROR
ALUMINIUM
DETECTOR
COMPUTER
TACH.
POT.
LOCAL
POSITION
SERVO.
ENCODER
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360° Flexible Arm
Frequency characteristics
Poles-Zeros
Unstable zeros !
(Identified Model)
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360° Flexible Arm
Sampling frequency : 20 Hz
A ( q − 1 ) = 1 − 2 .1049 q − 1 + 1 .04851 q − 2 + 0 . 33836 q − 3 + 0 .46 q − 4
Model
− 1 .5142 q − 5 + 0 . 7987 q − 6
B ( q −1 ) = 0 .0064 q −1 + 0 . 0146 q − 2 − 0 .0697 q − 3 + 0 .044 q − 4
+ 0 . 0382 q − 5 − 0 . 007 q − 6
d =0
Vibration modes:
ω1 = 2.617rad / sec, ζ 1 = 0.018; ω2 = 14.402rad / sec, ζ 2 = 0.025; ω3 = 48.117rad / sec, ζ 3 = 0.038
Specifications:
Tracking:
Dominant poles:
ω 0 = 2.6173 rad / sec, ζ = 0.9
ω 0 = 2.6173 rad / sec, ζ = 0.8
Robustness margins: ∆M ≥ 0 .5
Zero steady state error (integrator)
Constraints on Sup :
∆τ ≥ 2TS
Sup ≤ 15 dB for f < 4 Hz; Sup ≤ 0 dB for 4.5 ≤ f < 6.5 Hz;
S up < 15 dB for 6.5 ≤ f < 8 Hz; S up < 10 dB for 8 ≤ f ≤ 10 Hz
I.D. Landau A course on robust discrete time control, part1
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Robust Control. Basic concepts
I.D. Landau A course on robust discrete time control, part1
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Digital control in the presence of disturbances and noise
(disturbance)
p(t)
r(t)
+
T
1/S
-
v(t)
+
+
u(t)
B/A
y(t)
+
+
PLANT
+
R
Output sensitivity function
(p
y)
Input sensitivity function
(p
u)
+
b(t)
(measurement noise)
−1
S yp ( z ) =
−1
S up ( z ) =
Noise-output sensitivity function
(b
y)
−1
S yb ( z ) =
A( z −1 )S ( z −1 )
A( z −1 )S ( z −1 ) + B( z −1 )R( z −1 )
− A( z −1 )R (z −1 )
A( z −1 ) S ( z −1 ) + B( z −1 )R (z −1 )
− B( z −1 ) R( z −1 )
A( z −1 )S ( z −1 ) + B( z −1 ) R( z −1 )
B( z −1 )S ( z −1 )
Input disturbance-output sensitivity function S ( z −1 ) =
yv
(v
y)
A( z −1 )S ( z −1 ) + B( z −1 )R( z −1 )
All four sensitivity functions should be stable ! (see book pg.102 - 103)
I.D. Landau A course on robust discrete time control, part1
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All four sensitivity functions must be stable !
z − d B ( z −1 )
−1
Plant model:
A
(
z
) is unstable
−1
A( z )
Supose : R ( z −1 ) = A( z −1 ) (poles compensation by the controller zeros)
−1
S yp ( z ) =
A( z −1 ) S ( z −1 )
−1
−1
−1
−1
=
S ( z −1 )
S ( z −1 ) + B ( z −1 )
A( z −1 )
−1
Sup ( z ) = −
=−
−1
−1
−1
−1
A( z ) S ( z ) + B ( z ) A( z )
S ( z −1 ) + B( z −1 )
B( z −1 ) A( z −1 )
B ( z −1 )
−1
S yb ( z ) = −
=−
−1
−1
−1
−1
A( z ) S ( z ) + B( z ) A( z )
S ( z −1 ) + B ( z −1 )
B( z −1 ) S ( z −1 )
B( z −1 ) S ( z −1 )
−1
S yv ( z ) =
=
−1
−1
−1
−1
A( z ) S ( z ) + B( z ) A( z ) A( z −1 )[ S ( z −1 ) + B ( z −1 )]
A( z ) S ( z ) + B( z ) A( z )
A( z −1 ) A( z −1 )
Syv is unstable while Syp, Sup,and Syb may be stable if :
S ( z −1 ) + B ( z −1 ) = 0 ⇒ z < 1
I.D. Landau A course on robust discrete time control, part1
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Complementary sensitivity function
(disturbance)
p(t)
r(t)
+
T
1/S
-
v(t)
+
+
u(t)
B/A
+
+
y(t)
PLANT
+
R
+
b(t)
(measurement noise)
For T = R one has:
−1
S yr ( z ) =
B ( z −1 ) R ( z −1 )
−1
−1
−1
−1
A( z ) S ( z ) + B( z ) R( z )
= −S yb ( z −1 )
S yp ( z −1 ) − S yb ( z −1 ) = S yp ( z −1 ) + S yr ( z −1 ) = 1
I.D. Landau A course on robust discrete time control, part1
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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))
Im H
− jω
− jω
B
(
e
)
R
(
e
)
H OL (e − jω ) =
A(e − jω ) S (e − jω )
Critical point
ω=π
-1
-1
S yp = 1 + H (e -jω )
BO
A( z −1 ) S ( z −1 ) + B( z −1 ) R( z −1 )
S yp ( z ) = 1 + H OL ( z ) =
A( z −1 ) S ( z −1 )
−1
−1
−1
Re H
H (e -j ω )
BO
ω =0
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 growing
frequencies (from 0 to 0.5fS) leaves the critical point[-1, j0] on the left
I.D. Landau A course on robust discrete time control, part1
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Stability of closed loop discrete time systems
Nyquist criterion (discrete time –O.L. is unstable)
The Nyquist plot of the open loop transfer fct. HOL(e-jω) traversed in the sense of growing
frequencies (from 0 et f S) 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 of
unstable poles in open loop.
Remarks:
Im H
ω = 2π
1 unstable pole in Open Loop
Stable Closed Loop (a)
ω=π
-1
ω=π
ω =0
-The controller poles may become
unstable if high performances are
required without using an appropriate
design method
Re H
Stable Open Loop
Unstable Closed Loop(b)
-The Nyquist plot from 0.5fS to fS is the
symmetric with respect to the real axis
of the Nyquist plot from 0 to 0.5fS
ω =0
I.D. Landau A course on robust discrete time control, part1
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Marges de robustesse
The minimal distance with respect to the critical point
characterizes the robustness of the CL with respect to
uncertainties on the plant model parameters( or their variations)
Im H
1
∆G
1
-1
∆Μ
Crossover
frequency
∆Φ
Re H
-Gain margin ∆G
-Phase margin ∆φ
-Delay margin ∆τ
-Modulus margin ∆M
|H OL |=1
ωCR
I.D. Landau A course on robust discrete time control, part1
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Robustness margins
Gain margin
∆G =
1
H BO ( jω 180 )
pour
∠φ (ω180 ) = −180 o
Phase margin
∆φ = 180 0 − ∠φ (ω cr )
∆φ = min ∆ φ i
pour H BO ( jω cr ) = 1
If there several intersections with the unit circle
i
Delay margin
∆τ =
∆φ
ω cr
Several intersections points:
∆τ = min
i
∆φ i
ω cr
i
Modulus margin
∆M = 1 + H BO ( jω ) min =
−1
S yp
(
jω )
min

=  S yp ( jω )

max 
I.D. Landau A course on robust discrete time control, part1
−1
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Robustness margins – typical values
Gain margin : ∆G ≥ 2 (6 dB) [min : 1,6 (4 dB)]
Phase margin : 30° ≤ ∆φ ≤ 60°
Delay margin : fraction of system delay (10%) or
of time response (10%) (often 1.TS)
Modulus margin : ∆ M ≥ 0.5 (- 6 dB) [min : 0,4 (-8 dB)]
A modulus margin ∆ M ≥ 0.5 implies ∆G ≥ 2 et ∆φ > 29°
Attention ! The converse is not generally true
The modulus margin defines also the tolerance with respect
to nonlinearities
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Robustness margins
Good gain and phase margin
Bad modulus margin
Good gain and phase margin
Bad delay margin
I.D. Landau A course on robust discrete time control, part1
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Modulus margin and sensitivity function
∆M = 1 + H OL ( z ) min = S yp ( z ) min = ( S yp ( z ) max ) =
−1
−1
−1
−1


A( z ) S ( z )


−1
−1
−1
−1
 A( z ) S ( z ) + B( z ) R( z ) max 
−1
−1
S yp (e − jω )
dB
max
−1
−1
pour z −1 = e − j 2πf
dB = ∆M −1dB = − ∆M dB
-1
S yp
S yp
= - ∆M
max
0
ω
-1
S yp
S yp
= - S yp
min
= ∆M
max
I.D. Landau A course on robust discrete time control, part1
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Robust stability
To assure stability in the presence of uncertainties (or variations)
on the dynamic chatacteristics of the plant model
HOL – nominal F.T.; H’OL –Different from HOL (perturbed)
Robust stability
condition
(sufficient cond.):
H OL′ ( z −1 ) − H OL ( z −1 ) < 1 + H OL ( z −1 ) = S yp−1 ( z −1 ) =
A( z −1 ) S ( z −1 ) + B ( z −1 ) R( z −1 )
P( z −1 )
=
; z −1 = e − jω
−1
−1
−1
−1
A( z ) S ( z )
A( z ) S ( z )
dB
Im H
-1
1 + H OL
'
H OL
(*)
-1
S yp
S yp
Re H
= - ∆M
max
0
ω
HOL
-1
S yp
S yp
= - S yp
min
= ∆M
max
Size of the tolerated uncertainity on HOL
at each frequency (radius)
I.D. Landau A course on robust discrete time control, part1
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Tolerance to plant additive uncertainty
From previous slide :
B ′( z −1 ) R ( z −1 ) B ( z −1 ) R ( z −1 ) R ( z −1 )
−
=
⋅
−1
−1
−1
−1
−1
′
A ( z ) S ( z ) A( z ) S ( z ) S ( z )
H’OL
B′( z −1 ) B ( z −1 ) A( z −1 ) S ( z −1 ) + B ( z −1 ) R ( z −1 )
−
<
−1
−1
′
A ( z ) A( z )
A( z −1 )S ( z −1 )
H’
HOL
H
B ′( z −1 ) B ( z −1 ) A( z −1 ) S ( z −1 ) + B ( z −1 ) R ( z −1 )
P ( z −1 )
−
<
=
= S up−1 ( z −1 )
−1
−1
−1
−1
−1
−1
A′( z ) A( z )
A( z ) R ( z )
A( z ) R ( z )
H’
H
S
up
(*)
(**)
dB
Limitation of the actuator stress
0
0.5f
e
S up -1
(Size of tolerated uncertainties)
I.D. Landau A course on robust discrete time control, part1
38
Tolerance to plant normalized uncertainty
(multiplicative uncertainty)
From (**), previous slide:
B′( z −1 ) B ( z −1 )
−
A′( z −1 ) A( z −1 ) A( z −1 ) S ( z −1 ) + B ( z −1 ) R ( z −1 )
P ( z −1 )
<
=
= S yb−1 ( z −1 )
−1
−1
−1
−1
−1
B( z )
B( z ) R( z )
B( z ) R( z )
A( z −1 )
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:
H '−H
H ' = H + ( H '−H ) = H (1 +
)
H
I.D. Landau A course on robust discrete time control, part1
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Passivity (Hyperstability) Theorem
u1
H1
y1
strictly
positive real
-
H2
y2
passive
u2
H1: Strictly positive real transfer function (state x )
H2: linear or nonlinear, time invariant or time-varying
η 2 (0, t1 ) ≥ t∑= 0 y 2T (t )u2 (t ) ≥ −γ 2 ; γ 22 < ∞ ; ∇t1 ≥ 0
t1
2
Then :
lim
x(t ) = 0 ; lim
u1 (t ) = 0 ; lim
y1 (t ) = 0
t →∞
t →∞
t →∞
I.D. Landau A course on robust discrete time control, part1
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Strictly positive real transfer function (SPR)
Frequency domain
continuous
jω
s
Im H
s =j ω
H(j ω)
Re H
σ
discrete
Im H
Im z
+j
z
jω
z =e
1
-1
Re H < 0 Re H > 0
j
-
Re H
Re z
jω
z =e
jω
H(e )
Re H < 0 Re H > 0
- asimptotycally stable
- Re H (e ) > 0 for all e = 1, (0 < ω < π )
jω
jω
(discrete time case)
Input – output property (time domain)
η1 (0, t1 ) ≥ t∑=0 y1 (t )u1 (t ) ≥ −γ + κ u1
t1
T
2
2
1
2T
; γ 12 < ∞ ; κ > 0 ; ∇t1 ≥ 0
I.D. Landau A course on robust discrete time control, part1
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Stability criteria for nonlinear time/varying feedback system
Im H (e-j ω )
y2
ku2
- k1
Re H (e -j ω )
-1
u2
H (e -j ω )
Im H (e -j ω )
critical circle
u2
β u
α u
y2
1
-α
-1
Re H ( z ) > −1 / k ; ∀ z = 1
-1
β
H (e -j ω )
I.D. Landau A course on robust discrete time control, part1
Re H (e -j ω )
Circle criterion
42
Stability in the presence of nonlinearities
(tolerance of nonlinearities)
Im H ( e -j ω )
critical circle
u
βu
αu
H
BO
(z
-1
)
- 1
α
-
-1
-1
β
Re H ( e -j ω )
∆M
H ( e -j ω )
The modulus margin defines the tolerance with respect to
nonlinear and/or time varying elements.
The tolerance sector is defined by :
Min gain: 1 /(1 + ∆M )
Max gain: 1 /(1 − ∆M )
I.D. Landau A course on robust discrete time control, part1
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The circle criterion : a particular case
KM=1
-1
1
Km=-1
H(z) should lie inside the unit circle
The modulus of the H(z) should be smaller than 1 at all frequencies i.e.:
H ( z) ∞ < 1
This is the “small gain theorem”
I.D. Landau A course on robust discrete time control, part1
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Small gain theorem
u1
-
S1 ∞ < 1
S2
y2
y1
S1
S2
∞
≤1
u2
S1: linear time invariant (state x)
S1 ∞ < 1
S2:
S2
∞
≤1
Then:
lim
x(t ) = 0 ; lim
u1 (t ) = 0 ; lim
y1 (t ) = 0
t →∞
t →∞
t →∞
It will be used to characterize “robust stability”
I.D. Landau A course on robust discrete time control, part1
45
Relationship between passivity theorem and small gain theorem
H −1
S∞=
≤1
H +1 ∞
If S ∞ ≤ 1 ⇒ H = 1 + S
is passive
1− S
H −1
<1
If H is strictly passive , S ∞ =
H +1 ∞
1+ S
S
<
1
⇒
H
=
If
is strictly passive
∞
1− S
If H is passive,
Hint for a proof:
H (e − j ω ) − 1
H − 2 Re H + 1
4 Re H
− jω
S (e ) =
=
=
1
−
2
2
2
H (e − j ω ) + 1
H + 2 Re H + 1
H (e − jω ) + 1
2
2
2
Remark:
Correct writing in the general operator case
S ∞ = ( H − 1)( H + 1) −1 ∞ ≤ 1
I.D. Landau A course on robust discrete time control, part1
46
Relationship between passivity theorem and small gain theorem
S1 ∞ < 1
S1
I
u1
2I
-
H1
+
-
strictly
positive real
+
-
y1
I
I
y2
0.5I
+
H2
+
-
passive
u2
I
S2
S2
∞
≤1
I.D. Landau A course on robust discrete time control, part1
47
Description of uncertainties in the frequency domain
Im H
Re H
Uncertainty disk
(at a certain frequency)
1) It needs a description by a transfer function which may have any phase but a modulus < 1
2) The size of the radius will vary with the frequency and is characterized by a transfer function
I.D. Landau A course on robust discrete time control, part1
48
Additive uncertainty
H ' ( z −1 ) = H ( z −1 ) + δ ( z −1 )Wa ( z −1 )
δ ( z −1 )
any stable transfer function with
Wa ( z −1 )
a stable transfer function
δ ( z −1 ) ∞ ≤ 1
H ' ( z −1 ) − H ( z −1 ) max = H ' ( z −1 ) − H ( z −1 ) ∞ = Wa ( z −1 ) ∞
δ
-
K
Wa
H
+
+
-
− Sup
Wa
δ
Apply small gain theorem
K = R / S; H = z B / A
−d
Robust stability condition:
S up ( z −1 )Wa ( z −1 ) ∞ < 1
I.D. Landau A course on robust discrete time control, part1
49
Multiplicative uncertainties
H ' ( z −1 ) = H ( z −1 )[1 + δ ( z −1 )Wm ( z −1 )]
δ ( z −1 )
any stable transfer function with
Wm ( z −1 )
a stable transfer function
δ ( z −1 ) ∞ ≤ 1
Wa ( z −1 ) = H ( z −1 )Wm ( z −1 )
δ
Wm
+
-
K
+
Robust stability condition:
-
H
− S yb
Wm
δ
S yb ( z −1 )Wm ( z −1 ) ∞ < 1
I.D. Landau A course on robust discrete time control, part1
50
Feedback uncertainties on the input
δ ( z −1 )
−1
H
(
z
)
H ' ( z −1 ) =
[1 + δ ( z −1 )Wr ( z −1 )]
any stable transfer function with δ ( z −1 ) ≤ 1
∞
Wr ( z −1 )
a stable transfer function
δ
Wm
-
-
K
-
H
+
Robust stability condition:
− S yp
Wr
δ
S yp ( z −1 )Wr ( z −1 ) ∞ < 1
I.D. Landau A course on robust discrete time control, part1
51
Robust stability conditions
Family (set) of plant models
H , H '∈ Ρ(W , δ )
Robust stability :
The feedback system is asymptotically stable for all the
plant models belonging to the family Ρ(W ,δ )
• Additive uncertainties
S up ( z −1 )Wa ( z −1 ) ∞ < 1
• Multiplicative uncertainties
S yb ( z −1 )Wm ( z −1 ) ∞ < 1
Sup (e − jω ) < Wa (e − jω )
−1
S yb (e − jω ) < Wm (e − jω )
−1
0≤ω ≤π
0≤ω ≤π
• Feedback uncertainties on the input (or output)
−1
S yp ( z −1 )Wr ( z −1 ) ∞ < 1
S yp (e − jω ) < Wr (e − jω ) 0 ≤ ω ≤ π
I.D. Landau A course on robust discrete time control, part1
52
Robust stability and templates for the sensitivity functions
Robust stability condition:
S xy (e − jω ) < Wz (e − jω )
−1
0≤ω ≤π
−1
−1
•The functions W ( z ) (the inverse of the size of the uncertainties)
define an “upper” template for the sensitivity functions
• Conversely the frequency profile of S xy (e − jω ) can be interpreted in
terms of tolerated uncertainties
dB
-1
S
S yp
S yp
= - ∆M
ω
S yp
-1
yp
dB
max
0
S
up
= - S yp
min
= ∆M
max
0
0.5f
e
S up -1
Tolerated additive uncertainty
Tolerated feedback uncertainty on the input
I.D. Landau A course on robust discrete time control, part1
53
Modulus margin and robust stability
Modulus margin:
S yp (e − jω ) < ∆ M
Robust stability cond.:
S yp (e ) < Wr (e )
− jω
− jω
−1
0≤ω ≤π
Possible uncertainties characterized by:
Wr −1 ( z −1 ) = ∆ M
δ ( z −1 ) = λ f ( z −1 ) ; − 1 ≤ λ ≤ 1
−1
−2
z
+
z
f ( z −1 ) = 1, z −1 , z − 2 ,...,
2
Examples of families of plant models for which robust stability is guaranteed:
1
1 − λ∆ M
1
H ′( z −1 ) = H ( z −1 )
1 − λ∆ M z −1
H ′( z −1 ) = H ( z −1 )
I.D. Landau A course on robust discrete time control, part1
54
Delay margin and robust stability
∆τ = 1.Ts
H ′( z −1 ) − H ( z −1 )
−1
H (z )
−1
H (z ) =
= z −1 − 1
z −d B ( z −1 )
−1
A( z )
−1
H ′( z ) =
z − d −1B ( z −1 )
A( z −1 )
Can be interpreted as a multiplicative uncertainty
H ′( z −1 ) = H ( z −1 )[1 + δ ( z −1 )Wm ( z −1 ) ] = H ( z −1 )[1 + ( z −1 − 1)]
δ ( z −1 ) = 1 ; Wm ( z −1 ) = ( z −1 − 1)
Robust stability condition:
S yb (e − jω ) < Wm (e − jω ) 0 ≤ ω ≤ π
1
−1
S yb ( z ) < −1 ; z = e − jω , 0 ≤ ω ≤ π
z −1
S yb−1 ( z −1 ) dB < −20 log 1 − z −1 ; z = e − jω
−1
Define a template on Syb
I.D. Landau A course on robust discrete time control, part1
Syb
55
Delay margin template on Syp
−1
S yp ( z ) =
A( z −1 )S ( z −1 )
−1
S yb ( z ) =
A( z −1 )S ( z −1 ) + B( z −1 )R( z −1 )
− B( z −1 ) R( z −1 )
A( z −1 )S ( z −1 ) + B( z −1 ) R( z −1 )
S yp ( z −1 ) = 1 + S yb ( z −1 )
1 − S yb ( z −1 ) ≤ S yp ( z −1 ) ≤ 1 + S yb ( z −1 )
S yb ( z −1 ) <
1 − 1 − z −1
1
− jω
;
z
=
e
,0≤ω ≤π
−1
z −1
−1
≤ S yp ( z −1 ) ≤ 1 + 1 − z −1
−1
Approximate
condition
I.D. Landau A course on robust discrete time control, part1
56
Defintion of “templates” for the sensitivity functions
Nominal performance requirements and robust stability conditions
lead to the definition of desired templates on the sensitivity functions
The union of various templates W −1 (e− jω ) will define an upper and a lower template
i
Upper template:
W −1 (e − jω ) ,... W −1 (e − jω ) 
W −1 ( e − jω ) sup = min
S
i
 S

i
Lower template:
1
n
W −1 (e − jω ) ,... W −1 ( e − jω ) 
W −1 ( e − jω ) inf = max
I
i
 I

i
1
n
I.D. Landau A course on robust discrete time control, part1
57
Frequency templates on the sur sensitivity functions
The robust stability conditions allow to define frequency templates
on the sensitivity functions which guarantee the delay margin and
the modulus margin;
The templates are essential for the design a good controller
Frequency template on the noise-output
sensitivity function Syb for ∆τ = TS
Frequency template on the output sensitivity
function Syp for ∆τ = TS and ∆Μ = 0.5
I.D. Landau A course on robust discrete time control, part1
58
Templates for the Sensitivity Functions
Syp dB
Syp max= - ∆ M
delay
margin
nominal
perform.
0
Output Sensitivity
Function
0.5f s
S up dB
actuator effort
0
0.5f s
Input Sensitivity
Function
S up -1
size of the tolerated additive uncertaintyaW
( G ’ = G + δW a )
I.D. Landau A course on robust discrete time control, part1
59
Templates for the Sensitivity Functions
S yp dB
S yp max = - ∆ M
0
opening
the loop
0,5f e
Output Sensitivity
Function
Attenuation zone
S up dB
Limitation of the actuator
stress
0.5f e
0
Zone with uncertainty
upon the modelquality
Input Sensitivity
Function
Opening the loop at the
frequency f (< - 100 dB)
I.D. Landau A course on robust discrete time control, part1
60