Uploaded by David Bensoussan

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Improved Simultaneous Performance in the Time Domain
and in the Frequency Domain
Azeddine Ghodbane, David Bensoussan and Maher Hammami
École de technologie supérieure, University of Quebec, Montreal, Canada
Abstract— An innovative approach for controlling unstable and
invertible systems has demonstrated superior performance compared
to conventional controllers. It has been successfully applied to a
levitation system and drone control. Simulations have yielded
satisfactory performances when applied to a satellite antenna
controller. This design method, based on sensitivity analysis, has also
been extended to handle multivariable unstable and invertible systems
that exhibit dominant diagonal characteristics at high frequencies,
enabling decentralized control.
Furthermore, this control method has been expanded to the realm of
adaptive control. In this study, we introduce an alternative adaptive
architecture that enhances both time and frequency performance,
helpfully mitigating the effects of disturbances from the input plant and
external disturbances affecting the output. To facilitate superior
performance in both the time and frequency domains, we have
developed a user-friendly interactive design methods using the
GeoGebra platform.
Keywords— Control theory, decentralized control, sensitivity theory,
input-output stability theory, robust multivariable feedback control
design.
effects of plant input and plant output perturbations on the
feedback system output [11].
Moreover, it has been shown that B control can be extended
to multivariable unstable and invertible system and ensures
decentralized control [12].
II. HEURISTIC PRESENTATION OF THE SISO B CONTROLLER
The B compensator structure [1] is represented in Fig. 1.
Fig. 1. Structure of the compensator C(s).
Note that the closed loop transmission ๐‘‡(๐‘ ) and the
sensitivity ๐‘†(๐‘ ) are related to the output, the input and the error
signal as follows:
๐‘Œ(๐‘ ) = ๐‘‡(๐‘ )๐‘…(๐‘ ), with ๐‘‡(๐‘ ) = ๐‘ƒ(๐‘ )๐ถ(๐‘ )(1 + ๐‘ƒ(๐‘ )๐ถ(๐‘ ))
I. INTRODUCTION
T
HE principle of quasi linear control was presented in the
context of sensitivity minimization [1]. In a quasi linear
controller, the gain and the poles of the compensator are
interrelated. The principle can be illustrated by a Nichols chart:
when the gain is increased, the pole is also increased in such a
way that the critical point (0 ๐‘‘๐ต, 180° ) is avoided. Doing so,
stability with an acceptable gain margin is maintained. The
advantages of high gain feedback are improved sensitivity and
improved tracking.
The design of the quasi linear controller has been formalized
for the case of a second order system [2] allowing arbitrarily
fast and robust tracking by feedback. The result has been
extended to a transfer function of any order [3]. However,
implementation of a quasi linear controller has shown that it
applies perfectly only when the gain is increased unboundedly.
To remedy to this problem the B control method [4,5], has
been proposed. The gain to pole dependency is calibrated
differently. It has been shown that for the same gain, B control
offers better settling times while keeping the system stable. It
has been tested for the control of the arm of a hard disk [6], the
orientation of a satellite antenna [7], a levitation system [8] and
the control of a drone [9,10] Simulations have also shown that
integrating a B controller to a L1 adaptive controller results in
a better settling time, as well as an improved attenuation of the
Azeddine Ghodbane (email : azeddine.ghodbane@etsmtl.ca) and David
Bensoussan (e-mail: david.bensoussan@etsmtl.ca) are within the École de
๐ธ(๐‘ ) = ๐‘†(๐‘ )๐‘…(๐‘ ), with ๐‘†(๐‘ ) = (1 + ๐‘ƒ(๐‘ )๐ถ(๐‘ ))
−1
(1)
−1
(2)
๐‘‡(๐‘ ) + ๐‘†(๐‘ ) = 1
(3)
We will present the case where the plant P(s) is minimum
phase, i.e. it is stable and invertible [13] and has the attenuation
property, i.e. there exist constants ๐‘ and ๐‘ž at a frequency high
enough such that
๐‘
|๐‘ƒ(๐‘—๐œ”)| > ๐‘ž , for all |๐‘ | > ๐œ”0
(4)
|๐‘ |
|๐‘†(๐‘—๐œ”)|
Fig. 2. Modulus of the sensitivity in the SISO case.
The design of the series compensator aims at havening
sensitivity on a limited frequency range ω1 smaller than any
positive constant ๐œ€ and the sensitivity norm is less than any
constant ๐‘€๐‘  greater than unity (Fig. 2).
โ€–(1 + ๐‘ƒ๐ถ)−1 โ€–∞ < ๐‘€๐‘  , ๐‘€๐‘  > 1
(5)
โ€–(1 + ๐‘ƒ๐ถ)−1 โ€–๐œ”1 < ๐œ€, 0 < ๐œ€ < 1
(6)
๐ถ(๐‘ ) = ๐‘ƒ −1 (๐‘ )๐ฝ1 (๐‘ )๐ฝ3 (๐‘ ) = ๐‘ƒ −1 (๐‘ ) (
๐‘˜1
๐‘ +๐œ”1
)(
๐œ”2
๐‘ +๐œ”2
)
๐‘˜
(7)
๐‘˜, ๐‘˜1 , ๐œ”1 and ๐œ”2 are chosen according to the following
criteria. for ๐ฝ2 (๐‘ ) = 1, keeping in mind that the addition of
technologie supérieure, Montreal, Canada. Maher Hammami (email:
hammami_maher@yahoo.fr) is with University of Sfax, Tunisia.
phase circuits in the mid frequencies allows to better tune the
time response.
๐‘˜≥๐‘›
(8)
1+๐œ€
๐‘€๐‘  −1
(๐‘˜+1)⁄2
)]
๐‘˜1 ≥ max [2
๐œ”1 ( ) , ๐œ”1 (
(9)
๐œ€
๐‘€
๐œ”๐‘ > ๐œ”1 [
๐œ”12 (1−
1
๐‘€๐‘ 
๐œ”
)
2
๐Ž = ๐Ÿ๐ŸŽ
− 1]
(10)
๐œ”
๐œ‹
๐œ”1
2
๐‘˜ tan−1 ( ๐‘) + tan−1 ( ๐‘) <
๐œ”2
๐’Œ = ๐Ÿ๐ŸŽ๐Ÿ“
๐‘ 
1⁄
2
๐‘˜12
๐‘ท(๐Ž)๐‘ฎ(๐Ž)
๐Ž = ๐Ÿ๐ŸŽ๐ŸŽ
(11)
๐œ”2 > max(๐œ”๐‘ , ๐‘ 0 )
(12)
X: Phase (degrees) Y: Magnitude (dB)
III. THE QUASI LINEAR CONTROLLER
Fig. 4: Nichols chart using the quasi-linear controller [3].
Given the plant represented by rational functions of the
๐‘ (๐‘ )
complex variable ๐‘  with the form ๐‘ƒ(๐‘ ) = ๐‘ƒ
and a
compensator represented by ๐ถ(๐‘ ) = ๐‘˜
๐‘๐ถ (๐‘ )
๐ท๐ถ (๐‘ )
๐ท๐‘ƒ (๐‘ )
, the closed loop is
represented by:
๐‘Œ(๐‘ )
๐‘…(๐‘ )
= ๐‘‡(๐‘ ) = ๐‘˜
๐‘๐‘ƒ (๐‘ )๐‘๐ถ (๐‘ )
IV. THE STABLE AND INVERTIBLE CASE
We apply the B control to the design of the controller of a hard
disk (Fig. 5).
(13)
1 + ๐‘˜๐ท๐‘ƒ (๐‘ )๐ท๐ถ (๐‘ )
Let ๐‘‘ represent the excess of poles over zeros of ๐‘‡(๐‘ ) and ๐‘“
be a real number (๐‘‘ − 1)๐‘“ < 1 < ๐‘‘๐‘“.
๐น๐‘œ๐‘ก ๐‘˜ > 0 large the closed loop transfer function is:
๐‘‡๐‘˜ (๐‘ ) = ๐‘‡๐‘ง๐‘˜ (๐‘ )๐‘‡๐‘‘๐‘˜ (๐‘ )
(14)
(๐‘ +๐‘ง )(๐‘ +๐‘ง )…(๐‘ +๐‘ง )
๐‘‡๐‘ง๐‘˜ (๐‘ ) = (๐‘ +๐‘ง1)(๐‘ +๐‘ง2)…(๐‘ +๐‘ง๐‘š) , Re[z1 ] ≤ โ‹ฏ ≤ Re[z๐‘š ]
(15)
ฬƒ
ฬƒ
ฬƒ
1
2
๐‘˜
,
1 )…(๐‘ +๐‘ฬƒ๐‘‘ )
๐‘‡๐‘‘๐‘˜ (๐‘ ) = (๐‘ +๐‘ฬƒ
๐‘š
๐‘…๐‘’[๐‘ฬƒ1 ] ≤ โ‹ฏ ≤ ๐‘…๐‘’[๐‘ฬƒ๐‘‘ ]
(16)
It has been shown [2] that when ๐‘˜ approaches infinity, the
poles ๐‘งฬƒ1 are approaching the zeros ๐‘ง๐‘– so that we can concentrate
on ๐‘‡๐‘‘๐‘˜ (๐‘ ), the poles of which take negative real values, the
module of which becomes bigger as the gain increases, ensuring
the stability and the fast response of the feedback system.
The compensator takes the form:
๐‘˜(๐‘ +๐‘ง1 )(๐‘ +๐‘ง2 )…(๐‘ +๐‘ง๐‘‘−1 )
๐ถ๐‘˜ (๐‘ ) = (๐‘ +๐‘Ž ๐‘“ )(๐‘ +๐‘Ž
(17)
๐‘˜
๐‘˜ ๐‘“ )…(๐‘ +๐‘Ž
๐‘˜ ๐‘“)
1
2
๐‘‘−1
Where the constants ๐‘Ž1 is given by:
๐‘˜1−(๐‘‘−1)๐‘“ ๐‘ฬƒ1 =
1
๐‘Ž1 ...๐‘Ž๐‘‘−1
, ๐‘ฬƒ๐‘– = ๐‘Ž๐‘– ๐‘˜ ๐‘“ , ๐‘– > 1
As an example, for ๐‘ƒ(๐‘ ) =
๐ถ(๐‘ ) =
๐‘˜(๐‘ +1)
(๐‘ +2)
1
๐‘ 2
(18)
The arm of a hard disk model is given by [14]:
๐‘ƒ(๐‘ ) =
6.4013×107
1 1
๐ถ๐‘˜ (๐‘ ) = (๐‘ +2๐‘˜ 0.6) , ๐‘‘ = 2, ๐‘“ = 0,6 ∈ ( , )
๐‘ƒ๐‘Ÿ,1 (๐‘ ) =
๐‘ƒ๐‘Ÿ,2 (๐‘ ) =
๐‘ƒ๐‘Ÿ,3 (๐‘ ) =
(19)
2 1
Fig. 3 and Fig. 4 show how the stability the stability of the
feedback system is maintained after the introduction of a quasi
linear controller: as the gain increases, so does the pole ๐‘ฬƒ1 so
that the Nichols chart keeps a secure distance from the critical
point (0 ๐‘‘๐ต, 180° ).
๐‘ท(๐Ž)๐‘ฎ(๐Ž)
๐’Œ = ๐Ÿ๐ŸŽ๐ŸŽ๐ŸŽ
๐Ž = ๐Ÿ๐ŸŽ
๐Ž = ๐Ÿ๐ŸŽ๐ŸŽ
X: Phase (degrees) Y: Magnitude
(dB)
Fig. 3: Nichols chart without using the quasi linear controller [3].
๐‘ 2
∏4๐‘–=1 ๐‘ƒ๐‘Ÿ,๐‘– (๐‘ )
(20)
with:
, the linear compensator
is modified to become gain dependent:
๐‘˜(๐‘ +1)
Fig. 5: B controller design of the control of the arm of a hard disk.
๐‘ƒ๐‘Ÿ,4 (๐‘ ) =
0.912๐‘  2 +457.4๐‘ +1.433×108
๐‘  2 +359.2๐‘ +1.433×108
0.7586๐‘  2 +962.2๐‘ +2.491×108
๐‘  2 +789.1๐‘ +2.491×108
9.917×108
๐‘  2 +1575๐‘ +9.917×108
2.731×109
๐‘  2 +2613๐‘ +2.731×109
(21)
(22)
(23)
(24)
๐ถ(๐‘ ) = ๐‘ƒ −1 (๐‘ )๐ฝ(๐‘ )
๐‘ƒ(๐‘ )๐ถ(๐‘ ) = ๐ฝ(๐‘ ) = ๐ฝ1 (๐‘ )๐ฝ2 (๐‘ )๐ฝ3 (๐‘ )
(25)
(26)
๐‘˜1 ๐ฝ1 (๐‘ ) is the transfer function of a high gain filter having
an fast time response. For example, ๐ฝ1 (๐‘ ) can have the
following form:
๐œ”
๐ฝ1 (๐‘ ) = ๐‘˜1 ( 1 )
(27)
๐‘ +๐œ”
1
The ๐ฝ2 (๐‘ ) compensator contains a set of lead/lag
compensator elements operating in the intermediate frequency
range. For example, ๐ฝ2 (๐‘ ) can have the following form:
๐‘ +๐‘ง
๐ฝ2 (๐‘ ) = ∏๐‘›๐‘–=1 ๐‘–
(28)
๐‘ +๐‘
๐‘–
๐ฝ3 (๐‘ ) is the transfer function of a low-pass filter acting at a
very high frequency so as to ensure that the controller ๐ถ(๐‘ )
remains strictly proper. It can have the general form:
๐œ”
๐ฝ3 (๐‘ ) = ๐›ฑ๐‘–๐‘˜ =1 2๐‘–
, ๐‘˜≥๐‘ž
(29)
๐‘ +๐œ”
2๐‘–
Note that the choice of ๐œ”2๐‘– could be reduced in various ways
to improve the implementation, such as a reduction in energy
requirements. The compensator parameters [15] are given
by ๐‘› = 2, ๐œ”1 = 50, ๐œ”๐ต = 13 × 104 , ๐‘1 = ๐‘2 = 50, ๐‘ง1 = ๐‘ง2 =
33, ๐‘ž = ๐‘˜ = 6, ๐‘˜1 = 6223. ๐œ”2i = ๐œ”2 is a fraction of ๐œ”๐‘ .
Our design leads to the following results. Fig.6 shows the
temporal performances when tuning ๐œ”2 . It allows the designer
to study the effect of the increased bandwidth of the
compensator.
System Respose to a Step Function
Fig. 8: Closed loop control of the orientation of a satellite [16].
1.2
Y(t) (um)
1
0.8
0.6
0.4
t=0.001 s
No Corrector w2=wb
No Corrector w2=2/3wb
No Corrector w2=1/3wb
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
-3
Time (sec) (seconds)
x 10
Fig. 6 Temporal response to a step input vs ๐œ”2 [15].
Fig. 7 shows the temporal performances when tuning ๐‘˜.
To compare the B controller to the existing control methods,
simulations for the velocity feedback and the position feedback
are compared to those obtained by PID controllers and were
published in the literature.
The PID is defined as follows [7]:
1
๐ถ๐‘ƒ๐ผ๐ท (๐‘ ) = 99.6827 + 0.03
− 903.9207. ๐‘ 
(31)
๐‘ +9.8203
The parameters of the B controller follow: ๐œ”1 = 42, ๐œ”2 =
10750, ๐‘˜1 = 37, ๐‘˜ = 3.
Fig. 9 and Fig. 10 present the improvements achieved for
the step time response for the velocity feedback and the position
feedback loops.
System Respose to a Step Function
0.25
Y(t) (radians)
0.2
0.15
B Control
PID Control
Reconstructed PID Control
0.1
K=50
K=200
K=1000
K=2500
K=5000
0.05
0
0
0.01
0.02
0.03
Time (sec)
0.04
0.05
Fig. 9: PID vs B controller for closed loop velocity control [7].
0.06
Fig. 7: Temporal response to a step input vs the exponent ๐‘˜ [15].
Applying the quasi linear compensator to a hard disk, the
settling time is greater than the one that can be obtained with B
control for the same compensator gain. However, improvement
of the settling time of the quasi linear compensator can be
obtained with a much greater and impracticable gain.
The performances of B controller are compared to existing
control methods in Table I. ๐บ๐‘€ and ๐‘ƒ๐‘€ represent the Gain
Margin and the Phase margin, ๐‘ก๐‘Ÿ and ๐‘ก๐‘  are the rise time and the
settling time.
TABLE I
Control of the arm of the hard disk [6]
Type of control
๐บ๐‘€ (๐‘‘๐ต)
๐‘ƒ๐‘€ (°)
๐‘ก๐‘Ÿ (๐‘ )
Proportional
11.9
12
0.0158
PID
54.1
47.8
0.0156
State feedback
22.8
49
0.0219
Quasi Linear
89.8
91.3
4.39
Present method
13.2
66
0.00019
๐‘ก๐‘  (๐‘ )
0.406
0.0637
0.0738
5.95
0.00027
Similarly, the control for positioning satellite antennas is
handled. The model of the rigid satellite antenna presented in
Figure 8 and modelled by the transfer function [7]:
1
๐‘ƒ(๐‘ ) = 2
(30)
๐‘  +1.72๐‘ +1.9
Fig. 10: PID vs B controller for closed loop position control [7].
Table II presents the time and frequency performances of
these two controllers.
Type of control
๐‘ƒ๐‘€ (°)
๐บ๐‘€ (๐‘‘๐ต)
๐‘‚๐‘† (%)
๐‘ก๐‘  (s)
TABLE II
Control of a satellite antenna [7]
Velocity feedback
Position feedback
PID
B
PID
B
46.7
70.3
147
63
∞
14.9
∞
12
30
0.06
15.2
0.12
4
1.5
3.5
0.45
V. THE UNSTABLE AND INVERTIBLE CASE
The unstable process can be decomposed into the product of
its minimum-phase part ๐‘ƒ1 (๐‘ ) and its unstable part ๐‘ƒ2 (๐‘ ), so
that the process ๐‘ƒ(๐‘ ) is represented by:
๐‘ƒ(๐‘ ) = ๐‘ƒ1 (๐‘ )๐‘ƒ2 (๐‘ )
(32)
A transfer function ๐ป(๐‘ ) can be defined such that, for some
value ๐‘ 0 :
๐‘
๐‘
๐ป(๐‘ ) = [(๐‘ +๐‘  )๐‘ž] ๐‘ƒ2−1 (๐‘ )๐‘ƒ(๐‘ ) = (๐‘ +๐‘  )๐‘ž ๐‘ƒ1 (๐‘ )
(33)
0
0
So that ๐ป(๐‘ ) has the same behavior as ๐‘ƒ(๐‘ ) at high
frequency.
โ€–๐‘ƒ(๐‘ )๐ป(๐‘ )−1 − 1โ€– < ๐›ผ < 1
(34)
Where ๐›ผ is a value less than unity. Note that ๐ป(๐‘ ) is
holomorphic by its design and that its inverse is
holomorphically invertible in ๐‘…๐‘’(๐‘ ) ≥ 0. The controller ๐ถ(๐‘ )
is designed as follows:
1
๐ถ(๐‘ ) = ๐ป −1 (๐‘ )๐ฝ(๐‘ ) = ( ) (๐‘  + ๐‘ 0 )๐‘ž ๐‘ƒ1−1 (๐‘ )๐ฝ(๐‘ )
(35)
๐‘
−1 (๐‘ )๐ฝ(๐‘ ).
So that the loop gain is ๐‘ƒ(๐‘ )๐ถ(๐‘ ) = ๐‘ƒ(๐‘ )๐ป
For example, we could choose:
(๐‘ +๐‘ง๐‘–)
๐‘˜ ๐œ”
1 1
∏๐‘›
๐ฝ(๐‘ ) = ๐ฝ1 (๐‘ )๐ฝ2 (๐‘ )๐ฝ3 (๐‘ ) = (๐‘ +๐œ”
[
) ๐‘–=1 (๐‘ +๐‘๐‘–) (
1
๐œ”2
๐‘ +๐œ”2
๐‘˜
]
)
(36)
Fig. 12: Time response of B controller for different values of ๐‘› [8].
Applying the proposed compensator with ๐œ”2 = 3000
๐‘Ÿ๐‘Ž๐‘‘/๐‘ , ๐‘˜ = 3, ๐‘ 0 = 3, ๐œ”1 = 0.019, ๐‘ง1 = 70, ๐‘1 = 100 and
๐พ๐‘ ๐‘ = −6100:
s ๏ƒถ๏ƒฆ
s
s ๏ƒถ
๏ƒฆ s ๏ƒถ๏ƒฆ
๏ƒถ๏ƒฆ
−6100 ๏ƒง1 + ๏ƒท๏ƒง1 + ๏ƒท๏ƒง1 +
๏ƒท๏ƒง 1 +
๏ƒท
3 ๏ƒธ๏ƒจ 70 ๏ƒธ๏ƒจ 31.3407 ๏ƒธ๏ƒจ 184.378 ๏ƒธ
๏ƒจ
C (s) =
3
s ๏ƒถ๏ƒฆ
s ๏ƒถ๏ƒฆ
s ๏ƒถ
๏ƒฆ
๏ƒง1 +
๏ƒท๏ƒง1 +
๏ƒท๏ƒง 1 +
๏ƒท
๏ƒจ 0.019 ๏ƒธ๏ƒจ 100 ๏ƒธ๏ƒจ 3000 ๏ƒธ
(42)
The simulation results using (42) lead to the following
optimal values [8, 17]: Static gain of the compensator ๐พ๐‘ ๐‘ =
6100, ๐‘ก๐‘Ÿ = 60๐‘š๐‘ , ๐‘ก๐‘  = 60๐‘š๐‘ , overshoot: ๐ท % = 1.25%,
๐บ๐‘€ = 4.92 ๐‘‘๐ต, and ๐‘ƒ๐‘€ = 68.6° .
For a sampling time of 10 ๐‘š๐‘ , the digital compensator ๐ถ(๐‘ง)
is:
๐ถ(๐‘ง) =
๐ถ(๐‘ง) =
Fig. 11: Magnetic levitation system: Experimental setup.
We apply the B control design to a levitation system which
is open loop unstable (Fig. 11):
−7817.5
๐‘ƒ(๐‘ ) = ๐‘ƒ1 (๐‘ )๐‘ƒ2 (๐‘ ) = (๐‘ −30.51)(๐‘ +31.34)(๐‘ +184.38)
๐‘›4 ๐‘ง 4 +๐‘›3 ๐‘ง 3 +๐‘›2 ๐‘ง 2 +๐‘›1 ๐‘ง+๐‘›0
๐‘ง 5 +๐‘‘4 ๐‘ง 4 +๐‘‘3 ๐‘ง 3 +๐‘‘2 ๐‘ง 2 +๐‘‘1 ๐‘ง+๐‘‘0
−37.9๐‘ง 4 +46.54๐‘ง 3 −9.372๐‘ง 2 −1.159×10−6 ๐‘ง−1.25×10−19
๐‘ง 5 −1.368๐‘ง 4 +0.368๐‘ง 3 −1.033×10−13 ๐‘ง 2 +9.611×10−27 ๐‘ง−3.09×10−40
with:
−7817.5
(38)
1
(๐‘ −30.51)
(39)
๐‘ƒ2 (๐‘ ) =
1
๐‘ƒ(๐‘ )๐ถ(๐‘ ) = ๐‘ƒ(๐‘ )๐ป −1 (๐‘ )๐ฝ(๐‘ ) = (๐‘  + ๐‘ 0 )๐‘ž ๐ฝ(๐‘ )๐‘ƒ2 (๐‘ )
๐‘
(40)
which leads to
− (5000 + 1000
๐ถ(๐‘ ) =
(๐‘› − 1) 2
๐‘ 
๐‘ 
๐‘ 
๐‘ 
) (1 +
)
) (1 + ) (1 + ) (1 +
2
70
3
31.38
184.38
๐‘ 
๐‘ 
๐‘ 
(1 +
) (1 +
) (1 +
)
100
0.019
3000
(41)
The index ๐‘› is modified to increase the value of the gain.
Fig. 12 shows the time response using B controller.
(44)
The performances of time response using the B controller
are compared to the PID controller. These controllers have been
applied to the same experimental setup. Fig. 13 and Fig. 14
illustrate this comparison.
(37)
๐‘ƒ1 (๐‘ ) = (๐‘ +31.34)(๐‘ +184.38)
(43)
Fig. 13. PID control of a levitation system [17].
error signal ๐‘ฅฬƒ = ๐‘ง − ๐‘ฅฬ‚. The input of the feedback system ๐‘Ÿ is
multiplied by a constant gain ๐‘˜๐‘Ÿ .
Fig. 16: Architecture L1 [18].
Application of Masson formula (Fig.16) leads to the
following relations:
Fig. 14. B control of a levitation system [17].
Fig. 15 shows the effects of the various Nyquist diagrams
plotted with GeoGebra software for:
๐ฝ2 (๐‘ ) =
๐‘ 
๐‘ 
๐‘ง1
๐‘ง2
๐‘ 
๐‘ 
(1+ )(1+ )
๐‘1
๐‘2
(1+ )(1+ )
(45)
๏ƒฉ x ๏ƒน ๏ƒฉ H zr ( s )
๏ƒช z ๏ƒบ ๏ƒช H (s)
zr
๏ƒช ๏ƒบ=๏ƒช
๏ƒชu ๏ƒบ ๏ƒช P −1 ( s ) H zr ( s )
๏ƒช ๏ƒบ ๏ƒช −1
๏ƒซ v ๏ƒป ๏ƒซ P ( s ) H zr ( s )
H zn ( s ) − 1
H z๏ณ ( s )
H zn ( s )
๏ƒน
๏ƒบ๏ƒฉr ๏ƒน
๏ƒบ ๏ƒช๏ณ ๏ƒบ
−1
−1
P ( s ) H z๏ณ ( s ) − 1 P ( s ) ( H zn ( s ) − 1) ๏ƒบ ๏ƒช ๏ƒบ
๏ƒบ ๏ƒชn๏ƒบ
P −1 ( s ) H z๏ณ ( s )
P −1 ( s) ( H zn ( s) − 1) ๏ƒป ๏ƒซ ๏ƒป
(46)
with:
Hxr (s) =
๐ป๐‘ฅ๐œŽ (๐‘ ) =
with: ๐‘ง1 = 3.7, ๐‘1 = 3.9, ๐‘ง2 = 8.5, ๐‘2 = 10.8
H z๏ณ ( s )
๐ป๐‘ฅ๐‘› (๐‘ ) =
kr (1+Pm (s)Γ(s))P(s)
(47)
1+Pm (s)Γ(s)+C(s)Γ(s)(P(s)−Pm (s))
(1+๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ )−๐ถ(๐‘ )๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ ))๐‘ƒ(๐‘ )
(48)
1+๐‘ƒ๐‘š (๐‘ )๐›ค(๐‘ )+๐ถ(๐‘ )๐›ค(๐‘ )(๐‘ƒ(๐‘ )−๐‘ƒ๐‘š (๐‘ ))
−๐ถ(๐‘ )๐›ค(๐‘ )๐‘ƒ(๐‘ )
(49)
1+๐‘ƒ๐‘š (๐‘ )๐›ค(๐‘ )+๐ถ(๐‘ )๐›ค(๐‘ )(๐‘ƒ(๐‘ )−๐‘ƒ๐‘š (๐‘ ))
For the particular case:
๐‘ƒ(๐‘ ) = ๐‘ƒ๐‘š (๐‘ ) =
๐‘
๐‘ +๐‘Ž๐‘š
, Γ(๐‘ ) =
๐›พ
๐‘ 
and ๐‘ƒ1 (๐‘ ) = ๐‘  2 + ๐‘Ž๐‘š ๐‘  + ๐‘๐›พ,
we get:
๏ƒฉ kr b
๏ƒช
๏ƒช s + am
๏ƒฉ x ๏ƒน ๏ƒช kr b
๏ƒชz๏ƒบ ๏ƒช
๏ƒช ๏ƒบ = ๏ƒช s + am
๏ƒชu ๏ƒบ ๏ƒช
๏ƒช ๏ƒบ ๏ƒช kr
๏ƒซv ๏ƒป ๏ƒช
๏ƒช
๏ƒช k
๏ƒช๏ƒซ r
๏ƒน
๏ƒบ
๏ƒบ
๏ƒบ
๏ƒฆ b๏ง C ( s ) ๏ƒถ
b๏ง C ( s ) ๏ƒบ r
๏ƒฉ ๏ƒน
1−
๏ƒง1 −
๏ƒท Pm (s)
P1 ( s ) ๏ƒธ
P1 ( s )
๏ƒบ๏ƒช ๏ƒบ
๏ƒจ
๏ƒบ ๏ƒช๏ณ ๏ƒบ
๏ง C ( s )( s + am ) ๏ƒบ ๏ƒช n ๏ƒบ
−b๏ง C ( s )
−
๏ƒซ ๏ƒป
๏ƒบ
P1 ( s )
P1 ( s )
๏ƒบ
๏ง C ( s )( s + am ) ๏ƒบ
b๏ง C ( s )
1−
−
๏ƒบ๏ƒป
P1 ( s )
P1 ( s )
๏ƒฆ b๏ง C ( s ) ๏ƒถ
๏ƒง1 −
๏ƒท Pm (s)
P1 ( s ) ๏ƒธ
๏ƒจ
−
b๏ง C ( s )
P1 ( s )
(50)
It is wise to study the stability of the loop ๐ฟ๐‘ข1๐‘ข2 (Fig. 16).
๐ฟ๐‘ข1๐‘ข2 =
๐‘ข1
๐‘ข2
=−
๐ถ(๐‘ )(1+๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ ))−1 ๐›ค(๐‘ )๐‘ƒ(๐‘ )
1−๐ถ(๐‘ )(1+๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ ))−1 ๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ )
(51)
which leads to:
Fig. 15: Nyquist diagram of ๐ฝ(๐‘ ) [4].
In violet, ๐ป −1 (๐‘ )๐ฝ(๐‘ ); in blue: the normalized gain curves
of the compensator ๐ถ(๐‘ )/๐ถ๐‘š๐‘Ž๐‘ฅ and in brown: the modulus of
the transmittance ๐‘‡(๐‘ ).
Similarly, B control has been successfully applied to the
control of a drone.
๐ฟ๐‘ข1๐‘ข2 = −
๐›พ๐‘๐ถ(๐‘ )
๐‘ (๐‘ +๐‘Ž๐‘š )+๐‘๐›พ(1−๐ถ(๐‘ ))
=−
๐›พ๐‘๐ถ(๐‘ )
๐‘ƒ1 (๐‘ )−๐›พ๐‘๐ถ(๐‘ )
(52)
One example of integration of B control and L1 adaptive
control is the BL1 architecture (Fig. 17).
VI. THE L1 ADAPTIVE CONTROL CASE
The insertion of the low pass filter ๐ถ(๐‘ ) decouples the high
frequency adaptation feedback from the feedback system loop.
๐ถ(๐‘ ) is usually a simple pole low pass filter. In Fig. 15, ๐‘ง is the
output of the plant that takes into consideration the effects of
input and output plant perturbation ๐‘› and ๐œŽ. The plant input ๐‘ฃ
takes in consideration the plant input perturbation ๐œŽ. The
๐›พ
adaptation feedback gain (the MIT gain) ๐œŽฬ‚ = amplifies the
๐‘ 
Fig. 17: BL1 architecture [11].
๐ถ๐ต (๐‘ ) is a serial B controller applied to the L1 adaptive
system of Fig. 17. This scheme leads to the following relations:
๐ป๐‘ง๐‘ (๐‘ ) =
๐ป๐‘ง๐œŽ (๐‘ ) =
๐ป๐‘ง๐‘› (๐‘ ) =
๐‘ƒ(๐‘ )๐ถ๐ต (๐‘ )
1+๐‘ƒ(๐‘ )๐ถ๐ต (๐‘ )+๐ถ(๐‘ )๐›ค(๐‘ )(1+๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ ))−1 (๐‘ƒ(๐‘ )−๐‘ƒ๐‘š (๐‘ ))
1−๐ถ(๐‘ )(1+๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ ))−1 ๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ )
1+๐‘ƒ(๐‘ )๐ถ๐ต (๐‘ )+๐ถ(๐‘ )๐›ค(๐‘ )(1+๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ ))−1 (๐‘ƒ(๐‘ )−๐‘ƒ๐‘š (๐‘ ))
1−๐ถ(๐‘ )(1+๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ ))−1 ๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ )
1+๐‘ƒ(๐‘ )๐ถ๐ต (๐‘ )+๐ถ(๐‘ )๐›ค(๐‘ )(1+๐›ค(๐‘ )๐‘ƒ๐‘š (๐‘ ))−1 (๐‘ƒ(๐‘ )−๐‘ƒ๐‘š (๐‘ ))
(53)
๐‘ƒ(๐‘ )
(54)
(55)
The gang of six of the system described in Fig. 17) is:
๏ƒฉ x ๏ƒน ๏ƒฉ H zc ( s )
๏ƒช z ๏ƒบ ๏ƒช H ( s)
zc
๏ƒช ๏ƒบ=๏ƒช
๏ƒชu ๏ƒบ ๏ƒช P −1 ( s ) H zc ( s )
๏ƒช ๏ƒบ ๏ƒช −1
๏ƒซ v ๏ƒป ๏ƒซ P ( s ) H zc ( s )
๏ƒน
๏ƒบ๏ƒฉc๏ƒน
H z๏ณ ( s )
H zn ( s)
๏ƒบ ๏ƒช๏ณ ๏ƒบ
−1
−1
P ( s ) H z๏ณ ( s ) − 1 P ( s ) ( H zn ( s ) − 1) ๏ƒบ ๏ƒช ๏ƒบ
๏ƒบ ๏ƒชn๏ƒบ
P −1 ( s ) H z๏ณ ( s )
P −1 ( s) ( H zn ( s) − 1) ๏ƒป ๏ƒซ ๏ƒป
[6]
(56)
For the particular case:
๐‘
๐›พ
๐‘ƒ(๐‘ ) = ๐‘ƒ๐‘š (๐‘ ) =
, Γ(๐‘ ) = and ๐‘ƒ1 (๐‘ ) = ๐‘  2 + ๐‘Ž๐‘š ๐‘  + ๐‘๐›พ,
[7]
๐‘ 
the new open loop gain is given by:
๐ฝ(๐‘ ) = ๐ป๐‘ฅ๐‘Ÿ (๐‘ )๐ถ๐ต (๐‘ ) =
[5]
H zn ( s ) − 1
H z๏ณ ( s )
๐‘ +๐‘Ž๐‘š
[4] Bensoussan David, Benkhellat Lyes, Geogebra as a tool of design of
๐‘˜๐‘Ÿ (1 + ๐‘ƒ๐‘š (๐‘ )๐›ค(๐‘ ))๐‘ƒ(๐‘ )
1 + ๐‘ƒ๐‘š (๐‘ )๐›ค(๐‘ ) + ๐ถ(๐‘ )๐›ค(๐‘ )(๐‘ƒ(๐‘ ) − ๐‘ƒ๐‘š (๐‘ ))
= ๐‘˜๐‘Ÿ ๐‘ƒ(๐‘ )๐ถ๐ต (๐‘ )
(57)
Simulation of the plant output versus the feedback system
input c , the plant perturbation input ๏ณ and the plant output n
respectively applied at times 0, 1 and 10 seconds. Fig. 18 shows
the time response improvement of BL1 architecture over the L1
architecture.
[8]
[9]
[10]
๐œธ = ๐Ÿ๐ŸŽ๐ŸŽ๐ŸŽ
L1 Response
BL1 Response
[11]
[12]
[13]
[14]
Fig. 18: The effect of plant perturbations - L1 vs BL1 [11].
[15]
VII. CONCLUSION
B control exhibits tangible advantages for the minimum
phase case such as the control of a hard disk or a satellite
antenna, for the unstable and invertible SISO case exemplified
by the levitation experiment, as well as the MIMO unstable (but
invertible) case [12]. It can also improve the L1 adaptive
control. Time response improvements were obtained while
maintaining appreciable gain and phase margins.
Research on the B control method can be extended to time
delay systems, internal model control, state space formulations
and nonlinear systems.
VIII. REFERENCES
[1] Bensoussan, David. System and method for feedback control, P, US Patent
8,831,755, Deposited September 9, 2014.
[2] Kelemen, Mattei, 2002, Arbitrarily fast and robust tracking by feedback,
International Journal of Control, 75, 443-465
Bensoussan David, “On the Design, Robustness,
Implementation and Use of Quasi-Linear Feedback Compensator”,
International Journal of Control, 15 avril 2004, Vol 77, No 6, pp 527-545
- 77, no. 6 (2004): 527-545
[3] Kelemen,
[16]
[17]
[18]
ultrafast and robust controller, 2017 IEEE International Conference on
Industrial Technology, Toronto, March 22-25, 2017.
Bensoussan David, Robust and Ultrafast Response Compensator for
Unstable Invertible Plants, Automatica, Volume 60, Octobre 2015 pp. 4347, 2015.
Bensoussan David, Sun Yulan, Hammami Maher, “Robust and ultrafast
design of a control system based on optimal sensitivity and optimal
complementary sensitivity”, The 14th international conference on
Sciences and Techniques of Automatic control & computer engineering,
December 20โ€22, 2013, Sousse, Tunisia.
Bensoussan David, Houimdi Amine, Hammami Maher, Sun Yulan, Loo
Darren, Brossard Jeremy, Demonstration of a New Control Method for
Positionning Satellite Antennas, Internal Report INNOV B, École de
technologie supérieure, 2020.
Sun. Yulan, Bensoussan David, Hammami M, Wang Tao., and Houimdi
Amine, Robust and Ultrafast Response Compensator Applied to a
Levitation System, European Control Conference ECC15, Linz, Austria,
July 15-17, 2015.
Brossard Jeremy, Bensoussan David, Landry René, Hammami Maher, A
new fast compensator design applied to a quadcopter, 2019 8th
International Conference on Systems and Control, Octobre 2019,
Marrakech, Maroc.
Brossard Jeremy, Bensoussan David, Landry René, Hammami Maher,
Robustness Studies on Quadrotor Control, International Conference on
Unmanned Aircraft Systems, ICUAS'19, Atlanta, GA, USA, on June 1114 2019.
Ghodbane Azeddine, Madali Adem Soufiane, Bensoussan David, and
Hammami Maher An improved L1 Adaptive controller, 2022 10th
International Conference on Systems and Control (ICSC), 23-25
novembre 2022.
Bensoussan David, Brossard Jérémy, Decentalized and ultrafast control,
21st International Conference on System Theory, Control and Computing,
Sinaia, Romania, October 19 – 21, 2017.
Bensoussan, D. Sensitivity reduction in single-input single-output
systems, International Journal of Control, Vol. 39, p. 321-335, 1984.
https://doi.org/10.1080/00207178408933168
Chen Ben M., Lee T. H., Peng K., Venkataramanan V., Hard Disk Drive
Servo Systems, Springer, 2006.
M. Hammami, D. Bensoussan and Y. Sun, "Fine tuning the time response
of a robust and ultrafast compensator," 22nd Mediterranean Conference
on Control and Automation, Palermo, Italy, 2014, pp. 1400-1405, doi:
10.1109/MED.2014.6961572.
Wodek Gawronski. “Modeling and Control of Antenna and Telescopes”,
Springer Science, 2008.
M.-A. Houimdi, « Ajustement d’un nouvel algorithme d’optimisation
simultanée des réponses temporelles et fréquentielles », École de
technologie supérieure, MSc. Thesis, 2015.
Kharisov, Evgeny et al. “Comparison of architectures and robustness of
model
reference
adaptive
controllers
and
L1
adaptive
controllers.” International Journal of Adaptive Control and Signal
Processing 28 (2014): 633 - 663.
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