V L A D I M Í R K U Č E R A , T O M Á Š V Y H L Í D A L , M A R T I N H R O M Č Í K

D E P A R T M E N T O F I N S T R U M E N T A T I O N A N D C O N T R O L E N G I N E E R I N G , F M E

A N D

D E P A R T M E N T O F C O N T R O L E N G I N E E R I N G , F E E

Summary of recent results on design and application of signal shapers

Contain

Basics of signal shapers

Classical approach and design

Spectral feature of shaper with distributed delay

Distributed delay based shaper

Design and spectral feature

Robustness

Application examples

Case study of blended wing body aircraft

Portal crane

Flexible link

ZV shaper

𝐺 𝑠 =𝜔2

𝑠2+2𝜉𝜔𝑠+𝜔2

𝑉 𝜉,𝜔 = 𝑒−𝜉𝜔𝑡𝑖 𝐶 𝜉,𝜔 2 + 𝑆 𝜉,𝜔 2

𝐶 𝜉,𝜔 = 𝐴𝑖𝑒𝜉𝜔𝑡𝑖cos(𝜔 1 − 𝜉2𝑡𝑖)

𝑛

𝑖=0

S 𝜉, 𝜔 = 𝐴𝑖𝑒𝜉𝜔𝑡𝑖sin(𝜔 1 − 𝜉2𝑡𝑖)

𝑛𝑖=0

𝐴𝑖𝑡𝑖=

1

1+𝐾

𝐾

1+𝐾

0𝑇𝑑

2

, 𝐴𝑖 > 0, 𝐴𝑖 = 1𝑛𝑖=1

𝐾 = 𝑒−𝜉𝜋

1−𝜉

*

t(s) t(s) t(s)

r(t) sr(t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t [s]

Am

plitu

da [-

]

Přechodová charakteristika Posicastu

Systém druhého řádu

Reference

Posicast a systém druhého řádu

ZV ZV

Convolution with reference command Convolution with reference command

𝑣 𝑡 = 𝐴𝑖𝑤(𝑡 − 𝑡𝑖)

𝑁

𝑖=1

Multi modal shapers

𝑍𝑉 𝑠 = A + 1 − 𝐴 e−st

𝑍𝑉𝐷 𝑠 = 𝑍𝑉 𝑠 ∗ 𝑍𝑉(𝑠)

𝐸𝐼 𝑠 = 𝑍𝑉1 𝑠 ∗ 𝑍𝑉2(𝑠)

Dominant Zeros of time delay system

Dominant Zeros of time delay system

𝑠𝑘 = −1

𝜏ln𝐴

1 − 𝐴± 𝑗𝜋

𝜏2𝑘 + 1 , 𝑘 = 0

Zeros of shapers Zeros of shapers

Robustness of Multimodal shapers

SensitivityfunctionFromequationV ξ, ω SensitivityfunctionFromequationV ξ, ω

Spectral feature

𝑆𝑍𝑉 = 𝐴 + 1 − 𝐴 𝑒−𝑠𝜏

𝑠𝑘 = −1

𝜏ln𝐴

1 − 𝐴± 𝑗𝜋

𝜏2𝑘 + 1 , 𝑘 = 0,1, … ,∞

Spectral feature of ZVD with non−commensurable delays Spectral feature of ZVD with non−commensurable delays

DZV shaper

𝐵𝜗𝑠 + 1 − 𝐵 1 − 𝑒−𝑠𝜗 = 0

𝑆𝐷𝑍𝑉 𝑠 = 𝐵 + 1 − 𝐵1−𝑒−𝑠𝜗

𝑠𝜗=𝐵𝜗+(1−𝐵)(1−𝑒−𝑠𝜗)

𝑠𝜗

𝑤ℎ𝑒𝑟𝑒𝐵 ∈ 𝑅+, B<1

Characteristic Equation

Zeros 𝑠 −𝑘,𝑘 of Characteristic Eq. can be evaluate numerricaly or via the Lambert

W function

−𝛽 ± 𝑗Ω =1

𝜗(𝑊 1,

1 − 𝐵

𝐵𝑒1−𝐵𝐵 −

1 − 𝐵

𝐵)

𝝑 from 1st. nonzero root of 𝑩 from

Ω𝑒𝛽𝜗 − 𝛽𝑠𝑖𝑛Ω𝜗- Ωcos Ω𝜗 = 0 𝐵 =1−𝑒−𝛽𝜗cosΩ𝜗

𝛽−1+𝑒−𝛽𝜗cosΩ𝜗

Spectral feature DZV

𝑠 −𝑘,𝑘 = −1

𝜗𝑙𝑛𝐵3

2+2 𝑘−1 𝜋

1−𝐵± 𝑗

3

2+ 2 𝑘 − 1

𝜋

𝜗, 𝑘 = 1,2, . . , ∞

ZerosofDZVshaper ZerosofDZVshaper

Spectral feature DZV

Spectral feature of DZVD with non−commensurable delays Spectral feature of DZVD with non−commensurable delays

Robustness

Sensitivity function for ZV and DZV shaper Sensitivity function for ZV and DZV shaper

BWB aircraft (M.Hromčík,T.Haniš,V.Kučera)

ACFA 2020 EU project FP7

FUEL consuption

Noice

ACFA 2020

Design plant

G=Act*Aircraft*Sen

Total of 36 states

Feedback: robust H_inf

fixed, low order (3) controller

CMD shaped by ZV shaper

ZV as CMD filter ZV as CMD filter

ACFA 2020

First and second bending mode compensation OL

Mag

nitud

e (

dB

)

Bode Diagram

Frequency (rad/sec)

Step Response

Time (sec)

Am

plitu

de

Nz law lateral Nz law lateral Modal sensors of NZ law Modal sensors of NZ law

ACFA 2020

Step Response

Time (sec)

Am

plit

ude

Roll reference command Roll reference command

Portal crane

Experimental verification of DZV shaper

Portal crane results

ZV shaper ZV shaper DZV shaper DZV shaper

Flexible link

The Quanser Inc. laboratory experiment

Position - angle of the table - is measured and

Actuated by feedback servo (designed again using

the mixedsensitivity H∞ approach

Shaped references Shaped references

Flexible link results

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

-0.8

-0.4

-0.2

0

0.2

0.4

0.8

Time(s)

Am

plitu

de

(°)

Vibration of the flexible structure

2 4 6 8 10 12-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time(s)

Am

plitu

de

(°)

Vibration of the flexible structure

2.5 3 3.5

-0.1

0

0.1

Nominal case Nominal case non−Nominal case non−Nominal case

Flexible link results

Coclusions

Signal shapers as an efficient tool for vibration compensation.

The new DZV shaper is presented as more robust alternatives than ZV shaper

Signal shapers for filtering reference command of BWB aircraft

Two laboratory experiments verification of DZV shaper are presented

FUTURE WORK- Signal shapers in feedback loop.

- Delay resonator, KONTAKT II LH12066

Thanks for attention

References

[1] Vyhlidal T., and Kucera V., (2012) Input shapers with uniformly distributed delays.accepted to TDS IFAC, Boston, 2012

[2] Vyhlidal T., and Kucera V.Signal shapers with distributed de-lays, spectral analysis and design, submited to Automatica 2012,

[3] Kucera V.,Hromcik M., (2011)Delay-Based input commandshapers: frequency properties and finite-dimensional alternatives, IFAC, Milan, 2011

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Acknowledgement: The presented research has been supported by the Ministry of Education of the Czech Republic under program KONTAKT II LH12066, by the Grant Agency of Czech Technical University in Prague. Grant No. SGS11/150/OHK2/3T/12 and by EC project ACFA 2020-Active Control for Flexible 2020 Aircraft under No. 213321