adaptronics

Ad t iAdaptronics

P f D I T N ićProf. Dr.-Ing. Tamara Nestorović

Ruhr-University Bochum, Mechanics of Adaptive Systems

E-mail: [email protected]; www.rub.de/mas

Fundamentals of ActiveStructural Control

Prof. Dr.-Ing. Tamara Nestorovićwww.rub.de/mas

Adaptronics

Recommended literature and links• Lecture Notes – Course slides

• Matlab online Helpdesk and Tutorialshttp://www.mathworks.com/academia/student center/tutorials/http://www.mathworks.com/academia/student_center/tutorials/http://www.mathworks.com/academia/student_center/tutorials/mltutorial_launchpad.html#http://www.mathworks.com/academia/student_center/tutorials/sltutorial_launchpad.htmlhttp://www.mathworks.com/academia/student_center/tutorials/controls-tutorial-launchpad.htmlp _ phttp://www.mathworks.com/access/helpdesk/help/helpdesk.html

• Fuller C. R., Elliott S. J., Nelson P. A.: Active Control of Vibration, Academic Press Ltd, London, 1996

• Franklin G. F., Powell J. D., Emami-Naeini A.: Feedback Control of Dynamic Systems, second edition,Addison-Wesley Publishing Company, 1991

• Franklin G. F., Powell J. D., Workman M. L.: Digital Control of Dynamic Systems, third edition, Addison-Wesley Longman, Inc., 1998

• Richard J. Vaccaro: Digital Control, A State-Space Approach, McGraw-Hill Inc., 1995

• Preumont A.: Vibration Control of Active Structures: An Introduction, Kluwer Academic Publishers,Dordrecht, Boston, London, 1997



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Active Structural Control – Motivation

Motivation for development of smart structures andactive structural control:

Growing interest and application possibilities in a broad field of engineering such as automotive industry civil engineeringengineering, such as automotive industry, civil engineering,aero-space industry, robotics, mechatronics, etc.

Overall design of smart systems(modeling, control, simulation, experimental testing, implementation)

Possibility for investigation in early development phaseswhen real system or a prototype is not available

Cost reduction (material, space, energy savings) through optimization



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Active Structural Control – Tasks

Smart structural control challenges and tasks:

- to achieve desired behavior and stabile smart structures in a noisyyenvironment or in the presence of excitations

- a need to design a robust and stable control systemg yusing minimal prior knowledge about the controlled plant

- solution of different optimization problems,solution of different optimization problems,e.g. vibration suppression and noise control:

- active vibration control AVC,active vibration control AVC,- active noise control ANC,- active structural acoustic control ASAC

- design and application of control laws for the vibration and noisesuppression of piezoelectric smart structures



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Smart Structures and Adaptronics - Terms

Terms

Several terms are used to denote structures and systems with ability to adapt themselves to changing environmental conditions.

Most often used terms

- English terminology:English terminology:smart structures, intelligent structures, adaptive structures,active structures;

- German terminology: adaptronics, adaptronic systems (Adaptronik)



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Smart Structures and Adaptronics - Definitions

Definitions

Definition suggested by the Fraunhofer SocietySource: http://www.fraunhofer.de/fhg/research/index/perspektiven_4.jsp

Under the term smart structures or adaptronics one can understand structures which enable a direct actively controlled adaptation of mechanical properties in accordance to changing environmentalmechanical properties in accordance to changing environmental operating conditions through integrated functionalities within the structure itself.



Adaptronics

Smart Structures and Adaptronics - DefinitionsDefinition of DLR – Deutsches Zentrum für Luft- und Raumfahrt (Leitprojekt Adaptronik),Source: http://www.mechatronik-portal.de/mechatronik_definition.php

Adaptronics stands for a field of technology which provides a new class of so-called intelligent structures. This concept imposes the development of adaptive systems, which are able to adapt to p p y pdifferent operating conditions through autonomous self-regulating mechanisms. The premise for that is an optimal integration of sensors and actuators based on new active functional materials (like e.g.and actuators based on new active functional materials (like e.g. piezoelectric fibers or films) as well as of adaptive controllers. These active materials play at the same time the carrying or supporting role of mechanical structure as well as the the actuating/sensing role andmechanical structure, as well as the the actuating/sensing role, and therefore they are multi-functional.



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Structure of smart systems

Comparison of different structural systems

inoiseless vibrationinstable

ExcitationsGoal:

Passivestructure

Excitations

- vibration elimination- stable- lightweightless noise

Adaptive controller

adaptivelightweight

- external controller

Passivestructure

Excitations

less noiseless vibrationstable, heavy

Excitations- vibration elimination- stable

structure

SensorsActuators active

adaptive

- stable- lightweight- integrated controller

Controller



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Examples of Adaptive Structures and SystemsICE bogie



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Hierarchy of Smart Structures and Systems

+

+

+



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Development of active structures – goals and tasks

GoalsIncreasing the efficiency, safety and comfort by:comfort by: active vibration suppression active noise attenuation active shape control active shape control active position control health monitoring

i d ti noise reduction

Development tasksDevelopment tasks structure conform integrated

actuator / sensor systems

discrete real time controllers discrete real-time controllers

up-to-date design andoptimization methods



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What are we studying in the field of active vibration and noise control?









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GoalsIncreasing the efficiency, safety and comfort by:comfort by: active vibration suppression active noise attenuation active shape control

Multi-functional materials:

• Piezo ceramics

active shape control active position control health monitoring

i d ti • Piezo ceramics

• Shape memory alloys (SMA)

• Electrorheological fluids (ERF)

noise reduction

Development tasks • Electrorheological fluids (ERF)Development tasks structure conform integrated






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GoalsIncreasing the efficiency, safety and comfort by:comfort by: active vibration suppression active noise attenuation active shape control

Controller design:

• Optimal LQ control

active shape control active position control health monitoring

i d ti Optimal LQ control(SISO and MIMO control systems)

• LQ controller with additionaldynamics

noise reduction

Development tasks dynamics

• Adaptive control (MRAC)

Development tasks structure conform integrated






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Development tasks

• Modeling of adaptive structures:− FE approach

E perimental identificationDevelopment tasks structure conform integrated


discrete real time controllers

− Experimental identification

• Optimal placement of actuators and sensors

discrete real-time controllers




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Examples of active structural control – automotive

actuators

90110

dB]

305070

Am

plitu

de [d

sensorsactuatorsdisturbances

-1010

0 25 50 75 100

F [H ]

A controlled uncontrolled



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Frequency [Hz]

Examples of active structural control – medical solutionsd isturbance

actuators

sen sors

100

120controlled uncontrolled

60

80

100

mpl

itude

[dB]

Piezoelectric films cut down noise generated by the MRI scanner

20

40

0 20 40 60 80

Am



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Frequency [Hz]

Example of an actively controlled piezoelectric beam structure

Basic elements of an active structural system

Example:Vib ti i i t l t h iVibration suppression using control techniques



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Classification of control lawsModel-based control: - FE-Model

- Model identificationAdaptive controllers: - Model Reference Adaptive Control (MRAC)

- Self-tuning regulators (STR)Classification of model-based controller design techniques

M d l b d C t llModel-based Controller

Linear Time Invariant Controller (LTI) Nonlinear Controller PID ControlLinear Time-Invariant Controller (LTI)

Output FeedbackState Feedback

Nonlinear Controller PID Control

Output FeedbackState Feedback

DiscreteContinuousDigital TrackingDigital Tracking

Systems

LQR-ControllerPole Placement

LQR-ControllerPole Placement

Disturbance RejectionReference Model Design

Disturbance Cancellation



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Pole Placement Pole Placement Disturbance Cancellation

Design of piezoelectric smart structures for active vibration suppression and noise reduction

Piezoceramic materials as multifunctional materialsIn 1880, Jacques and Pierre Curie discovered an unusual characteristic of certain crystalline minerals: when subjected to a mechanical force, the crystals became electrically polarized.

Piezoelectricity is based on the ability of certain crystals to generate an electrical charge when mechanically loaded with pressure or tension (direct piezo effect).

Conversely, these crystals undergo a controlled deformation when exposed to an electric field – a behavior referred to as the inverse piezo effect.

Si th i l t i ff t hibit d b t l t lli t i l h tSince the piezoelectric effect exhibited by natural mono-crystalline materials such as quartz, tourmaline, Rochelle salt, etc. is very small, polycrystalline piezoelectric ceramic materials, such aslead (plumbum) zirconate titanate (PZT), with improved properties have been developed.

PZT ceramics are a ailable in man ariations and arePZT ceramics are available in many variations and areby far the most widely used materials for piezoelectricactuator applications today.Lead zirconate titanate (PZT) is a ceramic material

d f l d (Pb) (O) d i i (Ti)made of lead (Pb), oxygen (O) and titanium (Ti) or zirconium (Zr).

Its unit cell has a cubic lattice, the so called Perovskite structure Pb (Zr/Ti)O (see the picture)



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Perovskite structure Pb (Zr/Ti)O3 (see the picture)

The piezoelectric effect (monocrystal and polycrystal structures)

A piezoelectric substance is one that produces an electric charge when a mechanical stress is applied (thesubstance is squeezed or stretched). Conversely, a mechanical deformation (the substance shrinks orexpands) is produced when an electric field is applied. This effect is formed in crystals that have no centerp ) p pp yof symmetry. To explain this, we have to look at the individual molecules that make up the crystal. Eachmolecule has a polarization, one end is more negatively charged and the other end is positively charged,and is called a dipole. This is a result of the atoms that make up the molecule and the way the moleculesare shaped. The polar axis is an imaginary line that runs through the center of both charges on thep p g y g gmolecule. In a monocrystal the polar axes of all of the dipoles lie in one direction. The crystal is said to besymmetrical because if you were to cut the crystal at any point, the resultant polar axes of the two pieceswould lie in the same direction as the original. In a polycrystal, there are different regions within thematerial that have a different polar axis. It is asymmetrical because there is no point at which the crystalp y p ycould be cut that would leave the two remaining pieces with the same resultant polar axis. Figure 1illustrates this concept.

Monocrystal with single polar axis Polycrystal with random polar axis



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The piezoelectric effect (Curie temperature)

An important parameter of the piezoelectric materials is the co-called Curie temperature (Tc).

Fig. (1) T>Tcg ( )The unit cell of a piezoelectirc material subjected to a temperature higher than the Curie temperature will have a symmetric cubic structure. In this case the piezoelectric properties of the material are lost. Heating the piezoelectric material above the Curie temperature destroys the piezoelectric effectmaterial above the Curie temperature destroys the piezoelectric effect.

Fig. (2) T<Tc

B l h C i h i i l i i hBelow the Curie temperature the ionic lattice structure in the PZT crystallites becomes distorted and asymmetric (with an axis of polarity) and, additionally, exhibits spontaneous polarization. One result is that the discrete PZT crystallites become piezoelectric. However, the statistical distribution of the

i i t ti i th i ill th i b h i t bgrain orientations in the ceramic will cause the macroscopic behaviour to be non-piezoelectric.

An additional property, the ferroelectric nature of the PZT material, will help to l thi blsolve this problem.



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The piezoelectric effect (polarization)

When an intense electric field is applied to the ceramic, the different lattice orientations of the individual ceramic grains can be permanentlyceramic grains can be permanently altered.

In order to produce the piezoelectric effect, the (1) (2) (3)

polycrystal is usually heated under the application of a strong electric DC field (>2000 V). The heat allows the molecules to move more freely and the electric field forces all of the dipoles to become aligned in the direction of field. They will maintain this orientation even when the DC field is no longer applied (remanent polarization). Due to this property the piezoelectric ceramics belong to Remanent elongation

Elongation due to

Polarization of ceramic materialto generate piezoelectric effect

ferroelectric materials.

As a result of this “poling process” the ceramic is accorded a net orientation of its internal, sponta-neous polarization in the direction of the poling

Elongation due toelectric field

to generate piezoelectric effect

(1) before polarization(2) during polarization(3) after polarization

neous polarization in the direction of the poling field and shows an overall piezoelectric effect.

For some PZT ceramics, it is necessary to perform the poling process at elevated temperatures.



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( ) p

Direct and inverse piezoelectric effect

Direct piezoelectric effect is used in sensors.

Mechanical deformations of piezoelectric materials cause (generate) electric charge.Piezoelectric sensors are therefore also called piezo generators.

Inverse piezoelectric effect is used in actuatorsInverse piezoelectric effect is used in actuators.

An electric field applied to piezo material causes mechanical deformation (or displacement).

Piezoelectric actuators are therefore also called piezo motors.

Now we are going to consider the inverse piezo effect first.



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Inverse piezoelectric effect (actuator effect or piezo motor)

Inverse piezoelectric effect: Electric field causes mechanical deformation.

On a crystal level: Due to an external electric field the geometric dimensions

On macroscopic level: Electric field of the same polarity and orientation as original

E = 0

electric field the geometric dimensions of the crystal will be changed. polarization (or “poling”) field will expand the

ceramics in direction along the axis of polarization.

Original state without applied voltage

E

applied voltage++ ++ + ++

E

U P l0 + u0l0

+ + + + + + +

Th ld id ti i t d fi th + + + + + + +

Poling direction

The worldwide convention is to define the direction of the electric field vector as the direction that a positive test charge is pushed or pulled when in the presence of th l t i fi ld

Th l i t t !

Orientation of the electric fieldsame as original poling directionduring manufacturing of piezo material

Poling directionthe electric field.



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The volume remains constant !during manufacturing of piezo material

Inverse piezoelectric effect (actuator effect or piezo motor)



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Inverse piezoelectric effect (longitudinal and transverse motors)Source: http://www piezo com/catalog7C pdfSource: http://www.piezo.com/catalog7C.pdf

When an electric field having the same polarity and orientation as the original polarization field is placed across the thickness of a single sheet of piezoceramic, the piece expands in the thickness or “longitudinal” direction (i e along the axis of polarization) and contracts in the transverse direction (perpendicular to thedirection (i.e. along the axis of polarization) and contracts in the transverse direction (perpendicular to the axis of polarization). See Figures-1 and 2. When the field is reversed, the motions are reversed. Sheets and plates utilize this effect. However, the motion of a sheet in the thickness direction is extremely small (on the order of tens of nanometers). On the other hand, the transverse motion along the length is generally larger (on the order of microns to tens of microns) since the length dimension is oftengenerally larger (on the order of microns to tens of microns) since the length dimension is often substantially greater than the thickness. The transverse motion of a sheet laminated to the surface of a structure can induce the structure to stretch or bend, a feature often exploited in structural control systems.

Important:pd31 effect is used in piezo films



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See also: http://www.morganelectroceramics.com/resources/piezo-ceramic-tutorials/piezoelectric-voltage/

Direct piezoelectric effect (sensor effect or piezo generator)

Direct piezoelectric effect: Mechanical deformations cause electric charge

On a macroscopic level: The electric field induced due to forming of electric charge on the surface of the

On a crystal level: Under mechanical deformation of some types of aniso-

F 0 FTitanium / Zirconium (+)

piezo ceramics will act against the mechanical deformation, i.e. it will try to bring it to undeformed state. In other words, if a piezoelectric ceramics is compressed, the induced electric field would try to

tropic crystals an electric charge is being formed on the crystal surface.

Lead (+)F = 0 FTitanium / Zirconium (+)

F

extend it and vice versa.Original state without

applied compressing force

D

+ + + + + + +

Oxygen (-) P

+ + + + + + +

The volume remains constant !F



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See also: http://www.morganelectroceramics.com/resources/piezo-ceramic-tutorials/piezoelectric-voltage/

Direct piezoelectric effect (longitudinal and transverse generators)Source: http://www piezo com/catalog7C pdf

When a mechanical stress is applied to a single sheet of piezoceramic in the longitudinal direction (parallel to polarization), a voltage is generated which tries to return the piece to its original thickness (Figure 1). Similarly when a stress is applied to a sheet in a transverse direction (perpendicular to polarization) a

Source: http://www.piezo.com/catalog7C.pdf

Similarly, when a stress is applied to a sheet in a transverse direction (perpendicular to polarization), a voltage is generated which tries to return the piece to its original length and width (Figure 2).

A piezo sheet bonded to a structural member which is stretched or flexed will induce electrical generation.

Figure 2: Transverse (d31) generator, compressed on sides

Figure 1: Longitudinal (d33) generator

compressed on sides

Important: d31 effect is usedin piezo sensor-patches



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Hysteresis curve of piezoelectric materials

The remanent polarization produced by the polarization process is illustrated b h t i i th fi

+by hysteresis curve in the figure.The dielectric displacement density D is plotted over the applied electric fieldstrength E. When a field is applied, D i ith E l th “i iti l

P

-

increases with E along the “initial curve“ (starting from 0) until saturation is achieved.

If E i d d t thi i t D illIf E is reduced at this point, D will decrease insignificantly; at the remanence point it retains a finite value for E = 0 which corresponds to the

t l i ti l l P

-

+

P

remanent polarization level Pr.In other words, the material exhibits piezoelectric properties.

When an opposed field is applied, D will decrease and disappear at a given field strength the coercive field strength –Ec. As E becomes more negative, D will reach a negative saturation point. If the material goes through several positive and negative field cycles, it becomes apparent that the forward and return graphs do not coincide. The resulting forward/return loop is called the hysteresis curve.



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g p g p y

Depolarization of piezoelectric materials

A full or partial elimination of the domain alignment achieved by the polarizing process (depolarization) will degrade the piezoelectric properties of the material.

Depolarization may be the result of three factors:

• Thermal depolarization due to heat exposure. In application environments, the component temperature should not exceed one-half the Curie temperature stated in the data sheet. Storage temperatures should also not exceed this temperature by a significant margin.

• Electric depolarization due to electric fields acting against the original polarization direction Electric depolarization due to electric fields acting against the original polarization direction.

• Mechanical depolarization caused by high-pressure loads, especially with short-circuited electrodes. The maximum permissible pressure level varies largely according to the material used.

Since most applications will involve some form of superimposed loading (e.g., exposure to an electric field opposed to the polarizationdirection at elevated operating temperatures) and theelectric field opposed to the polarizationdirection at elevated operating temperatures) and the depolarization behaviour cannot be estimated in this case. It is recommended to run lifetime tests under close-to-application conditions at the project planning stage.



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Piezoelectric material behaviour

ε3, σ3F- -

U+ -

u0

F, u-

ε1, σ1F, uU

l0

ε5, σ5F, u

l0 P E3U

F, uUPE3 PE1 u0

,

+l0 u0

+

Longitudinal effect

E || P || F, u

Transversal effect

E || P ⊥ F, u

Sheer effect

E ⊥ P || F, u



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Electro-mechanical equations of piezoelectric materials

T

The equations of electro mechanical behaviour at constant temperature can be written in the index notation in the following way:

Ti i ik iD d Eλ λσ ε= +

Inverse piezoelectric effect (actuator effect)

Direct piezoelectric effect (sensor effect)

Ei is d Eλ λμ μ λε σ= +

, 1...6, 1...3i k

λ μ ==

i iλ λμ μ λ

P,

These equations are written in engineering notation,

P

q g g ,where σ stands for stress and ε for strain.

In the literature, especially in solid physics notation or in crystallography notation as ell as in the catalog es of man fact rers the follo ingnotation, as well as in the catalogues of manufacturers, the following notation can be found:

Tλ which stands for stress and Sλ which stands for strain.



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Tλ which stands for stress and Sλ which stands for strain.

Inverse piezoelectric effect

E Es dε σ +ii

E EdTsS λμλμλ += (Notation in solid body physics)

, 1...6λ μ = Pi iEs d λλλ μμε σ= + ,

1...3iμ=

Electric field [V/m]

Piezoelectric constants [m/V]

Stress [N/m2]Compliance coefficients [m2/N]Compliance coefficients [m /N]

Strain [-]

First index of compliance coefficients s First index of piezoelectric constants d denotesFirst index of compliance coefficients sdenotes the direction of extension (compression), second index denotesthe stress direction

First index of piezoelectric constants d denotes direction of electric field, the second index denotes direction of performed extension (compression).

Es11Constant electric fieldStress in direction 1 31d Extension (compression) in direction 1

Example: Example:



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11Strain in direction 1 Electric field in direction 3

Direct piezoelectric effect

(Notation in solid body physics)

, 1...6λ μ = PdD Eσ

Ti i ik iD d T Eλ λ ε= +

,1...3i

μ=

Electric field [V/m]

ii ikidD Eλσ

λ εσ= +

Dielectric constants at σ=const [F/V]

Stress [N/m2]Piezoelectric constants [m/V]Piezoelectric constants [m/V]

Electric displacement [C/m2]

First index of dielectric constants εFirst index of dielectric constants εdenotes the direction of electric displacement, the second index denotes direction of electric field

First index of piezoelectric constants d denotes direction of generated electric displacement, the second index denotes direction of applied stress

σε11Constant stressElectric field in direction 1Electric displacement in direction 1

31d Applied stress in direction 1Generated electric displacement in direction 3

Example:Example:



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Electric displacement in direction 1 Generated electric displacement in direction 3

Electromechanical material matrix

The equations of electro mechanical behaviour at constant temperature can be written in matrix notation in the following way:

11 12 13 31

112 11 13 31

0 0 0 0 00 0 0 0 0

E E E

E E E

s s s ds s s dε

1σ

12 11 13 31

213 13 33 33

3

44 154

0 0 0 0 00 0 0 0 0 0 0

E E E

E

s s s ds d

εεε

2

3

4

σσσ

44 154

5 44 15

6 11 12

0 0 0 0 0 0 00 0 0 0 0 2( ) 0 0 0

E

E E

s ds s

εεε

= −

4

5

6

σσσ

11 12

115 11

215 113

( )0 0 0 0 0 0 00 0 0 0 0 0 0

D dDdD

σ

σ

εε

1

2

3

EEE

15 113

31 31 33 330 0 0 0 0D

d d d σε 3E



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Formulation of actuation effect

Usual actuator forms

11 1 31E

x s d Eε σ= +d31- Effect

ε1, σ1F, u

U

-

11 1 31x

11 2 31E

y s d Eε σ= +Transverse effect

U

l0 u0

E3 P

+

d33-Effect l0 u0

U+ -+

33 3 33E

z s d Eε σ= + Longitudinal effect εz, σ3F, uP

E3

d15-EffectE d Eε σ +

l0

U+ -

4 44 4 15Es d Eε σ= +

5 44 5 15Es d Eε σ= +

Sheer effectPE1 u0

ε5, σ5F, u



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5 44 5 15s d Eε σ + 0

Forms and conversion of constitutive equations

In the literature one of the following 4 forms of constitutive equations can be found:

EdTsS ⋅+⋅= t

Strain-charge form

EeScT = t

Stress-charge form

DgTsS += t

Strain-voltage form

DqScT = t

Stress-voltage form

In the literature one of the following 4 forms of constitutive equations can be found:

EεTdDEdTsS⋅+⋅=⋅+⋅=

T

E

EεSeDEeScT

⋅+⋅=⋅−⋅=

S

E

DεTgE

DgTsS

⋅+⋅−=

⋅+⋅=−1T

D

DεSqE

DqScT

⋅+⋅−=

⋅−⋅=−1S

D

Strain-charge to stress-charge

Conversions

Strain-voltage to stress-voltage

t

sde

sc

⋅=

=−

−

1

1E

1EE

sgq

sc

⋅=

=−

−

1D

1DD

tdsdεD ⋅⋅−= −1ET tgsgεε ⋅⋅+= −−− 1

D1

T1

S

Strain-charge to strain-voltage Strain-charge to stress-voltage

dεg

dεdss

⋅=

⋅⋅−=−

−

1T

1TED

t

eεq

eεecc

⋅=

⋅⋅−=−

−

1S

1SED

t



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Symbol definitions

Source:http://www.efunda.com/materials/piezo/piezo_math/symbol_definition.cfm



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Manufacturing of piezoelectric ceramics (example - PZT)

Material preparation

Mixing/MillingCalciningSeparation of gas-

MixingHomogenisation and

PbO

TiOHomogenising to a

predefined particle size

p geous by-products at

temp. ~900°

gmilling in liquid

mediumZrO2

Half-product manufacturing

PressingAdhesion of

ShapingSpray-drying

Dopants

Adhesion of granulates using adhesive agents

Mechanical forming by pressing

Separation of liquid phase

Sintering and finishing

Sintering PolarisationApplying electrodes1000-1300°C SinteringCompression in 3

temperature intervals 1100°-1300°

Polarisation

Orientation of piezoelectric domains

Applying electrodes

Screen printing, vacuum metalizing etc.



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Manufacturing process of piezoelectric ceramics

Production normally involves a powder preparation process in which oxide raw materials are mixed to obtain the defined chemical product composition. Further steps include the

ti f th diti d d d f i t h d i Thcompaction of the conditioned powder and a forming stage such as dry-pressing. The ceramic blank is then sintered at temperatures between 1000 °C and 1300 °C in a continuous tunnel style electric furnace. In the course of this sintering step the product develops its polycrystalline ceramic structure. The sintered piezoceramic component isdevelops its polycrystalline ceramic structure. The sintered piezoceramic component is mechanically finished by grinding, lapping, polishing, and sawing to assure its geometrical dimensions and surface finish remain within the specified narrow tolerances. The geometrical dimensions of the product, apart from material coefficients, determine it f ti l h t i ti h it t f El t i lits functional characteristics such as capacitance or resonant frequency. Electrical connections are usually made by applying silver electrodes in a screen printing process with subsequent sintering at approx. 600 °C. This is followed by the polarization step in which the product is exposed to an electric DC field (2 to 3 kV/ mm) at temperatureswhich the product is exposed to an electric DC field (2 to 3 kV/ mm) at temperatures between 80 °C and 140 °C to achieve the appropriate dipole orientation within the ceramic. In the final outgoing product inspection, the component is tested for a wide range of parameters (e.g., geometrical dimensions, adhesion strength of the silver

t lli ti l ) It i l t i d t d d i t hi imetallization layer). Its piezoelectric data are recorded prior to shipping.



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Piezoelectric actuators and sensors Piezoelectric active materials

• integrated in composite materialsPTZ films PTZ fibres

• i t t d ithi t t /PTZ layers

Multifunctionalcomposite materials

• integrated within actuators/sensors

Stack actuators

Source: www.globalspec.comSource: http://www.duraact.net/



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Source: www.piezojena.comSource: www.telcona.com/pages/piezoelectronic/piezo.html Source: RUB Mechanik adaptiver Systeme

Example: P-876 DuraAct™ Piezoelectric Patch Transducer

Design principle

d31 effect (Actuator)

Lateral contraction when voltage is applied

+

-

-+



Adaptronics

Source: PI Ceramics; http://www.duraact.net/

Example: P-876 DuraAct™ Technical Data Sheet



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Design of piezoelectric smart structures – modelling

Modelling possibilities:

Modelling – the finite element method FEM approach

• analytical

• FEM approach

For development of adaptive piezoelectric structures the finite element method

• experimental model identification (e.g. Subspace Identification)

For development of adaptive piezoelectric structures the finite element method (FEM) can be very convenient even at early design stages, when no real structure or a prototype is available.Modelling with the FEM approach results in a structural model, which can be represented in the so called state space form, which is convenient for the controller design.Another approach to modelling is an experimental model identification. It is possible if a real structure is availableif a real structure is available. In the following we shall concern the basics of the FEM approach to modelling of piezoelectric structures.



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FEM-based modelling of piezoelectric smart structures

Additional relationsMechanical and electric balance equations Structural

Constitutive equations

Strain – displacement Equations of motion

equations

Charge equations of l t t ti

Electric field – potential

Electrodynamics

dynamics

Mechanical and electric boundary conditions

electrostatics

Approximation with shape functions

Variational statement of governing equationsfor coupled electro-mechanical problem

Approximation with shape functions(mechanical displacement – electric potential)

Equations of motion of the coupled electromechanical problem of a single element in a semi-discrete form

Coupling of all elements,Equations of motion of the whole structure



Adaptronics

Equations of motion of the whole structure




Strain – displacementEquations of motion

equations



Electrodynamics

dynamics


electrostatics

Linearized constitutiveequations

σ stress vectorC symmetric elasticity matrixε strain vector

eECεσ −=

κEεeD += T

ε strain vectore piezoelectric matrixE electric field vector

D vector of electric displacements(electric charge density displacement)

κ symmetric dielectric matrix (electric permittivity)



Adaptronics





equations



Electrodynamics

dynamics


electrostatics

∂∂ 00

Equations of motion:

∂∂∂

∂

y

x

00

00

P uuPσD ρ+T

∂∂∂∂

∂∂

∂=

xy

zu

0

00D

=

z

y

x

PPP

P

=

wvu

uuPσD ρ=+u

P body force vector

∂∂

∂∂

∂∂

∂∂

xz

yz

0

0P body force vectoru mechanical displacementsρ mass densityDu differentiation matrix



Adaptronics

∂∂ xz




Strain – displacement

equations


Electrodynamics

dynamicsEquations of motion



electrostatics

uPσD ρ+T

Equations of motion:

∂∂∂x

D

Electric balance equations:

uPσD ρ=+u

∂∂∂

=

z

yϕD

0DD =Tϕ Dϕ = div(D) – differentiation matrix



Adaptronics





equations



Electrodynamics

dynamics


electrostatics

uPσD ρ+T

Equations of motion

uDε =Strain – Displacement:


uPσD ρ=+uuDε u

0DD =Tϕ



Adaptronics




Equations of motion

equations


Electrodynamics

dynamicsStrain – displacement



electrostatics

uPσD ρ+T

Equations of motion Strain – Displacement:

uDε =


uPσD ρ=+u

Electric field – Potential:

uDε u

0DD =Tϕ

ϕϕDE −=



Adaptronics





equations



Electrodynamics

dynamics


electrostatics








Adaptronics



StructuralAdditional relationsMechanical and electric balance equations


Electrodynamics

dynamicsStrain – displacementEquations of motion

equations




electrostatics








Adaptronics





Electrodynamics


equations




electrostatics





Zusammensetzen aller finiten Elemente,Bewegungsgleichungen der gesamten Strukturjkjkjkj FqKqDqM =++

= k

k ϕu

q



Adaptronics

Bewegungsgleichungen der gesamten Strukturjjjj kϕ




Electrodynamics


equations




electrostatics








Adaptronics


Modeling of vibro-acoustic problems using the FEM approach


Additional relationsMechanical and electricbalance equations Structural


balance equations

Charge equationsf l t t ti


Electrodynamics

dynamicsAcoustics


of electrostatics

Approximation with shape functions( h i l di l t l t i t ti l l it t ti l)

Variational statement of the governing equations

(mechanical displacements, electric potential, velocity potential)

Equations of motion of a coupledelectromechanical problem

FE formulation of theacoustic problemelectromechanical problem acoustic problem

Vibro-acoustic coupling, approximation by finite elements,element assembly and adding up all element contributions



Adaptronics

y g p


Linear wave equation

StructuralFluid particle velocity –

velocity potential relation

Sound pressure –

Electrodynamics

dynamicsAcoustics










Adaptronics

y g p


Linear wave equationConstitutive equations

Additional relationsMechanical and electricbalance equations Fluid particle velocity –


Sound pressure –


balance equations


Electric field – potentialvelocity potential relation


of electrostatics









Adaptronics

y g p





Sound pressure –


balance equations




of electrostatics






=

−

+

+

ϕϕϕϕϕ

ϕ

ff

φw

KKKK

φw

000C

φw

000M uu

Tu

uuuuuuu

w – vector of mechanical displacements

ϕ – electric displacements(charge or potential/voltage)



Adaptronics

ϕϕϕϕϕ φφφ u (charge or potential/voltage)





Sound pressure –


balance equations




of electrostatics






=

−

+

+

ϕϕϕϕϕ

ϕ

ff

φw

KKKK

φw

000C

φw

000M uu

Tu

uuuuuuu

aaaa fΦKΦCΦM =++



Adaptronics

ϕϕϕϕϕ φφφ u





Sound pressure –


balance equations




of electrostatics









Adaptronics

element assembly and adding up all element contributions


Coupled vibro-acoustic state-space models

aaaa fΦKΦCΦM =++

=

−

+

+

ϕ

ff

φw

KKKK

φw

000C

φw

000M uu

Tuuuuuuu

ΦCf ucuc = wCf acac =

ϕϕϕϕϕ fφKKφ00φ00 u

Structural load vector due to fluid pressure

Acoustic load vector due to structural vibrations

− ϕuuu fw0KKwC0Cw00M

Symmetric semi-discrete equations of motion of the coupled smart piezoelectric vibro-acoustic field

ρ−=

ρ−−+

ρ−−+

ρ− 0

ϕϕ

0

ϕϕϕ

ϕ

00 a

uu

a

Tu

uuu

aTuc

ucuu

a

uu

fff

Φφw

K000KK0KK

Φφw

C0C000C0C

Φφw

M0000000M

Model reduction (modal truncation)

State space vector: [ ]TΦwΦwx ϕϕ=

Modally reduced state space model: EfBuAxx ++= FfDuCxy ++=



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Modally reduced state space model: EfBuAxx ++= FfDuCxy ++

Basic control concepts

Open-loop system

Open-loop feed-forward system

Closed-loop control system

Feedback control system Adaptive feedforwardcontrol system



Adaptronics

y

Obtaining the reduced state space models from FE models

Modelling of piezoelectric structures using the FE approach results in state space models with large number of degrees of freedom (even (>100.000)with large number of degrees of freedom (even ( 100.000)

For the controller design purposes, the order of the plant models, i.e. the number of equations describing the motion of a structure, should be reduced (usually < 100)

Through the modal reduction only the dominant eigen modes and corresponding eigen frequencies are considered and kept within the model.

It is assumed, that the eigen modes, which dominantly influence the structural behaviour, are known. They can be determined through the modal analysis procedure, either based on the FE model or experimentally.

Modelling of piezoelectric structures and modal analysis can be performed using some of the standard FE software tools (e.g. ANSYS, ABAQUS, COSAR). A convenient software tool for the controller design is Matlab/Simulink software.g



Adaptronics


)()(d tt uBfEFKqqDqM +==++ Assembled equations of motion: (1)

First we consider the solution of the eigen-value problem without damping

)()( xxtt uBfEKqqM +=+ (2)

q – general displacement vectorf – external forcesu – control forcesϕk – eigen vectors

2

),,()( xxtt uBfEKqqM +=+

The solution of the characteristic equation:

(2) ϕk eigen vectorsωk – eigen frequenciesζ – damping

(3)0MK =ϕω− kk )( 2

results in: modal matrix ][ 21 nm ϕϕϕ= Φ

(3)

(4)

)( 2kdiag ω=Λspectral matrix

By introducing zΦq m=modal coordinates(5)

y g q m

the equations of motion (1) can be written in the following form:

)()( tt uBfEzKΦzMΦ +=+ (6)



Adaptronics

)()( ttmm uBfEzKΦzMΦ ++


)()( ttmm uBfEzKΦzMΦ +=+ (7)

Multiplying equation (2) withTmΦ

)()( tt TTTT uBΦfEΦzKΦΦzMΦΦ +=+

on the left hand side we obtain:

)()( tt mmmmmm uBΦfEΦzKΦΦzMΦΦΛI

+=+

Using the orthogonal properties of the eigenmodes

)1(diagmTm == IMΦΦ

)()( 2T didi ωλΛKΦΦ

(8)

(9))()( 2kkm

Tm diagdiag ωλ === ΛKΦΦ (9)

we obtain )()( tt Tm

Tm uBΦfEΦzΛz +=+ (10)we obtain )()( mm (10)

This system of equations contains only the reduced number of decoupled equations, which corresponds to the number of considered eigen-frequencies of interest.



Adaptronics


),,()( xxttd uBfEKqqDqM +=++In the FE modelling damping is taken into account in the damping matrix Dd, like in equation (1).

In this case transformation into modal coordinates does not lead directly to a diagonal system of equations.

By applying modal coordinates (5) and orthogonal transformation (8), (9) we obtain:

),,()( xxtt Tm

Tmmd

Tm uBΦfEΦΛzzΦDΦz +=++

Δ

uBΦfEΦzΛzz Tm

Tm +=+Δ+ (11)

)2(dT

kkmm diag ωζ== ΔΦDΦwith

In a general case Δ is not a diagonal matrix, and therefore the equations (11) are in a general case not decoupled.



Adaptronics


In order to obtain a diagonal matrix Δ, in the FE modelling the so called Rayleigh-damping is assumed. The Rayleigh-damping is proportional to matrices M and K.

KMD βα +=d

α and β are parameters which can be used to obtain damping models, which provide a good agreement between measurements and numerical simulation using the FE approach.

Finally by introducing the state space coordinates equation (11)TT ][ zzx =y y g p q ( )

uBΦfEΦzΛzz Tm

Tm +=+Δ+

][

can be written in the state space form:

EfBA

Modally reduced state space model

)()()()( TT ttttmm

fEΦ

0u

BΦ0

xΔΛI0

x

+

+

−−

=EfBuAxx ++=FfDuCxy ++=



Adaptronics

General representation of a continuous-time linear state-space model

)()()()( tttt EfBuAxx ++=)()()()( tttt FfDuCxy ++=

State equation

Output equation

Notation and matrix dimensions:

State vector ( ×1)State vector x (n×1)Input vector (control input) u (m×1)Output vector y (r×1)System matrix A (n× n)Input matrix B ( × )Input matrix B (n× m)Output matrix C (r× n)

Matrix of direct couplingbetween input and output D (r× m)p p ( )

Disturbance (Excitation) f (p×1)

Matrix of coupling betweenthe state vector and disturbances E (n× p)( p)

Matrix of coupling betweenoutput and disturbances F (r× p)

First control of the state-space model: matrix dimensions must be appropriate!



Adaptronics

p pp p

Block diagram of a continuous-time linear state-space model

)()()()( tttt EfBuAxx ++= )()()()( tttt EfBuAxx ++=)()()()( tttt FfDuCxy ++=



Adaptronics

General mathematical approach to a state space representationof a system of differential equationsof a system of differential equations

In the general case the form of the state equations is:g q

In matrix form this system of equations may be written as:



Adaptronics


Further we restrict our consideration only to linear time-invariant (LTI) systems that is systems

State equations

Further we restrict our consideration only to linear time-invariant (LTI) systems, that is systemsdescribed by linear differential equations with constant coefficients.

This system of equations may be written compactly in a matrix form:

or shorter:BuAxx +=



Adaptronics


A system output is defined to be any system variable of interest. A description of a physical system in terms of a set of state variables does not necessarily include all of the variables of

Output equations

system in terms of a set of state variables does not necessarily include all of the variables of direct engineering interest. An important property of the linear state equation description is that all system variables may be represented by a linear combination of the state variables xi and the system inputs ui. An arbitrary output variable in a system of order n with r inputs may be written:written:

If a system has m outputs:

Matrix form:

or: DuCxy += Cxy =In control theory often only shorter form:



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or: DuCxy += CxyIn control theory often only shorter form:

Example of a state variable modellingMass-spring-damper system (mechanical translation system)

Differential equation of motion of the mass-spring-damper system

)(f (1)K

y(t)

)(tfKyyByM =++

)()()()(2

2tKytdyBtftydM −−=

(1)

(2)

Mf(t))()(2 y

dtf

dt

Using the state space approach we can define the following state variables:

( )B

)()(1 tytx =

)()( 1 tdxtdy

Position of the mass(3)

)()()()( 11

2 txdt

tdxdt

tdytx === Velocity(4)

Then, from (1):

)(1)()()()()(122

22

2

tfM

txMKtx

MBtx

dttdx

dttyd +−−=== (5)



Adaptronics

Example of a state variable modelling

We can rearrange the previous equationsto obtain the system of equations, whichcorrespond to a state space form:

)()( txtx

Written in the matrix notation the equations become:

)(10

)()(10

)()( 11 tf

ttxBK

ttx

+

=

)()( 21 txtx =

)(1)()()( 122 tfM

txMKtx

MBtx +−−=

)()()( 22

fMtxMMtx

−−

[ ]

=)(

01)( 1 txty

)()( 1 txty =[ ]

)(

01)(2 tx

ty

which corresponds to the state space representation:

ubAxx += Single-Input Single-Output system (SISO)

cx=yubAxx +

)(1 tx )(f

g p g p y ( )Input – uOutput - y

with:

=

)()(

2

1

txtx

x )(tfu = Notation art which is often usedin books on control theory:

matrix – capital letters, bold, e.g. A

−−=

MB

MK

10A

=

M10

b [ ]01=cvector – small letters, bold, e.g. b, c, xscalar – regular letters (italic), e.g. u, y



Adaptronics

See: Matlab tutorials (general)



Adaptronics

General representation of a continuous-time linear state-space model

)()()()( tttt EfBuAxx ++=)()()()( tttt FfDuCxy ++=

State equation

Output equation

Notation and matrix dimensions:

State vector ( ×1)State vector x (n×1)Input vector (control input) u (m×1)Output vector y (r×1)System matrix A (n× n)Input matrix B ( × )Input matrix B (n× m)Output matrix C (r× n)

Matrix of direct couplingbetween input and output D (r× m)p p ( )

Disturbance (Excitation) f (p×1)

Matrix of coupling betweenthe state vector and disturbances E (n× p)( p)

Matrix of coupling betweenoutput and disturbances F (r× p)

First control of the state-space model: matrix dimensions must be appropriate!



Adaptronics

p pp p

Block diagram of a state-space model (simplified form)

In control theory is often used the following form of the state space model, where the state equation is not influenced by disturbances and the output equation is influenced only by the system states, vector x (compare with general representation of a continuous-time linear state-space model). Disturbances or excitations can then be taken into consideration in the controller deign procedure, g p ,like in the tracking system design (which will be addressed later).

)()()( ]11[]1[]1[][]1[ tutt nnnnn ××××× += bxAx )()()( ]1[][]1[][]1[ ttt mmnnnnn ××××× += uBxAx)()( ]1[]1[]11[ tty nn ××× = xc )()( ]1[][]1[ tt nnrr ××× = xCy

b cu y

Multiple-Input Multiple-Output system (MIMO)Single-Input Single-Output system (SISO)

matrix – capital letters, bold, e.g. A, B, Cvector – small letters, bold, e.g. b, c, xscalar – regular letters (italic), e.g. u, y



Adaptronics

Transfer function

Transfer function representation is convenient for SISO systems. The SISO system from the previous slide can also be represented in the equivalent form by the following diagram:

ubAxx +=u y

cx=yubAxx +=

b cu y

⇔

11 mmnn

The state space form of the model can be obtained from a system description by a linear differential equation with constant coefficients, with appropriate choice of the state variables:

)()(...)()()()(...)()(011

1

1011

1

1 tubdt

tdubdt

tudbdt

tudbtyadt

tdyadt

tydadt

tyda m

m

mm

m

mn

n

nn

n

n ++++=++++ −

−

−−

−

−

(∗)

If we consider any function y(t) as a continuous-time signal,the Laplace transform Y(s) of the function y(t) is defined in the following way:

∞

−==0

)()}({)( dtetytyLsY stdef

where s represent the Laplace operator



Adaptronics

Transfer function

An important property of the Laplace transform (assuming zero initial conditions), which we use here is:

)()}({)()( ssYtyLsYstydL pp

==

)()}({)( ydt p

Applying the Laplace transform to both sides of the differential equation (∗)

)()(...)()()()(...)()(011

1

1011

1

1 tubdt

tdubdt

tudbdt

tudbtyadt

tdyadt

tydadt

tyda m

m

mm

m

mn

n

nn

n

n ++++=++++ −

−

−−

−

−

bt ione obtains:

)()...()()...( 011

1011

1 sUbsbsbsbsYasasasa mm

mm

nn

nn ++++=++++ −

−−

−

oror

011

1

011

1 ...)()(:)(

asasasabsbsbsb

sUsYsG nn

mm

mm

++++++++

== −

−− (∗∗)

011 ...)( asasasasU nn ++++ −

)()()( sUsGsY =Transfer function G(s) is the linear mapping of the Laplace transform of the input U(s) to the output Y(s).



Adaptronics

Typical inputs considered in control systems and their Laplace transforms

Dirac impulse

≠=∞+

=0,00,

)(δtt

t 1)()()( == sUttu δ

Heaviside unit step

≥<

=0,10,0

)(tt

th sR

sUthRtu 00 )()()( =⋅=

Ramp input 20

0 )()(sV

sUtVtu =⋅=

Periodic input function (sine) 220

0 )()sin()(ωωω+

=⋅=s

AsUtAtu

)()()( sUsGsY =System response to such inputs is obtained from

Response y(t) in time domain is obtained by applying the inverse Laplace transform to Y(s) or by solving analytical differential equations, which describe the system behaviour.



Adaptronics

Table of Laplace transforms (1)



Adaptronics

Table of Laplace transforms (2)



Adaptronics



Adaptronics

Matlab functions for Laplace and inverse Laplace transforms

laplace(F)

L = laplace(F)

ilaplace(F)

F = ilaplace(L)

is the Laplace transform of the scalar symbol F with default independent variable t. The default return is a function of s. The

is the Laplace transform of the scalar symbol F with default independent variable t. The default return is a function of s. The L l t f i li d t f tiLaplace transform is applied to a function

of t and returns a function of s.

)()( sLLtFF ==

Laplace transform is applied to a function of t and returns a function of s.

)()( tFFsLL ==)()( sLLtFF ==

∞

−= )()( dtetFsL stdef

)()(

∞+

=ic

ic

stdef

dsesLtF )()(0

Example:

4)( ttf =f = t^4laplace(f)

∞−ic

Example:)( ttf =

∞

−=0

4)( dtetsL st

laplace(f)

returns24/s^5

ilaplace(1/(s-1)) returns exp(t)



Adaptronics

0

Using Laplace transform to solve differential equations (1)

Example:

Using the 5th property from the Table of Laplace transform properties we obtain:

Find the solution to the differential equation:

0)()( =+ tyty where βα == )0(,)0( yy

Using the 5th property from the Table of Laplace transform properties, we obtain:

0)()()()0()0()( 2012 =+−−=+−− sYssYssYysyssYs βα

βY )()1( 2 βα +=+ ssYs )()1( 2

)1()1()( 22 +

++

=ss

ssY βα)()(

Using the table of Laplace transforms (18, 17, respectively) to find the inverse Laplace transform, we obtain:

ttty sincos)( βα += ttty sincos)( βα +

The same solution can be obtained using Matlab funcitions ilaplace:

syms s a b ← declaration of symbolic variables (see Matlab Help)syms s a b ← declaration of symbolic variables (see Matlab Help)

ilaplace(a*s/(s^2+1)) returns a*cos(t)ilaplace(b/(s^2+1)) returns b*sin(t)



Adaptronics

Using Matlab functions to solve differential equations (2)

Now we are going to solve the same problem using Matlab functions.

Matlab function dsolve for solving differential equations (see Matlab Help):

d l ('D2 0' ' (0) ' 'D (0) b') t b* i (t) * (t)dsolve('D2y+y=0','y(0)=a','Dy(0)=b') returns b*sin(t)+a*cos(t)

The same problem can be solved in Matlab using Laplace and inverse Laplace transform:

>> syms t>> Seq = laplace(diff(sym('y(t)'),2)+sym('y(t)'))

Seq =s^2*laplace(y(t),t,s)-D(y)(0)-s*y(0)+laplace(y(t),t,s)

>> syms Ys a b syms Ys a b

>> w=subs(Seq, {'laplace(y(t),t,s)', 'y(0)', 'D(y)(0)'}, {Ys, a, b}) returnsw =s^2*Ys-b-s*a+Ys

>> Y=solve(w,Ys) returns Y =(b+s*a)/(s^2+1)

>> y=ilaplace(Y) returns y =b*sin(t)+a*cos(t)



Adaptronics

Relation between the transfer function and the state space representation (SISO)

Laplace transform can be applied to the state space system in matrix form as follows:

)()()( sUsss bAXX += (2-1)(1)

)()()( tutt bAxx +=)()()( sdUssY += cX (2-2)

(1))()()( tdutty += cx

Laplace transform enables simpler solution of the systems of differential equations.

From (2-1) one obtains: )()()( 1 sUss bAIX −−=

The system (2) can be solved as a system of algebraic equations.

And from (2-2) and (2-1): ( ) )()()( 1 sUdssY +−= − bAIc

The transfer function of the whole system is then:

dssadjds

sUsYsG +

−−=+−== − b

AIAIcbAIc

)det()()(

)()()( 1 (3)

The transfer function of the whole system is then:

ssU AI )det()(The right-hand side of the equation (3) represents the ratio of the two polynomials in s, like in (∗∗). Real plants/systems must fulfil the condition m<n.



Adaptronics

Relation between the transfer function and the state space representation (MIMO)

)()()( ttt BuAxx +=),()( tt Cxy =

(3-1)

(3-2)

For a MIMO system transfer functions are formed from each input to each output. If a MIMO system (3-1), (3-2) has m inputs u[m× 1] and r outputs y[r× 1] , then all transfer functions from input i to output j can be organized in a transfer matrix G(s) which has m× r elements G (s)input i to output j can be organized in a transfer matrix G(s), which has m× r elements Gij(s).The transfer matrix is then:

)()()( 11211 sGsGsG r

=)()()(

)( 22221 sGsGsGs r

G

and it can be calculated from the state space model in the following way:

)()()( 21 sGsGsG mrmm

BAIAICBAIC

UYG

)det()()(

)()()( 1

−−=−== −

ssadjs

sss

p g y



Adaptronics

Example: conversion from state space to transfer function representation

For a system represented in a state space form:

0,]12[,01

,0123

==

=

−−= dcbA find an equivalent transfer function.001

q

dssUsYsG +−== − bAIc 1)()()()()(

−+

=

−−−

=−

ss

ss

s1

230123

00

)( AI

−+=−

ss

s2

13)(

T

AI

−

=−=− − 21)()( 1 ssadjs AIAI

+++− 312)3()det(

)(ssss

sAI

AI

)()( 1 =−=− − sadjs AIbAIc

2)3(12

01

]112[2)3(

101

312

]12[2)3(

1

)det()(

+++=

++

++=

+−

++

−

sssss

ssss

ss

ss

AIbc



Adaptronics

)()()(

Matlab functions for transfer function and state space representations

tf Creation of transfer functions or conversion to transfer function.

C iCreation:

SYS = TF(NUM,DEN) creates a continuous-time transfer function SYS with

numerator(s) NUM and denominator(s) DEN. The output SYS is a TF object.

Data format:

For SISO models, NUM and DEN are row vectors listing the numerator and denominatorFor SISO models, NUM and DEN are row vectors listing the numerator and denominator coefficients in descending powers of s.

For MIMO models with NY outputs and NU inputs, NUM and DEN are NY-by-NUFor MIMO models with NY outputs and NU inputs, NUM and DEN are NY by NU

cell arrays of row vectors where NUM{i,j} and DEN{i,j} specify the

transfer function from input j to output i.



Adaptronics


ltimodels tf

Transfer function (TF) models are created with the TF command.

To create a SISO continuous-time transfer function, you can specify its numerator and denominator as row , y p yvectors of coefficients, e.g.,

H = tf([1 2],[1 3 5])

specifies H(s) = (s+2)/(s^2+3s+5). You can also define the Laplace variable s as a special TF model by p ( ) ( ) ( ) p p y

s = tf('s')

and then enter your transfer function as a rational expression in s:

H ( +2)/( ^2+3* +5)H = (s+2)/(s^2+3*s+5) .

To create a MIMO continuous-time transfer function, you need to specify a numerator and denominator for each I/O pair You can collect this data in a cell array of row vectors e geach I/O pair. You can collect this data in a cell array of row vectors, e.g.,

H = tf({1 ; [1 2]} , {[1 0] ; [1 3 5]})

Alternatively, you can build this transfer matrix by concatenation of SISO transfer functions, as in

H = [1/s ; (s+2)/(s^2+3*s+5)] (where s = tf('s')),

or equivalently,

H = [tf(1,[1 0]) ; tf([1 2],[1 3 5])] .



Adaptronics

[ ( ,[ ]) ; ([ ],[ ])]

E l

Matlab functions for transfer function and state space representationsExamples:

H = tf({1 ; [1 2]} , {[1 0] ; [1 3 5]})

Alternatively, you can build this transfer matrix by concatenation of SISO transfer functions, as in

H = [1/s ; (s+2)/(s^2+3*s+5)] (where s = tf('s')),

or equivalently,

H = [tf(1,[1 0]) ; tf([1 2],[1 3 5])] . [ ( ,[ ]) ; ([ ],[ ])]



Adaptronics


ss Creates state-space model or converts model to state space.

SYS = SS(A,B,C,D) creates a SS object SYS representing the ( , , , ) j p g

continuous-time state-space model

dx/dt = Ax(t) + Bu(t)

(t) C (t) + D (t)y(t) = Cx(t) + Du(t)

You can set D=0 to mean the zero matrix of appropriate dimensions.



Adaptronics

Matlab function for conversion: transfer function to state-space

tf2ss Transfer function to state-space conversion.

[A,B,C,D] = TF2SS(NUM,DEN) calculates the state-space

representation:representation:

.x = Ax + Bu

y = Cx + Duy Cx Du

of the system:

NUM(s)

H( )H(s) = --------

DEN(s)

from a single input. Vector DEN must contain the coefficients o the denominator in descending powers of M t i NUM t t i th t ffi i t ith th t t Ths. Matrix NUM must contain the numerator coefficients with as many rows as there are outputs y. The

A,B,C,D matrices are returned in controller canonical form. This calculation also works for discrete systems.

For discrete-time transfer functions, it is highly recommended to make the length of the numerator andFor discrete time transfer functions, it is highly recommended to make the length of the numerator and denominator equal to ensure correct results. You can do this using the function EQTFLENGTH in the Signal Processing Toolbox. However, this function only handles single-input single-output systems.

See also tf2zp, ss2tf, zp2ss, zp2tf.



Adaptronics

Matlab function for conversion: state-space to transfer function

ss2tf State-space to transfer function conversion.

[NUM,DEN] = SS2TF(A,B,C,D,iu) calculates the transfer function:

NUM(s) -1

H(s) = -------- = C(sI-A) B + D

DEN( )DEN(s)

of the system:

.A Bx = Ax + Bu

y = Cx + Du

from the iu'th input. Vector DEN contains the coefficients of the

denominator in descending powers of s. The numerator coefficients

are returned in matrix NUM with as many rows as there are outputs y.are returned in matrix NUM with as many rows as there are outputs y.

See also tf2ss, zp2tf, zp2ss.



Adaptronics

Poles and zeros of the transfer function

011

1

011

1

......)(

asasasabsbsbsbsG n

nn

n

mm

mm

++++++++= −

−

−−

Roots of the numerator polynomial of the transfer function G(s) are the zeros of the transfer function.Roots of the denominator polynomial of the transfer function G(s) are the poles of the transfer function.

sadjsY AI )()( dssadjds

sUsYsG +

−−=+−== − b

AIAIcbAIc

)det()()(

)()()( 1From:

One can obtain the poles of the transfer function by solving the equation 0)det( =−AIsOne can obtain the poles of the transfer function by solving the equation )(

A system or a plant is stable if the poles have negative real parts

Plant/system stability

0, <±= σωσ jsA system or a plant is stable if the poles have negative real parts.

mZero-pole-gain representation of a transfer function

∏

∏=

−

−=

−−−−−−

= n

m

ii

n

m

ps

zsK

pspspszszszs

KsG 1

21

21

)(

)(

))...()(())...()((

)(zi zerospi polesK gain



Adaptronics

∏=i

ips1

)( K gain


Zero-pole-gain (ZPK) models are created with the ZPK command.

To create SISO continuous-time ZPK models, specify the vector of poles, the vector of zeros, and the gain as input arguments to ZPK. For example,

H = zpk(1,[2-i 2+i],3)

specifies H(s) = 3(s-1)/(s^2-4s+5).

You can also define the Laplace variable s as a special ZPK model by p p y

s = zpk('s')

and then enter your model as a rational expression in s:

H 3*( 2)/( ^2 4* +5)H = 3*(s-2)/(s^2-4*s+5) .

To create MIMO continuous-time ZPK models, you need to specify a vector of zeros, a vector of poles, and a scalar gain for each I/O pair You can collect this data into two cell arrays for the zeros and poles and aa scalar gain for each I/O pair. You can collect this data into two cell arrays for the zeros and poles,and a matri for the gains, e.g.,

H = zpk({[] ; [1 -1]} , {0 ; [1-i 1+i]} , [3 ; -1])

Alternatively you can build this MIMO model by concatenation of SISO models as inAlternatively, you can build this MIMO model by concatenation of SISO models as in

H = [3/s ; -(s-1)*(s+1)/(s^2-2*s+2)] (where s = zpk('s')),

or equivalently,



Adaptronics

H = [zpk([],0,3) ; zpk([1 -1],[1-i 1+i],-1)] .

Examples



Adaptronics

Controllability and observability criteria

Plant/system states given by the state equation BuAxx +=are controllable if it is possible - by admissible inputs - to steer the states from any initial value to any final value within some time window.

A continuous time-invariant state-space model is controllable if and only if the rank of the controllability matrix Sco equals n.

Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs

,][ 1co BAABBS −= n nrank =coS

knowledge of its external outputs.

For a plant/system given by the state space model ,BuAxx += Cxy =)(tthe observability matrix Sob is defined as follows:

2CACAC

SA continuous time-invariant state-space

d l i b bl if d l if

=

−1

2

n

ob

CA

CAS nrank ob =S

model is observable if and only if



Adaptronics

CA

Matlab function for controllability matrix

ctrb

CTRB Compute the controllability matrixCTRB Compute the controllability matrix.

CO = CTRB(A,B) returns the controllability matrix [B AB A^2B ...].

CO = CTRB(SYS) returns the controllability matrix of the

d l SYS i h li i (A B C D) Thi istate-space model SYS with realization (A,B,C,D). This is

equivalent to CTRB(sys.a,sys.b).

For ND arrays of state-space models SYS, CO is an array with N+2

dimensions where CO(:,:,j1,...,jN) contains the controllability

matrix of the state-space model SYS(:,:,j1,...,jN).



Adaptronics

Matlab function for observability matrix

obsv

OBSV Compute the observability matrix.

OB = OBSV(A,C) returns the observability matrix [C; CA; CA^2 ...]

CO = OBSV(SYS) returns the observability matrix of the

state-space model SYS with realization (A,B,C,D). This is p ( )

equivalent to OBSV(sys.a,sys.c).

For ND arrays of state-space models SYS, OB is an array with N+2

dimensions where OB(:,:,j1,...,jN) contains the observability

matrix of the state space model SYS(: : j1 jN)matrix of the state-space model SYS(:,:,j1,...,jN).

See also OBSVF, SS.



Adaptronics

See: T1 Simulink tutorial



Adaptronics

Controller design using pole placement pole placement = pole assignment (Engl.)

The main idea behind the pole placement for the controller design is to introduce the control input in the form of a state feedback (1) in such a way that the poles of the closed-loop control system have desired locations. In this way the plant can be stabilized or it can perform a

fdesired response defined by desired poles location.

)()( tt Lxu −= (1)

Substituting (1) into state equation:

(L – feedback gain matrix)

Open-loop (plant only)

)()()( ttt BLxAxx −=BuAxx += one obtains:

)()()( tt GxAxBLA =−=

with the state matrix of the closed-loop system

,BLΑA −=G

With appropriate design of the feedback gain matrix L, the eigenvalues of the matrix AG

are equal to desired poles of the closed loop system

Closed-loop (plant and controller)



Adaptronics

are equal to desired poles of the closed-loop system.

Controller design using pole placement

The controller design task is thus to obtain the control input u(t)= - Lx(t)through the pole placement procedure in such a way thatthe state space matrix of the closed-loop system AG=A-BLp p y Ghas predefined eigenvalues, which are equal to desired closed-loop poles

The motivation for this procedure is based on the fact that the eigenvalues i eThe motivation for this procedure is based on the fact that the eigenvalues, i.e. the poles of the closed-loop system influence crucially the system behaviour, which can be caused either by initial conditions or in a transient response.

As closed-loop pole locations the left-hand half-plane of the s-plane should be chosen, as shown in picture.

Damping

In this way the necessary stability degree, damping and suppression of measurement noise can be achieved.

Stability degreeSuppression of

measurement noise



Adaptronics

measurement noise


The poles of the closed-loop system )()()( tt xBLAx −= can be chosen to have

desired locations if the plant )()()( ttt BuAxx += is fully controllable, i.e.

if the plant controllability matrix has the rank n (see the controllability criterion).

If the plant is not fully controllable i e if:If the plant is not fully controllable, i.e. if:

,][)( 1 nnrankrank cn

co <== − BAABBS

then, n – nc poles of the matrix A represent at the same time the closed-loop poles, i.e. the poles of A – BL, for an arbitrary matrix L.

Th i i l f th l d l t bt i d i d l tiThe remaining nc poles of the closed-loop system can obtain desired locations by an appropriate choice of the feedback gain matrix L.

Therefore when choosing the locations of the closed-loop poles one should g p ptake care that the poles of the closed-loop system must not be equal to the poles of the plant.



Adaptronics


In order to explain the procedure for controller design using the pole placement we are going first to explain the control canonical form of the plant. We start again from a general plant representation (system process structure ) described by a differential equation DE (∗)

Control canonical form of the plant

(system, process, structure…) described by a differential equation DE (∗)

)(...)(... 011

1

1011

1

1 tubdtdub

dtudb

dtudbtya

dtdya

dtyda

dtyda q

q

qq

q

qn

n

nn

n

n ++++=++++ −

−

−−

−

− (∗) (1)

The control input on the right-hand side of DE (1) is in the control theory usually given in the formb0 u(t), and the DE (1) transformed so that the coefficient with the first term on the left-hand side is an=1 can be rewritten in the following form:

)()(... tubtyadtdya

dtyda

dtyd

n

n

nn

n

0011

1

1 =++++ −

−

− (2)

xyx

yx

==

=1

Then we can define the vector of state variables in the following way:

n yd

xyx

==

−1

12

(3) x1

nn

n

nnn

xd

yd

xdt

ydx

=

== −− 11

=

x

xt

2

1

)(x



Adaptronics

nndt nx


With the previous definition of the state variables, eq. (2) can be rewritten in the following way:

f

)(... 010211 tubxaxaxax nnn +−−−−= −

We can assemble all state variables to obtain the system in the state space form:

xxxx

==

32

21

(5)(4)nn xx =−1

32

duty

ut

+=

+=

cx

bAxx

)(

)(⇔

)(... 010211 tubxaxaxax nnn +−−−−= −

1xy =

with .0,]0001[,000

,01000010

==

=

= d

cbA (6)with ,][,,1000

01210

−−−− − baaaa n

( )



Adaptronics


bIn a general case, for eq. (1) if all coefficients bi are known, we can writethe control canonical form of a state space system in the following way:

00010

,00

,1000

0100

cc

=

=

bAIn control theory a set of matrices/vectors Ac, Bc, Cc, Dcrepresenting the model in control canonical form is often called

11000

1210 naaaa

−−−− −

canonical form is often called

realisation in controlcanonical formand denoted with (Ac, Bc, Cc, Dc ).

(5)

,],,[][

111100

21

nnnnn

nc

bdabbabbabb

ccc

=−−−=

=

−−c Index “c” stands for

“control canonical form”

Characteristic polynomial of the plant in control canonical form is obtained as:

(6)

.nc bd

011

1 ...)det()( asasasssp nn

nc ++++=−= −

−AI

Plant poles s1, s2, ... , sn are the roots of the characteristic polynomial

011nc

0)det()( =−= ssp AI



Adaptronics

0)det()( =−= cssp AI


For the simplicity, the pole placement procedure will first be explained for a plant in the control

Ackerman’s formula

canonical form.

The task of the pole placement is to find a feedback law (7)xLcu −=

in such a way that the closed-loop system has the poles with desired locations.

The closed-loop system is given by the following state-space model:

)()()()()()( ttttt Gcccccc xAxLbAxLbxAx =−=−=

cccG LbΑA −=

The feedback gain vector is calculated in the form:

][ 21 lll =L

Then, the closed-loop state matrix can be obtained in the following way:

][ 21 cnccc lllL



Adaptronics


00

01000010

⋅

−

−−−−

=−= cncccccG lll

aaaa

21 ][

1

01000

LbAA

−naaaa 1210

00000000

01000010

1

=

−

−−−−

=

− cncccn llllaaaa 3211210

00001000

=01000010

−−−−−−−− − cnnccc lalalala 1322110

1000



Adaptronics


nsss ,...,, 21On the other hand, if the desired closed-loop poles are predefined

then they are at the same time the eigenvalues of the closed-loop state matrix AG. i.e. they represent the roots of the characteristic polynomial of closed loop systemthe roots of the characteristic polynomial of closed-loop system.

011

121 ...)(...))(()det()( asasasssssssssp nn

nnG ++++=−−−=−= −

−AI

=

=−=01000010

01000010

cccG

LbAA

−−−−

−−−−−−−− −− 12101322110

10001000

ncnnccc aaaalalalala

c

c

c

laalaalaa

−−=−−−=−−−=−

322

211

100

][][][ 11011021 −− −== nncnccc aaaaaalll L

cnnn laa −−=− −− 11

][][][ 11011021 nncnccc



Adaptronics


Now, the procedure will be extended to a general case, when a system may not be represented in a control canonical form, but in an arbitrary state-space form:

dutyut +=+= cxbAxx )()( dutyut +=+= cxbAxx )(,)(

xTx 1−=

Using the following matrix transform, the previous state space system can be transformed into control canonical form:

(8)xTx = cc

Where xc represents the state vector in control canonical form.

1−

( )

2

= r

r

r

c AsAs

s

Twhere sr represents the last row of

the inverse controllability matrix 1−coS

1]1000[ −= cor Ss

][ 12 bAbAAbbS −= n

1−

n

r As

the inverse controllability matrix co ][ bAbAAbbSco

LxxTLxL −=−=−= −1ccccu

Substituting (8) into control law for the system in control canonical form, one obtains:



Adaptronics


( )][][ 1110110

1 −−−

−

−==

r

r

r

r

cnncc aaaaaa

Ass

Ass

TTLL

][][

1

2110

1

2110

−

−

−

−

−

=

nr

rn

nr

rn aaaaaa

As

As

As

As

][][ 2110

2110 −−

−

= nrr

r

r

n aaaaaa AAI

sAsAs

s

11 −−

r

nnr

s

AAs

rs

)...(][ 111

00

2110

−−− +++−

= nnrr

r

n aaaaaa AAAsAsAs

=+

= −n

rr

r

naaa AsAsAs

2110 ][

1−

nr As

nnnaaa AAAA −=+++ −1

110

0 ...

−nr As 1

Cayley-Hamilton-Theorem



Adaptronics

naaa −110 ...


r

r

AAs

s

2

r

r

Ass

−

−

n

nr

rnaaaa

AAs

As

1

2

1210 ]1[

=+

=

−

−n

r

n

rnaaa As

As

As

1

2110 ][

n

r As r As

( )nnnr aaa AAAIs ++++= −−

1110 ...

Finally, the Ackerman’s formula reads:

( )nnnr aaa AAAIsL ++++= −−

1110 ...

See:01_Examples-Tasks.zip !!!

011

121 ...)(...))(()det()( asasasssssssssp nn

nnG ++++=−−−=−= −

−AI

where the coefficients are obtained from the predefined closed-loop poleska nsss ,...,, 21



Adaptronics

Matlab functions for pole placement

place Pole placement technique

K = PLACE(A,B,P) computes a state-feedback matrix K such that

the eigenvalues of A-B*K are those specified in vector P.

No eigenvalue should have a multiplicity greater than the

number of inputsnumber of inputs.

acker Pole placement gain selection using Ackermann's formula.acker Pole placement gain selection using Ackermann s formula.

K = ACKER(A,B,P) calculates the feedback gain matrix K such that

the single input system

.x = Ax + Bu

with a feedback law of u = -Kx has closed loop poles at thewith a feedback law of u Kx has closed loop poles at the

values specified in vector P, i.e., P = eig(A-B*K).



Adaptronics

Example – controller design using pole placement

For a given plant:

See: 02_Example.zip

)()()()( tftutt ebAxx ++=

)()()( tdutty += cx )()()(y

[ ]cbA =

=

−

−= ,001,

010

,340

120201

be ==

−

,0

0340

d

The closed-loop control system should be designed in such a way that closedloop poles are: –1, – 2, – 4. ]421[ −−−=p



Adaptronics

>> A=[-1 0 2; 0 -2 1; 0 4 3]; >> b=[0; 1; 0] >> c=[1 0 0]

>> A=[-1 0 2;

0 2 1

b =

0

1

c =

1 0 0

0 -2 1;

0 4 3]

A =

1

0

>> eig(A)-1 0 2

0 -2 1

>> eig(A)

ans =0 4 3 -1.00000000000000

-2.70156211871642Th l t i t bl !!!

help eig

EIG Eigenvalues and eigenvectors

3.70156211871642 The plant is unstable !!!

Matlab command pzmap(sys)EIG Eigenvalues and eigenvectors.

E = EIG(X) is a vector containing the eigenvalues of a square matrix X.



Adaptronics

Simulink model of the plant (open-loop)

← Response of the plant to a step input



Adaptronics

← Response of the plant to a step input

Step and impulse response using Matlab commands

x 105 Step Response

10

12

14x 10 p p

A=[-1 0 2; 0 -2 1; 0 4 3];4

6

8

Ampl

itude

b=[0; 1; 0]

c=[1 0 0]

d=00 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

2

Time (sec)d 0

sys=ss(A,b,c,d)

step(sys) 7

8x 10

5 Impulse Response

Impulse (sys)

3

4

5

6

Ampl

itude

0

1

2

3



Adaptronics

0 0.5 1 1.5 2 2.5 3 3.5 40

Time (sec)

Pole placement applying Ackerman’s formula



Adaptronics

L=place(A,b,p) L=acker(A,b,p)

L = [0 7 9.75][ ]

eig(A-b*L)

ans =

-1.00000000000000

-4.00000000000000

-2.00000000000000 Step response of theclosed loop system →closed-loop system →

Closed-loop system



Adaptronics

Closed loop system

Responses to sinusoidal inputResponses to sinusoidal input

Response of the plant (open-loop) Response of the closed-loop system



Adaptronics

Transfer function representation

Review

Laplace transform of a derivative of function f(t) with zero initial condition:

)()}({ ssFtfL ==

Transfer function of a plant in polynomial num-den form

11

21

11

21

...

...)()(:)(

+−

+−

++++++++

==nn

nnmm

mm

asasasabsbsbsb

sUsYsG )()()( sUsGsY =

121)( +nn

][den],[num 121121 ++ == nm aaabbb ai, bi coefficients of the numerator and denominator polynomials of transfer function

∏m

)(

Zero-pole-gain representation of a transfer function

polynomials of transfer function

∏

∏=

−

−=

−−−−−−

= n

ii

ii

n

m

ps

zsK

pspspszszszs

KsG

1

1

21

21

)(

)(

))...()(())...()((

)(zi zerospi polesK gain



Adaptronics

=i 1

Pole properties of a general rational function

11

21 ...)( +

− ++++= mm

mm bsbsbsbsF

Consider the general form for the rational function F(s) consisting of a ratio of two polynomials

or))...()((

)( 21 mzszszsKsF

−−−=

11

21 ...)(

+− ++++ nn

nn asasasasF

Assuming that poles pi are real or complex, but distinct, this function can be transformed

or))...()((

)(21 npspsps

KsF−−−

using the partial-fraction expansion (see Matlab function residue) as follows:

)(...

)()()( 21 nCCCsF +++= To determine the set of constants Ci we

multiply the previous equation with (s p ):)()()( 21 npspsps −−− multiply the previous equation with (s−pi):

)(...)()()( 112

11n psCpsCCsFps

−++−+=−

Inverse Laplace transform of each of the terms:

)(...

)()()(

211

npspsCsFps

−++

−+

Then, each of the coefficients Ci can be obtained as:

FC )()(

)(1)(

ii ps

sF−

= will be:

tpitf )( See the table of Laplaceipsii sFpsC

=−= )()(tp

iietf =)( See the table of Laplace

transforms (Nr. 7)

For Re(pi) > 0 → fi(t) will grow exponentialy,



Adaptronics

so it will be unbounded Instability !!!

Influence of pole locations – real poles

First we are going to consider the influence or real poles.

Since the impulse response is given by the time function corresponding to the transfer function, we call the impulse the natural response of the system (also important for modal analysis).

)()()( sUsGsY =Recall:

and for )()( sGsY =1

1)()()( == sUttu δ

)}({)( 1 sGLty −=)σ(

1)(+

=s

sG

F 0 l i l t d t 0 th ti l i d d th i l i t bl

tety σ)( −=For σ>0 pole is located at s<0, the exponential expression decays and the impulse response is stable.

Example:num=[2 1];den=[1 3 2];[R,P,K] = residue(num,den);

3112 +s Poles farther to the -----------------numG=[2 1];denG=[1 3 2];

sysG=tf(numG,denG);

23

11

2312)( 2 +

++

−=++

+=ssss

ssGPoles farther to the left in the s-plane are associated with natural signals that decay faster that

figure;impulse(sysG);-----------------ff3=@(x) -exp(-x)+3*exp(-2*x);

tt eety 23)( −− +−=

decays faster

decay faster that those associated with poles closer to the imaginary axis.



Adaptronics

ff3=@(x) exp( x)+3 exp( 2 x);fplot(ff3, [0,10])faster pole

Influence of pole locations – real poles

2

Zeros and poles of the transfer function

0.8

1Pole-Zero Map

Faster pole tt eety 23)( −− +−=2

0

0.2

0.4

0.6

gina

ry A

xis

Faster pole

1

.5

-0.8

-0.6

-0.4

-0.2Imag

0.9

1 exp(-x)exp(-2x)

0

0.5-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-1

Real Axis

Impulse response0.5

0.6

0.7

0.8

0 1 2 3 4 5 60.5

p p

0.1

0.2

0.3

0.4



Adaptronics

0 1 2 3 4 5 6 7 8 9 100

Influence of pole locations – complex conjugate poles

Complex poles can be defined in terms of their real and imaginary parts:Complex poles can be defined in terms of their real and imaginary parts:

(1)

If σ > 0 then a pole has a negative real part

djs ωσ ±−=

If σ > 0, then a pole has a negative real part.Complex poles always come in conjugate pairs.Denominator corresponding to a complex conjugate pair will be:

)]([)]([ jsjsden ωσωσ +

Such poles can be interpreted as poles of a mass-spring-damper-system:

)()()]([)]([

dd

dd

jsjsjsjsden

ωσωσωσωσ

++−+=−−−+−−=

Such poles can be interpreted as poles of a mass spring damper system:

Differential equation of the mass-spring-damper-system

)()()()( tftkxtxbtxm =++ kcan be transformed using direct Laplace transform with zero initial-conditions into:

)()()()( tftkxtxbtxm ++

m

k

f(t)

The transfer function of interest is

)()()()(2 sFskXsbsXsXms =++

)()(

sFsXb x(t)



Adaptronics

)(sF


1)(sX)(

1)()(

2 kbsmssFsX

++=

The poles can be found from: 02 =++ kbsms

)()()( 2 sFsXkbsms =++

mk

mb

mb

mmkbbs −

±−=−±−=

22

2/1 2224

(*)

Now we are going to consider the undamped system (consisting of a mass and spring only)in order to adopt abbreviations usual in control and dynamic.

From a differential equation of the mass spring system we obtain:From a differential equation of the mass-spring-system we obtain:

)()()( tftkxtxm =+)()()(2 sFskXsXms =+

mk f(t)

)(1

)()()()()(

)()()(

22

kmssFsXsFsXkms

sFskXsXms

+==+

+x(t)

The poles of this undamped system can be determined from:kskms −==+ 22 0 02/1 ωj

mkjs =±=



Adaptronics

m m


So we use notation: where represents the natural frequency of the undamped system2ω=k ωSo we use notation: , where represents the natural frequency of the undamped system.

Let us consider the poles of the damped mass-spring-damper system again:We can write from (*):

0ω=m 0ω

We are interested for now only in the complex-conjugate solution for which is the case if:mk

mb

mbs −

±−=

2

22sWe are interested for now only in the complex-conjugate solution for , which is the case if:2/1s

02

2

<−

mk

mb

That means: and the solution for the poles can be written in the following way:02

2

>

−

mb

mk

2/1s

2 bkb

22

−±−=

mb

mkj

mbs

And we want to consider the influence of the complex-conjugate poles of some general transferAnd we want to consider the influence of the complex-conjugate poles of some general transferfunction, in the form (1) given in the beginning

(1)djs ωσ ±−=



Adaptronics

(1)dj


Comparing the last two equations, we can introduce the following notation:

mb

2=σ 2

0ω=k

2

2

2

−= bk

dω −= 220

2 σωωdor (*)m2 m 2 mmd

−=

2

0

20

2 1ωσωωd

already introduced

Introducing damping ratio0ωσξ = 2

0 1 ξωω −=d

js ωσ ±−= with andξωσ =

Finally, we can write a complex-conjugate pair of poles as:

21 ξωω −=

which can be considered as poles of a general transfer function in the form:

djs ωσ ±=2/1 with and0ξωσ = 0 1 ξωω =d

200

2

20

2)(

ωξωω

++=

sssG



Adaptronics


Now we can consider these complex-conjugate poles in s-plane and using the introducedNow we can consider these complex conjugate poles in s plane and using the introduced relations we can make conclusions about the influence of the complex-conjugate poles on the time-response.

ξσθσθ iii

ξω

θω

θ arcsinarcsinsin00

=

=→=

ξθ arcsin= ξθ =sinξ ξ

Damping ratio is determined by the angle θ

Im(s) Im(s) Im(s)

45o30o

17.5o

Re(s) Re(s) Re(s)

7070ξ 50ξ 30ξ



Adaptronics

707.0=ξ 5.0=ξ 3.0=ξ


We can use the table of Laplace transforms to determine the time response of the systemWe can use the table of Laplace transforms to determine the time-response of the system. For that purpose the transfer function can be written in the following form:

20)(sG =

ω

2

220

20 )1()(

)(

*

ssG

C

−++

ξ

ξωξω

22

*22

02

0

20

20

)()1()(

1

1 basbC

sa

++=

−++

−⋅

−=

ξωξωξω

ξ

ω

a

Then using entry No. 20 from the Table of Laplace transforms we obtain:

t )i ()( 20 ξω ξω− tetx t )1sin(1

)( 202

0 0 ξωξ

ω ξω −−

= −

tetx dt ωsin

1

ω)( σ

20 −

−=

ξ)cos()( 0

σ* ϕω +=⇔ − teCtx dt



Adaptronics

Resume:


tetx dt ωsin

1

ω)( σ

20 −

−=

ξ te σ2

0

1−

ξω

0.8

1

1−ξ

0

0.2

0.4

0.6

-1

-0.8

-0.6

-0.4

-0.2

Important relations:

0 2 4 6 8 10 12 14

Poles which are further left fromthe imaginary axis are faster

0ξωσ =determines the decay rate of the envelope that multiplies the

→σthe imaginary axis are faster.

↑= ξθξ sin

the envelope that multiplies thesinusoid. ↑θ

If the real part of the pole is positive, the plant will be instable !

Explanation: for such poles

↑= σξωσ 0



Adaptronics

↑= ξθξ sinthe plant will be instable ! ↑= σξωσ 0


St f dI l f d Step responses of second order systems versus ζ

Impulse responses of second order systems versus ζ



Adaptronics


Resume



Adaptronics

Time domain specifications



Adaptronics

Time domain specifications

Commonly used approximations for the second-order case with no zeros are:

8.1=t0ω

=rt

6.46.4tσωξ 0

=⋅

=st

2ξ1/ξ 1ξ02ξ-1/πξ <≤≈ −eM p

%16)0 5ξ(M

%5)0.7ξ(

%16)0.5ξ(

==

==

p

p

M

M

These approximations should provide only a starting point for the design iteration and the time response should always be checked after the control design is complete.



Adaptronics

Open-loop control systems

Why use feedback control?Why use feedback control?(open-loop vs. closed-loop architecture)

d



Adaptronics

Feedback control systems

d

r = 0Closed-loop TF from d to y input

disturbance rejection



Adaptronics

Connecting LTI systems (series connection)

)()()(11

11 sUGsUsUG ⋅== )()(

)( 111 sUGsUsU

G

)(sY )()()()()(

12121

2 sUGGsUGsYsUsYG ⋅⋅=⋅==

)(Y21)(

)( GGsUsYGs ⋅==



Adaptronics

Connecting LTI systems (parallel connection)

)( )(Y)()(1

1 sUsYG =

)()(2

2 sUsYG =

)()()()()( 2121 sUGGsYsYsY +=+=

)(Y21)(

)( GGsUsYGp +==



Adaptronics

Connecting LTI systems (feedback connection)

Negative feedback Positive feedback

)]()()()[()()()( sYsHsRsGsEsGsY ==

)()(1)(

)()(

fb sGsHsG

sRsYG

+== )()(1

)()()(

fb sGsHsG

sRsYG

−==

)()(1)( sGsHsR +Negative feedback

)()(1)( sGsHsRPositive feedback



Adaptronics

Connecting LTI systems (unity feedback with a compensator)

Based on negative feedback for H=1 (unity feedback)and the transfer function in the forward path D*G we obtain:

)()()( sGsDsYG ==)()(1)(ufb sGsDsR

G+

==



Adaptronics

Connecting LTI systems (resume)



Adaptronics

PID controller as a compensator in the forward path

sKKKsUsD DI

P ++== )()( sKs

KsE

sD DPc ++)(

)(

1

++= sT

sTKsD D

IPc

11)(



Adaptronics

See: T2 Control system tutorial



Adaptronics

Digital (discrete-time) control

Example: Control aim → Vibration suppression

EfBuAxx ++=FfDuCxy ++=

Plant

Computer

Control law

p



Adaptronics

F ti l t ll i l t ti di it l t i ft d

Digital (discrete-time) control For practical controller implementations a digital computer is often used.Controlled plant is in reality a continuous-time system and it can have some scalar inputs and outputs.These input and output vales can be measured at discrete time instants tk.

210kTk ...2,1,0, == kTkt dkMeasured continuous-time plant outputs (signals) can be converted using an AD (analog-to-digital) converter. In this way a series of discrete-time measured values is obtained:

yk = y(tk)Using an appropriate control algorithm the computer generates a series of discrete-time control values uk, which represent the discrete-time (digital) control law. These discrete-time control signalsvalues uk, which represent the discrete time (digital) control law. These discrete time control signals can be converted into analog (or continuous-time) signals using a DA (digital-to-analog) converter.

PlantPlant

C

Computer

Digital control system

Control law



Adaptronics

Digital control system

Discrete-time equivalent of continuous models

In order to design a model-based digital controller, the model of the plant must be appropriately represented in a discrete-time form. In other words, a continuous-time representation of the plant should be replaced by a corresponding equivalent discrete-time representation.

Discrete-time (digital) systemPlant

Sampler Hold

Control law

Computer

Sampler Hold

PlantController

Control law

Plant



Adaptronics

Sample and Hold Systems

– Sampler –The analog signal is continuous in time and it is necessary to convert it into a flow of digital values. Conversion of a continuous-time signal y(t) into a series of discrete-time values y(k) is performed by a sampler of analog to digital converter (ADC)sampler of analog-to-digital converter (ADC).

The signal values at intervals of time Td (the sampling time) i.e. at each time instant t=kTd are measured. Then they can be stored as instant values of the signal y(kTd) and transferred to the controller as an input for the control algorithm.

The sampling frequency ωT (or sampling rate) can be calculated from the sampling time as:

Td – sampling time

The sampling frequency ωT (or sampling rate) can be calculated from the sampling time as:

dT T

π=ω 2

d

A/D t (S l )



Adaptronics

A/D converter (Sampler)

Sampling theorem

During sampling, a continuous-time signal is being converted into a series of discrete-time values at sampling instants. Therefore a discrete-time signal represents antime values at sampling instants. Therefore a discrete time signal represents anincomplete picture of the analog signal. By storing the sampled values it should bepossible to reproduce exactly the original signal from the discrete-time values by aninterpolation formula. However, this faithful reproduction is only possible if the sampling

t i hi h th t i th hi h t f f th i l Thi i th f thrate is higher than twice the highest frequency of the signal. This is the essence of theShannon-Nyquist sampling theorem.

Therefore it is very important to chose the sampling frequency properly. Through thee e o e s e y po a o c ose e sa p g eque cy p ope y oug esampling procedure one obtains an incomplete information about considered analogsignal. Many different continuous functions y(t), can result in the same series ofsampled values y(k) depending on the sampling rate.

If the digital values produced by the ADC are, at some later stage in the system,converted back to analog values by a digital to analog converter or DAC, it is desirablethat the output of the DAC be a faithful representation of the original signal. If the inputp p g g psignal is changing much faster than the sample rate, then this will not be the case.



Adaptronics

Sampling theorem

Let’s consider, for example, two sinusoids with different frequencies (ω1 < ω2), which aresampled with the same sampling frequency (ωT < ω2). The original analog signal with thehigher frequency ω cannot be “seen” behind the sampled data (one can assume that the

Sampling theorem (or the Shannon-Nyquist

higher frequency ω2 cannot be seen behind the sampled data (one can assume that theoriginal analog signal has the lower frequency ω1 and reconstruct it in this way). Thisproblem is called aliasing.

Sampling theorem (or the Shannon-Nyquist sampling theorem) requires the sampling rate which can guarantee, that the aliasing does not occur.

In essence the theorem shows that an analog signal that has been sampled can be exactly reconstructed from the samples if the samplingreconstructed from the samples if the sampling rate exceeds 2ωmax (samples per second), where ωmax is the highest frequency in the original signal.

The sampling theorem is quoted here without the proof.

Sampling theorem: ωT >2 ωmax



Adaptronics

Recommendations for choosing the sampling rate

• From the controller design point of view the sampling frequency should be as high as possible in order for the discrete-time system to have the same properties as the p y p pcontinuous-time system.

• On the other hand, the control and calculation effort should be as low as possible, and for that reason the sampling frequency should be as low as possible provided that thefor that reason the sampling frequency should be as low as possible, provided that the necessary accuracy of the system can be maintained. In this case one could use slower AD and DA converters, so that the implemented computer can control more then one control loop.

• For the choice of the sampling frequency the essential role plays the maximal frequency ωmax which occurs in the control system. According to the sampling theorem, the sampling frequency must exceed 2ωsampling frequency must exceed 2ωmax.

ωT > 2ωmax

• A continuous-time controller could be used in a discrete-time control system without essential changes if the sampling rate is:

ωT ≈ 20ωmax



Adaptronics

Sinusoidal signals with three different sampling frequencies

ωT = 6 ωmax

ωT = 20 ωmax

ωT = 40 ωmax



Adaptronics

Digital-to-analog conversion

The zero-order hold (ZOH) is a mathematicalmodel of the practical signal reconstructiond b ti l di it l t l

Zero-order-hold

The first-order hold (FOH) is a mathematicalmodel of the practical reconstruction of sampledsignals that could be done by a conventional

First-order-hold

done by a conventional digital-to-analogconverter (DAC). That is, it describes the effectof converting a discrete-time signal, a series ofsamples u(0), u(1), u(2), u(3), ... into to acontinuous time signal u(t) by holding each

signals that could be done by a conventionaldigital-to-analog converter (DAC) and an analogcircuit called an integrator. For the FOH, thesignal is reconstructed as a piecewise linearapproximation to the original signal that wascontinuous-time signal u(t) by holding each

sample value for one sample interval

u(t)=u(k)=konst. for kT ≤ t < (k+1)T.

approximation to the original signal that wassampled.



Adaptronics

Reconstruction of a continuous-time signal using ZOH or FOH

Relation between continuous-time and discrete-time systems

In order to analyse relation between continuous and discrete-time (or digital systems), y ( g y ),let us consider first the following block-diagrams of a continuous system (a) and continuous system with controller implemented using a digital computer (b).

(a) Continuoussystem

)()()(

Fig.1

)()(1)()(

)()(

sGsDsGsD

sRsY

+=

b) Continuousplant with

Fig.1 (plant with digital control computer



Adaptronics

Relation between continuous-time and discrete-time systems

Both the continuous and the discrete-time (digital) systems in the previous slide represent ( g ) y p pclosed-loop systems with unity feedback. Continuous controller is represented by its transfer function D(s) (compensator transfer function), while a discrete-time controller is represented by algebraic recursive equations called difference equations (equivalent to diff ti l ti i ti ti d i )differential equations in continuous time domain).

For a continuous-time closed loop system (represented in the next figure) with the plant transfer function G(s) and transfer function in the feedback loop H(s), the transfer function f th l d l t i bt i d i th f ll iof the closed-loop system is obtained in the following way:

)]()()()[()()()( sYsHsRsGsEsGsY −==

)()(1)(

)()(

sGsHsG

sRsY

+=

By the analogy the transfer function of the continuous closed-loop system with unity feedback, Fig. 1(a) in the previous slide, is determined as:

)()(1)()(

)()(

sGsDsGsD

sRsY

+=



Adaptronics

Euler’s method – forward rectangular rule

In a continuous time system the controller (compensator) is represented by its transfer functionIn a continuous-time system the controller (compensator) is represented by its transfer function D(s), which actually represents in Laplace domain original differential equations describing dynamics of the compensator. An equivalent discrete-time compensator is represented by appropriate difference equations.pp p q

A simple way to make a digital computer approximate the real time solution of differential equations is to use Euler’s method. It follows from the definition of a derivative that:

xδli where δx is the change in x over a time interval δt Even if δt is nottxx

t δδlim

0δ →= where δx is the change in x over a time interval δt. Even if δt is not

quite equal to zero, this relationship will be approximately true, and

kxkxkx )()1()( −+≅kk ttT −= +1 sample interval

Tkx )( ≅

kTtk = for a constant sample interval

k is an integer

x(k) is the value of x at tForward rectangular rule

x(k) is the value of x at tk

x(k+1) is the value of x at tk+1Tkxkxkx )1()()( −−≅

x(k-1) is the value of x at tk 1



Adaptronics

Backward rectangular rulex(k 1) is the value of x at tk-1

Difference equations using Euler’s method

Example 1 For a controller D(s) we are going to find difference equations to be programmedExample 1. For a controller D(s) we are going to find difference equations to be programmed into control computer as in Fig.1(b)

Let the transfer function of the controller be:bsasK

sEsUsD

++== 0)(

)()(bssE +)(

First we find the differential equation that corresponds to D(s).

)()()()( 0 sEasKsUbs +=+ )()()()( 0

This equation in s-domain corresponds to the following differential equation (recall the properties of the Laplace transform):

)(0 aeeKbuu +=+ We are going to use Euler’s method to get the approximating difference equation:

+−+=+−+ )()()1()()()1(

0 kaeT

kekeKkbuT

kuku

Aft i bt i )1()()1()()1()1( ++++ kKkTKkbTkAfter rearranging we obtain: )1()()1()()1()1( 00 ++−+−=+ keKkeaTKkubTku

The difference equation shows how to compute the new value of the control u(k+1), given the past value of the control u(k) and the new and past values of the error signal e(k+1) and e(k).



Adaptronics

past value of the control u(k) and the new and past values of the error signal e(k+1) and e(k).

Lead compensation using a digital computer

Example 2 Lead compensation D(s) for the plant G(s) should be implemented as:Example 2. Lead compensation D(s) for the plant G(s) should be implemented as:

a) continuous-time controller;b) digital controller with sampling rate 20Hz,c) digital controller with sampling rate 40Hz )1(

1)(;10270)(

+=

++=

sssG

sssD

Step responses of the three control systems should be compared.

c) digital controller with sampling rate 40Hz

)(Y

)(

)()()()( sGsD

sEsY =

For the continuous-time system the step response can be obtained using the following summarized algorithm (for more details see m-file Fig0302.m in 05_Examples)

numD=70*[1 2], denD=[1 10]numG=1, denG=[1 1 0]sys1=tf(numD denD)*tf(numG denG)sys1=tf(numD,denD) tf(numG,denG)sysCL=feedback(sys1, 1)step(sysCL)



Adaptronics

Lead compensation using a digital computer

For digital system with the sampling rate 20 Hz T = 1/20 sec = 0 05 sec and theFor digital system with the sampling rate 20 Hz, T = 1/20 sec = 0.05 sec, and the difference equation of the lead compensator is:

)](9.0)1([70)(5.0)1( kekekuku −++=+

1.5Fig. 3.2 Continuous and digital response using Eulers methodFor digital system with the sampling

rate 40 Hz, T = 1/40 sec = 0.025 sec,

0.5

1

outp

ut y

analog control

(a) 20 Hz

and the difference equation of the lead compensator is:

)(750)1( kuku +=+0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0*-----*-----* digital control------------- analog control

1 5

)](95.0)1([70

)(75.0)1(

keke

kuku

−+

+=+

1

1.5tp

ut y

( ) 40 H

Comparison of the step responses is given in the figure.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5out

*-----*-----* digital control------------- analog control

(a) 40 HzFor more details on obtaining the diagrams see m-file Fig0302.m (05_Examples)



Adaptronics

time (sec)

Discrete-time PID controller

PID controller is implemented as a compensator with proportional integral and derivativePID controller is implemented as a compensator with proportional integral and derivative action (recall the continuous PID controller). If we want to implement a PID controller on a digital computer, an appropriate difference equation of the controller should be derived.

Proportional term

Integral term

)()( tKetu = )()( kKeku =

=T

deKtu )()( ηη (apply Euler’s integration rules; inverse to derivation)Integral term =I

deT

tu0

)()( ηη

KBackward rectangular rule

KForward rectangular rule

(apply Euler s integration rules; inverse to derivation)

)()1()( kTeTKkukuI

+−= )1()1()( −+−= kTeTKkukuI

Derivative term

)()( teKTtu D = )]1()([)( −−= kekeT

KTku Dapplied Euler’s approximaion

Backward rectangular rule



Adaptronics

T


Combined transfer function of a PID controller:

sKs

KKsEsUsD D

I ++==)()()(

ssE )(

++= sT

sTKsD D

I

11)( sTI

Differential equation relating u(t) and e(t) is:

1 )1( eTeT

eKu DI

++=

Backward rectangular rule:

( )∗

Tkekeke )1()()( −−≅

Backward rectangular rule:



Adaptronics

T


( )

−−−−−−=−−=≅

•

Tkeke

Tkeke

TTkekekeke )2()1()1()(1)1()()()(

Apply Euler’s backward rectangular rule once more to the previous expression:

TTTT

[ ])2()1(2)(1)( 2 −+−−≅ kekekeT

ke ( )∗∗2T

)1( eTeT

eKu DI

++= ( )∗From:

applying Euler’s method twice like in we obtain the difference equation of a discrete-time PID controller:

( )∗∗

[ ] [ ] [ ]

−+−−++−−=−− )2()1(2)()(1)1()(1)1()(12 kekeke

TTke

Tkeke

TKkuku

TD

I

−+−

+−

+++−= )2()1(21)(1)1()( ke

TTke

TTke

TT

TTKkuku DDD

I

Finally:



Adaptronics

TTTTI

Relation between continuous and discrete models – z-transform

We have seen that difference equations can be evaluated directly by a digital computer and that they can represent models of physical processes and approximations to integration.If difference equations are linear with constant coefficients, we can describe the relation between u and e (compensator output and input) by a transfer function and thereby gainbetween u and e (compensator output and input) by a transfer function, and thereby gaina great aid to analysis and also to the design of linear, constant discrete controls.

z-TransformI d t i t d th t f th di t ti t f f ti fi t i tIn order to introduce the term of the discrete-time transfer function, first we are going to introduce the term of the z-transform.

Let’s consider a signal y(t). If the signal has discrete values y0, y1, y2, . . . , yk, …thenthe z-transform of the signal is defined in the following way.

The z-transform of a series of discrete-time values y[k] is defined as the function

k

k

defzkykyZzY −

∞

===

0][]}[{)(

where k represents notation for kT (T denotes the sampling period ).

If we recall the definition of the Laplace transform (s-domain) of a continuous function y(t):

∞

−== )()}({)( dtetytyLsY stdef

we can observe an analogy between the two representations



Adaptronics

0

the two representations.



Adaptronics

Relation between continuous and discrete models – discrete transfer function

The z-transform has the same role in discrete systems that the Laplace transform has in analysis of y p ycontinuous systems. The use of the z-transform permits rapid solution of linear difference equations with constant coefficients, in a similar way as the Laplace transform enables the solution of differential equations by solving algebraic equations.

)(]}[{ zYznkyZ n−=− (∗)

Here we are going to use the following important property of the z-transform:

Discrete-time transfer function H(z) is defined as the ratio of the z-transform of the output to the z-transform of the input.For a general difference equation depending only on past inputs and outputs:o a ge e a d e e ce equa o depe d g o y o pas pu s a d ou pu s

the transfer function as previously defined will be:

mkmkknknkkk ebebebuauauau −−−−− ++++−−−−= ...... 1102211 (*)

nn

mm

zazazazbzbb

zEzUzH −−−

−−

+++++++

==...1

...)()()( 2

21

1

110 (**)

And if n≥m, we can write the transfer function as a ratio of polynomials in z as:

)()(...

)( 21

110

zazbzbzbzb

zH nnn

mnm

nn=

+++++++

= −−

−−(***)



Adaptronics

)(...22

11 zaazazaz n

nnn ++++

Discrete transfer function – Matlab representation

Discrete-time transfer function is represented in Matlab in the tf form similarly to the continuous case.

][num 210 mbbbb =

]1[den 21 naaa =

sys=tf (num, den, T)

Where T represents the sampling period.

By default, in tf the transfer function is assumed with positive powers of z (***).

By setting the variable properties, the ascending powers of z-1 can also be considered (see the Matlab help for tf).



Adaptronics

Physical meaning of the variable z

Recall the equations (*) and (**).

mkmkknknkkk ebebebuauauau −−−−− ++++−−−−= ...... 1102211 (*)

nn

mm

zazazazbzbb

zEzUzH −−−

−−

+++++++

==...1

...)()()( 2

21

1

110 (**)

Suppose all the coefficients in the equation (**) are equal to zero except b1 and b1 =1.

)()( 1 zEzzU −=

1−= kk euOn the other hand, it means that the equation (**) reduces to:

)()(

1kk

The present value of the output uk equals the input delayed by one period. Thus we see that a

transfer function of z−1 is a delay on one time unit.

1−= kk eu1+= kk ue

)()( 1 zEzzU −=)(zE1−z



Adaptronics

)()( zEzzU)(

A discrete-time equivalent of the continuous-time state-space model

)()()( ttt FfDuCxy ++=),()()( ttt EfBuAxx ++=Discrete-time equivalent:

Plant

][][][][][][][]1[

kkkkkkkk

FfDCεfΓuΦxx

++++=+

][][][][ kkkk FfDuCxy ++=

Corresponding matrices of the discrete-time model are calculated in the following way (for derivation [V 1995])

τ=τ== ττdd

d

TTT dedee

00,, EεBΓΦ AAA

see [Vaccaro, 1995]):

3322

T li ti kT

where ...!32

3322++++= ttte

deft AAAIA represents matrix exponential.



Adaptronics

Td – sampling time k represents notation for → kTd

Matlab function C2D for conversion of continuous-time models to discrete-time ones

SYSD = c2d(SYSC,Ts,METHOD) converts the continuous-time LTI model SYSC to a discrete-time model SYSD with sample time Ts. The string METHOD selects the discretization method among the following:

'zoh' Zero-order hold on the inputs'foh' Linear interpolation of inputs (triangle appx.)'tustin' Bilinear (Tustin) approximation'prewarp' Tustin approximation with frequency prewarping.

The critical frequency Wc (in rad/sec) is specifiedas fourth input by

SYSD = C2D(SYSC,Ts,'prewarp',Wc)(f S SO )'matched' Matched pole-zero method (for SISO systems only).

The default is 'zoh' when METHOD is omitted.

For state-space models,[SYSD G] C2D(SYSC T METHOD)[SYSD,G] = C2D(SYSC,Ts,METHOD)

also returns a matrix G that maps continuous initial conditionsinto discrete initial conditions. Specifically, if x0,u0 areinitial states and inputs for SYSC, then equivalent initial

diti f SYSD i bconditions for SYSD are given byxd[0] = G * [x0;u0], ud[0] = u0 .

See also D2C, D2D, LTIMODELS.



Adaptronics

C2DM Conversion of continuous LTI systems to discrete-time.

[Ad,Bd,Cd,Dd] = c2dm(A,B,C,D,Ts,'method')

converts the continuous-time state-space system (A,B,C,D) to discrete time i ' th d'using 'method':

'zoh' Convert to discrete time assuming a zero orderhold on the inputs.

'f h' C t t di t ti i fi t d'foh' Convert to discrete time assuming a first order hold on the inputs.

'tustin' Convert to discrete time using the bilinear (Tustin) approximation to the derivative.

' ' C t t di t ti i th bili'prewarp' Convert to discrete time using the bilinear (Tustin) approximation with frequency prewarping.Specify the critical frequency with an additionalargument, i.e. C2DM(A,B,C,D,Ts,'prewarp',Wc)

' t h d' C t th SISO t t di t ti i th'matched' Convert the SISO system to discrete time using thematched pole-zero method.

[NUMd,DENd] = c2dm(NUM,DEN,Ts,'method')

converts the continuous-time polynomial transfer function G(s) = NUM(s)/DEN(s) to discrete

time, G(z) = NUMd(z)/DENd(z), using 'method'.



Adaptronics

Example c2d (05_Examples)

010

b340120201

A

=

−

−=

[ ] 0d001c

0340

==

Ts=0.001;

sysc=ss(A,b,c,d);

sysd=c2d(sysc, Ts, 'zoh');

[phi,gamma,c,d]=ssdata(sysd)

phi =

0.9990 0.0000 0.0020

gamma =

1.0e-003 *

0.00000 0.9980 0.0010

0 0.0040 1.00300.9990

0.0020



Adaptronics

Relation between the poles in s-domain and in z-domain

For a pole s0 in s-domain, a corresponding pole in z-domain is:

dTsez 0=0A)-det(sI eig(A) =⇔

ez0 =

>> eig(A)

ans = 0.9990001.011

1 === ⋅−eez dTs

-1.00000000000000

-2.70156211871642

3.70156211871642

0.99901 eez0.9973001.070156.2

22 === ⋅−eez dTs

1.0037001.070156.33

2 === ⋅eez dTs

phi =

0 9990 0 0000 0 0020

3

>> eig(phi)

ans =0.9990 0.0000 0.0020

0 0.9980 0.0010

0 0.0040 1.0030

0.9990

0.9973

1.0037



Adaptronics

The ZOH pole-mapping formula

When the plant is replaced by its ZOH equivalent, the design model is a discrete-timesystem. The performance specifications for the design model are thus discrete-timespecifications. However, the actual control system operates in continuous time, and it isnatural to use continuous-time performance specifications (for example desired closed-looppoles). A convenient way to achieve performance specifications is to specify the polelocations that the closed-loop system should have. Because the design model is a discrete-time system however the desired closed-loop s-plane poles must be mapped into antime system, however, the desired closed loop s plane poles must be mapped into anequivalent set of desired z-plane poles. The poles of the ZOH equivalent are related to thepoles of the continuous-time plant by the previous relation:

Ts dTsez 00 =

Since the state matrix Φ of the discrete-time equivalent is related to the state matrix A of the

dTeAΦ =continuous-time plant by:

the eigenvalues of Φ and A are related by the same exponential function. This statement isgiven here without the proof, which can be established based on the Cayley-Hamiltontheorem and on the fact that for two matrices A and B, where B=f(A), if λi is an eigenvalueof A, then f(λi) is an eigenvalue of B. In addition see the analogy between the s and ztransforms (relation between continuous time and discrete time models )



Adaptronics

transforms (relation between continuous-time and discrete-time models ).

Mapping of the s-plane poles into the z-plane and stability region in z-plane



Adaptronics

Pole placement for discrete-time systems

Pole placement for discrete-time systems can be performed in a similar way as the pole placement for continuous-time plants.p p

In order to introduce the discrete-time transfer function (SISO case) or transfer matrix (MIMO case) of a discrete-time plant and closed-loop system let us recall de definition of the z transform of a discrete time series of values y[k] which corresponds to a discretizedthe z-transform of a discrete-time series of values y[k], which corresponds to a discretized (sampled) continuous-time function y(t):

kdef

kkZY −∞

][]}[{)( k

kzkykyZzY

===

0][]}[{)(

An important property of the z-transform is:

)(]}[{ zYznkyZ n−=−

If a discrete-time equivalent of a continuous-time plant is described by the following state-

(∗)

space representation:

][][][][][]1[

kkkkkk

DuCxyBuAxx

+=+=+



Adaptronics

][][][ kkk DuCxy +

then taking into account property (∗), the equations of the discrete-time plant can be written in the following way.g y

)()()(),()()( zzzzzzz uDxCYUBXAX +=+=

Y 1)()()( zDiscrete-time transfer function of the plant is then: DBAIC

UYG +−== −1)(

)()()( z

zzz

Recall the transfer function derivation for a continuous-time plant!

The control task of the pole placement in discrete-time is to find for a discrete-time plant:

][][]1[ kkk BuAxx +=+

][][][ kkk DuCxy +=

such a feedback control law: that the closed-loop system][][ kk Lxu −=such a feedback control-law: that the closed loop system][][ kk Lxu =

][)(]1[ kk xBLAx −=+

][][][ kkk DuCxy +=

has desired discrete-time closed-loop poles.



Adaptronics

Observer

PlantA prerequisite for the feedback gain controller design is thatall state variables in the state-space model must bemeasurable. However, in many practical examples this is notthe case The role of the observer in a control system is to

Observer

the case. The role of the observer in a control system is toestimate the unmeasurable elements of the state vector,based on the measured input and output values.

0)0()()()( xxBuAxx =+= ttt )()( tt Cxy =

We consider the plant model in the following form,with the initial state vector x0

0)0(),()()( xxBuAxx + ttt )()( tt Cxy

For the observer consideration the plant model is augmented with additional input value uBwhich guaranties the convergence of the observer:

0ˆ)0(ˆ),()()(ˆ)(ˆxxuBuxAx =++= ttt

dttd

B )(ˆ)(ˆ tt xCy =

Correction term u is fed back as a difference between the measured plant output and theCorrection term uB is fed back as a difference between the measured plant output and the output of the plant model, multiplied by the feedback gain matrix LB

))(ˆ)(()( ttt BB yyLu −=



Adaptronics

BB

The observer equation is then: ))(ˆ)(()()(ˆ)(ˆtttt

dttd

B xxCLBuxAx −++=dt

and the model operates in the same manner as the real plant. If a difference between the As long as the last term in the previous equation will be equal to zero and),(ˆ)( tt xx =

state vector and its estimate appears, the behaviour of the model will be influenced by the feedback. In that case the matrix LB must be chosen in such a way that the difference

)(ˆ)( tt xx − tends to zero in time.

B

Observer



Adaptronics

Observer error:

d )ˆ()()ˆ(ˆ)ˆ( xxCLAxxCLBuxABuAxxxe −−=−−−−+=−= BBdtd

If all eigenvalues of the matrix (A L C) have negative real parts the observer error willIf all eigenvalues of the matrix (A-LBC) have negative real parts, the observer error will tend to zero:

0lim =∞→

et ∞→t

Recommendations for choice of the observer poles

The observer error should tend to zero faster then the transient response of the plant In orderThe observer error should tend to zero faster then the transient response of the plant. In order to fulfil this condition, the eigenvalues of the matrix A–LBC (observer poles) should be places as far as possible from the imaginary axis in the left-hand s half plane. On the other hand, the more distant from the imaginary axis these poles are, the higher will be the controller gain. Therefore, the observer matrix LB should be designed in such a way that the eigenvalues ofA–LBC are placed left from the dominant poles of the matrix A (dominant poles are a pair of complex-conjugate poles, which are closest to the imaginary axis).

If an observer is used within a closed-loop feedback system with the feedback gain matrix L, then the observer poles should be chosen with respect to the closed loop poles (eigenvalues of A–BL), and not with respect only to the plant poles (eigenvalues of A).



Adaptronics

Plant

Observer



Adaptronics

Kalman Filter as an observer for a discrete-time system

][][][]1[

kkk

kkkk w++=+ εuΓxΦx

Discrete-time state space model

][][][ kkk v+= xCy

zero meanw[k] process noise

0== ]}[{]}[{ kk vw EE

C i f th

w[k] process noisev[k] measurement noise

withsequencesrandom

.,0)}()({)}()({ TT jijiji ≠== vvww EEno time correlation

Covariances of the processand measurement noise: .}][][{}][][{ TT

vw vvww RR == kkkk E,E

][][ˆ]1[])[][]([][][ˆ kkkkkkkk ΓuxΦxxCyLxx +++Kaman Filter:

1T][][ −= vRCPL kkest

][][]1[]),[][]([][][ kkkkkkkk est ΓuxΦxxCyLxx +=+−+=Kaman Filter:

][)][(][][][ 1TT kkkkk CMRCCMCMMP −+−= v

.][]1[ TT εεRΦΦPM w+=+ kk



Adaptronics

Matlab command KALMAN

kalman Continuous- or discrete-time Kalman estimator.

[KEST,L,P] = KALMAN(SYS,QN,RN,NN) designs a Kalman estimator KEST

for the continuous- or discrete-time plant with state-space modelfor the continuous or discrete time plant with state space model

SYS. For a continuous-time model

.

x = Ax + Bu + Gw {State equation}

y = Cx + Du + Hw + v {Measurements}

with known inputs u, process noise w, measurement noise v, andwith known inputs u, process noise w, measurement noise v, and

noise covariances

E{ww'} = QN, E{vv'} = RN, E{wv'} = NN,

the estimator KEST has input [u;y] and generates the optimal

estimates y_e,x_e of y,x by:

..x_e = Ax_e + Bu + L(y - Cx_e - Du)

|y_e| = | C | x_e + | D | u

|x e| | I | | 0 |



Adaptronics

|x_e| | I | | 0 |



Adaptronics

Closed-loop systems without and with observer



Adaptronics

Example – clamped beam

4 x actuators235 passive Semiloof shell elements80 active Semiloof shell elements

10

25

50

2050

30010

Sensor

F(t)F(t)

30

Material:Plate: E = 2.00⋅105 N/mm2 Actuator/sensor: E11 = E22=3.77⋅104 N/mm2 d31 = 2.1⋅10−7 mm/V

ν = 0.3 G12 = 1.3⋅104 N/mm2 κ33 = 3.36⋅10−9 F/m

ρ = 7.86⋅10−9 Ns2/mm4 ν = 0.38 t = 0.4 mm (thickness)

t = 2.0 mm (thickness) ρ = 7.85⋅10−9 Ns2/mm4



Adaptronics

Active vibration suppression of a clamped beam

EfBuAxx ++=FfDuCxy ++=

4 actuators

PlPlant

Controller

Experimental setup with the beam and dSPACE System

Controller



Adaptronics

State-space model of the beam obtained through FEM modelling and modal reduction

00 00000010000000000.04658171-000000000.000000100000

⋅=

000.00000232-0000.0465817-00000.0000016-0000.00117118-

0.00000010000000000.00000010000000

107A

0.000001220001.45035088-00000.00000080-0000.36249919-00

00000000

0000000000000000

,

3.232977813.232977781.11557656-1.11557656-0.181388560.18138856

0010,

1.21732161-1.217321590.342195420.34219542-0.13713482-0.137134820.137539310.13753931-0.003661870.00366187-0.006083670.00608366-

000010 53

⋅=

⋅= EB

6.253134646.253045393.232977813.23297778

0.10482390-0.104824081.088613611.08861338-1.217321611.217321590.342195420.34219542

[ ] [ ] [ ].00,0000,000004332433.00893192.023953924.054877435.1 ==−= FDC



Adaptronics

Controllability and observability criteria

Plant/system states given by the state equation BuAxx +=are controllable if it is possible - by admissible inputs - to steer the states from any initial value to any final value within some time window.

A continuous time-invariant state-space model is controllable if and only if the rank of the controllability matrix Sco equals n.

Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs

,][ 1co BAABBS −= n nrank =coS

knowledge of its external outputs.

For a plant/system given by the state space model ,BuAxx += Cxy =)(tthe observability matrix Sob is defined as follows:

2CACAC

SA continuous time-invariant state-space

d l i b bl if d l if

=

−1

2

n

ob

CA

CAS nrank ob =S

model is observable if and only if



Adaptronics

CA

Eigenfroms of the beam

Eigenform 1 Eigenform 4

f1 = 17 224 Hzf1 17.224 Hz f4 = 303.022 Hz

Eigenform 2 Eigenform 6

f2 = 108.625 Hz f6 = 606.117 Hz



Adaptronics



Adaptronics

Simulation with Matlab/Simulink (optimal LQ controller)

Sensorsignalsee Example-beam

Aktorsignale



Adaptronics

Modeling and identification of the smart acoustic box

Piezoelectricactuators (inside)

Geometryl1 = 600 mm, l2 = 400 mm, l3 = 600 mm,Plate thickness h = 2 mmPlate thickness h 2 mmSize of PZT patches: 100 mm × 50 mm,Patch thickness 0,2 mm

Wooden box sides (outside) layeredwith steel plates (inside)

Material date of the aluminium plateE = 70000 N/mm², ν = 0,3,ρ = 2,63 × 10-9 Ns²/mm4

with steel plates (inside)Material date of the PZT patchesC11 = C22 = 107601 N/mm², C12 = 63129 N/mm²,C66 = 22236 N/mm²,

9 4ρ = 7,76 × 10-9 Ns²/mm4,e31 = -9,52 × 10-6 N/(mV)mm,κ33 = 1,87 × 10-14 N/(mV)²

Aluminium plate

Material date of the fluidc = 340000 mm/s, ρ0 = 1,29 × 10-12 Ns²/mm4



Adaptronics

Aluminium plate


Outside Inside

Experimental set-up (inner and outer view of the plate)

2

3

5

6

8

9

11

12

14

15

1471013

Acceleration transducer

Laser scanning vibrometer



Adaptronics


Scheme of the experimental rig for modal analysis



Adaptronics


Modelling of the box → FEM approach with the FE formulation of the acoustic field

FE-Simulation:

Experiment

1. Mode 2. Mode 3. Mode 4. Mode

structural acoustic

66,7 Hz 68,0 Hz

106,2 Hz 200,8 Hz

163,8 Hz 204,0 Hz

172,1 Hz 278,1 Hz

201,2 Hz 291,5 Hz

Fist 5 eigenfrequencies



Adaptronics


Scheme of the experimental rig for subspace identification



Adaptronics


Excitation by shaker (valid for all three sensor constellations)



Adaptronics

Excitation by shaker (valid for all three sensor constellations)


Excitation by shaker (constellation 2) Excitation by actuator-patch (constell. 3)



Adaptronics


MIMO case – constellation 1

Excitation by shaker and actuator-patches



Adaptronics

Excitation by shaker and actuator patches


Comparison of the frequency responses (constellation 1)

FRFs: sensor 1 – shaker, determined experimentally, numerically andfrom the identified model

FRFs: microphone – shaker, determined experimentally, numerically andfrom the identified model



Adaptronics

Noise control of the smart acoustic box

0 4

0.6

0.8

1x 10-3 Uncontrolled and controlled output (LQ control)

-0 4

-0.2

0

0.2

0.4

y [V

] yp uncontrolled ym

0 5 10 15 20 25 30-1

-0.8

-0.6

-0.4

Time [s]Time [s]

0.6

0.8

1x 10

-3 Uncontrolled and controlled output

t ll d

0.1

0.15

Control input up

Reference model:-0.2

0

0.2

0.4

y [V

]

uncontrolled

controlled MRAC 0

0.05

u p [V]

Qm=0.001⋅I12×12, Rm=1

Tc=0.01, σc=0.1f=0.01⋅sin(2πfw2)

0 5 10 15 20 25 30-1

-0.8

-0.6

-0.4ym (controlled LQ)

0 5 10 15 20 25 30

-0.1

-0.05



Adaptronics

0 5 10 15 20 25 30Time [s]

0 5 10 15 20 25 30Time [s]

Noise control of the smart acoustic box

=

3

1)sin(

iwifExcitation Random excitation

=1i

Optimal LQ tracking system with additional dynamics: uncontrolled and controlled microphone output signal



Adaptronics

Modeling and control of an MRI tomograph

MRI tomograph Funnel shaped inlet of the tomographg p Funnel shaped inlet of the tomograph



Adaptronics

61

1R1L

Actuatorgroup

2R2L

Sensor

35

3R3L



Adaptronics

dSPACE Low-pass filterPower

DS2102

±5 ±10V, VDS2001ADC Board±5V Actuator

patch-group

Kemo VBF21M PowerAmplifier

35 61

DAC Board

Funnel

Sensor-patchAmplifier / Low-pass filter



Adaptronics

help n4sid

N4SID Estimates a state-space model using a sub-space method.

MODEL = N4SID(DATA) or MODEL = N4SID(DATA,ORDER)

MODEL: Returned as the estimated state-space model in the IDSS format.

DATA: The output input data as an IDDATA object. See HELP IDDATA.

ORDER: The order of the model (Dimension of state vector). If entered

as a vector (e.g. 3:10) information about all these orders will be

given in a plot. (Note that input delays (NK, see below) largergiven in a plot. (Note that input delays (NK, see below) larger

than 1 will be appended as extra states, giving a resulting model

of higher order.) If ORDER is entered as 'best', the default order among

1 10 i h Thi i th d f lt h i1:10 is chosen. This is the default choice.

ORDER can also be an IDSS model object, in which case all model structure

and algorithm properties are taken from this object.



Adaptronics

help iddata

IDDATA Create DATA OBJECT to be used for Identification routines

Very Basic Use:

DAT = IDDATA(Y,U,Ts) to create a data object with output Y andDAT IDDATA(Y,U,Ts) to create a data object with output Y and

input U and sampling interval Ts. Default Ts=1. If U=[ ],

or not assigned, DAT defines a signal or a time series.

With Y [] DAT d ib j t th i tWith Y =[], DAT describes just the input.

Y is a N-by-Ny matrix with N being the number of data and Ny

the number of output channels, and similarly

for U. Y and U must have the same number of rows.

Retrieve data by DAT.y, DAT.u and DAT.Ts

Select portions by DAT1 = DAT(1:300) etc.p y ( )



Adaptronics



Adaptronics

gama =

1.0e-003 *ee =

0.09050905971996

0.04599678005752

-0.04187063275436

-0.02735109753846

-0.02362063754278-0.10894746312406

-0.30926536398667

-0.22738491669319

0 02679593207987

-0.01054969106658

-0.01067178928435

0.02379752891331D = 0

0.02679593207987

0.41572192517877

0.02137413211198

0 11096458201883

-0.03547190888155

0.00163335782438

-0.011784803790450.11096458201883

0.25503073013999

C =

-0.03071487667334

C

Columns 1 through 4

-1.41880329184396 -2.06703717591548 1.24926599998795 -2.76552060996599

Columns 5 through 8 g

0.29513547587073 0.57354385554845 -1.11512221075853 1.23919546581217

Columns 9 through 10

-0.67902838288362 1.45824258878514



Adaptronics

Subspace based identification of the modelSingle-Input Single-Output case

Actuator/sensor pair A2R S1R Actuator/sensor pair A2R S2LActuator/sensor pair A2R – S1R Actuator/sensor pair A2R – S2L

model order n=60

Frequency responses from the state-space models obtained using the subspace identification

model order n 60



Adaptronics

Multiple-Input Multiple-Output case(model order n=50)One actuator A2R – two sensors S1R, S2L

Two actuators A1R A2R – two sensors S1R S2LTwo actuators A1R, A2R – two sensors S1R, S2L



Adaptronics

Vibration suppression of the funnel – optimal LQ control (SISO)

Response of controlled system: actuator/sensor pair A2R–S1RSinusoidal excitation f1=9.573 Hz



Adaptronics

1

Vibration suppression of the funnel – optimal LQ control (MIMO)

Excitation: → sin(2π f1t)Excitation:sin(2π f1t)+ sin(2π f2 t)+ sin(2π f3 t)

f1=9.573 Hzf2=23.333 Hzf3=31.439 Hzf3

MIMO control system: actuators A2R and A2L sensors S1R und S1L



Adaptronics

MIMO control system: actuators A2R and A2L, sensors S1R und S1L

MIMO control: A2R, S1R, S2L

Excitation: frequency f1 Excitation: frequencies f1, f2, f3



Adaptronics

Vibration suppression of the car roof

Car roof with Sources:

Control aim: vibration suppression

FE modelModal analysis

piezoelectric patches

Sources:LP Adaptronik, DLR

CAD model

Car with adaptivepiezoelectric roof Control

Signal processing

piezoelectric roof



Adaptronics


Control aim: vibration suppression

FE modelModal analysis

CAD model

Car with adaptivepiezoelectric roofpiezoelectric roof



Adaptronics


Experimental set-up

Test car: passenger compartment and inner surface of the car roofwith attached piezoelectric actuators/sensors and exciting shakers



Adaptronics


FEM modelling and modal analysis

b)

a)

a) CAD model and the FEM mesh of the car roof;b) Selected eigenforms of the car roof obtained on the basis of the FEM model

Selected eigenfrequencies:

f =48 45Hz f =51 12Hz f =63 23Hz



Adaptronics

f1 48.45Hz f2 51.12Hz f3 63.23Hz




Adaptronics

Time responses of the controlled and uncontrolled vibrations

adaptronics

Documents

active control of vibration

active vibration control

application of control

active noise control

stabile smart structures

vibration suppression

aerospace industry

terms english terminology