auto e7 lecture2

8/12/2019 AUTO E7 Lecture2

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Control ofdynamical systems

O.Sename

Introduction

Methods for systemmodelling

Physical examples

Hydraulic tanks

Satellite attitude control

model

The DVD player

The suspension system

The wind tunnel

Energy and comfort

management in

intelligent building

State spacerepresentation

Physical examples

Linearisation

Conversion to transfer

function

Methods for analysis and control of dynamical systems

Lecture 2: Modelling of dynamical systems

O. Sename1

1Gipsa-lab, CNRS-INPG, FRANCE

[email protected]

www.gipsa-lab.fr/o.sename

2nd February 2014
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Outline

Introduction

Methods for system modelling

Physical examples

Hydraulic tanks

Satellite attitude control modelThe DVD player


The wind tunnel

Energy and comfort management in intelligent building

State space representationPhysical examples

Linearisation

Conversion to transfer function
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

References

Some interesting books:

G. Franklin, J. Powell, A. Emami-Naeini, Feedback Control of

Dynamic Systems, Prentice Hall, 2005

R.C. Dorf and R.H. Bishop,Modern Control Systems, Prentice

Hall, USA, 2005. G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System

Design, Prentice Hall, New Jersey, 2001.

K.J. Astrom and B. Wittenmark,Computer-Controlled Systems,

Information and systems sciences series. Prentice Hall, New

Jersey, 3rd edition, 1997.
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Types of models

Finite state models (Petri nets, Grafcet) of logical systems:

discrete-event systems

Graph models : Bond graph. Allow a physical description in a

unique way whatever the physical domain is.

Experimental models : allow to reproduce an input-output behavior. State models: A mathematical description of the system in terms

of a minimum set of variablesxi(t),i=1, . . . , n, together withknowledge of those variables at an initial timet0 and the system

inputs for timet t0, are sufficient to predict the future systemstate and outputs for all timet

t0

.
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Different issues for modelling (1)

Identification based method- Black box models

System excitations using step inputs, sinusodal signals, or PRBS

(Pseudo Random Binary Signal)

Determination of a transfer function reproducing the input/ouputsystem behavior

Method : direct identification (Strejc) or by optimization.

Objective: determination of the set of model parameters.
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Different issues for modelling (2)

Knowledge-based method - White box models

Represent the system behavior using differential and/or algebraic

equations, based on physical knowledge.

Formulate a nonlinear state-space model, i.e. a matrix differentialequation of order 1.

Determine the steady-state operating point about which to

linearize.

Introduce deviation variables and linearize the model.
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Why knowledge-based method are of interest?

dynamical systems where physical equations can be derived :

electrical engineering, mechanical engineering, aerospace

engineering, microsystems, process plants ....

includephysical parameters: easy to use when parameters are

changed for design

State variables have physical meaning.

Allow for including non linearities (state constraints )

Easy to extend toMulti-Input Multi-Output(MIMO) systems

Advanced control design methodsare based on state space

equations (reliable numerical optimisation tools)
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8/34


O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Height control of a single Tank

Consider a water tank of area S, heightH, feeded by an input flow Qe,

with an output flowQs

Figure:Bac.

Usually the flow is considered to be proportional to the square root of

the pressure difference, then

Qs=kt

H

and

SdH

dt =QeQs=Qekt

H
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

A satellite attitude control model

A simple model for a one-axis system is :

I=MD+ Fcd

where is the inertia,the angular position,MDa small disturbancemoment on the satellite,Fcthe control force that comes from the

reaction jets,dthe distance from the jet to the center of gravity.
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

the DVD player
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Modelling the DVD player

Useo of physical principles.Focus and radial actuators: are constituted by a lens attached to the

pick-up body by two parallel leaf spring, and moved in vertical and

radial direction by a voice coil and a magnet.
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Modelling the DVD player (2)

Actuator : voltagev(t). Controls the pick-up voice coil

Output signal:laser spot positionx(t)

Electrical part: the voltagev(t)applied to the R-L circuit makes flow init a currenti(t):

Li(t)

t + Ri(t) =v(t)Kex(t)

t (1)

Magnetic part:

f(t) =Kei(t) (2)

whereKe is the back-emf constant,

Mechanical part: The forcef(t) [N]acts on the objective lens mass M[Kg], making the actuator moves:

M2x(t)

t2 + D

x(t)

t + kx(t) =f(t) (3)

C l f
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Data of the DVD player 1

Table: Values of the physical parameters of Pick-up 1 (for focus and trackingactuators), from Pioneer.

Name Description Value Focus Value TrackingR DC resistance of coil 5.41.1 5.91.2L Inductance of coil 156H 96HM Moving mass 0.7g 0.7g

SDC DC Sensitivity 2.69mm/V 0.63mm/Vf0 Resonance frequency 307Hz 477Hz

QdB Resonance peak 15dB 15dB

C t l f
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Data of the DVD player 2

Table: Values of the physical parameters of Pick-up 2 (for focus and trackingactuators), from Sanyo

Name Description Value Focus Value TrackingR DC resistance of coil 6.51 6.51L Inductance of coil 256H 186HM Moving mass 0.33g 0.33g

SDC DC Sensitivity 0.94mm/V 0.27mm/Vf0 Resonance frequency 527Hz 527Hz

QdB Resonance peak 20dB 20dB

Control ofC l l i f h


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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Calculation of the parameters

The elastic constantk ([N/m]), the dumping factorD([Ns/m]), and theelectro-magnetic constantKe([Wb/m]) are :

wn = 2f0

k = Mw2n

D = wnM211 1Q2Ke = kRSDC

whereQdenotes the absolute value of the actuator amplitude peak, at

the resonance frequencyf0,SDC ([mm/V]) is the value of the actuatorDC sensitivity ,M([kg]) is the objective lens massR([]) andL ([H])the resistance and inductance of the coil.

Control ofC l l ti f th t f f ti
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Calculation of the transfer function

Electrical part:

I(s) = 1

Ls+ R[V(s)KesX(s)] (4)

Magnetic part:

F(s) =KeI(s) (5)

Mechanical part: X(s)

F(s)=

1

Ms2 + Ds+ k (6)

Combining these equations , it leads

H(s) =

X(s)

V(s)=

KeML

s3 +

RL +

DM

s2 +

DRML+

kM+

K2eML

s+ kRML

(7)

Control ofS i t
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Suspension system

A simplified quarter vehicle model with semi-active suspension.

zs and zus) are the relative position of the

chassis and of the wheel,

ms (resp. mus) the mass of the chassis

(resp. of the wheel),ks (resp. kt) the spring coefficient of the

suspension (of the tire),

uthe active damper force,

zris the road profile.

Control ofSuspension system (2)
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O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Suspension system (2)

The mechanical equations are:

mszs =

Fk(zdef)

Fc(zdef)

muszus = Fk(zdef) + Fc(zdef)kt(zuszr)zdef

zdef zdef

(8)whereFk(zdef)and Fc(zdef)(withzdef=zszusand zdef=zs zus)are the nonlinear forces provided by the spring and damper

respectively.

0.1 0.05 0 0.054000

3000

2000

1000

0

1000

2000

3000

4000

Stifness coefficient

zdef

[m]

Fk

[N]

1 0.5 0 0.5 11000

500

0

500

1000

1500

Damping coefficient

zdef

[m/s]

Fc

[N]

Figure:Nonlinear forces provided by the Spring (left) and the Damper (right).

Control ofSuspension system (3)
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dynamical systems

O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Suspension system (3)

The involved model parameters have been identified on a "RenaultMgane Coup" car and are given below.

Symbol Value Description

ms 315kg sprung mass

mus 37.5kg unsprung mass

k 29500N/m suspension linearized stiffnessc 1500N/m/s suspension linearized dampingkp 210000N/m tire stiffness[zdef, zdef] [8, 6]cm suspensions deflection limits

Table:Parameters model of a "Renault Mgane Coup".

Control ofA wind tunnel
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dynamical systems

O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

A wind tunnel

Objective: feedback control of the Mach number in a wind tunnel

(NASA)

In steady-state operating conditions (some constant fan speed, liquid

nitrogen injection rate, and gaseous-nitrogen vent rate), the dynamic

response of the Mach number perturbations Mto small perturbationsin the guide vane angle actuator A

M(t) +M(t) = k(th)(t) + 2(t) +2(t) = 2A(t)

(t)is the guide vane angle.Time-delay h: transportation time between the guide vanes of the fan

and the test section of the tunnelhvaries as a function of the temperature and is such that

0.288 h 0.455s.

Control ofSome issues
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dynamical systems

O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Some issues

Why intelligent control systems (Energy Management System)?

Use several actuators : lights, window opening, shading,

heating/cooling (air conditioning)... Control objectives:

Air quality: CO2, particule matter, Volatile Organic Compounds Comfort: humidity, temperature, luminance

Energy savings: consumption

Heating Ventilating and Air Conditioning (HVAC)

A system complex to be modelled:

A Multi-Zone system

Wireless Sensor Network Air flow (thermodynamics: fans, ducts, doors,.. ) and Thermal

models (temperature, humidity

Control ofd i l tRoom temperature
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dynamical systems

O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Room temperature

Lquation fondamentale reprsentant la variation de temprature de

part et dautres dun mur est la suivante:

.cv.VdTdt

= wallx

.A.(ToutT) + Qsources (9)

o

T (K) temprature de la pice

Tout (K) temprature extrieurecv (J/kgK) capacit thermique de lair volume constant = 719cp(J/kgK) capacit thermique de lair pression constante = 1010

V (m3) Volume de la pice

(kg/m3) densit de lair = 1.169

x (m) paisseur

A(m2) Surface du mur

Qsources Sources (extrieures + contrle)

Les coefficients de conductivitwalldpendent des matriauxcomposant les murs, et sont donns :

Control ofdynamical systemsGeneral dynamical system
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dynamical systems

O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

General dynamical system

Many dynamical systems can be represented byOrdinary Differential

Equations(ODE) as

x(t) =f((x(t), u(t), t), x(0) =x0y(t) =g((x(t), u(t), t)

(10)

wheref andgare non linear functions.

Control ofdynamical systemsDefinition of state space representations
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dynamical systems

O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Definition of state space representations

Acontinuous-timeLINEAR state space system is given as : x(t) =Ax(t) + Bu(t), x(0) =x0

y(t) =Cx(t) + Du(t)(11)

x(t) Rn is the system state (vector of state variables), u(t) Rm the control input y(t) Rp the measured output A,B,CandDare real matrices of appropriate dimensions

x0 is the initial condition.

nis the order of the state space representation.Matlab : ss(A,B,C,D) creates a SS object SYS

representing a continuous-time state-space model

Control ofdynamical systemsHeight control of a single Tank
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dynamical systems

O.Sename

Introduction


Physical examples

Hydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Height control of a single Tank

In steady state : Qe=Q0,H= H0Consider the variationsqe, qs, haround the steady state as:

Qe= Q0+ qe; Qs=Q0+ qs; H= H0+ h.This leads to the equation :

Sdh

dt =qekt(

H0+ h

H0)

Using the first order approximation(1 + x) =1 +x, it leads

Sdh

dt =qeh

Denoting the state variablex=h, the control inputu=qe, the outputy=h, we get

x = Ax+ Bu (12)

y = Cx (13)

withA = kt2

H0,B= 1

S andC= 1.

Control ofdynamical systemsSome exampes
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dynamical systems

O.Sename

Introduction


Physical examplesHydraulic tanks


model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Some exampes

Suspension system

Choose the state variables and give the state space representation of

the system, with input zr(not controlled) and output zs

zusorzs

SatelliteChoose the state variables and give the state space representation of

the system, with controlled inputFc, disturbance inputMDand output.

Control ofdynamical systemsA wind tunnel
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y y

O.Sename

Introduction




model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

In steady-state operating conditions (fan speed, liquid nitrogen injectionrate and gaseous-nitrogen vent rate) the dynamic response of the Mach

number is given by the following system:

x(t) =

0.5091 0 00 0 1

0 36 9.6

x(t)

+

0 0.005956 00 0 00 0 0

x(th)

+ 00

36

u(t) + 001

w(t)y(t) =

1 0 0

x(t) + w(t)

z(t) =

1 1 1

x(t)

x(t) =(t); t [h, 0]whereh=0.33sec.,x1 is the Mach number,x2 is the guide vane angleandx3= x2.

Control ofdynamical systemsExample : Wind turbine
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y y

O.Sename

Introduction




model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

p

An complete model (ADAMS) includes 193 DOFs.

Control ofdynamical systemsSome important issues
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O.Sename

Introduction




model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

p

A complete ADAMS model includes 193 DOFs to represent fully

flexible tower, drive-train, and blade components

simulation

model Different operating conditions according to the wind speed

Control objectives: maximize power , enhance damping in the first

drive train torsion mode, design a smooth transition different

modes

A Generator torque controller to enhance drive train torsiondamping in Regions 2 and 3

The control model is obtained by linearisation of a non linear

electro-mechanical model:

x(t) =Ax(t) + Bu(t) + Ed(t)y(t) =Cx(t)

wherex1 = rotor-speed x2 = drive-train torsion spring force,x3=

rotational generator speed

u= generator torque,d : wind speed

Control ofdynamical systemsMore generally


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O.Sename

Introduction




model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Reformulate Nth-order differential equation into N simultaneous

first-order differential equations

dny

dtn + an1

dn1ydtn1

+ . . . + a1y+ a0y=f

Define the state variables :

x1=..., , x2=...., , . . . xn=...,

and give the according state space representation.

Remark : Knowledge of state variables allows one to determine every

possible output of the system

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O.Sename

Introduction




model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Linearisation

Control ofdynamical systemsLinearisation
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O.Sename

Introduction




model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

The linearisation can be done around an equilibrium point or around a

particular point defined by:

xeq(t) =f((xeq(t), ueq(t), t), givenxeq(0)yeq(t) =g((xeq(t), ueq(t), t)

(14)

Defining

x=xxeq, u=uueq, y=yyeqthis leads to a linear state space representation of the system, around

the equilibrium point: x(t) =Ax(t) + Bu(t),

y(t) =Cx(t) + Du(t)(15)

withA = fx|x=xeq,u=ueq,B= fu|x=xeq,u=ueq,

C= gx|x=xeq,u=ueq andD=

gu|x=xeq,u=ueq

Usual caseUsually an equilibrium point satisfies:

0 =f((xeq(t), ueq(t), t) (16)

For the pendulum, we can choosey==f=0.

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O.Sename

Introduction




model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Linear systems : transfer function

Control ofdynamical systemsEquivalence transfer function - state space representation


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O.Sename

Introduction




model

The DVD player


The wind tunnel

Energy and comfort

management in



Physical examples

Linearisation


function

Consider a linear system given by: x(t) =Ax(t) + Bu(t), x(0) =x0y(t) =Cx(t) + Du(t)

(17)

Using the Laplace transform (and assuming zero initial condition

x0=0), (17)becomes:

s.x(s) =Ax(s) + Bu(s) (s.InA)x(s) =Bu(s)Then the transfer function matrix of system (17)is given by

G(s) =C(sInA)1B+ D= N(s)D(s)

(18)

Matlab: if SYS is an SS object, thentf(SYS)gives the associated

transfer matrix. Equivalent totf(N,D)
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auto e7 lecture2

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