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8/12/2019 AUTO E7 Lecture2
1/34
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
Olivier.Sename@gipsa-lab.inpg.fr
www.gipsa-lab.fr/o.sename
2nd February 2014
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2/34
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
Outline
Introduction
Methods for system modelling
Physical examples
Hydraulic tanks
Satellite attitude control modelThe DVD player
The suspension system
The wind tunnel
Energy and comfort management in intelligent building
State space representationPhysical examples
Linearisation
Conversion to transfer function
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3/34
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
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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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
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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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
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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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
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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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
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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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
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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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
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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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
the DVD player
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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
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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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
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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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
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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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
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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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
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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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
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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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
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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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Methods for systemmodelling
Physical examplesHydraulic 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
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
Methods for systemmodelling
Physical examplesHydraulic 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
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
Methods for systemmodelling
Physical examplesHydraulic 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
p
An complete model (ADAMS) includes 193 DOFs.
Control ofdynamical systemsSome important issues
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O.Sename
Introduction
Methods for systemmodelling
Physical examplesHydraulic 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
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
Methods for systemmodelling
Physical examplesHydraulic 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
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
Control ofdynamical systems
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O.Sename
Introduction
Methods for systemmodelling
Physical examplesHydraulic 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
Linearisation
Control ofdynamical systemsLinearisation
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O.Sename
Introduction
Methods for systemmodelling
Physical examplesHydraulic 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
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.
Control ofdynamical systems
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O.Sename
Introduction
Methods for systemmodelling
Physical examplesHydraulic 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
Linear systems : transfer function
Control ofdynamical systemsEquivalence transfer function - state space representation
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34/34
O.Sename
Introduction
Methods for systemmodelling
Physical examplesHydraulic 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
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)
http://find/