Lecture2 Models

8/2/2019 Lecture2 Models

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EE392m - Winter 2003 Control Engineering 2-1

Lecture 2 - Modeling and Simulation Model types: ODE, PDE, State Machines, Hybrid

Modeling approaches: physics based (white box)

input-output models (black box)

Linear systems

Simulation

Modeling uncertainty


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Goals Review dynamical modeling approaches used for control

analysis and simulation Most of the material us assumed to be known

Target audience

people specializing in controls - practical


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Modeling in Control Engineering Control in a

system

perspective

Physical system

Measurement

system

Sensors

Control

computing

Control

handles

Actuators

Physicalsystem

Control analysis

perspective

Control

computing

System model Controlhandle

model

Measurement

model


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Models Model is a mathematical representations of a system

Models allow simulating and analyzing the system Models are never exact

Modeling depends on your goal

A single system may have many models

Always understand what is thepurpose of the model

Large libraries of standard model templates exist

A conceptually new model is a big deal

Main goals of modeling in control engineering

conceptual analysis

detailed simulation


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),,(

),,(

tuxgy

tuxfx

=

=&

Modeling approaches Controls analysis uses deterministic models. Randomness and

uncertainty are usually not dominant.

White box models: physics described by ODE and/or PDE

Dynamics, Newton mechanics

Space flight: add control inputs u and measured outputs y

),( txfx =&


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vr

tFr

rmv pert

=

+=

&

& )(3

Orbital mechanics example

Newtons mechanics

fundamental laws

dynamics

=

3

2

1

3

2

1

v

v

v

r

r

r

x),( txfx =&

Laplace

computational dynamics(pencil & paper computations)

deterministic model-based

prediction1749-1827

1643-1736 rv


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Orbital mechanics example

Space flight mechanics

Control problems: u - ?

=

3

2

1

3

2

1

v

v

v

r

r

r

x

),,(

),,(

tuxgy

tuxfx

=

=&

=

)(

)(

r

ry

vr

tutFr

rmv pert

=

++=

&

& )()(3

Thrust

state

model

observations /

measurementscontrol


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Gene

expression

model


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),,(

),,()(

tuxgy

tuxfdtx

=

=+

Sampled Time Models Time is often sampled because of the digital computer use

computations, numerical integration of continuous-time ODE

digital (sampled time) control system

Time can be sampled because this is how a system works

Example: bank account balance

x(t) - balance in the end of day t

u(t) - total of deposits and withdrawals that day

y(t) - displayed in a daily statement

Unit delay operatorz-1:z-1x(t) =x(t-1)

( ) kdttuxfdtxdtx =++ ),,,()(

xytutxtx

=

+=+ )()()1(


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Finite state

machines TCP/IP State Machine


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Hybrid systems Combination of continuous-time dynamics and a state machine

Thermostat example

Tools are not fully established yet

off on

72=x

75=x

70=x

70

=

xKxx&

75

)(

=

x

xxhKx&


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PDE models Include functions of spatial variables

electromagnetic fields

mass and heat transfer

fluid dynamics

structural deformations

Example: sideways heat equation

1

2

2

0)1(;)0(

=

=

==

=

xx

Ty

TuT

x

Tk

t

T

yheat flux

x

Toutside=0Tinside=u


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Black-box models Black-box models - describe P as an operator

AA, ME, Physics - state space, ODE and PDE

EE - black-box, ChE - use anything

CS - state machines, probablistic models, neural networks

P

x

u

input data

y

output data

internal state


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Linear Systems Impulse response

FIR model IIR model

State space model

Frequency domain Transfer functions

Sampled vs. continuous time

Linearization


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Linear System (black-box) Linearity

Linear Time-Invariant systems - LTI

)()( 11 yuP

)()( TyTuP

)()()()( 2121 ++ byaybuauP

)()( 22 yuP

Pu

t

y

t


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Impulse response Response to an input impulse

Sampled time: t= 1, 2, ...

Control history = linear combination of the impulses system response = linear combination of the impulse responses

( ) )(*)()()(

)()()(

0

0

tuhkukthty

kukttu

k

k

==

=

=

=

)()( hP

u

t t


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Linear PDE System Example Heat transfer equation,

boundary temperature input u

heat flux outputy

Pulse response and step response

00. 2

0. 40. 6

0. 81 0

0. 2

0 .4

0. 6

0 .8

1

0

0. 2

0. 4

0. 6

0. 8

1

TIME

TEMPER ATURE

COORDINATE

0)1()0(

2

2

==

=

TTux

Tk

t

T

1=

=

xx

Ty

0 20 40 60 80 1000

2

4

6x 10

-2

TIME

HEATFLUX

PULSE RESP ONSE

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

TIME

HEATFLUX

STEP RESPONSE


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FIR model

FIR = Finite Impulse Response

Cut off the trailing part of the pulse response to obtain FIR

FIR filter statex. Shift register

),(

),()1(

uxgy

uxftx

=

=+

h0

h1

h2

h3

u(t)

x1=u(t-1)

y(t)

x2=u(t-2)

x3=u(t-3)

z-1

z-1

z-1

( ) )(*)()()(0

tuhkukthty FIR

N

k

FIR ===


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IIR model IIR model:

Filter states: y(t-1), ,y(t-na), u(t-1), , u(t-nb)

==

+=ba n

k

k

n

k

k ktubktyaty01

)()()(

u(t) b0

b1

b2

u(t-1)

u(t-2)

u(t-3)

z

-1

z-1

z-1

-a1

-a2

y(t-1)

y(t-2)

y(t-3)

z

-1

z-1

z-1

y(t)

b3-a3


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IIR model Matlab implementation of an IIR model: filter

Transfer function realization: unit delay operatorz-1

( ) ( ) )(...)(...1...

...

...1

...

)(

)()(

)()()(

)(

1

10

)(

1

1

1

1

1

10

1

1

1

10

tuzbzbbtyzazaazaz

bzbzb

zaza

zbzbb

zA

zBzH

tuzHty

zB

N

N

zA

N

N

N

NN

N

NN

N

N

N

N

4444 34444 21444 3444 21

+++=+++

+++

+++=

+++

+++==

=

FIR model is a special case of an IIR with A(z) =1 (orzN)


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IIR approximation example Low order IIR approximation of impulse response:

(prony in Matlab Signal Processing Toolbox)

Fewer parameters than a FIR model

Example: sideways heat transfer

pulse response h(t)

approximation with IIR filter a = [a1 a2 ], b=[b0 b1 b2 b3 b4 ]

0 20 40 60 80 1000

0.02

0.04

0.06

TIME

IMPULSE RESPONSE

2

2

1

1

4

4

3

3

2

2

1

10

1)(

++

++++=

zaza

zbzbzbzbbzH


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Linear state space model Generic state space model:

LTI state space model

another form of IIR model

physics-based linear system model

Transfer function of an LTI model

defines an IIR representation

Matlab commands for model conversion: help ltimodels

( )[ ]

( ) DBAIzzH

uDBAIzy

+=

+=

1

1

)(

)()()(

)()()1(

tDutCxty

tButAxtx

+=

+=+

),,(

),,()1(

tuxgy

tuxftx

=

=+


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Frequency domain description Sinusoids are eigenfunctions of an LTI system:

LTIPlant

tiititieeeez

==

)1(1

Frequency domain analysis

uzHy )(=

==

deueHydeuu

ti

y

iti

43421)(~

)(~)()(~

)(~

u

e tiu

Packet

of

sinusoids)(~

y

e tiPacketof

sinusoids

)( ieH y


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Frequency domain description Bode plots:

tii

ti

eeHy

eu

)(==

Example:

)(arg)(

)()(

i

i

eH

eHM

=

=

7.0

1)(

=

zzH

Bode Diagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB)

-5

0

5

10

15

10-2

10-1

100

-180

-135

-90

-45

0

|H| is often measured

in dB


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Black-box model from data Linear black-box model can be determined from the data,

e.g., step response data

This is called model identification

Lecture 8


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z-transform, Laplace transform Formal description of the transfer function:

function of complex variablez

analytical outside the circle |z|r

for a stable system r1

k

kzkhzH

== 0 )()(

Laplace transform:

function of complex variable s

analytical in a half plane Re s a for a stable system a1

= dtethsHst)()(

)()()( susHsy =


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Stability analysis Transfer function poles tell you everything about stability

Model-based analysis for a simple feedback example:

)(

)(

d

yyKu

uzHy

=

=dd yzLy

KzH

KzHy )(

)(1

)(=

+=

IfH(z) is a rational transfer function describing an IIR

model

ThenL(z) also is a rational transfer function describing anIIR model


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Poles and Zeros System not quite so!

Example:

7.0)(

==

zzuzHy

19

171819

0011400016280...4907.0)(z

.z.z.zzuzHy FIR +++++==

Impulse Response

Time (sec)

Am

plitude

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Impulse Response

Time (sec)

Amplitude

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

FIR model - truncated IIR


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IIR/FIR example - contd Feedback control:

Closed loop:

)()(7.0

)(

dd yyyyKuz

zuzHy

==

==

Impulse Response

Time (sec )

Amplitude

0 5 10 15 20 250

0.2

0.4

0.6

0.8

uzLuzH

zHy )(

)(1

)(=

+=

uzLuzH

zH

y FIRFIR

FIR

)()(1

)(

=+=


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-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

IIR/FIR example - contd

Poles and zeros Blue: Loop

with IIRmodel poles x

and zeros o

Red: Loopwith FIRmodel poles x

and zeros o


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LTI models - summary Linear system can be described by impulse response

Linear system can be described by frequency response =Fourier transform of the impulse response

FIR, IIR, State-space models can be used to obtain close

approximations of a linear system

A pattern of poles and zeros can be very different for asmall change in approximation error.

Approximation error model uncertainty


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Nonlinear map linearization Nonlinear - detailed

model

Linear - conceptualdesign model

Static map, gain

range, sector

linearity

Differentiation,

secant method

)()( 0uuufufy =


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Nonlinear state space model

linearization

Linearize the r.h.s. map

Secant method

Or capture a response to small step and build an

impulse response model

BvAqq

uu

u

fxx

x

fuxfx

vq

+=

+

=

&

4342143421& )()(),( 00

{]0...1...0[

)(

# j

j

j

j

j

s

ssxf

xf

=

+=


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Sampled time vs. continuous time Continuous time analysis (Digital implementation of

continuous time controller)

Tustins method = trapezoidal rule of integration for

Matched Zero Pole: map each zero and a pole in accordance with

Sampled time analysis (Sampling of continuous signalsand system)

+

==

1

1

1

12)()(

z

z

TsHzHsH s

ssH

1)( =

sTes =


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Sampled and continuous time Sampled and continuous time together

Continuous time physical system + digital controller

ZOH = Zero Order Hold

Sensors

Control

computing

ActuatorsPhysicalsystem

D/A, ZOHA/D, Sample


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Signal sampling, aliasing

Nyquist frequency:

N= S; S= 2/T

Frequency folding: kS map to the same frequency

Sampling Theorem: sampling is OK if there are no frequency

components above

N

Practical approach to anti-aliasing: low pass filter (LPF)

Sampledcontinuous: impostoring

Digitalcomputing

D/A, ZOHA/D, Sample

Low

Pass

Filter

LowPass

Filter


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Simulation ODE solution

dynamical model:

Euler integration method: Runge-Kutta: ode45 in Matlab

Can do simple problems by integrating ODEs

Issues: mixture of continuous and sampled time

hybrid logic (conditions)

state machines

stiff systems, algebraic loops systems integrated out of many subsystems

large projects, many people contribute different subsystems

),( txfx =&

( )ttxfdtxdtx ),()()( +=+


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Simulation environment

Block libraries

Subsystem blocks

developed independently

Engineered for developing

large simulation models

Supports code generation

Simulink by Mathworks

Matlab functions and analysis

Stateflow state machines

Ptolemeus -

UC Berkeley


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Model block development Look up around for available conceptual models

Physics - conceptual modeling

Science (analysis, simple conceptual abstraction) vs.

engineering (design, detailed models - out of simple blocks)


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Modeling uncertainty Modeling uncertainty:

unknown signals

model errors

Controllers work with real systems:

Signal processing: data algorithm data

Control: algorithms in a feedback loop witha realsystem

BIG question: Why controller designed for a model would

everwork with a real system?

Robustness, gain and phase margins,

Control design model, vs. control analysis model Monte-Carlo analysis - a fancy name for a desperate approach

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