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1 ECPS-S2013-L3 (Stefan Resmerita) Embedded and Cyber-Physical Systems - Modeling continuous behavior - Reference book: Edward A. Lee and Pravin Varaiya, Structure and Interpretation of Signals and Systems , Second Edition, LeeVaraiya.org, ISBN 978-0-578-07719-2, 2011.
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Embedded and Cyber-Physical Systems -selected topics-

Mar 19, 2022

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Page 1: Embedded and Cyber-Physical Systems -selected topics-

1 ECPS-S2013-L3 (Stefan Resmerita)

Embedded and Cyber-Physical Systems

- Modeling continuous behavior -

Reference book: Edward A. Lee and Pravin Varaiya, Structure and Interpretation of Signals and Systems, Second Edition, LeeVaraiya.org, ISBN 978-0-578-07719-2, 2011.

Page 2: Embedded and Cyber-Physical Systems -selected topics-

2 ECPS-S2013-L3 (Stefan Resmerita) Slide from Lee & Seshia

Actor Model of Systems

•A system is a function that accepts an input signal and yields an output signal.

•The domain and range of the system function are sets of signals, which themselves are functions.

•Parameters may affect the definition of the function S.

Page 3: Embedded and Cyber-Physical Systems -selected topics-

3 ECPS-S2013-L3 (Stefan Resmerita)

Helicopter example contd.

A helicopter without a tail rotor, like the one below, will spin uncontrollably

Control system problem: Apply torque using the tail rotor

Page 4: Embedded and Cyber-Physical Systems -selected topics-

4 ECPS-S2013-L3 (Stefan Resmerita) Slide from Lee & Seshia

Actor model of the helicopter

•Input is the net torque of the tail rotor. Output is the angular velocity around the y axis.

Parameters of the model are shown in the box. The input and output relation is given by the equation to the right.

Page 5: Embedded and Cyber-Physical Systems -selected topics-

5 ECPS-S2013-L3 (Stefan Resmerita) Slide from Lee & Seshia

Composition of actor models

Page 6: Embedded and Cyber-Physical Systems -selected topics-

6 ECPS-S2013-L3 (Stefan Resmerita) Slide from Lee & Seshia

Actor models with multiple inputs

Page 7: Embedded and Cyber-Physical Systems -selected topics-

7 ECPS-S2013-L3 (Stefan Resmerita) Slide from Lee & Seshia

Proportional controller

desired angular velocity

error signal

net torque

Note that the angular velocity appears on both sides, so this equation is not trivial to solve.

Page 8: Embedded and Cyber-Physical Systems -selected topics-

8 ECPS-S2013-L3 (Stefan Resmerita)

Feedback Control

0 ,1

0 ,0)(

t

ttu

Let‘s see how to derive the solution!

Page 9: Embedded and Cyber-Physical Systems -selected topics-

9 ECPS-S2013-L3 (Stefan Resmerita)

Convolution function

• Definition:

• Properties (on a need-to-know basis):

dtgftgf )()())((

)(1

))(*()()(

)()( c)

)()( b)

a)

tue

tgftutg

tuetf

hgfhgf

fggf

tt

Page 10: Embedded and Cyber-Physical Systems -selected topics-

10 ECPS-S2013-L3 (Stefan Resmerita)

Laplace transform

• Laplace transform:

• Properties:

integrable absolutely is )( where

)()(ˆ

t

st

etf|is

dtetfsF

C

)(ˆ)(ˆ))((

1)(

1)(

sGsFtgf

stue

stu

t

s

sFtf

s

nt

n

n

)(ˆ)(

!1

if is )( integrable absolutelytf finite is and exists )(

dttf

Page 11: Embedded and Cyber-Physical Systems -selected topics-

11 ECPS-S2013-L3 (Stefan Resmerita)

Your solution here

Page 12: Embedded and Cyber-Physical Systems -selected topics-

12 ECPS-S2013-L3 (Stefan Resmerita)

How about the main rotor?

Page 13: Embedded and Cyber-Physical Systems -selected topics-

13 ECPS-S2013-L3 (Stefan Resmerita)

Solution with top rotor included

• Suppose that the torque given by the top rotor is

• The desired angular rotation is zero

• The solution is then

• Does the controller achieve its objective?

)(tbuTt

)1)(()( yyIKt

y etuK

bt

Page 14: Embedded and Cyber-Physical Systems -selected topics-

14 ECPS-S2013-L3 (Stefan Resmerita)

Discrete model of systems

State

Machine

Page 15: Embedded and Cyber-Physical Systems -selected topics-

15 ECPS-S2013-L3 (Stefan Resmerita)

The discrete state space model

• Linear system:

– Initial state is zero

– The output and nextState functions are linear

• Time-invariant: the two functions don‘t change with time

• LTI: linear time-invariant

Page 16: Embedded and Cyber-Physical Systems -selected topics-

16 ECPS-S2013-L3 (Stefan Resmerita)

State space model of LTI systems

• The functions are represented by matrices

Page 17: Embedded and Cyber-Physical Systems -selected topics-

17 ECPS-S2013-L3 (Stefan Resmerita)

Continuous time state space models

For an LTI single-input, single-output system:

Page 18: Embedded and Cyber-Physical Systems -selected topics-

18 ECPS-S2013-L3 (Stefan Resmerita)

The Dirac delta (generalized) function

Page 19: Embedded and Cyber-Physical Systems -selected topics-

19 ECPS-S2013-L3 (Stefan Resmerita)

Impulse response and I/O

• Impulse response:

Page 20: Embedded and Cyber-Physical Systems -selected topics-

20 ECPS-S2013-L3 (Stefan Resmerita)

The transfer function

)(ˆ)(ˆ)(ˆ))(()( SXsHSYtxhty

)(ˆ)(

)(ˆ)(

)(ˆ)(

sHth

sYty

sXtx

The transfer function is the Laplace transform of the impulse response! • Write the transfer function of the helicopter system.

Page 21: Embedded and Cyber-Physical Systems -selected topics-

21 ECPS-S2013-L3 (Stefan Resmerita)

Stability

• A continuous time LTI system is stable if and only if its impulse response is absolutely integrable

• For rational Laplace transforms:

• A continuous time causal system is stable if and only if all the roots of B(s) have negative real parts

Page 22: Embedded and Cyber-Physical Systems -selected topics-

22 ECPS-S2013-L3 (Stefan Resmerita)

Model composition: Cascade

• Write y(t) as a function of x(t)

• What is the transfer function of the composition?

• Can we use this to stabilize a system?

Page 23: Embedded and Cyber-Physical Systems -selected topics-

23 ECPS-S2013-L3 (Stefan Resmerita)

Model composition: Parallel

• Write y(t) as a function of x(t)

• What is the transfer function of the composition?

• Can we use this for stabilization?

Page 24: Embedded and Cyber-Physical Systems -selected topics-

24 ECPS-S2013-L3 (Stefan Resmerita)

Model composition: Feedback

• What is the transfer function of the composition?

• Can we use this for stabilization?

Page 25: Embedded and Cyber-Physical Systems -selected topics-

25 ECPS-S2013-L3 (Stefan Resmerita)

Helicopter questions

• What is the tf of the helicopter model?

• What is the tf of the proportional controller?

• What is the tf of the closed-loop systems?

• Are the three systems above stable?

• What other effect has the tail rotor torque?

• Think about how to counteract that effect

– New control strategy, new actuator on the helicopter

Page 26: Embedded and Cyber-Physical Systems -selected topics-

26 ECPS-S2013-L3 (Stefan Resmerita)

Food for thought: helicopter spinning

• Watch the two video examples

• The helicopters are similar in construction

• Why do they behave differently?