Lecture9b Discontinuous Systems

8/14/2019 Lecture9b Discontinuous Systems

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Discontinuous Systems

Harry G. KwatnyDepartment of Mechanical Engineering & Mechanics

Drexel University


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Outline Simple Examples

Bouncing ball

Heating system

Gearbox/cruise control

Simulating Hybrid Systems Simulink with Stateflow Stateflow/Simulink Bouncing ball

Power conditioning system

Inverted pendulum

Brockets Necessary Condition Some systems cannot be stabilized by smooth state feedback Extensions to BNC

Solutions to Discontinuous Differential Equations Various notions of solution may be appropriate


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SIMPLE EXAMPLES


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Bouncing Ball

( ) ( )

( ) ( ) [ ]

Free fall:

0Collision:

, 0,1

y g

y t y t

y t cy t c

+

+

=

= =

=


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Bouncing Ball

0

2 00

10 0 0

1 2

1 10 0 0

1 1 1

0

1 20

2

2 2 2

, , ,

2 2 2 1

1

2 1For 0 1, lim

1

Zeno phenomenon transitions in finite time

i

i

NN N Ni i

N ii i i

NN

v v gt

vx v t gt t

g

v v v

t t c t cg g g

v v v cT t c c

g g g c

vc T

g c

= = =

=

= = =

= = =

= = = =

< =

Time for N

transitions


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Heater

( ) ( )

{ }

( ) ( )( ) ( )

continuous state: ,

50, , ,

100

discrete state: , ,

, 73

, 731 , , ,

, 77

, 77

x R

x q offx f x q f x q

x q on

q on off

on q off x

off q off xq k x q k x q

off q on x

on q on x

+ == =

+ =

= = >

+ = ==

=


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Automobile Gearbox Control

Problem 1: Automated gearbox - coordinateautomatic gearshift with throttle command

Problem 2: Cruise control automate throttle andgearbox to maintain speed

Background R. W. Brockett, "Hybrid Models for Motion Control," in Perspectives in

Control, H. L. Trentelman and J. C. Willems, Ed. Boston: Birkhauser, 1993,pp. 29-54.

S. Hedlund and A. Rantzer, "Optimal Control of Hybrid Systems," presentedat Conference on Decision and Control, Phoenix, AZ, pp. 3972-3977, 1999.

F. D. Torrisi and A. Bemporad, "HYSDEL-A Tool for GeneratingComputational Hybrid Models for Analysis and Synthesis Problems," IEEETransactions on Control Systems Technology, vol. 12, pp. 235-249, 2004.


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Transmission

( ) ( )

[ ]

( )

{ }

2

1 2 3 4 5

,

sin

0,1 throttle position

engine speedengine torque-speed characteristic

, , , , transmission state

corresponding gear ratios1, ,5

rear gear

i

i

i

q fin

q eng b veh

wheel

eng

q

fin

R RM v f u F cv M g

r

u

f u

q q q q q q

R i

R

=

=

ratio

wheel radius

brake force

wheel

b

r

F


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Cruise Control

2

2 , gear2

P I I

q fin

wheel

q

v k v k v k v

R Rv

r

+ + =

=

3

3, gear3

P I I

q fin

wheel

q

v k v k v k v

R Rv

r

+ + =

=

5

5 , gear5

P I I

q fin

wheel

q

v k v k v k v

R Rv

r

+ + =

=

4

4 , gear4

P I I

q fin

wheel

q

v k v k v k v

R Rv

r

+ + =

=

1

1, gear1

P I I

q fin

wheel

q

v k v k v k v

R Rv

r

+ + =

=

0 , neutralq

u u u u

l >


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Example 1

1

2 1

x u

x x u

=

=

1.0 0.5 0.5 1.0x1

1.0

0.5

0.5

1.0

x2

1.0 0.5 0.5 1.0F1

1.2

1.0

0.8

0.6

0.4

0.2

0.2

F2

Points on F2 axisclose to F1 axis are

outside image.


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Example 2

1 3 1

2 3 1

3 2

2 1

cos cos

sin sin

Notice that points on the axis close to axis

are not in the image of the mapping.

x

x

x v x x ud

y v x x udt

x u

F F

= = =

=


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Notions of Solution forDiscontinuous Dynamics


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Solutions of ODEs

Classical Solutions

Caratheodory Solutions

Satisfies the ode almost everywhere on [0,t]

Filippov Solutions (differential inclusion a set)

( ) ( )( ) ( ) 0, , 0x t f x t t x x= =

( ) ( )( )00

,t

x t x f x s s ds= +

( ) ( )( ) ( ) 0, , 0x t x t t x x =F


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Classical Solutions

( ) ( )( )( )

,

: is continuously differenticlassical solution able.

x t f x t t

x t

=

1 2 3 4t

0.2

0.4

0.6

0.8

1.0

v t

Example: brick on rampwith stiction.

sgn sinmv v cv mg = +Not a classical solution


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Caratheodory Solutions

( ) ( )( )

[ ]

is satisfied at almost all points on every

interval , ,

Stopping solutions for the brick on ramp problems are not

Caratheodory solutions. For these solutions the brick is

stopped on a finite

x t f x t

t a b a b

=

>

( ) [ ]( ) [ ]

interval, i.e, 0 on ,

0 on ,

sgn 0 sin 0

v t t a b

v t t a b

mg

=

=

+ =


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Brick Example try something

else

sgn sin

sgn sin

sin 0

sin , sin 0

sin 0

mv v cv mg

cv v v g

m m

cv v g v

m m

v g g vm m

cv v g v

m m

= +

= +

= + >

+ + =

= +

= F

}:),(


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Example: nearest neighbor

3 agents moving in square

Q

Rule: move diametrically

away from nearestneighbor

{ } { }{ }1 2 3

Nearest neighbor to

arg min , , \

Action

i

i i i

i i

i

i i

p

p q q Q p p p p

pp

p

=

=

N

N

N


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Example: nearest neighbor,

contd

Consider 1 agent - in which case

the only obstacles are the walls.

The nearest neighbor is easily

identified on the nearest wall.

The vector field is well defined

everywhere except on the

diagonals where it is not defined

because there are multiple

nearest neighbors.

1

1

1

p qp

p q

=


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Example: nearest neighbor,

contd

( )

( )

( )

( )

( )

2 1 2

1 2 1

1 2

2 1 2

1 2 1

0, 1

1,0,

0,1

1,0

x x x

x x xx f x x

x x x

x x x

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Lecture9b Discontinuous Systems