CIS 4930/6930: Principles of Cyber-Physical Systems Chapter 4: - - PowerPoint PPT Presentation

CIS 4930/6930: Principles of Cyber-Physical Systems

Chapter 4: Hybrid Systems Hao Zheng

Department of Computer Science and Engineering University of South Florida

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Hybrid Automata

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Timed Automaton Model of a Thermostat h

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Possible Execution of the Timed Thermostat Model

h(t) t ... (a) (b) (c) s(t) t ... τ(t) t ... 20

t1 t1 + Th Tc

1

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Higher Order Dynamics: Bouncing Ball

t1 t2

t

t1 t2

t

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Higher Order Dynamics: Bouncing Ball

t1 t2

t

t1 t2

t

y(0)

=

h0

˙

y(t)

= −gt

y(t)

=

y(0)+

t

• ˙

y(τ)dτ = h0 − 1

2gt2

• H. Zheng (CSE USF)

CIS 4930/6930: Principles of CPS 5 / 19

Higher Order Dynamics: Bouncing Ball

t1 t2

t

t1 t2

t

y(0)

=

h0

˙

y(t)

= −gt

y(t)

=

y(0)+

t

• ˙

y(τ)dτ = h0 − 1

2gt2

y(t1) = 0, h0 − 1 2gt2

1 = 0, thus t1 =

• 2h0/g
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CIS 4930/6930: Principles of CPS 5 / 19

Higher Order Dynamics: Bouncing Ball

t1 t2

t

t1 t2

t

At t1, y(t)1 = 0. The bump transition takes place with new speed

−a˙

y(t1).

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Higher Order Dynamics: Bouncing Ball

t1 t2

t

t1 t2

t

At t1, y(t)1 = 0. The bump transition takes place with new speed

−a˙

y(t1).

˙

y(t) = −a˙ y(t1)− gt (t > t1)

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Sticky Masses Example

y1(t) y2(t)

y1(t) y2(t) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 5 10 15 20 25 30 35 40 45 50 Displacement of Masses time

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Sticky Masses Example System Dynamics

• Let p1 and p2 be neutral places of the two springs.
• The forces due to the springs are zero.
• Suppose the spring force is proportional to the displacement.
• When apart, forces due to the springs:

F1

=

k1(p1 − y1(t)) F2

=

k2(p2 − y2(t))

• Under Newton’s 2nd Law (i.e., F = ma):

¨

y1(t)

=

k1(p1 − y1(t))/m1

¨

y2(t)

=

k2(p2 − y2(t))/m2

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Sticky Masses Example System Dynamics

• When stuck together, pulled in opposite directions by two springs:

F

=

F1 + F2 m

=

m1 + m2 y(t) = y1(t) = y2(t)

¨

y(t) = k1p1 + k2p2 −(k1 + k2)y(t) m1 + m2

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CIS 4930/6930: Principles of CPS 9 / 19

Sticky Masses Example System Dynamics

• Guard on the apart to together transition is: y1(t) = y2(t).
• Initial velocity of combined mass, ˙

y(t), set by conservation of momentum:

˙

y(t)(m1 + m2)

= ˙

y1(t)m1 + ˙ y2(t)m2

˙

y(t)

= ˙

y1(t)m1 + ˙ y2(t)m2

(m1 + m2)

• Guard on the together to apart transition is:

F2 − F1 = (k1 − k2)y(t)+ k2p2 − k1p1 > s where s represents the stickiness of the two masses.

• This transition occurs when the right-pulling force, k2(p2 − y(t)),

exceeds the left-pulling force, k1(p1 − y(t)), by the stickiness s.

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Hybrid System Model for Sticky Masses

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Control Systems

• A control system includes:
• The plant - the physical process that is to be controlled.
• The environment.
• The sensors.
• The controller.
• The controller has two levels:
• Supervisory control determines the mode transition structure.
• Low-level control determines the time-based inputs to the plant.
• Supervisory controller determines the strategy while the low-level

controller implements the strategy.

• Hybrid systems are ideal for modeling control systems.
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Automated Guided Vehicle (AGV) Example

track AGV global coordinate frame

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AGV Dynamics

• The speed is u(t) is restricted to:

0 ≤ u(t) ≤ 10 mph

• The angular speed is ω(t) is restricted to:

−π ≤ ω(t) ≤ π radians/second

• Position is (x(t),y(t)) ∈ R2 and angle is θ(t) ∈ (−π,π].
• The motion of the AGV is defined by the differential equations:

˙

x(t)

=

u(t) cos θ(t)

˙

y(t)

=

u(t) sin θ(t)

˙ θ(t) = ω(t)

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Determining the Error in Position

photodiode track ATV

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Hybrid System Model for the AGV Example

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A Trajectory for the AGV Example

initial position straight right straight left track

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AGV Example Summary

• Plant is the differential equations governing the AGV motion.
• Environment is the closed track.
• Sensor is e(t) which gives the AGV position relative to the track.
• Supervisory controller are the four modes and guards to switch

b/w them.

• Low-level controller is the specification of inputs to the plant u and

ω.

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Concluding Remarks

• Hybrid systems are a bridge between state-based and time-based

models which allow for the description of real-world systems.

• Discrete transitions are used to change the mode of operation.
• These transitions are taken when guards are satisfied that include

both inputs and predicates on continuous variables.

• The change in mode may result in a change in continuous

behavior.

• Analysis of hybrid systems is complicated by the fact that both

state-based and time-based analysis is required.

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