Integrated Scheduling and Synthesis of Control Applications on Distributed

Embedded Systems

Soheil Samii1, Anton Cervin2, Petru Eles1, Zebo Peng1

1 Dept. of Computer and Information Science

Linköping University

Sweden

2 Dept. of Automatic Control

Lund University

Sweden

2

Motivation

• Many embedded control systems are distributed

• Typical example: the modern car

• Timing delays

• Sampling, computation, and actuation

• Sharing of computation and communication resources

• Problem: Degradation of control performance

• System scheduling

• Controller design

3

Outline

• Motivation

• System model

• Example and problem formulation

• Scheduling and synthesis approach

• Experimental results

• Summary and contribution

4

System model

Linear plant model:

• dx(t)/dt = Ax(t) + Bu(t)

• y(t) = Cx(t)

Internal-state vector x(t)

Input u(t)Output y(t)

Measurement noise e(t)

Plant disturbance v(t)

Application model:

• Periodic tasks

• Data dependencies

Linear plant model:

• dx(t)/dt = Ax(t) + Bu(t) + v(t)

• y(t) = Cx(t) + e(t)

Controller

A/D D/AWhat is a good sampling period?

What is a good control law u?

5

Control performance

• Quadratic cost: J = E{ xTQ1x + uTQ2u }

• Depends on

• the sampling period,

• the control law, and

• the distribution of the delay between sampling and actuation of the control signal

• Synthesis of optimal control-law for given

• sampling period and

• constant delay

• Toolbox “Jitterbug”, developed at Lund University in Sweden

6

Example: Control of two pendulums

u

y

u

y

• Measure the angle y

• Stabilize in upright position y=0

• Control the acceleration u of the cart

0 1 0

/ 0.2 0 / 0.2x x u v

g g

1 0y x e

0.2 m0.1 m

J = E{y2 + 0.002u2}

7

Example: Platform

S

A

C

S

A

C

Decide

(1) sampling periods,

(2) design control laws, and

(3) schedule the tasks and messages

8

Example: Ideal control

S

A

C

S

A

C

• Control laws synthesized for the constant delays of each application (9 and 13)

• J1=0.9, J2=2.4, Total=3.3 (achieved for the ideal runtime scenario: dedicated resources)

Sample 20 ms Sample 30 ms

9

Example: Scheduling

S

A

C

S

A

C

S

CS

C S A

S C

S C

A C

A

A

A

20 4010 30 50• Delay distribution

• Application 1: 32, 29, 14

• Application 2: 44, 24

• J1=4.2, J2=6.4, Total=10.6

Sample 20 ms Sample 30 ms• Ideal case

• J1=0.9, J2=2.4, Total=3.3

10

Example: Scheduling

S

A

C

S

A

C

S

CS

C S A

S C

S C

A C

A

A

A

20 4010 30 50

S

CS

C SA

SC

S C

A C

A

A

A

20 4010 30 50• Delay distribution

• Application 1: 14 (constant)

• Application 2: 18, 24

• J1=1.1, J2=5.6, Total=6.7

Sample 20 ms Sample 30 ms

• Compensate for the delays in the schedule (14 and 21)

• J1=1.0, J2=3.7, Total=4.7

• Ideal case

• J1=0.9, J2=2.4, Total=3.3

• First schedule

• J1=4.2, J2=6.4, Total=10.6

11

Example: Change periods

S

A

C

S

A

C

Sample 30 ms Sample 20 ms

S

CS

C SA

S C

C

A

A

A

20 4010 30 50S

C

A

• Delay distribution

• Application 1: 13, 23

• Application 2: 18

• J1=1.3, J2=2.1, Total=3.4 (with delay compensation)

Good selection of periods combined with integrated

scheduling and control-law synthesis is important!

• With periods 20 ms and 30 ms:

• J1=1.0, J2=3.7, Total=4.7

12

Problem formulation

Available sampling periods

Deadlines

Execution-time specifications

Scheduling and synthesis tool

i iw JPeriods

Control laws

?Minimize

13

Approach (Static-cyclic scheduling)

Select controller periods

Task periods

Schedule the tasks and messages

Delay distributions

Synthesize control-laws and compute cost

Stop?No

Yes Done!

Cost

• Constraint logic programming (CLP)

• Minimize delay and jitter

• CLP solver ”ECLiPSe”

What if we have priority-based scheduling?

• Genetic algorithm for period assignment

14

Approach (Priority-based scheduling)

Select task and message priorities

Synthesize control-laws and compute cost

No

Cost

Delay distributions

Schedulable?

Priorities

SimulateYes

Stop?Yes

CostNo

• Run response-time analysis to obtain worst-case delays

• Bounded delays?

• Deadlines met?

• Genetic algorithm for priority assignment

15

Experimental results

Number of plants

Ave

rag

e co

st i

mp

rove

men

t [%

]

Isolated scheduling and control-law synthesisIntegrated approach

Straightforward period assignment

16

Summary and contribution

• Problem: Sharing of computation and communication resources degrades the control performance

• Solution: Integrate scheduling with control design (period assignment and control-law synthesis)

• Contribution:

• A tool for such integrated design of distributed embedded control systems with

–static-cyclic scheduling or

–priority-based scheduling

17

EXTRA SLIDES

18

Evaluation

Period optimization with genetic algorithms

Integrated control-law synthesis and

scheduling

Straightforward period assignment

Isolated control-law synthesis and

scheduling

1. Select smallest periods for all applications

2. Schedule system and synthesize control-laws

3. If not schedulable, increase the period of the application with highest resource demand and then go back to Step 2

1. Synthesize control-law with neglecting the implementation aspects (traditional design)

2. ”As soon as possible” or ”rate-monotonic” scheduling

19

Experiments

Period optimization with genetic algorithms

Integrated control-law synthesis and

scheduling

Straightforward period assignment

Isolated control-law synthesis and

scheduling

• Straightforward approach as a baseline, JSF

• Compute relative cost improvement

• (JSF – J) / JSF

• Evaluate each part of the optimization in isolation

• Generated benchmarks with inverted pendulums, servos, and other examples of unstable plants

• 6 to 45 tasks, 2 to 7 computation nodes

20

Static-cyclic scheduling

Number of plants

Ave

rag

e co

st i

mp

rove

men

t [%

]

21

Priority-based scheduling

Number of plants

Ave

rag

e co

st i

mp

rove

men

t [%

]

22

Optimization time

Number of plants

Ave

rag

e ru

nti

me

[sec

on

ds]