Dynamic Scheduling and Control-Quality
Optimization of Self-Triggered Control
Applications
Soheil Samii, Petru Eles, Zebo Peng
Department of Computer and Information Science
Linköping University, Sweden
Paulo Tabuada
Anton Cervin
Department of Electrical Engineering
Department of Automatic Control
University of California, Los Angeles, USA
Lund University, Sweden
CPU
Motivation
Act
P1
Sen
Act
P2
Sen
Act
P3
Sen
 Periodic implementation
 Conservative CPU usage when
the control error is small
 Constraints in the number of controller
executions
 Sensing, computation, actuation
2
Motivation
 Self-triggered control: Control ”when needed”
[Velasco et al., 2003], [Anta and Tabuada, 2008]
 More often when the control error is large, and
vice versa
 Reduced number of controller executions
 Ensures stability
3
Motivation
 This paper:
 Scheduling of multiple self-triggered tasks on a
CPU
 Optimize control quality and reduce CPU usage
4
Outline
 Self-triggered control
 System model
 Motivational example
 Scheduling heuristic
 Experimental results
 Conclusions
5
Self-triggered control
Act
P
Sen
Periodic execution
Self-triggered execution
 Deadline based on
 control law
 plant state
 plant model
time
 Deadline must be met to ensure
stability
 Executing earlier than the
deadline can give better control
6
Scheduler
CPU
System model
dx1/dt = Ax1(t) + Bu1(t)
d1
u1=K1x1
u1
Act
d2
u2
d3
Act
x1
P1
x2
P2
Sen
x3
u3
Act
Sen
P3
Sen
 Control quality
 Deadline constraints
 Reduce CPU usage
7
Scheduling – Control quality
time
10
13
Deadline
J(t start ) 
T
x(t)
Qx(t)dt

Control cost
Completion time
 Control cost
 Latest measurements

 Difference between deadline
and completion time
 Normalized!
Next execution
8
Scheduling – CPU cost
time
13
CPU cost
10
Next execution
9
Scheduling – Trade-off!
time
10
13
+ρ
(CPU demand)
=
Total cost
Combined cost of a task
Next execution
10
Scheduling – How?
?
10
time
13
 How does the scheduler
1. find the best trade-off?
2. decide whether and where to move
scheduled executions?
3. guarantee that all deadlines are met?
11
Problem formulation
 When a task completes execution:
 Schedule its next execution before its deadline
 The other tasks are already scheduled for
execution
 Can be moved!
 Cost to minimize:

all tasks
J
control

J
CPU
all tasks
12
Scheduler outline
Optimization of start time
Candidate start times
Schedule realization
Set of candidate schedules
Not empty
Choose best solution
Empty
Emergency schedule
13
Optimization of start time
time
10
13
Total cost
 Golden-section search
 Cost evaluation for a
limited number of points
 Precalculated values
 Interpolation
Next execution
14
Schedule realization
A
B
B
C
A
Candidate
start time
D
C
D
E
F
time
E
F
time
1. Are deadlines for tasks C and D violated?
2. Compute new costs for tasks C and D
 Golden-section search has been performed at
previous scheduling points
15
Emergency schedule
 Can we guarantee that all deadlines are met?
 Yes, if
Σ WCET ≤ minimum deadline!
 How?
time
?
16
Scheduler – summary
1. Golden-section optimization of start times
2. Conflicting executions are moved forward
3. Stability is guaranteed by construction of the
heuristic and an offline verification
17
Total control cost
Comparison to periodic control
Periodic
Our approach
CPU usage [%]
 Time overhead of the scheduler has been
considered!
18
Conclusion
 Runtime scheduling of multiple control tasks
 Adaptive implementation of self-triggered
controllers achieving the required trade-off
between control quality and CPU usage
19