1. 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 Department of Electrical Engineering University of California, Los Angeles, USA Anton Cervin Department of Automatic Control Lund University, Sweden

2. 2 Motivation ActSen P1P1 ActSen P2P2 ActSen P3P3 CPU  Periodic implementation  Conservative CPU usage when the control error is small  Constraints in the number of controller executions  Sensing, computation, actuation

3. 3 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

4. 4 Motivation  This paper:  Scheduling of multiple self-triggered tasks on a CPU  Optimize control quality and reduce CPU usage

5. 5 Outline  Self-triggered control  System model  Motivational example  Scheduling heuristic  Experimental results  Conclusions

6. 6 ActSen P Self-triggered control  Deadline based on  control law  plant state  plant model  Deadline must be met to ensure stability  Executing earlier than the deadline can give better control time Periodic execution Self-triggered execution

7. 7 System model ActSen P1P1 ActSen P2P2 ActSen P3P3 CPU Scheduler d1d1 d2d2 d3d3  Control quality  Deadline constraints  Reduce CPU usage dx 1 /dt = Ax 1 (t) + Bu 1 (t) u1u1 x1x1 u 1 =K 1 x 1 u2u2 u3u3 x2x2 x3x3

8. 8 Scheduling – Control quality time 1013 Next executionControl cost  Control cost  Latest measurements  Difference between deadline and completion time  Normalized!

9. 9 Scheduling – CPU cost time 13 Next execution CPU cost 10

10. Scheduling – Trade-off! time 13 Total costNext execution (CPU demand) + ρ = Combined cost of a task 10

11. 11 Scheduling – How? time 1013  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? ?

12. 12 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:

13. 13 Scheduler outline Optimization of start timeSchedule realization Candidate start timesSet of candidate schedules Choose best solutionEmergency schedule Not emptyEmpty

14. 14 Optimization of start time Total costNext execution  Golden-section search  Cost evaluation for a limited number of points  Precalculated values  Interpolation time 1310

15. 15 D A EF C A BDEFB C Schedule realization 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 time Candidate start time

16. 16 Emergency schedule  Can we guarantee that all deadlines are met?  Yes, if Σ WCET ≤ minimum deadline!  How? time ?

17. 17 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

18. 18 Comparison to periodic control  Time overhead of the scheduler has been considered! CPU usage [%] Total control cost Periodic Our approach

19. 19 Conclusion  Runtime scheduling of multiple control tasks  Adaptive implementation of self-triggered controllers achieving the required trade-off between control quality and CPU usage