Deadlines misses and their Implication on Feedback Control Loops. Dip Goswami Eindhoven University of Technology (TU/e) The Netherlands

Transcription

1 Deadlines misses and their Implication on Feedback Control Loops Dip Goswami Eindhoven University of Technology (TU/e) The Netherlands

2 Periodic tasks dd ii 2

3 Periodic tasks dd ii Hard deadlines 3

4 Periodic tasks dd ii Some jobs miss deadlines -- no restriction on order of misses: soft deadline 4

5 Firm deadlines (m,k)-firm deadline: A periodic task is said to have an (m,k)- firm guarantee if it is adequate to meet the deadlines of m out of k consecutive instances of the task (jobs), where m k. 5

6 (m,k)-firm deadline A periodic task is said to have an (m,k)-firm guarantee if it is adequate to meet the deadlines of m out of k consecutive instances of the task (jobs), where m k. miss=1 dd ii Window of size k=3 6

7 (m,k)-firm deadline A periodic task is said to have an (m,k)-firm guarantee if it is adequate to meet the deadlines of m out of k consecutive instances of the task (jobs), where m k. miss=1 dd ii Window of size k=3 7

8 (m,k)-firm deadline A periodic task is said to have an (m,k)-firm guarantee if it is adequate to meet the deadlines of m out of k consecutive instances of the task (jobs), where m k. miss=1 dd ii Window of size k=3 8

9 (m,k)-firm deadline A periodic task is said to have an (m,k)-firm guarantee if it is adequate to meet the deadlines of m out of k consecutive instances of the task (jobs), where m k. miss=1 dd ii Window of size k=3 9

10 (m,k)-firm deadline A periodic task is said to have an (m,k)-firm guarantee if it is adequate to meet the deadlines of m out of k consecutive instances of the task (jobs), where m k. miss=1 dd ii Window of size k=3 10

11 (m,k)-firm deadline A periodic task is said to have an (m,k)-firm guarantee if it is adequate to meet the deadlines of m out of k consecutive instances of the task (jobs), where m k. miss=0 dd ii Window of size k=3 11

12 (m,k)-firm deadline A periodic task is said to have an (m,k)-firm guarantee if it is adequate to meet the deadlines of m out of k consecutive instances of the task (jobs), where m k. miss=1 dd ii Window of size k=3 m=1 (1,3)-firm deadline 12

13 Outline Task model and deadlines in feedback control Typical application scenario Implication of an firm-deadline Analytical outlook Some results 13

14 ...control task model and deadlines... 14

15 Feedback loop T m : measure Loop start u = K*[x1(i);x2(i)] + r*f; xkp1 = A*[x1(i);x2(i)]+ B*u; T c : compute T a : actuate x1(i+1) = xkp1(1); x2(i+1) = xkp1(2); Loop repeats T m sensor task T c controller task T a actuator task 15

16 Control loop Feedback loop Sensor reading τ = sensor-to-actuator delay T m : measure T m T c : compute T c T a : actuate T a h= sampling period Actuation Ideal design assumes: or 16

17 Control task triggering In general, T m and T a tasks consume negligible computational time and are time-triggered T c needs finite computation time and can be preemptive When multiple tasks are running on a processor, T c can be preempted T m T c T a τ τ τ τ Sensor-to-actuator delay: τ 17

18 Deadline for control tasks T c preemption wait T m Deadline D c T a T m Sampling period = h 18

19 T m T c T a T m D c T m T m T a T a pp ii =h dd ii =D c 19

20 Constant delay: sampled-data model x(t k ) x(t k+1 ) D c xx kk + 1 = xx kk + BB 1 (DD cc )uu kk 1 + BB 0 (DD cc )uu kk yy kk = CCCC[kk]...will be illustrated in Part 4 20

21 ...automotive application scenario... Based on the article: Dip Goswami, Samarjit Chakraborty, Reinhard Schneider Relaxing Signal Delay Constraints in Distributed Embedded Controllers IEEE Trans. On Control Systems Technology,

22 Distributed automotive control loops actuator Physical System sensor actuate() Plant ECU measure() T p Kx[k- ] m c FlexRay/CAN x[k] m x Other ECUs Kx[.] x[.] T c compute() Controller ECU 22

23 Task partitioning Feedback loop T p : measure T c : compute Loop start xkp1 = A*[x1(i);x2(i)]+ B*u; x1(i+1) = xkp1(1); x2(i+1) = xkp1(2); u = K*[x1(i+1);x2(i+1)] + r*f; Loop repeats T p plant task T c controller task 23

24 Timing diagram and deadline h Plant ECU T p T p FlexRay Controller ECU m x T c m c T c τ = sensor-to-actuator delay Deadline sampling period 24

25 FlexRay: typical delay profile τ (ms) Sampling instants k 25

26 Delay based classification Non-ideal τ (ms) Ideal 20 h Sampling instants k 26

27 measure: x[k] x[0] x[1] x[2] x[3] x[4] x[5] τ h =1 τ h =1 τ h =1 τ h = 2 τ h =1 τ h =1 actuate: u[k] -- f(x[0]) f(x[1]) f(x[2]) f(x[2]) f(x[4]) sampling instant: k 0 ideal ideal ideal non-ideal ideal time

28 Control scheme u[ k] = = Kx[ k F r 2 1] + F r 1 non -ideal ideal samples samples xx kk + 1 = AAAA kk + BBBB[kk] Ideal: x[ k Non -ideal : x[ k + 1] = Ax[ k] + BKx[ k + 1] = Ax[ k] + BF r 1] + BF r 2 1 z[ k z[ k + 1] = + 1] = A o z[ k] A cl z[ k] 28

29 Back to firm-deadline setting k k+1 k+2 k+3 k+4 k+5... zz kk + 1 = AA cccc zz kk zz kk + 2 = AA oo zz kk + 1 = AA oo AA cccc zz kk zz kk + 3 = AA cccc zz kk + 2 = AA cccc AA oo AA cccc zz kk zz kk + 4 = AA cccc zz kk + 3 = AA cccc AA cccc AA oo AA cccc zz kk zz kk + 5 = AA oo zz kk + 4 = AA oo AA cccc AA cccc AA oo AA cccc zz kk... 29

30 Performance: fast disturbance rejection SS zz[kk + αα] zz[kk] zz[kk] A cl α k kk + αα Hard deadlines 30

31 Performance and deadline miss zz[kk + αα]= AA oo AA cccc AA cccc AA oo AA cccc...z[k] zz[kk] SS zz[kk + αα] zz[kk] AA oo AA cccc AA cccc AA oo AA cccc... k kk + αα Firm deadlines 31

32 Simplified special case SS AA oo AA cccc AA cccc AA oo AA cccc... For any matrix A, the following property holds AA kk ccγγ kk where c and γγ are constants AA cccc kk cc 1 γγ 1 kk AA oo kk cc 2 γγ 2 kk Special case of (m,k)-firmness: m=1 1 deadline miss is allowed by within αα samples SS (cc 1 ) 2 γγ 1 αα 1 cc 2 γγ 2 Compute minimum αα 32

33 Some results Cruise control systems: Sampling period = 40ms Performance requirement: Change in reference speed should be adapted asap 95% of the disturbance must be rejected in 5 sec Implies αα=125 Allowed deadline miss: 7 out of

34 Simulation based results Number of samples τ(ms) v 1 [k] Time (sec) Number of samples v 1 [k] τ (ms) Time (sec) 34

35 Summary Task model and deadlines in feedback control Typical application scenario Implication of an firm-deadline Analytical outlook Some results 35