Delay-Aware Period Assignment in Control Systems

Anton Cervin Lund UniversitySweden

Enrico Bini Scuola Superiore Sant’Anna, Pisa, Italy

Real-Time Systems Symposium, 2008

Matlab routine available at: http://retis.sssup.it/~bini/

Overview

We find an analytical solution!!!

• n independent plants to be controlled by n controllers

• one CPU to schedule themProblem: assign the periods {Ti} such that

overall control cost is minimized

1 2 n

T1 T2 Tn CPU

The execution of a controller

•a set of n control tasks {1,...,n}•scheduled by FP

A task i is modeled by (Ci, Ti), where:• Ci is the worst-case computation time;• Ti is the period;• Ui=Ci/Ti is the utilization;• Rij is the job response time;• no deadline is enforced.

The cost of a controller

iTi

iJ

The cost is standard Linear Quadratic Gaussian (LQG) that is

We make a linearization

iiiii TJ

iTi

iJ

iTi

iJ

It applies to any other domain where period, delay minimization is required

The linear cost

“The cost Ji is ”iiiii TJ Let Ji(T,) be any smooth cost function, Ti0 and i0 be nominal values of period and delay of i (for example Ti0=i0=Ci)1. make a linear approx of Ji at (Tik,ik) and we set ik

and ik as the partial derivative of Ji;2. Solve (very efficiently) the linear problem and set Tik+1 as the solution and ik+1 accordingly

3. Go back to 1.The convergence is not demonstrated, but always obtained.

Modeling the delay

In control, i is the separation between sampling and actuationHow do we model i as function of T1,...,Tn?

• separation from activation to completion varies job by job

• i=Ri? Overestimate, no expression func of Ti• i=Riub? Exists expression of Ti, larger overestimate• i=avgj{Rij}? good estimate, no expression

Average response time seems the good one

The delay approximation

•seems a good approximation of avgj{Rij}

• it’s the resp time if hp tasks were fluid

• if, a posteriori, we see good performance then it’s a good delay approximation

We claim that average Rij is “something like”

1

1

approx

1i

j j

ii

U

CR

(2,4)

(3,6)

jj R2avg approx2R jj R2inf jj R2sup

7 5 6 6

Formalization of the problem

0 ,1subject to

1minimize

1

1 1

1

i

n

ii

n

i i

j j

iiii

UU

U

CT

Seto et al [RTSS96] solved with i=0

Notice we don’t care of classic FP schedulability!

We aggregate the overall cost by sum Ji

Solution of the problem

Analytical solutionof the problem

Lagrange multipliers +

Partial differentiation +

Luck =

At pag. 296, Equations (34)--(37), you can see the expression of the solution

The priority assignment

• RM is not guaranteed to be optimal.• What is the control-optimal assignment?Intuition: since the cost is

1

11

i

j j

iiii

U

CT

Since we found the analytical (fast) solution, then we can try to enumerate all possible (n!) priority orderings

• large i high priority to i• two “orthogonal” problems: priority assignment

(based on i) and period assignment (based on i)

Introducing experiments

Tested methods:

• FirstVertex, by Bini, Di Natale [RTSS05]

• RiApprox, this method

• Seto96(1), by Seto et al [RTSS96], U=1

• Seto96(Ulub), Seto96, U=Ulub

One experiment

Ci 2 5 10 1 15

U Jii 4 4 4 4 4

i 20 6 3 4 2

Seto96(Ulub) 22.227 35.145 49.702 15.717 60.872 0.7435 913.4

Seto96(1) 16.526 26.130 36.953 11.686 45.258 1 764.1

FirstVertex 18 36 36 18 36 1 770.2

RiApprox 18.905 29.097 39.364 12.263 38.781 1 754.8

Matlab code available on my homepage

• priority assigned by brute force (n!)

One experiment

Ci 2 5 10 1 15

U Jii 4 4 4 4 4

i 20 6 3 4 2

Seto96(Ulub) 22.227 35.145 49.702 15.717 60.872 0.7435 913.4

Seto96(1) 16.526 26.130 36.953 11.686 45.258 1 764.1

FirstVertex 18 36 36 18 36 1 770.2

RiApprox 18.905 29.097 39.364 12.263 38.781 1 754.8

Matlab code available on my homepage

• priority assigned by brute force (n!)

Quality of delay approx

task priority

• random Ci, i, i, n{3,7,19}• We compute periods according

RiApprox• For all tasks we evaluate Riapprox/Riavg

• Riavg is computed by simulatingthe schedule

• on x axis, tasks by decreasing pri

highest lowest

Riap

prox

/Riav

g

Quality of delay approx

task priorityhighest lowest

Riap

prox

/Riav

g

Comparing the costs

n=2 n=5 n=10 n=20

Seto96(Ulub) 208 393 497 673

FirstVertex 169 353 5012 595

RiApprox 167 333 426 554

n=2 n=5 n=10 n=20

Seto96(Ulub) 246 519 6011 14

FirstVertex 226 429 576 7610

RiApprox 195 406 488 664

n=2 n=5 n=10 n=20

Seto96(Ulub) 273 8124 12113 1559

FirstVertex 278 6114 11 1469

RiApprox 244 10279 10012 1287

2 s

tab

le p

ole

s2

sta

ble

, 1

uns

tab

le p

ole

s3

sta

ble

, u

nsta

ble

po

les

• cost is LQG (not synthetic)• 3 kinds of plant generated randomly

• 2 stable poles• 2 stable pole, 1 unstable• 3 stable/ustable poles

• reference is Ti=Di=Ci• costs are normalized with reference

Comparing the costs

n=2 n=5 n=10 n=20

Seto96(Ulub) 208 393 497 673

FirstVertex 169 353 5012 595

RiApprox 167 333 426 554

n=2 n=5 n=10 n=20

Seto96(Ulub) 246 519 6011 14

FirstVertex 226 429 576 7610

RiApprox 195 406 488 664

n=2 n=5 n=10 n=20

Seto96(Ulub) 273 8124 12113 1559

FirstVertex 278 6114 11 1469

RiApprox 244 10279 10012 1287

2 s

tab

le p

ole

s2

sta

ble

, 1

uns

tab

le p

ole

s3

sta

ble

, u

nsta

ble

po

les

Future works

• Extension in presence of a virtual processor.

• More efficient priority assignment.

• A delay sequence of 2, 4, 2, 4, 2, 4,... may not perform the same as a constant delay of 3. Account for it.

?

?

Quality of delay approx 2

i magnutude (sensitivity to delay)

cost

/Se

to9

6(1

)

• 5 tasks• random Ci, i• we increased the average i• cost is synthetic• reference is Seto(1)