Feedback Control for Real-Time Systems Chenyang Lu Cyber-Physical - - PowerPoint PPT Presentation

CPS Week 2013

Feedback Control for Real-Time Systems

Chenyang Lu

Cyber-Physical Systems Laboratory Department of Computer Science and Engineering

CPS Week 2013

Outline

q CPU UClizaCon Control for Distributed Real-Time Systems

q Model PredicCve Control

q Thermal Control for Real-Time Systems

q Nested Control Design

2

CPS Week 2013

Outline

q CPU UClizaCon Control for Distributed Real-Time Systems

q Model PredicCve Control

q Thermal Control for Real-Time Systems

q Nested Control Design

3

CPS Week 2013

4

Control for Distributed Real-Time Systems

q Common characterisCcs of compuCng problems

q MIMO: mulC-input (knobs), mulC-output (objecCves) q Coupling between objecCves. q Constraints on knobs.

q Model PredicCve Control

q OpCmizaCon + PredicCon + Feedback

CPS Week 2013

5

Why CPU U?liza?on Control?

q Overload protecCon

q CPU over-uClizaCon à system crash

q Meet response Cme requirement

q CPU uClizaCon < bound à meet deadlines

CPS Week 2013

6

Challenge: Uncertain?es

q ExecuCon Cmes?

q Unknown sensor data or user input

q Request arrival rate?

q Aperiodic events q Bursty service requests

q Disturbance?

q Denial of Service a[acks

Control-theoreCc approach à Robust uClizaCon control in face of workload uncertainty

CPS Week 2013

7

End-to-End Tasks in Distributed Systems

q Task Ti: sequence of subtasks {Tij} on different processors

q Periodic: All the subtasks of a task run at a same rate.

q Task rate can be adjusted

q Within a range q Higher rate à higher uClity

Remote Invocation Subtask

T1 T2 T3 T11 T12 T13

P1 P2 P3

CPS Week 2013

8

q Bi: UClizaCon set point of processor Pi (1 ≤ i ≤ n) q ui(k): UClizaCon of Pi in the kth sampling period q rj(k): Rate of task Tj (1 ≤ j ≤ m) in the kth sampling period subject to rate constraint: Rmin,j ≤ rj(k) ≤ Rmax,j (1 ≤ j ≤ m)

Problem Formula?on

min

{rj (k)|1≤ j≤n}

(Bi − ui(k))2

i=1 n

∑

CPS Week 2013

9

Single-Input-Single-Output (SISO) Control

Single Processor

Monitor Processor OS Application Sensor Inputs Set point Us = 69% Task Rates R1: [1, 5] Hz R2: [10, 20] Hz Middleware Actuator Controller u(k) {r(k+1)}

• C. Lu, X. Wang, and C. Gill, Feedback Control Real-Time Scheduling in

ORB Middleware, IEEE Real-Time and Embedded Technology and Applications Symposium (RTAS'03), May 2003.

CPS Week 2013

10

New in Distributed Systems

q Need to control uClizaCon of mulCple CPUs q UClizaCon of CPUs are coupled due to end-to-end tasks

à ReplicaCng a SISO controller on all processors does not work!

q Constraints on task rates

T1 T2 T3 T11 T12 T13

P1 P2 P3

CPS Week 2013

11

EUCON: Mul?-Input-Mul?-Output Control

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Δ Δ ) ( ) (

1

k r k r

m

• Model

Predictive Controller B

1

 Bn ! " # # # # $ % & & & & , Rmin,1  Rmin,m Rmax,1  Rmax,m ! " # # # # $ % & & & &

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ) ( ) (

1

k u k u

n

• Distributed System

(m tasks, n processors)

Utilization Monitor Rate Modulator RM UM UM RM Feedback Loop Remote Invocation Subtask

Control Input Measured Output

• C. Lu, X. Wang and X. Koutsoukos, Feedback Utilization Control in

Distributed Real-Time Systems with End-to-End Tasks, IEEE Transactions

• n Parallel and Distributed Systems, 16(6): 550-561, June 2005.

CPS Week 2013

12

Control Design Methodology

1. Derive a dynamic model of the system 2. Design a controller 3. Analyze stability

CPS Week 2013

13

Dynamic Model: One Processor

q Si: set of subtasks on Pi q cjl: esCmated execuCon Cme of Til q gi: uClizaCon gain of Pi

q raCo between actual and esCmated change in uClizaCon q models uncertainty in execuCon Cmes

ui(k) = ui(k −1) + gi c jlΔrj(k −1)

T jl ∈Si

∑

CPS Week 2013

14

Dynamic Model: Mul?ple Processors

q G: diagonal matrix of uClizaCon gains q F: subtask allocaCon matrix q models the coupling among processors

q fij = cjl if task Tj has a subtask Tjl on processor Pi q fij = 0 if Tj has no subtask on Pi

u(k) = u(k-1) + GFΔr(k-1)

T1 T2 T11

P1 P2

T21 T22 T3 T31

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =

31 22 21 11

c c c c F

CPS Week 2013

15

Model Predic?ve Control

q Suitable for coupled MIMO control problems with constraints. q Compute input to minimize cost over a future interval.

q Cost funcCon: tracking error and control cost. q Predict cost based on a system model and feedback. q Compute input subject to constraints.

q OpCmizaCon + PredicCon + Feedback

CPS Week 2013

q Cost q Reference trajectory: exponenCal convergence to B

Cost Func?on

16

V(k) = u(k + i) − ref (k + i)

2 i=1 P

∑

+ Δr(k + i) − Δr(k + i −1)

2 i= 0 M −1

∑

Tracking Error Control Cost

ref (k +i) = B −e

− Ts Tref i

(B −u(k))

CPS Week 2013

17

Model Predic?ve Controller

At the end of each sampling period

q Compute inputs in future sampling periods Δr(k), Δr(k+1), ... Δr(k+M-1) to minimize the cost funcCon q Cost is predicted using (1) feedback u(k-1) (2) approximate dynamic model q Apply Δr(k) to the system

At the end of the next sampling period

q Shic Cme window and re-compute Δr(k+1), Δr(k+2), ... Δr(k+M) based

• n feedback

CPS Week 2013

18

EUCON Controller

Least Squares Solver

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ) ( ) (

1

k u k u

n

• ⎥

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Δ Δ ) ( ) (

1

k r k r

m

• Model Predictive Controller

System Model Cost Function Reference Trajectory

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡

n

B B

• 1

Rate Constraints Least Squares Solver

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ) ( ) (

1

k u k u

n

• ⎥

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Δ Δ ) ( ) (

1

k r k r

m

• Model Predictive Controller

System Model Cost Function Reference Trajectory

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡

n

B B

• 1

Rate Constraints

Difference from reference trajectory Desired trajectory for u(k) to converge to B Constrained

• p?miza?on

solver

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + Δ + Δ ) 1 ( ) 1 (

1

k r k r

m

CPS Week 2013

19

Stability Analysis

q Stability: uClizaCon of all processors converge to set points q Derive stability condiCon à range of G q Tolerable variaCon of execuCon Cmes à Provides analyCcal assurance despite uncertainty

CPS Week 2013

20

Stable System

0.2 0.4 0.6 0.8 1

50 100 150 200 250 300

Time (sampling period) CPU utilization P1 P2 Set Point

execuCon Cme factor = 0.5 (actual execuCon Cmes = ½ of esCmates)

CPS Week 2013

21

Unstable System

execuCon Cme factor = 7 (actual execuCon Cmes = 7 Cmes esCmates)

0.2 0.4 0.6 0.8 1 100 200 300 Time (sampling period) CPU utilization P1 P2 Set Point

CPS Week 2013

22

Stability

q Stability condiCon à tolerable range of execuCon Cmes AnalyCcal assurance on uClizaCons despite uncertainty

actual execution time / estimation Predicted bound for stability

CPS Week 2013

23

FC-ORB Middleware

Feedback lane

Remote request lanes

Priority Manager Rate Modulator Model Predic?ve Controller

Remote request lanes

U?liza?on Monitor

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ) ( ) ( ) (

3 2 1

k u k u k u

Measured Output

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ) ( ) (

2 1

k r k r

Control Input Priority Manager Rate Modulator U?liza?on Monitor Priority Manager Rate Modulator U?liza?on Monitor

• X. Wang, C. Lu and X. Koutsoukos, Enhancing the Robustness of

Distributed Real-Time Middleware via End-to-End Utilization Control, IEEE Real-Time Systems Symposium (RTSS'05), December 2005.

CPS Week 2013

24

Workload Uncertainty

Cme-varying execuCon Cmes disturbance from periodic tasks

CPS Week 2013

25

Processor Failure

1. Norbert fails. 2. move its tasks to other processors. 3. reconfigure controller 4. control u?liza?on by adjus?ng task rates

CPS Week 2013

26

Summary: Model Predic?ve Control

q ApplicaCon to CPU uClizaCon control

q Robust uClizaCon control for distributed systems q Handle coupling among processors q Enforce constraints on task rates q Analyze tolerable range of execuCon Cmes

q Applicable to many compuCng problems

q MIMO: mulC-input (knobs), mulC-output (objecCves) q Coupling between objecCves q Constraints on knobs

CPS Week 2013

Outline

q CPU UClizaCon Control for Distributed Real-Time Systems

q Model PredicCve Control

q Thermal Control for Real-Time Systems

q Nested Control Design

27

CPS Week 2013

Nested Control

q MulCple control objecCves

q Coupling between objecCves q Dynamics at different Cme scales

q Approach: Nested feedback control loop

28

CPS Week 2013

Thermal Control for Real-Time Systems

q Temperature control

q Prevent processor overheaCng q Avoid hardware thro[ling à unpredictable slowdown

q UClizaCon control

q Maintain real-Cme performance q Enforce schedulable uClizaCon bound

q UncertainCes

q Power, ambient temperature, thermal faults, execuCon Cme

29

CPS Week 2013

Goals

q Enforce both thermal and real-Cme constraints

q Temperature bound Tb < hardware thro[ling threshold q CPU uClizaCon bound Ub < schedulable uClizaCon bound

q Robust against uncertainCes q Run-Cme efficiency

30

CPS Week 2013

Control-theore?c Approach

q Deal with uncertainCes through feedback control

q Rate adaptaCon based on temperature and uClizaCon feedback

q Nested control structure

q Modular: separate controllers for temperature and uClizaCon q Efficiency control algorithms: O(1) complexity q Rigorous stability and sensiCvity analysis

31

• Y. Fu, N. Kottenstette, Y. Chen, C. Lu, X. Koutsoukos and H. Wang,

Feedback Thermal Control for Real-time Systems, IEEE Real-Time and Embedded Technology and Applications Symposium (RTAS'10), April 2010.

CPS Week 2013

Dynamic System Model

32

Tasks rates (control input) Temperature (controlled variable)

Power

U(k +1) = U(k) + Gu ciΔr

i i

∑

P(k) = GpP

aU(k) + P idle(1−U(k))

dT(t) dt = −c2(T(t) − T0) + c1P(t) Thermal Control UClizaCon Control

UClizaCon (control input) (controlled variable)

CPS Week 2013

TCUB: Thermal Control under U?liza?on Bound

q Outer loop: thermal control

q Handle slower thermal dynamics

q Inner loop: CPU uClizaCon control

q Handle faster load dynamics

33

U(k’)

TCUB

Thermal Controller

Processor

Rate Actuator UClizaCon Monitor Thermal Monitor UClizaCon Controller Tb Ub T(k) U(k’) Us(k) {△ri(k’) } Tasks

CPS Week 2013

Varying Power

Ac?ve power = 2 x es?mate

34

TCUB UClizaCon Control

CPS Week 2013

Varying Execu?on Times

Execu?on ?me = 2 x es?mate

35

TCUB Thermal Control

CPS Week 2013

Summary: Nested Control

q Example: Thermal control for real-Cme systems

q Control both temperature and uClizaCon bounds q Robust against uncertainCes

q Nested control approach

q Control variables with dynamics at different Cme scales q Modular design q Efficient control algorithm

36

CPS Week 2013

37

References

q Centralized Control (EUCON): C. Lu, X. Wang and X. Koutsoukos, Feedback UClizaCon Control in Distributed Real-Time Systems with End-to-End Tasks, IEEE TransacCons on Parallel and Distributed Systems, 16(6): 550-561, June 2005. q Middleware (FC-ORB): X. Wang, C. Lu and X. Koutsoukos, Enhancing the Robustness of Distributed Real-Time Middleware via End-to-End UClizaCon Control, IEEE Real-Time Systems Symposium (RTSS'05), December 2005. q Decentralized Control (DEUCON): X. Wang, D. Jia, C. Lu and X. Koutsoukos, DEUCON: Decentralized End-to-End UClizaCon Control for Distributed Real-Time Systems, IEEE TransacCons on Parallel and Distributed Systems, 18(7): 996-1009, July 2007. q Thermal Control (Single Core): Y. Fu, N. Ko[enste[e, Y. Chen, C. Lu, X. Koutsoukos and H. Wang, Feedback Thermal Control for Real-Cme Systems, IEEE Real-Time and Embedded Technology and ApplicaCons Symposium (RTAS'10), April 2010. q Thermal Control (Mul?core): Y. Fu, N. Ko[enste[e, C. Lu and X. Koutsoukos, Feedback Thermal Control of Real-Cme Systems on MulCcore Processors, ACM InternaConal Conference on Embedded Socware (EMSOFT'12), October 2012. q Model Predic?ve Control: J.M. Maciejowski, PredicCve Control with Constraints, PrenCce Hall, 2002. q Adap?ve QoS Control Project: h\p://www.cse.wustl.edu/~lu/control.html