Towards Petascale Computing

© 2004 HellersteinFeedback Control of Computing Systems

Feedback Control of Computing SystemsM6: Control Design

Joseph L. HellersteinIBM Thomas J Watson Research Center, [email protected]

September 23, 2004

2 © 2004 HellersteinFeedback Control of Computing Systems: M6 – Control Design

Motivating Example

( )r k ( )y k

( )e kController

NotesServer

NotesSensor

( )u k

( 1) (0.43) ( ) (0.47) ( )w k w k u k ( 1) 0.8 ( ) 0.72 ( ) 0.66 ( 1)y k y k w k w k

( )w k

( ) ( ) ( )e k r k y k

The problemDesign a control system that is stable, accurate, settles quickly, and has small

overshoot.

u(k) = u(k-1)+(KP+KI)e(k)+KPe(k-1)

Determine the values of control gains that achieve the design objectives.


M6:Lecture


Outline

Control design methodology Simple application In-class lab: Designing a load balancing system

Reference: “Feedback Control of Computer Systems”, Chapters 8,9.


Key Results for LTI Systems

G(z)aa

kS of polelargest theis || where,||log

4

)1()()( G

uy

ssg of G(z) is

Stable if |a|<1, where a is the largest pole of G(z)

0 10 20 300

0.5

1

1.5

2

k

u(k

)

0 10 20 300

0.1

0.2

0.3

0.4

0.5

k

y(k

)

G(z) Y(z)U(z)

A(z)

C(z)

B(z)

Adding signals:If {a(k)} and {b(k)} are signals, then

{c(k)=a(k)+b(k)} has Z-Transform A(z)+B(z).

G(z) W(z)U(z)H(z) Y(z)

G(z)H(z)Y(z)U(z)is equivalent to

Transfer functions in series

)()()(1)()(

)()()(

zGzKzHzGzK

zRzTzFR


Control Design

Purpose Determine the value of the control gains

Steps1. Specify the control objectives, such as settling time, maximum overshoot.2. Determine the desired poles from the objectives3. Find the modelled transfer function of the system for which the controller is being

designed The denominator is expressed in terms of the control gains

4. Solve for the control gains by equating terms in the denominator of step 3 with the expanded polynomial from step 2.


Example of Applying the Design Methodology: Step 1

Specify the control objectivesSettling times should be no more than 3 time units.

( )r k ( )y k

( )e kController

NotesServer

( )u k

u(k) = u(k-1)+(KP+KI)e(k)+KPe(k-1) y(k) = 0.43y(k-1)+0.47u(k-1)



Determine the desired poles from the objectivesMust specify two poles since the closed loop system is second order.

26.0344

eea Sk Use 0.1 as the second pole. Settling time is determined by the largest pole.

( )r k ( )y k

( )e kController

NotesServer

( )u k


)()()(1)()(

)()()(

zGzKzHzGzK

zRzTzFR

G(z)aa

kS of polelargest theis || where,||log

4


Example of Applying the Design Methodology:Step 3

Find the modelled transfer function in terms of the control gains

( )r k ( )y k

( )e kController

NotesServer

( )u k


)()()(1)()(

)()()(

zGzKzHzGzK

zRzYzFR

PIP

PIP

PIP

PIPR

PIP

KzKKzKzKK

KzKKzzKzKKzF

zHz

zG

zKzKK

zEzUzK

47.043.0)43.147.047.0(47.0)(

47.0)()43.0)(1(47.0)()(

1)(43.0

47.0)(

1)(

)()()(

2



Solve for the control gains

( )r k ( )y k

( )e kController

NotesServer

( )u k


42.1,86.0026.047.043.0

36.043.147.047.0 of degreeby tscoefficien Equate

47.043.0)43.147.047.0( :function transfer modelled theofr Denominato

026.036.0)1.0)(26.0( :function transfer ofr denominato Desired2

2

IP

P

IP

PIP

KKKKK

zKzKKz

zzzz


M6:Group Lab


Methodology for Building Control Systems

1. Draw and describe the system (e.g., components, interactions)2. Determine the control objectives (regulation? disturbance rejection?)3. Draw a control block diagram of the system4. Do a simple simulation of the control system (comparing with the real system if

possible)5. Find the transfer functions and validate with the simulation6. Do a control design7. Check the design with the simulation and the real system


Designing a Load Balancing System for 2 Servers

WorkQueue

Balancer 1

Balancer 2

Requests

Queue Depths

Server 2

Server 1Completions

Completionsrequests

requests

queue depth

queue depth

Operation1. Incoming requests are placed in work queue.2. Balancers take requests at a rate so as to balance queue depths between the

servers.3. Controllers know the queue depth of the other server (although information

may be delayed).4. Servers may have additional work that adds to their queue depths. But

servers do not drop requests (at least in the load range studied).


Designing a Load Balancing System for 2 Servers

WorkQueue

Balancer 1

Balancer 2

Requests

Queue Depths

Server 2

Server 1Completions

Completionsrequests

requests

queue depth

queue depth

Questions1. How should the balancers be designed so that loads are balanced and steady

state is achieved quickly?2. How does additional work in one server affect the other server?3. What is the impact of delays in exchanging queue depth information?4. What happens if one system is faster than the other system?


Step 2: Determining the Control Objectives

What is the control objective?1. Regulation?2. Disturbance rejection?3. Optimization?

WorkQueue

Balancer 1

Balancer 2

RequestsQueue Depths

Server 2

Server 1Completions

Completionsrequests

requests

queue depth

queue depth


Step 3: Construct the Control Block Diagram

Target system? Controller? Reference input? Measured output? Control input?

WorkQueue

Balancer 1

Balancer 2

RequestsQueue Depths

Server 2

Server 1Completions

Completionsrequests

requests

queue depth

queue depth

What do we know about the transfer functions for the servers?


A Possible Control Diagram

K1(z) G1(z)

0.5

K2(z) G2(z)

D2(z)

D1(z)

Server 1

Server 2Balancer 2

Y1(z)

Y2(z)

Balancer 1

E2(z)

E1(z)R1(z)

R2(z)

Y(z)

V2(z) U2(z)

U1(z)V1(z)


Step 4: Construct a Simple Simulation

See ShortClass-M6 Assume first order system for servers Consider P and I control


Step 5a: Identify the transfer functions to study

K1(z) G1(z)

0.5

K2(z) G2(z)

D2(z)

D1(z)

Server 1

Server 2Balancer 2

Y1(z)

Y2(z)

Balancer 1

E2(z)

E1(z)

What are the input signals? What are the output signals?


Step 5a (continued): Identify the Transfer Functions

K1(z) G1(z)

0.5

K2(z) G2(z)

D2(z)

D1(z)

Server 1

Server 2Balancer 2

Y1(z)

Y2(z)

Balancer 1

E2(z)

E1(z)

From Di(z) to Ei(z): Effect of disturbance on the system on which the disturbance occurred

From Di(z) to Ej(z): Effect of disturbance on the other system.

2)()(

)(

,)()(

)(

zYzYzE

zDzE

zF

iji

i

jij


Step 5b: Construct the Transfer Functions

K1(z) G1(z)

0.5

K2(z) G2(z)

D2(z)

D1(z)

Server 1

Server 2Balancer 2

Y1(z)

Y2(z)

Balancer 1

E2(z)

E1(z)

Express Yi(z) interms of Yj(z) and the transfer functions and then do a lot of algebra.

)()()()(2

)()()( zGzDzGzK

zYzYzY iiii

iji

Ei(z)


Step 5b: Construct the Transfer Functions

)()()()()( thatRecall

)()(5.0)()(5.01)(5.0

)(

)()(5.0)()(5.01)(5.0)(

zFzDzFzDzE

zGzKzGzKzG

zF

zGzKzGzKzGzF

jijiiii

jjii

jji

jjii

iii

K1(z) G1(z)

0.5

K2(z) G2(z)

D2(z)

D1(z)

Server 1

Server 2Balancer 2

Y1(z)

Y2(z)

Balancer 1

E2(z)

E1(z)

Questions1. What is the control error if

the servers, balancers, and distrubances are identical?

2. What controller will ensure a zero control error even if the systems and disturbances are different? Why?

)()( that Observe zFzF ijii


Step 5b: Why I-Control Ensures Balanced Systems

?)1( isWhat

)(5.0)(5.01)1)((5.0

)()(5.0)()(5.01)(5.0)(

,,

ii

jjIiiI

j

jjii

iii

F

zzGKzzGKzzzG

zGzKzGzKzGzF

K1(z) G1(z)

0.5

K2(z) G2(z)

D2(z)

D1(z)

Server 1

Server 2Balancer 2

Y1(z)

Y2(z)

Balancer 1

E2(z)

E1(z)

See how theory compares with simulation.

1)( ,

zzKzK iIi


Assignment: 2 Weeks

1. Find the poles of the load balancing system for identical servers that have poles at 0.5.

2. Now change the controller and do a PI design.3. Expand to more systems4. Study the effect of delays, especially if delays are not the same

between systems

.1

)( and 1)( that Suppose ,

zzKzK

azazG iIi

ii


Summary

Approach to control design Limited to systems that are second order in closed loop (More general approaches typically recall MIMO models.)

Load balancing case study Optimization instead of regulation

But still uses linear time invariant models Systematic methodology makes use of simulation, analysis, and (hopefully) real

systems

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Towards Petascale Computing