1. Presentation subtitle: 20pt Arial Regular, teal R045 | G182 | B179 Recommended maximum length: 2 lines Confidentiality/date line: 13pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Disclaimer information may also be appear in this area. Place flush left, aligned at bottom, 8-10pt Arial Regular, white IBM logo must not be moved, added to, or altered in any way. Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Presentation title: 28pt Arial Regular, black Recommended maximum length: 2 lines Group name: 17pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white © 2004 Hellerstein Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations Feedback Control of Computing Systems Feedback Control of Computing Systems M6: Control Design Joseph L. Hellerstein IBM Thomas J Watson Research Center, NY hellers@us.ibm.com September 23, 2004

2. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 2 Motivating Example Controller Notes Server Notes Sensor   The problem Design a control system that is stable, accurate, settles quickly, and has small overshoot. u(k) = u(k-1)+(K P +K I )e(k)+K P e(k-1) Determine the values of control gains that achieve the design objectives.

3. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 3 M6: Lecture

4. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 4 Outline Control design methodology Simple application In-class lab: Designing a load balancing system Reference: “Feedback Control of Computer Systems”, Chapters 8,9.

5. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 5 Key Results for LTI Systems ssg of G(z) is Stable if |a|<1, where a is the largest pole of G(z) 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

6. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 6 Control Design Purpose  Determine the value of the control gains Steps 1. Specify the control objectives, such as settling time, maximum overshoot. 2. Determine the desired poles from the objectives 3. 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.

7. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 7 Example of Applying the Design Methodology: Step 1 Specify the control objectives Settling times should be no more than 3 time units. Controller Notes Server   u(k) = u(k-1)+(K P +K I )e(k)+K P e(k-1) y(k) = 0.43y(k-1)+0.47u(k-1)

8. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 8 Example of Applying the Design Methodology: Step 2 Determine the desired poles from the objectives Must specify two poles since the closed loop system is second order. Use 0.1 as the second pole. Settling time is determined by the largest pole. Controller Notes Server   u(k) = u(k-1)+(K P +K I )e(k)+K P e(k-1) y(k) = 0.43y(k-1)+0.47u(k-1)

9. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 9 Example of Applying the Design Methodology:Step 3 Find the modelled transfer function in terms of the control gains Controller Notes Server   u(k) = u(k-1)+(K P +K I )e(k)+K P e(k-1) y(k) = 0.43y(k-1)+0.47u(k-1)

10. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 10 Example of Applying the Design Methodology: Step 4 Solve for the control gains Controller Notes Server   u(k) = u(k-1)+(K P +K I )e(k)+K P e(k-1) y(k) = 0.43y(k-1)+0.47u(k-1)

11. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 11 M6: Group Lab

12. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 12 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 system 4. Do a simple simulation of the control system (comparing with the real system if possible) 5. Find the transfer functions and validate with the simulation 6. Do a control design 7. Check the design with the simulation and the real system

13. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 13 Designing a Load Balancing System for 2 Servers Work Queue Balancer 1 Balancer 2 Requests Queue Depths Server 2 Server 1 Completions requests queue depth Operation 1. 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).

14. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 14 Designing a Load Balancing System for 2 Servers Work Queue Balancer 1 Balancer 2 Requests Queue Depths Server 2 Server 1 Completions requests queue depth Questions 1. 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?

15. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 15 Step 2: Determining the Control Objectives What is the control objective? 1. Regulation? 2. Disturbance rejection? 3. Optimization? Work Queue Balancer 1 Balancer 2 Requests Queue Depths Server 2 Server 1 Completions requests queue depth

16. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 16 Step 3: Construct the Control Block Diagram Target system? Controller? Reference input? Measured output? Control input? Work Queue Balancer 1 Balancer 2 Requests Queue Depths Server 2 Server 1 Completions requests queue depth What do we know about the transfer functions for the servers?

17. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 17 A Possible Control Diagram K 1 (z) G 1 (z) 0.5 K 2 (z) G 2 (z)      D 2 (z) D 1 (z) Server 1 Server 2 Balancer 2 Y 1 (z) Y 2 (z)     Balancer 1 E 2 (z) E 1 (z) R 1 (z) R 2 (z) Y(z) V 2 (z)U 2 (z) U 1 (z) V 1 (z)

18. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 18 Step 4: Construct a Simple Simulation See ShortClass-M6  Assume first order system for servers  Consider P and I control

19. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 19 Step 5a: Identify the transfer functions to study K 1 (z) G 1 (z) 0.5 K 2 (z) G 2 (z)       D 2 (z) D 1 (z) Server 1 Server 2Balancer 2 Y 1 (z) Y 2 (z)     Balancer 1 E 2 (z) E 1 (z) What are the input signals? What are the output signals?

20. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 20 Step 5a (continued): Identify the Transfer Functions K 1 (z) G 1 (z) 0.5 K 2 (z) G 2 (z)       D 2 (z) D 1 (z) Server 1 Server 2Balancer 2 Y 1 (z) Y 2 (z)     Balancer 1 E 2 (z) E 1 (z) From D i (z) to E i (z): Effect of disturbance on the system on which the disturbance occurred From D i (z) to E j (z): Effect of disturbance on the other system.

21. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 21 Step 5b: Construct the Transfer Functions K 1 (z) G 1 (z) 0.5 K 2 (z) G 2 (z)       D 2 (z) D 1 (z) Server 1 Server 2Balancer 2 Y 1 (z) Y 2 (z)     Balancer 1 E 2 (z) E 1 (z) Express Y i (z) interms of Y j (z) and the transfer functions and then do a lot of algebra. E i (z)

22. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 22 Step 5b: Construct the Transfer Functions K 1 (z) G 1 (z) 0.5 K 2 (z) G 2 (z)       D 2 (z) D 1 (z) Server 1 Server 2Balancer 2 Y 1 (z) Y 2 (z)     Balancer 1 E 2 (z) E 1 (z) Questions 1. 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?

23. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 23 Step 5b: Why I-Control Ensures Balanced Systems K 1 (z) G 1 (z) 0.5 K 2 (z) G 2 (z)       D 2 (z) D 1 (z) Server 1 Server 2Balancer 2 Y 1 (z) Y 2 (z)     Balancer 1 E 2 (z) E 1 (z) See how theory compares with simulation.

24. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 24 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 systems 4. Study the effect of delays, especially if delays are not the same between systems

25. IBM logo must not be moved, added to, or altered in any way. Background should not be modified. Title/subtitle/confidentiality line: 10pt Arial Regular, white Maximum length: 1 line Information separated by vertical strokes, with two spaces on either side Slide heading: 28pt Arial Regular, blue R120 | G137 | B251 Maximum length: 2 lines Slide body: 18pt Arial Regular, black Square bullet color: teal R045 | G182 | B179 Recommended maximum text length: 5 principal points Group name: 14pt Arial Regular, white Maximum length: 1 line Copyright: 10pt Arial Regular, white Template release: Oct 02 For the latest, go to http://w3.ibm.com/ibm/presentations © 2004 Hellerstein Optional slide number: 10pt Arial Bold, white Indications in green = Live content Indications in white = Edit in master Indications in blue = Locked elements Indications in black = Optional elements Feedback Control of Computing Systems: M6 – Control Design 25 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