Feedback Control of Computing Systems M 5 Control

Feedback Control of Computing Systems M 5: Control Analysis Joseph L. Hellerstein IBM Thomas J Watson Research Center, NY hellers@us. ibm. com September 23, 2004 Feedback Control of Computing Systems © 2004 Hellerstein

Motivating Example + - Controller Notes Server Notes Sensor The problem Design a control system that is stable, accurate, settles quickly, and has small overshoot. Take a holistic approach Design a control system, not just a controller 2 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

M 5: Lecture 3 Feedback Control of Computing Systems: M 5 – Control Analysis

Outline ¢ ¢ ¢ P & I controllers Accuracy of I control Pre-compensation Control system with filter PI control Reference: “Feedback Control of Computer Systems”, Chapters 8, 9. 4 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Key Results for LTI Systems U(z) G(z) Y(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). A(z) + C(z) + B(z) Transfer functions in series U(z) G(z) W(z) H(z) Y(z) is equivalent to Stable if |a|<1, where a is the largest pole of G(z) U(z) G(z)H(z) Y(z) ssg of G(z) is 5 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Basic Controllers E(z) R(z) + - Proportional (P) Control K(z) U(z) Y(z) G(z) Integral (I) Control KP and KI are called control gains. 6 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Group Lab ¢ ¢ ¢ 7 Create P and I control systems for Lotus Notes Server Compare accuracy and settling times Explain results using control analysis Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Analysis E(z) R(z) + - K(z) U(z) Y(z) G(z) Use MATLAB to find the poles and steady state gain Compare with the spreadsheet 8 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Conclusions from P vs. I Comparison KP=2. 3 KI=0. 8 r(k)=200 Conclusions: P is fast I is accurate and has less overshoot. Design challenge: Make P accurate. Reduce P’s overshoot. 9 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Making P Control Accurate E(z) R(z) + - U(z) K(z) Y(z) G(z) Precompensation: Adjusts the reference input so that the right output is obtained. R(z) 10 P(z) E(z) + - K(z) U(z) Y(z) G(z) Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Group Lab ¢ ¢ Modify P control to include pre-compensation Find a value for the precompensator that makes P control accurate v v ¢ ¢ 11 Trial and error Adjust based on ratio between reference and output What happens if the reference input changes? What if the control gain changes? What is the general rule for the value of the precompensator? Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Computing Value of Precompensator R(z) P(z) E(z) + - K(z) U(z) Y(z) G(z) Try on spreadsheet. See if it works for other reference inputs. 12 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Reducing P’s Overshoot R(z) P(z) E(z) + - K(z) U(z) Y(z) G(z) Filter: Smooths values over time. R(z) P(z) E(z) + - K(z) t(k+1)=ct(k)+y(k+1) U(z) Y(z) G(z) H(z) T(z) Design of H(z) filter Choose c to get right amount of smoothing. c – Weight past history (make it smoother) 13 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Group Lab ¢ ¢ ¢ 14 Add a filter to precompensated P control What values of c produce smooth t(k)? What are the other effects of the filter? Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Results of Filter Design w/o filter with filter: c = 0. 75 r(k)=200 The good news about the filter: Can eliminate overshoot The bad news: Inaccurate and slower. Why? 15 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Analysis of the Filter t(k+1)=ct(k)+y(k+1) R(z) P(z) E(z) + - K(z) U(z) Y(z) G(z) H(z) T(z) Analysis 1: Why does H(z) cause the system to be inaccurate? Check the spreadsheet. 16 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Designing a Better Filter t(k+1)=ct(k)+y(k+1) R(z) P(z) E(z) + - K(z) U(z) Y(z) G(z) H(z) T(z) Check the spreadsheet. 17 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Analysis of the Filter t(k+1)=ct(k)+(1 -c)y(k+1) R(z) P(z) E(z) + - K(z) U(z) Y(z) G(z) H(z) T(z) Analysis 2: Why does H(z) cause the system to be slower? So, the filter adds a closed loop pole at c. Check the spreadsheet. 18 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

PI Control 19 Feedback Control of Computing Systems: M 5 – Control Analysis ©

Derivative Control ¢ Motivation v. Anticipate ¢ 20 trends by looking at the change in the control error What is the ARX (time domain) model? Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

M 5: Labs 21 Feedback Control of Computing Systems: M 5 – Control Analysis

PID Control 4. Construct a block diagram for a PID controller that combines P, I, D control What is the transfer function for a PID controller? What is its ARX model? What are some implications of going from PI to PID control? ¢ Hints 1. 2. 3. v v 22 How should the PI block diagram be changed to include D control? How should the PI T. F. be changed? Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein

Design a PI Control System E(z) R(z) + - K(z) U(z) Y(z) G(z) Steps: 1. Find the time series model for the PI transfer function 2. Simulate the system 3. Show that PI control is always accurate for a step input 4. How fast can you make the system? 23 Feedback Control of Computing Systems: M 5 – Control Analysis © 2004 Hellerstein