Feedback Control of Computing Systems M4: Analyzing Composed Systems

1. Feedback Control of Computing SystemsM4: Analyzing Composed Systems Joseph L. Hellerstein IBM Thomas J Watson Research Center, NY hellers@us.ibm.com September 21, 2004

2. Adding signals: If {a(k)} and {b(k)} are signals, then {c(k)=a(k)+b(k)} has Z-Transform A(z)+B(z). U(z) Y(z) G(z) A(z) + C(z) + U(z) U(z) W(z) Y(z) Y(z) B(z) G(z) G(z)H(z) H(z) Transfer functions in series is equivalent to ssg of G(z) is Key Results for LTI Systems Stable if |a|<1, where a is the largest pole of G(z)

3. + Controller Notes Server Notes Sensor - This module addresses the analysis of composed systems: steady state gain, stability, settling time Motivating Example The problem Want to find y(k) in terms of KI so can design control system that is stable, accurate, settles quickly, and has small overshoot. But this is difficult to do with ARX models. The Solution Use a different representation

4. M4: Lecture

5. Outline • Canonical feedback loop and its transfer functions • Generalized canonical feedback loop • Complex structures Reference: “Feedback Control of Computer Systems”, Chapter 4.

6. Block Diagram Basics D(z) • Functional block: component of the system • Arrow: signal • Summation point: addition of signals • Branching point: signal with multiple destinations Target System Controller R(z) U(z) + V(z) Y(z) E(z) G(z) K(z) + + - W(z) H(z) Transducer Permitted operations Summing signals Cascading systems

7. Canonical Feedback Loop Noise Input Disturbance Input N(z) D(z) Reference Input Measured Output Target System Controller + Y(z) R(z) U(z) + V(z) E(z) T(z) G(z) K(z) + + + - W(z) H(z) Transducer Want to analyze characteristics of the entire system, like its stability, settling time, and accuracy (ability to achieve the reference input). It’s all done with transfer functions!

8. Canonical Feedback Loop – Reference Input FR(z) N(z)=0 D(z)=0 Reference Input Measured Output Target System Controller + Y(z) R(z) U(z) + V(z) E(z) T(z) G(z) K(z) + + + - Transducer W(z) H(z) • View the dark rectangle as a large transfer function FR(z) with input R(z) and output T(z). • System is stable if largest pole of FR(z) has an absolute value that is less than 1 • System is accurate if FR(1)=1 • System’s settling time is determined by the largest pole of FR(z)

9. Canonical Feedback Loop Has Many T.F. Noise Input Disturbance Input N(z) D(z) Reference Input Measured Output Target System Controller + Y(z) R(z) U(z) + V(z) E(z) T(z) G(z) K(z) + + + - W(z) H(z) A transfer function is specified in terms of its input and output. Assess noise robustness Assess Accuracy Assess disturbance robustness Transducer Transfer function from the reference input to the measured output Transfer function from the disturbance input to the measured output Transfer function from the noise input to the measured output

10. 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) + U(z) U(z) W(z) Y(z) Y(z) B(z) G(z) G(z)H(z) H(z) Transfer functions in series is equivalent to Results Used to Find T.F.

11. Computing FR(z) The only non-zero input is R(z). R(z) U(z) T(z) E(z) G(z) K(z) + - Simplified B.D. since D(z)=0=N(z) W(z) H(z) A set of equations relates R(z) to T(z) based on our previous results W(z) = H(z)T(z) by the definition of a transfer function. E(z) = R(z)-W(z) since this is an addition of signals. T(z) = E(z)K(z)G(z) since K(z) and G(z) are in series. T(z) = (R(z)-H(z)T(z))K(z)G(z) by substitution.

12. Computing FD(z)=T(z)/D(z) The only non-zero input is D(z). N(z)=0 D(z) R(z)=0 + Y(z) U(z) + V(z) E(z) T(z) G(z) K(z) + + + - W(z) H(z) W(z) = H(z)T(z) by the definition of a transfer function. E(z) = -W(z) since this is an addition of signals. T(z) = E(z)K(z)G(z) + D(z)G(z). Y(z) = (-H(z)T(z))K(z)G(z) + D(z)G(z) by substitution.

13. U(z) Y(z) G(z) ssg of G(z) is Results Used Next Stable if |a|<1, where a is the largest pole of G(z)

14. N(z) D(z) Properties of Canonical Loop R(z) + Y(z) U(z) + V(z) E(z) T(z) G(z) K(z) + + + - W(z) H(z) Reference to Output Disturbance to Output Noise to Output? What can we say about the stability and settling times of these three transfer functions? They are the same! When is the system accurate in the sense that T(z)=R(z)? FR(1)=1 When is the system robust to disturbances and noise? FD(1)=0= FN(1)

15. N(z) D(z) Predicting Measured Outputs R(z) + Y(z) U(z) + V(z) E(z) T(z) G(z) K(z) + + + - W(z) H(z) What is T(z) if R(z), D(z), and N(z) are all non-zero?

16. ? What is Generalized Canonical Feedback Loop I2(z) In-1(z) In+1(z) I1(z) + T(z) Gn(z) G1(z) + + + - Hm(z) H1(z)

17. is equivalent to R(z) Y(z) K(z) + - H(z) Nested Structures - R(z) Y(z) K1(z) K2(z) G(z) + + - Find: H(z) Approach 1: Solve directly. Approach 2: a. Transform into a canonical control loop. b. Apply result for T.F. of canonical control loop.

18. Summary • Control systems have many transfer functions of interest • FR(z) – From reference input to measured output • FD(z) – From distrubance input to measured output • FN(z) – From noise input to measured output • Key properties of system are indicated by the transfer functions • Stability, accuracy, settling time, robustness to disturbance, robustness to noise • Find transfer functions by • Solving a (simple) set of equations • Transforming into a canonical control loop and solving

19. Adding signals: If {a(k)} and {b(k)} are signals, then {c(k)=a(k)+b(k)} has Z-Transform A(z)+B(z). U(z) Y(z) G(z) A(z) + C(z) + U(z) U(z) W(z) Y(z) Y(z) B(z) G(z) G(z)H(z) H(z) Transfer functions in series is equivalent to ssg of G(z) is Key Results for LTI Systems Stable if |a|<1, where a is the largest pole of G(z)

20. M4: Labs

21. Analytic Solution to First Lab Exercise Block Diagram ARX Models Reference RIS MaxUsers Actual RIS Control error: e(k)=r*-y(k) e(k) + u(k) y(k) P controller: u(k)=Ke(k) Controller Notes Server r* System model: y(k)=(0.43)y(k-1) +(0.47)u(k-1) - Assignment Find FR(z) Find the steady state error K, K = .1, 1, 3 Find the poles for these same values of K. For each, determine the settling times. How do they compare with simulation results?

22. Solution ARX Models Control error: e(k)=r*-y(k) P controller: u(k)=Ke(k) System model: y(k)=(0.43)y(k-1)+(0.47)u(k-1) e(k) Controller Notes Server + r* u(k) y(k) - Compare with simulations in M1.

23. Homework: Due 9/23/2004 D(z) w(k+1)=0.43w(k)+0.47v(k) u(k)=u(k-1)+KIe(k) V(z) + Controller Notes Server Notes Sensor + - Assignment Find FD(z) Assess the robustness of this control system to noise. Use FD(z) to find the KI that produces the smallest settling time. (Hint: plot the largest pole versus KI. Verify the results of analysis using simulations.