Feedback Control of Computing Systems M4: Analyzing Composed Systems Joseph L. Hellerstein IBM Thomas J Watson Research Center, NY hellers@us.ibm.com September 21, 2004

Key Results for LTI Systems + 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). Key Results for LTI Systems G(z) Y(z) U(z) G(z) W(z) U(z) H(z) Y(z) G(z)H(z) is equivalent to Transfer functions in series Stable if |a|<1, where a is the largest pole of G(z) ssg of G(z) is

Motivating Example Controller Notes Server Sensor - + The problem This module addresses the analysis of composed systems: steady state gain, stability, settling time Controller Notes Server Sensor - + 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

M4: Lecture

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

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

Canonical Feedback Loop Noise Input Disturbance Input D(z) N(z) Reference Input Measured Output Controller Target System R(z) E(z) U(z) + V(z) Y(z) + G(z) T(z) + K(z) + - + W(z) H(z) May also want transfer functions to the control error, which adds another 3. Many transfer functions are potentially of interest. For each, we are interested in stability, ssg, settling time. 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!

Canonical Feedback Loop – Reference Input FR(z) D(z)=0 N(z)=0 Reference Input Measured Output Controller Target System R(z) E(z) U(z) + V(z) Y(z) + T(z) + K(z) G(z) - + + Transducer W(z) H(z) May also want transfer functions to the control error, which adds another 3. Many transfer functions are potentially of interest. For each, we are interested in stability, ssg, settling time. 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)

Canonical Feedback Loop Has Many T.F. Noise Input Disturbance Input D(z) N(z) Reference Input Measured Output Controller Target System R(z) E(z) U(z) + V(z) Y(z) + + K(z) G(z) T(z) + + - W(z) H(z) A transfer function is specified in terms of its input and output. May also want transfer functions to the control error, which adds another 3. Many transfer functions are potentially of interest. For each, we are interested in stability, ssg, settling time. Assess Accuracy Assess disturbance robustness Transducer Assess noise robustness 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

Transfer functions in series + 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). Results Used to Find T.F. G(z) W(z) U(z) H(z) Y(z) G(z)H(z) is equivalent to Transfer functions in series We’ll use these result

The only non-zero input is R(z). Computing FR(z) The only non-zero input is R(z). R(z) E(z) U(z) T(z) + K(z) G(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 This is a transfer function and so can be analyzed just like any other. 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.

Computing FD(z)=T(z)/D(z) The only non-zero input is D(z). D(z) N(z)=0 R(z)=0 E(z) U(z) + V(z) Y(z) + + K(z) G(z) T(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.

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

Properties of Canonical Loop D(z) N(z) Properties of Canonical Loop R(z) U(z) + V(z) Y(z) + E(z) + K(z) G(z) T(z) - + + W(z) H(z) Reference to Output Disturbance to Output Noise to Output? As what the T.F. 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)

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

Generalized Canonical Feedback Loop In-1(z) In+1(z) I1(z) + G1(z) Gn(z) T(z) + + - + Hm(z) H1(z) What is ? How can this be simplified in terms of the series of transducers? (Make into a single transducer.)

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

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

Key Results for LTI Systems + 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). Key Results for LTI Systems G(z) Y(z) U(z) G(z) W(z) U(z) H(z) Y(z) G(z)H(z) is equivalent to Transfer functions in series Stable if |a|<1, where a is the largest pole of G(z) ssg of G(z) is

M4: Labs

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? Key insights for this system Accuracy requires a larger K Stability & short settling times require a smaller K Overshoot is less (or non-existent for smaller K) Why are these important properties?

Compare with simulations in M1. 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.

Homework: Due 9/23/2004 Controller Notes Server Notes Sensor - D(z) w(k+1)=0.43w(k)+0.47v(k) u(k)=u(k-1)+KIe(k) Controller + V(z) 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.