System Model Construction Prof. J. Stankovic University of Virginia Spring 2015

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 © 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 Canonical Feedback Loop K(z) G(z)   Controller Target System   D(z) T(z) R(z) E(z) U(z)V(z) H(z) Transducer W(z)   N(z) Y(z) Measured Output Noise Input Disturbance Input Reference Input Equations for system

Discrete Systems Linear difference equations –First principles (like laws of physics for electromechanical systems) –System ID

First Principles Any First Principles for Computer Systems? –Little’s result from Q.T. –Other results from Q.T. –EDF, RM Utilization bounds –…–…

Input-Output Relationships Use difference equation System Control Input u(k) Output y(k+1) y(k+1) = ay(k) + bu(k)

Model Structure and Parameters y(k+1) = ay(k) + bu(k) first order model structure Model Structure: Order (first, second, third, …) y(k+1) = a(1)y(k) + a(2)y(k-1) + b(1)u(k) + b(2)u(k-1) second order Model Parameters: a, b, a(1), a(2), b(1), b(2) Single Input Single Output Model Model: y(k)=(0.43)y(k-1) +(0.47)u(k-1)

Multiple Input-Output Models MIMO System Control Inputs u1(k) u2(k) u3(k) … Outputs y1(k+1) Y2(k+1) … y1(k+1) = a11y1(k) + a12y2(k) + b11u1(k) + b12u2(k) y2(k+1) = a12y1(k) + a22y2(k) + b21u1(k) + b22u2(k) NOTE: First Order

System ID 1.Choose inputs and outputs 2.Decide the model structure First order, second order, etc 3.Design (and run) experiments to collect data to estimate model parameters 4.Estimate the values of the parameters using least squares (Matlab) 5.Evaluate the quality of the model fit Poor - try 2 nd order model ; repeat

Comments - System ID Don’t overfit – use simplest order model that works Choose expected workloads when running experiments –Have good coverage of workloads –Mimic expected dynamics Choose good sample times when running experiments

Example From text Example done on blackboard in class regarding IBM Lotus Domino Server