Feedback Control Theory a Computer Systemʼs Perspective

Transcription

1 Feedback Corol Theory a Compuer Syemʼ Perpecve Corol Iroduco Wha feedback corol? Why do compuer yem eed feedback corol? Corol deg mehodology Syem modelg Performace pec/merc Coroller deg Summary Applyg pu o caue yem varable o coform o dered value called he referece. Crue-corol car: f_ege(? peed6 mph E-commerce erver: Reource allocao? T_repoe5 ec Embedded ework: Flow rae? Delay ec Compuer yem: QoS guaraee Ope-loop corol Feedback (cloe-loop) Corol Compue corol pu whou couou varable meaureme Smple Need o kow EVERYTHING ACCURATELY o work rgh Crue-corol car: frco(, ramp_agle( E-commerce erver: Workload (reque arrval rae? reource coumpo?); yem (ervce me? falure?) Ope-loop corol fal whe We doʼ kow everyhg We make error emao/modelg Thg chage Coroller corol fuco + - error corol pu Acuaor Moor ample Corolled Syem mapulaed varable corolled varable referece Feedback (cloe-loop) Corol Meaure varable ad ue o compue corol pu More complcaed (o we eed corol heory) Cououly meaure & correc Crue-corol car: meaure peed & chage ege force Ecommerce erver: meaure repoe me & admo corol Embedded ework: meaure collo & chage backoff wdow Feedback corol heory make poble o corol well eve f We doʼ kow everyhg We make error emao/modelg Thg chage Why feedback corol? Ope, upredcable evrome Deeply embedded ework: eraco wh phycal evrome Number of workg ode Number of ereg eve Number of hop Coecvy Avalable badwdh Cogeed area Iere: E-bue, o-le ock broker Upredcable off-he-helf hardware

2 Why feedback corol? We wa QoS guaraee Advaage of feedback corol heory Deeply embedded ework Updae ruder poo every 3 ec Repor fre < m E-bue erver Purchae compleo me < 5 ec Throughpu > raaco/ec The problem: provde QoS guaraee ope, upredcable evrome Adapve reource maageme heurc Laborou deg/ug/eg erao No eough cofdece face of ueed workload Queug heory Doeʼ hadle feedback No good a characerzg rae behavor overload Feedback corol heory Syemac heorecal approach for aaly ad deg Predc yem repoe ad ably o pu Oule Corol deg mehodology Iroduco Wha feedback corol? Why do odayʼ compuer yem eed feedback corol? Corol deg mehodology Syem modelg Performace pec/merc Coroller deg Summary Modelg aalycal yem ID Dyamc model Coroller Deg Roo-Locu PI Corol Corol algorhm Safy Requreme Aaly Performace Specfcao Syem Model Dyamc Model Lear v.. o-lear (dffereal eq Deermc v.. Sochac Tme-vara v.. Tme-varyg Compuer yem are dyamc Curre oupu deped o hory Characerze relaohp amog yem varable Dffereal equao (me doma) Are coeffce fuco of me? Couou-me v.. Dcree-me Syem ID v.. Fr Prcple ( a y + a y( + a y( b u( + b u( ) Trafer fuco (frequecy doma) Y( G(U( b + b c c G( + a + a + a! p! p Block dagram (pcoral) R( - C( G( Y(

3 Example Ulzao corol a vdeo erver Perodc ak T correpodg o each vdeo ream c[]: proceg me, p[]: perod Sream ʼ requeed CPU ulzao: u[]c[]/p[] Toal CPU ulzao: U(Σ {k} u[k], {k} he e of acve ream Compleo rae: R c ( (Σ {kc kc}u[m])/ u[m])/δ, where {m} he e of ermaed vdeo ream durg [, +Δ] Ukow Admo rae: R ( (Σ a {ka} u[j])/δ, where {j} he e of admed ream durg [, +Δ] Problem: deg a admo coroller o guaraee U(U regardle of R c ( Model Dffereal equao Error: E(U -U( Model (dffereal equao): U (!( (# ) " Rc (# )) d# Coroller C? E( R a ( C? - U U( CPU R a ( # R c ( A Dvero o Mah Syem repreeao Three way of yem modelg A Dvero o Mah Laplace raform Laplace raform of a gal f( Tme doma: covoluo; dffereal equao. u( g( y( y( g( * u(! g( "# ) u( # ) d# (frequecy) doma: mulplcao U( G( Y( Y ( G( U ( Block dagram: pcoral -doma a mple & powerful laguage for corol aaly F " # ( L[ f ( ]! f ( e d # where σ+ω a complex varable. Laplace raform a ralao from me-doma o -doma Dffereal equao Polyomal fuco ( a y + a y( + a y( b u( + b u( ) b + b! Y ( U ( a + a + a A Dvero o Mah Laplace raform Bac ralao Impule fuco f(δ( F( Sep gal f(a( F(/ mp gal f(a F(a/ Exp gal f(e a F(/(-a) Suod gal f((a F(a/( +a ) Compoo rule Leary Dffereao Iegrao L[ f(τ)d L[af(+bg(] )] al[f(]+bl[g(] L[df(/d] F( f( - ) )dτ] F(/ A Dvero o Mah Trafer fuco Modelg a lear me-vara (LTI) yem G( Y(/U( Y( G(U( U( G( Y( E.g., a ecod order yem wh pole p ad p b + b G( a + a + a c c +! p! p 3

4 A Dvero o Mah Pole ad Zero The repoe of a lear me-vara (LTI) yem bm F( a $ K $ # f ( m m! + b + a m" m" " " b a ( " z ) C C + ( " p ) " p " p C e p C " p A Dvero o Mah Tme repoe v. pole locao Sable Uable {p } are pole of he fuco ad decde he yem behavor f ( e p, p a+bj A Dvero o Mah Block dagram A pcoral ool o repree a yem baed o rafer fuco ad gal flow Repree a feedback corol yem R( R( - C( G c ( G o ( Y( Y( C( Go ( Gc + C( G ( Y ( G ( R( c o Back o Our ulzao corol example Error: E(U -U( Model (dffereal equao): U (!( (# ) " Rc (# )) d# Coroller C? E( R a ( C? - U U( CPU R a ( # R c ( Model Trafer fuc. & block dag. Corol deg mehodology CPU modeled a a egraor ( " ( U ( #( ( $ ) " Rc ( $ )) d$! U (! Go ( Ipu: $ referece U ( U /; compleo rae R c ( Cloe-loop yem rafer fuco U ( a pu: G o o R c ( G o o Oupu: U(G (U /+G (R c ( Modelg aalycal yem ID Dyamc model Coroller Deg Roo-Locu PI Corol Corol algorhm Safy R c ( Requreme Aaly Performace Specfcao U / C( R a ( G o U( 4

5 Deg Goal Performace Specfcao Sably Trae repoe Seady-ae error Robue Durbace rejeco Sevy Performace Spec: bouded pu,bouded oupu ably BIBO ably: bouded pu reul bouded oupu. A LTI yem BIBO able f all pole of rafer fuco are he LHP ( p, Re[p ]<). & Y ( G( U ( K & $ y( ' Noe: C e p C e p m #! " "" #! ( % z ) C C C ( % p ) % p % p % p f Re[ p ] > Performace Spec Sably Performace pecfcao Corolled varable Sable Uable Referece Overhoo Seady ae error ±ε% Trae Sae Seady Sae Selg me Tme Example: Corol & Repoe a Emal Server (IBM) Repoe (queue legh) Performace Spec Seady-ae error Seady ae (rackg) error of a able yem Good Corol (MaxUer Bad e lme( lm( r(! y( ) #" #" r( he referece pu, y( he yem oupu. How accuraely ca a yem acheve he dered ae? Fal value heorem: f all pole of F( are he ope lef-half of he -plae, he Slow Uele lm f ( lm F(!"! Eay o evaluae yem log erm behavor whou olvg e lme( lm E(!"! 5

6 Performace Spec Seady-ae error Performace Spec Robue Seady ae error of a CPU-ulzao corol yem Durbace rejeco: eady-ae error caued by exeral durbace U( U e -% Ca a yem rack he referece pu depe of exeral durbace? Deal-of-ervce aack Sevy: relave chage eady-ae oupu dvded by he relave chage of a yem parameer Ca a yem rack he referece pu depe of varao he yem? Icreaed ak execuo me Devce falure Performace Spec Goal of Feedback Corol Corol deg mehodology Guaraee ably Improve rae repoe Shor elg me Small overhoo Small eady ae error Improve robue wr ucerae Durbace rejeco Low evy Modelg aalycal yem ID Dyamc model Coroller Deg Roo-Locu PID Corol Corol algorhm Safy Requreme Aaly Performace Specfcao Coroller Deg PID corol Proporoal-Iegral-Dervave (PID) Corol Proporoal Corol Iegral corol Dervave corol x ( Ke(! C( K x( KK " e( # ) d#! C( x( KK d e(! C( KK d KK Clacal coroller wh well-uded propere ad ug rule Coroller Deg CPU Ulzao Corol CPU modeled a a egraor ( " ( U ( #( ( $ ) " Rc ( $ )) d$! U (! Go ( Ipu: $ e-po U ( U / ; ak compleo R c ( Cloe-loop yem rafer fuco U ( a pu: G o o R c ( G o o C(? o acheve zero eady-ae error: U( U R c ( R( E( C( - X( G o ( Y( U / C( R a ( G o U( 6

7 Proporoal Corol Sably Proporoal Coroller r a (Ke(; C( K Trafer fuco U / a pu: G ( K/(+K) R c ( a pu: G ( /(+K) Sably Pole p -K< Syem BIBO able ff K> Noe: Syem may hoo o % f K<! U / C( R a ( R c ( G o U( Proporoal Corol Seady-ae error Aume compleo rae R c ( keep coa for a me perod loger ha he elg me: R (R c c / Syem repoe U G ( RcG ( KU! Rc U ( + ( + K) Compue eady-ae err ug fal value heorem, KU! Rc Rc lmu ( lm U ( lm U! $ # # + K K " e Rc! < # K P-corol cao acheve he dered CPU ulzao U ; ead wll ed up lower by R c /K Oop! The larger he proporoal ga K, he cloer wll CPU ulzao approach o U U( U CPU Ulzao Proporoal Corol e -% Proporoal-Iegral Corol Sably Proporoal Coroller r (K(e(+K a e(τ)d )dτ) Trafer fuco C( K(+K / U / a pu: G ( (K+KK )/( +K+KK ) R c ( a pu: G ( /( +K+KK ) Sably Pole Re[p ]<, Re[p ]< Syem BIBO able ff K> & K > R c ( U / C( R a ( G o U( Proporoal Corol Seady-ae error CPU Ulzao Proporoal-Iegral Corol Aume compleo rae R c ( keep coa for a me perod loger ha he elg me: R (R c c / Syem repoe U G ( RcG ( ( KU + Rc ) + KKU U ( + ( + K + KK ) U( U M p e Compue eady-ae err ug fal value heorem, ( KU + Rc ) + KKU lmu ( lm U ( lm U "# " " + K + KK! e PI corol ca accuraely acheve he dered CPU ulzao U Corol aaly gve deg gudace r p 7

8 Coroller Deg Summary & poer PID corol: mple, work well may yem P corol: may have o-zero eady-ae error I corol: mprove eady-ae rackg D corol: may mprove ably & rae repoe Lear couou me corol Roo-locu deg Frequecy-repoe deg Sae-pace deg G. F. Frakl e. al., Feedback corol of dyamc yem Dcree Corol More ueful for compuer yem Tme dcree; ampled yem deoed k ead of Ma ool z-raform f(k) F(z), where z complex Aalogou o Laplace raform for -doma Z[ f ( k)] F( z)! " k f ( k) z # k Dcree Modelg Dfferece equao V(m) ) a V(m-) + a V(m-) + b U(m-) + b U(m-) z doma: V(z) ) a z - V(z) ) + a z - V(z) ) + b z - U(z) ) + b z - U(z) Trafer fuco G(z) ) (b z + b )/(z -a z - a ) V(m): oupu m h amplg wdow U(m): pu m h amplg wdow Order : : #amplg-perod hory affec curre performace SP 3 ec, ad Curre yem performace deped o prevou 6 ec Roo Locu aaly of Dcree Syem Sably boudary: z (U crcle) Selg me dace from Org Speed locao relave o Im ax Rgh half lower Lef half faer Effec of dcree pole Feedback corol work CS Hgher-frequecy repoe Uable Im( Sable z Loger elg me Re( U.Ma: ework flow coroller (TCP/IP RED) IBM: Lou Noe admo corol UIUC: Drbued vual rackg UVA Web Cachg QoS Apache Web Server QoS dffereao Acve queue maageme ework Proceor hermal corol Ole daa mgrao ework orage (wh HP) Real-me embedded eworkg Corol mddleware Feedback corol real-me chedulg T Iuo : z e 8

9 Advaced Corol Topc Robu Corol Ca he yem olerae oe? Adapve Corol Coroller chage over me (adap MIMO Corol Mulple pu ad/or oupu Sochac Corol Coroller mmze varace Opmal Corol Coroller mmze a co fuco of error ad corol eergy Nolear yem Neuro-fuzzy corol Challegg o derve aalyc reul Iue for Compuer Scece Mo yem are o-lear Bu lear approxmao may do eg, flud approxmao Fr-prcple modelg dffcul Ue emprcal echque Mappg corol objecve o feedback corol loop CorolWare paper Deeply embedded eworkg Mavely deceralzed corol problem Modellg Node falure 9