Control Theory and Congestion

Transcription

1 Control Theor and Congetion Glenn Vinnicombe and Fernando Paganini Cambridge/Caltech and UCLA IPAM Tutorial March Outline of econd part: 1. Performance in feedback loop: tracking, diturbance rejection, tranient repone. Integral control. 2. Fundamental deign tradeoff. The role of dela. Bode Integral formula 3. Extenion to multivariable control.

2 Performance of feedback loop ( ) P ( ) Stabilit and it robutne are eential propertie; however, the are onl half of the tor. The cloed loop tem mut alo atif ome notion of performance: Stead-tate conideration (e.g. tracking error). Diturbance rejection. Speed of repone (tranient, bandwidth of tracking). Performance and tabilit/robutne are often at odd. For ingle input-output tem, frequenc domain tool (Nquit, Bode) are well uited for handling thi tradeoff.

3 Performance pec 1: Stead-tate tracking r e ( ) u P ( ) et () = rt () t () Error between reference ignal r and output. Tracking mean thi error i kept mall. Suppoe that rt ( ) = r 0,contant, and that the tem i table. Then a t, e( t) e( ), tead-tate error. Ideall, we would like the tead-tate error to be zero.

4 Tracking, enitivit and loop gain r e The mapping from rt ( ) to et ( ) ha L( ) 1 tranfer function S ( ) =. 1 + L ( ) That i, R ( ) = SE ( ) ( ) in Laplace. S ( ) i called the enitivit function of the tem. Under tabilit, S ( ) ha no pole in Re[ ] 0. 1 Then for rt ( ) r, we have e( ) = S(0) r= r 1 + L(0) Good tead-tate tracking S(0) mall L(0) l arge.

5 r e L( ) Integral control Suppoe L( ) ha a pole at = 0. 1 Then S(0) = = L(0) Zero tead-tate error! Example: L ( ) =. () t = ( r ). Loop i table for > 0, and ha a pole at = 0. Therefore, it ha zero tead-tate tracking error. ( e ) In the time domain: for ( ), t () = 1 t rt r r r 0 0 0

6 Simple congetion control example : Tranmiion rate (pkt/ec) c: Capacit of the link (pkt/ec) Source q q: Queue ize; aume it i fed back to ource. c Suppoe ource control i = f( q), where f i a decreaing function. Single link/ource, no dela for now. Model: q = c= f( q) c Equilibrium for c= c: f( q ) = = c. Linearize around it: c= c0+ δc, = 0+ δ, q= q0+ δq. f δ= δq, = ( q0 ) > 0. δc 1 δ q q = c. δq δ δ δ Integral Control Perfect tead-tate tracking for contant δc.

7 Performance pec 2: tracking of low-frequenc reference ignal. r e L( ) Tranfer function from rt ( ) to et ( ) 1 i the enitivit S ( ) =. 1 + L ( ) j S ( ) Let S( jω) = S( jω) e φ ω be the polar decompoition. Aume the tem i table: then the tead-tate repone to a inuoidal reference rt ( ) = rco( ω t) i et () = r S( jω ) co( ω t+ φ ( ω )) S Good tead-tate tracking S( jω ) mall L( jω ) large. 0 0

8 Repreentation of frequenc function Bode plot L( jω ) S( jω ) log L( jω ) log S( jω ) φl( ω ) φs ( ω ) log( ω) log( ω) Tracking Large L( jω ) Small S( jω ) in frequenc range of interet.

9 Example 2: tracking of variable reference δc 1 δ q δ Source q Variation in capacit (e.g. Available Bit Rate) c log L( jω ) L( jω ) = jω log S( jω ) A grow, track higher bandwidth logω

10 Performance pec 3: diturbance rejection. r ( ) d P ( ) Input diturbance: P () Td () = 1 + L ( ) r ( ) P ( ) d Output diturbance: 1 Td () = 1 + L( ) To reject diturbance, we need attenuation in the frequenc range of interet Large L( jω ).

11 δc Example 3: diturbance rejection 1 δ q δ δ ' Source Noie traffic generated b other uncontrolled ource. q c ' log L( jω ) L( jω ) = jω log S( jω ) logω A grow, reject diturbance over a higher bandwidth

12 Performance pec 4: peed of repone r e L( ) i i Superimpoed to the tead-tate olution dicued before, t we have tranient term of the form Ce i. Here the mode are the root of 1 + L( ) = 0. i For fat repone, Re[ ] mut be a negative a poible. i Example: L( ) =. 1 + L( ) = 0 1 =. The higher, the fater our tranient repone. For intance for rt ( ), olution i t ( ) = i i ( t ) r r e

13 r e L( ) Heuritic look baed on Fourier: frequencie where L( jω ) << 1 log L( jω ) cannot occur (filtered out). So the peed of repone i roughl the bandwidth where L( jω ) 1. Tranient deca in a 1 time of the order of ω c ω c For L ( ) = (e.g. our congetion control with queue feedback) 1 c = deca in the order of econd. For fater repone, increae the open loop bandwidth. ω

14 Performance pecification: recap Tracking of contant, or varing reference ignal. Diturbance rejection. Tranient repone. Rule of thumb for all: increae the gain or bandwidth of the loop tranfer function L( jω ). What top u from arbitraril good performance? Anwer: tabilit/robutne.

15 Example: loop with integrator and dela. () t = ( r() t ( t τ )) r e τ Source Example: q L ( ) Our = e τ e τ jω L( jω) = L( jω) = (independent of dela!) jω ω earlier rule a: increae L e τ (arie if we conider round- trip dela) for performance. Stabilit? 1 + ( ) = 0 + = 0. Trancendental equation. However, ue Nquit. c

16 r Stabilit anali via Nquit: e τ e τ Loop function L ( ) = π L( jω ) =, φω ( ) = ωτ. ω 2 To avoid encirclement, impoe L( jω ) < 1 at ω where φ( ω ) = π Nquit plot of L( jω ) : 1 2 πτ Not much harder than anali without dela! Much impler than other alternative (trancendental equation, Lapunov functional, ) Stable for < π 2τ

17 Stabilit in the Bode plot π L( jω ) =, φω ( ) = ωτ. ω 2 log L( jω ) Impoe L( jω ) < 1 at ω : φω ( ) = π 0 0 Increaing move the top plot upward. Contraint on 0 φl( ω ) for tabilit. Concluion: dela limit the achievable performance. Alo, other dnamic of the plant (known or uncertain) produce a imilar effect. P() () ω 0

18 The performance/robutne tradeoff ( ) P ( ) A we have een, we can improve performance b increaing the gain and bandwidth of the loop tranfer function L(jw). L() can be deigned through (). B canceling off P(), one could think L() would be arbitraril choen. However: Untable dnamic cannot be canceled. Dela cannot be canceled (othewie () would not be caual). Cancellation i not robut to variation in P(). Therefore, the given plant poe eential limit to the performance that can be achieved through feedback. Good deign addre thi baic tradeoff. For ingle I-O tem, loophaping the Bode plot i an effective method.

19 The Bode Integral formula r e Recall: the mapping from rt () to et () L( ) 1 ha tranfer function S ( ) =. 1 + L ( ) For tracking, we want the enitivit S( jω ) to be mall, for a large a frequenc range a poible. How large can it be? Theorem { } 0 n ( ) (Bode): Suppoe L ( ) =, a rational function d( ) with deg( d( )) deg( n( )) 2. Let p be the et of pole of L( ) in Re[ ] > 0. Then i log S( jω) dω = π log( e) Re[ pi ]

20 The Bode Integral formula. log S( jω ) dω = log S( jω ) 0 π log( e) Re[ p ] i ω The untable pole that come from the p i plant P ( ) cannot be eliminated b ( ) () P() Integral of enitivit i a conerved quantit over all tabilizing feedback. Small enitvit at low frequencie mut be paid b a larger than 1 enitivit at ome higher frequencie.

21 But all thi i onl linear! The above tradeoff i of coure preent in nonlinear tem, but harder to characterize, due to the lack of a frequenc domain (partial extenion exit). So mot ucceful deign are linear baed, followed up b nonlinear anali or imulation. Beware of claim of uperiorit of trul nonlinear deign. The rarel addre thi tradeoff, o ma have poor performance or poor robutne (or both). A baic tet: linearized around equilibrium, the nonlinear controller hould not be wore than a linear deign.

22 r Multivariable control e ( ) u P( ) Signal are now vector-valued (man input and output). Tranfer function are matrix-valued. 1() P11() P1n () u1() ( ) = = = P( ) u( ) m() Pm1 () Pm n() un() L () () = Pe () ()() [ I + L ()] e () = r () e () = r () ()

23 r e ( ) u P( ) [ ] 1 e () = I+ L () r () S ( ) [ ] 1 () = L() I + L() r() 1 [ I L ] [ I L ] Stabilit: pole of + ( ) (i.e., root of det + ( ) = 0) mut have negative real part. Multivariable Nquit criterion: tud encirclement of the origin of det [ I + L( jω) ] = ( 1 + λ( )) i jω, where λ( jω) are the eigenvalue of L( jω). i

24 Performance of multivariable loop r e ( ) u P ( ) [ ] 1 e( jω ) = S( jω) r( jω) = I + L( jω) r( jω) The tracking error will depend on frequenc, but alo on the direction of the vector r( jω ). The wort-cae direction i captured b the maximum ingular value: { C n } σ( S( jω)) = max S( jω) v : v, v = 1.

25 Network congetion control example L communication link hared b S ource-detination pair. R li 1 if link l erve ource 0 otherwie = Routing matrix: i x: Rate of i th ource (pkt/ec) i l l cl b = i ue l i ue l i : Total rate of l th link (pkt/ec) : Capacit of the l th link (pkt/ec) : Backlog of the l th link (pkt) q= b : Total backlog for i th ource (pkt) i x l db dt l = Rx = c q= ( ) l T l R b Suppoe ource receive q b feedback, and et x = f ( q) i i i i

26 Linearized multivariable model, around equilibrium. q x : δ x = δ q ource rate i i i : aggregate backlog per ource δc δ I SOURCES T q= R δ b δb T R δ q R T R - δ x LINS R δ = Rδ x b: link backlog δ : aggregate flow per link δ b L ( ) = l = 1 1 δ RR l T

27 1 T Now: L ( ) = RR i eail diagonalized. λ1 0 0 λ1 0 0 T T T RR = V 0 0 V, λl 0 L( ) = V 0 0 V. 0 0 λ L λ L 0 0 i Mode: root of det( I + L ( )) = 0 = λ, l= 1,..., L. T Therefore: table if RR i full rank. Tranient repone T dominated b lowet mode, λmin( RR ). 1 i Singular value of S( jω)= ( I + L( jω)) are 1 λl ω ω 1+ = σ ( S( jω) ) = jω λ + jω λ + jω l Performance anali reduce to the calar cae. min l

28 Now, conider tabilit in the preence of dela. For implicit, ue a common dela (RTT) for all loop. L( ) = e τ RR T λ min ω Diagonalize and appl Nquit: Stable for T π λmax ( RR ) < 2 τ λ max ω T Summar: performance defined b λ ( RR ), dela robutne T T b λ ( RR ). Tradeoff i harder for ill-conditioned RR! max min

29 More generall, eigenvalue don t tell the full tor. r e ( ) u P ( ) i Performance: for tranfer function which are not elf-adjoint, σ( S( jω)) can be much larger-than the maximum eigenvalue. i Robut tabilit: conider a ball of plant P ( ) = P0 ( ) + ( ), σ( ( jω)) 1. Nquit not ver ueful to etablih tabilit αω ( ) for all, ince det( I+ P) depend on it in a complicated wa. However, it can be hown that the condition σ( S( jω) ( jω) α( ω)) < 1 ω give robut tabilit. Singular value are more important than eigenvalue.

30 Summar A well deigned feedback will repond a quickl a poible to regulate, track reference or reject diturbance. The fundamental limit to the above feature i the potential for intabilit, and it enitivit to error in the model. A good deign mut balance thi tradeoff (robut performance). In SISO, linear cae, tradeoff i well undertood b frequenc domain method. Thi explain their prevalence in deign. Nonlinear apect uuall handled a poteriori. Nonlinear control can potentiall (but not necearil) do better. A baic tet: linearization around an operating point hould match up with linear deign. In multivariable tem, frequenc domain tool extend with ome complication (ill conditioning, ingular value veru eigenvalue, ) All of thi i relevant to network flow control: performance v dela/robutne, ill-conditioning, Nonlinearit eem mild.