Control Theory and Congestion

Glenn Vinnicombe and Fernando Paganini Cambridge/Caltech and UCLA

IPAM Tutorial – March 2002.

Outline of second part:1. Performance in feedback loops: tracking, disturbance

rejection, transient response. Integral control.2. Fundamental design tradeoffs. The role of delay. Bode

Integral formula3. Extensions to multivariable control.

Performance of feedback loops

• Stability and its robustness are essential properties; however, they are only half of the story.

• The closed loop system must also satisfy some notion of performance:– Steady-state considerations (e.g. tracking errors).– Disturbance rejection.– Speed of response (transients, bandwidth of tracking).

• Performance and stability/robustness are often at odds.• For single input-output systems, frequency domain tools

(Nyquist, Bode) are well suited for handling this tradeoff.

( )P s( )K s

Performance specs 1: Steady-state tracking

( ) ( ) ( )

Error between reference signal and output .

Tracking means this error is kept small.

e t r t y t

r y

0Suppose that ( ) ,constant, and that the system is

stable. Then as , ( ) stea( ), dy-state error.

r t r

t e t e

Ideally, we would like the steady-state error to be zero.

( )P s( )K s u yr e

Tracking, sensitivity and loop gain

( )L syr e The mapping from ( ) to ( ) has

1 transfer function ( ) .

1 ( )That is, ( ) ( ) ( ) in Laplace.

r t e t

S sL s

R s S s E s

0 0 0

Under stability, ( ) has no poles in Re[ ] 0.

1Then for ( ) , we have ( ) (0)

1 (0)

S s s

r t r e S r rL

Good steady-state tracking (0) small (0) l .argeS L

sensitivity function( ) is called the of the system. S s

Integral control

( )L syr e

Suppose ( ) has a pole at 0.

1Then (0) = 0.

1 (0)

Zero steady-state error!

L s s

SL

( ) .

.

Example: ( ) .

Loop is stable for 0, and has a pole at 0.Therefore, it has zero steady-state tracking error

t K rKL s ysK s

y

0 0 0( ) 1In the time domain: for ( ) , Kty tr t r r re

Simple congestion control exampleSingle link/source, no delays for now.

: Transmission rate (pkts/sec): Capacity of the link (pkts/sec): Queue size; assume it is fed back to source.

ycq

Source

yq

cSuppose source control is ( ), where is a decreasing function.y f q f

0 0 0 0Model: ( ) Equilibrium for : ( ) .y c f q c c c f q y cq

0 0 0

0

Linearize around it: , , .

, ( ) 0.

.

c c c y y y q q qf

y K q K qq

y cq

K

1

s

qc y

Integral Control Perfect steady-state tracking for constant .c

Performance specs 2: tracking of low-frequency reference signals.

( )L syr e Transfer function from ( ) to ( )

1is the sensitivity ( ) .

1 ( )

r t e t

S sL s

0 0

0 0 0 0

Assume the system is stable: then the steady-state

response to a sinuoidal reference ( ) cos( ) is

( ) | ( ) | cos( ( )).S

r t r t

e t r S j t

( )Let ( ) | ( ) | be the polar decomposition.SjS j S j e

0 0Good steady-state tracking | ( ) | small | ( ) .| largeS j L j

( )L j ( )S j

Tracking Large | ( ) | Small | ( ) |

in frequency range of interest.

L j S j

log | ( ) |L j log | ( ) |S j

( )L ( )S

Representation of frequency functions Bode plot

log( ) log( )

Example 2: tracking of variable references

Source

yq

c

K1

s

qc y

log | ( ) |L j

log | ( ) |S j

log

( )K

L jj

As grows,

track higher

bandwidth

K

Variations in capacity (e.g. Available Bit Rate)

Performance specs 3: disturbance rejection.

( )P s( )K s yrd

( )P s( )K s yrd

Input disturbance:

( )( )

1 ( )d yP s

sL s

T

Output disturbance:

1( )

1 ( )d y s L sT

To reject disturbances, we need attenuation in the

frequency range of interest Large | ( ) | .L j

Example 3: disturbance rejection

Source

yq

c

K1

s

qc y

log | ( ) |L j

log | ( ) |S j

log

( )K

L jj

As grows, reject

disturbances over

a higher bandwidth

K

Noise traffic generated by other uncontrolled sources.

'y'y

Performance specs 4: speed of response

Superimposed to the steady-state solutions discussed before, we have transient terms of the form Here the

modes are the roots of 1 ( ) 0.

For fast response, Re[ ] must be as ne

.ii

i

i

is tC

s L s

s

e

gative as possible.

1

Example: ( ) . 1 ( ) 0 .

The higher , the faster our transient response.

KL s L s s K

sK

0 0For instance for ( ) , solution is ( ) 1 Ktr t y tr r e

( )L syr e

( )L syr e

log | ( ) |L j

c

Transient decays in a 1

time of the order of c

For ( ) (e.g. our congestion control with queue feedback)1

decays in the order of seconds.c

KL s

sK

K

For faster response, increase the open loop bandwidth.

Heuristic look based on Fourier:

frequencies where | ( ) | 1

cannot occur (filtered out). So

the speed of response is roughly

the bandwidth where | ( ) | 1.

L j

L j

Performance specifications: recap• Tracking of constant, or varying reference signals.

• Disturbance rejection.

• Transient response.

Rule of thumb for all: increase

the gain or bandwidth of the

loop transfer function ( ).L j

What stops us from arbitrarilygood performance?Answer: stability/robustness.

Example: loop with integrator and delay. ( ) ( ) ( )t K r t ty y

se K

syr

( )s

L ss

Ke

Our

( ) ( ) (independent of d

earlier rule says: increase for pe

el

rformance.

ay!)

j

L j L jj

K

Ke K

Stability? 1 ( ) 0 0.

Transcendental equation. However, use Nyquist.

sL s s Ke

Source

yq

c(arises if we consider round- trip delay)

Example:

Stability analysis via Nyquist:

Loop function ( )sK

L ss

e

( ) , ( ) .2

KL j

0 0 0

To avoid encirclements, impose

( ) 1 at where ( )L j

2K

1

Nyquist plot of ( ) :L j

Stable for

2

K

Not much harder than analysis without delay! Much simpler than other alternatives (transcendental equations, Lyapunov functionals,…)

se K

syr

Stability in the Bode plot

0

0 0

Impose ( ) 1

at : ( )

L j

Conclusion: delay limits the achievable performance. Also, other dynamics of the plant (known or uncertain) produce a similar effect. ( )P s( )K s

log | ( ) |L j

( )L 0

( ) , ( ) .2

KL j

Increasing moves the top plot upwards.

Constraint on for stability.

K

K

The performance/robustness tradeoff

• As we have seen, we can improve performance by increasing the gain and bandwidth of the loop transfer function L(jw).

• L(s) can be designed through K(s). By canceling off P(s), one could think L(s) would be arbitrarily chosen. However:– Unstable dynamics cannot be canceled.– Delay cannot be canceled (othewise K(s) would not be causal). – Cancellation is not robust to variations in P(s).

• Therefore, the given plant poses essential limits to the performance that can be achieved through feedback.

• Good designs address this basic tradeoff. For single I-O systems,“loopshaping” the Bode plot is an effective method.

( )P s( )K s

The Bode Integral formula

( )L syr e Recall: the mapping from ( ) to ( )

1has transfer function ( ) .

1 ( )

r t e t

S sL s

For tracking, we want the sensitivity | ( ) | to be small,

for as large a frequency range as possible. How large can it be?

S j

0

( ) (Bode): Suppose ( ) , a rational function

( )

with deg( ( )) deg( ( )) 2.

Let be the set of poles of ( ) in Re[ ] 0. Then

log ( ) log( ) Re[ ]

Theorem

i

i

n sL s

d s

d s n s

p L s s

S j d e p

The Bode Integral formula.

0

log ( )

log( ) Re[ ]i

S j d

e p

log | ( ) |S j

( )P s( )K sThe unstable poles that come from the

plant ( ) cannot be eliminated by ( )

Integral of sensitivity is a conserved

quantity over all stabilizing feedbacks.

ip

P s K s

Small sensitvity at low frequencies must be “paid” by alarger than 1 sensitivity at some higher frequencies.

But all this is only linear!• The above tradeoff is of course present in nonlinear

systems, but harder to characterize, due to the lack of a frequency domain (partial extensions exist).

• So most successful designs are linear based, followed up by nonlinear analysis or simulation.

• Beware of claims of superiority of “truly nonlinear” designs. They rarely address this tradeoff, so may have poor performance or poor robustness (or both).

• A basic test: linearized around equilibrium, the nonlinear controller should not be worse than a linear design.

Multivariable control( )P s( )K s u yr e

Signals are now vector-valued (many inputs and outputs).Transfer functions are matrix-valued.

1 11 1 1

1

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

n

m m m n n

y s P s P s u s

y s P s u s

y s P s P s u s

( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

L s

y s P s K s e s I L s e s r s

e s r s y s

1

( )

( ) ( ) ( )

S s

e s I L s r s

( )P s( )K s u yr e

1( ) ( ) ( ) ( )y s L s I L s r s

1Stability: poles of ( ) (i.e., roots of det ( ) 0)

must have negative real part.

I L s I L s

Multivariable Nyquist criterion: study encirclements of

the origin of det ( ) 1 ( ) ,

where ( ) are the eigenvalues of ( ).

i

i

I L j j

j L j

Performance of multivariable loops

( )P s( )K s u yr e

1( ) ( ) ( ) ( ) ( )e j S j r j I L j r j

The tracking error will depend on frequency, but also

on the direction of the vector ( ). The worst-case

direction is captured by the maximum singular value:

( ( )) max | ( ) |: , 1 .n

r j

S j S j v v v

C

Network congestion control exampleL communication links shared by S source-destination pairs.

1 if link serves source

0 otherwise li

l iR

Routing matrix:

i uses l

i uses l

: Rate of th source (pkts/sec)

: Total rate of th link (pkts/sec)

: Capacity of the th link (pkts/sec): Backlog of the th link (pkts)

: Total backl

=

og for s= th

i

l

l

l

i

li

x i

y l

c

x

lb l

q ib

ource (pkts)

y Rx

ll l

db

dty c

Tq R b

Suppose sources receive by feedback, and set ( )i i i iq x f q

Linearized multivariable model, around equilibrium.

LINKSSOURCES

R

TR

source rates

:x: aggregate

flows per linky

: aggregate

backlog per source

q

b: link backlogs

y R x

Tq R b

i i ix K q

1l lb ys

R-KI

sTR

b q x yc

1( ) TL s RKR

s

1Now: ( ) is easily diagonalized.TL s RKR

s

, 0 ( ) .T T Tl

L L

sRKR V V L s V V

s

min

Modes: roots of det( ( )) 0 , 1,..., .

Therefore: stable if is full rank. Transient response

dominated by slowest mode, ( ).

l

T

T

I L s s l L

RKR

RKR

1

1

min

Singular values of ( ( ( )) are

1+ (l

l

S j I L j

S jj j j

Performance analysis reduces to the scalar case.

( ) Ts

L s RKRs

e

Now, consider stability in the presence of delay. For simplicity, use a common delay (RTT) for all loops.

min

max

max

Diagonalize and apply

Nyquist: Stable for

( ) 2

TRKR

min

max

Summary: performance defined by ( ), delay robustness

by ( ). Tradeoff is harder for ill-conditioned !

T

T T

RKR

RKR RKR

More generally, eigenvalues don’t tell the full story.

( )P s( )K s u yr e

Performance: for transfer functions which are not self-adjoint,( ( )) can be much larger-than the maximum eigenvalue.S j

0 Robust stability: consider a ball of plants ( ) ( ) ( ),1( ( )) . Nyquist not very useful to establish stability ( )

for all , since det( ) depends on it in a complicated way.

P s P s s

j

I KP

However, it can be shown that the condition ( ( ) ( ) ( )) < 1 gives robust stability.S j K j

Singular values are more important than eigenvalues.

Summary• A well designed feedback will respond as quickly as possible to

regulate, track references or reject disturbances.

• The fundamental limit to the above features is the potential for instability, and its sensitivity to errors in the model. A good design must balance this tradeoff (robust performance).

• In SISO, linear case, tradeoff is well understood by frequency domain methods. This explains their prevalence in design.

• Nonlinear aspects usually handled a posteriori. Nonlinear control can potentially (but not necessarily) do better. A basic test: linearization around any operating point should match up with linear designs.

• In multivariable systems, frequency domain tools extend with some complications (ill conditioning, singular values versus eigenvalues,…)

• All of this is relevant to network flow control: performance vs delay/robustness, ill-conditioning,… Nonlinearity seems mild.