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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.