slides sjtu2008

8/6/2019 Slides SJTU2008

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Controlling

AcrossNetworks

Controlling Across Networks

Chaouki T. Abdallah

Professor & ChairECE Department, The University of New Mexico

Talk Given at Shanghai Jiao Tong University

http://find/http://goback/


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Controlling

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The Really Big Picture



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Controlling

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The Big Picture

We live and operate in a networked world.

Networks provide a powerful metaphor for describing

system behavior in biology, computer science, physics,

social science, and engineering.Complex networks are being studied for the purpose of

gaining insight into how properties such as community

structure and small-world effects emerge.

In modern cars and airplanes, as well as in networked

homes and office buildings, modern control systems

are increasingly incorporating communication networks

in feedback loops.



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Controlling

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Networks and Control

Control engineers are thus forced to expand their

application domain by incorporating the communication

infrastructure into their designs, and by considering the

impact of link capacity, latency, and packet loss on the

performance of feedback control.

In this talk, ...



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Controlling

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The Need to Study Networks

The increasing complexity and interconnectedness of

energy, telecommunications, transportation, and financial

infrastructures pose new challenges for secure, reliable

management and operation.1

Complex interactive networks are omnipresent and critical

to economic and social well-being.

M. Amin, National Infrastructures as Complex Interactive

Networks, Chapter 14 in Automation, Control, andComplexity: An Integrated Approach, T. Samad and J.

Weyrauch, Eds. (John Wiley and Sons, NY, 2000).



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Controlling

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Obvious Questions

How many of us would choose an old and more

expensive computer over a modern and cheaper one,

IF the first can be networked while the second can not?

It is the obviously the network!

What connects us makes us stronger (the whole is

more than the sum of it parts), but also more vulnerable

(viruses, marketing, etc.)

BUT, suppose you understand how networks come to

be, their structure, and some of their hidden properties:Can you use that knowledge to design better processes

over the network?



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Controlling

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Networks as Communications Media

Connectedness: which expresses the existence of a

path between the information transmitter and the

information receiver.

Navigability: quantified by the difficulty of finding a

connecting path. Typically, this difficulty depends onwhether the path is predetermined, or whether it is

discovered in an ad hoc fashion.

Efficiency: as represented by the latency (delays) of

each utilized path. This latency, usually a function ofthe number of hops and the individual link latencies,

must be sufficient to guarantee desired end-to-end

communication latencies.



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Controlling

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Connectedness

Connectedness is identical to the mathematical notion of

percolation. Percolation is illustrated by a wildfire, initiated

at a source vertex, spreading across an edge connected toa burning vertex with a fixed probability By analyzing the

number of vertices reached by the process, it is possible to

determine whether there exists a path connecting a given

pair of nodes.



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Controlling

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Power Laws in Networks

One common feature of many real world networks is a

power-law degree distribution, in which the probability of a

randomly chosen vertex having k neighbors scales as

p(k) k

(1)

The ubiquity of the power-law degree distribution has led

network theorists to focus on graph models that exhibit this

feature, but whose topological structure is otherwise

random. Obviously, a network with many redundant pathsbetween all pairs of vertices becomes more robust to node

and edge failures of all kinds.



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Controlling

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Random Newtorks

Figure: Random Networks



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Controlling

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Power Laws

Figure: Random Networks



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Controlling

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Navigability

Given that a network is connected, several paths may

typically connect a transmitter with a receiver. In network

theory, a networks navigability is determined both by how

easily such a path can be found, and how many hops

(delays!) such a path ultimately requires. Solutions can be

grouped into two categories: Central authorities, in which

the communication path between two vertices is determined

by an external source, later mirrored by the networks

routers, and Decentralized techniques, in which routingdecisions are made independently by network routers,

possibly in an ad-hoc fashion.



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Controlling

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Efficiency

As far as control design is concerned, a communication

channel is merely a medium for obtaining or sending

information (measurement signals, or control commands).

From this perspective, what seems to be important is: how

much information can be carried, and how fast can it be

transferred.



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Controlling

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Our Focus

In the remainder of the talk, I will focus on the issues of

source encoding in order to send output and control signalsacross a packet network.

I will not discuss the issues of connectedness or navigability



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Controlling

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Closed-Loop Type I

LTI

ENCODER

NETWORK

DECODER

CONTROL

Rate: Rppackets/time_unit

Figure: Closed-loop network control system: Type 1



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Controlling

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Closed-Loop Type II

ENCODER

NETWORK

1

DECODER

ENCODER

NETWORK

2

DECODER

CONTROL

LTI

Rate: Rp1 packets/time_unit

Rate: Rp2 packets/time_unit

Figure: Closed-loop network control system: Type 2



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Controlling

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Fundamental Results

Let the system be

x(k+ 1) = Ax(k) + Bu(k); y(k) = Cx(k)

Theorem (Data Rate Theorem)The minimum required rate for stabilization is given by

R >n

i=1

log2(|i(A)|); (2)

wherei are the eigenvalues of an open-loop discrete linearsystem.



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Controlling

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Fundamental Results

Theorem (Tatikonda)

For system with(A,B) a stabilizable pair, a necessary

condition on the channel capacity for almost surelyasymptotic stabilizability is that

C (A)

max { 0, log2 |(A)| }

.



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Controlling

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Specific Problem

Consider the discrete LTI system given by

x(k+ 1) = Ax(k) + Bu(k) (3)

where A is n n and we assume that it is diagonalA = diag(1, . . . , n) and |j| 1, j {1, . . . , n}, andi = j if j = i, x(k) is n 1, B is n m and u(k) is m 1.



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ControllingAcross

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Assumptions

Assumptions

The packet based network considers a packet size of

Dmax bits used for data.

Noiseless network.

The controller does not saturate.

There are not packet losses in the network.

Synchronization between encoder and decoder: the

decoder knows exactly both the sign and the position ofeach significant bit when it is encoded.



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ControllingAcross

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Notation

log() represents log2().

The norm symbol (.) denotes the Euclidean norm.

. is the ceil function.

The variable to denote the controllability index.



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ControllingAcross

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Background: Previous Results

Theorem

(L. Shi, R. Murray) Assuming B, C are invertible and the

system dimension isn. Then a sufficient condition for the

closed loop exponential stability is that the networkparameters and the system parameters satisfy the

inequality below

|A| 2R1

n + |B| B1A 2R2

n < 1



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ControllingAcross

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Disadvantage of the Result

Disadvantages

The assumption of an invertible B is very conservative.

Moreover, the idea of state augmentation for the

time-delay consideration is not longer valid since the

augmented B is, in general, not invertible.

We focus our work in removing this constraint.



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ControllingAcross

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Results: NCS Type I

For a Closed-Loop Type I, we have the following result.

Theorem

Assuming an equal allocation of bits per state component, a

network rate, Rp of packets/bits, and(A,B) is a controllable

pair with controllability index, a sufficient condition forsystem (3) to be asymptotically stabilizable is

Rp

R

DMax

,

where R = n log (A) + 1 and every state can allocate Rn

bits/sample.


R l NCS T I


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ControllingAcross

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Results: NCS Type I

Corollary

Assuming an equal allocation of bits per state component

and(A,B) is a controllable pair, where B isn 1 and the

control law, u(k), is1 1, a sufficient condition for system(3) to be asymptotically stabilizable is

Rp

R

DMax

,

where R = n log (An) + 1 and every state allocates Rn

bits/sample.


R lt NCS T I Ti D l


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ControllingAcross

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Results: NCS Type I: Time Delay

Let us consider the network control System type 1 and the

discrete LTI system given by the following equation:

x(k+ 1) = Ax(k) + Bu(k p) (4)

where A = diag(1, . . . , n) and |j| 1, j {1, . . . , n}, andi = j if j = i, x(k) is n 1, B is n 1 and u(k) is 1 1 andp N is the time delay. We assume here that the delay is a

constant equal to p time-steps even though that the networkprobably imparts a time-varying and random delay.


R lt NCS T I Ti D l


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ControllingAcross

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Results: NCS Type I: Time Delay

Theorem

Assuming an e.a.b per state component, a network rate of

Rp =

RDMax

packets/time-step, and(A,B) is a controllable

pair. A sufficient condition for system (4) to be

asymptotically stabilizable is

R (n + p)

log(An+p) + 1

whereA =

A B 0 . . . 00 0 1 . . . 00 0 0 . . . 0

1

0 0... . . . 0

andB =

0

0

0...

1

and every state

can allocate Rn+p bits/sample..


R lt NCS T II


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ControllingAcross

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Results: NCS Type II

Theorem

Assuming an equal allocation of bits per mode and (A,B) isa controllable pair, where B isn 1 and the control law, u(k),

is1 1, a sufficient condition for system (3) to beasymptotically stabilizable is

An 2R1

n+1 +

1A

2R2+1 < 1

where =B| AB| . . . | An1B

.


Simulations: Example 1


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ControllingAcross

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First, we tested the results of Theorem 1 for the system:

x(k+ 1) =

1 0 0

0 3 0

0 0 4

x(k) +

1 0

1 1

0 1

u(k) (5)

With initial condition x(0) =

1.333.7688.44

.

The rate obtained is R/n = 6 bit/time-step and thesimulation for such a rate is shown in the next figure.




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ControllingAcross

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0 2 4 6 8 10 12

0

10

20

30

40

50

60

Time Step

States

System Evolution (Using R/n = 6 bit/timestep)

x1(k)

x2(k)

x3(k)

Figure: (Type 1): Multi-Input Case using Rn

= 6 bits/time-step.




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We considered a single-input system given by:

z(k+ 1) =

20 0 100 10 0

0 10 30

z(k) +

11

1

u(k) (6)

Using a state-space transformation, we diagonalized thesystem to obtain:

x(k+ 1) =

20 0 0

0 10 0

0 0 30

x(k) +

1.000

2.1211.225

u(k) (7)

We assume the initial condition to be x(0) =

1.333.768

8.44

.

We get R/n = 16 bit/time-step.




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0 5 10 156000

4000

2000

0

2000

4000

6000

8000

10000

12000

14000

Time Step

States

System Evolution (R/n = 16 bit/timestep)

x1(k)

x2(k)

x3(k)

Figure: (Type 1): Single Input Case using Rn

= 16. bit/time-step.


Is the result conservative?


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ControllingAcross

Networks

Is the result conservative?

0 5 10 15 20 256

4

2

0

2

4

6

8

10

12

x 104

Time Step

States

System Evolution (Using R/n = 14 bits/timestep)

x1(k)

x2(k)

x3(k)

Figure: (Type 1): Single Input Case using Rn

= 14. bit/time-step.


Simulations: Example 3 (Time Delay)


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ControllingAcross

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Simulations: Example 3 (Time Delay)

Let us finally consider a system with time-delay p = 2evolving according to the following dynamics:

x(k+ 1) =2 0

0 1.5

x(k) +1

1

u(k 2) (8)

with the initial condition state vector x(0) =

1.33

30.768

.

For this system, our result gives a rate bounded below byR/n = 6 bit/time-step.




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ControllingAcross

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0 2 4 6 8 10 12 14 16 1850

0

50

100

150

200

250

300

350

400

TimeStep

State

s

System Evolution (Using R/(n+p) = 6 bit/timestep)

x1(k)

x2(k)

Figure: Closed-loop NCS with Time-Delay: Type 1


Playing Communications against Control


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ControllingAcross

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Playing Communications against Control

By modifying the communication protocol, Tanner shows

that velocity synchronization in a connected group of

autonomous mobile agents, may still be achieved when the

agent controllers use delayed information, regardless of the

size of this delay, if control and communication are properly

interleaved.

Tanner used the composition properties of graphs are used

to show that under certain assumptions on the

communication topology, delays may have no effect onstability.


Designing with Delays


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ControllingAcross

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Consider the following second-order linear, time-invariant

plant:

H(s) =Y(s)

U(s)=

1

s2 + w2n(9)

where wn is the natural frequency of the system. Let astatic, output-feedback delay compensator be given by:

U(s) = C(s)U(s); C(s) = kes (10)

where k and are both design parameters. The closed-looptransfer function is given by:

Y(s)

R(s)=

kes

s2 + w2n kes

(11)




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ControllingAcross

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Let us consider the Nyquist plot of the open-loop system

H(s)C(s) given by:

H(jw)C(jw) =

Kejw

w2n w2 (12)

We divide the Nyquist graph of (12) into three regions: the

first, when the frequency is less than the natural frequency,

or w < wn; the second, when they are equal, i.e. w = wn;

and the third when the frequency is greater than the naturalfrequency, so that w > wn.




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Note that the magnitude H(jw)C(jw)w=wn is infinite, so ouranalysis focuses on the cases were w = wn. For thoseregions, the magnitude of the open-loop gain is:

|G(jw)C(jw)| =K

w2n w

2; 0 w < wn

|G(jw)C(jw)| =K

w2 w2n; w > wn

(13)

and its phase is given by

(w) = w;for0 w < wn

(w) = w;for w > wn(14)




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ControllingAcross

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s g g ays

The intersections of the polar plot with the negative real axis

take place at the frequencies wc where

wc =

2n

, 0 w

c wn

(15)

In order to guarantee asymptotic stability of closed-loop

system, the magnitude |G(jw)C(jw)| evaluated at wc must beless than 1 so that the -1 point is not encircled.




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ControllingAcross

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g g y

Therefore:

k

w2n (2n)2/2

< 1, for0 2n/ < wn

k

((2n + 1))2/2 w2n

< 1, for(2n + 1)/ > wn

(16)

Combining the last two conditions we find the lower and

upper bounds of the stability region for positive gain k > 0

2nw2n k < >(2n 1)

w2n + k

0 > k 1 + 4n

1 + 4n + 8n2w2n

(18)

Combining the stability regions for positive and negative

gains (and positive-negtaive delays) , we obtain the graphshown below for w2n = 1.



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Figure: Stability Regions with Positive and Negative Delay




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g g y

As expected, the stabilizing gain regions decrease as the

delay increases along the vertical axis. Using this simple

graphical tool, we may choose the delay and gain values to

guarantee that the closed-loop system is stable. In order toillustrate the process, assume that the open-loop plant is

described by the transfer function:

H(s) =1

s2

+ 1

(19)




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ControllingAcross

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Due to the connecting network, the output signal is subject

to a delay that is randomly distributed between 0 and 3

seconds according to a uniform distribution. Let the initial

conditions be y(0) = y(0) = 0.1. We the show via a Matlabsimulation, how the closed-loop system is stable with a

choice of a small gain k = 0.1 despite the fact that the delayis randomly varying. Note that this simply illustrates that the

delayed feedback controller is somewhat robust to changes

in the delay as may be encountered across a sharedcommunication network.




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ControllingAcross

Networks

0 50 100 150 200 250 300 350 4000.2

0.15

0.1

0.05

0

0.05

0.1

0.15

Figure: Simulation of Second Order System with Uniform VariableDelay, Positive Feedback


CDC 2008-Cancun


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ControllingAcross

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Figure: See you there, December 9-11, 2008.


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