Control over Communication Networks

- Trends and Challenges

Lihua Xie

School of Electrical and Electronic Engineering

Nanyang Technological University, Singapore

30th Chinese Control ConferenceYantai, 23 July 2011

1

Why Networked Control?

• D2H2• Elderly care

• Sensor networks for indoor/outdoor environments

• Multi-level control for energy efficiency

• Diagnostics, control and safety networks

• Wireless manufacturing

• Cooperative detection/tracking

• Cooperative actions

• Smart metering / real-time pricing

• Renewable energy• Smart power

networks

• Situation reactive multi-junction traffic lightscontrol

• Real-time vehicle routing and road pricing for congestion control

2

Devices are spatially

distributed and may operate in an asynchronous but coordinated manner.

Multi control loops are closed over a network of hierarchical, multi-layer control structure connected via digital networks

Different types of networks (CAN based such as DeviceNet, SCADA, Ethernet based networks ) for control, diagnostics and safety

Basic Features of Networked Control Systems

NCS is a control system wherein the control loops are closed through a real-time network.

3

K. Johansson (NecSys10)4

Point-to-Point Connections

Wired Fieldbuses

Wireless Networks

Reduced wiring and cost.

Enhanced flexibility, efficiency, etc.

Main challenges for wireless control networks:• Existing wireless networks are designed for applications

such as data comm (wireless LAN), voice comm (mobile phone), sensor networks (Zigbee)

• Determinism of data transmission, interferences, fading and time-varying throughput, reliability and security, etc

• Conflict between contention based communications and time based control

• Lack of standards for industrial control and rigorous control design methods

Towards Wireless Sensor/Actuator Networks

5

Example of Existing Wireless Control Networks

Wireless HART

To support wide range of applications from monitoring to closed loop control

Mesh topology Mixed TDMA and CSMA protocol

Time delay: average 20 msec per hop Accuracy of time synchronization: 1 msec Testing results show that the Wireless

HART allows for secure, highly reliable and low latency control

Provide built-in 99.9% end-to-end reliability in all industrial environments.

Others include WIA-PA ( ) and ISA100. 6

Two paradigm shifts:

Network must now be explicitly considered in feedback system

Renewed emphasis on distributed control due to wide availability of low cost processing and comm devices.

7

Research Issues in NCS

• Sensor sampling strategies (periodic, event-driven sampling, or random sampling)

• Sensor data processing• Communication strategies such

as packet generation and scheduling

Plant N

Plant 1

Controller 1Controller N

Network Manager

8

• Network design for control, e.g., network topology, MAC protocol design (e.g., TDMA, CSMA/CA, or hybrid), etc

Plant N

Plant 1

Controller 1Controller N

Network Manager

Research Issues in NCS

9

• Communication constrained control design

• Communication and control co-design

• Robustness

Plant N

Plant 1

Controller 1

Controller N

Network Manager

Research Issues in NCS

10

Outline

Why networked controlBasic features of networked control systemsShannon channel capacity and channel capacity in controlStabilization over fading channels- State feedback- Output feedback w/o channel feedbackNetwork topology for multi-agent consensusConclusions

The focus is on minimal information and information flow for control

11

Outline

Why networked controlBasic features of networked control systemsShannon channel capacity and channel capacity in controlStabilization over fading channels- State feedback- Output feedback w/o channel feedbackNetwork topology for multi-agent consensusConclusions

The focus is on minimal information and information flow for control

12

Shannon Channel Capacity

EncoderChannel

Decoder( | )p y xMessage

nX nY WMessage estimate

W

Channel capacity: Highest rate in bits per channel use at which information can be transmitted reliably over channel.

Shannon channel capacity:

( ) ( )max ( , ) max( ( ) ( | ))

p x p xC I X Y H X H X Y= = −

Shannon’s capacity is a universal upper bound for all comm. schemes.

Shannon capacity may be achieved by long LDP code whichresults in significant delay.

If code length is limited, a scaled capacity can be achieved. Compromise between delay and capacity is to be made.

There is no constraint on the coding complexity.

Zero-error capacity : 0C zero probability of estimation error13

Configuration of Communication System

AWGN Channel (satellite comm, air-to-air, optical comm)

1 log(1 )2

C γ= +

2/ . (constant)nPγ σ=

Fading Channel (urban, underwater comm)

n(t) is white Gaussian with variance

Power of channel input is bounded by

Signal-to-noise ratio (SNR):

2.nσn(t) is white Gaussian with variance

Power of channel input is bounded by

Channel side information:

Instantaneous SNR:

2.nσ

.P

2

2

( ) . (time-varying) n

P tξγσ

=

( ).tξ.P

14

Channel Capacity of Fading Channel

If only the distribution of is known at the transmitter and receiver, the achievable channel capacity remains open in general. With CSI at receiver, Shannon channel capacity:

γ

20

1 log (1 ) ( )2

C p dγ γ γ∞

= +∫

Based on Jensen’s inequality,

i.e., capacity is upper bounded by the capacity of AWGN channel with the average SNR

2 21 1[log (1 )] log (1 [ ])2 2

C E Eγ γ= + ≤ +

15

Channel capacity in control

What is the required network resource for control ? How various network conditions affect the capability of effective information transmission for control ?How to allocate network resources ?How to design coder/decoder/controller to achieve control objectives ?

16

Mahler measure (Kurt Mahler, 1960)

Mahler measure is related to the minimal energy control:

| ( )| 1

( ) | ( ) |i

iA

M A Aλ

λ≥

= ∏

Mahler Measure

2 22:

inf || || ( )duK A BK stableT M A

+=

K (A,B)d

u

x

Mahler measure is also related to optimal H-infinity performance (Gu and Qiu, 2008):

:inf || || ( )duK A BK stable

T M A∞+=

We focus on its relation to constraint of information flow and patterns of information flow for control.

17

Network

Noise Noisy outputSystem

Controller

1k k k kx Ax Bu w+ = + +

Channel Capacity in ControlStabilization over noiseless discrete channel (Nair and Evans, SICON, 2004)

For any and satisfying some mild conditions, the system is M.S. stabilizable

0x kw

2log ( )R M A>

Perfect channel: no transmission error, no packet drop, no delay

What determines the minimal data rate for stabilization ?

Related work: Wong and Brockett (1995, 1997); Baillieul (1999); Delchamps (1990);Brockett and Liberzon (2000); Elia and Mitter (2001); Fu and Xie (2005); etc 18

Stabilization over digital erasure channel

Channel packet drop characterized by an i.i.d. processwith dropout rate

Question is that can we exactly quantify the additional bit rate required?

p

Single input system is M.S. stabilizable

2

1 ;( )

pM A

<

2 221 1lolog .2 )

( g1 (

)R M A ppM A−

−> +

Channel with i.i.i. Packet drop (You and Xie, TAC10)

19

kUSystemController

/encoder Channel Decoderkσ

kXkσ∅

kγ

Maximum pack drop out rate for MS stabilizability under no constraint on data rate (Sinopoli et al 2004)

Additional bit rate to overcome the effect of packet drop

Scalar system is M.S. stabilizable

2

2

2 2

1 (1 ) | | ( 1lo )log2 1 (1 ) | |

g | | .p p qq

Rλ

λ λ− + + −− −

> +

2

11 ;| |

qλ

> −

Channel conditions are usually correlated

Use Markov packet drop to characterize channel correlations

Assume that there exists a feedback channel

Can we exactly quantify the additional amount of information required?

Packet drop correlations make the problem much more complicated

Channel with Markovian Packet drop (You and Xie, TAC11)

0 1T − 1 1T − 1kT −

kγ

Time

20

Key to solution: To show that the packet receiving interval is i.i.d.

Stabilization over AWGN Channel

n is zero mean, Gaussian white and v is zero mean, stationary.Channel Capacity:

( )21 log 1 SNR ;2cC C= = + 2

n

PSNRσ

=

(Braslavsky, Middleton and Freudenberg, TAC 2007)

The system is M.S. stabilizable 2log ( )cC M A>

21

Outline

Why networked controlBasic features of networked control systemsShannon channel capacity and channel capacity in controlStabilization over fading channels- State feedback- Output feedback w/o channel feedbackNetwork topology for stabilization – a multi-agent perspectiveConclusions

22

Real fading is often used as the phase component can be estimated and compensated for at the receiver.

is white with mean and variance

(Elia, SCL, 2005) Mean Square Capacity:

Different from capacity defined in information theory.

Examples: Rayleigh and Nakagami fading, packet drop.

Deterministic case: logarithmic quantization (Fu and Xie, 2005)

2

MS 2 2

1 log 1 .2cC C ξ

ξ

µσ

= = +

ξ ξµ 2.ξσ

u vξ=

Channel ModelStochastic multiplicative model (fading)

23

Control Over Fading Channel

( ) ( ) ( ).h t t g tξ=

Plant G(z): strictly proper with realization (A,B,C), where A is unstable, (A,B,C) is stabilizable & detectable.Design parameter: K(z)Fading channel(s):Single (series transmission) or multiple (parallel transmission) channels Channel feedback, pre and post channel processing

Controller K(z)

Fading Channel(s)

Plant G(z)

(Xiao, Xie, Qiu, TAC 11)

24

If all input signals are sent over the same channel one by one whose fading is constant within one time step,

( ) ( ), , .i jt t i jξ ξ= ∀

Stabilization over Single Transmission Channel

Capacity: 1

1

2

,1 2 2

1 log 1 .2MSC ξ

ξ

µσ

= +

Theorem: The minimal overall capacity for stabilizability is

MS,1 2logC υ>

10,

( ) 1( )

infS Y

M A mA m n

m ng

υ ρ

>

== = ≠

25

are scalar-valued white noise processes with ( ), 1, 2, ,i t i mξ =

( ) 0, ( ( ) )( ( ) )i i i i j j ijE t E t tξ µ ξ µ ξ µ σ= ≠ − − =

In general MIMO comm., the numbers of transmitters and receivers could be different.

Fading between channels could be uncorrelated if an orthogonal access scheme is adopted (Goldsmith, 2005).

We focus on independent fading channel

Stabilization over Multiple Fading Channels

Multiple fading channels: 1 2( ) ( ( ), ( ), , ( )).mt diag t t tξ ξ ξ ξ=

2

MS MS 2 21 1

1 log 1 .2

i

i

m m

ii i

C C ξ

ξ

µσ= =

= = +

∑ ∑

Overall capacity:

26

M.S. stabilization with independent fading is equivalent to

where is the maximal row sum of For MI cases, the search of the minimal overall capacity for stabilization is non-convex and has no explicit solution in general.

21, MS

inf 1.K T−Θ Θ Θ <

2

MST

1( ) ( ( ) ( ) ) ( ) ( )T z I K z G z K z G z−= − Π Λ

2

2.ijT

K(z) 1−Λ Π

( )t∆

( )G z Λ

w z

( )t∆

( )T z

[ ]1( ) ( )t tξ−∆ = Λ −Π

Stabilizability via State Feedback over Independent Fading Channel

2 21 2 1, , , , , ,m mdiag diagµ µ µ σ σΠ = Λ =

27

1( ), , ( )mdiag t tδ δ

( ) :i tδ white noise of zero-mean and unit-variance

Communications & Control Co-design

Communication resource (e.g., overall capacity) is allocatable towards control objectives (e.g., stabilization)

Question: What is the minimal resource requirement and how to distribute it?

Assumption 1: The fading channels are independent, and the overall capacity can be allocated

among the fading channels.

Theorem: Under Assumption 1, the minimal overall capacity for stabilizability is

MS ,1

m

MS ii

C C=

=∑

MS 2log ( )C M A>28

Stabilizability via Dynamic Output Feedback(without channel feedback)

29

Theorem: Minimal capacity for stabilizability is

where and are nonnegative and are functions of the anti-stable poles, NMP zeros and relative degree of .

2MS 2

1 log ( ) ( ) ( )2

C M G G Gη ϕ> + +

( )Gη ( )Gϕ

Single Input Case

22 2

1 log ( ) ( ) ( ) log ( )2

M G G G M Gη ϕ+ + ≥Note:

Equality holds only when G(z) is MP with relative degree one.The requirement on capacity in the output feedback case is generally more stringent than that in the state feedback.

Will output feedback require additional network resource for stabilizability ?

Can one exactly quantify the additional resource ?

( )G z

30

There always exists a unimodular matrix UN such that

where LG is either lower or upper triangular.

Assume has no or has distinct NMP zeros.

Theorem: Under Assumption 1, necessary overall capacity for stabilizability is

and sufficient overall capacity for stabilizability is

Multiple Input Case

Assume is tall and can be triangularly decoupled (e.g., there exists a controller such that is triangular).

( )G z

2MS 2

1

1 log ([ ] ) ([ ] ) ([ ] ) .2

q

G ii G ii G iii

C M L L Lη ϕ=

> + +∑

MS 2log ( ),C M G>

( )T z

[ ]( )ii

L G

0G

N

LU G

=

31

Dynamic Output Feedback with Coding & Channel Feedback

Theorem: Under Assumptions 1, the minimal overall capacity for stabilizability is

MS 2log ( ).C M G>

The scaling factor and the decoder are redundant.The channel feedback plays a key role in eliminating thelimitation induced by NMP zeros and high relative degree. 32

Example: Vehicle Platooning

LeaderFollowerFollower

Remote Controller

33

LeaderFollowerFollower

Vehicle model: .Vehicle separation: , and is the target separation.Fading channel: .Define , then

with

Note:G(z) has a lower triangular structure.

34

Simulation Result

Number of vehicles: 5

Dynamics of the followers:

Fading model:

2

1( ) , 1, 2,3, 4.( 1)iG z iz

= =−

( ) ( ) ( ), 1, 2,3, 4.i i it t t iξ = Ω ϒ =

( )i tΩ

( )i tϒ

0 10 20 30 40 50 60 70 80 90 100-20

0

20

40

60

80

100

Time (second)Tr

ajec

torie

s (m

eter

)

x0

x1

x2x3

x4

The trajectories of vehicles in the platoon with sufficient capacity.

is the packet- loss process with loss rate 0.2

is Nakagami distributed with mean power gain 2 and severity of fading 2

35

Channel Model One: Channel Model Two:

u v nξ= + ( )u v nξ= +

n is zero mean, white with variance ; the power of vis bounded by Further let

is white with mean and variance

A necessary and sufficient condition for stabilizability is

for channel model one;

for channel model two.

ξ ξµ2.ξσ

.P

• Suitable for general analog fading • Suitable for digital erasure2nσ

22

2 1 ( ) ,1

M Aξ

ξ

µ γσ γ

+ >+

22

2 2 2 1 ( ) ,M Aξ

ξ ξ ξ

µ γσ γ µ σ

+ >+ +

2 .nPγ σ=

Channel Fading and SNR Constraint Co-exist

The scaling factor is essential in satisfying the power constraint.

36

Outline

Why networked controlBasic features of networked controlShannon channel capacity and channel capacity in controlStabilization over fading channels- State feedback- Output feedback w/o channel feedbackNetwork topology for stabilization – a multi-agent perspective

Conclusions

37

Networked agents.

Each agent can only talk to its neighboring agents. Information may propagate across the network. The goal is to carry out coordinated tasks.We are interested in network topology (information flow)

for distributed consensus (simultaneous stabilization)

38

Cooperative Control and Consensus

Graph Representation

Both data rate and network topology are are important for cooperative control performanceWeighted digraphNeighborhood

Weighted adjacency matrixDegree

Degree matrix Laplacian matrix

=G V,E,A

| ( , ) iN j j i= ∈ ∈V E

1 1deg ( ) , deg ( )

N N

in ij out jij j

i a i a= =

= =∑ ∑

[ ]ij N Na ×=A 0, 0ij ij ia a j N≥ > ⇔ ∈

diag(deg (1),...,deg ( ))in in ND =L = D - A

Crucial for MAS cooperation( ), 1, 2, ,i L i Nλ = 39

Identical linear agent dynamic:

Static distributed protocol with perfect local states

Consensusable if for any finite , there exists a control gain such that:

Assuming no constraint on data rate, what are conditions on the network topology and the agent dynamics so that consensus can be achieved?

( 1) ( ) ( ), 0,1, ,i i iX k AX k Bu k k N+ = + = …

1( ) ( ( ) ( ))

N

i ij j ij

Ku k a X k X k=

= −∑(0)iX

lim ( ) ( ) 0, , .i jkX k X k i j

→∞− = ∀

40

Distributed consensus (You and Xie, TAC 11)

K

Theorem: Assume A has all poles or outside unit circle. The multi-agent system is consensusable if and only if

is a stabilizable pair.

Protocol gain:

P is a positive definite solution to

( , )A B

2

2

1 /( ) ,1 /

N

N

M A λ λλ λ

+<

−

2

2 T

TN

B PAKB PBλ λ

=+

0.T T

TT

A PBB PAP A PAB PB

− + >41

Network synchronizability

Consensusability via state feedback

Consensus can be shown to be equivalent to simultaneous

stabilization of

Like channel capacity, network topology needs to meet

certain condition related to Mahler measure of agent.

, 2,3, ,iA BK i Nλ+ =

Mahler measure imposes a lower bound on network synchronizability for distributed consensus, similar to its constraint on channel capacity in single loop control.

For an unstable agent, consensusability implies , i.e. the graph must be connected.

For the complete network, , consensusability can be achieved for any unstable system.

The protocol requires global knowledge on network topology, what if the information is unavailable?

What if there is also a constraint on data rate?

2 1Nλ λ= =

42

Key observations:

2 0λ >

i encoding transmission decoding j

States are real-valued，but only finite bits of information are

transmitted at each time-step

( )ijx t( )ix t

, , =G V E A

(Li, Fu, Xie and Zhang, TAC, 2010; Li and Xie; Automatica, 2011)

43

Consensus with Limited Data Rate

Problem 1:How many bits does each pair of neighbors need to exchange at each time step to achieve consensus of the whole network ?

Problem 2:What is the relationship between the consensus convergence rate and bit rate ?

Consensus vs Data Rate

Key techniques• Quantized innovation for bit rate efficiency

Z-1)1(

1−tg q

)1( −tg

)(tx j

)(tjξ

)1( −tjξ )(tj∆

−

+

innovation

channel(j,i)

( ) ( ( ) ( )), 0,1,...i

jii ij ij N

u t a x t t tξ∈

= − =∑

• Self-compensation to achieve exact consensus

For a connected network, average-consensus can be achieved

with exponential convergence ratebased on a single-bit exchangebetween each pair of neighbors

at each time step

Scalar multi-agent systems

Asymptotic convergence rate :

( , )2

inflim 1

exp2

Kh

NNKQN

γ γ∈Ω

→∞=

−

Synchronizability

Barahona & Pecora PRL 2002

Donetti et al. PRL 2005

2N

N

Q λλ

=

Networked control simulation platform

Aim to simulate and visualize control performance with incorporation of comm network for data transmissions

OMNet++ can simulate data transmissions under specified environment

UNREAL engine enables visualizing control system behavior

47

Integrated simulation platform

Test Omnet

Communicator

48

Integrated Simulation of UAV Formation

49

Cooperative Control of Multi-robot System

51

Cooperative Control of Multi-robot System

52

Target Tracking with Sensor Network

54

There has been much work on interplay between stability theory and information theory for single loop systemsMulti-loop control over networks are of practical importanceMore research is needed for performance control over realistic communication channels and associated robustnessNetwork design for control and control and communication co-design are interesting research problemsComplex networked systems remain challenging.

Concluding Remarks