Chess Review May 11, 2005 Berkeley, CA Closing the loop around Sensor Networks Bruno Sinopoli...
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Chess ReviewMay 11, 2005Berkeley, CA
Closing the loop around Sensor Networks
Bruno SinopoliShankar Sastry Dept of Electrical Engineering,UC Berkeley
Chess Review, May 11, 2005 2
Conceptual Issues
• Given a certain wireless sensor network can we successfully design a particular application?
• How does the application impose constraints on the network?
• Can we derive important metrics from those constraints?
• How do we measure network parameters?
Chess Review, May 11, 2005 3
What can you do with a sensor network?
• Literature provides key asymptotic results
• We are interested in answering different semantic questions, e.g.:
• At the algorithmic level:– How much packet loss can a tracking
algorithm tolerate?
• At the network level:– How many objects can a particular sensor
network reliably track?
Chess Review, May 11, 2005 4
Wireless Sensor Networks
• It’s a network of devices:– Many nodes: 103-105
– Multi-hop wireless communication with adjacent nodes
– Cheap sensors– Cheap CPU
• Issues w/ Sensor Networks and Data Networks ?– Random time delay– Random arrival sequence– Packet loss– Limited Bandwidth
Chess Review, May 11, 2005 5
Control Applications with Sensor Networks
Pegs
HVAC systems
Power Grids
Human body
Chess Review, May 11, 2005 6
Sensor net increases visibility
Control and communication over Sensor Networks
Observationsy(t)
Controlinputs
u(t)
Computationalunit
Chess Review, May 11, 2005 7
Experimental results: Pursuit evasion games
Chess Review, May 11, 2005 8
Problem Statement
Given a control systems where Given a control systems where components, i.e. plant, sensors, components, i.e. plant, sensors, controllers, actuators, are connected via controllers, actuators, are connected via a specified communication network, a specified communication network, design an design an ““optimal optimal ““ controller for the controller for the systemsystem
Chess Review, May 11, 2005 9
Outline
• Problem Statement• Optimal Estimation with intermittent
observations• Optimal control with both intermittent
obs and control– TCP-like protocols– UDP-like protocols
• Conclusions
Chess Review, May 11, 2005 10
plant
estimator+
controller
y(t)
u(t)
Network
LGQ scenario for lossy network
Bernoulli independent
switches
Modeling
Chess Review, May 11, 2005 11
Assumptions
• System:– Discrete time linear time invariant– Additive white gaussian noise on both the dynamics and
the observation
• Communication network:– Packets either arrive or are lost within a sampling period
following a bernoulli process.– A Delay longer than sampling time is considered lost.– Packet Acknowledgement depends on the specific
communication protocol
Chess Review, May 11, 2005 12
Optimal estimation with intermittent observations
PlantAggregate
SensorState
estimatorCommunication
Network
• Main Results (IEEE TAC September 2004)• Kalman Filter is still the optimal estimator• We proved the existence of a threshold
phenomenon:
maxmin
cmax
cPt
ctt
PtMPE
PPE
|)2| (
11
0condition initialany and 1for ][
0condition initial some and 0for ][lim
0
0
0
Kalman FilterKalman Filter
Chess Review, May 11, 2005 13
Optimal control with both intermittent observations and control packets
• What is the minimum arrival probability that guarantees “acceptable” performance of estimator and controller?
• How is the arrival rate related to the system dynamics?• Can we design estimator and controller independently?• Are the optimal estimator and controllers still linear?• Can we provide design guidelines?
PlantAggregate
Sensor
ControllerState
estimator
CommunicationNetwork
CommunicationNetwork
Chess Review, May 11, 2005 14
Control approach
• The problem of control is traditionally subdivided in two sub-problems:– Estimation
• Allows to recover state information from observations– Control
• Given current state information, control inputs are provided to the actuators
• The separation principle:– allows, under observability conditions, to design
estimator and controller independently.• If separation principle holds, optimal estimator (in
the minimum variance sense) and optimal controller (LQG) are linear and independent
Chess Review, May 11, 2005 15
LQG control with intermittent observations and control
PlantAggregate
Sensor
ControllerState
estimator
CommunicationNetwork
CommunicationNetwork
Ack is always
present Ack is
relevant
We’ll group all communication protocols in two classes: TCP-like (acknowledgement is available) UDP-like (acknowledgement is absent)
Chess Review, May 11, 2005 16
, – Bernoulli, indep.
Minimize J_N subject to
• TCP – Transmission Control Protocol– PRO: feedback information on packet delivery– CONS: more expensive to implement
• UDP – User Datagram Protocol– PRO: simpler communication infrastructure– CONS: less information available
;;
LQG mathematical modeling
Chess Review, May 11, 2005 17
Estimator Design
TCP UDPPrediction Step
Correction Step
Chess Review, May 11, 2005 18
LQG Controller Design: TCP case
Solution via Dynamic Programming:
1. Compute the Value Function t=N and move backward
2. Find Infinite Horizon by taking N+1Vt(xt) – minimum cost-to-go if in state xt at time t
Chess Review, May 11, 2005 19
LQG Controller Design: TCP case
We can prove that for TCP the value function can be written as:
with:
Minimization of v(t) yields:
Chess Review, May 11, 2005 20
Stochastic variable !!
LQG Controller Design: TCP case
Chess Review, May 11, 2005 21
unbounded
LQG averaged cost is bounded for all N if the following Modified Algebraic Riccati Equations exist:
1
1bounded
time-varying estimator gain
estimator controller
constant controller gain
OPTIMAL LQG CONTROL
Infinite Horizon: TCP case
Chess Review, May 11, 2005 22
PlantAggregate
Sensor
ControllerState
estimator
CommunicationNetwork
Special Case: LQG with intermittent observations,
unbounded
=1
1bounded
11
Chess Review, May 11, 2005 23
LQG Controller Design: UDP case
Scalar system, i.e. x2R
t=N
t=N-1
Chess Review, May 11, 2005 24
LQG Controller Design: UDP case
t=N-2
NONLINEAR FUNCTION OF INFORMATION SET It
Chess Review, May 11, 2005 25
prediction
correction
Without loss of generality I can assume C=I
UDP controller: Estimator designSpecial case: C invertible, R=0
Chess Review, May 11, 2005 26
UDP special case:C invertible, R=0
It is possible to show that:
Chess Review, May 11, 2005 27
UDP Infinite Horizon:C invertible, R=0
It is possible to show that:
unbounded
1
1bounded
Necessary condition for boundedness:
Sufficient only if B invertible
estimator/controller coupling
Chess Review, May 11, 2005 28
Conclusions
• Closed the loop around sensor networks• General framework applies to
networked control system• Solved the optimal control problem for
full state feedback linear control problems
• Bounds on the cost function • Transition from state boundedness to
instability appears• Critical network values for this transition
Chess ReviewMay 11, 2005Berkeley, CA
Thank you !!!For more info: [email protected] publications:•Kalman Filtering with Intermittent Observations -IEEE TAC September 2004
•Time Varying Optimal Control with Packet Losses -IEEE CDC 2004
•Optimal Control with Unreliable Communication: the TCP Case -ACC 2005
•LQG Control with Missing Observation and Control Packets -IFAC 2005