On Systems with Limited
Communication
PhD Thesis Defense
Jian Zou
May 6, 2004
Motivation I
 Information theoretical issues are traditionally
decoupled from consideration of decision and control
problems by ignoring communication constraints.
 Many newly emerged control systems are distributed,
asynchronous and networked. We are interested in
integrating communication constraints into
consideration of control system.
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Examples
• MEMS
• UAV
• Biological
System
Picture courtesy: Aeronautical
Systems
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Theoretical framework for systems with
limited communication
 A theoretical framework for systems with limited
communication should answer many important questions
(state estimation, stability and controllability, optimal
control and robust control).
 The effort just begins. It is still a long road ahead.
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State Estimation
 Communication constraints cause time delay and
quantization of analog measurements.
 Two steps in considering state estimation problem
from quantized measurement. First, for a class of
given underlying systems and quantizers, we seek
effective state estimator from quantized
measurement. Second, we try to find optimal
quantizer with respect to those state estimators.
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Motivation II
 Optimal reconstruction of a Gauss-Markov process
from its quantized version requires exploration of the
power spectrum (autocorrelation function) of the
process.
 Mathematical models for this problem is similar to that
of state estimation from quantized measurement.
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Major contributions
 We found effective state estimators from quantized
measurements, namely quantized measurement
sequential Monte Carlo method and finite state
approximation for two broad classes of systems.
 We studied numerical methods to seek optimal
quantizer with respect to those state estimators.
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Systems with limited
communication
Motivation
Mathematical
Models
(Chapter 2)
Sub optimal
State
Estimator
(Chapter 3, 4
and 5)
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Reconstruction of
a Gauss-Markov
process
Noiseless
Measurement
Noisy Measurement
Quantized
Measurement
Kalman Filter
( or Extend
Kalman Filter)
Quantized
Measurement
Sequential
Monte Carlo
method
Quantized
Measurement
Kalman
Filter
PhD Thesis Defense, Jian Zou
Finite
State
Approximation
8
System Block Diagram
Figure 2.1
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Assumptions
 We only consider systems which can be modeled
as block diagram in Figure 2.1.
 Assumptions regarding underlying physical object
or process, information to be transmitted, type of
communication channels, protocols are made.
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Mathematical Model
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State Estimation from Quantized
Measurement
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Optimal Reconstruction of Colored
Stochastic Process
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Systems with limited
communication
Motivation
Mathematical
Models
(Chapter 2)
Sub optimal
State
Estimator
(Chapter 3, 4
and 5)
5/6/2004
Reconstruction of
a Gauss-Markov
process
Noiseless
Measurement
Noisy Measurement
Quantized
Measurement
Kalman Filter
( or Extend
Kalman Filter)
Quantized
Measurement
Sequential
Monte Carlo
method
Quantized
Measurement
Kalman
Filter
PhD Thesis Defense, Jian Zou
Finite
State
Approximation
14
Noisy Measurement
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Two approaches
 Treating quantization as
additive noise + Kalman
Filter (Extended
Kalman Filter)
 We call them Quantized
measurement Kalman
filter (extended Kalman
filter) respectively.
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 Applying sequential
Monte Carlo method
(particle filter).
 We call the method
Quantized measurement
sequential Monte Carlo
method (QMSMC).
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Treating quantization as additive noise
 Definition 3.3.1 (Reverse map
and quantization function )
 Definition 3.3.2 (Quantization
noise function n)
 Definition 3.3.3 (Quantization
noise sequence)
 Impose Assumptions on
statistics of quantization noise.
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Quantized Measurement Kalman filter
(Extend Kalman filter)
 Kalman filter is modified to incorporate the
artificially made-up quantization noise. The
statistics of quantization noise depends on the
distribution of measurement being quantized.
 Extend Kalman filter is modified in a similar way.
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QMSMC algorithm
Samples of
step k-1
Prior Samples
Evaluation of
Likelihood
…
…
…
…
…
…
Resampling and
sample of step k
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Diagram for General Convergence
Theorem
Evolution of a posterior distribution
Evolution of approximate distribution
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Properties of QMSMC

complexity at each iteration. Parallel
Computation can effectively reduce the computational
time.
The resulted random variable sequence indexed by
number of samples used converges to the conditional
mean in probability. This is the meaning of
asymptotical optimality.
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Simulation Results
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Simulation Results
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Simulation Results
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Simulation results for navigation model of
MIT instrumented X-60 helicopter
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Systems with limited
communication
Motivation
Mathematical
Models
(Chapter 2)
Sub optimal
State
Estimator
(Chapter 3, 4
and 5)
5/6/2004
Reconstruction of
a Gauss-Markov
process
Noiseless
Measurement
Noisy Measurement
Quantized
Measurement
Kalman Filter
( or Extend
Kalman Filter)
Quantized
Measurement
Sequential
Monte Carlo
method
Quantized
Measurement
Kalman
Filter
PhD Thesis Defense, Jian Zou
Finite
State
Approximation
26
Noiseless Measurement
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Two approaches
 Treating quantization as
additive noise + Kalman
Filter (Extended
Kalman Filter)
 Discretize the state
space and apply the
formula for partially
observed HMM.
 We call the method
finite state
approximation.
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Finite State Approximation
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Finite State Approximation
 We assume that the evolution of
obeys
time invariant linear rule. We also assume this rule
can be obtained from evolution of underlying
systems.
 Under this assumption, we apply formula for
partially observed HMM for state estimation.
 Computational complexity
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Finite State Approximation
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Optimal quantizer
For Standard Normal
Distribution
Numerical methods
searching for
optimal quantizer for
Second-order Gauss
Markov process
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Properties of Optimal Quantizer for
Standard Normal Distribution
 Theorem 6.1.1, 6.1.2 establish bounds on conditional
mean in the tail
of standard
normal distribution.
 Theorem 6.1.3 proposes an upper bound on
quantization error contributed by the tail.
 After assuming conjecture 6.1.1, we obtain upper
bounds of error associated with optimal N-level
quantizer for standard normal distribution.
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Numerical Methods Searching for
Optimal Quantizer for Second-order
Gauss Markov Process
 For Gauss-Markov underlying process, define cost
function of an quantizer to be square root of mean
squared estimation error by Quantized
measurement Kalman filter.
 Algorithm 6.2.1 search for local minimum of cost
function using gradient descent method with
respect to parameters in quantizer.
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Numerical Results
 For second order systems with different damping
ratios, optimal quantizers are indistinguishable
based on our criteria.
 Lower damping ratio will reduce error associated
with optimal quantizer.
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Conclusions
 We considered systems with limited communication and
optimal reconstruction of a Gauss-Markov process.
 Effective sub optimal state estimators from quantized
measurements.
 Study of properties of optimal quantizer for standard
normal distribution and numerical methods to seek optimal
quantizer for Gauss-Markov process.
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Systems with limited
communication
Motivation
Mathematical
Models
(Chapter 2)
Sub optimal
State
Estimator
(Chapter 3, 4
and 5)
5/6/2004
Reconstruction of
a Gauss-Markov
process
Noiseless
Measurement
Noisy Measurement
Quantized
Measurement
Kalman Filter
( or Extend
Kalman Filter)
Quantized
Measurement
Sequential
Monte Carlo
method
Quantized
Measurement
Kalman
Filter
PhD Thesis Defense, Jian Zou
Finite
State
Approximation
38
Optimal quantizer
For Standard Normal
Distribution
Optimal
Quantizer
(Chapter 6)
Numerical methods
searching for
optimal quantizer for
Second-order Gauss
Markov process
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Future Work
 Other topics regarding systems with limited
communication such as controllability, stability, optimal
control with respect to new cost function and robust
control.
 Improving QMSMC and finite state approximation
methods and related theoretical work.
 New methods to search optimal quantizer for GaussMarkov process.
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Acknowledgements
 Prof. Roger Brockett.
 Prof. Alek Kavcic, Prof. Garrett Stanley and Prof. Navin
Khaneja
 Haidong Yuan and Dan Crisan
 Michael, Ben, Ali, Jason, Sean, Randy, Mark, Manuela.
 NSF and U.S. Army Research Office
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