On Systems with Limited Communication PhD Thesis Defense

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On Systems with Limited CommunicationPhD Thesis
Defense

• Jian Zou
• May 6, 2004

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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
Reconstruction of a Gauss-Markov process
Mathematical Models (Chapter 2)
Noisy Measurement
Noiseless Measurement
Quantized Measurement Kalman Filter ( or
Extend Kalman Filter)
Quantized Measurement Sequential Monte
Carlo method
Quantized Measurement Kalman Filter
Finite State Approximation
Sub optimal State Estimator (Chapter 3, 4 and 5)
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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
Reconstruction of a Gauss-Markov process
Mathematical Models (Chapter 2)
Noisy Measurement
Noiseless Measurement
Quantized Measurement Kalman Filter ( or
Extend Kalman Filter)
Quantized Measurement Sequential Monte
Carlo method
Quantized Measurement Kalman Filter
Finite State Approximation
Sub optimal State Estimator (Chapter 3, 4 and 5)
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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.

• 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
Reconstruction of a Gauss-Markov process
Mathematical Models (Chapter 2)
Noisy Measurement
Noiseless Measurement
Quantized Measurement Kalman Filter ( or
Extend Kalman Filter)
Quantized Measurement Sequential Monte
Carlo method
Quantized Measurement Kalman Filter
Finite State Approximation
Sub optimal State Estimator (Chapter 3, 4 and 5)
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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
Reconstruction of a Gauss-Markov process
Mathematical Models (Chapter 2)
Noisy Measurement
Noiseless Measurement
Quantized Measurement Kalman Filter ( or
Extend Kalman Filter)
Quantized Measurement Sequential Monte
Carlo method
Quantized Measurement Kalman Filter
Finite State Approximation
Sub optimal State Estimator (Chapter 3, 4 and 5)
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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
  Gauss-Markov 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