On Systems with Limited Communication PhD Thesis Defense Jian Zou May 6, 2004.

On Systems with Limited Communication

PhD Thesis Defense

Jian ZouMay 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

Picture courtesy: Aeronautical Systems

• Biological System

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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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Reconstruction of a Gauss-Markov

process

Systems with limited communication

Noisy Measurement Noiseless

Measurement

QuantizedMeasurement Kalman Filter

( or ExtendKalman Filter)

QuantizedMeasurement

SequentialMonte Carlo

method

QuantizedMeasurement

KalmanFilter

FiniteState

Approximation

Motivation

MathematicalModels

(Chapter 2)

Sub optimalState

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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Reconstruction of a Gauss-Markov

process

Noisy Measurement Noiseless

Measurement

QuantizedMeasurement Kalman Filter

( or ExtendKalman Filter)

QuantizedMeasurement

SequentialMonte Carlo

method

QuantizedMeasurement

KalmanFilter

FiniteState

Approximation

Motivation

MathematicalModels

(Chapter 2)

Sub optimalState

Estimator(Chapter 3, 4

and 5)

Systems with limited communication

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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 approximate distribution

Evolution of a posterior 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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Reconstruction of a Gauss-Markov

process

Noisy Measurement Noiseless

Measurement

QuantizedMeasurement Kalman Filter

( or ExtendKalman Filter)

QuantizedMeasurement

SequentialMonte Carlo

method

QuantizedMeasurement

KalmanFilter

FiniteState

Approximation

Motivation

MathematicalModels

(Chapter 2)

Sub optimalState

Estimator(Chapter 3, 4

and 5)

Systems with limited communication

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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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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

Finite State Approximation

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Finite State Approximation

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Optimal quantizerFor Standard Normal

Distribution

Numerical methods searching for

optimal quantizer forSecond-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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Reconstruction of a Gauss-Markov

process

Systems with limited communication

Noisy Measurement Noiseless

Measurement

QuantizedMeasurement Kalman Filter

( or ExtendKalman Filter)

QuantizedMeasurement

SequentialMonte Carlo

method

QuantizedMeasurement

KalmanFilter

FiniteState

Approximation

Motivation

MathematicalModels

(Chapter 2)

Sub optimalState

Estimator(Chapter 3, 4

and 5)

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Optimal quantizerFor Standard Normal

Distribution

Numerical methods searching for

optimal quantizer forSecond-order Gauss

Markov process

Optimal Quantizer

(Chapter 6)

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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