On Systems with Limited Communication PhD Thesis Defense Jian Zou May 6, 2004
Jan 15, 2016
On Systems with Limited Communication
PhD Thesis Defense
Jian ZouMay 6, 2004
5/6/2004 PhD Thesis Defense, Jian Zou 2
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.
5/6/2004 PhD Thesis Defense, Jian Zou 3
Examples
• MEMS • UAV
Picture courtesy: Aeronautical Systems
• Biological System
5/6/2004 PhD Thesis Defense, Jian Zou 4
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.
5/6/2004 PhD Thesis Defense, Jian Zou 5
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.
5/6/2004 PhD Thesis Defense, Jian Zou 6
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.
5/6/2004 PhD Thesis Defense, Jian Zou 7
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.
5/6/2004 PhD Thesis Defense, Jian Zou 8
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)
5/6/2004 PhD Thesis Defense, Jian Zou 9
System Block Diagram
Figure 2.1
5/6/2004 PhD Thesis Defense, Jian Zou 10
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.
5/6/2004 PhD Thesis Defense, Jian Zou 11
Mathematical Model
5/6/2004 PhD Thesis Defense, Jian Zou 12
State Estimation from Quantized Measurement
5/6/2004 PhD Thesis Defense, Jian Zou 13
Optimal Reconstruction of Colored Stochastic Process
5/6/2004 PhD Thesis Defense, Jian Zou 14
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
5/6/2004 PhD Thesis Defense, Jian Zou 15
Noisy Measurement
5/6/2004 PhD Thesis Defense, Jian Zou 16
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).
5/6/2004 PhD Thesis Defense, Jian Zou 17
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.
5/6/2004 PhD Thesis Defense, Jian Zou 18
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.
5/6/2004 PhD Thesis Defense, Jian Zou 19
QMSMC algorithm
Samples of
step k-1
Prior Samples
Evaluation of Likelihood
… … … …… …
Resampling and sample of step k
5/6/2004 PhD Thesis Defense, Jian Zou 20
Diagram for General Convergence Theorem
Evolution of approximate distribution
Evolution of a posterior distribution
5/6/2004 PhD Thesis Defense, Jian Zou 21
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.
5/6/2004 PhD Thesis Defense, Jian Zou 22
Simulation Results
5/6/2004 PhD Thesis Defense, Jian Zou 23
Simulation Results
5/6/2004 PhD Thesis Defense, Jian Zou 24
Simulation Results
5/6/2004 PhD Thesis Defense, Jian Zou 25
Simulation results for navigation model of MIT instrumented X-60 helicopter
5/6/2004 PhD Thesis Defense, Jian Zou 26
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
5/6/2004 PhD Thesis Defense, Jian Zou 27
Noiseless Measurement
5/6/2004 PhD Thesis Defense, Jian Zou 28
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.
5/6/2004 PhD Thesis Defense, Jian Zou 29
Finite State Approximation
5/6/2004 PhD Thesis Defense, Jian Zou 30
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
5/6/2004 PhD Thesis Defense, Jian Zou 31
Finite State Approximation
5/6/2004 PhD Thesis Defense, Jian Zou 32
Optimal quantizerFor Standard Normal
Distribution
Numerical methods searching for
optimal quantizer forSecond-order Gauss
Markov process
5/6/2004 PhD Thesis Defense, Jian Zou 33
5/6/2004 PhD Thesis Defense, Jian Zou 34
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.
5/6/2004 PhD Thesis Defense, Jian Zou 35
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.
5/6/2004 PhD Thesis Defense, Jian Zou 36
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.
5/6/2004 PhD Thesis Defense, Jian Zou 37
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.
5/6/2004 PhD Thesis Defense, Jian Zou 38
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)
5/6/2004 PhD Thesis Defense, Jian Zou 39
Optimal quantizerFor Standard Normal
Distribution
Numerical methods searching for
optimal quantizer forSecond-order Gauss
Markov process
Optimal Quantizer
(Chapter 6)
5/6/2004 PhD Thesis Defense, Jian Zou 40
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.
5/6/2004 PhD Thesis Defense, Jian Zou 41
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