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

5/6/2004PhD Thesis Defense, Jian Zou2 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.

5/6/2004PhD Thesis Defense, Jian Zou3 Examples MEMSUAV Picture courtesy: Aeronautical Systems Biological System

5/6/2004PhD Thesis Defense, Jian Zou4 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.

5/6/2004PhD Thesis Defense, Jian Zou5 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.

5/6/2004PhD Thesis Defense, Jian Zou6 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.

5/6/2004PhD Thesis Defense, Jian Zou7 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.

5/6/2004PhD Thesis Defense, Jian Zou8 Reconstruction of a Gauss-Markov process Systems with limited communication 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 Motivation Mathematical Models (Chapter 2) Sub optimal State Estimator (Chapter 3, 4 and 5)

5/6/2004PhD Thesis Defense, Jian Zou9 System Block Diagram Figure 2.1

5/6/2004PhD Thesis Defense, Jian Zou10 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.

5/6/2004PhD Thesis Defense, Jian Zou11 Mathematical Model

5/6/2004PhD Thesis Defense, Jian Zou12 State Estimation from Quantized Measurement

5/6/2004PhD Thesis Defense, Jian Zou13 Optimal Reconstruction of Colored Stochastic Process

5/6/2004PhD Thesis Defense, Jian Zou14 Reconstruction of a Gauss-Markov process 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 Motivation Mathematical Models (Chapter 2) Sub optimal State Estimator (Chapter 3, 4 and 5) Systems with limited communication

5/6/2004PhD Thesis Defense, Jian Zou15 Noisy Measurement

5/6/2004PhD Thesis Defense, Jian Zou16 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).

5/6/2004PhD Thesis Defense, Jian Zou17 Treating quantization as additive noise  Definition (Reverse map and quantization function )  Definition (Quantization noise function n)  Definition (Quantization noise sequence)  Impose Assumptions on statistics of quantization noise.

5/6/2004PhD Thesis Defense, Jian Zou18 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.

5/6/2004PhD Thesis Defense, Jian Zou19 QMSMC algorithm Samples of step k-1 Prior Samples Evaluation of Likelihood … Resampling and sample of step k

5/6/2004PhD Thesis Defense, Jian Zou20 Diagram for General Convergence Theorem Evolution of approximate distribution Evolution of a posterior distribution

5/6/2004PhD Thesis Defense, Jian Zou21 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.

5/6/2004PhD Thesis Defense, Jian Zou22 Simulation Results

5/6/2004PhD Thesis Defense, Jian Zou23 Simulation Results

5/6/2004PhD Thesis Defense, Jian Zou24 Simulation Results

5/6/2004PhD Thesis Defense, Jian Zou25 Simulation results for navigation model of MIT instrumented X-60 helicopter

5/6/2004PhD Thesis Defense, Jian Zou26 Reconstruction of a Gauss-Markov process 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 Motivation Mathematical Models (Chapter 2) Sub optimal State Estimator (Chapter 3, 4 and 5) Systems with limited communication

5/6/2004PhD Thesis Defense, Jian Zou27 Noiseless Measurement

5/6/2004PhD Thesis Defense, Jian Zou28 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.

5/6/2004PhD Thesis Defense, Jian Zou29 Finite State Approximation

5/6/2004PhD Thesis Defense, Jian Zou30  We assume that the evolution ofobeys 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

5/6/2004PhD Thesis Defense, Jian Zou31 Finite State Approximation

5/6/2004PhD Thesis Defense, Jian Zou32 Optimal quantizer For Standard Normal Distribution Numerical methods searching for optimal quantizer for Second-order Gauss Markov process

5/6/2004PhD Thesis Defense, Jian Zou33

5/6/2004PhD Thesis Defense, Jian Zou34 Properties of Optimal Quantizer for Standard Normal Distribution  Theorem 6.1.1, establish bounds on conditional mean in the tail of standard normal distribution.  Theorem 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.

5/6/2004PhD Thesis Defense, Jian Zou35 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 search for local minimum of cost function using gradient descent method with respect to parameters in quantizer.

5/6/2004PhD Thesis Defense, Jian Zou36 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.

5/6/2004PhD Thesis Defense, Jian Zou37 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.

5/6/2004PhD Thesis Defense, Jian Zou38 Reconstruction of a Gauss-Markov process Systems with limited communication 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 Motivation Mathematical Models (Chapter 2) Sub optimal State Estimator (Chapter 3, 4 and 5)

5/6/2004PhD Thesis Defense, Jian Zou39 Optimal quantizer For Standard Normal Distribution Numerical methods searching for optimal quantizer for Second-order Gauss Markov process Optimal Quantizer (Chapter 6)

5/6/2004PhD Thesis Defense, Jian Zou40 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.

5/6/2004PhD Thesis Defense, Jian Zou41 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