On Systems with Limited Communication PhD Thesis Defense

1. On Systems with Limited CommunicationPhD Thesis Defense Jian Zou May 6, 2004

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. PhD Thesis Defense, Jian Zou

3. Examples • UAV • Biological System • MEMS Picture courtesy: Aeronautical Systems 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. 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. 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. 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. PhD Thesis Defense, Jian Zou

8. 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) PhD Thesis Defense, Jian Zou

9. System Block Diagram Figure 2.1 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. PhD Thesis Defense, Jian Zou

11. Mathematical Model PhD Thesis Defense, Jian Zou

12. State Estimation from Quantized Measurement PhD Thesis Defense, Jian Zou

13. Optimal Reconstruction of Colored Stochastic Process PhD Thesis Defense, Jian Zou

14. 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) PhD Thesis Defense, Jian Zou

15. Noisy Measurement PhD Thesis Defense, Jian Zou

16. 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). Two approaches 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. 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. PhD Thesis Defense, Jian Zou

19. QMSMC algorithm Samples of step k-1 Prior Samples Evaluation of Likelihood … … … … … … Resampling and sample of step k PhD Thesis Defense, Jian Zou

20. Diagram for General Convergence Theorem Evolution of a posterior distribution Evolution of approximate distribution 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. PhD Thesis Defense, Jian Zou

22. Simulation Results PhD Thesis Defense, Jian Zou

23. Simulation Results PhD Thesis Defense, Jian Zou

24. Simulation Results PhD Thesis Defense, Jian Zou

25. Simulation results for navigation model of MIT instrumented X-60 helicopter PhD Thesis Defense, Jian Zou

26. 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) PhD Thesis Defense, Jian Zou

27. Noiseless Measurement PhD Thesis Defense, Jian Zou

28. 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. Two approaches PhD Thesis Defense, Jian Zou

29. Finite State Approximation PhD Thesis Defense, Jian Zou

30. 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 PhD Thesis Defense, Jian Zou

31. Finite State Approximation PhD Thesis Defense, Jian Zou

32. Optimal quantizer For Standard Normal Distribution Numerical methods searching for optimal quantizer for Second-order Gauss Markov process PhD Thesis Defense, Jian Zou

33. 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. 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. 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. 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. PhD Thesis Defense, Jian Zou

38. 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) PhD Thesis Defense, Jian Zou

39. Optimal quantizer For Standard Normal Distribution Optimal Quantizer (Chapter 6) Numerical methods searching for optimal quantizer for Second-order Gauss Markov process 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. 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 PhD Thesis Defense, Jian Zou