Module 1: Introduction to Probability (2 hours)
Module 2: Random Variables: Discrete and continuous random variables, Probability Mass and Density Functions, Joint and Conditional Distribution, Expectation, Variance and Moments, Function of Random Variables, Limit Theorems (9 hours)
Module 3: Random Processes: Discrete and continuous random processes, Brownian motion, Gaussian process, Markov Process, White noise, Stochastic difference and differential equation, Strict Sense and Wide Sense Stationary Processes. (6 hours)
Module 4: Goodness of estimator: Bias, variance, efficiency and sufficiency of estimator. (2 hours)
Module 5: Classical Estimation: Minimum Variance Unbiased Estimator (MVUE), Cramer Rao Lower Bound (CRLB), Best Linear Unbiased Estimator (BLUE), Maximum Likelihood Estimation (MLE) Least Squares (LS) and Weighted Least Squares (WLS) estimation, Orthogonality principle. (6 hours)
Module 6: Parameter Estimation for dynamical systems: Least Square (LS) estimation and Yule-Walker equation for AR, ARX, ARMAX model, One step ahead estimation. (4 hours)
Module 7: Recursive Estimation: Recursive Least Square (RLS) estimation, RLS with forgetting factor (2 hours)
Module 8: Bayesian Estimation: Minimum Mean Squared Error (MMSE) estimation, Wiener Filer, Kalman Filter- discrete, continuous, hybrid, Propagation of mean and variance, Extended Kalman Filter (EKF). (6 hours)

The course will provide an understanding of uncertainty or stochasticity in dynamical systems.

The course will provide an understanding of estimation problems using central mathematical technique of probability.

The course will provide an understanding of the main results in estimation theory and how they are used in various applications.

At the end of the course, students will be able to

CO1: Demonstrate basic knowledge of probability theory and random variables.
CO2: Demonstrate ability to model stochastic processes.
CO3: Apply the concepts of conditional probability and independence in random variables and random processes.
CO4: Formulate estimation problems and find state and parameter estimates of dynamical systems.
CO5: Evaluate performance of different estimators for given systems.
CO6: Use computational tools to implement various estimation algorithms in MATLAB/Python.

Dimitri P. Bertsekas and John N. Tsitsiklis, Introduction to Probability, 2nd Ed., Athena Scientific , https://ocw.mit.edu/courses/res-6-012-introduction-to-probability-spring-2018/pages/part-i-the-fundamentals/

Steven M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall

Athanasios Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, McGraw Hill

Dan Simon, Optimal State Estimation: Kalman, H8, and Nonlinear Approaches, Wiley

Sorenson, H. W. (1970). "Least-squares estimation: from Gauss to Kalman". IEEE spectrum, 7(7), 63-68.

Kalman, R. E. (1960). "A New Approach to Linear Filtering and Prediction Problems." ASME. J. Basic Eng. March 1960 82(1): 35–45. https://doi.org/10.1115/1.3662552