### Estimation Theory

#### Course Description

Methods for system model estimation (system identification) and system states identification are elaborated. System identification based on deterministic excitation signals. Nonparametric system identification methods in time and frequency domain. Estimation of the system model parameters: least-squares method, instrumental variables method, maximal likelihood method. Selection of the model structure and verification of the obtained model. State estimation of the deterministic systems. State estimation of the stochastic systems: linear and nonlinear Kalman filters. Simultaneous estimation of system parameters and states. Examples of estimation techniques application to system control, system fault detection, signal processing etc.

#### General Competencies

This course qualifies students for identification of mathematical models of dynamical systems and for the design of system states and system parameters estimators. Additionally, students gain practical skills to apply these models and estimators in various fields.

#### Learning Outcomes

1. apply basics of probability theory in system identification and system state estimation
2. explain application ways for non-parametric system identification methods and suitability of their use in corresponding system surrounding
3. combine tools for carrying out non-parametric system identification methods and validation of the obtained system models
4. explain application ways for parametric system identification methods and suitability of their use in corresponding system surrounding
5. combine tools for carrying out parametric system identification methods and validation of the obtained system models
6. design state estimator for lienar deterministic and stochastic syastems
7. apply Kalman filter in various variants for system state and parametar estiamtions
8. modify Kalman filter equations according to the system and implementation constraints

#### Forms of Teaching

Lectures

Lectures are organized in two cycles.

Consultations

Upon request.

Other Forms of Group and Self Study

Homework assignments.

Continuous Assessment Exam
Homeworks 50 % 8 % 50 % 8 %
Mid Term Exam: Written 30 % 40 % 0 %
Final Exam: Written 30 % 40 %
Final Exam: Oral 12 %
Exam: Written 50 % 80 %
Exam: Oral 12 %
##### Comment:

By a regular exam the 50% threshold relates to the sum of the points obtained from homeworks and the written exam.

#### Week by Week Schedule

1. Lecture 00 -- Course organization and administration; Lecture 01 -- Introduction -- course overview and motivation; Lecture 02 -- Short overview: probability theory, stochastic processes (3 hours)
2. Lecture 02 -- Short overview: probability theory, stochastic processes (continuation) Lecture 03 -- Non-parametric identification methods (using Fourier analysis and correlation analysis) (4 hours)
3. Lecture 03 -- Non-parametric identification methods (correlation analysis) (4 hours)
4. Lecture 04 -- Parametric identification methods (parametric models and least squares method) (4 hours)
5. Lecture 04 -- Parametric identification methods -- continuation (instrumental variable method, maximum likelihood method, using forgetting factor in identification) (4 hours)
6. Lecture 05 -- Practical aspects in system identification; Lecture 06 -- Choice of the model structure and model validation (4 hours)
7. Solving theoretical and numerical tasks in system identification as a preparation for the midterm exam (2 hours)
8. Midterm exam
9. Lecture 07 -- Short introductio to state estimation and State estimation of deterministic linear systems (4 hours)
10. Lecture 08 -- Discrete-time Kalman filter (4 hours). Covariance and state propagation. Derivation of a discrete-time Kalman filter. Kalman filter properties. One-step formulation of Kalman equations. Alternative expressions for covariance propagation. Divergence problems.
11. Lecture 09 -- Continuous-time and nonlinear forms of Kalman filter (4 hours). Continuous linearized Kalman filter. Extended Kalman filter. Iterative extended Kalman filter. Secon order extended Kalman filter. Weighted sum of multiple Kalman filters. Kalman filter application to state estimation.
12. Lecture 10 -- Implementation forms of Kalman filter (4 hours). Sequential Kalman filter. Information filter. Steady-state Kalman filter. Alpha-beta and Alpha-beta-gamma filter.
13. Lecture 11 -- Some additional aspects in Kalman filter application (2 hours). Kalman filter performance check. Kalamn filter for state estimation in systems with correlated process and measurement noises. Kalman filter for state estimation with coloured process and measurement noises.
14. Solving theoretical and numerical tasks in system identification as a preparation for the final exam (2 hours)
15. Final exam

#### Study Programmes

Control Engineering and Automation -> Electrical Engineering and Information Technology (Profile)

Information Processing -> Information and Communication Technology (Profile)

#### Literature

Perić, N., Petrović, I (2005.), Identifikacija procesa-predavanja, Skripta, FER, Zagreb
Ljung, L. (1998.), System Identification - Theory for the User (2nd Edition), Prentice-Hall
Haykin, S. (Editor) (2001.), Kalman Filtering and Neural Networks, John Wiley & Sons
Gustafsson, F. (2001.), Adaptive Filtering and Change Detection, John Wiley & Sons
Dan Simon (2006.), Optimal State Estimation - Kalman, Hinf, and Nonlinear Approaches, John Wiley & Sons, Inc.

#### Lecturers in Charge

5 ECTS
L1 English Level
L1 e-Learning
45 Lecturers
0 Exercises
0 Laboratory exercises