Optimal Control and Identification

• 26 hrs lectures
• 13 hrs projects

• 100 pts final exam

Knowledge of the tasks of optimal control, static and dynamic optimization of deterministic, stochastic and adaptive control. Introductory knowledge of dynamac optimization, forms of loss functions, border conditions, Euler-Lagrange equation. Knowledge of limitation of shapes of control non-equations and Pontrjagin principle of minimum. Understanding of  dynamic programming, design of loss functions, Hamiltona-Jakobiho-Bellman equation. Introductory knowledge of linear controller, design of loss function, Riccati equation. Repeating of characteristics of random processes, mean values, dispersion, correlation, covariation, Wiener-Chincin relationships, Parceval theorem, while and "color" noise, transformation of random signals in linear system. Overview of Bayesovs estimations, loss and risk functions, general principle of dynamic filtration. Knowledge of linear dynamic (Kalman) filter, its design, conversion to discrete filter, generalization of dynamic filter, Wiener filter. Knowledge of parallel identification of system and trajectory as well as generalized state vector, linearized Kalman filter, contruction of selected non-linear filters. Overview of stochastic control, linear quadratic Gauss problem, continuous and discrete stochastic state regulator and servo mechanism. Knowledge of adaptive systems, parallel identification of status, parameters, and control, most frequent structures of adaptive systems.

THe goal of the course is to, using suitable forms, explain principles of automatic control. The tasks of optimal control will be formulated in a general way as optimization tasks. Similarly, the stochastic methods of control and identification are explained. Classical methods of control will be considered as partial general tasks solved using contemporary mathoematics apparatus and possibilities.

Basic knowledge of signal processing and mathematical statistics.

• Astrom,K.J.-Wittenmark,B.: Computer Controlled Systems. Prentice-Hall,1990.
• E.B. Lee and L. Markus, Foundations of Optimal Control Theory, Wiley, New York 1967.
• Fleming W. H., Rishel R. W.: Deterministic and Stochastic Optimal Control. Springer, New York, 1975, sec. edition 2001.
• Sage, A.P.: Estimation Theory with Application to Communication and control. N.Y. 1972.
• Sage, A.P.: Optimum Systems Control. New Jersey 1982.
• Dimitri Bertsekas. Reinforcement Learning and Optimal Control. Athena Scientific, 1st Ed., 2019.
• Dimitri Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific, 4th Ed., 2017.
• Frank L. Lewis, Draguna Vrabie, Vassilis L. Syrmos. Optimal Control. Wiley, 3rd Ed., 2012.

The indicative outline of the course is shown below. The topics of the lectures will be adjusted based in the initial knowledge of students. The end of the course is expected in a form of seminars and individual presentations.

1. Tasks of optimal control, static and dynamic optimization of deterministic, stochastic and adaptive control.
2. Dynamac optimization, forms of loss functions, border conditions, Euler-Lagrange equation.
3. Limitation of shapes of control non-equations and Pontrjagin principle of minimum.
4. Dynamic programming, design of loss functions, Hamiltona-Jakobiho-Bellman equation.
5. Linear controller, design of loss function, Riccati equation.
6. Repeating of characteristics of random processes, mean values, dispersion, correlation, covariation, Wiener-Chincin relationships, Parceval theorem, while and "color" noise, transformation of random signals in linear system.
7. Overview of Bayesovs estimations, loss and risk functions, general principle of dynamic filtration.
8. Linear dynamic (Kalman) filter, its design, conversion to discrete filter, generalization of dynamic filter, Wiener filter.
9. Parallel identification of system and trajectory as well as generalized state vector, linearized Kalman filter, contruction of selected non-linear filters.
10. Stochastic control, linear quadratic Gauss problem, continuous and discrete stochastic state regulator and servo mechanism.
11. Adaptive systems, parallel identification of status, parameters, and control, most frequent structures of adaptive systems.

Individual projects whose results will be presented in a form of seminar in at end of the course.

Presentations in the form of a seminar.

Oral exam.