Dynamic systems and control
Examiners: Katalin Hangos, Ferenc Hartung, Attila Magyar, Mihály Pituk
Exam material: 6 freely chosen topics from the following list, which the candidate agrees with the examiner in advance
1. Fundamentals of system and control theory: Signals and systems, operator description of dynamic systems, the most important system classes, input-output and state space representations and their mathematical properties, deterministic and stochastic systems, the concept and methods of determining the most important system properties (stability, observability, controllability) for LTI systems, principle and properties of the most basic control and identification and diagnostic methods.
2. Nonlinear systems: Input-affine system model, observability and controllability of nonlinear systems, stability concepts and stability analysis methods of nonlinear systems, input-output and feedback linearization. Stability analysis with linearization.
3. Discrete event and hybrid systems: Concept and description methods of discrete event systems (automaton, Petri net), solving models of discrete event systems (simulation), examination of the most important system properties (availability, dead points, infinity, limitation, invariants), control and diagnostics of discrete event systems; concept of hybrid systems, methods of description, solution of models and examination of their properties.
4. Modern control methods: Stabilizing, disturbance rejection and robust control methods, pole-placement and its extensions, LQR and its extensions, controllers based on feedback and input-output linearization for nonlinear systems, direct passivation for nonlinear systems, fuzzy control.
5. Intelligent control systems: The software architecture of intelligent control systems, the properties of and the cooperation between the intelligent and real-time subsystem, rule-based expert systems, real-time expert systems, the relationship of dynamic and intelligent system models, qualitative differential and difference equations, Petri-nets, fuzzy control systems, analysis of the properties of intelligent system models.
6. Identification and filtering: Parameter and structure estimation of dynamical systems, the general parameter estimation problem and its properties, least squares estimations and their properties, maximum likelihood estimations and their properties, Bayesian estimations and their properties, the method and properties of auxiliary variables, identification of nonlinear systems, recursive parameter estimation methods and their properties, signal filtering and change detection methods, state estimation methods of dynamic systems, the Kalman filter and its extensions. Curve fitting using the method of least squares.
7. Modeling and diagnostics of dynamic systems: The method of building dynamic system models, the determination of the state variables, input and output of the system, system models in the form of differential-algebraic equations and their properties, verification and validation of dynamic system models, structural system properties, the basics of diagnostics of dynamic systems: diagnostics based on parameter estimation and prediction, modeling of dynamic systems for diagnostic purposes, fault-sensitive filters.
8. Linear differential equations: Solutions of linear autonomous differential equations, general theory of linear differential equations (fundamental set of solutions, variations of constants formula). Solutions of higher-order differential equations, asymptotic behavior. Laplace-transform method and its application. Mathematical characterization of input-output and state-space models of continuous-time LTI systems, transfer function and properties, relationship with models in differential equations.
9. Stability theory: Lyapunov stability, asymptotic stability, uniform stability. Lyapunov functions and basic Lyapunov stability theorems. Conditions for stability and asymptotic stability of linear systems. Principle of linearized stability. Theorems for instability. Lyapunov functions and Lyapunov theorem for LTI systems.
10. Difference equations: Description of solutions of linear difference equations using characteristic roots. Theorems concerning the boundedness of the solutions of linear autonomous differential equations and their convergence to zero. Definition, properties, and application of the z-transform. Theorems concerning the asymptotic behavior of solutions of difference equations. Basic concepts of control theory for difference equations. Mathematical characterization of input-output and state space models of discrete-time LTI systems, the pulse transfer operator and its properties, and its relationship with models in difference equations.
11. Delay differential equations: Existence and uniqueness, continuation of the solutions, differentiability, stability, linear systems, perturbed linear systems, periodic solutions, applications.
12. Numerical methods: Solution of nonlinear equations and systems (fixed point iteration, Newton’s method, order of convergence). Solution of ODEs (Euler-, Taylor-, multistep methods). Minimization (simplex method, gradient method, Newton’s method). Approximation of eigenvalues and eigenvectors of matrices. Application of minimization methods for parameter estimation in nonlinear ODEs.
13. Partial differential equations: Fourier’s method in one and higher dimensions (heat equation, Laplace’s equation, wave equation). Method of characteristics in wave equation. Wave equation on infinite interval.