1. Why are models of dynamical systems created using differential equations? What are the inputs and outputs of a dynamical system. When can a variable in a system of differential equations be considered either an input or an output, even when there is no specific dynamical system underlying the equations? Explain, using an example, what modelling from basic principles consists in.
2. What is the general form of the input/state/output representation of a nonlinear dynamical system? State the theorem for the local existence and uniqueness of the solution of a differential equation in such a form. What does this theorem imply for the presence of causality in this equation?
3. What is the equilibrium point of a dynamical system? Derive and illustrate the equilibrium point(s) using a specific example.
4. Give the definitions of stability and uniform stability of an equilibrium point. Explain the difference between these concepts. Give an example of a dynamical system with a uniformly stable equilibrium point.
5. Give the definitions of asyptotic stability and uniform asymptotic stability of an equilibrium point. Explain the difference between these concepts. Give an example of a dynamical system with a uniformly asymptotically stable equilibrium point. Give a definition of exponential stability of an equilibrium point.
6. Lyapunov’s Indirect Method. Stability theorems for equilibrium points of linear, stationary and non-stationary systems.
7. Lyapunov’s Direct Method: the theorem and an example to illustrate it.
8. Lyapunov’s Direct Method and its complementary theorem (referring to the invariant set due to the system equation) for stationary systems. Illustrative example.
9. A dynamical system as an operator, the induced norm of a linear dynamical system, the 2nd norm and the ∞ norm of a linear dynamical system.
10. Input-output stability (L_p stability). Input-output stability theorems for linear stationary and non-stationary systems.
11. Internal stability of the system. The small gain theorem.
12. Stablization of the equilibrium point of a nonlinear system on the ground of the Lyapunov’s Indirect Method. Pole placement control problem for a linear system; a pole placement control law for a system with one input - Ackermann's formula, extension of this control law to the case with a system with multiple inputs.
13. Mathematical tools for the feedback linearization of a nonlinear system: a vector field, a Lie derivative, a Lie bracket, a diffeomorphism: definitions and properties.
14. Mathematical tools for the feedback linearization of a nonlinear system: completely integrable set of vector fields, involutivity of a set of vector fields, Frobenius theorem
15. Definition of input-state feedback linearization for nonlinear systems with one input. Necessary and sufficient conditions for the existence of state and input transformations, linearizing a nonlinear input-state system.
16. An input-state feedback linearization procedure for a nonlinear system with a single input. Stabilization and trajectory tracking algorithms for a nonlinear system based on its linearized form.
17. Feedback linearization of SISO systems: a feedback linearization procedure, the definition of a relative degree, the normal form of a system, the zero dynamics of a system.
18. Feedback linearization of SISO systems: a definition of an asymptotic minimum-phase nonlinear system, stabilization and trajectory tracking for a nonlinear system based on a linearized system.
19. The optimal control problem. Pontriagin's maximum principle.
20. Application of Pontriagin's maximum principle: minimum-energy control.