To show the organization of the course that includes this module, follow this link:  Course organization

Learning outcomes

The course objective is to present the material related to dynamical system theory, the specific properties of the sub-class of linear, time invariant systems (LTI(, and the methods to design control algorithms for LTI systems.
The course aims at explaining the analysis and synthesis methods of the "modern" control theory, based on the introduction of concept of state. The course will also present a few advanced topics, such as observability, controllability, state estimation and control, and the general concept of system stability, based on Lyapunov theory.

NOTE: to achieve good results in the course, students are expected to be familiar with the following concepts:
- Algebra and Matrix calculus
- Laplace, Fourier and Zeta transforms
- Basic concepts of analysis of linear systems.

Program

Review of the basic concepts of system analysis:
- Definitions and properties of linear, time invariant (LTI) systems,
- models in time, frequency and "s" and "z" domains,
- the transfer function
- main properties of LTI systems in "t", "f", "s" and "z",
- discrete time systems and Z trasnform
- main properties of feedback systems.

State models:
- AR, MA, ARMA models,
- input-state-output representation,
- definitions of state, causality, algebraic equivalence,
- state and output update map,
- exponential matrix and its properties,
- Jordan canonical form, characteristic polynomial, algebraic and geometric multiplicity,
- modes, their characteristics, simple/asymptotic/BIBO stability,
- Relation between state representation and Laplace and Z transforms,
- Transfer functions, eigenvalues and poles.

Stability in state models:
- equilibrium state,
- stability of an equilibrium state,
- Lyapunov stability criterion,
- Lyapunov equation,
- linearization and reduced Lyapunov criterion.

Reachability:
- main concepts and the reachability Gramian,
- state space control,
- standard form of reachability, canonical control form,
- PBH criterion of reachability,
- state feedback.

Observability:
- main concepts and observability Gramian,
- State estimation (open and closed loop),
- standard form of observability, canonical observation form,
- PBH criterion of observability.

Duality:
- overview of discrete time Kalman filter,
- overview of optimal linear, quadratic controller in discrete time domain.

Examination Methods

The exam will consist of a written test on the course topics. In special cases, the student can ask to be tested on the teorethical part of the course with an oral exam.