Linear control systems. Continuos-time and discrete-time systems. Input-output differential equations, state space. Stability, controllability, observability. Regulators, open-loop and closed-loop systems.

System response. Solution of a continuous-time sysstem. Numerical computation of matrix exponential using Taylor series, Padé approximation, and matrix factorizations.

Numerical tests for controllability and observability. Distance to the nearest uncontrollable system. Distance to the nearest unstable system.

Numerical methods for and stability of Lyapunov and Sylvester matrix equations. Application of Jordan canonical form, Bartels-Stewart algorithm, Hessenberg-Schur method, generalized Schur methods.

Numerical methods for and stability of Riccati matrix equations. Application of eigendecomposition, Schur method, Newton method, generalized Schur methods.

Internal balancing. Model reduction. State-feedback stabilization and eigenvalue assignment problem. Stabilizable system. Pole assignment.

# Numerical methods for linear control systems

•K. J. Åström, R. M. Murray: Feedback Systems: An Introduction for Scientists and Engineers,Princeton University Press, Princeton, 2008.

•B. N. Datta: Numerical Methods for Linear Control Systems, Academic Press, San Diego, 2004.

•P. Hr. Petkov, N. D. Christov, M. M. Konstantinov: Computational Methods for Linear Control Systems, Prentice Hall, New York, 1991.

Student learns basics of linear control systems with emphasis on numerical methods for various related matrix problems. The acquired knowledge is consolidated by homework assignements and solving problems using computer programs.

Knowledge and understanding: Understanding of basics of control linear systems. The knowledge of basic numerical methods for related problems. Knowledge of computer programming package Matlab or other similar software for solving such problems.

Application: Numerical computation of problems from linear control theory.

Reflection: Understanding of the theory from the applications.

Transferable skills: The ability to solve mathematical problems using a computer.

Lectures, exercises, homeworks, consultations, projects

Type (homeworks, examination, oral, coursework, project):

homeworks or project

written exam

oral exam

Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Bor Plestenjak:

HOCHSTENBACH, Michiel E., KOŠIR, Tomaž, PLESTENJAK, Bor. A Jacobi-Davidson type method for the two-parameter eigenvalue problem. SIAM journal on matrix analysis and applications, ISSN 0895-4798, 2005, vol. 26, no. 2, str. 477-497. [COBISS-SI-ID 13613401]

HOCHSTENBACH, Michiel E., PLESTENJAK, Bor. Backward error, condition numbers, and pseudospectra for the multiparamerer eigenvalue problem. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2003, vol. 375, str. 63-81. [COBISS-SI-ID 12778841]

PLESTENJAK, Bor. A continuation method for a weakly elliptic two-parameter eigenvalue problem. IMA journal of numerical analysis, ISSN 0272-4979, 2001, vol. 21, no. 1, str. 199-216. [COBISS-SI-ID 10497369]