Discrete-Time, Linear Periodic Time-Varying System Norm Estimation Using Finite Time Horizon Transfer Operators

Abstract

The norm is one of the fundamental concepts of linear algebra and functional analysis. The notion of the norm is often employed in engineering, e.g. in control engineering, where main application is calculating the norm of the transfer function. Unfortunately existing methods are applicable for systems that can be described using Laplace transform, i.e. linear time-invariant (LTI) systems. An operational equivalent of the transfer function for linear time-varying systems is transfer operator. The transfer operator defined for finite time horizon can be described by finite dimensional matrix. Although for infinite time horizon the operator is infinite dimensional. In the paper a method for norm estimation of transfer operator defined on infinite time horizon is proposed. The method is applicable for linear time-varying, discrete-time systems given in general state-space form. The method takes advantage of the properties of the transfer operator norm on a finite time horizon. Theoretical considerations are complemented by numerical examples.

Norma je jedan od osnovnih koncepata linearne algebre i funkcionalne analize. Pojam norme se često koristi kod inžinjera, npr. kod upravljanja, gdje je jedna od glavnih aplikacija računanje norma prijenosne funkcije. Nažalost postojeće metode su primjenjive samo na sustave koji se mogu opisati koristeći Laplaceovu transformaciju, tj. linearne vremenski nepromjenjive sustave. Ekvivalent prijenosnoj funkciji za linearne vremenski promjenjive sustave je prijenosni operator. Prijenosni operator definiran za konačni vremenski horizont može se opisati konačno dimenzionalnom matricom. Iako je za beskonačni vremenski horizont operator beskonačno dimenzionalan. U radu je predložena metoda za estimaciju norme prijenosnog operatora definiranog na beskonačnom vremenskom horizontu. Metodaje primjenjiva na linearne vremenski promjenjive diskretne sustave zadane u obliku prostora stanja. Metoda koristi svojstva norme prijenosnog operatora za konačni vremenski horizont. Teoretska promišljanja nadopunjena su numeričkim primjerima.

Key words:

• Norm estimation
• Discrete-time systems
• Time-varying systems
• Non-stationary systems

Ključne riječi:

• estimacija norme
• diskretni sustavi
• vremenski promjenjivi sustavi
• nestacionarni sustavi

Additional information

Notes on contributors

Przemysław Orłowski

Przemysław Orłowski is assistant professor in Department of Control and Measurements at the Faculty of Electrical Engineering at the West Pomeranian University of Technology, Szczecin (Poland) where he received his M.Sc. and Ph.D. degrees in Electrical Engineering in 1999 and 2002, respectively. He is involved in teaching lectures in undergraduate and graduate courses e.g. on “Computer aided engineering”, “Theory and methods of optimization”. He worked as a grant manager on the two scientific projects supported by Ministry of Science and Higher Education of Poland. Dr Przemyslaw Orlowski is the author/co-author of 41 papers: 11 journal papers, 2 book chapters and 28 conference papers. His main research interests are in the fields of discrete-time systems, time-varying systems.