Controllability and Observability of Linear Nabla Discrete Fractional Systems

The main purpose of this thesis to examine the controllability and observability of the linear discrete fractional systems. First we introduce the problem and continue with the review of some basic definitions and concepts of fractional calculus which are widely used to develop the theory of this subject. In Chapter 3, we give the unique solution of the fractional difference equation involving the Riemann-Liouville operator of real order between zero and one. Additionally we study the sequential fractional difference equations and describe the way to obtain the state-space repre- sentation of the sequential fractional difference equations. In Chapter 4, we study the controllability and observability of time-invariant linear nabla fractional systems.We investigate the time-variant case in Chapter 5 and we define the state transition matrix in fractional calculus. In the last chapter, the results are summarized and directions for future work are stated

Automatic Determining of a Modulating Function

This article shows how modulating functions can be generated automatically and used to identify selected parameters of commensurable fractional systems. Existing methods use universal modulating functions. These have to be adjusted to each single problem. The proposed method uses a model based auxiliary system which is derived from the actual system, but has a fixed structure. The problem of applying a suitable modulating function is transformed into a control problem. Taking an additional precondition into account, each parameter can be separately identified. A study of the practical applicability and a numerical example complete the article

Control theory for nonlinear fractional dispersive systems

We consider a terminal control problem for processes governed by a nonlinear system of fractional ODEs. In order to show existence of the control, we first consider the linear counterpart of the system and reprove a number of classical theorems in the fractional setting (representation of the solution through the Gramian type matrix, Kalman's principle, equivalence of the controllability and observability). We are then in the position to use a fixed point theorem approach and various techniques from the fractional calculus theory to get the desired result

Fractional Systems’ Identification Based on Implicit Modulating Functions

This paper presents a new method for parameter identification based on the modulating function method for commensurable fractional-order models. The novelty of the method lies in the automatic determination of a specific modulating function by controlling a model-based auxiliary system, instead of applying and parameterizing a generic modulating function. The input signal of the model-based auxiliary system used to determine the modulating function is designed such that a separate identification of each individual parameter of the fractional-order model is enabled. This eliminates the shortcomings of the common modulating function method in which a modulating function must be adapted to the investigated system heuristically

Qualitative aspects of Volterra integro-dynamic system on time scales

This paper deals with the resolvent, asymptotic stability and boundedness of the solution of time-varying Volterra integro-dynamic system on time scales in which the coefficient matrix is not necessarily stable. We generalize at time scale some known properties about asymptotic behavior and boundedness from the continuous case. Some new results for discrete case are obtained

Nelinearni problemi upravljanja sa i bez frakcionih izvoda

The main subject of research of this thesis are nonlinear control problems with the Caputo fractional derivative of order α between 0 and 1, which in the case α=1 reduces to the classical first order derivative. We consider the question of global controllability. More precisely, we examine which conditions need to be satisfied so that, for any given initial and final data, one can find a control function for which the solution of the system reaches the desired state at the end of the given interval. In order to do so, we derive several auxiliary results regarding fractional differential equations and linear time-varying fractional control problems. We define the Riemann-Liouville and the Caputo state-transition matrices, which are essential part of the solutions of linear fractional systems, and derive estimates of those matrices. Further, we consider linear time-varying fractional control problems, introduce the controllability Gramian matrix, prove the equivalence between controllability and regularity of the Gramian, introduce the associated adjoint problem and prove the equivalence between controllability of the control problem and observability of the associated adjoint problem. Moreover, we apply the Hilbert uniqueness method and techniques from the calculus of variations to obtain the optimal control function in the weighted L2-space. Then, using properties of the solution of the linearized control problem and the Leray-Schauder fixed point theorem, we derive controllability result for one class of nonlinear control problems with unbounded dynamics. We consider the cases with fractional and with the integer-order derivative, since in the non-integer case the construction of the solution requires to take into account the memory embedded in the fractional derivative

Asymptotic properties of solutions to wave equations with time-dependent dissipation

Gegenstand der Dissertation ist die Untersuchung der asymptotischen Eigenschaften von Lösungen des Cauchy-Problems für eine Wellengleichung mit zeitabhängiger Dämpfung $b=b(t)$ und das Wechselspiel zwischen dem Verhalten des Koeffizienten $b(t)ge0$ und sich ergebenden Abschätzungen der Energie auf der Basis von $L^q$, $qge2$. Dabei stellt sich heraus, dass zwischen zwei Szenarien, dem der nicht-effektiven und dem der effektiven Dämpfung zu unterscheiden ist. In beiden Fällen werden die Hauptterme der Lösungsdarstellung konstruiert und davon ausgehend erstmalig $L^p$--$L^q$ Abschätzung für die Lösung und ihre Ableitungen angegeben. Ebenso wird die Schärfe der Abschätzungen diskutiert und in Form einer modifizierten Scattering-Theorie beziehungsweise des Diffusionsphänomens konkretisiert

Modellbasierte Identifikation fraktionaler Systeme und ihre Anwendung auf die Lithium-Ionen-Zelle

In dieser Arbeit werden modellbasierte Verfahren zur Online-Identifikation physikalischer Alterungsparameter von Batteriezellen entworfen und auf die Lithium-Ionen-Zelle angewendet. Die neuartigen Methoden basieren auf fraktionalen Impedanzmodellen und agieren, im Unterschied zum State-of-the-Art, erstmals als late-lumping Verfahren. Zudem wird in diesem Beitrag die zeitvariante fraktionale Systemtheorie um eine Steuerbarkeitsanalyse und eine energieoptimale Steuerung erweitert