Minimum Energy Problem in the Sense of Caputo for Fractional Neutral Evolution Systems in Banach Spaces

We investigate a class of fractional neutral evolution equations on Banach spaces involving Caputo derivatives. Main results establish conditions for the controllability of the fractional-order system and conditions for existence of a solution to an optimal control problem of minimum energy. The results are proved with the help of fixed-point and semigroup theories.Comment: This is a preprint of a paper whose final and definite form is published Open Access in 'Axioms' at [https://doi.org/10.3390/axioms11080379

Digital Filters and Signal Processing

Digital filters, together with signal processing, are being employed in the new technologies and information systems, and are implemented in different areas and applications. Digital filters and signal processing are used with no costs and they can be adapted to different cases with great flexibility and reliability. This book presents advanced developments in digital filters and signal process methods covering different cases studies. They present the main essence of the subject, with the principal approaches to the most recent mathematical models that are being employed worldwide

Applications of equivalent representations of fractional- and integer-order linear time-invariant systems

Nicht-ganzzahlige - fraktionale - Ableitungsoperatoren beschreiben Prozesse mit Gedächtniseffekten, deshalb werden sie zur Modellierung verschiedenster Phänomene, z.B. viskoelastischen Verhaltens, genutzt. In der Regelungstechnik wird das Konzept vor allem wegen des erhöhten Freiheitsgrades im Frequenzbereich verwendet. Deshalb wurden in den vergangenen Dekaden neben einer Verallgemeinerung des PID-Reglers auch fortgeschrittenere Regelungskonzepte auf nicht-ganzzahlige Operatoren erweitert. Das Gedächtnis der nicht-ganzzahligen Ableitung ist zwar essentiell für die Modellbildung, hat jedoch Nachteile, wenn z.B. Zustände geschätzt oder Regler implementiert werden müssen: Das Gedächtnis führt zu einer langsamen, algebraischen Konvergenz der Transienten und da eine numerische Approximation ist speicherintensiv. Im Zentrum der Arbeit steht die Frage, mit welchen Maßnahmen sich das Konvergenzverhalten dieser nicht ganzzahligen Systeme beeinflussen lässt. Es wird vorgeschlagen, die Ordnung der nicht ganzzahligen Ableitung zu ändern. Zunächst werden Beobachter für verschiedene Klassen linearer zeitinvarianter Systeme entworfen. Die Entwurfsmethodik basiert dabei auf einer assoziierten Systemdarstellung, welche einen Differenzialoperator mit höherer Ordnung verwendet. Basierend auf dieser Systembeschreibung können Beobachter entworfen werden, welche das Gedächtnis besser mit einbeziehen und so schneller konvergieren. Anschließend werden ganzzahlige lineare zeitinvariante Systeme mit Hilfe nicht-ganzzahliger Operatoren dargestellt. Dies ermöglicht eine erhöhte Konvergenz im Zeitintervall direkt nach dem Anfangszeitpunkt auf Grund einer unbeschränkten ersten Ableitung. Die periodische Löschung des so eingeführten Gedächtnisses wird erzielt, indem die nicht ganzzahlige Dynamik periodisch zurückgesetzt wird. Damit wird der algebraischen Konvergenz entgegen gewirkt und exponentielle Stabilität erzielt. Der Reset reduziert den Speicherbedarf und induziert eine unterlagerte zeitdiskrete Dynamik. Diese bestimmt die Stabilität des hybriden nicht-ganzzahligen Systems und kann genutzt werden um den Frequenzgang für niedrige Frequenzen zu bestimmen. So lassen sich Beobachter und Regler für ganzzahlige System entwerfen. Im Rahmen des Reglerentwurfs können durch den Resets das Verhalten für niedrige und hohe Frequenzen in gewissen Grenzen getrennt voneinander entworfen werden.Non-integer, so-called fractional-order derivative operators allow to describe systems with infinite memory. Hence they are attractive to model various phenomena, e.g. viscoelastic deformation. In the field of control theory, both the higher degree of freedom in the frequency domain as well as the easy generalization of PID control have been the main motivation to extend various advanced control concepts to the fractional-order domain. The long term memory of these operators which helps to model real life phenomena, has, however, negative effects regarding the application as controllers or observers. Due to the infinite memory, the transients only decay algebraically and the implementation requires a lot of physical memory. The main focus of this thesis is the question of how to influence the convergence rates of these fractional-order systems by changing the type of convergence. The first part is concerned with the observer design for different classes of linear time-invariant fractional-order systems. We derive associated system representations with an increased order of differentiation. Based on these systems, the observers are designed to take the unknown memory into account and lead to higher convergence rates. The second part explores the representation of integer-order linear time-invariant systems in terms of fractional-order derivatives. The application of the fractional-order operator introduces an unbounded first-order derivative at the initial time. This accelerates the convergence for a short time interval. With periodic deletion of the memory - a reset of the fractional-order dynamics - the slow algebraic decay is avoided and exponential stability can be achieved despite the fractional-order terms. The periodic reset leads to a reduced implementation demand and also induces underlying discrete time dynamics which can be used to prove stability of the hybrid fractional-order system and to give an interpretation of the reset in the frequency domain for the low frequency signals. This concept of memory reset is applied to design an observer and improve fractional-order controllers for integer-order processes. For the controller design this gives us the possibility to design the high-frequency response independently from the behavior at lower frequencies within certain limits

Stability of fractional order systems

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled

Crackling Noise in Fractional Percolation -- Randomly distributed discontinuous jumps in explosive percolation

Crackling noise is a common feature in many systems that are pushed slowly, the most familiar instance of which is the sound made by a sheet of paper when crumpled. In percolation and regular aggregation clusters of any size merge until a giant component dominates the entire system. Here we establish `fractional percolation' where the coalescence of clusters that substantially differ in size are systematically suppressed. We identify and study percolation models that exhibit multiple jumps in the order parameter where the position and magnitude of the jumps are randomly distributed - characteristic of crackling noise. This enables us to express crackling noise as a result of the simple concept of fractional percolation. In particular, the framework allows us to link percolation with phenomena exhibiting non-self-averaging and power law fluctuations such as Barkhausen noise in ferromagnets.Comment: non-final version, for final see Nature Communications homepag