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

Nonlinear observer design with application to the synchronization of chaotic systems

Ankara : Department of Electrical and Electronics Engineering and Institute of Engineering and Sciences, Bilkent University, 1996.Thesis (Master's) -- Bilkent University, 1996.Includes bibliographical references leaves 57-60.Observers cire used to estimate the states of dynamiccil systems whenever are not available through direct measurements. Although the design of linear observers is a well-developed branch of control theory, its counterpart for nonlinear systems is a relatively new field. In this thesis, an observer construction methodology is proposed lor a chiss of nonlinear systems satisfying certain conditions. Then, the problem of synchronizing chaotic systems, which has found recent appliccitions in the area of secure message transmission, is addressed from the observer design point of view. In the design, we exploited one of the essential properties of the chaotic systems that the trajectories remain in a bounded region of the state space. It is also shown that, for certciin well-known chaotic systems, the system structure enables one to nse linecir observer schemes in order to have global synchronization.Solak, ErcanM.S

Generalized FM

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (p. 127-128).In frequency modulation (FM) systems, a continuous-time information signal is modulated onto a sinusoidal carrier wave by using the information signal to modulate the frequency of the carrier wave. In this thesis, a more general type of modulation is developed, of which FM is a special case, that we refer to as rate modulation. A rate modulation system consists of a dynamical system whose rate of evolution is varied in proportion to an information signal. The rate-modulated carrier wave is a scalar function of the state variables of the modulator. The thesis is focused on three aspects of rate modulation and demodulation systems. First, explicit expressions are derived for the power density spectrum of the rate modulated carrier wave for sinusoidal modulation. Second, a systematic procedure is derived for constructing demodulators. This procedure requires that the dynamical system used in the modulator has a known exponentially convergent observer. Assuming such an observer is known, a systematic procedure for constructing demodulators is given that depends on the underlying dynamical system in a simple manner. Finally, the quasi-moment neglect closure technique is used to approximate the signal-to-noise ratio when the carrier wave is corrupted by additive white-noise.by Wade P. Torres.Ph.D

Output feedback rotational maneuvers and vibration damping of Nasa Scole system by inversion theory

We treat the question of large rotational maneuver and vibration stabilization of NASA Spacecraft Control Laboratory Experiment System (SCOLE). The mathematical model of SCOLE System includes the dynamical equations for rigid body slew maneuver and three dimensional vibration of the rigid space shuttle, the flexible beam and the reflector with an offset mass. The contribution of this thesis lies in the development of two control systems based on (i) Euler angles or (ii) angular velocity components as output variables for large rotational maneuvers and vibration suppression of the SCOLE System. Nonlinear inversion theory is used for the design. The design approach taken here is to decompose the rigid mode control from vibration stabilization. (Abstract shortened with permission of author.)