Estimation and control of non-linear and hybrid systems with applications to air-to-air guidance

Issued as Progress report, and Final report, Project no. E-21-67

Control of a network of Euler–Bernoulli beams

AbstractThe aim is to study the boundary controllability of a system modeling the vibrations of a network of N Euler–Bernoulli beams connected by n vibrating point masses. Using the classical Hilbert Uniqueness Method, the control problem is reduced to the obtention of an observability inequality. The solution is then expressed in terms of Fourier series so that it is also enough to show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds as soon as the roots of a function denoted by f∞ (and giving the asymptotic behaviour of the eigenvalues) are all simple. For a network of N=2 different beams, this assumption on the multiplicity of the roots of f∞ (denoted by (A)) is proved to be satisfied and controllability follows. For higher values of N, a numerical approach allows one to prove (A) in many situations and no counterexample has been found but the problem of giving a general proof of controllability remains open

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

Observability and observer design for switched linear systems

Hybrid vehicles, HVAC systems in new/old buildings, power networks, and the like require safe, robust control that includes switching the mode of operation to meet environmental and performance objectives. Such switched systems consist of a set of continuous-time dynamical behaviors whose sequence of operational modes is driven by an underlying decision process. This thesis investigates feasibility conditions and a methodology for state and mode reconstruction given input-output measurements (not including mode sequence). An application herein considers insulation failures in permanent magnet synchronous machines (PMSMs) used in heavy hybrid vehicles. Leveraging the feasibility literature for switched linear time-invariant systems, this thesis introduces two additional feasibility results: 1) detecting switches from safe modes into failure modes and 2) state and mode estimation for switched linear time-varying systems. This thesis also addresses the robust observability problem of computing the smallest structured perturbations to system matrices that causes observer infeasibility (with respect to the Frobenius norm). This robustness framework is sufficiently general to solve related robustness problems including controllability, stabilizability, and detectability. Having established feasibility, real-time observer reconstruction of the state and mode sequence becomes possible. We propose the embedded moving horizon observer (EMHO), which re-poses the reconstruction as an optimization using an embedded state model which relaxes the range of the mode sequence estimates into a continuous space. Optimal state and mode estimates minimize an L2-norm between the measured output and estimated output of the associated embedded state model. Necessary conditions for observer convergence are developed. The EMHO is adapted to solve the surface PMSM fault detection problem