Modelling & analysis of hybrid dynamic systems using a bond graph approach

Hybrid models are those containing continuous and discontinuous behaviour. In constructing dynamic systems models, it is frequently desirable to abstract rapidly changing, highly nonlinear behaviour to a discontinuity. Bond graphs lend themselves to systems modelling by being multi-disciplinary and reflecting the physics of the system. One advantage is that they can produce a mathematical model in a form that simulates quickly and efficiently. Hybrid bond graphs are a logical development which could further improve speed and efficiency. A range of hybrid bond graph forms have been proposed which are suitable for either simulation or further analysis, but not both. None have reached common usage. A Hybrid bond graph method is proposed here which is suitable for simulation as well as providing engineering insight through analysis. This new method features a distinction between structural and parametric switching. The controlled junction is used for the former, and gives rise to dynamic causality. A controlled element is developed for the latter. Dynamic causality is unconstrained so as to aid insight, and a new notation is proposed. The junction structure matrix for the hybrid bond graph features Boolean terms to reflect the controlled junctions in the graph structure. This hybrid JSM is used to generate a mixed-Boolean state equation. When storage elements are in dynamic causality, the resulting system equation is implicit. The focus of this thesis is the exploitation of the model. The implicit form enables application of matrix-rank criteria from control theory, and control properties can be seen in the structure and causal assignment. An impulsive mode may occur when storage elements are in dynamic causality, but otherwise there are no energy losses associated with commutation because this method dictates the way discontinuities are abstracted. The main contribution is therefore a Hybrid Bond Graph which reflects the physics of commutating systems and offers engineering insight through the choice of controlled elements and dynamic causality. It generates a unique, implicit, mixed-Boolean system equation, describing all modes of operation. This form is suitable for both simulation and analysis

Controllability of linear impulsive systems – an eigenvalue approach

summary:This article considers a class of finite-dimensional linear impulsive time-varying systems for which various sufficient and necessary algebraic criteria for complete controllability, including matrix rank conditions are established. The obtained controllability results are further synthesised for the time-invariant case, and under some special conditions on the system parameters, we obtain a Popov-Belevitch-Hautus (PBH)-type rank condition which employs eigenvalues of the system matrix for the investigation of their controllability. Numerical examples are provided that demonstrate--for the linear impulsive systems, null controllability need not imply their complete controllability, unlike for the non-impulsive linear systems

Reduced realizations and model reduction for switched linear systems:a time-varying approach

In the last decades, switched systems gained much interest as a modeling framework in many applications. Due to a large number of subsystems and their high-dimensional dynamics, such systems result in high complexity and challenges. This motivates to find suitable reduction methods that produce simplified models which can be used in simulation and optimization instead of the original (large) system. In general, the study aims to find a reduced model for a given switched system with a fixed switching signal and known mode sequence. This thesis concerns first the reduced realization of switched systems with known mode sequence which has the same input-output behavior as original switched systems. It is conjectured that the proposed reduced system has the smallest order for almost all switching time duration. Secondly, a model reduction method is proposed for switched systems with known switching signals which provide a good model with suitable thresholds for the given switched system. The quantitative information for each mode is carried out by defining suitable Gramians and, these Gramians are exploited at the midpoint of the given switching time duration. Finally, balanced truncation leads to a modewise reduction. Later, a model reduction method for switched differential-algebraic equations in continuous time is proposed. Thereto, a switched linear system with jumps and impulses is constructed which has the identical input-output behavior as original systems. Finally, a model reduction approach for singular linear switched systems in discrete time is studied. The choice of initial/final values of the reachability and observability Gramians are also investigated

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

Algebraic characterization of controllability and observability for second order descriptor systems

We analyze controllability and observability conditions for second order descriptor systems and show how the classical conditions for first order systems can be generalized to this case. We show that performing a classical transformation to first order form may destroy some controllability and observability properties. To avoid this, we will derive a canonical form and new first order formulations that do not destroy the controllability and observability properties. As an example, we demonstrate that the loss of impulse controllability in constrained multi-body systems is due to the representation as first order system

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