Modelling of biochemical systems has received considerable attention over the last decade from bioengineering, biochemistry, computer science, and mathematics. This thesis investigates the applications of computational techniques to computational systems biology, for the construction of biochemical models in terms of topology and kinetic rates. Due to the complexity of biochemical systems, it is natural to construct models representing the biochemical systems incrementally in a piecewise manner. Syntax and semantics of two patterns are defined for the instantiation of components which are extendable, reusable and fundamental building blocks for models composition. We propose and implement a set of genetic operators and composition rules to tackle issues of piecewise composing models from scratch. Quantitative Petri nets are evolved by the genetic operators, and evolutionary process of modelling are guided by the composition rules. Metaheuristic algorithms are widely applied in BioModel Engineering to support intelligent and heuristic analysis of biochemical systems in terms of structure and kinetic rates. We illustrate parameters of biochemical models based on Biochemical Systems Theory, and then the topology and kinetic rates of the models are manipulated by employing evolution strategy and simulated annealing respectively. A new hybrid modelling framework is proposed and implemented for the models construction. Two heuristic algorithms are performed on two embedded layers in the hybrid framework: an outer layer for topology mutation and an inner layer for rates optimization. Moreover, variants of the hybrid piecewise modelling framework are investigated. Regarding flexibility of these variants, various combinations of evolutionary operators, evaluation criteria and design principles can be taken into account. We examine performance of five sets of the variants on specific aspects of modelling. The comparison of variants is not to explicitly show that one variant clearly outperforms the others, but it provides an indication of considering important features for various aspects of the modelling. Because of the very heavy computational demands, the process of modelling is paralleled by employing a grid environment, GridGain. Application of the GridGain and heuristic algorithms to analyze biological processes can support modelling of biochemical systems in a computational manner, which can also benefit mathematical modelling in computer science and bioengineering. We apply our proposed modelling framework to model biochemical systems in a hybrid piecewise manner. Modelling variants of the framework are comparatively studied on specific aims of modelling. Simulation results show that our modelling framework can compose synthetic models exhibiting similar species behaviour, generate models with alternative topologies and obtain general knowledge about key modelling features.

A method is proposed to infer stability properties for non-linear switching under continuous state feedback. Continuous-time systems which are dissipative in the multiple storage function sense are considered. A partition of the state space, induced by the cross-supply rates and the feedback function, is used to derive a restriction on switching. Then, conditions are proposed, under which, systems controlled by the feedback function and switching according to the rule are stable. In particular, Lyapunov and asymptotic stability are proved, both in a local and in a global context. Further, it is shown that the approach can be extended when one uses multiple controllers, and, therefore, is able to construct multiple partitions; conditions for this case are also presented. Finally, it is shown that, for the switching families that satisfy the switching rule posited by the results, one is able to find elements (that is, stabilising switching laws for the system) which are non-Zeno. Additional rule-sets that allow this are provided. It is argued that the conditions proposed here are easier to verify and apply, and that they offer additional flexibility when compared to those proposed by other approaches in the literature. The same infrastructure is used in the study of hybrid systems. For a general class of non-linear hybrid systems, a new property is proposed, that retains some of the properties of dissipativity, but it differs from it, crucially in the fact that it is not purely input-output. For systems having this property, it is shown that the partition used in the switching case can also be used. This, along with a set of conditions allows for the characterisation of the system behaviour in two scenaria. First, when the continuous behaviours and the jumping scheme act co-operatively, leading the system to lower energy levels (from the dissipativity point of view). Second, when the continuous behaviours are allowed to increase the stored energy, but the jumping is able to 6 compensate this increase. In the first case, it is shown that the equilibrium point under study is stable; in the second, it is shown that the system exhibits a type of attractivity, and, under additional conditions, it is asymptotically stable. Besides stability, a collection of stabilisation results are given for the case of dissipative switching systems. It is shown that one may design state feedback functions (controllers) with the objective that they satisfy the conditions of the stability theorems in this work. Then, systems under the designed controllers are shown to be stable, provided that the switching adheres to a specific switching rule. This problem is approached using a variety of tools taken from analysis, multi-valued functions and the space of non-switching stabilisation. In addition to the main results, an extensive overview of the literature in the area of switching and hybrid systems is offered, with emphasis on the topics of stability and dissipativity. Finally, a collection of numerical examples are given, validating the results presented here.

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La thèse se situe à la charnière de l'informatique théorique et de la logique mathématique. Elle se concentre en particulier sur les aspects mathématiques de la vérification et du contrôle. La thèse se focalise sur l'étude de deux sous-classes d'automates hybrides: les automates temporisés pondérés et les automates hybrides o-minimaux. Concernant les automates temporisés pondérés, en introduisant un nouvel algorithme, nous donnons une caractérisation exacte de la complexité du problème d'atteignabilité optimal en prouvant qu'il est PSpace-complet. Nous prouvons que le model-checking de la logique WCTL est en général indécidable. Nous nous intéressons alors à une restriction de la logique WCTL. Nous montrons que la décidabilité du model-checking de WCTL restreint dépend de la dimension de l'automate et du fait que le temps soit discret ou dense. Finalement pour, nous prouvons que le problème de contrôle optimal est en général indécidable. Nous prouvons cependant que ce même problème est décidable pour les automates temporisés pondérés de dimension 1. En ce qui concerne les automates hybrides o-minimaux, à l'aide d'un encodage symbolique des trajectoires par des mots, nous sommes parvenus à prouver l'existence d'une bisimulation finie pour ces automates. De plus (toujours en utilisant nos encodages des trajectoires par des mots), nous avons obtenu des résultats de décidabilité pour des problèmes de jeux sur ces mêmes automates hybrides o-minimaux. Pour une classe d'automates hybrides o-minimaux, nous avons prouvé (i) que l'existence de stratégie gagnante pouvait être décidée et (ii) que ces stratégies gagnantes pouvaient être synthétisées.

The estimation approaches examined in this dissertation focus on manipulating system dynamical models to allow the well-known form of the continuous-discrete extended Kalman filter (CDEKF) to accommodate constrained and hybrid systems. This estimation algorithm filters sequential discrete measurements for nonlinear continuous systems modeled with ordinary differential equations. The aim of the research is to broaden the class of systems for which this common tool can be easily applied. Equality constraints, holonomic or nonholonomic, or both, are commonly found in the system dynamics for vehicles, spacecraft, and robotics. These systems are frequently modeled with differential algebraic equations. In this dissertation, three tools for adapting the dynamics of constrained systems for implementation in the CDEKF are presented. These strategies address (1) constrained systems with quasivelocities, (2) kinematically constrained redundant coordinate systems, and (3) systems for which an equality constraint can be broken. The direct linearization work for constrained systems modeled with quasi-velocities is demonstrated to be particularly useful for systems subject to nonholonomic constraints. Concerning redundant coordinate systems, the "constraint force" perspective is shown to be an effective approximation for facilitating implementation of the CDEKF while providing similar performance to that of the fully developed estimation scheme. For systems subject to constraint violation, constraint monitoring methods are presented that allow the CDEKF to autonomously switch between constrained and unconstrained models. The efficacy of each of these approaches is shown through illustrative examples. Hybrid dynamical systems are those modeled with both finite- and infinite-dimensional coordinates. The associated governing equations are integro-partial differential equations. As with constrained systems, these governing equations must be transformed in order to employ the CDEKF. Here, this transformation is accomplished through two finite-dimensional representations of the infinite-dimensional coordinate. The application of these two assumed modes methods to hybrid dynamical systems is outlined, and the performance of the approaches within the CDEKF are compared. Initial simulation results indicate that a quadratic assumed modes approach is more advantageous than a linear assumed modes approach for implementation in the CDEKF. The dissertation concludes with a direct estimation methodology that constructs the Kalman filter directly from the system kinematics, potential energy, and measurement model. This derivation provides a straightforward method for building the CDEKF for discrete systems and relates these direct estimation ideas to the other work presented throughout the dissertation. Together, this collection of estimation strategies provides methods for expanding the class of systems for which a proven, well-known estimation algorithm, the extended Kalman filter, can be applied. The accompanying illustrative examples and simulation results demonstrate the utility of the methods proposed herein.

Hybrid singularly perturbed systems (SPSs) with time delay are considered and exponential stability of these systems is investigated. This work mainly covers switched and impulsive switched delay SPSs . Multiple Lyapunov functions technique as a tool is applied to these systems. Dwell and average dwell time approaches are used to organize the switching between subsystems (modes) so that the hybrid system is stable. Systems with all stable modes are first discussed and, after developing lemmas to ensure existence of growth rates of unstable modes, these systems are then extended to include, in addition, unstable modes. Sufficient conditions showing that impulses contribute to yield stability properties of impulsive switched systems that consist of all unstable subsystems are also established. A number of illustrative examples are presented to help motivate the study of these systems.

Hybrid singularly perturbed systems (SPSs) with time delay are considered and exponential stability of these systems is investigated. This work mainly covers switched and impulsive switched delay SPSs . Multiple Lyapunov functions technique as a tool is applied to these systems. Dwell and average dwell time approaches are used to organize the switching between subsystems (modes) so that the hybrid system is stable. Systems with all stable modes are first discussed and, after developing lemmas to ensure existence of growth rates of unstable modes, these systems are then extended to include, in addition, unstable modes. Sufficient conditions showing that impulses contribute to yield stability properties of impulsive switched systems that consist of all unstable subsystems are also established. A number of illustrative examples are presented to help motivate the study of these systems.

This thesis introduces the notion of using stability analysis in terms of two measures for impulsive switched systems. Impulsive switched systems are defined in the context of hybrid system theory and the motivation for the study of these systems is presented. The motivation for studying stability in two measures is also given, along with the definitions of stability, uniform stability, and uniform asymptotic stability in one and two measures. The results presented are a sets of sufficient stability criteria for linear and nonlinear systems. For autonomous linear systems, there are criteria for stability and asymptotic stability using a particular family of choices for the two measures. There is an additional stronger set of criteria for asymptotic stability using one measure, for comparison. There is also a proposed method for finding the asymptotic stability of a non-autonomous system in one measure. The method for extending these criteria to linearized systems is also presented, along with stability criteria for such systems. The criteria for nonlinear systems cover stability, uniform stability, and uniform asymptotic stability, considering state-based and time-based switching rules in different ways. The sufficient stability criteria that were found were used to solve four instructive examples. These examples show how the criteria are applied, how they compare, and what the shortcomings are in certain situations. It was found that the method of using two measures produced stricter stability requirements than a similar method for one measure. It was still found to be a useful result that could be applied to the stability analysis of an actual impulsive switched system.

This thesis introduces the notion of using stability analysis in terms of two measures for impulsive switched systems. Impulsive switched systems are defined in the context of hybrid system theory and the motivation for the study of these systems is presented. The motivation for studying stability in two measures is also given, along with the definitions of stability, uniform stability, and uniform asymptotic stability in one and two measures. The results presented are a sets of sufficient stability criteria for linear and nonlinear systems. For autonomous linear systems, there are criteria for stability and asymptotic stability using a particular family of choices for the two measures. There is an additional stronger set of criteria for asymptotic stability using one measure, for comparison. There is also a proposed method for finding the asymptotic stability of a non-autonomous system in one measure. The method for extending these criteria to linearized systems is also presented, along with stability criteria for such systems. The criteria for nonlinear systems cover stability, uniform stability, and uniform asymptotic stability, considering state-based and time-based switching rules in different ways. The sufficient stability criteria that were found were used to solve four instructive examples. These examples show how the criteria are applied, how they compare, and what the shortcomings are in certain situations. It was found that the method of using two measures produced stricter stability requirements than a similar method for one measure. It was still found to be a useful result that could be applied to the stability analysis of an actual impulsive switched system.

Στην παρούσα διπλωματική εργασία εξετάζεται η δυνατότητα δύο υβριδικών συστημάτων να καλύψουν το ηλεκτρικό φορτίο ενός συγκροτήματος πέντε βιομηχανικών κτηρίων συνδεδεμένων στο εθνικό δίκτυο. / Aim of this project is to examine the possibility of two hybrid energy systems to cover the electric load of a a block of five industrial buildings which are grid connected