The objective of this dissertation is to provide a framework for constructing a transitional behavior, connecting any two trajectories from a set with a particular characteristic, in such a way that the transition is as inconspicuous as possible. By this we mean that the connection is such that the characteristic behavior persists during the transition. These special classes include stationary solutions, limit cycles etc. We call this framework the Gluskabi raccordation. This problem is motivated from physical applications where it is often desired to steer a system from one stationary solution or periodic orbit to another in a ̒smooth̕ way. Examples include motion control in robotics, chemical process control and quasi-stationary processes in thermodynamics, etc. Before discussing the Gluskabi raccordations of periodic behaviors, we first study several periodic phenomena. Specifically, we study the self- propulsion of a number of legless, toy creatures based on differential friction under periodic excitations. This friction model is based on viscous friction which is predominant in a wet environment. We investigate the effects of periodic and optimal periodic control on locomotion. Subsequently, we consider a control problem of a stochastic system, under the basic constraint that the feedback control signal and the observations from the system cannot use the communication channel simultaneously. Hence, two modes of operation result: an observation mode and a control mode. We seek an optimal periodic regime in a statistical steady state by switching between the observation and the control mode. For this, the duty cycle and the optimal gains for the controller and observer in either mode are determined. We then investigate the simplest special case of the Gluskabi raccordation, namely the quasi-stationary optimal control problem. This forces us to revisit the classical terminal controller. We analyze the performance index as the control horizon increases to infinity. This problem gives a good example where the limiting operation and integration do not commute. Such a misinterpretation can lead to an apparent paradox. We use symmetrical components (the parity operator) to shed light on the correct solution. The main part of thesis is the Gluskabi raccordation problem. We first use several simple examples to introduce the general framework. We then consider the signal Gluskabi raccordation or the Gluskabi raccordation without a dynamical system. Specifically, we present the quasi-periodic raccordation where we seek the maximally ̒smooth̕ transitions between two periodic signals. We provide two methods, the direct and indirect method, to construct these transitions. Detailed algorithms for generating the raccordations based on the direct method are also provided. Next, we extend the signal Gluskabi raccordation to the dynamic case by considering the dynamical system as a hard constraint. The behavioral modeling of dynamical system pioneered by Willems provides the right language for this generalization. All algorithms of the signal Gluskabi raccordation are extended accordingly to produce these ̒smooth̕ transition behaviors.

Το Πι-κριτήριο των Bittanti et al. (1973) έχει χρησιμοποιηθεί εκτενώς σε εφαρμογές με στόχο την πρόβλεψη ενδεχόμενης βελτίωσης της απόδοσης ενός μη γραμμικού συστήματος υπό περιοδική είσοδο. Το κριτήριο όμως έχει τοπική ισχύ και περιορίζεται σε περιοδικές διαταραχές μικρού πλάτους. Η παρούσα εργασία αναπτύσσει μια μέθοδο προσδιορισμού διορθώσεων υψηλότερης τάξης στο πι-κριτήριο, προερχόμενη από βασικά αποτελέσματα της θεωρίας κεντρικής πολλαπλότητας (Center Manifold theory). Η προτεινόμενη μέθοδος βασίζεται στην επίλυση της μερικής διαφορικής εξίσωσης της κεντρικής πολλαπλότητας με χρήση δυναμοσειρών. Το τελικό αποτέλεσμα της προτεινόμενης προσέγγισης είναι ο κατά προσέγγιση υπολογισμός του δείκτη απόδοσης υπό μορφή σειράς, η οποία παρέχει ακριβή αποτελέσματα σε μεγαλύτερα εύρη. Η προτεινόμενη μέθοδος εφαρμόζεται σε έναν συνεχή αντιδραστήρα πλήρους ανάδευσης (CSTR), όπου στόχος είναι η μεγιστοποίηση της παραγωγής του επιθυμητού προϊόντος. Κατασκευάστηκε αλγόριθμος που προβλέπει την μόνιμη κατάσταση στην οποία καταλήγει ένα σύστημα απομάκρυνσης αζώτου που αποτελείται από αντιδραστήρα εμβολικής ροής και δεξαμενή δευτεροβάθμιας καθίζησης με ανακύκλωση. Με χρήση υπολογιστικού μοντέλου βασιζόμενο στο ASM3 υπολογίστηκαν οι μόνιμες καταστάσεις αυτού του συστήματος για ένα εύρος καταστάσεων λειτουργίας. Βρέθηκαν οι βέλτιστες τιμές των βαθμών ελευθερίας για την ελαχιστοποίηση του συνολικού αερισμού και για την ελαχιστοποίηση του συνολικού αζώτου στην απορροή. Και στις δύο περιπτώσεις στις βέλτιστες μόνιμες καταστάσεις παρατηρήθηκε έκπλυση των Nitrobacter δηλαδή παράκαμψη της παραγωγής των νιτρικών. / The frequency-dependent Pi criterion of Bittanti et al. (1973) has been used extensively in applications to predict potential performance improvement under periodic forcing in a nonlinear system. The criterion, however, is local in nature and is limited to periodic forcing functions of small magnitude. The present work develops a method to determine higher-order corrections to the pi criterion, derived from basic results of Center Manifold theory. The proposed method is based on solving the Center Manifold partial differential equation via power series. The end result of the proposed approach is the approximate calculation of the performance index in the form of a series expansion, which provides accurate results under larger amplitudes. The proposed method is applied to a continuous stirred tank reactor, where the yield of the desired product must be maximized. An algorithm was constructed, that predicts the steady state of a nitrogen removal system consisting of a plug flow reactor and a secondary clarifier with recycle. Using a numerical model based on ASM3 and a grid of degrees of freedom, the steady states of this system were calculated. The optimal values for minimizing the total aeration were found, as well as those for minimizing the total nitrogen exit flow. In both cases the Nitrobacter bacteria were washed out thus indicating the bypassing of nitrate production.