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Combining Acoustic Echo Cancellation and Suppression / Att kombinera akustisk ekoutsläckning och ekodämpningWallin, Fredrik January 2003 (has links)
The acoustic echo problem arises whenever there is acoustic coupling between a loudspeaker and a microphone, such as in a teleconference system. This problem is traditionally solved by using an acoustic echo canceler (AEC), which models the echo path with adaptive filters. Long adaptive filters are necessary for satisfactory echo cancellation, which makes AEC highly computationally complex. Recently, a low-complexity echo suppression scheme was presented, the perceptual acoustic echo suppressor (PAES). Spectral modification is used to suppress the echoes, and the complexity is reduced by incorporating perceptual theories. However, under ideal conditions AEC performs better than PAES. This thesis considers a hybrid system, which combines AEC and PAES. AEC is used to cancel low-frequency echo components, while PAES suppresses high-frequency echo components. The hybrid system is simulated and assessed, both through subjective listening tests and objective evaluations. The hybrid scheme is shown to have virtually the same perceived quality as a full-band AEC, while having a significantly lower complexity and a higher degree of robustness.
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Optimally-Sized Design of a Wind/Diesel/Fuel Cell Hybrid System for a Remote CommunityVafaei, Mehdi 29 September 2011 (has links)
Remote communities, characterized by no connection to the main power grid, traditionally get their power from diesel generators. Long geographical distances and lack of suitable roads make the fuel transportation difficult and costly, increasing the final cost of electricity. A microgrid using renewable energy as the main source can serve as a viable solution for this problem with considerable economical and environmental benefits. The focus of this research is to develop a microgrid for a remote community in northern Ontario (Canada) that combines wind, as a renewable source of energy, and a hydrogen-based energy storage system, with the goal of meeting the demand, while minimizing the cost of energy and adverse effect on the environment. The existing diesel generators remain in the system, but their use is minimized.
The microgrid system studied in this research uses a wind turbine to generate electricity, an electrolyser to absorb the excess power from the wind source, a hydrogen tank to store the hydrogen generated by the electrolyser, a fuel cell to supply the demand when the wind resource is not adequate, and a diesel generator as a backup power.
Two scenarios for unit-sizing are defined and their pros. and cons. are discussed. The economic evaluation of scenarios is performed and a cost function for the system is defined. The optimization problem thus formulated is solved by solvers in GAMS. The inputs are wind profile of the area, load profile of the community, existing sources of energy in the area, operating voltage of the grid, and sale price of electricity in the area. The outputs are the size of the fuel cell and electrolyser units that should be used in the microgrid, the capital and running costs of each system, the payback period of the system, and cost of generated electricity. Following this, the best option for the microgrid structure and component sizes for the target community is determined.
Finally, a MATLAB-based dynamic simulation platform for the system under study with similar load/wind profile and sizing obtained in optimization problem is developed and the dynamic behaviour of microgrid at different cases is studied.
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Hybrid intelligent machine systems : design, modeling and controlOuyang, Puren 02 September 2005 (has links)
To further improve performances of machine systems, mechatronics offers some opportunities. Traditionally, mechatronics deals with how to integrate mechanics and electronics without a systematic approach. This thesis generalizes the concept of mechatronics into a new concept called hybrid intelligent machine system. A hybrid intelligent machine system is a system where two or more elements combine to play at least one of the roles such as sensor, actuator, or control mechanism, and contribute to the system behaviour. The common feature with the hybrid intelligent machine system is thus the presence of two or more entities responsible for the system behaviour with each having its different strength complementary to the others. The hybrid intelligent machine system is further viewed from the systems structure, behaviour, function, and principle, which has led to the distinction of (1) the hybrid actuation system, (2) the hybrid motion system (mechanism), and (3) the hybrid control system. <p>This thesis describes a comprehensive study on three hybrid intelligent machine systems. In the case of the hybrid actuation system, the study has developed a control method for the true hybrid actuation configuration in which the constant velocity motor is not mimicked by the servomotor which is treated in literature. In the case of the hybrid motion system, the study has resulted in a novel mechanism structure based on the compliant mechanism which allows the micro- and macro-motions to be integrated within a common framework. It should be noted that the existing designs in literature all take a serial structure for micro- and macro-motions. In the case of hybrid control system, a novel family of control laws is developed, which is primarily based on the iterative learning of the previous driving torque (as a feedforward part) and various feedback control laws. This new family of control laws is rooted in the computer-torque-control (CTC) law with an off-line learned torque in replacement of an analytically formulated torque in the forward part of the CTC law. This thesis also presents the verification of these novel developments by both simulation and experiments. Simulation studies are presented for the hybrid actuation system and the hybrid motion system while experimental studies are carried out for the hybrid control system.
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Maximally smooth transition: the Gluskabi raccordationYeung, Deryck 24 August 2011 (has links)
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.
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Stochastic Hybrid Dynamic Systems: Modeling, Estimation and SimulationSiu, Daniel 01 January 2012 (has links)
Stochastic hybrid dynamic systems that incorporate both continuous and discrete dynamics have been an area of great interest over the recent years. In view of applications, stochastic hybrid dynamic systems have been employed to diverse fields of studies, such as communication networks, air traffic management, and insurance risk models. The aim of the present study is to investigate properties of some classes of stochastic hybrid dynamic systems.
The class of stochastic hybrid dynamic systems investigated has random jumps driven by a non-homogeneous Poisson process and deterministic jumps triggered by hitting the boundary. Its real-valued continuous dynamic between jumps is described by stochastic differential equations of the It\^o-Doob type. Existing results of piecewise deterministic models are extended to obtain the infinitesimal generator of the stochastic hybrid dynamic systems through a martingale approach. Based on results of the infinitesimal generator, some stochastic stability results are derived. The infinitesimal generator and stochastic stability results can be used to compute the higher moments of the solution process and find a bound of the solution.
Next, the study focuses on a class of multidimensional stochastic hybrid dynamic systems. The continuous dynamic of the systems under investigation is described by a linear non-homogeneous systems of It\^o-Doob type of stochastic differential equations with switching coefficients. The switching takes place at random jump times which are governed by a non-homogeneous Poisson process. Closed form solutions of the stochastic hybrid dynamic systems are obtained. Two important special cases for the above systems are the geometric Brownian motion process with jumps and the Ornstein-Uhlenbeck process with jumps. Based on the closed form solutions, the probability distributions of the solution processes for these two special cases are derived. The derivation employs the use of the modal matrix and transformations.
In addition, the parameter estimation problem for the one-dimensional cases of the geometric Brownian motion and Ornstein-Uhlenbeck processes with jumps are investigated. Through some existing and modified methods, the estimation procedure is presented by first estimating the parameters of the discrete dynamic and subsequently examining the continuous dynamic piecewisely.
Finally, some simulated stochastic hybrid dynamic processes are presented to illustrate the aforementioned parameter-estimation methods. One simulated insurance example is given to demonstrate the use of the estimation and simulation techniques to obtain some desired quantities.
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Ανίχνευση νέων πρωτεϊνικών αλληλεπιδράσεων της μυοειδικής πρωτεΐνης δεσμίνης στα καρδιακά μυϊκά κύτταρα και προτάσεις νέων μηχανισμών δράσης της. / Novel protein-proteinΚυριακόπουλος, Ανδρέας 28 June 2007 (has links)
Η μυο-ειδική πρωτεΐνη δεσμίνη, αποτελεί μέλος των πρωτεϊνών του κυτταροσκελετού των ενδιαμέσων ινιδίων και εκφράζεται στους λείους και τους γραμμωτούς μυς. Στους συσταλτούς μύες, το πλέγμα του κυτταροσκελετού της Δεσμίνης περιβάλλει τους Ζ-δίσκους διασυνδέοντάς τους, ενώ παράλληλα συνδέει μεταξύ τους τις συσταλτές περιοχές της μυικής ίνας με την σαρκοπλασματική μεμβράνη, με διάφορα οργανίδια και με τον πυρήνα. Για να προσδιορίσουμε τους ακριβείς μηχανισμούς δράσης της δεσμίνης χρησιμοποιήσαμε το σύστημα υβριδισμού των ζυμών – yeast two hybrid screen system – προκειμένου να ανιχνεύσουμε πρωτεΐνες που αλληλεπιδρούν με τη δεσμίνη. Χρησιμοποιήσαμε ως «δόλωμα» αλληλουχίες των άκρων του μορίου της δεσμίνης του αμινο-τελικού και το καρβόξυ-τελικού. Μελετώντας τις πρωτεΐνες που προέκυψαν, διαπιστώσαμε ότι το αμινο-τελικό άκρο της δεσμίνης αλληλεπιδρά με διάφορες μιτοχονδριακές πρωτεΐνες. Με το ίδιο σύστημα αποκαλύψαμε αλληλεπιδράσεις της δεσμίνης με λυοσωματικές πρωτεΐνες όπως η καθεψίνη D και η προσαποσίνη οι οποίες αλληλεπιδρούν με το αμινοτελικό άκρο της δεσμίνης. Η καθεψίνη D είναι μια λυοσωματική πρωτεάση, που οδηγείται και ωριμάζει πλήρως στα λυοσώματα ενώ η προσαποσίνη είναι ένα πρόδρομο λυοσωματικό μόριο με πρωτεόλυση του οποίου, εντός του λυοσώματος, προκύπτουν οι σαποσίνες Α έως D. Η καθεψίνη D αποτελεί δείκτη καταστάσεων αυτοφαγία και τελευταία φαίνεται ότι επεμβαίνει σε φαινόμενα απόπτωσης επάγωντάς την κατά περίπτωση. Η αλληλεπίδραση της δεσμίνης με την καθεψίνη D επιβεβαιώθηκε και με βιοχημικές τεχνικές (in vitro) όπως η συνεργιστική ανοσοκαθίζηση /ανοσοκατακρήμνιση (co-immuno-precipitation) και η τεχνική GST pull-down. Μετά και από αυτές τις in vitro αποδείξεις, φαίνεται πως μάλλον συμβαίνει ευθεία αλληλεπίδραση μεταξύ της δεσμίνης και της καθεψίνης D. Γι’ αυτό, και με βάση όσα είναι γνωστά για την καθεψίνη D, προτείνουμε μια νέα λειτουργία του κυτταροσκελετού της δεσμίνης πιθανόν στην μετακίνησης και τη δημιουργία των λυοσωμάτων αλλά και έναν νέο ρυθμιστικό ίσως ρόλο της, σε διαδικασίες αυτοφαγίας και απόπτωσης, μέσω της πρόσδεσής της με σημαντικά μόρια ρυθμιστές τέτοιων διαδικασιών.................... / Desmin is the muscle - specific member of the intermediate filament family of cytoskeletal proteins, expressed both in striated and smooth muscle tissues. In mature striated muscle fibers, the desmin filament lattice surrounds the Z-discs, interconnects them to each other and links the entire contractile apparatus to the sarcolemmal cytoskeleton, cytoplasmic organelles and the nucleus. In order to identify the exact mechanisms of desmin’s action, we performed a yeast two-hybrid screen for desmin-interacting proteins. For this purpose, we used as baits the two non helical terminal regions of the desmin molecule, the amino (head)- and the carboxy (tail)- terminal domain. We have found that the head domain of desmin potentially interacts with two new groups of proteins, mitochondria and lysosome related. Specifically, in the second category, we have revealed an association of the head domain of desmin with Cathepsin D (one of the lysosomal proteinases) and prosaposin (a single precursor which gives rise to Saposins A-D by proteolytic cleavage in lysosomes and is also referred to as sphigolipid activator proteins). In addition to its targeting to lysosomes, Cathepsin D is also involved in apoptosis and autophagy processes. This protein interaction result has been retested. The interaction between cathepsin D and desmin has also been further confirmed both with reverse yeast transformation as well as biochemical assays such as co-immunoprecipitation and GST pull down assay. The above described strong evidence of direct interaction between desmin and cathepsin D, has allowed us to propose a novel function of desmin IFs in lysosomal trafficking and/or as a new regulator of autophagy and apoptotic cell death.
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Optimally-Sized Design of a Wind/Diesel/Fuel Cell Hybrid System for a Remote CommunityVafaei, Mehdi 29 September 2011 (has links)
Remote communities, characterized by no connection to the main power grid, traditionally get their power from diesel generators. Long geographical distances and lack of suitable roads make the fuel transportation difficult and costly, increasing the final cost of electricity. A microgrid using renewable energy as the main source can serve as a viable solution for this problem with considerable economical and environmental benefits. The focus of this research is to develop a microgrid for a remote community in northern Ontario (Canada) that combines wind, as a renewable source of energy, and a hydrogen-based energy storage system, with the goal of meeting the demand, while minimizing the cost of energy and adverse effect on the environment. The existing diesel generators remain in the system, but their use is minimized.
The microgrid system studied in this research uses a wind turbine to generate electricity, an electrolyser to absorb the excess power from the wind source, a hydrogen tank to store the hydrogen generated by the electrolyser, a fuel cell to supply the demand when the wind resource is not adequate, and a diesel generator as a backup power.
Two scenarios for unit-sizing are defined and their pros. and cons. are discussed. The economic evaluation of scenarios is performed and a cost function for the system is defined. The optimization problem thus formulated is solved by solvers in GAMS. The inputs are wind profile of the area, load profile of the community, existing sources of energy in the area, operating voltage of the grid, and sale price of electricity in the area. The outputs are the size of the fuel cell and electrolyser units that should be used in the microgrid, the capital and running costs of each system, the payback period of the system, and cost of generated electricity. Following this, the best option for the microgrid structure and component sizes for the target community is determined.
Finally, a MATLAB-based dynamic simulation platform for the system under study with similar load/wind profile and sizing obtained in optimization problem is developed and the dynamic behaviour of microgrid at different cases is studied.
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Modified yeast two-hybrid screening identifies SKAP-HOM as a novel substrate of PTP-PESTScott, Adam Matthew. January 1900 (has links)
Thesis (M.Sc.). / Written for the Dept. of Biochemistry. Title from title page of PDF (viewed 2008/12/09). Includes bibliographical references.
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Habillage mécanique d'un nanofil par un champ de force : de la mesure vectorielle ultrasensible aux systèmes quantiques hybrides / Mechanical dressing of a nanowire by a force field : from ultrasensitive vectorial measurement to hybrid quantum systemsMercier de Lépinay, Laure 30 June 2017 (has links)
L'étude du couplage hybride entre les vibrations d'un résonateur mécanique et un degré de liberté quantique requiert une très grande sensibilité en force. Ceci a motivé le développement récent de nano-oscillateurs ultra-légers, qui constituent des sondes de force ultra-sensibles opérant désormais au niveau de l'attoNewton.Le premier volet de ce travail traite de l'habillage mécanique d'un nanofil suspendu de carbure de silicium, oscillant dans deux directions transverses. Son évolution Brownienne ou excitée dans un champ de force externe perturbe ses propriétés : fréquences et directions propres de vibration. Une technique de mesure optique des vibrations du nanofil en deux dimensions a été développée. Elle a permis, dans un premier temps, de cartographier un champ de force électrostatique, révélant une phénoménologie de l'habillage spécifique aux dimensions supérieures à un. En particulier, les composantes cisaillantes du champ induisent une rotation des directions propres de vibration. Dans un deuxième temps, la technique de mesure a été testée dans un champ de force optique, produit par un laser focalisé sur le nanofil, et qui contient une composante rotationnelle, non conservative. On rapporte l'observation d'une brisure de l'orthogonalité des modes propres, d'une distorsion et d'une amplification des spectres de mouvement Brownien, toutes en accord avec le modèle décrivant l'habillage mécanique du nanofil par ce champ de force externe non potentiel. A partir d'un protocole adapté à la multi-dimensionnalité du système, une déviation à la relation fluctuation-dissipation a également été mesurée : il s'agit d'une conséquence de la sortie de l'équilibre induite par la force optique non-conservative. L'étude des propriétés thermodynamiques du système suggère des corrections à la relation fluctuation-dissipation et prédit enfin un phénomène de compression du bruit thermique de l'oscillateur en champ de force rotationnel.Le deuxième volet de la thèse concerne le système hybride constitué d'un nanofil et d'un qubit de spin, un centre coloré NV du diamant attaché à son extrémité. Les propriétés optiques de cette source de photons uniques oscillante sont caractérisées à partir de la mesure de corrélations spatio-temporelles de la fluorescence du centre NV sur lesquelles les vibrations de l'oscillateur se retrouvent encodées. Nous avons ainsi développé un système de mesure du mouvement compatible avec de très faibles flux de photons, inférieurs en particulier au taux de décohérence mécanique. La dernière partie de ce manuscrit présente une étude préliminaire du couplage du spin-mécanique. Après l'observation d'un triplet de Mollow phononique, les développements expérimentaux de la première partie de la thèse ont été intégrés à une expérience hybride de seconde génération stabilisée amenant la mesure de force ultra-sensible à la portée du système hybride. / The study of the hybrid coupling between the vibrations of a mechanical resonator and a quantum degree of freedom requires extremely high force sensitivities. This was one of the motivations for the recent development of ultra-light nano-oscillators which are ultra-sensitive force probes now routinely operating at the attoNewton level.The first part of this work deals with the mechanical dressing of a silicon carbide suspended nanowire oscillating in two transverse directions. Its Brownian or driven evolution in an external force field modifies its mechanical properties: eigen-frequencies and eigen-directions of oscillation. An optical technique to measure the nanowire vibrations in two dimensions was developed. First, this technique enabled to map out an electrostatic force field, which revealed a dressing phenomenology specific to dimensions greater than one. In particular, shearing components of the force field are responsible for a rotation of the eigen-directions of vibration. Second, the measurement technique was tested in an optical force field applied by a laser focused on the nanowire. This field contains a rotational, non-conservative component. The reported experimental observations: eigenmodes orthogonality breaking, distorsion and amplification of Brownian motion spectra, are all in good agreement with the model of the mechanical dressing. Using a protocol adapted to the multidimensionality of the system, a deviation to the fluctuation-dissipation relation has also been measured, as a consequence of the non-conservative force bringing the system out of equilibrium. The study of this system's thermodynamic properties suggests corrections to the fluctuation-dissipation relation and predicts a squeezing of the oscillator's thermal noise in rotational force fields.The second part of the thesis concerns the hybrid system composed of a nanowire and a spin qubit: a colored NV center in diamond bound to its extremity. The optical properties of this oscillating single photon source are characterized through the measurement of space-time fluorescence correlations on which the oscillator's vibrations are encoded. We then developed a motion measurement technique compatible with very low photon fluxes, in particular inferior to the mechanical decoherence rate. The last part of the manuscript presents a preliminary study of the spin-mechanical coupling. After the observation of a phononic Mollow triplet, the experimental developments of the first part of the thesis were integrated in a second-generation stabilized hybrid experiment bringing the ultrasensitive force measurement within the reach of the hybrid system.
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A Calculus of Complex Zonotopes for Invariance and Stability Verification of Hybrid Systems / Un calcul des zonotopes complexes pour l'invariance et la vérification de la stabilité des systèmes hybridesAdimoolam, Santosh Arvind 16 May 2018 (has links)
Le calcul des ensembles atteignables est une approche de facto utilisée dans de nombreuses méthodes de vérification formelles pour les systèmes hybrides. Mais le calcul exact de l'ensemble atteignable est un problème insurmontable pour de nombreux types de systèmes hybrides, soit en raison de l'indécidabilité ou de la complexité de calcul élevée. Alternativement, beaucoup de recherches ont été axées sur l'utilisation de représentations d'ensembles qui peuvent être manipulées efficacement pour calculer une surestimation suffisamment précise de l'ensemble atteignable. Les zonotopes sont une représentation utile de l'ensemble dans l'analyse de l'accessibilité en raison de leur fermeture et de leur faible complexité pour le calcul de la transformation linéaire et des opérations sommaires de Minkowski. Mais pour approximer les ensembles de temps non bornés atteignables par des invariants positifs, les zonotopes ont l'inconvénient suivant. L'efficacité d'une représentation d'ensemble pour calculer un invariant positif dépend de l'encodage efficace des directions de convergence des états vers un équilibre. Dans un système hybride affine, certaines des directions de convergence peuvent être codées par les vecteurs propres à valeur complexe des matrices de transformation. Mais la représentation zonotopique ne peut pas exploiter la structure propre complexe des matrices de transformation car elle n'a que des générateurs à valeur réelle.Par conséquent, nous étendons les zonotopes réels au domaine de valeur complexe d'une manière qui peut capturer la contraction le long de vecteurs évalués complexes. Cela donne une nouvelle représentation d'ensemble appelée zonotope complexe. Géométriquement, les zonotopes complexes représentent une classe plus large d'ensembles qui comprennent des ensembles non polytopiques ainsi que des zonotopes polytopiques. Ils conservent le mérite des zonotopes réels que nous pouvons effectuer efficacement la transformation linéaire et les opérations sommaires de Minkowski et calculer la fonction de support. De plus, nous montrons qu'ils peuvent capturer la contraction le long de vecteurs propres complexes. De plus, nous développons des approximations traitables par calcul pour la vérification d'inclusion et l'intersection avec des demi-espaces. En utilisant ces opérations sur des zonotopes complexes, nous développons des programmes convexes pour vérifier les propriétés d'invariance linéaire des systèmes hybrides affines à temps discret et la stabilité exponentielle des systèmes impulsifs linéaires. Nos expériences sur certains exemples de benchmarks démontrent l'efficacité des techniques de vérification basées sur des zonotopes complexes. / Computing reachable sets is a de facto approach used in many formal verification methods for hybrid systems. But exact computation of the reachable set is an in- tractable problem for many kinds of hybrid systems, either due to undecidability or high computational complexity. Alternatively, quite a lot of research has been focused on using set representations that can be efficiently manipulated to com- pute sufficiently accurate over-approximation of the reachable set. Zonotopes are a useful set representation in reachability analysis because of their closure and low complexity for computing linear transformation and Minkowski sum operations. But for approximating the unbounded time reachable sets by positive invariants, zonotopes have the following drawback. The effectiveness of a set representation for computing a positive invariant depends on efficiently encoding the directions for convergence of the states to an equilibrium. In an affine hybrid system, some of the directions for convergence can be encoded by the complex valued eigen- vectors of the transformation matrices. But the zonotope representation can not exploit the complex eigenstructure of the transformation matrices because it only has real valued generators.Therefore, we extend real zonotopes to the complex valued domain in a way that can capture contraction along complex valued vectors. This yields a new set representation called complex zonotope. Geometrically, complex zonotopes repre- sent a wider class of sets that include some non-polytopic sets as well as polytopic zonotopes. They retain the merit of real zonotopes that we can efficiently perform linear transformation and Minkowski sum operations and compute the support function. Additionally, we show that they can capture contraction along complex valued eigenvectors. Furthermore, we develop computationally tractable approx- imations for inclusion-checking and intersection with half-spaces. Using these set operations on complex zonotopes, we develop convex programs to verify lin- ear invariance properties of discrete time affine hybrid systems and exponential stability of linear impulsive systems. Our experiments on some benchmark exam- ples demonstrate the efficiency of the verification techniques based on complex zonotopes.
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