Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Architectures and design patterns for functional design of logic control and diagnostics in industrial automation

Recently in most of the industrial automation process an ever increasing degree of automation has been observed. This increasing is motivated by the higher requirement of systems with great performance in terms of quality of products/services generated, productivity, efficiency and low costs in the design, realization and maintenance. This trend in the growth of complex automation systems is rapidly spreading over automated manufacturing systems (AMS), where the integration of the mechanical and electronic technology, typical of the Mechatronics, is merging with other technologies such as Informatics and the communication networks. An AMS is a very complex system that can be thought constituted by a set of flexible working stations, one or more transportation systems. To understand how this machine are important in our society let considerate that every day most of us use bottles of water or soda, buy product in box like food or cigarets and so on. Another important consideration from its complexity derive from the fact that the the consortium of machine producers has estimated around 350 types of manufacturing machine. A large number of manufacturing machine industry are presented in Italy and notably packaging machine industry,in particular a great concentration of this kind of industry is located in Bologna area; for this reason the Bologna area is called “packaging valley”. Usually, the various parts of the AMS interact among them in a concurrent and asynchronous way, and coordinate the parts of the machine to obtain a desiderated overall behaviour is an hard task. Often, this is the case in large scale systems, organized in a modular and distributed manner. Even if the success of a modern AMS from a functional and behavioural point of view is still to attribute to the design choices operated in the definition of the mechanical structure and electrical electronic architecture, the system that governs the control of the plant is becoming crucial, because of the large number of duties associated to it. Apart from the activity inherent to the automation of themachine cycles, the supervisory system is called to perform other main functions such as: emulating the behaviour of traditional mechanical members thus allowing a drastic constructive simplification of the machine and a crucial functional flexibility; dynamically adapting the control strategies according to the different productive needs and to the different operational scenarios; obtaining a high quality of the final product through the verification of the correctness of the processing; addressing the operator devoted to themachine to promptly and carefully take the actions devoted to establish or restore the optimal operating conditions; managing in real time information on diagnostics, as a support of the maintenance operations of the machine. The kind of facilities that designers can directly find on themarket, in terms of software component libraries provides in fact an adequate support as regard the implementation of either top-level or bottom-level functionalities, typically pertaining to the domains of user-friendly HMIs, closed-loop regulation and motion control, fieldbus-based interconnection of remote smart devices. What is still lacking is a reference framework comprising a comprehensive set of highly reusable logic control components that, focussing on the cross-cutting functionalities characterizing the automation domain, may help the designers in the process of modelling and structuring their applications according to the specific needs. Historically, the design and verification process for complex automated industrial systems is performed in empirical way, without a clear distinction between functional and technological-implementation concepts and without a systematic method to organically deal with the complete system. Traditionally, in the field of analog and digital control design and verification through formal and simulation tools have been adopted since a long time ago, at least for multivariable and/or nonlinear controllers for complex time-driven dynamics as in the fields of vehicles, aircrafts, robots, electric drives and complex power electronics equipments. Moving to the field of logic control, typical for industrial manufacturing automation, the design and verification process is approached in a completely different way, usually very “unstructured”. No clear distinction between functions and implementations, between functional architectures and technological architectures and platforms is considered. Probably this difference is due to the different “dynamical framework”of logic control with respect to analog/digital control. As a matter of facts, in logic control discrete-events dynamics replace time-driven dynamics; hence most of the formal and mathematical tools of analog/digital control cannot be directly migrated to logic control to enlighten the distinction between functions and implementations. In addition, in the common view of application technicians, logic control design is strictly connected to the adopted implementation technology (relays in the past, software nowadays), leading again to a deep confusion among functional view and technological view. In Industrial automation software engineering, concepts as modularity, encapsulation, composability and reusability are strongly emphasized and profitably realized in the so-calledobject-oriented methodologies. Industrial automation is receiving lately this approach, as testified by some IEC standards IEC 611313, IEC 61499 which have been considered in commercial products only recently. On the other hand, in the scientific and technical literature many contributions have been already proposed to establish a suitable modelling framework for industrial automation. During last years it was possible to note a considerable growth in the exploitation of innovative concepts and technologies from ICT world in industrial automation systems. For what concerns the logic control design, Model Based Design (MBD) is being imported in industrial automation from software engineering field. Another key-point in industrial automated systems is the growth of requirements in terms of availability, reliability and safety for technological systems. In other words, the control system should not only deal with the nominal behaviour, but should also deal with other important duties, such as diagnosis and faults isolations, recovery and safety management. Indeed, together with high performance, in complex systems fault occurrences increase. This is a consequence of the fact that, as it typically occurs in reliable mechatronic systems, in complex systems such as AMS, together with reliable mechanical elements, an increasing number of electronic devices are also present, that are more vulnerable by their own nature. The diagnosis problem and the faults isolation in a generic dynamical system consists in the design of an elaboration unit that, appropriately processing the inputs and outputs of the dynamical system, is also capable of detecting incipient faults on the plant devices, reconfiguring the control system so as to guarantee satisfactory performance. The designer should be able to formally verify the product, certifying that, in its final implementation, it will perform itsrequired function guarantying the desired level of reliability and safety; the next step is that of preventing faults and eventually reconfiguring the control system so that faults are tolerated. On this topic an important improvement to formal verification of logic control, fault diagnosis and fault tolerant control results derive from Discrete Event Systems theory. The aimof this work is to define a design pattern and a control architecture to help the designer of control logic in industrial automated systems. The work starts with a brief discussion on main characteristics and description of industrial automated systems on Chapter 1. In Chapter 2 a survey on the state of the software engineering paradigm applied to industrial automation is discussed. Chapter 3 presentes a architecture for industrial automated systems based on the new concept of Generalized Actuator showing its benefits, while in Chapter 4 this architecture is refined using a novel entity, the Generalized Device in order to have a better reusability and modularity of the control logic. In Chapter 5 a new approach will be present based on Discrete Event Systems for the problemof software formal verification and an active fault tolerant control architecture using online diagnostic. Finally conclusive remarks and some ideas on new directions to explore are given. In Appendix A are briefly reported some concepts and results about Discrete Event Systems which should help the reader in understanding some crucial points in chapter 5; while in Appendix B an overview on the experimental testbed of the Laboratory of Automation of University of Bologna, is reported to validated the approach presented in chapter 3, chapter 4 and chapter 5. In Appendix C some components model used in chapter 5 for formal verification are reported.

Essays in Financial Econometrics

The first paper sheds light on the informational content of high frequency data and daily data. I assess the economic value of the two family models comparing their performance in forecasting asset volatility through the Value at Risk metric. In running the comparison this paper introduces two key assumptions: jumps in prices and leverage effect in volatility dynamics. Findings suggest that high frequency data models do not exhibit a superior performance over daily data models. In the second paper, building on Majewski et al. (2015), I propose an affine-discrete time model, labeled VARG-J, which is characterized by a multifactor volatility specification. In the VARG-J model volatility experiences periods of extreme movements through a jump factor modeled as an Autoregressive Gamma Zero process. The estimation under historical measure is done by quasi-maximum likelihood and the Extended Kalman Filter. This strategy allows to filter out both volatility factors introducing a measurement equation that relates the Realized Volatility to latent volatility. The risk premia parameters are calibrated using call options written on S&P500 Index. The results clearly illustrate the important contribution of the jump factor in the pricing performance of options and the economic significance of the volatility jump risk premia. In the third paper, I analyze whether there is empirical evidence of contagion at the bank level, measuring the direction and the size of contagion transmission between European markets. In order to understand and quantify the contagion transmission on banking market, I estimate the econometric model by Aït-Sahalia et al. (2015) in which contagion is defined as the within and between countries transmission of shocks and asset returns are directly modeled as a Hawkes jump diffusion process. The empirical analysis indicates that there is a clear evidence of contagion from Greece to European countries as well as self-contagion in all countries.

Spectrum shaping and its application: spread-spectrum clock generator and circuits for ultra wide band

Electromagnetic spectrum can be identified as a resource for the designer, as well as for the manufacturer, from two complementary points of view: first, because it is a good in great demand by many different kind of applications; second, because despite its scarce availability, it may be advantageous to use more spectrum than necessary. This is the case of Spread-Spectrum Systems, those systems in which the transmitted signal is spread over a wide frequency band, much wider, in fact, than the minimum bandwidth required to transmit the information being sent. Part I of this dissertation deals with Spread-Spectrum Clock Generators (SSCG) aiming at reducing Electro Magnetic Interference (EMI) of clock signals in integrated circuits (IC) design. In particular, the modulation of the clock and the consequent spreading of its spectrum are obtained through a random modulating signal outputted by a chaotic map, i.e. a discrete-time dynamical system showing chaotic behavior. The advantages offered by this kind of modulation are highlighted. Three different prototypes of chaos-based SSCG are presented in all their aspects: design, simulation, and post-fabrication measurements. The third one, operating at a frequency equal to 3GHz, aims at being applied to Serial ATA, standard de facto for fast data transmission to and from Hard Disk Drives. The most extreme example of spread-spectrum signalling is the emerging ultra-wideband (UWB) technology, which proposes the use of large sections of the radio spectrum at low amplitudes to transmit high-bandwidth digital data. In part II of the dissertation, two UWB applications are presented, both dealing with the advantages as well as with the challenges of a wide-band system, namely: a chaos-based sequence generation method for reducing Multiple Access Interference (MAI) in Direct Sequence UWB Wireless-Sensor-Networks (WSNs), and design and simulations of a Low-Noise Amplifier (LNA) for impulse radio UWB. This latter topic was studied during a study-abroad period in collaboration with Delft University of Technology, Delft, Netherlands.

Enabling Blocks for Integrated CMOS UWB Transceivers

The last decades have seen an unrivaled growth and diffusion of mobile telecommunications. Several standards have been developed to this purposes, from GSM mobile phone communications to WLAN IEEE 802.11, providing different services for the the transmission of signals ranging from voice to high data rate digital communications and Digital Video Broadcasting (DVB). In this wide research and market field, this thesis focuses on Ultra Wideband (UWB) communications, an emerging technology for providing very high data rate transmissions over very short distances. In particular the presented research deals with the circuit design of enabling blocks for MB-OFDM UWB CMOS single-chip transceivers, namely the frequency synthesizer and the transmission mixer and power amplifier. First we discuss three different models for the simulation of chargepump phase-locked loops, namely the continuous time s-domain and discrete time z-domain approximations and the exact semi-analytical time-domain model. The limitations of the two approximated models are analyzed in terms of error in the computed settling time as a function of loop parameters, deriving practical conditions under which the different models are reliable for fast settling PLLs up to fourth order. Besides, a phase noise analysis method based upon the time-domain model is introduced and compared to the results obtained by means of the s-domain model. We compare the three models over the simulation of a fast switching PLL to be integrated in a frequency synthesizer for WiMedia MB-OFDM UWB systems. In the second part, the theoretical analysis is applied to the design of a 60mW 3.4 to 9.2GHz 12 Bands frequency synthesizer for MB-OFDM UWB based on two wide-band PLLs. The design is presented and discussed up to layout level. A test chip has been implemented in TSMC CMOS 90nm technology, measured data is provided. The functionality of the circuit is proved and specifications are met with state-of-the-art area occupation and power consumption. The last part of the thesis deals with the design of a transmission mixer and a power amplifier for MB-OFDM UWB band group 1. The design has been carried on up to layout level in ST Microlectronics 65nm CMOS technology. Main characteristics of the systems are the wideband behavior (1.6 GHz of bandwidth) and the constant behavior over process parameters, temperature and supply voltage thanks to the design of dedicated adaptive biasing circuits.

Advanced CMOS Interfaces for Bio-Nanosensors

The improvement of devices provided by Nanotechnology has put forward new classes of sensors, called bio-nanosensors, which are very promising for the detection of biochemical molecules in a large variety of applications. Their use in lab-on-a-chip could gives rise to new opportunities in many fields, from health-care and bio-warfare to environmental and high-throughput screening for pharmaceutical industry. Bio-nanosensors have great advantages in terms of cost, performance, and parallelization. Indeed, they require very low quantities of reagents and improve the overall signal-to-noise-ratio due to increase of binding signal variations vs. area and reduction of stray capacitances. Additionally, they give rise to new challenges, such as the need to design high-performance low-noise integrated electronic interfaces. This thesis is related to the design of high-performance advanced CMOS interfaces for electrochemical bio-nanosensors. The main focus of the thesis is: 1) critical analysis of noise in sensing interfaces, 2) devising new techniques for noise reduction in discrete-time approaches, 3) developing new architectures for low-noise, low-power sensing interfaces. The manuscript reports a multi-project activity focusing on low-noise design and presents two developed integrated circuits (ICs) as examples of advanced CMOS interfaces for bio-nanosensors. The first project concerns low-noise current-sensing interface for DC and transient measurements of electrophysiological signals. The focus of this research activity is on the noise optimization of the electronic interface. A new noise reduction technique has been developed so as to realize an integrated CMOS interfaces with performance comparable with state-of-the-art instrumentations. The second project intends to realize a stand-alone, high-accuracy electrochemical impedance spectroscopy interface. The system is tailored for conductivity-temperature-depth sensors in environmental applications, as well as for bio-nanosensors. It is based on a band-pass delta-sigma technique and combines low-noise performance with low-power requirements.

Propagation modelling and resource allocation in mobile radio communications

Over the past years, ray tracing (RT) models popularity has been increasing. From the nineties, RT has been used for field prediction in environment such as indoor and urban environments. Nevertheless, with the advent of new technologies, the channel model has become decidedly more dynamic and to perform RT simulations at each discrete time instant become computationally expensive. In this thesis, a new dynamic ray tracing (DRT) approach is presented in which from a single ray tracing simulation at an initial time t0, through analytical formulas we are able to track the motion of the interaction points. The benefits that this approach bring are that Doppler frequencies and channel prediction can be derived at every time instant, without recurring to multiple RT runs and therefore shortening the computation time. DRT performance was studied on two case studies and the results shows the accuracy and the computational gain that derives from this approach. Another issue that has been addressed in this thesis is the licensed band exhaustion of some frequency bands. To deal with this problem, a novel unselfish spectrum leasing scheme in cognitive radio networks (CRNs) is proposed that offers an energy-efficient solution minimizing the environmental impact of the network. In addition, a network management architecture is introduced and resource allocation is proposed as a constrained sum energy efficiency maximization problem. System simulations demonstrate an increment in the energy efficiency of the primary users’ network compared with previously proposed algorithms.

Numerical simulations of magma chamber dynamics at Campi Flegrei, and associated seismicity, deformation and gravity changes

Understanding the complex relationships between quantities measured by volcanic monitoring network and shallow magma processes is a crucial headway for the comprehension of volcanic processes and a more realistic evaluation of the associated hazard. This question is very relevant at Campi Flegrei, a volcanic quiescent caldera immediately north-west of Napoli (Italy). The system activity shows a high fumarole release and periodic ground slow movement (bradyseism) with high seismicity. This activity, with the high people density and the presence of military and industrial buildings, makes Campi Flegrei one of the areas with higher volcanic hazard in the world. In such a context my thesis has been focused on magma dynamics due to the refilling of shallow magma chambers, and on the geophysical signals detectable by seismic, deformative and gravimetric monitoring networks that are associated with this phenomenologies. Indeed, the refilling of magma chambers is a process frequently occurring just before a volcanic eruption; therefore, the faculty of identifying this dynamics by means of recorded signal analysis is important to evaluate the short term volcanic hazard. The space-time evolution of dynamics due to injection of new magma in the magma chamber has been studied performing numerical simulations with, and implementing additional features in, the code GALES (Longo et al., 2006), recently developed and still on the upgrade at the Istituto Nazionale di Geofisica e Vulcanologia in Pisa (Italy). GALES is a finite element code based on a physico-mathematical two dimensional, transient model able to treat fluids as multiphase homogeneous mixtures, compressible to incompressible. The fundamental equations of mass, momentum and energy balance are discretised both in time and space using the Galerkin Least-Squares and discontinuity-capturing stabilisation technique. The physical properties of the mixture are computed as a function of local conditions of magma composition, pressure and temperature.The model features enable to study a broad range of phenomenologies characterizing pre and sin-eruptive magma dynamics in a wide domain from the volcanic crater to deep magma feeding zones. The study of displacement field associated with the simulated fluid dynamics has been carried out with a numerical code developed by the Geophysical group at the University College Dublin (O’Brien and Bean, 2004b), with whom we started a very profitable collaboration. In this code, the seismic wave propagation in heterogeneous media with free surface (e.g. the Earth’s surface) is simulated using a discrete elastic lattice where particle interactions are controlled by the Hooke’s law. This method allows to consider medium heterogeneities and complex topography. The initial and boundary conditions for the simulations have been defined within a coordinate project (INGV-DPC 2004-06 V3_2 “Research on active volcanoes, precursors, scenarios, hazard and risk - Campi Flegrei”), to which this thesis contributes, and many researchers experienced on Campi Flegrei in volcanological, seismic, petrological, geochemical fields, etc. collaborate. Numerical simulations of magma and rock dynamis have been coupled as described in the thesis. The first part of the thesis consists of a parametric study aimed at understanding the eect of the presence in magma of carbon dioxide in magma in the convection dynamics. Indeed, the presence of this volatile was relevant in many Campi Flegrei eruptions, including some eruptions commonly considered as reference for a future activity of this volcano. A set of simulations considering an elliptical magma chamber, compositionally uniform, refilled from below by a magma with volatile content equal or dierent from that of the resident magma has been performed. To do this, a multicomponent non-ideal magma saturation model (Papale et al., 2006) that considers the simultaneous presence of CO2 and H2O, has been implemented in GALES. Results show that the presence of CO2 in the incoming magma increases its buoyancy force promoting convection ad mixing. The simulated dynamics produce pressure transients with frequency and amplitude in the sensitivity range of modern geophysical monitoring networks such as the one installed at Campi Flegrei . In the second part, simulations more related with the Campi Flegrei volcanic system have been performed. The simulated system has been defined on the basis of conditions consistent with the bulk of knowledge of Campi Flegrei and in particular of the Agnano-Monte Spina eruption (4100 B.P.), commonly considered as reference for a future high intensity eruption in this area. The magmatic system has been modelled as a long dyke refilling a small shallow magma chamber; magmas with trachytic and phonolitic composition and variable volatile content of H2O and CO2 have been considered. The simulations have been carried out changing the condition of magma injection, the system configuration (magma chamber geometry, dyke size) and the resident and refilling magma composition and volatile content, in order to study the influence of these factors on the simulated dynamics. Simulation results allow to follow each step of the gas-rich magma ascent in the denser magma, highlighting the details of magma convection and mixing. In particular, the presence of more CO2 in the deep magma results in more ecient and faster dynamics. Through this simulations the variation of the gravimetric field has been determined. Afterward, the space-time distribution of stress resulting from numerical simulations have been used as boundary conditions for the simulations of the displacement field imposed by the magmatic dynamics on rocks. The properties of the simulated domain (rock density, P and S wave velocities) have been based on data from literature on active and passive tomographic experiments, obtained through a collaboration with A. Zollo at the Dept. of Physics of the Federici II Univeristy in Napoli. The elasto-dynamics simulations allow to determine the variations of the space-time distribution of deformation and the seismic signal associated with the studied magmatic dynamics. In particular, results show that these dynamics induce deformations similar to those measured at Campi Flegrei and seismic signals with energies concentrated on the typical frequency bands observed in volcanic areas. The present work shows that an approach based on the solution of equations describing the physics of processes within a magmatic fluid and the surrounding rock system is able to recognise and describe the relationships between geophysical signals detectable on the surface and deep magma dynamics. Therefore, the results suggest that the combined study of geophysical data and informations from numerical simulations can allow in a near future a more ecient evaluation of the short term volcanic hazard.

Dynamic identification of structures: experimental assessment of modal parameteres through methods in frequency domain, in time-frequency domain and model updating

Among the experimental methods commonly used to define the behaviour of a full scale system, dynamic tests are the most complete and efficient procedures. A dynamic test is an experimental process, which would define a set of characteristic parameters of the dynamic behaviour of the system, such as natural frequencies of the structure, mode shapes and the corresponding modal damping values associated. An assessment of these modal characteristics can be used both to verify the theoretical assumptions of the project, to monitor the performance of the structural system during its operational use. The thesis is structured in the following chapters: The first introductive chapter recalls some basic notions of dynamics of structure, focusing the discussion on the problem of systems with multiply degrees of freedom (MDOF), which can represent a generic real system under study, when it is excited with harmonic force or in free vibration. The second chapter is entirely centred on to the problem of dynamic identification process of a structure, if it is subjected to an experimental test in forced vibrations. It first describes the construction of FRF through classical FFT of the recorded signal. A different method, also in the frequency domain, is subsequently introduced; it allows accurately to compute the FRF using the geometric characteristics of the ellipse that represents the direct input-output comparison. The two methods are compared and then the attention is focused on some advantages of the proposed methodology. The third chapter focuses on the study of real structures when they are subjected to experimental test, where the force is not known, like in an ambient or impact test. In this analysis we decided to use the CWT, which allows a simultaneous investigation in the time and frequency domain of a generic signal x(t). The CWT is first introduced to process free oscillations, with excellent results both in terms of frequencies, dampings and vibration modes. The application in the case of ambient vibrations defines accurate modal parameters of the system, although on the damping some important observations should be made. The fourth chapter is still on the problem of post processing data acquired after a vibration test, but this time through the application of discrete wavelet transform (DWT). In the first part the results obtained by the DWT are compared with those obtained by the application of CWT. Particular attention is given to the use of DWT as a tool for filtering the recorded signal, in fact in case of ambient vibrations the signals are often affected by the presence of a significant level of noise. The fifth chapter focuses on another important aspect of the identification process: the model updating. In this chapter, starting from the modal parameters obtained from some environmental vibration tests, performed by the University of Porto in 2008 and the University of Sheffild on the Humber Bridge in England, a FE model of the bridge is defined, in order to define what type of model is able to capture more accurately the real dynamic behaviour of the bridge. The sixth chapter outlines the necessary conclusions of the presented research. They concern the application of a method in the frequency domain in order to evaluate the modal parameters of a structure and its advantages, the advantages in applying a procedure based on the use of wavelet transforms in the process of identification in tests with unknown input and finally the problem of 3D modeling of systems with many degrees of freedom and with different types of uncertainty.