Authors: Ali Fuat Ergenc; Nejat Olgac

Abstract: A new concept is presented for determining the kernel and the offspring hypersurfaces for general LTI dynamics with multiple delays. These hypersurfaces, as they are very recently introduced in a concept paper (Sipahi and Olgac 2005), form the basis of the overriding paradigm which is called the "Cluster Treatment of Characteristic Roots (CTCR)". In fact, these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots. To determine these kernel and offspring we use the extraordinary features of "Extended Kronecker Summation" operation in this paper. The end result is that the infinite dimensional problem reduces to a finite dimensional one (and preferably into an eigenvalue problem). Following the procedure described in this paper we are able to shorten the computational time considerably in determining these hypersurfaces. We demonstrate these concepts via some example case studies.

Keywords: time-delay system,Kronecker sum,stability

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00026

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 157-162

Authors: Dimitri Breda; Stefano Maset; Rossana Vermiglio

Abstract: In this work we address the question of asymptotic stability of linear delay differential equations (DDEs) with time periodic coefficients, a class which is recognized to be fundamental in machining tool. Since the dynamics of such a class of delay systems is governed by the dominant eigenvalues (multipliers) of the monodromy operator associated to the system of DDEs, i.e. the solution operator over the period of the coefficients, we discretize it by using pseudospectral differencing techniques based on collocation and approximate the dominant multipliers by the eigenvalues of the resulting matrix. The use of pseudospectral methods has already been proposed in the context of simpler DDEs. Here we fully generalize the method to the class of linear time periodic coefficients DDEs with arbitrary period and multiple discrete and distributed delays. The scheme is shown to have spectral accuracy by means of several numerical examples.

Keywords: delay differential equations,periodic coefficients,characteristic multipliers,monodromy operator,pseudospectral methods

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00027

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 163-168

Authors: Tomáš Vyhlídal; Pavel Zítek

Abstract: An original method for computing the spectrum of a retarded time delay system over a large region in the complex plane is presented. The method is based on mapping the quasi-polynomial characteristic function and on utilizing asymptotic properties of the root chains. First, asymptotic exponentials of the root chains are assessed and large areas free of the roots are determined. These areas are then omitted in systematic mapping of the roots location. The roots are located as the intersection points of the contours determined by real and imaginary parts of the characteristic function. The theoretical explanation of the method is supplemented ith a discussion of implementation issues and an application example is added.

Keywords: retarded system, quasi-polynomial,spectrum distribution,root-finding techniques

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00028

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 169-174

Authors: Nejat Olgac; Tomáš Vyhlídal; Rifat Sipahi

Abstract: The stability of neutral systems with two cross-talking delays is investigated using the method of cluster treatment of characteristic roots (CTCR). here are two main outcomes of this study: (a) Generation of the well-known delay stabilizability condition as a by-product of the CTCR procedure. This is achieved by a small delay stability treatment over the system. We also demonstrate for the delay-stabilizable systems the exact bounds of the stability regions in the domain of the delays. (b) Validation of these stability regions using an alternative point-wise algorithm, which computes the right-most roots of the characteristic quasi-polynomial.

Keywords: time-delay system,neutral system,delay-dependent stability

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00029

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 175-180

Authors: Silviu-Iulian Niculescu; Peilin Fu; Jie Chen

Abstract: In this paper we study the stability properties of a class of lossless propagation systems. Roughly speaking, a lossless propagation model is defined by a system of semiexplicit delay differential algebraic equations, that is a system of differential equations coupled with a system of (continuous-time) difference equations. We show that the stability analysis in the commensurate delay case can be performed by computing the generalized eigenvalues of certain matrix pencils, which can be executed efficiently and with high precision. The results extend previously known work on retarded, and neutral systems, and demonstrate that similar stability tests can be derived for such systems.

Keywords: propagation,delay,stability,switches,matrix pencil

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00030

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 181-186

Authors: Rifat Sipahi; Silviu-Iulian Niculescu

Abstract: Analytical stability study of some deterministic car following models under time-delay influences is presented and various case studies are demonstrated. Interestingly, for some control law deployed by human drivers, more than one stability interval in the domain of time delay is revealed. Physical interpretations along with comparisons conclude the study.

Keywords: car following model,delay,crossing curves,stability,sensitivity

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00031

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 187-192

Authors: Gábor Stépán; Gábor Orosz

Abstract: A nonlinear car-following model that includes the reaction-time delay of drivers is considered. When investigating the linear stability of the uniform flow solution, boundaries of Hopf bifurcations are determined in the parameter space. Crossing these boundaries, oscillations may appear corresponding to travelling wave solutions. Hopf normal form calculations prove robustly subcritical behavior which leads to bistability between the stable uniform traffic flow and the stop-and-go waves travelling against the flow of vehicles. Analogies with wheel shimmy dynamics and machine tool vibrations are presented.

Keywords: vehicular traffic,reaction-time delay,translational symmetry,subcritical Hopf bifurcation, bistability

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00032

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 193-198

Authors: Gábor Orosz; Bernd Krauskopf; R. Eddie Wilson

Abstract: We consider an optimal-velocity car-following model with reaction-time delay of drivers, where the characteristics of the drivers change according to a suitably calibrated random walk. In the absence of this stochasticity we find stable and almost stable oscillations that correspond to stop-and-go traffic jams that eventually merge or disperse. We study how the distribution of the merging times depends on the parameters of the random walk. Our numerical simulations suggest that the motion of the fronts into and out of traffic jams may be subject to a 'macroscopic' random walk.

Keywords: optimal velocity model,reaction-time delay,stop-and-go waves,random walk

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00033

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 199-204

Authors: Arne Kesting; Martin Treiber

Abstract: We investigate two causes for the instability of traffic flow: The time lag caused by finite accelerations of the vehicles, and the delay caused by the finite reaction times of the drivers. Furthermore, we simulate to which degree drivers may compensate for these delays by looking several vehicles ahead and anticipate future traffic situations. Since vehicular traffic flow is a multi-particle system with many degrees of freedom, two concepts of linear stability have to be considered: Local stability of a car following a leader that drives at constant velocity, and string (chain) stability of a "platoon" of several vehicles following each other. Typically, string stability is a much more restrictive criterion than local stability. We simulate both types of stability with the human driver model (HDM)[M. Treiber et al., Physica A, Vol. 360 (1), 71-88 (2006)], which includes all the features above. We found several remarkable results: (i) with a suitable anticipation, we obtained string stability for reaction times exceeding the "safe time headway", which, to date, has not yet been obtained for any other car-following model; (ii) parameter changes that destabilize the model variant with zero reaction time may stabilize the model with finite reaction times and vice versa, (iii) distributed reaction times (every driver has a different reaction time) can stabilize the system compared to drivers with identical reaction times that are equal to the mean.

Keywords: car-following model,microscopic traffic simulation,human reaction time,time-delayed system,local and string stability

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00034

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 205-210

Authors: Alfredo Bellen; Nicola Guglielmi

Abstract: For a delay differential equation of neutral type (in short NDDE) the usual approach for solving delay equations, based on the use of a continuous ODE method, must take care of the additional term involving the derivative of the solution at some previous instant. Therefore, besides the continuous extension for approximating the solution, the method must provide a continuous approximation for the derivative of the solution as well. Starting from this point we present a general class of numerical methods for solving NDDEs and stress the peculiarities and the differences with respect to the standard case of DDEs. First we analyze some basic aspects of convergence theory. Then we focus our attention on the regularity of the solutions; since for NDDEs the smoothing of solutions does not take place, taking care of jump discontinuities is much more important with respect to DDEs. Finally we explain why uniqueness and existence of the solution is not anymore guaranteed for general state-dependent problems and how to detect these occurrences automatically.

Keywords: neutral delay differential equations,Runge-Kutta methods,Radau IIA methods,breaking points,non-existence and non-uniqueness of solutions,error control,numerical well-posedness

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00035

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 211-216

Authors: A. E. Ruehli; G. Antonini; J. Ekman

Abstract: The purpose of this paper is twofold. First, we present the Partial Element Equivalent Circuit (PEEC) formulation of Maxwell's equations. Second, the resultant system of neutral delay differential equations (NDDEs), arising from the time domain solution of PEEC models, and the stability of these are considered. In the classical PEEC model, single delay approximations were used for the partial elements describing the electric- and magnetic field couplings. It was shown that the resultant system of delay differential equations may become unstable for some time domain applications. In this paper, we present a systematic process for the stability and accuracy enhancement of the time domain PEEC model by improving the evaluation of the partial element based on the subdivision technique and by a filtering process.

Keywords: neutral delay differential equations (NDDE),partial element equivalent circuit (PEEC) method

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00036

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 217-222

Authors: Lucio Torelli

Abstract: This paper is devoted to stability analysis of Delay Differential Equations of Neutral type. In particular we present two reformulations of the problem, for which the neutral system is transformed into an ordinary differential equation coupled to an algebraic recursion. Hypotheses on the original equation allow to study contractivity and asymptotic stability of the true solution and of the numerical schemes by the theory of stability with respect to a forcing term.

Keywords: delay differential equations of neutral type,contractivity,asymptotic stability,Hale's form,numerical methods

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00037

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 223-228

Authors: P. Palumbo; S. Panunzi; A. De Gaetano

Abstract: A discrete delay-differential model of the glucose-insulin system is presented, representing the Intra-Venous Glucose Tolerance Test and allied experimental procedures of diabetological interest. The model ensures unique positive bounded solutions for any positive initial condition, and provides a unique equilibrium point. Conditions are given on the physical parameters in order to ensure the local asymptotic stability of the equilibrium point. These conditions are always satisfied, given the actual parameter estimates obtained experimentally. A study of the global stability properties is also performed.

Keywords: delay-differential systems,glucose-insulin homeostasis,stability analysis

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00038

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 229-234

Authors: Pavel Ettler; Ivan Puchr

Abstract: An interesting control problem was to be solved for an array of annealing furnaces with a simple construction. Temperature of the heated material should follow a prescribed diagram while no enforced cooling is available, furnace heating output is limited and a load-dependent time delay is present. The solution combines several control techniques including feedback using predicted values of temperature.

Keywords: time delay,prediction,industry automation,on-off control,annealing furnace

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00039

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 235-239

Authors: Alireza Alfi; Mohammad Farrokhi

Abstract: In this paper, a new structure for teleoperated systems with time-varying delay in communication channel will be proposed, which provides transparency for the system. The key features of this structure are its simple design as well as the ability to analyze the stability of the closed-loop system using the property of the stable scalar functions and the small gain theorem. In the proposed structure, two local controllers will be designed, such that the transparency of the teleoperated system as well as the local stabilities is guaranteed. One local controller will be designated for position tracking of the slave system and the other one, whilst ensuring the stability of the closed-loop system in presence of time-varying delay in communication channel, performs the force tracking.

Keywords: bilateral teleoperation,transparency,time delay

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00040

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 240-245

Authors: Winston García-Gabín; Julio E. Normey-Rico; Eduardo F. Camacho

Abstract: This paper presents the application of a sliding mode predictive control (SMPC) algorithm on the benchmark control problem of the isothermal operation of a continuous stirred tank reactor (CSTR) with Van der Vusse reactions and an on-line concentration analyzer that introduces a dead-time to the non-linear behavior of the process. The SMPC showed a considerable robustness improvement with respect to MPC in the presence non-linearities and time delay even when using a linear model to compute the predictions in the controller, avoiding the use of non-linear optimization and allowing a low cost consuming algorithm.

Keywords: process control,predictive control,sliding-mode control,time delay

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00041

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 246-251

Authors: A. N. Gündeš; H. Özbay; A. B. Özgüler

Abstract: Conditions are presented for closed-loop stabilizability of linear time-invariant (LTI) multi-input, multi-output (MIMO) plants with I/O delays (time delays in the input and/or output channels) using PID (Proportional + Integral +Derivative) controllers. We show that systems with at most two unstable poles can be stabilized by PID controllers provided a small gain condition is satisfied. For systems with only one unstable pole, this condition is equivalent to having sufficiently small delay-unstable pole product. Our method of synthesis of such controllers identify some free parameters that can be used to satisfy further design criteria than stability.

Keywords: PID control,time delay,unstable systems,MIMO systems

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00042

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 252-257

Authors: H. Özbay; A. N. Gündeş

Abstract: Recently (Gündes et al., 2006) obtained stabilizing PID controllers for a class of MIMO unstable plants with time delays in the input and output channels (I/O delays). Using this approach, for plants with one unstable pole, we investigate resilient PI and PD controllers. Specifically, for PD controllers, optimal derivative action gain is determined to maximize a lower bound of the largest allowable controller gain. For PI controllers, optimal proportional gain is determined to maximize a lower bound of the largest allowable integral action gain.

Keywords: PID control,time delay,unstable systems,MIMO systems

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00043

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 258-263

Authors: Constantin-Irinel Morărescu; Silviu-Iulian Niculescu; Keqin Gu

Abstract: In this paper we develop a method to obtain the stability crossing curves of a Smith Predictor control scheme. More explicitly, we compute the crossing set, which consists of all frequencies corresponding to all points on the stability crossing curve, and we give their complete classification. Furthermore, the directions in which the zeros cross the imaginary axis are explicitly expressed.

Keywords: Smith Predictor,delay uncertainty,stability

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00044

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 264-269

Authors: L. Zaccarian; P. Pepe; D. Nesic

Abstract: In this paper we show the effectiveness of a novel control scheme for dead-time systems recently proposed in (Zaccarian and Nešić, 2006 American Control Conference). In that paper, the prediction problem is formally stated and shown to be achievable for any type of dead-time system, as long as it is possible to determine a stabilizing control law. We propose here the employment of an LMI-based control strategy for this stabilization task and show the advantages of the novel modified Smith predictor scheme on a MIMO example. In particular, satisfactory simulation results are given both for the linearized model and for the nonlinear model of an inverted pendulum subject to disturbances.

Keywords:

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00045

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 270-275

Authors: Tsuneo Yoshikawa

Abstract: Several works we have completed in coping with time delay problems in robotic systems and haptic interfaces are presented: ground-space teleoperation of a robot arm on a space satellite, passivity-based adaptive control of flexible robot arms and power assist devices, and partially continuous-time impedance display for haptic interface systems. For each topic, background, overview, the approach taken, and experimental results are discussed. Passivity is utilized directly or indirectly in all these works.

Keywords: time delay,robot,teleoperation,interface,virtual reality,passivity,PD-control

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00046

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 276-283

Authors: M.-N. Alpaslan Parlakçi

Abstract: In this paper a new class of augmented quasi full size Lypunov-Krasovskii functional is introduced for the robust stability of linear uncertain neutral systems. The nonlinear parameter perturbations and norm bounded uncertainties are taken into consideration separately. Delay-dependent robust stability criteria are derived in the form of linear matrix inequalities. Numerical examples are presented to illustrate the significant improvement on the conservativeness of the delay bound over some reported results in the literature.

Keywords: neutral systems,nonlinear parameter perturbations,norm-bounded uncertainty,delay-dependent stability,augmented quasi full size Lyapunov functional

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00047

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 284-289

Authors: Toshiki Oguchi; Henk Nijmeijer

Abstract: This paper considers the state prediction of chaotic behaviour using a master-slave synchronization scheme. In this paper, we consider the robust convergency of the prediction error of anticipating synchronization of nonlinear systems with model uncertainties and sufficient conditions for anticipatory synchronization are derived. The effectiveness of the obtained result is illustrated by a simulation example.

Keywords: nonlinear systems,time-delay,synchronization,state predictor

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00048

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 290-295

Authors: Elias Jarlebring

Abstract: In this work we analyze stability properties of retarded linear time invariant multi-dimensional, multi-delay, time delay systems with respect to perturbations in the delay parameters. We analyze two methods which allow the computation of the critical delays, i.e., the points in delay-space which causes the system to have a purely imaginary eigenvalue. The critical delays are potential stability boundaries as the boundaries of the stability region is necessarily a subset of the critical delays. The two methods originates from a Lyapunov-type condition, which is completely self-contained in this work. The first method corresponds to the case of commensurate delays, for which the the Lyapunov-type condition reduces to a polynomial eigenvalue problem for which the first companion form is exactly the eigenvalue problem occurring in Chen et al. (1995). The second method is the result of a simple substitution which allows the computation of the critical delays of an incommensurate system by solving a quadratic eigenvalue problem. For the scalar multi-delay case we find a closed expression for the critical curves using this method. We confirm the methods by comparing it to previous work and published examples.

Keywords: time-delay systems,critical delays,quadratic eigenproblems,Kronecker products,stability,stability chart

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00049

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 296-301

Authors: Mauro Boccadoro; Francesco Martinelli; Paolo Valigi

Abstract: In this paper we develop a dynamical model of a distributed production system taking into account the lead times of each site and the transportation among different sites, which introduce a delay on the effect of control actions on the inventory levels of the sites. We linearize about typical operative conditions, and show the intuitive result that it is not possible to decouple the buffer levels from the external (unknown) demand. Structural properties of this model are presented and different policies are defined in order to prevent undesired phenomena and to guarantee the optimal management of the overall system.

Keywords:

Digital Object Identifier (DOI): 10.3182/20060710-3-IT-4901.00050

Conference: 6th IFAC Workshop on Time Delay Systems (2006)

Location: Hotel Duca degli Abruzzi, L’Aquila, Italy

Start Date: Mon Jul 10 2006 - End Date: Wed Jul 12 2006

Page Numbers: 302-307