1

Abstract An overview of our latest research and development is presented on structural system identification of damage detection devoted to reinforced concrete beams and steel-concrete composite beam-to-column joints under monotonic and cyclic lateral loads. Moreover, a real-time compatible algorithm for an adaptive technique in view of the control of structural systems is introduced. In detail, the use of Fibre Bragg Grating sensors for health monitoring of smart reinforced concrete elements and of accelerometers for detecting the variation of curvature and of stiffness in members and beam-to-column joints, respectively, by means of displacement mode shapes, demonstrate the feasibility and versatility of localised identification techniques and sensors to determine damage in view of the functional safety of structures. Moreover, a real-time compatible algorithm based on a Rosenbrock method is suggested for adaptive control systems which are generally used to control plants whose parameters are unknown or uncertain. The adaptive minimal control synthesis algorithm which is a direct adaptive controller is formulated in discrete form in view of a stability analysis of non-linear controllers. Keywords: structural system identification, health monitoring, curvature and displacement mode shape, adaptive control, model reference, stability analysis. 1 Introduction The modelling of complex structures subjected to earthquake or dynamic loading sometimes experimentally validated as well as active control has been a subject of intense research in the last decades [1, 2]. Due to the governing assumptions of linearity in the analysis, to the statistical variability in the properties of structural components owing to fabrication tolerance, and to the increased heterogeneity of the structures being modelled, a systematic improvement in the reliability of finite

Identification and Control of Structural Systems

O.S. Bursi and L. Vulcan Department of Mechanical and Structural Engineering University of Trento, Italy W. Salvatore and L. Nardini Department of Structural Engineering University of Pisa, Italy

2

element (FE) analysis is difficult to achieve. Therefore, test-validated FE models still play a key role in the design of high reliable structures. Modal testing is perhaps the most versatile form of structural model validation testing [3]. Nonetheless, modal testing is not a straightforward technique as the verification of the dynamic performance of a structure remains strongly dependent on the choice of analytical models. Then, unlike mass property measurements, modal testing captures global response measures which characterize mass, damping and stiffness behaviour, simultaneously. As a result, the use of multiple simultaneous random excitations with multiple output readings, arguably the multiple input/output (MIMO) testing, is the most important modal testing method used today.

An important area of application of modal testing based on system identification is structural health monitoring, in view of detection, location and quantification of damage [4]. This practice has taken on increased importance in civil applications owing to the increased use of structures far beyond their original life expectancy, which underlies the concept of open-loop smart structures developed in Subsection 2.1. In detail, vibration-based health monitoring algorithms can be classified into two categories: i) model-based approaches; ii) non-model-based approaches. Model-based techniques use changes in response functions or modal parameters such as natural frequencies, mode shapes, or their derivatives, in order to identify damage locations and levels. Analysis of changes of parameters between sequential tests over time is used to determine damage characteristics [4]. Some of the algorithmic approaches include mode shape-base techniques [5] and flexibility methods [6]. Relevant applications will be shown in Subsections 3.1 and 3.2. The major drawback of these methods is that the structural variability due to the changing of boundary conditions, environmental conditions and slightly different excitation levels, may cause response characteristic changes larger than those induced by damage. Conversely, non-model-based schemes determine direct changes in the sensor output signal in order to locate damage in the structure [7]. These can be thought of as signal-processing solutions to the problem.

Model-based damage detection methods can also be classified with regard to global or local response characteristics. In a greater detail, the changes in a structure modal parameters could be detected either using the global modes directly, or using subassemblies of the structure. This localised approach avoids to consider the entire system provided that enough measurements to characterize the subassembly under consideration are available. An application of this concept will be illustrated in Subsection 3.2.

Some novel measuring technologies of interests are Fibre Optic Sensors (FOSs) and Micro-Electro-Mechanical Systems (MEMSs). Many applications of FOSs have been recently proposed in the field of structural engineering, in order to measure quantities such as strains, temperatures, moisture content. The possibility to embed FOSs into the structure during the construction process allows one to monitor the strains or other parameters at critical locations where high values are expected. These advantages, together with excellent durability properties, make the FOS an effective tool in the lifelong monitoring of structural elements and structures as a whole. With regard to FOSs, many applications of Fibre Bragg Grating (FBG) sensors are reported in the monitoring of bridges and viaducts [8].

3

A key evolution in sensors is toward miniaturisation employing MEMSs. The sensor itself consists of a thin silicon foil, even if commonly comes mounted in the form of a micro-chip. Today, a number of MEMS sensors is available off-the-shelf, including transducers accelerometers, pressure gauges, load cells, gyroscopes and chemical gauges [9]. Moreover, they can be wireless, i.e. they need power, but no cables for signals. MEMSs application in civil engineering is theoretically feasible, even though few applications can be found in the literature.

The subject of structural control offers opportunities to design new structures and to retrofit existing ones by the application of counter-forces, smart materials, frictional devices, etc. instead of just increasing the strength of the structure at greater cost [10]. A variety of applications has already been installed in building structures to control wind induced motions that are objectionable to the occupants; and many applications of passive control, such as base isolations have been installed to reduce structural acceleration produced by strong earthquake ground shaking. There is a consensus that structural control has the potential for improving the performance of structures, new or existing, if appropriate research and experimentation are undertaken.

The remainder of the paper is organised as follows. Section 2 introduces concepts relevant to smart structures and presents an example of open-loop smart element that can be employed in structural engineering. In Section 3, two applications of model-based structural health detection techniques are shown. In detail, the use of FOSs and classical accelerometers in conjunction with curvature mode shapes and localised damage techniques are used, respectively, for damage detection and quantification. Section 4 illustrates the theory of adaptive control and introduces a real-time compatible algorithm based on a Rosenbrock method for an adaptive minimal control synthesis algorithm formulated in discrete form in view of a stability analysis of a non-linear controller. Finally, conclusions are drawn in Section 5 with some remarks on future developments. 2 Development of Smart Structures Within the past decade, technological advances in the understanding of materials have led to the advent of many so-called smart materials at the nano/micro scale. With the availability of advanced computing capabilities and new developments in material sciences, researches can now characterise processes, design, model and manufacture materials with desirable performance and properties. One of the challenges is to model short-term micro-scale material behaviour through the meso-scale and macro-scale behaviour into long-term structural system performance. Supercomputers and/or workstations used in parallel are useful tools to solve these scaling problems by allowing the development of models that take into account the large number of variables and unknowns needed to project micro-behaviour into structure systems performance [11].

We introduce the concept of a complete structure that can be the basis, in the new millennium, to design structural components. Therefore, functions of sensing, diagnosing and assessing the health of structures and actuating structures should be

4

integrated to form the complete structure, and this philosophy forms the basis for developing smart structures. It is important to highlight that smartness can be inherent in a structure by virtue of the material microstructure or the design itself. In a greater detail, smart structures or smart structural components have an extraordinary ability in performing their design functions: to sense, process and diagnose in critical zones any variation of selected variables such as strain, temperature, pressure and so forth, and to take an appropriate action to preserve the structural integrity and to continue to perform the design functions. 2.1 Some Concepts We introduce some terms relevant to a recent design philosophy of engineering structures that has moved from a conservative safe-life approach to a more rational damage tolerant approach. Open-loop smart structure: unusual micro- or macro-structural design that enhances the structural integrity.

For instance, we can design a structural element according to a damage tolerant approach in the field of high-cycle fatigue: the ability to monitor cracks and their dynamics will allow additional years of useful service of the structural member. Embedded FOSs are invaluable in such monitoring and diagnostic tasks. This is in contrast to the basic design philosophy of the conservative safe-life of a structural member that is set on the basis of fatigue test data, the extent of whose scatter determined the setting of the safe limit below the lowest values. Close-loop smart structure: ability to sense a selected variable such as strain, temperature, pressure and so forth, to diagnose the nature and the extent of the problem, to initiate an appropriate action to address the identified problem, and to store the processes in memory and learn to use the actions taken as a basis next time around. The attribute of smartness thus includes the abilities to self-diagnose, recover, report and learn.

Examples are active control of bridges and tall structures subjected to wind and earthquakes; use of control systems based on neural networks in order to improve the prediction of computer programs relevant to crack distributions on the basis of measured cracks. Smart structural design: consider the functions a structure needs to perform, under the influence of the environment, in order to select and balance a material organization with the selection of the cross section, profile or size of a structure or of a structural component where are incorporated instruments that have the ability to diagnose, assess and initiate an action appropriate to meet a design intent. 2.2 An open-loop smart element

Along the lines stated above, we are developing an innovative low-cost distributed construction system based on smart prefabricated concrete structural elements, capable of real-time assessment of the condition state and of the safety level of bridge structures. This element, shown schematically in Figure 1, could be produced with an embedded intensive low-cost high-durability sensing system based on novel

5

Project Level Network Level

DAQ Unit

StructuralModel

Tel Com Signal Processing

Modal Analysis

Model Updating

Damage Detection

Element Level

DamageAnalysis

Internet

passive smart bar

smart element

Sensor Level

Figure 1. A prefabricated reinforced concrete smart element with an open-loop scheme.

FOS strategies, including Optical Time Domain Reflectometry (OTDR) and multifunctional MEMS sensors for measures of strain and acceleration. The sensors are conceived as an integral part of the product. Sensor media will be capable to detect point to point physical quantities such as: strain and vibration response; cracking location and extension. An open, non-proprietary Internet-based Network of Tools will be able to translate these quantities in terms of condition state and safety level, and will make these available to the users, i.e. producers, constructors, owner, manager, etc. As the sensor is activated, all of the actors involved in the production/maintenance process line of the structure will be capable at any time, from anywhere to connect to the Internet and to directly watch, in a comprehensive and user-friendly fashion, the actual behaviour of the structural element. During the construction phase, users can directly monitor the behaviour of each single element, as well as of the entire structure, and control all the mounting operations including assembling, post-tensioning and finishing grouting. During the operation of the structure, the owner will be provided with tools for the day-by-day measurement of the condition state, and the real-time control of the safety of the structure. In detail, the features of the developing technology are indicated hereafter.

Real-Time: this technology allows the detailed monitoring and control of the performance and safety of structural elements and of the whole bridge structure, starting from the production phase, the assembling phase and the live phase.

Lifelong durable: the technology is conceived in such a way to warrant its operation during the whole life of the bridge structure estimated in approximately 80 years for new Prestressed Reinforced Concrete (PRC) constructions; for this reason only non-electronic sensor technologies will be utilized in the embedded part of the system; for MEMS based components, whose duration is expected of the order of 20 years, special packaging and system provision will be designed, which will warrant an easy-to-manage, low-cost, semi-automatic failure detection and replacement of the components.

Cost effective: a large market is expected for multi-sensors MEMSs, and the long-term cost is expected of the order of 8 to 30 € per sensor in 6 years. OTDR

6

technologies are likely to reduce the cost of the sensors system to the mere packaging. In the medium-long term the construction cost of a bridge made of smart structural elements is estimated to be 3% to 5% higher with respect to the corresponding bridge made of classical PRC members. Today, the total management and maintenance cost of a Reinforced Concrete (RC) and PRC bridge during the whole life is estimated to be of the same order of the construction cost. The new technology will reduce the management and maintenance cost up to the 30%, i.e. roughly to the 30% of the construction cost, thus resulting in a significant saving for the owner.

Modular: PRC elements producers will be able to offer in their catalogue standard series of smart elements; in turn and in the same way, sensors suppliers will offer a variety of smart bars suitable for their direct integration into the prefabricated concrete production.

Flexible: the current potential market demand of smart prefabricated elements concerns low-to-medium span bridges from 15 to 50 m. Therefore, products cover the construction and maintenance of this type of structures. 3 Damage Detection Using Vibration-based Health Monitoring Two applications of model-based techniques which exploit changes in modal parameters such curvature and displacement mode shapes, respectively, will be addressed in this section to identify damage locations and damage levels. Analysis of changes seen in parameters between sequential tests over time is used to determine damage characteristics. In detail, damage detection techniques will be applied to a prototype RC beam instrumented with FOSs; and to a steel-concrete composite building instrumented with piezoelectric accelerometers subjected to increasing levels of peak ground acceleration (pga). 3.1 An FBG-based Dynamic Measurement System for Real-Time Monitoring of Reinforced Concrete Elements

FOSs find today widespread use in long-term monitoring of civil structures, although their employment is typically restricted to static measurements [12]. Major advantages of FOSs include: small dimensions, so they are suited to direct embedding in structural materials such as concrete and composites; the expected durability; and their insensitivity to electric and magnetic fields. FO sensing techniques are based on measuring concepts such as Fibre Bragg Grating (FBG). An FBG-based system generally includes a broadband source, i.e. a light emission device, a set of optical fibres with prewritten grating sensors and an Interrogation Unit with an optical spectrum analyser. A Bragg grating sensor is a segment of the optical fibre in which a periodic modulation of effective refractive index neff with grating pitch Λ has been formed by exposing the core to intense ultraviolet light. The regions having different refractive indices reflect the beam propagating in a

7

narrow band centred about the Bragg wavelength λbragg = 2 neff Λ. Any strain variations of the grating region result in changes in the grating pitch and refractive index, and can therefore be determined by observing the wavelength shift of the reflected beam. As the sensors exhibit a linear strain relationship to the wavelength shift within the elastic limit of the fibre, the axial strain ε of the grating can be evaluated as follows:

refGF λλε ∆

=1 (1)

where GF is the gauge factor obtained by specific calibration, λref is the reference wavelength and ∆λ is the wavelength shift. In order to develop a cost-effective technology for producing PRC elements embedding a sensing system for lifelong real-time monitoring in civil structures, like the one depicted in Figure 1, FOSs have been chosen as the most suitable sensing technology, in view of their durability and their expected future low cost [13]. The identified target sensing system is a multiplexed system with a sample frequency of about 500 Hz per channel, capable of measuring finite displacements. At the time the research started, most of the commercial Interrogation Units allowed either multiplexed but low-frequency measurements with a scan rate typically lower than 50 Hz or high-frequency measurements with a single channel.

A scheme of the system developed is depicted in Figure 2. Optical components are all provided by AOS GmbH, and include a sensing module, an optical switch and a calibrator. The core of the system is the sensing module. It generates the light beam and analyses the signal reflected by sensors, converting the peak wavelength data in a double electrical tension value, UA and UB, respectively, by using a pair of photodiodes. A National Instruments DAQ-card mounted on an external PC acquires the voltage signal as well as the signal coming from other external devices. As the photodiodes operate continuously, when the system is in single-channel mode, the maximum sampling rate basically depends on the DAQ-card performance, which is of the order of 100 kHz with the current set-up. In a multi-channel mode, acquisition occurs by multiplexing the optical signal through the optical switch. In this case, the voltage output of the sensing module takes the form of stepped signal of the type shown in Figure 3, which needs to be decoupled via software. In detail, the post-processing algorithm recognizes and discards the rise-time Tr and the fall-time Tf branches of the signal owing to the switching operation, and averages the response of each channel over the corresponding plateau. The switch control is software-based, and also operates through the DAQ card. In multi-channel mode, the sample frequency is limited by the maximum switching frequency, which is of the order of 2.5 kHz for the model under use. Hence, with a 4-channel set-up, the system is capable of acquiring data at a sampling rate of 625 Hz per channel.

8

PC UA

UB

DAQ

Hammer Shaker

SensingModule

Calibrator

Opt

. Sw

itch

FBG FOS

PC UA

UB

DAQ

Hammer Shaker

SensingModule

Calibrator

Opt

. Sw

itch

FBG FOS

Figure 2. Lay-out of the system.

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4[ms]

[V]

UaUb

ch. 1

ch. 2

ch. 3

ch. 4

Tr2 Tf2

switch cycle time

a b

Figure 3. a) Voltage signal in a multi-channel acquisition; b) interrogation unit.

A long-gauge sensor model has been specifically developed in order to use FBGs

in direct displacement measurements as illustrated in Figure 4a. The sensor basically consists of a 600 mm-long protected fibre including a grating, fixed at the edges inside two segments of a threaded bar, and provided on one side with an optical connection. The sensor looks like a flexible wire that can be easily handled and coupled to the monitored structure by means of simple metal supports illustrated in Figure 4b. When the sensor is pretensioned, the displacement measure ∆l is linearly related to the strain measured at the FBG through a constant, labelled effective length lm. This constant approximately corresponds to the physical distance between the inner edges of the two bars, but may also depend on the stiffness characteristics and the construction technology of each sensor component. In practice, it is convenient to calibrate lm through a simple static test by comparing the FOS measurement to that of a micrometric gauge. The precision of the FOS is directly related to the nominal resolution of the Bragg grating, which is of the order of 1 µε. The dynamic performance of the FOS has been tested by comparing the response of the fibre to a Dirac δ function with that of two piezoelectric accelerometers.

We are validating the effectiveness of the system for structural monitoring in the laboratory, in two experimental prototypes of instrumented structural elements. Each

9

a b

Figure 4. Prototype of a FO system: a) scheme; b) anchorage system.

Figure 5. Detail of the reinforcement with an anchorage plate of the prototype beam and

beam view during a vibration-based experiment. specimen consists of a 0.2x0.3x5.6 m RC beam specifically designed for permanent instrumentation with 8 long-gauge FOSs at the lower edge. The sensors are arranged inside a 30x50 mm cavity, and connected to the structure by means of steel supports. These consist of small drilled steel plates which have been welded to the stirrup reinforcements before concrete casting as illustrated in Figure 5.

The scope of the experiment is to recognize the changes in dynamic response of each RC beam to different damage levels and configurations through direct modal strain measurements [5,6]. In detail, the testing protocol includes three sequential static load tests for each specimen, in which the relevant increasing of damage is produced by means of hydraulic actuators. At each stage, the specimen is dynamically characterized using both shock tests and stepped-sine tests [3]. During the dynamic test, the beam is freely supported on springs and instrumented with 19 additional accelerometers arranged for vertical measurements and spaced by 0.3 m as depicted in Figure 6. So far, only the results of the first characterization phase are available, and they are reported in detail hereafter. Some of the FRFs obtained through stepped-sine test are depicted in Figure 7, using as excitation source the electromagnetic shaker illustrated in Figure 6. In detail, the FRFs based on the strain dynamic response of FOSs are illustrated in Figure 7a, while the corresponding FRFs provided by the acceleromenters are plotted in Figure 7b. Consistently, with

10

( ) ( ) ( )21 2X X Xω ω ω ω− ∆ = + (2)

which relates the receptance ∆X(ω) of the FOSs to the inertances ( )ω1X and ( )ω2X of the accelerometers, respectively, the characterization obtained through FOSs yields good results only in the low-frequency range, as higher frequency peaks are covered by noise. Thus, a highest excitation energy is required to obtain better results through FOSs.

Shaker

Accelerometers

FOS

Actuator

Steel beam

Figure 6. Set-up of the reinforced concrete beam model.

0.0E+00

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

1.2E-01

10 20 30 40 50 60 70 80 90 100

[Hz]

[kg-1

]

Acc 1Acc 3Acc 5Acc 7Acc 9Acc 11

0.0E+00

5.0E-07

1.0E-06

1.5E-06

2.0E-06

2.5E-06

3.0E-06

10 20 30 40 50 60 70 80 90 100

[Hz]

[m N

-1]

Fos A1

Fos B1

Fos C1Fos D1

a b

Figure 7. Frequency Response Functions obtained by means of: a) FOSs; b) accelerometers.

Classical modal parameters, i.e. frequency fk, damping ξk and k-th mode shape kφ

were extracted from acceleration signals using a classical SIMO curve-fitting method [3]. Figure 8 shows the results of the extraction restricted to the first 3 vertical flexural modes, compared with those obtained by means of a FE model. The two sets of shapes agree with each other. Nonetheless, we should keep in mind that the scope of the experiment was to perform a damage detection based on changes in strain mode shapes and that modal strains can be related to modal curvature. In detail, approximate curvatures [5] can be computed from displacements using the k-th discrete form:

1 12

2k k kk i i i

i hφ φ φφ − +− +

= (3)

11

where data come from three adjacent instruments, and h = 0.3 m defines the distance between accelerometers. When h is small, (3) is likely to yield inconsistent data owing to the propagation of measurement errors and noise owing to the double derivation. The errors can be reduced performing a smoothing over curvature data. In any case, this results in a loss in the spatial resolution. This is evident in Figure 9, where the appearance of the first theoretical curvature mode shape is compared with those experimentally obtained through (3) without smoothing and through a 5-point least-square parabolic fitting scheme. Alternatively, the curvature mode shape kφ was evaluated on the basis of the strain mode shapes extracted from FOSs strain-based FRFs. In this case, the curvature was directly calculated, for each section, as the ratio between the strain measured at the sensor and hy = 144 mm, viz. the distance between the fibre and the neutral axis of the beam. The first curvature mode shape resulting from FOSs data is also represented in Figure 9 and this curve agrees with the FE prediction. Moreover, the spatial resolution of the measurement is obviously related to the number and position of sensors, thus highly accurate results could be achieved without any numerical adjustment.

-0.08

-0.04

0.00

0.04

0.08

0 1 2 3 4 5[m]

[kg-1

/2]

0 1 2 3 4 5[m]

0 1 2 3 4 5[m]

F1 = 21.1 Hz, ξ1 = 2.5% F2 = 67.2 Hz, ξ2 = 1.7% F3 = 138.3 Hz, ξ3 = 1.5%

Figure 8. Beam mode shapes obtained through FOSs, accelerometers and FE predictions at relevant FE frequencies

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5[m]

[kg-1

/2m

-2]

FEM prediction FOS Curvature Shape

Acc. without Smoothing Acc. Least-squares parabola

Figure 9. First curvature mode shape obtained through a FE prediction, FOSs and

accelerometers measurements without and with smoothing.

12

3.2 Damage detection in a steel-concrete composite moment resisting frame structure

In the framework of civil engineering, dynamic monitoring has been successfully applied to the verification of bridges or single structural elements [14]. However, at-tempts to extend its application to buildings have revealed to be more difficult, as the responses of such structures are often not easy to model.

In general, it is possible to utilize model updating techniques and simulate damage through variations in parameters: reductions in the resisting section; reductions in elastic modulus [15]; and modifications to the degree of connection between structural elements [16]. To date, little research has been conducted in order to validate numerical simulations via experimental tests, and nearly no studies have ever utilised full-scale specimens. Among these, some noteworthy examples are the PRESSS project [17], in which the dynamic response obtained from a PRC structure was analysed before and after pseudo-dynamic (PsD) tests, utilizing techniques based on modal curvature variations [18].

Along this line the present part, based on the results of two research projects funded by the European Community [19, 20], describes some results on the dynamic identification of a steel-concrete composite frame structure with partial strength beam-to-column joints. The structure, built at the ELSA Laboratory of the Joint Research Centre at Ispra, was subjected to four PsD tests with increasing pga and a final cyclic test in order to verify the feasibility of its design, which was carried out in conformity with Eurocode 8 (EC8) rules [21]. Moreover, vibration tests were initially performed on the intact, undamaged structure; then again after a PsD test corresponding to the ultimate limit state of the structure; and finally after a final cyclic test following the collapse limit state.

Based on design assumptions, which call for concentrating inelastic phenomena entirely in correspondence to partial strength beam-to-column joints and column bases, two types of tests were planned: tests with a global configuration, in order to characterize the behaviour of the entire structure; and tests with local configurations, in order to analyse the damage of beam-to-column joints. In particular, starting from modal shape displacements identified via local measurements, it was possible to evaluate modal forces acting on elements collaborating in a joint and therefore, the stress and strain acting on the joint itself, given that beams and columns remain in the elastic range both during PsD and cyclic tests. The expressly developed and tested experimental procedure has thereby enabled determination of the variations in stiffness undergone by beam-to-column joints following PsD and cyclic tests, a technique which reveals to be effective and easy to use in view of damage estimates.

The moment resisting (MR) frame structure was made up of three identical MR frames arranged at a spacing of 3.0 meters, with two bays, 5.0 and 7.0 meters in length, respectively, and two storeys 3.5 meters in height as depicted in Figure 10. It is structurally symmetric in the main direction and it was equipped with X-shaped braces in the transverse direction. 3D and plan views of the whole structure are shown in Figures 11 and 12, respectively. Material properties and details of the prototype structure are available in [20]. The total weights of the bottom and top

13

HEB

260

HEB

260

HEB

260

HEB

260

IPE 300 IPE 300

IPE 300 IPE 300

HEB

280

HEB

280

4005000 7000

400

3500

3500

7000

3.5

m3.

5 m

Secondary beams

Main moment resisting frames

Figure 10. Side view of the test structure. Figure 11. 3D view of the frame structure.

B

A

C

321

IPE 240IPE 240

IPE 240

IPE 240IPE 240

IPE 300IPE 300

IPE 300IPE 300

IPE 300IPE 300

IPE 240

12800

4005000 7000

400

7400

CONCRETE SLABthickness : 15cm

700

3000

3000

700

Figure 12. Plan view of the test structure

Table 1. Vibration, PsD and cyclic tests.

PsD testPga [g] Performance objective

-- Identification at the undamaged state 0.10 Elastic behaviour 0.25 Serviceability Limit State (SLS) 1.40 Ultimate Limit State (ULS)

-- Identification at the ULS 1.80 Collapse Limit State (CLS)

Cyclic Maximum top displacement 300mm -- Identification at the CLS

storeys of the bare structure subjected to vibration-based tests are 453.7 kN and 415.4 kN, respectively. However, additional loads of 518.4 kN and 496.2 kN at the bottom and top storeys were applied during PsD and the final cyclic test, in order to reproduce initial stresses in the composite slab owing to permanent and variable loads. Partial strength beam-to-column joints were designed to provide a joint rotation of 35 mrad, associated to a residual strength of at least 80 % of the maximum value [21]. Similar performance was guaranteed by column base joints, composed of a thick extended plate welded to the columns and connected to the reinforced concrete foundation blocks by hooked rebars threaded at their upper ends. As shown in Table 1, four PsD and one final cyclic test were carried out only in the main direction of the structure. Some spectrum compatible accelerograms were generated as described in [20]. In order to perform PsD tests, one of these was chosen based on the greatest degree of damage to beam-to-column joints and a limited amount of damage in column base joints. The duration of the input accelerogram was set at 17.5 seconds with an additional free vibration period of 2.5 seconds.

The first PsD test, characterized by a pga of 0.10 g, was conducted to check the test procedure and to characterize the elastic structural behaviour. The second PsD test, run at a pga level of 0.25 g, brought the structure to its serviceability limit state. In the third and fourth PsD tests, a pga of 1.4 g and 1.8 g were applied in order to approach the ultimate limit and the collapse limit state, respectively. The final cyclic

14

test was performed by imposing increasing amplitude displacements to the top storey up to a maximum of 300 mm. The ratio of the reaction forces applied to the structure at the bottom and at the top storey was fixed equal to 0.97.

The response of the structure to environmental forced and impact actions was measured on the undamaged structure at two different damage levels, as reported in Table 1. The Stepped Sine Test (SST) involved the application of sinusoidal-type forces with constant frequency and amplitude imparted by an electromagnetic shaker ELECTRO-SEIS 400 with 33.79 kg of total vibrating mass. Both the application point and direction of the sinusoidal force are reported in Figure 13. An instrumented sledge-hammer PCB 086D50 characterized by a total mass of 10.77 kg was employed in the Shock Hammer Test (SHT); the direction of applied forces is indicated in Figure 13.

B10

B17

B3

B8B15

B9B11

B7

B13

B22

B18

B1

B2

B4

B16B20

B21B12

B19

B5

B14

B6

B23

Frame B

Figure 14. Location of accelerometers in configuration B for the interior joint in frame B.

Shaker

A1 A2

A3Hammer

X1

Y1

d1d1

d2

Accelerometer

+

Hammer

+

Shaker

d2

A6

X2

d1d1

A4 Y2 A5

Direction 3Direction 2Hammer

SECOND FLOOR

FIRST FLOOR

C10

C9

C8

C7

Frame B

C14C13

C6

C19

C11 C12

C5

C4

C3C20

C15

C2

C16

C17

C1

C18

Figure 13. Location of shaker, hammer impact points and accelerometers.

Figure 15. Location of accelerometers in configuration C for the exterior joint in frame B.

Signals produced by the piezoelectric accelerometers PCB 393C, endowed with a

sensitivity of 9.8*10-4 m/s2 and the PCB 393 B12 with a sensitivity of 8*10-5 m/s2, were processed and acquired with a PCB 584 amplifier and National Instruments PCI-6031E multiplexing acquisition device. A sampling frequency of 800 Hz was chosen for data recording. Three different accelerometer configurations were considered.

15

A global configuration A aimed at describing the overall dynamic behaviour of the structure conceived as a system with two translational DoFs and one rotational DoF per storey. Rigid floor decks and inextensible columns were implicitly assumed. A total of 6 accelerometers, three per floor, labelled A1 - A6, respectively, were utilized, as illustrated in Figure 13. Two local configurations B and C aimed at investigating the behaviour of both the interior and exterior bottom storey beam-to-column joints of the interior frame. The interior joint was equipped with a maximum of 23 accelerometers, as illustrated in Figure 14, while 22 accelerometers were exploited for the exterior joint as depicted in Figure 15. The FRFs and time histories of the dynamic tests have been presented and commented upon in [22]. As example, the FRF relevant to the accelerometer A3 at the top storey, see Figure 13, is illustrated in Figure 16.

Dynamic experimental analyses allow dynamic properties of the structure, such as natural frequencies, damping and modal shapes to be estimated. Three different stages were considered: the undamaged state; a moderately damaged one; and a heavily damaged condition. Two different techniques for the extraction of modal parameters were applied in sequence. The first one, viz. the Inverse Receptance Method, was applied to experimental FRFs, providing a first estimate of natural frequency values and damping ratios. Later, the aforementioned values were utilized as primers for the non-linear regression procedure, the Non-Linear Least Square Procedure, which allows modal shapes and more accurate frequency and damping values to be tracked [3].

In order to verify usual assumptions regarding damping ratios employed in frame structures, a non-proportional viscous model was adopted in the analyses, as it is more general than the Rayleigh’s model. The extracted complex values of modal shapes for configuration A were arranged approximately along a straight line inclined -45° in the Nyquist plot as depicted in Figure 17. This property assures that damping can be considered to be proportional [23].

Therefore, it was possible to extract the actual values of modal shapes. For instance, the first six modal shapes of the structure in Stage I, i.e. the undamaged condition, are represented in Figure 18; while the natural frequencies and damping value of the above-mentioned six modes for Stages I, II and III are reported in

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1 2 3 4 5 6 7 8 9 10 11 12 13[Hz]

[kg-1

]

test Itest IItest IiI

-5.00E-04

-2.50E-04

0.00E+00

2.50E-04

5.00E-04

-5.00E-04 -2.50E-04 0.00E+00 2.50E-04 5.00E-04

Re [kg-1/2]

Im [k

g-1/2

]

mode_1mode_2mode_3mode_4mode_5mode_6

Figure 16. FRF obtained from the SST at the maximum force of the shaker for acceler. A3.

Figure 17. Complex plan representation of the first normalized six mode shapes in phase I.

16

Table 2. Frequencies of the first six mode shapes.

Test I [Hz]

Test II [Hz]

Test III [Hz]

1 3.41 2.45 2.07 2 5.13 4.49 4.24 3 7.02 6.29 5.77 4 11.00 9.06 8.18 5 16.50 13.87 13.71

Mod

es

6 22.60 19.90 19.10

Table 3. Damping values of six mode shapes.

Test I [%]

Test II [%]

Test III [%]

1 0.47 0.74 1.23 2 0.64 1.05 0.73 3 0.60 0.59 0.76 4 0.55 0.57 0.42 5 0.57 0.34 0.55

Mod

es

6 0.70 0.44 0.46

a) First flexural mode in the main frame direction.

d) Second flexural mode in the main frame direction.

b) First flexural mode in the transverse direction.

e) Second flexural mode in the transverse direction.

c) First torsional mode.

f) Second torsional mode.

Figure 18. Modal shapes of the frame structure in Stage I.

Tables 2 and 3, respectively. A clear general reduction in natural frequencies is evident owing to damage of partial strength beam-to-column joints and column base joints as argued in [24].

The structural identification was extended to the two analysed local configurations via the same techniques employed for the global test [25]. For brevity, we only show a local analysis relevant to the instrumented exterior joint depicted in Figure 15. The first and second modal shapes of the aforementioned joint provided by dynamic tests in Stage I are represented in Figures 19 and 20, respectively. Local measurements furnished more effective information than simple modal data. In fact, on the basis of the observation that damage was localised in partial strength connections and shear panels of joints, as enforced by design [19], it was possible to evaluate internal forces acting on beams and columns, still elastic, starting with modal displacements obtained from measurements. Thus, by referring

17

to the structural model proposed in [26] and reported in Figure 21, representing a mechanical idealization involving 4 rigid bars connected together by pins with translational and rotational springs, it was also possible, by imposing internal equilibrium, to estimate actions acting on joints and, hence, their stiffnesses. Note that in the joint mechanical model, a translational spring simulates the stiffness of a column web panel under shear; while a rotational spring represents the flexural stiffness of a beam-to-column connection.

Figure 22 illustrates the model relevant to measured modal displacements and related internal actions. For each test and each modal shape, equilibrium was imposed on the whole substructure represented in Figure 22, by updating the stiffness of the beam and of the upper and lower columns in such a way as to minimize relevant errors.

Figure 19. First mode shape of the exterior joint at Stage I.

Figure 20. Second mode shape of the exterior joint at Stage I.

2

5

67

M1 T1

M2 T2

M6 T6

M5 T5

4-5

M4 T4

M7 T7

6-7

4

Figure 21. a) Exterior partial strength beam-to-

column joint; b) mechanical idealization Figure 22. Internal actions obtained by

equilibrium for the exterior joint.

18

On the basis of the aforementioned considerations, Tests I, II and III yielded the stiffness values reported in Table 4. The comparison between values reported in the above-mentioned table with those collected in Table 5 and obtained via monotonic and cyclic tests on substructures reproducing an exterior joint performed at the University of Pisa [26], clearly shows a good agreement between results. It must be emphasized that the slab effective width in the substructure tests was fixed at 140 cm by specimen design, executed as per EC8 guidelines [21]; while in the Ispra tests slab continuity allowed a greater width in the elastic range to be reached, as confirmed by the aforementioned update, i.e. almost 300 cm. The decrease in stiffness after the first test must be attributed to failure of concrete slab against the column after a few cycles, as confirmed by tests both on substructures and on the complete structure.

Table 4. Estimated stiffness of the exterior joint on the basis of local test data.

Test I Test II Test III

Connection [kN m / mrad]

49.83 12.43 5.72

Column web panel [kN / m] 2.43 106 8.65 105 9.05 105

Table 5. Measured stiffness of the exterior joint on the basis of Pisa tests on substructures.

Initial stiffness

8th cycle

16th cycle

Connection [kN m / mrad]

49.00 12.85 6.15

Column web panel [kN / m] 2.28 106 1.06 106 9.01 105

Finally, note that the proposed technique is able to track stiffness and strength

changes being equilibrium involved in the identification procedure. As a result, also strength degradation could be evaluated with suitable excitation levels in view of a residual structural integrity assessment. 4 Adaptive Control of Structural Systems It is clear that the introduction of a controller to generate actuator commands and a feedback to realize a closed-loop smart structure can drastically alter the dynamics of a structural system, affecting its natural frequencies and modes, its transient response and even its stability. However, we can take advantage of the work developed in the field of structural control which has matured over the last two decades. To begin, we underline the fact that structural control is not the same as control theory which has been developed in electrical engineering and applied mechanics. The essence of structural control is the satisfactory management of the performance of relatively massive structures by physical means which require the application of large forces but do not require a high degree of accuracy. Conversely, control theory has developed knowledge which, to some degree, provides information of value to structural control but does not solve the problems of structural control. In the following subsections we will sum up some aspects relevant to adaptive control, viz. a controller that can adapt to changes in the plant.

19

Adaptive control is generally used to control plant whose parameters are unknown or uncertain [27]. In detail, we will focus our attention on discrete-time systems and we will introduce a real-time compatible algorithm based on a Rosenbrock method in view of stability, sensitivity and robustness. 4.1 Adaptive Control Systems with a Reference Model We can define an adaptive controller as a controller with adjustable parameters and a mechanism for adjusting the parameters. This technique is introduced to overcome the limits of classical control systems, as the reduced knowledge of the dynamic characteristics, the presence of large uncertainties or unpredictable plant variations. This solution has to be used carefully, because the controller becomes non-linear for the parameter adjustment mechanism even if the system is linear and time invariant [2]. Hence, the design and analysis of adaptive controllers is more involved than that of fixed gain controllers [27]. The adaptive controllers are generally divided into direct and indirect methods. In direct methods the controller parameters are adjusted directly based on the error between measured and desired outputs. In indirect methods, the parameters of a model for the unknown plant are estimated on-line, and the controller parameters are calculated as the solution of an underlying controller design problem based on the estimated plant parameters.

The adaptive algorithm measures a system parameter, called performance index, through inputs, outputs and system states. Comparing this parameter with a designed index, the adaptive mechanism modifies the adjustable system parameters or produces a modified input, to keep the measured index near the designed value.

In order to describe the properties of adaptive algorithms we need to introduce an optimal control policy applied to the controlled plant, described by a state-vector differential equation:

z Az Bu= + (4) where z is the state vector, u the control input, and A and B are constant matrices. It is possible under certain assumptions to find the optimal control u which minimizes the quadratic performance index J:

( )1

0

t

T TJ z Q z u R u dt= +∫ (5)

where Q and R are positive definite matrices, named weight matrices. Several comments can be made on adaptive algorithms:

• the number of systems described by linear and time unchangeable differential equations is quite limited;

• it is supposed that the controller properties are characterized by the choice of the performance index and that the matrices Q and R are given; but these assumptions are far from being evident in many cases;

20

• the identification of plant parameters A and B is difficult in system dynamics, because these are seldom directly measurable;

• the design of an optimal control law needs the access to the whole state vector, which usually is not accessible or its measure is too expensive.

To realize the optimal control design, the choice of weight matrices in (5) becomes more difficult with the increasing of system complexity and dimension; moreover, there is no explicit reference to the designed dynamic characteristics, as rise time, overshoot or damping ratio. These difficulties are overcome introducing a reference model, which specifies the desired system characteristics, and the direct adaptive technique that applies this property is named Model Reference Adaptive Control System, called MRAC [28]. This control system minimizes J expressed as function of the difference of state vector and control signal between system and reference model; while the choice of weight matrices rules how the plant model follows the reference model. The performance index J is defined as:

( ) ( )( )1

0

tT T

m mJ z z Q z z u Ru dt= − − +∫ (6)

The absolute stability of MRAC controllers is based on the hyperstability theory

and Popov’s criterion [29], which is guaranteed for every system represented as two subsystems: the first linear and time invariable, called feedforward block; and the second non-linear and time variable, called feedback block. The adaptive laws, based on the Popov hyperstability theory [29], have been verified in numerical simulations and experimental tests with external disturbances, parameters variation or inaccurate system models. But MRAC methods require the identification of system parameters, the controller synthesis and a Lyapunov’s equation solution.

The equations that rule the MRAC method commence with the plant equation:

( ) ( ) ( ) ( )z t Az t Bu t d t= + + (7) where A and B are assumed to be constant and d the disturbance, which includes any unmodelled term, non-linearity, external disturbance and parameter variation.

The exact linear reference model, that defines the required state trajectory zm, is ruled by:

( ) ( ) ( )m m m mz t A z t B r t= + (8) where r is the reference signal of the same dimension as u; zm is also of the same dimension of z. We define the error vector ze as the difference between the reference model and plant states: e mz z z= − (9)

The aim of control is the minimization of the error vector ze, hence when time goes to infinity the limit of the error would have to be zero. Using (7)-(9) and the feedback control law:

21

( ) ( ) ( )ru t Kz t K r t= − + (10) we obtain the differential equation: ( ) ( ) ( ) ( ) ( ) ( ) ( )e m e m m rz t A z t A A BK z t B BK r t d t= + − + + − − (11)

It is possible to reduce the problem supposing fixed values of gain matrices, of plant and reference model parameters, so that only the first term in the rhs of (11) does not cancel out. The values which allow these simplifications are:

( )

( ) 1

r m

m

T T

K B B

K B A A

B B B B

+

+

−+

=

= −

=

(12)

Hence for a linear and time invariable system, i.e. d(t)=0, the error vector is

asymptotically stable if the reference model is an asymptotically stable system. This entails that the eigenvalues of Am are strictly inside the unit disk.

The choice of K and Kr assumes that the reference model satisfies the Erzberger‘s conditions [30]:

( )( )

( ) ,

0

0

n m

n m n m

I BB A A

I BB B

+

+

− − =

− = (13)

If the system is non-linear or time-variable, to compensate the parameter variation, the controller has to be time-variable, therefore, assuming the control law: ( ) ( )( ) ( ) ( ) ( )r ru t K K t x t K K r tδ δ= − − + + (14) with the value of gain matrices K and Kr expressed in (12), the error equation becomes: ( ) ( ) ( ) ( ) ( ) ( )( ) ( )m re t A e t B K t r t K t z t d tδ δ= − + − (15) The progress of the disturbance d is not known, hence the value of δK and δKr are not known explicitly, but can be obtained through the Popov’s hyperstability theory [29]. A solution for the problem provides the following values:

22

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

T Te e e e

T TR e e e e

K t C z t z t dt C z t z t

K t C z t r t dt C z t r t

δ α β

δ α β

= +

= +

∫∫

(16)

where Ce is the error output matrix, which represents the weight matrix for the state-vector components, while α and β are adaptive parameters.

The optimisation of the control law is completed by the choice of these parameters. The error output matrix Ce must satisfy the hyperstability condition; the adaptive parameters α and β are arbitrary and chosen proceeding by trial and error; the ratio α / β changes the system damping and increasing β one reduces the algorithm adaptive time. The reference model has to be selected to be stable and its performance can be expressed through intuitive indices as settling time or overshoot. 4.2 The Adaptive Minimal Control Synthesis Algorithm

The adaptive Minimal Control Synthesis (MCS) algorithm is a direct adaptive controller, developed from the MRAC methods [31, 32], which assumes the value of the fixed gain matrices K and Kr equal to zero and minimises the parameter synthesis. However, the controlled system has to be known at least as number of degrees of freedom and state vector dimension. Hence, the control law u(t) is assumed as: ( ) ( ) ( ) ( ) ( )Ru t K t z t K t r tδ δ= + (17) where the adaptive gain parameters are the same of the MRAC method introduced in (16). The choice of the reference model parameter characteristics rules the global performances of a controlled system, because the controller imposes an input signal which guarantees that the system output follows the reference model output.

For instance, the reference model used for dynamic problems can be assumed to be a second-order system with two poles and no zero, unitary static gain and is usually designed to be critically damped, so to reduce overshoot and guarantee the maximum convergence velocity of the system. The rise time, which is the time spent by the system with a step signal to reach the 90% of the steady-state response from the 10% of its value, establishes the maximum velocity and the bandwidth of an ideal system. The ideal system would have to have a reduced rise time, but this is bounded by the sampling frequency, usually fixed by the acquisition board at about 1 kHz. So, to avoid the aliasing phenomenon and introducing a safety factor of 5, usually the rise time has a minimum value of 10 ms. The MCS algorithm, in its classic form, is represented in Figure 23.

23

r

+C

z=Az+Bu+

+

e

zm=Amzm+Bmr

u=Kz+KRr

zm

z

ze ye

Figure 23. The classical Minimal Control Synthesis algorithm.

A further development of the MCS algorithm is the error-based with integral

action algorithm, called Er-MCSI algorithm, solely driven by the error signal that is generated within the closed-loop system, and contains an explicit integral gain term [33]. The purpose of this new structure is to remove the problem of variable adaptive effort with changes in the operating set point and gain wind-up effects owing to plant disturbances and signal offsets.

The modified control law u(t) is as follows:

( ) ( ) ( ) ( ) ( )E E I Iu t K t z t K t z t= + (18) where zE(t) is the state error, zI(t) is the scalar integral of output error, KE(t) is the state-error feedback gain matrix and KI(t) is the scalar integral gain. These parameters are ruled by the following relations:

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

E m

I m

T TE e E E e E E

I e e I e e I

z t z t z t

z t y t y t dt

K t C z t z t dt C z t z t

K t C z t z t dt C z t z t

α β

α β

= −

= −

= +

= +

∫∫∫

(19)

The controller Er-MCSI scheme is represented in Figure 24.

24

r

z=Az+Buu=KEzE+KIzI

+

+

zm=Amzm+Bmr

zE+

z

zm

C/s

ye Ce

zI

Figure 24. The Error-based Minimal Control Synthesis algorithm with Integral action.

4.3 Discrete numerical stability analysis of the MCS adaptive control

To analyse the properties of the MCS algorithm in its discrete form we suppose that the system under observation is completely known, so that the exact solution of the system is available for any input force. For instance, we introduce two linear systems, one of first- and the other of second-order. We must say that any superior order system can be transformed into an equivalent first-order system through a phase canonical transformation.

The systems are represented in Figure 25 and ruled by the following relations:

( ) ( ) ( )

( ) ( ) ( ) ( )2 2

1System a):

System b): 2

mm m

m m

x t x t u tT T

x t x t x t u t

λ

ξω ω λω

+ =

+ + = (20)

with the constant parameters Tm, the time constant, λm and λ, low frequency gains, ξ, the damping ratio and ω the natural frequency. Substituting the state expression for displacement, velocity and acceleration with a state vector z, both the systems can be expressed through a vector-matrix differential equation of first-order: ( ) ( ) ( )c cz t A z t B u t= + (21) where the independent variable z is called the system state and the equation itself is called the state equation. The relevant solution with the initial condition z(0)=z0 can be computed by means of the Laplace transform or by using the exponential matrix approach. The solution through the latter method reads:

( ) ( ) ( )00

cc

tA tA t

cz t e z e B u dτ τ τ−= + ∫ (22)

25

a)

u

x

k

m ~ 0

cTm=c/kλm=1

b)

m

k

u

c

x

ω=⌦(k/m)ξ=c/(2⌦(km))

λ=1/k

Figure 25. a) Translational spring/damper system; b) translational mass/spring/damper system.

Now we analyse how a continuous-time state-space model can be converted into a discrete-time representation for computer-controlled systems. We will consider a sample and hold device, called zero-order hold (ZOH), which takes a continuous signal and turns it into a stepwise signal sampled and held for a certain interval of time ∆t, called sampling interval. With this assumption, the solution of (21) calculated at the end of the time interval from t0 = k ∆t to t = (k+1) ∆t reads:

( ) ( ) ( )'

0

1 'c c

tA t A

cz k e z k e d B u kτ τ∆

∆+ = + ∫ (23)

Now with this simplified notation to hand, we have a discrete-time representation of the original continuous-time state space model:

( ) ( ) ( )

'

0

1 ' '

with ' and ' 'c c

tA t A

c

z k A z k B u k

A e B e d Bτ τ∆

∆

+ = +

= = ∫ (24)

Matrices A’ and B’ can be evaluated in closed form for linear and time-invariable systems as functions of system parameters and sampling time ∆t. The discrete forms of the gain matrices K and KR for the MCS algorithm are calculated by the z-transform and read:

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1

1 1 1

T Te e e e

T TR R e e e e

K k K k C z k z k t C z k z k

K k K k C z k r k t C z k r k

β β α

β β α

= − + − − ∆ − −

= − + − − ∆ − − (25)

26

For the stability analysis of the algorithm, we need to combine (24) and (25), augmenting the state vector with appropriate functions, and analysing the stability of the resulting amplification matrix [34].

An alternative approach for stability analysis of the MCS algorithm relies on the study of its continuous expressions and the development of the Jacobian matrix about the equilibrium solution for the system under a step signal. A third approach that can be used for the aforementioned scope is based on the notion of passivity which is a formulation able to link the energy dissipation in physics with the operational input-output relation in systems theory [27].

It is important to underline that if the system is too complicated and the exact solution or the discrete solution with the ZOH device are unavailable, we need to introduce a numerical integrator to solve the equations stated in (21). In this respect, several choices are made which range from a fourth-order Runge-Kutta scheme [35] to the second-order trapezoidal rule [36]. Both of them are not real-time compatible and do not entail dissipation of high-frequency components of the response. A good candidate to tackle (21) appears to be the single step two-stage L-stable Rosenbrock method [37], which is endowed with second-order accuracy and is a true single-step method. This algorithm is ideal for stiff problems, because it is L-stable and nearly eliminates the response of a high-frequency mode in a single time step. Moreover, this algorithm is real-time compatible.

The application of this numerical integrator to a generic non-autonomous system ruled by: ( ) ( ),z t f t z= (26) with initial condition ( )0 0z t z= , letting J0 be the Jacobian of f evaluated at 0z z= from time t0 = k ∆t to time t = (k+1) ∆t is given by:

( ) ( )

( ) ( )( )

21 0 0 0 1 0 0

2 0 0 1 0 2 1

0 0 2

, ,

/ 2, / 2

k t f t z tJ k t f t z

k t f t t z k tJ k k

z t t z k

γ γ

γ

= ∆ ⋅ + ∆ + ∆

= ∆ ⋅ + ∆ + + ∆ −

+ ∆ = +

(27)

where 11

2γ = − .

One can observe how the algorithm is relatively easy to programme. The analysis of its performance combined with the MCS algorithm is on going. 5 Concluding Remarks and Future Perspectives The paper has surveyed some recent research and development results relevant to structural health monitoring of structural systems and some aspects of adaptive control in view of closed-loop smart structures.

27

Today, vibration-based damage detection techniques are mostly based on strain measurements and make use of few low-frequency modes. This suggests addressing the technological development toward a system capable of direct strain measurements in dynamics. The reported experiment demonstrates how a dynamic fibre-optic-based technology is feasible and technically effective when compared with the classical acceleration-based approach. In addition, optical fibre sensors exhibit features, such as durability and stability, which render them definitively more attractive than electrical gauges for long-term monitoring of smart structures.

An original procedure for evaluating damage on partial strength beam-to-column joints was illustrated and discussed in this paper. The proposed method, based on a purposely designed dynamic experimental series, furnishes simple and rapid assessment procedures for estimating the stiffness of joint components in which inelastic phenomena are localized. Application of the procedure to the results of dynamic measurements performed on a steel-concrete composite frame structure both in the undamaged and damaged states has confirmed its validity and efficiency. In this study, the parameter identification of the stiffness was performed by means of equilibrium as the parameters exhibited a clear physical meaning. This implies that also the variation of strength of components could be assessed with suitable excitation levels. Nevertheless, the adoption of a model updating technique which takes into account the relationships between the structure and the substructures might be a very interesting solution in view of an automatic procedure. Arguably, the evaluation of the residual structural integrity is still an objective to pursue further. One of the difficulties is due in relating stiffness deterioration to strength degradation.

We have also dealt with model reference adaptive control technique suggested in its basic and error control version for possible application in structural control. The stability analysis that we are pursuing in conjunction with a real-time compatible single step two-stage L-stable Rosenbrock method is promising, though many problems remains owing to the non-linearities caused by the controller.

The implementation in a digital environment of the delay time and successively of non-linearities deserves further studies. Finally, the computational tools of modern control cannot be expected to be so robust on a high number of degrees of freedom. Acknowledgments These research projects are sponsored by grants from the Autonomous Province of Trento, the Italian Ministry of Education, University and Research (M.I.U.R.) and from the European Union through the ECOLEADER HPR-CT-1999-00059 as well as the ECSC 7210-PR-250 project for which the authors are grateful. Opinions expressed in this paper are those of the writers and do not necessarily reflect those of the sponsors.

28

References [1] Juang, J.-N., “Applied system identification”, Prentice-Hall, Englewood

Cliffs, NJ, 1994. [2] Housner, G.W., Bergman, L.A., Caughey, T.K., Chassiakos, A.G., Claus,

R.O., Masri, S.F., Skelton, R.E., Soong, T.T., Spencer, B.F., Yao, J.T.P., “Structural control: past, present and future”, Journal of Engineering Mechanics, ASCE, 123, 9, 1997.

[3] Maia, N. and Silva, J., “Theoretical and experimental modal analysis”, Research Studies Press Ltd., Taunton, Somerset, England, 1997.

[4] Doebling, S.W., Farrar, C. R., Prime, M. B., "A summary review of vibration-based damage identification methods" The Shock and Vibration Digest, 30(2), 91-105, 1998.

[5] Pandey, A. K., Biswas, M., Sammam, M., "Damage detection from changes in curvature mode shapes", J. of Sound & Vibration, 145(2), 321-332, 1991.

[6] Yao, G. C., Chang, K. C., Lee, G. C., "Damage diagnosis of steel frames using vibrational signature analysis", J. Engrg. Mech., ASCE, 118(9), 1949-1961, 1992.

[7] Ray, L.R., Ramasubramanian, A., Townsend, J., "Adaptive friction compensation using extended Kalman-Bucy filter friction estimation", Control Engineering Practice, 9, 169-179, 2001.

[8] Liu, J.-G., Schmidt-Hattenberger, C., Borm, G., “Dynamic strain measurement with a fibre Bragg grating sensor system”, Measurement, 32, 151-161, 2002.

[9] Van der Auweraer H., Peeters, B., “Sensors and systems for structural health monitoring.” Journal of Structural Control, 10, 117–125, 2003.

[10] Soong, T.T., Spencer Jr, B. F., “Supplemental energy dissipation: state-of-the-art and state-of-the-practice.” Engineering Structures, 24, 243–259, 2002.

[11] Flatau, A. B., Chong, K. P., “Dynamic smart material and structural systems.” Engineering Structures, 24, 261-270, 2002.

[12] Udd, E., “Fiber optic sensor”, Wiley-Interscience, New York, 1991. [13] Zonta, D., Bursi, O.S., Pozzi, M., “Development of an FBG-Based Dynamic

Measurement System for Real-Time Monitoring of RC Elements”, Proceedings of the Second European Workshop on Structural Health Monitoring, Munich, July 7-9, 2004.

[14] Maeck, J., Wahab, M. A., Peeters, B., De Roeck, G., De Visscher, J., De Wilde, W. P., Ndambi, J. M., Vantomme, J., “Damage identification in reinforced concrete structures by dynamic stiffness determination.” Engineering Structures, 22, 1339–1349, 2000.

[15] Brownjohn, J. M. W., Xia, P. Q., Hao, H., Xia, Y. “Civil structure condition assessment by FE model updating: methodology and case studies.” Finite Elements in Analysis and Design, 37, 761-775, 2001.

[16] Lam, H. F., Ko, J. M., Wong, C. W., “Localization of damaged structural connections based on experimental modal and sensitivity analysis.” Journal of Sound and Vibration, 210(1), 91-115, 1998.

29

[17] Priestley, M. J. N., Sritharan, S., Conley, J. R., Pampanini, S., “Preliminary result from the PRESSS Five-Storey Precast Concrete test building.” PCI Journal, 2(159), 42–67, 1999.

[18] Zonta, D., Elgamal, A., Fraser, M., Priestley, M. J. N., “A vibrational based methodology for after-earthquake damage assessment of multistorey frame-resistant buildings. Proc. of 12th European Conference on Earthquake Engineering, London, 2002.

[19] Caramelli, S., Salvatore, W., et al., “Applicability of Composite Structure to Sway Frames.” ECSC Project n. 7210-PR-250, EUR Report, European Community., 2004.

[20] Bursi, O. S., Caramelli, S., Fabbrocino, G., Molina, J., Salvatore, W., Taucer, F., Zandonini, R., “3D Full Scale Seismic Testing of a Steel-Concrete Composite Building at ELSA”. EUR Report, Joint Research Centre, European Community, 2004.

[21] CEN, European Committee for Standardization. 2003. prEN 1998-1. Eurocode 8: Design of structures for earthquake resistance. Part 1: General rules, seismic actions and rules for buildings. European Community, Brussels, Belgium, 2003.

[22] Bursi, O. S., Molina, J., Salvatore, W., Taucer, F., “Dynamic characterization of a 3-D full scale steel-concrete composite building at ELSA”. EUR Report, Joint Research Centre, European Community, 2004.

[23] Balmes, E., “New results on the identification of normal modes from experimental complex modes.” Mechanical Systems and Signal Processing, 11(2), 229-243, 1997.

[24] Salvatore, W., Bursi, O. S., Molinari, M., Nardini, L., Zonta, D., “Vibration -based identification and assessment of a 3D composite frame structure at different damage levels”. 13th World Conference on Earthquake Engineering, Vancouver, B.C. Canada, August 1-6, 2004. Paper n. 2812.

[25] Chellini, G., Salvatore, W., Bursi, O. S., “Identifying the mechanical parameters of a steel-concrete composite moment resisting frame structure in undamaged and damaged states.” Proc. of the 18th Autraliasian Conference on the Mechanics of Structures and Materials. Perth, Australia, December 1-3, 2004.

[26] Salvatore, W., Braconi, A., Bursi, O. S., Tremblay, R., “Modelling of a partial-strength beam-to-column joint for high ductile steel-concrete composite MR frames.”, Proc. of the 13th World Conference on Earthquake Engineering. Vancouver, B.C., Canada, August 1-6, 2004. Paper n. 2929.

[27] Åström, K.J., Wittenmark, B., “Adaptive Control”, Addison-Wesley Publishing Company, New York, 1995.

[28] Landau, Y.D., “Adaptive Control: The Model Reference Approach”, Marcel Dekker, New York, 1979.

[29] Popov, V.M., “Hyperstability of Control System”, Springler-Verlag, New York, 1973.

[30] Erzberger, P., “Analysis and design of model following control systems by state space techniques”, Proceeding of the Joint Automatic Control Conference, Ann Arbor, 572-581, 1968.

30

[31] Stoten, D.P., Benchoubane, H., “Empirical studies of an MRAC algorithm with minimal controller synthesis”, International Journal of Control, 51(4), 823-849, 1990.

[32] Stoten, D.P., Benchoubane, H., “Robustness of a minimal controller synthesis algorithm”, International Journal of Control, 51(4), 851-861, 1990.

[33] Stoten, D.P., Neild, S.A., “The error-based minimal control synthesis algorithm with integral action”, Proceedings of the IMECH E Part I Journal of Systems and Control Engineering, 217, 187-201, 2003.

[34] Åström, K.J., Wittenmark, B., “Computer-Controlled Systems”, Prentice Hall, Upper Saddle River, NJ, 1997.

[35] Huang, K., Betti, R., “Predictive optimal control for seismic analysis of non-linear and hysteretic structures”. Earthquake Engineering and Structural Dynamics, 28, 585-607, 1999.

[36] Wong, K. K. F., Yang, R., “Predictive instantaneous optimal control of elastic structures during earthquakes”. Earthquake Engineering and Structural Dynamics, 32, 2161-2177, 2003.

[37] Piché, L., “An L-stable Rosenbrock method for step-by-step time integration in structural dynamics”, Computer Methods in Applied Mechanics and Engineering, 126, 343-354, 1995.