Adaptive Strategy to Damage-Tolerant Active Control

9th IFAC Symposium on Fault Detection, Supervision and 9th IFAC on Fault Detection, Supervision and Safety of Symposium Technical Processes Available online and at www.sciencedirect.com 9th IFAC on Fault Detection, Supervision Safety of Symposium Technical Processes September 2-4, 2015. Arts et Métiers ParisTech, Paris, France Safety of Technical Processes September 2-4, 2015. Arts et Métiers ParisTech, Paris, France September 2-4, 2015. Arts et Métiers ParisTech, Paris, France

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Adaptive Strategy to Damage-Tolerant Active Control Adaptive Adaptive Strategy Strategy to to Damage-Tolerant Damage-Tolerant Active Active Control Control Nazih Mechbal* and Eurípedes G. O. Nóbrega** Nazih Mechbal* and Eurípedes G. O. Nóbrega**  Nazih Mechbal* and Eurípedes G. O. Nóbrega** 

 Paris, France (e-mail: [email protected]) *PIMM (UMR – CNRS), Arts et Métiers ParisTech *PIMM (UMR – CNRS), Arts et Métiers ParisTech Paris, France (e-mail: [email protected]) *PIMM (UMR – CNRS),of Arts et Métiers ParisTech Paris, France (e-mail: [email protected]) **Department Computational Mechanics, Universidade Estadual de Campinas, **Department of Computational Mechanics, Universidade Estadual de Campinas, SP, Brasil (e-mail: [email protected]) **Department of Computational Mechanics, Universidade Estadual de Campinas, SP, Brasil (e-mail: [email protected]) SP, Brasil (e-mail: [email protected]) Abstract: Smart structure development is a recent trend that has been intensely researched, including Abstract: Smart new structure development is a recent trend that has active been intensely including multidisciplinary technologies and methods. Damage-tolerant control is researched, a new research area, Abstract: Smart structure development is a recent trend that has been intensely researched, including multidisciplinary new technologies and methods. Damage-tolerant active control is a new research area, applying fault-tolerant methods mechanical flexible structures, which demands spatial constraints multidisciplinary new control technologies andtomethods. Damage-tolerant active control is a new research area, applying fault-tolerant control methodscontrol to mechanical flexible structures, which demands not commonly found in the general approach. A proposed strategy, applied tospatial activeconstraints vibration applying fault-tolerant control methods to mechanical flexible structures, which demands spatial constraints not commonly found in the general control approach. A proposed strategy, applied to active vibration controller design and considering the possibility to face damage consequences, are here introduced and not commonly found in the general control approach. A proposed strategy, applied to active vibration controller design and considering the possibility to face damage consequences, are here introduced and assessed through finite elements models of healthy and damaged structures, used to simulate achieved controller design and considering the possibility to face damage consequences, are here introduced and assessed through finite elements models of healthy and damaged structures, used to simulate achieved performance. A Lamb technique is adopted to detect and localize damage,used associated to spatial norm assessed through finitewave elements models of healthy and damaged structures, to simulate achieved performance. A Lamb wave techniquetoisattain adopted to detect and localize damage, associated to spatial norm controller design and reconfiguration, an acceptable performance to the damaged structure. Results performance. A Lamb wave technique is adopted to detect and localize damage, associated to spatial norm controller design and reconfiguration, to attain an acceptable performance to the damaged structure. Results illustrate pertinent concepts and permit to expect successful applicationtoofthe thedamaged proposedstructure. approach. controllerthe design and reconfiguration, to attain an acceptable performance Results illustrate the pertinent concepts and permit to expect successful application of the proposed approach. illustrate the pertinent concepts and permit to expectControl) successful application of the proposed approach. © 2015, IFAC (International Federation of Automatic Hosting by Elsevier Ltd. All rights reserved. Keywords: Damage-tolerant active control, fault-tolerant control, structural health monitoring, active Keywords: active control, structural health monitoring, active control, faultDamage-tolerant detection and diagnosis, vibrationfault-tolerant control, finitecontrol, elements. Keywords: Damage-tolerant active control, fault-tolerant control, structural health monitoring, active control, fault detection and diagnosis, vibration control, finite elements. control, fault detection and diagnosis, vibration control, finite elements.   

1 INTRODUCTION 1 INTRODUCTION 1 INTRODUCTION Recent years has witnessed a growing interest in the field of Recent years hasmaterials witnessedand a growing interest the field of multifunctional structures. Thisin Recent years has witnessed a growing interest indevelopment the field of multifunctional materials and structures. This development trend is driven materials by the need for structures perform multifunctional and structures. This that development trend is driven structural by the need for structures functions. that perform simultaneously and non-structural An trend is driven by the need for structures that perform simultaneously structural and non-structural functions. An example would be a load-bearing structure that has the simultaneously structural and non-structural functions. An example would be a load-bearing structure that has the capability of providing own noise orstructure vibration that control, example would be a its load-bearing has selfthe capability providing its own noise or vibration control, repair, andof harvesting/storage. A response to selfthis capability of energy providing its own noise or vibration control, selfrepair, energy harvesting/storage. A response this demand and is today a laminate composite smart structure,to repair, and energy harvesting/storage. A response towhich this demand is today a laminate composite smart structure, which depends is ontoday sensors and actuators to properly its which goals. demand a laminate composite smartachieve structure, depends on sensors can and actuators properly achieve its goals. Different materials used onto structures to depends on sensors andbe actuators tomultifunctional properly achieve its goals. Different materials can be used on multifunctional structures to act and sense. However, among the current most common Different materials can be used on multifunctional structures to act and sense. However, among the current most common transducers, those based on piezoelectric effect widely act and sense. However, among the current mostarecommon transducers, those based on piezoelectric effect are widely used, due to their adaptable properties. A smart layer, transducers, those based on piezoelectric effect are widely used, due oftoa network their adaptable properties.may A smart layer, composed of piezo elements, expected to used, due to their adaptable properties. A besmart layer, composed of a network of piezo elements, may be expected to be found on every smartelements, laminatemay structure, because composed of aalmost network of piezo be expected to be found on almost every smart laminate structure, because piezoelectric can besmart used laminate as sensorsstructure, and/or actuators, be found on patches almost every because piezoelectric patches can be used as sensors and/or actuators, configuring several different possibilities of interrogation piezoelectric patches can be used as sensors and/or actuators, configuring several schemes. different This possibilities of isinterrogation sensing and control adaptability part of the configuring several different possibilities of interrogation sensing and control schemes. Thishere adaptability is part of the novel damage control method proposed, combining sensing and control schemes. This adaptability is part of the novel damage control method here proposed, combining structural health monitoring (SHM)here and adaptive tolerant active novel damage control method proposed, combining structural health monitoring (SHM) and adaptive tolerant active control (ATAC). structural health monitoring (SHM) and adaptive tolerant active control Recently(ATAC). Damage-Tolerant Active Control (DTAC) concepts control (ATAC). Recently Damage-Tolerant Control (DTAC) concepts were introduced (Mechbal &Active Nobrega, 2012), as an adaptation Recently Damage-Tolerant Active Control (DTAC) concepts were introduced (Mechbal & Nobrega, 2012), as an adaptation of Fault-Tolerant Control& Nobrega, (FTC) and SHM to were introduced (Mechbal 2012), as anmethods adaptation of Fault-Tolerant Controlaiming (FTC) and SHM methods to configure active controllers performance goals of Fault-Tolerant Control (FTC)minimal and SHM methods to configure active controllers aiming minimal performance goals established during the design phase of the structure. configure active controllers aiming minimal performance goals established during the design phase of the structure. established thesystems design phase of the astructure. Specifically,during DTAC encompass set of methods to Specifically, systems set of methods to design activeDTAC controllers of encompass mechanical aavibrations, to face Specifically, DTAC systems encompass set of methods to design active controllers of mechanical vibrations, to face eventualactive damage on smart transducerto layer design controllers of structures. mechanical Avibrations, face eventual damage on smart structures. transducer layer produces vibration measurement data, usedA feed specialized eventual damage on smart structures. Atotransducer layer produces vibration measurement data, used to feed specialized modules implementing structural data, integrity but also produces vibration measurement usedtechniques, to feed specialized modules implementing structural integrity techniques, but also modules implementing structural integrity techniques, but also

Fig. 1: DTAC scheme in a reconfigurable design Fig. 1: DTAC scheme in a reconfigurable design Fig. 1: DTAC scheme in a reconfigurable design to feedback a control loop, in order to guarantee DTAC

to feedback a control in order to guarantee DTAC controller goals. Fig. 1 loop, presents a generic DTAC scheme, to feedback a control loop, in order to guarantee DTAC controller goals. Fig. 1 presents a generic DTAC scheme, where the active is adapted on line, DTAC using data from controller goals. controller Fig. 1 presents a generic scheme, where themodule active controller is adapted on line, using data from an SHM a reconfiguration mechanism. where the active and controller is adapted on line, using data from an SHM module and a reconfiguration mechanism. an SHMstrategies module and a reconfiguration mechanism. Control depend if the structure is a new and healthy, Control strategies depend if the structure is a new and healthy, or if it is a working one where damage has detected. In Control strategies depend if the structure is a been new and healthy, or if it is a working one where damage has been detected. general, monitoring system has is need, part ofIn or if it isaacondition working one where damage been as detected. Ina general, a condition monitoring system is of need, part of a SHM module. Considering the complexity the as application, general, a condition monitoring system is need, as part of a SHM module. Considering the complexity of thedata. application, an expert operator must analyze the generated But the SHM module. Considering the complexity of the application, an expert operator must analyze the generated data. But the main architecture an automatic schemedata. achieving its an expert operatoris must analyze control the generated But the main architecture is an automatic control scheme achieving goals architecture without human interference. Thereby, damage its is main is an automatic control schemeifachieving its goals without human interference. Thereby, if damage is identified two control schemes could be applied: evolving goals without human interference. Thereby, if damage is identified two control schemes could be applied: evolving damage tolerant active identifiedactive two control (EDAC) schemes and couldadaptive be applied: evolving damage active control (EDAC) andobjective adaptive istolerant active control (ATAC). In the first case, the to prevent the damage active control (EDAC) and adaptive tolerant active control (ATAC). In the first case, the objective is to prevent the damage(ATAC). evolution; case, the SHM module must control Infor thethe firstsecond case, the objective is to prevent the damage evolution; for the second case, the SHM module must provide evolution; data to for be the interpreted bythethe reconfiguration damage second case, SHM module must provide data to be online interpreted by parameters the reconfiguration mechanism to change controller to attend provide data to be interpreted by the reconfiguration mechanism to change online controller parameters to attend their designed means that all these functions, ergo, mechanism to goals. changeThis online controller parameters to attend their designed goals. This means that all these functions, ergo, detect, localizegoals. and inform, fromthat the all SHM interpret their designed This means thesemodule, functions, ergo, detect, localize and inform, from the SHM module, interpret data andlocalize recalculate of interpret damage, detect, and parameters, inform, fromand theactive SHMcontrol module, data and recalculate parameters, and active design, control of must be intertwined since their respective in damage, order to data and recalculate parameters, and active control of damage, must be intertwined since their respective design, in order to achieve the expected results. Considering that vibration control must be intertwined since their respective design, in order to achieve the expected results. Considering that vibration control achieve the expected results. Considering that vibration control

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involves also spatial constraints, the methodology is necessarily apart from the FTC case, where in general only time dependence is considered. The aim of this paper is to introduce an ATAC scheme to a composite smart structure. The adopted approach is to accommodate a detected and localized damage maintaining an adequate level of performance. We consider a smart structure with some sensors and actuators, and including an SHM module to detect and localize damage. A spatial 𝐻𝐻∞ controller is initially designed considering an uniform spatial distribution weighing function, which means that the vibration is attenuated at the complete structure (Halim & Moheimani, 2002; Barrault et al., 2008). After detection, the controller design is recalculated, now considering a Gaussian function distribution for the spatial constraint, over the region module where damage is supposed to be, according to SHM data. The idea here is to illustrate ATAC concept and strategy; the connection between the reconfiguration modules is the subject of another publication.

others are used as sensors. Comparison between signals recorded on healthy and damaged states, can relate scattered information contained in the signals to damage, such as location, size, orientation, type among others. Time-of-flight (ToF), which corresponds to the time taken by the wave packet to travel from the damage to a sensor through a given path, is an extracted feature widely used for damage localization (Buli et al., 2009; Ihn & Chang, 2008). Knowing the ToF values and the wave velocity for different sensors, damage localization problem becomes to solve a set of deterministic nonlinear equations that describe the relationship between the coordinates of the damage, the ToF, and the wave velocity. Two kind of equations can be used to do so: those relying on time-of-arrival (ToA), which provide damage location as the intersection of several ellipses; and those based on time-difference-of-arrival (TDoA), which provide damage location as the intersection of hyperbolas. These strategies have been successfully used for damage localization (Su & Ye, 2009). In this paper an SHM methodology based on ToA and TDoA is proposed, based on a combination of both approaches avoiding switching between all the actuator/sensor pairs in order to account for feedback constrains.

In the next section, we briefly introduce some SHM concepts; in the third section a description of the smart structure and its finite element (FE) model are presented. In the forth section, Lamb wave method used to localize damage is described, and spatial norm control approach and simulated results are presented and analyzed in the fifth section. Concluding remarks are presented in the last section. 2

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3 3.1

COMPOSITE SMART STRUCTURE

Description of the structure

Nowadays, composite materials are increasingly used due to their lightness and strength properties, which is particularly welcome in the aeronautical industry. Due to its multilayer structure, they are suitable to host smart materials. Embedded sensors and actuators could constitute a smart layer to become part of the structure, which may be easily done during the manufacturing phase of the composite panels. An active cantilevered laminate plate in a horizontal position is adopted in this study, according to dimensions and elements in Fig. 2. A FE model is developed considering six degree-offreedom (DOF) at each node in a dense FE grid. The structure consists in a four epoxy/carbon layers with orientation angle [0°/−45°/+45°/0°]. The actuators are circular PZT patches from NOLIAC® and sensors are flexible macro fiber ceramic (MFC), from Smart Materials®.

SHM METHODOLOGY

SHM has been object of great research interest in the last decades. It is a multidisciplinary field, dedicated to real-time detection and identification of damage in mechanical structures. SHM is the key technology to enable the transition from schedule-driven maintenance to condition-based maintenance (Chang, 2011). It integrates sensor and actuator networks and structures with embedded hardware and software to process, manage and interpret the respective signals. In order to carry on all these functions, a SHM system results can be classified into four different sequential levels, detection, identification, quantification, and prognosis (Worden et al., 2007). To perform damage monitoring, a variety of techniques have been developed (Hajrya & Mechbal, 2013). Evaluation of wave propagation on solids is one of the most successful techniques, based on an interrogation scheme. A piezoelectric actuator emits periodic bursts, exciting Lamb waves in the structure, and a set of receiving sensors generate signals that are processed to extract structure condition and damage related information (Su & Ye, 2009). An outstanding advantage of Lamb wave techniques is that such waves can be used to monitor various types of damage (delaminations, disbands fibber breaking, impact, holes, etc...). However, wave dispersion and anisotropy (dependence of the wave velocity with respect to frequency and propagation direction), mode conversion (change in propagation velocity when the wave interacts with structural discontinuities or boundaries) as well as changing environment and operational conditions, make reliable damage localization a challenging task (Zhongqing et al., 2006). Considering a piezoelectric transducer network, the actuator role may be attributed successively to each element while the

Fig. 2: Adopted composite structure with active elements

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qi  d m,i qi  km,i qi  bm,i Fa ,i ,

Flexible structure models

(5) where 𝑖𝑖 = 1,2, . . . 𝑚𝑚, is the mode index, 𝑑𝑑𝑚𝑚 and 𝑘𝑘𝑚𝑚 are the damping and stiffness coefficients for the respective mode. Adopting the state vector definition 𝑥𝑥(𝑡𝑡) = [𝑞𝑞(𝑡𝑡) 𝑞𝑞̇ (𝑡𝑡)]𝑇𝑇 , corresponding to modal displacements and velocities, the respective state-space model derived from (5) may be written as x (t )  Ax (t )  Bw w(t )  Bu u (t ) (6) y (t )  Cx(t ) where 𝑢𝑢(𝑡𝑡) is the control input, 𝑦𝑦(𝑡𝑡) the measured output and 𝑤𝑤(𝑡𝑡) is the disturbance input. The system matrix 𝐴𝐴 has dimension 2𝑚𝑚 × 2𝑚𝑚 and is given by 0 𝐼𝐼 (7) 𝐴𝐴 = [ 𝑚𝑚×𝑚𝑚 𝑚𝑚×𝑚𝑚 ] 𝐾𝐾𝑚𝑚 𝐷𝐷𝑚𝑚 and the 𝐵𝐵𝑤𝑤 , 𝐵𝐵𝑢𝑢 and 𝐶𝐶 matrices must reflect the position of the respective applied forces, representing disturbance and control law input. These two transfer functions, respectively 𝑇𝑇𝑦𝑦𝑦𝑦 and 𝑇𝑇𝑦𝑦𝑦𝑦 , based on the combination of each considered mode, are given as (8) and (9),

The dynamic behavior of a generic flexible structure can be described by a set of second-order differential matrix equation based on concentrated parameters 𝑀𝑀𝑝𝑝 𝑝𝑝̈ + 𝐷𝐷𝑝𝑝 𝑝𝑝̇ + 𝐾𝐾𝑝𝑝 𝑝𝑝 = 𝐵𝐵0 𝐹𝐹𝑎𝑎 (1) where 𝑝𝑝 is the displacement vector, 𝑀𝑀𝑝𝑝 , 𝐷𝐷𝑝𝑝 and 𝐾𝐾𝑝𝑝 are respectively the inertial, damping and stiffness matrices, 𝐹𝐹𝑎𝑎 is the vector of external forces acting on the structure and 𝐵𝐵0 reflects its position distribution. Equation (1) is called the spatial model, and a very convenient form to represent an FE model, but modern controller design is based on state space formulation, with the system homogenous equation written as 𝑥𝑥̇ (𝑡𝑡) = 𝐴𝐴𝐴𝐴(𝑡𝑡) where 𝑥𝑥(𝑡𝑡) is the space vector and 𝐴𝐴 is the system matrix. This means that we have to convert (1) to the second form in a convenient way, based on a modal model (Ewins, 2000). Considering the main vibration characteristics of a mechanical structure frequency response, it presents natural frequencies corresponding to resonant amplitude peaks, which may be independently excited, permitting that total response be obtained through summation of each contribution, called the vibration modes. Each mode corresponds to a complex pole with small negative real parts, which implies a controllable and observable system (Gawronski, 2004). These poles are the roots of the structure characteristic equation, corresponding to the eigenvalues of the system matrix. In consequence, decoupled second-order differential equations, describing each mode response, may be written as (2) 𝑚𝑚𝑖𝑖 𝑝𝑝̈𝑖𝑖 + 𝑑𝑑𝑖𝑖 𝑝𝑝̇𝑖𝑖 + 𝑘𝑘𝑖𝑖 𝑝𝑝𝑖𝑖 = 𝑓𝑓𝑖𝑖 where 𝑝𝑝𝑖𝑖 , 𝑚𝑚𝑖𝑖 , 𝑑𝑑𝑖𝑖 and 𝑘𝑘𝑖𝑖 are respectively the displacement, mass, damping and stiffness of the i-mode, and 𝑓𝑓𝑖𝑖 is the external correspondent applied force. Every mechanical structure has an infinite number of vibration modes, however, considering that there are always a limited frequency range of interest, we adopt the number of modes that is adequate to the problem at hand. To decouple the modal equations, we use the modal matrix defined as Φ = [𝜙𝜙1 𝜙𝜙2 … 𝜙𝜙𝑚𝑚 ] where 𝜙𝜙𝑖𝑖 𝑖𝑖 = 1, … 𝑚𝑚 are the respective eigenvectors for the first m adopted modes. Considering the transformation using 𝑝𝑝 = Φ𝑞𝑞 and pre-multiplying (2) by Φ𝑇𝑇 , it results T

T

T

T

 M p q   D p q   K p q   B0 Fa

Tyw ( s, r )  C ( sI  A) 1 Bw 

iT B0 w  2 i 1 s  d m, i s  k m, i

(8)

Tyu ( s, r )  C ( sI  A) 1 Bu 

iT B0u  2 i 1 s  d m, i s  k m, i

(9)

n

n

where the matrices 𝐵𝐵0𝑤𝑤 and 𝐵𝐵0𝑢𝑢 correspond to the respective position of actuators. 3.3

Finite element simulations

The FE models developed in this work use the Structural Dynamics Toolbox (SDT) for MATLAB®, (Balmes, 2012). The adopted formulation is based on piezoelectric Mindlin shells, taking into account the viscoelasticity of the composite core, the glue, and the piezoelectric coupling equations. Electrical degrees of freedom (DOF) are included in addition to the nodal displacement. For piezoelectric shell elements, electrical DOFs correspond to the difference of potential on the electrodes while the corresponding load is the electrical charge, see (Balmes & Deramaeker, 2013) for more details. 3.4

(3)

Damage simulation

To introduce damage, we have developed, using SDT software, a damage-patch with specific meshes. Patch dimension and properties could then be changed and adjusted, reflecting damage-caused variation. The introduction of patch before damage simulation does not alter the modal structure response. Moreover, nodes inside the patch could be removed to simulate cracks. It is possible to see in Fig. 3 two examples of this approach to represent damage in FE models. Notice also in Fig. 3 that there is a mesh for the structure, a different one for the damage boundaries and a third for the damage itself. It is worth mention that this FEM model was experimentally validated with probe specimens (Balmes et al., 2014). For the simulated studies here presented, a circular patch is adopted to represent damage, with diameter ∅ = 20 𝑚𝑚𝑚𝑚, centered at position (570,140) 𝑚𝑚𝑚𝑚 (see Fig. 11). It represents the effect of an impact that decreases the material properties

that may be written as (4) q  M m1 Dm q  M m1 K m q  M m1 Bm Fa . 𝑇𝑇 𝑇𝑇 𝑇𝑇 where 𝑀𝑀𝑚𝑚 = Φ 𝑀𝑀𝑝𝑝 Φ, 𝐷𝐷𝑚𝑚 = Φ 𝐷𝐷𝑝𝑝 Φ and 𝐾𝐾𝑚𝑚 = Φ 𝐾𝐾𝑝𝑝 Φ, are diagonal matrices and 𝐵𝐵𝑚𝑚 = Φ𝑇𝑇 𝐵𝐵0 . In order to get a modal damping matrix also diagonal, it is common to adopt the proportional form, 𝐷𝐷𝑝𝑝 = 𝛼𝛼𝑀𝑀𝑝𝑝 + 𝛽𝛽𝐾𝐾𝑝𝑝 , where 𝛼𝛼 and 𝛽𝛽 ≥ 0. This assumption gives rather approximate values, considering that flexible structures have very small damping factors, as already stated. Proportional damping is convenient but not a mandatory hypothesis, since the control techniques are not dependent of proportional damping values. Considering these assumptions, Equation (4) represents uncoupled modal displacements and it can be written as a set of 𝑚𝑚 independent equations. It is always possible to have the modal matrix Φ such as the modal mass matrix results an identity matrix, leading to (5), for each mode, 660

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(𝒂𝒂)

(𝒃𝒃)

Fig. 3: Example of damage-patches: (𝑎𝑎) circular (∅ = 20 𝑚𝑚𝑚𝑚); (𝑏𝑏) rectangular (10 × 80 𝑚𝑚𝑚𝑚).

by a factor 𝚼𝚼 (here 𝚼𝚼 = 2 × 10−3 ). Different response effects may result from different damages, which depend on several factors. It is possible for instance that the damage can lead to a decrease in the structure vibration on some regions. 4 4.1

STRUCTURAL HEALTH MONITORING

Damage localization approach

Lamb wave-based damage localization in structures relies on the fundamental idea that damage causes signal scattering. When a propagating wave in a solid medium interacts with any structure discontinuity, the wave reflects in various directions, depending on the discontinuity form. The scattered signal is obtained using baseline subtraction from the current signal. This procedure eliminates the complexity of the signal, which otherwise would present reflections from all structural boundaries and direct arrivals (Su & Ye, 2009). Measured ToF gives a good estimation of distance between damage and sensor. ToF algorithms usually fall into two groups: the time of arrival-based algorithm (ToA - ellipse method) and the time difference of arrival-based algorithm (TDoA - hyperbola method). Both methods use the triangulation principle for damage localization (Zhongqing et al., 2006).

661

where (𝑥𝑥𝑑𝑑 , 𝑦𝑦𝑑𝑑 ), is the damage coordinate, 𝑉𝑉𝑔𝑔 and 𝑉𝑉𝑔𝑔′ are the group velocities of the Lamb wave. We have considered here a general model where 𝑉𝑉𝑔𝑔 and 𝑉𝑉𝑔𝑔′ could be different due propagation direction in anisotropic composite structures and also mode conversion that could happen when the wave is scattered. The solution to (10) is an ellipse, which gives the coordinates of the probable position of the damage. These coordinates represent the outline of the ellipse. For all pairs sensors / actuators considered, one can draw an ellipse and the area of damage is given by the intersection of all ellipses. The TDoA method considers piezoelectrics in groups of three with one acting as actuator (transmitter, noted 𝑎𝑎) and the others two as sensors (receivers, noted 𝑠𝑠1 and 𝑠𝑠2 ). If there is damage at the point (𝑥𝑥𝑑𝑑 , 𝑦𝑦𝑑𝑑 ), the difference in the times of arrival of the scattered signals at the two receivers is 𝑇𝑇𝑇𝑇𝑇𝑇𝑎𝑎→𝑠𝑠 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑎𝑎→𝑠𝑠 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑠𝑠 −𝑠𝑠 (11) 1 2 1 2 Since the distance from the actuator to the damage is the same for both receivers, then for each three piezoelectrics, solution of (11) form a hyperbola. The damage location is then obtained as the point of intersection of all the resulting hyperbolas for all the possible triplets (𝑎𝑎, 𝑠𝑠1 , 𝑠𝑠2 ) in the transducers network, with 𝑖𝑖 ≠ 𝑗𝑗. As the strategy here is to process simultaneously the control and the localization signals, the time needed for this latter step should be consistent with the sample frequency used by the controller. A filter process is necessary to separate the two parts of the signal, and use a multi-thread algorithm. Therefore, we propose here to mix the ToA and TDoA approach in order to avoid switching between all the actuator/sensor (Fig. 5).

Fig. 5: Damage localization based on the ellipse and hyperbola methods (one actuator and 2 sensors)

4.2

Damage detection and localization results

Fig. 4: Time of flight based damage localization principle

The geometric relationship for damage localization in plate-like structures using ToA model is schematically shown in Fig. 4. Here a single piezoelectric actuator-sensor pair is presented. The coordinates of the actuator are (𝑥𝑥𝑎𝑎 , 𝑦𝑦𝑎𝑎 ), and (𝑥𝑥𝑠𝑠 , 𝑦𝑦𝑠𝑠 ), for the sensor. When a wave travels from the actuator across a pointlike damage to the sensor, the ToA of the scattered wave corresponds to the wave travel time of the actuator-damagesensor path. This wave travel time can be expressed as L𝑎𝑎→𝐷𝐷 L𝐷𝐷→𝑠𝑠 + ′ = 𝑇𝑇𝑇𝑇𝑇𝑇𝑎𝑎→𝑠𝑠 (10) 𝑉𝑉𝑔𝑔 𝑉𝑉𝑔𝑔 with L𝑎𝑎→𝐷𝐷 = √(𝑥𝑥𝑑𝑑 − 𝑥𝑥𝑎𝑎 ) L𝐷𝐷→𝑠𝑠 = √(𝑥𝑥𝑑𝑑 − 𝑥𝑥𝑠𝑠 )

2 2

+ (𝑦𝑦𝑑𝑑 − 𝑦𝑦𝑎𝑎 ) + (𝑦𝑦𝑑𝑑 − 𝑦𝑦𝑠𝑠 )

2 2

The proposed strategy has been simulated. PZT 4 is used to generate a Lamb wave burst and data are collected on PZT 2 and 3.We have considered a burst signal described in Fig. 6.

Fig. 6: Burst signal (a) and its spectrum (b) centered at 200 𝑘𝑘𝑘𝑘𝑘𝑘

Considering the hyperbolas obtained by the PZT triplets (4 ; 2,3) damage localization result depicted in Fig. 7 shows that the relative error for damage localization, the difference between the real damage position (550,140) 𝑚𝑚𝑚𝑚 and the estimated one (557,142) 𝑚𝑚𝑚𝑚, is small.

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662 SAFEPROCESS 2015 September 2-4, 2015. Paris, France

Damage imaging: ellipses

Y-axis

𝐽𝐽∞ =

100

Piezo 1

Piezo 4

50

Piezo 3 100

200

300

400

500

600

700

800

900

X-axis Damage imaging: hyperbola

Piezo 2

Y-axis

150

100

Piezo 1

Piezo 4

50

Piezo 3 100

200

300

400

500

600

700

800

900

X-axis Sum damage imaging

Piezo 2

150

100

O:(557,142)

Piezo 1

Piezo 4

50

Piezo 3 100

200

300

400

500

600

700

800

900

Fig. 7: Damage localization imaging result. Black circle real damage, white circle estimate damage.

5 5.1

∞

∫0 ∫Ω 𝑧𝑧(𝑡𝑡)𝑇𝑇 𝑄𝑄(𝑟𝑟)𝑧𝑧(𝑡𝑡)𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

(14) ∞ ∫0 𝑤𝑤(𝑡𝑡)𝑇𝑇 𝑤𝑤(𝑡𝑡)𝑑𝑑𝑑𝑑 Considering the objective function in (14), the new time and spatial dependent performance vector may be written as

Piezo 2

150

ADAPTIVE SPATIAL 𝐻𝐻∞ CONTROLLER

Spatial H∞ controller design

Consider the general MIMO system framework seen in Fig. 8 presenting two blocks and four signals, respectively the structural plant and controller blocks and disturbance and control signal vectors input signals 𝑤𝑤(𝑡𝑡) and 𝑢𝑢(𝑡𝑡), and the time and spatial dependent performance index vector 𝑧𝑧(𝑟𝑟, 𝑡𝑡) and measured output vector 𝑦𝑦(𝑡𝑡).

Fig. 8: Active control problem framework

To introduce briefly the spatial norm control approach, the state space model for the smart structure to be controlled may be represented by (12), where 𝑥𝑥(𝑡𝑡) is the state vector and all vectors and matrices must have adequate dimensions, regarding the number of inputs and outputs and the order of the plant. Considering initially only the time dependence, the state-space formulation is represented as

𝑧𝑧(𝑟𝑟, 𝑡𝑡) = 𝐶𝐶𝑧𝑧 (𝑟𝑟)𝑥𝑥𝑝𝑝 (𝑡𝑡) + 𝐷𝐷𝑧𝑧𝑧𝑧 (𝑟𝑟)𝑤𝑤(𝑡𝑡) + 𝐷𝐷𝑧𝑧𝑧𝑧 (𝑟𝑟)𝑢𝑢(𝑡𝑡) (15) To consolidate the performance index as dependent only on time, we need to isolate the spatial dependence using an auxiliary matrix. We may write 𝑧𝑧𝑟𝑟 (𝑡𝑡) = Π𝑥𝑥𝑝𝑝 (𝑡𝑡) + Θ𝑤𝑤 𝑤𝑤(𝑡𝑡) + Θ𝑢𝑢 𝑢𝑢(𝑡𝑡). where Γ = [Π Θ𝑤𝑤 Θ𝑢𝑢 ] is this matrix, such as Γ 𝑇𝑇 Γ = 𝑇𝑇 (𝑟𝑟)𝐷𝐷 (𝑟𝑟)𝐷𝐷 (𝑟𝑟)] [𝐶𝐶 𝑄𝑄(𝑟𝑟)[𝐶𝐶𝑧𝑧 (𝑟𝑟)𝐷𝐷𝑧𝑧𝑧𝑧 (𝑟𝑟)𝐷𝐷𝑧𝑧𝑧𝑧 (𝑟𝑟)]𝑑𝑑𝑑𝑑 ∫Ω 𝑧𝑧 (16) 𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧 . This results in a objective function 𝐽𝐽∞ ∞ ∫0 [𝑥𝑥𝑝𝑝 (𝑡𝑡) 𝑤𝑤(𝑡𝑡) 𝑢𝑢(𝑡𝑡)]Γ 𝑇𝑇 Γ[𝑥𝑥𝑝𝑝 (𝑡𝑡) 𝑤𝑤(𝑡𝑡) 𝑢𝑢(𝑡𝑡)]𝑇𝑇 𝑑𝑑𝑑𝑑 (17 = , ) ∞ ∫0 𝑤𝑤(𝑡𝑡)𝑇𝑇 𝑤𝑤(𝑡𝑡)𝑑𝑑𝑑𝑑 conducting to (18), ∞ ∫0 𝑧𝑧𝑟𝑟 (𝑡𝑡)𝑇𝑇 𝑧𝑧𝑟𝑟 (𝑡𝑡)𝑑𝑑𝑑𝑑 (18) . 𝐽𝐽∞ = ∞ ∫0 𝑤𝑤(𝑡𝑡)𝑇𝑇 𝑤𝑤(𝑡𝑡)𝑑𝑑𝑑𝑑 which is similar to a regular 𝐻𝐻∞ objective function. Then, the spatial 𝐻𝐻∞ problem may be solved using known methods for the 𝐻𝐻∞ problem. 5.2

Application to the composite structure

Two models are adopted for the numerical simulation: for the first one, which is used to design the controllers, only the 12 first modes are adopted, and it is here called the nominal model; and the second one, which is based on the first 40 modes and is used to simulate the performance of the designed controllers, we called the complete model. Fig. 9 shows results for the healthy structure, for transfer functions between disturbance input 𝑤𝑤 and performance output 𝑧𝑧, and also for measured outputs 𝑦𝑦1 and 𝑦𝑦2 , both open loop and closed loop. This first controller is designed using an unitary weighing function. It is clear the attenuation result on the nominal model bandwidth. Fig. 10 shows the 𝐻𝐻∞ norm on the spatial perspective, where the controller effect is also easily seen, from the clamped to the free side of the plate.

x p (t )  Ap x p (t )  Bw w(t )  Bu u (t ) z (t )  C z x p (t )  Dzw w(t )  Dzu u (t )

(12)

y (t )  C y x p (t )  Dyw w(t )

The 𝐻𝐻∞ problem is to find a controller which satisfy the infinity norm, stated as an optimization objective function for the closed loop transfer matrix 𝑇𝑇𝑧𝑧𝑧𝑧 according 𝐽𝐽∞ =

∞

∫0 𝑧𝑧(𝑡𝑡)𝑇𝑇 𝑧𝑧(𝑡𝑡)𝑑𝑑𝑑𝑑

(13) ∞ ∫0 𝑤𝑤(𝑡𝑡)𝑇𝑇 𝑤𝑤(𝑡𝑡)𝑑𝑑𝑑𝑑 To take into account a spatial region Ω of the structure where we want to minimize the 𝐻𝐻∞ spatial norm, a space dependent weighing matrix 𝑄𝑄(𝑟𝑟), where r is the spatial vector, is introduced according to (13):

Fig. 9: Bode diagram of the uncontrolled and controlled healthy structure with 𝑧𝑧(𝑡𝑡, 𝑟𝑟 = 650𝑚𝑚𝑚𝑚, 150𝑚𝑚𝑚𝑚).

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concentration on the damaged zone comes at the expense of relaxing control on the region near the clamped side. 6. CONCLUSIONS AND PERSPECTIVES A novel paradigm to design damage-tolerant controllers, dedicated to flexible mechanical structures, is introduced and examined through FE based simulations, used to design two controllers and to assess respective performances facing damage. The first controller, based on a spatial 𝐻𝐻∞ norm approach, presents good vibration attenuation, however, after damage was inflicted, the performance is shown to deteriorate significantly. A Lamb wave based SHM module is proposed to detect and localize the damaged region in order to redesign the controller. The new controller design approach effectively concentrates the control law energy on the damaged region of the structure, therefore recovering performance, despite damage. An experimental setup to integrate the several modules in real time is the next step to consolidate the DTAC approach here proposed.

Fig. 10: Spatial 𝐻𝐻∞ norm of the controlled and uncontrolled system

REFERENCES Fig. 11: Gauss spatial weighing functions 𝑄𝑄(𝑟𝑟)

Fig. 12: Closed-loop 𝐻𝐻∞ norm: (𝑎𝑎) healthy; (𝑏𝑏) Controller with uniform weighing function; (𝑐𝑐) Reconfigured controller with Gaussian weighing function.

In Fig. 11 it is seen the Gaussian gate function used to define the new spatial weighing to design the second controller, after damage is detected and localized in the structure. In Fig. 12 the closed loop spatial 𝐻𝐻∞ norm of the healthy and the damaged plate with the two controllers are presented. In the upper panel it is shown the healthy closed loop, in the middle panel, it may be seen the first controller facing the damage and in the lower panel, the second controller after reconfiguration with Gaussian spatial weighing functions 𝑄𝑄(𝑟𝑟). It may be seen the damage effect on the performance of the first controller in the middle panel, and that the second controller solves this problem showing a good performance on the damage region compared to the first controller. It is visible also that this energy

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