Ninon Burgos
email: ninon.burgos@cnrs.fr

Medical image synthesis using segmentation and registration

Keywords: Image synthesis, Segmentation, Registration, Atlas database

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Medical image synthesis using segmentation and registration

Introduction

Many of the early works on medical image synthesis had for objective to improve subsequent image processing steps such as segmentation or registration. In 2005, van de Kraats et al. [START_REF] Van De Kraats | Multispectral MR to X-ray registration of vertebral bodies by generating CT-like data[END_REF] presented a new method to register (preoperative) magnetic resonance (MR) images to (intraoperative) X-ray images. MR to X-ray registration is very challenging because of the different underlying contrast mechanisms. A consequence of these differences is that several tissues, especially bone, have dissimilar appearances on both modalities, which makes it difficult to directly apply gradient-based or intensity-based registration methods. Their idea was thus to convert multispectral MR images into computed tomography (CT) images by constructing a look-up table, and to then register the synthetic CT to the X-ray, which is an easier task as both modalities have the same underlying contrast mechanisms. They showed that registering synthetic CTs to X-ray data using a gradient-based approach outperformed direct registration from the MR data. Five years later, Roy et al. [START_REF] Roy | MR contrast synthesis for lesion segmentation[END_REF] investigated whether lesions could be detected in the absence of an appropriate lesion-distinguishing MR sequence (a FLAIR) by synthesising a FLAIR from other sequences (T1-weighted and T2-weighted) using a patch-based approach (see Chapter 5), and showed that it was indeed the case. Other examples are described in Chapter 10.

In both these examples, image synthesis was used to improve segmentation or registration. Instead, what will be described in this chapter is how segmentation and registration can be used to synthesise medical images of a particular modality from images of another modality. The use case throughout the chapter will be the synthesis of CT from MR images as almost all the works reported target two very concrete applications: attenuation correction of positron emission tomography (PET) data, particularly for PET/MR scanners, and radiotherapy treatment planning from MRI only (more details are provided in Chapters 19 and 20.

One of the factors limiting the widespread use of PET/MR scanners is probably the imperfect attenuation correction, leading to a bias of the PET activity, particularly in the brain. Before the emergence of PET/CT scanners, attenuation correction was mostly based on transmission measurements. However, because of the restricted space and the strong magnetic field, installing a rotating radiation source in a PET/MR scanner is challenging. With PET/CT systems, the attenuation coefficients are derived from CT scans. However, in contrast with CT images, MR image intensities do not directly provide information about the tissue attenuation properties.

The aim of radiotherapy treatment planning (RTP) is to optimise the therapeutic ratio by delivering an optimal dose of radiation over the tumorous area while sparing the normal tissues. RTP first requires contouring the tumour and organs at risk. Once these volumes have been defined, the attenuation properties of the different tissues are used as parameters in an optimisation process calculating the optimal dose distribution to treat the tumour. Most radiotherapy treatments are planned using a CT scan of the patient as its acquisition is fast and tissue attenuation coefficients can easily be derived from the CT intensity values in Hounsfield unit. However, CT images have a low soft-tissue contrast, which can lead to large organ delineation errors, particularly when located in the brain, head & neck, or pelvic regions. MRI is often preferred over CT as a structural imaging modality, mainly for its excellent soft-tissue contrast. Although MRI is increasingly used in clinical practice, its role in RTP is limited by the fact that it does not readily provide tissue attenuation properties, hampering the calculation of dose distributions. This is a critical limitation for the clinical deployment of the MR-linac devices combining an MR scanner and a linear accelerator-the machine delivering the radiation dose.

To answer these two problems, numerous methods have been proposed. Both the general principles and particular examples of segmentation-based and registration-based image synthesis approaches will be described. The reasons why these two families of methods have almost exclusively been applied to the MR to CT synthesis task and why they have been supplanted by other methods will then be discussed.

Segmentation-based image synthesis

Segmentation-based approaches can generally be decomposed into two components: the first one consists in segmenting the source image and the second one consists in assigning intensity values to the different tissue classes obtained to generate the desired image. The segmentation can be manual or automatic, and the intensities can be assigned in bulk (i.e. predefined values are assigned to each tissue class) or in a subject-specific manner. This process is illustrated in Figure 1. Note that segmentation-based image synthesis can also be called classification-based image synthesis [START_REF] Prince | Image synthesis and superresolution in nedical imaging[END_REF].

Segmentation approaches

Manual segmentation

Manual segmentation is not uncommon in domains such as RTP [START_REF] Jonsson | Treatment planning using MRI data: An analysis of the dose calculation accuracy for different treatment regions[END_REF][START_REF] Korhonen | A dual model HU conversion from MRI intensity values within and outside of bone segment for MRI-based radiotherapy treatment planning of prostate cancer[END_REF][START_REF] Guerreiro | Evaluation of a multi-atlas CT synthesis approach for MRI-only radiotherapy treatment planning[END_REF]. In their early work on the use of MRI for RTP, Jonsson et al. [START_REF] Jonsson | Treatment planning using MRI data: An analysis of the dose calculation accuracy for different treatment regions[END_REF] manually segmented the patient outline, bones, lungs and air cavities. An example of a pelvis MRI and synthetic CT, overlaid with patient outline and bone contours, is shown in Figure 2. Body and bone outlines were also contoured in [START_REF] Korhonen | A dual model HU conversion from MRI intensity values within and outside of bone segment for MRI-based radiotherapy treatment planning of prostate cancer[END_REF] and bone and air in [START_REF] Guerreiro | Evaluation of a multi-atlas CT synthesis approach for MRI-only radiotherapy treatment planning[END_REF]. Manual segmentation is very time-consuming and thus automatic segmentation methods have naturally been employed.

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Automatic segmentation

Segmenting MR images for PET attenuation correction was already a topic of interest in the 1990s: Le Goff et al. [START_REF] Goff-Rougetet | Segmented MR images for brain attenuation correction in PET[END_REF] segmented brain T1-weighted (T1w) MR images based on thresholding and mathematical morphology tools. Later, Martínez-Möller et al. [START_REF] Martinez-Möller | Tissue classification as a Biomedical Image Synthesis and Simulation, Chapter 4 potential approach for attenuation correction in whole-body PET/MRI: Evaluation with PET/CT data[END_REF] used thresholding to segment the body into four classes: background, lungs, fat and soft-tissues. They relied on a specific MRI sequence, the Dixon method, which exploits the fact that water and fat protons have slightly different resonance frequencies. By acquiring images when their spins are in-and out-of-phase with each other, water and fat images can be derived [START_REF] Dixon | Simple proton spectroscopic imaging[END_REF]. In this work [START_REF] Martinez-Möller | Tissue classification as a Biomedical Image Synthesis and Simulation, Chapter 4 potential approach for attenuation correction in whole-body PET/MRI: Evaluation with PET/CT data[END_REF], a threshold was applied to the water and fat images to separate soft-tissue and fat from the background. Connectedcomponent analysis of the air in the inner part of the body was used to define the lung class. The tissue/air classification was refined using a morphologic closing filter. Catana et al. [START_REF] Catana | Toward implementing an MRI-based PET attenuation-correction method for neurologic studies on the MR-PET brain prototype[END_REF] and Keereman et al. [START_REF] Keereman | MRI-based attenuation correction for PET/MRI using ultrashort echo time sequences[END_REF] also used a particular MRI sequence to segment head images into three classes (bone, soft-tissue and air), the ultra-short echo time (UTE) sequence. Two images, UTE 1 and UTE 2 , were acquired at different echo times, TE 1 and TE 2 respectively, such that the bone signal is present in UTE 1 but not in UTE 2 , while the signals from other tissues are similar in both images. In [START_REF] Catana | Toward implementing an MRI-based PET attenuation-correction method for neurologic studies on the MR-PET brain prototype[END_REF], a head mask is first obtained by applying a morphologic closing operation to the UTE 2 image. Bone tissue was segmented from a combination of the two UTE images UTE 1 -UTE 2 UTE 2 2 using an empirically determined threshold. The air cavities were segmented in the same way but from another combination of the UTE images

UTE 1 +UTE 2 UTE 1 2
. Finally, the voxels included in the head mask that were not identified as bone or air were classified as soft-tissue. From the two UTE images, in [START_REF] Keereman | MRI-based attenuation correction for PET/MRI using ultrashort echo time sequences[END_REF] the R2* value, which is the inverse of the T2* relaxation time, was first calculated for each voxel as follows: R2* = ln UTE 1 -ln UTE 2 TE 2 -TE 1

. It was then possible to distinguish between cortical bone, which is expected to have high R2* values, from softtissue, expected to have low R2* values. The distinction between air and tissue is however more difficult because of possible artefacts or noise in the UTE images. To overcome this problem, a binary air mask was created from the first-echo image (UTE 1 ) and applied to the R2* map. Finally, the corrected R2* map was segmented into three tissue classes (bone, soft-tissue and air) using thresholds. The method is illustrated in Figure 3. Refinements for the R2* map segmentation were later proposed [START_REF] Cabello | MR-Based attenuation correction using ultrashort-echo-time pulse sequences in dementia patients[END_REF]. Combining cortical bone segmentation and water-fat decomposition, Berker et al. [START_REF] Berker | MRI-based attenuation correction for hybrid PET/MRI systems: A 4-class tissue segmentation technique using a combined ultrashort-echo-time/Dixon MRI sequence[END_REF] proposed a 4-class (bone, soft-tissue, fat and air) segmentation technique using a UTE triple-echo sequence. The combination of the first and third echoes, both in-phase, was used to segment cortical bone, while the combination of the second echo, outof-phase, with the in-phase echoes enabled the separation of the fat and water signals and the calculation of the water-fat fraction. Bone was then segmented The fuzzy c-means (FCM) algorithm was implemented in several works. This unsupervised method generalises the k-means algorithm by allowing each data point to belong to multiple clusters with varying degrees of membership [START_REF] Bezdek | Pattern Recognition with Fuzzy Objective Function Algorithms[END_REF]. This is for example the case of Zaidi et al. [START_REF] Zaidi | Magnetic resonance imagingguided attenuation and scatter corrections in three-dimensional brain positron emission tomography[END_REF], who applied the FCM algorithm on T1w spin-echo images to segment air, brain tissue, skull, nasal sinuses, and scalp. Some manual intervention of the operator was required to refine the segmentation of the skull. Later, Hsu et al. [START_REF] Hsu | Investigation of a method for generating synthetic CT models from MRI scans of the head and neck for radiation therapy[END_REF] applied it to segment a set of MR images (T1w, T2w, Dixon and UTE) into five tissue classes (fluid, fat, white matter, grey matter and bone). Su et al. [START_REF] Su | Generation of brain pseudo-CTs using an undersampled, single-acquisition UTE-mDixon pulse sequence and unsupervised clustering[END_REF] followed a similar idea but reduced the inputs to a single acquisition UTE-mDixon sequence. The three images derived from this sequence (fat, water and R2*) were used as inputs for an FCM algorithm aiming at identifying five clusters (air, brain, fat, fluid, and bone). They later extended the approach to thorax synthetic CT generation [START_REF] Su | UTE-mDixon-based thorax synthetic CT generation[END_REF]. Instead of the UTE sequence, Khateri et al. [START_REF] Khateri | Generation of a four-class attenuation Biomedical Image Synthesis and Simulation, Chapter 4 map for MRI-based attenuation correction of PET data in the head area using a novel combination of STE/Dixon-MRI and FCM clustering[END_REF] proposed to use a short echo time sequence, combined with a Dixon sequence, to segment the head in four tissue classes using FCM. Active learning-based classification was proposed in [START_REF] Qian | mDixon-based synthetic CT generation for PET attenuation correction on abdomen and pelvis jointly using transfer fuzzy clustering and active learning-based classification[END_REF] to refine segmentations obtained with fuzzy clustering.

Other segmentation approaches have also been used. A knowledge-based segmentation approach consisting of three steps was proposed in [START_REF] Wagenknecht | Knowledge-based segmentation of attenuation-relevant regions of the head in T1-weighted MR images for attenuation correction in MR/PET systems[END_REF]. First, tissues were segmented using a supervised neural network-based classification. The brain and extracerebral region were then separated based on anatomical knowledge: it is known that the brain soft-tissue is surrounded by cerebrospinal Biomedical Image Synthesis and Simulation, Chapter 4 fluid (CSF) which is in turn surrounded by the extracerebral region, while the extracerebral region is surrounded by the background region. Anatomical knowledge was finally used to segment the extracerebral region into the mastoid process, craniofacial cavities, bone, and soft-tissue, based for example on the fact that air-filled cavities are compact and connected by small junctions, while bone regions are narrow and elongated. Schulz et al. [START_REF] Schulz | Automatic, three-segment, MR-based attenuation correction for whole-body PET/MR data[END_REF] based their 3-class (air, soft-tissue and lung) segmentation method on a T1w sequence. The body was first extracted from the background using slice-wise region growing with an automatically determined threshold, and another region growing was performed to segment the lungs. Instead of segmenting MR images, the use of non-attenuation corrected (NAC)-PET images was investigated in [START_REF] Chang | Investigating the use of nonattenuation corrected PET images for the attenuation correction of PET data[END_REF]. The 3-class segmentation consisted of segmenting the NAC-PET to derive a first attenuation map, using this map to correct the raw PET data for attenuation, and refining the segmentation using the newly reconstructed PET image. This process was repeated to first segment the body contour, then segment the lungs, and finally refine the lung segmentation. Instead of segmenting the skull in image space, Yang et al. [START_REF] Yang | Multiscale segmentation of the skull in MR images for MRI-based attenuation correction of combined MR/PET[END_REF] proposed a skull segmentation method for T1w MR images via a multiscale bilateral filtering processing of MRI sinogram data in the Radon domain. This segmentation method was combined with a multiscale FCM approach to classify the other tissues. In [START_REF] An | MRI-based attenuation correction for PET/MRI using multiphase level-set method[END_REF], air, bone and soft-tissue were segmented from brain UTE images using a multi-phase levelset approach. This active contour strategy allows segmenting images with more than two regions [START_REF] Vese | A multiphase level set framework for image segmentation using the Mumford and Shah model[END_REF]. Finally, a 3D deep convolutional neural network (DeepMedic [START_REF] Kamnitsas | Efficient multi-scale 3D CNN with fully connected CRF for accurate brain lesion segmentation[END_REF]) was trained in [START_REF] Bradshaw | Feasibility of deep learning-based PET/MR attenuation correction in the pelvis using only diagnostic MR images[END_REF] to segment air, water, fat, and bone from T2w and Dixon images of the pelvis.

Intensity assignment approaches

Once the input image has been segmented, meaningful intensities must be assigned to each tissue class in order to obtain the synthetic image of interest.

Segmentation methods with bulk assignment

The simplest way to obtain a synthetic image from a segmented image is to assign predefined intensity values to each tissue class. This can be easily done when generating CT images as their voxels are displayed in terms of relative radiodensity expressed in Hounsfield unit (HU). For example, air has an attenuation of -1000 HU, fat between -120 and -90 HU, water of 0 HU, and cortical bone between 500 and 1900 HU. In their work illustrated in Figure 2, Jonsson et al. [START_REF] Jonsson | Treatment planning using MRI data: An analysis of the dose calculation accuracy for different treatment regions[END_REF] assigned mass densities of 1.33 g/cm 3 to femoral bone, 1.025 g/cm 3 to soft-tissue and 0.001 g/cm 3 to air.

Segmentation methods with subject-specific assignment

The main drawback of assigning predefined values to the different tissue classes is that subject variability is not taken into account, which limits synthesis accuracy. This is particularly important in the MRI to CT image synthesis scenario for the bone region as it is known that several factors, such as age, can affect bone density [START_REF] Fehily | Factors affecting bone density in young adults[END_REF][START_REF] Schulte-Geers | Age and gender-dependent bone density changes of the human skull disclosed by high-resolution flat-panel computed tomography[END_REF]. To overcome this limitation, several groups explored the possible existence of a mapping between the CT HU and MRI intensities, particularly R2* values in the bone region. If the ability of CT imaging to determine bone mineral content has been known for a long time [START_REF] Reich | Determination of bone mineral content using CT scanning[END_REF], several studies showed that MR imaging could also provide some useful information. It was for example demonstrated using the UTE sequence that the water bound to the collagen in cortical bone (higher density) has a shorter spin-spin relaxation time T2 than the water present in the porous regions of cortical bone (lower density) [START_REF] Horch | Characterization of 1 H NMR signal in human cortical bone for magnetic resonance imaging[END_REF]. As T2* includes T2 relaxation, it is expected that changes in R2* values (R2* = 1/T2*) would reflect changes in bone density.

Kapanen et al. [START_REF] Kapanen | T1/T2*-weighted MRI provides clinically relevant pseudo-CT density data for the pelvic bones in MRI-only based radiotherapy treatment planning[END_REF] exploited in-phase images from a Dixon sequence to model the relation between MR intensity and HU within bone by fitting a second order polynomial. Pseudo CTs were generated by manually contouring the MR images, converting MR intensity values into HU within bone, and assigning to the other tissue classes a value of 0 HU. Juttukonda et al. [START_REF] Juttukonda | MR-based attenuation correction for PET/MRI neurological studies with continuous-valued attenuation coefficients for bone through a conversion from R2* to CT-Hounsfield units[END_REF] acquired dual UTE and Dixon sequences from which they derived R2*, fat, water, and UTE 1 images used to segment bone, fat, soft-tissue, and air, respectively, based on simple thresholding. Predefined linear attenuation coefficients were then assigned to the air, fat and soft-tissues classes while tissues classified as bone were converted to HU using a regression model between the R2* values and HU. Several steps were necessary to obtain this model. First, the CT and R2* images of several subjects were aligned, and bone was segmented in both modalities using thresholds. To reduce noise, the R2* values were divided into bins. The mean R2* values and HU were then computed and plotted for each bin. Finally, a sigmoid model was fitted to the data.

Ladefoged et al. [START_REF] Ladefoged | Region specific optimization of continuous linear attenuation coefficients based on UTE (RESOLUTE): Application to PET/MR brain imaging[END_REF] also acquired and used UTE images to extract a continuous bone signal for each patient and map R2* values to HU. The model was however different as a third order polynomial was fitted instead of a sigmoid, see Figure 4. This could be explained by the lower threshold chosen for included bone and the different formula used to compute the R2* map. In addition to the R2* to HU mapping, they proposed to improve the attenuation map generation using regional masks to separately treat complex areas with mixed air and tissue. These regional masks, defined on an atlas, were transported into to subject's space and used, in combination with the R2* values, to assign predefined attenuation coefficients to the frontal sinus, nasal septa and ethmoidal sinus, mastoid process, and skull base. Other sequences have been exploited to convert MR intensities into CT HUs, mainly the zero echo-time (ZTE) sequence that allows capturing bone density variations [START_REF] Wiesinger | Zero TE MR bone imaging in the head[END_REF][START_REF] Huang | Continuous MR bone density measurement using water-and fat-suppressed projection imaging (WASPI) for PET attenuation correction in PET-MR[END_REF][START_REF] Khalifé | Subject-specific bone attenuation correction for brain PET/MR: Can ZTE-MRI substitute CT scan accurately?[END_REF][START_REF] Leynes | Hybrid ZTE/Dixon MR-based attenuation correction for quantitative uptake estimation of pelvic lesions in PET/MRI[END_REF]. A linear regression was used to convert ZTE-normalized to HU values in the skull [START_REF] Khalifé | Subject-specific bone attenuation correction for brain PET/MR: Can ZTE-MRI substitute CT scan accurately?[END_REF] while a two-segment piecewise linear model was used in pelvic bone regions [START_REF] Leynes | Hybrid ZTE/Dixon MR-based attenuation correction for quantitative uptake estimation of pelvic lesions in PET/MRI[END_REF]. Examples of brain pseudo CT images obtained with this approach are shown in Figure 5.

Registration-based image synthesis

In registration-based methods, an atlas is deformed to match the subject's anatomy using non-rigid registration. If we keep as use case MRI to CT image synthesis, the atlas can be composed of a single CT image, of a pair of MR and CT images, or of multiple pairs of MR and CT images, as illustrated in Figure 6. The principle was first introduced in 1993 by Miller et al [START_REF] Miller | Mathematical textbook of deformable neuroanatomies[END_REF].

Single-atlas registration approaches

Direct multimodal registration

When the atlas is composed of only one image of the target modality, a synthetic image is generated by directly registering the atlas to the source image of the subject of interest. The registration is thus multimodal. Schreibmann et al. [START_REF] Schreibmann | MR-based attenuation correction for hybrid PET-MR brain imaging systems using deformable image registration[END_REF] developed a multimodality optical flow deformable model mapping a representative CT template directly to the subject's MR image, thus generating a simulated CT image that matches the patient's anatomy. Multimodal inter-subject being the most challenging registration scenario, very few studies implemented a direct registration image synthesis approach.

Indirect unimodal registration

The majority of single-atlas registration approaches use an atlas composed of a pair of images: an image of the source contrast/modality and an image of the target contrast/modality. The source atlas image is registered to the Biomedical Image Synthesis and Simulation, Chapter 4 subject's source image and the resulting transformation is applied to the target atlas image to generate the subject's target image. The atlas can be obtained from a single individual but is most often obtained by averaging the images of multiple individuals. Long before the introduction of PET/MR scanners, researchers were already developing image synthesis approaches for attenuation correction. Huang et al. [START_REF] Huang | Three-dimensional PET emission scan registration and transmission scan synthesis[END_REF] did so in 1997 using a pair of transmission and emission scans as atlas. They proposed an inter-subject registration method to align the emission scan of the atlas to that of the target subject. The resulting transformation was then used to align the transmission scan of the atlas and generate a transmission scan for the subject under investigation. Ten years later, Kops et al. [START_REF] Kops | Alternative methods for attenuation correction for PET images in MR-PET scanners[END_REF] built a transmission-MRI atlas from the transmission and MR images of normal volunteers that were spatially normalised and averaged using the Statistical Parametric Mapping version 2 (SPM2)1 software. To generate the subject's transmission image, the MRI atlas was registered to the subject's MR image and the same transformation was applied to the transmission atlas. With this method, it is possible to take into account morphological differences related to gender using female and male templates [START_REF] Kops | Template based attenuation correction for PET in MR-PET scanners[END_REF]. Downling et al. [START_REF] Dowling | An atlas-based electron density mapping method for magnetic resonance imaging (MRI)-alone treatment planning and adaptive MRI-based prostate radiation therapy[END_REF] proposed to generate a pseudo CT image by registering an MRI atlas, result of a groupwise registration, to the subject's MR image and applying the same transformation to a CT atlas aligned to the MRI atlas. Izquierdo-Garcia et al. [START_REF] Izquierdo-Garcia | An SPM8-based approach for attenuation correction combining segmentation and nonrigid template formation: Application to simultaneous PET/MR brain imaging[END_REF] generated attenuation maps using the SPM8 software. The initial step of the method consisted of creating MRI and CT atlases. To do so, T1w MR images were first segmented into six tissue classes (grey matter, white matter, cerebrospinal fluid, bone, soft-tissue, and air) using SPM. The segmented images were then non-rigidly co-registered using Dartel [START_REF] Ashburner | A fast diffeomorphic image registration algorithm[END_REF] to form the MRI atlas. The same transformations were applied to the CT images, previously affinely aligned with the T1w images, and the CT atlas was created by averaging the co-registered CT images. To generate a pseudo CT, the MR image of the subject of interest is segmented into six tissue classes and non-rigidly registered to the MRI atlas. The associated CT atlas is finally mapped into the subject's space by applying the inverse transformation.

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Multi-atlas registration approaches

The methods described in the previous section rely on a single atlas and strongly depend on the accuracy of the mapping between the atlas and subject spaces and on the representativeness of the atlas. A possible solution to improve the representativeness of the atlas and limit the consequences of mis-registrations is to use multiple atlases.

The multi-atlas methods reported below rely on a database of CT and MRI pairs. Each pair is created by registering the CT and MR images of a subject. To generate a pseudo CT from the subject's MR image, the MR images from Biomedical Image Synthesis and Simulation, Chapter 4 the database are non-rigidly registered to the subject's MRI. The CTs in the database are then mapped to the target subject using the transformation that maps the subject's corresponding MRI in the atlas database to the subject's MRI. A final step consists in fusing the deformed CT images. The approach is illustrated in Figure 7.

To fuse the deformed CTs, Sjolund et al. [START_REF] Sjölund | Generating patient specific pseudo-CT of the head from MR using atlas-based regression[END_REF] proposed an iterative registration to the mean. The consistency of the deformed CTs was improved by iteratively forming their voxelwise mean, registering each deformed CT to the mean, and creating a new mean estimate. However, it appeared that better results were reached by directly taking the voxelwise median of the deformed CTs. Instead of computing the voxelwise median, Burgos et al. [START_REF] Burgos | Attenuation correction synthesis for hybrid PET-MR scanners: Application to brain studies[END_REF] obtained a pseudo CT by computing a weighted-average of the deformed CTs. The weights were derived from a local similarity measure computed between the MR image of the subject of interest and the MR images of the atlases. This image similarity measure reflects the morphological similarity, meaning that more weight is given to the atlases that are the most similar in terms of morphology to the subject of interest. This approach can generate pseudo CT images from a single [START_REF] Burgos | Attenuation correction synthesis for hybrid PET-MR scanners: Application to brain studies[END_REF] or multiple [START_REF] Burgos | Multi-contrast attenuation map synthesis for PET/MR scanners: Assessment on FDG and Florbetapir PET tracers[END_REF] MRI contrasts in the brain [START_REF] Burgos | Attenuation correction synthesis for hybrid PET-MR scanners: Application to brain studies[END_REF][START_REF] Burgos | Multi-contrast attenuation map synthesis for PET/MR scanners: Assessment on FDG and Florbetapir PET tracers[END_REF], head & neck [START_REF] Guerreiro | Evaluation of a multi-atlas CT synthesis approach for MRI-only radiotherapy treatment planning[END_REF][START_REF] Burgos | Robust CT synthesis for radiotherapy planning: Application to the head & neck region[END_REF] and pelvic [START_REF] Guerreiro | Evaluation of a multi-atlas CT synthesis approach for MRI-only radiotherapy treatment planning[END_REF] regions. See examples in Figure 8. A similar approach was implemented in [START_REF] Dowling | Automatic substitute computed tomography generation and contouring for magnetic resonance imaging (MRI)-alone external beam radiation therapy from standard MRI sequences[END_REF]. Merida et al. [START_REF] Mérida | Multi-atlas attenuation correction supports full quantification of static and dynamic brain PET data in PET-MR[END_REF] looked at the probability of a voxel in the subject space to belong to a certain tissue class. Three tissue classes (air, soft-tissue and bone) were defined by intensity thresholding the deformed CTs. The intensity value of the pseudo CT was determined by averaging for each voxel the CT intensities of the atlases belonging to the maximum probability class.

Combination of registration and regression approaches

To increase synthesis accuracy, several groups proposed to combine multi-atlas registration and regression approaches.

In the seminal work of Hofmann et al. [START_REF] Hofmann | MRI-based attenuation correction for PET/MRI: A novel approach combining pattern recognition and atlas registration[END_REF], local information derived from a supervised learning technique and global information obtained from multiatlas registration were combined to predict a pseudo CT image: multi-atlas registration was used as prior knowledge for a Gaussian process regression. The global step of the method consisted of non-rigidly registering the MR images from the training dataset to the subject's MR image and applying the same transformations to the corresponding CT images in the training dataset. The local step aimed at determining a mapping between MR image patches centred at a voxel of interest and the real CT HU values using Gaussian process regression. Training pairs comprising the position of the voxel in the subject coordinate system and the patch surrounding it were extracted from the deformed images in the training dataset. Prior knowledge from the atlas registration was included by setting the mean function of the Gaussian process to the average value of the registered CT images. For an unseen patch, Gaussian process regression returned a Gaussian-distributed predictive distribution for the mapping of the MR patches to the CT HU. Finally, to predict a pseudo CT image for a new subject, for each voxel of the MR image a surrounding patch was extracted and the mean of the Gaussian-distributed predictive distribution was defined as the estimated CT value. The method was also adapted to whole-body attenuation correction [START_REF] Hofmann | MRI-based attenuation correction for whole-body PET/MRI: Quantitative evaluation of segmentation-and atlas-based methods[END_REF].

A similar approach combining intensity and geometry information into a unifying probabilistic Bayesian framework was later proposed [START_REF] Gudur | A unifying probabilistic Bayesian approach to derive electron density from MRI for radiation therapy treatment planning[END_REF]. The method implemented in [START_REF] Chen | Probabilistic air segmentation and sparse regression estimated pseudo CT for PET/MR attenuation correction[END_REF] relies on a database of co-registered T1w MR images, CT images and air masks segmented from the CT images using kmeans clustering. Pseudo CT generation required three steps. First, each atlas was non-rigidly registered to the subject's MR image. A probabilistic air mask was then computed as the percentage of the aligned CT atlases that labelled a voxel as air, and refined using a hidden Markov random field segmentation on the T1w MR image. Each voxel labelled as air was assigned the mean value of the atlas CTs belonging to the air mask. For the rest of the tissues, small neighbourhoods centred around each voxel were defined in the target. Patches of the same size were defined in the atlases in the vicinity of the central voxel. Sparse regression was then used to select the most similar patches, in terms of intensity, among all the atlas patches within the search window. Finally, the sparse coefficients were applied to combine the atlas CTs and generate the pseudo CT. More details on the use of dictionary learning for image synthesis can be found in Chapter 5.

Hybrid approaches combining segmentation and registration

The methods described in this section combine both image segmentation and registration to generate synthetic images. Bezrukov et al. [START_REF] Bezrukov | MR-based attenuation correction methods for improved PET quantification in lesions within bone and susceptibility artifact regions[END_REF] proposed a method combining segmentation, multiatlas registration and supervised learning techniques, which relied on the subject's Dixon images and on an atlas database. Each atlas comprised a 5-tissue class segmentation (outer air, lung, fat tissue, non-fat tissue, and fat/non-fat tissue mix), a map of potential artefact locations extracted from regions with low MR intensity and differentiated into air-filled regions not part of the lungs and soft-tissue regions, a CT image, and a bone location map. After segmentation of the subject's MR image into five tissue classes, the segmented MR images from the atlas database were registered to the subject's image and the resulting transformations were applied to the images forming the atlases. A subject-specific map of potential artefact locations was created by averaging the atlas artefact maps. This map was used to classify the low MRI intensity regions as inside air or soft-tissue masked by artefacts. Predefined linear attenuation coefficients were assigned to the segmented regions of the MR image to create an intermediate attenuation map. For the prediction of bone tissue, a map of potential bone locations was created from the atlas CT images by combining the individual bone location maps. This map determined the areas where Gaussian process regression was applied. For each voxel within these areas, a pseudo CT value was computed from the CT atlases and converted to linear attenuation coefficients using a piecewise linear transformation. The final attenuation map was created by augmenting the intermediate attenuation map with the predicted bone linear attenuation coefficients.

Combining the segmentation of an MR image and the use of a CT image database, Marshall et al. [START_REF] Marshall | Description and assessment of a registration-based approach to include bones for attenuation correction of whole-body PET/MRI[END_REF] proposed a method where the subject's MR image was compared to an atlas database of CT images using a set of 19 similarity metrics such as gender, age, and body, lung and bone geometries. The most similar CT image was selected and non-rigidly aligned to the subject's MR image. Bones from the registered CT image were then added to the MR image previously segmented into four tissue classes (air, lung, fat and lean tissue).

Exploiting multiple MRI sequences, Anazodo et al. [START_REF] Anazodo | Feasibility of simultaneous Biomedical Image Synthesis and Simulation, Chapter 4 whole-brain imaging on an integrated PET-MRI system using an enhanced 2point Dixon attenuation correction method[END_REF] proposed to overlay a bone mask to the attenuation map obtained from the segmentation of Dixon images [START_REF] Martinez-Möller | Tissue classification as a Biomedical Image Synthesis and Simulation, Chapter 4 potential approach for attenuation correction in whole-body PET/MRI: Evaluation with PET/CT data[END_REF]. The bone mask was created by segmenting a T1w magnetisationprepared rapid gradient-echo image using the New Segment function from SPM8 [START_REF] Ashburner | Unified segmentation[END_REF]. SPM has also been used in [START_REF] Teuho | Tissue probability-based attenuation correction for brain PET/MR by using SPM8[END_REF].

The method implemented by Paulus et al. [START_REF] Paulus | Whole-body PET/MR imaging: Quantitative evaluation of a novel model-based MR attenuation correction method including bone[END_REF], illustrated in Figure 9, was based on a regular 4-class segmentation from a Dixon sequence and atlas-based bone segmentation algorithm. The atlas was composed of pairs of pre-aligned MR image and bone mask (containing bone densities) for each major bone in the body, including left and right upper femur, left and right hip, spine, and skull. To generate a subject-specific attenuation map, the subject's Dixon images were first segmented into four classes (air, lung, soft-tissue and fat). The bone class was then added by registering the MR model to the subject for each bone individually and applying the same transformation to the bone mask. The method was then validated in the brain region [START_REF] Koesters | Dixon sequence with superimposed model-based bone compartment provides highly accurate PET/MR attenuation correction of the brain[END_REF].

In their work, Arabi et al. [START_REF] Arabi | Atlas-guided generation of pseudo-ct images for mri-only and hybrid pet-mri-guided radiotherapy treatment planning[END_REF] combined multi-atlas segmentation and synthesis with the aim to improve the synthesis of CT images from MRI in the bone region. Their framework consisted of two steps. In the first one, bone was segmented using multi-atlas segmentation. In the second one, a pseudo CT was generated using a multi-atlas approach, the atlases being fused based on the resemblance of both the bone map obtained at the first step and the morphological similarity to the MR image of the subject being analysed.

Biomedical Image Synthesis and Simulation, Chapter 4

Future directions and research challenges

In this chapter, we covered image synthesis approaches that rely on wellestablished image processing techniques: segmentation and registration. The vast majority of these approaches have been applied to the synthesis of CT from MR images. This can be explained by two reasons. First, these image synthesis approaches were developed to answer particular needs: that of accurate electron density estimation when only MR images are available, which occurs in the context of both PET/MR attenuation correction (see Chapter 19) and MRI-only RTP (see Chapter 20). The second reason is that their accuracy appears limited on other tasks, which is why they have been supplanted by better performing methods.

The development of MR to CT image synthesis approaches followed the development of new technologies. Regarding attenuation correction, even though first attempts of synthesising attenuation maps date back to the 1990s [START_REF] Huang | Three-dimensional PET emission scan registration and transmission scan synthesis[END_REF], many works have accompanied the release of PET/MR scanners (2010 for Philips and Siemens Healthineers, 2011 for GE Healthcare). Prototypes of MR-Linac date back to the same period [START_REF] Lagendijk | MRI/linac integration[END_REF]. These technological developments led to new problems, namely the fact that MR image intensities do not directly provide information about the tissue attenuation properties, to which image synthesis was proposed as a solution.

Segmentation-based methods essentially rely on the subject's MR images. This explains why specific sequences such as the Dixon, UTE or ZTE, often only acquired for the purpose of image synthesis, have been employed. Various segmentation strategies have been implemented: manual segmentation, mainly in the context of RTP and/or as proof of concept, and a large range of automatic methods that include mainly intensity thresholding and fuzzy c-means. Initially, once segmented, predefined values were often assigned to each tissue class, but subject-specific assignments have shown their superiority, especially in bone regions. A limitation of segmentation methods is the difficulty in delineating tissues in regions with air/tissue and air/bone interfaces. Another one is their susceptibility to MR image artefacts.

Registration-based methods rely on both the subject's MR images and on an atlas or database of atlases. The first methods developed relied on a single atlas to generate a subject-specific attenuation map. Their accuracy was limited as a single template can hardly be representative of all the potential patients, even if they do not present morphological abnormalities. These methods also strongly depend on the accuracy of the registration used to map the atlas to the target subject. A solution to both improve the representativeness of the atlas and reduce the impact of registration inaccuracies has been to use a database of atlases. However, this comes with the price of having to run multiple, time-consuming, non-rigid registrations. Another limitation is the limited ability of these methods to handle anatomical abnormalities (e.g. unusual bone density).

The second reason that can explain the use of segmentation and registrationbased approaches to only MR to CT image synthesis is that their accuracy appears limited on other tasks. Jog et al. [START_REF] Jog | Random forest regression for magnetic resonance image synthesis[END_REF] for example compared a multiatlas registration approach with their random forest regression method for the synthesis of T2w from T1w MR images and of FLAIR from T1-, T2-, and proton density-weighted MR images, and observed that the multi-atlas registration approach led to blurrier images from which lesions were barely visible. Multi-atlas registration has nonetheless successfully been used to generate pseudo-healthy PET images from T1w MR images [START_REF] Burgos | Subject-specific models for the analysis of pathological FDG PET data[END_REF][START_REF] Burgos | Anomaly detection for the individual analysis of brain PET images[END_REF]. Many techniques have now supplanted segmentation and registration-based approaches, starting with patch-based methods that do not require non-rigid alignment between the training and target subjects (more in Chapter 5) until the nowadays highly predominant deep learning approaches (more in Chapters 6, 7 and 8).

Interested readers can find further information in several reviews and comparison articles. In the context of RTP, approaches that exist to generate synthetic CT images have been reviewed for example in [START_REF] Edmund | A review of substitute CT generation for MRIonly radiation therapy[END_REF][START_REF] Johnstone | Systematic review of synthetic computed tomography generation methodologies for use in magnetic resonance imaging-only radiation therapy[END_REF][START_REF] Dowling | MR-only methodology[END_REF] while their performance has been compared in [START_REF] Arabi | Comparative study of algorithms for synthetic CT generation from MRI: consequences for MRI-guided radiation planning in the pelvic region[END_REF][START_REF] Largent | Pseudo-CT generation for MRI-only radiation therapy treatment planning: comparison among patch-based, atlasbased, and bulk density methods[END_REF]. Several reviews have been published over the years regarding image synthesis for attenuation correction [START_REF] Hofmann | Towards quantitative PET/MRI: A review of MR-based attenuation correction techniques[END_REF][START_REF] Bezrukov | MRbased PET attenuation correction for PET/MR imaging[END_REF][START_REF] Keereman | Challenges and current methods for attenuation correction in PET/MR[END_REF][START_REF] Wagenknecht | MRI for attenuation correction in PET: Methods and challenges[END_REF][START_REF] Izquierdo-Garcia | MR imaging-guided attenuation correction of PET data in PET/MR imaging[END_REF][START_REF] Mehranian | Vision 20/20: magnetic resonance imaging-guided attenuation correction in PET/MRI: Challenges, solutions, and opportunities[END_REF], and Ladefoged et al. [START_REF] Ladefoged | A multi-centre evaluation of eleven clinically feasible brain pet/mri attenuation correction techniques using a large cohort of patients[END_REF] published a multi-centre study aiming at evaluating the performance of various attenuation correction methods, including five segmentation-based [START_REF] Martinez-Möller | Tissue classification as a Biomedical Image Synthesis and Simulation, Chapter 4 potential approach for attenuation correction in whole-body PET/MRI: Evaluation with PET/CT data[END_REF][START_REF] Catana | Toward implementing an MRI-based PET attenuation-correction method for neurologic studies on the MR-PET brain prototype[END_REF][START_REF] Cabello | MR-Based attenuation correction using ultrashort-echo-time pulse sequences in dementia patients[END_REF][START_REF] Ladefoged | Region specific optimization of continuous linear attenuation coefficients based on UTE (RESOLUTE): Application to PET/MR brain imaging[END_REF][START_REF] Juttukonda | MR-based attenuation correction for PET/MRI neurological studies with continuous-valued attenuation coefficients for bone through a conversion from R2* to CT-Hounsfield units[END_REF], three atlasbased [START_REF] Izquierdo-Garcia | An SPM8-based approach for attenuation correction combining segmentation and nonrigid template formation: Application to simultaneous PET/MR brain imaging[END_REF][START_REF] Burgos | Attenuation correction synthesis for hybrid PET-MR scanners: Application to brain studies[END_REF][START_REF] Mérida | Multi-atlas attenuation correction supports full quantification of static and dynamic brain PET data in PET-MR[END_REF] and two hybrid approaches combining segmentation and use of an atlas [START_REF] Anazodo | Feasibility of simultaneous Biomedical Image Synthesis and Simulation, Chapter 4 whole-brain imaging on an integrated PET-MRI system using an enhanced 2point Dixon attenuation correction method[END_REF][START_REF] Koesters | Dixon sequence with superimposed model-based bone compartment provides highly accurate PET/MR attenuation correction of the brain[END_REF].

Summary

Even though medical image synthesis can be used to improve subsequent image processing techniques such as segmentation or registration, in this chapter we saw how segmentation and registration could themselves be used for image synthesis. Segmentation-based approaches consist in segmenting the source image, either manually or automatically, and assigning intensity values to the different tissue classes obtained, using bulk or subject-specific assignment, to generate the desired image. Registration-based methods consist in deforming an atlas, composed of a single image, a pair of images or multiple pairs, to match the subject's anatomy using non-rigid registration. Most segmentationand registration-based image synthesis approaches have been used to synthesise CT from MR images in the context of PET(/MR) attenuation correction and MRI-only radiotherapy treatment planning. Even though these techniques helped answer concrete needs, they have now been supplanted by deep learning methods.

Figure 1 :

 1 Figure 1: Segmentation-based approaches can be decomposed into two parts: the first one consists in segmenting, manually or automatically, the source image (here an MR image) and the second one consists in assigning intensity values to the different tissue classes obtained to generate the target image (here a CT image in Hounsfield unit [HU]). Intensities can be assigned in bulk (i.e. predefined values assigned to each tissue class) or in a subject-specific manner.

Figure 2 :

 2 Figure 2: Example of a manually segmented MR image (left) and the resulting synthetic CT with assigned mass densities (right) [4]. Jonsson et al. 2010 Radiat. Oncol. 5 62 doi:10.1186/1748-717X-5-62 © 2010, Jonsson et al; licensee BioMed Central Ltd.

Figure 3 :

 3 Figure 3: Automatic segmentation of dual ultra-short echo time (UTE) images using a thresholding approach [11]. The uncorrected R2 map is calculated from the UTE images acquired at echo times TE 1 and TE 2 . The R2 map is masked with an air mask derived from the first-echo image to generate the corrected R2 map. The corrected R2 map can be segmented into bone, soft-tissue, and air. Keereman et al. 2010 J. Nucl. Med. 51(5) 812 doi:10.2967/jnumed.109.065425

Figure 4 :

 4 Figure 4: Illustration of how MR intensity values can be converted into CT Hounsfield units (HU) for subject-specific intensity assignment [33]. R2* versus HU scatter plot with sampled points are shown for each of the ten patients in grey. Every third median bin-value is shown with black squares. Standard deviation to the mean value is shown with black error bars. A 3 rd order polynomial fit to the median points is shown with the blue line (goodness of fit r 2 = 0.93). Ladefoged et al. 2015 Phys. Med. Biol. 60 8047 doi:10.1088/0031-9155/60/20/8047 © Institute of Physics and Engineering in Medicine. Reproduced by permission of IOP Publishing. All rights reserved.

Figure 5 :

 5 Figure 5: Three examples of CT images and ZTE-based pseudo-CTs generated using the segmentation algorithm and the linear relationship between normalized ZTE intensity and CT density in Hounsfield unit (HU) described in [38]. Khalifé et al. 2017 Phys. Med. Biol. 62 7814 doi:10.1088/0031-9155/60/20/8047 © Institute of Physics and Engineering in Medicine. Reproduced by permission of IOP Publishing. All rights reserved.

Figure 6 :

 6 Figure 6: Registration-based approaches for image synthesis: direct multimodal (A) and indirect unimodal (B) single-atlas registration, and (C) multi-atlas registration. The atlas is composed of (A) a single CT image, (B) a pair of MRI and CT images, and (C) multiple pairs of MRI and CT images.

  %&'() *+, *+,-./01'21)031'1#1)% 4%5678%3 *+,0193 ./ 1'21)%) !"#$"%&#' ()%&*" !"#$%&'()0 :)%"36 ./ !"#$+,%)-.$/#0+$1&%#+2#&*"030 4*526&#'07+*7212#&*"

Figure 7 :

 7 Figure 7: Multi-atlas registration approach to generate a pseudo CT from an MR image. All the MR images in the atlas database are registered to the subject's MR image. The CTs in the atlas database are then mapped using the same transformation to the subject's MR image. The subject's pseudo CT image is obtained by fusing the deformed atlas CT images.

Figure 8 :

 8 Figure 8: Examples of pseudo CT images obtained from MR images with a multiatlas registration approach [6, 49, 51] in the brain (top), head & neck (bottom left) and pelvis (bottom right).

Figure 9 :

 9 Figure 9: Schematic drawing of the hybrid segmentation/registration-based approach for considering bone in whole-body PET/MR attenuation correction [63]. The subject-specific attenuation map (µ-map) is generated by first segmenting the subject's Dixon images into four classes (air, lung, soft-tissue and fat). The bone class is then added by registering the MR model to the subject for each bone individually and applying the same transformation to the bone mask. Paulus et al. 2015 J. Nucl. Med. 56 1061 doi:10.2967/jnumed.115.156000

Biomedical Image Synthesis and Simulation, Chapter 4

http://www.fil.ion.ucl.ac.uk Biomedical Image Synthesis and Simulation, Chapter 4

Acknowledgments

The research leading to these results has received funding from the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, ref. ANR-19-P3IA-0001 (PRAIRIE 3IA Institute) and ANR-10-IAIHU-0006 (Agence Nationale de la Recherche -10-IA Institut Hospitalo-Universitaire-6).


Dr Hayder F N Al-Shuka
email: hayder.al-shuka@rwth-aachen.de

Ehab N Abbas

Keywords: Nonlinear vibrations, feedback linearization, passivity-based control, backstepping control, adaptive approximation control

Function approximation technique (FAT)-based nonlinear control strategies for smart thin plates with cubic nonlinearities

The current work introduces three nonlinear control solutions for the regulation of a vibrating nonlinear plate considering model uncertainty. These solutions are feedback linearization control (FBL), virtual velocity error-based control (VVEC), and backstepping control (BSC). In the FBL control, a nonlinear control law is designed with linear closed-loop dynamics such that dynamic stability is ensured. Whereas, by the VVEC (or so-called passivity-based approach in the robotics community) the limitations of the feedback linearization are overcome. On the other side, the BSC selects virtual control variables with stabilized intermediate control laws based on Lyapunov theory. Systematic modelling for the target vibrating plate with piezo-patches is described. In effect, considering the nonlinear influence makes the resulted mode shapes for the vibrating structure are highly coupled and careful control design is required. Using the Galerkin approach, the partial differential equation for the smart plate is transformed into definite ordinary differential equations; the multi-input multi-output model is established. The aforementioned control strategies are evaluated and investigated in detail. In essence, they are powerful tools for dealing with nonlinear dynamic systems, however, the VVEC could be considered superior in comparison with the FBL control and the BSC since the designed control structure does not include inertia inverse matrix and modal coordinate acceleration that could make computational problems. As a result, simulation experiments were focused on the VVEC strategy and the latter was implemented on a simply supported thin plate structure with collocated piezo-patches. The results show the validity of the proposed control architecture.

INTRODUCTION

Piezoelectric materials can transform electrical energy into mechanical energy or vice versa. Much attention is devoted to using them in vibration suppression of flexible structures such as strings, beams, plates, shells, etc. [START_REF] Moheimani | Spatial control of vibration: theory and experiments[END_REF]. Integration of smart materials, e.g. piezoelectric patches, with flexible components (structures) makes them adaptively accommodate external disturbances and attenuate vibrational motions if exist. A suitable control law is always required to get perfect performance. Consequently, this paper is concerned with the function approximation technique (FAT)-based adaptive control approaches for vibration suppression of a piezo-electrically actuated/sensed thin plate considering nonlinear effects. For modelling of transverse vibration of a plate structure, the following points should be noticed [START_REF] Vepa | Dynamics of smart structures[END_REF]. [START_REF] Moheimani | Spatial control of vibration: theory and experiments[END_REF] In a thick plate structure, both the shear deformation and bending deflection are considered. [START_REF] Vepa | Dynamics of smart structures[END_REF] In a membrane structure, the thickness is thinner than a thin plate with a negligible bending stiffness while the bending loadings are resisted by the membrane action. (3) The bending deflection is considered for a thin plate considering the effect of the axial stretching (loading) that complicates the dynamics and control tasks. The current work is interested in modelling and control of this category of plates.

The key cause of structural nonlinearity comes from in-plane stretching of the flexible structure, however, the nonlinear behaviour for thin structures could originate from nonlinear geometry, inelastic behaviour of the structure, the nonlinear applied excitation loadings, etc. These structural phenomena cannot be sufficiently investigated by linear theory since the deflection of the target structure is large in comparison with the thickness. Moreover, considering the effect of such a large deflection on the response of the flexible structure is essential [START_REF] Wagg | Control and exploitation of nonlinearity in smart structures[END_REF][START_REF] Teng | Nonlinear forced vibration analysis of the rectangular plates by the Fourier series method[END_REF]. In effect, considering the nonlinear influence makes the resulted mode shapes for the vibrating structure is highly coupled and careful control design is required. Despite there being different types of nonlinearity sources, see e.g. [START_REF] Bajović | A New General Approach to Airplane Rotation[END_REF][START_REF] Ilic | Impact of changing quality of air/fuel mixture during flight of a piston engine aircraft with respect to vibration low frequency spectrum[END_REF], this study is focused on the nonlinearity resulting from axial stretching of plates. Due to the accompanying complexity of the nonlinear equation of motion for the vibrating plates, a little work reported in the literature has been focused on the analytical derivation of the nonlinear dynamics for the plates and the finite element method FEM is the potential tool for modelling and analysis. . Plates and plate-like structures play important roles in miscellaneous application such as floor and foundation slabs, lock-gates, offshore structures, the wing structure of an aircraft, and aircraft fuselage, see e.g. [START_REF] Szilard | Theories and applications of plate analysis: classical, numerical and engineering methods[END_REF][START_REF] Garinis | Dynamic analysis of modified composite helicopter blade[END_REF][START_REF] Dinulović | Numerical modeling of nomex honeycomb core composite plates at meso scale level[END_REF]. The reader is referred to [START_REF] Dowell | Aeroelasticity of plates and shells[END_REF][START_REF] Amabili | Nonlinear Vibrations and Stability of Shells and Plates[END_REF][START_REF] Awrejcewicz | Spatio-temporal chaos and solitons exhibited by von Kármán model[END_REF][START_REF] Gordnier | Development of a three-dimensional viscous aeroelastic solver for nonlinear panel flutter[END_REF][START_REF] Sathyamoorthy | Nonlinear vibrations of plates: an update of recent research developments[END_REF] for more details on applications and modelling of plate-like structures. To model a thin plate structure with piezopatches, a partial differential equation PDF is obtained based on the Newton-Euler formulation. It is an infinitedimensional system that has infinite degrees-of freedom DOFs. This complicates the control problem since it may theoretically require several distributed piezoactuators to fully actuate and regulate the vibrating plate. However, the first mode shapes are considered since they are dominant in the low-frequency region such that high-frequency amplitudes would be neglected. Galerkin approach is used to transform the infinite-dimensional PDF of the target plate into finite N-ordinary differential equations ODEs. The resulting nonlinear ODEs for the vibration plate are represented as a multi-input multi-output dynamic system. Cubic nonlinear stiffness terms are resulted due to the coupling of the bending-tension effect.

In effect, two essential strategies that are possible to damp vibration of structures are passive vibration damping [START_REF] Ilic | The efficiency of passive vibration damping on the pilot seat of piston propeller aircraft[END_REF][START_REF] Ilic | Potential connections of cockpit floor -seat on passive vibration reduction at a piston propelled airplane[END_REF] and active vibration damping. This study is focused on adaptive vibration damping of platetype structures considering disturbance and complex nonlinearities. To suppress the nonlinear vibrations of adaptive plate-like structures, piezo-patches are bonded on the surface of the plates that can generate counteract moments and sense the resulted deflection for stabilization of the target vibrating plates. As a result, the objective of the controller is to regulate the vibrational motion of the target system under parameter uncertainty and disturbances, see e.g., [START_REF] Lee | Active vibration control of a piezoelectric laminate plate using spatial control approach[END_REF][START_REF] Gupta | Active vibration control of a smart plate using a piezoelectric sensor-actuator pair at elevated temperatures[END_REF][START_REF] Yaman | Active Vibration Control of a Smart Beam[END_REF][START_REF] Dimitriadis | Piezoelectric actuators for distributed vibration excitation of thin plates[END_REF][START_REF] Abdullah | Active control of strain in a composite plate using shape memory alloy actuators[END_REF][START_REF] Tairidis | Fine tuning of a fuzzy controller for vibration suppression of smart plates using genetic algorithms[END_REF][START_REF] Vinyas | Vibration control of skew magnetoelectro-elastic plates using active constrained layer damping[END_REF]. However, if the dynamic system undergoes uncertain complex nonlinear terms, the linear control architectures e.g. PID family controllers are no longer useful and hence the best solution tools are nonlinear control strategies. In effect, there are two substantial techniques for dealing with the abovementioned problem: adaptive control and sliding mode control [START_REF] Farrell | Adaptive approximation based control: unifying neural, fuzzy and traditional adaptive approximation approaches[END_REF]. Adaptive control is a powerful tool to control dynamic systems with unknown constant parameters [START_REF] Al-Shuka | On local approximation-based adaptive control with applications to robotic manipulators and biped robots[END_REF]. The Lyapunov theory is used as a basis for guaranteeing the stability of the proposed control law and the corresponding updating laws associated with unknown parameters. For unknown time-varying parameters, adaptive approximation control is used instead to comprise model-free techniques or so-called intelligent control [START_REF] Al-Shuka | On local approximation-based adaptive control with applications to robotic manipulators and biped robots[END_REF][START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF][START_REF] Al-Shuka | Regressor-free adaptive vibration control of constrained smart beams with axial stretching[END_REF][START_REF] Al-Shuka | FAT-Based Adaptive Backstepping Control of an Electromechanical System with an Unknown Input Coefficient[END_REF][START_REF] Al-Shuka | Proxy-Based Sliding Mode Vibration Control with an Adaptive Approximation Compensator for Euler-Bernoulli Smart Beams[END_REF][START_REF] Al-Shuka | Function approximation technique-based adaptive virtual decomposition control for a serial-chain manipulator[END_REF][START_REF] Al-Shuka | Hybrid Regressor and Approximation-Based Adaptive Control of Robotic Manipulators with Contact-Free Motion[END_REF][START_REF] Kaleel | Adaptive Approximation-Based Feedback Linearization Control for a Nonlinear Smart Thin Plate[END_REF][START_REF] Al-Hamadani | Proportional-derivative pd vibration control with adaptive approximation compensator for a nonlinear smart thin beam interacting with fluid[END_REF][START_REF] Al-Shuka | Decentralized adaptive partitioned approximation control of high degreesof-freedom robotic manipulators considering three actuator control modes[END_REF]. On the other hand, robust control requires knowledge of limits of uncertainty. One of the wellknown robust techniques used is the sliding mode control SMC that uses a signum function associated with a suitable sliding surface. However, a signumbased term can be easily integrated with adaptive control algorithms for compensating modelling error if exists.

The current work introduces three nonlinear control solutions for the regulation of the vibrating nonlinear plate considering the model uncertainty. These solutions are feedback linearization control FBL, virtual velocity error-based control VVEC, and backstepping control BSC. These control algorithms are strong tools to control complex dynamic systems such as the miniature robot described in [START_REF] Stevanović | Development of a Miniature Robot Based on Experience Inspired by Nature[END_REF]. In FBL control, a nonlinear control law is designed with linear closed-loop dynamics such that dynamic stability is ensured [START_REF] Wagg | Nonlinear vibration with control, for flexible and adaptive structures[END_REF][START_REF] Slotine | Applied nonlinear control[END_REF]. Whereas, by the VVEC (or so-called passivity-based approach in the robotics community) the limitations of the feedback linearization are resolved [START_REF] Al-Shuka | On local approximation-based adaptive control with applications to robotic manipulators and biped robots[END_REF][START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF]. On the other side, the BSC assumes that virtual control variables with stabilized intermediate control laws based on Lyapunov theory [START_REF] Farrell | Adaptive approximation based control: unifying neural, fuzzy and traditional adaptive approximation approaches[END_REF]. The above-mentioned control algorithms are reformulated based on FAT in connection with a robust sliding term for compensating modelling/approximation errors if exist. Systematic modelling for the target vibrating plate with piezopatches is described. In effect, considering the nonlinear influence makes the resulted mode shapes for the vibrating structure are highly coupled and careful control design is required. Using the Galerkin approach, the partial differential equation for the smart plate is transformed into definite ordinary differential equations; the multi-input multi-output model is established. The aforementioned control strategies are evaluated and investigated in detail. In essence, they are powerful tools for dealing with nonlinear dynamic systems, however, the VVEC could be considered superior in comparison with the FBL control and the BSC since the designed control structure does not include inertia inverse matrix and modal coordinate acceleration that could make computational problems. As a result, simulation experiments were focused on the VVEC strategy and the latter was implemented on a simply supported thin plate structure with collocated piezopatches. The results show the validity of the proposed control architecture. The contributions of the current paper are:

1. Considering the axial stretching in the plate model that complicates the dynamic behaviour and control task since coupled cubic nonlinearity terms occur. To our knowledge, this point has been a little considered in previous work. 2. Introducing three powerful nonlinear control strategies and describing features and shortcomings of every technique in detail.

3. Regressor-free techniques have been presented as a basis for control laws and the associated closed-loop dynamics. The remainder of the paper is organized as follows. Section 2 introduces the dynamic modelling of the smart thin plate while the nonlinear control structures are described in section 3. Section 4 introduces simulation results and discussions. Section 5 concludes.

Remark 1.

In effect, we avoid using the terminology '' passivity-based control'' for the VVEC since the passivity property may not hold for the current modelling of the vibrating plate. The idea of passivitybased control is to exploit the skew-symmetric matrix property such that (

C M 2  
, where M is the inertia matrix and C is the damping matrix). Fortunately, the target plate has an inertia matrix with constant-value elements while the elements of the damping matrix are assumed to be positive ( ij c ) and these ease the proof of the stability for the VVEC away from the passivity concept. In effect, the damping term will be integrated into the complex stiffness terms and hence it will not make an obstacle for proving the dynamic stability for the investigated system.

DYNAMIC MODELLING OF THE SMART PLATE

This section is focused on modelling a vibrating thin plate with surface bonded piezo-patches. The effect of the piezo-patches can be added as moment loadings. To this end, consider a thin plate with m collocated piezopatches, see Fig. 1. The following assumptions are considered [START_REF] Moheimani | Spatial control of vibration: theory and experiments[END_REF]:

1. The plate is thin with a uniform thickness.

2. The dynamics of the piezo-patches are neglected.

3. The plate undergoes large deflection that imposes axial stretching on it.

                            (1a) 2 2 2 2 2 2 4 y w x w y x w Eh F                   (1b) with ) 1 ( 12 , 2 2 3 4 4 2 2 4 4 4 4 v Eh D y y x x             
where F is the Airy stress function, m is the mass of the plate, 4  is the biharmonic operator, h refers to the plate thickness, p is the external transverse load per unit area, (.) p M is the piezo-actuator external moment per unit length, D is the flexural rigidity, E is the Young's modulus, and v is the Poisson's ratio. Now it is necessary to transform the resulted PDFs of (1) into ODEs using the Galerkin technique, hence expanding w and F in terms of mode shape function and modal amplitudes

      1 1 ) ( ) , ( ) , , ( m n mn mn t q y x t y x w  (2a)       1
      mn s r n m rsmnij mn rs ij mnij mn mn mnij L q a p K q q M , , ,   dxdy y x y M x M ij a b py px ) , ( 0 0 2 2 2 2                (3a)    s r l k n m mnkltu mn kl rstu rs B q q A a , , , , (3b)
                          2 2 2 2 2 2 dxdy y x y x y x x y L ij mn rs mn rs mn rs rsmnij                                        2 2 2 2 2 2 2 2 2 2 2
By substituting (3b) into (3a), complex cubic nonlinear terms appear

dxdy y x y M x M p q q q q k q m ij a b py px ij ij ij ij ij ij ) , ( ) , , ( 0 0 2 2 2 2 (.) (.) (.) (.) (.)                         (4)
As we see, (4) includes a cross-coupling term represented by (.)

 and this complicates the controller task; hence advanced nonlinear control techniques are required to regulate the vibration motion of the plate. Now, let us expand the right-hand side of (4) related to piezo moments. Assuming equal piezoelectric charge constants in x and y directions, then

   ) ( ) ( ) ( ) ( 2 1 1 t v y y H y y H x x H M M a py px        (5)
where  is a constant that depends on the physical parameters of the regular plate and the piezo-actuators, see [START_REF] Moheimani | Spatial control of vibration: theory and experiments[END_REF] for more details.

(.) H is a Heaviside unit step function and ) (t v a is the piezo-actuator voltage. Taking the second derivatives for px M and py M and summing the result with further simplifications, we can get

) ( ) , ( 1 2 2 2 2 t v y x y M x M ak N k k py px a          (6) with     ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , ( 2 1 2 1 2 1 2 1 k k k k k k k k k k x x H x x H y y y y y y y y H y y H x x x x x x y x                                             
and a N refers to the number of piezo-actuators used that is equal to the number of piezo-sensors. So the the right hand side of (4) can be reformulated as

) ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( 1 0 0 0 0 1 0 0 2 2 2 2 t v dydx y x y x dydx y x t v y x dydx y x y M x M ak N k ij a b k ij a b ak N k k ij a b py px a a                                     (7)
Using the following Dirac-Delta function property

0 )) ( ( ) 1 ( ) ( ) ( 0 x x n n n n n dx x d dx x dx x x d            Then (7) becomes              a a N k ak ijk k ak N k ij a b k t v t v dydx y x y x 1 1 0 0 ) ( ) ( ) , ( ) , (     (8) where                                    
     Equation (4) can be reformulated as . ,..., 3 , 2 , 1 ,..., 3 , 2 , 1 , M j N i u d q k q m ij ij ij ij ij ij ij          (9) where       a N k ak ijk k ij a b ij ij t v u dydx y x y x t p d 1 0 0 ) ( , ) , ( ) , , (    Writing (9) in matrix form to obtain u d σ Kq q M       (10)
                                                       NM NM NM NM NM u u u d d d σ k k K m m M               
At this stage, it is suitable to consider the damping effect and a viscous damping term is used

u d Kq q C q M          (11a) with            NM c c C      0 0 11
For control purpose, (11a) can be reformulated as

d Kq q C t q q u t q q q M               ) , , ( , ) , , ( (11b)
Remark 2. In effect, the damping term can help suppress the model vibration; however, it should have positive values for each element of the damping matrix ( C ). Therefore, to avoid this constraint, it is integrated within the nonlinear term ) , , ( t q q   such that the damping matrix does not appear while proving the dynamic stability of the investigated plate. Remark 3. For detailed derivation for dynamics of a simply supported plate considering axial stretching due to large deflection, the reader is referred to [START_REF] Wagg | Control and exploitation of nonlinearity in smart structures[END_REF][START_REF] Teng | Nonlinear forced vibration analysis of the rectangular plates by the Fourier series method[END_REF].

Remark 4.

For simplicity, we will refer to the dimension of the dynamic matrices in terms of the number of the mode shapes ) (l . Therefore, the dimension of the dynamic matrices can be represented as . , , , 0 0 , 0

0 1 1 1 1 1                                                        l l l l l u u u d d d σ k k K m m M               

CONTROL STRUCTURE

The proposed controller aims at suppressing the vibrational motion of the target smart plate. The input control is provided by the piezo-actuator while the beam deflection (or equivalently modal amplitude) is sensed indirectly via the sensor voltages. This section introduces three powerful nonlinear control strategies considering the model uncertainty and the nonlinear effect due to large deflection. These strategies are: 1) feedback linearization FBL, 2) virtual velocity errorbased control VVEC, and 3) backstepping control BSC.

The key idea of FBL is to cancel the nonlinearity embedded in the dynamic system by selecting a nonlinear input control such that we obtain closed-loop dynamics. However, the problem associated with FBL is the presence of an inertia inverse matrix in the control structure that could complicate the solution if an adaptive control algorithm is used. Therefore, VVEC is used alternatively to overcome these drawbacks by introducing filtered errors. The last approach includes applying intermediate control law in a recursive style until we obtain the actual control law and this scheme is called the BSC.

FAT-based FBL

As aforementioned, the FBL strategy includes designing a nonlinear control law such that the closed-loop dynamics is linear, and hence the intuitive controller can be expressed as [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF]  

) sgn( ) ( ) ( ˆx P B κ η q q K q q K q M u T T d p d d d            (12a)
where the symbol )

.

( refers to the estimation, l d R q  is the desired value for the modal vector,

l l d R K   and l l p R K  
are both diagonal positive definite matrices,

l l R κ   is a robust sliding gain, l l l R I 0 B          2 ,   l T T T R e e x 2    , l l T R P P 2 2   
is a symmetric positive definite matrix satisfying the Lyapunov equation

    PA P A T (12b) with l l d p l R K K I 0 A 2 2            and l l T R 2 2     
is also a symmetric positive definite matrix, and

d q q e  
. Substituting (12a) into [START_REF] Amabili | Nonlinear Vibrations and Stability of Shells and Plates[END_REF] results in the following closed-loop dynamics

d 1 T T p d q q e ε η q M ( M x P B κ e K e K e           , ) ~) sgn(      (13) where
) . ( (.) ) . (   and l R ε 
represents the modelling/approximation error. Equation ( 13) is basically a linear closed-loop dynamics (if it is without the robust sliding term), however, due to the presence of the robust term ) sgn(

x P B κ T T the system is no longer linear. Using the FAT, the inertia and nonlinear matrices/vectors can be represented as

M M T M ε W M    (14a)     ε W η T   (14b)
where

l l M R W    , and l l R W     are weighting matrices, l l M R     and    l R 
are matrices of basis function, with  referring to the number of basis function terms. Using the same set of basis functions, the corresponding estimates can be represented as

M T M W M   (15a)    T W η  (15b)
Thus, the controller (12a) becomes

  ) sgn( ) ( ) ( ˆx P B κ W q q K q q K q W u T T T d p d d d M T M                (16)
and the closed-loop dynamics (13) become

ε W q W ( M x P B κ e K e K e T M T M 1 T T p d         ) ~) sgn(          (17)
Representing ( 17) in a state space form

    ) sgn( 1 x P B κ ε W q W M B Ax x T T T M T M              (18)
Let us select the following update laws

) ( ˆ1    M PB x q Q W T M M M     (19a) ) ( ˆ1    M PB x Q W T      (19b)
where

  l l R Q   (.)
is a positive definite adaptation matrix.

Theorem 1. The dynamics of the vibrating plate modelled in [START_REF] Amabili | Nonlinear Vibrations and Stability of Shells and Plates[END_REF] with the control law, closed-loop dynamics and the associated updating laws described in ( 16)- [START_REF] Yaman | Active Vibration Control of a Smart Beam[END_REF] are stable in the sense of the Lyapunov theory.

Proof.

Let us select the following Lyapunov-like function along the closed-loop dynamics (18)

     W Q W W Q W tr Px x V T M M T M T 2 1 2 1 1 1      (20)
Taking the time-derivative of (20) and substituting [START_REF] Gupta | Active vibration control of a smart plate using a piezoelectric sensor-actuator pair at elevated temperatures[END_REF] leads to

                                   W Q W tr W Q W tr x P B Bκ Bε W q W M B P x x x V T M M T M T T T M T M T T      ˆ) sgn( 2 1 1 1 1 (21)
Equation ( 21) can be re-written as

    )) sgn( ( ˆ2 1 1 1 1 1 x P B κ ε PB x W Φ M PB x W tr W Φ M PB x q W tr x x V T T T T T M M T M T M T                                                  (22)
Substituting [START_REF] Yaman | Active Vibration Control of a Smart Beam[END_REF] into above equation to get

            i i i T T T T T T ε χ x x x P B κ ε PB x x x V   2 1 )) sgn( ( 2 1  (23) where x P B χ T T  . Selecting the components i  such that i i i      ( 24
)
where i  is a positive constant and hence [START_REF] Vinyas | Vibration control of skew magnetoelectro-elastic plates using active constrained layer damping[END_REF] becomes

       l 1 i i T x x 2 1 V  (25)
Equation ( 25) is stable in the sense of Lyapunov theory.

Remark 5.

A good way for selecting the positive definite matrices P and Q in (12b) is choosing Q first then solving (12b) to determine P for a given A . If the produced P is positive definite, then the system stability is ensured.

Remark 6. Two disadvantages can be noted for the adaptive FBL technique which are: (1) the regressor matrix depends on angular link acceleration, and (2) the update adaptive law of parameter vector requires inversion of inertia matrix which could be subject to singularity and computationally intense. These shortcomings will be resolved by using the control strategy mentioned in the next subsection.

FAT-based VVEC

Although this approach does not linearize the motion equation of the target system, it has two distinct advantages:

(1) it does not require measurement/estimation of manipulator acceleration, and (2) calculation of inertia matrix inverse is not requested [3826]. In effect, this technique has been invented by Slotine and Li [START_REF] Slotine | Applied nonlinear control[END_REF] and it depends on the sliding mode concept. Consequently, the following control law can be selected as

) sgn( ˆs s K r M u ds        ( 26
)
where .

, , e q r q q e e e r q s

d d              (27)
where

l l ds R K   is a positive definite feedback gain matrix, s l R  is the virtual velocity error, r l R 
is the required velocity and is equal to the desired velocity minus the scaled position error, and

 l l R  
is a diagonal positive definite matrix that is related to the time constant of the closed-loop dynamics. Substituting ( 27) and ( 26) into Eq. ( 11) to obtain

        ) sgn( r M s s K s M ds   ( 28
)
Using the FAT, we can get the following closed-loop dynamics

           T M T M ds W r W s s K s M ) sgn(   (29)
As noted in Eq. ( 29), the inertia matrix inverse and the modal acceleration disappear from the right-hand side.

Let us select the following update laws of the unknown parameter vector that can ensure convergence of tracking errors.

T T M M M s Q W s r Q W           ˆ ( 30
)
where

  l l R Q   (.) , l l M R W    , and l l R W     are weighting matrices, l l M R     and    l R  are
matrices of basis function, with  referring to the number of basis function terms.

Theorem 2. The dynamics of the vibrating plate described in Eq. ( 11) with the control law, closed-loop dynamics and the associated updating laws for the weighting matrices introduced in ( 26)-( 30) respectively is stable the sense of Lyapunov theory.

Proof.

) ( 2 1 1 1    W Q W W Q W tr Ms s V M M M T      (31)
Taking the time-derivative of (31) to obtain

) ( 2 1 1 1    W Q W W Q W tr s M s s M s V T M M T M T T            (32)
Knowing that 0  M  and Substituting (29) into above equation to get

                                            W Q rs W tr W Q s r W tr s s s s K s V T T M M T M T M T T ds T     ˆˆ) sgn( 1 1 (33)
Substituting [START_REF] Al-Shuka | Function approximation technique-based adaptive virtual decomposition control for a serial-chain manipulator[END_REF] in [START_REF] Al-Hamadani | Proportional-derivative pd vibration control with adaptive approximation compensator for a nonlinear smart thin beam interacting with fluid[END_REF]

leads to ) ( 1       l i i i T ds T s s s K s V    ( 34
)
Selecting the components

i  such that i i i      ( 35
)
where i  is a positive constant and hence (34) becomes

     l i i i ds T s s K s V 1   (36)
Equation ( 34) is stable in the sense of Lyapunov theory.

FAT-based BSC

The key idea of backstepping control is to design a control law recursively by considering some of the state variables as virtual control variables and designing intermediate control laws in backstepping manner. To this end, let us design a systematically backstepping control framework for the target plate dynamics described in [START_REF] Amabili | Nonlinear Vibrations and Stability of Shells and Plates[END_REF] considering the following state variables

q x q x    2 1 ( 37
)
The dynamics can be reformulated in terms of statespace form as follows.

gu f u M x x x       ) ( 1 2 2 1    (38) where, 1 1 , 
   M g M f



Two steps are required to design the target control architecture: 1) reformulating [START_REF] Spong | Robot modelling and control[END_REF] in terms of error dynamics, and 2) capturing the control law and the associated updating laws based on a Lyapunov function.

Step 1. Let us denote the error dynamics x ~ for 1 x as

d y x x   1 1 ~ (39)
where d d q y  . Hence the associated dynamics with (39) that ensure stability are

2 1 1 1 x x k x     (40a)
where

d d y y x k x x         ) ( , ~1 1 2 2   (40b) with l R  
referring to the virtual input control variable that stabilize (40a), and

l l R k   1
is a postive definite gain matrix. Now, let us describe the dynamics of 2

x as

            u W W u G f x g T g f T f 2 (41)
where Step 2. Now define the following Lyapunov function for the modified dynamics described in (40a) and ( 41)

l l R W    (.)
  T g g T g T f f T f T T W Q W W Q W Tr x x x x V ( 2 1 1 1 2 2 1 1       (42)
Where

  l l R Q   (.)
is an adaptation matrix. Taking the time-derivative of (42) and substituting (40a) and ( 41) to obtain

                    T g g T g T f f T f g T g f T f T W Q W W Q W Tr u W W u G f x x x k x V     ˆ1 1 1 2 1 1 1    (43)
Expanding [START_REF] Shaw | Normal modes for nonlinear vibratory systems[END_REF] with some manipulations to get

                                   T g g T g T g T f f T f T f T T W Q x u W W Q x W Tr u G f x k x x x k x x k x V     1 2 1 2 2 2 1 2 2 2 2 1 1 1     (44)
Choosing the following control law with associated updating laws

  2 2 2 2 2 1 1 ~) sgn( ) ( ˆx u Q W x Q W x κ x k x f G u g g T g f f T f                (45) Then we can obtain            i i i T T T x κ x x k x x k x V x κ x x x k x x k x V 2 2 2 2 2 1 1 1 2 2 2 2 2 2 1 1 1 ~) sgn(     (46)
We see that choosing the component i κ such that

i i i κ    
, where i  is strictly positive leads to

     i i i x x k x x k x V 2 2 2 2 1 1 1   (47)
Table I makes a detailed comparison for the aforementioned three control algorithms.

FAT-FBL



Control law: Eq. ( 12).



Closed-loop dynamics: Eq. ( 13).



Updating laws: Eq. ( 19).



Although the control law is nonlinear, the closed-loop dynamics are linear.



The closed-loop dynamics and the updating laws include acceleration variable and inertia inverse matrix that make computational problems such as singularity.



One of the preferable solutions to above is to assume that the inertia matrix is known (identification procedure is required).



It does not need satisfaction of passivity condition to ensure stability.

FAT-VVEC



Control law: Eq. ( 26).



Closed-loop dynamics: Eq. ( 29).



Updating laws: Eq. ( 30).



Both the control law and the closedloop dynamics are nonlinear.



The closed-loop dynamics and the associated updating laws do not include inertia inverse matrix or modal acceleration.



However, there is a constraint associated with design of this controller is that the passivity condition should be satisfied to ensure stability. This includes 0

) 2 (   s C M s T 
. However, for our plate 0  M  while 0  Cs s T and hence the stability can be still guaranteed. In effect, the damping term is integrated with the nonlinear term to avoid the constraint 0  Cs s T .

FAT-BSC



Control law: Eq. (45a).



Closed-loop dynamics: Eqs. (40a) and [START_REF] Nayfeh | Nonlinear normal modes of buckled beams: Threeto-one and one-to-one internal resonances[END_REF].



Updating laws: Eq. (45b).



Two disadvantages associated with control law and the related updating laws that are: 1) presence of 1 ˆ g in the control law that would make singularity problem, and 2) presence of the input control in the updating law associated with the unknown term g ˆ.



Due to the recursive technique required, for a dynamic system having higher orders the computations would be tedious and complex expressions would result.



The above-mentioned problems can be solved if we assume that g ˆ is known (system identification is required to estimate g ˆ).

 It does not need satisfaction of passivity condition to ensure stability.

SIMULATION RESULTS AND DISCUSSIONS

A flexible plate (shown in Fig. 1) with all edges are simply supported is simulated using MATLAB/Simulink package for verification of the proposed control structure, see Table II

. 0  d u Kq q C q M         (48)
From Eq. (48), we can see that the dynamic equation includes convention stiffness term (the fourth term) and a nonlinearly coupled cubic stiffness term that complicates the frequency response. By dividing (49) by

M , the coefficient ) ( 1 K M 
represents the linearized natural frequency. However, the following points should be noted [START_REF] Wagg | Nonlinear vibration with control, for flexible and adaptive structures[END_REF]:

I K I Q I Q I ds M       1.
If the excitation force term is added, peak resonances in natural frequencies may occur. 2. Due to the presence of nonlinear cubic stiffness term, the natural frequency may not be the same as the linearized natural frequency and hence nonlinear resonances may occur. In effect, if we consider only one mode shape for (49), then the equation is similar to Duffing's equation where the system stiffness varies as a function of the modal coordinate. Here there will be a jump in the frequency spectrum and as well as the type of nonlinear stiffness is determined based on the coefficient of cubic modal coordinate, see page 48 of [START_REF] Jauregui | Parameter identification and monitoring of mechanical systems under nonlinear vibration[END_REF] for more details.

The coefficient

C M 1 
represents the damping term that can enhance vibration suppression of the target dynamic system. It should be noted that this work assumes that the damping coefficient is constant while the plate has constant mass distribution and hence the damping mass proportional, i.e. there is no dynamic coupling associated with damping term, see page 14 of [START_REF] Thomsen | Vibrations and stability: advanced theory, analysis, and tools[END_REF] for more details.

4. There are different methods for modal analysis of nonlinear vibration, however, this topic is beyond the scope of this paper and the reader is referred to [START_REF] Nayfeh | Nonlinear normal modes of buckled beams: Threeto-one and one-to-one internal resonances[END_REF][START_REF] Vakakis | Normal modes and localization in nonlinear systems[END_REF][START_REF] Shaw | Normal modes for nonlinear vibratory systems[END_REF] for more details. Now let us come back to the implementation of control algorithms for the target vibrating plate. As mentioned previously in Table II, the FAT-based VVEC has two significant features: 1) the control structure does not include the inertia inverse matrix, and also 2) it does not require the calculation of modal acceleration and hence this technique is a powerful tool comparing with FATbased FBL and FAT-based BSC. In view of the above, the simulation experiments are focused on FAT-VVEC under the conditions that the system parameters are fully unknown. One intrinsic disadvantage for this technique is that the term s C M s T ) 2 (   should be equal or less than zero to ensure stability. This constraint is equivalent to satisfying the passivity condition for the target dynamic system. Fortunately for our plate system, the time-derivative for the inertia matrix equals zero 0  M  , besides, the damping term of the plate system is integrated with the nonlinear term  and hence the stability of the vibrating plate can be proved to overcome the condition of satisfaction of passivity property. The control structure for FAT-based VVEC includes feedforward terms that are updated based on FAT and a PD feedback term plus a sliding term for compensating any modelling error if exists. The orthogonal Chebyshev polynomials are used as approximators for the FAT. The number of basis functions used is 15 and the initial values for the weighting coefficient matrix are assumed zero. For simplicity, it is assumed that modelling error is neglected and hence the robust sliding term can also be neglected. Figures 2 and3 show the responses of modal amplitudes and sensor voltages for the vibrating smart plate considering the first two-mode shapes. As expected, for an open-loop system the modal coordinates undergo oscillation due to the application of the impulse force. Whereas, the proposed VVEC strategy would suppress the modal vibrations precisely despite the presence of model uncertainty. The feedback and adaptation gains used in the simulation are listed in Table II. The input control response is shown in Fig. 4. The Pseudo-inverse matrix plays an important role in the calculation of optimal input voltage if the number of piezo-actuators is not equal to the investigated number of mode shapes, see [START_REF] Yaman | Active Vibration Control of a Smart Beam[END_REF] for details.

CONCLUSIONS

This work makes a comprehensive study on modelling and vibration control of smart thin plate-like structures considering cubic nonlinearities related to axial stretching loads. Three control strategies are investigated in detail considering features and shortcomings. The function approximation techniques are used as a basis for control structures that make the control task easily dealing with complex robotic systems. The VVEC shows superior benefits in comparison with others; the inertia inverse matrix and the modal acceleration are avoided in the controller structure. Therefore, the simulation experiments here are focused on this control structure. In effect, the VVEC is called passivity-based control in the robotic community and it is a strong tool for stabilization and tracking control of complex dynamic systems. An important point should be mentioned that the passivity property associated with the VVEC strategy is guaranteed here since the inertia matrix is assumed constant in modelling and the Coriolis components are negative. However, integration of piezoelectric materials inertias with plate inertia matrix makes the system is coupled, besides complex motions of plate structure may include variant-time inertia terms that complicate the proof of passivity constraint. However, future work could be focused on the following points:

1

Fig. 1 :

 1 Fig.1: A simply supported thin plate with surface bonded piezo-transducers

Fig. 2 :Fig. 3 :Fig. 4 :

 234 Fig. 2: Modal amplitude response

Table 1 . A Comparison between the FBL, the VVEC and the BSC strategies

 1

Table 2 . Physical parameters and feedback gains used in simulation experiments

 2

		a			  , 400 mm	b		  , 350 mm
	Plate	h				  , 3 mm	E		  , 210 GPa
				  , / 7800 3  m kg		. 3 . 0
		a		p		  , 25 mm	b	p		  , 25 mm
	Piezo-materials	h	p		  , 25 . 0 mm	E	p		3 . 6		10 10	  . / 2 m N
	Control gains					75 20	2	22 ,	,			, , 2 22 300 01 . 0

  . Modelling and control of 3D plate-like structure considering different smart actuation/sensing scenarios. 2. Modelling and control of floating base robots with integrated flexible structures. 3. Extending the current control algorithms for vibration suppression of smart shells; here the nonlinear restoring force term is very complex. 4. Integration of adaptive approximation observer with the control architecture is important for flexible structures interacting with external environments such as aero-elastic nonlinear structures.

	NOMENCLATURE
	a rs	(t	)	Time function associated with Airy stress function
	D				Flexural rigidity
	E				Young's modulus
	F				Airy stress function
	h				Plate thickness
	H	(.)		Heaviside step function
	K				A positive definite feedback gain matrix
		d
					associated with derivative-control term,
						l R 	l
	K				A positive definite feedback gain matrix
		p			associated with proportional-control
					term,		l R 	l
	K				A positive definite feedback gain
	ds
					matrix,		l R 	l
	m				Mass of the plate
	N				Number of piezo-actuators
		a
	M	(.)	Actuator external moment per unit
		p
					length
	p				External transverse load per unit area
	P				A symmetric positive definite matrix
					satisfying the Lyapunov equation,
						l R 2 2 	l
	q		(t	)

n I Identity matrix of (n,n) dimension mn Modal amplitude d q desired value for the modal vector (.) Q An adaptation matrix