Zhongzhe Xiao
email: zhongzhe.xiao@ec-lyon.fr

Emmanuel Dellandrea
email: emmanuel.dellandrea@ec-lyon.fr

Weibei Dou

Liming Chen
email: liming.chen@ec-lyon.fr

ESFS: A new embedded feature selection method based on SFS

Keywords: feature selection, emotion classification, evidence theory, audio, speech, music

Feature subset selection is an important subject when training classifiers in Machine Learning (ML) problems. Too many input features in a ML problem may lead to the so-called "curse of dimensionality", which describes the fact that the complexity of the classifier parameters adjustment during training increases exponentially with the number of features. Thus, ML algorithms are known to suffer from important decrease of the prediction accuracy when faced with many features that are not necessary. In this paper, we introduce a novel embedded feature selection method, called ESFS, which is inspired from the wrapper method SFS since it relies on the simple principle to add incrementally most relevant features. Its originality concerns the use of mass functions from the evidence theory that allows to merge elegantly the information carried by features, in an embedded way, and so leading to a lower computational cost than original SFS. This approach has successfully been applied to the emergent domain of emotion classification in audio signals.

Introduction

When a classification problem has to be solved, the common approach is to compute a wide variety of features that will carry as much as possible different information to perform the classification of samples. Thus, numerous features are used whereas, generally, only a few of them are relevant for the classification task. Including the other in the feature set used to represent the samples to classify, may lead to a slower execution of the classifier, less understandable results, and much reduced accuracy (Hal, 1997). In this context, the objective of feature selection is three-fold: improving the prediction performance of the predictors, providing faster and more cost-effective predictors, and gaining a deeper insight into the underlying processes that generated the data.

Thus, a feature selection method aims at finding the most relevant features. There exist considerable works in the literature on the question. Interesting overviews include [START_REF] Kohavi | Wrappers for Feature Subset Selection[END_REF][START_REF] Guyon | An introduction to variable and feature selection[END_REF]. However, the relevance notion is not perfectly defined and may depend on the feature selection method. One of these definitions [START_REF] Blum | Selection of relevant features and examples in machine learning[END_REF] is to consider that a feature f is relevant if it is incremental useful to a learning algorithm L with respect to a feature subset S: the accuracy that L produces an hypothesis using the feature set f∪S is higher than the accuracy achieved only using S. In the case of classification problems, the accuracy can be the correct classification rate.

Feature selection methods can be categorized into three main categories according to the dependence to the classifiers: filter approaches, wrapper approaches and embedded approaches [START_REF] Kojadinovic | Comparison between a filter and a wrapper approach to variable subset selection in regression problems[END_REF].

Filter methods include Relief method (Arauzo-Azofra, 2004), Focus algorithm [START_REF] Almuallim | Learning with many irrelevant features[END_REF], and normally evaluate the statistical performance of the features over the data without considering the proper classifiers. The irrelevant features are filtered out before the classification process (Hal, 1997). Their main advantage is their low computational complexity which makes them very fast. Their main drawback is that they are not optimized to be used with a particular classifier as they are completely independent of the classification stage.

Wrapper methods on the contrary evaluate feature subsets with the classification algorithm in order to measure their efficiency according to the correct classification rate [START_REF] Kohavi | Wrappers for Feature Subset Selection[END_REF]. Thus, feature subsets are generated thanks to some search strategy, and the feature subset which leads to the best correct classification rate is kept. Among algorithms widely used, we can mention Genetic Algorithm (GA) and Sequential Forward Selection (SFS) methods. The computational complexity is higher than the one of filter methods but selected subsets are generally more efficient, even if they remain sub-optimal [START_REF] Spence | The role of feature selection in building pattern recognizers for computer-aided diagnosis[END_REF].

In embedded feature selection methods, similarly to wrapper methods, feature selection is linked to the classification stage, this link being in this case much stronger as the feature selection in embedded methods is included into the classifier construction. Recursive partitioning methods for decision trees such as ID3, C4.5 and CART are examples of such method. Embedded methods offer the same advantages as wrapper methods concerning the interaction between the feature selection and the classification. Moreover, they present a better computational complexity since the selection of features is directly included in the classifier construction during training process.

In our work, we introduce a new embedded feature selection method we have developed and called ESFS, inspired from the wrapper method SFS since it relies on the simple principle to add incrementally most relevant features, and making use of the term of mass function which is introduced from the evidence theory which allows elegantly to merge feature information in an embedded way, leading to a lower computational cost than original SFS.

This approach has been evaluated on the problem of emotion classification in audio signals. We consider two types of data: speech and music samples. As speech samples present different signal properties than music samples, two different feature sets are considered. The speech feature set includes 226 features, whereas the music feature set includes 188. The high number of features compared to the relatively low number of samples available for training classifiers (about 500 samples) suggests the use of a feature selection method to improve classification accuracy.

The reminder of this paper is organized as follows. In section 2, we introduce the evidence theory on which our feature selection method is based, and detailed in section 3. Experimental results are presented in section 4. Finally, conclusions and perspectives are drawn in section 5.

Overview of the evidence theory

In our feature selection scheme, the term "belief mass" from the evidence theory is introduced into the processing of features.

Dempster and Shafer wanted in the 1970's to calculate a general uncertainty level from the Bayesian theory. They developed the concept of "uncertainty mapping" to measure the uncertainty between a lower limit and an upper limit [START_REF] Dempster | A generalization of Bayesian inference[END_REF]. Similar to the probabilities in the Bayesian theory, they presented a combination rule of the belief masses (or mass function) m().

The evidence theory was completed and presented by Shafer in [START_REF] Shafer | A mathematical theory of evidence[END_REF]. It relies on the definition of a set of n hypothesis Ω which have to be exclusive and exhaustive. In this theory, the reasoning concerns the frame of discernment 2 Ω which is the set composed of the 2 n subsets of Ω. In order to express the degree of confidence we have in a source of information for an event A of 2 Ω , we associate to it an elementary mass of evidence m(A).

The elementary mass function or belief mass which presents the chance of being a true statement is defined as: !: 2 Ω → &0,1*,1] which satisfies:

!+Φ-= 0 and 3 !+4- The belief function shows the lower bound on the chances, and it corresponds to the mass function with the following formulaes

89:+4-= 3 !+8- N⊆5 ∀ 4 ⊂ Ω !+4-= 3+-1-|5QN| 89:+8- N⊆5
where |X| means the number of elements in the subset X.

The doubt function is defined as RST+4-= 89:+4 Uand the upper probability function is defined as

V:+4-= 1 -RST+4-
The true belief in A should be between Bel(A) and Pl(A). The Dempster's combination rule can combine two or more independent sets of mass assignments by using orthogonal sum. For the case of two mass functions, let m 1 and m 2 be mass functions on the same frame Ω, the orthogonal sum is defined as m=m1⊕ m2, to be m(Φ)=0, and

!+4-= W 3 ! ; +X-. Y∩[\5 ! 7 +]- with W = 1 1 -∑ ! ; +X-.
Y∩[\M ! 7 +]-For the case with more than two mass functions, let m=m1⊕…⊕ m2. It satisfies m(Φ)=0 and

!+4-= W 3 c ! F +4 F - ;dFd> e \ f g 5 with W = 1 1 -∑ ∏ ! F +4 F - ;dFd> e \ f g

M

This definition of mass functions from the evidence is used in our model in order to represent the source of information given by each feature, and to combine them easily and to consider the as a classifier whose recognition value is given by the mass function.

ESFS scheme

Recall that an exhaustive search of the best subset of features, leading to explore a space of 2 n subsets, is impractical, we turn to a heuristic approach for the feature selection. The SFS is selected as the basic of our feature selection. For this classifier dependent sub-optimal selection method, we have provided in this work two innovations. First, the range of subsets to be evaluated in the forward process is extended to multiple subsets for each size, and the feature set is reduced according to a certain threshold before the selection in order to decrease the computational burden caused by the extension of the subsets in the evaluation. Second, since the SFS is a classifier dependent method, the concept of belief masses which comes from the evidence theory is introduced to consider the audio feature as a classifier which leads to an embedded feature selection method.

Method overview

Heuristic feature selection algorithm can be characterized by its stance on four basic issues that determine the nature of the heuristic search process. First, one must determine the starting point in the space of feature subsets, which influences the direction of search and operators used to generate successor states. Second decision involves the organization of the search. As an exhaustive search in a space of 2 n feature subsets is impractical, one needs to rely on a more realistic approach such as greedy methods to traverse the space. At each point of the search, one considers local changes to the current state of the features, selects one and iterates. The third issue concerns the strategy used to evaluate alternative subsets of features. Finally, one must decide on some criterion for halting the search. In the following, we bring our answers to the previous four questions.

The SFS algorithm begins with an empty subset of features. The new subset S k with k features is obtained by adding a single new feature to the subset S k-1 which performs the best among the subsets with k-1 features. The correct classification rate achieved by the selected feature subset is used as the selection criterion. In the original algorithm of SFS, there are totally n*(n+1)/2 subsets which need to be evaluated and the optimal subset may be missing in the searching.

In order to avoid departure too far from the optimal performance, we proposed an improvement of the original SFS method by extending the subsets to be evaluated. In each step of forward selection, instead of keeping only one subset for each size of subsets, a threshold is set according to the compromise between the performance and the computational burden (which is decided from the performance from experiments with a small amount of data in our work) and all the subsets with the performance above the threshold are kept to enter the evaluation in the next step. Since remaining multiple subsets in each step may lead to heavy computational burden, only the features selected in the first step (subsets with single feature), thus having the best abilities to discriminate among classes that occur in the training data, are used in the evaluation in posterior steps.

As the features are added to the potential subsets one by one in the SFS process, the forward process of creating a feature subset with size k can be seen as a combination between two elements: a subset with size k-1 and a single feature. Thus, if we consider each subset as a feature itself, the process of creating a new feature subset can be interpreted as generating a new feature from two features.

A wrapper feature selection scheme such as the SFS needs to specify a classifier in order to evaluate improvement of classification accuracy as feature selection criterion. In our case, the classifier used in this feature selection method is simply based on the belief masses of the features which are modeled from the distribution of the features for each class obtained from the training data. The belief masses of samples in the testing sets are calculated with the model of the belief masses. The class with the highest belief mass is then taken as the output of the classification. This classifier is repeated for every subset in evaluation for searching the best feature subset. The procedure is detailed in the next subsection.

Feature selection procedure

The feature selection procedure is introduced in this section with its four steps.

Step 1: Calculation of the belief masses of the single features. Before the feature selection starts, all features are normalized into [0, 1]. For each feature, i9j > = i9j >k -min +i9j >k -max+i9j >k --min +i9j >kwhere Fea n0 is the set of original value of the n th feature, and Fea n is the normalized value of the n th feature.

By definition of the belief masses, the mass can be obtained by different ways which can represent the chance for a statement to be true. In this paper, the PDFs (probability density functions) of the features computed from the training data are used to represent the masses of the single features.

The curves of PDFs of the features are obtained by applying polynomial interpolation to the statistics of the distribution of the feature values from the training data.

Taking the case of a 2-class classifier as example, the classes are defined as subset A and subset A C . First, the probability densities of the features in each of the 2 subsets are estimated from the training samples by the statistics of the values of the features in each class. We define the probability density of the k th feature Fea k in subset A as Pr k (A, f k ) and the probability density in subset A C as Pr k (A C , f k ), where the f k is the value of the feature Fea k .According to the probability densities, the masses of feature Fea k on these two subsets can be defined as

! n +4, o n -= Vp n +4, o n - Vp n +4, o n -+ Vp n +4 r , o n - ! n +4 r , o n -= Vp n +4 r , o n - Vp n +4, o n -+ Vp n +4 r , o n -
where at any possible value of the

k th feature f k , m k (A, f k )+m k (A C , f k )=1.
In the case of N classes, the classes are defined as A 1 , A 2 ,…,A N . The masses of feature F k of the i th class A i can be obtained as

! n +4 F , o n -= Vp n +4 F , o n - ∑ Vp n +4 > , o n - s >\; which satisfies 3 ! n s F\; +4 F , o n -= 1
Step 2: Evaluation of the single features and selection of the initial set of potential features.

When the distribution model of the belief masses of the single features for the different classes have been extracted from the training data, the single features are evaluated by passing the distribution model derived from the training data. For each sample, its belief mass value can be extracted from feature mass functions. The samples are assigned to the class which has the highest belief mass and thus performances of correct classification rates can be obtained.

Within this process, the single features can then be ordered according to the correct classification rate given by mass functions and thus the best features can be selected.

The features are ordered in descending order according to the correct classification rates Rsingle(Fk) as {Fs1, Fs2,…, FsN}, where N means the total number of features in the whole feature set.

In order to reduce the computational burden in the feature selection, an initial feature set FSini is constructed with the best K features in the re-ordered feature set according to a certain threshold in classification rates as FSini={Fs1, Fs2,…, FsK}.

The threshold of the classification rates is decided according to the best classification rate as:

Rsingle(Fs_K)≥thres_1* Rbest_1
where Rbest_1=Rsingle(Fs_1). In our work on emotion analysis, the threshold value thres_1 is set to 0.8 according to a balance between the overall performance and the calculation time by experiments. This threshold may vary with different problems, and around 30 features are kept in our applications above the threshold of 0.8.

Only the features selected in the set FSini will attend in the latter steps of feature selection process. The elements (features) in FS ini are seen as subsets with size 1 at the same time.

Step 3: Combination of features for the generation of the feature subsets.

For the iterations with subsets with size k (k≥2), the generation of a subset is converted into the creation of a new feature by using an operator of combination from two original features, and the subsets are selected according to a threshold similar to the case with single features for each size of subsets.

We note the set of all the feature subsets in the evaluation with size k as FSk and the set of the selected subsets with size k as FS ' k. Thus, FS1 equals to the original whole feature set, and FS ' 1=FSini. From k=2, the set of the feature subsets FSk is noted as:

FSk=Combine(FS ' k-1, FSini)={Fc01_k, Fc02_k,…, Fc0Nk_k}
where the function "Combine" means to generate new features by combining features from each of the two sets FS ' k-1 and FSini with all the possible combinations except the case in which the element from FSini appears in the original features during the generation process of the element from FS ' k-1; Fc0n_k represents the generated new features; and Nk is the number of elements in the set FSk.

The creation of a new feature from two features is implemented by combining the contribution of the belief masses of the two features, making use of an operator of combination. The combining process works as follows.

Assume that N classes are considered in the classifier. For the i th class Ai, the preprocessed mass m * for the new feature Fc0t_k, which is generated with Fcx_k-1 from FS ' k-1 and Fsy from FSini, Fc0t_k= Combine (Fcx_k-1, Fsy), is calculated as

)) , ( ), , ( ( ) 0 , ( 1 _
_ * y i k x i k t i fs A m fc A m T fc A m - =
where f x is the value of the feature F x , and T(x,y) is an operator of combination that corresponds to a t-norm operator, being a generalization of the conjunctive 'AND' [START_REF] Schweizer | Probabilistic metric spaces[END_REF]. The sum of m * s may not be 1 according to different operators. In order to meet the definition of belief masses, the m * s can then be normalized as the masses for the new feature:

∑ = = N n k t n k t i k t i fc A m fc A m fc A m 1 _ * _ * _ ) 0 , ( ) 0 , ( ) 0 , (
The performance of the combined new feature may be better than both two features in the combination. However, the combined new feature may even performance worse than any of the two original features, which will be eliminated in the selection.

The correct classification rates of the combined new features can be obtained with the belief masses by assigning the class with the highest belief mass to the data samples, and the combined new features can then be ordered in descending order according to the correct classification rates as with the single features:

FSk={Fc01_k, Fc02_k,…, Fc0Nk_k}={Fc1_k, Fc2_k,…, FcNk_k}
The best feature with size k is noted as Fcbest_k=Fc1_k, and the recognition rate of feature Fcbest_k is recorded as Rbest_k.

Similar to the selection of FSini in the evaluation of the single features, a threshold is set to select a certain number of subsets with size k to take part to the next step of forward selection. The set of the subsets remained is noted as

FS ' k={Fc1_k, Fc2_k,…, FcN0k_k}
which satisfies R(FcN0k_k)≥thres_k* Rbest_k. In order to simplify the selection, the threshold value thres_k is set in our work to the same value as 0.8 in every step without any adaptation to each step.

Step 4: Stop criterion and the selection of the best feature subset.

The stop criterion of ESFS occurs when the best classification rate begins to decrease while increasing the size of the feature subsets. In order to avoid missing the real peak of the classification performance, the forward selection stops when the classification performance continues to decrease in two steps, Rbest_k<min(Rbest_k-1, Rbest_k-2).

Experimental Results

The feature selection method proposed in previous section has been evaluated on the problem of emotion classification in speech and music.

Datasets

Our experiments are performed on two datasets, presented below. The Berlin emotional speech database is developed by Professor Sendlmeier and his fellows in Department of Communication Science, Institute for Speech and Communication, Berlin Technical University (Sendlmeier). This database contains 535 speech samples (302 from female voices and 233 from male voices) belonging to 7 kinds of emotions: anger, boredom, disgust, fear, happiness, sadness and neutral.

As there is no public music emotion dataset available, we have built a dataset for music emotion recognition. It contains 603 samples of classical music labeled according to four emotions: exuberance, anxious, contentment and depression.

Feature extraction

A total number of 226 features have been computed to represent each speech sample from Berlin dataset. The corresponding feature set thus includes harmonic features, frequency features, energy features, MFCC features and Zipf features. As speech signal present different signal properties than music, a second set of features has been computed to represent each music sample. This feature set is composed of 188 features, including rhythmic features, tonality features, timbre features and octave-based features.

Results

Three groups of experiments are made with different features on Berlin dataset: one with all the features without selection, the second with features selected with fisher filter method (Narendran, 1977), and the third with the best features selected by the ESFS.

Five types of one step global classifiers are tested: Multi-layer Perception (Neural Network, marked as MP in the following text), C4.5, Linear Discriminant Analysis (LDA), K-NN, and Naive Bayes (NB). Each classifier is tested with several parameter configurations, and only the best results are kept. The experiments are carried out on TANAGRA platform [START_REF] Rakotomalala | TANAGRA : un logiciel gratuit pour l'enseignement et la recherche[END_REF] with 10-folds cross-validation. The experimental results are listed in Table 1.

The features selected by the embedded method ESFS are actually working in a filter way on the several classifiers in this experiment. The result show that for most of the classifiers tested in this experiment, the features selected by ESFS work better than the features selected by fisher filtering criterion. Especially, the features selected by ESFS fit the LDA very well, and classification rate on the LDA with these features is even better than the result from the ESFS itself on female voice samples. This result shows that the ESFS method is able to select the most discriminative features on the problem of classification of emotional speech, and the features selected with this method are more suitable to be used in the linear classifier methods than the nonlinear ones. We also made experiments on the problem of classification of music emotion with four classes (Table 2). In order to test the ESFS itself without the effects of the structure of the classifiers, global classifiers with one step in the classification of the four classes are applied. The same classifiers on the TANAGRA platform -MP, C4.5, LDA, K-NN, and NB -as used on the Berlin dataset are also tested on the music emotion dataset. The result of ESFS on the problem of classification of music emotion with global classifier is 72.80%, which is 2% higher than that obtained from the experiments on TANAGRA as 70.80% with Naïve Bayes. Although the superiority of the result of the ESFS is not so obvious, the ESFS still shows a better performance than the popular used classification schemes, and with lower computational complexity because the feature selection and the classification processes are implemented simultaneously.

Conclusion and Future Work

In this paper, we have presented a novel feature selection method, ESFS, which relies on the simple principle to add incrementally most relevant features. To this purpose, we represent each feature thanks to mass functions, from the evidence theory, which allows to merge the information carried by features, in an embedded way, and so leading to a lower computational cost than wrapper method. Indeed, our ESFS scheme allows simultaneously to select most relevant features and to perform classification, with no need of an extra classifier.

Experimental results on the problems of emotion classification in speech and music have shown that selecting relevant features improves the classification accuracy, and for this purpose, ESFS, used as a filter selection method, performs better than the traditional filter method, namely Fisher algorithm. Moreover, ESFS, when used as both feature selector and classifier, allows to obtain a better classification accuracy than representative state of the art classifiers, such as neural networks, or decision trees.

We envisage in our future work to use ESFS as the basis of a hierarchical classifier, which will be represented by a binary classification tree where ESFS will be nodes. The purpose of this hierarchical structure is to allow to better separate classes by first separating classes far away from each other and then concentrating on closer classes. Moreover, thanks to ESFS, each subclassifier could have at its disposal its own feature set.

Table 1 .

 1 Comparison between the result without feature selection and with the features selected by ESFS on Berlin dataset.

			F EMALE			MALE
		NO	FILTER	ESFS	NO	FILTER	ESFS
		SELECTION	SELECTION	SELECTION	SELECTION	SELECTION	SELECTION
	MP	65.73±2.85	66.38±2.73	71.03±1.39	61.78±2.93	65.75±3.19	66.44±2.50
	C4.5	55.46±2.7	56.22±2.95	55.73±3.38	55.75±0.66	54.66±2.32	56.51±3.53
	LDA	60.92±2.56	70.16±3.14	74.00±2.08	51.16±3.05	70.62±2.37	71.97±1.57
	K-NN	60.14±2.37	64.16±3.44	67.41±1.42	57.88±2.85	61.51±2.23	66.23±2.31
	NB	62.67±1.45	59.78±1.10	67.41±1.46	56.30±1.12	57.60±0.59	62.81±2.79
	Best	65.73	70.16	74.00	61.78	70.62	71.97
	ESFS		71.75%±3.10%			73.77%±2.33%

Table 2 .

 2 Comparison of classification accuracy between ESFS and other classifiers (%).

	ESFS	MP	C4.5	LDA	K-NN	NB
	72.80 69.14±2.2 57.33±2.6 60.86±3.1 68.99±2.0 70.80±1.7


Ning Dang

Lingyu Liu

Eric Maire
email: eric.maire@insa-lyon.fr

Jé Rôme Adrien

Sophie Cazottes

Wenlong Xiao
email: wlxiao@buaa.edu.cn

Chaoli Ma

Lian Zhou

Analysis of shear stress promoting void evolution behavior in an α/β Ti alloy with fully lamellar microstructure

Keywords: Ti-6Al-4V, Lamellar microstructure, X-ray tomography, Shear behavior, Damage development

HAL is

Introduction

Titanium and its alloys play an important role in modern industries i.e. aerospace, biomedical, and chemical processing and synthesis, due to the integrated properties including excellent mechanical properties, unrivalled corrosion resistance and outstanding biocompatibility [START_REF] Lütjering | Introduction, Titanium[END_REF]. Ti-6Al-4V(TA6V), because of its integrated properties and lower cost, is considered as the most common titanium alloy [START_REF] Banerjee | Perspectives on Titanium Science and Technology[END_REF][START_REF] Peters | Titanium Alloys for Aerospace Applications, Titanium and Titanium Alloys[END_REF]. Depending on the thermo-mechanical treatment or heat treatment of such (α+β) titanium alloy, the microstructure morphologies or mechanical properties may vary in a wide range [START_REF] Peters | Titanium Alloys for Aerospace Applications, Titanium and Titanium Alloys[END_REF]. It is well recognized that the mechanical properties are mainly dependant on microstructural features; furthermore, failure behavior of Ti alloys could be affected by these features significantly [START_REF] Peters | Titanium Alloys for Aerospace Applications, Titanium and Titanium Alloys[END_REF][START_REF] Wu | Effect of microstructure on the fatigue properties of Ti-6Al-4V titanium alloys[END_REF].

Depending on the thermo-mechanical treatment of such (α+β) titanium alloy, different microstructures can be obtained, which can lead to different mechanical properties [START_REF] Attanasio | Influence of Material Microstructures in Micromilling of Ti6Al4V Alloy[END_REF][START_REF] Jaffery | Wear Mechanism Analysis in Milling of Ti-6Al-4V alloy[END_REF][START_REF] Fan | Relationships among the Microstructure, Mechanical Properties, and Fatigue Behavior in Thin Ti6Al4V[END_REF]. Two extreme microstructure morphologies are fully lamellar microstructure and equiaxed microstructure. The fully lamellar microstructure could present a lower strength and lower ductility, but better fatigue crack propagation resistance; whereas the equiaxed microstructure can show a better initiation fatigue crack resistance. Any bimodal microstructure is a combination of equiaxed and lamellar microstructures that combine both mechanical properties and can present well-balanced fatigue properties.

From the literature it can be noticed that most of researchers' attention was paid on fully equiaxed [START_REF] Katani | Micromechanical modelling of damage behaviour of Ti-6Al-4V[END_REF][START_REF] Choi | Microstructure-based constitutive modeling of TRIP steel: Prediction of ductility and failure modes under different loading conditions[END_REF][START_REF] Sun | Predicting failure modes and ductility of dual phase steels using plastic strain localization[END_REF][START_REF] Shi | Elastic plastic deformation of TC6 titanium alloy analyzed by in-situ synchrotron based X-ray diffraction and microstructure based finite element modeling[END_REF][START_REF] Helbert | The influence of stress trlaxiality on the damage mechanisms in an equiaxed α/β Ti-6AI-4V alloy[END_REF][START_REF] Lecarme | Heterogenous void growth revealed by in situ 3-D X-ray microtomography using automatic cavity tracking[END_REF] or bimodal microstructure [START_REF] Zhang | Failure mechanism and mode of Ti-6Al-4V alloy under uniaxial tensile loading: Experiments and micromechanical modeling[END_REF][START_REF] Kumar | Stress triaxiality effect on fracture behavior of IMI-834 titanium alloy: A micromechanics approach[END_REF][START_REF] Chapman | Characterisation of short fatigue cracks in titanium alloy IMI 834 using X-ray microtomography[END_REF] rather than fully lamellar microstructure [START_REF] Semiatin | Cavitation during hot tension testing of Ti-6Al-4V[END_REF][START_REF] Bieler | An experimental and theoretical investigation of the effect of local colony orientations and misorientation on cavitation during hot working of Ti-6Al-4V[END_REF]. However, after casting or welding process, the existence of fully lamellar microstructure cannot be avoided. In order to enhance the mechanical properties of components with this structure, it is necessary to focus on the failure mode of TA6V presenting a fully lamellar microstructure. Because TA6V is consisting of HCP α phase and BCC β phase, microstructural inhomogeneity is inevitable. This may induce an inhomogeneous stress/strain distribution, which could play a key role in the ductile damage evolution [START_REF] Pineau | Failure of metals I: Brittle and ductile fracture[END_REF].

In order to describe this microstructure inhomogeneity and consequently, heterogeneous stress/strain distribution at micro-scale, microstructure-based micromechanical modeling approach can be adopted [START_REF] Lecarme | Heterogenous void growth revealed by in situ 3-D X-ray microtomography using automatic cavity tracking[END_REF][START_REF] Paul | Real microstructure based micromechanical model to simulate microstructural level deformation behavior and failure initiation in DP 590 steel[END_REF][START_REF] Paul | Micromechanics based modeling to predict flow behavior and plastic strain localization of dual phase steels[END_REF][START_REF] Sodjit | Microstructure based prediction of strain hardening behavior of dual phase steels[END_REF][START_REF] Ohata | 3D-Simulation of ductile failure in twophase structural steel with heterogeneous microstructure[END_REF][START_REF] Paul | Micromechanics based modeling of Dual Phase steels: Prediction of ductility and failure modes[END_REF]. Consequently, numerous models were proposed and validated to elucidate micromechanical behavior with microstructure features for dual phase metals, especially for dual phase steels [START_REF] Paul | Real microstructure based micromechanical model to simulate microstructural level deformation behavior and failure initiation in DP 590 steel[END_REF][START_REF] Paul | Micromechanics based modeling to predict flow behavior and plastic strain localization of dual phase steels[END_REF][START_REF] Sodjit | Microstructure based prediction of strain hardening behavior of dual phase steels[END_REF][START_REF] Ohata | 3D-Simulation of ductile failure in twophase structural steel with heterogeneous microstructure[END_REF][START_REF] Paul | Micromechanics based modeling of Dual Phase steels: Prediction of ductility and failure modes[END_REF].

As indicated by Q.Xue et al [START_REF] Xue | Self-organization of shear bands in titanium and Ti-6Al-4V alloy[END_REF], microstructure inhomogeneity, internal boundary and favorable orientation of grains or colonies could affect the distribution of strains which could determine the site for shear band formation, on which void nucleation occurs. Because of local constraints induced by microstructure distribution heterogeneities, heterogeneous void growth can be found for Ti-6Al-4V alloy during damage development [START_REF] Lecarme | Heterogenous void growth revealed by in situ 3-D X-ray microtomography using automatic cavity tracking[END_REF]. Unfortunately, up till now, for TA6V dual phase titanium alloys, the micro-mechanism of heterogeneous stress/strain distribution and its acting on damage evolution is not completely known.

This work aims at deepening our understanding in the origins of plastic deformation inhomogeneities and its role in the failure processing of TA6V alloy. Based on the authors' previous work [START_REF] Dang | 3D X-ray Tomographic analysis of crack nucleation and growth by void coalescence in an α/β Ti alloy lamellar microstructure[END_REF], ductile damage, a process driven by plastic deformation, is the main failure mode for such alloy. It is commonly proved that the procedure can be divided into three steps [START_REF] Chapman | Characterisation of short fatigue cracks in titanium alloy IMI 834 using X-ray microtomography[END_REF], void nucleation, void growth and void coalescence. With the help of 3D tomography technology, it is possible to investigate the action of such heterogeneities in the development of ductile failure. [START_REF] Lecarme | Heterogenous void growth revealed by in situ 3-D X-ray microtomography using automatic cavity tracking[END_REF][START_REF] Dang | 3D X-ray Tomographic analysis of crack nucleation and growth by void coalescence in an α/β Ti alloy lamellar microstructure[END_REF][START_REF] Maire | Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography[END_REF][START_REF] Herbig | 3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast X-ray tomography[END_REF][START_REF] Landron | Validation of void growth models using X-ray microtomography characterization of damage in dual phase steels[END_REF].

Hence, in the present work, in situ tensile tests are carried out based on synchrotron X-ray micro-tomography for Ti-6Al-4V alloy with a fully lamellar microstructure in order to track and analyze void evolution. Then, post mortem procedure are performed by scanning electron microscopy(SEM) and electron backscatter diffraction (EBSD) in order to analyze the micro-mechanism of plastic deforming heterogeneities in terms of microstructure inhomogeneities. Then, shear behavior, correlated with such plastic deforming heterogeneities, is analyzed at the microscale and grain-scale. This leads to the formation of micro-shear band, shear band initiation or neighbor grains /colonies rotation, respectively.

The influence of these shear induced mechanisms on void nucleation and propagation are also analyzed and interpreted in this work.

Experiments

Materials

In order to obtain fully lamellar microstructure, commercially available casting Titanium Ti-6Al-4V alloy, fabricated by BaoTi Group Co.Ltd was investigated in the form of plates, with a dimension of 300mm×200mm×10mm. The chemical composition of the material, in wt% is 6.43Al, 4,21V, 0.40Fe, 0.15Si, 0.10C, 0.05N, 0.015H and bal. Ti. In the subsequent heat treatment, the solution-aging parameters were selected based on the β-transus temperature(995℃±5℃). This TA6V plate was treated at 960℃ for 1 hour followed by air cooling and aging treatment in order to obtain fully lamellar structure. Then the plate was aged at 540℃ for 8 hours following air cooling to ensure equilibrium distribution of α/β phase.

For the subsequent analysis, TA6V samples were cut from the plate with the size of 35mm×10mm×10mm for manufacturing the samples for microstructure observation and in-situ tensile test. Phase composition and microstructure morphology produced by this treatment are illustrated in Fig. 1. Only α phase could be noticed in the result of XRD(Fig. 1

X-ray tomography

Tomography measurements were performed at the ID19 beamline at the European Synchrotron Radiation Facility(ESRF) in Grenoble, France.

More details on the set-up can be found in Ref. [27.30]

In situ tensile test

For in situ tensile test based on X-ray tomography, the shape of the samples, inspired by Landron et al [START_REF] Landron | Validation of void growth models using X-ray microtomography characterization of damage in dual phase steels[END_REF], were cut into 2.5 mm radius notched samples. The geometry of the samples is given in Fig. 2. sample,(b) 3D view [START_REF] Bron | Ductile rupture in thin sheets of two grades of 2024 aluminum alloy[END_REF] The local strain, ε loc , during deformation was calculated as

0 = ln S S     (1)
where S 0 is the area of initial cross-section in the undeformed state, and S is the solid area of minimal cross-section at each damage step. This means that in the measurement of S, the eventual area of the voids in the current section is subtracted. The minimal cross-section S was also used to evaluate the true stress in this section,

true F S   ( 2
)
where F is the measured force during the tensile test.

Microscopic measurement for X-ray tomography data

After this tensile test, damage was quantified according to the postprocessing procedure for dual-phase steel presented in [START_REF] Maire | Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography[END_REF][START_REF] Landron | Validation of void growth models using X-ray microtomography characterization of damage in dual phase steels[END_REF][START_REF] Landron | Characterization and modeling of void nucleation by interface decohesion in dual phase steels[END_REF]. The raw data obtained from reconstructed volumes were median filtering and thresholding, based on the approach described in [START_REF] Landron | Validation of void growth models using X-ray microtomography characterization of damage in dual phase steels[END_REF]. This procedure allows us to distinguish the voids from the TA6V matrix.

Only the central region of the tensile specimen is selected for damage analysis, according to T. Pardoen and F. Delannays' work [START_REF] Pardoen | Assessment of void growth models from porosity measurements in cold-drawn copper bars[END_REF], which indicates that the central volume undergoes the largest stress triaxiality and highest strain that leads to the fastest damage rates. The post treatment and 3D rendering work was managed by using open source image analysis software, Image J and Tomviz.

(b) (a)

Post mortem based on SEM

After fracture, the samples were embedded into epoxy matrix parallel to their tensile direction, followed by polishing down to the central plane.

The surfaces were grinded by using SiC paper up to 2500 grade. Then, the surfaces were polished and etched by a mixed solution of OP-S and H 2 O 2 and Kroll's reagent (1 ml HF+ 3ml HNO 3 + 96ml water). Both microstructure and voids, were observed by SEM (Zeiss Supra 55VP).

High resolution EBSD analysis was conducted in this SEM equipped with

Oxford Instrument Nordlys detector and the Aztec online acquisition software package. A free tool --Analysis Tools for orientation Mapping‖(ATOM) was used for EBSD data post-processing [START_REF] Beausir | ATOM -Analysis Tools for Orientation Maps[END_REF].

Results

Microstructure morphologies

Before deformation, a full lamellar microstructure morphology can be observed (seen in Fig. 1 (b)&(c) ). The samples comprises colony microstructure with a prior-β grain size (and comparable colony size) of 500~800 μm and a grain boundary α layer approximately 3~5 μm thick.

After deformation, as shown in Fig. 4, the sample still tends to maintain fully lamellar microstructure morphology. However, microstructure distortion can be clearly observed which means that severe plastic deformation exists within this local region. The tensile direction was parallel to the horizontal direction of this Figure. It is interesting to note that in previous work concerning damage analysis of other ductile metals, such as dual phase steels [START_REF] Maire | Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography[END_REF][START_REF] Landron | Validation of void growth models using X-ray microtomography characterization of damage in dual phase steels[END_REF][START_REF] Landron | Characterization and modeling of void nucleation by interface decohesion in dual phase steels[END_REF] or Al alloys [START_REF] Maire | Fast X-ray tomography and acoustic emission study of damage in metals during continuous tensile tests[END_REF][START_REF] Sué Ry | Fast in-situ X-ray micro tomography characterisation of microstructural evolution and strain-induced damage in alloys at various temperatures[END_REF][START_REF] Bron | Ductile rupture in thin sheets of two grades of 2024 aluminum alloy[END_REF], obvious necking behavior can be noticed during the deformation procedure. While slight necking phenomenon can be found within Fig. 5 & Fig. 6, it may symbolize the discrepancy of damage evolution between TA6V in this current work and other ductile metals.

X-ray tomography

In addition, comparing with voids number of DP steels [START_REF] Maire | Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography[END_REF][START_REF] Landron | Validation of void growth models using X-ray microtomography characterization of damage in dual phase steels[END_REF][START_REF] Landron | Characterization and modeling of void nucleation by interface decohesion in dual phase steels[END_REF] or Al alloys [START_REF] Maire | Fast X-ray tomography and acoustic emission study of damage in metals during continuous tensile tests[END_REF][START_REF] Sué Ry | Fast in-situ X-ray micro tomography characterisation of microstructural evolution and strain-induced damage in alloys at various temperatures[END_REF][START_REF] Bron | Ductile rupture in thin sheets of two grades of 2024 aluminum alloy[END_REF] in the literatures, only few voids could be observed in Fig. 6.

This could be associated with microstructure morphology and micromechanism of deformation for this alloy. Details will be given in a following section of this paper.

Microscopic damage measurement

The curvature radius of outer surface R and the radius of the minimal section r can be used to calculate the average stress triaxiality T in the

(d) (c) (b) (a)
center of the minimal cross-section with the help of Bridgman formula [START_REF] Bridgman | Effects of High Hydrostatic Pressure on the Plastic Properties of Metals[END_REF],

1 T= ln 1 32 r R     (3)
It has been proved that local strain calculated from Eq.( 1) and stress triaxiality calculated from Eq.( 3) exhibit acceptable agreement when comparing with the results from finite elements simulations [START_REF] Landron | Validation of void growth models using X-ray microtomography characterization of damage in dual phase steels[END_REF]. In modern damage models, including commonly used Rice-Tracey model(R-T model) [START_REF] Rice | On the ductile enlargement of voids in triaxial stress fields[END_REF] and Gurson-Tvergarrd-Needleman model(GTN model) [START_REF] Gurson | Continuum Theory of Ductile Rupture by Void Nucleation and Growth : Part 1-Yield Criteria and Flow Rules for Porous Ductilr Media[END_REF][START_REF] Tvergaard | Analysis of the cup-cone fracture in a round tensile bar[END_REF], stress triaxiality(T) is considered as an important factor in void evolution processing. According to Helbert's work [START_REF] Helbert | The influence of stress trlaxiality on the damage mechanisms in an equiaxed α/β Ti-6AI-4V alloy[END_REF], for Ti-6Al-4V alloy, stress triaxiality, which could affect void evolution significantly during void development, can be reflected by the angle between stress axis and void growing direction. This angle could increase with the increase of triaxiality value, which can lead to the damage mode transformation, from plastic instability to cleavage mode.

As shown in Fig. 8, it can be clearly seen that the angle tends to maintain a constant value during damage development. In Fig. 8(a), one void is observed tilting at about 23° to the stress axis. While in Fig. 8 Such regions of high deformation could promote the nucleation and growth of voids (pointed out by blue arrows) along shear bands [START_REF] Benzerga | Anisotropic ductile fracture : Part I: experiments[END_REF][START_REF] Anand | Initiation of localized shear bands in plane strain[END_REF].

In the literature, it well documented that slip or shear bands play an important role during plastic deformation processes [START_REF] Qin | Transgranular shearing introduced brittlement of Ti-5Al-5V-5Mo-3Cr alloy with full lamellar structure at room temperature[END_REF][START_REF] Zhang | Simulation of slip band evolution in duplex Ti-6Al-4V[END_REF][START_REF] Huang | Effect of microstructure on tensile properties of Ti-5Al-5Mo-5V-3Cr-1Zr alloy[END_REF][START_REF] Castany | In situ transmission electron microscopy deformation of the titanium alloy Ti-6Al-4V: Interface behavior[END_REF][START_REF] Sun | Shear banding in commercial pure titanium deformed by dynamic compression[END_REF][START_REF] Wang | Study of twinning/detwinning behaviors of Ti by interrupted in situ tensile tests[END_REF][START_REF] Ren | Microstructure and deformation behavior of Ti-6Al-4V alloy by high-power laser solid forming[END_REF]. For instance, at the micro scale, the slip bands [START_REF] Zhang | Simulation of slip band evolution in duplex Ti-6Al-4V[END_REF][START_REF] Sun | Shear banding in commercial pure titanium deformed by dynamic compression[END_REF][START_REF] Ren | Microstructure and deformation behavior of Ti-6Al-4V alloy by high-power laser solid forming[END_REF] and shear bands [START_REF] Zhang | Simulation of slip band evolution in duplex Ti-6Al-4V[END_REF][START_REF] Castany | In situ transmission electron microscopy deformation of the titanium alloy Ti-6Al-4V: Interface behavior[END_REF][START_REF] Sun | Shear banding in commercial pure titanium deformed by dynamic compression[END_REF][START_REF] Wang | Study of twinning/detwinning behaviors of Ti by interrupted in situ tensile tests[END_REF][START_REF] Ren | Microstructure and deformation behavior of Ti-6Al-4V alloy by high-power laser solid forming[END_REF] can be treated as a symbol of plastic strain localization in Ti alloys, which may relate to nucleation and propagation of local voids. Furthermore, it has been proved that shear bands can be treated as vulnerable sites for nucleation, growth and coalescence of voids in the way of numerical simulation [START_REF] Xue | Self-organization of shear bands in titanium and Ti-6Al-4V alloy[END_REF] and experiments [START_REF] Ren | Microstructure and deformation behavior of Ti-6Al-4V alloy by high-power laser solid forming[END_REF][START_REF] Antolovich | Plastic strain localization in metals: origins and consequences[END_REF][START_REF] Qin | Tensile deformation and fracture of Ti-5Al-5V-5Mo-3Cr-1.5Zr-0.5Fe alloy at room temperature[END_REF].

As shown in Fig. 9 In order to understand the interplay between local inhomogeneities and void evolution, EBSD mapping was performed for the regions adjacent to fracture surface. Within these maps (See Fig. 11), distinct orientation gradient can be observed, which means that slip is the main deforming mechanism during deformation process.

From Fig. 11(a)~(c), it can be seen that the fracture trace tends to

(b) (a)
follow the lath boundaries, accompanied with microstructure distortion in the vicinity of ruptured surface. Besides, one crack(marked as crack A) exists in the distorting zone. As shown in Fig. 11(d)&(e), three grains were selected and marked as Grain1, 2 and 3 adjacent to the fracture surface. These grains present a close orientation and tend to align their <0001> axis with the Normal Direction(ND). Moreover, the distortion of the α-colonies close to the crack surface can be clearly observed in these three grains, which is in line with SEM image (shown in Fig. 11(a) ). Fig. 13 presents the view of the map focusing on the region neighboring to fracture trace (marked by yellow dash line). One grain, which was marked as Grain 4, shows no obvious orientation gradient (grain or colony rotation) inside. This could indicate slip may not occur within in this grain, which may reveal that deformation performs in term of twinning rather than slip in Grain 4. This could be verified by the appearance of deformation twins within this grain(indicated by yellow arrows). But it is worth to notice that no twins can be found in the grains adjacent to the fracture trace(Grain 1, 2 and 3). In other words, slip is the dominating approach during plastic deformation.

Discussion

Alloys with different microstructure morphologies certainly exhibit different damage mechanisms. From simple morphological arguments (spherical inclusions induce less stress concentration than lamellar ones). It was shown in [START_REF] Attanasio | Influence of Material Microstructures in Micromilling of Ti6Al4V Alloy[END_REF][START_REF] Jaffery | Wear Mechanism Analysis in Milling of Ti-6Al-4V alloy[END_REF][START_REF] Fan | Relationships among the Microstructure, Mechanical Properties, and Fatigue Behavior in Thin Ti6Al4V[END_REF] that α Ti alloys or (α+β) Ti alloys with bimodal or equiaxed microstructures exhibit better strength and ductility than the material with a fully lamellar microstructure.

For the alloys with bimodal [START_REF] Shi | Elastic plastic deformation of TC6 titanium alloy analyzed by in-situ synchrotron based X-ray diffraction and microstructure based finite element modeling[END_REF][START_REF] Helbert | The influence of stress trlaxiality on the damage mechanisms in an equiaxed α/β Ti-6AI-4V alloy[END_REF][START_REF] Lecarme | Heterogenous void growth revealed by in situ 3-D X-ray microtomography using automatic cavity tracking[END_REF][START_REF] Zhang | Failure mechanism and mode of Ti-6Al-4V alloy under uniaxial tensile loading: Experiments and micromechanical modeling[END_REF][START_REF] Kumar | Stress triaxiality effect on fracture behavior of IMI-834 titanium alloy: A micromechanics approach[END_REF][START_REF] Chapman | Characterisation of short fatigue cracks in titanium alloy IMI 834 using X-ray microtomography[END_REF] or equiaxed microstructures [START_REF] Semiatin | Cavitation during hot tension testing of Ti-6Al-4V[END_REF][START_REF] Bieler | An experimental and theoretical investigation of the effect of local colony orientations and misorientation on cavitation during hot working of Ti-6Al-4V[END_REF][START_REF] Pineau | Failure of metals I: Brittle and ductile fracture[END_REF], it is known that the precursors of microcracks, voids, tend to nucleate at α/β interfaces due to plastic strain incompatibilitites between the two phases. In this situation, stress triaxiality, a significant parameter in damage procedure, may vary seriously during damage development. Besides, Helbert et al [START_REF] Helbert | The influence of stress trlaxiality on the damage mechanisms in an equiaxed α/β Ti-6AI-4V alloy[END_REF] have suggested that for Ti-6Al-4V alloy with equiaxed microstructure, under low triaxiality (T<0.7), voids appear and grow within α particles. For high triaxiality (T>0.7), voids prefer to nucleate at the interface of α/β phases.

Whereas, in this work, the result is not consistent with Helbert et al's suggestion [START_REF] Helbert | The influence of stress trlaxiality on the damage mechanisms in an equiaxed α/β Ti-6AI-4V alloy[END_REF]. During the whole deformation procedure, stress triaxiality tends to maintain a low and nearly constant value (0.45<T< 0.55, see Fig. 7). But voids prefer to nucleate and propagate at the interface of α/β phases, within shear bands and grains/colonies boundaries instead of within α phase.

Hence, it is rational to consider that stress triaxiality could not be the dominating factor during damage development in this work. According to experimental results, multi-scale shear stress may play a significant role in void/micro-crack evolution. The influence of shear stress on voids/micro-crack evolution will be interpreted at different scales.

Voids initiation and propagation at the interface of α/β phase(micro-scale).

Because of different crystallographic structure for α phase (HCP structure) and β phase (BCC structure), different numbers of slip systems can be activated for α phase and β phase. This can result in the so-called -plastic deformation incompatibility‖ and could stimulate the initiation of voids in local region, especially at the interface between α phase and β phase. Strain gradient and local stress concentration could evolve during plastic deformation for dual phase Ti alloys due to this plastic deformation incompatibility of α phase and β phase [START_REF] Ankem | The role of elastic interaction stresses on the onset of plastic flow for oriented two ductile phase structures[END_REF][START_REF] Ankem | A rationalization of stress-strain behavior of two-ductile phase alloys[END_REF][START_REF] Ashby | The deformation of plastically non-homogeneous materials[END_REF].

From the literature [START_REF] Tan | Slip transmission behavior across α/β interface and strength prediction with a modified rule of mixtures in TC21 titanium alloy[END_REF][START_REF] Chan | Effects of Deformation-Induced Constraint on High-Cycle Fatigue in Ti Alloys with a Duplex Microstructure[END_REF][START_REF] Tan | Cyclic Deformation and Microcrack Initiation During Stress Controlled High Cycle Fatigue of a Titanium Alloy[END_REF][START_REF] Chan | Changes in fatigue life mechanism due to soft grains and hard particles[END_REF][START_REF] Hashimoto | The role of elastic interaction stresses on the onset of slip in polycrystalline alpha brass-I. Experimental determination of operating slip systems and qualitative analysis[END_REF][START_REF] Tan | Effect of α-phase morphology on low-cycle fatigue behavior of TC21 alloy[END_REF], the soft α phase can undertake higher plastic strain than β phase. This could lead to dislocation glide and high local strain concentration at the interface. Local shear stress and subsequent local shear band could be generated at the interface, which shows in terms of obscure microstructure kink and distortion (see As presented in Fig. 9(c)~(e), it can be noticed that voids locate within shear band which is in the vicinity of the boundary. As pointed by Q.Xue et al [START_REF] Xue | Self-organization of shear bands in titanium and Ti-6Al-4V alloy[END_REF], the nucleation of voids can be treated as a result of tensile stress inside shear band. Furthermore, the directions of the shear bands in Fig. 9(c) & (e) tend to tilt from the tensile deforming direction(Z axis) at an angle of 51° and 48°, respectively. These are approximately equal to the angle between Z axis and the direction of highest shear stress in this situation, 45°. Thus, it is rational to infer that the plastic strain localization (shear band) can act as a stimulant to void nucleating at or adjacent to the boundaries.

As for the voids within the shear band, which are shown in between these neighbor grains/colonies, could induce void evolution at the boundaries(see crack A in Fig. 11 ).

In the authors' previous work [START_REF] Dang | 3D X-ray Tomographic analysis of crack nucleation and growth by void coalescence in an α/β Ti alloy lamellar microstructure[END_REF], voids prefer to nucleate at junction of α colonies and grow along the boundary. When plastic deformation occurs, neighbor grains/colonies tend to rotate in order to respect this deformation and to activate slip systems. Since these neighbor grains/colonies have distinct initial crystallographic orientations, they have to rotate to different direction in order to activate slip systems.

Hence, severe distorting and shear stress, which could lead to the nucleation of voids, could be generated along the boundary of such

Conclusions

The damage development of Ti-6Al-4V alloy with lamellar microstructure was analyzed by employing in situ tensile test based on Synchrotron X-ray micro tomography. Furthermore, the origins of shear stress generation and role of shear playing in void initiation and propagation was conducted by post mortem using SEM and EBSD techniques. The main conclusions are as follows.

1) For Ti-6Al-4V titanium alloy with full lamellar microstructure, slip is the dominant approach during uniaxial tensile deformation, as a symbol of clear orientation gradient from EBSD map. After deforming, obvious microstructure distortion and kinking can be found, which refers to the existence of severe plastic stress/strain localization in local region.

2) Void, locating at the central region, initiate as soon as the alloy starts to yield. Then, such voids grow towards the direction with an inclining angle from the tensile direction of 23°. Besides, the value of stress triaxiality nearly keeps as a constant during the whole damage development. Therefore, it could be considered shear stress could play a significant role rather than stress triaxiality.

3) Origin of shear stress generation is due to microstructure

  Fig.1. Only α phase could be noticed in the result of XRD(Fig.1(a) ), this

Figure1

  Figure1. X-ray diffraction spectrum(a), microstructure morphology at low

Figure. 2

 2 Figure.2 Schematic illustration of tensile sample, (a) geometry of the

Fig. 3 (

 3 Fig.3(a) and (b) present the force-displacement and true stress-strain

Figure. 3

 3 Figure.3 Force-displacement curve (a) and scanning points (b) of TA6V

Figure. 4

 4 Figure.4 Microstructure morphology for the Ti-6Al-4V sample after

Fig. 5

 5 Fig.5 presents the typical evolution of the reconstructed vertical

Figure. 5

 5 Figure.5 Tomographic slices parallel to the tensile axis at different

Figure.6 3 -

 3 Figure.6 3-D presentation of the reconstructed volume for the sample in

Fig. 7

 7 Fig.7 shows development of stress triaxiality T as a function of local

Figure. 7

 7 Figure.7 The stress striaxiality as a function of the local strain, calculated

  Fig.8(c) ). Comparing the result of Fig.8(b) with (c), it can be found that

Figure. 8 3 . 4

 834 Figure.8 presents the variation in angle between stress axis and void

  (b), it can be noticed that voids locate at the inner part of shear band(pointed out by blue arrows), which means that formation of shear band could induce heterogeneous strain distribution and thus motivate inner void nucleation. Moreover, these voids tend to advance along this shear band, which may be induced by occurrence of local shear deformation within this region. If focusing on the area of shear band (rectangle area in Fig.9(b) ), as shown in Fig.9(c), it can be clearly observed that voids prefer to be produced either at the interface of α lamellae and β lath (pointed by blue arrows) or the boundary between α colonies (pointed by yellow arrows).The same void nucleating behavior could be found in the other part of this specimen, which is shown in Fig.9(d)&(e). Hence, it can be considered that microscopic inhomogeneities (α phase and β phase) and local plastic incompatibilities (α colonies with different crystallographic orientations) may trigger the formation of shear bands, and lead to the initiation of voids.

Figure. 9

 9 Figure.9 SEM images for central plane of the fractured sample. (a) zig-

Figure. 10

 10 Figure.10 SEM images for the distribution of the voids within one grain.

Figure. 11

 11 Figure.11 EBSD maps for the fracture region. (a) linked SEM image, (b)

Fig. 12

 12 Fig.12 depicts the rotation of these three grains in terms of Inverse

Grain 3 ,

 3 distinct rotating deviation could be noticed in IPF LD and IPF TD, which could refer to different rotating levels for these two grains, even they have rotated to activate the same slip system. This may be resulted from different constraint for grains rotation; and could promote the damage initiation at the boundary of Grain 1 and Grain 3(Crack A).

Figure. 12

 12 Figure.12 Inverse Pole Figure of Grain 1, 2 and 3 located at the vicinity

Figure. 13

 13 Figure.13 Closer view of the grain neighboring on the fracture trace, IPF

Fig. 10 (

 10 Fig.10(a)&(b) ). It was treated as -micro-shear band‖ in order to

Fig. 14 (

 14 Fig.14 (a)~(c) illustrate the formation of micro-shear band and

Figure. 14

 14 Figure.14 Schematic diagrams of micro-shear band formation and voids

Fig. 9 (

 9 Fig.9(d)&(e), they tend to elongated to elliptical shape along the

Figure. 15 4 . 3

 1543 Figure.15 Schematic diagrams of grain-scale shear behavior and its

  neighbor grains/colonies(See Fig.9(b)~(d) ). Schematically, the generation of shear stress resulting from grains/colonies and formation of voids was illustrated in Fig.16.

Figure. 16

 16 Figure.16 Schematic diagrams of shear behavior formation at boundaries

  inhomogeneity and consequently, plastic strain localization. Within one grain/colony, this shear stress, which results from deformation incompatibility of α phase and β phase, can promote voids initiation along the interface of α / β phases. Within the range of one grain, shear band can become the preferred site for nucleation, growth and coalescence of voids. Within the range of 2~3 neighbor grains, during the deformation, formation of shear stress at the boundary could be attributed to the heterogeneous rotation of neighbor grains which exhibit distinct initial crystallographic orientation. This shear stress could -tear‖ the boundary in order to accommodate voids nucleating along it.

Acknowledgement

The authors (Ning DANG, Chaoli MA and Lian ZHOU) are grateful to the financial support by International Science and Technology Cooperation Program of China [grant numbers 2015DFA51430], National Natural Science Foundation of China [grant numbers 51671012, 51671007and 51401010] and Aeronautical Science Foundation of China [grant numbers 2015ZF51069] and to carry out this work. Also the author, Ning DANG, would like to thank the China Scholarship Council [grant numbers 201406020005] for financial support. Besides, the authors would like to appreciate Prof. Yongmei WANG in BaoTi Group Co.Ltd and Prof. Yingli YANG in Northwest Institute For Non-ferrous Metal Research of providing Ti-6Al-4V alloy. ESRF beam line ID19 is acknowledged for providing beam time.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.


Daniel-Mihail Ioan

Florin Stoican
email: florin.stoican@acse.pub.ro

Karl Worthmann
email: karl.worthmann@tu-ilmenau.de

Active Fault Detection and Isolation in a Zonotopic Framework

Keywords: active FDI, switching dynamics, zonotopes

We consider a plant affected by multiple faults (modeled through a piecewise affine formalism). Using a bank of finite-window observers and an artificially-induced feedback delay we provide an exact fault detection and isolation (FDI) mechanism, integrated into the overall fault tolerant control scheme. We use zonotopic characterizations for the sets of interest in order to provide explicit conditions for FDI exactness (such that a fault occurrence can be signaled unambiguously) and to alleviate the numerical issues specific to set operations.

I. INTRODUCTION

Fault tolerant control (FTC) is one of today's main topics of interest in the control community [START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF]. The recent proliferation of large-scale, complex systems has raised the chances of fault occurrences (either at actuator, sensor or plant dynamics level). Thus, a reliable FTC scheme with an exact fault detection and isolation (FDI) together with a robust reconfiguration mechanism is becoming essential.

In what follows, we consider a set-based approach [START_REF] Seron | Multisensor switching control strategy with fault tolerance guarantees[END_REF], [START_REF] Olaru | Positive invariant sets for fault tolerant multisensor control schemes[END_REF]. Having bounding sets for noises, disturbances and model variations it is possible to bound the signals of interest and thus, to characterize explicitly separation conditions which ensure FDI. To this end we use the set-theoretic notions of robust positive invariance and reachability. The former allows offline computations and gives a priori stability guarantees while the later allows to handle the transitional behavior sparked by fault(s) occurrences [START_REF] Seron | Fault tolerant control allowing sensor healthy-to-faulty and faulty-to-healthy transitions[END_REF]. There are several issues in the literature which are not usually tackled:

i) The FDI separation condition may not be verified for the current operation conditions. ii) The FDI mechanism may need a non-zero observation window to asses a fault occurrence. This means that the closed-loop scheme may use faulty information. iii) The computation of invariant / reachable sets is cumbersome for large dimensions and/or for long intervals [START_REF] Blanchini | Set-theoretic methods in control[END_REF]. For issue i) the key is to observe that the sets involved are parametrized after variables influenced by the control scheme [START_REF] Stoican | Reference governor design for tracking problems with fault detection guarantees[END_REF]. Therefore, choosing them such that the condition is always respected avoids the issue of false alarms or missed faults.

To simplify the set computations and help with issue ii) we induce a delay in the feedback generation. Assuming fault persistence and given reference dynamics, 'steady' and transitional sets which bound the tracking error and estimation error dynamics are computed. Residual sets (coming from the output estimation sets) are used to check the FDI condition. Since these sets are parametrized after the reference state and input, we use these variables to guarantee exact FDI.

Lastly, to handle issue iii), we consider zonotopic sets. A sub-class of the polyhedral sets, they are increasingly used due to their resilience to the "dimensionality curse' [START_REF] Xu | Sensor-fault tolerance using robust mpc with set-based state estimation and active fault isolation[END_REF], e.g, for reachability analysis [START_REF] Althoff | Computing reachable sets of hybrid systems using a combination of zonotopes and polytopes[END_REF], collision detection [START_REF] Guibas | Zonotopes as bounding volumes[END_REF] or guaranteed state estimation [START_REF] Guibas | Zonotopes as bounding volumes[END_REF]. In contrast to polytopes, the operations with zonotopic sets are significantly less computationally demanding, and, also, do not raise numerical instabilities [START_REF] Althoff | Zonotope bundles for the efficient computation of reachable sets[END_REF], [START_REF] Althoff | Cora 2015 manual[END_REF]. Toolboxes like CORA [START_REF] Althoff | Cora 2015 manual[END_REF] handle zonotopic sets representations and operations efficiently.

Combining all these elements we reach an explicit formulation which ensures FDI and involves reference inputs and states. Due to the nonlinear nature of the problem (a bilevel formulation), we consider the mixed integer formalism proposed in [START_REF] Scott | Input design for guaranteed fault diagnosis using zonotopes[END_REF] but, instead of directly controlling the inputs applied to the control scheme, we control the reference inputs and assume the feedback law already given. Notation: The Minkowski sum of two sets, A and B is denoted as A ⊕ B = {x : x = a + b, a ∈ A, b ∈ B}. conv(S) denotes the convex hull of set S.

II. PRELIMINARIES

In the rest of the paper sets will be used to bound various signals with the end goal of detecting and isolating unambiguously a fault occurrence. This implies the use of (robust) invariance, set projection, set addition and all the other tools used in set-theoretic methods.

While there are multiple choices for representing a system (e.g., via polyhedral or ellipsoidal sets [START_REF] Blanchini | Set-theoretic methods in control[END_REF]), in this paper we consider zonotopes as they provide an excellent compromise between numerical complexity and fidelity of representation: Definition 1 ( [START_REF] Kühn | Rigorously computed orbits of dynamical systems without the wrapping effect[END_REF]): A zonotope is a centrally symmetric polytope, which can be described as a Minkowski sum of line segments. In its generator representation a zonotope Z is described by center c ∈ R n and generators g 1 , . . . , g ng ∈ R n :

Z = {c + ng ∑ i=1 ξ i g i : ∥ξ∥ ∞ ≤ 1} (1)
or, compactly with G = [g 1 , . . . , g ng ], as

Z = {Gξ + c|ξ ∈ R ng , ∥ξ∥ ∞ ≤ 1} (2)
In the next sections, we use notation Z(G, c) to denote [START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF]. Let us consider two zonotopes:

Z 1 = Z(G 1 , c 1 ) ⊂ R n , Z 1 = Z(G 2 , c 2 ) ⊂ R n
and a matrix R ∈ R m×n . Then the following propeties hold [START_REF] Fukuda | From the zonotope construction to the Minkowski addition of convex polytopes[END_REF]: i) is closed under linear transformation:

RZ(G 1 , c 1 ) = Z(RG 1 , Rc 1 ); (3)
ii) is closed under Minkowski sum:

Z(G 1 , c 1 )⊕Z(G 2 , c 2 ) = Z( [ G 1 G 2 ] , c 1 +c 2 ); (4)
iii) is symmetric, up to its center:

-Z 1 = -Z{G 1 , c 1 } = Z{G 1 , -c 1 }. ( 5
)
In particular, the Minkowski addition and projection operations greatly simplify for zonotopic sets with respect to their polyhedral counterparts and make large-scale computations feasible and numerically robust [START_REF] Althoff | Computing reachable sets of hybrid systems using a combination of zonotopes and polytopes[END_REF].

For further use, we define the notion of robust positive invariance (RPI).

Definition 2: For LTI dynamics

x k+1 = Ax k + δ k with δ k ∈ ∆, the set Ω is called RPI iff the set inclusion AΩ ⊕ ∆ ⊆ λΩ, (6)
holds for 0 < λ ≤ 1. When 0 < λ < 1 we call Ω contractive and convergence inside it in a finite time is guaranteed. It is well-known [START_REF] Kolmanovsky | Theory and computation of disturbance invariant sets for discrete-time linear systems[END_REF] that the minimal RPI (mRPI) set Ω ∞ can be obtained through the set recurrence:

Ω 0 = {0}, Ω k+1 = AΩ k ⊕ ∆. (7)
In general, the limit set of (7) cannot be obtained explicitly. Instead, various methods exist for computing arbitrarily close approximations [START_REF] Rakovic | Invariant approximations of robustly positively invariant sets for constrained linear discrete-time systems subject to bounded disturbances[END_REF]. The repeated additions in [START_REF] Xu | Sensor-fault tolerance using robust mpc with set-based state estimation and active fault isolation[END_REF] are manageable for zonotopic sets. In fact, due to (3) and (4), and assuming that ∆ = Z(G, c) we have that

Ω k = Z( [ G . . . A k-1 G ] , c + . . . A k-1 c) (8)
Using [START_REF] Althoff | Computing reachable sets of hybrid systems using a combination of zonotopes and polytopes[END_REF] for k → k +1 and introducing it in [START_REF] Stoican | Reference governor design for tracking problems with fault detection guarantees[END_REF] we have that, for a desired λ, the minimal value of k for which [START_REF] Stoican | Reference governor design for tracking problems with fault detection guarantees[END_REF] holds is

k * = arg min k A k Z(G, c) ⊆ λZ(G, c). (9)

III. PROBLEM DESCRIPTION

Let us consider the dynamics

x k+1 = A(i k )x k + B(i k )u k + r(i k ) + B w (i k )w k (10a) y k = C(i k )x k + s(i k ) + D v (i k )v k (10b)
where x k , x k+1 ∈ R n denote the current and successor states, 10) is both general enough to handle realistic dynamics (subject to model variation, noises, faults, etc.) and restrictive enough to provide, coupled with the tools from Section II, numerically manageable constructions. In particular, we use [START_REF] Althoff | Zonotope bundles for the efficient computation of reachable sets[END_REF] to model dynamics which alternate between a healthy mode (by convention, i k = 0) and various pre-defined1 faulty modes (i k ̸ = 0). What remains to be assessed is the fault occurrence (via a FDI mechanism).

u k ∈ R m the input, y k ∈ R p the output, w k ∈ W ⊂ R

A. State estimation

Let us assume a bank of observers, one per mode of functioning. That is, we define the j-th observer as a finitewindow Luenberger observer whose internal model is given by the the j-th mode of dynamics [START_REF] Althoff | Zonotope bundles for the efficient computation of reachable sets[END_REF]:

xj k-τ +1 = xk-τ+1 (11a)
xj ℓ+1 = A(j)x j ℓ + B(j)u ℓ + r(j) + L j (y ℓ -ŷj ℓ ) (11b) ŷj ℓ = C(j)x j ℓ + s(j), (11c)
for ℓ = k -τ + 1, . . . , k and xk-τ+1 , the state reference at time instant k -τ + 1, to be defined later. Dynamics ( 11) can be interpreted as follows: at the current moment 'k', all the available information of the last τ instants of time (u k-τ +1 , . . . , u k , y k-τ +1 , . . . , y k ), together with initialization xj k-τ +1 = xk-τ+1 provide a state estimation xj k and output estimation ŷj k . While arguably this construction discards some information (i.e., older information from the estimation), it will prove to be useful for the FDI mechanism.

Combining [START_REF] Althoff | Zonotope bundles for the efficient computation of reachable sets[END_REF] and [START_REF] Althoff | Cora 2015 manual[END_REF] we get the state and output estimation error, xj k x k -x j k and ỹj k y k -ŷ j k , respectively, associated to the j-th observer:

xj k-τ +1 = z k-τ +1 (12a) xj ℓ+1 = [ A(j) -L j C(j) ] xj ℓ + [ (A(i ℓ ) -A(j)) -L j (C(i ℓ ) -C(j)) ] x ℓ + [B(i ℓ ) -B(j)] u ℓ + r(i ℓ ) -r(j) + B w (i ℓ )w ℓ -L j [s(i ℓ ) -s(j)] -L j D v (i ℓ )v ℓ , ( 12b
)
ỹj ℓ =C(j)x j ℓ + [C(i ℓ ) -C(j)] x ℓ + s(i ℓ ) -s(j) + D v v ℓ , (12c)
for ℓ = k -τ + 1, . . . , k and z k-τ +1 , the tracking error at time instant k -τ + 1, to be defined later. Matrix L j is taken such that the closed-loop state-matrix

A L j = A(j) -L j C(j) is stable (always possible if the pair (A(j), C(j)) is observable).
Note that when the model used in the observer and the active dynamics coincide (i.e., i k = j), [START_REF] Scott | Input design for guaranteed fault diagnosis using zonotopes[END_REF] reduces to:

xj k-τ +1 = z k-τ +1 (13a) xj ℓ+1 =A L j xj ℓ + B w (j)w ℓ -L j D v (j)v ℓ , ( 13b
)
ỹj ℓ =C(j)x j ℓ + D v (j)v ℓ . ( 13c
)
for ℓ = k -τ + 1, . . . , k and z k-τ +1 , the tracking error at time instant k -τ + 1, to be defined later.

B. Tracking error

Let us assume the reference dynamics (with ūj k persistent references a priori given):

xj k+1 = A(j)x j k + B(j)ū j k + r(j), (14a)
ȳj k = C(j)x j k + s(j). ( 14b
)
Remark 1: The reference models [START_REF] Fukuda | From the zonotope construction to the Minkowski addition of convex polytopes[END_REF] are in fact the nominal models (with noises w k and v k discarded) from [START_REF] Althoff | Zonotope bundles for the efficient computation of reachable sets[END_REF].

To close the loop we take

u k = ūi ′ k k + K i ′ k (x i ′ k k-τ +1 - xi ′ k k-τ +1 ) (15)
where the selection of feedback matrix, state estimation and reference values

(K i ′ k , xi ′ k k-τ +1 , ūi ′ k k , xi ′ k k
) will be done by the fault tolerant scheme described in Section IV-B. Note that the use of an artificially delayed information (the state estimation

xi ′ k k-τ +1
) is required by the FDI construction). A typical measure of performance and stability is the analysis and bounding of the state and output tracking errors 10), ( 14) and ( 15) we have:

z j k x k -xj k and ξ j k y k -ȳj k respectively. Gathering (
z k+1 = A(i k )z k + B(i k )K i ′ k z k-τ +1 + [r(i k ) -r(i ′ k )] + [A(i k ) -A(i ′ k )] xi ′ k k + [B(i k ) -B(i ′ k )] ūi ′ k k -B(i k )K i ′ k xi ′ k k-τ +1 + B w (i k )w k (16a) ξ k = C(i k )z k + [C(i k ) -C(i ′ k )] xi ′ k k + D v (i k )v k + [s(i k ) -s(i ′ k )] . ( 16b
)
Matrices K i ′ k are taken such that the closed-loop dynamics are stable [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach[END_REF]. Assuming that the active dynamics [START_REF] Althoff | Zonotope bundles for the efficient computation of reachable sets[END_REF] and the selected gain and state estimation coincide (i.e., i k = i ′ k = j), the tracking error dynamics [START_REF] Rakovic | Invariant approximations of robustly positively invariant sets for constrained linear discrete-time systems subject to bounded disturbances[END_REF] become

z k+1 =A(j)z k + B(j)K j z k-τ +1 -B(j)K j xj k-τ +1 + B w (j)w k (17a) ξ k =C(j)z k + D v (j)v k . ( 17b
)
With [START_REF] Kolmanovsky | Theory and computation of disturbance invariant sets for discrete-time linear systems[END_REF] we revisit [START_REF] Scott | Input design for guaranteed fault diagnosis using zonotopes[END_REF] and highlight the reference state and input:

xj k-τ +1 =z k-τ +1 (18a) xj ℓ+1 =A L j xj ℓ + [( A(i ′ ℓ ) -A(j)) -L j (C(i ′ ℓ ) -C(j)) ] (x i ′ ℓ ℓ + z ℓ ) + [B(i ′ ℓ ) -B(j)] (ū ℓ + K i ′ ℓ (z ℓ-τ +1 - xi ′ ℓ ℓ-τ +1 )) + r(i ′ ℓ ) -r(j) + B w (i ′ ℓ )w ℓ -L j [s(i ′ ℓ ) -s(j)] -L j D v (i ′ ℓ )v ℓ , ( 18b
)
ỹj ℓ =C(j)x j ℓ + [C(i ′ ℓ ) -C(j)] (x i ′ ℓ ℓ + z ℓ ) + s(i ′ ℓ ) -s(j) + D v (i ′ ℓ )v ℓ , ( 18c
)
for ℓ = k -τ + 1, . . . , k. We use index i ′ ℓ when rewriting u ℓ : for now we make no assumption on the selected index in [START_REF] Kolmanovsky | Theory and computation of disturbance invariant sets for discrete-time linear systems[END_REF].

C. Set characterizations

For further use we require the sets which characterize the state estimation error [START_REF] Scott | Input design for guaranteed fault diagnosis using zonotopes[END_REF] and tracking error ( 16) dynamics.

Recall that by construction, observers (11) have a finite window (i.e., the k-th state and output estimations are derived only from the information available in the interval k -τ + 1, . . . , k). Hence, whatever happens outside of this interval has no influence in the estimations and, implicitly, on the sets which bound them.

First, several assumptions are necessary: A1) any two consecutive switches are separated by at least τ + τ c instants of time; A2) (15) uses correct information (the indices of the state estimation and of the dynamics are matched, i k = i ′ k = j); Assuming that dynamics [START_REF] Althoff | Zonotope bundles for the efficient computation of reachable sets[END_REF] have been under the j-th mode for at least τ consecutive time instants (Assumption A1)), the estimation error of the j-th observer follows [START_REF] Kühn | Rigorously computed orbits of dynamical systems without the wrapping effect[END_REF], thus allowing to compute the corresponding bounding sets:

Xj,j k-τ +1 ={z k-τ +1 }, (19a) Xj,j ℓ+1 =A L j Xj,j ℓ ⊕ B w (j)W ⊕ {-L j D v (j)V}, (19b) Ỹj,j ℓ =C(j) Xj,j ℓ ⊕ D v (j)V. ( 19c
)
for ℓ = k -τ + 1, . . . , k. Consequently, we have inclusions xj ℓ+1 ∈ Xj,j ℓ+1 and ỹj ℓ ∈ Ỹj,j ℓ . Assuming that the correct estimation is used in (15) -Assumption A2) the tracking error follows [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach[END_REF]. Further assuming that xj k-τ +1 ∈ Xj,j , w k ∈ W hold, an invariant set Z j,j is computed as follows: using the extended tracking error state

z k = [ z ⊤ k z ⊤ k-1 . . . z ⊤ k-τ
] ⊤ , we obtain an invariant set which guarantees z k ∈ Z j . Projecting along the subspace associated with z k we obtain the set which bounds z k while under the selected index j:

z k ∈ Z j,j . ( 20
)
Up to now we have shown the sets bounding the 'steady' estimation and tracking error (when the observer and plant model coincide). For further use, we have to consider the case of model mismatch as well.

Assuming that dynamics [START_REF] Althoff | Zonotope bundles for the efficient computation of reachable sets[END_REF] have been under the j'th mode for at least τ consecutive time instants (Assumption A1)), the estimation error of the j-th observer follows (18) with the additions i ℓ = i ′ ℓ = j ′ and xj ′ ℓ-τ ∈ Xj ′ ,j ′ . This leads to bounding sets for the state/output estimation parametrized after the tracking error and state/input references:

Xj,j ′ k-τ +1 ={z k-τ +1 } (21a) Xj,j ′ ℓ+1 =A L j Xj,j ′ ℓ ⊕{[( A (j ′ ) -A(j)) -L j (C(j ′ ) -C(j)) ] (x j ′ ℓ + z ℓ )} ⊕ [ B(j ′ ) -B(j)] ({ū j ′ ℓ } ⊕ K j ′ ({z ℓ-τ } ⊕ {-Xj ′ ,j ′ })) ⊕{r(j ′ ) -r(j)} ⊕ B w (j ′ )W ⊕{-L j [s(j ′ ) -s(j)]} ⊕ {-L j D v (j ′ )V}, (21b) Ỹ j,j ′ ℓ =C(j) Xj,j ′ ℓ ⊕ {[C(j ′ ) -C(j)] (x j ′ ℓ + z ℓ )} ⊕{s(j ′ ) -s(j)} ⊕ D v (j ′ )V, (21c)
for ℓ = k -τ + 1, . . . , k. As expected, (21) reduces to (19) when j = j ′ . Lastly, we evaluate the behavior of the tracking error in the interval after a fault and while the correct observer is not yet correctly selected. Using [START_REF] Rakovic | Invariant approximations of robustly positively invariant sets for constrained linear discrete-time systems subject to bounded disturbances[END_REF] and with the notations i k → j, i ′ k → j ′ and assuming that the switch j → j ′ happens at k and that xj ′ ℓ-τ ∈ Xj ′ j ′ we have the set recurrence:

Z j,j ′ k = {z k } (22a) Z j,j ′ ℓ+1 = A(j)Z j,j ′ ℓ ⊕ {B(j)K j ′ z ℓ-τ +1 } ⊕ {-B(j)K j ′ Xj ′ j ′ } ⊕ B w (j)W ⊕ {[A(j) -A(j ′ )] xj ′ ℓ-τ +1 } ⊕ {[B(j) -B(j ′ )] ūj ′ k }. ( 22b
) for ℓ = k + 1, . . . , k + τ . Remark 2:
The sets defined here are parametrized after the tracking error and after the reference state and input values 2 . The tracking errors appearing in (19a), (21a)-( 21b) and (22a) will be replaced with bounding sets as resulted from the FTC scheme. The reference state and input appearing in (21b)-( 21c) and (22b) will be used as decision variables to ensure FDI.

IV. FAULT TOLERANT CONTROL SCHEME

The use of bounding sets allows an exact FDI implementation (without missed faults and false alarms) but it requires that the set inclusions assumed in the construction of ( 19), ( 21), ( 20) and ( 22) hold at all times.

First we discuss the set inclusions and their convergence to 'steady' values and second we show how to explicitly choose the reference states and inputs such that FDI is guaranteed.

A. Fault tolerant control scheme implementation

As a first step we revisit the sets obtained earlier and show the stability of the closed-loop scheme. To better clarify these issues, let us consider the following fault scenario, also illustrated in Fig. 1 where, at k = k 1 + 1 the plant switches from functioning mode j to mode j ′ ': I) For k ≤ k 1 the plant dynamics are in the j-th mode for a sufficiently long time and the plant/control models are matched (i k = i ′ k = j). Therefore, the tracking error and the estimation errors are in their respective sets (z k ∈ Z j,j , xj k ∈ Xj,j (Z j,j ), xj ′ k ∈ Xj ′ ,j (Z j,j ). II) From k = k 1 + 1 onwards the dynamics switch to mode i k = j ′ (i.e., a (different) fault occurs). Hence, the tracking error is no longer guaranteed to use a matched reference model (i ′ k ̸ = j ′ ) and thus z k is given by dynamics [START_REF] Rakovic | Invariant approximations of robustly positively invariant sets for constrained linear discrete-time systems subject to bounded disturbances[END_REF] and its bounding set enlarges (z k ∈ Z j ′ ,j k ). The estimation errors are indefinite since they are based on mixed information (the observation window contains data from both the j and j ′ modes of functioning). III) From k = k 1 + τ + 1 and until k = k 1 + τ + τ c the j ′ -th observer retrieves the correct estimation and the 2 To keep the notation simple we ignored this aspect but whenever necessary, we will employ the full form. For example, Ỹk closed-loop is guaranteed to use the correct information

(i k = i ′ k = j ′ )
. The tracking error is not yet in its invariant set (Z j ′ ,j ′

) but converges towards it (in τ c time instants, computed a priori). The estimation error for the mismatched observers (for any j ̸ = j ′ ) is computed iteratively via (21) but is cumbersome due to the terms z k1+τ +1 . . . z k1+τ +τc which are bounded by transitional sets. At k = k 1 + τ + τ c we have inclusion z k ∈ Z j ′ ,j ′ and hence the estimation errors (and associated output errors) are again within their 'steady' sets. Recalling Remark 2, note that in step I) we use Xj,j (Z j,j ), Xj ′ ,j (Z j,j ). By this, we mean that in [START_REF] Kühn | Rigorously computed orbits of dynamical systems without the wrapping effect[END_REF] and (21) we replace z k-τ +1 and, z k-2τ +1 . . . z k , respectively, with Z j,j .

B. Active fault detection and isolation

Recall that within our framework, the plant is called "underfault" whenever the active index in [START_REF] Althoff | Zonotope bundles for the efficient computation of reachable sets[END_REF] is i k ̸ = 0. Thus, we require so-called residual signals which are sensible to specific faults occurrences and whose behavior can unambiguously state whether the system is under fault (detection) and, if so, which is the active fault (isolation). In here we take the observer outputs [START_REF] Althoff | Cora 2015 manual[END_REF] as the residuals.

From ( 19) and ( 21) we have that the output (12b) can stay in one of N possible output sets (in Ỹj,j whenever the plant and observer models coincide and in one of the remaining N-1 sets, Ỹj,j ′ when j ̸ = j ′ ). Hence, a sufficient condition to unambiguously decide whether the plant is in the j-th mode of functioning is to have

Ỹj,j (Z j,j ) ∩ Ỹj,j ′ (Z j,j ) = ∅, ∀j ̸ = j ′ , (23)
Then, the set inclusion ỹj k ∈ Ỹj,j is unambiguous (in the sense that ỹj k ∈ Ỹj,j implies ỹj k / ∈ Ỹj,j ′ , ∀j ′ ̸ = j). Repeating (23) for each of the observers allows to uniquely identify the fault (out of the N observers just the one where the internal model and the dynamics model coincide will respect ỹj k ∈ Ỹj,j ):

i ′ k = arg min j: ỹj k ∈ Ỹj,j ||ỹ j k || 2 . ( 24
)
Using the index from (24) into (15) closes the loop and guarantees that the tracking error dynamic remains in a predefined domain. Several remarks are in order. Remark 3: Any fault could be isolated from a single observer as long as j ′ ̸ = j ′′ ↔ Ỹj,j ′ ∩ Ỹj,j ′′ = ∅. However, in general this is hard to check and since we already have a bank of observers at our disposal, we prefer to check (23).

Remark 4: Usually the problem of "closing the loop" is ignored when devising the FDI mechanism, i.e., the effects of the fault are ignored when analyzing the residuals. As it can be clearly seen in ( 18) and ( 16) these effects cannot be ignored since τ cannot be chosen arbitrarily large.

Remark 5: Relation (23) assumes that z k ∈ Z j,j holds. This corresponds to step I) from Fig. 1. Consequently, we avoid checking the FDI condition during steps II) and III). Instead, having detected the jump j → j ′ at k = k 1 + 1 we know (via Assumption A1)) that no other switch will happen in this interval. Hence, we let the estimation error and tracking error sets to settle and re-activate the FDI mechanism only after the lapsing of τ + τ c instants. Thus, we avoid the cumbersome calculations of the interconnected transitional estimation and tracking error sets resulting from ( 21) and ( 22).

k 1 k 1 + τ k 1 + τ + τ c k 1 -τ k 1 + 1 k 1 + τ + 1 j j' k k I: z k ∈ Z j , xj k ∈ X j,j , xj k ∈ X j ,j II: z k ∈ Z j,j k , xj k , xj k indefinite III: z k → Z j , xj k ∈ X j ,j , xj k ∈ X j,j z k1+τ ∈ Z j,j k1+τ → z k1+τ +τc ∈ Z j
Up to now, we have only stated the constraints which ensure FDI. The more interesting question is whether it is possible (and if so, how) to force the validation of the FDI condition (23). The key is to notice that Ỹj,j ′ is parametrized after reference variables 3 xj k-τ +1 , ūj k-τ +1 . . . ūj k . What remains is to extract the 'variable' part out of Ỹj,j ′ . To this end, let us define the following operators:

A N A =      A A 2 . . . A N      , B N A,B =      B . . . 0 AB . . . 0 . . . . . . . . . A N -1 B . . . B      (25a) C N C,A,B = [ CA N -1 B . . . CB ] . (25b)
Further denoting

xj k = [ xj,⊥ k-τ +1 . . . xj,⊥ k ] ⊥ , ūj k = [ ūj,⊥ k-τ +1 . . . ūj,⊥ k ] ⊥
and X(j ′ , j) = X(j ′ ) -X(j) we have that the variable part in Ỹj,j ′ is given by

C τ C(j),A L j ,B(j ′ ,j) ūj k + C τ C(j),A L j ,A L j,j ′ xj k + C(j ′ , j)x j k (26)
with A L j,j ′ = A(j ′ , j) -L j C(j ′ , j). Using [START_REF] Fukuda | From the zonotope construction to the Minkowski addition of convex polytopes[END_REF] we have that

xj k =A τ A(j) x j k-τ +1 + B τ A(j),B(j) ūj k + B τ A(j),I rj (27a) xj k =A τ (j)x j k-τ +1 + C τ I,A(j),B(j) ūj k (27b) Introducing (27) in (26) we have [ C τ C(j),A L j ,A L j,j ′ A τ A(j) + C(j ′ , j)A τ (j) ] xj k-τ +1 + [ C τ C(j),A L j ,B(j ′ ,j) + C τ C(j),A L j ,A L j,j ′ B τ A(j),B(j) +C(j ′ , j)C τ I,A(j),B(j) ] ūj k . ( 28
)
Let us denote with F j,j ′ and G j,j ′ the matrices which multiply vectors xj k-τ +1 and ūj k respectively, in (28). Further, we 3 Variables xj k-τ +2 . . . xj k do not appear since they can be expressed, via [START_REF] Fukuda | From the zonotope construction to the Minkowski addition of convex polytopes[END_REF], through the former.

denote the fixed part of Ỹj,j ′ by Ỹj,j ′ , * . This includes the fixed offset B τ A(j),I r j which was extracted from (28). All these notations allow to reformulate (23) as follows:

F j,j ′ xj k-τ +1 + G j,j ′ ūj k / ∈ Ỹj,j (Z j,j ) ⊕ {-Ỹj,j ′ , * (Z j,j )}, (29)
for all j ′ ̸ = j.

The resulting problem is non-convex: the vector of elements

[ (x j k-τ +1 ) ⊤ ( ūj k ) ⊤ ] ⊤
has to reside outside of a union of convex sets. The solution proposed here is to use the construction from [START_REF] Scott | Input design for guaranteed fault diagnosis using zonotopes[END_REF] which exploits the zonotopic nature of the sets for a compact representation of the problem. A mixed integer formulation was employed to reformulate the complementarity condition resulting from the rewriting of the bilevel problem.

V. ILLUSTRATIVE EXAMPLE

We take the example from [START_REF] Scott | Input design for guaranteed fault diagnosis using zonotopes[END_REF], where a second-order piecewise affine system is considered:

A(1) = [ 0.6 0.2 -0.4 -0.2 ] , B(1) = [ 1 0 0 1 ] C(1) = [ 1 0 ] , s(1) = [ 1 ] B w (1) = [ 1 0 0 0 ] , D v (1) = [ 1 ] , r(1) = [ 0 0
] with the next modifications for the four faulty models (i = 2, . . . , 5):

B(2) = [ 1 0 0 0 ] , B(3) = [ 0 0 0 1 ] A(4) = [ 1.2 0.2 -0.4 -0.2 ] , A(5) = [ 2.0 0.2 -0.4 -0.7
] .

The necessary sets, in zonotopic notation, are: W = Z(0.5I, 0), V = Z(0.2, 0), the disturbance noises and Ū = Z(9I, 0) and X = Z(0.2I, 0), the bounds for the references.

We take the observation window τ = 5 and proceed iteratively in order to reach the separation problem (29).

First, for each observer [START_REF] Althoff | Cora 2015 manual[END_REF] we choose gain matrices L j via a discrete Riccati equation (with penalty matrices Q = I and R = I). Next, using the set-recurrence (19), we obtain the state and output estimation error bounds in the matched case. Note that, in order to simplify the construction we overapproximate the initialization value ({z k-τ +1 } → X).

Next, to obtain bounding sets (20) we proceed as follows: using the results from [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach[END_REF] we obtain the feedback matrices K j ; compute the invariant set associated to the lifted tracking error z k as in 4 (8) and lastly, we project to the z k subspace.

We illustrate in Fig. 2 the resulted sets for each mode of functioning. The first figure shows the Xjj sets obtained in the case τ = 5. The second figure shows the sets Zjj which incorporate the former sets into their description and show that the closed-loop system is stable in closed-loop. We have now the tools to compute the transitional sets (21). Applying the set-recurrence and taking the tracking error bounds obtained earlier we construct the fixed part of the output sets, Ỹj,j ′ , * , as they appear in (29).

Computing matrices F j,j ′ , G j,j ′ and putting (29) in the mixed-integer formalism of [START_REF] Scott | Input design for guaranteed fault diagnosis using zonotopes[END_REF] we obtain a non-convex optimization problem whose solution is a feasible initial reference state x j k-τ +1 and sequence u j k-τ +1 . . . u j k which ensures FDI form the point of view of the j-th observer.

j

  ,j ′ from (21) may be written as Ỹk j,j ′ (z k-2τ +1 . . . z k , ūk-τ+1 . . . ūk , xk-τ+1 . . . xk ).

Fig. 1 :

 1 Fig. 1: Timeline for a fault scenario with set inclusion illustration.

Fig. 2 :

 2 Fig. 2: Illustration of bounding sets for state estimation and tracking errors (for τ = 5).

D. Ioan is currently with the Laboratory of Signals and Systems, Univ. Paris-Sud -CentraleSupelec-CNRS, Université Paris Saclay.

Note that we assume both the fault models and their magnitudes known.

We fix the value k * =

in[START_REF] Guibas | Zonotopes as bounding volumes[END_REF] and obtain the corresponding scaling factors λ.

VI. CONCLUSIONS

We have used here a bank of finite-window observers and an artificially-induced feedback delay to simplify the set construction and guarantee the exactness of an FDI mechanism. Zonotopes have been used extensively due to their theoretical and numerical properties. Further work will be done for the implementation of a FDI-aware reference governor and for the bounding of the transitional sets sparked by a fault occurrence.

ACKNOWLEDGMENT

The work has been funded by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-2713.