Emilie Morvant

Amaury Habrard

Stéphane Ayache

Majority Vote of Diverse Classifiers for Late Fusion

Keywords: Multimedia analysis, Classifier fusion, Majority vote, Ranking

In the past few years, a lot of attention has been devoted to multimedia indexing by fusing multimodal informations. Two kinds of fusion schemes are generally considered: The early fusion and the late fusion. We focus on late classifier fusion, where one combines the scores of each modality at the decision level.

To tackle this problem, we investigate a recent and elegant well-founded quadratic program named MinCq coming from the machine learning PAC-Bayesian theory. MinCq looks for the weighted combination, over a set of real-valued functions seen as voters, leading to the lowest misclassification rate, while maximizing the voters' diversity. We propose an extension of MinCq tailored to multimedia indexing. Our method is based on an order-preserving pairwise loss adapted to ranking that allows us to improve Mean Averaged Precision measure while taking into account the diversity of the voters that we want to fuse. We provide evidence that this method is naturally adapted to late fusion procedures and confirm the good behavior of our approach on the challenging PASCAL VOC'07 benchmark.

Introduction

Combining multimodal information is an important issue in pattern recognition. Indeed, the fusion of multimodal inputs can bring complementary information from various sources, useful for improving the quality of the final decision. In this paper, we focus on multimodal fusion for image analysis in multimedia systems (see [START_REF] Atrey | Multimodal fusion for multimedia analysis: a survey[END_REF] for a survey). The different modalities correspond generally to a relevant set of features that can be grouped into views. Once these features have been extracted, another step consists in using machine learning methods in order to build voters-or classifiers-able to discriminate a given concept. In this context, two main schemes are generally considered [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF]. On the one hand, in the early fusion approach, all the available features are merged into one feature vector before the learning and classification steps. This can be seen as a unimodal classification. However, this kind of approach has to deal with many heterogeneous features which are sometimes hard to combine. On the other hand, the late fusion 4 works at the decision level by combining the prediction scores available for each modality (see Fig. 1). Even if late fusion may not always outperform early fusion 5 , it tends to give better results in multimedia analysis [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF]. Several methods based on a fixed decision rule have been proposed for combining classifiers such as max, min, sum, etc [START_REF] Kittler | On combining classifiers[END_REF]. Other approaches, often referred to as stacking [START_REF] Wolpert | Stacked generalization[END_REF], need of an extra learning step.

In this paper, we address the problem of late fusion with stacking. Let h i be the function that gives the score associated with the i th modality for any instance x. A classical method consists in looking for a weighted linear combination of the scores seen as a majority vote and defined by: H(x)= n i=1 q i h i (x), where q i is the weight associated with h i . This approach is widely used because of its robustness, simplicity and scalability due to small computational costs [START_REF] Atrey | Multimodal fusion for multimedia analysis: a survey[END_REF]. It is also more appropriate when there exist dependencies between the views through the classifiers [START_REF] Wu | Optimal multimodal fusion for multimedia data analysis[END_REF][START_REF] Ma | Linear dependency modeling for classifier fusion and feature combination[END_REF]. The objective is then to find an optimal combination of the classifiers' scores by taking into account these dependencies. One solution is to use machine learning methods to assess automatically the weights [START_REF] Kuncheva | Combining Pattern Classifiers: Methods and Algorithms[END_REF][START_REF] Dietterich | Ensemble methods in machine learning[END_REF][START_REF] Re | Ensemble methods: a review[END_REF][START_REF] Sun | A survey of multi-view machine learning[END_REF]. Indeed, from a theoretical machine learning standpoint: considering a set of classifiers with a high diversity is a desirable property [START_REF] Dietterich | Ensemble methods in machine learning[END_REF]. One illustration is given by the algorithm AdaBoost [START_REF] Freund | Experiments with a new boosting algorithm[END_REF] that weights weak classifiers according to different distributions of the training data, introducing some diversity. However, AdaBoost degrades the fusion performance when combining strong classifiers [START_REF] Wickramaratna | Performance degradation in boosting[END_REF].

To tackle the late fusion by taking into account the diversity between score functions of strong classifiers, we propose a new framework based on a recent machine learning algorithm called MinCq [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF]. MinCq is expressed as a quadratic program for learning a weighted majority vote over real-valued functions called voters (such as score functions of classifiers). The algorithm is based on the minimization of a generalization bound that takes into account both the risk of committing an error and the diversity of the voters, offering strong theoretical guarantees on the learned majority vote. In this article, our aim is to show the usefulness of MinCq-based methods for classifier fusion. We provide evidence that they are able to find good linear weightings, and also performing non-linear combination with an extra kernel layer over the scores. Moreover, since in multimedia retrieval, the performance measure is related to the rank of positive examples, we extend MinCq to optimize the Mean Average Precision. We base this extension on an additional order-preserving loss for verifying ranking pairwise constraints.

The paper is organized as follows. The framework of MinCq is introduced in Section 2. Our extension for late classifier fusion is presented in Section 3 and it is evaluated on an image annotation task in Section 4. We conclude in Section 5.

MinCq: A Quadratic Program for Majority Votes

We start from the presentation of MinCq [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF], a quadratic program for learning a weighted majority vote of real-valued functions for binary classification. Note that this method is based on the machine learning PAC-Bayesian theory, first introduced in [START_REF] Mcallester | PAC-bayesian model averaging[END_REF].

We consider binary classification tasks over a feature space X ⊆ R d of dimension d. The label space (or output space) is Y = {-1, 1}. The training sample of size m is S = {(x i , y i )} m i=1 where each example (x i , y i ) is drawn i.i.d. from a fixed-but unknown-probability distribution D defined over X ×Y . We consider a set of n realvalued voters H, such that: ∀h i ∈ H, h i : X → R. Given a voter h i ∈ H, the predicted label of x ∈ X is given by sign[h i (x)], where sign[a] = 1 if a ≥ 0 and -1 otherwise. Then, the learner aims at choosing the weights q i , leading to the Q-weighted majority vote B Q with the lowest risk. In the specific setting of MinCq6 , B Q is defined by,

B Q (x) = sign [H Q (x)] , with H Q (x) = n i=1 q i h i (x),
where ∀i ∈ {1, . . . , n},

n i=1 |q i | = 1 and -1 ≤ q i ≤ 1. Its true risk R D (B Q ) is defined as the probability that B Q misclassifies an example drawn according to D, R D (B Q ) = P (x,y)∼D (B Q (x) = y) .
The core of MinCq is the minimization of the empirical version of a bound-the C-Bound [START_REF] Lacasse | PAC-Bayes bounds for the risk of the majority vote and the variance of the gibbs classifier[END_REF][START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF]-over R D (B Q ). The C-Bound is based on the notion of Q-margin, which is defined for every example (x, y) ∼ D by: yH Q (x), and models the confidence on its label. Before expounding the C-Bound, we introduce the following notations respectively for the first moment M D Q and for the second moment

M D Q 2 of the Q-margin, M D Q = E (x,y)∼D yH Q (x) = E (x,y)∼D n i=1 yq i h i (x), M D Q 2 = E (x,y)∼D (yH Q (x)) 2 = E (x,y)∼D n i=1 n i ′ =1 q i q i ′ h i (x)h i ′ (x). (1)
By definition, B Q correctly classifies an example x if the Q-margin is strictly positive. Thus, under the convention that if yE h∼Q h(x) = 0, then B Q errs on (x, y), we have:

∀D over X × Y, R D (B Q ) = Pr (x,y)∼D yH Q (x) ≤ 0 .
Knowing this, the authors of [START_REF] Lacasse | PAC-Bayes bounds for the risk of the majority vote and the variance of the gibbs classifier[END_REF][START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF] have proven the following C-bound over R D (B Q ) by making use of the Cantelli-Chebitchev inequality.

Theorem 1 (The C-bound). Given H a class of n functions, for any weights {q i } n i=1 , and any distribution

D over X×Y , if E (x,y)∼D H Q (x) > 0 then R D (B Q ) ≤ C D Q where, C D Q = Var (x,y)∼D (yH Q (x)) E (x,y)∼D (yH Q (x)) 2 = 1 - (M D Q ) 2 M D Q 2 .
In the supervised binary classification setting, [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF] have then proposed to minimize the empirical counterpart of the C-bound for learning a good majority vote over H, justified by an elegant PAC-Bayesian generalization bound. Following this principle the authors have derived the following quadratic program called MinCq.

argmin Q Q t S M S Q -A t S Q, (2)
s.t. m t S Q = µ 2 + 1 nm m j=1 n i=1 y j h i (x j ), (3)
and ∀i ∈ {1, . . . , n}, 0

≤ q ′ i ≤ 1 n , (4)
(M inCq) where t is the transposed function, Q = (q ′ 1 , . . . , q ′ n ) t is the vector of the first n weights q i , M S is the n×n matrix formed by 1 m m j=1 h i (x j )h i ′ (x j ) for (i, i ′ ) in {1, . . . , n} 2 , A S = 1 nm n i=1 m j=1 h 1 (x j )h i (x j ), . . . , 1 nm n i=1 m j=1 h n (x j )h i (x j ) t
, and,

m S = 1 m m j=1 y j h 1 (x j ), . . . , 1 m m j=1 y j h n (x j ) t .
Finally, the majority vote learned by MinCq is

B Q (x) = sign[H Q (x)], with, H Q (x) = n i=1 2q ′ i -1 n qi h i (x).
Concretely, MinCq minimizes the denominator of the C-bound (Eq. ( 2)), given a fixed numerator, i.e. a fixed Q-margin (Eq. ( 3)), under a particular regularization (Eq. ( 4)) 7 .

Note that, MinCq has showed good performances for binary classification.

A New Framework for Classifier Late Fusion

MinCq stands in the particular context of machine learning binary classification. In this section, we propose to extend it for designing a new framework for multimedia late fusion. We actually consider two extensions for dealing with ranking, one with pairwise preferences and a second based on a relaxation of these pairwise preferences to lighten the process. First of all, we discuss in the next section the usefulness of MinCq in the context of multimedia late fusion.

Justification of MinCq as a Classifier Late Fusion Algorithm

It is well known that diversity is a key element in the success of classifier combination [START_REF] Atrey | Multimodal fusion for multimedia analysis: a survey[END_REF][START_REF] Kuncheva | Combining Pattern Classifiers: Methods and Algorithms[END_REF][START_REF] Dietterich | Ensemble methods in machine learning[END_REF][START_REF] Fakeri-Tabrizi | Multiview semi-supervised ranking for automatic image annotation[END_REF]. From a multimedia indexing standpoint, fuzing diverse voters is thus necessary for leading to good performances. We quickly justify that this is exactly what MinCq does by favoring majority votes with maximally uncorrelated voters.

In the literature, a general definition of diversity does not exist. However, there are popular diversity metrics based on pairwise difference on every pair of individual classifiers, such as Q-statistics, correlation coefficient, disagreement measure, etc. [START_REF] Kuncheva | Combining Pattern Classifiers: Methods and Algorithms[END_REF][START_REF] Leonard | Applying machine learning diversity metrics to data fusion in information retrieval[END_REF] We consider the following diversity measure assessing the disagreement between the predictions of a pair of voters according to the distribution D,

diff D (h i , h i ′ ) = E (x,y)∼D h i (x)h i ′ (x).
7 For more technical details on MinCq please see [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF].

We then can rewrite the second moment of the Q-margin (see Eq.( 1)),

M D Q 2 = n i=1 n i ′ =1 qiq i ′ diff D (h i , h i ′ ). (5)
The first objective of MinCq is to reduce this second moment, implying a direct optimization of Eq. ( 5). This implies a maximization of the diversity between voters: MinCq favors maximally uncorrelated voters and appears to be a natural way for late fusion to combine the predictions of classifiers separately trained from various modalities.

MinCq for Ranking

In many applications, such as information retrieval, it is well known that ranking documents is a key point to help users browsing results. The traditional measures to evaluate the ranking ability of algorithms are related to precision and recall. Since a low-error vote is not necessarily a good ranker, we propose in this section an adaptation of MinCq to allow optimization of the Mean Averaged Precision (MAP) measure.

Concretely, given a training sample of size 2m we split it randomly into two subsets S ′ and S = {(x j , y j )} m j=1 of the same size. Let n be the number of modalities. For each modality i, we train a classifier h i from S ′ . Let H = {h 1 , . . . , h n } be the set of the n associated prediction functions and their opposite. Now at this step, the fusion is achieved by MinCq: We learn from S the weighted majority vote over H with the lowest risk. We now recall the definition of the MAP measured on S for a given real-valued function h. Let S + = {(x j , y j ) : (x j , y j ) ∈ S ∧y j = 1} = {(x j + , 1)} m + j + =1 be the set of the m + positive examples from S and S -= {(x j , y j ) :

(x j , y j ) ∈ S ∧ y j = -1} = {(x j -, -1)} m - j -=1
the set of the m -negative examples from S (m + +m -= m). For evaluating the MAP, one ranks the examples in descending order of the scores. The MAP of h over S is,

M AP S (h) = 1 |m + | j:y j =1
P rec@j, where P rec@j is the percentage of positive examples in the top j. The intuition is that we prefer positive examples with a score higher than negative ones.

MinCq with Pairwise Preference. To achieve this goal, we propose to make use of pairwise preferences [START_REF]Preference Learning[END_REF] on pairs of positive-negative instances. Indeed, pairwise methods are known to be a good compromise between accuracy and more complex performance measure like MAP. Especially, the notion of order-preserving pairwise loss was introduced in [START_REF] Zhang | Statistical analysis of some multi-category large margin classification methods[END_REF] in the context of multiclass classification. Following this idea, [START_REF] Yue | A support vector method for optimizing average precision[END_REF] have proposed a SVM-based method with a hinge-loss relaxation of a MAP-loss. In our specific case of MinCq for late fusion, we design an order-preserving pairwise loss for correctly ranking the positive examples. For each pair (x j +, x j -) ∈ S + ×S -, we want,

H Q (x j + ) > H Q (x j -) ⇔ H Q (x j -)-H Q (x j + ) < 0.
This can be forced by minimizing (according to the weights) the following hinge-loss relaxation of the previous equation (where [a] + = max(a, 0) is the hinge-loss),

1 m + m - m + j + =1 m - j -=1 n i=1 2qi -1 n hi(x j -) -hi(x j + ) H Q (x j -)-H Q (x j + ) + . (6)
To deal with the hinge-loss of ( 6), we consider m + × m -additional slack variables ξ S + ×S -= (ξ j + j -) 1≤j + ≤m + ,1≤j -≤m -weighted by a parameter β > 0. We make a little abuse of notation to highlight the difference with (M inCq): Since ξ S + ×S -appear only in the linear term, we obtain the following quadratic program (M inCq P W ),

argmin Q,ξ S + ×S -Q t S MSQ -A t S Q + β Id t ξ S + ×S -, s.t. m t S Q = µ 2 + 1 nm m j=1 n i=1 yjhi(xj), ∀(j + , j -) ∈ {1, .., m + }×{1, .., m -}, ξ j + j -≥ 0, ξ j + j -≥ 1 m + m - n i=1 2q ′ i -1 n hi(x j -)-hi(x j + ) ,
and ∀i ∈ {1, . . . , n}, 0

≤ q ′ i ≤ 1 n , (M inCqP W )
where Id = (1, . . . , 1) of size (m + ×m -). However, one drawback of this method is the incorporation of a quadratic number of additive variables (m + ×m -) which makes the problem harder to solve. To overcome this problem, we relax this approach as follows.

MinCq with Average Pairwise Preference. We relax the constraints by considering the average score over the negative examples: we force the positive ones to be higher than the average negative scores. This leads us to the following alternative problem (M inCq P W av ) with only m + additional variables.

argmin Q,ξ S + Q t S MSQ -A t S Q + β Id t ξ S + , s.t. m t S Q = µ 2 + 1 nm m j=1 n i=1 yjhi(xj), ∀j + ∈ {1, . . . , m + }, ξ j + ≥ 0, ξ j + ≥ 1 m + m - m - j -=1 n i=1 2q ′ i -1 n hi(x j -)-hi(x j + ) ,
and ∀i ∈ {1, . . . , n}, 0

≤ q ′ i ≤ 1 n , (M inCqP W av )
where Id = (1, . . . , 1) of size m + . Note that the two approaches stand in the original framework of MinCq. In fact, we regularize the search of the weights for majority vote leading to an higher MAP. To conclude, our extension of MinCq aims at favoring Q-majority vote implying a good trade-off between classifiers maximally uncorrelated and leading to a relevant ranking.

Experiments on PascalVOC'07 benchmark

Protocol. In this section, we show empirically the usefulness of late fusion MinCqbased methods with stacking. We experiment these approaches on the PascalVOC'07 benchmark [START_REF] Everingham | The PASCAL visual object classes challenge[END_REF], where the objective is to perform the classification for 20 concepts. The corpus is constituted of 10, 000 images split into train, validation and test sets. For most of concepts, the ratio between positive and negative examples is less than 10%, which leads to unbalanced dataset and requires to carefully train each classifier. For simplicity reasons, we generate a training set constituted of all the training positive examples and negative examples independently drawn such that the positive ratio is 1/3. We keep the original test set. Indeed, our objective is not to provide the best results on this benchmark but rather to evaluate if the MinCq-based methods could be helpful for the late fusion step in multimedia indexing. We consider 9 different visual features, that are • LCH are 3×3×3 histogram on a grid of 8×6 or 4×3 blocs. Color Moments are represented by the two first moments on a grid of 8×6 or 4×3 blocs. • HOG is computed on a grid of 4 × 3 blocs. Each bin is defined as the sum of the magnitude gradients from 50 orientations. Thus, overall EDH feature has 600 dimensions. HOG feature is known to be invariant to scale and translation. • LBP is computed on grid of 2×2 blocs, leading to a 1, 024 dimensional vector. The LBP operator labels the pixels of an image by thresholding the 3×3-neighborhood of each pixel with the center value and considering the result as a decimal number.

LBP is known to be invariant to any monotonic change in gray level. • Percept features are similar to SIFT codebook where visual words are related to semantic classes at local level. There are 15 semantic classes such as 'sky', 'skin', 'greenery', 'rock', etc. We also considered SIFT features from a dense grid, then map it on a codebook of 1000 visual words generated with Kmeans.

We train a SVM-classifier for each feature with the LibSVM library [START_REF] Chang | LIBSVM: a library for support vector machines[END_REF] and a RBF kernel with parameters tuned by cross-validation. The set H is then constituted by the 9 score functions associated with the SVM-classifiers.

In a first series of experiments, the set of voters H is constituted by the 9 SVMclassifiers. We compare our 3 MinCq-based methods to the following 4 baselines:

• The best classifier of H:

h best = argmax hi∈H M AP S (h i ).
• The one with the highest confidence: best(x) = argmax hi∈H |h i (x)|.

• The sum of the classifiers (unweighted vote): • The MAP-weighted vote:

Σ(x) = hi∈H h i (x).
Σ M AP (x) = h i ∈H M AP S (hi) h i ′ ∈H M AP S (h i ′ ) h i (x).
In a second series, we propose to introduce non-linear information with a RBF kernel layer for increasing the diversity over the set H. We consider a larger H as follows. Each example is represented by the vector of its scores with the 9 SVM-classifiers and H is now the set of kernels over the sample S: Each x ∈ S is seen as a voter k(•, x). We compare this approach to classical stacking with SVM.

Finally, for tuning the hyperparameters we use a 5-folds cross-validation process, where instead of selecting the parameters leading to the lowest risk, we select the ones leading to the best MAP. The MAP-performances are reported on Tab. 1 for the first series and on Tab. 2 for the second series. Results. Firstly, the performance of Σ M AP fusion is lower than Σ, which means that the performance of single classifiers is not correlated linearly with its importance on the fusion step. On Tab. 1, for the first experiments, we clearly see that the linear MinCqbased algorithms outperform on average the linear baselines. MinCq-based method produces the highest MAP for 16 out of 20 concepts. Using a Student paired t-test, this result is statistically confirmed with a p-value < 0.001 in comparison with Σ M AP , best and h best . In comparison of Σ, the p-values respectively associated with (M inCq P W av ), (M inCq P W ) and (M inCq P W ) are 0.0139, 0.0232 and 0.0088. We can remark that (M inCq P W ) implies lower performances than its relaxation (M inCq P W av ). A Student test leads to a p-value of 0.223, which statistically means that the two approaches produce similar results. Thus, when our objective is to rank the positive examples before the negative examples, the average constraints appear to be a good solution. However, we note that the order-preserving hinge-loss is not really helpful: The classical (M inCq) shows the best MAP results (with a p-value of 0.2574). Indeed, the trade-off between diversity and ranking is difficult to apply here since the 9 voters are probably not enough expressive. On the one hand, the preference constraints appear hard to satisfy, on the other hand, the voters' diversity do not really vary. The addition of a kernel layer allows us to increase this expressivity. Indeed, Tab. 2 shows that the MinCq-based methods achieve the highest MAP for every concept in comparison with SVM classifier. This confirms that the diversity between voters is well modeled by MinCq algorithm. Especially, M inCq rbf P W av with the averaged pairwise preference is significantly the best: a Student paired test implies a p-value of 0.0003 when we compare M inCq rbf P W av to SVM, and the p-value is 0.0038 when it is compared to M inCq rbf . Thus, the the order-preserving loss is a good compromise between improving the MAP and keeping a reasonable computational cost. Note that we do not report the results for (M inCq P W ) in this context, because the computational cost is much higher and the performance is lower. The full pairwise version implies too many variables which penalize the resolution of (M inCq P W ). Finally, it appears that at least one MinCq-based approach is the best for each concept, showing that MinCq methods outperform the other compared methods. Moreover, a Student test implies a pvalue < 0.001 when we compare M inCq rbf P W av to the approaches without kernel layer. M inCq rbf P W av is significantly then the best approach in our experiments. We conclude from these experiments that MinCq-based approaches are a good alternative for late classifiers fusion as it takes into account the diversity of the voters. In the context of multimedia documents retrieval, the diversity of the voters comes from either the variability of input features or by the variability of first layer classifiers.

Conclusion and Perspectives

In this paper, we proposed to make use of a well-founded learning quadratic program called MinCq for multimedia late fusion tasks. MinCq was originally developed for binary classification, aiming at minimizing the error rate of the weighted majority vote by considering the diversity of the voters [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF]. We designed an adaptation of MinCq able to deal with ranking problems by considering pairwise preferences while taking into account the diversity of the models. In the context of multimedia indexing, this extension of MinCq appears naturally appropriate for combining the predictions of classifiers trained from various modalities in a late classifier fusion setting. Our experiments have confirmed that MinCq is a very competitive alternative for classifier fusion in the context of an image indexing task. Beyond these results, this work gives rise to many interesting remarks, among which the following ones. Taking advantage of a margin constraint for late classifier fusion may allow us to prove a new C-bound specific to ranking problems, and thus to derive other algorithms for classifier fusion by maximizing the diversity between the classifiers. This could be done by investigating some theoretical results using the Cantelli-Chebychev's inequality [START_REF] Devroye | A Probabilistic Theory of Pattern Recognition[END_REF] as in [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF]. Additionally, it might be interesting to study the impact of using other diversity metrics [START_REF] Kuncheva | Combining Pattern Classifiers: Methods and Algorithms[END_REF] on performances for image and video retrieval. Such an analysis would be useful for assessing a trade-off between the quality of the ranking results and the diversity of the inputs for information retrieval. Finally, another perspective, directly founded on the general PAC-Bayes theory [START_REF] Mcallester | PAC-bayesian model averaging[END_REF], could be to take into account a prior belief on the classifiers of H. Indeed, general PAC-Bayesian theory allows one to obtain theoretical guarantees on majority votes with respect to the distance between the considered vote and the prior belief measured by the Kullback-Leibler divergence. The idea is then to take into account prior information for learning good majority votes for ranking problems.

Fig. 1 .

 1 Fig. 1. Classical late classifier fusion scheme.

Table 1 .

 1 MAP obtained on the PascalVOC'07 test sample.

	concept	M inCq P W av	M inCq P W	M inCq	Σ	Σ M AP	best	h best
	aeroplane	0.487	0.486	0.526	0.460	0.241	0.287	0.382
	bicycle	0.195	0.204	0.221	0.077	0.086	0.051	0.121
	bird	0.169	0.137	0.204	0.110	0.093	0.113	0.123
	boat	0.159	0.154	0.159	0.206	0.132	0.079 0.258
	bottle	0.112	0.126	0.118	0.023	0.025	0.017	0.066
	bus	0.167	0.166	0.168	0.161	0.098	0.089	0.116
	car	0.521	0.465	0.495	0.227	0.161	0.208	0.214
	cat	0.230	0.219	0.220	0.074	0.075	0.065	0.116
	chair	0.257	0.193	0.230	0.242	0.129	0.178	0.227
	cow	0.102	0.101	0.118	0.078	0.068	0.06	0.101
	diningtable	0.118	0.131	0.149	0.153	0.091	0.093	0.124
	dog	0.260	0.259	0.253	0.004	0.064	0.028	0.126
	horse	0.301	0.259	0.303	0.364	0.195	0.141	0.221
	motorbike	0.141	0.113	0.162	0.193	0.115	0.076	0.130
	person	0.624	0.617	0.604	0.001	0.053	0.037	0.246
	pottedplant	0.067	0.061	0.061	0.057	0.04	0.046	0.073
	sheep	0.067	0.096	0.0695	0.128	0.062	0.064	0.083
	sofa	0.204	0.208	0.201	0.137	0.087	0.108	0.147
	train	0.331	0.332	0.335	0.314	0.164	0.197	0.248
	tvmonitor	0.281	0.281	0.256	0.015	0.102	0.069	0.171
	Average	0.240	0.231	0.243	0.151	0.104	0.100	0.165

SIFT, Local Binary Pattern (LBP), Percepts, 2 Histograms Of Gradient (HOG), 2 Local Color Histograms (LCH) and 2 Color Moments (CM):

Table 2 .

 2 MAP obtained on the PascalVOC'07 test sample with a RBF kernel layer.

	concept	M inCq rbf P W av	M inCq rbf	SVM rbf
	aeroplane	0.513	0.513	0.497
	bicycle	0.273	0.219	0.232
	bird	0.266	0.264	0.196
	boat	0.267	0.242	0.240
	bottle	0.103	0.099	0.042
	bus	0.261	0.261	0.212
	car	0.530	0.530	0.399
	cat	0.253	0.245	0.160
	chair	0.397	0.397	0.312
	cow	0.158	0.177	0.117
	diningtable	0.263	0.227	0.245
	dog	0.261	0.179	0.152
	horse	0.495	0.450	0.437
	motorbike	0.295	0.284	0.207
	person	0.630	0.614	0.237
	pottedplant	0.102	0.116	0.065
	sheep	0.184	0.175	0.144
	sofa	0.246	0.211	0.162
	train	0.399	0.385	0.397
	tvmonitor	0.272	0.257	0.230
	Average	0.301	0.292	0.234

The late fusion is sometimes called multimodal classification or classifier fusion.

For example, when one modality provides significantly better results than others.

In PAC-Bayes these weights are modeled by a distribution Q over H s.t. ∀hi ∈ H, qi = Q(hi).

Acknowledgments. This work was in parts funded by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 308036. the authors would like to thanks Thomas Peel for useful comments.


Kai Zhang

Gérard Scorletti

Mohamed Ichchou
email: mohamed.ichchou@ec-lyon.fr

Fabien Mieyeville

Quantitative robust LPV H ∞ vibration control of flexible structures for saving the control energy

Keywords: LPV H ∞ control, phase and gain control policies, saving the control energy, parametric and dynamic uncertainties

In this article, a general and systematical quantitative robust linear parameter varying (LPV) control method is proposed for active vibration control of LPV flexible structures such that a complete set of control objectives can be considered, especially, the reduction of necessarily required control energy. To achieve this goal, phase and gain control policies are employed in LPV H ∞ control designs for suitable selection of weighting functions.

Problem statement

Since lightweight components are widely used in practical structures for miniaturization and efficiency, these structures become more flexible and more susceptible to vibrations, which may cause significant noises, harmful stresses, malfunctions and even failures. As a result, the flexible structures have naturally become suitable candidates for active vibration control. However, in the presence of random variations in structural properties and/or the errors in the system identification process, the obtained dynamical models inevitably have parametric uncertainties. Besides, since the flexible structures have an infinite number of resonant modes and only the first few ones can be modeled and employed in the controller designs, the high-frequency neglected dynamics are usually represented by a dynamic uncertainty, which could lead to the spillover problem1 [START_REF] Balas | Feedback control of flexible systems[END_REF][START_REF] Iorga | H ∞ control with µ-analysis of a piezoelectric actuated plate[END_REF]. Considering parametria and dynamic uncertainties, in the field of robust active vibration control of flexible structures, a quantitative robust control method is proposed in [START_REF] Zhang | Robust active vibration control of piezoelectric flexible structures using deterministic and probabilistic analysis[END_REF]. It employs phase-and-gain-control-policiesbased output feedback H ∞ control [START_REF] Zhang | Phase and gain control policies for robust active vibration control of flexible structures[END_REF] and reliable robustness analyses to take into account a complete set of control objectives, e.g. the a priori determined specification of vibration reduction and the control energy are explicitly considered, and the robustness properties of the closed-loop system to parametric and dynamic uncertainties are quantitatively verified. This control method is most suitable for Linear Time-Invariant (LTI) plants, where LTI dynamical models are used for the system modeling and an LTI H ∞ controller is correspondingly designed. Besides, all the parametric uncertainties are assumed to be within certain ranges but not measurable.

However, in practice, some plants have measurable time-varying parameters, and thus recently Linear Parameter Varying (LPV) systems have received a rapidly increasing attention to model the dynamics of these plants, due to the fact that they can provide an interesting framework for gain-scheduling control by means of convex optimization [START_REF] Rugh | Research on gain scheduling[END_REF][START_REF] Boyd | Convex Optimization[END_REF]. The LPV systems constitute a class of linear systems whose dynamics usually depend on physical time-varying parameters, which are not a priori known but assumed to be measurable on-line. Such parameters are restricted to vary in predetermined sets and can be used as extra information in the control designs to synthesize parameter-dependent controllers, thus leading to improved control performances compared to classical robust control designs. The LPV system modeling and control designs have been used in a variety of applications as introduced in Mohammadpour and Scherer (2012) and the references therein.

In general, in the presence of parametric and dynamic uncertainties, there exist two approaches to the design of robust controllers for LPV systems: the controllers that do not depend on the variation of the changing parameter, but guarantee the control objectives for all possible dynamical models, e.g. the classical robust or the worst-case controllers as used for LTI systems; the controllers that change according to the variations of the changing parameters, i.e. the parameter-dependent controllers are designed. Using worst-case control designs, the dynamics of LPV systems are modeled with norm bounded uncertainties and no exact knowledge of the uncertain parameters can be considered, even it is available. In contrast, with LPV control designs, the time-varying parameters are assumed to be measured on-line and used in the LPV controller synthesis, which could provide better control performances.

It is notable that, for some particular cases as investigated in this article, both the worst-case controller and the LPV one can satisfy the specification of vibration reduction and a certain level of robustness properties. But, in addition to these normal control objectives, the designed controllers are required to consume as little control energy as possible for their practical implementations [START_REF] Skogestad | Multivariable Feedback Control-Analysis and Design[END_REF], since in some applications very little energy is available for active control, yet passive and semi-active methods cannot meet the control objectives, especially when the control energy is obtained from harvesting systems, e.g. [START_REF] Ichchou | A reduced energy supply strategy in active vibration control[END_REF]; Wang and Inman (2013a,b), and/or low-power storage devices (batteries or super capacitors) as often desirable in aerospace systems, e.g. [START_REF] Moreira | Design of a reduced-order H ∞ controller for smart structure satellite applications[END_REF]; [START_REF] Yang | Mixed H 2 /H ∞ state-feedback design for microsatellite attitude control[END_REF]. As a result, if the control energy is not well considered or even totally neglected in the control designs, the active vibration control systems may eventually be powered off of harvested energy and/or low power storage devices. Moreover, due to the hardware limitations, the control input must be restricted by a prescribed upper bound to avoid the controller saturation and exceeding the actuator operated voltage, e.g. [START_REF] Saberi | Control of Linear Systems with Regulation and Input Constraints[END_REF]; [START_REF] Materazzi | Robust structural control with system constraints[END_REF]. Exceeding the upper bound could cause unexpected behavior of the closed-loop system such as actuator damages, large overshoots, loss of control effectiveness or even a dynamic instability. In addition, as claimed in [START_REF] Assadian | A comparative study of optimal linear controllers for vibration suppression[END_REF], usually the vibration control capability of various controllers is measured using their effects on the sensitivity transfer function in the frequency domain. This fails to provide the control designers a physical measure for comparisons, but ranking controllers based on their energy requirements or control inputs provides an supplement and important physical measure for the controller selection.

Therefore, an important constraint in practical active vibration control designs is the required control energy and the control input. To achieve effective robust controllers, this constraint is critical and really deserves enough attention. In the following, we have an extensive review of various techniques for saving the control energy and reducing the control input:

• [START_REF] Kondoh | The positioning of sensors and actuators in the vibration control of flexible systems[END_REF] propose an optimization criterion for the location selection of actuators and sensors to obtain effective vibration reduction and minimize the control energy. [START_REF] Bardou | Active power minimization and power absorption in a plate with force and moment excitation[END_REF] focus on physical parameter optimization of the plate and the locations of the excitation and the actuator forces to minimize the control energy. In [START_REF] Lee | Optimal control of vibration suppression in flexible systems via dislocated sensor/actuator positioning[END_REF] and [START_REF] Baz | Performance of an active control system with piezoelectric actuators[END_REF], to reduce the required control energy for active vibration control of flexible structures, an optimal direct velocity feedback (DVF) control and a modified independent modal space control are respectively used to determine the optimal locations of the actuators and sensors and the control gains. [START_REF] Kumar | Active vibration control of beams with optimal placement of piezoelectric sensor/actuator pairs[END_REF] numerically reveal that, by optimal placement of collocated piezoelectric actuators and sensors, the designed linear quadratic regulator (LQR) optimal controller can achieve effective vibration reduction of the flexible beam, while requiring a smaller control input compared to DVF control. For vibration control of a thin-walled composite beam, [START_REF] Zorić | Optimal vibration control of smart composite beams with optimal size and location of piezoelectric sensing and actuation[END_REF] employ the fuzzy optimization strategy to determine the size and the location of piezoelectric actuators and sensors. The particle swarm optimization (PSO) based LQR controller is then designed to maximize the closed-loop damping ratios and minimize the control input. Besides, a literature review about optimal placement of piezoelectric actuators and sensors for minimizing the control energy can be found in [START_REF] Gupta | Optimization criteria for optimal placement of piezoelectric sensors and actuators on a smart structure: a technical review[END_REF].

• [START_REF] Assadian | A comparative study of optimal linear controllers for vibration suppression[END_REF] computes the control energy for active vibration control of an vibratory system and investigates the effects of control methods on the control energy, where nonoptimal DVF control, classical The REC law is then implemented in Wang and Inman (2013a,b) to improve unmanned aerial vehicle performance in wind gusts and reduce the control energy which is limited and harvested from ambient wing vibration. In [START_REF] Kumar | Adaptive vibration control of smart structures: a comparative study[END_REF], for active vibration control of an in-verted L structure, the LQR based adaptive controller achieves robust performance and requires smaller control input compared to the pole placement method. [START_REF] Materazzi | Robust structural control with system constraints[END_REF] employ the 'Statedependent Riccati Equation' to reduce the control input, which consists of solving online the LQR problem with adaptive weighting functions and system matrices. In [START_REF] Qiu | Experiments on vibration suppression for a piezoelectric flexible cantilever plate using nonlinear controllers[END_REF], nonlinear controllers are proposed for active vibration control of a piezoelectric cantilever plate, where the control gains are computed with three nonlinear functions to adapt to the measured vibration amplitudes and regulate the control input in real-time for effective vibration reduction and avoiding the control saturation.

H ∞ control
• With classical H ∞ control, related weighting functions are used to tune the bandwidth of the H ∞ controller, thus imposing constraints on the control energy, e.g. the frequency-independent weighting functions are used in [START_REF] Zhang | Robust H ∞ vibration control for flexible linkage mechanism systems with piezoelectric sensors and actuators[END_REF]; [START_REF] Huo | H ∞ robust control design of active structural vibration suppression using an active mass damper[END_REF], and the frequencydependent ones are used in [START_REF] Sivrioglu | Low power consumption nonlinear control with H ∞ compensator for a zero-bias flywheel AMB system[END_REF]; [START_REF] Zhang | Phase and gain control policies for robust active vibration control of flexible structures[END_REF][START_REF] Zhang | Robust active vibration control of piezoelectric flexible structures using deterministic and probabilistic analysis[END_REF]. Based on H ∞ loop shaping designs, [START_REF] Reinelt | H ∞ loop shaping for systems with hard bounds[END_REF][START_REF] Reinelt | Robust control of a Two-Mass-Spring system subject to its input constraints[END_REF][START_REF] Reinelt | Loop shaping of multivariable systems with hard constraints on the control signal[END_REF] investigates active control of multivariable systems with hard bounded control input to avoid the control saturation. This control method assumes the reference signal and its first derivative to be norm bounded, and focuses on the selection of weighting functions which are explicitly related to the upper bound on the control input. The selection procedure is fulfilled until the prescribed upper bound is met and indeed user iterative as performed in [START_REF] Forrai | Robust controller design with hard constraints on the control signal[END_REF] and [START_REF] Forrai | Robust active vibration suppression control with constraint on the control signal: application to flexible structures[END_REF] for active vibration control of a three-storey flexible structure.

In [START_REF] Kumar | Enhanced active constrained layer damping (ACLD) treatment using stand-off-layer: robust controllers design, experimental implementation and comparison[END_REF], LQR control, classical mixed sensitivity H ∞ control, H ∞ loop shaping design and µ synthesis are used for active vibration control of a flexible beam with variable boundary conditions. These controllers are compared in terms of the required control energy and the closed-loop robust performance evaluated with µ analysis [START_REF] Skogestad | Multivariable Feedback Control-Analysis and Design[END_REF]. It shows that, for this specifical case, the H ∞ loop shaping based controller outperforms others in terms of the control energy utilization.

Above literature review proves that, for practical active vibration control designs, it is critical to consider the constraint on the control energy and the control input. It is also shown that, in most of these researches, the constraint is achieved by kinds of optimizations of the placement and sizing of the actuators and sensors, the structural parameters, and the parameters of fixed controllers such as DVF, PID and PPF. However, as claimed in [START_REF] Darivandi | An algorithm for LQ optimal actuator location[END_REF], these optimization methods are generally non-convex and the dynamical models of flexible structures usually have a large number of degrees of freedom. Consequently, these optimization based methods could be inaccurate or computationally impractical. Furthermore, due to physical and installation limitations, sometimes there exists little flexibility for such optimization, for instance, although non-collocated actuators and sensors are not desirable for the closed-loop robust stability, they are unavoidable due to installation restrictions and even recommendable for high degrees of observability and controllability (Bayon de [START_REF] Bayon De Noyer | Single actuator and multi-mode acceleration feedback control[END_REF][START_REF] Kim | A modal filter approach to non-collocated vibration control of structures[END_REF]. Besides, the measurement of all state variables required by LQR is not always practically available, and the specification of vibration reduction and the robustness properties cannot be quantitatively investigated with DVF, PPF, LQR, PID or nonlinear controllers.

On the other hand, the H ∞ loop shaping designs do not directly consider the control energy and only enforce the constraint on the control signal with the following inequality [START_REF] Reinelt | Robust control of a Two-Mass-Spring system subject to its input constraints[END_REF]:

u(s) ∞ ≤ 2n T ud (s) ∞ d(s) ∞ (1)
where, as shown in Figure 1, T ud (s) is the closed-loop transfer function form the disturbance signal d(s) to the control signal u(s), u(s) ∞ represents the maximum amplitude of u(s) and n denotes the McMillan degree of T ud (s) [START_REF] Saberi | Control of Linear Systems with Regulation and Input Constraints[END_REF]. This inequality shows that decreasing T ud (s) ∞ reduces the upper bound for the maximum control input. Therefore, the weighting functions such as W 1 (s) and W 2 (s) are used in the H ∞ loop shaping design

to adjust the open-loop transfer function L(s) = G p (s)K(s) so as to reduce T ud (s) ∞ according the following relationship:

|T ud (jω)| = |G d (jω)K(jω)(1 + G p (jω)K(jω)) -1 | ≈ |G d (jω)K(jω)|, at frequency |L(jω)| = |G p (jω)K(jω)| 1 = |G d (jω)W 1 (jω) K∞ (jω)W 2 (jω)|
where the controller K∞ (s) is designed based on the shaped plant dynamical

model Ĝp (s) = W 2 (s)G p (s)W 1 (s).
These formulations provide a relationship between the upper bound for the maximum control input and related weighting functions. However, in many H ∞ loop shaping designs, e.g. [START_REF] Forrai | Robust controller design with hard constraints on the control signal[END_REF][START_REF] Forrai | Robust active vibration suppression control with constraint on the control signal: application to flexible structures[END_REF], the magnitudes of related weighting functions, e.g. |W 1 (jω)| and |W 2 (jω)|, are tuned in the whole frequency range, that is, the selection is frequency-independent. This selection is relatively simpler than the phase-and-gain-control-policies-based frequency-dependent selection [START_REF] Zhang | Phase and gain control policies for robust active vibration control of flexible structures[END_REF]. But, the gain of the corresponding controller could be very small not only at high frequencies for avoiding the spillover problem and saving the control energy, but also around the controlled resonant frequencies, thus failing to have effective vibration reduction. This implies that the frequency-independent weighting functions cannot provide a good trade-off among various control objectives. Based on above discussions, in order to tackle these critical drawbacks, the main focus of this article is placed on the application of LPV control techniques to develop a general quantitative robust active vibration control method for flexible structures such that the complete set of control objectives could be satisfied, particularly the required control energy and the control input could be reduced. In Section 2, to develop this control method, the Linear Fractional Representation (LFR) [START_REF] Hecker | Enhanced LFR-toolbox for Matlab[END_REF][START_REF] Hecker | Generation of low order LFT representations for robust control applications[END_REF] is used to give a systematical approach for the LPV system modeling, where the scheduled variables, parametric and dynamic uncertainties can be considered H ∞ controller such that the complete set of control objectives are satisfied, especially the required control energy is reduced. To quantitatively verify the robustness properties of the closed-loop system, various robustness analyses are conducted [START_REF] Zhang | Robust active vibration control of piezoelectric flexible structures using deterministic and probabilistic analysis[END_REF]. In Section 3, the design processes and the effectiveness of the proposed control method are illustrated by active vibration control of a non-collocated piezoelectric cantilever beam, where the considered scheduled variable is the position of the external force. This is representative of the systems with parameter-dependent dynamics as investigated in [START_REF] Wood | Control of parameter-dependent mechanical systems[END_REF]; [START_REF] Paijmans | A gain-scheduling-control technique for mechatronic systems with positiondependent dynamics[END_REF], which could be modeled as LPV systems. In addition to the LPV H ∞ control, classical robust H ∞ control is also used for this numerical case. Their nominal control performances and the robustness properties are compared. In Section 4, the effectiveness of these controllers is compared in terms of the control energy, the control input and the system output in the time domain, which cannot be translated precisely to anything tractable in the frequency domain [START_REF] Boyd | Linear Controller Design: Limits of Performance[END_REF] and are not fully investigated in previous active vibration control designs [START_REF] Kumar | Enhanced active constrained layer damping (ACLD) treatment using stand-off-layer: robust controllers design, experimental implementation and comparison[END_REF]. Conclusions and perspectives are present in Section 5.

Proposed quantitative robust LPV control method

2.1. LPV system modeling

An LPV system is a linear system whose dynamics, e.g. defined by a state space representation, depend on time-varying exogenous parameters whose trajectories are a priori unknown. Nevertheless, some information is available such as the intervals to which the parameters and sometimes their derivatives belong to. More formally, let R denote the field of real numbers, an LPV system can be defined as following (Scorletti and Fromion, 2008b):

Definition 2.1. LPV system Let the set Θ t ∈ R n θ be a compact set, Θ be a set of measurable functions from [0, ∞) to R n θ such that for θ(•) ∈ Θ, for all t ≥ 0, θ(t) ∈ Θ t and   A(θ) B(θ) C(θ) D(θ)   (2)
be a continuous matrix function defined from Θ t ∈ R (n+no)×(n+n i ) . A Linear Parameter Varying (LPV) system is defined as

q = Σ LP V (p)              ẋ(t) = A(θ(t))x(t) + B(θ(t))p(t) q(t) = C(θ(t))x(t) + D(θ(t))p(t) x(t 0 ) = x 0 , ∃θ(•) ∈ Θ (3) where x(t) ∈ R n is the state vector, p(t) ∈ R n i the disturbance input, q(t) ∈
R no the output and θ(t) ∈ R n θ the exogenous parameter vector assumed to be measured on-line:

θ(t) = [θ(t), • • • , θ n θ (t)] T .
An LPV system is thus defined by Equation ( 2) and a set Θ. The LPV systems can be usually classified along the class of the set Θ and the class of the state space matrix functions of Σ LP V (s, θ) on θ. In this article, we focus on below class of state space matrices.

Set Θ: The compact set Θ t is a polytope (more precisely an hyperrectangle):

Θ t = θ = [θ 1 , • • • , θ n θ ] T | ∀i = 1, . . . , n θ
The set Θ is defined from Θ t and unbounded parameter rates of variation is used [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related control problems[END_REF][START_REF] Scherer | LPV control and full block multipliers[END_REF]:

Θ = {θ(•) | for all t ≥ 0, θ(t) ∈ Θ t }
There are mainly two kinds of state space matrices dependence on θ: one is that the state space matrices are affine functions of θ and the other one is that the state space matrices are rational functions of θ (Scorletti and Fromion, 2008b). The later one is focused in this article, that is, any rational matrix function in Θ has an Linear Fractional Representation (LFR): there exists

four matrices A Σ , B Σ , C Σ and D Σ of compatible dimensions such that   A(θ) B(θ) C(θ) D(θ)   = D Σ + C Σ ∆ Σ (θ(t))(I -A Σ ∆ Σ (θ(t))) -1 B Σ (4) with ∆ Σ (θ(t)) =            θ 1 (t)I r1 0 • • • • • • 0 0 . . . . . . . . . . . . . . . θ i (t)I r i . . . . . . . . . . . . . . . 0 0 • • • • • • 0 θ n θ (t)I rn θ           
for some r i , i = 1, ..., n θ . Such LPV systems are also referred to as Linear

Fractional Transformation (LFT) systems.

Proposed LPV H ∞ control design

As performed in classical LTI H ∞ control designs, suitable weighting functions are necessary to augment the LPV plant Σ LP V (s, θ), thus developing an augmented LPV plant P au (s, θ) for the controller synthesis. As the complete set of control objectives usually have conflicting requirements on the controller, the selection of the weighting functions has to consider a trade-off among these control objectives in a rational and systematic way.

Besides, in contrast with LTI H ∞ control designs, some weighting functions have to depend on θ to improve some control objectives, e.g. to reduce the required control energy by adapting the bandwidth of the controller to θ.

As shown in Figure 2, with the parameter-dependent input and output weighting functions, i.e. W in (s, θ) and W out (s, θ), the augmented LPV plant P au (s, θ) can be developed and defined as

   z y    = P au       w u                      ẋ(t) = A(θ(t))x(t) + B w (θ(t))w(t) + B u (θ(t))u(t) z(t) = C z (θ(t))x(t) + D zw (θ(t))w(t) + D zu (θ(t))u(t) y(t) = C y (θ(t))x(t) + D yw (θ(t))w(t) (5)
where x(t) ∈ R np is the state vector, u(t) ∈ R nu the control input, y(t) ∈ R ny the measured output, z(t) ∈ R nz the weighted regulated output, w(t) ∈ R nw the exogenous input. The state space matrices of P au (s, θ) are assumed to be rational functions of θ. Based on the definition of P au (s, θ), we consider the LPV control problem:

Design an LPV controller u = K LP V (y) such that with the closed-loop system of Figure 2 denoted by the lower LFT F l (P au , K LP V ) [START_REF] Zhou | Robust and Optimal Control[END_REF]:

• F l (P au , K LP V ) is asymptotically stable;

• F l (P au , K LP V ) satisfies a performance specification, for example, F l (P au , K LP V ) has an L 2 gain less than a given γ, where the L 2 gain is defined as the smallest γ such that for any input w,

T 0 z(t) T z(t)dt ≤ γ 2 T 0 w(t) T w(t)dt, ∀T ≥ 0. For LTI systems, the L 2 gain is equal to the H ∞ norm. Moreover, if the L 2 gain of F l (P au , K LP V ) is no larger than γ, necessarily we have F l (P au (s, θ i ), K LP V (s, θ i )) ∞ ≤ γ, ∀θ i . ) , (  s LPV  ) , (  s K ) , ( 1  s W i ) , (  s W z on ) , (  s W w in ) , ( 1  s W o ) , (  s W ij ) , (  s W ok 1 w j w w n w 1 z k z z n z ) , (  s P au 1 p j p w n p 1 q k q z n q ) , (  s W in ) , (  s W out Figure 2: Augmented LPV plant P au (s, θ)
Evidently, the weighting functions representing the complete set of control objectives are critical to have an efficient K LV P (s, θ) and have to be appropriately determined. In this article, the phase and gain control policies proposed in [START_REF] Zhang | Phase and gain control policies for robust active vibration control of flexible structures[END_REF] are employed for the selection. Numerous LPV controller design approaches have been proposed since last 90's with different levels of conservatism or computational efficiency. A classification of LPV controllers can be obtained based on the following features: the controller parameters, the feedback structure and the dependence of the state matrices of the controller on the parameters. The detailed classification of existing LPV controller, different cases of parameter dependence and available feedback structures can be found in Scorletti and Fromion (2008b). In this article, the controller state space matrices only depend on θ(t) and the output feedback control is used, that is, the output y(t) of the plant P au (s, θ) is assumed to be measured on-line [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related control problems[END_REF]:

u = K LP V (y)      ẋK (t) = A K (θ(t))x K (t) + B K (θ(t))y(t) u(t) = C K (θ(t))x K (t) + D K (θ(t))y(t) (6) where x K (t) ∈ R n K and the matrices A K (θ(t)), B K (θ(t)), C K (θ(t)), D K (θ(t))
have to be synthesized. In this case, we obtain the following state space representation for the closed-loop system

  A cl (θ(t)) B cl (θ(t)) C cl (θ(t)) D cl (θ(t))   =      A(θ(t)) 0 B w (θ(t)) 0 0 0 C z (θ(t)) 0 D zw (θ(t))      +      0 B u (θ(t)) I n 0 0 D zu (θ(t))        A K (θ(t)) B K (θ(t)) C K (θ(t)) D K (θ(t))     A K (θ(t)) I n 0 C y (θ(t)) 0 D yw (θ(t))  
Considering the conservatism and computational efficiency, the LPV control technique proposed in [START_REF] Scorletti | Some results about the stabilization and the L2-gain control of LFT systems[END_REF]; Scorletti and L. EI Ghaoui (1998) is employed for the LPV controller synthesis, which can be solved with LMI constraints as briefly presented in Appendix A.

With the designed LPV controller, reliable deterministic and probabilistic robustness analyses have to be conducted with µ/ν analysis and the random algorithm respectively [START_REF] Zhou | Robust and Optimal Control[END_REF][START_REF] Calafiore | Randomized algorithms for probabilistic robustness with real and complex structured uncertainty[END_REF]. They can take into account the probabilistic information of parametric uncertainties and quantitatively verify the robustness properties both in the deterministic sense and the probabilistic one. According to the results of the robustness analyses, if necessary, the weighting functions used in the control design can be retuned and a trade-off could be made among various control objectives.

The LPV system modeling, the LPV controller design and the robustness analyses consist of the proposed quantitative robust LPV control method, which is general and allows to satisfy the complete set of control objectives.

In this article, the design processes and effectiveness of the control method are subsequently illustrated with active vibration control of a piezoelectric cantilever beam excited by an external position-varying force, which has position-dependent dynamics and is modeled as an LPV system.

Application of the proposed control method

The proposed quantitative robust LPV control is applied to active vibration of a piezoelectric cantilever beam, as shown in Figure 3. It is excited by an external position-varying force F (t, x f ) , i.e. x f is varying within a bounded range and assumed to be measurable in real-time. This is representative of the systems with parameter-dependent dynamics and could be modeled as an LPV system.

Based on the above discussion, some main steps are outlined for the design of a quantitative robust LPV H ∞ controller:

Step 1: Focus on the LPV system modeling to determine the schedule parameter θ and develop the LPV model Σ LP V (s, θ) for the position-dependent dynamics using LFR.

Step 2: According to the complete set of control objectives such as the fixed specification of vibration reduction and the modulus margin, necessary weighting functions are appropriately employed based on phase and gain control policies. Especially, to fully employ the information of θ and improve some control objectives, one or several weighting functions have to depend on θ, for instance, the gain of W i (s, θ), i.e. k W i (θ), depends on θ to reduce the control energy. It is critical to determine k W i (θ) in the controller design: first a finite number of allowable θ j are chosen, which provides the correspond-

ing LTI plant Σ LP V (s, θ j ). Based on Σ LP V (s, θ j ), the corresponding k W i (θ j )
and other weighting functions are selected to develop P au (s, θ j ). Then one LTI H ∞ controller K ∞ (s, θ j ) is achieved to satisfy these control objectives, e.g. F l (P au (s, θ j ), K ∞ (s, θ j )) ∞ ≤ 1. Lastly, based on the chosen θ j and the selected k W i (θ j ), the interpolation of k W i (θ) can be obtained using least mean square method to have k W i (θ) for the infinite number of allowable θ.

Step 3: Based on Σ LP V (s, θ) and the weighting functions, the augmented LPV plant P au (s, θ) is well developed using LFR. Then with the employed LPV control technique, the LPV controller

K LP V (s, θ) can be synthesized, that is, the matrices A K (θ(t)), B K (θ(t)), C K (θ(t)), D K (θ(t)) of Equation (6)
are achieved.

Step 4: Verify that the complete set of control objectives are satisfied with the designed K LP V (s, θ) for any allowable value of θ. With the weighting functions, these control objectives are reduced to F l (P au (s, θ), K LP V (s, θ)) ∞ ≤ 1, ∀θ. As above discussed, when the L 2 gain of F l (P au (s, θ), K LP V (s, θ)) is no larger than one, necessarily we have

F l (P au (s, θ), K LP V (s, θ)) ∞ ≤ 1, ∀θ,
that is, the set of control objectives are satisfied with K LP V (s, θ). Besides, in the presence of parametric and dynamic uncertainties, the robustness properties of the closed-loop system using K LP V (s, θ) are quantitatively verified with deterministic and probabilistic robustness analyses. If some control objectives are not satisfied, return to Step 2 to make a better trade-off among various control objectives by adjusting the weighting functions and employ more values of θ j for a better interpolation of k W i (θ).

LPV modeling of the position-dependent dynamics

As shown in Figure 3, the location of the accelerometer sensor and that of the piezoelectric actuator are determinant, but the location of the external force is varying within a certain range, i.e. x s and x a are fixed and the scheduled variable θ can be introduced for x f such that

x f = θL beam , θ ∈ [θ min , θ max ], 0 < θ min < θ max < 1
where L beam is the total length of the cantilever beam and θ min , θ max determine the allowable position of the force.

Based on modal analysis approach [START_REF] Meirovitch | Elements of Vibration Analysis[END_REF] and the modeling of piezoelectric actuators [START_REF] Moheimani | Piezoelectric Transducers for Vibration Control and Damping[END_REF], applying Laplace voltage applied on the piezoelectric actuator V a (x a , s) to the beam acceleration Ÿ (s, x s ), that is,

G d (s) = Ÿ (s, x s ) F (s, x f ) = n i=1 G di (s) = n i=1 k di (x s , x f )s 2 s 2 + 2ζ i ω i + ω 2 i G p (s) = Ÿ (s, x s ) V a (s, x a ) = n i=1 G pi (s) = n i=1 k pi (x s , x a )s 2 s 2 + 2ζ i ω i + ω 2 i
To determine G d (s) and G p (s), we have to obtain the modal parameters such as the damping ratio ζ i , the natural frequency ω i and the gain k pi/di . Based on the analytical formulations for the Euler-Bernoulli beam bounded with piezoelectric actuators [START_REF] Moheimani | Piezoelectric Transducers for Vibration Control and Damping[END_REF], ω i and k pi depend on x s , x a and the structural properties, e.g. the material properties and the geometrical dimensions. Since these elements are fixed in this case, G p (s) is determined and independent on θ. On the other hand, k di depends on

x f = θL beam , that is, k di (θ) = g i [sinh(λ i θ) -sin(λ i θ)] + h i [cosh(λ i θ) -cos(λ i θ)] (7)
where g i , λ i , h i depend on the determinant structural properties. As shown in Figure 4, for i = 1, 2, 3, the gain k di (θ) has particularly severe dependence on θ such that small variations in θ can generate large variations in the magnitude and the phase of G di (s).

Note that, for a given structure, G di (s, θ) and G pi (s) have the same ω i , and for the sake of simplicity, their damping ratio ζ i is also assumed to be the same. To consider this fact and be readily employed in the control design, for the i th resonant mode, it is desirable to consider the transfer function vector

[G di (s, θ), G pi (s)] with the state space form:

A i =   -2ζ i ω i 1 -ω 2 i 0   ∈ R 2×2 , B i (θ) =   -2ζ i ω i -ω 2 i   [k di (θ) k pi ] ∈ R 2×2 C i = 1 0 ∈ R 1×2 , D i (θ) = [k di (θ) k pi ] ∈ R 1×2
Naturally, when the first n resonant modes of [G d (s, θ), G p (s)] have to be investigated, we have the state space matrices:

A(θ) =         A 1 0 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 A n         ∈ R 2n×2n B(θ) =      B 1 (θ) . . . B n (θ)      =         b 1 0 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 b n         k d (θ) k p ∈ R 2n×2 C(θ) = C 1 , • • • , C n ∈ R 1×2n D(θ) = [1, • • • , 1] k d (θ), k p ∈ R 1×2 (8)
where 0 represents the zero matrix of a compatible dimension,

k p = [k p1 , • • • , k pn ] T ∈ R n×1 , k d (θ) = [k d1 (θ), • • • , k dn (θ)] T ∈ R n×1 and b i =   -2ζ i ω i -ω 2 i   ∈ R 2×1 .
To appropriately consider the dependence of k d (θ) on θ, the LFR of k d (θ) is used:

k d (θ) = [k d1 (θ), • • • , k dn (θ)] T = θ   A k d B k d C k d D k d   (9)
where is the Redheffer star product [START_REF] Zhou | Robust and Optimal Control[END_REF], the matrices

A k d ∈ R m×m , B k d ∈ R m×1 , C k d ∈ R n×m and D k d ∈ R n×1 have
[G d (s, θ), G p (s)] = 1 s                                    A 1 0 • • • 0 b 1 0 • • • 0 0 . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . 0 0 • • • 0 A n 0 • • • 0 b n C 1 , • • • • • • , C n 1, . . . . . . , 1               I 0 0 [k d (θ), k p ]                          = ( 1 s , θ)   Â B Ĉ D   (10)
where I represents the identity matrix of a compatible dimension, the con- and Fromion, 2008a). This is desirable for the controller synthesis and the robustness analysis.

stant matrices  ∈ R (2n+m)×(2n+m) , B ∈ R (2n+m)×2 , Ĉ ∈ R 1×(2n+m
In this article, using x a = 3.5mm, x s = 223.2mm and the structural properties listed in Table 1, we have the nominal modal parameters for the first three resonant modes:

ω i = [295.2, 1850.1, 5180.2], i = 1, 2, 3 k pi = [-8
.9 × 10 -3 , 20.0 × 10 -3 , -10.4 × 10 -3 ], i = 1, 2, 3

ζ i = [20.0 × 10 -3 , 8.0 × 10 -3 , 5.0 × 10 -3 ], i = 1, 2, 3
With θ min = 0.4 and θ max = 0.8, the corresponding matrices for the LFR of

LPV and LTI H ∞ control designs

Both the proposed LPV H ∞ control design and the worst-case H ∞ control design as employed in [START_REF] Zhang | Phase and gain control policies for robust active vibration control of flexible structures[END_REF]

(ω) = 40dB, ∀ ω ∈ R, that is, |T yd (jω, θ j )| ≤ U (ω) = 40dB, ∀ ω ∈ R, ∀θ j ∈ [0.4, 0.8] ( 11
)
where T yd (s) is the closed-loop transfer function from the disturbance d(s)

to the output y(s), as shown in Figure 5.

LPV H ∞ control design

Based on the typical feedback control structure of Figure 1, the augmented LPV plant P au (s, θ) can be well constructed by using a set of necessary and suitable weighting functions W i (s, θ), as shown in Figure 5, where the mea-

surement noise n(s) = W a (s, θ)w 1 (s), the disturbance d(s) = W b (s, θ)w 2 (s), the regulated signals z 1 (s) = W 1 (s, θ)v(s) and z 2 (s) = W 2 (s, θ)u(s). By partitioning P au (s, θ) according to the sizes of z(s) = [z 1 (s), z 2 (s)] T and d n 1 w v u y ) , (  s G d ) , (  s K ) , (  s W a 2 w ) , (  s W b ) (s G p ) , ( 1  s W ) , ( 2  s W 2 z 1 z ) , (  s P au
Figure 5: LPV H ∞ control structure with parameter-dependent weighting functions w(s) = [w 1 (s), w 2 (s)] T , P au (s, θ) can be described as

     z 1 (s) z 2 (s) y(s)      = P au (s, θ)      w 1 (s) w 2 (s) u(s)     
To have the smallest order of P au (s, θ), we have

P au (s, θ) = W out (s, θ)               1 0 0 0 0 1 -1 0 0      +      -1 0 -1      [G d (s, θ), G p (s)]   0 1 0 0 0 1            W in (s, θ) (12) with W out (s, θ) =      W 1 (s, θ) 0 0 0 W 2 (s, θ) 0 0 0 1      and W in (s, θ) =      W a (s, θ) 0 0 0 W b (s, θ) 0 0 0 1      Substituting [G d (s, θ), G p (
M m = inf ω |1 + L(jω)| = 1 sup ω 1 |1+L(jω)| = 1 sup ω |S(jω)| , ∀ω ∈ R (13)
where S(jω) = (1 + L(jω)) -1 is the sensitivity function of the closed-loop system. Based on the Nyquist stability criterion, the larger M m , the better stability robustness [START_REF] Skogestad | Multivariable Feedback Control-Analysis and Design[END_REF].

Based on the principle of phase and gain control policies, a second order

W 2 (s, θ) is used: W 2 (s, θ) = k W 2 (θ) × (s + M ω b )(s + f M ω b ) (s + )(s + f M 2 ω b ) ( 14
)
where M , ω b , , f are constants and the gain k W 2 (θ) determines the depen-dance of W 2 (s, θ) on θ. With LFR, W 2 (s, θ) can be represented as

W 2 (s, θ) = 1 s I 2      0 1 0 -f M 2 ω b -( + f M 2 ω b ) 1 (M ω b ) 2 f -f M 2 ω b M ω b (1 + f ) -( + f M 2 ω b ) 1      × • • •      1 0 0 0 1 0 0 0 k W 2 (θ)      (15)
and k W 2 (θ) can be represented as

k W 2 (θ) = θ   A k W 2 B k W 2 C k W 2 D k W 2   ( 16
)
where the parameter-independent matrices

A k W 2 ∈ R l×l , B k d ∈ R l×1 , C k d ∈ R 1×l and D k d ∈ R 1×1
have to be determined, and l is the necessary fractional order for k W 2 (θ). As the determination of k d (θ), for some values of θ j ∈ [0.4, 0.8], we select the corresponding value of k W 2 (θ j ) to satisfy the complete set of control objectives, as shown in Table 2. Then, these data can be used for the interpolation of k W 2 (θ), ∀θ ∈ [0.4, 0.8] with the least mean square method, that is,

A k W 2 = 4.044, B k W 2 = 4.00, C k W 2 = -3.637, D k W 2 =
-3.709 with the fractional order l = 1. The other parameters of W 2 (s, θ) are θ j 0.4 0.5 0.6 0.7 0.8 k W 2 (θ j ) 5.4 3.7 2.4 1.8 1.5 By incorporating these weighting functions into Equation ( 12), the simplest LFR of P au (s, θ) is obtained, which is then used for the K LP V (s, θ)

synthesis with the LPV control technique listed in Appendix A. The LFR realization of the designed

K LP V (s, θ) is presented in Appendix B. With the designed K LP V (s, θ), the L 2 gain of F l (P au (s, θ), K LP V (s, θ)
) is smaller than one, necessarily we have F l (P au (s, θ j ), K LP V (s, θ j )) ∞ < 1, that is, for any

||T yd (s, θ j )|| ∞ < 1 W 1 (s)W b (s) ∞ = 40dB M m (θ j ) = 1 S(s, θ j ) ∞ ≥ W 1 (s)W a (s) ∞ = 0.866
This implies that a priori considered control objectives are simultaneously satisfied with the designed K LP V (s, θ).

Worst-case H ∞ control design

In addition to These comparisons are consistent with the principle of phase and gain control policies. For this parameter-dependent system, the schedule variable θ only exists in G d (s, θ) and G p (s) is independent on θ. From the phase control policy, to satisfy the fixed specification of vibration reduction while saving the control energy, |L(jω, θ) = K(s, θ)G p (s)| has to change with θ. On the other hand, for the stability robustness to parametric uncertainties, since the phase of G p (s) does not depend on θ, the phase of K(s, θ) can also be independent on θ. As illustrated in Figure 9, the Nyquist plot of L(s, θ j ) verifies that, around the controlled resonant frequencies, |L(jω, θ j )| is large enough for effective vibration reduction and L(s, θ j ) stays in right half plane to have qualitative stability robustness to parametric uncertainties. The vibration reduction of the closed-loop system using K LP V (s, θ) is shown in

K LP V (s, θ), a worst-case H ∞ controller K w (s)

Quantitative robustness analysis of the closed-loop system

Although, in the designs of K LP V (s, θ) and K w (s), qualitative robustness properties of the closed-loop system are considered, both deterministic and probabilistic robustness analyses are necessary to quantitatively verify the robustness properties to parametric and dynamic uncertainties. In this article, the natural frequencies and damping ratios are assumed to have 20% variations, that is,

ω i = ω i0 + ω i1 δ ω i ; |δ ω i | ≤ 1, ω i1 = 0.2ω i0 , i = 1, 2, 3 ζ i = ζ i0 + ζ i1 δ ζ i ; |δ ζ i | ≤ 1, ζ i1 = 0.2ζ i0 , i = 1, 2, 3
where ω i0 , ζ i0 are the nominal values of these modal parameters. In addition, the scheduled variable θ ∈ [0.4, 0.8] is normalized such that

θ = θ 0 + θ 1 δ θ ; |δ θ | ≤ 1
with θ 0 = 0.6 and θ 1 = 0.2. Thus, the gain k di (θ) can be represented as

k di (θ) = k di0 + k di1 δ θ ; |δ θ | ≤ 1, i = 1, 2, 3
where k di0 is obtained with δ θ = 0. Note that, in this article, θ is assumed to be a bounded time-invariant uncertain parameter in the robustness analysis.

As shown in Figure 11, the additive dynamic uncertainty ∆ Dyn (s) is used with a suitable dynamic normalization function W Dyn (s) to represent the neglected high-frequency dynamics of G p (s), that is,

G p (s) = G p0 (s) + W Dyn (s)∆ Dyn (s), ∆ Dyn (s) ∞ ≤ 1
where G p0 (s) is the reduced nominal plant dynamical model including the first three resonant modes. To consider the robust performance, a fictitious unit-normalized performance uncertainty ∆ Perf (s) is also used with the corresponding performance normalization function W Perf (s) [START_REF] Skogestad | Multivariable Feedback Control-Analysis and Design[END_REF].

With above uncertainty modeling, the unit-normalized diagonal augmented uncertainty ∆ = diag(∆ 1 , ∆ 2 ) ∈ B ∆ can be used, where B ∆ is the norm bounded diagonal uncertainty block as defined in [START_REF] Zhang | Robust active vibration control of piezoelectric flexible structures using deterministic and probabilistic analysis[END_REF].

The ∆ 1 = diag(∆ Para , ∆ Dyn ) represents the parametric uncertainty and the dynamic one, and ∆ 2 = ∆ Perf is the norm bounded fictitious performance uncertainty. Particularly, in this article, for the designed K LP V (s, θ), we have

∆ Para = diag (δ ω 1 I 2 , δ ω 2 I 2 , δ ω 3 I 2 , δ ζ 1 , δ ζ 2 , δ ζ 3 , δ θ I 5 )
where δ θ I 5 is due to the fact that δ θ occurs three times in K LP V (s, θ) and two times in G d (s, θ).

As performed in [START_REF] Zhang | Robust active vibration control of piezoelectric flexible structures using deterministic and probabilistic analysis[END_REF], reliable µ analysis is used to obtain the deterministic robustness margin k DRM of the closed-loop system, as shown in Table 3. Since the upper and lower bounds of k DRM coincide well, the estimated k DRM is reliable, in other words, the closed-loop system remains stable for any ∆ ∈ 1.02∆ 1 with K w (s) and for any ∆ ∈ 1.35∆ 1 with K LP V (s, θ). By ν analysis [START_REF] Skogestad | Multivariable Feedback Control-Analysis and Design[END_REF], we have the Gaussian distributed ω i , if a 10.0% loss of probabilistic robust stability is tolerated, the corresponding k PRM = 1.75 is increased by 71.6% with respect to its deterministic counterpart k DRM = 1.02 and increased by 9.37% with respect to the result for uniformly distributed ω i . Probabilistic worst-case performance analysis is also performed, as summarized in Table 5. It shows that, with probability 99.0%, the specification of vibration reduction is fulfilled for all sampled ∆ 1 ∈ 1.00B ∆ 1 with K w (s) and K LP V (s, θ), and when ∆ 1 ∈ 1.20B ∆ 1 , a few perturbations can be found to violate the specification of vibration reduction. This is consistent with the result from ν analysis.

K LP V (s, θ)
controllers for effective vibration reduction in the presence of parametric and dynamic uncertainties, but also to save the necessarily required control energy and reduce the control input. In fact, the specification of vibration reduction can be achieved with relatively simpler acceleration feedback control (AFC), for example, based on the worst-case disturbance dynamical model G wd (s), K AF C (s) can be designed for comparison purpose with the cross-over point method [START_REF] Bayon De Noyer | Single actuator and multi-mode acceleration feedback control[END_REF]:

K AF C (s) =
-8.0 × 10 7 (s 2 + 2025.0s + 1.1 × 10 6 ) (s 2 + 165.3s + 8.7 × 10 4 )(s 2 + 1080s + 3.4 × 10 6 ) × (s 2 -926.1s + 6.4 × 10 5 ) (s 2 + 2020.0s + 2.7 × 10 7 )

As numerically verified, K AF C (s) can also satisfy the specification of vibration reduction as K w (s) and K LP V (s, θ) do.

To emphasize the advantages of K LP V (s) in terms of the control energy and the control input, within MATLAB/Simulink R2012 environment, a unit step signal is used as the external force and several numerical simulations are evaluated in the time domain. As shown in Figure 13, compared to K AF C (s), less control energy is required by K w (s). As explained in [START_REF] Zhang | Phase and gain control policies for robust active vibration control of flexible structures[END_REF], this is mainly due to the fixed structure of AFC that makes |K AF C (jω)| too large at very low frequencies, where no control energy is actually required.

Furthermore, as G d (s, θ) depend on θ ∈ [0.4, 0.8], the required control energy to satisfy the fixed specification of vibration reduction greatly varies, and K LP V (s, θ) has the ability to adapt its bandwidth to θ such that K LP V (s, θ)

consumes less control energy than K w (s) does for any θ ∈ [0.4, 0.78] and K AF C (s) does for any θ ∈ [0.4, 0.8]. The fact that K LP V (s, θ) could save the control energy is beneficial in avoiding the phenomenon of insufficient control energy and quite desirable for practical implementation, for instance, the control energy is obtained from harvesting systems or low-power storage devices (batteries or super capacitors) as often used in aerospace applications.

On the other hand, as shown in Figure 14, for any θ ∈ [0.4, 0.8], the required control input using K LP V (s, θ) is smaller than that using K w (s) or K AF C (s). This is useful to avoid exceeding the control saturation and the actuator operated voltage. Furthermore, K LP V (s, θ), K w (s) and K AF C (s) can achieve not only the same specification of vibration reduction in the frequency domain and the satisfactory robustness properties as shown in Table 3 -5, but also the system output in the time domain, as illustrated in Figure 15 where the cases with θ = 0.4, 0.8 are used for the sake of simplicity.

In these numerical simulations, the unit step-signal is used to simulate external disturbances. The most importance of the PSD for an unit step is at

Conclusions and perspectives

This research builds off of our previous researches on the quantitative robust control method for LTI systems using classical H ∞ control designs and reliable robustness analyses, and focuses on exploiting the benefits of efficient LPV H ∞ control technique in saving the required control energy and reducing the control input. With this proposed LPV H ∞ control method, the varying parameters of the LPV system represented by θ can be fully investigated and the trade-off among various control objectives, e.g. the specification of vibration reduction, the closed-loop robustness properties and the saving of required control energy, can be achieved by systematical adjustments of the weighting functions which could also depend on θ. Compared to AFC and the classical H ∞ control, due to the dependence of the controller on θ, the proposed control method can explicitly save the required control energy and, in some extend, reduce the control input, while maintaining almost the same control performances both in the frequency and time domains.

In this article, some parameter-independent Lyapunov functions are used for the synthesis of K LP V (s, θ). It provides a satisfactory LPV controller for the investigated case. If, in the applications under consideration, the employed parameter-independent approach appears to be very conservative, parameter dependent LMI formulations can be used for the synthesis of K LP V (s, θ), which is expected to be less conservative. The details of the approach can be found in [START_REF] Dinh | Parameter dependent H ∞ control by finite dimensional LMI optimization: application to trade-off dependent control[END_REF][START_REF] Dinh | Synthèse dépendant de paramètres par optimisation LMI de dimension finie: Application à la synthèse de correcteurs reréglables[END_REF].

Although the motivation of this article is strongly influenced by practical application to quantitative robust active vibration control of flexible structures, it is important to appreciate that most of the design processes and employed techniques are general. In fact, many practical control problems involve the systems whose dynamics depend on some measurable exogenous parameters. For example, many vibration control systems are required to function across a variety of different temperatures, however, the variation of ambient temperature can change the structural natural frequencies and piezoelectric stress and permittivity coefficients, thus the applied control effort has to consider such temperature dependence [START_REF] Hegewald | Vibration suppression via smart structures across a temperature range[END_REF][START_REF] Chettah | Dynamic mechanical properties and weight optimization of vibrated ground recycled rubber[END_REF][START_REF] Gupta | Active structural vibration control: Robust to temperature variations[END_REF]. This kind of control problem is readily to be handled with the proposed quantitative robust LPV H ∞ control method which considers the time-varying temperature as the scheduled variable. Furthermore, the main interest of this research is not only for active vibration control as illustrated by the studied numerical example, but rather for the fundamental issues involved in practical active control designs, for example, with few modifications, this research will be applicable to active noise control [START_REF] Jemai | Double panel partition (AVNC) by means of optimized piezoceramic structural boundary control[END_REF], active suspensions to adapt road conditions [START_REF] Fialho | Road adaptive active suspension design using linear parameter-varying gain-scheduling[END_REF], active control of heating, ventilation and air condition In addition to linear systems, LPV control methods are firstly proposed for nonlinear systems and viewed as available alternatives to classical gainscheduling designs for controlling nonlinear systems [START_REF] Carter | Linear parameter varying representations for nonlinear control design[END_REF][START_REF] Rugh | Research on gain scheduling[END_REF]. In particular, LPV control methods offer advantages over classical gain-scheduled control in that the resulting LPV controllers are automatically gain-scheduled, and no ad hoc methods of interpolation of gains are needed. Therefore, the proposed control method can also be used for active vibration control of nonlinear systems, e.g. [START_REF] Zhou | Active vibration control of nonlinear giant magnetostrictive actuators[END_REF]; [START_REF] Ho | Vibration isolation using nonlinear damping implemented by a feedback-controlled MR damper[END_REF].

Appendix A. Employed LPV control technique

In this article, we use the LPV control method proposed in [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related control problems[END_REF], which models the augmented LPV plant P au (s, θ) with LFR and uses parameter-independent Lyapunov functions. By the scalings selection, this method allows us to make a trade-off between conservatism and computational complexity. With LFR, the P au (s, θ) of Equation ( 5 Note that from the lemma 3.2.1 in J-F. [START_REF] Magni | User manual of the linear fractional representation toolbox[END_REF], it is known that the input-output LFR realization of K LP V (s, θ), that is, With this realization, the related matrices are listed as below:

y = F u           A B 1 B 2 C 1 D 11 D 12
A =                     -4

  and LQR control are compared. The trade-off curves of the control energy versus the closed-loop control performance are investigated. P. Van Phuoc et al. (2009) employ a genetic algorithm for the parameter optimization of a positive position feedback (PPF) controller to minimize the control energy for active vibration reduction of a flexible robot manipulator. Similarly, Chen et al. (2011) use P-SO to determine the parameters of the proportional-integral-derivative (PID) controller such that the control energy for a mass-damper-spring system is minimized. Wang and Inman (2011) introduce a reduced energy control (REC) law by employing a saturation control to switch the control system from one state to another one, providing conventional active controllers with a limited voltage boundary. Both experimental and numerical comparisons are performed in terms of the control energy and the setting time with PPF control, PID control, nonlinear control and LQR control.

Figure 1 :

 1 Figure 1: A typical feedback control structure for active vibration control

  uniformly. As proposed in[START_REF] Dinh | Parameter dependent H ∞ control by finite dimensional LMI optimization: application to trade-off dependent control[END_REF], for a LTI plant considering a set of performance trade-offs parameterized by a scalar θ, several weighting functions depending on θ are incorporated into the LTI plant to develop an augmented LPV system, and an trade-off dependent H ∞ controller is synthesized by solving the finite dimensional Linear Matrix Inequality (LMI) optimization problem. In this article, an LPV plant with position-dependent dynamics has to be considered, and to save the control energy, a param-eter (force position) dependent controller is designed thanks to the introduction of parameter-dependent weighting functions. Based on the phase and gain control policies, the weighting functions can be appropriately determined, thus developing the augmented LPV system. Then, an efficient LPV H ∞ control technique, e.g. Scorletti and L. EI Ghaoui (1998);[START_REF] Dinh | Parameter dependent H ∞ control by finite dimensional LMI optimization: application to trade-off dependent control[END_REF], is used to synthesize a qualitative robust parameter-dependent

Figure 3 :

 3 Figure 3: A piezoelectric cantilever beam with position-dependent dynamics

  ) and D ∈ R 1×2 . It is notable that the vector [k d (θ), k p ] in both B(θ) and D(θ) has to be pulled out to have the simplest LFR of [G d (s, θ), G p (s)] (Scorletti

Figure 4 :

 4 Figure 4: Analytical and LFR of k di (θ), θ ∈ [0.4, 0.8], i = [1, 2, 3]

  s)] of Equation (10) into Equation (12), we have the simplest LFR of P au (s, θ), where either [G d (s, θ), G p (s)] or W i (s, θ) occurs just one time. It is then used for the controller synthesis and the robustness analysis. With the LPV H ∞ control design of Figure 5, W 2 (s, θ) can be used to enforce constraints on the magnitudes of |K(jω)S(jω)| and |G d (jω, θ)K(jω)S(jω)|, which are closely related to the control energy. Therefore, to adapt the control energy to θ, W 2 (s, θ) has to depend on θ, and other parameterindependent weighting functions are used to determine the fixed specification of vibration reduction and the requirement on the modulus margin M m , which is closely related to the stability robustness and defined as:

Figure 6 :

 6 Figure 6: The dependence of |W 2 (jω, θ)| on θ ∈ [0.4, 0.8]

Figure 7 :

 7 Figure 7: The worst-case G d (s, θ) for θ ∈ [0.4, 0.8]

Figure

  Figure Comparisons between K LP V (s, θ) and K w (s) for θ ∈ [0.4, 0.8]

Figure 9 :

 9 Figure 9: Phase and gain control policies with K LP V (s, θ) and K w (s): ω i represents the i th controlled resonant frequency

Figure 10 :

 10 Figure 10: Closed-loop system with K LP V (s, θ) for θ ∈ [0.4, 0.8]: the blue solid curves represent the open-loop systems and the red dashed curves represent the closed-loop systems by gridded θ

Figure 11 :

 11 Figure 11: The additive dynamic uncertainty normalized by W Dyn (s)

  Figure 12: Deterministic worst-case performance analysis with ∆ ∈ ∆ 1

KFigure 13 :

 13 Figure 13: Comparisons of the control energy consumption using K w (s), K AF C (s) and

Figure 14 :Figure 15 :

 1415 Figure14: Comparisons of the control input using K w (s), K AF C (s) and K LP V (s, θ)

(

  HVAC) systems Rasmussen and Alleyne (2010); Zhao et al. (2013), active control of machine tools N.J.M. van Dijk et al. (2010) and so on.

XS

  Figure B.16: LFR realization of K LP V (s, θ)

  the complexity of θ in K LP V (s, θ), since θ is not repeated in A K (θ(t)), B(θ(t)), C(θ(t)) and D(θ(t)) but occurs only once.

  7395 × 10 -3 2.4222 × 10 -1 -4.7265 × 10 -1 -1.4990 × 10 -3 5.7731 × 10 -1 -7.4509 × 10 -1 0.8770 511.01 25.722 -38.625 -19.679 -9.3001 7.9401 1.8002 D 21 = -1.0393 -1.6099 1.8877 , D 22 = 0.04935

  (θ j ), and then the least mean square method is used for the interpolation of k d (θ), θ ∈ [θ min , θ max ]. With the Equation (8) and Equation (9), we have the LFR of [G d (s, θ), G p (s)], that is,

	to be determined, and
	m is the necessary fractional order for k d (θ). Since the Equation (7) reveals
	that k di (θ) is not a rational function of θ, in order to obtain the LFR of k d (θ),
	it is necessary to approximate k d (θ) by a rational function. For this purpose,

enough samples of θ j ∈ [θ min , θ max ] are used to have the corresponding values of k d

Table 1 :

 1 are used to achieve the same Nominal geometrical and mechanical properties of the piezoelectric cantilever

	Property Beam	PZT	Unit
	E	50.0	140.0	Gpa
	l	248.0	45.0	mm
	w	20.5	20.5	mm
	t	4.0	1.5	mm
	ρ	2500.0	/	kg/m 3
	k d31	/	-1.23 × 10 -10	/

beam fixed specification of vibration reduction defined by a frequency-dependent function U (ω). In this article, for the sake of simplicity, U

Table 2 :

 2 The chosen θ

  Controller Uniformly distributed ω i Gaussian distributed ω i

	K w (s)	pn (1.35) = 100%	pn (1.35) = 100%
	K w (s)	pn (1.60) = 90%	pn (1.65) = 90.0%
	K LP V (s, θ)	pn (1.02) = 100%	pn (1.02) = 100%
	K LP V (s, θ)	pn (1.60) = 90%	pn (1.75) = 90%

Table 4 :

 4 Probabilistic stability analysis: = 0.01, δ = 0.01

	Controller	Uniformly distributed ω i	Gaussian distributed ω i

  .5301 -2612.1 -131.06 193.44 98.995 47.589 -41.269 -8.4422 2640.2 -37395 77398 -116221 -59215 -27983 23891 5416.7 -80.853 468.79 -4075.4 -9911.2 -1816.2 2061.2 -1600.5 1170.1

			 -0.4918 -286.57 -14.425 21.660	11.036	5.2154 -4.4528 -1.0095
	C 1 =	   	0.0076 -0.0904 2.5651 -2.8923 -1.9579 -0.9170 0.3092 0.0070 -0.1192 2.3009 -2.6212 -1.6963 -0.7777 0.2801	0.6601 0.5827
					
		-129.76 1293.5 249.50 -1779.2 -1149.4 16932 5756.9 -11144 -4433.0 511.88 -1975.4 2172.6 6095.2 1643.2 3384.2 -3267.4 -328.35 2874.3 5640.6 -28393 -13090 -829.59 -4561.9 5234.0 -1097.0 8725.4 7915.2 -91020 -36269 1798.3 -16248 17843                  
		278.66 -2243.3 -2280.2 25220.3	10344 -619.34 4012.3 -4840.8 
		 -0.09651 8.9632 -10.206 	 -0.2321 
	B 1 =	                   -45.7728 17186 -12987 55.657 -4844.1 5680.0 -1.5235 414.87 11.311 -5.9535 2440.4 -2100.0 8.8184 -3867.4 3229.1 -15.140 5572.3 -4309.4 12.308 -4306.7 3159.4                   	, B 2 =	                   -130.58 148.51 41.497 58.073 -85.43 132.77 467.92                   

the sensor outputs are contaminated by the neglected dynamics, which we called observation spillover, and the feedback control excites the neglected dynamics, which is termed as control spillover

deterministic worst-case performance, as illustrated in Figure 12. It shows that the specification of vibration reduction is fulfilled for any ∆ ∈ 1.0∆ 1 with K w (s) and K LP V (s, θ). Above µ and ν analyses quantitatively ensure that the closed-loop stability and the specification of vibration reduction are satisfied in the presence of 20% variation on the modal parameters and the assumed dynamic uncertainty.

Bounds on k

Lower bound on k DRM 1.355 1.020

Upper bound on k DRM 1.360 1.026 Besides, the probabilistic robustness analysis using random algorithm is performed to consider probabilistic information of parametric uncertainties and provide complements and comparisons to the deterministic robustness analysis. For this numerical case, both uniformly and Gaussian distributed ω i are considered and ζ k is assumed to have uniform distribution. As performed in [START_REF] Zhang | Robust active vibration control of piezoelectric flexible structures using deterministic and probabilistic analysis[END_REF], using Monte Carlo Simulation, the results from probabilistic stability analysis are illustrated in Table 4 with = 0.01, δ = 0.01.

They show that, with probability 1 -δ = 99%, for either uniformly or Gaussian distributed ω i , the closed-loop system remains stable for all sampled ∆ ∈ 1.02∆ 1 using K w (s) and for all sampled ∆ ∈ 1.35∆ 1 using K LP V (s, θ).

Additionally, a few destabilizing perturbations ∆ des ∈ 1.15∆ 1 are found using K LP V (s, θ), which means that there exist little conservatism in the probabilistic stability analysis. These results also demonstrate that the k DRM estimated from µ analysis is reliable. On the other hand, it shows that for and

then there exist an LPV controller such that the closed-loop system is internally stable with an L 2 gain less than γ.

This theorem actually presents a set of LMI constraints: first, a given γ is used to test the conditions of the previous theorem; then, the smallest γ is searched to satisfy the conditions of the theorem. If theses conditions can be satisfied, the matrices of the LFR representation of K LP V (s, θ) can be using a feasibility optimization problem. Explicit formulations of this optimization problem can be found in [START_REF] Scorletti | Improved LMI conditions for gain scheduling and related control problems[END_REF].

Appendix B. LFR realization of the designed K LP V (s, θ)

As illustrated with the Figure B.16, the input-output realization of the designed K LP V (s, θ) is y = F u (M, ∆)u with ∆ = diag(I 8 /s, I 3 θ), where F u is the upper LFT, the matrix M is defined on the page 19 of J-F. [START_REF] Magni | User manual of the linear fractional representation toolbox[END_REF] and can be appropriately partitioned according to the order of the controller and the size of θ, e.g. 8 is equal to the order of P au (s, θ) and 3 is the sum of m = 2 and l = 1.

By directly closing the θ loop of Figure B.16, the matrices defined in