Traoré Massiré

Gentil Christel

Benedetto Chiara

Hogrel Jean-Yves

De

Grange Pierre

Cadot Bruno

Benkhelifa-Ziyyat Sofia

Julien Laura

Lemaitre Mégane

Arnaud Ferry

Piétri-Rouxel France
email: france.pietri-rouxel@upmc.fr.

Falcone Sestina
email: sestinaf@gmail.com

An embryonic CaVβ1 isoform promotes muscle mass maintenance via GDF5 signaling in adult mouse

signaling to counteract muscle mass loss in

Introduction

A decrease in electrical activity, such as during neuromuscular disease, disuse, or aging can cause massive muscle atrophy [START_REF] Gundersen | Excitation-transcription coupling in skeletal muscle: the molecular pathways of exercise[END_REF]. Disuse atrophy following muscle denervation or immobilization is characterized by the activation of a compensatory response to counteract mass loss. In particular, the induction of Growth Differentiation Factor 5 (GDF5)/SMAD4 (an acronym from the fusion of Caenorhabditis elegans Sma genes and the Drosophila Mad, Mothers against decapentaplegic) pathway is essential for avoiding excessive muscle mass loss but also for promoting reinnervation after nerve crush [START_REF] Sartori | BMP signaling controls muscle mass[END_REF][START_REF] Macpherson | Dach2-Hdac9 signaling regulates reinnervation of muscle endplates[END_REF].

Aging muscles are characterized by progressive loss of mass and strength [START_REF] Pannérec | A robust neuromuscular system protects rat and human skeletal muscle from sarcopenia[END_REF], suggesting an impairment of compensatory mechanisms. Currently, the best approaches to maintain aged muscle function and size are caloric restriction (limits autophagy and DNA damage) and exercise (restores muscle activity) [START_REF] Valdez | Attenuation of age-related changes in mouse neuromuscular synapses by caloric restriction and exercise[END_REF][START_REF] Kern | Atrophy, ultra-structural disorders, severe atrophy and degeneration of denervated human muscle in SCI and Aging. Implications for their recovery by Functional Electrical Stimulation, updated 2017[END_REF][START_REF] Kern | Electrical stimulation counteracts muscle decline in seniors[END_REF][START_REF] Mosole | Long-term high-level exercise promotes muscle reinnervation with age[END_REF][START_REF] Vinel | The exerkine apelin reverses age-associated sarcopenia[END_REF]. Yet, the evidence that muscle stimulation improves mass maintenance suggests that proteins sensing sarcolemmal depolarization might be suitable candidates for triggering a compensatory response. Among the five subunits of the L-type Ca 2+ channel CaV1.1, CaVα1s subunit displays voltage sensor activity [START_REF] Schredelseker | The beta 1a subunit is essential for the assembly of dihydropyridine-receptor arrays in skeletal muscle[END_REF][START_REF] Dayal | Skeletal muscle excitation-contraction coupling is independent of a conserved heptad repeat motif in the C-terminus of the DHPRbeta(1a) subunit[END_REF]. The intracellular subunit of the complex, CaVβ1 (encoded by Cacnb1 gene), plays a role in (excitationcontraction coupling) ECC by targeting CaVα1s to the membrane and regulating its activity [START_REF] Dayal | Domain cooperativity in the β1a subunit is essential for dihydropyridine receptor voltage sensing in skeletal muscle[END_REF].

Currently, CaVβ1A is considered as the skeletal muscle specific isoform [START_REF] Gregg | Absence of the β subunit (cchb1) of the skeletal muscle dihydropyridine receptor alters expression of the α1 subunit and eliminates excitation-contraction coupling[END_REF]. Furthermore, CaVβ1 can be anchored to membrane, free in the cytoplasm, or located to nucleus of proliferating muscle cells where it acts as regulator of transcription factors [START_REF] Taylor | The Cavβ1a subunit regulates gene expression and suppresses myogenin in muscle progenitor cells[END_REF]. CaVβ1 is essential for the development of embryonic mouse muscles and for the pre-patterning and development of neuromuscular junctions (NMJ), independently of ECC [START_REF] Taylor | The Cavβ1a subunit regulates gene expression and suppresses myogenin in muscle progenitor cells[END_REF][START_REF] Chen | Neuromuscular synaptic patterning requires the function of skeletal muscle dihydropyridine receptors[END_REF][START_REF] Dayal | The Ca2+ influx through the mammalian skeletal muscle dihydropyridine receptor is irrelevant for muscle performance[END_REF]). Yet, no ECC-independent pathway involving CaVβ1 is known in adult muscle.

Here, we provided the link between electrical activity sensing protein and muscle adaptation that compensates for atrophy. First, we demonstrated how the innervation state of skeletal muscle regulates the CaVβ1 isoform transition epigenetically. We showed that CaVβ1D is the constitutively expressed isoform in adult mouse muscle, whereas CaVβ1E is the embryonic variant up-regulated upon impairment of electrical activity. Second, we highlighted CaVβ1E as a major player in limiting muscle mass loss in mice through its ability to trigger GDF5 signaling in denervated muscle. In addition, we revealed that muscle aging is associated with altered CaVβ1E/GDF5 axis and that over-expression of CaVβ1E drives the pathway necessary to counterbalance age-related muscle atrophy in rodents. Finally, we provided evidence of a human hCaVβ1E isoform expressed in adult muscle and tightly linked to muscle decline in aging.

Results

Embryonic CaVβ1E expression in adult mouse muscle

Resection of mouse sciatic nerve is a suitable model for measuring the response of adult skeletal muscle to alterations in activity [START_REF] Gundersen | Excitation-transcription coupling in skeletal muscle: the molecular pathways of exercise[END_REF][START_REF] Méjat | Histone deacetylase 9 couples neuronal activity to muscle chromatin acetylation and gene expression[END_REF]. To establish if the expression of Cacnb1 could change in this model we quantified its mRNA by RT-qPCR using primers in exons 2 and 3. We observed a time-dependent increase in the amplification of this region in denervated tibialis anterior muscles The Cacnb1 gene (GSMG0007319) has 14 exons that can be spliced to give 6 transcript variants (Fig 1D). To identify potential Cacnb1 splicing events occurring in denervated muscle, we performed a genome-wide transcriptomic analysis at the exon level on RNA extracted from innervated and denervated mouse TAs. We found 1022 differentially regulated alternative splicing events (from 706 distinct genes) (Datafile S1). Among these, Cacnb1 displayed a first exon splicing event showing that the transcript started in a putative non-coding sequence at the 5' end of exon 3 in innervated muscles. In denervated muscles, another transcript of Cacnb1 starting at exon 1 was found to be upregulated (Fig 1E By searching Cacnb1 variants in the National Center for Biotechnology Information (NCBI) database, we found that Cacnb1-D is the only one starting immediately upstream the exon 3.

Based on these results, our data suggest that the specific CaVβ1 isoform expressed in the adult mouse skeletal muscle is CaVβ1D and not CaVβ1A, as expected [START_REF] Taylor | The Cavβ1a subunit regulates gene expression and suppresses myogenin in muscle progenitor cells[END_REF]. Moreover, the size of the extra CaVβ1 band appearing in denervated muscle suggested that it could be CaVβ1B or CaVβ1E Several embryonic proteins are induced after muscle denervation [START_REF] Macpherson | Dach2-Hdac9 signaling regulates reinnervation of muscle endplates[END_REF][START_REF] Tang | Activity-dependent gene regulation in skeletal muscle is mediated by a histone deacetylase (HDAC)-Dach2-myogenin signal transduction cascade[END_REF][START_REF] Adams | Adaptation of nicotinic acetylcholine receptor, myogenin, and MRF4 gene expression to long-term muscle denervation[END_REF][START_REF] Tang | A histone deacetylase 4/myogenin positive feedback loop coordinates denervation-dependent gene induction and suppression[END_REF][START_REF] Tang | CaM kinase IIdependent phosphorylation of myogenin contributes to activity-dependent suppression of nAChR gene expression in developing rat myotubes[END_REF][START_REF] Fischer | Expression of functional alpha7 nicotinic acetylcholine receptor during mammalian muscle development and denervation[END_REF]. We wondered whether CaVβ1E could be an embryonic isoform. RT-PCR (primers matching exon 14) (Fig 1K) In conclusion, our results show that an alternative first exon splicing is the source of the differential expression of mouse adult and embryonic Cacnb1 variants. CaVβ1D, not CaVβ1A, is expressed in innervated adult skeletal muscle, whereas embryonic muscle expresses CaVβ1E.

CaVβ1E is low in innervated adult mouse muscle but denervation specifically increases its expression. Moreover, CaVβ1D and CaVβ1E display different intracellular locations in adult muscle fibers.

CaVβ1E and GDF5 signaling activation after denervation

To evaluate whether CaVβ1E has a role in disuse atrophy, we injected into mouse TA an Associated-adeno-virus (AAV) vector carrying a short-hairpin RNA targeting a sequence in transcription after denervation was significantly (P<0.0001) reduced in absence of CaVβ1E, suggesting its involvement in modulating genes associated with endplate formation, independently of myogenin. A previous study showed that in proliferating muscle cells canonical and non-canonical DNA E-box sequences (CANNTG and CANNNTG) of several promoter regions could be targeted by CaVβ1 [START_REF] Taylor | The Cavβ1a subunit regulates gene expression and suppresses myogenin in muscle progenitor cells[END_REF]. Analysis of the 100 most differentially regulated genes in innervated and denervated TA revealed the presence of one or more E-box sequences in the promoter of all these genes (Table S2). However, among them, only Gdf5 and myogenin had a demonstrable role in muscle mass homeostasis [START_REF] Sartori | BMP signaling controls muscle mass[END_REF][START_REF] Moresi | Myogenin and class II HDACs control neurogenic muscle atrophy by inducing E3 ubiquitin ligases[END_REF].

To further analyze the role of CaVβ1E on Gdf5 expression, we used the myogenic cell line C2C12 as tool in vitro. C2C12 cells expressed increasing Cacnb1-E during differentiation (Fig

Aging muscles: a key role for CaVβ1E

During aging skeletal muscle shows denervation-like signs and progressive muscle wasting [START_REF] Pannérec | A robust neuromuscular system protects rat and human skeletal muscle from sarcopenia[END_REF][START_REF] Li | Changes in aging mouse neuromuscular junctions are explained by degeneration and regeneration of muscle fiber segments at the synapse[END_REF]. This suggests that impairment in compensatory response might occur. However, little is known about CaVβ1 expression and function [START_REF] Taylor | Increased CaVbeta1A expression with aging contributes to skeletal muscle weakness[END_REF] Similarly, we checked if age-related modifications in fiber type composition occurred, maybe explaining CaVβ1E down-regulation. Expression of myosin heavy chain (MyHC) isotypes IIA and IIX decreased significantly during aging (P= 0.00616652 and P= 0.01394505, respectively), as reported [START_REF] Vinel | The exerkine apelin reverses age-associated sarcopenia[END_REF][START_REF] Lamboley | Contractile properties and sarcoplasmic reticulum calcium content in type I and type II skeletal muscle fibres in active aged humans[END_REF][START_REF] Balagopal | Age effect on transcript levels and synthesis rate of muscle MHC and response to resistance exercise[END_REF], whereas MyHC-I and -IIB did not change between 12 and 78 weeks of age (Fig 4E-H). To elucidate if CaVβ1E expression could be associated to fiber type modifications, we analyzed its amount in MyHC-I and -II positive fibers by immunofluorescence. In young muscle, CaVβ1E was expressed at higher intensity in MyHC- Overall, these data show that the loss of CaVβ1E/GDF5 cross-talk observed during aging may be critical for muscle wasting and its rescue counteracts the process of age-related muscle decline.

Human muscle: a new CaVβE isoform implicated in skeletal muscle aging

Given the apparent importance of CaVβ1E in mouse skeletal muscle, we wondered whether an analogous mechanism might be conserved in humans and, thus, if another unidentified CACNB1 isoform was expressed in adult human skeletal muscle. Three human CACNB1 (hCACNB1) variants have been identified corresponding to the mouse isoforms A (specific of skeletal muscle), B, and C (Fig 6A) [START_REF] Pragnell | Cloning and tissue-specific expression of the brain calcium channel β-subunit[END_REF][START_REF] Powers | Skeletal muscle and brain isoforms of a beta-subunit of human voltage-dependent calcium channels are encoded by a single gene[END_REF][START_REF] Hogan | Structure and alternative splicing of the gene encoding the human beta1 subunit of voltage dependent calcium channels[END_REF][START_REF] Gregg | Assignment of the Human Gene for the β Subunit of the Voltage-Dependent Calcium Channel (CACNLB1) to Chromosome 17 Using Somatic Cell Hybrids and Linkage Mapping[END_REF]. Human mRNA extracted from one quadriceps and two fascia lata (FL) muscle biopsies of healthy adult subjects together with human mRNA extracted from the cervical spinal cord (Table S2), as positive control for hCACNB1-B (18-20), were probed for exons 13 and 5-9. Amplification of the sequence in exon 13 showed that all muscles expressed hCACNB1-A. As in mouse muscle, amplification of the region between exons 5 and 9 demonstrated that only a 380 bp corresponding to the putative hCACNB1-A appeared and that the exon 7B-containing isoform, hCACNB1-B (245 bp), was solely expressed in spinal cord (SC).

Furthermore, no transcription of hCACNB1-C, which also has the short 7B exon, was found in muscle (Fig 6B). We further checked if human muscle could express a hCACNB1 transcript analogous to mouse Cacnb1-E. Amplification of the region in exon 14 revealed that the human muscles expressed the previously unidentified variant that we have called hCACNB1-E (Fig 6C).

This isoform corresponded to the predicted XM_006722072.2 variant, having a codon start (ATG2) upstream the exon 3 (Fig 6A ,C). Western blot and immunofluorescence experiments confirmed its expression in two different human FLs (Fig 6D ,E), with localization similar to that of mouse CaVβ1E (Fig S2E,F).

Since we found that altered CaVβ1E/GDF5 axis was associated to muscle wasting during aging in mice, we compared characteristics indicating muscle mass (lean mass percentage) and function (power) [START_REF] Reid | Skeletal muscle power: a critical determinant of physical functioning in older adults[END_REF][START_REF] Mcphee | Physiological and functional evaluation of healthy young and older men and women: design of the European MyoAge study[END_REF] in a cohort of healthy young (20-42 years) and aged (70-81 years) volunteers, included in a previous study [START_REF] Mcphee | Physiological and functional evaluation of healthy young and older men and women: design of the European MyoAge study[END_REF] (Table 1). Aged group displayed significantly lower lean mass and power than young group (P= 0.003255 and P<0.0001, respectively) (Fig 6F ,G). We then measured hCACNB1-E transcript and found a significant reduction (P= 0.0045) of its expression in aged group (Fig 6H), whereas hCACNB1-A transcription did not differ between groups (Fig

6I). A low hCACNB1-E expression was also associated with a low lean mass percentage (Fig 6J).

We detected a weak amount of hGDF5 in muscle biopsies, however, in aged muscle samples we could associate low expression of both hCACNB1-E and hGDF5 with low lean mass percentage.

In addition, participants with higher lean mass percentage displayed higher expression of both hCACNB1-E and hGDF5 (Fig 6K ).

Overall these results suggest that CaVβ1E/GDF5 axis in compensatory response and aging (Fig S9) might be a conserved mechanism between mice and humans.

Discussion

In adult skeletal muscle, the mechanism connecting electrical activity sensing and changes in gene expression is unclear. Here, we showed that denervation boosts the expression of embryonic CaVβ1E isoform which mediates transcriptional reprogramming in adult muscle. Many years ago CaVβ1 isoforms were identified in humans [START_REF] Powers | Skeletal muscle and brain isoforms of a beta-subunit of human voltage-dependent calcium channels are encoded by a single gene[END_REF][START_REF] Hogan | Structure and alternative splicing of the gene encoding the human beta1 subunit of voltage dependent calcium channels[END_REF][START_REF] Buraei | The β Subunit of Voltage-Gated Ca2+ Channels[END_REF], rat [START_REF] Pragnell | Cloning and tissue-specific expression of the brain calcium channel β-subunit[END_REF] and, less clearly, in mouse [START_REF] Gregg | Absence of the β subunit (cchb1) of the skeletal muscle dihydropyridine receptor alters expression of the α1 subunit and eliminates excitation-contraction coupling[END_REF].

We anticipated that 55 kDa band corresponded to CaVβ1A, a previously identified musclespecific isoform with this molecular weight [START_REF] Pragnell | Cloning and tissue-specific expression of the brain calcium channel β-subunit[END_REF][START_REF] Powers | Skeletal muscle and brain isoforms of a beta-subunit of human voltage-dependent calcium channels are encoded by a single gene[END_REF]. However, we speculated that the 70 kDa band could be a previously unappreciated CaVβ1 isoform. Our results revealed that CaVβ1D, not CaVβ1A, is the main constitutive isoform specific to normal adult mouse muscle and that CaVβ1D and CaVβ1E have distinct roles. Indeed, in silico analysis using cNLS mapper confirmed the presence of a Nuclear Localization Signal at the N-terminus of CaVβ1E but not in CaVβ1D. CaVβA is not expressed in adult mouse muscle and C2C12, however, its N-terminus is putatively identical with those of CaVβE, suggesting that the mechanism by which it translocates to the nucleus is the same described in past publications [START_REF] Taylor | The Cavβ1a subunit regulates gene expression and suppresses myogenin in muscle progenitor cells[END_REF][START_REF] Zhang | Troponin T3 regulates nuclear localization of the calcium channel Cavβ1a subunit in skeletal muscle[END_REF]. This finding also implicates CaVβD in regulating calcium channel assembly in the cytoplasm, whereas CaVβE could have a role in both modulating nuclear transcription factors and cooperating with cytoskeletal or Ttubules at the Z-Lines, as demonstrated for others proteins [START_REF] Falcone | N-WASP is required for Amphiphysin-2/BIN1-dependent nuclear positioning and triad organization in skeletal muscle and is involved in the pathophysiology of centronuclear myopathy[END_REF][START_REF] Takano | Nebulin and N-WASP cooperate to cause IGF-1-induced sarcomeric actin filament formation[END_REF].

We also discovered that CaVβ1E is the main CaVβ1 isoform in proliferating and differentiating C2C12 cells, while CaVβ1D and CaVβ1A isoforms are undetectable. Previous studies reporting expression, transcriptional activity and nuclear localization of CaVβ1A in muscle fibers [START_REF] Zhang | Troponin T3 regulates nuclear localization of the calcium channel Cavβ1a subunit in skeletal muscle[END_REF], muscle precursors and C2C12 cells [START_REF] Taylor | The Cavβ1a subunit regulates gene expression and suppresses myogenin in muscle progenitor cells[END_REF] were likely evaluating the as-yet undiscovered CaVβ1E.

An interesting issue of our study could be deciphering mechanisms underlying the transition between CaVβ1E and CaVβ1D during myogenesis and identifying molecular factors regulating the implicated splicing events.

The major finding of this work is the central role of CaVβ1E in maintaining muscle mass. GDF5 is one of the main regulators of muscle mass homeostasis in response to disuse atrophy [START_REF] Sartori | BMP signaling controls muscle mass[END_REF], however few studies report progress on modulators of this factor [START_REF] Macpherson | Dach2-Hdac9 signaling regulates reinnervation of muscle endplates[END_REF][START_REF] Zhao | Muscle Yap Is a Regulator of Neuromuscular Junction Formation and Regeneration[END_REF][START_REF] Hatazawa | Reduced Dnmt3a increases Gdf5 expression with suppressed satellite cell differentiation and impaired skeletal muscle regeneration[END_REF]. Our data validate previously described roles of GDF5 [START_REF] Sartori | BMP signaling controls muscle mass[END_REF][START_REF] Sartori | Smad2 and 3 transcription factors control muscle mass in adulthood[END_REF] and add a positive upstream player controlling its signaling. Indeed, CaVβ1E expression is needed for the activation of Gdf5 promoter, probably by acting on specific E-box sequences as described for other target genes [START_REF] Taylor | The Cavβ1a subunit regulates gene expression and suppresses myogenin in muscle progenitor cells[END_REF].

In addition, we demonstrated that a compromised CaVβ1E/GDF5 axis during aging is associated to muscle wasting. In our study, CaVβ1E over-expression led to increased GDF5 expression and activation of its pathway improving age-related muscle decline. By contrast, CaVβ1E overexpression did not affect muscle mass in young mice. This suggests that endogenous CaVβ1E induction following nerve withdrawal is sufficient to trigger the maximal compensatory response.

In addition, when muscles express physiological amount of GDF5, feedback mechanisms could be activated to prevent hypertrophy. Interestingly, GDF5 exerted hypertrophic effects mostly on innervated muscle, suggesting that nerve is important for its trophic signal. In this hypothesis, GDF5 could stimulate pre-and/or post-synaptic NMJ partners improving neuro-transmission. Thus, increased GDF5 signaling, induced by over-expression of CaVβ1E or GDF5 itself, might counteract age-related muscle decline also by preventing the associated denervation occurring later. CaVβ1 has been shown as important for correct synaptic patterning in embryo [START_REF] Chen | Neuromuscular synaptic patterning requires the function of skeletal muscle dihydropyridine receptors[END_REF]. Even though a role for GDF5 in this process has not been established yet, its relevance to mediate reinnervation in adult muscle is proven [START_REF] Macpherson | Dach2-Hdac9 signaling regulates reinnervation of muscle endplates[END_REF]. Importantly, we also demonstrated that a hCACNB1-E variant is expressed in human muscle, indicating that mechanisms governing skeletal muscle homeostasis might be conserved across mammalian species. Although hCACNB1-E transcript is slightly different compared to mouse Cacnb1-E, we observed similar localization and hypothesize the same function. In fact, hCACNB1-E expression was higher in young compared to aged human muscle with reduced lean mass and power, strongly suggesting a defect in compensatory mechanism contributing to age-related muscle mass loss. Despite weak transcription of hGDF5 in undamaged young human muscle, we demonstrated an association between its expression and muscle mass.

Other studies identified hypertrophic and potentially druggable factors reduced in sarcopenic muscle as IGF-1 [START_REF] Kern | Electrical stimulation counteracts muscle decline in seniors[END_REF], GH [START_REF] Bartke | Growth Hormone and Aging: Updated Review[END_REF], leptin [START_REF] Hamrick | Role of the Cytokine-like Hormone Leptin in Muscle-bone Crosstalk with Aging[END_REF] and, recently, apelin [START_REF] Vinel | The exerkine apelin reverses age-associated sarcopenia[END_REF]. These molecules are dependent on muscle activity and aging. Here, we identified CaVβ1E as "sensor" of electrical activity defect which could be a central player, upstream of all these factors. Age-related muscle decline is a life-threatening condition (50) affecting a large population, which results in progressive loss of autonomy, increased mortality associated to frailty and risk of falls in elderly subjects [START_REF] Arango-Lopera | Mortality as an adverse outcome of sarcopenia[END_REF]. GDF5 or GDF5-like molecules can be considered as potential therapeutic compounds to prevent muscle mass wasting during senescence but also, in a wider application, to ameliorate consequences of neuromuscular defects. Finally, deciphering the fine regulation of CaVβ1E expression and GDF5 signaling would open a promising therapeutic field that will contribute to increasing the quality of life.

The study has some limitations. The events altering the correct basal expression of CaVβ1E in aging muscle remain unresolved. Previous studies have reported a link between elevated CaVβ1A and age-related muscle weakness [START_REF] Taylor | Increased CaVbeta1A expression with aging contributes to skeletal muscle weakness[END_REF][START_REF] Delbono | Expression and Regulation of Excitation-Contraction Coupling Proteins in Aging Skeletal Muscle[END_REF][START_REF] Lopez | Raptor ablation in skeletal muscle decreases Cav1.1 expression and affects the function of the excitation-contraction coupling supramolecular complex[END_REF] but Cacnb1 transcripts were not fully characterized nor antibodies used well defined. CaVβ1E expression is higher in young type IIA-IIX fibers compared to IIB. The fiber type II decrease occurring during aging may thus have a role in CaVβ1E reduction. Still, other mechanisms, dependent on muscle activity per se, modifying DNA methylation or mitochondrial metabolism, could participate in altering the CaVβ1E/GDF5 axis.

Materials and Methods

Study design

The primary objective of this study was to decipher the link between alteration of electrical activity and skeletal muscle mass maintenance. We focused on the role of CaVβ1, a subunit of the voltage sensor CaV1.1. RNAseq data of innervated and denervated TAs led to identify an embryonic variant, CaVβ1E, up-regulated in mouse muscle by denervation. Validation of RNAseq data and characterization of CaVβ1 isoforms in mouse and human muscles have been performed by RT-PCR, qPCR, western blot and immunostaining experiments. To investigate the role of CaVβ1E in muscle mass homeostasis in mouse, we knocked-down its local expression in vivo by AAV-shRNA. We quantified atrophy by measuring muscle weight and fiber size distribution. To determine fiber size, images were processed with a machine learning algorithm (WEKA) under Fiji for accurate segmentation. Muscle histology was performed by Haematoxilin&Eosin and Sirius Red staining. To evaluate the effect of CaVβ1E down-regulation on the transcriptional activity of Gdf5 promoter, we performed a Luciferase reporter-based assay in C2C12. To establish the effect of CaVβ1E or GDF5 in vivo on muscle mass, we overexpressed them in mouse by local AAV-gene transfer. In order to characterize the expression of CaVβ1 isoforms in humans, we used muscle biopsies of a cohort of healthy young and aged volunteers [START_REF] Mcphee | Physiological and functional evaluation of healthy young and older men and women: design of the European MyoAge study[END_REF]. Body composition and muscle function of volunteers included in the study [START_REF] Mcphee | Physiological and functional evaluation of healthy young and older men and women: design of the European MyoAge study[END_REF] had been measured previously and provided for correlation analysis with hCACNB1-E. We estimated the sample size on the basis of known variability of assays. All mouse experiments were performed twice. Outliers not following normal distribution of samples were removed based on Grubb's test. All experiments using animals and cells were done in a non-blinded manner, yet investigators were blinded to allocation in using human samples for RT-qPCR experiments.

Statistical Analysis

For comparison between two groups, two-tailed paired and unpaired Student's t-tests were performed to calculate P values and to determine statistically significant differences (significance was for P<0.05, as detailed in the figure legends). For comparison among more than two groups, ordinary one or two-way Anova tests followed by the appropriated multiple comparison tests (as detailed in the figure legends) were performed. All experiments have been done twice with the same results. All statistical analyses were performed with GraphPad Prism7 software. Young, from 20 -42 years old (Y) and aged, from 70 -81 years old (A) participants; gender male (M) and female (F). Lean mass (%) was assessed by dual-energy X-ray absorptiometry. Power is expressed by Watt on kilograms (W/kg); ND: not determined. H, I: Distribution of (H) hCACNB1-E or (I) hCACNB1-A expression in human quadriceps biopsies from healthy young and aged volunteers (Table 1).

Supplementary material list

Materials and Methods

J: Linear regression between hCACNB1-E expression and lean mass percentage in human quadriceps biopsies from healthy young and aged volunteers, (Table 1).

K: Distribution of hGDF5 (pink triangles, left y axis) and hCACNB1-E (blue circles, right y axis) expression in human quadriceps biopsies from healthy aged volunteers (Table 1) having increasing lean mass percentage. Dot black line indicates the average of lean mass percentage of the young group.

(

  TAs) compared to innervated control muscles (D0) (Fig 1A). Western blot of CaVβ1 revealed the appearance of an extra 70 kDa band whose intensity increased with time after denervation while the intensity of the band around 55 kDa was unchanged (Fig 1B, C).

  and Table S1), implying the transcription of two different splicing isoforms. RT-PCR confirmed that in innervated TAs the Cacnb1 open reading frame is at the 5' end of exon 3 (ATG1). By contrast, in denervated muscles, two Cacnb1 transcripts were expressed: one starting at exon 1 (ATG2) and another at ATG1-exon 3 (Fig 1F, G).

(

  66 and 70 kDa, respectively). However, CaVβ1 antibody targeting a central peptide of the protein (AbCaVβ1: sc-25689 (H-50) raised against amino acids 211-260 L-type CaVβ1A of human origin -NM_199247-NP_954855) recognized only CaVβ1E in mouse. To validate which isoform was upregulated after denervation, we designed specific primers matching the two sequences at the level of the 3'-end of exon 13 for Cacnb1-D and exon 14 for Cacnb1-E. RT-PCR data confirmed that only Cacnb1-E increased in denervated muscle (Fig 1H, I) confirming protein expression data. Next, we verified if CaVβ1E was the only isoform induced in denervated muscle. Amplification of the region between exons 5 and 8/9 of Cacnb-1 let the discrimination of the several variants depending on the amplicon size. RT-PCR showed that innervated and denervated muscle expressed only Cacnb1 variants containing exon 7A, corresponding to Cacnb1-D and Cacnb1-E (381 bp), and not Cacnb1-B (246 bp), expressed in spinal cord (SC) (18-20) nor Cacnb1-C (246 bp) and Cacnb1-F (222 pb) (Fig 1J).

  and RT-qPCR (primers in exons 2-3 for Cacnb1-E and exon 13 for Cacnb1-D) revealed that Cacnb1-E is the specific variant in embryonic and neonatal muscles, while Cacnb1-D is not expressed in embryo (Fig S1A, B). Cacnb1-E transcript in embryonic muscle started at exon 1 (Fig S1C, D) as Cacnb1-E expressed in denervated adult muscle. Western blotting showed the same bands in denervated adult and embryonic muscle protein extracts using AbCaVβ1 (Fig 1L) or an antibody specific to CaVβ1E (Fig S1E). Finally, we analyzed if CaVβ1D and CaVβ1E could have different localization in muscle fiber. Immunofluorescence analysis of innervated and denervated muscle sections and isolated fibers with AbCaVβ1 showed that CaVβ1 staining and triadic localization most likely reflected expression of CaVβ1D (Fig S2A-C). In contrast, CaVβ1E-specific staining was distributed at the Z-lines (Fig S2 D-F). Furthermore, CaVβ1Especific staining localized mostly at the nuclei, consistently with the presence of a Nuclear Localization Signal predicted by cNLS mapper, and its intensity increased in denervated muscle compared to CaVβ1 staining (Fig S2G-L).

Cacnb1 exon 2 (

 2 26) (AAV-ShCaVβ1E) and thereby abolishing specifically Cacnb1-E. Two months after injection, CaVβ1E expression induced after denervation was decreased by ~90% (Fig 2A-C; Fig S3A, B), without decreasing CaVβ1D protein expression (Fig 2B, C). Blocking the induction of CaVβ1E exacerbated muscle atrophy after denervation (Fig 2D-F), with decreased muscle fiber size (Fig 2F) and increased fibrosis (Fig S3C-F). This suggested a role for CaVβ1E in preserving disused muscle mass. The GDF5 pathway has been shown to be essential for limiting muscle loss under atrophic conditions (2). We thus evaluated Gdf5 expression in absence of CaVβ1E and found a significant (P<0.0001) reduction in the denervation-induced Gdf5 increase (Fig 2G). In contrast, Gdf8 or Bmp7 transcription, which could be both implicated in increased atrophy (27, 28), was not modified upon CaVβ1E ablation (Fig 2H, I). Furthermore, all components of the GDF5 pathway, SMAD1/5 phosphorylation, SMAD4 nuclear translocation, and Id-2 transcription (29, 30), were inhibited with CaVβ1E down-regulation suggesting the positive control of GDF5 signaling by CaVβ1E (Fig 2J-O). However, Id-1 expression was not modified after denervation in Scra or AAV-ShCaVβ1E treated muscle (Fig 2N), suggesting that its activation might be very weak or transient (31). CaVβ1 has been shown to inhibit myogenin signaling during myoblast proliferation (14), yet there are no data about this regulation in adult muscle. In innervated or denervated TAs we found no change in transcription of Myogenin after CaVβ1E knock-down (Fig S3G) and no alterations in Fbxo32, MuRF1 expression (Fig S3H-I), indicating that the myogenin pathway is unlikely to participate in the increased atrophy following CaVβ1E down-regulation. However, Chrna1

  S4A, B) and only CaVβ1E protein (band at 70kDa) as embryonic muscle. (Fig S4C). Moreover, the transcription of Gdf5, as well as Myogenin, also increased in differentiating C2C12 cells (Fig S4D, E). Inhibiting Cacnb1-E expression in differentiating C2C12 cells by transfecting a plasmid ShCaVβ1E (pCDNA3-ShCaVβ1E) (Fig S4F) prevented the expression of Gdf5 without affecting Myogenin transcription (Fig S4G, H), mimicking its effect in vivo. We then measured in C2C12 the consequence of CaVβ1E down-regulation on the promoter activity of Gdf5. Thus, the sequence from -312 to the Gdf5 transcription start site, which contains one CANNTG and two CANNNTG E-boxes, was cloned upstream of firefly luciferase in HSVTK-Luc3' modified plasmid and transfected into C2C12 cells. Firefly/renilla signal increased during differentiation, reflecting Gdf5 promoter activation, and this effect was abolished by the ShCaVβ1E (Fig 2P). These data indicated that CaVβ1E targets Gdf5 promoter, consistently with the effect observed in vivo and in vitro. CaVβ1E seems to precede CaVβ1D appearance in embryo and in C2C12. We thus checked if Cacnb1E and Cacnb1D mRNA expression could affect each other and have a role in muscle mass homeostasis via GDF5. Although AAV-ShCaVβ1E injection prevented Cacnb1E rise after denervation, it also induced a slight decrease of Cacnb1D mRNA expression (Fig S5A, B), yet with no impact on CaVβ1D protein expression. However, only the rescue of CaVβ1E, and not CaVβ1D, restored Gdf5 transcription and atrophy after denervation in AAV-ShCaVβ1E treated muscles (Fig S5C, D). To confirm that the increased atrophy upon CaVβ1E down-regulation was dependent on inadequate GDF5 activation, we overexpressed this factor in TAs together with ShCaVβ1E, via AAV-gene transfer (Fig S5E, F). The up-regulation of Id1 and Id2 proved the activation of GDF5-dependent pathway (fig S5G, H) in double treated muscles. As expected, GDF5 over-expression rescued the atrophy induced CaVβ1E ablation and improved muscle histology by decreasing fibrosis (Fig S5I-R).

  and nothing is known about Gdf5 amount in aging muscle. Because our data indicate the involvement of CaVβ1E via Gdf5 in muscle maintenance, we investigated its role in age-related muscle wasting. In C57bl/6 mice, we observed significant (P<0.0001) TA muscle mass loss at 78 weeks relative to adult mice at 12 weeks (Fig 3A). In addition, Cacnb1-E basal expression was significantly lower in TAs of 78 compared to 12 weekold mice (P<0.0001) (Fig 3B), whereas Cacnb1-D transcription did not change (Fig 3C). To evaluate if the cross-talk between Cacnb1-E and Gdf5 would also be affected, we quantified these transcripts after denervation in TAs from 12-, 52-and 78-week-old mice. Although CaVβ1E expression increased in denervated young muscle, the up-regulation of transcript and protein in response to denervation was impaired since 52 weeks of age (Fig 3D, G, H). Consequently, the Gdf5 increase was reduced (Fig 3F) affecting SMAD1/5 phosphorylation (Fig 3J, K). Id1 remained unchanged (Fig 3L) whereas Id-2 transcription followed the altered Gdf5 induction (Fig 3M). CaVβ1D expression did not change (Fig 3E, G, I). We explored some possible causes responsible for decreasing CaVβ1E during aging. Measuring the expression of NMJ components Chrna1, Chrne, Chrng and Musk in 78 week-old mice muscles, we could not detect changes indicating endplate alterations at this age (Fig 4A-D).

  IIA and -IIX positive fibers, although in old muscle it was decreased in all fiber types (Fig 4I-K).If CaVβ1E increased upon lack of nerve activity, for example denervation, modifications in muscle activity could potentially affect basal Cacnb1E transcription. Hence, we measured Cacnb1E mRNA after acute exercise of young and old mice[START_REF] Baar | Adaptations of skeletal muscle to exercise: rapid increase in the transcriptional coactivator PGC-1[END_REF][START_REF] Brandt | PGC-1α and exercise intensity dependent adaptations in mouse skeletal muscle[END_REF]. Exercise training restored Cacnb1E transcription in old muscles but not in young (Fig 4L), with Cacnb1D expression unaffected (Fig 4M), suggesting that increased muscle activity is able to rapidly normalize Cacnb1E transcript only when it is decreased. However, MyHC-IIA and IIX expressions were not restored, implying that contractile activity could regulate Cacnb1E expression independently of type IIA-X fiber abundance (Fig 4N, O). To evaluate whether CaVβ1E or GDF5 over-expression might mitigate age-related muscle mass loss, AAV vectors carrying CaVβ1E or GDF5 were injected into the TA of 78-80-week-old mice. A strong over-expression of CaVβ1E and GDF5 was observed 3 months after injection (Fig. 5A, B, D and S6A) without modifying Cacnb1-D expression (Fig 5C and S6C), yet increasing expression of each other reciprocally (Fig 5D and S6B). The up-regulation of Cacnb1-E in old muscle activated GDF5 signaling, as measured by SMAD1/5 phosphorylation and SMAD4 nuclear translocation. (Fig 5E-G). The increase of Gdf5 either by CaVβ1E or by its own over-expression induced Id-2 transcription, with no changes in Id-1, confirming a weaker or transient response of this factor to GDF5 (Fig 5H, I and S6D, E). In addition, the rise of GDF5 signaling following Cacnb1-E or Gdf5 over-expression was associated with preservation of aged muscle mass (Fig 5J) and a gain of specific force (Fig 5K) compared to control old muscles. Subsequently, we investigated if CaVβ1E over-expression could affect muscle mass of young mice (12 weeks). Ectopic expression of CaVβ1E (Fig S7A, D, E) had no effect on Cacnb1-D, yet it induced a slight increase of Gdf5 expression (Fig S7B, C, D, E). However GDF5 signaling, measured by Id-1 and Id-2 transcription, was not or poorly activated (Fig S7F, G). Consequently, muscle mass and the response to denervation were not different between Scra-and CaVβ1Etreated TAs of young mice (Fig S7H-L). We also overexpressed GDF5 in young TAs (Fig S8A, D, E) which induced Cacnb1-E transcription in innervated TAs compared to scrambled (Fig S8B, D, E) without affecting Cacnb1-D expression (Fig S8C). Nevertheless, GDF5 over-expression and its activated signaling, measured as Id-1 and Id-2 transcription (Fig S8F, G), increased mostly innervated muscle mass (Fig S8H-L).

Figure S1 :Figure S5 :

 S1S5 Figure S1: expression of embryonic CaVβ1 isoform

Fig 1 :D:

 1 Fig 1: Expression of an embryonic CaVβ1 isoform by alternative first exon splicing after

Fig 2 :F:J:

 2 Fig 2: CaVβ1E down-regulation and increased muscle atrophy by reduced GDF5 signaling

Fig 3 :

 3 Fig 3: Muscle mass loss during aging and alteration of CaVβ1E-GDF5 axis

Figure 4 :

 4 Figure 4: NMJ-associated genes and MyHC expression and effect of exercise on CaVβ1E

F- I :

 I Means ± s.e.m. (Young: n=8; Aged: n=17) ** P < 0.01, ***P < 0.001, (independent-samples t-test (two-tailed)).

Table S1 : Expression data Table S2: 100 top up-regulated genes in innervated and denervated samples Table S3: List of human samples Table S4: Primer list Datafile S1: List of 1022 differentially regulated alternative splicing events Datafile S2: Uncut Western Blotting raw data Datafile S3: Individual subject data confirms

 S1 that this manuscript has not been previously published and is not currently under consideration by any other journal. Additionally, all of the authors have approved the contents of this paper and have agreed to the Science Translational Medicine submission policies.

	Participant Gender	Age (years)	Height (m)	Weight (kg)	Lean mass (%)	Power (W/Kg)
	Y1	M	20.9	1.82	84.4	82.3	54.7
	Y2	F	24.5	1.63	56.2	78.4	51.1
	Y3	F	26.1	1.53	53.0	70.4	38.4
	Y4	M	26.4	1.76	56.7	89.2	48.6
	Y5	M	27.1	1.83	73.8	75.9	42.4
	Y6	M	37.0	ND	ND	ND	ND
	Y7	M	38.0	ND	ND	ND	ND
	Y8	M	42.0	ND	ND	ND	ND
	A1	F	70.8	1.68	70.2	58.9	22.3
	A2	M	70.9	1.58	65.1	70.4	34.5
	A3	M	71.4	1.76	90.7	64.5	29.7
	A4	M	71.4	1.68	84.4	66.3	27
	A5	F	71.5	1.61	53.3	67.6	21.2
	A6	M	71.7	1.67	69.4	79.2	35.3
	A7	M	72.5	1.67	74.3	70.4	34.7
	A8	M	72.9	1.74	91.6	64.2	28.1
	A9	M	73.8	1.67	76.0	71.7	36
	A10	F	74.2	1.54	58.3	64.0	26.3
	A11	M	75.0	1.69	80.0	68.3	29.4
	A12	M	76.4	1.58	67.7	74.7	25.6
	A13	M	76.7	1.76	85.5	67.4	25.8
	A14	F	77.8	1.58	55.5	70.8	22.2
	A15	M	78.0	1.75	80.3	68.5	22.6
	A16	M	79.6	1.67	67.9	80.9	26.3
	A17	F	80.6	1.59	56.3	71.2	29

Table 1 : Characteristics of human muscles:

 1

Acknowledgements

We thank Y. Marie for RNA sequencing data; N. Robil for the help in the RNA-Seq data analysis; S. Vasseur and M. Chapart for their help in collecting human muscle at the Myology Institute Myobank; V. Sazdovitch for the human spinal cord biopsy; H. Amthor and A. Stantzou for gathering negative control muscles; M. Guilbaud and F. Aurade for their expertise advices for the luciferase assay; E. Mouisel, S. Vassilopoulos, M.G. Biferi, A. Forand, E. Gargaun, P.

Funding

This study was funded by Malakoff Médéric (to F.P.R and S.F.), Association Institute of Myology (to F.P.R and S.F.) and Association Française contre les myopathies AFM-Telethon (to F.P.R and S.F.).

Data and materials availability

All data associated with this study are available in the main text or the supplementary materials.

Author Contributions

F, G: Distribution of (F) lean mass percentage and (G) Power in human quadriceps biopsies from healthy young and aged volunteers (Table 1).


Nigina Toktassynova

Hassen Fourati
email: hassen.fourati@gipsa-lab.fr

Batyrbek Suleimenov

Application of grey system theory to phosphorite sinter process: From modeling to control

Keywords: burn through point, control structure, grey system, optimization algorithm, predictive model, sinter process

The sintering process of phosphorite ore occurs with a large amount of return caused by untimely process control. The control task of phosphorite ore sintering is to regulate the parameters of the process to obtain a high quality sinter. The parameter clearly responsible for sinter quality is the temperature in the wind box. Therefore, in order to solve the control task, it is necessary to predict the highest temperature of the charge (also known as the burn through point (BTP)). In this paper, the theory of grey systems is used as a predictive model, which makes it possible to obtain an adequate model that uses a small number of initial samples of real temperature data. Based on the grey model GMC(1,n) a new optimal model is presented, which is constructed by using optimization algorithm. Optimal model predicts the BTP, and to establish an optimal regulation, a control synthesis is carried out through an optimization of the prediction according to the "particle swarm" algorithm.

INTRODUCTION

Sintering in metallurgy is a thermal process that occurs in a metallurgical charge, composed of ore pellets, concentrates, and fuel (coke). The technological process is shown in Figure 1. It begins with the preparation of the phosphorus ore in predetermined proportions, which includes the following components: phosphorus trifle, coke trifle, and secondary return. This return is the bad quality sinter produced after breaking in yellow phosphorus production. Next, the phosphorus ore is mixed with a cold return and other components in a special drum to form the charge.

Abbreviations: BTP, burn through point; GMC, convolution grey model; NN, neural network; GM, grey model; GMC, convolution integral grey model; FGMC, first-pair-of-data convolution integral grey model; IGDMC, interval grey dynamic model with convolution integral; DGDMC, deterministic grey dynamic model with convolution integral; OGDMC, optimized grey dynamic model with convolution integral. This latter is loaded onto the moving grate of the sintering machine and is ignited using natural gas or CO 2 under the horn. With the help of wind boxes under the grate, an underpressure is created and a stream of hot gases gradually passes through the layer of 260 mm in height from the top to the underlying layers, ensuring the burning of coke breeze particles and the formation of sinter. At the end of the sintering machine, the sinter is cooled and crushed, and the poorly sintered charge is sent to the beginning of the process.

Due to the non-linearity of the process, real time quality control is a complex task. In practice, the operator makes changes to the process after the product is obtained, which leads to a large quantity of return and expenses for re-sintering. In this regard, it becomes necessary to predict the sinter quality in advance and control the process based on a predictive model. One of the key indicators of sintering quality is the BTP that indicates the completion of the sintering process and represents the point with the highest temperature. The problem of BTP prediction and control is studied in many works using various algorithms, such as, neural networks (NN), genetic algorithms, cluster analysis, and fuzzy control. Due to the evolution and wide spread of NN, a large number of works related to prediction use this method in BTP prediction and control. NN is used in [START_REF] Peng | Sintering finish point intelligent control[END_REF], where the strand speed and BTP at the previous instants of time are given to the network inputs. The network, together with the training algorithm, update the BTP model. In [START_REF] Du | Intelligent coordinating control between burn-through point and mixture bunker level in an iron ore sintering process[END_REF], the current value of BTP, temperatures in the middle and end of the process, and the strand speed are inputs to the NN to predict the BTP, and an intelligent control of the iron ore sintering process is proposed. A fuzzy NN is used in BTP prediction [START_REF] Wang | BTP prediction of sintering process by using multiple models[END_REF]. Temperatures in wind boxes are chosen as inputs and the BTP temperature as an output of the model.

Other works [START_REF] Wu | Intelligent optimal control for lead-zinc sintering process state[END_REF][START_REF] Wu | Neural-network-based integrated model for predicting burn-through point in lead-zinc sintering process[END_REF] use two BTP predictive models: temporary and technological. Then, to predict the permeability of lead-zinc ore in [START_REF] Wu | Intelligent optimal control for lead-zinc sintering process state[END_REF] the temporary NN is supplied with 6 previous values for permeability, and the technological NN uses the temperature of fire, humidity, sulfur, lead, silicon dioxide content, and strand speed. In [START_REF] Wu | Neural-network-based integrated model for predicting burn-through point in lead-zinc sintering process[END_REF], the BTP grey model GM(1,1) of lead-zinc ore is applied, and the charge permeability and strand speed of the pallets are fed to the technological neural network input. The approach proposed in [START_REF] Wu | An intelligent control system based on prediction of the burn-through point for the sintering process of an iron and steel plant[END_REF] for BTP prediction of iron ore sintering uses the integration of grey model and NN. Here the result of the grey model GM(1,1), especially the gas temperature in wind boxes, is one of the inputs of the back propagation NN, along with the strand speed and BTP at current time. The three works, with two predictive models, use a fuzzy expert controller that maintains the BTP desired position within the specified boundaries.

In [START_REF] Du | A fuzzy control strategy of burn-through point based on the feature extraction of time series trend for iron ore sintering process[END_REF], a carbon efficiency optimization model and BTP expert-fuzzy controller are presented. The main idea is to improve carbon efficiency by using an intelligent integrated control strategy for the BTP. The strand velocity of the carbon efficiency optimization model and BTP expert-fuzzy controller are integrated using the fuzzy satisfaction degree method, which calculates the end strand velocity. The BTP prediction model developed in [START_REF] Chen | T-S Fuzzy logic based modeling and robust control for burning-through point in sintering process[END_REF] also uses 2 models. Temperatures in wind boxes are supplied to a BTP soft-sensor model (1st model) to predict the BTP position. The BTP temperature prediction model (2nd model) uses BTP temperatures, obtained in previous steps. The outputs of 2 models together with strand velocity are used as inputs of fuzzy robust control, which is based on an improved Takagi-Sugeno model.

Taking another perspective, genetic algorithms are used within this framework. Thus, in [START_REF] Cheng | An application of adaptive genetic-neural algorithm to sinter's BTP process[END_REF], an adaptive genetic algorithm was used to control BTP in the iron ore sintering process, where the input layer of the NN is parameters of the initial material, density, strand speed, and ignition temperature, and the output layer is the values of temperature and underpressure of sintering gases and gases in wind boxes. The genetic NN was used also for the BTP prediction of iron ore sintering in [START_REF] Cheng | Prediction system of burning through point (BTP) based on adaptive pattern clustering and feature map[END_REF], where 707 groups of data were given to the input after clustering and classification of temperature and underpressure vectors from 18 wind boxes. The adaptive structural clustering system is based on a spatial clustering of initial data, a self-organizing NN map for extracting data relevancy properties, and a Kohonen map for the learning network. The idea of clustering is used in [START_REF] Shang | Data-driven prediction of sintering burn-through point based on novel genetic programming[END_REF] for the BTP prediction model of iron ore sintering. The K-means clustering module, the inputs of which are the cold-charge permeability model, the ignition temperature, and the coke residue values, is used to form clusters. The predictive model based on the clusters from K-means module were fed to the dynamic model of temperature and underpressure, which is constructed using the novel genetic programming. A firework algorithm based on a genetic algorithm was proposed in [START_REF] Wang | Application research based on GA-FWA in prediction of sintering burning through point[END_REF] to optimize the sintering terminal prediction model of the support vector machine. Seven main parameters were used as inputs to predict the BTP.

Models that do not use the NN for BTP prediction are based on equations of temperature or BTP. In [START_REF] Kim | Estimation of burn-through point in the sinter process[END_REF], the authors used a curve fitting method and represent results using a cubic and fifth-order curves for the part near to the BTP position. The authors in [START_REF] Wang | Hierarchical intelligent control system and its application to the sintering process[END_REF] describe a two-level hierarchical control system of the BTP and the vertical sintering rate in the iron ore sintering process, where the BTP model, represented by a piecewise-quadratic temperature, depends on the position on the strand. In [START_REF] Cao | A dynamic subspace model for predicting burn-through point in iron sintering process[END_REF], the BTP can be approximated using a quadratic function of exhaust-gas temperature. A dynamic state space model is proposed to obtain an exhaust-gas temperature, with 6 inputs. A grid search algorithm optimizes parameters for the best combination in a certain range using an evaluation function. In [START_REF] Wu | Design and application of generalized predictive control strategy with closed-loop identification for burn-through point in sintering process[END_REF], a BTP control in iron ore sintering process is divided into 2 parts: a closed-loop identification model and a generalized predictive control one. In the first part, the BTP is found based on the dynamic autoregressive exogenous model, to calculate which strand speed, amount of moisture, height of the charge, air volume, and underpressure are used. The closed-system identification method is used to dynamically determine the model parameters. The state of BTP also is predicted using a particle swarm optimization algorithm (PSO) [START_REF] Shi | Soft sensing of the burning through point in iron-making process[END_REF], which include 4 influence parameters: suction pressure, air input, velocity of sintering machine, and ignition temperature. Obviously, the current BTP is also affected by its previous stage and [START_REF] Du | Intelligent integrated control for burn-through point to carbon efficiency optimization in iron ore sintering process[END_REF] carried out time series trend analysis for the BTP, which gives the opportunity to determine global and local trend feature variables. These variables are supplied as inputs of a fuzzy controller, constructed on the basis of an operator's experience. The above review of the literature means we can conclude the following drawbacks. First, the prediction of the BTP is based on many different parameters: data of vertical sintering rate, underpressure, permeability, height of the charge, temperature in wind boxes or sintering gases, strand speed and previous values of the BTP. Second, the high accuracy of predictive models is achieved by using a large initial sample or historical data.

Application of existing models for BTP prediction has the following drawbacks. In practice, due to a lack of automation, only the main process parameters are frequently measured. The values of permeability, volume of sintering gases, and properties of charge are obtained only by laboratory experiments. Therefore, it becomes necessary to construct a predictive model using the main measured process parameters. Obtaining a large initial sample is also difficult in terms of developing a prediction model while in production, and it requires a long time to collect and accumulate data to improve prediction accuracy. In this case, if certain conditions of the process change, it will be necessary to collect data and retrain the model. Therefore, it is necessary to use a predictive model that will be adequate for training on a small number of initial samples and will be built in real time. To solve this problem the theory of grey systems developed by Deng in 1982 is used, which is focused on solving prediction problems with a small volume of the original samples and with poor information. The idea of grey systems theory is to consider the process as a "generalized energy system", and emphasizes that non-negative smooth discrete functions can be transformed into a sequence having the approximate exponential law, that is, the so-called grey exponential law [START_REF] Deng | Introduction to grey system[END_REF]. The application of grey models is considered in [START_REF] Wu | Neural-network-based integrated model for predicting burn-through point in lead-zinc sintering process[END_REF][START_REF] Wu | An intelligent control system based on prediction of the burn-through point for the sintering process of an iron and steel plant[END_REF] for BTP prediction. In these studies, the GM(1,1) model is used, in which only the previous values of the predicted variable influence the subsequent prediction. The disadvantage of this approach is the lack of consideration of influencing factors that can improve the accuracy of the prediction model and results.

Therefore, in this paper, a convolution integral grey model GMC(1,n) is used [START_REF] Tien | The indirect measurement of tensile strength of material by the grey prediction model GMC(1, n)[END_REF], which allows us to take into account (n-1) influencing factors on the prediction of BTP. Since the model takes into account not only temperature, but also underpressure in wind boxes, the control task is based on changing two variables, the strand speed (standard approach) and the under-pressure. The main idea of this paper is focused on the development of a BTP prediction system based on a small amount of an initial sample to implement it in a future a cheap control system of sinter quality. To achieve this goal the following steps are proposed: (1) determination of parameters that influ-ence BTP and can be measured in real-time; (2) analysis of BTP prediction models based on grey system theory; (3) development of optimal BTP prediction model by using a particle swarm optimization algorithm; (4) proposition of control structure of BTP based on forecast optimization method.

The paper is organized as follows. Section 2 presents the sintering process and the samples of the main parameters used for the prediction process. Section 3 is devoted to the grey predictive models, determination of the most accurate model to predict BTP, and development of an algorithm to improve GMC (1,n). Section 4 presents the structure of the sintering process control based on a predictive optimization. Section 5 provides some concluding remarks.

SINTERING PROCESS AND MAIN PARAMETERS FOR PREDICTION

The phosphorite ore sintering process takes place in the Novozhambul phosphorus plant, which is located in the south of Kazakhstan, near the Karatau phosphor field. Sintering machine capacity reaches only 50-60%, since the control is performed with a time delay that relies on an operator's decision. The sintering process includes a lot of parameters, which influence BTP. Some of these can be used and measured only through laboratory studies, such as: initial charge composition, pellet size, ignition process parameters, coke content in charge, percent of sinter return and its characteristics; other parameters can be measured in real-time processing: temperature of exhaust-gases in wind boxes, pressure drop across the bed data, strand velocity, sinter height; and finally there are parameters that are calculated or saved in a database, such as previous BTP data. In this paper the temperature in wind boxes, which can be measured in real-time and gas velocity, which can be calculated through a pressure drop across the bed, are used. Sinter height is constant value in production. Temperature curves (Figure 2) are samples of data taken at different times from the process by experiment using thermocouples, under different initial conditions, but for a constant speed, so they are shown only until the desired temperature is reached in order to determine the duration of the process.

According to the literature, gas velocity has an influence on the charge temperature and could be determined through the pressure drop across the bed data Δp, then the gas velocity 𝜈 (Figure 3) in the lower part of the charge can be calculated using an Ergun equation: where L[m] is the height of the bed, 𝜇[Pa •s] is the dynamic viscosity of gas, 𝜌 is the density of the charge, 𝜀 is the porosity of porous matrix calculated following [START_REF] Kaymak | Iron ore sintering process model to study local permeability control[END_REF], and d[m] is the equivalent particle size.

Δp L = 150𝜇𝜈(1 -𝜀) 2 d 2 𝜀 3 + 1.75𝜌𝜈 2 (1 -𝜀) d𝜀 3 (1)

DETERMINATION OF BTP GREY MODEL

Building a data acquisition system with data storage, analysis, and continuous retraining requires tremendous resources both of time and money. Also, the initial data collection, training, and building of an effective system requires constant interruption on the main production process, which impacts its effectiveness. In the production considered for this study there is no data acquisition system, therefore, to obtain data, additional measurement systems were required that were not included in the main control loop. In such conditions, it becomes necessary to use models that require a small amount of the initial sample to train and construct a predictive model. The theory of grey systems satisfies these requirements.

Grey theory is a relatively new theory for prediction issues. One of the main characteristics of grey system theory is the accumulated generating operation (AGO), which is used to reduce the randomness of the data. The mathematical relation of first-order AGO data (1-AGO) is described as following:

X (1) (t)= t ∑ l=1 X (0) (l), Y (1) (t)= t ∑ l=1 Y (0) (l), (2)
where X (0) , Y (0) are the original series and X (1) , Y (1) are 1-AGO data.

The temperature in the wind boxes is considered as predicted series Y. The gas velocity 𝜈, calculated from the known pressure Δp (from (1)), which is measured in real time, was selected as an influencing factor X.

The influence of several factors on the predicted value for grey models can be considered only on the basis of model modifications GM(1,n), which is clearly demonstrated in [START_REF] Tien | The indirect measurement of tensile strength for a higher temperature by the new model IGDMC(1, n), Measurement[END_REF]. A continuous convolution integral grey model GMC(1,n) [START_REF] Kaymak | Iron ore sintering process model to study local permeability control[END_REF] is one of the basic models with (n-1) influencing factors. On the basis of this model, other continuous linear grey models with (n -1) influencing factors were developed. For example, an interval model with a convolution integral IGDMC(1,n) [START_REF] Tien | The indirect measurement of tensile strength for a higher temperature by the new model IGDMC(1, n), Measurement[END_REF] is intended for prediction of the interval of the variable. Model FGMC(1,n) [START_REF] Tien | The indirect measurement of tensile strength by the new model FGMC (1, n), Measurement[END_REF] is developed on the basis of effectiveness of the first pair of original data. The deterministic grey model with the convolution integral DGDMC(1,n) [START_REF] Tien | The deterministic grey dynamic model with convolution integral DGDMC(1, n)[END_REF] distinguishes from GMC(1,n) by the estimation of first derivative and parameters: first derivative is estimated numerically by the cubic curve of the spline and the parameters of the model, according to the scheme of deterministic convergence. The error in the strength prediction considered by Tien is 0.54% for FGMC(1,n), 1.25% for GMC(1,n), 1.85% for DGDMC(1,n), and 2.4% for IGDMC (1,n). Later in [START_REF] Wang | An optimized grey dynamic model for forecasting the output of high-tech industry in China[END_REF], the optimized model GDMC(1,n) [START_REF] Tien | The indirect measurement of tensile strength by the new model FGMC (1, n), Measurement[END_REF] -OGDMC(1,n) is shown, in which the grey derivative value dX (1) 1 (t)∕dt is determined not by the weighted mean value X (1) 1 (t) and X (1) 1 (t -1), but through the weight coefficient 𝜌 i , determined using the PSO algorithm [START_REF] Kennedy | Particle swarm optimization[END_REF]:

𝜌 i X (1) i (t)+(1 -𝜌 i )X (1) i (t) (3)
Here, a value for the root mean squared percentage error to the priori sample period (RMSPEPR) in the OGDMC(1,n) decreased by 3.8 times -from 7.07% to 1.86%. To select the most accurate model for predicting the BTP in the wind box, we will conduct experiments using models FGMC(1,n), GMC(1,n) and OGDMC(1,n), which have the lowest prediction error value.

GMC(1,n)

The gas velocity, calculated from the known pressure Δp, which is measured in real time was selected as an influencing factor. Convolution integral grey model GMC(1,2) with one influencing factor is a linear differential model:

dY (1) (t) dt + b 1 Y (1) (t)=b 2 X (1) (t)+u (4)
The grey derivative for the first-order AGO data in Equation ( 4) is conventionally represented as:

dY (1) (t) dt = lim Δt→∞ Y (1) (t +Δt)-Y (1) (t) Δt = Y (1) (t +Δt)-Y (1) (t) (5)
when Δt → 1. The parameters of Equation ( 4) are determined by least squares method:

[b 1 , b 2 , u] T =(B T B) -1 B T Y R ( 6
)
where t changes from 1 to r which is the number of initial sample to construct the model in Equation ( 4),

B = ⎡ ⎢ ⎢ ⎢ ⎣ -0.5W Y (1) 0.5W X (1) 1 -0.5W Y (2) 0.5W X (2) 1 .. . -0.5W Y (r -1) 0.5W X (r -1) 1 ⎤ ⎥ ⎥ ⎥ ⎦ (7)
where W Y (r)= Y (1) (r)+Y (1) (r + 1) W X (r)= X (1) (r)+X (1) (r + 1) (1) (2), Y (1) (3), … , Y (1)

Y R =[Y
(r)] T (8)
Prediction of the sinter temperature Ŷ (0) is:

Ŷ (0) (t)= Ŷ (1) (t)-Ŷ (1) (t -1) (9)
where

Ŷ (1) (t)=Y (0) (1)e -b 1 (t-1) + 1 2 e -b 1 (t-1) × b 2 X (1) (t)+u + 1 2 b 2 X (1) (t)+u + t-1 ∑ i=2 e -b 1 (t-i) b 2 X (1) (i)+u

FGMC(1,n)

The differential equation of grey prediction model FGMC (1,n) presented in [START_REF] Tien | The indirect measurement of tensile strength by the new model FGMC (1, n), Measurement[END_REF] is the same as for GMC(1,n), but modeled with data including the information from the first pair of original data. The parameters of Equation ( 4) are determined using the LSM Equation ( 6), where (1) (1), Y (1) (2), … , Y (1) (r)] T [START_REF] Shang | Data-driven prediction of sintering burn-through point based on novel genetic programming[END_REF] Prediction of the sinter temperature is:

B = ⎡ ⎢ ⎢ ⎢ ⎣ -0.5W Y (0) 0.5W X (0) 1 -0.5W Y (1) 0.5W X (1) 1 .. . -0.5W Y (r -1) 0.5W X (r -1) 1 ⎤ ⎥ ⎥ ⎥ ⎦ (10) Y R =[Y
Ŷ (0) (t)=Y (0) (0)e -b 1 t + u(t -1)× × t ∑ i=1 ( 1
2 e -b 1 (t-i+0.5) b 2 X (1) (i)-X (1) (i - 1
)

) ( 12
)

OGDMC(1,n)

Differential equation of the Grey dynamic model with convolution integral is:

dY (1) (t) dt + b 1 Y (1) (t)=b 2 + n ∑ i=2 ( b 2i-1 dX (1) (t) dt + b 2i X (1) (t) ) ( 13
)
The parameters of Equation ( 13) are determined using the LSM Equation ( 6), the vector Y R is calculated according to Equation [START_REF] Shang | Data-driven prediction of sintering burn-through point based on novel genetic programming[END_REF] and

B = ⎡ ⎢ ⎢ ⎢ ⎣ -0.5W Y (1) 1 X (0) (1) 0.5W X (1) -0.5W Y (2) 1 X (0) (2) 0.5W X (2) .. . . -0.5W Y (r -1) 1 X (0) (r) 0.5W X (r -1) ⎤ ⎥ ⎥ ⎥ ⎦ (14)
Prediction of the temperature is determined by Equation [START_REF] Cheng | An application of adaptive genetic-neural algorithm to sinter's BTP process[END_REF], where:

Ŷ (1) (t)=Y (0) (1)e -b 1 (t-1) + u(t -2)× × t ∑ 𝜏=2 { 1 2
e -b 1 (t+0.5-𝜏+0.5𝜆 1 ) ×[0.5 (𝑓 (𝜏)+𝑓 (𝜏 -1))

+ + 0.5𝜆 1 (𝑓 (𝜏)+𝑓 (𝜏 -1))]+ 1 2 e -b 1 (t+0.5-𝜏+0.5𝜆 2 ) × ×[0.5 (𝑓 (𝜏)+𝑓 (𝜏 -1)) + 0.5𝜆 2 (𝑓 (𝜏)+𝑓 (𝜏 -1))]} (15) where 𝜆 1 =-1∕ √ 3, 𝜆 2 = 1∕ √ 3 and 𝑓 (t)=b 2 + ∑ n i=2 b 2i-1 X (0) (t)+b 2i X (1) (t)

Size of samples to construct the model

The number of data to construct a predictive model is an important parameter, which leads to reduction in errors in temperature prediction. Also it allows for the definition of wind boxes, where thermocouples should be installed in the future and reduce their quantity for economy. The main feature of grey theory is its capability to use as few as n + 3 pairs of data by GMC(1,n) [START_REF] Tien | The indirect measurement of tensile strength of material by the grey prediction model GMC(1, n)[END_REF]. For each sample, the following experiments were performed: (1) based on the n-pair of series from the sample, the model was built; (2) based on the model, the prediction of the remaining values was made; (3) the prediction results were compared to the original samples and the Root Mean Squared Percentage Error (RMSPE) according to Equation ( 16) was calculated.

RMSPE = √ √ √ √ 1 n ∑ i Ŷi -Y i 2 Y 2 i × 100% (16)
During the experiments, 17 models were built for each sample ranging from 5 to 20 pair of series. The results of the predicted errors for each sample are presented in Figure 4. The predicted errors for each sample take the minimum values for different pairs of series, ranging from 14 to 17. The mean smallest predictive error for 3 samples is achieved with 15 pair of series.

BTP prediction results for considered models

The RMSPE for the post sample period of predictive models for 3 samples (Figure 2) are presented in the Table 1, where the volume of initial sample r to construct the model includes data from the beginning of the sintering process up to 7.5 min. The results of predictive models for sample 1 are shown in Figure 5 and the RMSPE of all samples are presented in Table 1. The best result for predicting the BTP of the charge was obtained as a result of using the GMC(1,n) model. The predictive results based on grey systems do not improve accuracy in comparison with the other models, but allow the building of an adequate prediction model of BTP in the absence of a large amount of historical temperature data. Additional advantage is the time saving in relation to data collection and model training.

Optimal GMC(1,n)

To improve the accuracy of the predictive model GMC(1,n) instead of the weighted mean to determine the grey derivative, we introduce the coefficient 𝜌 ∈[0; 1], then Equation (7) takes the following form:

B = ⎡ ⎢ ⎢ ⎢ ⎣ Q Y (1) Q X (1) 1 Q Y (2) Q X (2) 1 .. . Q Y (r -1)( Q X (r -1) 1 ⎤ ⎥ ⎥ ⎥ ⎦ (17)
where (1) (r)+𝜌Y (1) (r + 1)

Q Y (r)=-(1 -𝜌)Y
Q X (r)=(1 -𝜌)X (1) (r)+𝜌X (1) (r + 1)
Coefficient 𝜌 is determined by a PSO method [START_REF] Kennedy | Particle swarm optimization[END_REF]. This algorithm is a system of particles that move to optimal solutions, each particle contains the coordinates of the found best solution (pbest)a n dt h eb e s ts o l u t i o nf r o m all the particles in the swarm (gbest). The direction and length of the particle velocity vector is determined by the following formula:

𝜐 i = 𝜐 i + a 1 rnd() • (pbest i -x i )+a 2 rnd() • (gbest i -x i ) (18)
where 𝜐 i is the particle speed vector, a 1 , a 2 are the constant accelerations and x is the particle current position. In this case the particle current position is the 𝜌.Asanoptimality criterion, the minimum of Equation ( 16) is used. The obtained predictive models and the RMSPE are presented in Table 2, where it can be seen that the accuracy of the grey model is improved.

In [START_REF] Wang | An optimized grey dynamic model for forecasting the output of high-tech industry in China[END_REF], interpolation coefficients 𝜌 i are introduced into the background values of each of the variables in GDMC (1,n). Application of this algorithm to GMC(1,n) model will have the following changes in Equation ( 7):

B = ⎡ ⎢ ⎢ ⎢ ⎣ Q Y 1 (1) Q X2 (1) 1 Q Y 1 (2) Q X2 (2) 1 .. . Q Y 1 (r -1)( Q X2 (r -1) 1 ⎤ ⎥ ⎥ ⎥ ⎦ (19)
where (1) (r)+𝜌 i Y (1) (r + 1) (1) (r)+𝜌 i X (1) (r + 1)

Q Yi (r)=-(1 -𝜌 i )Y
Q Xi (r)=(1 -𝜌 i )X
The predictive results have changed slightly (Table 2), but the time to find the optimal values of the 𝜌 i increased. Increase of (n -1) dependent variables leads to increased optimization time exponentially and to a difficulty in using the algorithm with different coefficients for each variable in real-time control. Therefore to enhance the modelling and predictive accuracy one coefficient 𝜌 is used to control the task. The coefficients of the grey model (Table 2), which were found by Equation ( 6), differ significantly for each sample. This is due to the fact that each temperature curve was obtained under different initial conditions. Moreover, in production under real conditions, it is not possible to continuously control the composition of the charge. Therefore, it is necessary to build a predictive model, which uses only measured real time factors and dynamically constructs the model for a certain batch of sintered ore.

CONTROL STRUCTURE

Dynamic construction of the grey model uses only actual data, which represents real time information about the sintering process, and does not depend on historical data, FIGURE 6 Control structure for the BTP making it possible to obtain the prediction of the charge temperature. If the predictive temperature of the charge at the end of the strand does not reach the set point (the highest temperature in the wind boxes), then it is necessary to control the sinter process. The control can be carried out by changing the amount of fuel in the initial charge, the strand speed and the underpressure created in the wind boxes. The change in the amount of fuel affects only the charge that has not yet been fed to the sinter, so this effect is not considered in the work.

The idea to build the prediction model of the BTP in real time was also considered in [START_REF] Wu | Design and application of generalized predictive control strategy with closed-loop identification for burn-through point in sintering process[END_REF], where the model parameters are determined by the closed-loop identification method. High accuracy of the generalized prediction model [START_REF] Wu | Design and application of generalized predictive control strategy with closed-loop identification for burn-through point in sintering process[END_REF] was achieved by a real-time measurement of additional parameters, which makes the proposed model more expensive.

The proposed control structure for the BTP is presented in Figure 6 and has the following algorithm: after the charge is passed under the horn, the system begins collecting data on temperature, pressure in the wind boxes, and sinter machine speed. Based on the collected data, a forecast model is built on the basis of the grey model that predicts the temperature at the end of the sintering machine. If the temperature does not correspond to the BTP, a predictive optimization algorithm calculates the necessary values of the sinter velocity and pressure in the wind boxes. The resulting values are fed to the appropriate controllers. After the charge reaches the end of the sintering machine, the cycle is repeated for the next charge.

The PSO is used as a method of predictive optimization. Its distinguishing feature from many other methods is that, for the particle swarm method, we can calculate only the value of the function being optimized, but not its gradient. In terms of efficiency, it can compete with methods of global optimization, and its low algorithmic complexity contributes to the simplicity of its implementation. Particle current position Equation ( 18) includes two variables: number of samples that represents time of sinter process and gas velocity.

The PSO algorithm is composed of the following steps.

1. Generate m-particles (m = 2) in the n-dimensional space. Position and velocity are represented by x i = (x i1 , x i2 , .., x in ) and 𝜐 i =(𝜐 i1 ,𝜐 i2 , .., 𝜐 in ), i = 1, .., m.

Generate randomly the initial position of the particles

x i within the limits t min ⩽ x 1 ⩽ t max , X min ⩽ x 2 ⩽ X max and initial velocity vector 𝜐 i . t is the time step and X is the underpressure in the wind boxes. 3. According to Equation ( 6) substitute x 2 into the matrix B and according to Equation ( 9) put x 1 as parameter t. 4. Calculate the fitness value Equation ( 16) for each particle. 5. Compare each particle's fitness to the global best position pbest to adapt to the value; if it is the optimum, it can be taken as the best position; if not, turn to next step. 6. Update particles velocity and position according to the evolution Equation (18). 7. If stopping criteria are met, show the output and its fitness, which corresponds to the optimal parameters and the RMSPE, or else go back to step 3.

With the help of the obtained optimal sintering process time t value, the strand speed is determined, and through the optimum gas velocity value, the underpressure in wind boxes is found according to Equation (1). Modeling results using the BTP control system for the samples are represented in Figure 7. In the first sample, the sintering temperature of the charge is reached in 22 minutes. To increase line productivity, extended boundaries of inequalities are set,wherein, the optimization algorithm allows to find such values of strand speed and pressure in wind boxes at which the BTP is reached in 19.5 minutes. Sample 1 results of the forecast optimization are as follows: the speed values change from 3.5 m/min to 4.5 m/min and the pressure in wind boxes from 800 mmHg to 760 mmHg. In the second sample, the BTP is not reached. Therefore, as a result of predictive optimization, a new value is supplied to the pressure regulator, at which the sintering temperature is reached. In the third case, as a result of predictive optimization, the speed of sin- ter strand and pressure in wind boxes is increased, which allows for achievement of the optimum temperature in 19 min.

CONCLUSIONS

The proposed control process structure of phosphorite ore sintering considered in this paper is intended to improve the product quality and reduce the return. As an indicator responsible for sintering quality, the BTP was cho-sen, that is, the position at which the process reaches the highest temperature. Prediction of this variable was realized through the grey predictive model with some main contributions: (1) dynamic construction of the predictive model using a small sample size, obtained from the beginning of the process, which reduces the time for data acquisition and system training; (2) control of the sintering process not only based on the strand speed, but also on the underpressure in the wind boxes; (3) use of a grey model with (n-1) influencing factors that takes into account not only the effect of the predicted value, but also other process variables; (4) modeling different continuous grey models and development of optimal GMC(1,n) model; (5) use of predictive optimization algorithm to determine optimal process parameters. In the framework of this article, the initial amount of data for constructing the model was also determined, as a result of which the amount of costs for building the data acquisition system is reduced, while the predictive model error does not exceed 1.5%.

FIGURE 1

 1 FIGURE 1 Technological chain of the sintering process [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 2 FIGURE 3

 23 FIGURE 2 Temperature of 3 samples in wind boxes [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 4

 4 FIGURE 4 Predictive errors for different sample sizes [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 7

 7 FIGURE 7 Modeling results of the BTP using the control structure [Color figure can be viewed at wileyonlinelibrary.com]

TABLE 1

 1 RMSPE of grey models, %

No of sample GMC(1,n) FGMC(1,n) OGDMC(1,n)

	1	2.2413	4.1062	2.6575
	2	1.6697	3.4544	3.6921
	3	1.0952	3.8123	4.1833
	FIGURE 5 Modeling results of sample 1 [Color figure can be
	viewed at wileyonlinelibrary.com]

TABLE 2 Results

 2

			𝝆𝝆 ,%	𝝆 i	𝝆 i , %
		𝜌	0.2261		0 & 0.2283
	1	b 1	-0.0005	1.2120	-0.0005	1.2110
		b 2	56.3769		56.3554
		u	28.3447		28.4573
		𝜌	0.3669		1 & 0.3683
	2	b 1	0.0003	1.4266	0.0003	1.4249
		b 2	58.9057		58.8892
		u	30.6943		30.7584
		𝜌	0.4193		1 & 0.4384
	3	b 1	0.0025	0.9756	0.0023	0.9588
		b 2	62.1686		61.9513
		u	37.5064		38.5224

of optimal GMC(1,n) model No Coef GMC(1,n) RMSPE GMC(1,n) RMSPE

AUTHOR BIOGRAPHIES

Nigina Toktassynova is currently a Ph.D. student of automation and control at the Satbayev University, Almaty, Kazakhstan. She earned her bachelor and master's degree in automation and control at the Almaty University of Power Engi-neering and Telecommunication, Almaty, Kazakhstan, in 2012, and 2014, respectively. Her research interests include predicitive algorithms, grey system theory, and control systems in sintering process.

Hassen Fourati is currently an associate professor of electrical engineering and computer science at the University of Grenoble Alpes, Grenoble, France, and a member of the Dynamics and Con-