74:
manipulating the convex hull of polytopic forms, and, as a result has revealed and proved the fact that convex hull manipulation is a necessary and crucial step in achieving optimal solutions and decreasing conservativeness in modern LMI based control theory. Thus, although it is a transformation in a mathematical sense, it has established a conceptually new direction in control theory and has laid the ground for further new approaches towards optimality. Further details on the control theoretical aspects of the TP model transformation can be found here:
2625:
controller design (this is the widely adopted approach). It is worth noting that both the TP model transformation and the LMI-based control design methods are numerically executable one after the other, and this makes the resolution of a wide class of problems possible in a straightforward and tractable, numerical way.
439:
49:, etc.) into TP function form if such a transformation is possible. If an exact transformation is not possible, then the method determines a TP function that is an approximation of the given function. Hence, the TP model transformation can provide a trade-off between approximation accuracy and complexity.
2624:
The core step of the TP model transformation was extended to generate different types of convex TP functions or TP models (TP type polytopic qLPV models), in order to focus on the systematic (numerical and automatic) modification of the convex hull instead of developing new LMI equations for feasible
2502:
It can be executed uniformly (irrespective of whether the model is given in the form of analytical equations resulting from physical considerations, or as an outcome of soft computing based identification techniques (such as neural networks or fuzzy logic based methods, or as a result of a black-box
73:
Besides being a transformation of functions, the TP model transformation is also a new concept in qLPV based control which plays a central role in the providing a valuable means of bridging between identification and polytopic systems theories. The TP model transformation is uniquely effective in
2478:
where trade-off is offered by the TP model transformation between complexity (number of components in the core tensor or the number of weighting functions) and the approximation accuracy. The TP model can be generated according to various constrains. Typical TP models generated by the TP model
1465:
2471:
1205:
2210:
571:
190:
2503:
identification), without analytical interaction, within a reasonable amount of time. Thus, the transformation replaces the analytical and in many cases complex and not obvious conversions to numerical, tractable, straightforward operations.
96:
The TP model transformation has recently been extended in order to derive various types of convex TP functions and to manipulate them. This feature has led to new optimization approaches in qLPV system analysis and design, as described at
656:
1334:
2628:
The TP model transformation is capable of performing trade-off between complexity and accuracy of TP functions via discarding the higher-order singular values, in the same manner as the tensor HOSVD is used for complexity
1876:
2899:
A. SzöllĆsi and P. Baranyi: âImproved control performance of the 3âDoF aeroelastic wing section: a TP model based 2D parametric control performance optimization.â in Asian
Journal of Control, 19(2), 450-466. /
1339:
1757:
1257:
2069:
876:
1570:
782:
2110:
2890:
A.Szollosi, and
Baranyi, P. (2016). Influence of the Tensor Product model representation of qLPV models on the feasibility of Linear Matrix Inequality. Asian Journal of Control, 18(4), 1328-1342
2499:
It transforms the given function into finite element TP structure. If this structure does not exist, then the transformation gives an approximation under a constraint on the number of elements.
2369:
713:
2506:
It generates the HOSVD-based canonical form of TP functions, which is a unique representation. It was proven by Szeidl that the TP model transformation numerically reconstructs the
2285:
2017:
1953:
2379:
179:
932:
144:
1989:
2240:
1594:
1113:
2121:
434:{\displaystyle f(\mathbf {x} )=\sum _{i_{1}=1}^{I_{1}}\sum _{i_{2}=1}^{I_{2}}\ldots \sum _{i_{N}=1}^{I_{N}}\prod _{n=1}^{N}w_{n,i_{n}}(x_{n})s_{i_{1},i_{2},\ldots ,i_{N}},}
1616:
1048:
2599:
2317:
1485:
478:
1505:
468:
82:
1075:
1006:
959:
581:
2567:
2619:
1095:
1026:
979:
2510:
of functions. This form extracts the unique structure of a given TP function in the same sense as the HOSVD does for tensors and matrices, in a way such that:
1262:
81:
The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions", on which further information can be found
2520:
the weighting functions are one variable functions of the parameter vector in an orthonormed system for each parameter (singular functions);
1460:{\displaystyle {\mathcal {S}}\in {\mathcal {R}}^{I_{1}\times I_{2}\times \ldots \times I_{N}\times L_{1}\times L_{2}\times ...\times L_{O}}}
1775:
2960:
Baranyi, P. (2018). Extension of the Multi-TP Model
Transformation to Functions with Different Numbers of Variables. Complexity, 2018.
2759:
P. Baranyi; D. Tikk; Y. Yam; R. J. Patton (2003). "From
Differential Equations to PDC Controller Design via Numerical Transformation".
2816:
D. Tikk; P. Baranyi; R. J. Patton (2007). "Approximation
Properties of TP Model Forms and its Consequences to TPDC Design Framework".
2371:
does not have TP structure (i.e. it is not in the class of TP models), then the TP model transformation determines its approximation:
98:
75:
67:
34:
2112:, whose TP structure maybe unknown (e.g. it is given by neural networks). The TP model transformation determines its TP structure as
1621:
1215:
93:
of functions, which provides exact results if the given function has a TP function structure and approximative results otherwise.
2029:
2526:
the core tensor and the weighting functions are ordered according to the higher-order singular values of the parameter vector;
787:
2795:
2704:
2486:
Various kinds of TP type polytopic form or convex TP model forms (this advantage is used in qLPV system analysis and design).
1510:
718:
2074:
57:
2969:
2339:
2725:"The Generalized TP Model Transformation for TâS Fuzzy Model Manipulation and Generalized Stability Verification"
661:
2466:{\displaystyle {\mathcal {F}}(\mathbf {x} )\approx {\mathcal {S}}\boxtimes _{n=1}^{N}\mathbf {w} _{n}(x_{n}),}
1097:. For qLPV modelling and control applications a higher structure of TP functions are referred to as TP model.
2245:
1994:
1881:
2543:
based canonical form of qLPV model to order the main component of the qLPV model). Since the core tensor is
149:
89:
based canonical form. Thus, the TP model transformation can be viewed as a numerical method to compute the
2854:
Lieven De
Lathauwer; Bart De Moor; Joos Vandewalle (2000). "A Multilinear Singular Value Decomposition".
881:
1200:{\displaystyle {\mathcal {F}}(\mathbf {x} )={\mathcal {S}}\boxtimes _{n=1}^{N}\mathbf {w} _{n}(x_{n}).}
116:
2324:
2205:{\displaystyle {\mathcal {F}}(\mathbf {x} )={\mathcal {S}}\boxtimes _{n=1}^{N}\mathbf {w} _{n}(x_{n})}
1963:
2331:
2221:
1575:
85:. It has been proved that the TP model transformation is capable of numerically reconstructing this
64:
2517:
the number of weighting functions are minimized per dimensions (hence the size of the core tensor);
566:{\displaystyle f(\mathbf {x} )={\mathcal {S}}\mathop {\otimes } _{n=1}^{N}\mathbf {w} _{n}(x_{n}),}
2868:
1599:
1031:
2984:
2572:
2290:
1470:
2917:. 3rd International Conference on Mechatronics (ICM 2006). Budapest, Hungary. pp. 660â665.
2863:
38:
1490:
651:{\displaystyle {\mathcal {S}}\in {\mathcal {R}}^{I_{1}\times I_{2}\times \ldots \times I_{N}}}
453:
2532:
introduces and defines the rank of the TP function by the dimensions of the parameter vector;
2529:
it has a unique form (except for some special cases such as there are equal singular values);
2649:
P. Baranyi (April 2004). "TP model transformation as a way to LMI based controller design".
1053:
984:
937:
8:
2546:
2496:
It is a non-heuristic and tractable numerical method firstly proposed in control theory.
2833:
2829:
2666:
2604:
1329:{\displaystyle {\mathcal {Y}}\in {\mathcal {R}}^{L_{1}\times L_{2}\times \ldots L_{O}}}
1080:
1011:
964:
2772:
2837:
2791:
2700:
2944:
2873:
2825:
2768:
2736:
2692:
2670:
2658:
2949:
2932:
2741:
2724:
447:
42:
21:
2877:
2853:
2696:
2978:
2662:
2569:
dimensional, but the weighting functions are determined only for dimensions
2539:
The above point can be extended to TP models (qLPV models to determine the
2937:
Journal of
Advanced Computational Intelligence and Intelligent Informatics
46:
2915:
Definition of the HOSVD-based canonical form of polytopic dynamic models
1871:{\displaystyle \forall n:\sum _{i_{n}=1}^{I_{n}}w_{n,i_{n}}(x_{n})=1}
16:
Key concept in higher-order singular value decomposition of functions
2933:"HOSVD Based Canonical Form for Polytopic Models of Dynamic Systems"
2788:
Tensor
Product model transformation in polytopic model-based control
2621:
dimensional elements, therefore the resulting TP form is not unique.
2523:
the sub tensors of the core tensor are also in orthogonal positions;
2912:
2758:
1507:, but expresses the fact that the tensor product is applied on the
56:
implementation of the TP model transformation can be downloaded at
1766:
A TP function or model is convex if the weighting functions hold:
2483:
HOSVD canonical form of TP functions or TP model (qLPV models),
2327:
2320:
60:
53:
37:
of functions. It transforms a function (which can be given via
2718:
2716:
2815:
2540:
2507:
1991:
is inside the convex hull defined by the core tensor for all
1752:{\displaystyle \Omega =\times \times ...\times \subset R^{N}}
1252:{\displaystyle {\mathcal {Y}}={\mathcal {F}}({\mathbf {x} })}
90:
86:
2913:
P. Baranyi; L. Szeidl; P. VĂĄrlaki; Y. Yam (July 3â5, 2006).
2490:
2713:
2064:{\displaystyle {\mathcal {Y}}={\mathcal {F}}(\mathbf {x} )}
2785:
871:{\displaystyle w_{n,i_{n}}(x_{n}),(i_{n}=1\ldots I_{n})}
2689:
1565:{\displaystyle L_{1}\times L_{2}\times ...\times L_{O}}
2607:
2575:
2549:
2382:
2342:
2293:
2248:
2224:
2124:
2077:
2032:
1997:
1966:
1884:
1778:
1624:
1602:
1578:
1513:
1493:
1473:
1342:
1265:
1218:
1116:
1083:
1056:
1034:
1014:
987:
967:
940:
884:
790:
777:{\displaystyle \mathbf {w} _{n}(x_{n}),(n=1\ldots N)}
721:
664:
584:
481:
456:
193:
152:
119:
2790:. Boca Raton FL: Taylor & Francis. p. 240.
2105:{\displaystyle \mathbf {x} \in \Omega \subset R^{N}}
2930:
784:contains continuous univariate weighting functions
66:. A key underpinning of the transformation is the
33:was proposed by Baranyi and Yam as key concept for
2613:
2593:
2561:
2465:
2363:
2311:
2279:
2234:
2204:
2104:
2063:
2011:
1983:
1947:
1870:
1751:
1610:
1588:
1564:
1499:
1479:
1459:
1328:
1251:
1199:
1089:
1069:
1042:
1020:
1000:
973:
953:
926:
870:
776:
707:
650:
565:
462:
446:that is, using compact tensor notation (using the
433:
173:
138:
59:or an old version of the toolbox is available at
2976:
2682:
2680:
2601:, namely the core tensor is constructed from
2786:P. Baranyi; Y. Yam & P. VĂĄrlaki (2013).
2364:{\displaystyle {\mathcal {F}}(\mathbf {x} )}
2856:Journal on Matrix Analysis and Applications
2677:
2651:IEEE Transactions on Industrial Electronics
2644:
2642:
1101:Finite element TP model (TP model in short)
181:, is a TP function if it has the structure:
2648:
1763:Finite element convex TP function or model
1104:This is a higher structure of TP function:
708:{\displaystyle s_{i_{1}i_{2}\ldots i_{N}}}
2948:
2926:
2924:
2908:
2906:
2867:
2754:
2752:
2740:
2491:Properties of the TP model transformation
1572:sized tensor elements of the core tensor
99:TP model transformation in control theory
76:TP model transformation in control theory
68:higher-order singular value decomposition
35:higher-order singular value decomposition
2849:
2847:
2639:
2811:
2809:
2807:
2722:
2686:
2280:{\displaystyle \mathbf {w} _{n}(x_{n})}
2012:{\displaystyle \mathbf {x} \in \Omega }
1948:{\displaystyle w_{n,i_{n}}(x_{n})\in .}
2977:
2921:
2903:
2749:
1618:is an element of the closed hypercube
1336:, thus the size of the core tensor is
961:-th weighting function defined on the
2844:
174:{\displaystyle \mathbf {x} \in R^{N}}
2804:
2779:
2218:namely it generates the core tensor
2931:L. Szeidl & P. VĂĄrlaki (2009).
13:
2830:10.1111/j.1934-6093.2007.tb00410.x
2729:IEEE Transactions on Fuzzy Systems
2406:
2385:
2345:
2323:implementation is downloadable at
2227:
2148:
2127:
2086:
2045:
2035:
2006:
1779:
1625:
1581:
1356:
1345:
1279:
1268:
1231:
1221:
1140:
1119:
927:{\displaystyle w_{n,i_{n}}(x_{n})}
598:
587:
501:
14:
2996:
2963:
139:{\displaystyle f({\mathbf {x} })}
2434:
2394:
2354:
2251:
2176:
2136:
2079:
2054:
1999:
1974:
1604:
1241:
1168:
1128:
1036:
724:
534:
489:
201:
154:
128:
1984:{\displaystyle f(\mathbf {x} )}
2893:
2884:
2457:
2444:
2398:
2390:
2358:
2350:
2274:
2261:
2235:{\displaystyle {\mathcal {S}}}
2199:
2186:
2140:
2132:
2058:
2050:
1978:
1970:
1939:
1927:
1921:
1908:
1859:
1846:
1733:
1707:
1689:
1663:
1657:
1631:
1589:{\displaystyle {\mathcal {S}}}
1246:
1236:
1191:
1178:
1132:
1124:
921:
908:
865:
833:
827:
814:
771:
753:
747:
734:
557:
544:
493:
485:
376:
363:
205:
197:
133:
123:
104:
1:
2773:10.1016/s0166-3615(03)00058-7
2633:
2242:and the weighting functions
1611:{\displaystyle \mathbf {x} }
1050:. Finite element means that
1043:{\displaystyle \mathbf {x} }
7:
2594:{\displaystyle n=1\ldots N}
2312:{\displaystyle n=1\ldots N}
10:
3001:
2950:10.20965/jaciii.2009.p0052
2742:10.1109/TFUZZ.2013.2278982
1480:{\displaystyle \boxtimes }
110:Finite element TP function
2878:10.1137/s0895479896305696
2697:10.1007/978-3-319-19605-3
2818:Asian Journal of Control
2026:Assume a given TP model
1500:{\displaystyle \otimes }
463:{\displaystyle \otimes }
2723:Baranyi, Peter (2014).
2687:Baranyi, PĂ©ter (2016).
2663:10.1109/tie.2003.822037
2023:TP model transformation
1467:. The product operator
1028:-the element of vector
2615:
2595:
2563:
2467:
2365:
2313:
2281:
2236:
2206:
2106:
2065:
2013:
1985:
1949:
1872:
1822:
1753:
1612:
1590:
1566:
1501:
1481:
1461:
1330:
1253:
1201:
1091:
1071:
1044:
1022:
1002:
975:
955:
928:
872:
778:
709:
652:
567:
528:
464:
435:
339:
318:
280:
245:
175:
140:
2761:Computers in Industry
2616:
2596:
2564:
2468:
2366:
2314:
2282:
2237:
2207:
2107:
2066:
2014:
1986:
1950:
1873:
1788:
1754:
1613:
1591:
1567:
1502:
1487:has the same role as
1482:
1462:
1331:
1254:
1202:
1092:
1072:
1070:{\displaystyle I_{n}}
1045:
1023:
1003:
1001:{\displaystyle x_{n}}
976:
956:
954:{\displaystyle i_{n}}
929:
873:
779:
710:
653:
568:
506:
465:
436:
319:
284:
246:
211:
176:
141:
2605:
2573:
2547:
2479:transformation are:
2380:
2340:
2291:
2246:
2222:
2122:
2075:
2030:
1995:
1964:
1882:
1776:
1622:
1600:
1576:
1511:
1491:
1471:
1340:
1263:
1216:
1114:
1081:
1054:
1032:
1012:
985:
965:
938:
882:
788:
719:
662:
658:is constructed from
582:
479:
454:
191:
150:
117:
31:model transformation
19:In mathematics, the
2562:{\displaystyle N+O}
2431:
2173:
1165:
1077:is bounded for all
981:-th dimension, and
578:where core tensor
2611:
2591:
2559:
2463:
2411:
2361:
2309:
2277:
2232:
2202:
2153:
2102:
2061:
2009:
1981:
1945:
1868:
1749:
1608:
1586:
1562:
1497:
1477:
1457:
1326:
1249:
1197:
1145:
1087:
1067:
1040:
1018:
998:
971:
951:
924:
868:
774:
705:
648:
563:
460:
431:
171:
136:
2797:978-1-43-981816-9
2706:978-3-319-19604-6
2614:{\displaystyle O}
1090:{\displaystyle n}
1021:{\displaystyle n}
974:{\displaystyle n}
715:, and row vector
113:A given function
2992:
2955:
2954:
2952:
2928:
2919:
2918:
2910:
2901:
2897:
2891:
2888:
2882:
2881:
2871:
2862:(4): 1253â1278.
2851:
2842:
2841:
2813:
2802:
2801:
2783:
2777:
2776:
2756:
2747:
2746:
2744:
2720:
2711:
2710:
2684:
2675:
2674:
2646:
2620:
2618:
2617:
2612:
2600:
2598:
2597:
2592:
2568:
2566:
2565:
2560:
2472:
2470:
2469:
2464:
2456:
2455:
2443:
2442:
2437:
2430:
2425:
2410:
2409:
2397:
2389:
2388:
2370:
2368:
2367:
2362:
2357:
2349:
2348:
2318:
2316:
2315:
2310:
2286:
2284:
2283:
2278:
2273:
2272:
2260:
2259:
2254:
2241:
2239:
2238:
2233:
2231:
2230:
2211:
2209:
2208:
2203:
2198:
2197:
2185:
2184:
2179:
2172:
2167:
2152:
2151:
2139:
2131:
2130:
2111:
2109:
2108:
2103:
2101:
2100:
2082:
2070:
2068:
2067:
2062:
2057:
2049:
2048:
2039:
2038:
2018:
2016:
2015:
2010:
2002:
1990:
1988:
1987:
1982:
1977:
1960:This means that
1954:
1952:
1951:
1946:
1920:
1919:
1907:
1906:
1905:
1904:
1877:
1875:
1874:
1869:
1858:
1857:
1845:
1844:
1843:
1842:
1821:
1820:
1819:
1809:
1802:
1801:
1758:
1756:
1755:
1750:
1748:
1747:
1732:
1731:
1719:
1718:
1688:
1687:
1675:
1674:
1656:
1655:
1643:
1642:
1617:
1615:
1614:
1609:
1607:
1595:
1593:
1592:
1587:
1585:
1584:
1571:
1569:
1568:
1563:
1561:
1560:
1536:
1535:
1523:
1522:
1506:
1504:
1503:
1498:
1486:
1484:
1483:
1478:
1466:
1464:
1463:
1458:
1456:
1455:
1454:
1453:
1429:
1428:
1416:
1415:
1403:
1402:
1384:
1383:
1371:
1370:
1360:
1359:
1349:
1348:
1335:
1333:
1332:
1327:
1325:
1324:
1323:
1322:
1307:
1306:
1294:
1293:
1283:
1282:
1272:
1271:
1258:
1256:
1255:
1250:
1245:
1244:
1235:
1234:
1225:
1224:
1206:
1204:
1203:
1198:
1190:
1189:
1177:
1176:
1171:
1164:
1159:
1144:
1143:
1131:
1123:
1122:
1096:
1094:
1093:
1088:
1076:
1074:
1073:
1068:
1066:
1065:
1049:
1047:
1046:
1041:
1039:
1027:
1025:
1024:
1019:
1007:
1005:
1004:
999:
997:
996:
980:
978:
977:
972:
960:
958:
957:
952:
950:
949:
933:
931:
930:
925:
920:
919:
907:
906:
905:
904:
877:
875:
874:
869:
864:
863:
845:
844:
826:
825:
813:
812:
811:
810:
783:
781:
780:
775:
746:
745:
733:
732:
727:
714:
712:
711:
706:
704:
703:
702:
701:
689:
688:
679:
678:
657:
655:
654:
649:
647:
646:
645:
644:
626:
625:
613:
612:
602:
601:
591:
590:
572:
570:
569:
564:
556:
555:
543:
542:
537:
527:
522:
511:
505:
504:
492:
469:
467:
466:
461:
440:
438:
437:
432:
427:
426:
425:
424:
406:
405:
393:
392:
375:
374:
362:
361:
360:
359:
338:
333:
317:
316:
315:
305:
298:
297:
279:
278:
277:
267:
260:
259:
244:
243:
242:
232:
225:
224:
204:
180:
178:
177:
172:
170:
169:
157:
145:
143:
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2970:TPtoolBoxMATLAB
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1259:is a tensor as
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878:. The function
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107:
43:neural networks
39:closed formulas
17:
12:
11:
5:
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2985:Control theory
2973:
2972:
2965:
2964:External links
2962:
2957:
2956:
2920:
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2892:
2883:
2843:
2824:(3): 221â331.
2803:
2796:
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2767:(3): 281â297.
2748:
2735:(4): 934â948.
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2657:(2): 387â400.
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448:tensor product
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22:tensor product
15:
9:
6:
4:
3:
2:
2997:
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2869:10.1.1.3.4043
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2375:
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2373:
2372:
2336:If the given
2334:
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2329:
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2256:
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2040:
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2003:
1967:
1942:
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1930:
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1901:
1897:
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1701:
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1644:
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1543:
1540:
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1528:
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1515:
1494:
1474:
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1412:
1408:
1404:
1399:
1395:
1391:
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1380:
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1315:
1311:
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1303:
1299:
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1226:
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1103:
1100:
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1098:
1084:
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989:
968:
946:
942:
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893:
890:
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852:
849:
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837:
830:
822:
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807:
803:
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792:
768:
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762:
759:
756:
750:
742:
738:
729:
698:
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690:
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681:
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666:
641:
637:
633:
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627:
622:
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614:
609:
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592:
560:
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539:
529:
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516:
513:
508:
496:
482:
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457:
449:
428:
421:
417:
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407:
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389:
385:
380:
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367:
356:
352:
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341:
335:
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327:
324:
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312:
308:
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299:
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281:
274:
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264:
261:
256:
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247:
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229:
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221:
217:
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208:
194:
187:
186:
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166:
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158:
120:
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109:
108:
102:
100:
94:
92:
88:
84:
79:
77:
71:
69:
65:
62:
58:
55:
50:
48:
44:
40:
36:
32:
28:
24:
23:
2959:
2943:(1): 52â60.
2940:
2936:
2914:
2895:
2886:
2859:
2855:
2821:
2817:
2787:
2781:
2764:
2760:
2732:
2728:
2688:
2654:
2650:
2477:
2335:
2217:
1959:
1211:
577:
445:
95:
80:
72:
51:
30:
26:
20:
18:
2319:. Its free
105:Definitions
47:fuzzy logic
2634:References
2629:reduction.
450:operation
2864:CiteSeerX
2838:121716136
2586:…
2413:⊠
2402:≈
2304:…
2155:⊠
2090:⊂
2087:Ω
2084:∈
2007:Ω
2004:∈
1925:∈
1790:∑
1780:∀
1737:⊂
1705:×
1693:×
1661:×
1626:Ω
1596:. Vector
1550:×
1538:×
1525:×
1495:⊗
1475:⊠
1443:×
1431:×
1418:×
1405:×
1392:×
1389:…
1386:×
1373:×
1351:∈
1312:…
1309:×
1296:×
1274:∈
1147:⊠
853:…
766:…
691:…
634:×
631:…
628:×
615:×
593:∈
530:
509:⊗
458:⊗
411:…
321:∏
286:∑
282:…
248:∑
213:∑
159:∈
146:, where
2979:Category
2330:Central
2287:for all
2071:, where
470:of ):
63:Central
2671:7957799
2326:or at
1008:is the
934:is the
52:A free
2866:
2836:
2794:
2703:
2669:
2328:MATLAB
2321:MATLAB
61:MATLAB
54:MATLAB
2834:S2CID
2667:S2CID
2541:HOSVD
2508:HOSVD
1212:Here
91:HOSVD
87:HOSVD
2900:2017
2792:ISBN
2701:ISBN
1878:and
83:here
2945:doi
2874:doi
2826:doi
2769:doi
2737:doi
2693:doi
2659:doi
41:or
2981::
2941:13
2939:.
2935:.
2923:^
2905:^
2872:.
2860:21
2858:.
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2806:^
2765:51
2763:.
2751:^
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2727:.
2715:^
2699:.
2691:.
2679:^
2665:.
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2653:.
2641:^
2333:.
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1759:.
101:.
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27:TP
2953:.
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2880:.
2876::
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2775:.
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2673:.
2661::
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1529:L
1520:1
1516:L
1451:O
1447:L
1440:.
1437:.
1434:.
1426:2
1422:L
1413:1
1409:L
1400:N
1396:I
1381:2
1377:I
1368:1
1364:I
1357:R
1346:S
1320:O
1316:L
1304:2
1300:L
1291:1
1287:L
1280:R
1269:Y
1247:)
1242:x
1237:(
1232:F
1227:=
1222:Y
1195:.
1192:)
1187:n
1183:x
1179:(
1174:n
1169:w
1162:N
1157:1
1154:=
1151:n
1141:S
1136:=
1133:)
1129:x
1125:(
1120:F
1085:n
1063:n
1059:I
1037:x
1016:n
994:n
990:x
969:n
947:n
943:i
922:)
917:n
913:x
909:(
902:n
898:i
894:,
891:n
887:w
866:)
861:n
857:I
850:1
847:=
842:n
838:i
834:(
831:,
828:)
823:n
819:x
815:(
808:n
804:i
800:,
797:n
793:w
772:)
769:N
763:1
760:=
757:n
754:(
751:,
748:)
743:n
739:x
735:(
730:n
725:w
699:N
695:i
686:2
682:i
676:1
672:i
667:s
642:N
638:I
623:2
619:I
610:1
606:I
599:R
588:S
561:,
558:)
553:n
549:x
545:(
540:n
535:w
525:N
520:1
517:=
514:n
502:S
497:=
494:)
490:x
486:(
483:f
429:,
422:N
418:i
414:,
408:,
403:2
399:i
395:,
390:1
386:i
381:s
377:)
372:n
368:x
364:(
357:n
353:i
349:,
346:n
342:w
336:N
331:1
328:=
325:n
313:N
309:I
303:1
300:=
295:N
291:i
275:2
271:I
265:1
262:=
257:2
253:i
240:1
236:I
230:1
227:=
222:1
218:i
209:=
206:)
202:x
198:(
195:f
167:N
163:R
155:x
134:)
129:x
124:(
121:f
25:(
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