291:
59:
286:{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {f} (\mathbf {x} (t),\mathbf {u} (t)),\quad \mathbf {x} (0)=\mathbf {x} _{0},\quad \mathbf {u} (t)\in R^{m},\quad \mathbf {x} (t)\in R^{n},{\text{Rank}}{\frac {\partial \mathbf {f} (\mathbf {x} ,\mathbf {u} )}{\partial \mathbf {u} }}=m}
1221:
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The flatness property is useful for both the analysis of and controller synthesis for nonlinear dynamical systems. It is particularly advantageous for solving trajectory planning problems and asymptotical setpoint following control.
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A. Isidori, C.H. Moog et A. De Luca. A Sufficient
Condition for Full Linearization via Dynamic State Feedback. 25th CDC IEEE, Athens, Greece, pp. 203 - 208, 1986
1009:
1216:{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t),\quad \mathbf {x} (0)=\mathbf {x} _{0}}
46:, which can be used to explicitly express all states and inputs in terms of the flat output and a finite number of its derivatives.
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M. Fliess, J. L. Lévine, P. Martin and P. Rouchon: Flatness and defect of non-linear systems: introductory theory and examples.
300:
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732:{\displaystyle \mathbf {y} =\Phi (\mathbf {x} ,\mathbf {u} ,{\dot {\mathbf {u} }},...,\mathbf {u} ^{(\alpha )})}
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If these conditions are satisfied at least locally, then the (possibly fictitious) output is called
987:
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1076:{\displaystyle \phi (\mathbf {y} ,{\dot {\mathbf {y} }},\mathbf {y} ^{(\gamma )})=\mathbf {0} }
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are differentially independent, that is, they satisfy no differential equation of the form
8:
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and a finite number of derivatives with respect to time
383:{\displaystyle \mathbf {y} (t)=(y_{1}(t),...,y_{m}(t))}
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as the nonlinear system is flat, if and only if it is
38:. A system that has the flatness property is called a
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both properties are equivalent, hence exchangeable.
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23:is a system property that extends the notion of
635:{\displaystyle u_{i}^{(k)},k=1,...,\alpha _{i}}
856:are representable as functions of the outputs
1099:Relation to controllability of linear systems
453:are representable as functions of the states
913:and of its derivatives with respect to time
1246:{\displaystyle \mathbf {x} ,\mathbf {u} }
392:that satisfies the following conditions:
1327:
974:{\displaystyle y_{i}^{(k)},i=1,...,m}
1223:with the same signal dimensions for
295:is flat, if there exists an output
42:. Flat systems have a (fictitious)
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1281:International Journal of Control
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906:{\displaystyle y_{i},i=1,...,m}
849:{\displaystyle u_{i},i=1,...,m}
792:{\displaystyle x_{i},i=1,...,n}
560:{\displaystyle u_{i},i=1,...,m}
503:{\displaystyle x_{i},i=1,...,n}
446:{\displaystyle y_{i},i=1,...,m}
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1287:), pp. 1327-1361, 1995
999:{\displaystyle \mathbf {y} }
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54:A nonlinear system
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1105:linear system
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1265:Significance
1255:controllable
1102:
1092:
1088:
1086:
396:The signals
391:
297:
294:
56:
53:
43:
39:
16:
15:
1089:flat output
799:and inputs
742:The states
510:and inputs
44:flat output
40:flat system
1274:Literature
50:Definition
1122:˙
1055:γ
1036:˙
1014:ϕ
719:α
688:˙
658:Φ
624:α
267:∂
238:∂
214:∈
183:∈
72:˙
33:nonlinear
1329:Category
1299:See also
17:Flatness
1257:. For
27:from
1093:flat
231:Rank
1283:61(
31:to
19:in
1331::
1103:A
1095:.
642::
1285:6
1240:u
1236:,
1232:x
1209:0
1204:x
1199:=
1196:)
1193:0
1190:(
1186:x
1181:,
1178:)
1175:t
1172:(
1168:u
1163:B
1159:+
1156:)
1153:t
1150:(
1146:x
1141:A
1137:=
1134:)
1131:t
1128:(
1118:x
1083:.
1070:0
1066:=
1063:)
1058:)
1052:(
1047:y
1042:,
1032:y
1025:,
1021:y
1017:(
993:y
981:.
969:m
966:,
963:.
960:.
957:.
954:,
951:1
948:=
945:i
942:,
937:)
934:k
931:(
926:i
922:y
901:m
898:,
895:.
892:.
889:.
886:,
883:1
880:=
877:i
874:,
869:i
865:y
844:m
841:,
838:.
835:.
832:.
829:,
826:1
823:=
820:i
817:,
812:i
808:u
787:n
784:,
781:.
778:.
775:.
772:,
769:1
766:=
763:i
760:,
755:i
751:x
739:.
727:)
722:)
716:(
711:u
706:,
703:.
700:.
697:.
694:,
684:u
677:,
673:u
669:,
665:x
661:(
655:=
651:y
628:i
620:,
617:.
614:.
611:.
608:,
605:1
602:=
599:k
596:,
591:)
588:k
585:(
580:i
576:u
555:m
552:,
549:.
546:.
543:.
540:,
537:1
534:=
531:i
528:,
523:i
519:u
498:n
495:,
492:.
489:.
486:.
483:,
480:1
477:=
474:i
471:,
466:i
462:x
441:m
438:,
435:.
432:.
429:.
426:,
423:1
420:=
417:i
414:,
409:i
405:y
378:)
375:)
372:t
369:(
364:m
360:y
356:,
353:.
350:.
347:.
344:,
341:)
338:t
335:(
330:1
326:y
322:(
319:=
316:)
313:t
310:(
306:y
281:m
278:=
271:u
262:)
258:u
254:,
250:x
246:(
242:f
227:,
222:n
218:R
211:)
208:t
205:(
201:x
196:,
191:m
187:R
180:)
177:t
174:(
170:u
165:,
160:0
155:x
150:=
147:)
144:0
141:(
137:x
132:,
129:)
126:)
123:t
120:(
116:u
112:,
109:)
106:t
103:(
99:x
95:(
91:f
87:=
84:)
81:t
78:(
68:x
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