13th IEEE Workshop on Variable Structure Systems, VSS’14, June 29 -July 2, 2014, Nantes, France.
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ˆ2 + u + k2 VLJQ(e1 ). s˙ = c1 x
x˙ 1 = x2 x˙ 2 = u + ρ1
7KHUHIRUH WKH V\VWHP LQ WKH FRRUGLQDWH RI x1 DQG s E\ XVLQJ DQG FRXOG EH ZULWWHQ DV
y = x1
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s˙ = c1 x ˆ2 + u + k2 VLJQ(e1 ).
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ZKHUH λ1 DQG λ2 DUH WKH GHVLJQHG SDUDPHWHUV IRU WKH FRQWURO 7KH FORVHG ORRS V\VWHP DIWHU DSSO\LQJ WKH FRQWURO WR x˙ 1 = s − c1 x1
1
s˙ = −λ1 |s| 2 VLJQ(s) −
0
t
λ2 VLJQ(s)dτ + k2 VLJQ(e1 )
RU x˙ 1 = s − c1 x1 1
s˙ = −λ1 |s| 2 VLJQ(s) + L + k2 VLJQ(e1 ) L˙ = −λ2 VLJQ(s)
= e2 − z1 = −z2 + ρ1
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x˙ 1 = s − c1 x1
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ZKHUH L LV VRPH ¿FWLWLRXV VWDWH YDULDEOH B. Discussion of above Mathematical Transformation ,W LV FOHDU IURP WKH DERYH PDWKHPDWLFDO WUDQVIRUPDWLRQ WKDW ZKHQ RQH FDQ XVH VXSHU WZLVWLQJ REVHUYHU 672 WR HVWLPDWH WKH VWDWH RI VHFRQG RUGHU XQFHUWDLQ V\VWHP DQG IROORZ WKH VWDQGDUG ZD\ RI WKH 67& GHVLJQ DV E\ VHOHFWLQJ VOLGLQJ PDQLIROG DV 7KHQ VHFRQG RUGHU VOLGLQJ PRWLRQ LV QHYHU VWDUW LQ WKH EHFDXVH s˙ FRQWDLQV WKH QRQ GLIIHUHQWLDEOH WHUP k2 VLJQ(e1 ) ZKLFK H[FOXGH WKH SRVVLELOLW\ RI ORZHU WZR VXEV\VWHP RI WR DFW DV WKH VXSHUWZLVWLQJ DQG ¿QDOO\ VWDUW WKH VHFRQG RUGHU VOLGLQJ PRWLRQ VR WKDW s = s˙ = 0 LQ ¿QLWH WLPH ,Q WKH QH[W VXEVHFWLRQ ZH DUH JRLQJ WR SURSRVH WKH SRVVLEOH PHWKRGRORJ\ RI WKH FRQWURO GHVLJQ VXFK WKDW QRQ GLIIHUHQWLDEOH WHUP k2 VLJQ(e1 ) LV FDQFHO RXW DQG WKHQ ORZHU WZR VXEV\VWHP RI DFW DV WKH VXSHUWZLVWLQJ DQG ¿QDOO\ VHFRQG RUGHU VOLGLQJ LV DFKLHYHG
C. Existence Condition of Sliding Mode using SuperTwisting Algorithm 7KH PDLQ DLP KHUH LV WR GHVLJQ u VXFK WKDW VOLGLQJ PRWLRQ RFFXUV LQ ¿QLWH WLPH )RU WKLV SXUSRVH FRQWURO LV VHOHFWHG DFFRUGLQJ WR WKH IROORZLQJ 3URSRVLWLRQ Proposition 1: 7KH FRQWURO LQSXW u ZKLFK LV GH¿QHG DV t 1 2 u = −c1 x ˆ2 − k2 VLJQ(e1 ) − λ1 |s| VLJQ(s) − λ2 VLJQ(s)dτ 0
ZKHUH λ1 > 0 DQG λ2 > 0 DUH VHOHFWLQJ DFFRUGLQJ WR >@ >@ OHDGV WR WKH HVWDEOLVKPHQW VHFRQG RUGHU VOLGLQJ LQ ¿QLWH WLPH ZKLFK IXUWKHU LPSOLHV DV\PSWRWLF VWDELOLW\ RI x1 DQG x2 Proof: 7KH FORVHG ORRS V\VWHP DIWHU VXEVWLWXWLQJ LQWR x˙ 1 = s − c1 x1
1
s˙ = −λ1 |s| 2 VLJQ(s) −
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1
u = −c1 x2 − λ1 |s| 2 VLJQ(s) −
t
0
λ2 VLJQ(s)dτ
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RU t
0
1
λ2 VLJQ(s)dτ
RU x˙ 1 = s − c1 x1 1
s˙ = −λ1 |s| 2 VLJQ(s) + ν ν˙ = −λ2 VLJQ(s)
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ν˙ = −λ2 VLJQ(s) + ρ˙ 1 .
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x˙ 1 = −c1 x1 x ˆ2 = −c1 x1
s˙ = −λ1 |s| 2 VLJQ(s) + ν
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2
1
ZKHUH z1 = k1 |e1 | 3 VLJQ(e1 ) z2 = k2 |e1 | 3 VLJQ(e1 ) DQG z3 = k3 VLJQ(e1 ) DUH FRUUHFWLRQ WHUPV /HW XV GH¿QH WKH HUURU e1 = x1 − x ˆ1 DQG e2 = x2 − x ˆ2 DQG WKH HUURU G\QDPLFV LV e˙ 1 = e2 − z1 e˙ 2 = −ˆ x3 − z2 + ρ1 ˙x ˆ3 = z3
x3 +ρ1 LI ρ1 LV /LSVFKLW] 1RZ GH¿QH WKH QHZ YDULDEOH e3 = −ˆ DQG |ρ˙ 1 | < ρ0 WKHQ RQH FDQ IXUWKHU ZULWH DV e˙ 1 = e2 − z1 e˙ 2 = e3 − z2 e˙ 3 = −z3 + ρ˙ 1
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1 3
= c1 x ˆ2 + u + k2 |e1 | VLJQ(e1 ) +
t
0
k3 VLJQ(e1 )dτ
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1 3
s˙ = c1 x ˆ2 + u + k2 |e1 | VLJQ(e1 ) +
0
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