Applied Mathematical Modelling 37 (2013) 7889–7890
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Corrigendum
Corrigendum to ‘‘Design, modeling and simulation of a new nonlinear and full adaptive backstepping speed tracking controller for uncertain PMSM’’ [Appl. Math. Modell. 36 (11) (2012) 5199–5213] Murat Karabacak a,⇑, Halil Ibrahim Eskikurt b a b
Department of Electronics Technology, Duzce University, 81010 Duzce, Turkey Department of Electronics and Computer Education, Sakarya University, 54010 Sakarya, Turkey
The authors regret that in the above article the Eqs. (5), (7), (10), (11), (13), (15), (22), (25) as well as Theorem 3.1. have some mathematical sign errors. These equations and Theorem 3.1. with correct signs are provided as follows:
1 dx 1 a3 ¼ iq ða1 x þ a2 Þ P P dt diq ¼ b1 iq b2 xe id b3 xe þ V q b2 dt did b2 ¼ b1 id þ b2 xe iq þ V d dt 1 dx 1 ~1 x þ a ^1 x þ a ~2 þ a ^2 Þ a3 ¼ iq ða P P dt diq ~ 1 iq b ^1 iq b ~ 2 xe i b ^ 2 xe i b ~ 3 xe b ^ 3 xe þ V q b2 ¼ b d d dt did ~ i b ^ i þb ~ x i þb ^ x i þV b2 ¼ b 1 d 1 d 2 e q 2 e q d dt 1 ~1 x þ a ^1 x þ a ~2 þ a ^2 þ a ~3 x ^3 x _ dþa _ dÞ V_ 1 ¼ e iq ða P
ð5Þ
ð7Þ
ð10Þ
1 ~1 x þ a ^1 x þ a ~2 þ a ^2 þ a ~3 x _ dþa ^3 x _ d Þ iqdes V_ 1 ¼ e iq ða P
ð11Þ
1 ~1 x þ a ^1 x þ a ~2 þ a ^2 þ a ~3 x _ dþa ^3 x _ dÞ V_ 1 ¼ e eq þ iqdes ða P
ð13Þ
1 ~1 x þ a ~2 þ a ~3 x _ d Þ k1 e V_ 1 ¼ e eq ða P
ð15Þ
DOI of original article: http://dx.doi.org/10.1016/j.apm.2011.12.048
⇑ Corresponding author. Tel.: +90 380 524 00 98; fax: +90 380 524 00 97. E-mail addresses:
[email protected] (M. Karabacak),
[email protected] (H.I. Eskikurt). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.04.001
7890
M. Karabacak, H.I. Eskikurt / Applied Mathematical Modelling 37 (2013) 7889–7890
^1 i b ^2 xe iq k3 e Vd ¼ b d d h i 1 ~1 iq b ~ 2 xe i b ~ 3 xe b ~2 Diqdes k2 eq þ e b ~1 i þ b ~2 xe iq k3 e ~1 x þ a ~ þa ~3 x _ d Þ k1 e þ eq b V_ ¼ e ða d d d d P Dt 2 1 _ 1 _ 1 _ 1 ~_ ~ 1 ~ ~_ 1 ~ ~_ ~1 þ a ~2 þ a ~3 þ b1 b1 þ b2 b2 þ b3 b3 ~2 a ~3 a þ a~1 a h1 h2 h3 h4 h5 h6
ð22Þ
ð25Þ
~ ;b ~ ;b ~ Þ : R9 Rþ ! Rþ is ~1 ; a ~2 ; a ~3 ; b Theorem 3.1. Invoking LaSalle’s invariance set theorem, it is evident that Vðe; ed ; eq ; a 1 2 3 continuously differentiable scalar function of the states, positive definite and radially unbounded such that ~ ;b ~ ;b ~ Þ 2 R9 . Then, ninth dimensional state converge to the largest invariant set M of _ ~1 ; a ~2 ; a ~3 ; b Vðe; ed ; eq Þ 6 0; 8ðe; ed ; eq ; a 1 2 3 n o ~ ;b ~ ;b ~ 2 R9 jðe; e ; eq Þ ¼ ð0; 0; 0Þ where V_ ¼ 0, and ~1 ; a ~2 ; a ~3 ; b (15), (19), (23) and (27) contained in the set E ¼ e; ed ; eq ; a 1 2 3 d there is no solution that can stay forever in E except for ðe; ed ; eq Þ ð0; 0; 0Þ. On this invariant set M, we have ^_ ; b ^_ ; b ^_ Þ ¼ ð0; 0; 0; 0; 0; 0Þ. Setting ^_ 1 ; a ^_ 2 ; a ^_ 3 ; b _ e_ d ; e_ q Þ ¼ ð0; 0; 0; 0; 0; 0Þ. Setting ðe; ed ; eq Þ ¼ ð0; 0; 0Þ in (27), we obtain ða ðe; ed ; eq ; e; 1 2 3 ~ iq b ~ xe i b ~ xe b ~ ðDi =DtÞ and ~1 x a ~2 a ~3 x _ d , 0 ¼ b _ e_ d ; e_ q Þ ¼ ð0; 0; 0Þ in (15), (19) and (23), we have 0 ¼ a ðe; 1 2 3 2 d qdes h i T T ~ ~ ~ ~ ~ ~ and F x ¼ x 1 x _ ði i Þ ð x i x i D i = DtÞ xe 0 ¼ b1 id þ b2 xe iq . h ¼ a ~1 a ~2 a ~3 b q e q e b b d d d qdes 1 2 3 can be described in matrix form. It follows that ~ hF Tx ¼ 0 on M. Thus, the largest invariant set M in E is:
M¼
n o ~ ;b ~ ;b ~ 2 R9 jðe; e ; e Þ ¼ ð0; 0; 0Þ; ~hF T ¼ 0 ~1 ; a ~2 ; a ~3 ; b e; ed ; eq ; a 1 2 3 d q x
Hence, the adaptive controller (14), (18) and (22) with the update laws (27a–f) ensure that all the signals in the closed loop system are bounded and the equilibrium manifold ðx; iq ; id Þ ¼ ðxd ; iqdes ; id d Þ is globally asymptotically stable [19], or equivalently: The authors would like to apologise for any inconvenience caused.