Corrigendum to “Observer-Based Adaptive Iterative Learning Control for a Class of Nonlinear Time Delay Systems with Input Saturation”

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 294313, 5 pages http://dx.doi.org/10.1155/2015/294313 Corrigendum Corrigendum to (Observer-Based Adaptive Iterative Learning Control for a Class of Nonlinear Time Delay Systems with Input Saturation) Jian-ming Wei, Yun-an Hu, and Mei-mei Sun Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai 264001, China Correspondence should be addressed to Jian-ming Wei; Received 15 October 2015; Accepted 3 November 2015 Copyright © 2015 Jian-ming Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The purpose of this paper is to correct the errors in the paper titled “Observer-Based Adaptive Iterative Learning Control for a Class of Nonlinear Time Delay Systems with Input ̂𝑘 (𝑡) in Saturation” [1]. In the adaptive learning law of 𝑊 (44) of [1], the variable 𝑧𝑘 was used. However in fact, 𝑧𝑘 is difficult to be obtained as the states are unmeasurable. The correction and some consequent modifications in the technical derivations are detailed as follows, while the main results are unchanged. By Young’s inequality, we can have ̃𝑇 𝜙 (𝑥̂𝑘 ) − 2𝑧𝑘𝑇 𝑃𝐵𝑊 𝑘 + ̃𝑘 ̃𝑇𝑊 + 𝛿2 𝑙𝑊 𝑘 −1 ̃𝑇 𝜙 (𝑥̂𝑘 ) ≤ −𝑧𝑘𝑇 𝑄𝑧𝑘 − 2𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 + −1 ̃𝑘 ̃𝑇𝑊 + 𝛿2 𝑙𝑊 𝑘 −1 ̃𝑇 𝜙 (𝑥̂𝑘 ) − 2𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 −1 −1 ̃𝑇 𝜙 (𝑥̂𝑘 ) ≤ −2𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 𝑧𝑘𝑇 𝑃𝑃𝑇 𝑧𝑘 2 ̃𝑇 ̃ 󵄩2 + 𝛿 𝑙𝑊𝑘 𝑊𝑘 , +󵄩 󵄩󵄩𝐶𝐶𝑇 + 𝛿𝐼𝑛 󵄩󵄩 󵄩 󵄩 where 𝛿 > 0 is a small positive constant. Then (27) in [1] should be updated to ̇ 𝑉𝑧̇ 𝑘 + 𝑉𝑈 𝑘 ≤ 𝑧𝑘𝑇 (𝐴𝑇 𝑃 + 𝑃𝐴 + 𝑛+3 𝑇 𝑃𝑃𝑇 ) 𝑧𝑘 𝑃𝑃 + 󵄩 󵄩󵄩𝐶𝐶𝑇 + 𝛿𝐼𝑛 󵄩󵄩󵄩2 𝜆 󵄩 󵄩 −1 ̃𝑇 𝜙 (𝑥̂𝑘 ) − 2𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 𝑛 𝜆 2 + 𝜆𝜀02 ∑ 𝜌𝑗2 (𝑦𝑘 ) + 𝜆𝐷02 + 4𝜆𝑙𝜀𝑊 (1 − 𝜅) 𝑗=1 󵄩 󵄩2 ≤ −𝜆 min (𝑄) 󵄩󵄩󵄩𝑧𝑘 󵄩󵄩󵄩 ̃𝑇 𝜙 (𝑥̂𝑘 ) = −2𝑧𝑘𝑇 𝐶𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 ̃𝑇 𝜙 (𝑥̂𝑘 ) − 2𝑧𝑘𝑇 𝛿𝐼𝑛 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 𝑛 𝜆 2 + 𝜆𝜀02 ∑ 𝜌𝑗2 (𝑦𝑘 ) + 𝜆𝐷02 + 4𝜆𝑙𝜀𝑊 (1 − 𝜅) 𝑗=1 (1) + 𝑛 𝜆 2 + 𝜆𝜀02 ∑ 𝜌𝑗2 (𝑦𝑘 ) + 𝜆𝐷02 + 4𝜆𝑙𝜀𝑊 (1 − 𝜅) 𝑗=1 ̃𝑘 , ̃𝑇𝑊 + 𝛿2 𝑙𝑊 𝑘 (2) where using updated inequality (12) of [1] 𝑛+3 𝑇 𝑃𝑃𝑇 𝐴𝑇 𝑃 + 𝑃𝐴 + < −𝑄. 𝑃𝑃 + 󵄩 (3) 󵄩󵄩𝐶𝐶𝑇 + 𝛿𝐼𝑛 󵄩󵄩󵄩2 𝜆 󵄩 󵄩 Consequently, the derivative of 𝑉𝑘 should be revised as −1 󵄩 󵄩2 𝑉𝑘̇ ≤ −𝜆 min (𝑄) 󵄩󵄩󵄩𝑧𝑘 󵄩󵄩󵄩 − 2𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑛−1 ̃𝑘 + 𝑠𝑘 [ ∑ 𝜆 𝑗 𝑒𝑗+1,𝑘 ̃𝑇 𝜙 (𝑥̂𝑘 ) + 𝛿2 𝑙𝑊 ̃𝑇𝑊 ⋅ 𝑃𝐵𝑊 𝑘 𝑘 [𝑗=1 2 Mathematical Problems in Engineering ̂𝑇 𝜙 (𝑥̂𝑘 ) + V𝑘 + [Λ𝑇 1] 𝐾0 𝑧1,𝑘 + 𝐾𝑒𝑠𝑘 − 𝑦𝑑(𝑛) + 𝑊 𝑘 Accordingly, the Lyapunov-like CEF is updated to 𝑏 tanh2 (𝑠𝑘 /𝜂 (𝑡)) Ξ (𝑦𝑘 ) ] + + [1 𝑠𝑘 ] 𝐸𝑘 (𝑡) = − 𝑏 tanh2 ( (1 − 𝛾1 ) 𝑇 𝛾1 𝑡 ̃𝑇 ̃ ̃ 𝑊 ̃𝑘 𝑊 ∫ 𝑊𝑘 𝑊𝑘 d𝜎 + 𝑘 2𝑞1 0 2𝑞1 𝑡 (1 − 𝛾2 ) 𝑇 𝛾 ̃𝑇 𝑊 ̃ d𝜎 + ̃ 𝑊 ̃ 𝑊 + 2 ∫ 𝑊 2,𝑘 2,𝑘 . 2𝑞2 0 2,𝑘 2,𝑘 2𝑞2 𝑠𝑘 )] Ξ (𝑦𝑘 ) − 𝐾𝑠𝑘2 . 𝜂 (𝑡) (4) ̂𝑘 (𝑡) in the absence of 𝑧𝑘 , we design In order to update 𝑊 the following differential-difference type learning law: The difference of 𝐸𝑘 (𝑡) should be changed to Δ𝐸𝑘 (𝑡) = 𝐸𝑘 (𝑡) − 𝐸𝑘−1 (𝑡) = ̂̇ 𝑘 = −𝛾1 𝑊 ̂𝑘 − 𝛾1 𝛼1 𝑊 ̂𝑘 + 𝛾1 𝑊 ̂𝑘−1 (1 − 𝛾1 ) 𝑊 −1 + 2𝑞1 𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝜙 (𝑥̂𝑘 ) , ̂𝑘−1 (𝑇) , ̂𝑘 (0) = 𝑊 𝑊 (5) ̂0 (𝑡) = 0, 𝑊 𝑡 ∈ [0, 𝑇] . ̂2,𝑘 (𝑡) is not changed, but The adaptive learning law for 𝑊 some notations are updated, which is specified by ̂̇ 2,𝑘 = −𝛾2 𝑊 ̂2,𝑘 + 𝛾2 𝑊 ̂2,𝑘−1 (1 − 𝛾2 ) 𝑊 + 𝑞2 𝑏 tanh2 ( 𝑠𝑘 ) 𝜙 (𝑦 ) , 𝜂 (𝑡) 2 𝑘 where 0 < 𝛾1 , 𝛾2 < 1, and 𝛼1 > 0 are design parameters. By substituting the controller back into (4), (46)-(47) of [1] should be replaced by 𝑇 𝛾2 𝑡 ̃𝑇 ̃ ̃ ̃𝑇 𝑊 ∫ [𝑊2,𝑘 𝑊2,𝑘 − 𝑊 2,𝑘−1 2,𝑘−1 ] d𝜎 2𝑞2 0 + (1 − 𝛾2 ) ̃ ̃𝑇 𝑊 ̃𝑇 ̃ [𝑊 2,𝑘 2,𝑘 − 𝑊2,𝑘−1 𝑊2,𝑘−1 ] . 2𝑞2 (1 − 𝛾1 ) ̃𝑘 − 𝑊 ̃ ̃𝑇𝑊 ̃𝑇 𝑊 [𝑊 𝑘 𝑘−1 𝑘−1 ] 2𝑞1 = 𝛾1 𝑡 ̃𝑇 ̃ ̃𝑇 ̃ ∫ [𝑊𝑘 𝑊𝑘 − 𝑊𝑘−1 𝑊𝑘−1 ] d𝜎 2𝑞1 0 + (1 − 𝛾1 ) 𝑡 𝑇 ̇ ̃ 𝑘 d𝜎 ̃ 𝑊 ∫ 𝑊 𝑘 𝑞1 0 + (1 − 𝛾1 ) ̃𝑘 (0) − 𝑊 ̃ ̃𝑇 (0) 𝑊 ̃𝑇 𝑊 [𝑊 𝑘 𝑘−1 𝑘−1 ] 2𝑞1 −1 0 −1 𝑇 ̃𝑇 𝜙 (𝑥̂𝑘 ) 𝑃𝐵𝑊 𝑘 ̃𝑘 − 𝑏𝑊 ̃𝑇𝑊 ̃𝑇 𝜙2 (𝑦𝑘 ) tanh2 ( + 𝛿2 𝑙𝑊 𝑘 2,𝑘 𝑠𝑘 ) 𝜂 (𝑡) −1 ̃𝑇 𝜙 (𝑥̂𝑘 ) 2𝑞1 𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 ̃𝑇 𝜙2 (𝑦𝑘 ) tanh2 ( + 𝑏𝑊 2,𝑘 𝑠𝑘 ) ≤ −𝑉𝑘̇ 𝜂 (𝑡) 𝑠 󵄩 󵄩2 − 𝜆 min (𝑄) 󵄩󵄩󵄩𝑧𝑘 󵄩󵄩󵄩 + [1 − 𝑏 tanh2 ( 𝑘 )] Ξ (𝑦𝑘 ) 𝜂 (𝑡) − 𝛾1 𝑡 ̃𝑇 ̂ +𝑊 ̂𝑘 − 𝑊 ̂𝑘−1 ) d𝜎 ∫ 𝑊 (𝛼 𝑊 𝑞1 0 𝑘 1 𝑘 + 𝛾1 𝑡 ̃𝑇 ̃ ̃𝑇 ̃ ∫ [𝑊𝑘 𝑊𝑘 − 𝑊𝑘−1 𝑊𝑘−1 ] d𝜎 2𝑞1 0 + (1 − 𝛾1 ) ̃𝑘 (0) − 𝑊 ̃ ̃𝑇 (0) 𝑊 ̃𝑇 𝑊 [𝑊 𝑘 𝑘−1 𝑘−1 ] 2𝑞1 (7) 𝑠𝑘 )] Ξ (𝑦𝑘 ) − 𝐾𝑠𝑘2 , 𝜂 (𝑡) ̃𝑘 . ̃𝑇𝑊 − 𝐾𝑠𝑘2 + 𝛿2 𝑙𝑊 𝑘 + ̃𝑇 𝜙 (𝑥̂𝑘 ) d𝜎 = ∫ 2𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 − 2𝑞1 𝑧1,𝑘 𝐶 (𝐶𝐶 + 𝛿𝐼𝑛 ) + [1 − 𝑏 tanh2 ( (1 − 𝛾1 ) ̃𝑘 − 𝑊 ̃ ̃𝑇𝑊 ̃𝑇 𝑊 [𝑊 𝑘 𝑘−1 𝑘−1 ] 2𝑞1 + 𝑡 󵄩 󵄩2 𝑉𝑘̇ ≤ −𝜆 min (𝑄) 󵄩󵄩󵄩𝑧𝑘 󵄩󵄩󵄩 + 𝛾1 𝑡 ̃𝑇 ̃ ̃𝑇 ̃ ∫ [𝑊𝑘 𝑊𝑘 − 𝑊𝑘−1 𝑊𝑘−1 ] d𝜎 2𝑞1 0 ̂2,0 (𝑡) = 0, 𝑊 𝑡 ∈ [0, 𝑇] , 𝛾1 𝑡 ̃𝑇 ̃ ̃𝑇 ̃ ∫ [𝑊𝑘 𝑊𝑘 − 𝑊𝑘−1 𝑊𝑘−1 ] d𝜎 2𝑞1 0 𝑡 −1 ̃𝑇 𝜙 (𝑥̂𝑘 ) d𝜎 = ∫ 2𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 0 (8) (10) By using adaptive learning law (5) and inequality ̃𝑇𝑊 ̂𝑘 ≥ 𝑊 ̃𝑘 − 𝑊∗𝑇 𝑊∗ , we may have ̃𝑇𝑊 2𝑊 𝑘 𝑘 (6) ̂2,𝑘 (0) = 𝑊 ̂2,𝑘−1 (𝑇) , 𝑊 (9) − 𝛼1 𝛾1 𝑡 ̃𝑇 ̂ 𝛾 𝑡 ̃𝑇 ̃ ̃ (𝑊𝑘 − 𝑊𝑘−1 ) d𝜎 ∫ 𝑊𝑘 𝑊𝑘 d𝜎 − 1 ∫ 𝑊 𝑞1 0 𝑞1 0 𝑘 + 𝛾1 𝑡 ̃𝑇 ̃ ̃𝑇 ̃ ∫ [𝑊𝑘 𝑊𝑘 − 𝑊𝑘−1 𝑊𝑘−1 ] d𝜎 2𝑞1 0 Mathematical Problems in Engineering + 3 (1 − 𝛾1 ) ̃𝑘 (0) − 𝑊 ̃ ̃𝑇 (0) 𝑊 ̃𝑇 𝑊 [𝑊 𝑘 𝑘−1 𝑘−1 ] 2𝑞1 𝑡 Choose suitable design parameters such that 𝛼1 𝛾1 /2𝑞1 − 𝑙𝛿2 > 0. Then it follows from (12) that −1 Δ𝐸𝑘 (𝑡) ̃𝑇 𝜙 (𝑥̂𝑘 ) d𝜎 = ∫ 2𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 0 𝑡 ≤ −𝑉𝑘 (𝑡) + 𝑉𝑘 (0) − 𝐾 ∫ 𝑠𝑘2 d𝜎 𝛼 𝛾 𝑡 ̃𝑇 ̂ − 1 1∫ 𝑊 𝑊 d𝜎 𝑞1 0 𝑘 𝑘 0 𝑡 (1 − 𝛾1 ) ̃𝑘 (0) − 𝑊 ̃𝑇 𝑊 ̃ ̃𝑇 (0) 𝑊 + [𝑊 𝑘 𝑘−1 𝑘−1 ] 2𝑞1 𝑡 𝑇 −1 𝑇 ≤ ∫ 2𝑧1,𝑘 𝐶 (𝐶𝐶 + 𝛿𝐼𝑛 ) 0 − + ̃𝑇 𝜙 (𝑥̂𝑘 ) d𝜎 𝑃𝐵𝑊 𝑘 + (1 − 𝛾1 ) ̃𝑘 (0) − 𝑊 ̃ ̃𝑇 (0) 𝑊 ̃𝑇 𝑊 [𝑊 𝑘 𝑘−1 𝑘−1 ] 2𝑞1 + (1 − 𝛾2 ) ̃2,𝑘 (0) − 𝑊 ̃𝑇 𝑊 ̃ ̃𝑇 (0) 𝑊 [𝑊 2,𝑘 2,𝑘−1 2,𝑘−1 ] . 2𝑞2 𝑇 Δ𝐸𝑘 (𝑇) ≤ −𝑉𝑘 (𝑇) − 𝐾 ∫ 𝑠𝑘2 d𝜎 (1 − 𝛾1 ) ̃𝑘 (0) − 𝑊 ̃𝑇 𝑊 ̃ ̃𝑇 (0) 𝑊 [𝑊 𝑘 𝑘−1 𝑘−1 ] . 2𝑞1 0 𝑇 󵄩 󵄩2 − 𝜆 min (𝑄) ∫ 󵄩󵄩󵄩𝑧𝑘 󵄩󵄩󵄩 d𝜎 (11) 0 Recalling (8), inequality (B.8) of [1] is changed to + Δ𝐸𝑘 (𝑡) 𝛼1 𝛾1 𝑡 󵄩󵄩 ∗ 󵄩󵄩 ∫ 󵄩𝑊 (𝜎)󵄩󵄩 d𝜎 2𝑞1 0 󵄩 𝑇 ̃𝑇 𝜙 (𝑥̂𝑘 ) d𝜎 ≤ ∫ 2𝑧𝑘𝑇 𝑃𝐵𝑊 𝑘 0 + (1 − 𝛾1 ) ̃𝑘 (0) − 𝑊 ̃𝑇 𝑊 ̃ ̃𝑇 (0) 𝑊 [𝑊 𝑘 𝑘−1 𝑘−1 ] 2𝑞1 + (1 − 𝛾2 ) ̃2,𝑘 (0) − 𝑊 ̃ ̃𝑇 (0) 𝑊 ̃𝑇 𝑊 [𝑊 2,𝑘 2,𝑘−1 2,𝑘−1 ] 2𝑞2 𝑡 ≤ −𝑉𝑘 (𝑡) + 𝑉𝑘 (0) − 𝐾 ∫ 𝑠𝑘2 d𝜎 + 𝑡 = (12) −1 = (1 − 𝛾2 ) ̃2,𝑘 (0) − 𝑊 ̃𝑇 𝑊 ̃ ̃𝑇 (0) 𝑊 [𝑊 2,𝑘 2,𝑘−1 2,𝑘−1 ] . 2𝑞2 𝛼1 𝛾1 ̃𝑇 ̂ 𝑊 𝑊 𝑞1 1 1 −1 ̃𝑇 𝜙 (𝑥̂1 ) + 2𝑧1,1 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 1 𝑡 + 𝛾1 ̃𝑇 ̃ ̂1 + 𝑊 ̂1 ] − 𝛾1 𝑊 ̂𝑇𝑊 ̂ ̃𝑇𝑊 ̂𝑇𝑊 [𝑊1 𝑊1 − 2𝑊 1 1 2𝑞1 2𝑞1 1 1 − 𝛼𝛾 ̃𝑘 d𝜎 ̃𝑇𝑊 − ∫ ( 1 1 − 𝑙𝛿2 ) 𝑊 𝑘 2𝑞1 0 (1 − 𝛾1 ) ̃𝑘 (0) − 𝑊 ̃ ̃𝑇 (0) 𝑊 ̃𝑇 𝑊 [𝑊 𝑘 𝑘−1 𝑘−1 ] 2𝑞1 𝛾1 ̃𝑇 ̃ 𝛾1 ̃𝑇 ̂ 𝛼1 𝛾1 ̃𝑇 ̂ 𝑊 𝑊 − 𝑊 𝑊 − 𝑊 𝑊 2𝑞1 1 1 𝑞1 1 1 𝑞1 1 1 ̃𝑇 𝜙 (𝑥̂𝑘 ) + 2𝑧1,𝑘 𝐶𝑇 (𝐶𝐶𝑇 + 𝛿𝐼𝑛 ) 𝑃𝐵𝑊 𝑘 0 + 𝛼1 𝛾1 󵄩󵄩 ∗ 󵄩󵄩 ∫ 󵄩𝑊 (𝜎)󵄩󵄩 d𝜎. 2𝑞1 0 󵄩 𝛾1 ̃𝑇 ̃ (1 − 𝛾1 ) ̃𝑇 ̃̇ 𝑊 𝑊 + 𝑊1 𝑊1 2𝑞1 1 1 𝑞1 𝑡 𝛼1 𝛾1 󵄩󵄩 ∗ 󵄩󵄩 ∫ 󵄩𝑊 (𝜎)󵄩󵄩 d𝜎 2𝑞1 0 󵄩 0 𝑡 ̂̇ 1 = −𝛾1 𝑊 ̂1 − By using adaptive learning law (1 − 𝛾1 )𝑊 𝑇 𝑇 −1 ̂1 + 2𝑞1 𝑧1,1 𝐶 (𝐶𝐶 + 𝛿𝐼𝑛 ) 𝑃𝐵𝜙(𝑥̂1 ), it is clear that 𝛾1 𝛼1 𝑊 0 󵄩 󵄩2 − 𝜆 min (𝑄) ∫ 󵄩󵄩󵄩𝑧𝑘 󵄩󵄩󵄩 d𝜎 𝑇 0 𝑡 𝑠 ̃𝑇 + ∫ 𝑏 tanh2 ( 𝑘 ) 𝑊 2,𝑘 𝜙2 (𝑦𝑘 ) d𝜎 𝜂 0 𝛼 𝛾 𝑡 ̃𝑇 ̃ 𝛼1 𝛾1 𝑡 󵄩󵄩 ∗ 󵄩󵄩 𝑊 d𝜎 ∫ 󵄩󵄩𝑊 (𝜎)󵄩󵄩 d𝜎 − 1 1 ∫ 𝑊 2𝑞1 0 2𝑞1 0 𝑘 𝑘 (14) 󵄩 󵄩2 ≤ −𝐾 ∫ 𝑠𝑘2 d𝜎 − 𝜆 min (𝑄) ∫ 󵄩󵄩󵄩𝑧𝑘 󵄩󵄩󵄩 d𝜎 𝑡 + (13) Consequently, (B.9) of [1] is updated to 𝛼1 𝛾1 𝑡 ̃𝑇 ̃ 𝛼 𝛾 𝑡󵄩 󵄩 ∫ 𝑊𝑘 𝑊𝑘 d𝜎 + 1 1 ∫ 󵄩󵄩󵄩𝑊∗ (...truncated)