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

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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 Saturation” [1]. In the adaptive learning law of in (44) of [1], the variable was used. However in fact, 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 havewhere is a small positive constant.

Then (27) in [1] should be updated towhere using updated inequality (12) of [1]

Consequently, the derivative of should be revised as

In order to update in the absence of , we design the following differential-difference type learning law:

The adaptive learning law for is not changed, but some notations are updated, which is specified bywhere , , and are design parameters.

By substituting the controller back into (4), (46)-(47) of [1] should be replaced by

Accordingly, the Lyapunov-like CEF is updated to

The difference of should be changed to

By using adaptive learning law (5) and inequality , we may have

Recalling (8), inequality (B.8) of [1] is changed to

Choose suitable design parameters such that . Then it follows from (12) that

Consequently, (B.9) of [1] is updated to

By using adaptive learning law , it is clear that

Thus is changed to

Denote . The integral of over is updated as follows:

According to the new definition of CEF (9), should be computed as follows:

Hence is bounded by

We choose , with being a convergent series, which is defined by , where and are design parameters, , . has the following property.

Property 1 (see S. Zhu, M. X. Sun, and X. X. He, “Iterative learning control of strict-feedback nonlinear time-varying systems,” Acta Automatica Sinica, vol. 36, no. 3, pp. 454–458, 2010). .

Using (14), it is followed by

According to Property 1, we know , which implies the boundedness of .

In the derivation of finiteness of , the changes are specified as follows.

Separate into two parts:

The boundedness of and is guaranteed . Thus, there exist two positive constants and satisfying

Consequently,

On the other hand, it follows from (13) that

Combining (23) and (24) results in

Since we have proved the boundedness of , the finiteness of can be deduced by induction method.

Finally, we give the necessary revisions for the proof of convergence of tracking errors.

We can obtain from (20) that

Considering Property 1 and taking the limitation of the above two inequalities yield

Similarly, according to the convergence of the sum of series, we can obtain the convergence of errors. The other parts are not changed.