Abstract
An iterative learning control problem for nonlinear systems with delays is studied in detail in this paper. By introducing the λ-norm and being inspired by retarded Gronwall-like inequality, the novel sufficient conditions for robust convergence of the tracking error, whose initial states are not zero, with time delays are obtained. Finally, simulation example is given to illustrate the effectiveness of the proposed method.
1. Introduction
Since iterative learning control is proposed by Arimoto et al. in 1984 (see [1]), this feedforward control approach has become a major research area in recent years. Iterative learning control, which belongs to the intelligent control methodology, is an approach for fully utilizing the previous control information and improving the transient performance of studied systems that is suitable for repetitive movements. Its goal is to achieve full range of tracking tasks on finite interval (see [2–14]).
In practice, the control problem of systems with delays has always been an interesting research, since time delay can be often encountered in a wide range, such as aircraft systems, turbojet engines, microwave oscillators, nuclear reactors, and chemical processes. The existence of time delay in a system may degrade the control performance and even at worst may become a source of instability. Stabilization problem of control systems with delay has received much attention for several decades and some research results have been reported in the literature (see [3, 11, 13, 15–23]). However, only a few results are available for nonlinear systems, combining with the iterative learning control items, with time delays [11, 24–26]. In this paper, under the case that the th iterative state vector is different from the th iterative state vector , that is, , the iterative learning controller of nonlinear time-delayed systems is designed by using λ-norm and retarded Gronwall-like inequality.
Before ending this section, it is worth pointing out the main contributions of this paper as follows.
The iterative learning control problem for nonlinear systems with delays is investigated. That is, we consider the systemwhich is different from the mentioned systemin past literature.
Based on retarded Gronwall-like inequality and the convergence of tracking error , the practical output is determinate by the previous iterative learning control information , , .
2. Preliminaries
Throughout this paper, the -norm for the -dimensional vector is defined as , while the λ-norm for a function is defined as , where the superscript represents the transpose and . and represent the identity matrix and a zero matrix, respectively.
Lemma 1 (see [10, 27]). Consider
Lemma 2 (see [28] retarded Gronwall-like inequality). Consider such an inequalityand suppose that(1)all are continuous and nondecreasing functions on and are positive on such that ;(2) is continuously differentiable in and nonnegative on , where are constants and ;(3)all are continuously differentiable and nondecreasing such that on ;(4)all , , are continuous and nonnegative functions on Take the notation for , where is a given constant. It is denoted by simply when there is no confusion. If is a continuous and nonnegative function on satisfying (4), then where is determined recursively by and is the largest number such that
3. Main Results
Consider the following system with time delay:where , , and are the state vector, output vector, and input vector, respectively. is the number of iterations, and . , , are real constant matrices; , is a reference output.
Suppose that there exist the bounded constants and such that
Theorem 3. For system (8) and a given reference , if there exist matrices and functions and such thatwhere is a constant, then system (8) with the iterative learning control law can guarantee that is bounded but cannot track on and on for arbitrary initial state .
Proof. It is easy to know that, for any ,So we obtain from condition (9)In this paper, we use Lemma 2. Taking , , , , , and , then and . So we have :Using Lemma 1 and multiplying both sides of the above inequality (14) by and taking the λ-norm, we haveThus, condition (10) can guarantee by selecting sufficiently large, so we have for any . It follows from the equivalence of norms; we get that .
For any ,where
It is easy to know that by using Lemma 2.
ThenUsing Lemma 1 and multiplying both sides of the above inequality by and taking the λ-norm, we havewhere :Imitating the above proof, the result for any is obtained by selecting sufficiently large. So it is true that is bounded on .
Remark 4. When the number of iterations and , the tracking error satisfies that on for arbitrary initial state .
System (8) iswhere the delay satisfies .
Similar to the proof of Theorem 3, inequality (14) is written as
Then we have the following.
Theorem 5. For system (21) and a given reference , if there exist matrices , , and and functions and such that is bounded and , where is a constant, then system (21) with the iterative learning control law can guarantee that is bounded but cannot track on and on for arbitrary initial state .
When and satisfiesit is easy to obtain thatwhere ,
Consider the following system:Similar to the proof of Theorem 3, we get
So we have the following result.
Theorem 6. For system (25) and a given reference , if conditions (9) are true and there exist matrix and functions and such that , , is bounded, , where is a constant, then system (25) with the iterative learning control law can guarantee that is bounded but cannot track on and on for arbitrary initial state .
4. Numerical Example
For further illustration, we consider the following system:where andtaking the reference .
From the above numerical example, it can be easily proved that the conditions of Theorem 6 are satisfied.
5. Conclusion
In this paper, considering the iterative learning control problem for nonlinear systems with delays, the novel sufficient conditions for robust convergence of the tracking error have been addressed.
Competing Interests
The author declares that there are no competing interests.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (11071141), the Natural Science Foundation of Shandong Province of China (ZR2011AL018, ZR2011AQ008), and a Project of Shandong Province Higher Educational Science and Technology Program (J13LI02, J11LA06).