Abstract
A new adaptive iterative learning control scheme is proposed for complex dynamical networks with repetitive operation over a fixed time interval. By designing difference type updating laws for unknown time-varying parameters and coupling strength, the state of each node in complex dynamical networks can track the reference signal. By constructing a composite energy function, a sufficient condition of the convergence of tracking error sequence is achieved in the iteration domain. Finally, a numerical example is given to show the effectiveness of the designed method.
1. Introduction
Iterative learning control (ILC) is an effective control scheme suitable for dynamical systems with repetitive operation over a fixed time interval. It has received considerable attentions since it was introduced by Arimoto et al. [1]. The ILC system improves its control performance by updating the control input iteratively with the information of errors and inputs in the preceding trials. In view of methodology, the ILC can be classified into the PID-type algorithm and the adaptive iterative learning control (AILC) approach.
Up to now, AILC schemes have been widely used in nonlinear systems [2–10]. Using a composite energy function, Xu and Tan [3] proposed an adaptive learning control approach for nonlinear systems with time-varying parametric uncertainties. Chen and Zhang [5] designed an AILC scheme for nonlinearly parameterized systems with unknown time-varying delays. An adaptive ILC for a class of discrete-time systems with iteration-varying trajectory and random initial condition was shown in [8]. The AILC of systems with unknown control direction was proposed in [9, 10]. In view of control objective, the AILC system can track the iteration-invariant reference signal [2–6, 9, 10] and the iteration-variant one [7, 8].
Complex dynamical networks [11–14] exist in many fields such as science, nature, and human societies. A complex dynamical network is a large set of interconnected nodes, in which each node represents an individual element and the connection between every two nodes represents the relation. In this context, a complex dynamical network is a class of interconnected system, in which each node is a subsystem. Synchronization of complex dynamical networks is an important topic that has drawn a great deal of attention [15–23]. Various control schemes, such as impulsive control [15–17], pinning control [17, 18, 22], and adaptive control [19–22], were reported to achieve network synchronization. Many synchronization criteria have been derived for complex dynamical networks with different special features, such as time-delay [19–21], time-varying coupling parameter [20, 21], nonlinear coupling [22, 23], and nonidentical nodes [23]. It is worth noting that the tracking problem of complex dynamical networks is rarely mentioned.
Just as mentioned above, a complex dynamical network is a large-scale interconnected system. Some control schemes were designed for the stabilization and tracking of large-scale interconnected systems, such as decentralized adaptive robust control [24] and decentralized iterative learning control [25]. Thus, it is meaningful to apply the iterative learning control strategy to complex dynamical networks with repetitive operation over a fixed time interval.
Motivated by the above discussions, a new adaptive iterative learning control scheme is proposed for complex dynamical networks with nonidentical nodes and unknown time-varying parameters in this paper. The control objective is that the state of each node in complex dynamical networks can track the desired trajectory over a fixed time interval through the iterative learning process. As we all know, the network topology of complex dynamical networks is the difficulty in the control design. Compared with the simple plant without network connection, the controllers of complex dynamical networks are more complicated. In this paper, a simple feedback controller is designed by analyzing the network topology of complex dynamical networks, which makes the control scheme more simple and useful.
The rest of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. Section 3 gives the design of controllers and adaptive learning laws. In Section 4, the convergence property of the proposed adaptive iterative learning control method is given. In Section 5, an illustrative example is given to show the effectiveness of the designed method. Finally, conclusions are given in Section 6.
2. Problem Formulation and Preliminaries
Consider a controlled complex dynamical network consisting of nonidentical nodes and operating repetitively over a fixed time interval . The state equation of each node is given by where the superscript denotes the iteration number of learning, represents the state vector, is a known smooth nonlinear vector-valued function, and denotes the input. is the constant inner-coupling matrix; is the outer-coupling matrix, in which is defined as follows: if there is a connection between node and node (), then , else , and the diagonal elements of matrix are defined by
In network (1), is an unknown continuous time-varying parameter, which shows that there are some unknown terms in the dynamical equation of each node; is the unknown continuous time-varying coupling strength of the network.
Remark 1. The unknown parameters are widely considered in the analysis and control of systems. In this paper, the unknown parameters in the complex dynamical network make the network study as real as possible.
The control objective is to find a sequence of control input on and the updating laws for unknown time-varying parameters of each node, such that the tracking error sequence converges to zero as tends to infinity; that is,
where the tracking error sequence is with the given desired trajectory .
In order to achieve the above objective, the following mathematical preliminaries are needed.
Assumption 2. In network (1), the inner-coupling matrix and outer-coupling matrix satisfy and for , where and are positive constants.
Assumption 3. The continuous time-varying parameter is nonnegative on .
Assumption 4. The desired trajectory is a differentiable vector-valued function with respect to , so that the boundedness of it is achieved.
Assumption 5. The initial condition satisfies ; that is, for .
Lemma 6 (Young’s inequality). For any vectors and any , the following matrix inequality holds:
Lemma 7. If the derivative of the sequence is uniformly bounded with respect to and satisfies , then uniformly on .
Proof. By invoking the definition of and carefully calculating, we have which yields for . According to Lemma 2 of [6], we get that uniformly on . Hence, uniformly on .
Definition 8. For a scalar , a saturation function is defined as follows: where are the lower and upper bounds of and satisfy .
Lemma 9 (see [6]). Given scalars and , suppose that ; then
3. Design of Controllers and Adaptive Learning Laws
In this section, a new adaptive iterative learning control scheme is proposed for the network described by (1). The dynamical equation of tracking error sequence is expressed as follows:
To achieve the control objective (3), we design controllers by where and are estimations to and , respectively.
Remark 10. The designed controller (9) is a simple feedback controller for each node, in which is used to deal with the network topology of complex dynamical networks. If we proposed the AILC scheme for systems without network connection, this term is not needed.
Denoting and , which are the estimation errors, we obtain the following closed-loop error equation:
A Lyapunov functional at the th iteration is selected as
The derivative of is
According to Assumptions 2 and 3 and Lemma 6, one obtains where is a positive constant. Choosing , we have ; then
The proposed adaptive learning laws of and are where and are positive design parameters.
The saturation function in (15) is the same as Definition 8; that is, for any signal (), we have where are the lower and upper bounds. Without losing generality, we assume that and lie within the saturation bounds.
4. Convergence Analysis
The convergence property of the proposed adaptive iterative learning control scheme is summarized in the following theorem.
Theorem 11. Considering system (1) under Assumptions 2, 3, 4, and 5, the control law (9) and the adaptive learning laws (15) guarantee the convergence of the tracking error sequence ; that is, for . Furthermore, all signals in the closed-loop system are bounded.
Proof. To facilitate the convergence analysis, we choose a Lyapunov-Krasovskii functional at the th iteration as follows:
Firstly, we calculate the difference of at the th iteration as follows:
Using Assumption 5 and (14), we have
Using the algebraic relation and the adaptive laws (15), we get
According to Lemma 9 and by carefully calculating, we have
Substituting (19) and (21) into (18) yields
When , inequality (22) is expressed as
Secondly, we prove that all signals in the closed-loop system are bounded. Using (17) and choosing , we have
The derivative of satisfies
According to the adaptive learning laws (15), one obtains
Substituting (26) into (25) yields
Because of the continuity of and on , a positive constant exists so that
Using Assumption 5, we have . Applying into inequality (28) yields
which shows that is bounded on .
Moreover, by invoking the definition of and applying inequality (22), we have
Therefore,
It also follows from (30) that
which gives rise to
is thus a nonincreasing sequence. It then follows from (29), (31), and (33) that () is uniformly bounded with respect to . In addition, by invoking the definition of , is uniformly bounded with respect to . Therefore, and are obviously uniformly bounded with respect to . According to (15), we can prove that and are uniformly bounded with respect to . Hence, all signals in the closed-loop system are bounded.
Finally, we prove the convergence of tracking error sequence. It follows from (32) that
Since is monotone decreasing but lower bounded, converges. Therefore,
then,
According to Lemma 7, we get , which shows that the tracking error sequence of each node converges.
Remark 12. From the result of the theorem, we get that the state of each node tracks the desired trajectory over a given fixed time interval as the iteration index tends to infinity. Hence, it is also said that the synchronization objective for complex dynamical networks is obtained in the iteration domain.
5. Simulation Example
In this section, a numerical example is given to show the effectiveness of the designed method.
Example 1. Consider a complex dynamical network with repetitive operation over a fixed time interval as follows:
The parameters in the network (37) are
The desired trajectory is . Based on the dynamical equation (37), we have , , , and . The design parameters are selected as follows:
In the simulation, we select as the lower and upper bounds of the saturation function, so that the time-varying parameters are within the bounds. Because , we select as the initial state.
The simulation results are shown in the Figures 1–4. Figure 1 shows that a perfect tracking performance is achieved as the iteration index increases. Figure 2 depicts the curves of control inputs. Figure 3 shows the evolutions of and as iteration index . Figure 4 shows the evolutions of and as iteration index . Obviously, the controllers and estimated parameters are bounded on .
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Remark 13. From the simulation results shown in the above figures, we find the estimated parameters of identical to the true values, which shows that the unknown terms in the dynamical equation of each node can be identified very well. However, the estimated parameter of is different from the true value. This is because of the simple feedback controller (9) added in each node. In this paper, we analyze the network topology of complex dynamical networks and design the controller without the precise information of the network topology, and then we get the perfect tracking performance. Thus, the control objective can be achieved even if the time-varying coupling strength was not identified very well.
6. Conclusions
In this paper, for a complex dynamical network with repetitive operation over a fixed time interval, we propose a new adaptive iterative learning control scheme to ensure that the state of each node tracks the desired trajectory as the iteration index tends to infinity. In this context, it is also said that the synchronization for complex dynamical networks is achieved in the iteration domain. The simulation example verifies the theoretical results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant no. 60974139 and the Foundation of Ministry of Education of China under Grants no. JY0300137002 and no. 20130203110021.