1
Introduction

1.1 Introduction to Iterative Learning Control

Iterative learning control (ILC), as an effective control strategy, is designed to improve current control performance for unpredictable systems by fully utilizing past control experience. Specifically, ILC is designed for systems that complete tasks over a fixed time interval and perform them repeatedly. The underlying philosophy mimics the human learning process that "practice makes perfect." By synthesizing control inputs from previous control inputs and tracking errors, the controller is able to learn from past experience and improve current tracking performance. ILC was initially developed by Arimoto et al. (1984), and has been widely explored by the control community since then (Moore, 1993; Bien and Xu, 1998; Chen and Wen, 1999; Longman, 2000; Norrlof and Gunnarsson, 2002; Xu and Tan, 2003; Bristow et al. 2006; Moore et al. 2006; Ahn et al. 2007a; Rogers et al. 2007; Ahn et al. 2007b; Xu et al. 2008; Wang et al. 2009, 2014).

Figure 1.1 shows the schematic diagram of an ILC system, where the subscript i denotes the iteration index and yd denotes the reference trajectory. Based on the input signal, ui, at the i th iteration, as well as the tracking error , the input for the next iteration, namely the th iteration, is constructed. Meanwhile, the input signal will also be stored into memory for use in the th iteration. It is important to note that in Figure 1.1, a closed feedback loop is formed in the iteration domain rather than the time domain. Compared to other control methods such as proportional-integral-derivative (PID) control and sliding mode control, there are a number of distinctive features about ILC. First, ILC is designed to handle repetitive control tasks, while other control techniques don't typically take advantage of task repetition-under a repeatable control environment, repeating the same feedback would yield the same control performance. In contrast, by incorporating learning, ILC is able to improve the control performance iteratively. Second, the control objective is different. ILC aims at achieving perfect tracking over the whole operational interval. Most control methods aim to achieve asymptotic convergence in tracking accuracy over time. Third, ILC is a feedforward control method if viewed in the time domain. The plant shown in Figure 1.1 is a generalized plant, that is, it can actually include a feedback loop. ILC can be used to further improve the performance of the generalized plant. As such, the generalized plant could be made stable in the time domain, which is helpful in guaranteeing transient response while learning takes place. Last but not least, ILC is a partially model-free control method. As long as an appropriate learning gain is chosen, perfect tracking can be achieved without using a perfect plant model.

Figure 1.1 The framework of ILC.

Generally speaking, there are two main frameworks for ILC, namely contraction-mapping (CM)-based and composite energy function (CEF)-based approaches. A CM-based iterative learning controller has a very simple structure and is easy to implement. A correction term in the controller is constructed from the output tracking error; to ensure convergence, an appropriate learning gain is selected based on system gradient information in place of an accurate dynamic model. As a partially model-free control method, CM-based ILC is applicable to non-affine-in-input systems. These features are highly desirable in practice as there are plenty of data available in industry processes but there is a shortage of accurate system models. CM-based ILC has been adopted in many applications, for example X-Y tables, chemical batch reactors, laser cutting systems, motor control, water heating systems, freeway traffic control, wafer manufacturing, and so on (Ahn et al., 2007a). A limitation of CM-based ILC is that it is only applicable to global Lipschitz continuous (GLC) systems. The GLC condition is required by ILC in order to form a contractive mapping, and rule out the finite escape time phenomenon. In comparison, CEF-based ILC, a complementary approach to CM-based ILC, applies a Lyapunov-like method to design learning rules. CEF is an effective method to handle locally Lipschitz continuous (LLC) systems, because system dynamics is used in the design of learning and feedback mechanisms. It is, however, worthwhile pointing out that in CM-based ILC, the learning mechanism only requires output signals, while in CEF-based ILC, full state information is usually required. CEF-based ILC has been applied in satellite trajectory keeping (Ahn et al. 2010) and robotic manipulator control (Tayebi, 2004; Tayebi and Islam, 2006; Sun et al., 2006).

This book follows the two main frameworks and investigates the multi-agent coordination problem using ILC. To illustrate the underlying idea and properties of ILC, we start with a simple ILC system.

Consider the following linear time-invariant dynamics:

where i is the iteration index, a is an unknown constant parameter, and T is the trial length. Let the target trajectory be xd(t), which is generated by

with ud(t) is the desired control signal. The control objective is to tune ui(t) such that without any prior knowledge about the parameter a, the tracking error can converge to zero as the iteration number increases, that is, for .

We perform the ILC controller design and convergence analysis for this simple control problem under the frameworks of both CM-based and CEF-based approaches, in order to illustrate the basic concepts in ILC and analysis techniques. To restrict our discussion, the following assumptions are imposed on the dynamical system (1.1).

Assumption 1.1

The identical initialization condition holds for all iterations, that is, , .

Assumption 1.2

For , , there exists a ud(t), such that implies , .

1.1.1 Contraction-Mapping Approach

Under the framework of CM-based methodology, we apply the following D-type updating law to solve the trajectory tracking problem:

where is the learning gain to be determined. Our objective is to show that the ILC law (1.3) can converge to the desired ud, which implies the convergence of the tracking error ei(t), as i increases.

Define . First we can derive the relation

Furthermore, the state error dynamics is given by

Combining (1.4) and (1.5) gives:

Integrating both sides of the state error dynamics and using Assumption 1.1 yields

Then, substituting (1.7) into (1.6), we obtain

Taking ? -norm on both sides of (1.8) gives

where , and the ? -norm is defined as

The ? -norm is just a time weighted norm and is used to simplify the derivation. It will be formally defined in Section 1.4.

If in (1.9), it is possible to choose a sufficiently large such that . Therefore, (1.9) implies that , namely .

1.1.2 Composite Energy Function Approach

In this subsection, the ILC controller will be developed and analyzed under the framework of CEF-based approach. First of all, the error dynamics of the system (1.1) can be expressed as follows:

where xd is the target trajectory.

Let k be a positive constant. By applying the control law

and the parametric updating law ,

we can obtain the convergence of the tracking error ei as i tends to infinity.

In order to facilitate the convergence analysis of the proposed ILC scheme, we introduce the following CEF:

where is the estimation error of the unknown parameter a.

The difference of Ei is

By using the identical initialization condition as in Assumption 1.1, the error dynamics (1.10), and the control law (1.11), the first term on the right hand side of (1.14) can be calculated as

In addition, the second term on the right hand side of (1.14) can be expressed as

where the updating law (1.12) is applied. Clearly, ?ixiei appears in (1.15) and (1.16) with opposite signs. Combining (1.14), (1.15), and (1.16) yields

The function Ei is a monotonically decreasing sequence, hence is bounded if E0...