Introduction

Abstract

There are several examples of periodic phenomena, such as the orbital motion of heavenly bodies and heartbeats, that can be observed in nature. In practice, many control tasks are often considered to exhibit periodic behavior. Industrial manipulators are often used to track or reject periodic exogenous signals when performing picking, placing, or painting operations. Moreover, some special applications of these periodic exogenous signals include magnetic spacecraft attitude control, active control of helicopter vibrations, autonomous vertical landing in an oscillating platform, elimination of harmonics in aircraft power supply, satellite formation, light-emitting diode tracking, control of hydraulic servomechanisms, and control of lower limb exoskeletons. High-precision control performance can be realized for such periodic control tasks using repetitive control (RC, or repetitive controller, which is also designated as RC). RC was initially developed for continuous single-input, single-output (SISO) linear time-invariant (LTI) systems in [1] to accurately track periodic signals with a known period. RC was then extended to multiple-input multiple-output (MIMO) LTI systems in [2]. Since then, RC has been the subject of increasing attention, and applications that employ RC have become a special subject of focus in control theory.

There are several examples of periodic phenomena, such as the orbital motion of heavenly bodies and heartbeats, that can be observed in nature. In practice, many control tasks are often considered to exhibit periodic behavior. Industrial manipulators are often used to track or reject periodic exogenous signals when performing picking, placing, or painting operations. Moreover, some special applications of these periodic exogenous signals include magnetic spacecraft attitude control, active control of helicopter vibrations, autonomous vertical landing in an oscillating platform, elimination of harmonics in aircraft power supply, satellite formation, light-emitting diode tracking, control of hydraulic servomechanisms, and control of lower limb exoskeletons. High-precision control performance can be realized for such periodic control tasks using repetitive control (RC, or repetitive controller, which is also designated as RC). RC was initially developed for continuous single-input, single-output (SISO) linear time-invariant (LTI) systems in [1] to accurately track periodic signals with a known period. RC was then extended to multiple-input multiple-output (MIMO) LTI systems in [2]. Since then, RC has been the subject of increasing attention, and applications that employ RC have become a special subject of focus in control theory. Recent developments concerning RC have not been consistent, with limited research on RC in nonlinear systems. However, the use of frequency-based methods has significantly aided the development of theories and applications pertaining to LTI systems. This chapter aims to answer the following question:

What are the challenges in employing repetitive control for nonlinear systems?

To answer this question, it is essential to introduce the basic idea of RC and provide a brief overview of RC for linear and nonlinear systems. This chapter presents a revised and extended version of a paper that was published earlier [3].

1.1 Basic Idea of Repetitive Control

1.1.1 Basic Concept

Before discussing RC, the concept of iterative learning control (ILC, or iterative learning controller, which is also designated as ILC) must be introduced to avoid confusion between these two similar control methods.

1.1.1.1 Iterative Learning Control

ILC is used for repetitive tasks with multiple execution times. It focuses on improving task results by learning from previous executions [4,5,6,7,8,9,10]. This control method performs a repetitive task and can utilize past control (delayed) information for generating present control action, which makes it different from most existing control methods. The classic ILC comprises three steps for each trial: (i) storing past control information; (ii) suspending the plant and resetting to the initial state condition; and (iii) controlling the plant using stored past control information and current feedback. For example, a remote pilot practices the take-off movements of a multicopter from the ground to a predetermined height. During each take-off, the remote pilot observes the trajectory of the multicopter (first step). If the trajectory is not satisfactory, the remote pilot will land the multicopter and then start it again by setting the initial rotation speed of the propellers to the previously recorded values (second step). Finally, the remote pilot adjusts the operation based on previous data. As the pilot continues to practice, the correct operation is learned and ingrained into the muscle memory of the pilot so that the skill of the pilot can be improved iteratively, which is the principle of the ILC learning method.

1.1.1.2 Repetitive Control

RC is used for periodic signal tracking and rejection. It aims to improve task results by learning from previous executions. RC is a special tracking method intended for a class of special problems. The classic RC comprises two steps for each trial: (i) storing past control information and (ii) controlling the plant using stored past control information (the last trial) and current feedback. For example, a pilot attempts to land a helicopter on a periodic oscillating deck at sea. Given the periodicity of the deck, the pilot can adjust his or her operation based on the previous trajectory of the helicopter, which is the principle of the RC learning method. The most significant difference between RC and ILC is that during RC, the initial state of the current trial cannot be reset to the final state of the previous trial. The entire process is continuous without any interruption at the end of each trial.

1.1.1.3 Comparison

Let us consider a class of linear systems as follows:

$$\begin{aligned} \dot{\mathbf {x}}\left( t\right)&=\mathbf {Ax}\left( t\right) +\mathbf {Bu}\left( t\right) \\ \mathbf {y}\left( t\right)&=\mathbf {C}^{\text {T}}\mathbf {x}\left( t\right) , \end{aligned}$$

where $\mathbf {A}\in \mathbf {\mathbb {R}}^{n\times n},$ $\mathbf {B,C}\in \mathbf {\mathbb {R}}^{n\times m},$ $\mathbf {x}\in \mathbf {\mathbf {\mathbb {R}}}^{n},$ and $\mathbf {y}\left( t\right) , \mathbf {u}\left( t\right) \in \mathbf {\mathbf {\mathbb {R}}}^{m}.$ The control objective is to design $\mathbf {u}$ that enables $\mathbf {y}$ to track the desired trajectory $\mathbf {y}_{\text {d}}.$ For simplicity, resetting the time is ignored for ILC and the control variable $\mathbf {u}$ is defined as

$$ \mathbf {u}_{k}\left( t\right) \triangleq \mathbf {u}\left( kT+t\right) ,t\in [0,T], $$

where $T>0$ denotes the interval time of a trial for ILC and the period for RC, $k=0,1,2,\ldots$. A comparison is presented in Table 1.1, where the two controller forms are found to be the same but exhibit a major difference in terms of the initial condition setting.

Table 1.1 Comparison between ILC and RC

1.1.2 Internal Model Principle

The basic concept of RC originates from the internal model principle (IMP); this principle states that if any exogenous signal can be regarded as the output of an autonomous system, then the inclusion of this signal model, namely, internal model, in a stable closed-loop system can assure asymptotic tracking and asymptotic rejection of the signal [11]. If a given signal is composed of a certain number of harmonics, then a corresponding number of neutrally stable internal models (one for each harmonic) should be incorporated into the closed-loop based on the IMP to realize asymptotic tracking and asymptotic rejection. To further explain the IMP, the zero-pole cancelation viewpoint of the IMP is used to explain the role of the internal models in step signals, sine signals, and T-periodic signals.

1.1.2.1 Step Signal

It is well known that integral control can track and reject any external step signal; this can be explained using the IMP given that the models of an integrator and a step signal are the same, namely, 1/s. Based on the IMP, inclusion of the internal model 1/s into a stable closed-loop system can assure asymptotic tracking and asymptotic rejection of a step signal.

Fig. 1.1

Step signal tracking

According to Fig. 1.1, the transfer function from the desired signal to the tracking error can be written as follows:

$$\begin{aligned} e\left( s\right)&=\frac{1}{1+\frac{1}{s}G\left( s\right) }y_{\text {d} }\left( s\right) =\frac{1}{s+G\left( s\right) }\left( s\frac{a}{s}\right) \\&=\frac{a}{s+G\left( s\right) }, \end{aligned}$$

where $y_{\text {d}}\left( s\right) =a/s$ is the Laplace transformation of a step signal with amplitude $a\in \mathbb {R} .$ Therefore, it is sufficient to merely verify whether the roots of the equation $s+G\left( s\right) =0$ are all in the left s-plane, which confirms the stability of the closed-loop system. If all roots are in the left s-plane, then the tracking error tends to zero as $t\rightarrow \infty$. Therefore, the tracking problem is converted into a stability problem of the closed-loop system.

1.1.2.2 Sine Signal

Suppose that the external signal is in the form $a_{0}\sin \left( \omega t+\varphi _{0}\right)$, where $a_{0},\varphi _{0}$ are constants, and the Laplace transformation model of $a_{0}\sin \left( \omega t+\varphi _{0}\right)$ is $\left( b_{1}s+b_{0}\right) /\left. \left( s^{2}+\omega ^{2}\right) \right.$, where $b_{0},b_{1}\in \mathbb {R} .$ Precise tracking or complete rejection can then be achieved by incorporating the model $1/ \left( s^{2}+\omega ^{2}\right) .$ into the closed-loop system.

Fig. 1.2

Sine signal tracking

Figure 1.2 demonstrates that the transfer function from the desired signal to the tracking error can be expressed as follows:

$$\begin{aligned} e\left( s\right)&=\frac{1}{1+\frac{1}{s^{2}+\omega ^{2}}G\left( s\right) }y_{\text {d}}\left( s\right) \\&=\frac{1}{s^{2}+\omega ^{2}+G\left( s\right) }\left( \left( s^{2} +\omega ^{2}\right) \frac{b_{1}s+b_{0}}{s^{2}+\omega ^{2}}\right) \\&=\frac{b_{1}s+b_{0}}{s^{2}+\omega ^{2}+G\left( s\right) }. \end{aligned}$$

Thus, it is adequate to only verify whether the roots of the equation $s^{2}+\omega ^{2}+G\left( s\right) =0$ are all in the left s-plane, which confirms the stability of the closed-loop system. Therefore, the tracking problem is converted into a stability problem of the closed-loop system.

Based on the IMP, designing a tracking controller for a general periodic signal is challenging because any periodic signal may be a summation of finite or infinite harmonics with period T. The harmonics of a general periodic signal must first be analyzed. However, obtaining accurate harmonics is difficult or time-consuming. Second, according to the IMP, the controller will contain more neutrally stable internal models (one for each harmonic) as the number of harmonics increases. For example, a T-periodic signal consists of harmonics with frequencies 0,  $2\pi / T ,\ldots ,2\pi N / T ,$ where $N\in \mathbb {Z} _{+}.$ The corresponding internal model can then be written as

$$\begin{aligned} I_{M,\text {fin}}=\frac{1}{s\displaystyle \prod _{k=1}^{N}\left( 1+\frac{T^{2}s^{2}}{4\pi ^{2}k^{2}}\right) }. \end{aligned}$$

(1.1)

This will result in an extremely complex controller structure. Moreover, it will be time-consuming to solve these neutrally stable internal models (differential equations) to obtain the control output. However, these two drawbacks can be overcome by using the following internal model for the T-periodic signal.

1.1.2.3 T-Periodic Signal

The Laplace transformation of a signal $y_{\text {d}}\left( t\right)$ delayed by T is expressed as follows:

$$\begin{aligned} \mathscr {L}\left( y_{\text {d}}\left( t-T\right) \right) =e^{-sT} \mathscr {L}\left( y_{\text {d}}\left( t\right) \right) . \end{aligned}$$

Suppose that the external signal is of the form $y_{\text {d}}\left( t\right) =y_{\text {d}}\left( t-T\right)$, which can represent any T-periodic signal. Its Laplace transformation is $1/ \left( 1-e^{-sT}\right)$ with an initial condition on the interval $[-T,0].$ Based on the IMP, asymptotic tracking and asymptotic rejection can be achieved by incorporating the internal model $1/ \left( 1-e^{-sT}\right)$ into the closed-loop system. The internal model for any T-periodic signal can be rewritten as follows [12]:

$$\begin{aligned} \displaystyle I_{M,\inf }\triangleq \frac{1}{1-e^{-sT}}=\underset{N\rightarrow \infty }{\lim }\frac{1}{s\displaystyle \prod _{k=1}^{N}\left( 1+\frac{T^{2}s^{2}}{4\pi ^{2}k^{2}}\right) }. \end{aligned}$$

(1.2)

This internal model contains the internal models of all harmonics with period T, including the step signal. However, it is interesting to note the simple structure of RC with an infinite number of harmonics. This observation validates the Chinese proverb that states “things will develop in the opposite direction when they become extreme.”

Fig. 1.3

T-periodic signal tracking

Figure 1.3 shows that the transfer function from the desired signal to the tracking error can be written as follows:

$$\begin{aligned} e\left( s\right)&=\frac{1}{1+\frac{1}{1-e^{-sT}}G\left( s\right) }y_{\text {d}}\left( s\right) \\&=\frac{1}{1-e^{-sT}+G\left( s\right) }\left( \left( 1-e^{-sT}\right) \frac{1}{1-e^{-sT}}\right) \\&=\frac{1}{1-e^{-sT}+G\left( s\right) }. \end{aligned}$$

Therefore, it is sufficient to merely verify whether the roots of the equation $1-e^{-sT}+G\left( s\right) =0$ are all in the left s-plane. Consequently, the tracking problem is converted into a stability problem of the closed-loop system.

Based on the RC presented in Table 1.1, the Laplace transformation of the controller is expressed as follows:

$$ \mathbf {u}\left( s\right) =\frac{1}{1-e^{-sT}}\mathbf {L}\left( \mathbf {y}_{k-1}\left( s\right) \text { }{\small -}\text { }\mathbf {y} _{\text {d}}\left( s\right) \right) , $$

where the internal model of T-periodic signals is also incorporated. A controller that includes the internal model $I_{M,\inf }$ in (1.2) is called an RC, and a system that employs this controller is called an RC system [2].

1.2 Brief Overview of Repetitive Control for Linear System

RC is an internal model-based control method where the infinite-dimensional internal model $I_{M,\inf }$ gives rise to an infinite number of poles on the imaginary axis. [2] proved that for a class of general linear plants, the exponential stability of RC systems can be achieved only when the plant is proper, but not strictly proper. Moreover, the system may be destabilized by the internal model $I_{M,\inf }$. A linear RC system is a neutral-type system in a critical case [13, 14]. Consider the following simple RC system:

$$\begin{aligned} \dot{x}\left( t\right)&=-x\left( t\right) +u\left( t\right) \\ u\left( t\right)&=u\left( t-T\right) -x\left( t\right) , \end{aligned}$$

where $x\left( t\right) ,u\left( t\right) \in \mathbb {R}$. The RC system expressed above can also be written as a neutral-type system in a critical case as follows:

$$ \dot{x}\left( t\right) -\dot{x}\left( t-T\right) =-2x\left( t\right) +x\left( t-T\right) . $$

The above system is a neutral-type system in a critical case [13, 14]. Additional information is presented in Chap. 3.

To enhance stability, a suitable filter is introduced, as shown in Fig. 1.4, forming a filtered repetitive controller (FRC, or filtered repetitive control, which is also designated as FRC) where the loop gain is reduced at high frequencies.¹ Stable results can only be achieved by compromising on high-frequency performance. However, using an appropriate design, FRC can often achieve an acceptable trade-off between tracking performance and stability. This trade-off broadens the practical applications of RC. The plug-in RC system shown in Fig. 1.4 is a widely used structure. The objective of this structure is to design and optimize the filter $Q\left( s\right)$ and compensator $B\left( s\right)$.

Fig. 1.4

Plug-in RC system diagram

Given the developments over the past 30 years, it is evident that significant research efforts have been devoted toward developing theories and applications regarding RC for linear systems. Additional information on RC for linear systems has been included as part of [6, 10, 15,16,17] and the references therein. Current research mainly focuses on robust RC [18,19,20,21], spatial-based RC [22], or a combination of both [24]. Robust RC mainly includes two aspects: robustness against uncertain parameters of the considered systems [18,19,20] and robustness against uncertain or time-varying periods [21, 23,24,25]. Researchers are currently attempting to design better RCs to satisfy the increasing practical requirements.

1.3 Repetitive Control for Nonlinear System

For nonlinear systems, the concept of FRC is not difficult to follow because the relevant theories have been derived in the frequency domain; these can be applied with difficulty, if at all, only to nonlinear systems. Currently, RCs for nonlinear systems are designed using two methods, namely, the feedback linearization method and adaptive-control-like method.

1.3.1 Major Repetitive Controller Design Method

1.3.1.1 Linearization Method

One of the design methods involves transforming a nonlinear system into a linear system and then applying the existing design methods to the transformed linear system. Earlier, researchers often considered the following nonlinear system:

$$\begin{aligned} \dot{\mathbf {x}}\left( t\right)&=\mathbf {Ax}\left( t\right) +\mathbf {Bu}\left( t\right) +\varvec{\phi }\left( t,\mathbf {x}\right) \nonumber \\ \mathbf {y}\left( t\right)&=\mathbf {C}^{\text {T}}\mathbf {x}\left( t\right) +\mathbf {Du}\left( t\right) . \end{aligned}$$

(1.3)

This method is related to the early stages of research on nonlinear systems. RC design is often restricted on the nonlinear term $\varvec{\phi }\left( t,\mathbf {x}\right)$, including Lipschitz conditions [26] or sector conditions [27, 28]. Along with the development of feedback linearization and backstepping, RC design for nonlinear systems was further developed [29,30,31,32,33]. Using these new techniques, some nonlinear systems can be transformed into (1.3) with some restrictions, while some existing design methods can also be used directly.

Differential geometric techniques are combined with the IMP, resulting in a nonlinear RC strategy. A formulation is presented for the case of input–state linearizable and input–output linearizable systems in continuous time [29]. Using the input–output linearized method and the approximate input–output linearized method, the applicability of the finite-dimensional RC to nonlinear tracking control problems is investigated for three different classes of nonlinear systems: (1) systems with a well-defined relative degree, (2) systems without a well-defined relative degree, and (3) linear plants with small actuator nonlinearity [30]. Using feedback linearization and output redefinition, RC was developed to achieve precise periodic signal tracking control of single-input single-output nonlinear nonminimum-phase systems [31, 32]. Using backstepping, RC design and analyses were developed for backstepping-controlled nonlinear systems [33]. Backstepping control employs both feedback and feedforward actions to render linearized I/O plants; therefore, the outer loop of the RC design can be based on the compensated linear system.

Using these new techniques, some nonlinear systems can be more easily transformed into LTI systems subject to nonlinear terms. Based on these techniques, existing design methods can be used directly, thereby facilitating RC design. However, it is interesting to note that not all nonlinear systems can be transformed into a familiar form; the resulting nonlinear terms can still be difficult to handle, or the output can still exhibit a nonlinear relationship with respect to the new state.

1.3.1.2 Adaptive-Control-Like Method

Another design method involves converting a tracking problem of nonlinear systems into a rejection problem of nonlinear error dynamics, and then applying the existing adaptive-control-like method to the converted error dynamics. This technique includes two design methods, namely, the Lyapunov-based (LB) method [34,35,36,37,38,39,40,41,42,43,44,45,46,47] and the evaluation-function-based method [48,49,50,51,52,53,54,55]. The LB method is only applicable to RC design, but the evaluation-function-based method is applicable to both RC design and ILC design. To clarify the findings of previous works, it is assumed that $\hat{\mathbf {v}}$ is a learning variable, $\mathbf {v}$ is the desired signal, and $\tilde{\mathbf {v}}=\mathbf {v}-\hat{\mathbf {v}}$ is the learning error.

The LB method is similar to the traditional adaptive-control (AC) method, where $\mathbf {v}_{\text {d}}$ is a T-periodic signal for the former and is a constant for the latter. Therefore, the resulting controllers are called adaptive repetitive (learning) controllers [39]. To adopt the traditional AC method or LB method, the nonlinear error dynamics in the form of

$$\begin{aligned} \dot{\mathbf {e}}\left( t\right) =\mathbf {f}\left( t,\mathbf {e}\right) +\mathbf {b\left( t,\mathbf {e}\right) } \tilde{\mathbf{v}}\left( t\right) , \end{aligned}$$

(1.4)

must be derived first, where $\mathbf {e}$ is the error. Given the different desired signals, the chosen Lyapunov functions and the designed controllers are different, as observed in Table 1.2.

Table 1.2 Differences between the traditional AC method and LB method

The AC method is currently used as the leading method in designing RCs for nonlinear systems. This method was first applied to control robot manipulators [34]. Finally, an intermediate result (an assumption) was obtained [35] to establish the framework for the LB method.

Intermediate Result: The functions $\mathbf {f}$  :  $\left[ 0,\infty \right)$ $\times$ $\mathbb {R}^{n}$ $\rightarrow$ $\mathbb {R}^{n}$ and $\mathbf {b}$  :  $\left[ 0,\infty \right)$ $\times$ $\mathbb {R}^{n}$ $\rightarrow$ $\mathbb {R} ^{n\times m}$ are bounded when $\mathbf {e}\left( t\right)$ is bounded. Moreover, there exist a differentiable function $V:\left[ 0,\infty \right) \times \mathbb {R} ^{n}\rightarrow \left[ 0,\infty \right) ,$ a positive-definite matrix $\mathbf {M}\left( t\right) =\mathbf {M}^{\text {T}}\left( t\right) \in \mathbb {R} ^{n\times n}\$with $\mathbf {0}<\underline{\lambda }_{M}\mathbf {I} _{n}<\mathbf {M}\left( t\right) ,$ $\underline{\lambda }_{M}>0$, and a function $\mathbf {h}\left( t\mathbf {,e}\right) \in \mathbb {R} ^{m}$ such that

$$\begin{aligned} \dot{V}\left( t\mathbf {,e}\right) \le -\mathbf {e}^{\text {T}}\left( t\right) \mathbf {M}\left( t\right) \mathbf {e}\left( t\right) +\mathbf {h}^{\text {T}}\left( t\mathbf {,e}\right) \tilde{\mathbf {v}}\left( t\right) . \end{aligned}$$

(1.5)

Based on the intermediate result, the controller

$$\begin{aligned} \mathbf {v}\left( t\right) =\mathbf {v}\left( t-T\right) +\mathbf {h}\left( t\mathbf {,e}\right) \end{aligned}$$

can ensure that the tracking error approaches zero. The proof must employ the Lyapunov function

$$\begin{aligned} V\left( t\mathbf {,e}\right) +1 / 2 \int _{t-T}^{t} \tilde{\mathbf {v}}^{\text {T}}\left( s\right) \tilde{\mathbf {v}}\left( s\right) \text {d}s \end{aligned}$$

and Barbalat’s Lemma [36, p. 123].

A novel learning approach has been described in [37] for asymptotic state tracking in a class of nonlinear systems. Compared to the previous methods, the proposed RC method is advantageous because it is computationally simple and does not require solving any complicated equations based on full system dynamics. Hybrid control schemes were developed, which utilize an RC term to compensate for periodic dynamics and other methods to compensate for aperiodic dynamics [38]. An LB-adaptive RC was proposed for a class of nonlinearly parameterized systems [39]. Both partially and fully saturated RCs were analyzed in detail and compared. Results for a class of periodically time-varying nonlinear systems have been presented in [40]. Given that many RC schemes require the plant to be parameterizable, an RC is integrated with an adaptive robust control based on the backstepping design for a class of cascade systems without parametrization [41]. A continuous universal RC was proposed in [42] to track periodic trajectory in a class of nonlinear dynamical systems with nonparametric uncertainty and unknown state-dependent control direction matrices. An FRC was proposed to achieve a trade-off between tracking performance and stability for a class of nonlinear systems [43]. More importantly, the proposed FRC can handle small input delays, while the corresponding RC cannot. The classical PID$^{\rho -1}$ control combined with RC was used for output regulation in a class of minimum-phase, nonlinear systems with unknown output-dependent nonlinearities, unknown parameters, and known relative degrees [44]. Local results were obtained and further results have been presented in [45], extending the nonlinear systems considered in [44]. Researchers [44, 45] also worked on the spatial-based RC for nonlinear autonomous vehicles [46] and permanent magnet step motors [47].

The evaluation-function-based method can be used to design both ILCs [5] and RCs. The evaluation function is often formulated as follows [48,49,50,51,52,53,54,55]:

$$\begin{aligned} E_{k}=\int _{0}^{T}\tilde{\mathbf {v}}_{k}^{\text {T}}\left( \theta \right) \tilde{\mathbf {v}}_{k}\left( \theta \right) \text {d}\theta , \end{aligned}$$

where $\tilde{\mathbf {v}}_{k}\left( \theta \right) \triangleq \tilde{\mathbf {v}}\left( kT+\theta \right) ,$ $\theta \in \left[ 0,T\right] ,$ T is the period, and $k=0,1,2,\ldots$ is the iteration number. The objective is often to design ILCs for the resetting condition and RCs for the alignment condition, resulting in the following relationship:

$$\begin{aligned} \Delta E_{k}&=E_{k}-E_{k+1}\\&\le \alpha \left( \left\| \mathbf {e}_{k}\left( 0\right) \right\| ^{2}-\left\| \mathbf {e}_{k}\left( T\right) \right\| ^{2}\right) -\beta \int _{0}^{T}\left\| \mathbf {e}_{k}\left( \theta \right) \right\| ^{2}\text {d}\theta , \end{aligned}$$

where $\mathbf {e}_{k}\left( \theta \right) \triangleq \mathbf {e}\left( kT+\theta \right) ,$ $\theta \in \left[ 0,T\right] ,$ $\alpha ,\beta >0.$ Under the resetting condition, $\left\| \mathbf {e}_{k}\left( 0\right) \right\| =0.$ Then,

$$ \Delta E_{k}\le -\alpha \left\| \mathbf {e}_{k}\left( T\right) \right\| ^{2}-\beta \int _{0}^{T}\left\| \mathbf {e}_{k}\left( \theta \right) \right\| ^{2}\text {d}\theta . $$

In this case, the following inequality can be obtained:

$$ \beta \underset{i\rightarrow \infty }{\lim }\overset{i}{\underset{k=0}{ {\displaystyle \sum } }}\int _{0}^{T}\left\| \mathbf {e}_{k}\left( \theta \right) \right\| ^{2}\text {d}\theta \le E_{0}. $$

On the other hand, under the alignment condition, $\left\| \mathbf {e} _{k+1}\left( 0\right) \right\| =\left\| \mathbf {e}_{k}\left( T\right) \right\| .$ In this case, the following inequality can be obtained:

$$\begin{aligned} \overset{i}{\underset{k=0}{ {\displaystyle \sum } }}\Delta E_{k}&\le \alpha \overset{i}{\underset{k=0}{ {\displaystyle \sum } }}\left( \left\| \mathbf {e}_{k}\left( 0\right) \right\| ^{2}-\left\| \mathbf {e}_{k}\left( T\right) \right\| ^{2}\right) -\beta \overset{i}{\underset{k=0}{ {\displaystyle \sum } }}\int _{0}^{T}\left\| \mathbf {e}_{k}\left( \theta \right) \right\| ^{2}\text {d}\theta \\&\le \alpha \left\| \mathbf {e}_{0}\left( 0\right) \right\| ^{2} -\beta \overset{i}{\underset{k=0}{ {\displaystyle \sum } }}\int _{0}^{T}\left\| \mathbf {e}_{k}\left( \theta \right) \right\| ^{2}\text {d}\theta . \end{aligned}$$

Then,

$$ \beta \underset{i\rightarrow \infty }{\lim }\overset{i}{\underset{k=0}{ {\displaystyle \sum } }}\int _{0}^{T}\left\| \mathbf {e}_{k}\left( \theta \right) \right\| ^{2}d\theta \le E_{0}+\alpha \left\| \mathbf {e}_{0}\left( 0\right) \right\| . $$

Finally, using Barbalat’s Lemma, the tracking error $\int _{0}^{T}\left\| \mathbf {e}_{k}\left( \theta \right) \right\| ^{2}$d$\theta$ can be proved to approach zero as $k\rightarrow \infty$.

In the early time, researchers mainly considered the resetting condition based on the concept described above. The alignment condition was analyzed in [48]. This work significantly stimulated the development of both ILCs and RCs. In [51], the backstepping technique was combined with the learning control mechanism to develop a constructive control strategy to cope with nonlinear systems subject to both structured periodic and unstructured aperiodic uncertainties. RC schemes using a proportional–derivative (PD) feedback structure was proposed in [52], where an iterative term was added to cope with unknown parameters and disturbances. The proposed adaptive ILC of robot manipulators was further improved in [54]. Fully saturated adaptive RC for trajectory tracking of uncertain robotic manipulators has been presented in [55].

The adaptive-control-like method is the leading method used to design RCs for nonlinear systems. The structures of RCs obtained for linear and nonlinear systems are similar, but the methods for obtaining these controllers are different. As discussed in Sect. 1.1.2, the tracking problem can be converted into a stability problem of the closed-loop system. Therefore, the error dynamics do not have to be obtained for LTI systems. However, for nonlinear systems, error dynamics must often be derived to convert a tracking problem into a disturbance rejection problem as in (1.4). This follows the concept of general tracking controller design, whereas the special feature of periodic signals is underexploited. Therefore, the general tracking controller design will not only restrict the application of RC but will also fail to represent the special features and importance of RC. For nonlinear nonminimum-phase systems, ideal internal dynamics are required to derive the error dynamics as in (1.4), which is difficult and computationally expensive, particularly, when the internal dynamics are subject to an unknown disturbance [56]. Therefore, this is considered the reason for which few RCs that work on these systems are reported.

1.3.1.3 Contraction Mapping Method

Other design methods, such as the contraction mapping approach, are briefly discussed. Contraction mapping was often used in ILC during the early years when ILC was proposed. The controller design does not require additional information on the plant model. This is the biggest advantage of the contraction mapping approach over other methods. However, this tool is difficult to use for designing RCs without resetting the initial condition. Some researchers also attempted to use contraction mapping to design RCs. A formalism of ILC was used in [57] to solve an RC problem that forces a system to track a prescribed periodic reference signal. The proposed method adopts the concept of contraction mapping. However, the proposed method is only applicable to discrete-time systems. Moreover, it cannot be applied to the rejection of periodic disturbances. Based on the contraction mapping approach, a conditional learning control was proposed to track periodic signals for a class of nonlinear systems with unknown dynamics [58]. Learning is based on a steady-state output so that the updating law works only when a particular condition has been satisfied. Using this mechanism, monotonic convergence of the control sequence in the iteration domain can be achieved.

1.3.2 Existing Problem in Repetitive Control

A linear RC system is a neutral-type system in a critical case [13, 14]. The characteristic equation of a neutral-type system includes an infinite sequence of roots with negative real parts approaching zero; i.e., sup$\left\{ \text {Re}\left( s\right) \left| F\left( s\right) =0\right. \right\} =0$, where $F\left( s\right)$ is the characteristic equation. This implies that a sufficiently small uncertainty may lead to sup$\left\{ \text {Re}\left( s\right) \left| F\left( s\right) =0\right. \right\} >0.$ Reference [43] shows that a nonlinear RC system loses its stability when subject to a small input delay. In practice, input delays are very common. Therefore, it is important to design an RC to deal with small input delays. In addition to input delays, the following problems must also be considered to increase the practical applications of RC.

(i) Most studies on LTI systems focus on designing digital RCs. However, current research on digital RCs for continuous-time nonlinear systems is not producing the desired results. Currently, controllers are generally realized using digital computers. Considering the insufficient robustness of RC systems, it is still unknown if digital controllers destabilized the original systems. Moreover, a minimum-phase system may be transformed into a nonminimum-phase system after discretization. Therefore, it is important to establish theories for designing digital RCs for continuous-time nonlinear systems.

(ii) Most RC designs require the period to be known as a priori. In practice, the period cannot be exactly known. Moreover, the method for designing an RC to cope with an uncertain period is very practical. For LTI systems, some researchers attempted to improve RCs to deal with uncertain periods [21]. However, research on nonlinear RC systems subject to uncertain periods is limited.

(iii) In addition to the uncertain parameters, the presence of high unmodeled dynamics is one of the main reasons for which a closed-loop system loses stability; this is easy to analyze in LTI systems using frequency-domain methods but difficult for nonlinear systems.

(iv) In addition to stability, transient response and convergence performance are key factors that determine the applicability of RC in practical applications. Therefore, the optimal RC for nonlinear systems must also be investigated.

RC is a specific tracking control. Therefore, in addition to the problems described above, the method used to design RC for nonlinear nonminimum-phase systems and underactuated nonlinear systems is challenging.

1.4 Objective and Structure of This Book

According to the authors, developments related to RC are inconsistent, while there is a significant development in terms of the theories and applications of LTI systems. On the other hand, research on RC for nonlinear systems is limited. Currently, there are two major methods for designing RCs in nonlinear systems: the linearization method and the adaptive-control-like method. Some problems exist in each method, and new design methods are expected to be developed. In addition, books on RC for nonlinear systems are scarce, and many of the existing books mainly discuss LTI systems. However, many books are available on ILC for nonlinear systems [7, 59,60,61,62,63]. This scarcity has motivated the authors to write this book; comprising ten chapters, as outlined in Fig. 1.5.

Fig. 1.5

Structure of this book

In the first four chapters, the introduction (Chap. 1) and preliminaries (Chaps. 2–4) are presented. The preliminaries consist of mathematical preliminaries (Chap. 2), a brief introduction to RC for linear systems (Chap. 3), and the robustness problem of the RC system (Chap. 4). These preliminaries present an overview of RC and the reasons for using FRC. This book focuses on discussing the different basic methods for solving RC problems in some special nonlinear systems. Commonly used methods such as the linearization method (Chap. 5) and the adaptive-control-like method (Chap. 6) will be discussed separately. These chapters discuss both the authors’ work and classical work. In addition, three new methods parallel to the two methods mentioned above are proposed. For example, the additive-state-decomposition-based method in Chaps. 7, 8 will bridge LTI systems and nonlinear systems so that linear RC methods can be used in nonlinear systems. The actuator-focused design method described in Chap. 9 is based on another viewpoint of the IMP proposed by the authors. This method eliminates the need for deriving error dynamics so that the RC problem for nonminimum-phase systems and periodic linear systems will be easier. The proposed contraction mapping method is another attempt by the authors to solve the RC problem for nonlinear systems without requiring the corresponding Lyapunov functions (Chap. 10). This book provides additional methods and tools for researchers working on the development of RC.