Observer-based Iterative and Repetitive Learning Control for a Class of Nonlinear Systems
  IEEE/CAA Journal of Automatica Sinica  2018, Vol. 5 Issue(5): 990-998   PDF    
Observer-based Iterative and Repetitive Learning Control for a Class of Nonlinear Systems
Sheng Zhu, Xuejie Wang, Hong Liu     
Department of Information and Electrical Engineering, Zhejiang University City College, Hangzhou 310015, China
Abstract: In this paper, both output-feedback iterative learning control (ILC) and repetitive learning control (RLC) schemes are proposed for trajectory tracking of nonlinear systems with state-dependent time-varying uncertainties. An iterative learning controller, together with a state observer and a fully-saturated learning mechanism, through Lyapunov-like synthesis, is designed to deal with time-varying parametric uncertainties. The estimations for outputs, instead of system outputs themselves, are applied to form the error equation, which helps to establish convergence of the system outputs to the desired ones. This method is then extended to repetitive learning controller design. The boundedness of all the signals in the closed-loop is guaranteed and asymptotic convergence of both the state estimation error and the tracking error is established in both cases of ILC and RLC. Numerical results are presented to verify the effectiveness of the proposed methods.
Key words: Iterative learning control (ILC)     observers     repetitive learning control (RLC)     time-varying parametrization    
Ⅰ. INTRODUCTION

Iterative learning control (ILC), [1] and repetitive control(RC) [2] are two typical learning control strategies developed for systems performing tasks repetitively. ILC aims to achieve complete tracking of the system output to a desired trajectory over a pre-specified interval, through updating the control input cycle by cycle, while RC addresses the problem of the periodic reference tracking and periodic disturbance rejection. The contraction-mapping-based learning control [1] features simplicity, especially reflected in the only use of the output measurements. However, the learning gains are not easy to be determined because of the difficulty in solving norm inequalities, that may lead to obstacles to the applications of the conventional learning control method.

Early in 1990s, aiming to overcome the mentioned limitation of the contraction-mapping-based method, there have been intense researches in developing Lyapunov-like based designs of iterative learning control [3]-[5] and repetitive control [6], [7]. Recently, such Lyapunov-like approach has received further attention [8], [9], [20], [21]-[25]. In [8], [9], the learning control problems were formulated for broader classes of uncertain systems with local Lipschitz nonlinearities and time-varying parametric and norm bounded uncertainties. Note that in the mentioned works, the full state information is assumed to be available. However, in many applications, the system state may not be available for the controller design, where it is necessary to design output-based learning controllers in the framework of Lyapunov-like based learning control theory.

For linear systems, Kalman filter [10] and Luenberger observer [11] are two kinds of basic practical observers that adequately address the linear state estimation problem. Observers for nonlinear uncertain systems have recently received a great deal of attention, and there have been many designs such as adaptive observers [12], robust observers [13], sliding mode observers [14], neural observers [15], fuzzy observers [16], etc. In the published literature, works have been done for output-feedback-based learning control. In [17], a transformation to give an output feedback canonical form is taken for nonlinear uncertain systems with well-defined relative degree. But the uncertainties in the transformed dynamics should be state independent. In [18], an adaptive learning algorithm is presented for unknown constants, linking two consecutive iterations by making the initial value of the parameter estimation in the next iteration equal to the final value of the one in the current iteration. The results are extended to output feedback nonlinear systems. But the nonlinear function of the system is assumed to be concerned with the system output only.

The observers used in learning control are reported in [19] and [20]. The former addresses the finite interval learning control problem while the latter addresses the infinite interval learning control problem. The nonlinear functions in [19] are not parametrized and the observer based learning controller is designed in the framework of contraction mapping approach without the requirement of zero relative degree. The observer used in [20] is special and complicated. By virtue of the separation principle, the state estimation observer and the parameter estimation learning law are taken into account respectively. The Lyapunov-like functions are employed and two classes of nonlinearties, the global Lipschitz continuous function of state variables and the the local Lipschitz continuous function of output variables, are all considered.

As is known, repositioning is required at the beginning of each cycle in ILC. Repositioning errors will accumulate with iteration number increasing, which may lead the system to diverge finally. The variables to be learned are assumed to be repetitive. Repetitive control requires no repositioning, but the variables to be estimated need to be periodic. It is commonly seen that a repetitive signal may not be periodic. RLC has been developed recently [21]-[25], and formally formulated in [24], given as follows:

F1) every operation ends in the same finite time of duration;

F2) the desired trajectory is given a priori and is closed;

F3) the initial condition of the system at the beginning of each cycle is aligned with the final position of the preceding cycle;

F4) the time-varying variables to be learnt are iteration independent cycle;

F5) the system dynamics are invariant throughout all the cycles.

Unlike the ILC and RC, RLC can handle the finite time interval tracking without the repositioning. In the published literature, however, there are few results on observer-based RLC.

In this paper, through Lyapunov-like synthesis, the ILC problem is addressed for a class of nonlinear systems that only the system output measurements are available. Compared with the existing works, the main contributions of our paper are given as follows. Firstly, the parametric uncertainties discussed in this paper are state-dependent, while the uncertainties treated in the existing results [17]-[19] are assumed to be output-dependent. The state-dependent terms cannot be directly used in the output feedback controller design due to the lack of the state information. Secondly, a robust learning observer is given by simply using Luenberger observer design. Reference [20] pointed out that for learning control systems many conventional observers are difficult to apply. We clarify the possibility of designing observer-based learning controller by using a Luenberger observer. The estimation of output, instead of the system output itself, is applied to form the error equation, which helps to establish convergence of the system output to the desired one. Finally, the method used in output-feedback ILC design is extended to the RLC design. To the best of our knowledge, the output-feedback RLC problem is still open. In this paper, the fully saturated learning laws are developed for estimating time-varying unknowns. The boundedness of the estimations plays an important role in establishing stability and convergence results of the closed-loop system.

The rest of the paper is organized as follows. The problem formulation and preliminaries are given in Section Ⅱ. The main results of this paper are given in Section Ⅲ and Section Ⅳ, providing performance and convergence analysis of the observer based ILC and RLC, respectively. Section Ⅴ presents simulation results and gives the comparison of the ILC and RLC schemes. The final section draws the conclusion of this work.

Ⅱ. PROBLEM FORMULATION AND PRELIMINARIES

Consider a class of uncertain nonlinear systems described by

$ \begin{eqnarray} &&\dot{x}(t)=Ax(t)+B(u(t)+\Theta(t)\xi(x(t), t))\nonumber\\ &&y(t)=Cx(t) \end{eqnarray} $ (1)

where $t$ is the time. $x(t)\in \mathbb{R}^n$, $u(t)\in \mathbb{R}^m$ and $y(t)\in \mathbb{R}^m$ represent the state vector, the system output and the control input, respectively. $A\in \mathbb{R}^{n\times n}$, $B \in \mathbb{R}^{n\times m}$ and $C\in \mathbb{R}^{m\times n}$ are known constant matrices. $\Theta(t)\in \mathbb{R}^{m\times n_1}$ is the unknown continuous time-varying matrix-valued function and $\xi(x(t), t)\in \mathbb{R}^{n_1}$ is the known vector-valued function.

Remark 1: $||\Theta(t)||$ is bounded over $[0, T]$, hence let $\theta_m$ be the supremum of the $||\Theta(t)||$, but the value of $\theta_m$ is unknown.

Assume that the system operates repeatedly over a specified interval $[0, T]$. Let us denote by $k$ the repetition index, and system (1) can be rewritten as follows:

$ \begin{eqnarray} &&\dot{x}_k(t)=Ax_{k}(t)+B(u_k(t)+\Theta(t)\xi(x_k(t), t))\nonumber\\ &&y_k(t)=Cx_k(t). \end{eqnarray} $ (2)

Given a desired trajectory $y_d(t)=Cx_d(t)$ over the interval $[0, T]$, our objective is to design a learning control law $u_k(t)$, such that the output $y_k(t)$ converges to the desired output $y_d(t)$, for all $t\in [0, T]$, as $k\rightarrow \infty$. To achieve the perfect tracking, the following assumptions are made.

Assumption 1: For system (2), there exist positive matrices $P \in \mathbb{R}^{n\times n}$ and $Q \in \mathbb{R}^{n\times n}$ satisfying

$ \begin{eqnarray} PA+A^TP=-Q \end{eqnarray} $ (3)
$ B^TP=C. $ (4)

Assumption 2: Rank$(CB)=m.$

Assumption 3: The nonlinear function $\xi(x(t), t)$ satisfies global Lipschitz condition, i.e., $\forall x_1(t), x_2(t) \in \mathbb{R}^n$, $||\xi(x_1(t), t)-\xi(x_2(t), t)||\leq \gamma ||x_1(t)-x_2(t)||$, where $\gamma$ is the unknown Lipschitz constant.

Remark 2: Assumption 1 is the common strictly positive real(SPR) condition. It guarantees the asymptotic stability of the linear part of the system which helps us construct the Lyapunov-like function easily. It also indicates that if $(A, B)$ is completely controllable, $(A, C)=(A, B^TP)$ is observable because the matrix $P$ is positive, which makes that the observer can be constructed. Assumption 2 is needed to guarantee the existence of the learning gain in the controller design. The Lipschitz condition in Assumption 3 is commonly seen in observer design and ILC design for nonlinear systems.

For the learning controller design, saturation function $\mathit{\rm{sat}}(\cdot)$ which is defined in the following ensures the boundedness of the parameters estimation. For scalar $f$,

$ \begin{eqnarray} \mathit{\rm{sat}}(f)=\left\{ \begin{array}{ll} \bar{f}^1, &\mathit{\rm{if}} \ f<\bar{f}^1\\[0.5mm] f, &\mathit{\rm{if}} \ \bar{f}^1 \leq f \leq \bar{f}^2\\[0.5mm] \bar{f}^2, &\mathit{\rm{if}}\ f >\bar{f}^2 \end{array} \right. \end{eqnarray} $ (5)

where $\bar{f}^1$ and $\bar{f}^2$ are the lower and upper limits, respectively. For $m\times n_1$ matrix $F=\{f^{ij}\}_{m\times n_1}$, the function $\mathit{\rm{sat}}(F)$ is defined as $\{\mathit{\rm{sat}}(f^{ij})\}_{m\times n_1}$, where the saturation limits are set to be the same for each entry.

Lemma 1: For $m\times n_1$-dimensional matrixes $F_1$ and $F_2$, if $F_1=\mathit{\rm{sat}}(F_1)$, and the saturation limits of $\mathit{\rm{sat}}(F_1)$ and $\mathit{\rm{sat}}(F_2)$ are the same, then

$ \begin{eqnarray} {\text{tr}}((F_1-\mathit{\rm{sat}}(F_2))^T(F_2-\mathit{\rm{sat}}(F_2)))\leq 0. \end{eqnarray} $ (6)

Proof: It follows that for the matrices $F_1=(f_{1ij})_{m\times n_1}$ and $F_2=(f_{2ij})_{m\times n_1}$

$ \begin{eqnarray} &&{\text{tr}}((F_1-\mathit{\rm{sat}}(F_2))^T(F_2-\mathit{\rm{sat}}(F_2)))\nonumber\\ &&~~~~=\sum\limits_{j=1}^{n_1}\sum\limits_{i=1}^{m}(f_{1ij}-\mathit{\rm{sat}}(f_{2ij}))(f_{2ij}-\mathit{\rm{sat}}(f_{2ij})). \end{eqnarray} $ (7)

Let $\bar{f}^1$ and $\bar{f}^2$ are the lower and upper limits of the saturation function $\mathit{\rm{sat}}(\cdot)$ such that $\bar{f}^1 \leq f_{1ij}\leq \bar{f}^2$ and $\bar{f}^1 \leq f_{2ij}\leq \bar{f}^2$. If $f_{2ij}>\mathit{\rm{sat}}(f_{2ij})$, then $f_{1ij}-\mathit{\rm{sat}}(f_{2ij})=f_{1ij}-\bar{f}^2\leq 0$, if $f_{2ij}<\mathit{\rm{sat}}(f_{2ij})$, then $f_{1ij}-\mathit{\rm{sat}}(f_{2ij})=f_{1ij}-\bar{f}^1\geq 0$, and if $f_{2ij}=\mathit{\rm{sat}}(f_{2ij})$, then $f_{2ij}-\mathit{\rm{sat}}(f_{2ij})=0$. Therefore, we obtain

$ \begin{eqnarray} (f_{1ij}-\mathit{\rm{sat}}(f_{2ij}))(f_{2ij}-\mathit{\rm{sat}}(f_{2ij})) \leq 0. \end{eqnarray} $ (8)

The following property of trace is used as below

$ \begin{eqnarray} {\text{tr}}(G^Tg_2g_1^T)={\text{tr}}(G^Tg_2g_1^T)^T=g_2^TGg_1 \end{eqnarray} $ (9)

where $G\in \mathbb{R}^{m\times n_1}, g_1 \in \mathbb{R}^{n_1\times 1}$ and $g_2 \in \mathbb{R}^{m\times 1}$.

To establish stability and convergence of the repetitive learning control systems in Section Ⅳ, the following lemma is given.

Lemma 2: The sequence of nonnegative functions ${f_k(t)}$ defined on $[0, T]$ converges to zero uniformly on $[0, T]$, i.e.,

$ \begin{eqnarray}\lim\limits_{k\rightarrow \infty} f_k(t)=0, \forall t\in [0, T]\nonumber\end{eqnarray} $

if

$ \begin{eqnarray} \lim\limits_{k\rightarrow \infty}\int_0^T f_k(\tau)d\tau=0 \end{eqnarray} $ (10)

and $f_k(t)$ is equicontinuous on $[0, T]$.

The proof of lemma 2 can be found in [19].

Ⅲ. OBSERVER-BASED ILC A. State Estimation

Let $\hat{x}_k(t)$ represent the state estimation, and a robust learning observer is constructed in the following form:

$ \begin{eqnarray} \dot{\hat{x}}_k(t)&\hspace{-0.2cm}=&\hspace{-0.2cm}A\hat{x}_k(t)+Bu_k(t)+B\hat{\Theta}_k(t)\xi(\hat{x}_k(t))\nonumber\\ &&+\frac{1}{2}B\hat{\mu}_k(t)(y_k(t)-C\hat{x}_k(t)) \end{eqnarray} $ (11)

where $\hat{\Theta}_k(t)$ is the estimation of the unknown time-varying function $\Theta(t)$, and $\hat{\mu}_k(t)$ is introduced to approximate an unknown constant $\mu=(2\theta_m^2\gamma^2)/{\lambda_1}$, where $\lambda_1$ is the minimum eigenvalue of the matrix $Q$. $\hat{\Theta}_k(t)$ and $\hat{\mu}_k(t)$ are updated by the learning laws (22) and (23) respectively.

By defining the estimation error $\delta x_k(t)=x_k(t)-\hat{x}_k(t)$, it can be easily derived

$ \begin{eqnarray} ~~~~~~ \delta \dot{x}_k(t)&\hspace{-0.2cm}=&\hspace{-0.2cm}\dot{x}_k(t)-\dot{\hat{x}}_k(t) \nonumber\\ &&\hspace{-0.6cm}= A\delta x_k(t)\!+\!B\tilde{\Theta}_k\xi(\hat{x}_k)\!+\!B\Theta(t)(\xi(x_k)\!-\!\xi(\hat{x}_k))\nonumber\\ &&\hspace{-0.2cm}-\frac{1}{2}B\hat{\mu}_k(t)(y_k(t)\!-\!C\hat{x}_k(t)) \end{eqnarray} $ (12)

where $\tilde{\Theta}_k(t)=\Theta(t)-\hat{\Theta}_k(t)$. Using the following Lyapunov function candidate

$ \begin{eqnarray} W_k^1(t)=\delta x_k^T(t)P \delta x_k(t) \end{eqnarray} $ (13)

where $P$ is defined in Assumption 1, we obtain

$ \begin{eqnarray} \dot{W}_k^1(t)&\hspace{-0.2cm}=&\hspace{-0.2cm}2\delta x_k^T(t)P A\delta x_k(t) +2\delta x_k^T(t)PB\tilde{\Theta}_k\xi(\hat{x}_k)\nonumber\\ &&\hspace{-0.2cm}+2\delta x_k^T(t)PB\Theta(t)(\xi(x_k)-\xi(\hat{x}_k))\nonumber\\ &&\hspace{-0.2cm}-\delta x_k^T(t)PB\hat{\mu}_k(t)(y_k(t)-C\hat{x}_k(t)). \end{eqnarray} $ (14)

According to Assumptions 1 and 2, we have

$ \begin{eqnarray} \dot{W}_k^1(t)&\hspace{-0.2cm}\leq&\hspace{-0.2cm} -\lambda_1||\delta x_k||^2+2\delta x_k^T(t)PB\tilde{\Theta}_k\xi(\hat{x}_k)\nonumber\\ &&\hspace{-0.2cm}+2||y_k\!\!-\!\!C\hat{x}_k||\theta_m\gamma||\delta x_k||\!-\!\hat{\mu}_k||y_k\!\!-\!\!C\hat{x}_k||^2. \end{eqnarray} $ (15)

Using the inequality

$ \begin{eqnarray} 2||y_k&\hspace{-0.2cm}-&\hspace{-0.2cm}C\hat{x}_k||\theta_m\gamma||\delta x_k||\leq \frac{\lambda_1}{2}||\delta x_k||^2\nonumber\\ &&\hspace{-0.2cm}+\frac{2\theta_m^2\gamma^2}{\lambda_1}||y_k-C\hat{x}_k||^2 \end{eqnarray} $ (16)

it can be verified that

$ \begin{eqnarray} \dot{W}_k^1(t)&\hspace{-0.2cm}\leq&\hspace{-0.2cm}-\frac{\lambda_1}{2}||\delta x_k||^2+2\delta x_k^TPB\tilde{\Theta}_k\xi(\hat{x}_k)\nonumber\\ &&\hspace{-0.2cm}+\tilde{\mu}_k(t)||y_k-C\hat{x}_k||^2 \end{eqnarray} $ (17)

where $\tilde{\mu}_k(t)=\mu-\hat{\mu}_k(t)$. In order to counteract the second and third terms to the right of (17), full saturated parameter updating laws are given in the following part.

B. Full Saturated Iterative Learning Control

Let us define the novel error function $e_k(t)=\hat{y}_k(t)-y_d(t)$, where $\hat{y}_k(t)=C\hat{x}_k(t)$. The objective of the learning controller design is to make $y_k(t)\rightarrow y_d(t)$, $t\in [0, T]$, as $k\rightarrow \infty$. Now that the proposed robust learning observer(11) ensures $\hat{x}_k(t)\rightarrow x_k(t)$, $t\in [0, T]$, as $k\rightarrow \infty$, and the observer-based iterative learning controller will be designed to make $\hat{y}_k(t)\rightarrow y_d(t)$, $t\in [0, T]$, as $k\rightarrow \infty$ firstly, then the objective can also be achieved. Using (11), the derivative of $e_k(t)$ is

$ \begin{eqnarray} \hspace{-0.5cm}\dot{e}_k(t)&\hspace{-0.2cm}=&\hspace{-0.2cm}C\dot{\hat{x}}_k(t)-\dot{y}_d(t)= CA\hat{x}_k(t)+CB\hat{\Theta}_k(t)\xi(\hat{x}_k(t))\nonumber\\ &&\hspace{-0.2cm}+\frac{1}{2}CB\hat{\mu}_k(t)(y_k(t)\!-\!C\hat{x}_k(t))\!-\!\dot{y}_d(t)\!+\!CBu_k(t) \end{eqnarray} $ (18)

from which we can easily obtain the control law

$ \begin{eqnarray} \hspace{-0.5cm}u_k&\hspace{-0.2cm}=&\hspace{-0.2cm}-\hat{\Theta}_k(t)\xi(\hat{x}_k(t))+(CB)^{-1}(\dot{y}_d(t)-CA\hat{x}_k(t)\nonumber\\ &&\hspace{-0.2cm}-L_1e_k(t))-\frac{1}{2}\hat{\mu}_k(t)(y_k(t)-C\hat{x}_k(t)) \end{eqnarray} $ (19)

where $L_1\in \mathbb{R}^{m\times m}$ is a given positive matrix.

Using the following Lypunov function candidate

$ \begin{eqnarray} W_k^2(t)=\frac{1}{2}||e_k(t)||^2 \end{eqnarray} $ (20)

and considering the control input (19) and the error dynamic (18), we obtain

$ \begin{eqnarray} \dot{W}_k^2(t) = e_k^T(t)\dot{e}_k(t)=-e_k^T(t)L_1\dot{e}_k(t)=-\lambda_2 ||e_k||^2 \end{eqnarray} $ (21)

where $\lambda_2$ is the minimum eigenvalue of the matrix $L_1$.

It should be noted that the error dynamics (18) is independent of nonlinear uncertainties in system (1), and all variables in (18) are available for controller design. This is the reason why we use $\hat{y}_k(t)$ instead of $y_k(t)$ in error definition. The controller (19) and the observer (11) work concurrently, where $\hat{\Theta}_k(t)$ and $\hat{\mu}_k(t)$ are updated by the following learning laws

$ \begin{eqnarray} \left\{\begin{array}{l} \hat{\Theta}^{*}_k(t)=\hat{\Theta}_{k-1}(t)+2L_2(y_k(t)-C\hat{x}_k(t))\xi^T(\hat{x}_k)\\ \hat{\Theta}_k(t)=\mathit{\rm{sat}}(\hat{\Theta}^*_k(t))\\ \hat{\Theta}_{-1}(t)=\{0\}_{m\times n_1}, t\in [0, T] \end{array} \right. \end{eqnarray} $ (22)

and

$ \begin{eqnarray} \left\{\begin{array}{l} \hat{\mu}^*_k(t)=\hat{\mu}_{k-1}(t)+l_3||y_k-C\hat{x}_k||^2\\ \hat{\mu}_k(t)=\mathit{\rm{sat}}(\hat{\mu}^*_k(t))\\ \hat{\mu}_{-1}(t)=0, \forall t\in [0, T] \end{array} \right. \end{eqnarray} $ (23)

where $L_2=L_2^T \in \mathbb{R}^{m\times m}$ is a given positive matrix and $l_3>0$ is a constant. Let us define $\Theta(t)=\{\theta^{i, j}(t)\}_{m\times n_1}$. Since $\Theta(t)$ is bounded, we assume $\bar{\theta}^1 < \theta^{i, j}(t)<\bar{\theta}^2$, where $\bar{\theta}^1$ and $\bar{\theta}^2$ are the saturation limits of the matrix-valued function $\mathit{\rm{sat}}(\hat{\Theta}_k(t))$. The saturation limits of the scalar function $\mathit{\rm{sat}}(\hat{\mu}_k(t))$ are $\bar{\mu}^1$ and $\bar{\mu}^2$, and we assume $\bar{\mu}^1 <\mu< \bar{\mu}^2$.

Assumption 4: At the beginning of each cycle, $\hat{x}_k(0)$ $=x_k(0)=x_d(0)$.

Remark 3: Assumption 4 is about the initial states resetting condition. This part focuses on the design of observers. An extension of initial states condition is given in Section Ⅳ. See Assumptions 5 and 6.

C. Convergence and Boundedness

Theorem 1: For system(1) satisfying Assumptions 1-4, let controller (19) together with full saturated learning laws (22) and (23), where $\hat{x}_k$ is given by observer (11), be applied. Then,

1) all signals in the closed-loop are bounded on $[0, T]$, and

2) $ \lim_{k\rightarrow \infty}||\delta x_k||^2=0 $, $\lim_{k\rightarrow \infty}||e_k||^2=0$, for $t\in[0, T]$.

Proof: Let us consider the following Lyapunov-like function

$ \begin{eqnarray} W_k(t)&\hspace{-0.2cm}=&\hspace{-0.2cm}W_k^1(t)+W_k^2(t)+\frac{1}{2l_3}\int_0^t \tilde{\mu}_k^2(\tau) d\tau \nonumber\\ &&\hspace{-0.2cm}+\frac{1}{2}\int_0^t {\text{tr}}[\tilde{\Theta}^T_k(\tau) L_2^{-1}\tilde{\Theta}_k(\tau)] d\tau \end{eqnarray} $ (24)

where $W_k^1(t)$ and $W_k^2(t)$ are given by (13) and (20), respectively.

For $k=$1, 2, $\ldots$, and $t\in [0, T]$, the difference of (24) is

$ \begin{eqnarray} \Delta W_k(t)&\hspace{-0.2cm}=&\hspace{-0.2cm}W_k(t)-W_{k-1}(t)\nonumber\\ &&\hspace{-0.6cm}=W_k^1(t)+W_k^2(t)\nonumber\\ &&\hspace{-0.2cm}+\frac{1}{2}\int_0^t \{{\text{tr}}[\tilde{\Theta}^T_k(\tau) L_2^{-1}\tilde{\Theta}_k(\tau)]\nonumber\\ &&\hspace{-0.2cm}-{\text{tr}}[\tilde{\Theta}^T_{k-1}(\tau)L_2^{-1}\tilde{\Theta}_{k-1}(\tau)]\} d\tau\nonumber\\ &&\hspace{-0.2cm}+\frac{1}{2l_3}\int_0^t [\tilde{\mu}_k^2(\tau)- \tilde{\mu}_{k-1}^2(\tau)] d\tau-W_{k-1}^1(t)\nonumber\\ &&\hspace{-0.2cm}-W_{k-1}^2(t). \end{eqnarray} $ (25)

Assumption 4 implies $W_k^1(0)=0$ and $W_k^2(0)=0$. In view of (17) and (21), we obtain

$ \begin{eqnarray} W_k^1(t)&\hspace{-0.2cm}=&\hspace{-0.2cm}W_k^1(0)+\int_0^t\dot{W}_k^1(\tau)d\tau \leq\int_0^t(-\frac{\lambda_1}{2}||\delta x_k||^2\nonumber\\ &&\hspace{-0.2cm}+2\delta x_k^TPB\tilde{\Theta}_k\xi(\hat{x}_k)+\tilde{\mu}_k||y_k-C\hat{x}_k||^2) d\tau \end{eqnarray} $ (26)

and

$ \begin{eqnarray} W_k^2(t)=W_k^2(0)+\int_0^t\dot{W}_k^2(\tau)d\tau =-\lambda_2\int_0^t ||e_k||^2 d\tau. \end{eqnarray} $ (27)

Using the equalities

$ \begin{eqnarray} \frac{1}{2}[{\text{tr}}(\tilde{\Theta}^T_kL_2^{-1} \tilde{\Theta}_k)-{\text{tr}}(\tilde{\Theta}^T_{k-1}L_2^{-1} \tilde{\Theta}_{k-1})]\nonumber\\ =-{\text{tr}}[(\hat{\Theta}_k-\hat{\Theta}_{k-1})^TL_2^{-1} \tilde{\Theta}_k]~~~~~~~~~~~~~ \nonumber\\ -\frac{1}{2}{\text{tr}}[(\hat{\Theta}_k-\hat{\Theta}_{k-1})^T L_2^{-1}(\hat{\Theta}_k-\hat{\Theta}_{k-1})] \end{eqnarray} $ (28)

and

$ \begin{eqnarray} \frac{1}{2l_3}\int_0^t \tilde{\mu}_k^2-\tilde{\mu}_{k-1}^2 d\tau&\hspace{-0.2cm}=&\hspace{-0.2cm}-\frac{1}{l_3}\int_0^t \tilde{\mu}_k(\hat{\mu}_k-\hat{\mu}_{k-1})d\tau\nonumber\\ &&\hspace{-0.2cm}-\frac{1}{2l_3}\int_0^t(\hat{\mu}_k-\hat{\mu}_{k-1})^2d\tau \end{eqnarray} $ (29)

and substituting (26), (27) into (25), it can be verified that

$ \begin{eqnarray} \Delta W_k(t) \leq &\hspace{-0.2cm}-&\hspace{-0.2cm}\frac{\lambda_1}{2}\int_0^t ||\delta x_k||^2 d\tau-\lambda_2\int_0^t ||e_k||^2 d\tau \nonumber\\ &&\hspace{-0.6cm}+\int_0^t 2\delta x_k^TPB\tilde{\Theta}_k\xi(\hat{x}_k)d\tau\nonumber\\ &&\hspace{-0.6cm}+\int_0^t\tilde{\mu}_k||y_k-C\hat{x}_k||^2 d\tau\nonumber\\ &&\hspace{-0.6cm}-\int_0^t {\text{tr}}((\hat{\Theta}_k\!-\!\hat{\Theta}_{k-1})^TL_2^{-1} \tilde{\Theta}_k) d\tau\!-\!\frac{1}{l_1}\int_0^t \tilde{\mu}_k\nonumber\\ &&\hspace{-0.6cm}\times(\hat{\mu}_k\!-\!\hat{\mu}_{k-1}) d\tau-W_{k-1}^1(t)\!-\!W_{k-1}^2(t). \end{eqnarray} $ (30)

Applying learning laws (22) and (23), inequality (6) and Lemma 1, we obtain

$ \begin{eqnarray} &2\delta x_k^TPB\tilde{\Theta}_k\xi(\hat{x}_k)-{\text{tr}}[(\hat{\Theta}_k-\hat{\Theta}_{k-1})^T L_2^{-1} \tilde{\Theta}_k]\nonumber\\ =&\hspace{-0.6cm} {\text{tr}}[(\hat{\Theta}^*_k-\mathit{\rm{sat}}(\hat{\Theta}^*_k))^TL_2^{-1}(\Theta-\mathit{\rm{sat}}(\hat{\Theta}_k^*))]\leq 0 \end{eqnarray} $ (31)

and

$ \begin{eqnarray} &&\hspace{-0.5cm}\tilde{\mu}_k||y_k-C\hat{x}_k||^2-\frac{1}{l_3} \tilde{\mu}_k(\hat{\mu}_k-\hat{\mu}_{k-1}) \nonumber\\ &&=\frac{1}{l_3}(\mu_k-\mathit{\rm{sat}}(\hat{\mu}^*_k))(\hat{\mu}^*_k- \mathit{\rm{sat}}(\hat{\mu}^*_k))\leq 0. \end{eqnarray} $ (32)

Substituting (31) and (32) into (30) gives rise to

$ \begin{eqnarray} \Delta W_k(t) \leq &\hspace{-0.2cm}-&\hspace{-0.2cm}\frac{\lambda_1}{2}\int_0^t ||\delta x_k||^2 d\tau-\lambda_2\int_0^t ||e_k||^2 d\tau- W_{k-1}^1(t)\nonumber\\ &&\hspace{-0.6cm}-W_{k-1}^2(t)\nonumber\\ \leq &\hspace{-0.2cm}-&\hspace{-0.2cm}\frac{\lambda_1}{2}\int_0^t ||\delta x_k||^2 d\tau-\lambda_2\int_0^t ||e_k||^2 d\tau\nonumber\\ &&\hspace{-0.6cm}-\lambda_3||\delta x_{k-1}||^2\!\!-\!\!\frac{1}{2}||e_{k-1}(t)||^2\!\leq\!\! 0 \end{eqnarray} $ (33)

where $\lambda_3$ is minimum eigenvalue of matrix $P$. Obviously, $W_k(t)$ is a monotone non-increasing sequence over $[0, T]$, therefore, in order to prove the boundedness of $W_k(t), t\in [0, T]$, we need to prove the boundedness of $W_0(t)$ for all $t\in [0, T]$. From (24), when $k=0$, we have

$ \begin{eqnarray} W_0(t)&\hspace{-0.2cm}=&\hspace{-0.2cm}\delta x_0^TP \delta x_0+\frac{1}{2}||e_0||^2+\frac{1}{2}\int_0^t {\text{tr}}[\tilde{\Theta}^T_0L_2^{-1} \tilde{\Theta}_0] d\tau\nonumber\\ &&\hspace{-0.2cm}+\frac{1}{2l_3}\int_0^t \tilde{\mu}_0^2 d\tau. \end{eqnarray} $ (34)

Since $\hat{\Theta}_{-1}(t)=\{0\}_{m\times n_1}$, $\hat{\mu}_{-1}(t)=0$, $\forall t\in [0, T]$, from (22) and (23), we obtain

$ \left\{ \begin{align} & \hat{\Theta }_{0}^{*}(t)=2{{L}_{2}}({{y}_{0}}(t)-C{{{\hat{x}}}_{0}}(t)){{\xi }^{T}}({{{\hat{x}}}_{0}}) \\ & {{{\hat{\Theta }}}_{0}}(t)=\text{sat}(\hat{\Theta }_{0}^{*}(t)) \\ \end{align} \right. $ (35)
$ \left\{ \begin{align} & \hat{\mu }_{0}^{*}(t)={{l}_{3}}||{{y}_{0}}-C{{{\hat{x}}}_{0}}|{{|}^{2}} \\ & {{{\hat{\mu }}}_{0}}(t)=\text{sat}(\hat{\mu }_{0}^{*}(t)). \\ \end{align} \right. $ (36)

Taking the derivative of $W_0(t)$ and using (35), (36) and Lemma 1, we have

$ \begin{eqnarray} \dot{W}_0 \leq &\hspace{-0.2cm}-&\hspace{-0.2cm}\frac{\lambda_1}{2}||\delta x_0||^2-\lambda_2 ||e_0||^2-\frac{1}{2}{\text{tr}}(\hat{\Theta}_0^TL_2^{-1}\hat{\Theta}_0) \nonumber\\ &&\hspace{-0.6cm}+{\text{tr}}((\hat{\Theta}_0^*-\mathit{\rm{sat}}(\hat{\Theta}_0^*))^TL_2^{-1}(\Theta-\mathit{\rm{sat}}(\hat{\Theta}_0^*))) \nonumber\\ &&\hspace{-0.6cm}+\frac{1}{2} {\text{tr}}(\Theta^TL_2^{-1} \Theta) +\frac{1}{l_3}(\mu-{\text{sat}}(\hat{\mu}_0^*))(\hat{\mu}_0^*-{\text{sat}}(\hat{\mu}_0^*))\nonumber\\ &&\hspace{-0.6cm}+\frac{1}{2l_3} s^2-\frac{1}{2l_3}\hat{\mu}_0^2 \leq\frac{1}{2} {\text{tr}}(\Theta^TL_2^{-1} \Theta) +\frac{1}{2l_3} \mu^2. \end{eqnarray} $ (37)

Since $\bar{\theta}^1 < \theta^{i, j}(t)<\bar{\theta}^2$, $\bar{\mu}^1 <\mu< \bar{\mu}^2$, $\dot{W}_0$ has an upper bound. By the boundedness of $W_0(0)$ and continuity of $W_0(t)$ on $[0, T]$, $W_0(t)$ is bounded. From (33), $W_k(t)$ is uniformly bounded on $[0, T]$. Therefore, $||\delta x_k||$ and $||e_k||$ are all uniformly bounded on $[0, T]$. It is seen that the full saturated learning laws (22) and (23) ensure the boundedness of $\hat{\Theta}_k(t)$ and $\hat{\mu}_k(t)$. We can conclude that from (19) $u_k(t)$ is uniformly bounded on $[0, T]$, and from (18), $\dot{e}_k(t)$ is uniformly bounded on $[0, T]$, and from (12), $\delta \dot{x}_k(t)$ is uniformly bounded on $[0, T]$.

From (33), we obtain

$ \begin{eqnarray} W_k(t) &\hspace{-0.2cm}=&\hspace{-0.2cm} W_0(t)+ \sum\limits_{i=1}^{k}\Delta W_i(t)\nonumber\\ &&\hspace{-0.6cm}\leq W_0(t)-\frac{\lambda_1}{2}\sum\limits_{i=1}^{k}\int_0^t ||\delta x_i||^2 d\tau-\lambda_2 \sum\limits_{i=1}^{k}\int_0^t ||e_i||^2 d\tau\nonumber\\ &&\hspace{-0.2cm}-\lambda_3\sum\limits_{i=1}^{k-1}||\delta x_i||^2-\frac{1}{2}\sum\limits_{i=1}^{k-1}||e_i||^2. \end{eqnarray} $ (38)

Since $W_k(t)$ is a monotone non-increasing series with an upper bound, its limit exists such that

$ \begin{eqnarray} \lim\limits_{k\rightarrow \infty} W_k(t) \leq W_0(t)-\frac{\lambda_1}{2}\lim\limits_{k\rightarrow \infty}\sum\limits_{i=1}^{k}\int_0^t ||\delta x_i||^2 d\tau\nonumber\\ -\lambda_2 \lim\limits_{k\rightarrow \infty}\sum\limits_{i=1}^{k}\int_0^t ||e_i||^2 d\tau -\lambda_3\lim\limits_{k\rightarrow \infty}\sum\limits_{i=1}^{k-1}||\delta x_i||^2\nonumber\\ -\frac{1}{2}\lim\limits_{k\rightarrow \infty}\sum\limits_{i=1}^{k-1}||e_i||^2. \end{eqnarray} $ (39)

By the positiveness of $W_k(t)$ and the finiteness of $W_0(t)$, we have $\lim_{k\rightarrow \infty}||\delta x_k||^2=0 $, $\lim_{k\rightarrow \infty}||e_k||^2=0$, for $ t\in[0, T]$.

Ⅳ. OBSERVER-BASED RLC

In this section, we extend the observer-based ILC design into RLC design for uncertain nonlinear systems. The following properties are assumed according to the repetitive learning control formulation.

Assumption 5: The desired trajectory is given to satisfy

$ \begin{eqnarray} y_d(0)=y_d(T) \end{eqnarray} $ (40)

and $y_d(t)$ is bounded for $t\in [0, T]$.

Assumption 6: At the beginning of each cycle,

$ \begin{eqnarray} x_k(0)=x_{k-1}(T) \end{eqnarray} $ (41)
$ \begin{eqnarray} \hat{x}_k(0)=\hat{x}_{k-1}(T) \end{eqnarray} $ (42)

where $\hat{x}_k(t), t\in [0, T]$ is given by observer (11).

Remark 4: Assumptions 5 and 6 satisfy F2) and F3). The initial state estimation condition (42) is required for the observer (11). We do not need Assumption 3 which is a strict condition in practical system. No extra limits of the unknown time-varying function $\Theta(t)$ are needed in RLC, that implies $\Theta(t)$ is also repetitive over $[0, T]$ instead of being periodic in repetitive control.

Theorem 2: Considering system(1) with controller (19) and full saturated learning laws (22) and (23), where the states are given by the observer (11) over a specified time interval $[0, T]$, if Assumptions 1-3 and Assumptions 5 and 6 are satisfied,

1) all signals in the closed-loop are bounded on $[0, T]$, and

2) $ \lim_{k\rightarrow \infty}||\delta x_k||^2=0 $, $\lim_{k\rightarrow \infty}||e_k||^2=0$, for $t\in[0, T]$.

Proof: Assumptions 5 and 6 imply

$ \begin{eqnarray} &\hspace{-0.6cm}&\hspace{-0.2cm}W_k^1(0)+W_k^2(0)\nonumber\\ &&\hspace{-0.6cm}=(x_k(0)-\hat{x}_k(0))^T P (x_k(0)-\hat{x}_k(0))+\frac{1}{2}||\hat{y}_k(0)-y_d(0)||^2\nonumber\\ &&\hspace{-0.6cm}=(x_{k-1}(T)-\hat{x}_{k-1}(T))^T P (x_{k-1}(T) -\hat{x}_{k-1}(T))\nonumber\\ &&\hspace{-0.2cm}+\frac{1}{2}||C\hat{x}_{k-1}(T)-y_d(T)||^2\nonumber\\ &&\hspace{-0.6cm}=W_{k-1}^1(T)+W_{k-1}^2(T) \end{eqnarray} $ (43)

where $W_k^1(t)$ and $W_k^2(t)$ are given by (13) and (20), respectively. We choose the same Lyapunov-like function $W_k(t)$ as (24), and use the same control law $u_k(t)$ as (19), where $\hat{x}_k(t)$ is obtained by the observer (11). It follows that for $k=1, 2, \ldots, $

$ \begin{eqnarray} W_k(t)&\hspace{-0.2cm}=&\hspace{-0.2cm} W_k^1(t)+W_k^2(t)+\frac{1}{2}\int_0^t \{{\text{tr}}[\tilde{\Theta}^T_k(\tau)L_2^{-1} \tilde{\Theta}_k(\tau)\nonumber\\ &&\hspace{-0.2cm}-{\text{tr}}[\tilde{\Theta}^T_{k-1}(\tau)L_2^{-1} \tilde{\Theta}_{k-1}(\tau)]\} d\tau+\frac{1}{2l_1}\int_0^t [\tilde{\mu}_k^2(\tau)\nonumber\\ &&\hspace{-0.2cm}-\tilde{\mu}_{k-1}^2(\tau)] d\tau\!\!-\!\!W_{k-1}^1(t)\!\!-\!\!W_{k-1}^2(t)\!\!+\!\!W_{k-1}(t). \end{eqnarray} $ (44)

In view of (17) and (21), substituting (28) and (29) into (44), we obtain

$ \begin{eqnarray} W_k(t)&\hspace{-0.2cm} \leq&\hspace{-0.2cm} -\frac{\lambda_1}{2}\int_0^t ||\delta x_k||^2 d\tau-\lambda_2\int_0^t ||e_k||^2 d\tau+W_k^1(0)\nonumber\\ &&\hspace{-0.2cm} +\int_0^t 2\delta x_k^TPB\tilde{\Theta}_k\xi(\hat{x}_k)d\tau\!\! +\!\!\int_0^t\tilde{\mu}_k||y_k\!\!-\!\!C\hat{x}_k||^2 d\tau\nonumber\\ &&\hspace{-0.2cm}-\int_0^t {\text{tr}}((\hat{\Theta}_k(t)\!\!-\!\!\hat{\Theta}_{k-1}(t))^TL_2^{-1} \tilde{\Theta}_k) d\tau\!\!+\!\!W_k^2(0)\nonumber\\ &&\hspace{-0.2cm} -\frac{1}{l_1}\int_0^t \tilde{\mu}_k(\tau)(\hat{\mu}_k(\tau)-\hat{\mu}_{k-1}(\tau)) d\tau-W_{k-1}^1(t)\nonumber\\ &&\hspace{-0.2cm} -W_{k-1}^2(t)+W_{k-1}(t). \end{eqnarray} $ (45)

Applying inequalities (31) and (32), we have

$ \begin{eqnarray} W_k(t) &\hspace{-0.2cm}\leq&\hspace{-0.2cm} -\frac{\lambda_1}{2}\int_0^t ||\delta x_k||^2 d\tau-\lambda_2\int_0^t ||e_k||^2 d\tau+W_k^1(0)\nonumber\\ &&\hspace{-0.2cm}+W_k^2(0) -W_{k-1}^1(t)-W_{k-1}^2(t)+W_{k-1}(t). \end{eqnarray} $ (46)

In addition, by the definition of $W_k(t)$, we obtain

$ \begin{eqnarray} W_{k-1}(t)-W_{k-1}^1(t)-W_{k-1}^2(t)~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\ = \frac{1}{2}\int_0^t {\text{tr}}(\tilde{\Theta}^T_{k-1}L_2^{-1} \tilde{\Theta}_{k-1}) d\tau+\frac{1}{2l_3}\int_0^t \tilde{\mu}_{k-1}^2 d\tau. \end{eqnarray} $ (47)

Therefore, in view of (43), we have

$ \begin{eqnarray} W_k(t) &\hspace{-0.2cm}\leq&\hspace{-0.2cm} -\frac{\lambda_1}{2}\int_0^t ||\delta x_k||^2 d\tau-\lambda_2\int_0^t ||e_k||^2 d\tau\nonumber\\ &&\hspace{-0.2cm} +\frac{1}{2l_3}\int_0^t \tilde{\mu}_{k-1}^2 d\tau+W_{k-1}^1(T)+W_{k-1}^2(T)\nonumber\\ &&\hspace{-0.2cm}+\frac{1}{2}\int_0^t {\text{tr}}(\tilde{\Theta}^T_{k-1}L_2^{-1} \tilde{\Theta}_{k-1}) d\tau. \end{eqnarray} $ (48)

It follows that

$ \begin{eqnarray} W_k(t) &\hspace{-0.2cm}\leq&\hspace{-0.2cm} \frac{1}{2}\int_0^t {\text{tr}}(\tilde{\Theta}^T_{k-1}L_2^{-1} \tilde{\Theta}_{k-1}) d\tau+\frac{1}{2l_3}\int_0^t \tilde{\mu}_{k-1}^2 d\tau\notag\\ &&\hspace{-0.2cm} +W_{k-1}^1(T)+W_{k-1}^2(T)\notag\\ &&\hspace{-0.6cm} \leq \frac{1}{2}\int_0^T {\text{tr}}(\tilde{\Theta}^T_{k-1}L_2^{-1} \tilde{\Theta}_{k-1}) d\tau+\frac{1}{2l_3}\int_0^T \tilde{\mu}_{k-1}^2 d\tau\notag\\ &&\hspace{-0.2cm} +W_{k-1}^1(T)+W_{k-1}^2(T). % \end{array} \end{eqnarray} $ (49)

It is obvious that the right-hand side of the last inequality is actually the $W_{k-1}(T)$, which implies

$ \begin{eqnarray} W_k(t)\leq W_{k-1}(T) \end{eqnarray} $ (50)

for all $t\in [0, T]$. By setting $t=T$, we obtain

$ \begin{eqnarray} W_k(T)\leq W_{k-1}(T). \end{eqnarray} $ (51)

From above, it is clearly seen that $W_k(T)$ is monotonically decreasing. Taking the derivative of $W_0(t)$, we can obtain the same result as (37) such that $W_0(t)$ is bounded on $[0, T]$. Therefore, $W_k(T)$ is uniformly bounded. Using (50), $W_k(t)$ is uniformly bounded on $[0, T]$. Therefore, $||\delta x_k||$ and $||e_k||$ are all uniformly bounded on $[0, T]$. It is seen that the full saturated learning laws (22) and (23) ensure the boundedness of $\hat{\Theta}_k(t)$ and $\hat{\mu}_k(t)$. We can claim that from (19), $u_k(t)$ is uniformly bounded on $[0, T]$, and from (18), $\dot{e}_k(t)$ is uniformly bounded on $[0, T]$, and from (12), $\delta \dot{x}_k(t)$ is uniformly bounded on $[0, T]$.

Setting $t=T$ in (48) results in,

$ \begin{eqnarray} W_k(T) \leq &\hspace{-0.2cm}-&\hspace{-0.2cm}\frac{\lambda_1}{2}\int_0^T ||\delta x_k||^2 d\tau-\lambda_2\int_0^T ||e_k||^2 d\tau\nonumber\\ &&\hspace{-0.6cm}+\frac{1}{2}\int_0^T {\text{tr}}(\tilde{\Theta}^T_{k-1}L_2^{-1} \tilde{\Theta}_{k-1}) d\tau\nonumber\\ &&\hspace{-0.6cm} +\frac{1}{2l_3}\int_0^T \tilde{\mu}_{k-1}^2(\tau) d\tau +W_{k-1}^1(T)+W_{k-1}^2(T)\nonumber\\ \leq &\hspace{-0.2cm}-&\hspace{-0.2cm}\frac{\lambda_1}{2}\int_0^T ||\delta x_k||^2 d\tau\nonumber\\ &&\hspace{-0.6cm}-\lambda_2\int_0^T ||e_k||^2 d\tau+W_{k-1}(T). \end{eqnarray} $ (52)

Therefore,

$ \begin{eqnarray} W_k(T)-W_{k-1}(T) \leq &\hspace{-0.2cm}-&\hspace{-0.2cm}\frac{\lambda_1}{2}\int_0^T ||\delta x_k||^2 d\tau\nonumber\\[3mm] &&\hspace{-0.6cm} -\lambda_2\int_0^T ||e_k||^2 d\tau. \end{eqnarray} $ (53)

Since $W_k(T)\geq 0$ is monotonically decreasing and bounded, its limit exists such that

$ \begin{eqnarray} \lim\limits_{k\rightarrow \infty} \int_0^T ||\delta x_k||^2 d\tau=0 \end{eqnarray} $ (54)
$ \begin{eqnarray} \lim\limits_{k\rightarrow \infty}\int_0^T ||e_k||^2 d\tau=0. \end{eqnarray} $ (55)

Now, based on the above analysis and using Lemma 1, we can summarize the stability and convergence results as Theorem 2.

Ⅴ. ILLUSTRATIVE EXAMPLES

In this section, two illustrative examples are presented to show the design procedure and the performance of the proposed controller for the cases of ILC and RLC, respectively.

Example 1: Consider the following system

$ \begin{eqnarray} \left[\!\!\begin{array}{l} \dot{x}_{1k}\\\dot{x}_{2k} \end{array}\!\!\right]&\hspace{-0.2cm}=&\hspace{-0.2cm}\left[\!\!\begin{array}{ll}-1 \ \ \ \ \ 3\\ \ \ 2 \ \ -2 \end{array}\!\!\right]\left[\!\!\begin{array}{l} x_{1k}\\x_{2k} \end{array}\!\!\right]\nonumber\\[3mm]&&\hspace{-0.2cm}+ \left[\!\!\begin{array}{l} 0\\1 \end{array}\!\!\right](u_k(t) +\eta(t, x_{1k}, x_{2k})) \end{eqnarray} $ (56)
$ \begin{eqnarray} y_k(t)=[\!\begin{array}{l} 0\ \ 1 \end{array}\!]\left[\!\begin{array}{l} x_{1k}\\x_{2k} \end{array}\!\right] \end{eqnarray} $ (57)

where $t\in [0, 1]$, and $\eta(t, x_{1k}, x_{2k})\!=\![\!\!\begin{array}{l} t\ \ \mathit{\rm{sin}}(t) \end{array}\!\!]$ $\left[\begin{array}{l} \mathit{\rm{sin}}(x_{1k})\\[1mm]x_{2k} \end{array}\right]$.

Choosing $P=\left[\!\begin{array}{ll} 1 &0\\0 &1 \end{array}\!\right]$, we have

$ \begin{eqnarray} Q=-(PA+A^TP)= -\left[\!\!\begin{array}{ll}-2 &\ \ 5\\[1mm] \ \ 5 &-4 \end{array}\!\!\right] \end{eqnarray} $ (58)

and

$ \begin{eqnarray} B^TP=[\!\begin{array}{l} 0\ \ 1 \end{array}\!]\left[\!\begin{array}{ll} 1 &0\\0 &1 \end{array}\!\right]=C. \end{eqnarray} $ (59)

therefore, Assumption 1 is satisfied.

Observer (11), control law (19) and full saturated learning laws (22) and (23) are applied. The desired trajectory is given by $y_d(t)=12t^2(1.1-t)$, $t\in [0, 1]$. We choose $L_1=0.1$, $L_2={\text{diag}}\{10\}$, and $l_3=7$. Simulation results can be seen in Figs. 1-4. In Fig. 1, it can be easily seen that the system output $y_k(t)$ and the output estimation $y_k^*(t)=C\hat{x}_{k}(t)$ converge to the desired trajectory $y_d(t)$ over $[0, 1]$. The quantities on vertical axis in Fig. 2 and Fig. 3 represent $J_k^*$ $=\log_{10}(\mathit{\rm{max}}_{t\in [0, 1]}||\delta x_k||)$ and $J_k=\log_{10}(\mathit{\rm{max}}_{t\in [0, 1]}|e_k|)$, respectively. The learned control quantity $u_k(t)$ is shown in Fig. 4.

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Fig. 1 Desired trajectory, output estimation and system output in the case of ILC ($k=$ 50).
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Fig. 2 State estimation errors in the case of ILC.
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Fig. 3 Output tracking errors in the case of ILC.
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Fig. 4 Control input in the case of ILC ($k=$ 50).

Example 2: Consider the circuit [20] described by

$ \begin{eqnarray} \left[\!\!\begin{array}{l} \dot{x}_{1k}\\\dot{x}_{2k} \end{array}\!\!\right]&\hspace{-0.2cm}=&\hspace{-0.2cm}\left[\!\!\begin{array}{ll}-\frac{R_1M_2}{M_1M_2-M_3^2} &\ \ \frac{R_2M_3}{M_1M_2-M_3^2}\\ \ \ \frac{R_1M_3}{M_1M_2-M_3^2} &-\frac{R_2M_1}{M_1M_2-M_3^2} \end{array}\!\!\right]\left[\!\!\begin{array}{l} x_{1k}\\x_{2k} \end{array}\!\!\right]\nonumber\\ &&\hspace{-0.2cm} +\! \left[\!\!\begin{array}{l} \frac{M_2-M_3}{M_1M_2-M_3^2}\\ \frac{M_1-M_3}{M_1M_2-M_3^2} \end{array}\!\!\right](u_k(t)\!+\!\eta(t, x_{1k}, x_{2k})) \end{eqnarray} $ (60)
$ \begin{eqnarray} y_k(t)= [\!\begin{array}{l} 0\ \ 2 \end{array}]\!\left[\!\begin{array}{l} x_{1k}\\x_{2k} \end{array}\!\right] \end{eqnarray} $ (61)

where $R_1=1\Omega$ and $R_2=1\Omega$ are resistors, and $M_1=0.36H$, $M_2=0.5H$ are inductors, and the mutual inductors $M_3$ $=0.5H$. $x_{1k}=i_1$ and $x_{2k}=i_2$ are the loop currents. $\eta(t)$ $=x_{2k}\sin^3t+0.8\sin^2t\sin{x_{1k}}$ represents the input perturbation. The desired trajectory is given by $y_d(t)=12t^2(1-t)$, $t\in [0, 1]$. We set $L_1=0.5$, $L_2={\text{diag}}\{0.8\}$, and $l_3=1$. Simulation results can be seen in Figs. 5-8. Fig. 5 shows the ultimate tracking of the system output and the output estimation and the desired trajectory. In order to show the difference between the ILC and the RLC, we depict the state $x_{2k}(t)$ of the top 10 iterations in Fig. 6. It can be easily seen that the initial value of the $k$th cycle is equal to the final value of the $(k-1)$th cycle which satisfies the Assumption 6.

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Fig. 5 Desired trajectory, output estimation and system output in the case of RLC ($k=$ 40).
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Fig. 6 The changes of state $x_{2k}(t)$ in 10 iterations in the case of RLC.
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Fig. 7 Output tracking errors in the case of RLC.
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Fig. 8 Control input in the case of RLC ($k=$ 40).

The vertical quantities in Fig. 7 represent $J_k\, =\, \log_{10} $ $(\mathit{\rm{max}}_{t\in [0, 1]}|e_k|)$ which implies perfect tracking can be achieved after 50 iterations. The learned control quantity $u_k(t)$ is shown in Fig. 8.

Ⅵ. CONCLUSIONS

In this paper, an observer-based iterative learning controller has been presented for a class of nonlinear systems. The uncertainties treated is parameterized into two parts. One is the unknown time-varying matrix-valued parameters and the other is the Lipschitz continuous function, which is also unknown due to unmeasurable system states. The learning controller designed for trajectory tracking composes of parameter estimation and state estimation which is given by a robust learning observer. The parameter estimations are constructed by full saturated learning algorithms, by which the boundedness of the parameter estimations are guaranteed. Further, the extension to repetitive learning control is provided. The observer-based RLC avoids the initial repositioning and does not require the strict periodicity constraint in repetitive control. The global stability of the learning system and asymptotic convergence of the tracking error are established through theoretical derivations for both ILC and RLC schemes, respectively.

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