Persistence of the heteroclinic loop under periodic perturbation

1.   Introduction

Recently, fractional calculus emerged as a crucial tool for describing the dynamics of real-world problems ^[1,2,3]. Indeed, many fractional-order systems (FOSs) have been reported in the literature, focusing on stability, fault tolerant control, and sliding mode control, among other issues (see ^[4,5,6] and references therein). Iterative learning control (ILC) is an interesting approach to obtain trajectory tracking of repetitive systems operated over finite-time ^[7]. In recent years, FOSs and ILC have been merged with the goal of increasing tracking performance. In ^[8], a D$^\alpha$-type ILC scheme was designed and its convergence was addressed. In ^[9,10,11], both the P- and D-type learning schemes were adopted in FOSs with Lipschitz nonlinearities. In ^[12], fractional-order PID learning control was proposed for linear FOSs, and output convergence was analyzed using the Lebesgue-p norm. In ^[13], the ILC framework was adopted for FOSs with randomly varying trial lengths. In ^[14,15], ILC problems of multi-agent systems with fractional-order models were investigated. Despite many relevant contributions, it should be pointed out that the above mentioned works mainly address linear and Lipschitz FOSs. Moreover, a variety of control strategies have been proposed for nonlinear systems (NSs) to achieve the desired performance. In ^[16,17], the convergence analysis for locally Lipschitz NSs was addessed based on the contraction mapping approach. In ^[18], adaptive optimal control was investigated for NSs based on the policy iteration algorithm. In ^[19], zero-sum control for tidal turbine systems was studied though a reinforcement learning method.

Compared with classical Lipschitz nonlinearity, one-sided Lipschitz (OSL) nonlinearity possesses less conservatism. Therefore, in recent years, is has often been used in control systems. Moreover, in many practical problems, the OSL constant is much smaller than the Lipschitz one, which simplifies the estimation of the influence of nonlinearities. OSL systems are a wide class of NSs, which contain Lipschitz systems as particular cases. Practical examples are Chua's circuits, Lorenz systems, and electromechanical systems ^[20,21,22]. In ^[23,24,25], observer design issues for OSL NSs were investigated. In ^[26], the classical OSL was considered, and an observer was designed by introducing the quadratically inner-bounded (QIB) constraint. In recent years, observer design and control of OSL NSs has attracted considerable attention. In ^[27], full- and reduced-order observers were derived via the Riccati equation. In ^[28], exponential observer design was investigated. In ^[29], tracking control for OSL nonlinear differential inclusions was considered. In ^[30], $H_\infty$ attenuation control was considered for OSL NSs in the finite frequency domain. In ^[31], event-triggered sliding mode control was studied for OSL NSs with uncertainties. In ^[32,33], consensus control was discussed for OSL nonlinear multi-agent systems. Other meaningful results on ILC of OSL NSs have also been reported ^[34,35,36]. In particular, the QIB constraint was employed to reach perfect trajectory tracking ^[34,35]. Note that the above-mentioned results are about classical integer-order systems. To the best of the authors' knowledge, for FOSs with OSL nonlinearity, the problem of how to achieve exact trajectory tracking through appropriate ILC design has not yet been investigated, which motivates the present study.

This paper deals with the ILC of a family of Caputo FOSs, where the fractional derivative is in the interval 0 and 1. The considered nonlinearity satisfies the OSL condition, which encompasses the classical Lipschitz condition. Open- and closed-loop P-type learning control algorithms are adopted. The convergence of the tracking error is guaranteed with the generalized Gronwall inequality. The novelty of this paper is summarized in the next two points.

● Unlike the control methods in references ^{[18,19,29,30,31,32,33]}, the ILC method proposed in this paper can lead OSL nonlinear Caputo FOSs to exhibit perfect tracking capability;

● In contrast to the works of ^[34,35,36], the ILC theory is extended from integer-order OSL NSs to fractional-order OSL NSs.

This paper is divided into 5 sections. Section 2 establishes some elemental assumptions and formulates the ILC problem of fractional OSL NSs. Section 3 constructs the open- and closed-loop P-type control algorithms, and presents the corresponding convergence results. Section 4 includes a numerical example to show the suitability of the algorithms. Finally, Section 5 summarizes the conclusions.

2.   Preliminaries and problem description

Some relevant lemmas and definitions are introduced. Afterwards, the problem to be tackled is formulated.

Definition 1 ^[37]. The Riemann-Liouville integral of order $\alpha > 0$ of a function $x(t)$ is

where $\Gamma(\alpha)$ stands for the Gamma function.

Definition 2 ^[37]. The Caputo derivative of order $0 < \alpha < 1$ of a function $x(t)$ is

Lemma 1 ^[38]. Consider the differentiable vector $x(t)\in R^n$. It follows that, for any time instant $t \geq 0$, we have

where the superscript T denotes the vector (or matrix) transpose.

Lemma 2. (Generalized Gronwall Inequality) ^[9] Consider that the function $u(t)$ is continuous on the interval $t\in [0, T]$, and let $v(t-\xi)$ be nonnegative and continuous on $0\leq \xi\leq t\leq T$. Additionally, consider that the function $w(t)$ is positive continuous and nondecreasing on $t\in [0, T]$. If

then we have

To simplify the notation, in the following, we use ${\mathcal{D}^{\alpha}}$ to refer to the Caputo derivative ${_C\mathcal{D}^{\alpha}_{0, t}}$.

Let us consider the nonlinear FOS

where $\alpha\in(0, 1)$, $t \in [0, T],$ and $k = 0, 1, 2, \cdot\cdot\cdot$ is the repetition. Moreover, $x_k(t) \in{R}^{n}$, ${u_k}(t) \in {R}^m$, and ${y_k}(t) \in {R}^p$ represent the state, control, and output of (2.1), respectively; $f(x_k(t)) \in {R^n}$ stands for a continuous nonlinear function; and $A$, $B$, $C$, and $D$ are constant coefficients matrices.

Assumption 1. The nonlinear function $f(\cdot)$ is OSL, meaning that, for $\forall x(t), \hat x(t) \in {R^n}$,

where $\|\cdot\|$ denotes the Euclidean norm, $\langle \cdot, \cdot\rangle$ represents the inner product, and $\sigma\in R$ is the OSL constant.

Remark 1. Note that the above constant $\sigma$ can assume any real value, while the Lipschitz constant is positive. From ^[26], a Lipschitz function is OSL ($\sigma > 0$), but the converse may not hold.

Assumption 2. The desired trajectory $y_d{(t)}$ is possible, meaning that a control ${u_d}(t)$ exists, guaranteeing

with $x_d{(t)}$ being the desired state.

Assumption 3. The system defined by expression (2.1) meets the initial condition

where $x_d{(0)}$ represents the desired initial state.

Remark 2. Assumption 2 is a representative condition for OSL Caputo FOSs in control law design. Assumption 3 is the identical initialization condition, which has been widely used in ILC design to obtain perfect tracking ^[7].

The main objective herein is to design a control sequence $u_k{(t)}$ so that the output $y_k{(t)}$ of (1) can track the specified trajectory $y_d{(t)}$, with $t\in [0, T]$, as $k \to \infty$.

3.   The ILC design

For the nonlinear FOS (2.1), we design an open-loop P-type learning control algorithm

where the output tracking error at the $k$th iteration is defined as ${e_{k}}(t) = {y_{d}}(t) - {y_{k}}(t)$ and the learning gain matrix is $\Psi\in {{{R}}^{m\times p}}$.

Theorem 1. Let us assume that Assumptions 1–3 hold for the FOS (2.1) with algorithm (3.1). If $\Psi$ can be chosen such that

then ${y_k}(t)$ converges to ${y_d}(t)$ for $t\in[0, T]$.

Proof. Let us use $\delta{(\cdot)_k}(t) = {(\cdot)_{k+1}}(t) - {(\cdot)_k}(t)$, where $(\cdot)$ stands for the variables $x$, $u$, and $f$. It follows from (2.1) and (3.1) that

If we left-multiply (3.3) by ${(\delta {x_k}(t))^{\rm T}}$ and use Assumption 1, then we have

According to Lemma 1,

From (3.4) and (3.5), we get

where ${c_1} = 2\left\| A \right\| + 2\left|\sigma \right| + 1$ and ${c_2} = {\left\| {B\Psi } \right\|^2}$. Applying the $\alpha$-order integral on (3.6), we get

It follows from Assumption 3 that ${\left\| {\delta {x_k}(0)} \right\|^2} = 0$, and we further get

which, together with (3.7), leads to

We can see that

From (3.8) and (3.9), we have

Setting

and using Lemma 2, we get

Multiplying the above inequality by ${\rm e}^{-2\lambda t}$, and using the $\lambda$-norm ${\| \cdot\|_\lambda }$, we have

where ${\| \cdot\|_\lambda } = \mathop {\sup }_{t \in [0, T]} \{{\rm e}^{-\lambda t}\| {\cdot} \|\}$.

Therefore, we get

where

It is obvious that

It follows from (3.2), (3.10), and (3.11) that

where

As $0 \le {\rho _1} < 1$ by (3.2), we can select $\lambda$ as large as needed so that $\hat \rho_1 < 1$. Thus, we obtain

Note that ${\left\| {{e_k}} \right\|_s} \le {{\rm{e}}^{\lambda T}}{\left\| {{e_k}} \right\|_\lambda }$, with $\|\cdot\|_s = \mathop {\sup }_{t \in [0, T]} \left\| {\cdot} \right\|$ denoting the supremum norm. Therefore, $\mathop {\lim }\limits_{k \to \infty } {\left\| {{e_k}} \right\|_s} = 0$, meaning that

This ends the proof.

Now, we design a closed-loop P-type learning control algorithm, such that

where the learning gain is $\Phi\in {{{R}}^{m\times p}}$.

Theorem 2. Consider that Assumptions 1–3 hold for the FOS (2.1) with the learning algorithm (3.13). If the gain $\Phi$ can be chosen such that

then ${y_k}(t)$ converges to ${y_d}(t)$ for $t\in[0, T]$.

Proof. From (2.1) and (3.13), we get

Left multiplying (3.15) by ${(\delta {x_k}(t))^{\rm T}}$ and considering Assumption 1, we obtain

Obviously, (3.16) together with (3.5) implies

where ${c_4} = {\left\| {B\Phi } \right\|^2}$. Similarly to the procedure adopted in Theorem 1, we get

where

From expressions (2.1) and (3.13), we have

that is

where the symbol $I$ stands for the identity matrix. Since $I + D\Phi$ is nonsingular, we get

Furthermore, we derive

Substituting (3.18) into (3.19) yields

Taking $\lambda$ such that

then we have

where

As $0 \le {\rho _2} < 1$, we can choose $\lambda$ as large as needed so that $\hat \rho_2 < 1$. From expression (3.20), we can obtain

As ${\left\| {{e_k}} \right\|_s} \le {{\rm{e}}^{\lambda T}}{\left\| {{e_k}} \right\|_\lambda }$, we know that $\mathop {\lim }\limits_{k \to \infty } {\left\| {{e_k}} \right\|_s} = 0$, and it follows that

This completes the proof.

4.   An illustrative example

We illustrate the applicability of the P-type learning algorithms by means of a practical example.

Let us choose the following nonlinear FOS, which can be used to describe the motion of a moving object in Cartesian coordinates ^[39]

where ${x_k}(t) = [{x_{1k}}(t)\; \; {x_{2k}}(t)]^{\rm T}$, with $t \in [0, 1]$,

We know from ^[26] that the nonlinear function $f(\cdot)$ is globally OSL with $\sigma = 0$ in ${{R}^2}$. Let us use

and consider

i) Open-loop algorithm (3.1).

Using the gain matrix

we then have

Figures 1 and 2 depict the desired trajectories $y^{(1)}_{d}(t)$ and $y^{(2)}_{d}(t)$, and the outputs $y^{(1)}_{k}(t)$ and $y^{(2)}_{k}(t)$, respectively, at the 3rd, 5th, and 7th iterations, obtained with the learning algorithm (3.1). Figure 3 represents the errors, showing that perfect tracking is reached as the number of iterations increases.

ii) Closed-loop algorithm (3.13).

Using the gain matrix

then we have

meaning that the convergence is verified. Figures 4 and 5 illustrate that $y^{(1)}_{k}(t)$ and $y^{(2)}_{k}(t)$ follow the desired trajectories from the 6th iteration. Figure 6 shows that the error converges under algorithm (3.13).

We also verify that, at the 10th iteration, $||e^{(i)}_{k}||_s\; (i = 1, 2)$ {achieves} $[1.4\times10^{-3}$, $5.7\times10^{-3}]$ and $[0.5\times10^{-3}$, $1.4\times10^{-3}]$ when using algorithms (3.1) and (3.13), respectively. This confirms the results observed in Figures 3 and 6, meaning that algorithm (3.13) performs better than (3.1) in terms of convergence speed.

5.   Conclusions

The ILC for a class of Caputo FOSs with OSL nonlinearity was investigated. Open- and closed-loop P-type learning algorithms were designed to guarantee perfect tracking of a desired trajectory, and their convergence was verified using the generalized Gronwall inequality. An example was provided to verify the theoretical results. It should be noted that the QIB constraint was used in the framework of ILC for OSL NSs with irregular dynamics ^[34,35]. To some extent, this limits the applicability of NSs due to their need to simultaneously satisfy the OSL and QIB constraints. Therefore, the ILC for irregular OSL nonlinear FOSs needs to be further investigated by relaxing or removing the QIB constraint.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 62073114 and 11971032) and Anhui Provincial Key Research and Development Project (202304a05020060).

Conflict of interest

The authors declare there is no conflict of interest.