Bifurcation analysis and chaos control of a discrete-time prey-predator model with fear factor

1.   Introduction

Dynamic positioning system (DPS) is a system that can automatically control the position and course of a ship ^[1]. It only depends on its own propulsion system. The position and course of the ship's motion are measured by various sensors equipped by the ship itself, and the computer is used to control the ship's thrust device to generate propulsion force and torque to resist the interference force caused by the external environment. Then the ship can maintain the target's positions and heading. The DPS can fix the ship in a certain position, and can also automatically adjust the heading of the ship to the optimal desired position according to the real-time direction of wind, wave and current. Additionally, it has been widely used in marine investigation ship, drilling ship, salvage ship, mining ship, cable laying ship and so on. By far, severable important control methods have been reported for DPS (^{[2,3,4,5,6,7]}). Liang et al. ^[2] studies the state space model of DPS by using Cummins equation and convolution integral term. Based on this model, a static output feedback controller based on L_∞ is designed. Based neural fuzzy algorithm, Fang and Lee ^[3] use a self-tuning controller to control the rotation speed of the outboard thruster to achieve the goal of controlling the ship's position, heading and path tracking. In Zhang et al. ^[4], a robust neural control algorithm for DPS based on the combination of dynamic surface control, neural network and robust damping technology is proposed. This algorithm solves the strong coupling between state variables and the uncertainty of actuator gain, reduces the parameter requirements of the DP model. In Ngongi et al. ^[5], a robust fuzzy controller using optimal control technology is proposed. The controller uses T-S fuzzy model to estimate the nonlinear part of DPS, and uses the generalized eigenvalue method to solve the controller gain. To deal with the issue of the model uncertainties, Li et al. ^[6] investigated a new collaborative design for path tracking of hybrid USVs and UAVs in the presence of structural uncertainty and external disturbances. In order to control the convergence of USV-UAV to the desired path, an adaptive fuzzy control algorithm was designed by integrating DSC and backstepping technology. In Zhang et al. ^[7], by combining robust neural damping with DSC technology, the gain dependent adaptive laws for the ships have been developed to address constraints of gain uncertainty and environmental disturbances.

Due to the wide application of digital circuits, microelectronic devices and modern computers, the sampled-data system has developed rapidly. The sampled-data control system is a kind of hybrid system, which combines the characteristics of discrete system and continuous system. Therefore, it can effectively reduce the consumption of computing resources, reduce the burden of communication transmission and is easier to implement in engineering. Recently, the sampled-data control theory has become one of the research hotspots in multiagent systems ^[8,9], fuzzy systems ^[10,11,12], chaotic systems ^[13,14,15], neural networks systems ^[16,17,18] and so on (^{[19,20,21,22,23,24]}), and has wide applications in train systems ^[25], airship systems ^[26] and ship systems ^[27]. Scholars have made corresponding achievements in sampled-data control (^{[28,29,30,31,32,33]}). Qian et al. ^[29] discussed the global stabilization of a class of nonlinear systems with higher order powers based on output feedback sampling control. A linear output feedback sampled-data controller is constructed in Zhang et al. ^[30], which can semi globally asymptotically stabilize the system with high order and linear growth nonlinearity. Zou et al. ^[31] designed a sampling control protocol using the backstep-ping method to ensure the leader follower consistency of second-order multi-agent with external interference and unpredictable speed information. In Zhang et al. ^[32], the global output feedback decentralized control problem for a class of uncertain interconnected systems based on sampled-data control is studied. By constructing sampled high gain observers and controllers, the corresponding system is guaranteed to be globally asymptotically stable.

DPS is a system controller by computer. Recently, sampled-data theory has been used to solve the control issue of DPS. In Katayama ^[34], the nonlinear sampled-data control issue of DPS is discussed by using the integrator backstepping technique and a semiglobal stability controller is designed. In Katayama and Aoki ^[35], the linear trajectory tracking issue for sampled-data DPS is discussed. Additionally, a state-feedback controller and observer are designed based on the Euler approximation model. In Zheng et al. ^[36], robust sampled-data control of neutral system for DPS is discussed. the state-derivative control law is designed. By combining the delayed decomposition technique and Wirtinger based integral inequality, less conservative results are obtained. In Yang and Zheng ^[37], the fault-tolerant control for sampled-data DPS with actuator failures is discussed. Zheng et al. ^[38] discusses the nonlinear fuzzy sampled-data control issue for DPS, and a free weighting matrix is used to ensure the stability of the system. In Chen et al. ^[39], the tracking control of DPS with actuator-failure and aperiodic measurement information is studied. Additionally, aperiodic sampling data controller is designed to guarantee that the DPS has excellent tracking performance.

Although some researches have been reported on the sampled-data control DPS, there is still much room for improvement. One reason is that the term in LKF share common quadratic function, which may lead to conservative result. The other reason is that due to the neglect of some important and useful terms in LKF, the available features of the actual sampling mode are not fully captured, which may lead to conservatism to a certain extent.

Motivated by the discussion, the design and stabilization of sampled-data for LC with DPS is discussed in this paper. First, the design method for LC is given. Second, the motion model of LC with DPS is established. Third, a novel LKF is introduced to capture the sampling pattern fully. Then, the stability conditions of the system are derived by LMIs and corresponding design algorithm of the controller is given. Finally, a specific example verifies the effectiveness of the proposed method.

2.   Design of luxury cruise

To solve the current problem such as ship function, aesthetic and humanization, we should base on the related theories of product system design and integrate design art and ship engineering knowledge in a scientific, systematic way to obtain the method of ship design innovation.

Due to the highly humanization, highly esthetic, highly functional and highly technology-integrated characteristics, it is of typical significance that we take cruise ship as representatives to do research and for educational means.

Then, a 150 m five-star luxury cruise is designed as follow. Figure 1 shows the three-dimensional modelling, Figure 2 is the panoramic effect picture and Figure 3 is nightscape effect picture of the luxury cruise.

3.   Problem formulation

This article focuses on the fixed point control of ships on the sea surface, so the influence on heave, sway and pitch is ignored, and only the three degrees of freedom of heave, sway and yaw which have significant effects on ship are considered. Hence, the following mathematical model of the LC with DPS is considered.

where

where ${\mathit{\boldsymbol{\eta}}} (t) = {\left[{\begin{array}{*{20}{c}} {{x_a}(t)} & {{y_a}(t)} & {\psi (t)} \end{array}} \right]^T}$ shows the position and heading angle of the ship in the northeast coordinate system, ${\mathit{\boldsymbol{\upsilon}}} (t) = {\left[{\begin{array}{*{20}{c}} {p(t)} & {v(t)} & {r(t)} \end{array}} \right]^T}$ describes the speed and heading rotation rate of the ship in the hull coordinate system (see Figure 4). $J\left(\psi \right)$ is the rotation matrix, $u(t) = \left[{{u_1}(t), {u_2}(t), {u_3}(t)} \right]$ are the control force vectors provided by the thruster, where ${u_1}(t)$ is the forward control force, ${u_2}(t)$ is the lateral drift control force, and ${u_3}(t)$ is the yaw control torque. M is the inertial matrix, and D is damping matrix. $m$ is the ship's mass, and ${I_z}$ is the rotational inertia; ${X_u}, {Y_v}, {N_r}$ are the linear damping coefficients of the ship in three degrees of freedom, and ${X_{\dot u}}, {Y_{\dot v}}, {N_{\dot r}}$ are the additional mass generated by the ship in the surge, sway and yaw. $w(t) = \left[{{w_1}(t), {w_2}(t), {w_3}(t)} \right]$ is an unknown disturbance term that represents the environmental disturbances such as wind, waves and currents.

Assume that the environment disturbance is unknown but energy bounded, that is, $\left| {{w_i}(t)} \right| \leqslant h,$$h$ is upper bound of the disturbances.

Assume that yaw angle $\psi (t)$ is small enough, which means that $\cos \left({\psi (t)} \right) \approx 1, \sin \left({\psi (t)} \right) \approx 0$, then

Define

Substitute (3) into (1) that

where

Assume that the state variable of the DPS is sampled by the time $0 = t_0<t_1<\cdots<t_k<\cdots<\lim\limits_{k \rightarrow \infty} t_k = +\infty$. Assume the sampling period is

where $d$ is the upper bound of the sampling period. The sampled-data controller is designed

where K is controller gain matrix.

Substitute (5) into (4), it can be obtained

The paper's aim is to propose a sampled-data controller to guarantee the stability of the system. To derive our main theorems, the following lemma is introduced.

Lemma 1(^[40]): For scalars ${\tau _2} > {\tau _1}$, any constant matrix ${\mathit{\boldsymbol{Z}}} \in {R^n}, {\kern 1pt} {\mathit{\boldsymbol{Z}}} > 0$, vector function $w:\left[{{\tau _1}, {\tau _2}} \right] \in {R^n}$, the following inequalities are established

4.   Main results

In the section, the stability conditions for system (6) are proposed. Consequently, a sampled-data controller is developed. First, the notations are defined as follow:

Theorem 1: The system (6) with $w(t) = 0$ is asymptotically stable, if there exist symmetric positive matrices ${\mathit{\boldsymbol{P}}} > 0, \; {\mathit{\boldsymbol{Q}}} > 0, \; {\mathit{\boldsymbol{Z}}} > 0, {\mathit{\boldsymbol{R}}} > 0, {\mathit{\boldsymbol{U}}} > 0,$ ${{\mathit{\boldsymbol{X}}}_{11}}, {{\mathit{\boldsymbol{X}}}_{13}}, {{\mathit{\boldsymbol{X}}}_{22}}, {{\mathit{\boldsymbol{X}}}_{23}}, {{\mathit{\boldsymbol{X}}}_{33}}, {\mathit{\boldsymbol{G}}}, {{\mathit{\boldsymbol{M}}}_i}, {{\mathit{\boldsymbol{N}}}_i}\; (i = 1, 2, 3, 4)$ and scales $d > 0, {\varepsilon _i}\; (i = 1, 2, 3, 4)$, such that the following LMIs hold:

where

Proof. Consider the following LKF

where

By Lemma 1, it can be verified that

From LMI (9), ${V_1}(t) \geqslant 0$ can be concluded, which means that $V(t) \geqslant 0$.

Calculate the derivative of $V(t)$, it yields that

By Lemma 1, one can obtain

which implies

Combining (14) with (16) that

For any matrixes ${\mathit{\boldsymbol{G}}},$${\mathit{\boldsymbol{M}}},$ and scalars ${\varepsilon _i}, i = 1, 2, 3, 4$, we have

From (13), (14) and (17)–(19), we obtain that

where

Following the Schur complement, (21) implies $\hat \Psi _4^{ij} < 0$. which implies that $\dot{V}(t)<-\sigma\|x(t)\|^2$ when $x(t) \neq 0, \sigma>0$. Thus, the system (6) is asymptotically stable. This completed the proof.

Remark 1: Note that some novel terms such as $d(t)\int_{ - d(t)}^0 {\int_{t + \theta }^t {\dot x{{(s)}^T}Z\dot x(s)dsd\theta } }$ are added in the constructed LKF, which means that the characteristics about sampling patterns are fully captured.

Remark 2: If ${\mathit{\boldsymbol{Q}}} = {{\mathit{\boldsymbol{R}}}_{12}} = {{\mathit{\boldsymbol{R}}}_{22}} = {\mathit{\boldsymbol{Z}}} = 0$ and ${{\mathit{\boldsymbol{X}}}_{13}} = {{\mathit{\boldsymbol{X}}}_{23}} = {{\mathit{\boldsymbol{X}}}_{33}} = {{\mathit{\boldsymbol{U}}}_{13}} = {{\mathit{\boldsymbol{U}}}_{12}} = {{\mathit{\boldsymbol{U}}}_{22}} = 0$, the LKF is simplified to that in ^[38] and ^[39]. Therefore, the proposed LKF are general and have wider application scopes. Moreover, according to Newton-Leibniz formulas, free matrices $M$ and $N$ have been considered to handle the derivation of LKF to avoid using the bounding techniques. Therefore, the conservatism is reduced.

Remark 3: Compared with ^[38] and ^[39], the more relaxed constraint conditions are introduced by adding X-dependent term in ${V_1}(t)$, which means that the involved symmetric matrices are not required to be positive definite. Besides, the term $d(t)\int_{{t_k}}^t {{{\dot x}^T}(s){\mathit{\boldsymbol{Q}}}\dot x(s)ds}$ is added in the LKF to obtain more relaxed condition and longer sampling period. Hence, the conservativeness can be further reduced.

Remark 4: The number of decision variables in the paper is 12n² + 13n, which depends on the system order n. When n increase, it will consume longer central processing unit time when solving stability conditions. Furthermore, it is worth noting that the number of decision variables in ^[41] and ^[42] is 70n² + 12n and 142n² + 18n, which is bigger than that in the paper. This means that the solution of the condition in ^[41] and ^[42] will waste more time to obtain less conservative results. It illustrates that the proposed methodology has lower computational complexity than ^[41] and ^[42].

Furthermore, the controller design algorithm is introduced via the following theorem.

Theorem 2: The system (6) with $w(t) = 0$ is asymptotically stable, if there exist symmetric positive matrices $\tilde P > 0, \; \tilde Q > 0, \; \tilde Z > 0, \tilde R > 0, \tilde U > 0,$ ${\tilde X_{11}}, {\tilde X_{13}}, {\tilde X_{22}}, {\tilde X_{23}}, {\tilde X_{33}}, \tilde G, {\tilde M_i}, {\tilde N_i}\; (i = 1, 2, 3, 4)$ and scales $d > 0, {\varepsilon _i}\; (i = 1, 2, 3, 4)$, such that:

where

The controller is derived that

Proof: Denoting

Pre- and post-multiplying (9)–(11) by ${\Upsilon _1}, {\Upsilon _2}, {\Upsilon _3}$ and $\Upsilon _1^T, \Upsilon _2^T, \Upsilon _3^T$ respectively, (22)–(24) are obtained. This completed the proof.

5.   Numerical examples

In this section, to vertify the performance of the given method, a simulation example for a LC is given ^[43]. The LC model parameters are considered in Table 1 as follow.

And the M and D in model (1) are

Noted that ${\mathit{\boldsymbol{A}}} = \left[{\begin{array}{*{20}{c}} 0 & {\mathit{\boldsymbol{I}}} \\ 0 & { - {{\mathit{\boldsymbol{M}}}^{ - 1}}{\mathit{\boldsymbol{D}}}} \end{array}} \right], \quad {\mathit{\boldsymbol{B}}} = \left[{\begin{array}{*{20}{c}} 0 \\ {{{\mathit{\boldsymbol{M}}}^{ - 1}}} \end{array}} \right], \quad {{\mathit{\boldsymbol{B}}}_w} = \left[{\begin{array}{*{20}{c}} 0 \\ {{{\mathit{\boldsymbol{M}}}^{ - 1}}} \end{array}} \right],$ then

First, from Table 2, the maximum sampling period obtained by theorem 2 is $d = 0.652$, which improves ^[37,38,39] above 160.8,146.97 and 22.56% respectively. One can find that the controller in this paper can obtain longer sampling period than that in ^[37,38,39].

The initial value of system state is assumed as ${x_s}(t) = \left[{\begin{array}{*{20}{c}} {10} & {10} & {0.1} & 0 & 0 & 0 \end{array}} \right]$, and $d = 3.5513, {\varepsilon _1} = {\varepsilon _2} = {\varepsilon _3} = {\varepsilon _4} = 1,$ The desired value of system state is ${x_r}(t) = \left[{\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 \end{array}} \right].$ Then, the gain can be computed that

The external environment disturbance is considered that

Compared the results with the reference ^[39] which used the normal LKF to deal with the sampled-data issue of the DPS, the simulation results are shown in Figures 5–10 respectively. Figures 5–8 are the responses of the ship's positions and yaw angle. It can be seen that the stable time required to achieve the expected positions and yaw angle using normal LKF in ^[39] is 20 seconds, while the stable time based on improved LKF in the paper is 12 seconds, which improves the ^[39] about 40%.

Figures 8–10 are the responses of the velocities. It is shown that the stable time required to achieve the expected velocities using normal LKF in ^[39] is 22 seconds, while the stable time based on improved LKF in the paper is 15 seconds, which improves the ^[39] about 31.81%.

From Figures 5–10, we can see that in the presence of external environmental disturbances, the sampled-data controller designed in the paper can track the desired goal of the ship more smoothly and quickly. Besides, the fluctuations of the system are fewer. Therefore, the proposed sampled-data controller is more effective and robustness.

6.   Conclusions

This article focused on the sampled-data issue for the LC with DPS. By constructing an improved LKF, the information about sampling modes is fully captured. Then, the conditions of asymptotical stability are present by means of LMI. Additionally, the sampled-data controller for LC with DPS is designed by analyzing the stabilization conditions. A numerical example is provided to illustrate that the proposed methods are effective and less conservativeness can be achieved. It is worth mentioning that the ship motion model only considers the linear problems, which may result in approximate errors in the actual object and reduce the robustness performance. In future research, the nonlinear systems and the modeling errors will be considered in the control synthesis process. In addition, other practical situations such as input saturation and incomplete state information will also be studied.

Use of AI tools declaration

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

Acknowledgments

This study was funded by Undergraduate Education and Teaching Reform Research Project of Fujian Province in 2022 (FBJG20220194); Graduate Education and Teaching Reform Research project of Jimei University in 2021 (YJG2105); National Natural Science Foundation of China (51879119); Youth Innovation Foundation of Xiamen (3502Z20206019); The Natural Science Foundation of Fujian Province (2021J01822).

Conflict of interest

The authors declare there is no conflict of interest.