T-S fuzzy observer-based adaptive tracking control for biological system with stage structure

1.   Introduction

For a biological system, the biological individual development has a trend from infancy to maturity in the natural state. The corresponding structural and functional characteristics will also change significantly. When studying biological dynamic systems, researchers often assumed that there was little difference between infancy and maturity ^[1]. However, because different stages of the biological population have different viabilities, this characteristic will affect the population density to a certain extent. The biological model with stage structure is closer to reality. The prey-predator model with stage structure was studied in ^[2,3,4]. In addition, the equilibrium attraction and biological control were studied for this model in ^[4]. The dynamics of biological systems with stage structure were studied in ^[5].

From the above research, it can be seen that the exploration of biological systems with stage structure is deepening. Many studies have been done to analyze the stability of biological systems, which is also an important property in biological systems. However, considering the impact of external factors on the density of a biological population, it is necessary to explore the prey-predator model from multiple perspectives. For example, due to the threat posed by the spread of COVID-19, we need to implement a lockdown policy immediately. The sudden outbreak led many countries to implement lockdown policies ^[6,7,8,9]. To a certain extent, it limits human behavior and has a certain impact on the density of the biological population. Therefore, it is natural to study the effects of reducing human behavior on biological populations. In order to ensure the sustainable development of the biological environment under human intervention, intelligent control is needed to manage the prey-predator population.

In recent years, more and more people have begun to study adaptive control methods, and the research results are constantly emerging. By continuously monitoring the controlled object and adjusting the control parameters according to its changes, the adaptive control method can make the system run in the optimal or suboptimal state. Applying adaptive control to biological systems has important implications for managing population density. The problem of epidemic outbreak with adaptive gain was studied in ^[10], and a nonlinear robust adaptive integral sliding mode controller was proposed. On the basis of reference ^[10,11] not only studied the adaptive control of the influenza model but also studied the sliding mode control of the influenza model with uncertainty. For an active suspension system, an adaptive control method was proposed in ^[12], and a fault-tolerant controller was also designed for actuator faults. Aiming at the problems of unknown faults and actuator sticking in ^[13], a solution was proposed by combining the adaptive fault-tolerant control and boundary vibration method. Different from ^[12,13], an observer-based adaptive centralized control algorithm was proposed using adaptive backstepping control design technology. The adaptive control problem of a multiple-input multiple-output uncertain nonlinear system was studied in ^[14]. Same as in ^[14], a new adaptive fuzzy controller was proposed by using backstepping technology to study the finite time tracking problem of a nonlinear pure feedback system in ^[15]. In ^[16], an event-triggered adaptive control strategy was proposed in order to solve the tracking control problem of a strict feedback system with uncertainty in a fixed time. For uncertain nonlinear systems with time-varying disturbance, a sliding mode control strategy based on an adaptive reaching law was proposed in ^[17]. For a nonlinear system with uncertainty and external disturbance, a new adaptive fuzzy sliding mode controller was proposed in ^[18]. In the autopilot system, in order to improve the control performance, an adaptive controller was designed in ^[19]. Taking the same approach, an adaptive fuzzy control scheme using fuzzy logic system was designed in ^[20]. It can be seen that adaptive control can be widely used in aerospace and mechanical systems. Applying adaptive control to biological systems, the research on adaptive control of biological systems is relatively rare, which is an important reason for our research.

Many physical controlled objects can be portrayed using nonlinear systems in real life. For a stochastic nonlinear system, the classical quadratic function was used to construct an adaptive tracking controller in ^[21]. The problem of adaptive tracking control for a class of uncertain switched nonlinear systems was studied in ^[22]. Based on backstepping and Lyapunov functions, an adaptive tracking controller was proposed. An adaptive tracking control was developed based on fuzzy approximation technology and event-triggered for a nonlinear system with uncertainty in ^[23]. A novel event-triggered tracking controller was designed by introducing a fuzzy logic system to approximate the unknown nonlinear term in ^[24]. A new speed curve optimization tracking control method was proposed in ^[25], and a sliding mode controller was designed for the speed curve tracking problem under bounded disturbance. The tracking control problem of a nonlinear system was studied in ^[26], which introduced an adaptive control strategy and constructed an adaptive tracking controller using a new Lyapunov Krasovskii functional. An adaptive sliding mode control scheme based on backstepping was proposed in ^[27] to make the tracking error converge to zero in the presence of uncertain nonlinearity and disturbance. Based on the backstepping design, an adaptive control strategy was proposed to offset the error in ^[28]. Applying the adaptive tracking control to the biological system not only ensures the stability of the system, but also ensures the tracking performance.

The purpose of this paper is to control the density of predators to track a desired density through an adaptive tracking controller, which is undoubtedly a very meaningful and challenging work. Adaptive controllers for observer-based nonlinear systems were designed in ^[29,30]. Due to the difficulty of measuring density involved in this paper, the T-S fuzzy method is used to construct a fuzzy observer, which is used to estimate the population density in biological system. An adaptive tracking controller is designed to make sure the density of predators can track the desired density. The main contributions of this paper are as follows: 1) A fuzzy state observer is designed, which can effectively observe the state of the system. In biological systems, it is difficult to accurately determine the density of a biological population, and an observer is required to estimate the biological population density. 2) An adaptive controller is designed, which can guarantee the effective tracking of predator population density. This is convenient for people to accurately grasp the density of predators, and it is conducive to the stability of ecological balance.

This paper is organized as follows. In Section 2, a biological system with stage structure is established, and its stability is analyzed. In Section 3, a fuzzy state observer is designed to estimate the density of biological population. Then, an adaptive controller is proposed. In Section 4, the validity of the observer, and the stability and tracking convergence of the biological system under the regulation of the controller are verified. Finally, a numerical simulation is carried out to support viewpoints in this paper.

Notations: The superscript $T$ of a matrix represents its matrix transpose. $det \left(\cdot \right)$ stands for the determinant of a square matrix. $Re\left(c \right)$ means the real part of a complex number $c$. $\left\|. \right\|$ represents Euclidean norm.

2.   Model establishment

Based on ^[31], the following system is established:

where ${{x}_{1}}\left(t \right)$ and ${{x}_{2}}\left(t \right)$ represent the juveniles, density and adults, density of the prey population at time $t$, respectively; $y\left(t \right)$ represents the density of the predator population at time $t$; $a$ represents the birth rate of the juvenile population, and $\beta$ represents the transformation rate from the juvenile population to the adult population; ${{b}_{1}}, {{b}_{2}}, {{b}_{3}}$ represent the mortality rates of the juvenile prey population, adult prey population and predator population, respectively; ${{s}_{1}}, {{s}_{2}}$ represent the intraspecific competition intensities of the prey adult population and predator population in order to compete for food, respectively; $\sigma$ represents the predation rate of the predator population to the prey population.

In system (2.1), all constants involved are positive. Considering the limitations of the actual ecological environment, the juveniles and adults density of the prey population and the density of the predator cannot exceed the maximum carrying capacity of the environment. The state variables and parameters meet the following conditions:

where ${{x}_{1\max }}, {{x}_{2\max }}, {{y}_{\max }}$ represent the maximum environmental carrying capacities of the juvenile prey population, adult prey population and predator population, respectively.

It can be seen that system (2.1) has four equilibrium points ${{P}_{1}}\left(0, 0, 0 \right)$, ${{P}_{2}}\left({{x}_{12}}, {{x}_{22}}, {{y}_{2}} \right)$, ${{P}_{3}}\left({{x}_{13}}, {{x}_{23}}, 0 \right)$, ${{P}_{4}}\left(0, 0, -\frac{{{b}_{3}}}{{{s}_{2}}} \right)$, where

Because $-\frac{{{b}_{3}}}{{{s}_{2}}} < 0$, and the biological population density involved in this paper is positive, the equilibrium points ${{P}_{1}}, {{P}_{2}}, {{P}_{3}}$ are considered. At the equilibrium points ${{P}_{2}}, {{P}_{3}}$, in order to ensure the existence of biological populations, the following conditions should be satisfied:

that is, $a\beta \sigma -{{b}_{2}}{{b}_{3}}{{s}_{1}}-{{b}_{1}}{{b}_{2}}\sigma -{{b}_{2}}\beta \sigma > 0, a\beta -{{b}_{1}}{{b}_{2}}-{{b}_{2}}\beta > 0$.

Theorem 2.1. 1) System $(2.1)$ is unstable at the equilibrium points ${{P}_{1}}\left(0, 0, 0 \right)$.

2) System $(2.1)$ is stable at the equilibrium points ${{P}_{2}}\left({{x}_{12}}, {{x}_{22}}, {{y}_{2}} \right)$ when the following inequality holds.

3) System $(2.1)$ is stable at the equilibrium point ${{P}_{3}}\left({{x}_{13}}, {{x}_{23}}, 0 \right)$ when the following inequality holds.

Proof. The Jacobian matrix of system (2.1) is

It can be obtained that the characteristic polynomial of system (2.1) at the equilibrium point ${{P}_{1}}\left(0, 0, 0 \right)$ is

Since ${{b}_{1}}+{{b}_{2}}+\beta > 0, {{b}_{2}}\left({{b}_{1}}+\beta \right)-a\beta < 0$, it can be obtained that the system is unstable at the equilibrium point ${{P}_{1}}\left(0, 0, 0 \right)$.

It can be obtained that the characteristic polynomial of the system at the equilibrium point ${{P}_{2}}\left({{x}_{12}}, {{x}_{22}}, {{y}_{2}} \right)$ is

where ${{\Phi }_{1}} = {{b}_{1}}+\beta +2{{s}_{1}}{{x}_{12}}+\sigma {{y}_{2}}, {{\Phi }_{2}} = -\sigma {{x}_{12}}+{{b}_{3}}+2{{s}_{2}}{{y}_{2}}$. Integrating the above formula, we have

where

Since $-\sigma {{x}_{12}}+{{b}_{3}}+2{{s}_{2}}{{y}_{2}} > 0, {{\sigma }^{2}}{{x}_{12}}{{y}_{2}}-a\beta > 0, {{\sigma }^{2}}{{x}_{12}}{{y}_{2}}-a\beta \left(-\sigma {{x}_{12}}+{{b}_{3}}+2{{s}_{2}}{{y}_{2}} \right) > 0$, we have $Re{{\lambda }_{1}} < 0, Re{{\lambda }_{2}} < 0, Re{{\lambda }_{3}} < 0$, and ${{\Upsilon }_{1}} > 0, {{\Upsilon }_{2}} > 0, {{\Upsilon }_{3}} > 0, {{\Upsilon }_{1}}{{\Upsilon }_{2}}-{{\Upsilon }_{3}} > 0$. According to the Routh-Hurwitz theorem ^[32], system (2.1) is stable at the equilibrium point ${{P}_{2}}\left({{x}_{12}}, {{x}_{22}}, {{y}_{2}} \right)$.

Similarly, it can be obtained that the characteristic polynomial of the system at equilibrium point ${{P}_{3}}\left({{x}_{13}}, {{x}_{23}}, 0 \right)$ is

where ${{\Omega }_{1}} = {{b}_{1}}+\beta +2{{s}_{1}}{{x}_{13}}, {{\Omega }_{2}} = -\sigma {{x}_{13}}+{{b}_{3}}$.

Since $-\sigma {{x}_{13}}+{{b}_{3}} > 0, {{b}_{2}}\left({{b}_{1}}+\beta +2{{s}_{1}}{{x}_{13}} \right)-a\beta > 0$, according to the Routh-Hurwitz theorem ^[32], system (2.1) is stable at the equilibrium point ${{P}_{3}}\left({{x}_{13}}, {{x}_{23}}, 0 \right)$.

From a developmental perspective, we should adopt some appropriate strategies to control the density of the biological population and avoid the loss of biological population diversity. The following assumptions are proposed:

Assumption 2.1. The predator population mainly preys on the juvenile population ${{x}_{1}}\left(t \right)$, and the capture of the adult population ${{x}_{2}}\left(t \right)$ by predators can be ignored.

Assumption 2.2. Considering that reducing human activities will have a certain impact on the population of predators, $u\left(t \right)$ indicates that human activities are constrained at time $t$.

Under the above assumptions, the following biological system is established.

System (2.2) is obtained by applying control in the third equation of (2.1). When $u\left(t \right) = 0$, it implies that the biological system has not been interfered with by human behavior.

3.   Main results

3.1.   Observer design

In real life, the direct measurement of density is difficult, so the density of biological population must be estimated by introducing a fuzzy state observer. The T-S fuzzy method is used to construct an observer to estimate the state of the system.

For the design of the observer, we recall the following lemma.

Lemma 3.1. In ^[33]: Given constant matrices $Q$ and $S$, the symmetric constant matrices $Z, J$ with $J > 0$, and a time-varying matrix $L\left(t \right)$ with appropriate dimension, the following matrix inequality holds

if $L\left(t \right)$ satisfies ${{L}^{T}}\left(t \right)L\left(t \right)\le J$.

System (2.2) can be rewritten as

where

Let

we have

Using the principle of maximum and minimum, ${{\Gamma }_{1}}\left(t \right), {{\Gamma }_{2}}\left(t \right), {{\Gamma }_{3}}\left(t \right), {{\Gamma }_{4}}\left(t \right)$ can be expressed as

where ${{\mu}_{i1}}+{{\mu}_{i2}} = 1, i = 1, 2, 3, 4$ are used to denote the membership function. The fuzzy rules can be found in the Appendix.

Let $\Gamma (t) = {[ \begin{matrix} {{\Gamma }_{1}}(t) & {{\Gamma }_{2}}(t) & {{\Gamma }_{3}}(t) & {{\Gamma }_{4}}(t) \end{matrix}]^T}$, ${\Gamma _1}(t) \in [ {\min {\Gamma _1}(t), \max {\Gamma _1}(t)} ], {\Gamma _2}(t) \in [ {\min {\Gamma _2}(t), \max {\Gamma _2}(t)} ]$, ${\Gamma _3}(t) \in [ {\min {\Gamma _3}(t), \max {\Gamma _3}(t)} ]$, ${\Gamma _4}(t) \in [ {\min {\Gamma _4}(t), \max {\Gamma _4}(t)}]$. ${{k}_{ij}}({{\Gamma }_{j}}\left(t \right))$ represents the grade of membership of ${{\Gamma }_{j}}\left(t \right)$ in the fuzzy set ${{k}_{ij}}$. Using the standard fuzzy blending method, we have

Then, system (3.1) can be rewritten as

Let $\sum\limits_{i = 1}^{16}{{{R}_{i}}\left(\Gamma \left(t \right) \right){{A}_{i}}}, \sum\limits_{i = 1}^{16}{{{R}_{i}}\left(\Gamma \left(t \right) \right){{B}_{i}}}$ be $\bar{A}, \bar{B}$, respectively. Then, system (3.2) can be rewritten as

In order to facilitate the design of the observer, adding the output vector $Y$ to system (3.3), we have

where $C$ is a matrix with appropriate dimension. For system (3.4), an observer of the following form is designed

where $\tilde{x}\left(t \right) = {{\left[ \begin{matrix} {{{\tilde{x}}}_{1}}\left(t \right) & {{{\tilde{x}}}_{2}}\left(t \right) & \tilde{y}\left(t \right) \\ \end{matrix} \right]}^{T}}$ represents the state estimate of $\hat{x}\left(t \right)$, $\tilde{Y}\left(t \right)$ represents the output vector of the observer, ${{\left(\bar{A}-GC \right)}^{T}}\left(\bar{A}-GC \right)\le J$, and $J > 0$ is a symmetric constant matrix.

By Lemma 3.1, we can have

where $Z$ is a symmetric constant matrix. By choosing the matrices $G, C$ such that all characteristic values of $Z$ have negative real parts less than $-\varpi \left(\varpi > 0 \right)$.

The observer error $\varsigma \left(t \right)$ can be obtained as $\varsigma \left(t \right) = \hat{x}\left(t \right)-\tilde{x}\left(t \right)$. From (3.4) and (3.5), the observer error system can be obtained:

3.2.   Controller design

Set ${{y}_{d}}\left(t \right)$ as the desired density of the predator population. The goal of our proposed adaptive control strategy is to steer the state variable $y\left(t \right)$ to track the desired density.

Assumption 3.1. The desired density ${{y}_{d}}\left(t \right)$ is bounded and satisfies

Remark 1. Assumption 3.1 is reasonable, because ${{y}_{d}}\left(t \right)$ represents the actual biological meaning.

It can be obtained from system (2.2):

Let $e\left(t \right) = y\left(t \right)-{{y}_{d}}\left(t \right)$ represent the error between the predator population and the desired density. By introducing a variable $\varphi\left(t \right) = {{\dot{y}}_{d}}\left(t \right)-\theta e\left(t \right)$ without considering $\dot{y}\left(t \right)$, where $\theta > 0$ is a positive constant, we have

where $X$ is a vector about ${{x}_{1}}, y$. $\delta$ is a vector containing parameters, which can be obtained from (3.8).

The adaptive controller can be designed as

where ${{\tilde{x}}_{1}}\left(t \right)$ and $\tilde{y}\left(t \right)$ represent estimates of biological population density ${{x}_{1}}\left(t \right)$ and $y\left(t \right)$, respectively. In addition, $\hat{\lambda }\left(t \right)$ denotes the adaptive law to ensure the stability of the system when the model is uncertain.

Design an update law $\hat{\lambda}\left(t \right)$

where $\varepsilon$ is a positive constant, representing the update rate of tracking error $e\left(t \right)$.

Therefore, (3.9) can be rewritten as

where $\tilde{X} = \left[ \begin{matrix} -{{{\tilde{x}}}_{1}}\tilde{y} & {\tilde{y}} & {{{\tilde{y}}}^{2}} \end{matrix} \right]$.

Remark 2. Furthermore, in order to avoid the undesired chatter in the controller (3.11), the sign function can be replaced by an approximately continuous alternative in the implementation. We consider the tangent hyperbolic function $\tanh \left(\rho e\left(t \right) \right)$ or the smoothing function $sat\left(\frac{e}{\chi } \right)$ instead of ${\mathop{\rm sgn}} \left(e\left(t \right) \right)$, where

$\rho, \chi$ are two constants. Besides, we can also avoid chattering by adding a low-pass filter at the output of the controller to eliminate high-frequency signals at the control input.

4.   Stability analysis

In this section, a Lyapunov function is used to prove the effectiveness of the proposed observer and controller.

Firstly, using the proposed adaptive controller, (3.7) is substituted into (3.9):

Equation (4.1) can be rewritten as

where $\hat{X}\left({{{\hat{x}}}_{1}}, \hat{y} \right) = \tilde{X}\left({{{\tilde{x}}}_{1}}, \tilde{y} \right)-X\left({{x}_{1}}, y \right) = {{\left[ \begin{matrix} {{{\tilde{x}}}_{1}}\left(t \right)\tilde{y}\left(t \right)-{{x}_{1}}\left(t \right)y\left(t \right) & \tilde{y}\left(t \right)-y\left(t \right) & {{{\tilde{y}}}^{2}}\left(t \right)-{{y}^{2}}\left(t \right) \\ \end{matrix} \right]}^{T}}$ represents the error vector, which is bounded.

Since $\dot{e}\left(t \right) = \dot{y}\left(t \right)-{{\dot{y}}_{d}}\left(t \right)$, it can be obtained that

Since the parameter vector ${\delta }$ is bounded, and the estimation error $\hat{X}\left({{{\hat{x}}}_{1}}, \hat{y} \right)$ of the state variable is bounded, it can be assumed that there is a positive constant $\gamma$ such that $\hat{X}\left({{{\hat{x}}}_{1}}, \hat{y} \right)\delta$ is bounded, that is

Next, the Lyapunov function is established as

It can be seen from (4.5) that the Lyapunov function is a positive definite function. Substitute (3.6) into (4.5), and the time derivative of (4.5) can be obtained:

Substitute (4.3) into the above equation to obtain

Since $e\left(t \right){\mathop{\rm sgn}}\left(e\left(t \right) \right) = \left| e\left(t \right) \right|$, substitute (3.10) into (4.7), and we have

Substitute (4.4) into (4.8), and we can get

This can be obtained by integrating the above formula.

Since $\theta, \varpi$ are two positive constants, $\dot{V}\left(t \right)\le 0$.

Theorem 4.1. The proposed observer can effectively estimate the state of the system. The designed adaptive controller ensures the tracking convergence. When $t\to \infty$, $\varsigma \left(t \right)\to 0$, $e\left(t \right)\to 0$. That is the predator population will converge to the desired density, $y\left(t \right)\to {{y}_{d}}\left(t \right)$.

Proof. According to Barbalat's Lemma ^[34], when $t\ge 0$, if $\kappa$ is a uniformly continuous function, $\underset{t\to \infty }{\mathop{\lim\limits}}\, \int_{0}^{t}{\kappa \left(\tau \right)d\tau }$ exists and is limited, and we have $\underset{t\to \infty }{\mathop{\lim\limits}}\, \kappa \left(t \right) = 0$.

Suppose $\psi \left(t \right) = \theta {{e}^{2}}\left(t \right)+\varpi {{\left\| \varsigma \left(t \right) \right\|}^{2}}\ge 0$, and (4.10) can be rewritten as

Integrate (4.11) from 0 to $t\to \infty$ to obtain

that is,

$\dot{V}\left(t \right)\le 0$ is known from (4.10), and $V\left(\infty \right)-V\left(0 \right)\le 0$ can be obtained after integrating it, so $V\left(0 \right)-V\left(\infty \right)\ge 0$. Combined with (4.12), $\underset{t\to \infty }{\mathop{\lim\limits}}\, \int_{0}^{t}{\psi \left(\tau \right)d\tau }$ exists and is limited, and $\underset{t\to \infty }{\mathop{\lim\limits}}\, \int_{0}^{t}{\psi \left(\tau \right)d\tau }$ is positive due to $\psi \left(t \right)\ge 0$. From Barbalat's Lemma

that is,

Since $\theta, \varpi$ are two positive constants, ${{e}^{2}}\left(t \right)\ge 0, {{\left\| \varsigma \left(t \right) \right\|}^{2}}\ge 0$, and it can be seen from (4.13) that when $t\to \infty$, $e\left(t \right)\to0, \varsigma \left(t \right)\to 0$. Therefore, the proposed adaptive controller can track the desired density of the predator population, that is, $y\left(t \right)\to {{y}_{d}}\left(t \right)$.

5.   Simulation

In recent years, human activities have seriously damaged the ecological environment. People have gradually realized the importance of protecting the ecological environment. In some of the latest research, it can be seen that human activities have led to the destruction of the ecological environment of a large number of biological species. Among them, Lampetra japonica is affected by soil erosion, and the living environments of spawning grounds and young fish are damaged. In addition, water pollution affects the living environment. So, its resources are quite small and in a vulnerable state. Therefore, it is urgent to reduce the impact on biological population density by controlling human behavior. In this paper, considering the survival of Lampetra japonica, the following parameters are selected according to its actual situation:

Then, the biological system can be obtained as

Considering the actual situation of the biological system, it follows that

At this time,

When $u\left(t \right) = 0$, we have four equilibrium points of system (5.1) and take the positive equilibrium point as ${{P}_{2}} = \left(0.1816, 0.4929, 0.6492 \right)$.

After adding the controller, the system is

Then, we have

So, the fuzzy model is

where

The membership function can be obtained:

Let

The observer error can be obtained as

When $t < 100$, set the desired density of the predator population as ${{y}_{d}}\left(t \right) = 0.3$. The tracking error is $e\left(t \right) = y\left(t \right)-0.3$. When $t\ge 100$, set ${{y}_{d}}\left(t \right) = 0.75$. Then, the tracking error is $e\left(t \right) = y\left(t \right)-0.75$. Let $\theta = 0.15$, and then the adaptive controller is

If the parameter of the designed adaptive update law is $\varepsilon = 0.09$, the adaptive update law is

The initial values are as follows: ${{x}_{1}}\left(0 \right) = 1.9, {{x}_{2}}\left(0 \right) = 2.05, y\left(0 \right) = 5.03$.

Figure 1 shows the state response without control. It can be seen that the density of the biological species is divergent and unstable. Figure 2 shows the state response of the biological population after applying the controller. It can be seen that the biological model converges and tends to be stable with control. It can be verified that the designed adaptive controller is effective.

Figure 3 shows the tracking error between the predator population density and the desired density. It can be seen that the error almost converges to 0, which can be confirmed by the enlarged subgraph. Figure 4 shows the trajectory of $y$ tracking the desired density ${{y}_{d}}$ with control. It can be seen that $y$ can track well the desired density ${{y}_{d}}$ under the regulation of the adaptive controller.

The trajectory of the adaptive controller is shown in Figure 5. Figure 6 shows the observer error. From the above figures, it can be verified that under the action of the proposed controller and update law, the biological system can remain stable, and the density of predators can well follow the desired density. In other words, under the regulation of the adaptive controller designed in this paper, the ecosystem of Lampetra japonica will maintain natural balance and benign conditions. The adaptive controller designed in this paper can stabilize the ecosystem of Lampetra japonica. Lampetra japonica is small in size, inhabits the river all its life and has no migratory habit. Due to the influence of human activities, its population is small and in a vulnerable situation. This may lead to the destruction of the ecological balance. According to the data, lampetra japonica has been listed in the second level of China's national key protected wildlife list. In this case, we need to reduce human activities. Therefore, a controller is applied to the third differential equation in (2.1). Through the adaptive controller, we study the effect of limiting human activities on the density of Lampetra japonica, and we make the density of Lampetra japonica able to track the given density. The adaptive controller also ensures the stability of the whole ecosystem.

6.   Conclusions

In this paper, an adaptive tracking control scheme has been investigated for a biological model with stage structure. A state observer is constructed by the T-S fuzzy method, and an adaptive tracking controller is designed on this basis. The proposed control scheme not only guarantees the stability of the biological model but also ensures that the predator density can track a desired density. Finally, a simulation example is given to illustrate the effectiveness of the results. From the biological point of view, the proposed adaptive tracking control method can appropriately adjust the biological population density to stabilize the whole biological system. At the same time, it can realize effective tracking and finally achieve the sustainable development of biological resources. Therefore, our results have a far-reaching impact on the study of biological population density.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 62103289.

Conflict of interest

The authors declare there is no conflict of interest.

Appendix

The fuzzy rules are given as follows:

Rule 1: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{11}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{21}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{31}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{41}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_1\hat{x}\left(t \right)+B_1U\left(t \right)$.

Rule 2: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{11}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{21}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{31}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{42}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_2\hat{x}\left(t \right)+B_2U\left(t \right)$.

Rule 3: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{11}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{21}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{32}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{41}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_3\hat{x}\left(t \right)+B_3U\left(t \right)$.

Rule 4: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{11}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{21}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{32}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{42}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_4\hat{x}\left(t \right)+B_4U\left(t \right)$.

Rule 5: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{11}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{22}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{31}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{41}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_5\hat{x}\left(t \right)+B_5U\left(t \right)$.

Rule 6: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{11}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{22}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{31}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{42}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_6\hat{x}\left(t \right)+B_6U\left(t \right)$.

Rule 7: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{11}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{22}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{32}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{41}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_7\hat{x}\left(t \right)+B_7U\left(t \right)$.

Rule 8: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{11}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{22}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{32}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{42}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_8\hat{x}\left(t \right)+B_8U\left(t \right)$.

Rule 9: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{12}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{21}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{31}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{41}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_9\hat{x}\left(t \right)+B_9U\left(t \right)$.

Rule 10: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{12}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{21}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{31}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{42}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_{10}\hat{x}\left(t \right)+B_{10}U\left(t \right)$.

Rule 11: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{12}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{21}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{32}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{41}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_{11}\hat{x}\left(t \right)+B_{11}U\left(t \right)$.

Rule 12: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{12}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{21}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{32}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{42}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_{12}\hat{x}\left(t \right)+B_{12}U\left(t \right)$.

Rule 13: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{12}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{22}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{31}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{41}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_{13}\hat{x}\left(t \right)+B_{13}U\left(t \right)$.

Rule 14: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{12}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{22}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{31}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{42}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_{14}\hat{x}\left(t \right)+B_{14}U\left(t \right)$.

Rule 15: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{12}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{22}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{32}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{41}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_{15}\hat{x}\left(t \right)+B_{15}U\left(t \right)$.

Rule 16: If ${{\Gamma }_{1}}\left(t \right)$ is ${{\mu }_{12}}\left({{\Gamma }_{1}}\left(t \right) \right)$, ${{\Gamma }_{2}}\left(t \right)$ is ${{\mu }_{22}}\left({{\Gamma }_{2}}\left(t \right) \right)$, ${{\Gamma }_{3}}\left(t \right)$ is ${{\mu }_{32}}\left({{\Gamma }_{3}}\left(t \right) \right)$, ${{\Gamma }_{4}}\left(t \right)$ is ${{\mu }_{42}}\left({{\Gamma }_{3}}\left(t \right) \right)$, then $\dot{\hat{x}}\left(t \right) = A_{16}\hat{x}\left(t \right)+B_{16}U\left(t \right).$

In the above,