Adaptive fuzzy output-feedback event-triggered control for fractional-order nonlinear system

1.   Introduction

Fractional-order nonlinear systems (FONSs) are a class of complex systems modeled by fractional calculus. In recent years, the application of fractional calculus has attracted wide-ranging attention. In practice, examples such as the blood ethanol concentration system by Qureshi et al. ^[1], the dynamics of the TB virus by Ullah et al. ^[2], the fractional Brusselator reaction-diffusion system by Jena et al. ^[3], etc. can be modeled as fractional order systems. This kind of system can well describe the genetic effect and long memory effect of a real physical system. Nevertheless, it is noteworthy that conventional control scheme is no longer applicable to fractional-order systems due to some general rules for integer-order calculation, such as Chain rules and Leibniz rules, not being well built with regard to fractional derivatives. Hence, how to solve the stability analysis or controller of fractional order systems has attracted extensive attention of scholars. For example, in ^[4], a fractional order controller designed by Liu et al. to ensure that the synchronization errors of the fractional order chaotic system reached the specified performance whether there is external interference or not. Wei et al. ^[5] designed an adaptive tracking controller and extended the result to the case of $1 < \alpha < 2$ by the semigroup property of the fractional derivative and fractional tracking differentiator. Afterwards, Li et al. ^[6] studied the adaptive control issue for a type of commensurate FONS with parametric uncertainty and external interference, and unlike the discontinuous function, the auxiliary function is used to get smooth control input and achieve perfect property tracking in case of bounded interferences. However, the above literature demands that nonlinear functions in the plant are known. So, these techniques are not appropriate for resolving the control issue of FONSs with unknown nonlinear functions.

In practice, there are always completely unknown nonlinear functions in the modeled control systems, which cannot be ignored. Therefore, ^{[7,8,9,10,11]} used FLSs or neural networks (NNs) with approximation capability to solve this problem. The authors in ^[7,8,9] put forward adaptive intelligent (fuzzy and NN) control schemes for FONSs, in which Wang and Liang ^[8] and Li et al. ^[9] are robust to input saturation and fault, respectively. Because the above intelligent adaptive control methods adopt a conventional backstepping control technology, there is a problem of computational complexity. In order to avoid this problem, Ma and Ma ^[10] and Sui et al. ^[11] proposed several adaptive intelligent state feedback control methods by introducing fractional-order filters. However, the existing methods in ^{[7,8,9,10,11]} are merely applicable to those FONSs whose states are only measured. Therefore, the above scheme is not suitable for the control design of FONSs with unmeasurable state variables. Then, ^[12,13,14] presented some adaptive intelligent output-feedback DSC methods for FONSs with unmeasurable states. It should be noted that in ^[12], the authors established a new fractional-order reduced-order observer, which not only obtained the information of unmeasurable states, but also reduced errors caused by the full-order state observer in the estimation of some measurable states. As far as we know, there are few results on output-feedback and DSC control simultaneously for fractional-order nonlinear systems, which prompted us to study this problem.

Notably, that the above results adopt conventional periodic control methods, and the control signal needs to be transferred to the actuator in actual time, which will lead to unnecessary sampling and communication. Recently, the ETC theorem has been exploited rapidly, which can reduce the communication load of the controlled system ^[15,16,17]. An event-triggered pulse control method for impulsive systems is proposed in ^[15], which successfully solves the stabilization matter of nonlinear impulsive systems and eliminates Zeno behavior. Later, in ^[16,17], Sui et al. and Wang et al. designed intelligent adaptive ETC schemes for stochastic systems and multi-agent systems to achieve system stability while reducing the waste of communication resources. Note that the systems considered by above intelligent controllers are only applicable to integer-order nonlinear systems, not FONSs. Meanwhile, it is not easy to extend the direct Lyapunov algorithm and its related control schemes from integer order to FONSs. Although ^[18,19,20] proposed several intelligent adaptive control algorithms for the FONSs, they are designed for situation in which the state is completely measurable and there is a problem of computational complexity. To our knowledge, there is a short age of studies on the observer-based ETC control methods for FONSs, which inspires us to study this problem.

According to the above-mentioned discussions, we study the adaptive output-feedback event-triggered control for FONS with unmeasurable state variables. The proposed ETC scheme can significantly decrease the consumption of communication resources. The main innovations of this paper are as follows.

1) This article first designs an output-feedback event-triggered controller of the FONS. The proposed control algorithm erases the restrictive condition in ^[18,19,20], with which the state of the system must be completely measured.

2) Due to the DSC technique being used to control design, the put forward control method settles the computational complexity issue in current works ^[18,19,20].

3) In this paper, the adaptive control law and the event-triggered mechanism are designed together. The stability of the controlled system can be guaranteed by using the fractional-order Lyapunov criterion. Unlike ^[9,10,11,12], the control signal needs to be sampled and updated regularly. The system drive will be generated only when the preset conditions are met in this paper, which greatly reduces the consumption of network resources.

2.   Preliminaries

2.1.   System statements

Consider the following FONS:

where ${\bar x_n} = {[{x_1}, {x_2}, ..., {x_n}]^T} \in {R^n}$ are the system state vectors, and $y \in R$ and $u \in R$ denote the output variable and control input of the system. ${f_i}(\cdot) \in R, i = 1, \ldots, n$, denotes an unknown smooth nonlinear function. This paper assumes that only the output variable $y$ is measurable.

Assumption 1 ^{[9,10,11,12,13,14]}: The given reference signals ${y_d}, {}_0^CD_t^\alpha {y_d}$ and ${}_0^CD_t^\alpha ({}_0^CD_t^\alpha {y_d})$ are smooth and bounded. Furthermore, assumed that there exists known constant ${Z_0} > 0$ satisfying that $y_d^2 + {({}_0^CD_t^\alpha {y_d})^2} + {({}_0^CD_t^\alpha ({}_0^CD_t^\alpha {y_d}))^2} \le {Z_0}$.

Control Objectives: In this article, a fuzzy adaptive event-triggered controller is designed for System (1) such that all signals in the considered system are bounded, and the tracking error converges to the compact set of the origin.

2.2.   Preliminaries

Some useful definitions and lemmas are given first.

Definition 1 ^[21]: The $\alpha th$ Caputo derivative is defined as:

where $\omega - 1 < \alpha \le \omega,$ $\omega$ is a positive integer. $\Gamma (\cdot) = \int_0^{ + \infty } {{\tau ^{ - 1}}{e^{ - \tau }}} d\tau$ denotes Euler's gamma function with $\Gamma (1) = 1$.

Definition 2 ^[21]: The Mittag-Leffler function is formulated as

where $\alpha > 0, \phi > 0,$ and $\gamma$ is a complex number.

Lemma 1 ^[21]: Let $\alpha \in (0, 2),$ $\eta \in R$ and $\phi \in ({{\pi \alpha } / 2}, \min \{ \pi, \pi \alpha \})$, and then one has

In (2.4), $r > 0,$ $|\zeta | \ge 0$, and $\phi \le \, |\arg (\zeta)|\, \le \pi$.

Lemma 2: Suppose that$f(x)$ is a continuous function on a compact set $\Omega$. There exists an FLS such as

where $\varepsilon > 0$ is any positive constant.

3.   Fuzzy state observer

Write the FONS (2.1) as follows:

where $A = {\left[{\begin{array}{*{20}{c}} { - {k_1}} & 1 & \cdots & 0\\ { - {k_2}} & 0 & \cdots & 0\\ \vdots & \vdots & \vdots & \vdots \\ { - {k_n}} & 0 & \cdots & 0 \end{array}} \right]_{n \times n}},$ $K = {\left[{\begin{array}{*{20}{c}} {{k_1}}\\ {{k_2}}\\ \vdots \\ {{k_n}} \end{array}} \right]_{n \times 1}},$ $B = {\left[{\begin{array}{*{20}{c}} 0\\ 0\\ \vdots \\ 1 \end{array}} \right]_{n \times 1}}, {B_i} = {\left[{\underbrace {0\begin{array}{*{20}{c}} {} \end{array} \cdots \begin{array}{*{20}{c}} {} \end{array}1}_i\begin{array}{*{20}{c}} {} \end{array} \cdots \begin{array}{*{20}{c}} {} \end{array}0} \right]_{1 \times n}}, C = {\left[{\begin{array}{*{20}{c}} 1 & 0 & \cdots & 0 \end{array}} \right]^T}$$, K$ is chosen such that $(A + CK)$ is a Hurwitz matrix. $Q = Q_{}^T > 0$ is a positive definite matrix, and exist a positive definite matrix $P = P_{}^T > 0$ such that

It is worth noting that ${f_i}({\bar x_i})$ in (3.1) is an unknown continuous function, so it is necessary to approximate ${f_i}({\bar x_i})$ with the help of an FLS ${\hat f_i}({\bar x_i}{\rm{|}}{\xi _i}) = \xi _i^T{\phi _i}({\bar x_i}),$. In the bounded sets $\Omega$, the definition of ideal parameter vectors $\xi _i^*$ are described as:

The definition of the optimal approximation error${\varepsilon _i}$ is described as

with ${\varepsilon _i}({\bar x_i})$ is bounded by constant $\varepsilon _i^* > 0$.

Design the fuzzy state observer for (3.1) as

where $\hat x = {[{\hat x_1}, {\hat x_2}, \ldots, {\hat x_n}]^T}$ are the estimations of $x$.

Define the observer error as $e = x - \hat x$.

From (3.1), (3.4) and (3.5), one has

where $\varepsilon = {[{\varepsilon _1}, {\varepsilon _2}, \ldots, {\varepsilon _n}]^T}$$, {\xi _i}$ are the estimations of ideal parameters $\xi _i^*,$ and $\tilde \xi = {\xi ^*} - \xi$.

Construct the Lyapunov function candidate as ${V_0} = \frac{1}{2}{e^T}Pe$, and then the following Theorem can be obtained.

Theorem 1: For controlled System (2.1), the fuzzy state observer (3.3) has the following property:

where ${\lambda _0} = ({\lambda _{\min }}(Q) - 2) > 0$, and ${\delta _0} = ||P|{|^2}\sum\nolimits_{i = 1}^n {||\xi _i^*|{|^2}} + \frac{1}{2}||P|{|^2}\sum\nolimits_{i = 1}^n {\varepsilon _i^{*2}}$.

Proof: We choose the Lyapunov function candidate as ${V_0} = \frac{1}{2}{e^T}Pe$. From (3.5) and (3.6), by the inequality ${{{}_0^CD_t^\alpha ({x^T}(t)x(t))} / 2} \le {x^T}(t){}_0^CD_t^\alpha x(t)$, ${}_0^CD_t^\alpha {V_0}$ can be calculated as

By employing Young's inequality and $\psi _i^T({\hat x_i}){\psi _i}({\hat x_i}) \le 1$, we can gain

Substituting (3.9) into (3.8), one can have (3.7). This completes the proof of Theorem 1.

Remark 1: Theorem 1 shows that if $\tilde \xi _i^T{\tilde \xi _i}$ is bounded, smaller observation errors ${e_i}$ can be obtained by selecting a large enough ${\lambda _0}$. It is further concluded that the constructed fuzzy state observer (3.5) can better estimate the unknown states.

4.   ETC controller design and stability analysis

This part will use the adaptive fuzzy backstepping control algorithm to provide the observer-based adaptive ETC control design program and give its stability analysis.

4.1.   Design procedure

Make the coordinate transforms as

where ${S_1}$ is tracking error, ${S_i}$ are dynamic surface errors, ${\upsilon _i}$ are filter variables, and ${\eta _i}$ are filter output errors, and ${\tau _{i - 1}}$ are the virtual control functions.

Step1: Via (3.3), (4.1), and ${x_2} = {e_2} + {\hat x_2}$, one has

Choose the Lyapunov function as

where ${\gamma _1} > 0$ is a known constant.

The virtual controller ${\tau _1}$ and the adaptive law ${}_0^CD_t^\alpha {\xi _1}$ are designed as

where ${c_1} > 0$ and ${\kappa _1} > 0$ are known constants.

Introduce dynamic surface filter in ^[15] as

where ${\sigma _1}$ is a constant.

By using (4.2) and (4.6), one has

where ${W_1}(\cdot)$ is a continuous function.

Remark 2: In the backstepping ETC design of FONSs, it is difficult to obtain the mathematical analytical expression of the fractional derivative of the virtual controllers. To solve this problem, some authors used the packaged approximation technology in ^[18,19,20] to repeatedly approximate the virtual controllers. Because this method takes all the signals of the closed-loop system as the input variables of the NN or FLS, it will increase the dimensions of the adjusted parameter vector, resulting in the problem of computational complexity. Therefore, this paper adopts the DSC technology to effectively avoid this problem.

Step i: From (3.5) and (4.1), one has

The Lyapunov function candidate is chosen as

where ${\gamma _i} > 0$ is a known constant.

The virtual controller ${\tau _i}$ and the adaptive law ${}_0^CD_t^\alpha {\xi _i}$ are designed as

where ${c_i} > 0$ and ${\kappa _i} > 0$ are known constants.

Introduce dynamic surface filter as

where ${\sigma _i}$ is a constant.

By using (4.8) and (4.12), one can obtain

where ${W_i}(\cdot)$ is a continuous function.

Step n: We first devise an event-triggered controller as

where ${c_n} > 0$ is a known constant, and ${t_k}(k \in {z^ + })$ defines input updating time.

Thus, in order to get a lower communication rate, the event-triggered condition can be designed as

where $\bar \delta \in (0, 1), m > 0$ and $\bar m > [{m / {(1 - \bar \delta)]}}$ are given as the known parameters, and $\nu (t) = \omega (t) - u(t)$ is called as the measurement error. When (4.17) is triggered, the time will be marked as ${t_{k + 1}}$, and the controller $u({t_{k + 1}})$ will be utilized to the system. At the time $t \in [{t_k}, {t_{k + 1}})$ the control signal is always unchanging.

In order to discuss the event-triggered rule, we consider that the actuator normally operates, i.e., $\left| {\omega (t) - u(t)} \right| \le \bar \delta \left| {u(t)} \right| + m$.

If $v({t_k}) > 0$, the measured error can be rewritten as

where ${\lambda _1}(t) \in [-1, 1]$.

If $v({t_k}) < 0$, we can transform the event-triggered condition (4.17) as

where ${\lambda _2}(t) \in [-1, 1]$.

From (4.18) and (4.19), one obtains

where $\left| {{\lambda _i}} \right| \le 1, i = 1, 2$ are time-varying variables. Then, one has

Remark 3: The event-triggered parameters $\bar \delta$ and $m$ in (4.17) are determined according to the required communication rate. Therefore, in practical applications, while ensuring satisfactory tracking performance, we should try to reduce the communication burden.

According to (3.5), (4.1) and (4.21), one has

Choose the Lyapunov function as

where ${\gamma _n} > 0$ is a known constant.

Assumption 2 ^[10,12]: For any initial conditions, there exists a constant $q > 0$, such that $V(0) \le q$.

The adaptive law ${}_0^CD_t^\alpha {\xi _n}$ is designed as:

where ${\kappa _n} > 0$ is a known constant.

Remark 4: A backstepping control algorithm is indicated for FONSs in ^[17]. It applies the stability analysis of integer-order Lyapunov methods to known fractional-order systems. However, in this article, the system model may be completely unknown. In addition, the stability of the control algorithm is analyzed by the fractional order adaptive stability criterion.

4.2.   Closed-loop systems stability analysis

Theorem 2: Consider System (2.1), under Assumptions 1–2, and then the put forward adaptive fuzzy output feedback event-triggered controller (4.21) with the event-triggered mechanism (4.17) can keep that controlled fractional-order system is stable, and the tracking error is able to regulate to a small residual set of the origin. Meanwhile, Zeno behavior is removed effectively.

Proof: Construct the whole Lyapunov function as

By utilizing the inequality ${{{}_0^CD_t^\alpha ({x^T}(t)x(t))} / 2} \le {x^T}(t){}_0^CD_t^\alpha x(t)$, one can obtain

According to (4.4), (4.10), (4.14), (4.22), (4.23), (4.26) and adding and subtracting ${S_n}{\tau _n}$, $|{S_n}\bar m|$ in the right of (4.26), the following equality can be obtained:

In view of $|{\lambda _i}(t)| \le 1$ and ${S_n}\omega (t) \le 0$, we can obtain

According to (4.26), (4.30) and (4.31), (4.29) can be rewritten as

Then, utilizing the inequality $\left| \tau \right| - \tau \tanh ({\tau / {{\psi}}} \le \chi {\psi})$, $\chi = 0.2785$, one can get

For (4.14)–(4.17), (4.33) applying Young's inequality, one gets

Substituting (4.9)–(4.11), (4.14)–(4.17), (4.10)–(4.31), (4.34)–(4.36) into (4.33) yields

Note that ${\Xi _0} = \{ y_d^2 + {({}_0^CD_t^\alpha {y_d})^2} + {({}_0^CD_t^\alpha ({}_0^CD_t^\alpha {y_d}))^2} \le {Z_0}\}$ and $\Xi = \{ V(t) \le q\}$ are compact sets, and thus ${\Xi _0} \times \Xi$ is still a compact set. Since ${W_i}(\cdot)$ are continuous functions on ${\Xi _0} \times \Xi$, there exist positive constants ${K_i}$ such that $\left| {{W_i}(\cdot)} \right| \le {K_i}$.

Applying Young's inequality again, yields:

Substituting (4.38) and (4.39) into (4.37), one has

Then, we have

where $\mu = \min \{ 2({{\bar \lambda } / {{\lambda _{\max }}(P)), 2{c_h}, {\kappa _h} - {\gamma _h}(||P|{|^2} + 1), {2 / {{\sigma _h} - 2}}}}\},$ and $D = {\delta _1} + \sum\limits_{h = 1}^{n - 1} {({{K_h^2} / 2}}) + \sum\limits_{h = 1}^n {({{{\kappa _h}} / {2{\gamma _h}}}})||\xi _h^*|{|^2} + 0.557{\varepsilon ^*}$.

Now, from (4.41) and according to Liu et al. ^[7] and Gong and Lan ^[22], one obtains

where $\beta (t) > 0$.

Applying the Lyapunov transform on (4.42) yields

Using the inverse Laplace transform on (4.43), yields

where $*$ denotes the convolution operator.

Notably, $\beta (t) \ge 0$ and ${t^{\alpha - 1}}{E_{\alpha, \alpha }}(- \mu {t^\alpha }) \ge 0$ in (4.44), so $\beta (t) * {t^{\alpha - 1}}{E_{\alpha, \alpha }}(- \mu {t^\alpha }) \ge 0$. Thus, one has from (4.44)

From Lemma 1, it follows that

Therefore, for $t \ge 0$, it follows that

Due to Lemma 1, one obtains

Then, one writes (4.48) as follows:

From (4.49), it follows:

Therefore, one has

Based on (4.49), (4.52) and (4.53), we proved that the controlled system can be stable, and ${\lim _{t \to \infty }}\left| {{S_1}} \right| = \sqrt {{{2Dd} / \mu }}$ and ${\lim _{t \to \infty }}\left\| e \right\| = \sqrt {{{2Dd} / {(\mu ({\lambda _{\min }}(P)))}}}$, which means that ${S_i}(i = 1, ..., n)$, $e$ and $\tilde \xi$ are bounded. Furthermore, as $\xi _i^*$ is bounded, ${\xi _i}$ as also bounded. Finally, by choosing design parameters, the tracking error and observer error are reduced.

Remark 5: From (4.46), it can be concluded that smaller tracking error can be obtained by increasing $\mu$ or decreasing $D$. Consequently, for smaller $D$, we can appropriately decrease ${\kappa _h}$ or increase ${\gamma _h}$ according on the definition of $D = {\delta _1} + \sum\limits_{h = 1}^{n - 1} {({{K_h^2} / 2}}) + \sum\limits_{h = 1}^n {({{{\kappa _h}} / {2{\gamma _h}}}})||\xi _h^*|{|^2} + 0.557{\varepsilon ^*}$. So as to get larger $\mu$, we can appropriately increase ${c_h}$ and ${\kappa _h}$ or decrease ${\gamma _h}$ and ${\sigma _h}$ by the definition of $\mu = \min \{ 2({{\bar \lambda } / {{\lambda _{\max }}(P)), 2{c_h}, {\kappa _h} - {\gamma _h}(||P|{|^2} + 1), {2 / {{\sigma _h} - 2}}}}\}$.

Ultimately, Zeno behavior is removed through the following proof. By recalling the measurement error $\nu (t) = \omega (t) - u(t)$, we have

From (4.20) that $\omega (t)$ is a differentiable signal of order $\alpha$, and ${}_0^CD_t^\alpha \omega$ is a bounded function.

Thus, the existence of $\rho > 0$ makes $\left| {{}_0^CD_t^\alpha \omega } \right| \le \rho$ hold. According to $\nu ({t_k}) = 0$ and ${\lim _{t \to {t_{k + 1}}}}v(t) = {m}$, one can have ${t_{k + 1}} - {t_k} \ge {{{m}} / \rho }$. So, Zeno behavior does not occur.

Remark 6: It is worth noting that the control schemes designed by the authors in ^{[7,8,9,10,11,12,13,14]} are based on time triggered control. Because the control signal is sampled and updated periodically, it leads to a waste of communication resources. In order to solve this issue, an event-triggered mechanism is introduced in the backstepping technology. This mechanism enables the control signal to be sampled and updated only when the given conditions cannot be met, thus decreasing the communication load.

Remark 7: The repeated differentiation of virtual control function will lead to complexity explosion, so filter is introduced to solve this problem. By using event-triggered mechanism and dynamic surface filter at the same time, this paper greatly saves the waste of computing resources.

Remark 8: In ^[15,16], it is important to study the adaptive ETC for integer-order nonlinear systems. However, we have designed an event-triggered rule for FONSs with unmeasurable states. Within the framework of event-triggered DSC scheme, we solved the computing explosion caused by duplicate derivation of virtual controllers. Unlike event-triggered rule in Wang et al. ^[17], threshold is a function of system state or tracking error. This paper determines the improved event-triggered mechanism based on the size of the control signal itself.

5.   Simulation example

This part gives a simulation example to verify the availability of theoretical results.

Example: The fractional-order strict-feedback system is described as follows:

when ${f_1}({x_1}) = {x_1} - x_1^3, {f_2}({x_1}, {x_2}) = 20{x_1} - {x_2}$.

The membership functions can be chosen as

The given reference signal is ${y_d} = \cos (2t)$. The observer gain is selected as ${K_1} = {[{k_1}, {k_2}]^T} = {[10,220]^T}$. Then, the observer (3.3) can be written as

Further, given $Q = 3I$, a positive definite matrix $P = \left[{\begin{array}{*{20}{c}} {0.3014} & {0.0136} \\ {0.0136} & {66.4364} \\ \end{array}} \right]$ can be obtained by solving the Lyapunov Eq (3.2).

The control law can be given as

The event-triggered controller is devised as

The event-triggered condition can be designed as

where $\bar \delta \in (0, 1)$, $m > 0$ and $\bar m > [{m / {(1 - \bar \delta)]}}$. With the following adaptive laws:

In the simulation, choose the parameters as follows: $\alpha = 0.98$, ${c_1} = 5$, ${c_2} = 15$, ${\kappa _1} = 0.01$, ${\kappa _2} = 0.02$, $\bar \delta = 0.45$, $\bar m = 0.15$ and ${\psi} = 0.4$. The system states initial conditions are chosen as ${x_1}(0) = 0.4$, ${x_2}(0) = 0.5$, ${\hat x_1}(0) = 0.5$, ${\hat x_2}(0) = 0.5$, ${\xi _1}(0) = {[-0.1, -0.3, -0.5, - 0.7, - 0.9]^T}$ and ${\xi _2}(0) = {[-0.1, -0.3, -0.5, - 0.7, - 0.9]^T}$.

The simulation results are shown in Figures 1–6. Figure 1 shows the curves of reference signal ${y_d}$ and the system output $y$. Figure 2 shows the curves of reference signal ${y_d}$ and the system output $y$ without ETC. From Figures 1 and 2, we can see that ETC can ensure satisfactory system performance. The trajectories of the tracking error are shown in Figure 3. Figures 4 and 5 response of ${x_i}$ and ${\hat x_i}$, $i = 1, 2$. Figure 6 responses of $u$. Figure 7 shows the trigger time intervals with the event-triggered control. However, by calculating, we know that the controller executes 2000 times without event-triggered control; under the event-triggered control method, the number of samples is only 1007, which greatly reduces the waste of communication resources. From Figures 1–7, it is concluded that the proposed event-triggered controller can achieve the stability of the controlled system and effectively decrease the communication load.

6.   Conclusions

The observer-based adaptive ETC algorithm for FONS was investigated in this study. By employing FLS to model the unknown dynamics, a fuzzy state observer is constructed for the unmeasurable state vectors. Using an adaptive backstepping control algorithm, an observer-based adaptive fuzzy DSC method is proposed. The put forward control algorithm has avoided computational complexity problem resulted in the repeated iteration of virtual controllers in the inherent backstepping method. Additionally, it has reduced the burden of communication and removed the Zeno behavior. The simulation results testify the validity of the controller. The further research will focus on the intelligent adaptive ETC problem of fractional-order nonlinear impulsive systems based on this study and literature ^[23,24].

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (under Grant No. 62173172), and in part by the Doctoral Startup Fund of Liaoning University of Technology (under Grant No. XB2021015).

Conflict of interest

The authors declare that there is no conflict of interest.