Neuro-adaptive finite-time control of fractional-order nonlinear systems with multiple objective constraints

1.   Introduction

In recent years, fractional-order calculus has received increasing attention since it describes actual systems more effectively than integral order calculus. Fractional-order calculus is a generalized calculation of traditional integer-order calculus. Notably, fractional-order systems have potential benefits and design flexibility, which can accurately characterize many physical phenomena, such as electromagnetic waves, viscoelastic materials models and chaotic systems ^[1,2]. Recently, some research outcomes were reported on fractional-order nonlinear system control ^[3,4]. Due to the presence of fractional-order derivatives and uncertain nonlinearities, almost all control methods currently in effective use for integer-order nonlinear systems cannot be readily extended to fractional-order nonlinear systems.

The backstepping control methodology is one of the most commonly used approaches for nonlinear systems. Specifically, NNs and fuzzy logic systems (FLSs) have become popular tools for modeling and controlling nonlinear systems due to their superior approximation properties ^[5,6]. Motivated by these results, some advanced adaptive intelligent control approaches are applied to the control of fractional-order nonlinear systems ^[7,8]. In ^[7], an adaptive fuzzy controller was constructed using the backstepping recursive technique to deal with unknown external disturbances of strict-feedback fractional-order nonlinear systems. Motivated by ^[7], a fractional-order filter-based dynamic surface control scheme was presented in ^[8] to solve the "complexity explosion". Many efforts are being made to investigate intelligent adaptive control for fractional-order nonlinear systems, including output-feedback control ^[9], optimal control ^[10] and input nonlinearities (for example, input saturation, dead zone and hysteresis ^[11,12]). However, the aforementioned results can only guarantee the convergent time tends to infinity.

Generally speaking, for many practical engineering applications, the study of finite-time stability is of greater significance compared to infinite-time stability since finite-time control has better fast transient performance and better robustness to uncertainties ^[13,14]. Note that the studies of finite-time stability for fractional-order nonlinear systems has produced some achievements. In ^[15], a finite time adaptive fuzzy controller was constructed using a sliding mode technique to cope with the finite-time stabilization problem for fractional-order nonlinear systems. Then, ^[16] introduced a fractional command filter to reduce the influence of filtering errors on control performance based on ^[15]. A class of adaptive event triggered finite time control problems was investigated in ^[17], and a new fractional-order finite time stability criterion was established. It is noteworthy that the above research ignores the impact of constraints.

There always exist some constraints in engineering applications because of physical limitations or safety considerations. If these constraints are not handled appropriately, it might lead to the decline of control performance and even system stability. Some researchers have recently addressed the nonlinear constraint control problem using the barrier Lyapunov function (BLF) ^[18,19] or a nonlinear state-dependent transformation ^[20]. Note that ^[18,19,20] only applies to integer-order systems due to the complexity of fractional calculus. As a backdrop, the constraint problem of fractional-order nonlinear systems is a fascinating and challenging task. By incorporating a BLF to avoid full-state constraint violations, an event-triggered controller was devised in ^[21] to simultaneously reduce the communication burden on the controlled system. On this basis, the research in ^[22] was extended to uncertain fractional-order nonlinear nonstrict-feedback systems with input saturation and full-state constraints.

Furthermore, focusing on fractional-order systems with immeasurable states and full-state constraints, an adaptive fuzzy output feedback control was established in ^[23]. It should be noted that the process of constraint control would inevitably involve multi-objective constraints, such as reducing logistics cost and improving commodity transportation quality ^[24] or lowering production cost and increasing production rate ^[25], which cannot be solved by conventional constraint methods. Correspondingly, the adaptive control schemes in ^[26,27,28] were presented for such controlled systems under multiple objective constraints. In spite of these attempts, to the authors' best knowledge, no results has been reported on fractional-order nonlinear systems control with multi-objective constraints, which is one of the our research motivations.

Inspired by the above observation and analysis, this paper focuses on the neuro-adaptive finite-time control of fractional-order nonlinear systems in nonstrict-feedback form subject to multiple objective constraints which contain unknown nonlinear dynamics and external disturbances. The primary characteristics and innovations of this paper are listed below:

1) This study introduces the neuro-adaptive finite time control method of the multi-objective constraints for fractional-order nonlinear systems by introducing a novel barrier function. Different objective functions are constrained to the predefined acceptable range by selecting the appropriate weight coefficients. Note that the results on the multi-objective constrained nonlinear systems in ^[26,27,28] only apply to integer-order systems without relating to fractional-order systems. Besides, the control method for the constrained nonlinear system is studied in ^[26], while the finite time convergence is not considered. Unlike the constant value constraint in ^[27,28], this paper adopts a time-varying constraint, which is more in line with engineering applications.

2) The system plants considered in this work is more general than those in ^{[7,8,15,16,17,21]}. A new parameter adaptive law is designed to control the nonlinear fractional-order system in nonstrict-feedback form, which eliminates the effect of unknown external disturbance and the algebraic loop problem. The devised control scheme ensures that the system output tracks the desired signal in finite time. Compared with ^[7,8,21,22], the control algorithm proposed in this paper accelerates the tracking rate and improves convergence.

The rest of this article can be summarized as follows: Section 2 gives the formulation and preliminary preparation of the problem. Section 3 presents the main results and a finite-time neuro-adaptive control method based on the backstepping algorithm. Section 4 provides simulation example to elaborate on the feasibility of the developed control algorithms. Section 5 summarizes this paper.

2.   Problem formulation and preliminaries

Two definitions of fractional calculus are introduced as follows.

Definition 2.1. (^[29]) The Riemann-Liouville (RL) fractional integral of order $\alpha > 0$ for the function $f(t)\in {C}^n([t_0, +\infty], \mathbb{R})$ is defined as

where

represents the Gamma function, which satisfies

$C^n$ is the collection of all continuous functions whose derivatives of the integer orders $i = 1, 2, \cdots, n$ are all continuous. Then, the $\alpha$th Caputo fractional-order derivative is expressed by

where $^c_0 D^{\alpha}_t$ represents the $\alpha$th order Caputo differential operator whose time starts from zero. $n-1 \leq \alpha < n$, $n \in \mathbb{N}^+$. Here, the superscript "RL" denotes the Riemann-Liouville derivative and "c" denotes the Caputo derivative. In this paper, only the case $\alpha \in (0, 1)$ is considered.

Consider a class of nonstrict-feedback systems with external disturbances described as follows:

where

and $y \in \mathbb{R}$ are the state vector and the system output, respectively. $u \in \mathbb{R}$ is the input of the controlled system. $f_i(x) \in \mathbb{R}$ and ${g_i(x)\in \mathbb{R}}$ denote the unknown but smooth nonlinear functions. $d_i(t) \in \mathbb{R}$ represents an unknown continuous external disturbance in the subsystem. Additionally, consider the system (2.1) subject to multiple objective constraints

and

where $J_j(x_1)$ is the $j$th objective function and the $r_{j}$ denote the weighting coefficients. By selecting appropriate weighting coefficients, we ensure that all objective functions can be unified within the common boundary

with

and

Remark 2.1. It is worth indicating that system (2.1) is more general than those considered in results mentioned in the introduction. If $g_i(x) = 1$ and

system (2.1) will degenerate into the systems investigated in ^[8,15]. When $d_i(t) = 0$, system (2.1) becomes the system considered in ^{[7,16,17,21,22]}.

To facilitate the controller design, we make common assumptions in the following points.

Assumption 2.1. For $i = 1, 2, \cdots, n$, the functions $g_i(x)$ are continuous, and there exist unknown constants $g_m > 0$ and $g_M > 0$ satisfying

For notation conciseness, we denote $g_i(x)$ as $g_i$.

Assumption 2.2. For $i = 1, 2, \cdots, n$, the external disturbances are bounded such that $\lvert d_i(t) \lvert \leq \bar{d}_i$ with $\bar{d}_i$ being unknown positive constants.

Assumption 2.3. The desired signal $y_d$ and its fractional-order derivative are continuous and limited with $Y_{0, 1, 2}$, which satisfy that

where $Y_0$, $Y_1$ and $Y_2$ are positive constants.

The following definition of finite-time stability for fractional-order nonlinear systems is introduced.

Definition 2.2. (^[13]) Let $\xi = 0$ be the equilibrium point of a fractional-order system $^c_0 D^{\alpha}_t {\xi} = f(\xi, u)$. It is called to be practical finite-time stable if for all initial conditions $\xi(t_0) = \xi_0$, there exists a constant $\epsilon > 0$ and a settling time $T(\epsilon, \xi_0) < \infty$ which satisfy $\| \xi \| < \epsilon$, $\forall t\geq t_0 + T$.

Control objective: The aims of this paper are to construct a finite-time adaptive controller for system (2.1) to ensure that the system output $y$ is capable of tracking the desired signal $y_d$ in finite time, and that multiple different objective function constraints are not transgressed. Simultaneously, all closed-loop signals are guaranteed to be bounded.

Then, the following lemmas will be used in the stability analysis of fractional-order nonlinear system.

Lemma 2.1. (^[30]) Suppose that $f(t)$: $[t_0, \infty]\rightarrow \mathbb{R}$, the following inequality holds for all $t\geq t_0$,

Lemma 2.2. (Gronwall-Bellman integral inequality ^[31]) Let $\mathfrak{f}(t)$ satisfy

with a real function $p(t)$ and a differentiable real function $q(t)$. There is

where $t\geq t_0.$ In particular, if $q(t) = q$ is a constant, we have

Lemma 2.3. (^[32]) Suppose $0 < \alpha < 1$, $\varrho > 1$ and $f(t)\in C^1([0, +\infty), \mathbb{R})$. It holds that

Lemma 2.4. (^[33]) Assume that $x(t)\in \mathbb{R}^n$ is a smooth and differentiable function vector. Then,

Lemma 2.5. (^[34]) Let $y(x(t))$: $\Omega \rightarrow \mathbb{R}$ and $x(t)$: $[t_0, +\infty)\rightarrow \Omega$ be two continuously differentiable functions, where $\Omega \subset \mathbb{R}$ is a set. If $y(x(t))$ is convex over $\Omega$ (i.e., $\partial^2 y(x)/\partial x^2\geq 0)$, then

holds for any $t \geq t_0$ and constant $0 < \alpha < 1$.

Lemma 2.6. (^[14]) For $\iota_1 > 0, \ \iota_2 > 0, \ \iota_3 > 0, \ \mu_1\geq0, \ \mu_2\geq0,$ and $\mu_3\geq0$, the following inequality holds

Note that the radial basis NN function is used as a promising tool to approximate unknown nonlinear functions ^[5]. For an unknown continuous nonlinear function $F(Z)$: $\mathbb{R}^n \rightarrow \mathbb{R}$, we can approximate it using the NN function $y(Z) = \theta^T\varphi(Z)$, where $\theta$ is a weight vector, which can be represented as

where $\delta(Z)$ is the approximation error.

is the ideal weight vector defined as

where

is the basis function vector and $n$ is the number of neurons. $\varphi_i(Z)$ is a Gaussian function formed as

where $q_i$ and $p_i$ are the center of the receptive field and width of the $i$th hidden layer.

3.   Main results

3.1.   Controller design

To achieve the multi-objective constraints, different objective functions are guaranteed to be confined within a certain acceptable range. Inspired by ^[26], a barrier function is introduced as

where $l \in \Lambda = \{1, \cdots, m\}$ and $J_l(x_1)$ is the abstract function with respect to $x_1$ satisfying $\partial J_l(x_1)/\partial x_1 \neq 0$ in the open compact set $\Omega_J$. $K_1(t)$ and $K_2(t)$ represent the upper and lower bounds of $J_l(x_1)$ such that $K_1(t) < K_{1}^+$ and $K_2(t) > K_{2}^-$, respectively. For any initial value $J_l(0)\in\Omega_J$, if $J_l \rightarrow K_{1}^+$ or $J_l \rightarrow K_{2}^-$, then $\hbar\rightarrow \infty$. In general, if $J_l$ is to comply with the constraint, then it is sufficient to ensure that $\hbar$ is bounded. So, the issue of satisfying the objective function constraint can be turned into ensuring that $\hbar$ is bounded.

Remark 3.1. It is remarkable that multi-objective constraints are widespread in various practice systems. Nevertheless, in practical engineering applications, the multiple optimization objectives we consider may be contradictory, such as reducing logistics cost and improving commodity transportation quality, reducing production cost and increasing production rate or other physical scenarios. It is unrealistic that all objective functions are reaching their optimum simultaneously. Therefore, we take $K_1(t)$ and $K_2(t)$ as the maximum of the common lower bound and the minimum of the common upper bound of all objective functions, respectively. Note that $K_1(t)$ and $K_2(t)$ can be asymmetric and possibly time-varying functions, which is different from ^[26,27,28]. By this compromise scheme, all objective functions can be unified simultaneously in the same boundary.

Then, from Lemma 2.5, by taking $\alpha$th order derivative of $\hbar$, it yields

with

and

Then, $^c_0 D^{\alpha}_t \hbar$ is rewritten as

where $\varpi_{11} = \varpi_{10}$sign$(\varpi_{10})$. Since

it is deduced that $\varpi_{11} \neq 0$.

Let us define the errors as

where

and $\bar{\alpha}_i$ are the output of the fractional-order filters with virtual control signal $\alpha_{i-1}$ as the corresponding input signals, which is defined as

where $\tau_i$ is a positive time constant. $\chi_i = \bar{\alpha}_i-\alpha_{i-1}$ denotes the filter error cased by the fractional-order filter.

Step 1: Thus, according to (3.1) and (3.4), the $\alpha$th fractional-order derivative of $z^2_1/2$ is derived as

where

and

Since the continuous function $F_1(X_1)$ contains the unknown functions $f_1$ and $g_1$, the NN function $\theta^{*T}_1\varphi_1(X_1)$ is employed to approximate it. According to Lemma 2.6, for given $\varepsilon_1 > 0$, it is known that

where $\theta^{*T}_1$ and $\varphi_1(X_1)$ are the optimal weight vector and the NN Gaussian function vector, respectively. $\delta_1$ is the minimum approximation error satisfying $\|\delta_1\| \leq \varepsilon_1$.

Using Young's inequality and the property

we have

where

and

$\kappa_1$ is a positive constant and

Remark 3.2. Note that $F_1(X_1)$ contains all the state variables, which implies that system (2.1) is a nonstrict-feedback system. If the traditional backstepping design method is adopted, the algebraic loop problem arises. In order to avoid such problem, this paper uses the adaptive parameter $\Theta_1^*$ to participate in the construction of the controller. In addition, the adaptive parameter $\rho^*_1$ and the compensation function $(\cdot)/ \tanh({\cdot}/{\varsigma})$ with $\varsigma > 0$, are introduced to handle the affect brought by uncertainties, approximation errors and external disturbances.

Consider the following Lyapunov function

where $\gamma_1 > 0$ and $\Gamma_1 > 0$ are the design constants, $\tilde{\Theta}_1 = \Theta^*_1-\hat{\Theta}_1$ stands for the estimation error when $\hat{\Theta}_1$ is estimating $\Theta^*_1$. Define $\tilde{\rho}_1 = \rho^*_1-\hat{\rho}_1$ when $\hat{\rho}_1$ is used to estimate of $\rho^*_1$.

According to Lemma 2.4, we estimate the $\alpha$th Caputo fractional derivative of $V_1$ as

The virtual controller $\alpha_1$, the adaptive laws $\hat{\Theta}_1$ and $\hat{\rho}_1$ are

where $c_1, k_1, \mu_1, \pi_1$ are positive constants with $\frac{1}{2} < \beta < 1$.

Substituting (3.10)–(3.12) into (3.9) produces

Applying the property of the inequality $\lvert\cdot \lvert- (\cdot)\tanh(\cdot/\varsigma)\leq 0.2785\varsigma$, where $\varsigma > 0$, then (3.13) can be rewritten as

where $\Delta_1 = 0.2785\varsigma g_m\rho^*_1+\kappa_1$.

From (3.5), we have

where $i = 2, \cdots, n$, $H_{i}(\cdot)$ is the continuous function related to variables $z_1, \cdots, z_i$, $\hat{\Theta}_1, \cdots, \hat{\Theta}_i$, $\hat{\rho}_1, \cdots, \hat{\rho}_{i}$, $\chi_2, \cdots, \chi_i$, $y_d, {^c_0 D^{\alpha}_t} {y}_d$ and ${^c_0 D^{\alpha}_t} [{^c_0 D^{\alpha}_t} {y}_d]$. Meanwhile, there may exist positive constants $\Upsilon_i$ satisfying $\lvert H_{i} \lvert < \Upsilon_i$ in the compact set $\Xi$. Then, it yields

Step i $(i = 2, \cdots, n-1)$: Similar to previous steps, then, it follows from (2.1) and (3.4) that

Now, consider the following Lyapunov function

where $\gamma_i > 0$ and $\Gamma_i > 0$ are the design parameters. $\hat{\Theta}_i$ is the estimate of $\Theta^*_i$ with

Define

with $\hat{\rho}_i$ being the estimation of $\rho^*_i$.

Then, the $\alpha$th derivative of (3.18) gives

where

and

Based on the NN function approximation, some transformations are exploited

where

and $\kappa_i > 0$ is a design parameter.

Construct the $i$th intermediate control function $\alpha_i$, and the adaptation laws of $\hat{\Theta}_i$ and $\hat{\rho}_i$ as follows:

where $c_i, k_i, \mu_i$ and $\pi_i$ are positive constants.

Substituting (3.20)–(3.23) into (3.19) yields

where

Step n: Along with step $i$, the derivative of $z_n$ is

Similar to the previous steps, consider the Lyapunov function candidate as

where $\gamma_n$ and $\Gamma_n$ are the positive design constants.

Then, the $\alpha$th derivative of (3.26) is calculated as

where

Based on the NN approximation, it follows that

where

and $\kappa_n$ is a positive constant.

It follows from (3.27) and (3.28) that

Then, take the actual input $u$ and adaptive laws as

where $c_n, k_n, \mu_2, \pi_n$ are design positive constants.

Invoking (3.29)–(3.32), we have

where

3.2.   Stability analysis

The above control method can be outlined in the following theorem.

Theorem 3.1. Suppose that Assumptions 2.1–2.3 hold for the nonstrict feedback fractional-order nonlinear systems (2.1) subject to multiple objective constraints (3.1), the actual control law constructed in (3.30) is based on the virtual control laws (3.10) and (3.21), the parameter adaptive laws ((3.11), (3.22), (3.31)) and ((3.12), (3.23), (3.32)). Then, it ensures that the system output driven to track the desire signal in finite time, the different objective functions do not transgress their constrained sets and all signals in the closed-loop systems are guaranteed to be bounded.

Proof. Based on Young's inequality, it is derived that

Therefore, $^c_0 D^{\alpha}_t V_n$ is expressed as

Using Lemma 2.6, for

and $\mu_3 = 1$, we obtain

Similarly, we have

and

Through above analysis, it follows that

where

Choose the design parameters such that $g_m\mu_j-1 > 0$, $g_m\pi_j-1 > 0$ $(j = 1, \cdots, n)$ and ${1}/{\tau_j} -{1}/{2}-{g_m}/{2} > 0$ $(j = 2, \cdots, n)$. If we let

and

then the inequality (3.37) is stated as

For (3.38), two parts are discussed as follows.

Part 1: Defining $V^*(t) = V_n(t) - \Psi/a$, according to Lemma 2.2 and $^c_0 D^{\alpha}_t V_n \leq -a V_n + \Psi$, then $V^*(t)$ is transformed into

Thus, there exists $\ell(t) \in \mathbb{R}^+$ such that

Based on Lemma 2.1, taking the fractional integration $^{RL} I^{\alpha}_t$ of (3.40) from $t_0$ to $t$, we obtain

On the other hand, from Lemma 2.2, we know that

Substituting $V^*(t) = V_n(t) - \Psi/a$ into (3.42), the inequality (3.42) is computed as

According to the definition of $V_n$ and (3.43), the error variables satisfies

Therefore, from the above-mentioned inequality, we can conclude the boundedness of $V_n$, which indicates that $z_i$, $\tilde{\Theta}_i$, $\tilde{\rho}_i$ for $i = 1, \cdots, n$ and $\chi_j$ for $i = 2, \cdots, n$ are bounded.

Consider $z_1 = \hbar - \hat{y}_d$. Based on the boundedness of $\hat{y}_d$, the boundedness of $\hbar$ is ensured, and from (3.1), the $J_l$ is bounded in the constraints. Considering the definition of $J_l$, we can get that $x_1$ is bounded by $\lvert x_1 \lvert \le X_0$ with $X_0 > 0$. Moreover, according to Assumption 2.3, there exist the bounds of $y_d$, i.e., $Y_0$ such that $J(Y_0) - K_1 > 0$, and $K_2 - J(Y_0) > 0$. Then, combining it with the definition of the tracking error, we obtain

From th Eqs (3.10)–(3.12), variables $\alpha_1$, $\dot{\hat{\Theta}}_1$ and $\dot{\hat{\rho}}_1$ are bounded. In the following, we are summarizing that $x_i, \bar{\alpha}_i, \dot{\hat{\Theta}}_i, \dot{\hat{\rho}}_i\ (i = 2, \cdots, n)$, $\alpha_j\ (j = 2, \cdots, n-1)$ and the actual input $u$ are bounded. Hence, all the variables of the closed-loop system are bounded.

Part 2: We show that $z_i, \tilde{\theta}$ and $\tilde{\rho}$ have finite-time convergence. From (3.38), it is clear to obtain that

Note that there exists a constant $0 < \vartheta < 1$ such that the inequality (3.44) can be rewritten as

If $V_n > (\Psi/b(1-\vartheta))^{1/\beta}$, we have $^c_0 D^{\alpha}_t V_n \leq -b \vartheta V_n^{\beta}$. Choosing $\beta = 2p - 1$, $\bar{\varsigma} = 1 + \alpha$, applying Lemma 2.3, we show that

Defining $\bar{V} = V^{\alpha+1}_n$, we deduce that

Therefore, the above inequality leads to

Taking the integration of the inequality (3.47) from $0$ to $t$, we get

Then, the reaching time $T_r$ is evaluated as

If $V_n \leq (\Psi/b(1-\vartheta))^{1/\beta}$, based on previous analysis, the trajectory of of all variables in $V_n$ will not be greater than $(\Psi/b(1-\vartheta))^{1/\beta}$.

Therefore, it can be indicated that the controlled system is finite-time stable and the tracking error $e_1$ can be converged into a sufficiently small neighborhood around the origin in the setting time $T_r$. The proof is completed. □

A block diagram of our control design structure is given in Figure 1.

Remark 3.3. The design parameters $c_i, \ k_i, \ \omega_i, \ \gamma_i, \ \Gamma_i, \ \mu_i$ and $\pi_i$ are only a sufficient condition to ensure the finite time stability of the control system and they should be a trade-off between tracking performance and controlling costs. From the proof of Theorem 3.1, the selection of some key design parameters are provided as below:

1) Increasing $c_i$, $\gamma_i$, $\Gamma_i$, $\kappa_i$ and $\varsigma$ helps to increase $a$ in (3.38), subsequently reducing the error bound;

2) Increasing $k_i$, $\omega_i$ helps to increase $b$ in (3.38), subsequently accelerating the convergence rate;

3) Increasing or decreasing $\mu_i$, $\pi_i$ will affect the approximation performance of FLS and the estimation effect of disturbances and approximation errors.

Remark 3.4. It should be emphasized that the form of (3.38) is discussed in two parts. First, the fractional-order finite-time stability criterion is introduced to ensure that the tracking error converges to a compact set within a finite-time. Then, fractional-order Lyapunov theory is employed to ensure the stability of the closed-loop system. Note that the finite-time control scheme proposed in this paper has a faster convergence rate than ^[15] when the initial state is far away from the origin point.

Remark 3.5. It is worth noting that the establishment of time $T_r$ is not an explicit relationship, that is, $T_r$ is not easy to calculate in practical applications. Compared with finite time control, predefined time control ^[35] has some superior advantages, such as that the upper bound of the settling time can be preset and independent of the initial conditions and the convergence accuracy can be ensured. However, this usually means that the control input needs to be large enough to achieve its outstanding convergence performance. Therefore, how to make a tradeoff between the convergence accuracy/time and the excessive control input is a problem that researchers continue to consider. It is undoubtedly a good choice to consider the optimal control of a certain performance index.

4.   Simulations

In the following, we illustrate the validity of the proposed method through numerical example. The dynamic model of 2-order fractional-order nonlinear systems is as follows

where the fractional order $\alpha$ is chosen as 0.95, and the external disturbances are given as $d_1 = d_2 = 0.1\sin(t)$.

The controller parameters are chosen as

The reference signal is $y_d = 0.5 \sin(t)$ and it follows that

We employ 11 nodes for each dimension of the NN with the center spaced in [-10, 10] and with the width 2.

The objective function $J_l$ is bounded

and three cases of objective functions are executed as

where the constraining boundaries are set as $K_1 = -0.85-0.15 \sin(t)$ and $K_2 = 1.2 + 0.5 \sin(t)$. $m_1 = 1, m_2 = m_3 = 0$ means that $J_1 = x_1$, which is the case with output constraints.

The simulation results of system (4.1) are depicted in Figures 2–9 under the three cases. Figures 2, 4 and 6 show that the system output $y = x_1$, and the trajectories of desired signal $y_d$, and the values of the error signals converge to zero in finite time, which means that the system output is confined in the prescribed constraints and the trajectory of the state $x_2$ is bounded. Figures 3, 5 and 7 show the curve of adaptive laws and the control inputs. It can be seen from these figures that the trajectories of adaptive laws are bounded in all the three cases. The multiple objective functions, in view of Figure 8, have no violated constraints and approximate the desired objective trajectories $J_d$.

To show the superiority of the proposed method, we use the control approach without finite time control technique in ^[7] to control the nonlinear system (4.1). For fair comparison, the initial conditions and control parameters are the same as in Example 1. The contrasting simulation result is plotted in Figure 9. It can be seen from Figure 9 that the proposed finite time controller in this paper has more satisfactory convergence speed and tracking performance. The comparative results in Figure 9 indicate that the settling time of the proposed finite time control method is less than 0.5 s, whereas the settling time of without finite time control becomes larger.

5.   Conclusions

This article has explicitly focused on the finite-time adaptive tracking control problem for nonstrict-feedback fractional-order nonlinear systems under multiple objective constraints and unknown disturbances. The NNs are utilized to identify the unknown dynamics. Meanwhile, a novel barrier function is constructed to ensure that all objective functions are constrained within the preset range, while keeping the system output in the prescribed boundaries. The stability analysis and simulation results illustrated that all the signals in the fractional-order nonlinear system are bounded, and the tracking error is guaranteed to converge to a small neighborhood of the origin in finite time. Future research will extend the finite time method to fractional-order nonlinear systems with immeasurable states and input nonlinearities.

Use of AI tools declaration

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

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 62073189 and Grant 62173207, and in part by the Taishan Scholar Project of Shandong Province under Grant tsqn202211129.

Conflict of interest

All authors declare no conflicts of interest in this paper.