Neural network–based adaptive fractional-order terminal sliding mode control

Abstract

This paper proposes an adaptive fractional-order (FO) terminal sliding mode control (TSMC) scheme to the robust current control of active power filter (APF) using a recurrent meta-cognitive fuzzy neural network (RMCFNN). An FO TSMC is developed by considering that the parametric perturbations and the external disturbances of APF are bounded. Compared with conventional TSMC approach, the proposed scheme, with an FO sliding surface, can obtain enhanced finite-time high-precision tracking performance due to another degree of freedom. Then, a novel observer-based FO TSMC is derived to achieve an absorbing model-free feature arising from RMCFNN. To improve the capabilities in managing the uncertainties, the specific online updating schemes for the structure and parameters of RMCFNN are designed. Meanwhile, closed-loop stability and finite-time convergence characteristic can be achieved using Lyapunov theory. Finally, simulation and experimental results indicate that the proposed observer-based FO TSMC can be easily implemented by microcontroller and has superior control performance compared with other existing schemes.

Keywords

Neural network control sliding mode control terminal sliding mode control finite-time control recurrent neural network

Introduction

Nowadays, the problems with harmonic pollution give rise to constant trouble for industrial applications. Active power filter (APF) in Figure 1 has become the favorable solution for harmonic suppression due to its flexible features and convenient installation (Chu et al., 2018; Hekss et al., 2021). The significant advantages of APF have been demonstrated in many fields, including radial power distribution feeder, solar photovoltaic power generation system, and synchronous generator.

Considering parametric uncertainties and other nonlinearities in APF, controller design for APF is treated as a challenging task in the existing works. To obtain remarkable control performance and boost the development of APF, Wang et al. (2021a) have made much effort to design efficient control approaches, such as model predictive control, sliding mode control (SMC) (Guzman et al., 2016; Hou and Fei, 2020), fuzzy control (Ray et al., 2019), and neural network (NN) control (Chu et al., 2020; Fei and Chu, 2020) . Among the abovementioned schemes, SMC has caused increasing attention due to its superior property of disturbance insensitiveness. However, for investigating a practical SMC to APF, one has to solve the conflict between the smooth control signal and the robustness to uncertainties (Hou et al., 2021c). In the traditional SMC, sign function is combined into control law to construct a robustness term for compensating uncertainties (Hou et al., 2021b). However, for APF, the chattering phenomenon caused by robustness term will give rise to undesired high-frequency switch of insulated gate bipolar transistor (IGBT), and this will degenerate the control performance. To suppress chattering, many strategies have been proposed, that is, observer-based compensation, integral compensation, and variable-gain approach (Li et al., 2017). Besides the foregoing methods, the boundary layer is also an attractive technique to alleviate the effect of chattering using saturation function instead sign function (Nguyen et al., 2017). Admittedly, boundary layer technique can achieve smooth control signal at the expense of precision loss. In order to deal with this issue, terminal sliding mode control (TSMC) is introduced to enhance high-precision and fast-response control properties (Mu and He, 2018). High tracking precision performance of TSMC has been indicated by plentiful literature due to finite-time convergence (Feng et al., 2020; Zhang et al., 2021).

However, most existing TSMC approaches only contain integer-order (IO) integrator and differentiator. In fact, the mathematical study of fractional order (FO) has been going on for over 300 years, and it has been applied to the field of control (Balootaki et al., 2020). FO controllers can achieve superior control characteristic due to the introduced another degree of freedom (Kilbas et al., 2006). Motivated by this benefit, FO SMC has been successfully utilized for various industrial applications, such as gyroscope (Fei and Lu, 2018), robot manipulators (Nojavanzadeh and Badamchizadeh, 2016; Wang et al., 2016, 2020), and linear motor (Sun and Ma, 2017). Although the remarkable results have been obtained, most of the abovementioned approaches belong to model-based ones requiring detailed dynamics, which become the potential practical performance limitations. The dependence on prior information of SMC has received much attention (Wang and Chen, 2020; Wang et al., 2021b; Xu, 2018; Zhu et al., 2020). Generally, these studies can be classified into two groups. Part studies introduce NN for learning the supremum of uncertainties. The other studies focus on the direct approximation of uncertain dynamics by constructing an uncertainty estimator. Our study belongs to the latter one.

For the past few decades, fuzzy neural network (FNN) becomes a mature control scheme for extensive practical systems by representing nonlinear dynamics with complex uncertainties (Hou et al., 2022; Sun et al., 2021). Admittedly, most of the foregoing FNN control approaches cannot achieve structure and parameter adjustment simultaneously. Actually, uncertain dynamics including the grid structure, load current, and other parameters in APF are variable. Specifically, APF will also give rise to the load current variation indirectly. In this context, the exciting results using FNN with fixed structure will be limited (Lin et al., 2019). Therefore, in order to achieve high-precision approximation of uncertain dynamics for APF, the structure and parameters of FNN both are required to be online updated. Meta-cognitive FNN (MCFNN) can be a reliable solution to resolve the foregoing issue due to its specific characteristic of flexible network structure (Rong et al., 2017; Subramanian and Suresh, 2012). Specifically, fuzzy rules, hidden nodes, and other parameters in the MCFNN are not required to be defined in advance, and it will adjust structure and all parameters using data deletion and data learning strategies to achieve a whole learning process. In addition, compared with simple feed-forward NN, the generalization ability of NN including recurrent framework is enhanced (Lin et al., 2019). A recurrent NN was introduced for a class of dynamic systems, in which comparative simulation and experimental results demonstrated its superiority (Chu et al., 2020).

In order to tackle the above difficulties, this study proposes an adaptive FO TSMC for APF, by utilizing recurrent meta-cognitive fuzzy neural network (RMCFNN). The major contributions of this study are listed in the following form:

FO TSMC is designed to enhance the tracking performance of existing TSMC by adding adjustable order thanks to FO. Higher precision of current control system can be achieved thanks to finite-time convergence so that harmonic suppression capability of APF can be further improved.

A novel adaptive FO TSMC using RMCFNN is introduced to realize a highly efficient learning of uncertainties considering sensor fault in practical applications. Specifically, fuzzy rules and number of nodes can be generated or pruned through novelty or contribution of current input variable using data learning strategies, which is highly preferable for industry-oriented applications.

To the best of our knowledge, a robust current tracking strategy for APF constructed using FO TSMC with RMCFNN has not been explored until recently. In addition, such investigation can be applied for other various applications, such as robot manipulators, linear motor, and so on.

This remaining part of the study is structured as follows. In section “Problem formulation,” the dynamic model of APF, basic knowledge of FO, and FO TSMC is exploited. Section “RMCFNN” presents a novel adaptive FO TSMC including a RMCFNN estimator. The comparative simulation and experimental evaluations between the proposed FO TSMC and other existing schemes are presented in section “Simulation and experimental results.” Finally, section “Conclusion” gives the conclusions.

Problem formulation

Figure 1 presents the circuit of APF. Generally speaking, two control loops should be designed, that is, DC-side voltage loop and AC-side current loop. The voltage loop is the outer loop whose objective is to maintain a constant DC-side voltage. Traditional PI control is chosen for DC-side voltage loop. The objective of the AC-side current loop, inner loop, is to track the irregular and changing reference signals such that it is vital for purifying harmonics. This study focuses on the robust control design for AC-side current loop.

The dynamic model for AC-side current loop is formulated as (Hou et al., 2019, 2021a)

{\overset{\cdot}{i}}_{c} = f (i_{c}) + U + H

(1)

where $f (i_{c}) = - (R_{1} / L_{1}) i_{c} + (U_{s} / L_{1}) \in R$ , $U = - (U_{dc} / L_{1}) u \in R$ , $U_{s} \in R$ is the grid-side voltage, $i_{c} \in R$ is the compensation current, $R_{1}$ and $L_{1}$ are the nominal values of the resistance $R$ and inductance $L$ of the APF, respectively, $U_{dc} \in R$ is the DC-side capacitor voltage, and $H \in R$ represents the lumped uncertainties.

To achieve fast and high-precision control performance for APF, the following FO TSMC surface is designed as given by equation (2)

s = λ_{1} e + λ_{2} D^{α - 1} [sig (e)^{β}]

(2)

where $e = i_{c} - i_{c}^{*}$ is the tracking error between the compensation current and desired current $i_{c}^{*}$ , $sig (e)^{β} = {| e |}^{β} sgn (e)$ , $0 < α < 1$ , $β = q / p$ , $p$ , and $q$ are positive integers satisfying $1 < q / p < 2$ , $λ_{1}, and λ_{2}$ are given constants, $D^{α}$ is the FO operator, and some preliminary results should be given as follows:

Definition 1. The FO operator including FO derivative and FO integral is given by equation (3) (Kilbas et al., 2006)

{{}_{t_{0}}D}_{t}^{α} = {\begin{matrix} \frac{d^{α}}{{dt}^{α}} α > 0 \\ 1 α = 0 \\ I^{α} = \int_{0}^{t} (dt)^{- α} α < 0 \end{matrix}

(3)

where $α$ is the FO of the operator and $t_{0}$ is the initial time.

Definition 2. The $α th - order$ Riemann–Liouville (RL) fractional derivative and integral can be presented as equations (4) and (5) (Wang et al., 2020)

D^{α} f (t) = \frac{d^{α} f (t)}{d t^{α}} = \frac{1}{Γ (m - α)} \frac{d^{m}}{d t^{m}} \int_{t_{0}}^{t} \frac{f (τ)}{{(t - τ)}^{α - m + 1}} d τ

(4)

{{}_{t_{0}}I}_{t}^{α} f (t) = D^{- α} f (t) = \frac{1}{Γ (α)} \int_{t_{0}}^{t} \frac{f (τ)}{{(t - τ)}^{1 - α}} d τ

(5)

where $m - 1 < α < m$ , and $m \in N$ . It is worth noting that the advantage of RL definition is that it only requires $f (t)$ to be continuous, and there is no special requirement for the derivatives of $f (t)$ . More details can be found in Balootaki et al.’s study (2020).

Lemma 1 (Feng et al., 2020): If there exist a continuous positive-definite function $V (x)$ , and such inequality as $V (x) + γ V^{λ} (x) \leq 0$ is satisfied, it can hit $V (x) = 0$ in a finite time $t_{r}$ that is bounded by

t_{r} \leq \frac{1}{γ (1 - λ)} V^{1 - λ} (x (0))

(6)

where $γ > 0$ , and $0 < λ < 1$ are constant.

In fact, for practical application systems, the lumped uncertainties $H$ always are bounded due to energy constraints so that we can get the following assumption.

Assumption 1. The lumped uncertainties $H$ is bounded, and the following inequality can be satisfied:

| H | \leq H_{M}

(7)

where $H_{M}$ is constant.

Then the FO TSMC is chosen as follows:

U_{FOTSM} = - f (i_{c}) + {\overset{\cdot}{i}}_{c}^{*} - \frac{λ_{2}}{λ_{1}} D^{α} [sig (e)^{β}] - K sgn (s)

(8)

where $K = H_{M} + ρ_{1}$ , and $ρ_{1}$ is a given constant.

Theorem 1: Considering the dynamics (1) and the FOTSM surface (2), if the FO TSMC law is formulated as equation (8), then the finite-time convergence characteristic can be achieved.

Proof: Construct the following Lyapunov function as follows

V_{1} = \frac{1}{2} s^{2}

(9)

Then, one can obtain the derivative of $V_{1}$

\begin{matrix} {\overset{\cdot}{V}}_{1} = s \overset{\cdot}{s} \\ = s [λ_{1} (f (i_{c}) + U + H - {\overset{\cdot}{i}}_{c}^{*}) + λ_{2} D^{α} [sig (e)^{β}]] \end{matrix}

(10)

Substituting equation (8) into equation (10) leads to

\begin{matrix} {\overset{\cdot}{V}}_{1} = λ_{1} s [H - K sgn (s)] \\ = λ_{1} Hs - λ_{1} K | s | \\ \leq λ_{1} H_{M} | s | - λ_{1} K | s | \\ = - λ_{1} (K - H_{M}) | s | \\ = - λ_{1} ρ_{1} | s | \end{matrix}

(11)

that is,

{\overset{\cdot}{V}}_{1} \leq - \sqrt{2} ρ V_{1}^{1 / 2} \leq 0

(12)

where $ρ = λ_{1} ρ_{1}$ . According to equation (11), we can get $\int_{0}^{t} {\overset{\cdot}{V}}_{1} dt \leq \int_{0}^{t} - ρ | s | dt$ , then it goes further to get $\int_{0}^{t} | s | dt \leq (V_{1} (0) - V_{1} (t)) / ρ$ . Since $V_{1} (0)$ is bounded, and $V_{1} (t)$ is bounded and decreasing, and $\int_{0}^{t} | s | dt$ is bounded. According to Barbalat’s Lemma, one can obtain $lim_{t \to \infty} s (t) = 0$ . Using equation (2), we can also get that $e$ and $\overset{\cdot}{e}$ will converge to zero. Therefore, the asymptotic stability and convergence can be achieved. Moreover, using Lemma 1, the tracking error will converge to zero in a finite time $t_{r}$ as $t_{r} \leq | s (0) | / ρ$ .

However, there exists one disadvantage of the proposed FO TSMC, where certain system parameters are required. For example, the detected $U_{s}$ may not be a real one because of sensor fault such that $f (i_{c})$ will be uncertain to become the limitation to achieve high control performance. To cope with uncertain $f (i_{c})$ , a mature scheme is designing a uncertain estimator using FNN. Considering that the generalization performance of classical FNN is limited due to the missing structural adjustment capability, a novel FNN called RMCFNN is constructed in the next section.

Remark 1. Motivated by the above analysis in section “Introduction,” the boundary layer is an attractive technique to alleviate the effect of chattering using saturation function instead of sign function such that FO TSMC can be designed as

U_{FOTSM} = {\begin{matrix} - f (i_{c}) + {\overset{\cdot}{i}}_{c}^{*} - \frac{λ_{2}}{λ_{1}} D^{α} [sig {(e)}^{β}] - K sgn (s) & if | s | > δ \\ - f (i_{c}) + {\overset{\cdot}{i}}_{c}^{*} - \frac{λ_{2}}{λ_{1}} D^{α} [sig {(e)}^{β}] - K \frac{s}{δ} & if | s | \leq δ \end{matrix}

(13)

where $δ$ is a positive constant.

RMCFNN

In this section, the proposed novel MCFNN with recurrent framework is explained. After describing the signal propagation and updating rules of RMCFNN, the novel adaptive FO TSMC using RMCFNN for APF is presented.

Structure of RMCFNN

The detailed architecture of RMCFNN is given in Figure 2. The proposed FNN contains the following two parts: (1) cognitive component and (2) meta-cognitive component.

1. Cognitive component

A four-layer FNN structure, which is designed as the cognitive component in the proposed RMCFNN, consists of the input layer, the membership layer, the rule layer, and the output layer. The details of each layer are recited in the following form.

Layer 1—Input Layer: In this layer, each node transmits the input variables $x = [\begin{matrix} x_{1} & \dots & x_{i} & \dots & x_{m} \end{matrix}]$ to the next layer as follows

{node}_{i}^{1} (N) = x_{i}^{1} (N) ω_{i}^{1} (N) y_{o}^{4} (N - 1)

(14)

y_{i}^{1} (N) = f_{i}^{1} ({node}_{i}^{1} (N)) = {node}_{i}^{1} (N)

(15)

where $x_{i}^{1} (N)$ represents the $i th$ input of layer 1, ${node}_{i}^{1} (N)$ is the node function of input layer, N is the number of the sampling period, $ω_{i}^{1} (N)$ is the weight of output-feedback loop, which can be defined as $W_{r} = [\begin{matrix} ω_{1}^{1} (N) & . . . & ω_{m}^{1} (N) \end{matrix}]$ for simplify, and $y_{i}^{1} (N)$ is the output of layer 1.

Layer 2—Membership layer: The function of the node in Layer 2 is to execute fuzzification. Considering that the input range of the actual control system is generally bounded, the membership function of this layer is Gaussian function with better local characteristics. For every node in this layer, the relationship of the input and output can be described as follows

{node}_{j}^{2} (N) = - \sum_{i = 1}^{m} \frac{{‖ x_{i}^{2} (N) - c_{i}^{j} ‖}^{2}}{{(b_{i}^{j})}^{2}}

(16)

\begin{matrix} y_{j}^{2} (N) = f_{j}^{2} [{node}_{j}^{2} (N)] \\ = \exp [{node}_{j}^{2} (N)] \end{matrix}

(17)

where $x_{i}^{2} (N) = y_{i}^{1} (N)$ and $y_{j}^{2} (N)$ is the input and output of $j th$ node in this layer, $c_{i}^{j}$ ( $i = 1, . . ., m$ and $j = 1, . . ., n$ ) and $b_{i}^{j}$ , respectively, is the center and the width of the Gaussian function. For brevity, $b$ and $c$ including all center and width of Gaussian functions are denoted as

\begin{matrix} b = [\begin{matrix} b_{1} & b_{2} & \dots & b_{n} \end{matrix}]^{T} \\ = {[\begin{matrix} b_{1}^{1} \dots b_{m}^{1} & b_{1}^{2} \dots b_{m}^{2} \dots b_{1}^{n} \dots b_{m}^{n} \end{matrix}]}^{T} \in R^{N_{r} \times 1} \\ and c = [\begin{matrix} c_{1} & c_{2} & \dots & c_{n} \end{matrix}]^{T} \\ = {[\begin{matrix} c_{1}^{1} \dots c_{m}^{1} c_{1}^{2} \dots c_{m}^{2} \dots c_{1}^{n} \dots c_{m}^{n} \end{matrix}]}^{T} \in R^{N_{r} \times 1} \end{matrix}

where $N_{r} = m * n$ represents the total number of membership functions.

Layer 3—Rule Layer: The neuron of this layer, represented by $Π$ , multiplies the input variables using the fuzzy inference mechanism and output this product. For the $j th$ node, we have

{node}_{j}^{3} (N) = x_{j}^{3} (N)

(18)

\begin{matrix} y_{j}^{3} (N) = f_{j}^{3} ({node}_{j}^{3} (N)) \\ = Π_{i = 1}^{m} ω_{ij}^{3} (N) {node}_{j}^{3} (N) \end{matrix}

(19)

where $x_{j}^{3} (N) = y_{j}^{2} (N)$ and $y_{j}^{3} (N)$ , respectively, represent the $j th$ input and $j th$ output of Layer 3, $ω_{ij}^{3} (N)$ is the connection weights, which is set to 1 in real-time control.

Layer 4—Output Layer: Each node in layer 4, represented by $Σ$ , computes the summations of the overall outputs in previous layer as final output $y_{o} (o = 1, . . ., N_{o})$ of this layer in equations (20) and (21).

{node}_{o}^{4} (N) = \sum_{j = 1}^{n} ω_{j}^{4} (N) x_{j}^{4} (N)

(20)

y_{o}^{4} (N) = f_{o}^{4} ({node}_{o}^{4} (N)) = {node}_{o}^{4} (N)

(21)

In order to give a more concise form, the outputs of RMCFNN can be denoted as

Y = W Γ

(22)

where $W = [\begin{matrix} ω_{1}^{4} & ω_{2}^{4} & \dots & ω_{n}^{4} \end{matrix}]$ and $Γ = {[\begin{matrix} x_{1}^{4} & x_{2}^{4} & \dots & x_{n}^{4} \end{matrix}]}^{T}$ .

2. Meta-cognitive component

The aim of meta-cognitive component is to execute the online learning algorithm of RMCFNN including data learning strategy, data deleting strategy, and data reserving strategy (Lin et al., 2019). In contrast to the existing meta-cognitive component, specially, there does not exist data reserving strategy and data deleting strategy for the proposed RMCFNN in order to relax the restrictions in the real-time control. Thus, only data learning strategy are utilized in the proposed RMCFNN.

The data learning strategy comprises the structure self-adjusting and parameter learning. With considerations analogous to Lin et al. (2019) and Subramanian and Suresh (2012), spherical potential is regarded as the criterion to determine when to add a new rule. Different with classical error-based criteria, the novelty in the framework of spherical potential will be judged by the projection of the input variable on to a hyper-dimensional feature space. Generally, Gaussian function is selected for projection such that the spherical hyper-dimensional feature space can be represented by mean value and standard deviation of Gaussian function (Subramanian and Suresh, 2012). Let the origin of -dimensional space be ${node}_{o}^{3} = \frac{1}{n} \sum_{j = 1}^{n} {node}_{j}^{3} (x, c)$ . Then the squared distance from the hyper-dimensional feature centered at origin ${node}_{o}^{3}$ is used to depict the spherical potential, which is formulated as

ψ = ‖ {node}^{3} (x, c) - {node}_{o}^{3} ‖^{2}

(23)

According to Subramanian and Suresh (2012), one can get the expansion of equation (23) as follows

ψ = {node}^{3} (x, x) - \frac{2}{n} \sum_{j = 1}^{n} {node}_{j}^{3} (x, c) + \frac{1}{n^{2}} \sum_{j = 1}^{n} {node}_{j}^{3} (c, c)

(24)

For Gaussian function, it is obvious that ${node}^{3} (x, x)$ and $(1 / n^{2}) \sum_{j = 1}^{n} {node}_{j}^{3} (c, c)$ in equation (24) are constants so that they can be removed from the measure of novelty. Therefore, the spherical potential $ψ$ is denoted as

ψ = | - \frac{2}{n} \sum_{j = 1}^{n} {node}_{j}^{3} (x, c) |

(25)

If $ψ$ is higher, it indicates that the novelty of the current input information is lower, and the existing fuzzy rules are highly matched with the current input, so there is no need to add additional new fuzzy rules. On the contrary, if $ψ$ is lower, it indicates that the novelty of the current input information is higher, and the existing fuzzy rule is difficult to match the current input, so a new fuzzy rule needs to be considered. More details can be found in Subramanian and Suresh’s study (2012). According to the novelty measure, if $‖ x_{i} ‖ > E_{a}$ and $ψ < E_{s}$ , a new fuzzy rule is generated. $E_{s}$ and $E_{a}$ are the corresponding thresholds, which are important parameters. For example, higher $E_{a}$ means fewer rules, whereas lower $E_{a}$ means more rules. Therefore, a compromise between the network size and network generalization ability should be made.

Once a new fuzzy rule has been generated, the initial center, width, and corresponding weight should be assigned as follows

\begin{matrix} c_{n + 1} = x \\ b_{n + 1} = κ * min_{\forall j} ‖ x_{i} - c_{i}^{j} ‖, j = 1, . . ., n \\ ω_{n + 1}^{4} = 0 \end{matrix}

(26)

where $κ$ is a positive constant. In general, $κ$ is determined in the range [0.5, 0.9] based on our extensive experiments.

Besides rule generation approach, there also exists rule removal process under the following condition that the rule is insignificant for the network. From the condition

β_{j} = Γ_{j} | ω_{j}^{4} |, j = 1, . . ., n

(27)

where $β_{j}$ is utilized to measure the contribution of $j th$ rule, $ω_{j}^{4}$ is the output weight of the $j th$ rule in Layer 4. To design the measure criteria of contribution, full information of this rule should be considered. Thus, we can see that the contribution is determined by $Γ$ and the weights $ω_{j}^{4}$ . If $β_{j}$ is below a threshold value $E_{p}$ , it means that this rule has little impact on the final output, the corresponding rule will be determined as a redundant one, and it will be removed. In this study, $E_{p}$ is set to 0.2 by trial-and-error method.

After the structure of network has been determined, the parameter learning strategy will adjust the network parameters online. Considering real-time control, it is unnecessary to update the network parameters at every period such that the parameter update is triggered only if it satisfies $‖ x_{i} ‖ > E_{l}$ . In the proposed RMCFNN, the detailed learning approach for network parameters derived from Lyapunov framework is introduced in the next part.

FO TSMC using RMCFNN estimator

As mentioned above, the FO TSMC (8) including uncertain function $f (i_{c})$ cannot be implemented due to sensor fault and other uncertain circuit parameters. Within the proposed context, RMCFNN serves to estimate $f (i_{c})$ to assure high control performance, so the input of RMCFNN is $i_{c}$ , and the output is an estimate of $f (i_{c})$ . The block diagram of the developed FO TSMC is given in Figure 3.

Assumption 2. There exist optimal weight $W^{*}$ , center $c^{*}$ , width $b^{*}$ , and recurrent weight $W_{r}^{*}$ for estimating $f (i_{c})$ as $f (i_{c}) = W^{*} Γ^{*} + ε$ , where $Γ^{*} = Γ^{*} (x, c^{*}, b^{*}, W_{r}^{*})$ , $ε$ is the mapping error which is uniformly bounded as: $| ε | \leq ε_{b}$ , where $ε_{b}$ is a positive arbitrary small constant.

Theorem 2: Considering the dynamics (1), and the FOTSM surface (2), if the adaptive FO TSMC scheme is formulated as equation (28), and the corresponding updating strategies for RMCFNN parameters are chosen as equations (30)–(33), then the finite-time convergence characteristic should be achieved.

U_{FOTSM} = - \hat{f} (i_{c}) + {\overset{\cdot}{i}}_{c}^{*} - \frac{λ_{2}}{λ_{1}} D^{α} [sig (e)^{β}] - K sgn (s)

(28)

\hat{f} (i_{c}) = \hat{W} \hat{Γ}

(29)

{\overset{\cdot}{\tilde{W}}}^{T} = - η_{1} λ_{1} s \hat{Γ}

(30)

{\overset{\cdot}{\tilde{c}}}^{T} = - η_{2} λ_{1} s \hat{W} Γ_{c}

(31)

{\overset{\cdot}{\tilde{b}}}^{T} = - η_{3} λ_{1} s \hat{W} Γ_{b}

(32)

{\overset{\cdot}{\tilde{W}}}_{r}^{T} = - η_{4} λ_{1} s \hat{W} Γ_{W_{r}}

(33)

where $η_{1}$ , $η_{2}$ , $η_{3}$ , and $η_{4}$ are positive constants, $\hat{W}$ and $\hat{Γ}$ , denoted as $\hat{Γ} = \hat{Γ} (x, \hat{c}, \hat{b}, {\hat{W}}_{r})$ , is the estimate of $W^{*}$ and $Γ^{*}$ . Usually, it is necessary for a recurrent neural network (RNN) to be trained to obtain the weights. Nevertheless, the structure and parameters of trained RNN is fixed, and its generalization performance will deteriorate under dynamic condition. Thus, adaptive mechanism is designed, as shown in equations (30)–(33), to adjust the relevant parameters of RMCFNN.

Proof: Define the following estimation errors as $\tilde{c} = c^{*} - \hat{c}$ , $\tilde{b} = b^{*} - \hat{b}$ , ${\tilde{W}}_{r} = W_{r}^{*} - {\hat{W}}_{r}$ , respectively.

Using equation (28) and Taylor series expansion, one can get the following expression

\begin{matrix} f (i_{c}) - \hat{f} (i_{c}) = W^{*} Γ^{*} - \hat{W} \hat{Γ} + ε \\ = W^{*} (\hat{Γ} + \tilde{Γ}) - \hat{W} \hat{Γ} + ε \\ = W^{*} \hat{Γ} + W^{*} \tilde{Γ} - \hat{W} \hat{Γ} + ε \\ = \tilde{W} \hat{Γ} + \hat{W} \tilde{Γ} + \tilde{W} \tilde{Γ} + ε \\ = \tilde{W} \hat{Γ} + \hat{W} (Γ_{c} \cdot \tilde{c} + Γ_{b} \cdot \tilde{b} + Γ_{W_{r}} \cdot {\tilde{W}}_{r} + O_{h}) + \underset{ε_{0}}{\underset{︸}{\tilde{W} \tilde{Γ} + ε}} \end{matrix}

(34)

where $Γ_{c}$ , $Γ_{b}$ , and $Γ_{W_{r}}$ can be defined as equations (35)–(37)

Γ_{c} = {[\begin{matrix} \frac{\partial x_{1}^{4}}{\partial c_{1}^{1}} & \dots & \frac{\partial x_{1}^{4}}{\partial c_{m}^{n}} \\ ⋮ & ⋱ & ⋮ \\ \frac{\partial x_{n}^{4}}{\partial c_{1}^{1}} & \dots & \frac{\partial x_{n}^{4}}{\partial c_{m}^{n}} \end{matrix}]}_{c_{i}^{j} = {\hat{c}}_{i}^{j}}

(35)

Γ_{b} = {[\begin{matrix} \frac{\partial x_{1}^{4}}{\partial b_{1}^{1}} & \dots & \frac{\partial x_{1}^{4}}{\partial b_{i}^{j}} \\ ⋮ & ⋱ & ⋮ \\ \frac{\partial x_{n}^{4}}{\partial b_{1}^{1}} & \dots & \frac{\partial x_{n}^{4}}{\partial b_{i}^{j}} \end{matrix}]}_{b_{i}^{j} = {\hat{b}}_{i}^{j}}

(36)

Γ_{W_{r}} = {[\begin{matrix} \frac{\partial x_{1}^{4}}{\partial ω_{1}^{1}} & \dots & \frac{\partial x_{1}^{4}}{\partial ω_{m}^{1}} \\ ⋮ & ⋱ & ⋮ \\ \frac{\partial x_{n}^{4}}{\partial ω_{1}^{1}} & \dots & \frac{\partial x_{n}^{4}}{\partial ω_{m}^{1}} \end{matrix}]}_{ω_{i}^{1} = {\hat{ω}}_{i}^{1}}

(37)

Let us choose a Lyapunov function as

\begin{matrix} V_{2} = \frac{1}{2} s^{2} + \frac{1}{2 η_{1}} tr ({\tilde{W}}^{T} \tilde{W}) + \frac{1}{2 η_{2}} tr ({\tilde{c}}^{T} \tilde{c}) \\ + \frac{1}{2 η_{3}} tr ({\tilde{b}}^{T} \tilde{b}) + \frac{1}{2 η_{4}} tr ({\tilde{W}}_{r}^{T} {\tilde{W}}_{r}) \end{matrix}

(38)

Using equations (28) and (34) and differentiating $V_{2}$ leads to

\begin{matrix} {\overset{\cdot}{V}}_{2} = λ_{1} s \tilde{W} \hat{Γ} + λ_{1} s \hat{W} (Γ_{c} \cdot \tilde{c} + Γ_{b} \cdot \tilde{b} + Γ_{W_{r}} \cdot {\tilde{W}}_{r}) \\ + λ_{1} s (O_{ho} - K sgn (s) + H) + \frac{1}{η_{1}} tr ({\overset{\cdot}{\tilde{W}}}^{T} \tilde{W}) \\ + \frac{1}{η_{2}} tr ({\overset{\cdot}{\tilde{c}}}^{T} \tilde{c}) + \frac{1}{η_{3}} tr ({\overset{\cdot}{\tilde{b}}}^{T} \tilde{b}) + \frac{1}{η_{4}} tr ({\overset{\cdot}{\tilde{W}}}_{r}^{T} {\tilde{W}}_{r}) \end{matrix}

(39)

where $O_{ho} = ε_{0} + \hat{W} O_{h}$ .

Assuming $O_{ho}$ are uniformly bounded as: $| O_{ho} | \leq O_{M}$ , and using (30)–(33), (39) can be rewritten as

\begin{matrix} {\overset{\cdot}{V}}_{2} = λ_{1} s (H + O_{ho}) - λ_{1} K | s | \\ \leq λ_{1} | s | (H_{M} + O_{M} - K) \\ \leq - λ_{1} ρ_{2} | s | \\ \leq 0 \end{matrix}

(40)

where $K = H_{M} + O_{M} + ρ_{2}$ , and $ρ_{2}$ is a given constant. The negative semi-definite of ${\overset{\cdot}{V}}_{2}$ means that $s$ , $\tilde{W}$ , $\tilde{c}$ , $\tilde{b}$ , and ${\tilde{W}}_{r}$ are bounded to ensure the closed-loop stability. In addition, the finite-time convergence characteristic also can be achieved with Assumption 2 and Theorem 1. This completes the proof.

Remark 2: $\tilde{W}$ and $\tilde{Γ}$ represent the estimated errors for network parameters. They can be assumed to be bounded, which is a commonly handling approach for constructing NN estimator for real-world applications (Lin et al., 2019).

Simulation and experimental results

The performance of APF with the adaptive FO TSMC is demonstrated using simulation and experimental studies. A hardware prototype of APF is built, while the system parameters have been listed in Table 1. The proposed adaptive FO TSMC using RMCFNN in Matlab/Simulink using the “C” language is embedded into dSPACE 1104 with 20 kHz of sampling frequency. The results are observed and captured by Agilent DSO X3034A. Harmonic spectrum analysis is obtained by power-quality analysis module of the oscilloscope.

Table 1.

The parameters for simulation and experiment.

Grid voltage	$V_{s} = 24 V, f = 50 Hz$
Nonlinear loads	$R_{L} = 15 Ω$ , $C_{L} = 1 mF$
The main circuit	$L = 10 mH, R = 0.1 Ω, C = 2200 μ F$
Switching frequency	$f_{sw} = 20 kHz$

The parameters for the developed controller are specified as follows: $λ_{1} = 100$ , $λ_{2} = 1000$ , $λ_{3} = 1000$ , $α = 0.01$ , and $β = 15 / 13$ . The initial values of the center and the width of the Gaussian function are chosen as $c = [- 2 - 1 0 1 2]^{T}$ and $b = {[\begin{matrix} 1 & 1 & 1 & 1 & 1 \end{matrix}]}^{T}$ , the initial weights are set to random numbers between −1 and 1. $E_{a} = 5$ , $E_{s} = 0.02$ , $κ = 0.8$ , $E_{p} = 0.2$ , $η_{1} = 1,$ $η_{2} = 0.01,$ $η_{3} = 0.01$ , and $η_{4} = 0.1$ .

The performance of APF under steady state is depicted in Figures 4 –7, respectively. In Figure 4, there is grid voltage, load current, compensation current, and grid current. The total harmonic distortion (THD) of highly distorted load current is around 30.24%. The control system of APF can cancel harmonics to a large extent thanks to adaptive FO TSMC using RMCFNN. Figure 5 shows the satisfactory harmonic suppression performance, in which the grid current is purified into be sinusoidal with the THD of 3.08%. Figure 6 depicts the number of rules. One can see that the number of rules is changing to obtain the best results using RMCFNN. Figure 7 depicts the network parameters to demonstrate that the closed-loop system tends to be stable.

The performance of APF under load change are depicted in Figures 8 and 9. As the load increases, compensation current to be supplied by APF also increases such that grid current gets purified efficiently. The performance with load reduction also can be seen from Figure 9. Therefore, one can say that APF with FO TSMC using RMCFNN reveal superior dynamic and steady-state performances.

In addition, Figure 10 and Table 2 are given to further validate the control performance of the proposed methodology against other classical methods, for example, TSMC, FNN, and back-propagation neural network (BPNN). In Figure 10, $e_{RMCFNN}$ , $e_{TSMC}$ , $e_{FNN}$ , and $e_{BPNN}$ represent tracking errors using RMCFNN, TSMC, FNN, and BPNN. Compared with $e_{TSMC}$ and $e_{RMCFNN}$ can achieve a faster convergence speed owing to FO TSMC. Thus, one can conclude that the finite-time convergence performance can be enhanced using FO TSMC. According to the results of FNN and BPNN, the proposed RMCFNN also possesses better control performance than other similar methods using NN. Moreover, Table 2 gives a detailed statistical analysis among these strategies, in which THD and root mean square error (RMSE), defined as $RMSE = \sqrt{(1 / N) \sum_{i = 1}^{N} e^{2} (i)}$ , are selected as two indicators. From Table 2, one can find that using the proposed RMCFNN can get the smallest THD and RMSE. Therefore, the proposed control strategy possesses better control performance than other classical approaches.

Table 2.

THD and RMSE of different control methods.

Method	THD (%)	RMSE
BPNN	5.71	4.5277
FNN	4.60	4.3468
TSMC	3.35	3.5362
RMCFNN	1.66	1.7623

THD: total harmonic distortion; RMSE: root mean square error.

Figure 11 depicts the tracking errors under sensor fault, in which $e_{RMCFNN}$ , $e_{TSMC}$ , and $e_{TSMCSF}$ represent tracking errors using RMCFNN, TSMC with normal sensor, and TSMC with sensor fault, respectively. In practical applications, when there exists a fault of voltage sensor, we cannot obtain a correct $U_{s}$ . In this case, $f (i_{c}) = - (R_{1} / L_{1}) i_{c} + (U_{s} / L_{1})$ is not right. If we use incorrect $f (i_{c})$ to implement the controller, the tracking performance will be deteriorated as shown in Figure 11. By utilizing the proposed RMCFNN, $f (i_{c})$ is estimated without $U_{s}$ . It means that the control capability will not be affected even if voltage sensors run in an unnormal way as shown in Figure 11.

Remark 3: The initial number of neurons in each layer of RMCFNN can be selected as follows. The number of neurons in the input layer should be the same as the number of inputs, and the number of neurons in the output layer should be the same as the number of outputs. The middle two hidden layer neurons are related to each other, which is determined by the number of fuzzy rules. By using trial-and-error method and relevant experience, here we design five initial rules.

Remark 4: In general, RMCFNN can be termed as type-1 fuzzy system. It has been demonstrated that type-2 fuzzy system possesses higher control performance than type-1 fuzzy system (33)–(35) (Mohammadzadeh and Kumbasar, 2020; Mohammadzadeh and Rathinasamy, 2020; Mosavi et al., 2020). Therefore, using type-2 fuzzy system to construct RMCFNN is our future direction.

Remark 5: The working principle of APF is to generate appropriate compensation current by changing the switching state of PWM converter, and inject the compensation current into power grid to suppress harmonic current. The switching frequency of the PWM converter we use is 20 kHz, which makes the switching state change every 50 µs. The periodic switching of converter will inevitably cause unsmooth current signals, which is the reason for the fluctuation of trajectories in all figures. In addition, trajectories are obtained by the current probe. Due to the noise interference in measurement process, the trajectories have unavoidable fluctuations caused by noise signals.

Conclusion

In this paper, an observer-based FO TSMC using RMCFNN is designed for APF. The uncertainties are effectively processed using the proposed RMCFNN to estimate the uncertain function for APF. Based on the experimental results, it is evident that the proposed FO TSMC is able to enhance the dynamic performance of APF. In addition, the performance of the proposed FO TSMC is compared with the conventional and recently reported control approach to show its superiority in harmonic compensation. Despite the sudden change in source voltage, the proposed FO TSMC also can ensure the harmonic suppression ability. Furthermore, the THD of source current is well maintained below 5% in different situations. It is noteworthy that the parameters used in the controller are selected by trail-and-error methodology to achieve the best tracking performance. Nevertheless, numerous tests will be required to obtain these parameters. Thus, parameter selection strategy using adaptive mechanism will be explored in our future work. Moreover, Gaussian membership function also can be replaced by other feasible membership functions such that the generalization performance of the RMCFNN can be further enhanced.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (grant nos. 62103132 and 6200 3132), the Changzhou Sci&Tech Program (grant no. CJ2020 0067), and the Fundamental Research Funds for the Central Universities (grant nos. B200202215 and B220202022).

ORCID iD

Yundi Chu

References

Balootaki

Rahmani

Moeinkhah

, et al. (2020) On the synchronization and stabilization of fractional-order chaotic systems: Recent advances and future perspectives. Physica A: Statal Mechanics and Its Applications 551: 124203.

Chu

Fei

Hou

(2018) Dynamic global proportional integral derivative sliding mode control using radial basis function neural compensator for three-phase active power filter. Transactions of the Institute of Measurement and Control 40: 3549–3559.

Chu

Fei

Hou

(2020) Adaptive global sliding-mode control for dynamic systems using double hidden layer recurrent neural network structure. IEEE Transactions on Neural Networks and Learning Systems 31(4): 1297–1309.

Fei

Chu

(2020) Double hidden layer output feedback neural adaptive global sliding mode control of active power filter. IEEE Transactions on Power Electronics 35(3): 3069–3084.

Fei

(2018) Adaptive fractional order sliding mode controller with neural estimator. Journal of The Franklin Institute 355(5): 2369–2391.

Feng

Zhou

Han

, et al. (2020) Integral-type sliding-mode control for a class of mechatronic systems with gain adaptation. IEEE Transactions on Industrial Informatics 16(8): 5357–5368.

Guzman

Vicuna

Morales

, et al. (2016) Model-based control for a three-phase shunt active power filter. IEEE Transactions on Industrial Electronics 63(7): 3998–4007.

Hekss

Abouloifa

Lachkar

, et al. (2021) Nonlinear adaptive control design with average performance analysis for photovoltaic system based on half bridge shunt active power filter. International Journal of Electrical Power & Energy Systems 125: 106478.

Hou

Fei

(2020) A self-organizing global sliding mode control and its application to active power filter. IEEE Transactions on Power Electronics 35(7): 7640–7652.

10.

Hou

Chu

Fei

(2021c) Adaptive type-2 fuzzy neural network inherited terminal sliding mode control for power quality improvement. IEEE Transactions on Industrial Informatics 17(11): 7564–7574.

11.

Hou

Chu

Fei

(2021b) Intelligent global sliding mode control using recurrent feature selection neural network for active power filter. IEEE Transactions on Industrial Electronics 68(8): 7320–7329.

12.

Hou

Chu

Fei

(2021a) Robust intelligent control for a class of power-electronic converters using neuro-fuzzy learning mechanism. IEEE Transactions on Power Electronics 36: 9441–9452.

13.

Hou

Fei

Chen

, et al. (2019) Finite-time adaptive fuzzy-neural-network control of active power filter. IEEE Transactions on Power Electronics 34(10): 10298–10313.

14.

Hou

Wang

Chu

, et al. (2022) Neural-observer-based terminal sliding mode control: Design and application. IEEE Transactions on Fuzzy Systems. Epub ahead of print 18 March. DOI: 10.1109/TFUZZ.2022.316061.

15.

Kilbas

Srivastava

Trujillo

(2006) Fractional integrals and fractional derivatives in theory and applications of fractional differential equations. North-Holland Mathematics Studies 204: 69–133.

16.

Yang

Zhu

, et al. (2017) State/model-free variable-gain discrete sliding mode control for an ultraprecision wafer stage. IEEE Transactions on Industrial Electronics 64(8): 6695–6705.

17.

Lin

Chen

Hsu

(2019) Intelligent backstepping control using recurrent feature selection fuzzy neural network for synchronous reluctance motor position servo drive system. IEEE Transactions on Fuzzy Systems 27(3): 413–427.

18.

Mohammadzadeh

Kumbasar

(2020) A new fractional-order general type-2 fuzzy predictive control system and its application for glucose level regulation. Applied Soft Computing Journal 91: 106241.

19.

Mohammadzadeh

Rathinasamy

(2020) Energy management in photovoltaic battery hybrid systems: A novel type-2 fuzzy control. International Journal of Hydrogen Energy 45: 20970–20982.

20.

Mosavi

Qasem

Shokri

, et al. (2020) Fractional-order fuzzy control approach for photovoltaic/battery systems under unknown dynamics variable irradiation and temperature. Electronics 9: 1455.

21.

(2018) Dynamic behavior of terminal sliding mode control. IEEE Transactions on Industrial Electronics 65(4): 3480–3490.

22.

Nguyen

Chen

Fan

(2017) Discrete-time quasi-sliding-mode control with prescribed performance function and its application to Piezo-actuated positioning systems. IEEE Transactions on Industrial Electronics 65(1): 942–950.

23.

Nojavanzadeh

Badamchizadeh

(2016) Adaptive fractional-order non-singular fast terminal sliding mode control for robot manipulators. IET Control Theory & Applications 10(13): 1565–1572.

24.

Ray

Das

Mohanty

(2019) Fuzzy-controller-designed-PV-based custom power device for power quality enhancement. IEEE Transactions on Energy Conversion 34(1): 405–414.

25.

Rong

Yang

Wong

, et al. (2017) A novel meta-cognitive fuzzy-neural model with backstepping strategy for adaptive control of uncertain nonlinear systems. Neurocomputing 230: 332–344.

26.

Subramanian

Suresh

(2012) A meta-cognitive sequential learning algorithm for neuro-fuzzy inference systems. Applied Soft Computing Journal 12(11): 3603–3614.

27.

Sun

(2017) Practical tracking control of linear motor with adaptive fractional order terminal sliding mode control. IEEE/ASME Transactions on Mechatronics 22(6): 2643–2653.

28.

Sun

Qiu

Karimi

, et al. (2021) Event-triggered robust fuzzy adaptive finite-time control of nonlinear systems with prescribed performance. IEEE Transactions on Fuzzy Systems 29(6): 1460–1471.

29.

Wang

Chen

(2020) Adaptive fuzzy dynamic surface control of nonlinear constrained systems with unknown virtual control coefficients. IEEE Transactions on Fuzzy Systems 28(8): 1737–1747.

30.

Wang

Dragicevic

Gao

, et al. (2021a) Neural network based model predictive controllers for modular multilevel converters. IEEE Transactions on Energy Conversion 36(2): 1562–1571.

31.

Wang

Xia

Sun

, et al. (2021b) Observer-based adaptive event-triggered tracking control for nonlinear MIMO systems based on neural networks technique. Neurocomputing 433: 71–82.

32.

Wang

, et al. (2016) Practical tracking control of robot manipulators with continuous fractional-order nonsingular terminal sliding mode. IEEE Transactions on Industrial Electronics 63(10): 6194–6204.

33.

Wang

Liu

Wang

, et al. (2020) Time-delay control using a novel nonlinear adaptive law for accurate trajectory tracking of cable-driven robots. IEEE Transactions on Industrial Informatics 16(8): 5234–5243.

34.

(2018) Composite learning control of flexible-link manipulator using NN and DOB. IEEE Transactions on Systems, Man, and Cybernetics: Systems 48: 1979–1985.

35.

Zhang

Wang

Yan

(2021) Fast finite time adaptive neural network control for a class of uncertain nonlinear systems subject to unmodeled dynamics. Information Sciences 565: 306–325.

36.

Zhu

Liu

Wen

(2020) Sliding-mode-control-theory-based adaptive general type-2 fuzzy neural network control for power-line inspection robots. Neurocomputing 401: 281–294.