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Abstract

Due to the complexities existing in the electric load simulator, this article develops a high-performance nonlinear adaptive controller to improve the torque tracking performance of the electric load simulator, which mainly consists of an adaptive fuzzy self-recurrent wavelet neural network controller with variable structure (VSFSWC) and a complementary controller. The VSFSWC is clearly and easily used for real-time systems and greatly improves the convergence rate and control precision. The complementary controller is designed to eliminate the effect of the approximation error between the proposed neural network controller and the ideal feedback controller without chattering phenomena. Moreover, adaptive learning laws are derived to guarantee the system stability in the sense of the Lyapunov theory. Finally, the hardware-in-the-loop simulations are carried out to verify the feasibility and effectiveness of the proposed algorithms in different working styles.

Keywords

Complementary controller self-recurrent wavelet neural network Lyapunov theory electric load simulator

Introduction

The electric load simulator (ELS) is a crucial equipment in the hardware-in-the-loop (HIL) experiments, widely used in the gun, aerospace, and marine engineering,^1–3 which is designed to generate the torque to simulate the dynamic loads acting on the actuator system and greatly accelerates the development of the research and product of the actuator system. The ELS of steering engines is commonly studied, which is different from the ELS for the gun control. The former is designed to simulate resisting torque, such as the air and water resistance, the latter is used to simulate the resisting and inertia torque. Apart from the same resisting torque, the inertia torque is also time variable, which can be effectively simulated comparatively, because the variations of the barrel angle and the number of the shells can be computed. In a word, the nonlinearities and uncertainties are playing great negative effects on the performance of the ELS, like the coupling of the torque motor and the load motor, the friction and backlash.^4,5 Therefore, how to improve the ELS torque tracking performance has become a research hotspot recently. The ordinary methods adopt the feed-forward compensation using the actuator’s position and velocity signals,^6,7 or the dual-loop scheme,⁸ but the constructed model cannot wholly describe the friction and backlash nonlinearities, especially the approximation precision shows worse in the practical working conditions, so intelligent controllers with nonlinear characteristics and high precisions have been proposed to substitute the mechanism model.

The feed-forward neural networks (NN) with amazing learning ability, parallel computation, and remarkable generalization ability are widely applied to approximate the nonlinear and uncertain system, but they also have the drawbacks of the local minimum and low convergence rate.^9,10 In order to solve these problems, the activation functions and the learning structures of the NN must be selected appropriately, which keep the model with the high computation efficiency and short training time. The wavelet functions with the advantages of time–frequency localization are incorporated into the NN, and the constructed wavelet neural networks (WNN) are widely applied to the engineering research area as the function approximation and signal processing.^11,12 Considering the sudden changes and the external disturbance of the ELS, the self-recurrent WNN (SW) with storing the previous data is adopted to maintain the system stability.^13,14 Meantime, considering the Takagi–Sugeno–Kang (TSK) fuzzy model with SW has generated a new network named fuzzy SW (FSW), which does well in enhancing the function approximation accuracy and generalization ability.¹⁵ However, it is hard to determine the learning structure of the FSW because of the trade-off between the learning performance and the number of neurons. If the number of neurons chosen is too large, the computation loading may be too heavy for the practical applications although the learning performance is very good. To avoid this problem, many structure learning algorithms are proposed for the adaption of the NN,^16,17 and the number of neurons can be adjusted dynamically by the significant index or rules. In practical control applications, it is impatient duty to have a systematic method of ensuring the stability of the overall system. Recently, many NN methods are introduced based on the Lyapunov stability theory,^16–19 but what we need to resolve and focus on is the determination of the assumptions and adaptive learning laws. With the above-mentioned motivations, the radial basis function (RBF) NN with a self-recurrent and variable structure (VS) not only guarantees the stability of the control system,¹⁹ but also does not need the constrained conditions and knowledge of the controlled system.^20,21 In order to further decrease the approximation error, many compensation methods are proposed aiming at eliminating the external disturbances, which reduce the approximation error and improve the control precision.^22,23

In this article, the proposed VSFSWC and complementary controller (CC) is a novel control structure, which comprises the merits of the fuzzy, self-recurrent, VS, WNN, and compensation method for the ELS system. The VSFSWC used as a main controller is online estimating the dynamic system, and the CC dispels the effect of the approximation error existing in the ELS system. Furthermore, the VS learning algorithm effectively decreases the computation loading, and the adaptive learning laws based on the Lyapunov theory are developed to tune the parameters of the WNN. Not only the fast convergence of the torque tracking error can be achieved but also the system stability can be guaranteed. Finally, the proposed control algorithms are comparatively applied to the HIL simulation system for the ELS, and the experimental results show that the proposed VSFSWC and CC can achieve a favorable control performance.

Problem statement and ideal backstepping controller design

Problem statement

Consider the characteristics of the ELS system which are described as the nth order nonlinear equation in the following^24,25

y_{n} (t) = f (y, t) + g (y, t) u (t) + d (t)

(1)

where $y (t)$ and $u (t)$ denote the control input and output of the system, respectively; $y_{n} (t)$ is the nth order derivative of $y (t)$ ; $f (y, t)$ and $g (y, t)$ are the system dynamic variables; $d (t)$ is the unknown but bounded external disturbances, which is introduced by the self-coupling, friction, and backlash.

Ideal backstepping controller design

The control objective is to find a control law so that the state $y (t)$ can track the reference command $y_{d} (t)$ closely. The ideal backstepping controller can be constructed step-by-step as follows:

Step 1. The tracking error is defined as follows

e_{1} (t) = y_{1} (t) - y_{d} (t)

(2)

Then, a stabilization function is as follows

σ_{1} (t) = {\overset{\cdot}{y}}_{d} (t) - λ_{1} e_{1} (t)

(3)

where $λ_{1}$ is a positive constant.

Step 2. The second control input $e_{2} (t)$ is expressed as follows

e_{2} (t) = y_{2} (t) - σ_{1} (t)

(4)

σ_{2} (t) = {\overset{\cdot}{σ}}_{1} (t) - e_{1} (t) - λ_{2} e_{2} (t)

(5)

where $λ_{2}$ is a positive constant; thus, the derivative of the error $e_{1} (t)$ is obtained as follows

\begin{array}{l} {\dot{e}}_{1} (t) = {\dot{y}}_{1} (t) - {\dot{y}}_{d} (t) = {\dot{y}}_{1} (t) - σ_{1} (t) - λ_{1} e_{1} (t) \\ = e_{2} (t) - λ_{1} e_{1} (t) \end{array}

(6)

Step $i (3 \leq i \leq n - 1)$ . An error $e_{i} (t)$ and a stabilization function $σ_{i} (t)$ are defined

e_{i} (t) = y_{i} (t) - σ_{i - 1} (t)

(7)

σ_{i} (t) = {\overset{\cdot}{σ}}_{i - 1} (t) - e_{i - 1} (t) - λ_{i} e_{i} (t)

(8)

where $λ_{i}$ is a positive constant; thus, the derivative of the error $e_{i - 1} (t)$ is obtained as follows

\begin{matrix} {\overset{\cdot}{e}}_{i - 1} (t) = {\overset{\cdot}{y}}_{i - 1} (t) - {\overset{\cdot}{σ}}_{i - 2} (t) \\ = y_{i} (t) - σ_{i - 1} (t) - e_{i - 2} (t) - λ_{i - 1} e_{i - 1} (t) \\ = e_{i} (t) - λ_{i - 1} e_{i - 1} (t) - e_{i - 2} (t) \end{matrix}

(9)

Step n. An error $e_{n} (t)$ and a stabilization function $σ_{n} (t)$ are defined

e_{n} (t) = y_{n} (t) - σ_{n - 1} (t)

(10)

{\overset{\cdot}{e}}_{n} (t) = {\overset{\cdot}{y}}_{n} (t) - {\overset{\cdot}{σ}}_{n - 1} (t) = u (t) + χ (y, t)

(11)

where $y_{n} (t)$ is the nth order derivative of y(t), and the nonlinear function $χ (y, t)$ is in the following

χ (y, t) = - {\overset{\cdot}{σ}}_{n - 1} (t) + (1 - \frac{1}{g (y, t)}) {\overset{\cdot}{y}}_{n} (t) + \frac{f (y, t)}{g (y, t)}

(12)

Step $n + 1$ . The ideal backstepping controller can be defined

u_{IBC}^{*} = - χ (y, t) - e_{n - 1} (t) - λ_{n} e_{n} (t)

(13)

where $λ_{n}$ is a positive constant; substituting the ideal backstepping control law $u = u_{IBC}^{*}$ into (11), we can obtain the following equation

{\overset{\cdot}{e}}_{n} (t) = - e_{n - 1} (t) - λ_{n} e_{n} (t)

(14)

Step $n + 2$ . Consider the candidate Lyapunov function in the following form

V_{1} (t) = \frac{e_{1}^{2} (t)}{2} + \frac{e_{2}^{2} (t)}{2} + \dots + \frac{e_{n}^{2} (t)}{2}

(15)

Then the derivative of $V_{1} (t)$ will be

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) & = e_{1} (t) {\overset{\cdot}{e}}_{1} (t) + e_{2} (t) {\overset{\cdot}{e}}_{2} (t) + \dots + e_{n} (t) {\overset{\cdot}{e}}_{n} (t) \\ = e_{1} (t) (e_{2} (t) - λ_{1} e_{1} (t)) + e_{2} (t) (e_{3} (t) - λ_{2} e_{2} (t) - e_{1} (t)) \\ + \dots + e_{n} (t) (- e_{n - 1} (t) - λ_{n} e_{n} (t)) \\ = - λ_{1} e_{1}^{2} (t) - λ_{2} e_{2}^{2} (t) - \dots - λ_{n} e_{n}^{2} (t) \leq 0 \end{matrix}

(16)

Since ${\overset{\cdot}{V}}_{1} (t)$ is negative semi-definite, which means that the equation $V_{1} (t) \leq V_{1} (0)$ can be achieved, it implies that $e_{1} (t), e_{2} (t), \dots, e_{n} (t)$ are bounded. Defining the equation $Ξ (τ) \equiv λ_{1} e_{1}^{2} (τ) + λ_{2} e_{2}^{2} (τ) + \dots + λ_{n} e_{n}^{2} (τ) \leq - {\overset{\cdot}{V}}_{1} (t)$ and integrating $Ξ (τ)$ with respect to time, the following equation is obtained

\int_{0}^{t} Ξ (τ) d τ \leq V_{1} (t) - V_{1} (0)

(17)

Because $V_{1} (0)$ is bounded and $V_{1} (t)$ is non-increasing and bounded, the result can be obtained as follows

lim_{t \to \infty} \int_{0}^{t} Ξ (τ) d τ \leq \infty

(18)

In addition, since $\overset{\cdot}{Ξ} (τ)$ is bounded, by Barbalat’s Lemma, $lim_{t \to \infty} Ξ (t) = 0$ .²⁶ As a result, the asymptotically stability can be guaranteed in the sense of the Lyapunov theory. However, the dynamic variables $f (y, t)$ and $g (y, t)$ cannot be entirely known in practical applications. Moreover, the external disturbance $d (t)$ causes an outcome of large chattering phenomena, which deteriorates the control precision and the robustness of the ELS system. Therefore, the ideal backstepping controller cannot be achieved precisely.

Adaptive complementary fuzzy self-recurrent WNN controller design

Self-recurrent WNN

The structure of the self-recurrent WNN controller (SWC) is shown in Figure 1, which is comprised of an input layer, wavelet layer, and output layer. The input of the input layer is $e (t) = [e_{1} (t), e_{2} (t), \dots, e_{n} (t)]^{T}$ , where T is the transpose and n is the number of dimensions.

Layer 1—Input layer: No function is performed in this layer; the node only transmits input values to layer 2.

Layer 2—Wavelet layer: In this layer, each activation function can be represented²⁷

ϕ (x) = 2^{a / 2} ϕ (2^{a} x - b)

(19)

where a and b are the dilation and translation parameters, respectively, which are self-learning based on the following adaptive laws. The Mexican hat mother wavelet function is defined as follows

φ (x) = (1 - ‖ x ‖^{2}) \exp (\frac{- {‖ x ‖}^{2}}{2})

(20)

where $‖ x ‖ = x^{T} x$ . Thus, the activation function of the jth neuron connected with the input layer is described as

\begin{array}{l} ϕ_{i j} (e_{i}) = 2^{a_{i j} / 2} (1 - {‖ 2^{a_{i j}} e_{i} - b_{i j} ‖}^{2}) \exp (\frac{- {‖ 2^{a_{i j}} e_{i} - b_{i j} ‖}^{2}}{2}), \\ i = 1, 2, \dots, n; j = 1, 2, \dots, m \end{array}

(21)

where m is the number of the neurons.

Layer 3—Output layer: With the self-recurrent rules, the output of the SWC is as follows¹⁴

γ_{j} = Π_{i = 1}^{n} ϕ_{ij} + η_{0} γ_{j - 1}

(22)

where $η_{0} \in [0, 1]$ is the self-recurrent feedback gain. The self-recurrent WNN differs from the typical WNN by the self-feedback link with fixed coefficient $η_{0}$ in the context node.

Figure 1.

Structure of the fuzzy self-recurrent WNN.

Fuzzy self-recurrent WNN

In comparison with the Mamdani, the fuzzy logic systems (FLS), and the TSK, the TSK model is more useful for the application of real-time operations. However, the TSK cannot wholly reflect the characteristics of the ELS system with limited rules; the SWC is incorporated into the TSK model to improve the control rapidity and accuracy. A typical fuzzy WNN for approximating the nonlinear function can be depicted by the linguistic rule.¹¹

If $x_{1} is A_{1}^{k}, x_{2} is A_{2}^{k}, \dots, x_{i} is A_{i}^{k}$ , then $χ_{k} = μ_{k} γ_{k}, k = 1, 2, \dots, m$ , where $x_{i}$ is the ith input variable of the system for $i = 1$ ; $A_{i}^{k}$ is the linguistic term characterized by a Gaussian-type membership function $μ_{k}$ for $i = 1$ , which is calculated as follows

{\hat{μ}}_{k} = Π_{i = 1}^{n} \exp (\frac{- {(x_{i} - c_{ik})}^{2}}{d_{ik}^{2}})

(23)

where $c_{ik}$ and $d_{ik}$ denote the center and scaling parameters of the membership function associated with rule k, then¹⁵

μ_{k} = \frac{{\hat{μ}}_{k}}{\sum_{k = 1}^{m} {\hat{μ}}_{k}}

(24)

In summary, the output of the fuzzy self-recurrent WNN controller (FSWC) can be obtained

χ = \sum_{k = 1}^{m} χ_{k} = \sum_{k = 1}^{m} μ_{k} γ_{k} = μ^{T} γ (a, b)

(25)

where $μ = [μ_{1}, μ_{2}, \dots, μ_{m}]$ , $γ = [γ_{1}, γ_{2}, \dots, γ_{m}]$ , $a = [a_{11}, \dots, a_{1 m}, \dots, a_{n 1}, \dots, a_{nm}]$ , and $b = [b_{11}, \dots, b_{1 m}, \dots, b_{n 1}, \dots, b_{nm}]$ .

Considering the universal function approximation property of the FSWC, there exists an expansion of (25) so that it can uniformly approximate the ideal backstepping controller^28,29

χ^{*} = μ^{* T} γ^{*} (a^{*}, b^{*}) + Δ_{1}

(26)

where $Δ_{1}$ is the approximation error; $μ^{*}$ , $γ^{*}$ , $a^{*}$ , and $b^{*}$ are the ideal parameter vectors of $μ$ , $γ$ , a, and b, respectively. Since the ideal parameter vectors cannot be obtained, an estimation FSWC is defined as follows

\hat{χ} = {\hat{μ}}^{T} \hat{γ} (\hat{a}, \hat{b})

(27)

where $\hat{μ}$ , $\hat{γ}$ , $\hat{a}$ , and $\hat{b}$ are the estimation parameter vectors of $μ$ , $γ$ , a, and b, respectively. Subtracting equation (27) from equation (26), an estimation error is defined

\begin{matrix} \tilde{χ} & = χ^{*} - \hat{χ} = μ^{* T} γ^{*} (a^{*}, b^{*}) + Δ_{1} - {\hat{μ}}^{T} \hat{γ} (\hat{a}, \hat{b}) \\ = {\tilde{μ}}^{T} \hat{γ} + {\hat{μ}}^{T} \tilde{γ} + {\tilde{μ}}^{T} \tilde{γ} + Δ_{1} \end{matrix}

(28)

where $\tilde{μ} = μ^{*} - \hat{μ}$ and $\tilde{γ} = γ^{*} - \hat{γ}$ . The linearization technique is employed to transform the membership functions into a partially linear form so that the expansion of $\tilde{γ}$ in a Taylor series can be obtained³⁰

\begin{matrix} \tilde{γ} = [\begin{matrix} {\tilde{γ}}_{1} \\ ⋮ \\ {\tilde{γ}}_{j} \\ ⋮ \\ {\tilde{γ}}_{m} \end{matrix}] = [\begin{matrix} (\frac{\partial {\tilde{γ}}_{1}}{\partial a}) \\ ⋮ \\ (\frac{\partial {\tilde{γ}}_{j}}{\partial a}) \\ ⋮ \\ (\frac{\partial {\tilde{γ}}_{m}}{\partial a}) \end{matrix}] |_{a = \hat{a}} (a^{*} - \hat{a}) + [\begin{matrix} (\frac{\partial {\tilde{γ}}_{1}}{\partial b}) \\ ⋮ \\ (\frac{\partial {\tilde{γ}}_{j}}{\partial b}) \\ ⋮ \\ (\frac{\partial {\tilde{γ}}_{m}}{\partial b}) \end{matrix}] |_{b = \hat{b}} \\ (b^{*} - \hat{b}) + ξ \\ = γ_{a} \tilde{a} + γ_{b} \tilde{b} + ξ \end{matrix}

(29)

where $\tilde{a} = a^{*} - \hat{a}$ and $\tilde{b} = b^{*} - \hat{b}$ ; $ξ$ is a vector of higher-order terms. $(\partial {\tilde{γ}}_{j} / \partial a)$ and $(\partial {\tilde{γ}}_{j} / \partial b)$ are defined

{[\frac{\partial {\tilde{γ}}_{j}}{\partial a}]}^{T} = [\underset{(j - 1) \times n}{\underset{⏟}{0, \dots, 0}}, \frac{\partial {\tilde{γ}}_{j}}{\partial a_{1 j}}, \dots, \frac{\partial {\tilde{γ}}_{j}}{\partial a_{nj}}, \underset{(m - j) \times n}{\underset{⏟}{0, \dots, 0}}]

(30)

{[\frac{\partial {\tilde{γ}}_{j}}{\partial b}]}^{T} = [\underset{(j - 1) \times n}{\underset{⏟}{0, \dots, 0}}, \frac{\partial {\tilde{γ}}_{j}}{\partial b_{1 j}}, \dots, \frac{\partial {\tilde{γ}}_{j}}{\partial b_{nj}}, \underset{(m - j) \times n}{\underset{⏟}{0, \dots, 0}}]

(31)

Substituting equation (28) into equation (29) yields

\begin{matrix} \tilde{χ} = {\tilde{μ}}^{T} (\hat{γ} + γ_{a} \tilde{a} + γ_{b} \tilde{b} + ξ) + {\hat{μ}}^{T} (γ_{a} \tilde{a} + γ_{b} \tilde{b} + ξ) + Δ_{1} \\ = {\tilde{μ}}^{T} \hat{γ} + {\hat{μ}}^{T} (γ_{a} \tilde{a} + γ_{b} \tilde{b}) + Δ_{2} \end{matrix}

(32)

where $Δ_{2} = {\tilde{μ}}^{T} \tilde{γ} + {\tilde{μ}}^{T} ξ + Δ_{1}$ . $Δ_{2}$ denotes the lump of the approximation error and is limited by $| Δ_{2} | \leq E_{0}$ , where $E_{0}$ is a positive constant.

VS learning algorithm

Considering the trade-off between the computation loading and the learning performance, an online VS learning algorithm is proposed to determine whether or not to add and prune the number of the neurons. The output of the SWC is represented as the index to add a neuron

γ_{min} = min_{1 \leq j \leq m (k)} | γ_{j} |

(33)

where $n \times m (q)$ is the number of the existing neurons at the qth control interval. If $γ_{min}$ is even bigger than a pre-specified and positive threshold $γ_{th}$ , a new neuron should be added. The parameters of the new neuron layer are selected as follows

{\begin{matrix} a_{m (q + 1)} = a_{0} \\ b_{m (q + 1)} = b_{0} \\ μ_{m (q + 1)} = 0 \\ m (q + 1) = m (q) + 1 \end{matrix}

(34)

where $a_{0}$ and $b_{0}$ are pre-specified constants. In order to avoid the infinite growth of the number of neurons, the other significant index $I_{j} (q)$ is adopted to evaluate the importance of the jth neuron at the qth control interval and defined as follows²⁴

I_{j} (q + 1) = {\begin{matrix} \begin{matrix} I_{j} (q) \exp (- τ) \end{matrix} \\ I_{j} (q) \end{matrix} \begin{matrix} \begin{matrix} if & γ_{j} (q) > γ_{0} \end{matrix} \\ \begin{matrix} if & γ_{j} (q) \leq γ_{0} \end{matrix} \end{matrix}

(35)

where the initial value setting of $I_{j} (q)$ is 1; $γ_{0}$ is the reduction threshold; $τ$ is the reduction speed constant. If $I_{j} \leq I_{th}$ is satisfied, where Ith is a pre-specified threshold, the jth neuron will be pruned.

Stability analysis

The developed VSFSWC and CC is shown in Figure 2, which is composed of a VSFSWC and a CC. The controller output is given as follows³¹

u_{VC} = u_{VSFSWC} + u_{CC} = - \hat{χ} - e_{n - 1} (t) - λ_{n} e_{n} (t) + u_{CC}

(36)

where the output of VSFSWC $\hat{χ}$ is designed to approximate the controlled system dynamics $χ (y, t)$ ; the output of CC $u_{CC}$ is designed to eliminate the effect of the approximation error. Substituting equation (36) into equation (11) yields

\begin{matrix} {\overset{\cdot}{e}}_{n} (t) & = χ (y, t) - \hat{χ} - e_{n - 1} (t) - λ_{n} e_{n} (t) + u_{CC} \\ = \tilde{χ} - e_{n - 1} (t) - λ_{n} e_{n} (t) + u_{CC} \end{matrix}

(37)

Figure 2.

Block diagram of the VSFSWC and CC.

Then substituting equation (32) into equation (37), the equation (38) can be obtained as follows

\begin{array}{l} {\dot{e}}_{n} (t) = {\tilde{μ}}^{T} \hat{γ} + {\hat{μ}}^{T} (γ_{a} \tilde{a} + γ_{b} \tilde{b}) + \\ Δ_{2} - e_{n - 1} (t) - λ_{n} e_{n} (t) + u_{C C} \end{array}

(38)

In this article, the CC is designed as follows

u_{CC} = - {\hat{E}}_{0} sgn (e_{n})

(39)

where ${\hat{E}}_{0}$ is the estimation bound of $Δ_{2}$ and sgn() is a sign function.

In order to guarantee the system stability, the Lyapunov function candidate is given as follows

\begin{array}{l} V_{2} (t) = \frac{e_{1}^{2} (t)}{2} + \frac{e_{2}^{2} (t)}{2} + \dots + \frac{e_{n}^{2} (t)}{2} + \frac{{\tilde{μ}}^{T} \tilde{μ}}{2 η_{1}} \\ + \frac{{\tilde{a}}^{T} \tilde{a}}{2 η_{2}} + \frac{{\tilde{b}}^{T} \tilde{b}}{2 η_{3}} + \frac{{\tilde{E}}_{0}^{2}}{2 η_{4}} \end{array}

(40)

where ${\tilde{E}}_{0} = E_{0} - {\hat{E}}_{0}$ is the estimation error and bounded; $η_{1}$ , $η_{2}$ , $η_{3}$ , and $η_{4}$ are positive learning rates. Differentiating (40) with respect to time and using (39), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{2} (t) & = e_{1} {\overset{\cdot}{e}}_{1} + e_{2} {\overset{\cdot}{e}}_{2} + \dots + e_{n} {\overset{\cdot}{e}}_{n} \\ + \frac{{\tilde{μ}}^{T} \overset{\cdot}{\tilde{μ}}}{η_{1}} + \frac{{\tilde{a}}^{T} \overset{\cdot}{\tilde{a}}}{η_{2}} + \frac{{\tilde{b}}^{T} \overset{\cdot}{\tilde{b}}}{η_{3}} + \frac{{\tilde{E}}_{0} {\overset{\cdot}{\tilde{E}}}_{0}}{η_{4}} \\ = e_{1} (e_{2} - λ_{1} e_{1}) + e_{2} (e_{3} - λ_{2} e_{2} - e_{1}) \\ + \dots + e_{n} ({\tilde{μ}}^{T} \hat{γ} + {\hat{μ}}^{T} (γ_{a} \tilde{a} + γ_{b} \tilde{b}) \\ + Δ_{2} - e_{n - 1} (t) - λ_{n} e_{n} + u_{CC}) \\ + \frac{{\tilde{μ}}^{T} \overset{\cdot}{\tilde{μ}}}{η_{1}} + \frac{{\tilde{a}}^{T} \overset{\cdot}{\tilde{a}}}{η_{2}} + \frac{{\tilde{b}}^{T} \overset{\cdot}{\tilde{b}}}{η_{3}} + \frac{{\tilde{E}}_{0} {\overset{\cdot}{\tilde{E}}}_{0}}{η_{4}} \\ = - λ_{1} e_{1}^{2} - λ_{2} e_{2}^{2} - \dots - λ_{n} e_{n}^{2} + {\tilde{μ}}^{T} (e_{n} \hat{γ} + \frac{\overset{\cdot}{\tilde{μ}}}{η_{1}}) \\ + {\tilde{a}}^{T} (e_{n} γ_{a} \hat{γ} + \frac{\overset{\cdot}{\tilde{a}}}{η_{2}}) \\ + {\tilde{b}}^{T} (e_{n} γ_{b} \hat{γ} + \frac{\overset{\cdot}{\tilde{b}}}{η_{3}}) + e_{n} (Δ_{2} + u_{CC}) + \frac{{\tilde{E}}_{0} {\overset{\cdot}{\tilde{E}}}_{0}}{η_{4}} \end{matrix}

(41)

The adaptive learning laws of the VSFSWC and CC are given in the following so that the convergence of the proposed algorithm is guaranteed

\overset{\cdot}{\hat{μ}} = - \overset{\cdot}{\tilde{μ}} = η_{1} e_{n} \hat{γ}

(42)

\overset{\cdot}{\hat{a}} = - \overset{\cdot}{\tilde{a}} = η_{2} e_{n} γ_{a} \hat{γ}

(43)

\overset{\cdot}{\hat{b}} = - \overset{\cdot}{\tilde{b}} = η_{3} e_{n} γ_{b} \hat{γ}

(44)

Then taking advantage of equations (39) and (41), we can obtain the following

\begin{matrix} {\overset{\cdot}{V}}_{2} (t) & = - λ_{1} e_{1}^{2} - λ_{2} e_{2}^{2} - \dots - λ_{n} e_{n}^{2} + Δ_{2} e_{n} - {\hat{E}}_{0} | e_{n} | + \frac{{\tilde{E}}_{0} {\overset{\cdot}{\tilde{E}}}_{0}}{η_{4}} \\ \leq - λ_{1} e_{1}^{2} - λ_{2} e_{2}^{2} - \dots - λ_{n} e_{n}^{2} + | Δ_{2} | | e_{n} | - {\hat{E}}_{0} | e_{n} | + \frac{{\tilde{E}}_{0} {\overset{\cdot}{\tilde{E}}}_{0}}{η_{4}} \\ \leq - λ_{1} e_{1}^{2} - λ_{2} e_{2}^{2} - \dots - λ_{n} e_{n}^{2} + (E_{0} - {\hat{E}}_{0}) | e_{n} | + \frac{{\tilde{E}}_{0} {\overset{\cdot}{\tilde{E}}}_{0}}{η_{4}} \\ = - λ_{1} e_{1}^{2} - λ_{2} e_{2}^{2} - \dots - λ_{n} e_{n}^{2} + {\tilde{E}}_{0} (| e_{n} | + \frac{{\overset{\cdot}{\tilde{E}}}_{0}}{η_{4}}) \end{matrix}

(45)

For achieving ${\overset{\cdot}{V}}_{2} (t) \leq 0$ , the error estimation law is designed as follows

{\overset{\cdot}{\hat{E}}}_{0} = - {\overset{\cdot}{\tilde{E}}}_{0} = η_{4} | e_{n} |

(46)

Then equation (44) can be rewritten as

{\overset{\cdot}{V}}_{2} (t) = - λ_{1} e_{1}^{2} - λ_{2} e_{2}^{2} - \dots - λ_{n} e_{n}^{2} \leq 0

(47)

As a result, similar to the discussion in the above section, the asymptotical stability of the proposed control system can be guaranteed in the sense of the Lyapunov theory.

Simulation

In this section, the proposed control algorithms are applied to validate the feasibility and effectiveness, and the reference commands with different algorithms are applied to the HIL simulation platform for the ELS as shown in Figure 3, which are evaluated by the root mean square error of the output torque. The host control computer with the proposed algorithm to achieve torque tracking control of the ELS mainly analyzes and calculates the feedback torque of the torque sensor, the compensation torque of inertia disks, and the actual torque required. The signal conditioning center is mainly used for processing signals from the sensors and the configuration settings needed for the hardware switches. According to the output of the host control computer, the output of drive provides the appropriate current to drive the torque motor. The torque sensor is used to collect the output of the torque motor, which is fed back to the host control computer to form the closed-loop control for the loading torque. The function of the resolver is to gathering the angle position of the gear reducer, which can be used to calculate the real-time command torque for the torque motor. The function of the angular velocity sensor is to gathering the angular velocity of the gear reducer to provide more actual values for the identification methods and the control strategies. The angular acceleration sensor collects the angular acceleration at the end of the gear reducer to compute the inertia moment produced by the inertia disks. The number of inertia disks depends on the actual demand of the gun control system, which provides various rotational inertia values to simulate the inertia torque. The following simulations are carried out in an Inter core i5 CPU with 3.2 GHz rate, 4 GB RAM, and 64-bit operating system. The parameters of the actual gun system are as follows: the rotary inertia of the turret is 7000 kg m², the total friction torque $M_{f}$ is less than 1200 N m, and the center position of gravity is located in 0.35 m away from the rotation center of the turret, the reduction ratio is 360. Therefore, the relative parameters are selected as $γ_{th} = 0.65$ , $I_{th} = 0.12$ , $γ_{0} = 0.1$ , $τ = 0.01$ , $η_{0} = 0.25$ , $η_{1} = 8$ , and $η_{2} = η_{3} = η_{4} = 0.5$ , the center value c of $μ_{k}$ is $c = [\begin{matrix} - 1.080 & - 0.225 & 0 & 0.225 & 1.080 \end{matrix}]$ , the scaling value d of $μ_{k}$ is $d = [\begin{matrix} 0.50 & 0.35 & 0.30 & 0.35 & 0.50 \end{matrix}]$ , the input vector is chosen as $[\begin{matrix} NB & NS & Z & PS & PB \end{matrix}]$ , the rotary inertia of every inertia disk is 2.6 × 10⁻³ kg m², the control period is 2 ms, and the sample frequency is 10 kHz. Considering the experimental results, the parameters of $\hat{a}$ , $\hat{b}$ , and $\hat{μ}$ are changed with the $η_{1}$ , $η_{2}$ , $η_{3}$ , $e_{n} (t)$ , and so on, ${\hat{E}}_{0}$ is used in the CC, the function of the CC is too little to reduce the approximation error when the value of $η_{4}$ is set more than 1.5. $η_{1}$ plays a great effect on the coefficients of the $χ_{k}$ , the value of $η_{1}$ is set too small to enhance the influence of $χ_{k}$ , which leads to the low convergence rate of the control system. $η_{2}$ and $η_{3}$ are related to the dilation and translation parameters, which are set too big or small to achieve the low convergence rate of the control system. The important torque motor parameters are illustrated in Table 1. In order to compare the superiority of all proposed algorithms, the step response reference commands with external disturbances are applied to analyze the rapidity and stability. Next, the static and dynamic loading performance is evaluated by changing the motion styles of the load motor and torque motor, respectively. Then, the different connection stiffnesses of the ELS are carried out by the proposed algorithms. In the above experiments, the number of the inertia disks is chosen as 3. Finally, the robustness performance is evaluated by changing the number of inertia disks and the frequencies of the load motor.

Figure 3.

Components of the ELS.

Table 1.

Parameters of the torque motor.

Parameters	Values
Rated power $p_{r}$	2.8 kw
Biggest locked-rotor torque $T_{max}$	19.8 N
Continuous locked-rotor torque $T_{cont}$	12 N
Continuous current $I_{cont}$	80 A
Rated speed $v_{r}$	1200 r/min
Coefficient of electromagnetic torque $K_{t}$	2.6 N m/A
Coefficient of counter electromotive force $K_{e}$	9.6 mv/rpm
Equivalent inductance of armature circuit $L_{m}$	0.0036 H
Equivalent resistance of armature circuit $R_{m}$	1.2 Ω
Rotational inertia $J_{m}$	0.038 kg m²
Viscous friction coefficient $B_{m}$	0.22 N m s/rad

Step response with external disturbance

In order to evaluate the proposed algorithms, such as SWC, FSWC, FSWC and CC, VSFSWC, and VSFSWC and CC, the step response with external disturbances is applied to the ELS when the load motor is blocked as shown in Figure 4. When the amplitude of the reference command is selected as 10 N, the VSFSWC and CC shows the best performance among all the proposed algorithms, which reaches the steady state in only 96 ms. Although there always exists torque tracking errors using the SWC, FSWC, and VSFSWC, the VSFSWC with smallest torque errors can meet the control requirements. When the CC is introduced into the FSWC and VSFSWC, the torque tracking errors are decreased effectively. An external disturbance is exerted to the ELS at the moment of 180 ms, the similar conclusion above is obtained, which further verifies the rapidity and stability to some extent.

Figure 4.

Step response of the proposed algorithms.

Static loading

In order to analyze the influences of the load motor, the torque motor is working in the stationary state when the load motor is working in the amplitude of 1 rad and different frequencies, such as 1, 5, and 10 Hz. The control requirement of the load motor is 0.8 mil (1 rad = 954.93 mil). The VSFSWC, FSWC and CC, and VSFSWC and CC algorithms are carried out to show the existing extraneous torques in Figures 5 –7. The results indicate that the output of extraneous torque is increasing when the frequencies become bigger, especially the bigger output are happening at the moment of changing the rotary direction of the load motor. The maximum and mean absolute values of the output torque are described in detail in Table 2, which further show that the VSFSWC and CC algorithm has great superiority than the VSFSWC and FSWC and CC. The mean tracking errors of the load motor meet the gun control system requirements, which have nothing to do with the torque motor and the control algorithms when the torque motor works in the stationary state, although the load motor works in different frequencies.

Figure 5.

1 Hz extraneous torques.

Figure 6.

5 Hz extraneous torques.

Figure 7.

10 Hz extraneous torques.

Table 2.

Values of the output extraneous torque and track error.

Frequencies of the load motor (Hz)	VSFSWC (N m)		FSWC and CC (N m)		VSFSWC and CC (N m)		Mean tracking error of the load motor (mil)
Frequencies of the load motor (Hz)	Max	Mean	Max	Mean	Max	Mean	Mean tracking error of the load motor (mil)
1	0.5237	0.2315	0.3108	0.2518	0.0827	0.0371	0.2518
5	0.8247	0.3615	0.4982	0.2385	0.2327	0.0954	0.2385
10	1.3830	0.9672	0.5723	0.2741	0.2581	0.1126	0.2741

FSWC: fuzzy self-recurrent WNN controller, CC: complementary controller.

Dynamic loading

To validate the torque tracking performance with the load motor’s position disturbance, the torque motor is working in different amplitudes and frequencies when the load motor is working in the motion $Δ dit = \sin (0.5 π t) rad$ . Considering the practical working conditions, the reference commands of the torque motor are selected as $y_{d 1} (t) = 5 \sin 20 π t$ , $y_{d 2} (t) = 5 \sin 10 π t$ , $y_{d 3} (t) = \sin 2 π t$ , $y_{d 4} (t) = 5 \sin 2 π t$ , and $y_{d 5} (t) = 10 \sin 2 π t$ , the experimental results including the torque tracking and variations of the number of neurons are presented in Figures 8 –12 and Table 3, respectively.

Figure 8.

Sinusoidal tracking with $y_{d 1} (t) = 5 \sin 20 π t$ : (a) VSFSWC and CC means the actual output with the VSFSWC and CC algorithm and (b) the number of neuron variation with the VSFSWC and CC algorithm.

Figure 9.

Sinusoidal tracking with $y_{d 2} (t) = 5 \sin 10 π t$ : (a) VSFSWC and CC means the actual output with the VSFSWC and CC algorithm and (b) the number of neuron variation with the VSFSWC and CC algorithm.

Figure 10.

Sinusoidal tracking with $y_{d 3} (t) = \sin 2 π t$ : (a) VSFSWC and CC means the actual output with the VSFSWC and CC algorithm and (b) the number of neuron variation with the VSFSWC and CC algorithm.

Figure 11.

Sinusoidal tracking with $y_{d 4} (t) = 5 \sin 2 π t$ : (a) VSFSWC and CC means the actual output with the VSFSWC and CC algorithm and (b) the number of neuron variation with the VSFSWC and CC algorithm.

Figure 12.

Sinusoidal tracking with $y_{d 5} (t) = 10 \sin 2 π t$ : (a) VSFSWC and CC means the actual output with the VSFSWC and CC algorithm and (b) the number of neuron variation with the VSFSWC and CC algorithm.

Table 3.

Comparison of different reference commands and mean tracking errors.

Reference command (N m)	Mean tracking error of the load -motor (mil)	Torque motor		Number of neurons
Reference command (N m)	Mean tracking error of the load -motor (mil)	Amplitude error (%)	Phase error (°)	Maximum	Stable status
$5 \sin 20 π t$	0.5205	6.43	−8.9329	24	12
$5 \sin 10 π t$	0.4592	5.81	−5.6582	24	12
$\sin 2 π t$	0.3382	4.12	−3.8128	21	12
$5 \sin 2 π t$	0.2762	3.38	−2.0792	18	12
$10 \sin 2 π t$	0.2149	2.26	−1.0381	15	9

In Table 3, the mean tracking errors of the load motor meet system control requirements. However, the smaller the reference command amplitudes of the torque motor and the higher the reference command frequencies of the torque motor (within certain limits), the worse the control performance of the torque motor and the bigger the mean tracking errors of the load motor. In addition, the figures about the number of neurons further show that the better the control performance, the less the neuron number. However, what we need to remind is that the control performance is relatively worse when the direction of the torque motor is changed, which plays a negative effect on the overall performance and need to be further studied in the future.

Different connection stiffnesses

In order to further evaluate the proposed algorithms, we change the connection stiffnesses of the ELS system by different couplings and torque sensor. The different connection stiffnesses play effects on the ELS in Figure 13; $G_{f}$ is the connection stiffness coefficient.

Figure 13.

Bode comparison with increasing $G_{f}$ .

From Figure 13, the frequency characteristic curve is shown when the connection stiffness is increased by five multiples. The system shear frequency is increasing and the phase characteristic tends to be steep when the connection stiffness is improved, which also broadens the system bandwidth. However, the bigger resonance peak happens when the system bandwidth is over-extended, which leads to the system instability. Therefore, aiming at different connection stiffness, the proposed VSFSWC and CC is applied to the ELS when the torque motor is working in the motion $5 \sin 2 π t$ . The comparison of different connection stiffnesses is given in Table 4, which further verifies the effectiveness of the proposed controller.

Table 4.

Comparison of different connection stiffnesses.

Connection stiffness coefficient	Amplitude error (%)	Phase error (°)	Number of neurons
Connection stiffness coefficient	Amplitude error (%)	Phase error (°)	Maximum	Stable status
10 N m/rad	9.43	−9.1538	45	24
50 N m/rad	6.25	−5.8925	30	18
250 N m/rad	3.38	−2.0792	18	12
1250 N m/rad	3.89	−1.6267	24	15

Robustness performance

The robustness performance is evaluated by changing the number of inertia disks and the frequencies of the load motor with the proposed VSFSWC and CC algorithms when the torque motor is kept working in the stationary state. From Table 5, it is clear that the number of inertia disks has no specific relationship to do with the control performance, but with the same parameters of the above controller, there inertia disks show better control performance. When the frequencies of the load motor are becoming bigger, the control performance is relatively worse. However, all the experimental results meet the gun control system requirements, which further show that the proposed controller takes advantages of the robustness and application performance.

Table 5.

Comparison of robustness performance.

Number of inertia disks	Frequencies (Hz)
	Maximum extraneous torque (N m)			Mean extraneous torque (N m)
	1	5	10	1	5	10
2	0.0912	0.2617	0.3040	0.0409	0.1176	0.1338
3	0.0827	0.2327	0.2581	0.0371	0.0954	0.1126
4	0.0932	0.2831	0.3106	0.0391	0.1072	0.1286
5	0.0964	0.2781	0.3085	0.0412	0.1204	0.1385

Conclusion

In this article, the VSFSWC and CC is proposed to address the complexities in the ELS, which is composed of two terms, one is used to construct equivalent nonlinearities and uncertainties, the other is designed to eliminate the effect of the approximation error. Moreover, in combination with ideal backstepping controller, the adaptive learning laws are introduced to guarantee the control system stability in the sense of the Lyapunov theory. Finally, extensive comparative simulations and experiments prove that the ability of suppressing the negative influence of the extraneous torque and improving the dynamic torque tracking performance is significantly promoted. In future works, the authors intend to further study the effect of changing the direction of the torque motor.

Footnotes

Academic Editor: Yaguo Lei

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) received no financial support for the research, authorship, and/or publication of this article.

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Adaptive complementary fuzzy self-recurrent wavelet neural network controller for the electric load simulator system

Abstract

Keywords

Introduction

Problem statement and ideal backstepping controller design

Problem statement

Ideal backstepping controller design

Adaptive complementary fuzzy self-recurrent WNN controller design

Self-recurrent WNN

Fuzzy self-recurrent WNN

VS learning algorithm

Stability analysis

Simulation

Step response with external disturbance

Static loading

Dynamic loading

Different connection stiffnesses

Robustness performance

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References