Adaptive Nonsingular Fast Terminal Sliding Mode Control for Braking Systems with Electro-Mechanical Actuators Based on Radial Basis Function

Abstract

In this paper an adaptive non-singular fast terminal sliding mode (NFTSM) control scheme is proposed to control the electro-mechanical actuator (EMA) in an electric braking system which is a complex electro-mechanical system. In order to realize high-performance brake pressure servo control, a radial basis function (RBF) neural network method is adopted to deal with the difficulty of estimating the upper bound of the compound disturbance in the system, to reduce the conservatism of the design of sliding mode switching gain, and effectively eliminate sliding mode chattering. The simulation results show that, compared with a linear controller, the proposed control strategy is able to improve the servo performance and control precision. In addition the response speed of the braking actuator is enhanced significantly, without changing the traditional double-loop control structure.

1. Introduction

Electric braking systems are a new type of braking system widely and continually researched in the aviation field. It makes use of electro-mechanical actuators (EMAs) instead of classical hydraulic actuators, thus getting rid of the dependence on the main hydraulic system, significantly reducing the weight and volume of the braking system, in order to improving the safety of the system, which is the development direction of braking systems in the future. Braking systems of all-electric aircraft require EMAs driven by motors to output brake pressure accurately and rapidly, and have a good tracking of the reference pressure signal.

Brushless direct current motors (BLDCMs) have been widely used with the rapid development of permanent magnet material technology, power electronics technology and other supporting technologies, as well as the continuous improvement of micro-motor manufacturing processes. BLDCMs have advantages like compact structures, higher torque inertia ratio and larger power density, which are suitable for the driving part of EMAs. Driven by BLCDMs, the electric braking system is essentially a high performance servo system, and the special application situation of aircraft puts strict demands on the control strategy of the BLDCM servo system.

It should be noted that, although modeling seeks to be exhaustive, it is not accurate enough to describe the dynamic characteristics of EMAs. The main reason is that the EMA is a complex mechatronic system, which contains backlash, friction and other nonlinear characteristics. For that reason, it is very difficult to obtain a mathematical model which can be applied to describe the dynamic behavior of an EMA accurately. EMAs work in harsh environments, where large fluctuations in temperature cause significant changes in motor parameters. The above reasons also led to the difficulty to achieve fast response and high precision of the pressure tracking effect with linear controllers. Based on the essential characteristics of the brake pressure servo, the nonlinear block diagram model is simplified. The compound disturbance is a combination of the time-varying parameter, unmodeled dynamics and external disturbance, and sliding mode control is applied to the state space model of the brake system, so as to design the control law.

Mercorelli et al. proposed two different electromagnetic motors controlled using a sliding mode approach [1]. Su et al. addressed the problem of global finite-time stabilization of planar linear systems subject to actuator saturation [2]. A simple saturated proportional-derivative controller was proposed. Morshed et al. designed a fuzzy second order integral terminal sliding mode controllerin [3], which places the system on a sliding surface in the initial state based on [4]. Wu et al. designed a nonlinear dynamic sliding mode surface with terminal characteristics in [5], but the disadvantage is that if the control parameters are chosen improperly, singular problems may arise. Yu et al. extended the terminal sliding mode (TSM) design to multi-input uncertain systems [6]. Feng et al. subsequently improved the sliding surface, and came up with the non-singular terminal sliding mode [7]. Yu et al. [8] proposed a fast terminal sliding mode (FTSM) design method based on [7]. Compared with TSM the algorithm has advantages in the convergence rate. In fact, FTSM is also considered as a combination of exponential reaching law and power reaching law, which makes the convergence speed of the system become faster when the system state is far away from or near the sliding surface. In recent years, in addition to the second order TSM methods, for example the Super-Twisting algorithm, some new TSM structures have been proposed, such as a full order sliding mode surface with terminal characteristic proposed by Feng et al. [9], and an essential NFTSM designed by Yang et al [10]. Feng et al. [11] carried out low-pass filtering on the control switching terms which provided smooth control quantities and proved that the filter has no effect on the stability of the TSM when appropriate switching terms are used. Similarly, Wang et al. designed a control switch with filter, which ensures the global stability of the system and reduces the high frequency chattering [12]. In [9] is worth noting that the filtering method used to design the switching control term must be combined with the stability analysis method. Furuta et al. [13] proposed the concept of sliding sector, defining define two sliding surfaces, dividing the state space into different sectors, designing different switching control items, constructing a continuous switching controller so that the control signals are continuous. The sliding sector method greatly reduces the sliding mode chattering frequency, but its defect is that when the state trajectory is far from the equilibrium point, the saturation characteristic width is large, which lead to a poor robustness of the system. Suzuki et al. [14] and Pan et al. [15] studied design methods for the time invariant sector and the time variable sector, respectively. Powly et al. [16] combined the reaching law and a sliding sector method, and designed a sliding mode controller for missile attitude control. Li et al. used a discrete-time nonlinear observer (DNLO) for estimation the state of charge (SOC) of lithium-ion batteries [17]. The results verified that DNLO has better performance in reducing the computation cost than the extended Kalman filter (EKF) algorithm. Huangfu et al. proposed a super-twisting sliding mode algorithm to improve the system stability [18,19].

According to the characteristics of the EMA pressure servo system, this paper proposes an adaptive non-singular FTSM control scheme. On the basis of [10], a RBF neural network is used to deal with the difficulty of estimating the upper bound of the uncertain compound disturbance in the system. There is a singular problem in the design of the traditional FTSM, and the proposed NFTSM, in essence, avoids singularity and is applicable to system which has external disturbances and parameter uncertainties. RBF has strong input and output mapping functions, and the theory proves that the RBF network is the proper choice to complete the mapping function in the feed-forward network. The switching gain of the controller is adjusted automatically by the system state information, which improves the adaptive ability of the EMA controller, reduces the conservativeness of the sliding mode control design, and effectively suppresses the chattering of the sliding mode control, while improving the performance of the pressure servo and the adaptive ability of the system.

The remainder of the paper is organized as follows: Section 1 introduces the problem of the electric braking system and EMA system. Section 2 presents the braking system principle and the EMA structure, and then establishes the EMA mathematical model by combining with the transmission mechanism model, to get the overall the nonlinear block diagram model of the EMA brake system. Section 3 presents the adaptive NFTSM controller and RBF neural network design method, and the convergence property in finite time is analyzed by a Lyapunov method. Section 4 gives some results and conclusions.

The main contributions of this paper can be summarized as follows: firstly, the paper proposes a feasible nonlinear control method for brake EMA systems and obtains a better control effect. Secondly, because of the restraint of the model uncertainty influence and external disturbances, the proposed algorithm did not change the traditional brake control structure, which improves the control accuracy and response speed. Thirdly, the new brake control system is fully validated experimentally.

2. EMA Mathematical Model for the Electric Braking System

2.1. EMA Mathematical Model

An electric braking system contains an anti-skid braking controller, braking power driver, EMA, brake wheel and related sensors. The basic system structure is shown in Figure 1. The braking principle of the electric braking system is not substantially different from that of a hydraulic braking system. The anti-skid braking controller serves as the control unit of the braking system, receiving the braking command signals, the aircraft speed signals and the wheel speed signals. The anti-skid controller generates the corresponding pressure control signal input to the drive circuit to driving the motor, and then drives the output pressure of the ball screw to the disc, generating the corresponding torque, and making the actual pressure follow the desired pressure signal through the pressure feedback loop. The tracking of a given slip ratio is eventually realized, at the same time, this prevents the locking of the wheels when braking with great urgency. According to the whole structure, the whole electric braking system is a double closed-loop control system, which is composed of the slip ratio and the pressure feedback.

As the internal dynamic characteristics of BLDCM are not the main focus of this paper, the DC motor mathematical model used in the EMA is suitably simplified. Once the simulation model is established, in the EMA, the DC motor is controlled with a variable armature voltage, thus we can simplify the brushless DC motor as follows:

$$U_{\mathrm{c}}=R_{\mathrm{m}}{I}_{\mathrm{a}}+L\frac{d{I}_{\mathrm{a}}}{dt}+E_{\mathrm{a}}$$

(1)

$$E_{\mathrm{a}}=K_{\mathrm{e}}{\omega }_{\mathrm{m}}$$

(2)

$$T_{\mathrm{e}}=K_{\mathrm{T}}{I}_{\mathrm{a}}{E}_{\mathrm{a}}=K_{\mathrm{e}}{\omega }_{\mathrm{m}}$$

(3)

where U_c is armature voltage (V), R_m is armature resistance (Ω), I_a is armature current (A), L is armature inductance (H), E_a is induced electromotive force (N), K_e is back electromotive force constant, K_T is torque constant.

It can be seen from Equations (1)–(3) that the expression of electromagnetic torque of the BLDCM is the same as that of an ordinary DC motor. The electromagnetic torque of the BLDCM is proportional to the magnetic flux and current amplitude, so the torque of the motor can be controlled by the amplitude of the output square wave current of the inverter.

By taking into account the requirements of the mechanism transmission ratio, nominal diameter d₀ and ball screw lead L₀, the rated dynamic load is usually determined by the maximum axial thrust of the brake disc driven by the ball screw. The inertia torque of the ball screw rod and the driving torque which is used to overcome the axial load are of great significance for the parameter selection of the motor. The complicated nonlinear factors such as torsion, wear and bending vibration of ball screw are neglected in this model for simplification of the system.

Motor mechanical motion equation:

$$T_{\mathrm{e}}-T_L-B_{\mathrm{v}}{\omega }_{\mathrm{m}}=J{\dot{\omega }}_{\mathrm{m}}$$

(4)

where the load torque (T_L) can be calculated through the following equation:

$$T_L=\left(T_{PL}+T_D+T_f\right)/k_i\eta$$

(5)

where T_PL is the resistance torque of the ball screw (N), T_D is the drive torque of the ball screw (N), T_f is the friction torque of the drive system (N), k_i is the transmission ratio of ball screw and motor shaft gear, η is the transmission efficiency of transmission mechanism.

The friction torque of the drive system T_f is closely related to the speed of screw, which includes the friction moment of the supporting bearing and the sealing device. T_f is difficult to accurately measure, and its true numerical value is related to the working temperature and lubrication. Compared with other torque, the numerical value of T_f is small, so the model can be valued by experience. η, T_PL, T_D can be calculated respectively as follows:

$$\eta =\frac{\tan {\rho }_{\alpha }}{\tan \left({\rho }_{\alpha }+{\rho }_d\right)}$$

(6)

$$T_{PL}=\frac{F_P{L}_0}{2\pi \eta }\left(1-{\eta }^2\right)$$

(7)

$$T_D=\frac{P_{\mathrm{A}}{L}_0}{2\pi \eta }$$

(8)

where ρ_d is equivalent friction angle, ρ_α is the thread angle, F_p is the axial pretightening force of the screw, P_A is the screw axial working load (i.e., the brake pressure).

The relationship between the rotary motion of the motor and the linear motion of the ball screw is shown as follows:

$$\frac{{\omega }_{\mathrm{m}}t}{2\pi }=\frac{x_{\mathrm{E}}}{L_0}$$

(9)

where x_E is the axial displacement of the ball screw.

2.2. EMA Model Simplification

From essential characteristic of the braking system, the EMA output force (the brake pressure) P_A can be calculated as:

$$P_{\mathrm{A}}=c_{\mathrm{b}}\left(x_{\mathrm{E}}-x_{\mathrm{b}}\right)$$

(10)

where c_b is the stiffness coefficient of the brake disc, x_E is the ball screw displacement, x_b is the lateral displacement of the brake disc. It can be assumed that the brake disc only has elastic deformation during the braking process, and no lateral displacement (x_b = 0).

The load force and the equation of motion of the ball screw can be obtained as:

$$P_{\mathrm{A}}=2\pi T_{\mathrm{l}}/L_0{\dot{x}}_{\mathrm{E}}=L_0{\omega }_{\mathrm{m}}/2\pi$$

(11)

The derivation of the brake pressure can be obtained by taking (11) into (10) as:

$${\dot{P}}_{\mathrm{A}}=c_{\mathrm{b}}{L}_0{\omega }_{\mathrm{m}}/2\pi$$

(12)

The relationship between brake actuator displacement and the brake force is linear, which is the special point of the pressure servo. By taking consideration of Equations (4)–(8) and (12) and selecting the state variables x_i = [P_A, ω_m, I_a], the state space model of EMA can be established as follows:

$$\{\begin{array}{l}{\dot{P}}_{\mathrm{A}}=c_{\mathrm{b}}{L}_0{\omega }_{\mathrm{m}}/2\pi \\ {\dot{\omega }}_{\mathrm{m}}=-\frac{1}{J}\frac{L_0}{2\pi }{P}_{\mathrm{A}}-\frac{B_{\mathrm{v}}}{J}{\omega }_{\mathrm{m}}+\frac{K_{\mathrm{T}}}{J}{I}_{\mathrm{a}}+d_1\\ {\dot{I}}_{\mathrm{a}}=-\frac{K_{\mathrm{e}}}{L}{\omega }_{\mathrm{m}}-\frac{R_{\mathrm{m}}}{L}{I}_{\mathrm{a}}+\frac{1}{L}{U}_{\mathrm{c}}+d_2\end{array}$$

(13)

where d₁, d₂ represent the superposition of unknown compound disturbance, which can be decomposed into internal disturbance caused by time-varying parameters and external disturbance and unmodeled dynamics, respectively.

The above model ignores the complex nonlinear factors, which can basically meet the requirements of general control design and analysis. However, with the improvement of the control performance and reliability of the EMA actuator, the model has been shown to be limited. Despite the reasonable simplification, the EMA still has nonlinear characteristics, such as high order and time-varying nature. It is difficult to determine the convergence condition and the controller parameter selection range. Moreover, the control law synthesis and performance optimization cannot be easily obtained. In addition, modeling errors are a combination of the time-varying parameters, unmodeled dynamics and external disturbance, and its existence also affects the validity of the mathematical electro-mechanical actuator model.

Compared with the hydraulic actuation, the important advantage of the EMA lies in a higher frequency response. The actual braking process is usually an alternating state of braking and releasing action, which means the motor needs to reverse frequently, so the design of the current loop accelerates the motor starting process, improving the braking system’s ability to respond quickly. Because of the particularity of the aircraft braking application, it is difficult for the three closed-loop control methods (composed by the pressure loop, speed loop and current loop) to meet the requirement of rapidity. At present, the double closed-loop control structure (the pressure loop and current loop) is widely used. The sliding mode control algorithm also cancels the speed loop, and does not change the existing control structure. It should be noted that, because the core of the research focus on the analysis and design of the pressure loop, considering the current loop response speed of the BLDCM is faster than the pressure loop, the current loop is assumed to be ideal.

In Equation (13), the mathematical model of brake pressure loop as:

$$\{\begin{array}{l}{\dot{P}}_{\mathrm{A}}=c_{\mathrm{b}}{L}_0{\omega }_{\mathrm{m}}/2\pi \\ {\dot{\omega }}_{\mathrm{m}}=-\frac{1}{J}\frac{L_0}{2\pi }{P}_{\mathrm{A}}-\frac{B_{\mathrm{v}}}{J}{\omega }_{\mathrm{m}}+\frac{K_{\mathrm{T}}}{J}{I}_{\mathrm{a}}+d\end{array}$$

(14)

2.3. Control Target

We design the pressure controller of the EMA described in Equation (14), so that P_A (the braking pressure) can be used to track the desired braking pressure signal P_A* in finite time, and adaptively adjust the switching gain through RBF neural network to extremely reduce the chattering of the sliding mode.

The electric braking system is essentially a BLDCM-driven high performance pressure servo system, and the special application also places strict requirements on the control performance of the BLDCM servo system in the harsh environment. It can be seen from the mathematical model that the controller design needs to consider the disturbance from inside and outside the system. The disturbance mainly includes the internal perturbation caused by parameters perturbation and external disturbance caused by load changes, as well as the unmodeled dynamics of friction, backlash nonlinearity and so on. If the disturbance cannot be controlled and processed properly, the performance of the overall control system will be affected to a certain extent.

3. Controller Design

3.1. Adaptive Nonsingular Fast Terminal Sliding Mode Control

According to the characteristics of the EMA braking pressure servo system, this paper proposes an adaptive NFTSM control method, which uses a neural network to deal with the difficulty of estimating the upper bound of the complex disturbance in the system, to reduce the conservatism of the design of sliding mode switching, that effectively eliminates sliding mode chattering, and implement a high-performance pressure servo control.

Consider the following second-order nonlinear system shown as:

$$\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=f\left(x_1,x_2\right)+bu+d\left(x_1,x_2,t\right)\end{array}$$

(15)

where x₁ and x₂ are the state variables, b is the control gain, f(·) is the continuously differentiable nonlinear function about x₁, x₂, u is the controlled inputs, d(·) is the unknown disturbance acting on the system, ||d(·)|| ≤ ρ(x₁,x₂,t) ≤ $\overline{\rho }\left(t\right)$, $\overline{\rho }\left(t\right)$ are bounded functions.

Conventional terminal sliding mode (TSM) can be described as follows:

$$s=x_2+\beta x_1{}^{q_0/p_0}$$

(16)

where β is a positive real number, q₀ and p₀ are odd numbers and satisfy p₀ > q₀. The terminal sliding mode has a singularity problem, so its application is limited accordingly. Feng et al. [7] proposed the following Nonsingular Terminal Sliding Mode (NTSM) algorithm shown as:

$$s=x_1+\frac{1}{\beta }{x}_2{}^{p_0/q_0}$$

(17)

where the control parameters are the same as in Equation (10). NTSM avoids the control of the singular region, but the shortcoming is that the convergence time is relatively long when the system state is far from the equilibrium point. Yu et al. [6,8] proposed the FTSM algorithm to compensate for this shortcoming. The sliding surface is shown as:

$$s=x_2+\alpha x_1+\beta x_1{}^{q_0/p_0}$$

(18)

where α and β are positive real numbers, q₀ and p₀ are odd and satisfy p₀ > q₀. When x₁ is far from zero, s ≈ ${\dot{x}}_1$ + αx₁, it can be approximated as an exponential convergence. If x₁ is close to zero, $s\approx x_2+\beta x_1{}^{q_0/p_0}$ , at this point, the terminal attractor plays a major role, so FTSM keeps the terminal characteristic and has fast convergence speed in the initial stage. However, the FTSM still has singular control problem when applied to practical engineering.

The NFTSM proposed by Yang et al. [10] is shown as:

$$s=x_1+\alpha \mathrm{sgn}\left(x_1\right){\left|x_1\right|}^{{\gamma }_1}+\beta \mathrm{sgn}\left(x_2\right){\left|x_2\right|}^{{\gamma }_2}$$

(19)

where we take α > 0, β > 0, $1<{\gamma }_2<2$, ${\gamma }_1>{\gamma }_2$.

Referring to [10], the sliding surface in Equation (22) has a fast terminal characteristic when the parameters are selected reasonably.

From the sliding surface structure, NFTSM, which superposed the nonlinear terms related to x₁ to improve the convergence speed of the initial stage, is based on the structure of NTSM in essence. The convergence speed of NFTSM is analyzed in detail in [10]. The results show that the convergence rate is faster than that of TSM, but slower than that of FTSM. However, avoiding the singularity of control makes NFTSM have more engineering value.

The feedback control law of NFTSM is shown as:

$$\begin{array}{cc}\hfill u& =-b^{-1}[{\left(\beta {\gamma }_2\right)}^{-1}\mathrm{sgn}\left(x_2\right){\left|x_2\right|}^{2-{\gamma }_2}\left(1+\alpha {\gamma }_1{\left|x_1\right|}^{{\gamma }_1-1}\right)\hfill \\ & +f+\kappa s+K\mathrm{sgn}\left(s\right)]\hfill \end{array}$$

(20)

Lemma 1

[17]. If the Lyapunov function V(t) satisfies the following inequality shown as:

$$\dot{V}\left(t\right)\le -\alpha V\left(t\right)-\beta V^{\gamma }\left(t\right)\forall t\ge t_0$$

(21)

where $\alpha ,\beta >0$, $0<\gamma <1$, $V\left(t\right)$ can converge to zero in a finite time, the convergence time is shown as:

$$t_1\le t_0+\frac{1}{\alpha \left(1-\gamma \right)}\ln \frac{\alpha V^{1-\gamma }\left(t_0\right)+\beta }{\beta }$$

(22)

3.2. Design of the RBF Neural Network

An RBF network usually consists of input layer, hidden layer and output layer. A typical structure is shown in Figure 2.

In Figure 2, the input layer only completes information transfer, the hidden layer is the receptive field unit, which is composed of a set of radial basis functions, and the output of the output layer neuron is weighted by the linear combination of the hidden layer output. RBF is a scalar function which is symmetrical in the radial direction. It is defined as a monotonic function of the Euclidean distance between any point x in a space and a center c_i. The most commonly used radial basis function is the Gaussian basis function. The expression is shown as:

$$h_i\left(x\right)=\exp \left(-\frac{{\Vert x-c_i\Vert }^2}{2{\sigma }_i^2}\right)i=1,2,\cdots N$$

(23)

where c_i is the center of the i-th basis function, σ_i is the scale factor of the i-th basis function, which determines the width of the base function around the center point, N is the nodes number of the hidden layer, ||·|| is the Euclidean norm.

The output of the RBF network is an approximation to the upper bound of uncertainty ($\overline{\rho }$) shown as:

$$\stackrel{⌢}{\rho }={\widehat{\mathit{W}}}^{\mathit{T}}\mathit{h}\left(\mathit{x}\right)$$

(24)

where $\widehat{\mathit{W}}$ is the weight matrix, $h={\left[h_j\right]}^{\mathrm{T}}$ is the output of Gaussian basis function.

Assumption 1

[20]. Suppose that the RBF network optimal weight matrix W^*exists and satisfies the following expression:

$$\epsilon =W^{*\mathrm{T}}h\left(x\right)-\overline{\rho },\epsilon <{\epsilon }_N$$

(25)

Assumption 2

[20]. Assuming that the uncertainly upper bound of the system in (9) satisfies the following expression:

$$\overline{\rho }\left(t\right)-\left|\rho \left(t\right)\right|>{\epsilon }_0>{\epsilon }_N>0$$

(26)

Assumption 3.

The expected pressure signal $P_{\mathrm{A}}^{*}$ is continuous and its 2-order derivative exists.

Assumption 4.

$c_{\mathrm{b}}{L}_0\Vert d(\cdot )\Vert /2\pi \le \rho \left(t\right)\le \overline{\rho }\left(t\right)$, $\overline{\rho }\left(t\right)$ is the bounded function.

We design the NFTSM controller according to Equations (19) and (20), suppose $P_{\mathrm{A}}^{*}$ is reference pressure, and the error variance $z=P_{\mathrm{A}}-P_{\mathrm{A}}^{*}$, $\dot{z}={\dot{P}}_{\mathrm{A}}-{\dot{P}}_{\mathrm{A}}^{*}$. The sliding surface is designed as follows:

$$s=z+\alpha \mathrm{sgn}\left(z\right){\left|z\right|}^{{\gamma }_1}+\beta \mathrm{sgn}\left(\dot{z}\right){\left|\dot{z}\right|}^{{\gamma }_2}$$

(27)

where $\alpha >0$, $\beta >0$, $1<{\gamma }_2<2$, ${\gamma }_1>{\gamma }_2$.

Set the reference current value $I_{\mathrm{a}}^{*}$ as follows:

$$\begin{array}{l}{I}_{\mathrm{a}}^{*}=\frac{2\pi J}{c_{\mathrm{b}}{L}_0{K}_{\mathrm{T}}}\left(I_{\mathrm{a}eq}^{*}+I_{\mathrm{a}sw}^{*}\right)\\ I_{\mathrm{a}eq}^{*}=\frac{c_{\mathrm{b}}{L}_0}{2\pi }\left(\frac{1}{J}\frac{L_0}{2\pi }{P}_{\mathrm{A}}+\frac{B_{\mathrm{v}}}{J}{\omega }_{\mathrm{m}}\right)+{\ddot{P}}_{\mathrm{A}}^{*}\\ I_{\mathrm{a}sw}^{*}=-\left[{\left(\beta {\gamma }_2\right)}^{-1}\mathrm{sgn}\left(\dot{z}\right){\left|\dot{z}\right|}^{2-{\gamma }_2}\left(1+\alpha {\gamma }_1{\left|z\right|}^{{\gamma }_1-1}\right)+\kappa s+K\mathrm{sgn}\left(s\right)\right]\end{array}$$

(28)

where $2\pi J{I}_{\mathrm{a}eq}^{*}/c_{\mathrm{b}}{L}_0{K}_{\mathrm{T}}$ is the equivalent control term, and $2\pi J{I}_{\mathrm{a}sw}^{*}/c_{\mathrm{b}}{L}_0{K}_{\mathrm{T}}$ is the switching control term. In practical systems, the switch gain K is proportional to the uncertainties of the system and the upper bound of the disturbance. If the disturbance is large at some time, a larger K is needed to keep the system stable, but this will inevitably lead to strong chattering. Therefore, the RBF is used for real-time adjustment of the switch gain K, so as to reduce the influence of chattering and improve the system performance. The EMA servo control structure based on RBF network is shown in Figure 3.

3.3. Stability Analysis

Theorem 1.

For the EMA system described in Equation (14), we select the sliding surface shown in Equation (27) and the current control law shown in Equation (28). The switch gain K is adjusted on-line using the RBF network shown in Equations (22) and (23). If the system satisfies the hypothesis of Assumptions 1–4, with appropriate selection of sliding mode surface parameters, controller parameters and RBF network weight regulative law, the sliding mode surface scan be reached in a finite time, and the closed-loop system is asymptotically stability, and the brake pressure tracking error then converges to zero in a finite time.

Proof.

Take the differential of $\dot{z}={\dot{P}}_{\mathrm{A}}-{\dot{P}}_{\mathrm{A}}^{*}$ as follows [21]:

$$\begin{array}{cc}\hfill \ddot{z}& ={\ddot{P}}_{\mathrm{A}}-{\ddot{P}}_{\mathrm{A}}^{*}\hfill \\ & =\frac{c_{\mathrm{b}}{L}_0}{2\pi }\left(-\frac{1}{J}\frac{L_0}{2\pi }{P}_{\mathrm{A}}-\frac{B_{\mathrm{v}}}{J}{\omega }_{\mathrm{m}}\right)-{\ddot{P}}_{\mathrm{A}}^{*}+\frac{c_{\mathrm{b}}{L}_0}{2\pi }d+\frac{c_{\mathrm{b}}{L}_0}{2\pi }\frac{K_{\mathrm{T}}}{J}{I}_{\mathrm{a}}\hfill \end{array}$$

(29)

Get the derivative of Equation (27) and take Equations (28) and (29) into Equation (27) as:

$$\begin{array}{cc}\hfill \dot{s}& =\dot{z}+\alpha {\gamma }_1{\left|z\right|}^{{\gamma }_1-1}\dot{z}+\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\ddot{z}\hfill \\ & =\dot{z}+\alpha {\gamma }_1{\left|z\right|}^{{\gamma }_1-1}\dot{z}+\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left(\frac{c_{\mathrm{b}}{L}_0}{2\pi }d+I_{\mathrm{a}sw}^{*}\right)\hfill \end{array}$$

(30)

Select the Lyapunov function as follows:

$$V=\frac{1}{2}{s}^2+\frac{1}{2\eta }{\tilde{W}}^2$$

(31)

where $\tilde{W}=W^{*}-\widehat{W}$. □

Get the derivative of Equation (27) as:

$$\begin{array}{cc}\hfill \dot{V}& =s\dot{s}-\frac{1}{\eta }{\tilde{W}}^{\mathrm{T}}\dot{\widehat{W}}\hfill \\ & =s\left[\dot{z}+\alpha {\gamma }_1{\left|z\right|}^{{\gamma }_1-1}\dot{z}+\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left(\frac{c_{\mathrm{b}}{L}_0}{2\pi }d+I_{\mathrm{a}sw}^{*}\right)\right]-\frac{1}{\eta }{\tilde{W}}^{\mathrm{T}}\dot{\widehat{W}}\hfill \\ & =s\left[\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left(\frac{c_{\mathrm{b}}{L}_0}{2\pi }d-\kappa s-K\mathrm{sgn}\left(s\right)\right)\right]-\frac{1}{\eta }{\tilde{W}}^{\mathrm{T}}\dot{\widehat{W}}\hfill \end{array}$$

(32)

Set $K=\stackrel{⌢}{\rho }+k_T$, where $k_T>0$, From Assumption 4, Equation (31) satisfies the following expressions:

$$\begin{array}{cc}\hfill \dot{V}& \le s\left[\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left(\rho -\stackrel{⌢}{\rho }\mathrm{sgn}\left(s\right)-k_T\mathrm{sgn}\left(s\right)-\kappa s\right)\right]-\frac{1}{\eta }{\tilde{W}}^{\mathrm{T}}\dot{\widehat{W}}\hfill \\ & =\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left(s\rho -\stackrel{⌢}{\rho }\left|s\right|\right)-\frac{1}{\eta }{\tilde{W}}^{\mathrm{T}}\dot{\widehat{W}}-k_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|-\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}{s}^2\hfill \\ & =-\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left[\left|s\right|\left(\stackrel{⌢}{\rho }+\overline{\rho }-\overline{\rho }\right)+s\rho \right]-\frac{1}{\eta }{\tilde{W}}^{\mathrm{T}}\dot{\widehat{W}}-k_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|-\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}{s}^2\hfill \\ & \le -\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|\left[\left(\stackrel{⌢}{\rho }-\overline{\rho }\right)-\left(\overline{\rho }-\left|\rho \right|\right)\right]-\frac{1}{\eta }{\tilde{W}}^{\mathrm{T}}\dot{\widehat{W}}-k_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|-\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}{s}^2\hfill \end{array}$$

(33)

The RBF network weights update law is shown as follows:

$$\dot{\widehat{W}}=\eta \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|h$$

(34)

According to Assumption 1 and Equation (33), Equation (32) can be further simplified as follows:

$$\begin{array}{cc}\hfill \dot{V}& \le -\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|\left[\left({\widehat{W}}^{T}h-W^{*\mathrm{T}}h+\epsilon \right)-\left(\overline{\rho }-\left|\rho \right|\right)\right]-\frac{1}{\eta }{\tilde{W}}^{\mathrm{T}}\dot{\widehat{W}}\hfill \\ & -k_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|-\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}{s}^2\hfill \\ & =-\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|\left[\epsilon -\left(\overline{\rho }-\left|\rho \right|\right)\right]-k_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|-\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}{s}^2\hfill \\ & \le \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|\left|\epsilon \right|-\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|\left[\overline{\rho }-\left|\rho \right|\right]-k_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|-\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}{s}^2\hfill \\ & \le \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|\left(\left|\epsilon \right|-\overline{\rho }+\left|\rho \right|\right)-k_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|-\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}{s}^2\hfill \end{array}$$

(35)

From Assumption 2 we can know that $\left|\epsilon \right|-\overline{\rho }+\left|\rho \right|<0$, as follows:

$$\begin{array}{cc}\hfill \dot{V}& \le -k_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}\left|s\right|-\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}{s}^2\hfill \\ & \le -\sqrt{2}{k}_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}{V}^{1/2}-2\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}V\hfill \end{array}$$

(36)

Take ${\rho }_1\left(\dot{z}\right)=\sqrt{2}{k}_T\beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}$, ${\rho }_2\left(\dot{z}\right)=2\kappa \beta {\gamma }_2{\left|\dot{z}\right|}^{{\gamma }_2-1}$, Equation (35) can be expressed as follows [22]:

$$\dot{V}\le -{\rho }_1\left(\dot{z}\right)V^{1/2}-{\rho }_2\left(\dot{z}\right)V$$

(37)

According to Equation (37), when $\dot{z}\ne 0$, there is ${\rho }_1>0$, ${\rho }_2>0$, and $\dot{V}\le 0$. According to Lemma 1 we know that V(t) converges to zero at finite time. It is easy to choose that it is not the equilibrium state when $\dot{z}=0$ and $z\ne 0$. The system can reach the sliding surface in a finite time, then the tracking error converges to zero in a finite time.

4. Validation

4.1. Simulation Analysis

More stringent requirements for the performance of the braking mechanism and the design of the control law are put forward for landing on an icy runway. In the following, we take the EMA braking on any icy runway as an example, and a numerical simulation is carried out under the MATLAB2014a environment to verify the effectiveness of the designed controller. In order to facilitate the control goal setting and the result analysis [18], we use the following formula:

$$\mu \left(\lambda \right)=2{\mu }_{\max }\frac{{\lambda }^{*}\lambda }{{\lambda }^{*}{}^2+{\lambda }^2}$$

(38)

For the parameters with time-varying features or difficult to be accurately measured/identified in the actual system, the consistent comparison about the mathematical model, physical model and experimental results must be obtained in order to make sure the reasonable parameter nominal value. Such parameters include the torque coefficient of the brake disc, gear clearance, actuator dead zone.

The speed at the start of the aircraft braking is set to V_x(0) = 72 m/s, and the starting angular velocity of the main wheel is ω(0) = 180 rad/s, The expected slip ratio is λ^* = 0.15, with a upper limit of slip ratio is λ^p = 0.20, and the maximum binding coefficient is μ_max = 0.2. According to the proposed algorithm, the control law is designed and complemented to the EMA braking system. The simulation results are shown in Figure 4.

The braking process time is 30.6 s with the braking distance of 1139 m. From the simulation results, it can be seen that the optimal slip rate is well tracked. There is no significant overshoot, and the braking pressure is kept stable. Moreover, the braking system works in a small range on both sides of μ_max.

4.2. Hardware In Loop (HIL) Experiment

In order to verify the brake actuator pressure servo performance, a sine wave emitted by the function signal generator for motor drive controller is used as brake pressure reference signal. An oscilloscope is also used to record the brake pressure feedback signal. The output of the signal from the pressure sensor is then amplified and filtered by the signal conditioning circuit in order to observe the performance in an easy way. The HIL system structure is shown in Figure 5.

The hardware of the HIL test bench is composed by an anti-skid braking controller, the EMA and the aircraft wheel. The software includes the upper computer and the simulation unit. The aircraft model is built by RT-LAB and downloaded to the target board to achieve the simulation of the braking process. The non-slip controller uses a Texas Instruments TMS320F2812 chip to adjust the slip ratio in real-time. The braking EMA test bench is shown in Figure 6.

In this paper, the algorithm is used to carry out the HIL experiment, and the controller parameters are the same as those used in the numerical simulation. Before the HIL simulation, firstly we verify the pressure servo performance of the EMA. A sine wave used as the reference signal of the driving motor brake pressure controller is generated by the signal generator. The pressure sensor output signal is recorded by the oscilloscope, after amplification and filtering. The double closed-loop linear control algorithm is satisfactory at low amplitude or frequency of the brake pressure. Otherwise, the traditional method is hard to achieve the desired effect by increase pressure or actuating frequency. The results are shown in Figure 7.

It can be seen form Figure 7a,b that the pressure signal frequency is 10 Hz and the amplitude is 2500 N. Using double closed loop proportional integral derivative (PID) control and proposed control algorithm, it can be seen that the NFTSM algorithm in this paper has better tracking accuracy and smaller signal lag. In Figure 7c,d, the pressure signal frequency is 25 Hz, amplitude is 6000 N. The feedback, frequency and the minimum value of the given signal are basically the same by PID. However, the peak value and peak value of the peak have some degree of attenuation. There is no significant attenuation with the NFTSM proposed in this paper.

It can be seen form Figure 8a that the optimal slip rate with no significant overshoot is working in a small range of μ_max. The braking process lasts 31.6 s, with a braking distance of 1147 m.

Figure 8b shows the slip rate tracking error is gradually increasing during the braking process, which indicates the slip rate tends to show extraordinary characteristics when the plane tends to stop, but the brake point is still located in the stable region. It should be noticed that when the braking process is ended and V_x = 0 (the braking speed), the slip rate is singular. There is no divergent slip phenomenon tends in ideal mathematical model. By taking consideration of the changing wheel load, landing gear buffering, runway roughness, tire compression characteristics, a small changes of ω (the wheel speed) will lead inevitably to some perturbation of the wheel line speed influenced by the torque. In the high-speed stage, the aircraft speed is big enough to avoid dynamic perturbation effects on slip rate caused by ω with little fluctuation, however, it will cause a strong influence on the slip rate when ω is little, which is the inherent characteristics of the slip ratio control. Therefore, with the existing control method it is difficult to avoid the low speed slip rate from alternating. The curve seems to have a great influence on the control results, whereas this is not the core problem of slip rate control in practical applications. The target slip ratio tracking effect is not obvious at low speed. The braking wheel is allowed to lock at the end of the braking process in an icy runway. If the speed is low enough, the pilot can manually brake or use maximum pressure to brake the aircraft without an anti-skid braking system.

5. Conclusions

Based on the analysis of the working principle of all-electric braking system, a mathematical model of the EMA pressure servo system is established, and the state space model with matching uncertainty is obtained through reasonable simplifications. A new FTSM control strategy is put forward according to engineering practice to realize high-performance pressure servo control of anti-skid braking systems. The validity of the control strategy is verified by experiments. The following conclusions are obtained:

• The sliding mode surface in the algorithm is of non-singular nature, which avoids the singularity problem of the traditional NTSM control strategy. The RBF neural network is used to solve the estimation problem of the upper bound of the composite interference, which reduces the conservativeness of sliding mode control design. By reducing the sliding mode switching gain, the sliding mode chattering is effectively inhibited, and the braking pressure servo performance is greatly improved. The analysis of Lyapunov method shows that the FTSM control strategy proposed in this paper has terminal characteristics, which can achieve the finite time arrival of the sliding surface and the asymptotic stability of the closed-loop system. The tracking error of the pressure then converges to zero in finite time.
• The experimental results show that the proposed control strategy without changing the traditional double-closed-loop control structure can improve the servo performance, control precision and response speed of the actuator compared with a linear controller.