Sliding mode controller with neural network compensation for tank

2.1. Electromechanical coupled dynamics modeling

Fig. 1Principle of bidirectional movement of an all-electric tank

Stabilizers for all-electric tanks include horizontal stabilizer and vertical stabilizer. In the horizontal direction, the actuator is a servo motor driving the turret in the horizontal direction via a reducer. In the vertical direction, the actuator is an electric cylinder. The extension and retraction of the electric cylinder can be realized by controlling the servo motor in the electric cylinder, thus accomplishing the rotational movement of the barrel around the trunnion [7]. As a result, the dynamical equation for the bidirectional stabilizers on an all-electric tank is established as:

where $\mathbf{q}\in {\mathbf{R}}^n$, $\dot{\mathbf{q}}\in {\mathbf{R}}^n$, $\ddot{\mathbf{q}}\in {\mathbf{R}}^n$ represent muzzle angular displacement, angular velocity, and angular acceleration, respectively. $\mathbf{M}\left(\mathbf{q}\right)\in {\mathbf{R}}^{n\times n}$ is the inertia matrix. Consider the definition of $\mathbf{M}\left(\mathbf{q}\right)$, it is a positive definite matrix. $\mathbf{C}\left(\mathbf{q},\dot{\mathbf{q}}\right)\in {\mathbf{R}}^{n\times n}$ and $\mathbf{C}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}$ is the Koch force matrix. $\mathbf{G}\left(\mathbf{q}\right)\in {\mathbf{R}}^n$ is the gravity matrix. $\mathbf{F}\left(\dot{\mathbf{q}}\right)\in {\mathbf{R}}^n$ is the friction matrix. ${\mathbf{\tau }}_d\in {\mathbf{R}}^n$ is all unmodeled error and system disturbance. $\mathbf{\tau }\in {\mathbf{R}}^n$ is the control torque. They are defined as follows:

where the parameters involved are defined as:

For ease of description, subscript ${\cdot }_1$ denotes the parameters associated with the tank in the horizontal direction and subscript ${\cdot }_2$ denotes the parameters associated with the tank in the vertical direction. $q_i$ denotes the angular displacement of the muzzle, $m_i$ denotes the mass of the rotating part, $R_i$ denotes the distance from the center of mass of the rotating body to the center of rotation, and ${\tau }_{di}$ denotes all unmodeled errors and systematic disturbances, where $i=\mathrm{1,2}$.

2.2. SMC controller design

Define the tracking angular displacement error $\mathbf{e}={\left[e_1,e_2\right]}^T$, and the angular velocity tracking error $\dot{\mathbf{e}}={\left[{\dot{e}}_1,{\dot{e}}_2\right]}^T$ as:

the desired value of $\mathbf{q}$, and ${\dot{\mathbf{q}}}_d=\left[{\dot{q}}_{1d},{\dot{q}}_{2d}\right]$ is the desired value of $\dot{\mathbf{q}}$.

Define the sliding mode function as:

where $\mathbf{s}\in {\mathbf{R}}^n$ is the sliding mode vector function. $\mathbf{\Lambda }$ is the designed diagonal matrix and all of its diagonal elements are greater than zero.

The dynamic properties of the sliding mode control strategy require the system to converge to an equilibrium state when stability is obtained, i.e., $\mathbf{e}=\dot{\mathbf{e}}=0$. Therefore, the goal now is to ensure that when the system equilibrium point is reached, there is $\mathbf{s}=\dot{\mathbf{s}}=0$. Combining Eqs. (2) and (3) and taking the derivative for $\mathbf{s}$ yields:

Bringing Eq. (1) into (4) yields:

where $\mathbf{P}=\mathbf{M}\left({\ddot{\mathbf{q}}}_d+\mathbf{\Lambda }{\dot{\mathbf{q}}}_d-\mathbf{\Lambda }\dot{\mathbf{q}}\right)+\mathbf{C}\dot{\mathbf{q}}+\mathbf{G}+\mathbf{F}$.

Let $\dot{\mathbf{s}}=0$, the control torque can be obtained as:

In this paper, sliding mode control based on exponential convergence law is selected as:

where $\mathbf{\nu }$ is the sliding mode robust term, and $\mathit{\gamma }$ is a matrix that has all diagonal elements greater than 0.

Combining Eqs. (5) and (7), one gets:

where $\mathbf{\beta }$ is the robust term.

2.3. Design of neural network compensation

Define the generalized state of $\mathbf{P}$ as $\widehat{\mathbf{P}}$, and introduce the neural network as follows:

where $\sigma \left(\cdot \right)$ is the activation function, $\mathbf{x}$ is the neural network input, $\mathbf{W}$ is the neural network weight function, $\widehat{\mathbf{W}}$ is the estimate of $\mathbf{W}$, $\widehat{f}$ is the estimate of $f$, and $\epsilon \left(\mathbf{x}\right)$ is the neural network approximation error.

In order to make the weight error $\tilde{\mathbf{W}}=\mathbf{W}-\widehat{\mathbf{W}}$ move in the direction of negative gradient, this paper defines the neural network adaptive rate as:

where $\mathbf{\Gamma }$ is a diagonal matrix with diagonal elements greater than 0, one of the design variables of the algorithm.

Combining Eqs. (8) and (9), the controller is obtained:

2.4. Stability analysis

Define the Lyapunov function as:

Taking the derivative of $V$ gives:

Therefore, the robust term $\mathbf{\beta }$ is designed as $\mathbf{\beta }=-\left({\mathbf{\epsilon }}_N+{\mathbf{\tau }}_{dN}\right)\mathrm{s}\mathrm{g}\mathrm{n}\left(\mathbf{s}\right)$, where ${\mathbf{\epsilon }}_N$ and ${\mathbf{\tau }}_{dN}$ are design parameters. Choosing the right ${\mathbf{\epsilon }}_N$ and ${\mathbf{\tau }}_{dN}$ means that $\dot{V}\le 0$ can be guaranteed.

Due to the positive definite character of the inertia matrix $\mathbf{M}$, it is clear that there is $V\to \infty$ as $\mathbf{s}\to \infty$. Therefore, the controller proposed in this paper can achieve asymptotic stabilization.

The final controller designed in this paper is:

3.1. Principle of co-simulation

The actual tank structure is quite complex, this paper only considers the fire system and chassis system of the tank. The multibody dynamics model of the real tank is established by the software Recurdyn, as shown in Fig. 2. The 3D model of E-level road surface is built according to the literature [8], and it is imported into the Recurdyn software. The controller is written in Matlab/Simulink. The software interface is utilized to realize the real-time data transmission of the two software. The principle of co-simulation is shown in Fig. 3.

3.2. Parameter selection

The relevant dynamical parameters involved are set with $m_1=$5200.00 kg, $m_2=$2088.15 kg, $R_1=$1.05 m, $R_2=$5.12 m. In addition, the acceleration of gravity $g=$9.80 m⋅s^-2.

The neural network input is selected as $\mathbf{x}=[e_1,e_2,q_1,q_2{]}^T$, and the neural network activation function is selected as Gaussian function:

where $j$ denotes the $j$-th node in the hidden layer. ${\mathbf{c}}_j$ is the center point of the Gaussian function, designed as ${\mathbf{c}}_j$= [–3, –2, –1, 0, 1, 2, 3]. ${\mathbf{b}}_j$ is the width of the Gaussian function, designed as ${\mathbf{b}}_j=$ 8.

In addition, other design parameters are selected as $\mathbf{\Lambda }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\mathrm{80,80}\right)$, $\mathbf{\Gamma }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\mathrm{100,100}\right)$, $\mathit{\gamma }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\mathrm{10,10}\right)$, $\mathbf{\nu }=$10.

To validate the superiority of the proposed controller, conventional PID controller is selected for comparison. PID is widely used in tank stabilizers, which are mainly adjusted by $K_P$, $K_I$, $K_D$ three parameters. For the co-simulation, $K_{P1}=$1200, $K_{I1}=$800, $K_{D1}=$35, $K_{P2}=$900, $K_{I2}=$600, $K_{D2}=$17 are chosen.

The muzzle motion command is set to a step function of:

Fig. 2Multibody dynamics model built in Recurdyn

Fig. 3Schematic diagram of co-simulation

4.1. Tracking performance analysis

The co-simulation is successfully implemented. Fig. 4(a) and Fig. 5(a) show the angular displacements of the muzzle. Fig. 4(b) and Fig. 5(b) show their errors. It can be clearly seen that the proposed controller has a better control effect compared to the PID controller. The proposed controller spends 1.45 seconds to reach stabilization, while the PID controller spends 1.98 seconds, an improvement of 26.8 %. Fig. 4(c) and Fig. 5(c) show the motor output torque with two controllers. The motor torque output reaches its maximum value at the instant the step command is received by the tank bidirectional stabilizer. Comparing the maximum value of torque of the two controllers, the proposed controller reduces the maximum value of torque by 8.6 % in the horizontal direction and 44.3 % in the vertical direction than the PID controller.

4.2. Anti-interference performance analysis

The anti-jamming performance analysis is realized by the experiment of fixed angle commands. The experiment is to keep the tank gun at a 0 degree pointing angle both horizontally and vertically, with different controllers. In order to observe the effects of uncertainty, to external excitation and noise on the deliberative controller, we compare the tank driving at 20 km/h and 30 km/h on D-level and E-level roads. In order to quantify the effectiveness of the controller, the absolute average error within a finite period of time is commonly employed to characterize tank gun stability accuracy. The expression is:

where $n$ is the total number of sampling points and ${\theta }_i$ is the angular position error at $n=i$. The results of the above experiments are calculated and analyzed ranging from 6 to 10 s. Their comparison pairs with PID controllers are shown in Fig. 6.

Fig. 4Performance of the tank stabilizer in the horizontal direction

a) Angular displacement

b) Angular displacement error

c) Output torque

Fig. 5Performance of the tank stabilizer in the vertical direction

a) Angular displacement

b) Angular displacement error

c) Output torque

Fig. 6Tank gun stability accuracy against different excitation

a) Horizontal accuracy at speeds of 20 km/h

b) Vertical accuracy at speeds of 20 km/h

c) Horizontal accuracy at speeds of 30 km/h

d) Vertical accuracy at speeds of 30 km/h