Adaptive Backstepping Sliding Mode Control for the Vertical Launching Barrel-Cover of the Underwater Missile

Abstract

The vertical launching barrel-cover device is a symmetry mechanism, so we can simplify this symmetry mechanism into an electro-hydraulic servo problem. The vertical launching barrel-cover of the underwater missile has parameter uncertainty, load time-varying and strong nonlinear of motion and coupling during the submarine movement. Therefore, it is important to study the nonlinear adaptive antidisturbances control method of the vertical launching barrel-cover. For the vertical launching cover system of the underwater missile, an adaptive backstepping sliding mode control system based on disturbance observer is proposed. The backstepping sliding mode controller is used to solve the problem of nonlinearity and chattering in the system; the adaptive method is used to dynamically follow the changes of the system characteristics, adjust the controller parameters. At the same time, the disturbance observer is added to the system to reduce the sensitivity of the system to disturbance By completing the process of switching cover, we have experimented with the designed adaptive backstepping sliding mode controller to verify the effectiveness of the controller. The controller improves the stability of the whole switch cover system. The experimental results show that compared with traditional proportional integral (PI) controller and sliding mode controller, the controller solves the problem of over-limit when the cover is in place, and has excellent tracking performance in the process of the switch cover.

1. Introduction

The vertical launching barrel-cover system is an essential part of the submarine-launched strategic missile launcher, and the whole device is a symmetry mechanism [1]. The characteristics of the barrel-cover system mainly include the open-cover time and the open-cover angle. These two characteristic parameters are important tactical indicators of underwater launcher [2], especially the open-cover time of the barrel-cover system, which directly affects the rapid response capability of the missile weapon system [3].

The vertical launching barrel-cover of the underwater missile adopts electro-hydraulic servo system to drive the launching barrel-cover [4]. However, the symmetry device has the characteristics of parameter uncertainty [5], load time-varying [6], strong nonlinearity of motion [7,8], and coupling [9,10] during the submarine movement [11]. Therefore, the study of the nonlinear adaptive anti-interference control method is of considerable significance to the improvement of the performance of the barrel-cover system [12].

To ensure the tracking characteristics of the system, scholars have done a lot of research work on the parameter uncertainty and external time-varying disturbance of electro-hydraulic servo system [13,14]. Ahn studied the adaptive backstepping control strategy of the electro-hydraulic servo system [15]; Yao proposed an adaptive backstepping controller for hydraulic actuator [16]; Guo proposed an extended state observer based backstepping control strategy for electro-hydraulic actuator [17]; Yang proposed an adaptive integral robust controller for electro-hydraulic servo system [18]; Zerdali proposed an adaptive extended Kalman filter algorithm estimating the stator stationary axis components of stator currents [19]; Navvabi proposed an extended adaptive fuzzy sliding mode controller [20]; Wang proposed a novel high-gain extended state observer based adaptive sliding mode path following control scheme [21]; Gao proposed a forecasting-based data-driven model-free adaptive controller [22]; Hu proposed a fuzzy integral sliding mode control strategy for flexible air-breathing hypersonic vehicles with input nonlinearity [23]; Min proposed an improved adaptive fuzzy backstepping control for uncertain, mismatched nonlinear systems [24]. Some of the methods mentioned above can not achieve good tracking effect due to the limitation of control parameters under the condition of parameter uncertainty and external disturbance, and some of them have some limitations in dealing with parameter uncertainty of nonlinear dynamic model. Backstepping control is one of the most common control schemes in nonlinear systems. Considering the chattering problem in the system and enhancing the robustness of the control system, we choose an adaptive backstepping sliding mode controller.

To suppress the unknown disturbance [25] and ensure the open-cover time and open-cover angle [26], we combine the nonlinear observer [27], adaptive control [28,29,30] and backstepping sliding mode control [31,32]. Scholars have studied many disturbance observers to suppress external disturbances [33,34]. Kayacan proposed a novel sliding mode control algorithm to handle mismatched uncertainties in systems via a novel self-learning disturbance observer [35]; Chang proposed a dynamic output feedback controller that uses a disturbance estimation algorithm [36]; Lee proposes a composed control approach by combining a nonlinear disturbance observer and an asymptotic tracking control for spacecraft formation flying system subject to nonvanishing disturbances [37]; Ma proposes an adaptive dynamic surface controller based on nonlinear disturbance observer [38]. At the same time, adaptive control can follow the changes of system characteristics dynamically [39,40], adjust the parameters of the controller and has adaptability. Therefore, we can find that the controller can meet our experimental requirements and improve the stability of the system.

To sum up, an adaptive backstepping sliding mode controller based on disturbance observer is designed for the parameter uncertainty, load time-varying, strong nonlinearity and coupling of the vertical launcher cover of an underwater missile, which can realize the adaptability of the parameters and the suppression of external variable disturbances during the operation of the barrel-cover system.

2. Nonlinear Modeling of the Vertical Launching Barrel-cover System

As shown in Figure 1, the system uses a valve-controlled hydraulic cylinder. When establishing the flow equation, it is assumed that the valve has a zero opening structure and the system oil supply pressure is constant.

Ignoring the compressed flow, the flow equation is:

$$Q_L=K_q{x}_v-K_c{P}_L$$

(1)

In the formula (1), $K_q$ is the servo valve flow gain, $K_c$ is the flow pressure coefficient, $P_L$ is the load pressure drop, and $P_L=P_1-P_2$ is defined.

The flow Continuity Equation of Hydraulic Cylinder:

$$Q_L=A_p\frac{d{x}_p}{dt}+C_{tp}{P}_L+\frac{V_t}{4{\beta }_e}\frac{d{P}_L}{dt}$$

(2)

In the formula (2), $A_p$ is the effective area of the piston in the hydraulic cylinder, $x_p$ is the displacement of the piston, $C_{tp}$ is the total leakage coefficient of the hydraulic cylinder, $V_{\mathrm{t}}$ is the total volume of the two chambers in the hydraulic cylinder, and ${\beta }_e$ is the elastic modulus of the effective volume.

Ignoring the mass and friction of the fluid, the force balance equation of the system is:

$$m\frac{d^2{x}_p}{d{t}^2}=A_p{P}_L-B_p\frac{d{x}_p}{dt}-F_L$$

(3)

In the formula (3), $m$ is the total mass of the piston and the load converted to the piston, $B_p$ is the viscous damping coefficient of the piston and the load, and $F_L$ is the external load force acting on the load.

In order to use the adaptive backstepping sliding mode method to control the barrel-cover system, first of all, it is necessary to write the system state equation. According to the control principle of the adaptive controller, the model considering external disturbances and parameter uncertainties is set up as follows:

$$\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2={\theta }_1{x}_3-{\theta }_2{x}_2+D\\ {\dot{x}}_3=-{\theta }_3{x}_2-{\theta }_4{x}_3+{\theta }_{5}u\end{array}$$

(4)

In the formula (4), D is an unknown external disturbance, $x_1$, $x_2$, $x_3$ are the displacement, speed and differential pressure of the hydraulic cylinder, and

$$\{\begin{array}{l}{\theta }_1=A_p/m,\\ {\theta }_2=B_p/m,\\ {\theta }_3=4{\beta }_e{A}_p/V_t,\\ {\theta }_4=4{\beta }_e{C}_{tp}/V_t,\\ {\theta }_5=4{\beta }_e/V_t,\\ u=Q_L,\\ D=F_L/m\end{array}$$

3. Controller Design

3.1. Design of Disturbance Observer

The disturbance observer is designed as follows:

$$\{\begin{array}{l}\widehat{D}=z+p(x_1,x_2)\\ \dot{z}=-L(x_1,x_2)z+L(x_1,x_2)\cdot (-p(x_1,x_2)-{\theta }_1{x}_3+{\theta }_2{x}_2)\end{array}$$

(5)

In the formula (5), $p(x_1,x_2)$ is a nonlinear function to be designed; $L(x_1,x_2)$ is the gain of the nonlinear observer; and the relationship between the two is as follows:

$$L(x_1,x_2){\dot{x}}_2=\dot{p}(x_1,x_2)$$

(6)

$\widehat{D}$ is the estimated value of the observer, $\tilde{D}$ is the observation error of the observer, So there is:

$$\tilde{D}=D-\widehat{D}$$

(7)

Differentiate the two ends of the equation:

$$\begin{array}{l}\dot{\tilde{D}}=\dot{D}-\dot{\widehat{D}}=-\dot{z}-\dot{p}(x_1,x_2)\\ =L(x_1,x_2)(z+p(x_1,x_2))-L(x_1,x_2)({\dot{x}}_2-{\theta }_1{x}_3+{\theta }_2{x}_2)\\ =L(x_1,x_2)\widehat{D}-L(x_1,x_2)D=-L(x_1,x_2)\tilde{D}\end{array}$$

(8)

In the above formula, $\dot{D}=0$, because the change speed of disturbance $D$ is slower time-varying than that of observer. Therefore, as can be seen from the above formula, selecting a gain greater than zero is $L(x_1,x_2)>0$, the error value of the observer will converge according to the exponent.

Let $L(x_1,x_2)=b,b>0$, there are:

$$p(x_1,x_2)=b{x}_2$$

(9)

The disturbance estimate $\widehat{D}$ is obtained by the above calculation. From the state equation of the system, it can be observed that the disturbance term $D$ is unknown. Therefore, the unknown situation is compensated by the controller. The disturbance term $\widehat{D}$ is input into the adaptive backstepping sliding mode controller algorithm.

3.2. Adaptive Backstepping Controller Design

Designed an adaptive backstepping sliding mode controller system that defines three error signals:

$$\{\begin{array}{l}{e}_1=x_1-x_{1d}\\ e_2=x_2-x_{2d}\\ e_3=x_3-x_{3d}\end{array}$$

(10)

At the same time, considering the three parameters of the change ${\theta }_2,{\theta }_3,{\theta }_4$, ${\widehat{\theta }}_2,{\widehat{\theta }}_3,{\widehat{\theta }}_4$ are defined as the estimated values of these three parameters, and ${\tilde{\theta }}_2,{\tilde{\theta }}_3,{\tilde{\theta }}_4$ are the estimated error values of the parameters. The relationship is as follows:

$$\{\begin{array}{l}{\tilde{\theta }}_2={\widehat{\theta }}_2-{\theta }_2\\ {\tilde{\theta }}_3={\widehat{\theta }}_3-{\theta }_3\\ {\tilde{\theta }}_4={\widehat{\theta }}_4-{\theta }_4\end{array}$$

(11)

Step 1

To make the error $e_1$ close to zero, the Lyapunov function of this order is chosen as:

$$V_1=\frac{1}{2}{e}_1^2\ge 0$$

(12)

The derivation at both ends of formula (12) can be obtained as follows:

$$\begin{array}{l}{\dot{V}}_1=\dot{e}{}_1·e_1=e_1({\dot{x}}_1-{\dot{x}}_{1d}) =e_1(x_2-{\dot{x}}_{1d})\\ =e_1(e_2+x_{2d}-{\dot{x}}_{1d})\end{array}$$

(13)

where $x_{2d}$ is the virtual control quantity, take:

$$x_{2d}=-k_1{e}_1+{\dot{x}}_{1d}$$

(14)

where $k_1$ is a positive number greater than zero. By introducing formula (14) into the formula (13), we can get that:

$${\dot{V}}_1=-k_1{e}_1^2+e_1{e}_2$$

(15)

Among them:

$${\dot{x}}_{2d}={\ddot{x}}_{1d}-k_1(x_2-{\dot{x}}_{1d})$$

(16)

$$\begin{array}{ll}{\ddot{x}}_{2d}=& -k_1({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2+\widehat{d})+k_1{\ddot{x}}_{1d}+{\stackrel{⃛}{x}}_{1d}\\ & -k_1{\tilde{\theta }}_2{x}_2+k_1\tilde{d}\end{array}$$

(17)

$${\ddot{x}}_{2d}={\ddot{\widehat{x}}}_{2d}+{\ddot{\tilde{x}}}_{2d}$$

(18)

We defined:

$${\ddot{\widehat{x}}}_{2d}=-k_1({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2+\widehat{d})+k_1{\ddot{x}}_{1d}+{\stackrel{⃛}{x}}_{1d}$$

(19)

$${\ddot{\tilde{x}}}_{2d}=-k_1{\tilde{\theta }}_2{x}_2+k_1\tilde{d}$$

(20)

Step 2

To make $e_2$ close to zero and consider the existence of uncertain parameters ${\theta }_2$ and $d$, the second-order Lyapunov function is taken as follows:

$$V_2=V_1+\frac{1}{2}{e}_2^2+\frac{1}{2{\tau }_2}{\tilde{\theta }}_2^2+\frac{1}{2}{\tilde{d}}^2{}_{}$$

(21)

The derivation at both ends of formula (21) can be obtained as follows:

$$\begin{array}{ll}{\dot{V}}_2& ={\dot{V}}_1+e_2{\dot{e}}_2+{\tau }_2^{-1}{\tilde{\theta }}_2{\dot{\tilde{\theta }}}_2+{\tilde{d}}_2{\dot{\tilde{d}}}_2\\ & =e_2({\dot{x}}_2-{\dot{x}}_{2d})-k_1{e}_1^2+e_1{e}_2+{\tau }_2^{-1}{\tilde{\theta }}_2{\dot{\widehat{\theta }}}_2+{\tilde{d}}_2{\dot{\tilde{d}}}_2\\ & =e_2({\theta }_1(e_3+x_{3d})-{\widehat{\theta }}_2{x}_2+\widehat{d}+k_1(x_2-{\dot{x}}_{1d})-{\ddot{x}}_{1d})\\ & -k_1{e}_1^2+e_1{e}_2+{\tilde{\theta }}_2{e}_2{x}_2-\tilde{d}{e}_2+{\tau }_2^{-1}{\tilde{\theta }}_2{\dot{\widehat{\theta }}}_2+{\tilde{d}}_2{\dot{\tilde{d}}}_2\end{array}$$

(22)

where $x_{3d}$ is the virtual control quantity, take:

$$x_{3d}=\frac{1}{{\theta }_1}(-e_1-k_2{e}_2+{\widehat{\theta }}_2{x}_2+{\dot{x}}_{2d}-\widehat{d})$$

(23)

where $k_2$ is a positive number greater than zero. By introducing formula (23) into the formula (22), we can get that:

$$\begin{array}{ll}{\dot{V}}_2& =-k_1{e}_1^2-k_2{e}_2^2+{\theta }_1{e}_2{e}_3+{\tilde{\theta }}_2{e}_2{x}_2-\tilde{d}{e}_2\\ & +{\tilde{\theta }}_2{\tau }_2^{-1}{\dot{\tilde{\theta }}}_2+\tilde{d}\dot{\tilde{d}}\end{array}$$

(24)

Among them:

$$\begin{array}{ll}{\dot{x}}_{3d}& =\frac{1}{{\theta }_1}(-k_2{\dot{e}}_2-{\dot{e}}_1+{\dot{\widehat{\theta }}}_2{x}_2+{\widehat{\theta }}_2{\dot{x}}_2+{\ddot{x}}_{2d}-\widehat{d})\\ & =\frac{1}{{\theta }_1}(-k_2({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2+\widehat{d}-{\dot{x}}_{2d})-x_2+{\dot{x}}_{1d}+{\dot{\widehat{\theta }}}_2{x}_2\\ & +{\widehat{\theta }}_2({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2+\widehat{d})+{\ddot{\widehat{x}}}_{2d}+{\widehat{\theta }}_2{\tilde{\theta }}_2{x}_2-{\widehat{\theta }}_2\tilde{d}-k_2{\tilde{\theta }}_2{x}_2\\ & +k_2\tilde{d}+{\ddot{\tilde{x}}}_{2d})\end{array}$$

(25)

We defined:

$$\begin{array}{ll}{\dot{\widehat{x}}}_{3d}& =\frac{1}{{\theta }_1}(-k_2({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2+\widehat{d}-{\dot{x}}_{2d})-x_2+{\dot{x}}_{1d}+{\dot{\widehat{\theta }}}_2{x}_2\\ & +{\widehat{\theta }}_2({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2+\widehat{d})+{\ddot{\widehat{x}}}_{2d}\end{array}$$

(26)

$${\dot{\tilde{x}}}_{3d}=\frac{1}{{\theta }_1}({\widehat{\theta }}_2{\tilde{\theta }}_2{x}_2-{\widehat{\theta }}_2\tilde{d}-k_2{\tilde{\theta }}_2{x}_2+k_2\tilde{d}+{\ddot{\tilde{x}}}_{2d})$$

(27)

Step 3

The sliding mode controller is introduced into the third-order subsystem, and the sliding mode surface is designed as follows:

$$s=c_1{e}_1+c_2{e}_2+e_3$$

(28)

The derivation at both ends of formula (28) can be obtained as follows:

$$\begin{array}{lll}\dot{s}& =& c_1{\dot{e}}_1+c_2{\dot{e}}_2+{\dot{e}}_3\\ & =& c_1(x_2-{\dot{x}}_{1d})+c_2({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2+\widehat{d}\\ & & -{\dot{x}}_{2d}+{\tilde{\theta }}_2{x}_2-\tilde{d})+{\dot{e}}_3\end{array}$$

(29)

Among them, ${\tilde{\theta }}_3={\widehat{\theta }}_3-{\theta }_3$, ${\tilde{\theta }}_4={\widehat{\theta }}_4-{\theta }_4$, and ${\dot{e}}_3$ is:

$$\begin{array}{ll}{\dot{e}}_3& ={\dot{x}}_3-{\dot{x}}_{3d}\\ & =-{\widehat{\theta }}_3{x}_2+{\widehat{\theta }}_4{x}_3+{\theta }_{5}u-{\dot{\widehat{x}}}_{3d}-{\dot{\tilde{x}}}_{3d}+{\tilde{\theta }}_3{x}_2+{\tilde{\theta }}_4{x}_3\end{array}$$

(30)

By introducing formula (30) into formula (29), we can get that:

$$\begin{array}{lll}\dot{s}& =& c_1(x_2-{\dot{x}}_{1d})+c_2({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2+\widehat{d}-{\dot{x}}_{2d})\\ & & -{\widehat{\theta }}_3{x}_2-{\widehat{\theta }}_4{x}_3+{\theta }_{5}u-{\dot{\widehat{x}}}_{3d}+c_2{\tilde{\theta }}_2{x}_2+{\tilde{\theta }}_3{x}_2\\ & & +{\tilde{\theta }}_4{x}_3-{\dot{\tilde{x}}}_{3d}-c_2\tilde{d}\end{array}$$

(31)

Define the Lyapunov function of this order subsystem as:

$$V_3=V_2+\frac{1}{2}\frac{1}{{\tau }_3}{\tilde{\theta }}_3^2+\frac{1}{2}\frac{1}{{\tau }_4}{\tilde{\theta }}_4^2+\frac{1}{2}{s}^2$$

(32)

Derivatives of Formula (32) are obtained:

$$\begin{array}{lll}{\dot{V}}_3& =& -k_1{e}_1^2-k_2{e}_2^2+{\theta }_1{e}_2{e}_3+{\tilde{\theta }}_2{x}_2{e}_2-\tilde{d}{e}_2+{\tau }_2^{-1}{\tilde{\theta }}_2{\dot{\tilde{\theta }}}_2+{\tau }_3^{-1}{\tilde{\theta }}_3{\dot{\tilde{\theta }}}_3+{\tau }_4^{-1}{\tilde{\theta }}_4{\dot{\tilde{\theta }}}_4+\tilde{d}\dot{\tilde{d}}+s\dot{s}\\ & =& -k_1{e}_1^2-k_2{e}_2^2+{\theta }_1{e}_2{e}_3+s(c_1(x_2-{\dot{x}}_{1d})+c_2({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2+\widehat{d}-{\dot{x}}_{2d})-{\widehat{\theta }}_3{x}_2-{\widehat{\theta }}_4{x}_3\\ & & +{\theta }_{5}u-{\dot{\widehat{x}}}_{3d})+{\tilde{\theta }}_2({\tau }_2^{-1}{\dot{\tilde{\theta }}}_2+x_2{e}_2+c_2{x}_{2}s-\frac{1}{{\theta }_1}{\widehat{\theta }}_2{x}_{2}s+\frac{1}{{\theta }_1}{k}_1{x}_{2}s+\frac{1}{{\theta }_1}{k}_2{x}_{2}s)+{\tilde{\theta }}_3({\tau }_3^{-1}{\dot{\tilde{\theta }}}_3\\ & & +x_{2}s)+{\tilde{\theta }}_4({\tau }_4^{-1}{\dot{\tilde{\theta }}}_4+x_{3}s)-\tilde{d}{e}_2+\tilde{d}\dot{\tilde{d}}-c_2\tilde{d}s+\frac{1}{{\theta }_1}({\widehat{\theta }}_2\tilde{d}s-k_2\tilde{d}s-k_1\tilde{d}s)\end{array}$$

(33)

Take the control quantity of the controller as follows:

$$u=\frac{1}{{\theta }_5}(-k_{3}s-c_1(x_2-{\dot{x}}_{1d})-c_2({\theta }_1{x}_3-{\widehat{\theta }}_2{x}_2-{\dot{x}}_{2d}+\widehat{d})+{\widehat{\theta }}_3{x}_3+{\widehat{\theta }}_4{x}_3+{\dot{x}}_{3d})$$

(34)

By introducing formula (34) into formula (33), we can get that:

$$\begin{array}{ll}{\dot{V}}_3=& -k_1{e}_1^2-k_2{e}_2^2-k_3{s}^2+{\theta }_1{e}_2{e}_3-(\mu +\frac{(c_2-\frac{{\widehat{\theta }}_2}{{\theta }_1}+\frac{1}{{\theta }_1}{k}_2+\frac{1}{{\theta }_1}{k}_1)s+e_2}{\tilde{d}}){\tilde{d}}^2\\ & +{\tilde{\theta }}_2({\tau }_2^{-1}{\dot{\tilde{\theta }}}_2+x_2{e}_2+c_2{x}_{2}s-{\theta }_1^{-1}{\widehat{\theta }}_2{x}_{2}s+{\theta }_1^{-1}{k}_1{x}_{2}s+{\theta }_1^{-1}{k}_2{x}_{2}s)+{\tilde{\theta }}_3({\tau }_3^{-1}{\dot{\tilde{\theta }}}_3+x_{2}s)+{\tilde{\theta }}_4({\tau }_4^{-1}{\dot{\tilde{\theta }}}_4+x_{3}s)\\ \end{array}$$

(35)

The adaptive rate of each uncertain parameter is selected as follows:

$$\{\begin{array}{l}{\dot{\widehat{\theta }}}_2=-{\tau }_2(x_2{e}_2+c_2{x}_{2}s-{\theta }_1^{-1}{\widehat{\theta }}_2{x}_{2}s+{\theta }_1^{-1}{k}_1{x}_{2}s+{\theta }_1^{-1}{k}_2{x}_{2}s)\\ {\dot{\widehat{\theta }}}_3=-{\tau }_3{x}_2{s}_3\\ {\dot{\widehat{\theta }}}_4=-{\tau }_4{x}_3{s}_3\end{array}$$

(36)

By introducing formula (36) into formula (35), we can get that:

$$\begin{array}{ll}{\dot{V}}_3& =-k_1{e}_1^2-k_2{e}_2^2-k_3{s}^2+{\theta }_1{e}_2{e}_3-(\mu +\frac{(c_2-\frac{{\widehat{\theta }}_2}{{\theta }_1}+\frac{1}{{\theta }_1}{k}_2+\frac{1}{{\theta }_1}{k}_1)s+e_2}{\tilde{d}}){\tilde{d}}^2\\ & =-E^{T}QE-(\mu +\frac{(c_2-\frac{{\widehat{\theta }}_2}{{\theta }_1}+\frac{1}{{\theta }_1}{k}_2+\frac{1}{{\theta }_1}{k}_1)s+e_2}{\tilde{d}}){\tilde{d}}^2\end{array}$$

(37)

Among them:

$$Q=\left[\begin{array}{lll}{k}_1+k_3{c}_1^2& k_3{c}_1{c}_2& k_3{c}_1\\ k_3{c}_1{c}_2& k_2+k_3{c}_2^2& k_3{c}_2-\frac{{\theta }_1}{2}\\ k_3{c}_1& k_3{c}_2-\frac{{\theta }_1}{2}& k_3\end{array}\right]$$

(38)

In order to ensure that the subsystems of each order are stable, the parameters $k_1$, $k_2$, $k_3$, $c_1$ and $c_2$ of the system must meet the following conditions:

$$\{\begin{array}{l}{k}_1>0,k_2>0,k_3>0,c_1>0,c_2>0\\ k_1{k}_2+k_1{k}_3{c}_2^2+k_2{k}_3{c}_1^2>0\\ k_1{k}_2{k}_3+k_1{k}_3{c}_2-(k_1+k_3{c}_1^2)/4>0\end{array}$$

(39)

The system stability proves that when the controller parameters $k_1$, $k_2$, $k_3$, $c_1$ and $c_2$ satisfy the formula (39), the matrix $Q$ is a positive definite matrix.

Let $W=E^{T}QE$, from formula (37) we know that $\dot{V}\le -W$, since $e_1$, $e_2$, $e_3$ and ${\theta }_2$, ${\theta }_3$, ${\theta }_4$ are bounded, so $V$ is bounded. According to Barbbalat lemma, when $t\to \infty$, $e_i(i=1,2,3)\to 0$, that is, the entire system is progressively stable.

Figure 2 is a schematic diagram of an adaptive backstep sliding mode controller. The observer is used to input the external disturbance D into the backstep sliding mode controller to achieve the final disturbances suppression compensation. Considering the influence of physical factors on system parameters, a parameter adaptive method is introduced. Among them, ${\theta }_2=B_p/m$, ${\theta }_3=4{\beta }_e{A}_p/V_t$, ${\theta }_4=4{\beta }_e{C}_{tp}/V_t$. They correspond to the changes of physical parameters in the electro-hydraulic servo system, namely, the viscous damping coefficient of the hydraulic cylinder, the total leakage coefficient of the hydraulic cylinder and the elastic volume modulus of the hydraulic oil.

4. Simulation Verification

Simulink in MATLAB is used to simulate and verify the designed adaptive backstepping sliding mode controller. The simulation model is based on the system state equation shown in formula (4). The state equation in Simulink is implemented by c-function programming. The simulation parameters are as follows:

$$\{\begin{array}{l}{A}_p=1.88\times {10}^{-3}{m}^2\\ B_p=7261N/m{s}^{-1}\\ {\beta }_e=1.37\times {10}^{9}N/m^2\\ V_t=0.38\times {10}^{-3}{m}^3\\ m=350kg\\ C_{tp}=9.2\times {10}^{-13}{m}^3/s\cdot Pa\end{array}$$

Backstepping: $k_1=400$, $k_2=250$, $k_3=200$; Adaptive: ${\tau }_2=8\times {10}^{-20}$, ${\tau }_3=1\times {10}^{-21}$,${\tau }_4=1\times {10}^{-19}$; Sliding surface parameters: $c_1=120$, $c_2=100$; Observer: $b=800$.

4.1. Simulation Verification of Conventional Sliding Mode Controller

As can be seen from Figure 3 and Figure 4, the external disturbance force causes a large impact problem when the barrel-cover starts to operate. When the opening of the cover is over, the position is over-limit, and the tracking error of the entire opening process is large. The PID controller cannot suppress the disturbance force, resulting in the position of the hydraulic cylinder exceeding the limit when the barrel-cover is in place, damaging the mechanical structure of the barrel-cover device. After the disturbance observer is added to the traditional sliding mode controller, the error becomes smaller when the cover is opened, but the tracking accuracy is poor, and the stability of the switch cover process cannot be guaranteed.

4.2. Simulation Verification of Adaptive Backstepping Sliding Mode Controller

In order to further improve tracking accuracy and anti-disturbance ability to external disturbance, an adaptive backstepping sliding mode controller is designed. As can be seen from Figure 5 and Figure 6, the opening process has a good tracking effect, and the opening error when the opening is completed is 0.05°. Compared with the traditional sliding mode controller, the adaptive backstepping sliding mode controller can effectively improve the tracking accuracy and suppress external disturbance, and suppress the chattering problem in the sliding mode.

5. Experimental Verification

5.1. Experimental Set-Up

As shown in Figure 7, the device is a vertical launching barrel-cover of the underwater missile.

The parameters of the vertical launching barrel-cover of the underwater missile test bench are shown in

Table 1. The parameters of the vertical launching barrel-cover.

Parameter	Numerical Value
barrel-cover quality/kg	350
drive form	electro-hydraulic servo
control method	position control
pressure/MPa	20
Opening angle/°	94.8~95.2

.

The entire barrel-cover device consists of three parts: mechanical structure, hydraulic circuit and control system. When the barrel-cover is opened, the hydraulic cylinder pushes the cover to rotate, and the encoder transmits the angle signal. After the opening angle reaches 95° and the barrel-cover is in place, the system is opened, locks the barrel-cover and then carry out the missile launch. After the missile is launched, the barrel-cover is unlocked and the launching barrel-cover is closed.

Figure 8 is a schematic diagram of the control of the vertical launching barrel-cover of the underwater missile. The host computer runs human-computer interaction software, which is responsible for the issuance of control commands and the processing and display after information collection. The closed loop of the whole system is composed of a command signal generated by a control algorithm to generate a displacement signal, and an output signal of a change in the angle of the barrel-cover device; The controller outputs a drive signal of −10 V~+10 V, which is sent by the ACL-6126 board. The signal conditioning circuit and the power amplifier generate a current signal of −40 mA to +40 mA to drive the movement of the electro-hydraulic servo valve. According to the excitation signal, the servo valve controls the barrel-cover device to perform the corresponding switch cover movement.

Figure 9 is an experimental control program in which the adaptive backstepping sliding mode controller control parameters mainly include four parts: Backstep control gain coefficient, uncertain parameter adaptive rate gain, sliding surface parameters, and disturbance observer control parameters.

Backstepping: $k_1=240$, $k_2=150$, $k_3=60$; Adaptive: ${\tau }_2=5\times {10}^{-20}$, ${\tau }_3=3\times {10}^{-21}$, ${\tau }_4=2\times {10}^{-19}$; Sliding surface parameters: $c_1=40$, $c_2=80$; Observer: $b=500$.

5.2. Experimental Result and Analysis

In order to verify the superiority of the proposed adaptive backstepping sliding mode controller in the vertical launching barrel-cover of the underwater missile, the control effects of the traditional PI controller, sliding mode controller, disturbance observer based sliding mode controller and disturbance observer based adaptive backstepping sliding mode controller are compared. Among them, the design requirements of the controller: to ensure excellent tracking accuracy in the process of switching cover, and when the barrel-cover is in place, the opening angle can not exceed the limit to ensure the safety of the mechanical structure; the performance criteria: the open-cover time is 6 s and the open-cover angle time is 95°.

As shown in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, the tracking error of the PI controller is large, and there is a positioning error when the barrel-cover is in place, which cannot meet the test requirements; The tracking accuracy of the sliding mode controller is improved, but the overshoot phenomenon still exists; After adding the interference observer, the tracking error is reduced, and there is no angular overrun problem when the barrel-cover is in place. It can be seen that the PI controller and the traditional sliding mode controller can not effectively suppress the influence of the disturbance force. The sliding mode controller based on the disturbance observer satisfies the basic requirements of the test, but there is still an error of 0.35° when the barrel-cover is in place. The stability of the switch barrel-cover cannot be guaranteed.

As shown in Figure 15, Figure 16 and Figure 17, compared with the sliding mode controller based on disturbance observer, the adaptive backstepping sliding mode controller gradually corrects the tracking error, though the error is larger when the barrel-cover is opened, and the error is only 0.03 degree when the barrel-cover is in place. Compared with the sliding mode controller added to the disturbance observer, the adaptive backstep sliding mode controller more effectively ensures the stability of the vertical launching barrel-cover of the underwater missile system operation process.

Compared with PI controller, sliding mode controller and sliding mode controller based on disturbance observer, the adaptive backstepping sliding mode controller is based on the controlled object, which can make full use of the uncertain system parameters adjusted on-line to design the control rate, adjust the trajectory error of the switch cover, and improve the trajectory tracking accuracy of the switch cover. The adaptive adjustment process of uncertain parameters ${\theta }_2$, ${\theta }_3$ and ${\theta }_4$ in the control model is shown in Figure 18, Figure 19 and Figure 20.

5.3. Quantitative Analysis

Based on the analysis of the control accuracy of the switch barrel-cover, the root mean square error is used to quantify the trajectory tracking accuracy. The root mean square error(RMSE) of PI controller, sliding mode controller, sliding mode controller based on disturbance observer and adaptive backstepping sliding mode controller are shown in Figure 21. As can be seen from Figure 21, the root mean square error value of the sliding mode control added to the disturbance observer is reduced to 0.25%, and the root mean square error can be further reduced to 0.17% after the introduction of the adaptive system and the backstepping controller to form the hybrid controller. As can be seen from Figure 22 and Figure 23, the experimental data after using different controllers are analyzed from the angle error and time error. The opening time is 6s and the opening angle is 95°. The error of adaptive backstepping sliding mode controller can be further reduced, which are 0.61% and 1.67% respectively. We can find that the effect of the adaptive backstep sliding mode controller is better than that of the traditional controller, and the model parameters and external disturbance existing in the system are improved and suppressed to some extent.

5.4. Experimental Verification of Controller in Different Water Depth Environment

As shown in Figure 24 and Figure 25, the experimental data of the switch barrel-cover are 30 m and 50 m underwater respectively. Due to confidentiality requirements, the tank simulator could not be displayed.

Submarine launches missiles at different depths in the deep sea environment according to the actual situation. Underwater depth fluctuation and underwater ocean current impact will affect the process of switch cover. Therefore, the controller we designed needs to be able to adapt to various complex deep-sea environments, to achieve the performance indicators of the time and angle of the vertical launching barrel-cover in different depth environments. As can be seen from Figure 24 and Figure 25, with the increase of depth, the trajectory tracking error changes little during the whole process of the switch cover. When the barrel-cover is in place, the positioning accuracy of the opening angle can be satisfactorily satisfied. We can know that the adaptive backstepping sliding mode controller based on the nonlinear disturbance observer can well meet the performance indicators in different underwater depth environments.

6. Conclusions

From the test of the vertical launching barrel-cover of the underwater missile, the proposed adaptive backstepping sliding mode based on disturbance observer has obvious advantages over the traditional controller. The adaptive backstepping sliding mode controller based on the disturbance observer considers the multi-variable input and output, parameter uncertainty and the ability to suppress external disturbance. The experimental results show that the proposed control strategy effectively suppresses the influence of external disturbance force on the vertical launching barrel-cover device, greatly improves the tracking accuracy of the switch barrel-cover trajectory, and ensures the stability of the switch barrel-cover process. The control strategy is practical.