An adaptive fuzzy sliding mode controller for the depth control of an underactuated underwater vehicle

Abstract

An adaptive fuzzy sliding mode controller is proposed for the depth control of an underactuated underwater vehicle based on the state-dependent Riccati equation. An adaptive fuzzy control algorithm is embedded into the sliding mode controller to solve the buffering and mismatched uncertain problems in the robust sliding mode variable structure controller, where an auxiliary fuzzy control unit is designed to automatically adjust the scale factor of the main fuzzy controller output. Based on Lyapunov stability theory and final value bounded theorem, the stability and convergence properties of the closed-loop system are demonstrated. Numerical simulations are carried out to validate the effectiveness of the proposed controller.

Introduction

Autonomous underwater vehicles (AUVs) have been widely used to explore ocean resources. These vehicles usually contain several critical devices to overcome hazardous underwater environment and to accomplish challenging tasks required for those vehicles. However, the effective controller of underwater vehicle continues to pose great challenges to system designers in the controller due to the system’s highly nonlinear, time-varying, dynamic behaviors and uncertainties in hydrodynamic coefficients and external disturbances. The dependence of the dynamic model on the uncertain underwater environment also makes the precise control of the AUV depth more challenging. Control strategies with intelligence, adaptive capability, fast convergence, and robustness are also required.

It has already been shown ^1,2 that, in the case of underwater vehicles, the traditional control methodologies are not the most suitable choice and cannot guarantee the required tracking performance. On the other hand, sliding mode control, due to its attractiveness such as robustness to parametric uncertainty, insensitivity to unknown disturbances, fast dynamic response, order reduction, easier implementation, and design simplification, has proven to be a very attractive approach to cope with these problems. However, the practical application of sliding mode controller (SMC) is limited by the lack of knowledge on the uncertainties of the dynamic model, including plant uncertainties and external disturbance. Many adaptive fuzzy control schemes ^{3 –9} involving linguistic fuzzy information from human experts have been effectively integrated into SMC to circumvent the problem of uncertainties for robot. For instance, Lakhekar and Waghmare ³ developed an adaptive fuzzy proportional integral (PI) SMC for trajectory tracking control of AUV to achieve high precise maneuvering. Ma et al. ⁴ designed an adaptive sliding mode dynamical surface controller for a hybrid unmanned aerial underwater vehicles. Sarfraz et al. ⁵ presented a robust stabilizing controller via adaptive integral sliding mode for nonholonomic systems with uncertainties. Zhang et al. ⁶ proposed an adaptive quasi-sliding mode control method for the path following system of an underactuated unmanned underwater vehicle. Bessa et al. ⁷ presented an adaptive fuzzy sliding mode controller (AFSMC) for remotely operated underwater vehicles. Yang et al. ⁸ propose a sliding mode control law for asymptotically stabilizing a nonholonomic mobile robot to a desired trajectory. Savkin and Wang ⁹ presented a biologically inspired strategy for the navigation of a unicycle-like robot toward a target while avoiding collisions with moving obstacles. Although the aforementioned SMCs have been used to reduce the effects of unknown uncertainties on the control performances, those unknown uncertainties must be located exactly on the position of control output vector or in the intermediate variable in the state space equation. Such limitation on the position of the nonlinear uncertain function is usually known as the strict match condition. Unless an uncertain system satisfies the strict match condition, these aforementioned SMCs cannot be used to achieve their desired control performances for the underwater vehicle. Only few investigations ^10,11 have been conducted on the design of an SMC with loosen constraints on the position of the mismatched uncertainties.

Varieties of neural networks and proportional differentials coupled with the SMC have been used to address the chattering problem in the general SMC. Chatchanayuenyong et al. ¹² proposed a neural network-based time optimal SMC to determine the solution of the nonlinear curve. However, the tuning of neuron weights requires a large quantity of experimental datum. Wang et al. ¹³ designed an SMC that has multivariable output feedback adaptive terminal. Although the convergence of the closed loop system is achieved using Lyapunov stability theory, some tracking errors are not bounded within a small tolerance region. Ouyang et al. ¹⁴ developed a proportional differential plus SMC to compensate for the uncertainties. However, the controller is not robust enough in terms of the proportional–derivative (PD) control gain. Many fuzzy control schemes involving linguistic fuzzy information from human experts were also integrated into the SMC to circumvent the chattering problem. For instance, Chen and Zhang ¹⁵ applied an adaptive fuzzy sliding mode control algorithm to resolve the system chattering problem. Aloui et al. ¹⁶ designed an AFSMC with an adaptive PI term to minimize the chattering phenomenon. Lakhekar et al. ¹⁷ proposed two types of fuzzy approximations to cope with chattering problem. Lakhekar et al. ¹⁸ introduced two types of fuzzy self-turning techniques to eliminate the effects of chattering. Amer et al. ¹⁹ developed a reduced rule-based fuzzy proportional integral SMC to eliminate chattering. Esfahani et al. ²⁰ designed an AFSMC to address the problems of chattering under the condition of lacking training data and non-robust tracking errors.

In this article, an AFSMC is proposed for the depth control of an underactuated underwater vehicle based on the state-dependent Riccati equation. An adaptive fuzzy control algorithm is embedded into the SMC to solve the buffering and mismatched uncertain problems in the robust sliding mode variable structure controller, where an auxiliary fuzzy control unit is designed to automatically adjust the scale factor of the main fuzzy controller output. Based on Lyapunov stability theory and final value bounded theorem, the stability and convergence properties of the closed-loop system are demonstrated. Numerical simulations are carried out to validate the effectiveness of the proposed controller.

This article is organized as follows. In the second section, the nominal model and uncertainty model of an underwater vehicle are formulated. In the third section, an adaptive fuzzy robust SMC is designed along with the assessment in its convergence. In the fourth section, the numerical simulation results are presented and discussed. Conclusion is given in the fifth section.

Dynamic modeling of underwater vehicle

Modeling

The Remote Environmental Unit (REMUS) AUV ²¹ and its corresponding reference frames are illustrated in Figure 1. The inertial reference frame is considered a fixed frame, whose z-axis directed into the gravitational direction and the other two axes are perpendicular to the z-axis. In contrast, the body reference frame is set at the geometric center of the underwater vehicle. The longitudinal u-axis directed into the direction from the tail to the nose, and the vertical w-axis is along the direction from the top to the bottom. The underwater vehicle is assumed a rigid body with a symmetrical longitudinal plane.

Figure 1.

Schematic of REMUS underwater vehicle. REMUS: remote environmental unit.

The nonlinear dynamic model of a 3-DOF underactuated underwater vehicle in the inertial frame can be written in the following uniform matrix form

$\begin{array}{l}m\left[\dot{w}-uq-x_G\dot{q}-z_G{q}^2\right]=Z_{\dot{q}}\dot{q}+Z_{\dot{w}}\dot{w}+Z_{uq}uq+Z_{uw}uw\\ +Z_{w|w|}w|w|+Z_{q|q|}q|q|\\ +\left(w-B_o\right)\text{cos}\theta +u^2{Z}_{uu}{\delta }_s,\end{array}$

1

and its kinematic models in the vertical plane related to the inertial frame with the body frame are expressed as follows

$\{\begin{array}{l}\dot{\theta }=q\\ \dot{z}=w\text{cos}\theta -u\text{sin}\theta \end{array}$

2

where w, q, z, and θ represent the heave velocity, the pith velocity, the depth variable, the pith angle, respectively; δ_s denotes the control fin angle; [x_B, y_B, z_B ]^T is the vector representing the coordinates of the buoyancy center; [x_G, y_G, z_G ]^T is the vector representing the coordinate of the center of gravity of the vehicle; I_yy denotes the moment of inertia of the vehicle about the pitch axis; u is the longitudinal velocity; $Z_{\dot{q}}$ , $Z_{\dot{w}}$ , $Z_{uq}$ , $Z_{uw}$ , $Z_{q|q|}$ , $Z_{uu}$ , $Z_{ww}$ , $M_{\dot{q}}$ , $M_{\dot{w}}$ , $M_{w|w|}$ , $M_{q|q|}$ , $M_{uq}$ , $M_{uu}$ , $M_{uw}$ denote the hydrodynamics parameters; W and B ₀ represent the weight and buoyancy of the underwater vehicle, respectively.

Given the state variable of x = [w, q, θ, z]^T, equation (2) can be simplified as the following compact form by ignoring the sufficiently small second-order viscous damping coefficient

$\dot{x}=Ax+Bu+d$

3

where d denotes the unmodeled dynamics and external interference and it is a matrix of four rows and one column; the matrix A is equal with $T^{-1}{A}^{\text{'}}$ , and the input matrix B is equal with $T^{-1}{B}^{\text{'}}$ , where the three vectors of T, A′, and B′ have the following forms: $T=\left[\begin{array}{cccc}m-Z_{\dot{w}}^{}& -Z_{\dot{q}}^{}& 0& 0\\ -M_{\dot{w}}^{}& I_{yy}^{}-M_{\dot{q}}^{}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ , $A^{\prime }=\left[\begin{array}{cccc}{Z}_{uw}^{}u& Z_{uq}^{}+mu& 0& 0\\ M_{uw}^{}u& M_{uq}^{}u& 0& 0\\ 0& 1& 0& 0\\ 1& 0& -u& 0\end{array}\right]$ , $B^{\prime }=\left[\begin{array}{c}{Z}_{uu}^{}{u}^2\\ M_{uu}^{}u\\ 0\\ 0\end{array}\right]$ , in which the matrix T can be written as a block matrix: $T\text{=}\left[\begin{array}{cc}R& 0\\ 0& I\end{array}\right]$ and $R\text{=}\left[\begin{array}{cc}m-Z_{\dot{w}}^{}& -Z_{\dot{q}}^{}\\ -M_{\dot{w}}^{}& I_{yy}^{}-M_{\dot{q}}^{}\end{array}\right]$ . Thus, the inverse of matrix T is calculated as $T^{-1}\text{=}\left[\begin{array}{cc}{R}^{-1}& 0\\ 0& I\end{array}\right]$ . The necessary and sufficient condition for matrix T invertible is that the matrix R is invertible. Substituting these parameters m, $Z_{\dot{w}}$ , $Z_{\dot{q}}$ , $M_{\dot{w}}$ , and I_yy in the fourth section into matrix R yields that the matrix R is invertible. Therefore, the matrix T in this article is invertible.

When those unmodeled dynamics and external interference are neglected, the nominal dynamic model of equation (3) can be expressed as

$\dot{x}=Ax+Bu$

4

When the two uncertainties associated with system matrix and input matrix are considered in the kinematic model, the nonlinear dynamic model of equation (4) with uncertainty terms can be rewritten in the following form

$\dot{x}=\left({\rm A}+\Delta {\rm A}\left(a(t)\right)\right)x+\left({\rm B}+\Delta {\rm B}\left(b(t)\right)\right)u+N\cdot n(t)$

5

where a(t) denotes an uncertainty vector associated with the system matrix; b(t) represents an uncertainty vector associated with the input matrix; and n(t) is an unmodeled dynamics and external interference, which is matched disturbance.

Orthogonal transformation of dynamic model

Equation (5) can be transformed into the block matrix form using the linear transformation

$\widehat{x}=Mx={\left[\begin{array}{cc}{\widehat{x}}_1& {\widehat{x}}_2\end{array}\right]}^{\text{T}}$

6

where M denotes the transformation matrix. After transformation, equation (5) becomes

$\{\begin{array}{l}{\widehat{\dot{x}}}_1={{\rm A}}_{11}{\widehat{x}}_1+{{\rm A}}_{12}{\widehat{x}}_2+\Delta {{\rm A}}_{11}{\widehat{x}}_1+\Delta {{\rm A}}_{12}{\widehat{x}}_2+N_{1}n(t)\hfill \\ {\widehat{\dot{x}}}_2={{\rm A}}_{21}{\widehat{x}}_1+{{\rm A}}_{22}{\widehat{x}}_2+{{\rm B}}_{2}u+\Delta {{\rm A}}_{21}{\widehat{x}}_1+\Delta {{\rm A}}_{22}{\widehat{x}}_2+\Delta {\rm B}u+N_{2}n(t)\hfill \end{array}$

7

where $M\cdot A\cdot M^{\text{T}}=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]$ , $M\cdot \Delta {\rm A}\cdot M^{\text{T}}=\left[\begin{array}{cc}\Delta {{\rm A}}_{11}& \Delta {{\rm A}}_{12}\\ \Delta {{\rm A}}_{21}& \Delta {{\rm A}}_{22}\end{array}\right]$ , M·ΔB=[0 ΔB ₂]^T, and M·N =[N ₁ N ₂]^T.

Assumption 1

These vectors of parameter uncertainty, input uncertainty, and interference uncertainty are bounded. The uncertainty parameters in equation (7) take the form of ΔA ₁₁=A ₁₂ l ₁₁, ΔA ₁₂ = A ₁₂ D ₁, N ₁= A ₁₂ E ₁, 2A ₂₁ = B ₂ l ₂₁, 1A ₂₂ = B ₂ l ₂₂, ΔB ₂ = B ₂ D ₂, and N ₂=B₂ E ₂. These bounded values of the matrix parameters are used to represent the uncertainty.

Substituting those uncertain parameters in Assumption 1 into equation (7) yields

$\{\begin{array}{l}{\widehat{\dot{x}}}_1=A_{11}{\widehat{x}}_1+A_{12}\left[\left(I+D_1\right){\widehat{x}}_2+h_1\right]\hfill \\ {\widehat{\dot{x}}}_2=A_{21}{\widehat{x}}_1+A_{22}{\widehat{x}}_2+B_2\left[\left(I+D_2\right)u+h_2\right]\hfill \end{array}$

8

where ${\eta }_1=l_{11}{\widehat{x}}_1+E_{1}n(t)$ and ${\eta }_2=l_{21}{\widehat{x}}_1+l_{22}{\widehat{x}}_2+E_{2}n(t)$ represent uncertainty term in the dynamic model, and their corresponding absolute values are bounded as follows: $\Vert {\eta }_1\Vert \le \text{max}\left(\Vert l_{11}\Vert ·\Vert {\widehat{x}}_1\Vert \right)+\text{max}\left(\Vert E_{1}n(t)\Vert \right),\Vert {\eta }_2\Vert \le \text{max}\left(\Vert l_{21}\Vert ·\Vert {\widehat{x}}_1\Vert \right)+\text{max}\left(\Vert l_{22}\Vert ·\Vert {\widehat{x}}_2\Vert \right)+\text{max}\left(\Vert E_{2}n(t)\Vert \right)$ . Matrices D_i (i = 1, 2) are assumed to satisfy the following inequality: $1+\text{max}\left\{{\lambda }_{\text{min}}\left[\frac{D_i+D^{\text{T}}{}_i}{2}\right]\right\}\ge {\gamma }_{D_i}$ .

Controller design

Sliding mode controller

A variable structure robust controller is proposed to control the unmatched uncertainty linear system defined in equation (5). Let s(t) be a sliding surface defined by equation s( ${\widehat{x}}_1$ , ${\widehat{x}}_2$ ) = 0. Then, the sliding mode surface can be expressed as

$s=C\widehat{x}=C_1{\widehat{x}}_1+C_2{\widehat{x}}_2=0$

9

where C ₁ is calculated by equation |λI + (A ₁₁ − A ₁₂ C ₁)| = 0; C ₂ is an identity matrix and s( ${\widehat{x}}_1$ , ${\widehat{x}}_2$ ) = 0 is defined on the basis of the pole placement which places all poles of the closed system in a certain region in the left side of the complex plane to ensure the closed system stability.

Taking the time derivative of the sliding mode surface in equation (9) and then substituting equation (8) into the result of the time derivative of the sliding mode surface in equation (9) yield

$\dot{s}=C_1{\widehat{\dot{x}}}_1+C_2{\widehat{\dot{x}}}_2=Ks+L{\widehat{\dot{x}}}_1+B_2\left[(I+D_2)u+{\eta }_2\right]+C_1{{\rm A}}_{12}\left[{\eta }_1+D_1(s-C_1{\widehat{\dot{x}}}_1)\right]$

10

where K = A ₂₂ + C ₁ A ₁₂, L = A ₁₂ − A ₂₂ C ₁ + C ₁ H, H = A ₁₁ − A ₁₂ C ₁.

The robust SMC is designed as the following combinational parts of the equivalent control law and continue robust control law

$u=u_{\text{cr}}+u_{\text{eq}}$

11

and the equivalent control law in the SMC is designed as

$u_{\text{eq}}=-B_2^{-1}\left[L{\widehat{x}}_1+(K-K*)s\right]$

12

where K* is located in the negative side of the half plane on the complex plane. Substituting equation (12) into equation (10) yields

$\dot{s}=K*s+B_2\left[\left(I+D_2\right)u_{cr}+{\eta }_3\right]$

13

where ${\eta }_3=R_{1}s+R_2+R_{3}n(t), R_1=l_{22}+B_2^{-1}{C}_1{A}_{12}{D}_1-D_2{B}_2^{-1}(K-K*)$ , $R_2=l_{21}-l_{22}{C}_1+B_2^{-1}{C}_1{A}_{12}{l}_{11}-B_2^{-1}{C}_1{A}_{12}{C}_1-D_2{B}_2^{-1}L$ , $R_3=E_2+B_2^{-1}{C}_1{A}_{12}{E}_1$ .

A continue robust control law in the SMC is defined as

$u_{cr}=-r_{D_2}^{-1}\left(1+\frac{\rho }{\Vert B_2^{\text{T}}Ps\Vert +\sigma }\right)B_2^{\text{T}}Ps$

14

where $\rho \ge {\overline{k}}_1\Vert x_1\Vert +{\overline{k}}_2$ , ${\overline{k}}_1=\text{max}\Vert R_2\Vert$ , ${\overline{k}}_2=\text{max}\Vert R_{3}n(t)\Vert$ , and σ is a positive constant; the vector P in equation (14) can be calculated by solving the following Riccati equation

$K{*}^T{P}_{}+P_{}K*-P_{}{B}_2{B}_2^{\text{T}}{P}_{}+Q_{}=0$

15

Theorem 1. ²²

For system $\dot{x}=f(x,t)$ , there is one mapping function V as U × J→R satisfies the following conditions: there are some constants λ > 0 and a scalar function $\overline{\lambda }$ as U→R such that $\lambda {\Vert x\left(t\right)\Vert }^2\le \Vert V\left(t\right)\Vert \le \overline{\lambda }\left(x\right){\Vert x\left(t\right)\Vert }^2$ and $\dot{V}\text{(}x\text{, }t\text{)}\le -{\lambda }_V\overline{\lambda }\text{(t)}\hspace{0.17em}{\Vert x\left(t\right)\Vert }^2+\epsilon$ for all (x, t) ϵ U × J and x ϵ Rⁿ .

Then, the terminal value of the system is globally bounded as t→∞

$\Vert x\left(t\right)\Vert \le \frac{1}{\lambda }V\left(x_0,t_0\right)e^{-{\lambda }_V(t-t_0)}+\frac{\epsilon }{\lambda {\lambda }_V}\left(1-e^{-{\lambda }_V\left(t-t_0\right)}\right)\le \frac{\epsilon }{\lambda {\lambda }_V}$

16

where $\frac{\epsilon }{\lambda {\lambda }_V}$ is a positive constant. The bounded terminal value system is essential condition for the asymptotical stability of the system.

Theorem 2

Given that the dynamic model defined in equation (5) satisfies Assumption 1, the sliding mode control laws in equations (12) and (14), and the Riccati-like equation in equation (15) guarantee the global stability of the closed-loop system.

Proof

Now, the following Lyapunov function is defined

$V(t)=s^{\text{T}}{P}_{}s$

17

Taking the time derivative of V(t), and substituting the equation (13) into the time derivative yields

$\begin{array}{l}\dot{V}(t)={\dot{s}}^{\text{T}}{P}_{}s+s^{\text{T}}{P}_{}\dot{s}\\ =s^{\text{T}}P{K}^{*}s+s^{\text{T}}{K}^{\text{*T}}Ps+u_{\text{cr}}^{\text{T}}{(I+D_2)}^{\text{T}}{B}_2^{\text{T}}{P}_{}s\\ +s^{\text{T}}{P}_{}{B}_2(I+D_2)u_{\text{cr}}+{\eta }_3^{\text{T}}{B}_2^{\text{T}}{P}_{}s+s^{\text{T}}{P}_{}{B}_2{\eta }_3\end{array}$

18

Let ${\dot{V}}_1(t)=u_{\text{cr}}^{\text{T}}{(I+D_2)}^{\text{T}}{B}_2^{\text{T}}{P}_{}s+s^{\text{T}}{P}_{}{B}_2(I+D_2)u_{\text{cr}}$ and ${\dot{V}}_2(t)={\eta }_{{}_3}^{\text{T}}{B}_2^{\text{T}}{P}_{}s+s^{\text{T}}{P}_{}{B}_2{\eta }_3$ and substituting the continuous robust control law in equation (14) into ${\dot{V}}_1(t)$ yields

$\begin{array}{c}{\dot{V}}_1(t)=u_{\text{cr}}^{\text{T}}{(I+D_2)}^{\text{T}}{B}_2^{\text{T}}{P}_{}s+s^{\text{T}}{P}_{}{B}_2(I+D_2)u_{\text{cr}}\\ =-2r_{D_2}^{-1}\left[1+\frac{\rho }{\Vert B_2^{\text{T}}{P}_{}s\Vert +\sigma }\right]s^{\text{T}}{P}_{}{B}_2\left(I+\frac{D_2+D_2^{\text{T}}}{2}\right)B_2^{\text{T}}{P}_{}s\end{array}$

19

By using Assumption 1, equation (19) becomes

${\dot{V}}_1\left(t\right)\le -2\left[1+\frac{\rho }{\Vert B_2^T{P}_{}s\Vert +\sigma }\right]s^{\text{T}}{P}_{}{B}_2{B}_2^{\text{T}}{P}_{}s$

20

Substituting the η ₃ in equation (13) into ${\dot{\mathit{V}}}_{\text{2}}\text{(t)}$ yields

$\begin{array}{l}{\dot{V}}_2(t)={\eta }_{{}_3}^{\text{T}}{B}_2^{\text{T}}{P}_{}s+s^{\text{T}}{P}_{}{B}_2{\eta }_3\\ =s^{\text{T}}{R}_{{}_1}^{\text{T}}{B}_2^{\text{T}}{P}_{}s+s^{\text{T}}{P}_{}{B}_2{R}_{1}s+{\left(R_2{x}_1+R_{3}n\right)}^{\text{T}}{B}_2^{\text{T}}{P}_{}s+s^{\text{T}}{P}_{}{B}_2(R_2{x}_1+R_{3}n)\end{array}$

21

Since ${\left(R_{1}s\right)}^{\text{T}}B{}_2{}^{T}Ps+{\left(B{}_2{}^{T}Ps\right)}^{\text{T}}{R}_{1}s\le {\left(R_{1}s\right)}^T{R}_{1}s+{\left(B{}_2{}^{T}Ps\right)}^{\text{T}}B{}_2{}^{T}Ps$ , equation (21) can be rewritten as follows

$\begin{array}{l}{\dot{V}}_2(t)\le s^{\text{T}}{R}_1^{\text{T}}{R}_{1}s+s^{\text{T}}P{B}_2{B}_2^{\text{T}}Ps+2\Vert R_2{x}_1+R_{3}n\Vert \Vert B_2^{\text{T}}Ps\Vert \\ \le s^{\text{T}}{R}_1^{\text{T}}{R}_{1}s+s^{\text{T}}P{B}_2{B}_2^{\text{T}}Ps+2\rho \Vert B_2^{\text{T}}Ps\Vert \end{array}$

22

Substituting equations (20) and (22) into equation (18) results in

$\begin{array}{c}\dot{V}\le s^{\text{T}}(-P_{}{B}_2{B}_2^{\text{T}}{P}_{}+R_1^{\text{T}}{R}_1+K^{\text{*T}}P+P{K}^{\ast })s\\ -\frac{2\rho }{\Vert B_2^{\text{T}}{P}_{2}s\Vert +\sigma }{s}^{\text{T}}{P}_{}{B}_2{B}_2^{\text{T}}s+2\rho \Vert B_2^{\text{T}}{P}_{}s\Vert \end{array}$

23

Substituting equation (15) into equation (23) yields

$\dot{V}\le -s^{\text{T}}\left(Q-R_1^{\text{T}}{R}_1\right)s-\frac{2\rho }{\Vert B_2^{\text{T}}Ps\Vert +\sigma }{\Vert B_2^{\text{T}}Ps\Vert }^2+2\rho \Vert B_2^{\text{T}}Ps\Vert$

24

Since $-\frac{{\Vert B_2^{\text{T}}{P}_{}s\Vert }^2}{\Vert B_2^{\text{T}}{P}_{}s\Vert +\sigma }\le -\Vert B_2^{\text{T}}{P}_{}s\Vert +\sigma ,\hspace{0.17em}\hspace{0.17em}Q\ge R_1^{\text{T}}{R}_1$ , equation (24) can be rewritten as follows

$\begin{array}{c}\dot{V}\le -s^{\text{T}}\left(Q-R_1^{\text{T}}{R}_1\right)s+2\rho \sigma \\ \le -{\lambda }_{\text{min}}\left(Q-R_1^{\text{T}}{R}_1\right){\Vert s\Vert }^2+2\rho \sigma \end{array}$

25

From the equation (25), the second property in Theorem 1 can then be obtained as $\dot{V}(x, t)\le -{\lambda }_V\overline{\lambda }(t){\Vert x\left(t\right)\Vert }^2+\epsilon ,\hspace{0.17em}\forall x\hspace{0.17em}ϵ\hspace{0.17em}{R}^{\text{n}}$ , where ${\lambda }_V\overline{\lambda }={\lambda }_{\text{min}}\left(Q-R_1^{\text{T}}{R}_{\text{1}}\right)$ and ε = 2ρσ. Therefore, using Theorem 1, the robust SMC defined in equations (12) and (14) can guarantee the asymptotical convergence of the target depth.

Adaptive fuzzy sliding mode controller

To improve the control performances, an adaptive fuzzy algorithm is employed to reduce the chattering issues. Figure 2 illustrates the principle of adaptive SMC.

Figure 2.

Block diagram of the AFSMC. AFSMC: adaptive fuzzy sliding mode controller.

As shown in Figure 2, the AFSMC is composed of a continuous robust control law of u _cr and an equivalent control law of u _eq. The adaptive fuzzy controller consists of two components: a main fuzzy controller component and an auxiliary fuzzy controller component. The knowledge base for the main fuzzy controller component contains a collection of fuzzy if–then rules with the following form (R_j ): If x ₁ is as A _1j and x ₂ is as A _2j , then y is as B_j (j = 1, 2,…1 n), where x_i (i = 1, 2) are input parameters of the main fuzzy control component; y is the output parameter of the main fuzzy control component; fuzzy sets A_ij and B_j are associated with the membership functions μ_Aij (x_i ) and μ_Bj (y), respectively, and n is the number of fuzzy rules. The input membership function is defined as the following triangular and trapezoidal hybrid function

${\mu }_{{{\rm A}}_{ij}^{}}\left(x_i\right)=\{\begin{array}{cc}0& x_i<a_{ij}\text{ or }{x}_i>c_{ij}\\ \frac{x_i-a_{ij}}{b_{ij}-x_i}& a_{ij}\le x_i\le b_{ij}\text{ and }j\ne 1\\ \frac{c_{ij}-x_i}{c_{ij}-b_{ij}}& b_{ij}<x_i\le c_{ij}\text{ and }j\ne n\\ 1& a_{i1}\le x_i\le b_{i1}\text{ or }{b}_{in}<x_i\le c_{in}\end{array}$

26

where a_ij , b_ij , and c_ij are fuzzy segmentation points. The output membership function is defined as the triangular membership function. The output parameter value of the main fuzzy control component with a weighted-center defuzzifier and a singleton fuzzifier can be expressed as follows

$y={\displaystyle \sum _{j=1}^n{\varphi }_j}(x)w_j$

27

where ${\varphi }_{\text{j}}\left(x\right)= {\mu }_{A_{1j}}\left(x_1\right){\mu }_{A_{2j}}\left(x_2\right)$ and w_j is the critical parameter value at the maximum of the output membership function ${\mu }_{B_j}$ . Equation (27) can be rearranged as

$y=W_1^{\text{T}}{\Phi }_1(x)$

28

where $W_1^{\text{T}}$ = [w ₁ , $\cdots$ , w_n ] and the vector of the fuzzy basis function is ${\text{Φ}}_1^{\text{T}}\left(x\right)=\left[{\varphi }_{\text{1}}\left(x\right),\dots ,{\varphi }_{\text{n}}\left(x\right)\right]$ .

The two input parameters of x ₁ and x ₂ in the main fuzzy control component are k ₁ s(t) and $k_2\dot{s}\left(t\right)$ , respectively, and the output parameter of y is ${\dot{u}}_{\text{cr}}/k_3$ . The fuzzy rules of the main fuzzy controller component are summarized in Table 1. The output parameter ${\dot{u}}_{\text{cr}}/k_3$ is equal to $W_1^{\text{T}}{\text{Φ}}_1\left(k_{1}s\left(t\right),k_2\dot{s}\left(t\right)\right)$ . To improve the consistency of a fuzzy system under the interference of unknown dynamic parameters or bounded external disturbances, the fuzzy system should contain a strong self-adaptive ability. A scale coefficient k ₃ is selected as a freely adjustable parameter, and the auxiliary fuzzy control component is adopted to adjust scale coefficient k ₃. Two input parameters in the auxiliary control component are denoted as a ₁ s(t) and $a_2\dot{s}\left(t\right)$ , and its output parameter is defined as a ₃. Through the main and auxiliary fuzzy control component, the existence and elimination of chattering can be demonstrated for SMC using a proposed Mamdani-type fuzzy inference system. The output of adaptive fuzzy controller is determined by the normalized values of s(t) and $\dot{s}\left(t\right)$ . The fuzzy control rules can be represented as the mapping of the input linguistic variables s(t) and $\dot{s}\left(t\right)$ to the output linguistic variable u _cr. Through online learning, it can establish and regulate the fuzzy rules continuously for responding to the system’s parameters and external disturbances. The fuzzy membership function and the defuzzifier of the auxiliary fuzzy control component are the same as those in the main fuzzy control component. Thus, the output parameter a ₃ can be calculated as a ₃ = $W_2^{\text{T}}{\text{Φ}}_1$ (a ₁ s(t), $a_2\dot{s}\left(t\right)$ ). The fuzzy rules in the auxiliary fuzzy control component are illustrated in Table 2. Replacing scale coefficient k ₃ by the output parameter in the auxiliary fuzzy control component and integrating ${\dot{u}}_{\text{cr}}$ would result in

$u_{\text{cr}}={\displaystyle \int W_1^{\text{T}}}{\Phi }_1\left(k_{1}s(t),k_2\dot{s}(t)\right)W_2^{\text{T}}{\Phi }_2\left(a_{1}s(t),a_2\dot{s}(t)\right)\text{d}t$

29

Table 1.

Fuzzy control row set in the main fuzzy control component.

k 1 s(t) $k_2\dot{s}\left(t\right)$	NB	NM	NS	ZE	PS	PM	PB
NB	PB	PB	PB	PB	PM	PS	ZE
NM	PB	PB	PB	PM	PS	ZE	NS
NS	PB	PB	PM	PS	ZE	NS	NM
ZE	PB	PM	PS	ZE	NS	NM	NB
PS	PM	PS	ZE	NS	NM	NB	NB
PM	PS	ZE	NS	NM	NB	NB	NB
PB	ZE	NS	NM	NB	NB	NB	NB

Table 2.

Adaptive fuzzy control row set in the auxiliary fuzzy control component.

a 1 s(t) $a_2\dot{s}\left(t\right)$	NE	ZE	PE
NB	PB	PB	PB
NM	PB	PB	PB
NS	PB	PB	PM

The membership functions input parameters and output parameters of the adaptive fuzzy control are shown in Figures 3 and 4.

Figure 3.

Membership functions of fuzzy input and output parameters in the main fuzzy control law: (a) the input variable s and $\dot{s}$ ; (b) the output variable ${\dot{u}}_{\text{cr}}/k_3$ .

Figure 4.

Membership functions in the auxiliary control law: (a) the input variables s and $\dot{s}$ ; (b) the output variable a ₃.

Simulation results and discussions

A 3-DOF REMUS underactuated underwater vehicle is taken as an example to examine the depth control ability of the above proposed AFSMC. The objective of the simulations is to examine the ability of the proposed controller in the depth control by forcing the proposed controller to track a desired depth at z _r = 0 m. The weight of the underwater vehicle is calculated as W = mg = 299 N with m = 30.48 kg, g = 9.81 m/s², and its buoyancy force B ₀ = 306 N. The center of gravity is defined as [x_G , y_G , z_G ]^T = [0 m, 0 m, 0 m]^T, and the center of buoyancy is defined as [x_B , y_B , z_B ]^T = [0 m, 0 m, 0 m]^T, the inertia moment of the vehicle about the pitch axis $I_{yy}=3.45\text{ kg}\cdot {\text{m}}^2$ . The hydrodynamic coefficients were calculated using Fluent software (Fluent 14.0) by simulating both steady state and unsteady state. The obtained coefficients are listed as follows: M_q = −4.88 kg·m²/rad, $M_{\dot{\text{w}}}=-1.93\text{ kg·m}$ , M_{w |w|} = 3.18 kg, M_{q |q|} = −188 kg·m²/rad, M_{uu δ} = −6.15 kg/rad, M_uq = −2 kg·m/rad, M_uw = 24 kg, ${\text{Z}}_{\dot{\text{w}}}=-35.5 \text{kg}$ , ${\text{Z}}_{\dot{\text{q}}}=-1.93\text{ kg·m/rad}$ , Z_ww = −131 kg/m, Z_{q |q|} = −0.632 kg·m/rad², Z_uw = −28.6 kg/m, Z_uq = −5.22 kg/rad, and Z_{uu δ} = −6.15 kg/(m·rad).

The start and desired depths in the simulation case are given as z ₀ = 2 m and z_r = 0 m, respectively. The initial states set as w ₀ = 0.02 m/s, q ₀ = 0.04 rad/s, θ ₀ = 0.1 rad, z ₀ = 2 m. These uncertainty parameters in equation (7) take the following values: $\Delta A=\left[\begin{array}{cccc}0& 0& 0& 0.1\\ 0& 0& 0& 0.01\\ 0& 0& 0& 0.769\\ 0.1\times \text{sin}t& 0& 0& 0\end{array}\right]$ , ΔB = 0, $N=\left[\begin{array}{c}-0.0722\\ -0.722\\ 0\\ 0\end{array}\right]$ , n(t ) = 2. The two matrices of the sliding mode surface in equation (9) are defined as $C_1={\left[\begin{array}{ccc}1.02& -1& -1.46\end{array}\right]}^{\text{T}}$ and C ₂ = 1. The constants σ, r_{D 1}, and r_{D 2} in equation (14) are selected as σ = 1.5, r_{D 1} = 1.5 and r_{D 2} = 0.8. The two parameters Q and K* in the Riccati equation (15) are chosen as 2.3 and −1, respectively.

Case 1. The SMC in equation (11) is applied in the nominal system and uncertain model for the 3-DOF REMUS AUV. Its control law is defined as follows

$u=u_{\text{cr}}+u_{\text{eq}}=-r_{D_2}^{-1}\left(1+\frac{\rho }{\Vert B_2^{\text{T}}Ps\Vert +\sigma }\right)B_2^{\text{T}}Ps-B_2^{-1}\left[L{\widehat{x}}_1+(K-K*)s\right]$

30

Under the control action of SMC, the depth control performance of the SMC is illustrated in Figure 5. It can be observed from Figure 5 that the SMC performs much better and more satisfactory in the nominal system than in uncertain model. As observed from Figure 5, it takes about 15 s from the initial state to the desired depth position. However, a large steady-state error exists for the control of systems with parameter uncertainties and external disturbances. Therefore, some additional appropriate compensation control schemes and chattering elimination method need to be developed and incorporated in the SMC to minimize the impact of uncertainties on the depth control performances.

Figure 5.

Depth control performances when the SMC is applied in the nominal system and uncertainty dynamic model: (a) depth position; (b) sliding mode surface; (c) control action. SMC: sliding mode controller.

Case 2. An AFSMC is designed to minimize the impact of uncertainties on the depth control performances for the 3-DOF REMUS AUV. Its control law is expressed as follows

$u=u_{\text{cr}}+u_{\text{eq}}={\displaystyle \int W_1^{\text{T}}}{\Phi }_1\left(k_{1}s(t),k_2\dot{s}(t)\right)W_2^{\text{T}}{\Phi }_2\left(a_{1}s(t),a_2\dot{s}(t)\right)\text{d}t-B_2^{-1}\left[L{\widehat{x}}_1+(K-K*)s\right]$

31

In the main fuzzy controller component, the triangular membership functions with the central values {−2; −0.5; −0.33; −0.08; 0; 0.08; 0.33; 0.5; 2} are adopted for the two input parameters k ₁ s(t) and k ₂ $\dot{s}\left(t\right)$ . The triangular membership function with the central value {−2.4; −2; −1.67; −1.33; −0.01; 0; 0.01; 1.33; 1.67; 2; 2.4} is also adopted for the output parameter ${\dot{u}}_{\text{cr}}/k_3$ . At the same time, the triangular membership functions with the central values {−5; −1; −0.5; 0; 0.5; 1; 5} applied into the auxiliary fuzzy controller component. With selected constants k ₁ = 1.5, k ₂ = 1, k ₃ = 30, a ₁ = 1, a ₂ = 2, the depth control performances of the adaptive fuzzy sliding mode control law in equation (31) for the 3-DOF REMUS AUV with uncertain dynamic model are illustrated in Figure 6. It can be observed from Figure 6 that the proposed AFSMC performs extremely well in overshoots elimination and also converges very fast even with the external disturbance and parameter uncertainties.

Figure 6.

Depth control performances for AFSMC: (a) depth position; (b) sliding mode surface; (c) control action. AFSMC: adaptive fuzzy sliding mode controller.

Some comparative studies were also conducted between the SMC and the proposed AFSMC. The desired trajectories tracked by the two control algorithms are the same as those in the mentioned simulation cases. For the convenience of comparisons, the depth, the control action, and sliding mode surface tracking errors in those cases are calculated by the following tracking error equations over one training cycle of a trajectory

$E_z=\frac{1}{N}{\displaystyle \sum _{i=1}^N{{\displaystyle \Vert z_d-z_i\Vert }}^2}{\text{(m)}}^{\text{2}},E_u=\frac{1}{N}{\displaystyle \sum _{i=1}^N{{\displaystyle \Vert u_d-u_i\Vert }}^2}{\text{(rad/s)}}^{\text{2}},E_s=\frac{1}{N}{\displaystyle \sum _{i=1}^N{{\displaystyle \Vert s_d-s_i\Vert }}^2}{\text{(m)}}^{\text{2}}$

32

where E_z , E_u , and E_s represent the depth tracking error, the control action error, and the sliding mode surface error, respectively; N denotes the element number of the depth vector; $z_d,u_d,s_d$ are the desired trajectories and $z_i,u_i,s_i$ are the actual trajectories. The tracking errors of those cases after the convergence are summarized in Table 3. It is clearly demonstrated from Table 3 that the AFSMC scheme in case 2 has a superior performance in tracking errors minimization over the SMC scheme in case 1.

Table 3.

Tracking errors in three cases.

Control schemes	Errors
	Ez (m)2 Eu (rad/m)2		Es (m)2
Case 1 (SMC)	0.0201	0.0010	0.0203
Case 2 (AFSMC)	7.9704e−06	2.0950e−04	1.7104e−05

SMC: sliding mode controller; AFSMC: adaptive fuzzy sliding mode controller.

Conclusions

In this article, an AFSMC was proposed to deal with the depth control of an underactuated underwater vehicle based on the state-dependent Riccati equation. An adaptive fuzzy control algorithm is embedded into the SMC to reduce the static error in the transient and steady states in the existence of unknown system parameters and external disturbances. The adaptive fuzzy control algorithm is composed of a main fuzzy controller component and an auxiliary fuzzy controller component, where an auxiliary fuzzy control unit is designed to automatically adjust the scale factor of the main fuzzy controller output. Moreover, the stability and convergence properties of the closed-loop system are demonstrated using Lyapunov stability theory and final value bounded theorem. Numerical simulation results have clearly demonstrated that the proposed AFSMC is superior over the SMC in improving the dynamic and static performances of the entire control system including the errors reduction, rapid convergence, and chattering elimination.