Stabilization of an Underactuated Surface Vessel Based on Adaptive Sliding Mode and Backstepping Control

Abstract

The paper studied controlling problem of an underactuated surface vessel with unknown interferences. It proved that the control problem of underactuated surface vessel can be transformed into the stabilization analysis of two small subsystems. This controller was designed by backstepping method and adaptive sliding mode, was suitable for solving the problem of the control of higher systems, can keep the system global asymptotic stability, and can inhibit unknown interference, and boundary layer can weaken the buffeting generated by sliding mode. The unknown interference was estimated by adaptive function. Finally, the simulation results are given to demonstrate the effectiveness of the proposed control laws.

1. Introduction

With the rapid development of the modern shipping industry, the study about the controller of the underactuated surface vessel has received extensive attention. Since stabilization control problem of the underactuated surface vessel belongs to the field of nonlinear control, nonlinear control theory is the basic foundation of the underactuated stabilization control theory [1]. Underactuated ship system is a nonholonomic system, and the nonholonomic system which does not meet the sufficient condition of Brockett system linear approximation is not controllable; so, stabilization control of underactuated systems cannot be continuous and smooth [2]. Similarly, it cannot use output feedback or smooth state feedback on a linear system, and the linear control method is not satisfactory to the stabilization control of underactuated system.

So far, underactuated ship stabilization control problem is more studied without interferences, just like [3–7] for the case of noninterference. But for a bounded environment disturbance (or unknown disturbances) and the uncertainty of the ship systems, underactuated stabilization controller is few, except for [8], but from the research results of them shows that the antidisturbance (or uncertainty) ability of the designed controller is less than ideal. Backstepping and sliding mode control method is the combination of backstepping and sliding mode control and takes full advantages of their own characteristics in [9]. Boundedness and convergence can keep the equilibrium point; sliding mode control system can match perturbation and disturbance, and when using backstepping under less restrictive conditions, it can ensure that the closed-loop system has strong robustness. The controller exist sliding mode; so, it will cause system chattering and make control input change severely, and it cannot meet the requirements of the actual system, but boundary layer method can solve it. Adaptive function is used to cope with approximation of uncertain functions. The paper studies underactuated surface vessel, which under the conditions of bounded variable environmental interference effect. A new adaptive sliding mode and backstepping control method is designed to deal with the underactuated ship stabilization problem, and according to the experimental simulation, the designed controller is effective.

2. Model of an Underactuated Surface Vessel

We consider an underactuated surface vessel with simplified dynamics off-diagonal terms of the linear and nonlinear damping matrices which are ignored. Based on [9], the dynamics and kinematics of an underactuated surface vessel are described as follows: ̇𝜓𝜓,̇𝜓𝜓,̇̇𝑚𝑥=𝑢cos−𝑣sin𝑦=𝑢sin+𝑣cos𝜓=𝑟,𝑢=²²𝑚₁₁𝑑𝑣𝑟−¹¹𝑚₁₁1𝑢+𝑚₁₁𝜏₁+1𝑚₁₁𝜏_𝑤1,̇𝑚𝑣=−¹¹𝑚₂₂𝑑𝑢𝑟−²²𝑚₂₂1𝑣+𝑚₂₂𝜏_𝑤2,̇𝑚𝑟=¹¹−𝑚₂₂𝑚₃₃𝑑𝑢𝑣−³³𝑚₃₃1𝑟+𝑚₃₃𝜏₃+1𝑚₃₃𝜏_𝑤3, (1) where [𝑥𝑦𝜓]^𝑇 denote position and orientation in inertial frame, [𝑢𝑣𝑟]^𝑇 denote linear velocities in surge sway and angular velocity in yaw, 𝑚₁₁,𝑚₂₂,𝑚₃₃ are the constants of inertia matrix, and 𝑑₁₁,𝑑₂₂,𝑑₃₃ are the constants of damping matrix, and we consider the case that the surface vessel has no side thruster; so, the control input can be described as [𝜏₁0𝜏₃]^𝑇, and [𝜏_𝑤1𝜏_𝑤2𝜏_𝑤3]^𝑇 are the environmental disturbances in the body-fixed frame.

3. Controller Design

3.1. Coordinate and Feedback Transformation

In order to design the controller conveniently, the underactuated ships model will be converted to a suitable form. Based on [10], the coordinate and input transformations are as follows: 𝑧₁𝑧=𝑥cos𝜓+𝑦sin𝜓,₂𝑧=−𝑥sin𝜓+𝑦cos𝜓,₃𝑧=𝜓,₄𝑚=−¹¹𝑑₂₂𝑢−𝑧₁,𝑧₅𝑧=𝑣,₆𝑢=𝑟,₁=𝑑₁₁𝑑₂₂−1𝑢−𝑧₂𝑧₆−𝜏₁+𝜏_𝑤1𝑑₂₂,𝑢₂=𝑚₁₁−𝑚₂₂𝑚₃₃𝑑𝑢𝑣−³³𝑚₃₃𝜏𝑟+³+𝜏_𝑤3𝑚₃₃. (2)

By using (2), we can transform the differential equation as follows: 𝑧̇₁𝑑=−²²𝑚₁₁𝑧₁−𝑑₂₂𝑚₁₁𝑧₄+𝑧₂𝑧₆−𝑚₁₁𝑑₂₂𝑧₅𝑧₆,𝑧̇₅𝑑=−²²𝑚₂₂𝑧₅+𝑑₂₂𝑚₂₂𝑧₆𝑧₁+𝑧₄,𝑧̇₂=𝑧₄𝑧₆+𝜏_𝑤2𝑑₂₂,𝑧̇₃=𝑧₆,𝑧̇₄=𝑢₁+𝑑₁,𝑧̇₆=𝑢₂+𝑑₃, (3)(4) where 𝑑₁=𝜏_𝑤1/𝑑₂₂, and 𝑑₃=−𝜏_𝑤3/𝑚₃₃.

Remark 1. Because state transformation (2) is a global diffeomorphism transformation, so lim_𝑡→∞𝑧_𝑖=0 for 𝑖=1,2,…,6 is convergent by exponential form, and thus we imply that lim_𝑡→∞(𝜂,𝑉)=0.

Proposition 2. lim_𝑡→∞𝑧_𝑖=0,𝑖=1,2,…,6, is globally exponential stable if lim_𝑡→∞𝑧_𝑖=0,𝑖=2,3,4,6, is globally exponential stable.

Proof. The similar proof has been given in [1].

According to Proposition 2, we know that in order to make the stabilization of underactuated surface vessels from the initial position to the origin, the system only needs to make transformation (4). It can be seen as strict feedback nonlinear subsystem, and the cascade system consists of two components which are shown in Figure 1 as follows: Σ₁∶𝑧̇₃=𝑧₆,𝑧̇₆=𝑢₂+𝑑₃,Σ₂∶𝑧̇₂=𝑧₄𝑧₆+𝜏_𝑤2𝑑₂₂,𝑧̇₄=𝑢₁+𝑑₁. (5)(6)

Figure 1 

Block diagram of the system of (4).

3.2. Controller Design

Figure 1 shows that the controller of underactuated surface vessels can be transformed into two single-input and single-output systems. So, we only need to design the controller to satisfy the fact that 𝑢₁ and 𝑢₂ can be controlled in three directions of the ship.

3.2.1. Design the Control Law of System Σ₁

Firstly, we consider the subsystem 𝑧₃ and define Lyapunov function 𝑉₁(𝑧₃)=(1/2)𝑧²₃, and the time derivative of 𝑉₁(𝑧₃) is as follows: 𝑉̇₁𝑧₃=𝑧₃𝑧̇₃=𝑧₃𝑧₆. (7)

We take 𝑧₆ as a virtual control input of 𝑧₃ subsystem and define 𝑧₆=𝑒₁+𝛼₁(𝑧₃). In order to ensure that the subsystem 𝑧₃ is asymptotically stable when 𝑒₁=0, we choose 𝛼₁(𝑧₃)=−𝑘₁𝑧₃, where 𝛼₁(0)=0, and 𝑘₁>0. Then, we can get equations as follows: 𝑒₁̇=𝑧̇₆−𝛼̇₁𝑧₃=𝑢₂+𝑑₃+𝑘₁𝑧₆,𝑉̇₁𝑧₃=𝑧₃𝑧₆=−𝑘₁𝑧²₃+𝑧₃𝑒₁≤0. (8)

Secondly, we consider the whole system 𝑧̇₃=𝑒₁−𝑘₁𝑧₃,𝑒₁̇=𝑢₂+𝑑₃+𝑘₁𝑧₆. (9)

We can assume the uncertain parameters 𝑑₃ exists the upper bound and choose a linear sliding mode switch function 𝜎₁=𝑐₁𝑧₃+𝑒₁,𝑐₁>0. (10)

Define the Lyapunov function 𝑉₂(𝑧₃,𝑧₆)=(1/2)𝜎²₁+𝑉₁(𝑧₃); the time derivative of 𝑉₂(𝑧₃,𝑧₆) becomes 𝑉̇₂𝑧₃,𝑧₆=−𝑘₁𝑧²₃+𝑧₃𝑒₁+𝜎₁𝑐₁𝑒₁−𝑘₁𝑧₃+𝑢₂+𝑑₃+𝑘₁𝑧₆. (11)

Choose the control law 𝑢₂=−𝑐₁𝑒₁−𝑘₁𝑧₃−𝑘₁𝑧₆−𝑑₃sgn𝑒₁−ℎ₁𝜎₁+𝛽₁sgn𝜎₁, (12) where sgn(𝑥) is the sign function, ℎ₁≥0,𝛽₁≥0, and 𝑑₃ is the upper bound of uncertain parameter 𝑑₃. Combining (12) into (11), we can obtain 𝑉̇₂𝑧₃,𝑧₆=−𝑘₁𝑧²₃+𝑧₃𝑒₁−ℎ₁𝜎²₁−ℎ₁𝛽₁𝜎₁+𝑑₃𝜎₁−𝑑₃𝜎₁≤−𝑘₁𝑧²₃+𝑧₃𝑒₁−ℎ₁𝜎²₁−ℎ₁𝛽₁𝜎₁+𝑑₃−𝑑₃𝜎₁≤−𝑘₁𝑧²₃+𝑧₃𝑒₁−ℎ₁𝜎²₁−ℎ₁𝛽₁𝜎₁≤0. (13)

Finally, because the upper bound of the general object is difficult to predict, and in order to determine the uncertainties of the upper bound, we use the adaptive algorithm to estimate it. 𝑑₃ is the upper bound of uncertain parameter 𝑑₃. And its adaptive rate is ̇‌𝑑₃=𝛾₁𝜎₁.

3.2.2. Design the Control Law of System Σ₂

Firstly, consider the subsystem 𝑧₂, and define Lyapunov function 𝑉₃(𝑧₂)=(1/2)𝑧²₂. We see 𝑧₄ as a virtual control input of subsystem 𝑧₂ and define 𝑧₄=−𝑘₂𝑧₂sgn(𝑧₆). The time derivative of 𝑉₃(𝑧₂) is as follows: 𝑉̇₃𝑧₂=−𝑘₂𝑧²₂𝑧₆sgn𝑧₆=−𝑘₂𝑧²₂𝑧₆≤0,𝑘₂>0. (14)

Assuming that lateral disturbance force decrease with time, and |𝜏_𝑤2/𝑑₂₂|≤|𝑧₂𝑧₆|. In this paper, we can make 𝜏_𝑤2=0; so, 𝑧₂ is asymptotically convergent to zero when 𝑧₆(0)≠0 (or 𝑧₃(0)≠0). In order to make 𝑧₂ asymptotically converging to zero, we can adjust the 𝑘₂ (and 𝑘₁) such that the convergence rate of 𝑧₂ is greater than the convergence rate of 𝑧₆ (or 𝑧₃).

Define 𝑧₄=𝑒₂+𝛼₂(𝑧₂). In order to ensure that the subsystem 𝑧₂ is asymptotically stable when 𝑒₂=0, we choose 𝛼₂(𝑧₂)=−𝑘₂𝑧₂sgn(𝑧₆), where 𝛼₂(0)=0. Then, we can get equations as follows: 𝑒₂̇=𝑧̇₄−𝛼̇₂𝑧₂=𝑢₁+𝑑₁+𝑘₂𝑧₄𝑧₆sgn𝑧₆,𝑉̇₃𝑧₂=𝑧₂𝑧₆𝑒₂−𝑘₂𝑧₂sgn𝑧₆≤0. (15)

Secondly, consider the whole system 𝑧̇₂=𝑧₆𝑒₂−𝑘₂𝑧₂sgn𝑧₆,𝑒₂̇=𝑢₁+𝑑₁+𝑘₂𝑧₄𝑧₆sgn𝑧₆. (16)

We assume the uncertain parameters 𝑑₁ exists the upper bound. Choose a linear sliding mode switch function 𝜎₂=𝑐₂𝑧₂+𝑒₂,𝑐₂>0. Define the Lyapunov function 𝑉₄(𝑧₂,𝑧₄)=(1/2)𝜎²₂+𝑉₃(𝑧₂); the time derivative of 𝑉₄(𝑧₂,𝑧₄) is as follows: 𝑉̇₄𝑧₂,𝑧₄=𝑧₂𝑧₆𝑒₂−𝑘₂𝑧₂sgn𝑧₆+𝜎₂𝑐₂𝑧₆𝑒₂−𝑘₂𝑧₂sgn𝑧₆+𝜎₂𝑢₁+𝑑₁+𝑘₂𝑧₄𝑧₆sgn𝑧₆. (17)

Choose the control law 𝑢₁=−𝑐₂𝑧₆𝑒₂−𝑘₂𝑧₂sgn𝑧₆−𝑘₂𝑧₄𝑧₆sgn𝑧₆−𝑑₁sgn𝑒₂−ℎ₂𝜎₂+𝛽₂sgn𝜎₂, (18) where ℎ₂≥0,𝛽₂≥0, and 𝑑₁is the upper bound of uncertain parameter 𝑑₁.

Combining (18) into (17), we can obtain 𝑉̇₄𝑧₂,𝑧₄=𝑧₂𝑧₆𝑒₂−𝑘₂𝑧₂sgn𝑧₆−ℎ₂𝜎²₂−ℎ₂𝛽₂𝜎₂+𝑑₁𝜎₂−𝑑₁𝜎₂≤𝑧₂𝑧₆𝑒₂−𝑘₂𝑧₂sgn𝑧₆−ℎ₂𝜎²₂−ℎ₂𝛽₂𝜎₂−𝑑₁−𝑑₁𝜎₂≤𝑧₂𝑧₆𝑒₂−𝑘₂𝑧₂sgn𝑧₆−ℎ₂𝜎²₂−ℎ₂𝛽₂𝜎₂≤0. (19)

Finally, we use the adaptive algorithm to estimate 𝑑₃. And its adaptive rate is ̇‌𝑑₃=𝛾₁𝜎₁.

3.3. Stability Analysis

Firstly, we should analyze the stability of the system Σ₁, defined as follows: 𝑄₁=𝑘₁+ℎ₁𝑐²₁ℎ₁𝑐₁−12ℎ₁𝑐₁−12ℎ₁,𝑊₁=𝑧₃𝑒₁^𝑇. (20)

We can get 𝑄₁=ℎ₁𝑘₁+ℎ₁𝑐²₁−ℎ₁𝑐₁−12²=ℎ₁𝑘₁+𝑐₁−14,𝑊^𝑇₁𝑄₁𝑊₁=𝑧₃𝑒₁𝑘₁+ℎ₁𝑐²₁ℎ₁𝑐₁−12ℎ₁𝑐₁−12ℎ₁𝑧₃𝑒₁=𝑧²₃𝑘₁+ℎ₁𝑐²₁+2𝑒₁𝑧₃ℎ₁𝑐₁−12+𝑒²₁ℎ₁=𝑘₁𝑧²₃+ℎ₁𝑐²₁𝑧²₃+2ℎ₁𝑐₁𝑒₁𝑧₃−𝑒₁𝑧₃+𝑒²₁ℎ₁. (21)(22)

So, (13) can be written as follows: 𝑉̇₂𝑧₃,𝑧₆≤−𝑊^𝑇₁𝑄₁𝑊₁−ℎ₁𝛽₁𝜎₁. (23)

From (21), choosing the adaptive value of 𝑘₁,ℎ₁, and 𝑐₁, we can make |𝑄₁|>0; then, 𝑉̇₂(𝑧₃,𝑧₆)≤0. The wide range of Lyapunov asymptotic stability theorem shows that the system Σ₁ is stable.

Secondly, system Σ₁ and system Σ₂ use the same method; so, the stability of the system Σ₂ can refer to system Σ₁.

Proposition 3. The system shown in (1) can be controlled by the input of (12) and (18), and a wide range of asymptotically stabilization can be obtained by (13) and (19).

Proof. The certification process has been given in the design steps.

From field of the theory, the motion choosing sliding mode is that parameter variations and external interference have nothing to do with the systems; so, the robustness of the system using sliding mode controller is better than the general control system. But in fact, sliding mode control can cause the system chattering for the reason of noncontinuous switch; so, the boundary layer method will be introduced to weaken this vibration. Saturation function is used to replace the ideal relay functions in the appropriate boundary layer; in other words, sat(𝑠) is used instead of sgn(𝑠) as follows: sat(𝑠)=|𝑠|+1𝑠>Δ,𝑘𝑠≤Δ,−1𝑠<−Δ, (24) where Δ is the thickness of boundary layer; 𝑘 is a constant. And Δ=0.33; 𝑘=1.

4. Simulation

In order to verify that the controller is able to calm from the initial point to the origin, consider an underactuated surface vessel with the model parameters in [11], and assume the initial condition to be as follows: 𝑚₁₁=200kg, 𝑚₂₂=250kg, 𝑚₃₃=80kgm², 𝑑₁₁=70kgs⁻¹, 𝑑₂₂=100kgs⁻¹, 𝑑₃₃=50kgm²s⁻¹, and [𝑥₀,𝑦₀,𝜓₀,𝑢₀,𝑣₀,𝑟₀]^𝑇=[5m,5m,(𝜋/4)rad,0m/s,0m/s,0.5rad/s]^𝑇.

Take the disturbances as follows: 𝜏_𝑤1=50+10sin((𝜋/6)𝑡)N, and 𝜏_𝑤2=0N, 𝜏_𝑤3=10+10sin((𝜋/6)𝑡)N⋅m.

In order to make ship at low speed operation, the longitudinal limited value of thrust is ±400N, and torque limit value is ±200N⋅m. The parameters of the control law are designed with the following: 𝑘₁=5,𝑐₁=8,ℎ₁=0.2, 𝛽₁=0.1, 𝑘₂=10, 𝑐₂=0.1, ℎ₂=6, 𝛽₂=1.8, 𝛾₁=0.05, and 𝛾₂=0.08. Simulation results are shown as follows.

From Figure 2, we can see the trajectory of the system to be better when there exist disturbances. Designed controller can make the underactuated surface vessel stabilization. From Figures 3 and 4, although the response time is short, designed controller can make all the states of the system stable to the origin. From Figure 5, because of introduction of the boundary layer approach to weaken the chattering phenomenon of the control force and moment, there is no change when the system is balanced, and it can save the power as well.

Figure 2 

Curve of the ship’s trajectory.

Figure 3 

Curve of ship position and heading angle variables.

Figure 4 

Curve of ship speed.

Figure 5 

Curve of the longitudinal force and yawing moment.

5. Conclusions

In this paper, underactuated surface vessels use the combination of adaptive sliding mode and backstepping to design the controller with the consideration of environmental disturbance and use the boundary layer method to solve the chattering problem. From the analysis of stabilization, the system is state global stabilization to the origin. The designed controller overcomes the problems of anti-to perturbation poor difficulties of the other controllers [8]; so, the designed controller improves the robustness of the system.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (51209056) and the Fundamental Research Funds for the Central Universities (HEUCF 100420 and HEUCF 110430).