MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 324954 10.1155/2013/324954 324954 Research Article Stabilization of an Underactuated Surface Vessel Based on Adaptive Sliding Mode and Backstepping Control Ding Fuguang Wu Jing Wang Yuanhui Lin Tsung-Chih Automation College Harbin Engineering University Harbin 150001 China hrbeu.edu.cn 2013 27 3 2013 2013 15 11 2012 10 02 2013 2013 Copyright © 2013 Fuguang Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . 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 . 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  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 , 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 . 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 , the dynamics and kinematics of an underactuated surface vessel are described as follows: (1)x˙=ucos(ψ)-vsin(ψ),y˙=usin(ψ)+vcos(ψ),ψ˙=r,u˙=m22m11vr-d11m11u+1m11τ1+1m11τw1,v˙=-m11m22ur-d22m22v+1m22τw2,r˙=m11-m22m33uv-d33m33r+1m33τ3+1m33τw3, 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 [τw1τw2τw3]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 , the coordinate and input transformations are as follows: (2)z1=xcosψ+ysinψ,z2=-xsinψ+ycosψ,z3=ψ,z4=-m11d22u-z1,z5=v,z6=r,u1=(d11d22-1)u-z2z6-τ1+τw1d22,u2=m11-m22m33uv-d33m33r+τ3+τw3m33.

By using (2), we can transform the differential equation as follows: (3)z˙1=-d22m11z1-d22m11z4+z2z6-m11d22z5z6,z˙5=-d22m22z5+d22m22z6(z1+z4),(4)z˙2=z4z6+τw2d22,z˙3=z6,z˙4=u1+d1,z˙6=u2+d3, where d1=τw1/d22, and d3=-τw3/m33.

Remark 1.

Because state transformation (2) is a global diffeomorphism transformation, so limtzi=0 for i=1,2,,6 is convergent by exponential form, and thus we imply that limt(η,V)=0.

Proposition 2.

lim t z i = 0 , i = 1,2 , , 6 , is globally exponential stable if limtzi=0,i=2,3,4,6, is globally exponential stable.

Proof.

The similar proof has been given in .

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: (5)Σ1:{z˙3=z6,z˙6=u2+d3,(6)Σ2:{z˙2=z4z6+τw2d22,z˙4=u1+d1.

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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M20"><mml:mrow><mml:msub><mml:mrow><mml:mi>Σ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

Firstly, we consider the subsystem z3 and define Lyapunov function V1(z3)=(1/2)z32, and the time derivative of V1(z3) is as follows: (7)V˙1(z3)=z3z˙3=z3z6.

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: (8)e˙1=z˙6-α˙1(z3)=u2+d3+k1z6,V˙1(z3)=z3z6=-k1z32+z3e10.

Secondly, we consider the whole system (9)z˙3=e1-k1z3,e˙1=u2+d3+k1z6.

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

Define the Lyapunov function V2(z3,z6)=(1/2)σ12+V1(z3); the time derivative of V2(z3,z6) becomes (11)V˙2(z3,z6)=-k1z32+z3e1+σ1[c1(e1-k1z3)+u2+d3+k1z6].

Choose the control law (12)u2=-c1(e1-k1z3)-k1z6-d-3sgn(e1)-h1(σ1+β1sgn(σ1)), where sgn (x) is the sign function, h10,β10, and d-3 is the upper bound of uncertain parameter d3. Combining (12) into (11), we can obtain (13)V˙2(z3,z6)=-k1z32+z3e1-h1σ12-h1β1|σ1|+d3σ1-d-3|σ1|-k1z32+z3e1-h1σ12-h1β1|σ1|+(d3-d-3)|σ1|-k1z32+z3e1-h1σ12-h1β1|σ1|0.

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. d-3 is the upper bound of uncertain parameter d3. And its adaptive rate is d-˙3=γ1σ1.

3.2.2. Design the Control Law of System <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M49"><mml:mrow><mml:msub><mml:mrow><mml:mi>Σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

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: (14)V˙3(z2)=-k2z22z6sgn(z6)=-k2z22|z6|0,k2>0.

Assuming that lateral disturbance force decrease with time, and |τw2/d22||z2z6|. In this paper, we can make τw2=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: (15)e˙2=z˙4-α˙2(z2)=u1+d1+k2z4z6sgn(z6),V˙3(z2)=z2z6(e2-k2z2sgn(z6))0.

Secondly, consider the whole system (16)z˙2=z6(e2-k2z2sgn(z6)),e˙2=u1+d1+k2z4z6sgn(z6).

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: (17)V˙4(z2,z4)=z2z6(e2-k2z2sgn(z6))+σ2[c2z6(e2-k2z2sgn(z6))]+σ2[u1+d1+k2z4z6sgn(z6)].

Choose the control law (18)u1=-c2z6(e2-k2z2sgn(z6))-k2z4z6sgn(z6)-d-1sgn(e2)-h2(σ2+β2sgn(σ2)), where h20,β20, and d-1is the upper bound of uncertain parameter d1.

Combining (18) into (17), we can obtain (19)V˙4(z2,z4)=z2z6(e2-k2z2sgn(z6))-h2σ22-h2β2|σ2|+d1σ2-d-1|σ2|z2z6(e2-k2z2sgn(z6))-h2σ22-h2β2|σ2|-(d-1-d1)|σ2|z2z6(e2-k2z2sgn(z6))-h2σ22-h2β2|σ2|0.

Finally, we use the adaptive algorithm to estimate d-3. And its adaptive rate is d-˙3=γ1σ1.

3.3. Stability Analysis

Firstly, we should analyze the stability of the system Σ1, defined as follows: (20)Q1=[k1+h1c12h1c1-12h1c1-12h1],W1=[z3e1]T.

We can get (21)|Q1|=h1(k1+h1c12)-(h1c1-12)2=h1(k1+c1)-14,(22)W1TQ1W1=[z3e1][k1+h1c12h1c1-12h1c1-12h1][z3e1]=z32(k1+h1c12)+2e1z3(h1c1-12)+e12h1=k1z32+h1c12z32+2h1c1e1z3-e1z3+e12h1.

So, (13) can be written as follows: (23)V˙2(z3,z6)-W1TQ1W1-h1β1|σ1|.

From (21), choosing the adaptive value of k1,h1, and c1, we can make |Q1|>0; then, V˙2(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: (24)sat(s)={+1s>Δ,ks|s|Δ,-1s<-Δ, 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 , and assume the initial condition to be as follows: m11=200kg, m22=250kg, m33=80kg m2, d11=70kg s-1, d22=100kgs-1, d33=50kg m2s-1, and [x0,y0,ψ0,u0,v0,r0]T=[5m,5m,(π/4) rad,0m/s,0m/s,0.5 rad/s]T.

Take the disturbances as follows: τw1=50+10sin((π/6)t) N, and τw2=0 N, τw3=10+10sin((π/6)t) N·m.

In order to make ship at low speed operation, the longitudinal limited value of thrust is ±400 N, and torque limit value is ±200 N·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.

Curve of the ship’s trajectory.

Curve of ship position and heading angle variables.

Curve of ship speed.

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 ; 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).

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