Nonlinear Model Predictive Controller Design for Path Following of Underactuated Ships

WANG Xiao-fei,LI Tie-shan,ZOU Zao-jian,LUO Wei-lin

(a.State Key Laboratory of Ocean Engineering;b.School of Naval Architecture,Ocean and Civil Engineering,Shanghai Jiao Tong University,Shanghai 200030,China)

1 Introduction

Recently,path following of underactuated surface ships has received increasing attention from the control community.Path following control aims at forcing a vessel to follow a desired path.It is more suitable for practical implementation of the guidance and control of vehicles than trajectory tracking which needs a suitable virtual vehicle to generate a reference trajectory.This problem has been studied extensively by using different techniques in past years.Path following control along straight lines was explored by Pettersen and Lefeber[1].They redefined the output as a combination of the cross-track error and the heading angle error,and used a cascade approach and feedback linearization technique to control the cross-track y and the heading angle ψ converging to zero.Zhang et al[2]addressed the path following control problem in restricted waters by using input-output linearization and sliding mode control which provides robustness against environmental changes.As pointed out by Do and Pan[3],some con-trollers require yaw velocity to be nonzero,thus they can not be used to track straight lines.On the other hand,good performances of both straight and curve path following can be achieved by introducing the Serret-Frenet frame[3-5].Based on the transformation of ship kinematics to Serret-Frenet frame,some controllers for path following of underactuated ships have been proposed in the past few years.A path following controller was presented by Encarnação and Pascoal[4]for autonomous underwater vehicles in the presence of constant but unknown currents.The design was built on Lyapunov theory and backstepping technique.Combined the acceleration feedback approach with 3-step backstepping technique for interconnecting subsystems,a path following controller was proposed by Skjetne and Fossen[5],which can keep the ship on any feasible path.Based on Skjetne and Fossen’s work,Do and Pan[3]modified the out-redefinition to make it more reasonable.They developed state and output feedback controllers to steer an underactuated surface ship following a predefined path at a constant forward speed controlled by the main thruster in the presence of environmental disturbances.

However,there are two disadvantages in the above-mentioned control algorithms.Firstly,the control parameters ki（i=0,…,ρ-1）(where ρ is the relative degree)are chosen by numerical experiments which are random and are lack of theory foundation,and kihave no necessary relationship each other.Especially when ρ is large,it is difficult to choose appropriate kito guarantee the system’s stability.Secondly,as pointed out by Chen et al[6],the control order is restricted to zero.This implies that the nonlinear system can only be approximated by Taylor-series expansion to its relative degree.This limitation might result in the poor performance.In view of this,Chen et al[6]proposed an analytic model predictive controller to solve the abovementioned problems.On the one hand,kiof the analytic model predictive controller are computed by predictive time T,relative degree ρ and control order l,which provides a systematic method to get appropriate kito guarantee the stability of systems.On the other hand,the analytic model predictive controller’s control order can be chosen to be reasonably large,so the closed-loop model predictive controller for nonlinear systems might be approximated by Taylor-series expansion to arbitrarily chosen order[6].

It is noted that the relative degree plays an important role in model predictive control and the ill-defined relative degree is disgusting.Chen[7]proposed a switching analytic model predictive control for nonlinear systems with ill-defined relative degree.The solution is complicated and its performance is not ideal,because the controller needs switch among the different state spaces,which is not suitable for some nonlinear systems.If we apply the analytic model predictive control for path following of underactuated ships directly,the problem of ill-defined relative degree will occur and result in complicated controller design.This problem can be avoided by introducing Serret-Frenet frame and output-redefinition.On the other hand,the original SIMO path following control system can be transformed into an equivalent SISO system which can simplify the controller design.Compared with the original system to follow path（x,y,ψ）on the complicated desired path（xd,yd,ψd）,the transformed system only needs guarantee the output-redefinition to be zero.

In this paper,the analytic model predictive control is used for the design of path following controller steering an underactuated ship along a given path.The ship is assumed to travel along a path with a constant forward speed,and its total velocity is tangent to the path.The ship kinematics is described by using Serret-Frenet frame and the output-redefinition is introduced.The SIMO system is transformed into a SISO system and the ill-defined relative degree is avoided,thus the controller design is simplified.The stability of the closed-loop system is analyzed.The path following errors including the position error zeand the orientation error to zero.

The paper is organized as follows.The mathematical model of an underactuated ship based on Serret-Frenet frame is presented in Section 2.Section 3 is devoted to the design of the analytic model predictive controller.Some numerical simulations are given in Section 4 to validate the proposed controller.Section 5 concludes the paper.

2 Mathematical model of an underactuated ship based on Serret-Frenet frame

The general mathematical model of an underactuated ship in surge,sway and yaw motion can be written as

where x,y,ψ are the surge displacement,sway displacement and yaw angle in the earth fixed frame,u,v,r denote surge,sway and yaw velocities,respectively.mjj（j=1,2,3）denote the ship inertia including added mass in surge,sway and yaw motion,respectively.dii（i=2,3）represent the hydrodynamic damping in sway and yaw motion.τris the yaw moment.The surge velocity u is assumed to be positive constant controlled by the main thruster system.Since the sway control force is not available,the ship represented by the mathematical model(1)is underactuated[3].

The Serret-Frenet frame for ship path following is shown in Fig.1[3].

In Fig.1,Ω is the predefined path,M is the orthogonal projection of the garvitational centre of the ship P on Ω.s is the length along the path between some arbitrary fixed point on thepath and M.xnand xtare the normal and the tangent unit vectors at M,respectively.zeis the distance between M and P,ψdis the angle between xtandis the total velocity of the ship.φ is the angle between the surge velocity and the total velocity.

The transformation of the ship dynamics(1)to Serret-Frenet frame can be described as[3]:

Fig.1 General framework of ship path following

The equilibrium point of system(2)iswhich can guarantee the ship is on the path and the total velocity is tangent to the path.Our control objective is to design the yaw moment τrto force the underactuated ship to follow the specified path Ω.The task is to track the outputsto the reference outputs:

In this paper,we study the problem under the following assumption:

We introduce the output-redefinition[3]:

where k is a positive constant to be selected later.

With the output-redefinition(5),the system(2)is rewritten as:

This means that zecan converge to zero exponentially.It follows,

Now the objective of path following control becomes a problem of tracking outputto the reference output zd=0.

3 Design of model predictive controller for path following

In this section,we use analytic model predictive control to design the controller for path following of an underactuated ship.The controller’s main features are that an analytical form of the optimal predictive controller is given,on-line optimization is not required,and stability of the closed-loop system is guaranteed[6].

We rewrite the system(6)as:

where,

The analytic model predictive controller can be written as:

where

Assumption(4)implies that:

So LgLfh≠0 for all X,the relative degree is well-defined and ρ=2.

The reference output and its derivatives up to ρ are

where T is the predictive time,l is the control order.

From equations(11)-(17),the model predictive controller for path following can be written as:

We can rewrite the controller(21)in the form of

Now an important result can be given as follows.

Theorem:If the assumption(4)is satisfied,when the predictive control law(22)is applied to the ship system(2),the track errors zeandcan converge exponentially to zero,and v,r are bounded if d22＞m11·c（s）·u is satisfied.

Proof:

Substituting control law(22)intowe get

Namely,

2.（v,r）

It can be seen that the sway velocity v is bounded when

From equation(25),it is obvious that r is also bounded.

4 Simulations and results

In this section,the control law(22)is validated by simulations.The ship parameters are[8]:m11=120×103kg,m22=217.9×103kg,m33=636×105kg·m2,d22=117×103kg/s,d33=802×104kg·m2/s.The ship forward speed is u=10m/s.The control parameters are taken as k=0.002,T=37.5s,l=6.Correspondingly,from equations(18)-(20),k0=0.365,k1=1.026 7.

The desired path is a circle centered at the origin with a radius of 80m.The initial conditions are chosen asFig.2 shows the ship positions and orientation inplane.Fig.3 plots the path following errors and the corresponding output-redefinition.It can be seen that the position error zeand the orientation errorconverge to zero,the outputredefinitionalso converges to zero.The sway and yaw velocities and control torque τrare shown in Fig.4.

5 Conclusions

An analytic model predictive controller is presented for steering an underactuated ship along a desired path at a constant forward speed.The mathematical model of the ship motion is described using Serret-Frenet frame by which the well-defined relative degree is guaranteed.With the help of the output-redefinition,the SIMO system is transformed into an equivalent SISO system.The control parameters of the proposed controller are more systematic,and the stability of the closed-loop system is guaranteed.Numerical simulations are presented to validate the proposed control law.It should be noted that the controller proposed in this paper is developed with constant parameters.In the future work,efforts will be made to design the controller with uncertain parameters.

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Appendix

Relative degree is important in this paper.This section gives the concept of relative degree.Consider a SISO nonlinear system,described by:

where x∈Rn,u∈R and y∈R are state vector,control and output,respectively.

Definition[7]:A nonlinear system(equation 28)is said to have a relative degree ρ at a point xoif:

The relative degree ρ of the nonlinear system(28)is said to be well defined if its relative degree is uniform for all x in an operating set.