Nonlinear Model Predictive Control with Terminal Invariant Manifolds for Stabilization of Underactuated Surface Vessel

and Applied Analysis 3 the second component of (1) behaves as a nonholonomic constraint, which is a nonintegrable relation involving not only the generalized coordinates and velocities but also the generalized accelerations [28]. C(]) is the matrix of Coriolis and centripetal terms also including added mass. J(η) is the rotation matrix for the transformation between body-fixed and earth-fixed coordinates:


Introduction
The waterjet propulsor is widely used to thrust existing planning surface vessels.The conventional method to control planning surface vessels is indirectly achieved through the course control which is actuated by a steerable nozzle.If the planar position and course are controlled directly, we need to regulate the angle and the thrust force of the steering nozzle to control the movements in three degrees of freedom synchronously.Obviously, the control system of planning surface vessels is typical underactuated system.In order to ensure the safety of the planning surface vessel, many constraints such as the angle of the steering nozzle and radius of gyration must be in consideration in the design process of ship motion controller; otherwise it will undermine the performance of the planning surface vessels and even lead to collapse of the hull and other serious consequences.The underactuated system with constraints is essentially nonlinear system and cannot be stabilized by any smooth time-invariant control laws.Predictive control is an effective optimization control method to deal with constrains [1][2][3][4][5][6].
In general, the use of MPC in an underactuated structure system necessitates a means of switching among the available models to the one that best describes the current operating condition.A closely related work is the stability analysis of switched stochastic systems by [7] in which dissipativitybased sliding mode control was constructed.Since designing MPC controllers that stabilize underactuated system may not result in a stable global closed-loop system, closed-loop stability in switch model/control approaches has also been studied [8].The study in the paper is aimed to analyse the underactuated characteristics of planning surface vessels and guarantee safe movements of the vessels.Furthermore, based on predictive control approach, the problem of the stabilization and tracking control is solved under the condition of reserved safety constraints.
Control of underactuated systems has been one of the active research topics due to its intrinsic nonlinear nature and practical applications.As a typical example of underactuated systems, control of an underactuated ship has been focused on recently.The main difficulty in the control of underactuated ships is that they are not actuated in the sway axis.This configuration is the most common among the surface ships [9].Furthermore, unlike underactuated systems with nonintegrable constraints, the surface vessels under the consideration are a class of underactuated systems with nonintegrable dynamics and are not transformed into a driftless system [10].Nevertheless, several authors have studied the trajectory tracking control problem.A discontinuous control approach with two stage control laws switched on at given time is proposed based on the stability analysis of the global transformed system in [11].In [12], the authors proposed a kinematic tracking controller that achieved global exponential practical stability of an underactuated surface vessel.In [13], a continuous time-invariant control law was proposed to obtain semiglobal exponential position and orientation tracking, provided the desired angular trajectory remains positive.Based on the cascaded approach, a global tracking result was obtained in [14,15].The stability analysis relied on the stability theory of linear time-varying systems.An application of the recursive technique proposed in [16] for the standard chain form systems was used in [17] to provide exponential stability of the reference trajectory.Based on Lyapunov's direct method and passivity approach, two constructive tracking solutions were proposed in [18] for an underactuated ship.The constructive control design procedure exploited the inherent cascade-interconnected structure of the ship dynamics and actually generated an explicit Lyapunov function whose availability might suit the requirements of robust and adaptive control design.With the help of the backstepping design methodology, a nonlinear time-invariant control law was proposed in [19] for an underactuated surface vessel.But the control design in [19] was not complete and lacked a further step in the backstepping procedure.In the comment letter [20], the control laws in [19] were revised and the states decayed asymptotically to zero.
Model predictive control (MPC) is a popular technique for the control of slow dynamical system subject to input and state constraints.At any time instant, MPC requires the online solution of an optimization problem to compute optimal control inputs over a fixed number of future time instants, known as the finite horizon or quasi-infinite horizon.Using MPC, it is possible to handle inequality constraints on the manipulated and controlled variables in a systematic manner during the design and implementation of the controller.Perhaps the principal shortcoming of existing MPC-base control techniques is their inability to explicitly incorporate plant model uncertainty.The fact that the rudder actuation is limited in amplitude and rate makes MPC approach a natural choice for the design of the path following controller.In [21], a standard model predictive control approach for path following with roll constraints of marine surface vessels in calm water using the rudder as the control input has been proposed.The focus is on satisfying all the input (rudder) and state (roll) constraints while achieving satisfactory path following performance.For notational convenience the ship dynamics in [21] are written into linearized matrix form based on the assumption that surge velocity is constant and the yaw moment is proportional to the rudder angle.However, the former assumption is hardly admissible in engineering.In [22], an analytic model predictive controller is presented for path following of an underactuated ship maneuvering along a predefined path.The mathematical model of ship motion is described by using Serret-Frenet frame, and a systematic method is provided to guarantee the stability of the closed-loop system in terms of transforming the original single-input multiple-output (SIMO) into an equivalent single-input single-output (SISO) system.However, the MPC algorithm proposed in [22] is designed as an analytic model predictive controller which is affected by the accuracy of the parameters.This paper illuminates the stabilization approach for underactuated surface vessels with only a surge force and a yaw moment.A nonlinear model predictive controller is presented for steering the states of underactuated ship into a desired terminal invariant manifolds.After the terminal manifolds set is reached, a linear feedback control is used to stabilize the system.However, in the techniques mentioned above, plant modeling is the critical step to obtain feedback controller and the control result is strongly influenced by the model; the problems inherent to plant modeling are inevitable.Moreover, even complex models cannot cover all the system dynamics [23][24][25][26].In the future work, we will make efforts to design data-driven MPC controllers to overcome these problems.

Problem Formulation
In this paper, we consider the trajectory tracking control problem of a surface vessel shown in Figure 1.There is no side thruster, but two independent main thrusters are located at a distance from the center line in order to provide both surge force and yaw moment.
The dynamics and kinematics of an underactuated surface vessel are described as follows [27]: The inertia matrix  = diag { 11 ,  22 ,  33 } and the damping matrix  = diag { 11 ,  22 ,  33 } are constant and positive definite.The vector  = [ 1 ,  2 ,  3 ] denotes the control forces in surge and sway and control torque in yaw.In this paper, the surface vessel is assumed as the common thruster configuration that has no side thruster, such as  2 = 0.So the second component of (1) behaves as a nonholonomic constraint, which is a nonintegrable relation involving not only the generalized coordinates and velocities but also the generalized accelerations [28].(]) is the matrix of Coriolis and centripetal terms also including added mass.() is the rotation matrix for the transformation between body-fixed and earth-fixed coordinates: The vector  = [, , ]  denotes the North and East positions and orientation of the underactuated surface vessel in the earth-fixed coordinate system.The vector ] = [, V, ]  denotes the linear velocities in surge and sway and the angular velocity in yaw.
As a general accepted conclusion [29,30], there is no continuous time-invariant feedback control law that makes the zero origin an asymptotically stable equilibrium of the system (1) and ( 2), for the system does not satisfy Brocketts condition [31].Then time-varying and discontinuous control approaches are only taken into account in this paper.
Neglecting the motions in heave, roll, and pitch, the simplified kinematic model which describes the geometrical relationship between the earth-fixed (E-frame) and the bodyfixed (B-frame) motion is given as The following global coordinate transformation and feedback transformation are adopted before control design.Define It is proved that the state transformation ( 5) is a global diffeomorphism [32].The feedback transformation is With the state and feedback transformation ( 5)-( 6), the system (1)-( 2) is eventually transformed to The system (7) has the same diffeomorphism properties as the system (1) and (2) [32]; that is, if lim  → ∞   = 0 (1 ≤  ≤ 6) then (, , , , V, ) converges to zero as  → ∞.

Lemma 1.
If there exists a control law which globally uniformly asymptotically stabilizes the system then the system (7) under the control law is also globally uniformly asymptotically stabilized.
Proof.An approach to prove Lemma 1 based on Lyapunov stability theorem has been given in [28].Therefore, to stabilize the system (1) and ( 2), it is only needed to design a stabilizing control law for the system (8).In the following section, a discontinuous and time-varying control approach with MPC control method is proposed for the stabilization of the system (8).

Control Design
Let us recall the following definitions firstly.
Definition 2. Let Φ :   →   be a smooth map.A manifold  = { ∈   : Φ() = 0} is said to be invariant for the control system ẋ = (, ) if all system trajectories starting in  at  =  0 remain in this manifold for all  ≥  0 .In other words, the Lie derivative of Φ along the vector field  is zero (  Φ() = 0) for all  ∈ .
3.1.Construction of the Invariant Manifold.First, in order to construct the invariant manifold of the system (8), assume that is a linear state feedback such that Substitute ( 9) into (8); we obtain The resulting closed-loop system (10) can be successively integrated to obtain where  1 ,  2 ,  3 ,  3 ( 0 ), and  4 ( 0 ) are the integration constants, which can be determined, at  = 0, as a function of the initial conditions  40 ,  50 , and  60 .Besides, from (11), one can easily see that [ 5  6  4  3 ] tends to [0 0  4 ( 0 )  3 ( 0 )] when  tends to infinity.So, if we take the initial conditions such that  4 ( 0 ) = 0 and  3 ( 0 ) = 0, then the whole state tends to the origin.Setting  = 0 in (11),  4 () and  3 () can be determined.Substituting  0 by  in the previous functions leads to Let From ( 13), it appears clearly that if the state variables belong to the 2-dimensional manifold then the whole state  tends to the origin, since  5 and  6 decay exponentially to zero.Furthermore, this manifold is invariant under the linear state feedback (9), as it is shown in the following result.
Proof.Denote vector fields of system (10) under the linear state feedback ( 9) Evaluating the Lie derivatives of along the vector fields ( 16) yields It appears clearly that the state variables   belong to the 2dimensional invariant manifold.Hence, we can construct the invariant manifold of the system (10) as   , which has the following characters: (1)  = 0 ∈   ; (2) to stabilize system (8)  Now it appears clearly that if the initial conditions are locating in the invariant manifold   , the system variables decay exponentially to zero in terms of linear feedback control.If the initial conditions locate outside of   , in order to stabilize the system, we should force the system variables into   firstly and then use the feedback control to stabilize the system.
In this paper, an MPC method is proposed for system (8) with the initial conditions out of the manifolds   .Consider an initial state [ 5  6  4  3 ] and a control horizon of .At initial time, let the manifolds   be the terminal constraint set.The first objective of the proposed algorithm is to use a T-step control horizon to steer the terminal set-valued state prediction [( + )] into the terminal constraint set   .The detail solution can be achieved by the minimization problem in Section 3.2.Secondly, use the feedback control to stabilize the system to the origin.

Design of MPC Controller.
In this subsection we focus our objective on the determination of the first term to make the MPC controller steer the state into the terminal manifold   , asymptotically attractive.Once on it, the whole state [ 5  6  4  3 ] tends to zero under the residual linear state feedback (9).
MPC is an attractive strategy for systems subject to terminal constraints.Due to the USV systems with terminal constraints, we obtain the control input by minimizing a nominal cost [(,   )] over a finite predictive horizon as follows: s.t.
In sampling time   , the value function for (  ,    , ) is Choose sample time  <  small enough such that extending the process (,   ) to [  ,   +  + ],   [  + ,   +  + ] will satisfy (22).To this control it will correspond to the extended trajectory [  +,   ++].The condition (22) Due to (,   ) ≥ 0, we know the value function is decreasing on each interval [  ,   + ] for any  and the function  is smaller at  +1 than at   .Hence the close-loop system ( 8) is asymptotically stable.The stabilizing properties of this approach can be confirmed by the existence of an admissible solution to the openloop optimization (21) at initial time , and so stability is guaranteed provided that   is reachable in time.Then the condition (,   ) = 0 is guaranteed by using the linear feedback control   .Because (,   ) = 0, the condition Ẇ = 0 ≤ (,   ) = 0 is satisfied.It is clear that condition ( 22) is satisfied.(A thorough discussion of the previous problems can be found in [33].) The approach was first described in [33].In this case, outside the invariant manifolds centered at the origin, we have to solve the open-loop optimal control problem with (21).Before we reach   , we have a free time problem.After the set   is reached, we switch to a linear stabilizing feedback controller for the linearized system.The course of solution is described as follows: where  MPC denotes the control law by using nonlinear MPC and denotes the linear feedback control law.Reference [11] y (m) x (m)
Simulation results are shown in the following figures, and the simulation time is set as 150 s. Figure 2 gives the time response of the position , . Figure 3 gives the time response of the orientation  and orientation velocity . Figure 4 gives the time response of the velocities , V.The responses of the control inputs  1 and  3 are shown in Figure 5.
Figures 2-5 show that the three control laws all asymptotically stabilize the underactuated surface vessel to the zero origin.Since the MPC control law guarantees a faster convergence rate of the simulation system, the MPC method may be superior to the other two.Furthermore, Figure 5 shows that surge control force  1 given by MPC is always positive.Compared with reverse thrust for an actual ship, positive thrust is easier to achieve in practice.Therefore, the MPC approach gets a more favorable control effect.

Conclusion
This paper proposes the stabilization approach for underactuated surface vessels with only a surge force and a yaw moment.The invariant manifolds constraints are studied, and stability theory of MPC controller is further developed, which is applied to the stabilization control of underactuated surface vessel.For the stabilization control of underactuated surface vessel, a nonlinear MPC control law with terminal invariant manifolds constraints is designed through coordinate transformation and state feedback transformation based on diffeomorphism and Lyapunov stability theory.The simulation results show that the proposed control law can effectively deal with the problem of stabilization control of underactuated surface vessel.
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 data-driven controller with uncertain parameters to reduce the effect of noisy data and computational complexity.