This paper presents a twotime scale control structure for the course keeping of an advanced marine surface vehicle, namely, the fully submerged hydrofoil vessel. The mathematical model of course keeping control for the fully submerged hydrofoil vessel is firstly analyzed. The dynamics of the hydrofoil servo system is considered during control design. A twotime scale model is established so that the controllers of the fast and slow subsystems can be designed separately. A robust integral of the sign of the error (RISE) feedback control is proposed for the slow varying system and a disturbance observer based state feedback control is established for the fast varying system, which guarantees the disturbance rejection performance for the twotime scale systems. Asymptotic stability is achieved for the overall closedloop system based on Lyapunov stability theory. Simulation results show the effectiveness and robustness of the proposed methodology.
With the development of science and technology, maritime transport has entered a highspeed era. As an advanced marine surface vehicle, the fully submerged hydrofoil vessel (FSHV) can cruise at a high speed under rough sea wave. The lift force of the hydrofoils generated by the highspeed fluid can elevate the ship hull from the water, which highly reduces the wave resistance and friction to the ship [
In fact, the path of marine vessels is usually straight line or straight lines formed by waypoints, on which condition there is little coupling between the heave/pitch dynamics and the yaw/roll dynamics. Some literatures have been investigated about the riding control [
As for the nonlinear steering control for marine vessels, a series of control methodologies have been explored, such as advanced sliding mode control [
In the existing literatures about motion control of marine vehicles with actuator dynamics, the actuator system is always regarded as a firstorder inertial element [
In most cases, adding the servo system for the actuator mechanism into the control objective will lead to the increase of the relative degree of the overall system, which will cause the socalled “explosion of complexity,” such that backstepping based control strategies are no longer conventional.
The frequency response of the actuator servo system is different from that of the attitude tracking system. Therefore, it is not advisable to make the system control design using classical state feedback control methodologies just in a simply single scheme.
External disturbance and model uncertainties exist in both of the systems mentioned above, where their physical characteristics are different according to the time scales of the attitude tracking subsystem and the actuator subsystem. In order to guarantee the control precision and disturbance rejection performance, time scale separation is required to justify the implementation of the controller design for each subsystem.
When the actuator dynamics is considered in the control design for the course keeping control of the FSHV, multitime scale phenomenon tends to be particularly sensitive for the interconnection analysis of the overall system. Singular perturbation approach is such a method to analyze and separate different time scale motions in control problems [
The basis of singular perturbation theory can be referred to [
In this paper, a hierarchical robust control structure is proposed for the course keeping of the FSHV with actuator dynamics against composite disturbances. The main contributions are listed as follows.
Different from the CHADC methodology, singular perturbation theory is used for the time scale separation of the lateral dynamics of the FSHV and the actuator servo system to explore the interconnection of the slow dynamics and the fast dynamics.
A robust integral of the sign of the error (RISE) feedback control is presented for the stabilization of the slowtime scale subsystem and a DOB based feedback control is utilized for the fasttime scale subsystem. All control signals are continuous which are practical for engineering implementation.
Disturbance attenuation performance is guaranteed in both quasisteadystate subsystem and boundarylayer subsystem. Interconnection of the subsystems is analyzed and uniformly asymptotic stability is achieved at the equilibrium point.
The rest of the paper is organized as follows. In Section
A typical configuration of a fully submerged hydrofoil vessel is shown in Figure
Fully submerged hydrofoil vessel.
Assuming that the surge speed is controlled by an individual propulsion system, the maneuvering model of a typical marine vehicle is shown as follows:
By substituting (
Define
If the parameter uncertainties are considered in the modeling and control of the FSHV, the following notation is introduced:
As to the actuator system, AC motor based electric servo solution and hydraulic servo scheme are often used for the implementation of the hydrofoil servo system [
Define
The control objective is to stabilize the maneuvering dynamics of the FSHV for the task of course keeping based on time scale separation and singular perturbation control theory. The control scheme of the overall system is summarized in Figure
Control structure for the course keeping of FSHV.
The nonlinear functions
The generalized disturbances
In this section, the multitime scale decomposition of the full model (
The slowfast time scale character is often associated with a small parameter multiplying some of the state variables of the state equations describing a physical system. However, usually that parameter may not be identifiable at all and only by physical insight and experiences does one know the details of fast and slow dynamics for the system.
Experience implies that the yaw and roll dynamics are slow relative to the dynamics of the servo system for flap mechanism among the state variables of the proposed mathematical model of the FSHV, which is the motivation to establish a singular perturbation control scheme as follows:
According to the twotime scale structure of the FSHV system with actuator dynamics, a hierarchical control strategy can be deployed for the maneuvering control of the FSHV. Before the following control design, a brief introduction of the singular perturbation control theory is given for convenience.
Singular perturbations cause a multitime scale behavior of dynamical systems characterized by the presence of slow and fast transients in the system’s response to external stimuli. Loosely speaking, the slow response is appropriated by the reduced model, while the discrepancy between the response of the reduced model and the full model is the fast transient. To see this point, let us consider the following generalized state equations:
Stretching the time to
Since the system is separated into the quasisteadystate subsystem and the boundarylayer subsystem, the control strategy can be designed individually for each subsystem; namely, the hierarchical control law is integrated by two parts as follows:
In this section, the composite control law is designed individually for the slow subsystem and fast subsystem. Inspired by [
Because of the disturbance property of the actuator servo system, a disturbance observer based state feedback controller is designed for the fast subsystem as
Define
Sometimes the external disturbance is fast varying; namely, the derivative of the disturbance is not zero. In this condition, the error dynamics of disturbance estimation still can be ultimately uniformly bounded [
Despite the fast control law (
Rewriting subsystem
The reduced system
Define the following Lyapunov function candidate:
Hence the closedloop boundarylayer system has an asymptotically stable equilibrium point.
Then consider control design of the slow varying subsystem. Assume
To stabilize the slow varying system, an adaptive RISE feedback control strategy is proposed.
Define a stabilizing error
Two auxiliary tracking errors
Assuming that the generalized disturbances and the guidance command input are sufficiently smooth, the following inequalities can be obtained:
Define
Given the slow varying dynamics of a FSHV presented in (
Define the following Lyapunov function as
The expression in (
According to the inequalities in (
Consider the multitime scale mathematical model for course keeping of a FSHV with actuator dynamics in (
In order to analyze the closedloop stability of the overall system, we need to explore the interconnection between the slow varying subsystem and the fast varying subsystem.
Define a composite Lyapunov function candidate as follows:
Therefore, according to Theorem
In this section, a mathematical model of a fully submerged hydrofoil vessel is applied to validate the performance of the proposed control laws (
Model parameters of the FSHV.
Parameter  Value  SIunit 


23.15  m/s 


kg 


kgm^{2} 


kgm^{2} 


m 


kgm^{2}/s 


kgm^{2}/s 


kgm^{2}/s 


kgm^{2}/s 


kgm^{2}/s 


kgm^{2}/s 


kgm^{2}/s 


kgm^{2}/s 

22.18  kgm^{2}/s 

32.45  kgm^{2}/s 
In the simulation, the irregular wave disturbance is simulated based on trip theory and equivalent energy division method, with the significant wave height
The command course angle is set to be
Figures
Roll dynamics of the FSHV with command course angle of −30°.
Yaw dynamics of the FSHV with command course angle of −30°.
Roll dynamics of the FSHV with command course angle of 60°.
Yaw dynamics of the FSHV with command course angle of 60°.
At the steady state of the lateral dynamics, the proposed control acquires higher precision and better disturbance attenuation performance since the model uncertainties and external disturbances are considered in both slow dynamics and fast dynamics of the FSHV. The composite control law guarantees the robustness against the lumped disturbances while the control design of the standard singular perturbation approach does not take the disturbance effects into account.
From Figures
Roll response of different time scale with command course angle of −30°.
Yaw response of different time scale with command course angle of −30°.
Roll response of different time scale with command course angle of 60°.
Yaw response of different time scale with command course angle of 60°.
Figures
Composite control input of the FSHV with command course angle of −30°.
Composite control input of the FSHV with command course angle of 60°.
In this paper, a twotime scale robust control structure is proposed for the course keeping control of the FSHV with actuator dynamics. We first analyze the lateral model of the FSHV as well as the hydrofoil servo system. Then a twotime scale separated model is established for the hierarchical control design. A RISE feedback control is designed for the slow varying subsystem and a DOB based state feedback control is used for the fast varying subsystem to achieve disturbance attenuation performance. Uniformly asymptotic convergence is achieved for the overall system. Simulation results indicate the effectiveness of the time separation method, which shows the advantages for control design of complicated interconnection systems with different time scale. In future work, a hardwareinloop simulation testbed will be implemented so that further experiments can be assigned to verify the effectiveness of the proposed methodology. New structures of DOB will be discussed to relax the restrictions of the lumped disturbances. And adaptive estimator is to be considered to handle unknown hydrodynamics combined with the RISE feedback.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China under Grant 51579047.