Aiming at the problems of multivariable coupling, nonlinearity and delay in ship motion, this paper proposes an internal model controller design for ship dynamic positioning based on the inverse algorithm. The simulation results show that compared with the traditional PID, the internal model control has the characteristics of strong robustness, high positioning accuracy and green control.
According to [
MV + DV =τ + RT (ϕ )b + E w v v
Where:τ is the control force and moment; M is the inertia matrix (including hydrodynamic added
mass), which meeting the positive definite requirements; D is the damping matrix, R(φ) is the
transformation matrix, b is unmodeled external disturbance forces such as wind, wave and current,
which can be described by the first-order markov deviation model [
b b (1) (2)
Where: Tb is a three-dimensional diagonal matrix containing time constants; ∈ is a zero mean Gaussian white noise and ∈ is a three dimensional diagonal matrix indicating the amplitude range of environmental forces. M, D is defined as follows:
m − X u M = 0 0
0 m − Yv mxG − Yv 0
mxG − Yr IZ − Nr ，
− X u D = 0 0
0 − Yv − Nv − Nr ,
0 cosϕ − sinϕ 0 − Yr R(ϕ ) = sinϕ cosϕ 0 0 0 1
Where: X u ,Yv ,Yr , Nv , Nr refers to the added mass coefficient of hydrodynamic force caused by acceleration, X u ,Yv ,Yr , Nv , Nr represents the hydrodynamic linear damping coefficient.
Before the dynamic positioning control system is applied in the actual environment, the simulation test must be carried out in the simulated environment. The purpose of the simulation is to verify whether the control method used by the ship in the simulated bad environment can meet the requirements of engineering control. When some control performance does not meet the requirements, adjust the controller parameters or structure to make the whole control system more perfect.
Matlab simulation software is an open system, which can carry out matrix calculation, numerical
analysis and scientific data visualization in an easy to use windows environment. Matlab simulation
software supports the analysis of single input and single output systems and multiple input and
multiple output systems, as well as the analysis and simulation of discrete systems and continuous
systems. In Matlab / Simulink environment, many control algorithms can be simulated and verified,
such as classical PID control algorithm and other intelligent control algorithms. This paper mainly
carries out the simulation verification of dynamic positioning control for the commonly used ship
dynamic positioning control algorithms, such as PID control algorithm and internal model control
algorithm, and analyzes the advantages and disadvantages of the algorithm [
The longitudinal motion of a 2.8m experimental ship in the key laboratory of ocean engineering of
Guangdong university of technology is studied. During the simulation, the surge motion part of the
low-frequency model structure proposed by Fossen of Norway is selected as the MATLAB simulation
model of this paper [
Where
Classical PID control method is adopted, and PID control is adopted for speed loop and position loop. The Simulink simulation structure diagram of the position ring is as Fig.1 shown.
The subsystem is the speed loop subsystem, and the speed loop also adopts the same PID control structure. The given position signal is 2, and the response result of the PID control system to the position signal is shown in Fig.2.
It can be seen from Fig.2 that the system cannot track and respond effectively and quickly when the double loop PID control is applied to the ship model. The control system has a huge overshoot. In the actual ship dynamic positioning, this control method is time-consuming and labor-consuming.
In practical industrial control applications, most of the controlled objects are not linear models with single input and single output, but nonlinear models with multiple input and multiple output and mutual coupling. Especially for complex systems such as ship dynamic positioning, decoupling control is required during internal model control, which increases the complexity of internal model control.
The inverse system method is a kind of decoupling control method which deals with nonlinear model by feedback linearization. Internal model control (IMC) is applied to the pseudo linear system. The internal model control based on inverse system method has clear structure and simple algorithm.
Based on the inverse system theory, the α order inverse system of the original system is constructed to perform internal model control on the new system composed of the original system and the α order inverse system. The system structure diagram is as follows:
The structure diagram of internal model control under pseudo linear system is shown as Fig.3, in which r is the system input, d is the system disturbance, y is the control system output, and ym is the internal model output. The new system constructed by the inverse system and the original system has a simple model, and the internal model controller design is easy and convenient, which solves the problem of constructing the inverse model of the original nonlinear system.
(1) Basic Theory of the Inverse System Principle
As a new control method, the inverse system theory has established a relatively complete design theory for general nonlinear systems in recent years. The inverse system method has no special requirements for the form of equations, so it has extensive research significance. The essence of inverse system method is to use feedback linearization to realize decoupling linearization of nonlinear, multivariable and strongly coupled systems. In this method, the nonlinear, multivariable and strongly coupled system is linearized in the global by state feedback to realize decoupling. Inverse system theory does not need to introduce problems into differential geometry and other fields, so it does not require a deep mathematical theoretical basis, which is very suitable for engineering applications.
The basic idea of inverse system theory is to generate an α order integral inverse system for the
original system by using the mathematical model of the controlled object and the feedback method.
The inverse system refers to the system that realizes the transformation from the output of the
original system to the input. For the system with known and relatively accurate model, the α order
inverse system of the original system can be obtained by mathematical derivation method, and the
new system composed of the α order inverse system and the original system is pseudo linear system.
Finally, various design methods and theories of linear systems are used to complete the synthesis of
pseudo linear systems [
For a continuous nonlinear or linear system Σ , assuming that the input is u , the output is y , and the initial state is x0 , note that the operator describing the mapping relationship of the system is θ : u → y , that is y = u ⋅θ (4)
Construct another continuous nonlinear or linear system Π , assuming that the input is η and the output is ud , the mapping operator of the system is θ :η → ud , if η = y∂ , that is, η is α order derivative of y . If the operator θ satisfies: θ ⋅θ ⋅η = θ ⋅θ ⋅ y ∂ = θ ⋅ ud =η (5)
Then the system Π is the α order inverse system of the system Σ . α = 0 is a special case of α order inverse system, which is the unit inverse system.
The composite new system composed of the original system and its inverse system is called
pseudo linear system, which composed of θ ⋅θ . The relationship between input and output of the new
composite system is linear, showing an identity mapping relationship, but the internal structure of the
composite system may also be nonlinear and coupled. Only when the mapping operator θ of the
original system itself is linear, the internal structure of the new composite system can be linear.
Obviously, the new system composed of the inverse system and the original system can be controlled
according to the general linear control method, and the whole control becomes relatively simple and
clear [
The dynamic positioning control system of a ship at sea includes two parts, one is to control the
speed of the ship, and the other is to control the position of the ship. In the design of this paper, the
internal model control method is used to control these two parts, forming a double closed-loop
internal model control system, that is, the inner loop is the speed loop control system, and the outer
loop is the position loop control system. At the same time, for the ship model, the principle of inverse
system is adopted to decouple and linearize the coupled and nonlinear ship model [
Regardless of the coordinate conversion in the actual motion control of the ship, the structure diagram of the ship control system based on the inverse system and internal model control algorithm is shown in Fig.5:
Where: G1 is the pseudo linear system, which is the controlled object of the inner loop (speed loop)
control system; Δ is the inner loop (speed loop) control system; G1 is the pseudo linear system
model; G2 is the outer loop (position loop) controlled object; G2 is the reference model of the
controlled object of the outer loop (position loop) [
(3) Simulation analysis
According to the actual ship model, control the surge, sway and yaw of the ship. The set values of surge, sway and yaw are 3m, 2m and 2 rad respectively. The system response is shown as Fig.6.
It can be seen from the simulation results in Fig.6 and Fig.7 that the simulation effect of the ship dynamic positioning controller based on the inverse system principle and internal model control algorithm proposed in this paper is good. The ship has no overshoot and error in response, stable dynamic performance and fast response. At the same time, considering the limitation of propeller thrust capacity and the imprecision of ship model in actual conditions, the ship dynamic positioning controller can still work with maximum efficiency and no deviation in response. The simulation results show the effectiveness and availability of the ship dynamic positioning controller based on the inverse system principle and internal model control algorithm.
For nonlinear ships, this paper proposes an inverse system control algorithm based on the internal model algorithm, the simulation results show that compared with the formation of classical PID control, the ship direction dynamic positioning controller based on the inverse system theory and internal model control algorithm has better dynamic characteristics and faster response speed, with less overshoot and strong anti-interference ability, it is suitable for engineering practice.
This work was supported in part by China Youth Innovation Talents Project of Guangdong Education Department under Grant 2017GKQNCX001, in part by China Characteristic innovation projects of Guangdong Education Department under Grant 2019GKTSCX003, in part by Guangzhou science and technology plan project (Grant No. 202102020860), in part by 2020 youth innovative talents project of Guangdong province ordinary university (Grant No. 2020KQNCX142) .