In order to solve the problem of track straight line and curve tracking control for under-actuated ships, Bech model is selected to solve the non-linear problems such as course instability. With the help of hyperbolic tangent function, the expected heading equation is constructed, and the track control problem is transformed into the heading control problem. A third-order Tracking differentiator is designed to extract the desired heading and its differential signals accurately. The variable structure integral sliding surface function is used to design the nonlinear error feedback control law to accelerate the convergence speed of the system. A ship track integral sliding mode controller based on Linear Active Disturbance Rejection Control (LADRC) is proposed. The linear extended state observer of the controller estimates and compensates the total disturbances on-line, and introduces Hurwitz polynomial to reduce the parameters to be tuned.
The simulation results show that the track convergence is fast, thermostatic element accurate, no overshoot and robust to external disturbances. Controller parameters have a certain universality, so it has a wide range of applications. Usually, the study of ship motion needs to consider three degrees of freedom, namely, forward and backward motion, lateral motion and yaw motion.
Ordinary ships (ships without side thrusters) control their forward and backward motion and yaw motion respectively by propeller thrust and rudder rotating moment. The control input of their motion system is less than the number of degrees of freedom they need to control, that is to say, ordinary ships are mostly underactuated systems. In addition, ships are vulnerable to the interference of wind, current and other external environment in the actual navigation process, resulting in transverse motion. At the same time, the ship itself has a large inertia, and its motion has a long time lag and non-linearity, which makes the ship track tracking control problem become one of the important issues in the field of ship motion control.
In recent years, researchers at home and abroad have made many achievements in ship track control. Document [3] adopts the approximate linearized ship model and uses sliding mode control to design ship track controller; Document [4] designs ship longitudinal and transverse position tracking controllers in the appendage coordinate system by means of variable structure sliding mode control method according to the under-actuation characteristics of ships, and the tracking effect is better; Document [5] introduces Active Disturbance Rejection Control (Active Disturbance Disturbance Control). Bance Rejection Control (ADRC) algorithm, with the help of Proportional Derivative (PD) control law, designs the track controller, and carries out the simulation of the under-actuated ship’s straight and curve track control.
The controller has fast convergence speed and strong robustness. Document [6] chooses Bech model as the ship model to solve the ship’s transportation to a certain extent. For the dynamic nonlinearity problem, the ship course controller based on discrete variable structure control theory has strong robustness to parameter perturbation and external disturbance, but this method has not been applied to the track control problem. Document [7] uses nonlinear sliding mode iteration and incremental feedback control technology to design a non-linear feedback control law which does not depend on accurate ship model, and solves the problem of ship track straight line and curve control, but the controller has a large amount of calculation, which is not conducive to engineering implementation; Document [8] uses Bech model and ADRC method, and puts forward a suitable for the outside world. The steering angle buffeting is serious when the ship is disturbed by external disturbances, and the parameters of the designed controller are too many to be tuned.
Literature [9] uses linear extended state observer (LESO) to observe the sum of internal uncertain disturbances and external disturbances in real time, and makes active compensation in error feedback control law. ] The desired heading direction of the system converges to the planned track is obtained by constructing the dimension reduction equation. The variable structure linear sliding surface is introduced into the design of error feedback control law to solve the problem of ship track tracking control. In this paper, Bech model is used as ship model to solve the non-linear problems such as course instability.
Aiming at the problem of ship model trajectory tracking control, the expected heading equation is constructed by using hyperbolic tangent function, and the goal of trajectory tracking control is achieved by tracking the expected heading. An integrated sliding mode course tracking controller based on linear ADRC (LADRC) is proposed by introducing integral sliding surface function in the design of error feedback control law. It inherits the advantages of LADRC technology and reduces the time of system convergence. The combination of the two control algorithms improves the quality of track control. The model of ship motion control system usually adopts linear or non-linear first-order or second-order Nomoto equation. The derivation of the model is based on the condition of small rudder angle and constant speed, which has great limitations. _is the bow angle; R is the bow angular velocity; U and V are the longitudinal and transverse velocities of ships; X and y are the longitudinal and transverse displacements of ships in the geodetic coordinate system; w (t) is the external interference; T1, T2 and T3 are the ship’s following index; K is the ship’s gyration index; a0, a1, A2 and A3 are the non-linear coefficients, and their specific values can be confirmed by the reversible helix. It is verified that the angle of the rudder is delta, which is the control input of the system. Actuator servo system is generally composed of electric-hydraulic drive system, whose function is to drive the rudder angle to the desired angle of control system command. In the formula, delta C (t) is the command rudder angle of the input, Delta (t) is the actual output rudder angle, TR is the time constant of the rudder, generally about 2.5 s, and KR is the control gain of the rudder, generally about 1 [12]. · Less than 3 degrees/s. During the actual sailing process, ships are prone to deviate from the planned course due to the interference of wind and current, and there is a certain angle between the heading and the planned track (i.e. the wind-current pressure difference angle).
When the actual yaw angle is tracked to_d, the longitudinal deviation X and the lateral deviation y of the ship can reach zero, thus realizing the target of ship track tracking control. And they are all positive numbers. b2tanh (b1_y), _* (-pi 2, PI 2) make g (_*) = 0, that is, _=-b2tanh (b1_y). ·= y=-b2 ytanh (b1y)<0, so when g(_*)=0, limt, y=0, that is, _* can make the lateral deviation of ship track converge to 0. Incremental function on. Let_p = 0, then_ = _, _* = _*, _r = R. When_ < _ *, _ 0.
Combined with the actual operation of the ship, the above conclusions can be interpreted as follows: when the actual heading angle_ > _* (_* can converge to 0), when R < 0, under the effect of the angular velocity of the bow, _, _*, g (_), 0, limt, y = 0. When the actual heading angle_ 0, under the effect of the angular velocity of the heading, _, _ *, g (_), 0, then limt, y = 0. Certification is completed. When a ship deviates from the planned course, a course controller is designed to track the heading angle to the desired heading angle d, which can converge the lateral deviation to 0 and achieve the purpose of track tracking.
Han Jingqing [13] proposed ADRC technology based on the original PID control technology, which can actively and real-time estimate and compensate the internal and external uncertainties of the controlled system. The technology greatly simplifies the ADRC method and solves the problem of parameter tuning of traditional ADRC technology.
The control technology still shows good control quality for complex uncertain non-linear controlled objects. LADRC consists of three parts: Nonlinear Tracking Differentiator (TD), LESO and Nonlinear State Error Feedback Control Law (NLSEF). In this paper, a 3-order tracking differentiator is designed to track the target signal and its differential signal. In the traditional NLSEF, variable structure integral sliding mode switching function is introduced to make the system enter and stabilize on the sliding mode surface quickly and accurately, so as to accelerate the convergence speed of the system. In order to improve the tracking accuracy and speed of the tracking differentiator, the following three-order TD is designed to track the target signal with the help of SGN function. Among them: h and R are adjustable parameters; D is the setting value of the heading angle, that is, the target heading; V1 is the tracking signal of_d; V2 is the tracking signal of d; V3 is the tracking signal of D.
Formula: x1, x2 and X3 are the state variables of the system; u is the input of the system; y is the output of the system; fx1, x2, X3 are the internal uncertainties of the nonlinear system; w (t) is the external disturbance of the system; B is the control gain. LESO is the core part of LADRC technology. Its function is to estimate and compensate the total disturbance inside and outside the system actively and in real time.