EP2123588B1 - Crane control with active swell sequence - Google Patents

Description

The present invention relates to a crane control with active Seegangsfolge for a crane arranged on a float, which has a lifting mechanism for lifting a hanging on a rope load.

Such crane controls are needed in order for a float on such as. As a ship, a semi-submersible or a bark-mounted crane to compensate for the undesirable influences of the sea state on the movement of the load, which otherwise affect the safety and accuracy of the stroke.

There is an increasing demand for floating cranes for the installation of offshore wind farms and subsea production facilities, so that crane controls with swell gear take on a special role. Such a crane control should make it possible to operate the crane safely, accurately and efficiently, even under bad weather conditions with heavy seas, in order to minimize weather-related downtimes. In addition, the safety of the operating personnel and the equipment should be ensured.

When a crane is mounted on a float, movement of the float through the sea causes movement of the load suspension point of the load hanging on the crane. On the one hand, this leads to a corresponding movement of the load, which hinders the exact positioning of the load and endangers the assembly personnel. Should z. As a rotor to be mounted on an offshore wind turbine, this requires a very precise positioning of the rotor blades on the hub, where they must be screwed by fitters. Any uncontrolled movement of the rotor blade through the sea can have catastrophic consequences here. In addition, the movement of the load suspension point can lead to critical force peaks in the rope and in the crane, which must be taken into account especially in deep sea stretches.

Already in cranes according to the prior art is trying to compensate for the movement of the load at sea, at least partially. On the one hand passive systems are known, in which the swaying motion is to be compensated passively by the construction of crane and hoist. Active control systems are already known in which the movement of the load suspension point generated by the swaying motion is to be compensated by an active counter-control. However, none of the known systems has led to a truly satisfactory solution.

From the EP 1 757 554 A2 a crane control for a ship crane with load oscillation damping is known, which has a diagnostic component with integrated model and forecasting component. With the help of the prognosis component, a swing of the load is predicted on the basis of the present swell and fed to the crane control to compensate for the pendulum motion.

WO2005090226 A1 discloses an active sea gait crane control system for a crane-mounted crane, comprising a sensor that detects a current sea gait movement from sensor data, and a gaze device that predicts future vertical movement of the load suspension point based on the determined current sea gait movement and a model of sea gait movement.

Object of the present invention is therefore to provide an improved crane control with active Seegangsfolge available.

This object is achieved by a crane control according to claim 1. The present invention thus provides a crane control with active Seegangsfolge for a On a float arranged crane, which has a lifting mechanism for lifting a hanging on a rope load, available. The crane control in this case has a measuring device, which determines a current seaway movement from sensor data. Furthermore, a prediction device is provided, which determines the prevailing modes of the seaward movement from the data of the measuring device and based on the particular prevailing modes creates a model of the sea state and a future vertical movement of the load suspension point based on the determined current sea state movement and a model of sea state movement predicts. Furthermore, a path control of the load is provided, which compensates for the vertical movement of the load through the seaway at least partially by the control of the hoist of the crane due to the predicted movement of the load suspension point.

The forecasting device according to the invention thus makes it possible to take into account the future vertical movement of the load suspension point in the control of the lifting mechanism on the basis of the determined current seaward movement and a model of the swaying motion, so that this movement of the load suspension point is compensated by a change in the rope length and the load of the provided train follows. The path control based on the predicted by the forecasting device future movement of the load suspension point leads to a significantly improved Seegangsfolge compared to a path control, which is based solely on the current movement of the load suspension point. This is due, in particular, to the fact that the actuators of a crane have high dead times and considerable time constants of up to 0.5 seconds, especially in the case of heavy loads. A control solely based on the currently measured movement of the load suspension point would thus lead to a delayed reaction. According to the invention, the forecasting device therefore has a forecast horizon of more than 0.5 seconds, advantageously more than one, and more advantageously more than 2 seconds, so that despite the dead times and time constants of the hoist, a secure compensation of the movement of the load suspension point due to the swell of the Float can be made. Advantageously, the controller takes into account the predicted movement of the load suspension point and the dead times of the hoist when it is controlled.

In addition to the predicted movement of the load suspension point is in the path control of the load, of course, the desired trajectory of the load, which of a rail planning z. B. is generated due to control commands of an operator or due to an automatically scheduled expiration of the stroke. The web control now ensures according to the invention that of the railway planning provided path of the load in spite of the movement of the load suspension point, which is caused by the swaying motion of the float, is complied with. By the crane control according to the invention can thus ensure an exact positioning of the load. Furthermore, it is ensured that the hub does not overload the rope or the crane.

Advantageously, the model of the swell movement used in the prediction device is independent of the properties, in particular of the design and dynamics of the floating body. As a result, the crane control according to the invention can be used flexibly for a large number of floats. In particular, so that the crane can be mounted on different ships, without the Seegangsfolge the crane control would have to be adapted for each of these, which would be very costly in a dependent on the characteristics of the ship modeling. The model is thus created independently of the characteristics of the floating body based solely on the measured sea state movement, to which the periodic portions of the sea state movement are used. For this purpose, not only the current seaward movement, but the course of the seaward movement over a certain period of time is analyzed.

Advantageously, the prevailing modes of the swaying motion are determined via a frequency analysis from the data of the measuring device. The forecasting device thus analyzes the seaward movement and determines the frequencies which determine the movement of the floating body through the sea. For example, here a Fourier analysis of the Seegangsbewegung be performed, from which by peak detection, the prevailing modes are determined. Advantageously, at least the three strongest modes of the swaying motion are considered, furthermore advantageously up to ten modes. The modes are determined by a longer-term observation of the seaward movement, the analysis being based on a period of the preceding seaward movement of several minutes range, eg to the previous five minutes. Based on the prevailing modes, the forecasting device thus creates a preliminary model of the seaway, which is based on a longer-term observation of the seaway movement.

Advantageously, the model thus created is continuously parameterized on the basis of the data of the measuring device, in particular via an observer, wherein in particular the amplitude and phase of the modes are parameterized. In addition to the creation of a preliminary model by the longer-term determination of the prevailing modes, this model is constantly adapted to the current data of the measuring device. In this case, there is always a comparison between the model predicted and the measured sea state, wherein the forecasting device continuously updates the amplitudes and phases of the individual modes used in the model. Likewise, the weighting of the individual modes in the model can be continuously updated.

According to the invention, this results in a two-part prognosis, in which first, based on a long-term analysis, the prevailing modes of seaward motion are determined, which provide the basis for the model of the seaway movement. This model is then constantly updated via an observer circuit by re-parameterizing the amplitude and phase of the modes by comparing the model's predicted seaward motion and the measured seawall motion. The prevailing modes are not changed by the observer.

Advantageously, however, the model is updated whenever the prevailing modes of the sea state change. This change in the prevailing modes of the sea state is detected by a longer-term observation of the sea state movement, whereby the model is updated when the deviation of the modes used in the model with the actual prevailing modes has exceeded a certain threshold. For example, an update may be the prevailing one Modes can be provided in the model of the sea state every 20 seconds.

Further advantageously, the path control according to the invention on a pilot control, which is stabilized on the basis of sensor data. The path control thus controls the hoist based on the predicted movement of the load suspension point so that a planned path of the load is maintained as accurately as possible. In order to stabilize the precontrol, sensor data are used, so that a more precise control of the lifting mechanism is possible by means of an observer circuit.

Advantageously, the path control is based on a model of crane, rope and load, in which a change in the rope length due to the expansion of the rope is taken into account. Since rope lengths of up to 4,000 m can occur, in particular at depth strokes, a large expansion of the rope can occur, which is now taken into account in the path control according to the invention.

Further advantageously, the path control is based on a model of crane, rope and load, which takes into account the dynamics of the hoist and / or the rope and in particular based on a physical model of the dynamics of the system of hoist, rope and / or load. Advantageously, while the dynamics of the hoist is taken into account, so that the feedforward z. B. reaction times and inertia of the hoist taken into account. To account for the dynamics of the rope and load system, it is advantageously treated as a damped oscillator. The resulting dynamics is modeled in the system and enters the feedforward control of the path control according to the invention, whereby the dynamic change in length of the cable can be taken into account in the feedforward control.

Advantageously, according to the invention, a force sensor is provided for measuring the force acting in the cable and / or on the hoisting gear, the measured data of which enter into the path control and via which, in particular, determines the rope length becomes. A direct feedback of the position of the load on the path control for stabilization is not possible because the position of the load itself can be difficult to measure. According to the invention, therefore, the force is measured in the rope or on the hoist and used to stabilize the control. The rope length can be reconstructed from the force in the rope on the basis of the model of the dynamics of the rope and load system and the position of the load can be determined.

Further advantageously, the measuring device of the present invention comprises gyroscopes, acceleration sensors and / or GPS elements, from the measured data of the current sea state movement is determined. Besides measuring devices in which only one of these three sensor types is used, systems with a combination of two or three of these sensor types can also be used. In particular, gyroscopes are used according to the invention. An absolute position determination is not possible with such gyroscopes, but for the active Seegangsfolge but also not necessary, since only the relatively high-frequency movements of the float due to the sea state movement must be considered, while a slow drift is not significant. The angular velocities or the position of the measuring point on which the gyroscopes are arranged are then determined from the data of the gyroscopes by integration once or twice.

Advantageously, the sensors of the measuring device are arranged on the crane, in particular on the crane foundation, the measuring device advantageously determining the movement of the load suspension point on the basis of a model of the crane and the relative movement of load suspension point and measuring point. If the sensors are arranged on the foundation of the crane, they move along with the float and thus only measure the seaway movement of the float. Based on the model of the crane, the movement of the load suspension point can be determined from this swaying motion of the floating body.

Advantageously, in this case the swaying motion of the floating body in the forecasting device is used to predict the future movement of the floating body and from this the future movement of the load suspension point on the basis of this future movement of the floating body is determined on the basis of the model of the crane. The arrangement of the sensors of the measuring device on the crane thereby ensures that the crane control according to the invention can be used flexibly and independently of the properties of the floating body.

By way of example, the forecasting device merely determines the future movement of the load suspension point in the vertical. By this restriction to a degree of freedom, a particularly simple forecasting device is provided, which nevertheless provides the decisive data for compensating for the swaying motion with comparatively little constructive effort.

The present invention further comprises a crane with a crane control as described above. In particular, this is a ship crane. In addition to a hoist, the crane according to the invention advantageously comprises a slewing gear and a luffing gear, which are likewise controlled by the crane control according to the invention.

Furthermore, the present invention also includes a floating body with a crane, as described according to the invention. In particular, it is advantageously a ship with a ship's crane.

The present invention further comprises a method of controlling a crane mounted on a float, which comprises a hoist for lifting a load suspended from a cable, comprising the steps of: determining the current sway motion from sensor data, determining the prevailing modes of swell movement from the data of Measuring device and determining a model of the seaway based on the determined prevailing modes, predicting a future vertical movement of the load suspension point based on the determined current seaward movement and the model of the seaward movement, and at least partially offsetting the vertical movement of the load through the seaway by the control of the Hoist of the crane due to the predicted movement of the load suspension point. Obviously, the same advantages result from the method according to the invention as have already been described with regard to the crane control.

Further advantageously, in the method for controlling the crane, the procedure is as described above with regard to the crane control. In particular, the method according to the invention is carried out by means of a crane control, as described above.

Fig. 1

ein Ausführungsbeispiel eines Schiffskranes, bei welchem die vorliegende Erfindung zum Einsatz kommt,

Fig. 2

eine Prinzipdarstellung eines Messverfahrens zur Bestimmung einer Seegangsbewegung eines Schiffes,

Fig. 3

eine Prinzipdarstellung eines Verfahren, mit welchem aus der Seegangsbewegung des Schiffes und einer Relativbewegung zwischen Lastaufhängepunkt und Messpunkt die Seegangsbewegung des Lastaufhängepunktes bestimmt wird,

Fig. 4

eine Prinzipdarstellung eines Ausführungsbeispiels eines Prognoseverfahrens gemäß der vorliegenden Erfindung,

Fig. 5

eine Prinzipdarstellung einer Modellidentifikation und Vorparametrierung in dem Ausführungsbeispiel eines Prognoseverfahrens gemäß der vorliegenden Erfindung,

Fig. 6

eine Darstellung des i-ten Werts der Bildfolge und dessen komplex konjugiertem Wert an der Stelle N_DFT - i während der Phasenbestimmung zur Vorparametrierung in dem Ausführungsbeispiel eines Prognoseverfahrens gemäß der vorliegenden Erfindung,

Fig. 7

eine Prinzipdarstellung der Korrektur der Modellidentifikation und Vorparametrierung mittels eines Beobachters in dem Ausführungsbeispiel eines Prognoseverfahrens gemäß der vorliegenden Erfindung,

Fig. 8

eine Prinzipdarstellung eines Ausführungsbeispiels einer Kransteuerung gemäß der vorliegenden Erfindung,

Fig. 9

eine schematische Darstellung eines Modells für die Dynamik des Systems aus Seil und Last,

Fig. 10

eine schematische Darstellung eines Ausführungsbeispiels eines Prognoseverfahrens der Seegangsbewegung,

Fig. 11

eine Darstellung der Veränderung der vorherrschenden Moden der Seegangsbewegung über die Zeit,

Fig. 12

eine Darstellung einer vorhergesagten und einer tatsächlichen Seegangsbewegung,

Fig. 13

eine graphische Darstellung der Lastbewegung mit einer reinen Vorsteuerung ohne Rückkopplung und ohne Prognose,

Fig. 14

eine graphische Darstellung der Lastbewegung mit einem geschlossenen Regelkreislauf, aber ohne Prognose und

Fig. 15

eine graphische Darstellung der Lastbewegung unter Verwendung des Steuerungsverfahrens gemäß der vorliegenden Erfindung.

The present invention will now be described with reference to an embodiment and with reference to drawings. Showing:

Fig. 1

An embodiment of a ship's crane, in which the present invention is used,

Fig. 2

a schematic representation of a measuring method for determining a seaward movement of a ship,

Fig. 3

a schematic diagram of a method with which from the seaward movement of the ship and a relative movement between Lastaufhängepunkt and measuring point, the sea state movement of the load suspension point is determined

Fig. 4

3 is a schematic diagram of an embodiment of a prediction method according to the present invention,

Fig. 5

a schematic representation of a model identification and pre-parameterization in the embodiment of a prediction method according to the present invention,

Fig. 6

a representation of the i-th value of the image sequence and its complex conjugate value at the point N _DFT - i during the phase determination for pre-parameterization in the embodiment of a prediction method according to the present invention,

Fig. 7

a schematic representation of the correction of the model identification and pre-parameterization by means of an observer in the embodiment of a prediction method according to the present invention,

Fig. 8

3 is a schematic diagram of an embodiment of a crane control according to the present invention,

Fig. 9

a schematic representation of a model for the dynamics of the system of rope and load,

Fig. 10

a schematic representation of an embodiment of a prognosis method of seaway movement,

Fig. 11

a representation of the change in the prevailing modes of seaward movement over time,

Fig. 12

a representation of a predicted and an actual sea state movement,

Fig. 13

a graphic representation of the load movement with a pure precontrol without feedback and without prognosis,

Fig. 14

a graphical representation of the load movement with a closed loop, but without forecast and

Fig. 15

a graphical representation of the load movement using the control method according to the present invention.

First, an embodiment of a measuring method will now be described, which is based on the one hand on the measurement of the movement of the ship and on the other hand on the determination of the relative position of the cantilever tip of the crane system from its foundation. For the former measurement task, an inertial platform is used which measures the straight-line accelerations and rotational rates of rotation around all three axes of the ship. The following is to be completed by the sensors of the crane system. With this measuring arrangement, a drift-free measurement of the dipping motion, an extremely small phase shift in the significant frequency range of the dipping motion and a maximum measurement deviation of approximately 15% of the amplitude of the dipping motion is achieved. The embodiment of a method for predicting the dipping movement of the load suspension point is based on a model of this movement. However, since the model can not be created a priori, it must be identified and parameterized online based on the measured dive movement. The identification is achieved by means of a frequency analysis of the vertical movement of the load suspension point. In order to always describe them correctly with the model of the dipping movement, the identification takes place at regular intervals. An observer is used for the best possible parameterization of the modeled dipping motion. The predicted sway motion is then used to minimize the influence of the swell on the movement of the load by counter-steering with the hoist.

The deployment and recovery of unmanned research stations several thousand meters deep, which detect resource sources and provide scientific knowledge of oceanography, is not possible without a determination of the seaward movement of the mission ship. In addition, numerous constructions such as oil and gas rigs or wind farms with a few dozen wind turbines are built annually to meet the enormous energy needs of mankind. The installation of these systems is carried out by floating cranes, which are exposed to the sea state of the respective region. To avoid collisions of the load with the seabed or the existing ones Shell construction, the height change of the load caused by the ship's movement must be compensated by Seegangsfolgeeinrichtungen. Again, the knowledge of the vertical ship movement is of central importance.

For these application examples, the measurement of the seaward movement of the ship is sufficient. This is understood to mean the vertical deflection of the ship around its rest position. The resting position of a ship is defined as the current, mean level of the smooth sea level. Slow level changes that are below a fixed frequency limit are therefore not part of the seaway movement. This includes, for example, the level changes caused by the tides. These are clearly not to be associated with the seaway movement.

The present invention provides a measuring method for this purpose, which can be used in conjunction with any Active Seismic Sequence (AHC) crane system. On the one hand, the measuring method determines the swaying motion of the load suspension point and, on the other hand, calculates a short-term prognosis for the further, temporal course of this movement. As an overall system, the interconnection between the crane and the fixed measuring system, referred to as the active sea sequencer, can be mounted on a variety of vessels without requiring significant adaptation measures. Depending on the design of the crane this Seegangsfolgeeinrichtung is either as a floating crane, or, but on an emergency vehicle located, to use for Tiefseehub. For this purpose, the measurement process is completely autonomous and acts platform independent. The knowledge of ship-specific data such as displacement, hull shape, etc. or the placement of the crane system on the deck of the ship is deliberately omitted. Therefore, the term ship is also to be understood very broadly. It is synonymous with any floating body and therefore includes barges or semi-divers.

In this context, the term "sequencing device" is understood as meaning a technical system which is capable of reducing the vertical load oscillations excited by the sea state. Ideally, the load should be kept at an equidistant distance from the seabed, regardless of whether the floating crane is on a wave crest or in a wave trough. In addition, the tilting of the floating crane around the longitudinal and transverse axis, which is called rolling and pitching movement, should not affect the load height. If the compensation of the unwanted load oscillation is effected purely constructively, then there is a passive segregation sequence. On the other hand, one speaks of an active swell as soon as the load vibration is deliberately counteracted by means of actuators.

The present measuring method is able to determine the swaying motion of the load suspension point with high resolution and without time delay. This is also achieved in offshore use, where wave heights of up to 10 m are to be expected. Slow, absolute position changes of the rest position of the ship are not of interest.

The prognosis of the swaying motion of the load suspension point has the goal of minimizing the negative influence of the dead times of the actuators of swell followers on the load height. For the generation of the target trajectory of the load movement can thus be specified a position profile of the load suspension point, which is the dead time of the corresponding actuator in the future, whereby a constant dead time is at best completely compensated. Since the load masses in the deep-sea hoist are in the range of up to 100 t and can even be up to 14000 t for screed divers, dead times of approx. 0.2 - 0.5 s are the rule. These are based on the enormous energy which must be provided for the load movement. For the fulfillment of the required task, the forecast thus satisfies a time window of about 1 s.

In Fig. 1 is a crane ship to see, which is mainly used for installation tasks above the sea level. It can be clearly seen that floating cranes generally have a load suspension point that is far above of the sea level. Its position can be specified by the crane operator by means of operating lever, whereby the load can be accurately positioned. In Tiefseehub, however, mostly rigid crane designs are used, which have the lowest possible load suspension point. These have the advantage of not increasing the movements of the ship unnecessarily. Horizontal changes in position of the load are achieved either by actuators on the load hook, or by appropriate positioning of the mission ship.

Regarding the swell, the actual structure of the crane system is not important. Only the vertical position of the load suspension point must be able to be measured. However, as it is usually not possible to install the sensors directly at the load suspension point, an alternative mounting location of the sensors must be selected. Here a fortification near the crane foundation proves to be useful. On the one hand, the lowest vibrations of the crane system are to be expected here, which falsify the measurement results. On the other hand, a defined orientation of the sensors during operation is achieved here. This would not be the case for a positioning of the sensor on a moving part of the crane, for example.

For this invention, therefore, an inertial platform (IMU Initial Measurement Unit) is used to measure the ship's motion, which is attached to the crane foundation. This cost-effective and autonomous measuring unit contains three accelerometers for the measurement of straight-line ship movements, as well as three yaw-rate sensors for determining the rolling, pitching, and yawing movements of the ship. The sampling frequency of the measurements is 40 Hz. The relevant ship movements are, however, in a frequency range between 0.04 Hz and 1 Hz. Furthermore, even in rough seas, the measured variables in the entire application range of the ship's cranes are not within the range of the measurement size restriction. Thus, by means of the selected inertial platform, an accurate determination of the ship's movement in all 6 degrees of freedom is possible.

The ship movement measurement method used for the present invention is based on the measurement signals from a single inertial platform that calculates the desired position and angle signals with integral constant cutoff filters. If a more precise measurement is sought in the course of the swell, the clear separation between measurement and prognosis also enables the measurement process to be exchanged at any time without the need for further adjustments.

In order to obtain the tilt of the ship from the rotation rates measured by the gyroscopes of the inertial platform, a simple integration of the latter is necessary. In addition, it is important to compensate for typical measurement errors such as measurement noise or bias errors. This is done by using one simple integrating filter per direction of rotation. To obtain the position of the inertial platform, the acceleration data has to be integrated twice. Here, too, occurring measurement errors are to be eliminated as far as possible, which means that a two-fold integrating filter is to be used for the three straight-line directions of movement. This is schematically in Fig. 2 shown.

Using the just described signal processing to measure the ship's motion, the complete movement of the ship can be determined from the measurement signals of the inertial platform. Static bias deviations are completely eliminated and a slow drift in the measured signals is largely compensated. Due to the necessary integration of the measured values, high-frequency sensor noise is strongly suppressed, so that no additional low-pass filtering is necessary.

Since the distance between the sensor for measuring the ship's movement and the load suspension is also necessary to measure the swaying motion of the load suspension point, this is determined separately. However, the necessary sensors are known from conventional crane controls. From the measurement of the ship's motion and the knowledge of the distance between the sensor for measuring the ship's movement and the load suspension can thus, as in Fig. 3 shown, the current movement of the load suspension point are determined.

The model used to predict the seaward movement is not an a priori known description of the dynamics of the ship. Rather, the model depicts the dynamics of the measured seaward motion. This is determined during the duration of the swell sequence, whereby the model is constantly being re-identified and parameterized.

Structurally, the method according to the signal flow diagram according to Figure 4 built up. The seaward movement will be regarded as a periodic movement. Their model is thus formed from a superimposition of N sinusoidal oscillations, which are referred to below as modes. Each mode is described completely by its amplitude A, angular frequency ω _M and phase φ _M.

For the on-line identification of the model of the swell motion, the first step is a frequency analysis of the measured swell motion. On the basis of this, a provisional parameterization of the completely identified model is also carried out. This model subsequently serves as the basis of a linear or non-linear observer and is updated at fixed intervals. This performs the exact adaptation of the model parameters taking into account the currently measured seaward movement. With the knowledge of the model, as well as its parameters, it is the task of the prediction to calculate a prediction of the seaward motion for a future time.

The aim of the model identification is to determine the basic structure of the model of the seaway movement. The determination of the necessary number of modes N is based on an online performed, discrete Fourier analysis of the measured sea state movement at time t _i and subsequent evaluation. For this purpose, the significant frequencies of the swell motion are determined by the amplitude response determined. This evaluation of the amplitude response is performed at runtime of the measurement by means of the peak detection. In addition to the number of modes N to be used for the model identification, the peak detection provides the frequencies ω _{N of} the detected modes and a first estimate of the vector of the amplitudes. The phases of the modes are then determined separately from the phase of the discrete Fourier transform. If the model is provided with these parameters, it provides the modeled seaward movement.

Using the created state models of sea state motion, the desired parameter adaptation is equal to an estimate of the current system state. The problem of the model parameterization can therefore be formulated analogously to an observation task. An observer always has the task of estimating the complete state of this route from the measured output variables of a track with sensors. The sought state is determined by means of a model of the route, which is corrected on the basis of the differences between the real and simulated output signals.

With online comparative observers an accurate prediction of the measured seaward movement for forecast periods of less than 2 s can be carried out. If it is also considered that the desired task of the prediction is the compensation of dead times in the range of approx. 0.5 s, the presented prognosis method offers the optimal conditions for this objective.

In the following the modeling of the dipping movement is shown in more detail:
In order to predict the dipping movement of the load suspension, it is necessary to model this movement. As already assumed when measuring the movement of the ship, the dive movement can be considered as a periodic movement. Their model is thus formed from a superimposition of N _M sinusoids, which are referred to as modes hereinafter. Each mode is described completely by its amplitude A _{M, k} , angular frequency ω _{M, k} and phase φ _{M, k} . In addition, the model still has to add a static offset z _{LA, off} , since the rest position of the dipping movement does not have to be in the origin of the z-axis of the world coordinate system. The modeled dipping movement of the load suspension z _LA is thus described as follows, with the choice of the starting time t ₀ = 0, without limiting the generality: $$z_{\mathit{LA}}\left(t\right)={\displaystyle \underset{k=1}{\overset{N_M}{\Sigma }}{z}_{\mathit{LA}.k}\left(t\right)+z_{\mathit{LA}.\mathit{off}}={\displaystyle \underset{k=1}{\overset{N_M}{\Sigma }}{A}_{M.k}\sin \left({\omega }_{M.k}t+{\phi }_{M.k}\right)+z_{\mathit{LA}.\mathit{off}}}}$$

Since this model of the dipping movement, as briefly mentioned above, is to be used in a state observer, it is necessary to generate a state model from this.

Linear state model of the dipping motion

For the linear observer, a model structure is sought which corresponds to the general description of a linear system without direct penetration shown in equation 5.2. $$\begin{array}{lll}\dot{x}=\underset{\_}{\mathrm{A}x}+\underset{\_}{\mathrm{B}u}.& \underset{\_}{x}\left(0\right)={\underset{\_}{x}}_{\mathrm{0}}.& \underset{\_}{x}\in {ℝ}^n.\mathrm{}u\in {ℝ}^p\\ \underset{\_}{y}=\underset{\_}{\mathrm{C}x}.& & \underset{\_}{y}\in {ℝ}^m\end{array}$$

resultiert ein autonomes System mit nur einem Ausgang, dessen Systemgleichung wie folgt aufzustellen ist: $$\begin{array}{ll}{\underset{\_}{\dot{x}}}_k& ={\underset{\_}{A}}_k{\underset{\_}{x}}_k=\left[\begin{array}{cc}0& 1\\ -{\omega }_{M,k}^2& 0\end{array}\right]{\underset{\_}{x}}_k\\ y_k& ={\underset{\_}{C}}_k{\underset{\_}{x}}_k=\left[\begin{array}{cc}1& 0\end{array}\right]{\underset{\_}{x}}_k,\end{array}\begin{array}{ll}{\underset{\_}{x}}_{\mathrm{0,}k}& =\left[\begin{array}{c}{A}_{M,k}\sin \left({\phi }_{M,K}\right)\\ {\omega }_{M,k}{A}_{M,k}\cos \left({\phi }_{M,k}\right)\end{array}\right]\\ k& =\mathrm{1,}\dots ,N_M\end{array}$$

Here, x denotes the vector of the states of the system of order n with the initial conditions x ₀ at time t ₀ , which are chosen to be zero without limiting the generality. u stands for the p inputs of the system. The matrix A is called the system matrix, B the control matrix, and C the measurement matrix. y characterizes the system output, which consists of m different measurement signals. If a single mode z _{LA, k} from equation 5.1 is represented as a linear differential equation system analogous to equation 5.2, then this is to be modeled as a free, undamped oscillation. With the choice of conditions $$\begin{array}{lll}{\underset{\_}{x}}_k& ={\left[\begin{array}{cc}{x}_{\mathrm{1,}k}& x_{\mathrm{2,}k}\end{array}\right]}^T={\left[\begin{array}{cc}{z}_{\mathit{LA}.k}& {\dot{z}}_{\mathit{LA}.k}\end{array}\right]}^T.& k=\mathrm{1,}....N_M\\ & =\left[\begin{array}{c}{A}_{M.k}\sin \left({\omega }_{M.k}t+{\phi }_{M.k}\right)\\ {\omega }_{M.k}{A}_{M.k}\cos \left({\omega }_{M.k}t+{\phi }_{M.k}\right)\end{array}\right].& k=\mathrm{1,}....N_M\end{array}$$

results in an autonomous system with only one output, whose system equation is to be set up as follows: $$\begin{array}{ll}{\underset{\_}{\dot{x}}}_k& ={\underset{\_}{A}}_k{\underset{\_}{x}}_k=\left[\begin{array}{cc}0& 1\\ -{\omega }_{M.k}^2& 0\end{array}\right]{\underset{\_}{x}}_k\\ y_k& ={\underset{\_}{C}}_k{\underset{\_}{x}}_k=\left[\begin{array}{cc}1& 0\end{array}\right]{\underset{\_}{x}}_k.\end{array}\begin{array}{ll}{\underset{\_}{x}}_{0k}& =\left[\begin{array}{c}{A}_{M.k}\sin \left({\phi }_{M.K}\right)\\ {\omega }_{M.k}{A}_{M.k}\cos \left({\phi }_{M.k}\right)\end{array}\right]\\ k& =\mathrm{1,}....N_M\end{array}$$

The scalar output y _k describes the k-th mode. If the individual modes are added up and the static offset is added to the model as the last state of the system description, then the linear model of the load movement of the load suspension is composed of the individual modes according to Equation 5.5 as follows: $$\begin{array}{cc}\dot{\underset{\_}{x}}=\underset{\underset{\_}{\mathrm{A}}}{\underset{\}}{\left[\begin{array}{ccccc}{\underset{\_}{\mathrm{A}}}_1& \underset{\_}{0}& ...& ...& 0\\ \underset{\_}{0}& {\underset{\_}{\mathrm{A}}}_2& \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & {\underset{\_}{\mathrm{A}}}_{{\mathrm{<figure-callout\; id="0"\; label="N\; M"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">N\; </figure-callout>}}_M}& 0\\ \underset{\_}{0}& & & \underset{\_}{0}& 0\end{array}\right]}}\underset{\_}{x}& {\underset{\_}{x}}_0=\left[\begin{array}{c}{\underset{\_}{x}}_{0.1}\\ {\underset{\_}{x}}_{0.2}\\ ⋮\\ {\underset{\_}{x}}_{0N_M}\\ z_{\mathit{LA}.\mathit{off}}\end{array}\right]\\ y=\underset{\underset{\_}{\mathrm{C}}}{\underset{\}}{\left[\begin{array}{ccccc}{\underset{\_}{\mathrm{C}}}_1& {\underset{\_}{\mathrm{C}}}_2& ...& {\underset{\_}{C}}_{N_M}& 1\end{array}\right]}}\underset{\_}{x}& \end{array}$$

It should be noted that the system output y in Equation 5.6 is chosen so that it describes the plunge motion of the load suspension point.

A general, nonlinear SISO system without direct penetration is described as a state model by the following differential equation system: $$\begin{array}{llll}\dot{\underset{\_}{x}}=f\left(\underset{\_}{x}.u\right).& t>0& \underset{\_}{x}\left(0\right)={\underset{\_}{x}}_0.& \underset{\_}{x}\in M_n\subseteq {ℝ}^n.\mathrm{}u{\in }_1\subseteq {ℝ}^1\\ y=H\left(\underset{\_}{x}\right).& t\ge 0& & y{\in }_1\subseteq {\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0005.png,imgf0010.png"\; state="\{\{state\}\}">ℝ</figure-callout>}}^1\end{array}$$

ergibt sich das autonome nichtlineare Modell der k-ten Mode zu: $$\begin{array}{cc}{\dot{\underset{\_}{x}}}_k={\underset{\_}{f}}_{\mathrm{}k}\left({\underset{\_}{x}}_k\right)& =\left(\begin{array}{c}{x}_{\mathrm{2,}k}\\ -x_{\mathrm{1,}k}{x}_{\mathrm{3,}k}^2\\ 0\end{array}\right),\\ y_k=h_k\left({\underset{\_}{x}}_k\right)=x_{\mathrm{1,}k}& =x_{\mathrm{1,}k}\end{array}\begin{array}{ll}{\underset{\_}{x}}_{\mathrm{0,}k}& =\left[\begin{array}{c}{A}_{M,k}\sin \left({\phi }_{M,k}\right)\\ {\omega }_{M,k}{A}_{M,k}\cos \left({\phi }_{M,k}\right)\\ {\omega }_{M,k}\end{array}\right]\\ k& =\mathrm{1,}\dots ,N_M\end{array}$$

Where n is the order of the system with the output y. The states x and their initial conditions x ₀ are located in the natural working space of a nonlinear system M _n , which is described by the n-dimensional manifold. The input of the system u lies in the permissible quantity of the input functions U ₁ . The dynamics of the system is described by the vector field f (x), which is thus the nonlinear analog of the system matrix A of the linear systems. h (x) stands for the output function of the system and can be compared with the measurement matrix C of the linear systems. If the dipping movement of the load suspension point is to be specified in accordance with equation 5.1 in the form just described, it is again advisable to initially consider only a single mode. With the following definition of states $$\begin{array}{lll}{x}_k& ={\left[\begin{array}{ccc}{x}_{\mathrm{1,}k}& x_{\mathrm{2,}k}& x_{\mathrm{3,}k}\end{array}\right]}^T={\left[\begin{array}{ccc}{z}_{\mathit{LA}.k}& {\dot{z}}_{\mathit{LA}.k}& {\omega }_{M.k}\end{array}\right]}^T.& k=\mathrm{1,}....N_M\\ & =\left[\begin{array}{c}{A}_{M.k}\sin \left({\omega }_{M.k}t+{\phi }_{M.k}\right)\\ {\omega }_{M.k}{A}_{M.k}\cos \left({\omega }_M{}_{.k}t+{\phi }_{M.k}\right)\\ {\omega }_{M.k}\end{array}\right].& k=\mathrm{1,}....N_M\end{array}$$

the autonomous nonlinear model of the k-th mode results: $$\begin{array}{cc}{\dot{\underset{\_}{x}}}_k={\underset{\_}{f}}_{\mathrm{}k}\left({\underset{\_}{x}}_k\right)& =\left(\begin{array}{c}{x}_{\mathrm{2,}k}\\ -{\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="imgf0005.png,imgf0010.png"\; state="\{\{state\}\}">x</figure-callout>}}_{\mathrm{1,}k}{x}_{\mathrm{3,}k}^2\\ 0\end{array}\right).\\ y_k=H_k\left({\underset{\_}{x}}_k\right)=x_{\mathrm{1,}k}& =x_{\mathrm{1,}k}\end{array}\begin{array}{ll}{\underset{\_}{x}}_{0k}& =\left[\begin{array}{c}{A}_{M.k}\sin \left({\phi }_{M.k}\right)\\ {\omega }_{M.k}{A}_{M.k}\cos \left({\phi }_{M.k}\right)\\ {\omega }_{M.k}\end{array}\right]\\ k& =\mathrm{1,}....N_M\end{array}$$

The complete, nonlinear model of the load suspension of the load suspension point, in turn, results from the combination of the models of the single modes from equation 5.10 and the introduction of an offset state. Will this last and thus 3N _M + 1-th state is included in the model, the description of the overall system is as follows: $$\begin{array}{ll}\dot{\underset{\_}{x}}=\underset{\_}{f}\left(\underset{\_}{x}\right)=\left(\begin{array}{c}{\underset{\_}{f}}_{\mathrm{}1}\left({\underset{\_}{x}}_1\right)\\ ⋮\\ {\underset{\_}{f}}_{\mathrm{}{N}_M}\left({\underset{\_}{x}}_{N_M}\right)\\ 0\end{array}\right)& {\underset{\_}{x}}_0=\left(\begin{array}{c}{\underset{\_}{x}}_{0.1}\\ ⋮\\ {\underset{\_}{x}}_{0N_M}\\ z_{\mathit{LA}.\mathit{off}}\end{array}\right)\\ y=H\left(\underset{\_}{x}\right)={\displaystyle \underset{k=1}{\overset{N_M}{\Sigma }}{H}_k\left({\underset{\_}{x}}_k\right)+x_{3N_M+1}}& \end{array}$$

The only output of the overall system is chosen so that it describes the dipping movement of the load suspension.

Model identification and pre-parameterization

The aim of the model identification is to determine the basic structure of the model of the dipping movement. Since this is already specified except for the number of modes, it is only necessary to determine their number. The pre-parameterization of the model has the task of adapting the parameters of the identified model as accurately as possible.

Considering equation 5.1, the dive motion is fully described with knowledge of the parameters N _M , A _{M, k} , ω _{M, k} , φ _{M, k} and z _{LA, off} . The number of parameters to be determined thus results in 3N _M + 2. Thus, it is linearly dependent on the number of sinusoidal oscillations that are needed to simulate the dipping movement. The determination of N _M is therefore the first and most important task, since it is similar to the model identification. Once the number of modes, and thus the model of the dive movement, is known, the remaining 3N _M +1 parameters can be adapted bit by bit.

The identification and pre-parameterization of the model of the dipping movement is carried out on the basis of the measured, vertical movement of the load suspension point. The structural approach is in FIG. 5 shown. The determination of the necessary number of modes N _M is based on an online performed, discrete Fourier analysis of the measured dipping movement at time t _i and subsequent evaluation.

The significant frequencies of the dive movement are determined by means of the amplitude response. This evaluation of the amplitude response is performed at runtime of the measurement by means of the peak detection. In addition to the number of modes N _{M, DFT} to be used for the model identification, the peak detection supplies the frequencies of the detected modes ω _{M, DFT, k,} which are combined to the vector ω _{M, DFT} , and a first estimate of the vector of the amplitudes A _{M, DFT} . The phases of the modes φ _{M, DFT} are then determined separately from the phase of the discrete Fourier transform. If the model is provided with these parameters, it supplies the modeled dipping movement in the time range between t ₀ and T, which is denoted by z _{LA, DFT} .

By means of discrete Fourier transformation (DFT), the amplitude response A _{DFT, i} and the phase response φ _{DFT, i are} determined from z _LA (t) via the time-discrete signal z _{La, n} . For example, the discrete Fourier transform can be applied every 10 seconds to the actual load movement of the load suspension.

peak detection

The amplitude spectrum of the dipping movement of the load suspension point determined online with the discrete Fourier transformation must now be evaluated with the aid of the peak detection. Almost all information necessary for the identification and pre-parameterization of the model of the dipping movement can be obtained from this.

Objectives of peak recognition

The main task of the peak recognition is to identify the state model of the dipping motion.

• Online-Identifikation des Modells der Tauchbewegung
  • Bestimmung der Anzahl der Moden N_M,SE
  • Modellbildung unter Berücksichtigung der maximalen Anzahl der Moden N_M,Max
• Online-Parametrierung des Modells der Tauchbewegung
  • Bestimmung der Amplituden A_M,DFT der Moden
  • Bestimmung der Kreisfrequenzen ω_M,DFT der Moden

This includes the following goals:

• Online identification of the model of the dive movement
  • Determination of the number of modes N _{M, SE}
  • Modeling taking into account the maximum number of modes N _{M, Max}
• Online parameterization of the model of the dipping movement
  • Determination of the amplitudes A _{M, DFT of} the modes
  • Determination of the angular frequencies ω _{M, DFT of} the modes

The problem of the detection of the modes is solved, for example, by a MiniMax task with a secondary condition applied to the amplitude response. As a secondary condition, a so-called border sequence is used. It determines the minimum amplitude, which must exceed a maximum, so that this is recognized as a mode. The determination of the amplitude of the limit sequence A _{DFT, limit, i} takes place adaptively as a function of the respective amplitude spectrum of the current dipping movement according to the following equation: $$A_{\mathit{DFT}.\mathit{border}.i}=\underset{\mathrm{offset\; shift}}{\underset{\}}{c_{\mathit{border}}{A}_{\mathit{DFT}.\mathit{Max}}}}+\underset{\mathrm{Averaging\; calculation}}{\underset{\}}{\frac{1}{6}\left({\displaystyle \underset{j=1}{\overset{3}{\Sigma }}{A}_{\mathit{DFTj}}+A_{\mathit{DFT}.-j}}\right)}}\mathrm{}i=4.5....\frac{N_{\mathit{DFT}}}{2}$$

Structurally, this is thus calculated from an offset shift and an averaging. The offset shift defines a constant minimum amplitude of the limit sequence over the entire frequency spectrum. It is formed from the product between the freely selectable design parameter c _limit and the absolute maximum of the amplitude response A _{DFT, Max} , which is determined analogously to equation 5.24. $$\begin{array}{cc}{A}_{\mathit{DFT}.\mathit{Max}}=\mathrm{Max}\left\{A_{\mathit{DFT}.i}\right\}.& i=1.2....\frac{N_{\mathit{DFT}}}{2}\end{array}$$

ergeben. Die lokalen Maxima des Amplitudengangs der Tauchbewegung werden durch eine diskrete Differentiation dieses ermittelt. Eine Spitze des Amplitudengangs an der Stelle i wird somit als solche erkannt, wenn gilt: $$\left(A_{\mathit{DFT},i}-A_{\mathit{DFT},i-1}>0\right)\wedge \left(A_{\mathit{DFT},i+1}-A_{\mathit{DFT},i}<0\right),i=\mathrm{1,}\dots ,\frac{N_{\mathit{DFT}}}{2}-1$$

The second part is a moving averaging applied to a limited frequency band of the amplitude spectrum. The filter used for this purpose is designed similar to the filters used in image processing. Since it is not possible to calculate the first four amplitude values of the limit sequence analogously to the equation shown, these are to be determined separately. For the sake of simplicity, these were chosen according to the last determinable amplitude, with which the initial values of the limit sequence to $$\begin{array}{cc}{A}_{\mathit{DFT}.\mathit{border}.i}=A_{\mathit{DFT}.\mathit{border}\mathrm{,\; 4}}& i=\mathrm{0,1,2,3}\end{array}$$

result. The local maxima of the amplitude response of the dipping motion are determined by a discrete differentiation of this. A peak of the amplitude response at the point i is thus recognized as such if: $$\left(A_{\mathit{DFT}.i}-A_{\mathit{DFT}.i-1}>0\right)\wedge \left(A_{\mathit{DFT}.i+1}-A_{\mathit{DFT}.i}<0\right).i=\mathrm{1,}....\frac{N_{\mathit{DFT}}}{2}-1$$

If the amplitude of the peak thus detected also exceeds the amplitude of the limit sequence, this is recognized as mode M _{SE, i} . The set of all modes M _{M, SE} is thus determined as follows:

The number of modes N _{M, SE to} be determined can now be determined from the cardinality of the set M _{M, SE} .

If the number of detected modes N _{M, SE is} determined, it is necessary to check whether it is less than or equal to the selected maximum number of modes N _{M, Max} . If this case occurs, then a model of the dipping motion should be used that takes into account N _{M, SE} modes. Otherwise, the number of modes considered is limited to N _{M, Max} , thus determining the number of modes N _{M, DFT} used for modeling as follows: $$N_{M.\mathit{DFT}}=\min \left\{N_{M.\mathit{SE}}.N_{M.\mathit{Max}}\right\}$$

If the models according to Equations 5.1, 5.6 or 5.11 are used to model the dipping movement, then these are completely identified with the knowledge of the number of modes to be considered.

The pre-parameterization of the models is now to be carried out with the set of modes M _{M, DFT} used for the model identification. This is equal to the set of detected modes M _{M, SE} , if N _{M, SE} ≦ N _{M, Max} . Otherwise it is the subset containing the N _{M, Max} modes with the largest amplitude.

und die Amplituden der Moden des Modells zu

The amplitude of the k th mode A _{M, SE, k} is determined by its value in the amplitude response. As explained in the introduction of the amplitude response, it is distributed in the frequency spectrum to two points of identical height. It thus arises too

and the amplitudes of the modes of the model too

The selection of the dominant modes is carried out in the present work by a sorting algorithm applied to the amplitudes of the modes. It should be noted that the assignment between the amplitude, frequency and phase of a mode is not lost by resorting the modes. The last task of the peak detection is still the circuit frequencies ω _{M, DFT determine} the modes. These are determined on the basis of the frequency axis of the amplitude spectrum with the following conversion:

Determination of the static offset

For the determination of the static offset of the dipping movement of the load suspension point, the amplitude response of this movement, determined online, is again used. The DC component of the sequence of measurement data provided for the discrete Fourier transformation corresponds to the first value of the amplitude response. For the mathematical justification equation 5.16 is to be used. If i is chosen to zero, which computes the first value of the amplitude response, the result is: $$z_{\mathit{LA}.\mathit{off}.\mathit{DFT}}=A_{\mathit{DFT}\mathrm{,\; 0}}=\frac{1}{N_{\mathit{DFT}}}{\displaystyle \underset{n=0}{\overset{N_{\mathit{DFT}}-1}{\Sigma }}{z}_{\mathit{LA}.n}}$$

This corresponds to the arithmetic mean of the accumulated sequence of the measured data and thus the static offset of the dipping movement in the time interval considered.

phase determination

The determination of the phases of the individual modes completes the pre-parameterization of the model of the dipping movement. They are determined by evaluating the phase response.

For the determination of the phase, a back transformation of the image sequence into the time domain has to be carried out. If the complete image sequence of the measured dipping movement z _{LA, i is} converted into the time domain by applying the transformation rule according to equation 5.15, the starting value of the dipping movement z _{LA, 0} results in: $$z_{\mathit{LA}\mathrm{,\; 0}}={\mathrm{<figure-callout\; id="0"\; label="z\; LA"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">z\; </figure-callout>}}_{\mathit{LA}}\left(0\right)=\frac{1}{N_{\mathit{DFT}}}{\displaystyle \underset{i=0}{\overset{N_{\mathit{DFT}}-1}{\Sigma }}{Z}_{\mathit{LA}.i}}$$

For a single mode, this expression is greatly simplified and can ultimately be represented as a function of a single value of the amplitude response ADFT, i and the phase response φ _{DFT, i} . This simplification is based on the property that in the image area of the Fourier transformation, a pure sine wave is described by a complex conjugate number of numbers whose values are located at the ith and N _DFT -i th position of the sequence. To clarify the further steps, this pair of numbers is in FIG. 6 (Representation of the i-th value of the image sequence and its complex conjugate value at the point N _DFT - i).

The starting values z _{LA, 0, k of} the N _{M, DFT} modes are thus determined by the following equations:

aus Gleichung 5.1 verglichen, so resultieren die gesuchten Phasen der Moden ϕ_M,DFT zum Zeitpunkt t₀ zu

Become the starting values of the modes of the dipping motion determined in this way with the following starting values $$z_{\mathit{LA}.\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">k</figure-callout>}}\left(0\right)=\begin{array}{ccc}\underset{=2A_{\mathit{DFT}\mathrm{,1}}\mathrm{see.\; Gl.5.30}}{\underset{\}}{A_{M.\mathit{DFT}.k}}}& \sin \left({\phi }_{M,\mathit{DFT}.k}\right).& k=\mathrm{1,}....N_{M.\mathit{DFT}}\end{array}$$

from equation 5.1, the desired phases of the modes φ _{M, DFT result} at time t ₀

Observer-based adaptation of the model parameters

For the adaptation of amplitude, phase and possibly the frequency observer-based approaches are pursued. An observer always has the task of estimating the complete state of this route from the measured output variables of a track with sensors. The desired state x is determined by means of a model of the distance, which is corrected on the basis of the differences between the real y and simulated output signals. In FIG. 7 In this case, a signal flow diagram of such an observer is shown.

Linear observer design

The linear observer design is based on the state model of the dipping motion according to equation 5.6. The linear Kalman-Bucy filter is one of the most popular used observers which build on the structure of the Luenberger observer. For the observer design, the system noise w (t) and the measurement noise v (t) must be taken into account, using the following model for the design process: $$\begin{array}{lll}\dot{\underset{\_}{x}}=\underset{\_}{\mathit{Ax}}+\underset{\_}{\mathit{Bu}}+\underset{\_}{w}\left(t\right)& \underset{\_}{x}\left(0\right)={\underset{\_}{x}}_0.& \underset{\_}{x}\in {ℝ}^n.\mathrm{}\underset{\_}{w}\left(t\right)\in {ℝ}^n.\mathrm{}\underset{\_}{u}{ℝ}^p\\ \underset{\_}{y}=\underset{\_}{\mathit{Cr}}+\underset{\_}{v}\left(t\right).& & \underset{\_}{y}\in {ℝ}^m.\underset{\_}{v}\left(t\right)\in {ℝ}^m\end{array}$$

womit die Kovarianzmatrizen Q und R eindeutig durch die Rauschsignale beschreiben werden. Diese ergeben sich zu konstanten, symmetrischen Matrizen. Damit ergeben sich die folgenden Gleichungen für den Beobachter: $$\underset{\_}{\dot{\widehat{x}}}=\underset{\mathrm{Simulstionsteil}}{\underbrace{\underset{\_}{\mathrm{A}\widehat{x}}+\underset{\_}{\mathrm{Bu}}}}+\underset{\mathrm{Korrekturteil}\underset{\_}{r}}{\underbrace{\underset{\_}{\mathrm{L}}\left(\underset{\_}{\mathrm{y}}-\underset{\_}{\widehat{\mathrm{y}}}\right)}}=\left(\underset{\_}{\mathrm{A}}-\underset{\_}{\mathrm{LC}}\right)\underset{\_}{\widehat{x}}+\underset{\_}{\mathrm{B}u}+\underset{\_}{\mathrm{L}}y,\mathrm{}{\widehat{\underset{\_}{x}}}_0=\mathrm{E}\left\{{\underset{\_}{x}}_0\right\}$$ It is assumed that the noise signals are stationary, mean-free, normally distributed and at the same time uncorrelated. For this noise applies $$\begin{array}{lllll}\mathrm{e}\left\{\underset{\_}{w}\left(t_1\right){\underset{\_}{w}}^T\left(t_2\right)\right\}& =& \mathrm{cov}\left\{\underset{\_}{w}\left(t_1\right){\underset{\_}{w}}^T\left(t_2\right)\right\}& =& \underset{\_}{\mathrm{Q}}\delta \left(t_1-t_2\right)\\ \mathrm{e}\left\{\underset{\_}{v}\left(t_1\right){\underset{\_}{v}}^T\left(t_2\right)\right\}& =& \mathrm{cov}\left\{\underset{\_}{v}\left(t_1\right){\underset{\_}{v}}^T\left(t_2\right)\right\}& =& \underset{\_}{\mathrm{R}}\delta \left(t_1-t_2\right).\end{array}$$

whereby the covariance matrices Q and R are uniquely described by the noise signals. These result in constant, symmetrical matrices. This yields the following equations for the observer: $$\underset{\_}{\dot{\widehat{x}}}=\underset{\mathrm{Simulstionsteil}}{\underset{\}}{\underset{\_}{\mathrm{A}\widehat{x}}+\underset{\_}{\mathrm{Bu}}}}+\underset{\mathrm{correction\; part}\underset{\_}{r}}{\underset{\}}{\underset{\_}{\mathrm{L}}\left(\underset{\_}{\mathrm{y}}-\underset{\_}{\widehat{\mathrm{y}}}\right)}}=\left(\underset{\_}{\mathrm{A}}-\underset{\_}{\mathrm{LC}}\right)\underset{\_}{\widehat{x}}+\underset{\_}{\mathrm{B}u}+\underset{\_}{\mathrm{L}}y.\mathrm{}{\widehat{\underset{\_}{x}}}_0=\mathrm{e}\left\{{\underset{\_}{x}}_0\right\}$$

The correction matrix L of the Kalman-Bucy linear filter is calculated by solving the following quadratic quality criterion:

The model of the linear Kalman-Bucy filter is then: $$\dot{\widehat{\underset{\_}{x}}}=\left(\left[\begin{array}{ccccc}{\underset{\_}{\mathrm{A}}}_1& \underset{\_}{0}& ...& ...& 0\\ \underset{\_}{0}& {\underset{\_}{\mathrm{A}}}_2& \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & {\underset{\_}{\mathrm{A}}}_{N_{M.\mathit{<figure-callout\; id="0"\; label="DFT"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">DFT</figure-callout>}}}& 0\\ \underset{\_}{0}& ...& ...& \underset{\_}{0}& 0\end{array}\right]-\underset{\_}{\mathrm{L}}\left[\begin{array}{lllll}{\underset{\_}{\mathrm{C}}}_1& {\underset{\_}{\mathrm{C}}}_2& ...& {\underset{\_}{\mathrm{C}}}_{N_{M.\mathit{DFT}}}& 1\end{array}\right]\right)\widehat{\underset{\_}{x}}+\underset{\_}{\mathrm{L}}y$$

The individual block matrices, which make up the system matrix A and the measurement matrix C, are analogous to Equation 5.5: $$\begin{array}{ll}{\underset{\_}{A}}_k& =\left[\begin{array}{cc}0& 1\\ -{\omega }_{M.\mathit{DFT}.k}^2& 0\end{array}\right].\\ {\underset{\_}{C}}_k& =\left[\begin{array}{cc}1& 0\end{array}\right].\end{array}\begin{array}{cc}k& =\mathrm{1,}....N_{M.\mathit{DFT}}\\ k& =\mathrm{1,}....N_{M.\mathit{DFT}}\end{array}$$

ist.As the output of the distance y, the subsequence of the stored measurement data of the dipping movement is used, which corresponds to the selected observation interval, whereby $$\begin{array}{ccc}y\left(t\right)=z_{\mathit{LA}.n}\left(t_i\right).& t_n={\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+n\mathrm{\Delta }{\mathrm{<figure-callout\; id="0"\; label="T\; DFT\; t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">T\; </figure-callout>}}_{\mathit{DFT}}& t_{}{}_{0\mathit{obs}}\le t_n\le T\end{array}$$

is.

For a rapid transient response of the observer, these most appropriate initial conditions ^ x ₀ for the time t _{0, Obs are} to be given. These are calculated from the parameters of the individual modes determined by the discrete Fourier transformation and the static offset as follows: $${\widehat{\underset{\_}{x}}}_0=\widehat{\underset{\_}{x}}\left({\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">t</figure-callout>}}_{0\mathit{obs}}\right)\underset{\_}{=}\left[\begin{array}{c}{\underset{\_}{\widehat{x}}}_{0.1}\\ {\underset{\_}{\widehat{x}}}_{0.2}\\ ⋮\\ {\underset{\_}{\widehat{x}}}_{0N_{M.\mathit{DFT}}}\\ z_{\mathit{LA}.\mathit{off}.\mathit{DFT}}\end{array}\right]$$

With: $${\underset{\_}{\widehat{x}}}_{0k}=\left[\begin{array}{c}{A}_{M.\mathit{DFT}.k}\sin \left({\omega }_{M.\mathit{DFT}.\mathrm{<figure-callout\; id="0"\; label="k\; t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">k\; </figure-callout>}}{t}_{}{}_0{}_{.\mathit{obs}}+{\phi }_{M.\mathit{DFT}.k}\right)\\ {\omega }_{M.\mathit{DFT}.k}{A}_{M.\mathit{DFT}.k\mathrm{}}\cos \left({\omega }_{M.\mathit{DFT}.\mathrm{<figure-callout\; id="0"\; label="k\; t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">k\; </figure-callout>}}{t}_{}{}_0{}_{.\mathit{obs}}+{\phi }_{M.\mathit{DFT}.k}\right)\end{array}\right].\mathrm{}k=\mathrm{1,}....N_{M.\mathit{DFT}}$$

For the calculation of L the design parameters Q and R are symmetrical and positive definite. Its dimensions are determined by the number of system states and the outputs of the observer model. Thus, Q is to be chosen as a (2N _{M, DFT} + 1 x 2N _{M, DFT} + 1) matrix and R as a scalar. If only the diagonal elements of the covariance matrix Q are described, due to the prevailing structure of the system matrix A, the dynamics of the error correction for each mode can be specified separately. The larger the track of the k-th block matrix Q k is selected, the faster is a correction of the corresponding deviations of the states ^ x _{k of} the mode. The design parameter R, on the other hand, influences the dynamics of all states equally. The smaller R is selected, the more dynamic the observer reacts to deviations between the measured and simulated dive movement.

The covariance matrix Q of this work used for the estimation of the individual modes of the dive movement is constructed according to the following equation. $$\underset{\_}{\mathrm{Q}}=\left[\begin{array}{ccccc}{\underset{\_}{\mathrm{Q}}}_1& \underset{\_}{0}& ...& ...& 0\\ \underset{\_}{0}& {\underset{\_}{\mathrm{Q}}}_2& \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & {\underset{\_}{\mathrm{Q}}}_{N_{M.\mathit{<figure-callout\; id="0"\; label="DFT"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">DFT</figure-callout>}}}& 0\\ \underset{\_}{0}& ...& ...& \underset{\_}{0}& c_{\mathit{off}}\end{array}\right]$$

The individual block matrices Q _k are again constructed as diagonal matrices and are determined as follows: $$\begin{array}{cc}{\underset{\_}{\mathrm{Q}}}_k=\left[\begin{array}{cc}{\mathrm{<figure-callout\; id="0"\; label="c\; k"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">c\; </figure-callout>}}_k& 0\\ 0& c_k\end{array}\right].& k=\mathrm{1,}....N_{M.\mathit{DFT}}\end{array}$$

$\frac{2\mathrm{\pi }}{40}$

$\frac{2\mathrm{\pi }}{20}$

$\frac{2\mathrm{\pi }}{20}$

$\frac{2\mathrm{\pi }}{10}$

$\frac{2\mathrm{\pi }}{10}$

∞ c_k = 0,001 c_k = 0,01 c_k = 0,5 c_k = 3 The factor c _{k of} the covariance matrices Q _k is determined as a function of the angular frequency of the associated mode. Table 5.2: Entries of the covariance matrix <u> Q </ u> as a function of the angular frequency <i> ω <sub> M, DFT, k </ sub></i> ω _Min ≤ ω _{M, DFT, k} < ω _Max ω _min [ rad / s ] ω _Max [ rad / s ] ω _min [ rad / s ] ω _Max [ rad / s ] ω _min [ rad / s ] ω _Max [ rad / s ] ω _min [ rad / s ] ω _Max [ rad / s ] 0 $\frac{2\mathrm{\pi }}{10}$

$\frac{2\mathrm{<figure-callout\; id="40"\; label="\pi "\; filenames="imgf0012.png,imgf0013.png"\; state="\{\{state\}\}">\pi </figure-callout>}}{40}$

$\frac{2\mathrm{<figure-callout\; id="20"\; label="\pi "\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">\pi </figure-callout>}}{20}$

$\frac{2\mathrm{<figure-callout\; id="20"\; label="\pi "\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">\pi </figure-callout>}}{20}$

$\frac{2\mathrm{\pi }}{10}$

$\frac{2\mathrm{\pi }}{10}$

∞ c _k = 0.001 c _k = 0.01 c _k = 0.5 c _k = 3

Nonlinear observer design

For the nonlinear observer design, the state model of the dip motion as given in Equation 5.11 is to be used. The Extended Kalman Filter is a nonlinear extended variant of the Kalman-Bucy linear filter. The basis of the observer design is thus to formulate the nonlinear SISO system according to Equation 5.7 as follows:

The description of the covariance matrices Q and R in turn takes place according to equations 5.56 and 5.57 through the noise processes, which are assumed to be stationary, mean-free, normally distributed and at the same time uncorrelated. $$\begin{array}{lllll}\mathrm{e}\left\{\underset{\_}{w}\left(t_1\right){\underset{\_}{w}}^T\left(t_2\right)\right\}& =& \mathrm{cov}\left\{\underset{\_}{w}\left(t_1\right){\underset{\_}{w}}^T\left(t_2\right)\right\}& =& \underset{\_}{\mathrm{Q}}\delta \left(t_1-t_2\right)\\ \mathrm{e}\left\{v\left(t_1\right)v^T\left(t_2\right)\right\}& =& \mathrm{cov}\left\{v\left(t_1\right)v^T\left(t_2\right)\right\}& =& \underset{\_}{\mathrm{R}}\delta \left(t_1-t_2\right).\end{array}$$

If the system or measurement noise is not known, these two matrices should be used as design parameters. The extended Kalman filter associated with Equation 5.95 is described by the following nonlinear differential equation system: $$\underset{\_}{\dot{\widehat{x}}}=\underset{\mathrm{simulation\; part}}{\underset{\}}{\underset{\_}{f}\left(\widehat{\underset{\_}{x}}.u\right)}}+\underset{\mathrm{correction\; part}\underset{\_}{r}}{\underset{\}}{\underset{\_}{\mathrm{L}}\left(t\right)\left(y-\widehat{y}\right)}}=\underset{\_}{f}\left(\underset{\_}{\widehat{x}}.u\right)+\underset{\_}{\mathrm{L}}\left(t\right)\left(y-H\left(\underset{\_}{x}\right)\right).\mathrm{}{\underset{\_}{\widehat{x}}}_0=\mathrm{e}\left\{{\underset{\_}{x}}_0\right\}$$

For this observer differential equation, with the noise-free simulation part and the correction part r, it is necessary to determine the time-variant correction matrix L (t). This is calculated as a function of the covariance matrices Q and R from the following matrix-Riccati differential equation.

ist das Erweiterte Kalman-Filter vollständig bestimmt.With the choice of the initial condition P _{0 of} the covariance matrix P to $$\underset{\_}{\mathrm{P}}\left(0\right)={\underset{\_}{\mathrm{P}}}_0=\mathrm{e}\left\{\left({\underset{\_}{\widehat{x}}}_0-{\underset{\_}{x}}_0\right){\left({\underset{\_}{\widehat{x}}}_0-{\underset{\_}{x}}_0\right)}^T\right\}$$

the Advanced Kalman Filter is completely determined.

For the realization of the extended Kalman filter, it is therefore necessary to integrate the n nonlinear filter equations. In addition, for the determination of the correction matrix the Jacobi matrices H (t) and F (t) are to be calculated, as well as the n (n + 1) / 2 differential equations of the symmetric covariance matrix P to be solved. All of this has to be done online, which greatly increases the amount of computation required with the system's order. The filter differential equations according to 5.98 result from using the online identified, nonlinear model of dipping 5.11 to: $$\dot{\widehat{\underset{\_}{x}}}=\left(\begin{array}{c}{\underset{\_}{f}}_{\mathrm{}1}\left({\underset{\_}{x}}_1\right)\\ ⋮\\ {\underset{\_}{f}}_{N_{M.\mathit{DFT}}}\left({\underset{\_}{x}}_{N_{M.\mathit{<figure-callout\; id="0"\; label="DFT"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">DFT</figure-callout>}}}\right)\\ 0\end{array}\right)+\underset{\_}{\mathrm{L}}\left(t\right)\left(y-\left({\displaystyle \underset{k=1}{\overset{N_{M.\mathit{DFT}}}{\Sigma }}{H}_k\left({\underset{\_}{x}}_k\right)}+x_3{{}_N}_{{}_{M.\mathit{<figure-callout\; id="1"\; label="DFT\; +"\; filenames="imgf0005.png,imgf0010.png"\; state="\{\{state\}\}">DFT\; </figure-callout>}}+1}\right)\right)$$

The individual vector fields f (x _k ) of the N _{M, DFT} detected modes and the output functions h _k (x _k ) are to be described analogously to Equation 5.10. $$\begin{array}{ll}{\underset{\_}{f}}_{\mathrm{}k}\left({\underset{\_}{x}}_k\right)=\left(\begin{array}{c}{x}_{\mathrm{2,}k}\\ -{\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="imgf0005.png,imgf0010.png"\; state="\{\{state\}\}">x</figure-callout>}}_{\mathrm{1,}k}{x}_{\mathrm{3,}k}^2\\ 0\end{array}\right).& k=\mathrm{1,}....N_{M.\mathit{DFT}}\\ H_k\left({\underset{\_}{x}}_k\right)=x_{\mathrm{1,}k}.& k=\mathrm{1,}....N_{M.\mathit{DFT}}\end{array}$$

In addition, the initial conditions of the filter equation 5.102 with the parameters of the modes determined by means of the discrete Fourier transformation are calculated as follows: $${\widehat{\underset{\_}{x}}}_0=\left[\begin{array}{c}{\underset{\_}{\widehat{x}}}_{0.1}\\ {\underset{\_}{\widehat{x}}}_{0.2}\\ \cdots \\ {\widehat{\underset{\_}{x}}}_{0N_{M.\mathit{DFT}}}\end{array}\right]$$

With: $$\begin{array}{cc}{\underset{\_}{x}}_{0k}=\left[\begin{array}{c}{A}_{M.\mathit{DFT}.k}\sin \left({\phi }_{M.\mathit{DFT}.k}\right)\\ {\omega }_{M.\mathit{DFT}.k}{A}_{M.\mathit{DFT}.k}\cos \left({\phi }_{M.\mathit{DFT}.k}\right)\\ {\omega }_{M.\mathit{DFT}.k}\end{array}\right].& k=\mathrm{1,}....N_{M.\mathit{DFT}}\end{array}$$

zu bestimmen. Die Blockmatrizen H_k des Systemausgangs und die diagonal angeordneten Blockmatrizen F_k sind wie nachfolgend beschrieben aufgebaut. $$\begin{array}{ll}{\underset{\_}{\mathrm{H}}}_k\left(t\right)=\left[\begin{array}{ccc}1& 0& 0\end{array}\right],& k=\mathrm{1,}\dots ,N_{M,\mathit{DFT}}\\ {\underset{\_}{\mathrm{F}}}_k\left(t\right)=\left[\begin{array}{ccc}0& 1& 0\\ -{\widehat{\underset{\_}{x}}}_{\mathrm{3,}k}^2& 0& -2{\widehat{\underset{\_}{x}}}_{\mathrm{1,}k}{\widehat{\underset{\_}{x}}}_{\mathrm{3,}k}\\ 0& 0& 0\end{array}\right],& k=\mathrm{1,}\dots ,N_{M,\mathit{DFT}}\end{array}$$

For the calculation of the time-variant correction matrix L (t), the time-variant Jacobi matrices H (t) and F (t) from the state of the observer ^ x are also continuously after $$\underset{\_}{\mathrm{H}}\left(t\right)=\left[\begin{array}{lllll}{\underset{\_}{\mathrm{H}}}_1& {\underset{\_}{\mathrm{H}}}_2& ...& {\underset{\_}{\mathrm{H}}}_{N_{M.\mathit{<figure-callout\; id="1"\; label="DFT"\; filenames="imgf0005.png,imgf0010.png"\; state="\{\{state\}\}">DFT</figure-callout>}}}& 1\end{array}\right].\mathrm{}\underset{\_}{\mathrm{F}}\left(t\right)=\left[\begin{array}{ccccc}{\underset{\_}{\mathrm{F}}}_1& \underset{\_}{0}& ...& ...& 0\\ \underset{\_}{0}& {\underset{\_}{\mathrm{F}}}_2& \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & {\underset{\_}{\mathrm{F}}}_{N_{M.\mathit{<figure-callout\; id="0"\; label="DFT"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">DFT</figure-callout>}}}& 0\\ \underset{\_}{0}& ...& ...& \underset{\_}{0}& 0\end{array}\right]$$

to determine. The block matrices H _{k of} the system output and the diagonally arranged block matrices F _k are constructed as described below. $$\begin{array}{ll}{\underset{\_}{\mathrm{H}}}_k\left(t\right)=\left[\begin{array}{ccc}1& 0& 0\end{array}\right].& k=\mathrm{1,}....N_{M.\mathit{DFT}}\\ {\underset{\_}{\mathrm{F}}}_k\left(t\right)=\left[\begin{array}{ccc}0& 1& 0\\ -{\widehat{\underset{\_}{x}}}_{\mathrm{3,}k}^2& 0& -2{\widehat{\underset{\_}{x}}}_{\mathrm{1,}k}{\widehat{\underset{\_}{x}}}_{\mathrm{3,}\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">k</figure-callout>}}\\ 0& 0& 0\end{array}\right].& k=\mathrm{1,}....N_{M.\mathit{DFT}}\end{array}$$

Finally, the design parameters of the extended Kalman filter must be specified. These consist of the covariance matrices Q and R, which are to be chosen symmetrically and positively definite. In addition, it is necessary to set a suitable initial condition for P ₀ . Q is thus again set as a diagonal matrix, whose entries are weighted according to the frequency of the associated mode. The structure of the covariance matrix Q, as given in equation 5.110, thus equals the matrix Q used in the linear case. $$\underset{\_}{\mathrm{Q}}=\left[\begin{array}{ccccc}{\underset{\_}{\mathrm{Q}}}_1& \underset{\_}{0}& \cdots & \cdots & 0\\ \underset{\_}{0}& {\underset{\_}{\mathrm{Q}}}_2& \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & {\underset{\_}{\mathrm{Q}}}_{N_{M.\mathit{<figure-callout\; id="0"\; label="DFT"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">DFT</figure-callout>}}}& 0\\ \underset{\_}{0}& ...& ...& \underset{\_}{0}& c_{\mathit{off}}\end{array}\right]$$

The individual block matrices Q _k , however, differ in their number of elements, since the nonlinear model of the dipping motion has three states per mode. Plotted as a diagonal matrix, these result in: $$\begin{array}{cc}{\underset{\_}{Q}}_k=\left[\begin{array}{ccc}{\mathrm{<figure-callout\; id="0"\; label="c\; k"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">c\; </figure-callout>}}_k& 0& 0\\ 0& {\mathrm{<figure-callout\; id="0"\; label="c\; k"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">c\; </figure-callout>}}_k& 0\\ 0& 0& c_{w.k}\end{array}\right].& k=\mathrm{1,}....N_{M.\mathit{DFT}}\end{array}$$

mit den Abschätzungen der einzelnen Zustandsfehler ^∼x_0,k der Moden $${\underset{\_}{\tilde{x}}}_{\mathrm{0,}k}={\widehat{\underset{\_}{x}}}_{\mathrm{0,}k}-{\underset{\_}{x}}_{\mathrm{0,}k}=\left[\begin{array}{c}\mathrm{0,7}{A}_{M,\mathit{DFT},k}\\ 2\pi \mathrm{0,002}\cdot \mathrm{0,7}{A}_{M.\mathit{DFT},k}\\ 2\pi \mathrm{0,002}\end{array}\right],\mathrm{}k=\mathrm{1,}\dots ,N_{M,\mathit{DFT}}$$

Finally, there is still a suitable initial condition for the matrix-Riccati differential equation P ₀ pretend. This is formed according to equation 5.101 from the expected deviation between the state of the observer and the real system. Using the error estimates of the model identification, this is as follows $${\underset{\_}{\mathrm{P}}}_0=\mathrm{e}\left\{\left[\begin{array}{c}\left({\underset{\_}{\tilde{x}}}_{0.1}\right)\\ \left({\widetilde{\underset{\_}{x}}}_{0.2}\right)\\ ⋮\\ \left({\widetilde{\underset{\_}{x}}}_{0N_{M.\mathit{DFT}}}\right)\\ 0.5\end{array}\right]\right\}{\left[\begin{array}{ccccc}\left({\underset{\_}{\tilde{x}}}_{0.1}\right)& \left({\underset{\_}{\tilde{x}}}_{0.2}\right)& ...& \left({\underset{\_}{\tilde{x}}}_{0N_{M.\mathit{DFT}}}\right)& 0.5\end{array}\right]}^T$$

with the estimates of the individual state ^{errors ~} x _{0, k of} the modes $${\underset{\_}{\tilde{x}}}_{0k}={\widehat{\underset{\_}{x}}}_{0k}-{\underset{\_}{x}}_{0k}=\left[\begin{array}{c}0.7A_{M.\mathit{DFT}.k}\\ 2\pi \mathrm{0,002}\cdot 0.7A_{M,\mathit{DFT}.k}\\ 2\pi \mathrm{0,002}\end{array}\right].\mathrm{}k=\mathrm{1,}....N_{M.\mathit{DFT}}$$

The calculation of the parameters of the modes takes place inversely to the calculation of the initial state ^ x _{0 of} the linear, as well as the nonlinear state model. The calculation basis used is the estimated state of the model ^ x (T) at time T, ie the present. This is adapted over the entire time interval of the observation of t _{0, Obs} ≦ t ≦ T on the basis of the latest measurement data of the dipping motion. Thus, all changes occurring up to this time in the dynamics of the dipping motion are considered.

In the linear case, the two states of the kth mode according to equation 5.61 are defined as follows: $$\begin{array}{ll}{\underset{\_}{\widehat{x}}}_{\mathrm{1,}k}\left(t\right)=A_{M.\mathit{obs}.k}\sin \left({\omega }_{M.\mathit{DFT}.k}t+{\phi }_{M.\mathit{obs}.k}\right).& k=\mathrm{1,}....N_{M.\mathit{DFT}}\\ {\underset{\_}{\widehat{x}}}_{\mathrm{2,}k}\left(t\right)={\omega }_{M.\mathit{DFT}.k}{A}_{M.\mathit{obs}.k}\cos \left({\omega }_{M.\mathit{DFT}.k}t+{\phi }_{M.\mathit{obs}.k}\right).& k=\mathrm{1,}....N_{M.\mathit{DFT}}\\ {\underset{\_}{\widehat{x}}}_{2N_{M.\mathit{DFT}}+1}\left(t\right)=z_{\mathit{LA}.\mathit{obs}}\left(t\right)& \end{array}$$

If the two equations are solved for φ _{M, Obs, k} and A _{M, Obs, k} at time T, the result is: $$\begin{array}{ll}{\phi }_{M.\mathit{obs}.k}=\arctan \left({\omega }_{M.\mathit{DFT}.k}\frac{{\underset{\_}{\widehat{x}}}_{\mathrm{1,}k}\left(T\right)}{{\underset{\_}{\widehat{x}}}_{\mathrm{2,}k}\left(T\right)}\right)-{\omega }_{M.\mathit{DFT}.k}T.& k=\mathrm{1,}....N_{M.\mathit{DFT}}\\ A_{M.\mathit{obs}.k}=\frac{{\underset{\_}{\widehat{x}}}_{\mathrm{1,}k}\left(T\right)}{\sin \left({\omega }_{M.\mathit{DFT}.k}T+{\phi }_{M.\mathit{obs}.k}\right)}.& k=\mathrm{1,}....N_{M.\mathit{DFT}}\end{array}$$

It should be noted that the arctangent is uniquely defined only in an interval between ± π, whereby a case distinction for determining the phases becomes necessary. If you also want to catch the possible division by zero during the implementation, you have to calculate the phase as follows: $$\begin{array}{cc}{\mathrm{\Phi }}_k=\arctan \left({\omega }_{M.\mathit{DFT}.k}\frac{\left|{\underset{\_}{\widehat{x}}}_{\mathrm{1,}k}\left(T\right)\right|}{\left|{\underset{\_}{\widehat{x}}}_{\mathrm{2,}k}\left(T\right)\right|}\right)-{\omega }_{M.\mathit{DFT}.k}T.& k=\mathrm{1,}....N_{M.\mathit{DFT}}\end{array}$$

With: $$\begin{array}{llllll}\mathrm{case}1:& & & {\widehat{\underset{\_}{x}}}_{\mathrm{2,}k}\left(T\right)>0& \Rightarrow & {\phi }_{M.\mathit{obs}.k}={\mathrm{\Phi }}_k\\ \mathrm{case}2:& {\widehat{\underset{\_}{x}}}_{\mathrm{1,}k}\left(T\right)>0& \wedge & {\widehat{\underset{\_}{x}}}_{\mathrm{2,}k}\left(T\right)=0& \Rightarrow & {\phi }_{M.\mathit{obs}.k}=\mathrm{\pi }/2\\ \mathrm{case}3:& {\widehat{\underset{\_}{x}}}_{\mathrm{1,}k}\left(T\right)\ge 0& \wedge & {\widehat{\underset{\_}{x}}}_{\mathrm{2,}k}\left(T\right)<0& \Rightarrow & {\phi }_{M.\mathit{obs}.k}=\mathrm{\pi }-{\mathrm{\Phi }}_k\\ \mathrm{case}4:& {\widehat{\underset{\_}{x}}}_{\mathrm{1,}k}\left(T\right)<0& \wedge & {\widehat{\underset{\_}{x}}}_{\mathrm{2,}k}\left(T\right)<0& \Rightarrow & {\phi }_{M.\mathit{obs}.k}=\mathrm{\pi }+{\mathrm{\Phi }}_k\\ \mathrm{case}5:& {\widehat{\underset{\_}{x}}}_{\mathrm{1,}k}\left(T\right)<0& \wedge & {\widehat{\underset{\_}{x}}}_{\mathrm{2,}k}\left(T\right)=0& \Rightarrow & {\phi }_{M.\mathit{obs}.k}=3\mathrm{\pi }-{\mathrm{\Phi }}_k\end{array}$$

In this case, the phase of the modes φ _{M, Obs, k} , consistent with the modeling of the dipping motion, refers to the time t ₀ . The last parameter that can be parameterized with the linear Kalman-Bucy filter is the stationary offset of the dive movement. This is described by the 2N _{M, DFT} + 1st state of the observer model and thus determined according to the following equation. $$z_{\mathit{LA}.\mathit{off}.\mathit{obs}}={\widehat{\underset{\_}{x}}}_{2N_{M.\mathit{DFT}}+1}\left(T\right)$$

As already mentioned, with the linear observer design it is only possible to achieve an observer-based adaptation of the amplitudes A _{M, Obs} , the phases ω _{M, Obs} and the static offset z _{LA, Obs} . The angular frequencies are thus to be taken over by the identification method by means of the discrete Fourier transformation for the subsequent forecast of the dipping movement. For a complete, observer-based parameterization of the model of the dipping motion the non-linear approach is to be used. Using the presented nonlinear observer, the parameters of the modes are calculated analogously to the linear case.

According to Equation 5.102, the states of the extended Kalman filter are defined as shown below. $$\begin{array}{llll}{\underset{\_}{\widehat{x}}}_{\mathrm{1,}k}\left(t\right)& =A_{M.\mathit{obs}.k}\sin \left({\omega }_{M.\mathit{obs}.k}t+{\phi }_{M.\mathit{obs}.k}\right).& k& =\mathrm{1,}....N_{M.\mathit{DFT}}\\ {\underset{\_}{\widehat{x}}}_{\mathrm{2,}k}\left(t\right)& ={\omega }_{M.\mathit{obs}.k}{A}_{M.\mathit{obs}.k}\cos \left({\omega }_{M.\mathit{obs}.k}t+{\phi }_{M.\mathit{obs}.k}\right).& k& =\mathrm{1,}....N_{M.\mathit{DFT}}\\ {\underset{\_}{\widehat{x}}}_{\mathrm{3,}k}\left(t\right)& ={\omega }_{M.\mathit{obs}.k}.& k& =\mathrm{1,}....N_{M.\mathit{DFT}}\\ {\underset{\_}{\widehat{x}}}_{3N_{M.\mathit{DFT}}+1}\left(t\right)& =z_{\mathit{LA}.\mathit{obs}}\left(t\right)& & \end{array}$$

The parameters A _{M, Obs, k} , ω _{M, Obs, k} , φ _{M, Obs, k} and z _{LA, Obs} to be used for the prediction of the dive motion are thus to be calculated in the sequence given below. $$\begin{array}{llll}{\omega }_{M.\mathit{obs}.k}& ={\widehat{\underset{\_}{x}}}_{\mathrm{3,}k}\left(T\right).& k& =\mathrm{1,}....N_{M.\mathit{DFT}}\\ {\phi }_{M.\mathit{obs}.k}& =\arctan \left({\omega }_{M.\mathit{obs}.k}\frac{{\widehat{\underset{\_}{x}}}_{\mathrm{1,}k}\left(T\right)}{{\widehat{\underset{\_}{x}}}_{\mathrm{2,}k}\left(T\right)}\right)-{\omega }_{M.\mathit{obs}.k}T.& k& =\mathrm{1,}....N_{M.\mathit{DFT}}\\ A_{M.\mathit{obs}.k}& =\frac{{\widehat{\underset{\_}{x}}}_{\mathrm{1,}k}\left(T\right)}{\sin \left({\omega }_{M.\mathit{obs}.k}T+{\phi }_{M.\mathit{obs}.k}\right)}.& k& =\mathrm{1,}....N_{M.\mathit{DFT}}\\ z_{\mathit{LA}.\mathit{obs}}& ={\widehat{\underset{\_}{x}}}_{3N_{M.\mathit{DFT}}+1}\left(T\right)& & \end{array}$$

In the case of inversion of the tangent, in turn, the equations 5.121ff. to take account of the case study.

An exemplary embodiment of a control system in which the measurement and prognosis methods described above are used to control the hoist of a crane will now be briefly described below:
Offshore installations impose stringent safety and efficiency requirements on the affected crane system during rough sea conditions. Therefore, a sea-tracking system based on a forecast of seaward motion and an inversion-based control strategy is proposed. The The control objective is to allow the payload suspended on a rope to follow a desired reference path in a fixed earth coordinate system without being affected by the seaward motion of the ship or watercraft. Therefore, a combination of a control unit, which decouples the disturbance of the track tracing, and a forecasting algorithm presented and evaluated with simulation and measurement results.

Today, offshore installations such as subsea oil and gas or wind farms are becoming increasingly important. The processing facilities for the yield of oil and gas fields are already installed on the seabed. Therefore, the accessibility for maintenance, repair and replacement is lower compared to floating or fixed production platforms. The operation of such systems leads to the affected crane system (see Fig. 1 ) to stringent safety and efficiency requirements. The main objective is to ensure operation during harsh ocean conditions to minimize downtime. Furthermore, the safety of the workers on board is essential. There may be situations where the control of the payload is lost.

In addition to the navigation / positioning problem, the wave-induced movements of the ship / vessel cause critical tension on the rope. The tension should not be below zero to avoid slack rope situations. The peak value must not exceed a safety limit. Therefore, wake tracking systems are used to improve the operational readiness of offshore installations during rough sea conditions. In addition, the vertical movement of the payload can be significantly reduced, which makes an exact positioning of the load possible.

The present invention provides a sea tracking system based on the forecast of ship / vessel motion and an inversion-based control strategy. Basically, there are two requirements for the following systems for offshore cranes. The first is to let the load follow a desired reference trajectory resulting from the hand lever signals of the operator is generated in a fixed earth reference coordinate system. The load should move in this coordinate system at the assigned reference speed decoupled from the wave-induced motion of the ship. The second requirement is a modular crane with swell. This means that the crane systems used for offshore installations can be built on many different types of ships or vessels. In addition, the ship's / vessel's vertical motion estimation and prediction algorithm must be independent of the type of vessel / vessel.

For this, the dynamic model of the system, which consists of the hydraulic actuator (winch) and the flexible rope, is derived. Based on this model, a linearizing tax law is formulated. To stabilize the control system, a regulator is derived. Fig. 8 shows the general control structure.

Furthermore, the estimation and forecast of the movement of the ship / watercraft is presented. Therefore, a model is formulated based on the prevailing modes of sea state motion. The modes are obtained by a fast Fourier transform and a peak detection algorithm. The estimation and prognosis is done by a Kalman filter. Simulation and measurement results are presented.

The dynamic model

The presently considered Seegangsfolgesystem consists basically of a hydraulic-powered winch, a crane-like structure and hanging on a rope load. To model the system, it is assumed that the crane structure is a rigid body. The payload suspended from a rope can be approximated by a spring-mass-damper system (see Fig. 9 ).

σ(z) ist die Zugspannung des Seils, E ist das Young'sche Modul, F(z) die auf das Seil an der Position z wirkende statische Kraft, A_rope die Schnittfläche des Seils, g die Gravitationskonstante, depth der Abstand der Last zum Meeresspiegel, m_l,rope und m_load die Masse des Seils pro Meter bzw. die Masse der Nutzlast.To approximate the compliant rope, the equivalent mass m _eq and the stiffness of the spring c _{rope must} be calculated. With the help of Hooke's law the deformation ε ( z ) for a rope at an arbitrary position z can be obtained from: $$\epsilon \left(z\right)=\frac{\sigma \left(z\right)}{e}=\frac{F\left(z\right)}{e{A}_{\mathit{rope}}}=\frac{G}{e{A}_{\mathit{rope}}}\left(\left(\mathit{depth}-z\right)m_{l.\mathit{rope}}+m_{\mathit{load}}\right),$$

σ ( z ) is the tension of the rope, E is the Young's modulus, F ( z ) the static force acting on the rope at position z , A _rope the sectional area of the rope, g the gravitational constant, depth the distance of the load to the sea level, m _{l, rope} and m _load the mass of the rope per meter or the mass of the payload.

The extension of the entire rope Δ l _R is obtained by means of equation (2). $$\begin{array}{ll}\mathrm{\Delta }{l}_R& ={\displaystyle \underset{0}{\overset{\mathit{depth}}{\int }}\epsilon \left(z\right)\mathit{dz}}\\ & =\underset{rOpe\mathrm{}suspended\mathrm{}\mathit{load}}{\underset{\}}{G\left(\frac{\mathit{depth}}{2}{m}_{l.\mathit{rope}}+m_{\mathit{load}}\right)\frac{\mathit{depth}}{e{A}_{\mathit{rope}}}}}=\underset{\mathit{approximation}}{\underset{\}}{\frac{G{m}_{\mathit{eq}}}{c_{\mathit{rope}}}}}\end{array}$$

An evaluation (2) yields $$m_{\mathit{eq}}=\left(\frac{\mathit{depth}}{2}{m}_{l.\mathit{rope}}+m_{\mathit{load}}\right)\mathrm{and}{c}_{\mathit{rope}}=\frac{\mathit{depth}}{e{A}_{\mathit{rope}}},$$

Using the Newton / Euler method, the second-order differential equation for the movement of the payload hanging on a cable is obtained (see (4)). The load oscillations are terminated by wind accelerations φ̈ _w and the second derivative of the seaway movement ẅ . $$m_{\mathit{eq}}{l}_R+d_{\mathit{rope}}{\dot{l}}_R+\frac{e{A}_{\mathit{rope}}}{\mathit{depth}}\left(l_R-\mathit{depth}\right)=m_{\mathit{eq}}\left(r_W{\mathrm{.phi}}_W+W\right)$$

ϕ̈_w, und ϕ̇_w sind die Winkelbeschleunigung bzw. Geschwindigkeit der Winde, T_w die Zeitkonstante, V_mot,w das Volumen des Hydraulikmotors, u_w die Eingangsspannung des Servoventils und K_v,w die proportionale Konstante der Strömrate zu u_w. The actuator for the Seegangsfolgesystem is the hydraulic-powered winch. The dynamics of this actuator can be approximated with a first order system. $${\mathrm{.phi}}_W=-\frac{1}{T_W}{\dot{\phi }}_W+\frac{2\pi K_{V,W}}{i_W{V}_{\mathit{mot}.W}{T}_W}{u}_W$$

φ̈ _w , and φ̇ _w are the angular acceleration or velocity of the winds, T _w the time constant, V _{mot, w} the volume of the hydraulic motor, u _w the input voltage of the servo valve and K _{v, w} the proportional constant of the flow rate to u _w .

The tax strategy

To derive a tax law, the dynamic model of the system is derived in the following form. The disturbance d is defined as the 4th derivative of the seaway movement. Thus, the relative degree of the system is equal to the relative degree of disturbance, and isidori is able to decouple the disturbance. $$\begin{array}{ll}\underset{\_}{\dot{x}}& =f\left(\underset{\_}{x}\right)+G\left(\underset{\_}{x}\right)u_W+p\left(x\right)d\\ y& =H\left(\underset{\_}{x}\right)\end{array}$$

Gleichung (4) und (5) sowie der Modellausweitung werden die dynamischen Gleichungen wie folgt erhalten $$\begin{array}{ll}\underset{\_}{\dot{x}}& =\left[\begin{array}{c}{x}_2\\ -\frac{E{A}_{\mathit{rope}}}{m_{\mathit{eq}}\mathit{depth}}\left(x_1-\mathit{depth}\right)-\frac{d_{\mathit{rape}}}{m_{\mathit{eq}}}{x}_2-\frac{r_W}{T_W}{x}_4+x_5\\ x_4\\ -\frac{1}{T_W}{x}_4\\ x_6\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ \frac{2\pi r_W{K}_{V,W}}{i_W{V}_{\mathit{mot},W}{T}_W}\\ 0\\ \frac{2\pi K_{V,W}}{i_W{V}_{\mathit{mot},W}{T}_W}\\ 0\\ 0\end{array}\right]u_W+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]\stackrel{⃜}{w}\\ y& =x_1-r_W{x}_3=l_R-r_W{\varphi }_W\end{array}$$

With the states $\underset{\_}{x}={\left[\begin{array}{cccccc}{l}_R& l_R& {\phi }_W& {\dot{\phi }}_W& W& \stackrel{⃛}{\mathrm{w}}\end{array}\right]}^T.$

Equations (4) and (5) and the model expansion, the dynamic equations are obtained as follows $$\begin{array}{ll}\underset{\_}{\dot{x}}& =\left[\begin{array}{c}{x}_2\\ -\frac{e{A}_{\mathit{rope}}}{m_{\mathit{eq}}\mathit{depth}}\left(x_1-\mathit{depth}\right)-\frac{d_{\mathit{rape}}}{m_{\mathit{eq}}}{x}_2-\frac{r_W}{T_W}{x}_4+x_5\\ x_4\\ -\frac{1}{T_W}{x}_4\\ x_6\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ \frac{2\pi r_W{K}_{V.W}}{i_W{V}_{\mathit{mot}.\mathrm{<figure-callout\; id="0"\; label="W\; T\; W"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">W\; </figure-callout>}}{T}_W{}_{}}\\ 0\\ \frac{2\pi K_{V.W}}{i_W{V}_{\mathit{mot}.\mathrm{<figure-callout\; id="0"\; label="W\; T\; W"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">W\; </figure-callout>}}{T}_W{}_{}}\\ 0\\ 0\end{array}\right]u_W+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]\stackrel{⃜}{w}\\ y& =x_1-r_W{x}_3=l_R-r_W{\phi }_W\end{array}$$

To check the flatness property of the proposed model of the system the relative degree must be determined.

Relative degree

The relative degree of output of the system is defined by the following conditions $$\begin{array}{ll}{L}_G{L}_f^{i}H\left(\underset{\_}{x}\right)=0& \forall i=0...r-2\\ L_G{L}_f^{r-1}H\left(\underset{\_}{x}\right)\ne 0& \forall x\in R^n\end{array}$$

sondern auch die Zustände x ₅ = ẅ und $x_6=\stackrel{⃛}{w}$

durch das später erläuterte Verfahren geschätzt und prognostiziert werden. Dies macht die Simulation der internen Dynamik unnötig und es kann ein bahnverfolgendes und die Störgröße abkoppelndes Steuergerät abgeleitet werden.The operator L _f represents the Lie derivative along the vector field f and L _g along the vector field g . With the output y , a relative degree r = 4 is obtained. The relative degree of disturbance is obtained by using (8) with the vector field p instead of g with r _d = 4. Since the order of the system is n = 6, there is internal second order dynamics and y is not a flat output. It can be shown that this internal dynamics is the disturbance variable model. In our case, the internal dynamics consist of a double integrator chain. This means that the internal dynamics are unstable. Thus, solving the internal dynamics through online simulation is impossible. But for the application given here not only the disturbance variable $d=\stackrel{⃜}{w}.$

but also the states x ₅ = ẅ and $x_6=\stackrel{⃛}{w}$

estimated and predicted by the method explained later. This makes the simulation of the internal dynamics unnecessary and it can be derived a track-following and the disturbance decoupling controller.

The web sequence control unit

The track sequence controller decoupling the disturbance may be formulated based on the method of linearization of input / output. $$\begin{array}{ll}{u}_{W.\mathit{lin}}& =\frac{1}{L_G{L}_f^{r-1}H\left(\underset{\_}{x}\right)}\left(-L_f^{r}H\left(\underset{\_}{x}\right)-L_p{L}_f^{r-1}H\left(\underset{\_}{x}\right)\stackrel{⃜}{\mathrm{w}}+y_{\mathit{ref}}^{\left(r\right)}\right)\\ & =\frac{m_{\mathit{eq}}{i}_W{V}_{\mathit{mol}.W}{T}_W\mathit{depth}}{2\mathit{\pi E}{A}_{\mathit{rope}}{r}_W{K}_{V.W}}\cdot \left({\left(-\frac{e{A}_{\mathit{rope}}}{m_{\mathit{eq}}\mathit{depth}}\right)}^2\left(l_R-\mathit{depth}\right)+...\frac{r_W}{m_{\mathit{eq}}{T}_W\mathit{depth}}{\dot{\phi }}_W-\frac{e{A}_{\mathit{rope}}}{m_{\mathit{eq}}\mathit{depth}}W-\stackrel{⃜}{w}+{\stackrel{⃜}{y}}_{\mathit{ref}}\right)\end{array}$$

To stabilize the resulting controlled system, a control term is added. The term (Equation (10)) offsets the error between the reference tracks y _ref and the derivatives of the output y . $$u_{W.\mathit{FB}}=\frac{{\displaystyle \underset{i=0}{\overset{r-1}{\Sigma }}{k}_i\left[L_f^{i}H\left(\underset{\_}{x}\right)-y_{\mathit{ref}}^{\left(i\right)}\right]}}{L_G{L}_f^{r-1}{H}^{*}\left(\underset{\_}{x}\right)}$$

The gains of the feedback values k _i are obtained by the pole assignment method. The control structure is in Fig. 8 illustrated.

The estimation and forecast of the sea state movement

The first part of this section makes a proposal, as the entire movement of the ship / vessel, by measuring with an inertial platform (Initial M easurement U nit (IMU)) can be estimated. The key requirement for this estimate is that all ship-specific information should be used. The second part explains a short-term forecasting problem. Here only the seaward movement of the cranes is forecasted. This reduces the complexity of 6 degrees of freedom to just one without losing required information. As previously desired, the prognosis is also completely independent of a ship model.

Measurement of ship movement

The rigid body of the ship / watercraft has 6 degrees of freedom. With an IMU, the displacement of the ship from the steady state can be measured with high precision. These cost-effective stand-alone motion sensors feature 3 accelerometers for measuring surf, swings, and swell, as well as 3 yaw rate sensors for rolling, pounding, and yawing. To obtain the desired relative position of the ship, a double integration of the acceleration signals and a simple integration of the rotation signals are required. To reduce typical errors such as sensor noise, bias and misalignment of the accelerometers, and to ensure stable integration, the signals can be processed.

If the IMU is not attached to the suspension point of the payload, a simple transformation between the coordinate system of the sensor and the coordinate system of the suspension point of the payload will result in the desired seaward motion.

Forecast of moving the suspension point of the payload

The fact that the movement of the payload suspension point is not completely chaotic, but depends on the dynamics of the ship and the ocean condition, makes it possible to calculate a forecast of its movement. It is even a short-term forecast without any knowledge of the properties of the ship possible.

wobei A_i die Amplitude ist, f_i die Frequenz und ϕ_i die Phase der i-ten Mode ist. Das Ziel der Prognose ist das Schätzen, wie viele Moden für eine präzise Vorhersage der Länge T _Pred erforderlich sind, und das Anpassen der drei Parameter für jede Mode.The main idea of this prediction method is to detect the periodic components of the measured sea state motion and use it to calculate future sea state development. Therefore, the measured wave motion w ( t ) between two times t ₀ and T is decomposed into a set of N sine waves, the so-called modes, and an additional arbitrary term u ( t ). This results in a Seegangsbewegungsmodell, which is described by: $$\begin{array}{cc}w\left(t\right)=\left({\displaystyle \underset{i=1}{\overset{N}{\Sigma }}{A}_i\sin \left(2\pi f_{i}t+{\phi }_i\right)}\right)+\upsilon \left(t\right)& i=\mathrm{1,}....\mathrm{<figure-callout\; id="0"\; label="N\; t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">N\; </figure-callout>}\mathrm{}{t}_{}{}_0\le t\le T\end{array}$$

where A _{i is} the amplitude, f _{i is} the frequency and φ _{i is} the phase of the i th mode. The goal of the prediction is to estimate how many modes are required for a precise prediction of the length T _Pred , and to adjust the three parameters for each mode.

The structure of the forecasting procedure is in Fig. 10 shown. First, a Fast Fourier Transform (FFT) is applied to the measured sea state motion w ( t ). The analyzed length and sampling time of the input signal are chosen so that the maximum frequency of the wave motion can be detected and the desired resolution of the frequencies is achieved. The peaks of the resulting amplitude response over frequency A (f) are then extracted by a peak detector. This results in a first estimate of the amplitudes and frequencies of the mode stored in the respective parameter vectors A _FFT and f _FFT . The mode size N is equal to the number of detected peaks. By taking into account the phase response φ (f), the phases φ _{FFT of} the mode can also be defined. With these parameters being updated online, the model of seaward motion described in (11) can be parameterized. The evaluation of the real measured wave motion data shows the necessity of a constantly updated model (see Fig. 11 ).

Here are the detected peaks of the movement of a ship under rough sea conditions. It is clear that the modes change during the measurement.

In the next step, an observer adjusts the parameter vectors by comparing the measured seaward motion w ( t ) with the modeled seaward motion. This is necessary because the FFT only detects averages of a long period of time, while the observer can take the last changes into account. With these new parameter _vectors , denoted by A _Obs , f _Obs and φ _Obs the prognosis of the seaward movement can be made by reusing (11).

observer

The design of the observer depends on a Seegangsbewegungsmodell, which is described by a set of ordinary differential equations (ODEs, abbreviated to English Ordinary Differential Equations). There are two ways to convert the model (11) to a set of ODEs. On the one hand, the seaward movement can be modeled as a nonlinear system that allows the observer to estimate all the parameters required for the forecast of the seaway. However, due to the requirement to obtain an online forecast, this method is not usable on more modern computers. In contrast, a linear model can be used instead. Here only the frequencies of the mode are not adapted again. However, these are estimated by the FFT anyway with great precision. When choosing the linear method, a Kalman filter can be used. This results in an observer equation shown below. $$\begin{array}{ll}\underset{\_}{\dot{\widehat{x}}}=A\widehat{\underset{\_}{x}}+L\left(w-\widehat{w}\right)& \underset{\_}{\widehat{x}}\left({\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">t</figure-callout>}}_0\right)={\underset{\_}{x}}_0\\ \widehat{w}=C\underset{\_}{\widehat{x}}& {\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">t</figure-callout>}}_0\le t\le T\end{array}$$

The system matrices A and C result from the Seegangsbewegungsmodell described below, while the prognosis results also depend on the proper definition of the correction matrix.

To convert the swell model described in (11), which is suitable for an observer, a single mode can be defined by the ODE $$\begin{array}{ll}\underset{\_}{\dot{x}}=A_i\underset{\_}{x}=\left(\begin{array}{cc}0& 1\\ -{\left(2\pi f_i\right)}^2& 0\end{array}\right)\underset{\_}{x}& \underset{\_}{x}\left({\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">t</figure-callout>}}_0\right)={\underset{\_}{x}}_{0i}=\left(\begin{array}{c}{A}_i\sin \left({\phi }_i\right)\\ 2\pi \mathrm{}{A}_i{f}_i\cos \left({\phi }_i\right)\end{array}\right)\\ w_i=C_i\underset{\_}{x}=\left(\begin{array}{cc}1& 0\end{array}\right)\underset{\_}{x}& i=\mathrm{1,}....N,\end{array}$$

The application of the parameter vectors obtained by the FTT, the summing of all modes and the introduction of an offset state representing the non-periodic term u ( t ) results in the observer model of the sea $$\begin{array}{ll}\underset{\_}{\dot{x}}=A\underset{\_}{x}=\left[\begin{array}{ccccc}{A}_1& 0& \cdots & \cdots & 0\\ 0& A_2& \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & A_{\mathrm{<figure-callout\; id="0"\; label="N"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">N</figure-callout>}}& 0\\ 0& \cdots & \cdots & 0& 0\end{array}\right]\underset{\_}{x}& \underset{\_}{x}\left({\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="imgf0013.png,imgf0014.png"\; state="\{\{state\}\}">t</figure-callout>}}_0\right)={\underset{\_}{x}}_0=\left[\begin{array}{c}{\underset{\_}{x}}_{0.1}\\ {\underset{\_}{x}}_{0.2}\\ ⋮\\ {\underset{\_}{x}}_{0N}\\ 0\end{array}\right]\\ w\left(t\right)=C\underset{\_}{x}=\left[\begin{array}{ccccc}{C}_1& C_2& \cdots & C_N& 1\end{array}\right]\underset{\_}{x},& \end{array}$$

The choice of L matrix elements can be achieved using the filter design of Kalman and Bucy. This requires solving the Riccati equation (solution is P ) and calculating the gain matrix L , as described in (15). $$\begin{array}{l}P{C}^T{R}^{-1}\mathit{CP}-\mathit{AP}-P{A}^T-Q=0\\ L=P{C}^T{R}^{-1}\end{array}$$

Here, the Q used as a design parameter is chosen to be a diagonal matrix that offends fast modes more than slow ones, whereas R equally affects all modes.

können die neuen Parameter berechnet werden durch: $$\begin{array}{lll}{f}_{\mathit{Obs},i}& =f_{\mathit{FFT},i}& \\ {\varphi }_{\mathit{Obs},i}& =\arctan \left(\frac{{\widehat{x}}_{\mathrm{1,}i}\left(T\right)}{{\widehat{x}}_{\mathrm{2,}i}\left(T\right)}\right)-2\pi f_{\mathit{FFT},i}T& \\ A_{\mathit{Obs},i}& =\frac{{\widehat{x}}_{\mathrm{1,}i}\left(T\right)}{\sin \left(2\pi f_{\mathit{FFT},i}T+{\varphi }_{\mathit{Obs},i}\right)}& i=\mathrm{1,}\dots ,N.\end{array}$$

The parameters adapted by the observer can be extracted from their states. Based on the equations of a single fashion $$\begin{array}{l}{\widehat{x}}_{\mathrm{1,}i}\left(t\right)=A_{\mathit{obs}.i}\sin \left(2\pi f_{\mathit{FFT}.i}t+{\phi }_{\mathit{obs}.i}\right)\\ {\widehat{x}}_{\mathrm{2,}i}\left(t\right)=2\pi A_{\mathit{obs}.i}{f}_{\mathit{FFT}.i}\cos \left(2\pi \mathrm{}{f}_{\mathit{FFT}.i}t+{\phi }_{\mathit{obs}.i}\right)\end{array}$$

the new parameters can be calculated by: $$\begin{array}{lll}{f}_{\mathit{obs}.i}& =f_{\mathit{FFT}.i}& \\ {\phi }_{\mathit{obs}.i}& =\arctan \left(\frac{{\widehat{x}}_{\mathrm{1,}i}\left(T\right)}{{\widehat{x}}_{\mathrm{2,}i}\left(T\right)}\right)-2\pi f_{\mathit{FFT}.i}T& \\ A_{\mathit{obs}.i}& =\frac{{\widehat{x}}_{\mathrm{1,}i}\left(T\right)}{\sin \left(2\pi f_{\mathit{FFT}.i}T+{\phi }_{\mathit{obs}.i}\right)}& i=\mathrm{1,}....N,\end{array}$$

forecast

The final part of the forecasting process is the calculation of the forecast itself. Therefore, (11) can be used using the parameter vectors adjusted by the observer, which yields: $$\begin{array}{cc}{w}_{\mathrm{P}\mathit{red}}\left(t\right)=\left({\displaystyle \underset{i=1}{\overset{N}{\Sigma }}{A}_{\mathit{obs}.i}\sin \left(2\pi f_{\mathit{obs}.i}t+{\phi }_{\mathit{obs}.i}\right)}\right)+\upsilon \left(t\right)& i=\mathrm{1,}....N.\mathrm{}T\le t\le T_{\mathrm{P}\mathit{red}}\end{array}$$

als Konstante definiert werden.The progression of the non-periodic term u ( t ) can not be predicted. Since he is equal to the observer's offset state, he should be using $$\begin{array}{cc}\upsilon \left(t\right)=\mathit{const},={\widehat{x}}_{2N+1}\left(T\right)& T\le t\le T_{\mathrm{P}\mathit{red}}\end{array},$$

be defined as a constant.

To give a brief impression of the performance of the seaway forecast, simulation results are presented below. Therefore, real ship IMU signals were used under rough sea conditions to reproduce sea state motion. Fig. 12 shows the predicted and measured seaward movement over time. The forecast interval T _Pred was chosen to be 1 second. For better illustration, the predicted seaward movement was then postponed. Thus, an error-free predicted signal would match the measured signal.

Simulation and measurement results

In Fig. 13 is the simulated sequential behavior of the Seegangsfolgesystems visible. The reference track is generated by a hand lever signal and the crane is subjected to a seaward movement. For this simulation, only a linearizing controller without stabilization was used. With this construction, the excitation of the suspension point movement of the payload, which in the first graph of Fig. 14 is shown to be reduced by a factor of 5. Of the Reason why these vibrations are not completely suppressed, is that the pump / motor system was simulated with a dead time, which is not taken into account in the design of the control unit.

Using the observer and closing the circuit of the control system dramatically improves follow-up behavior. As in Fig. 14 can be seen, the simulated position shift never exceeds ± 3cm.

For the first two simulations, the seaway forecast was switched off. Fig. 15 shows the follow-up behavior of the payload position in open circuit with a seaway forecast in the range of dead time of the actuator (0.2 seconds). It can be seen that good sea balance results are achieved as soon as the linearizing controller is activated, which occurs at 250 seconds. There is a clear improvement in comparing the consequences with and without a seaway forecast.

To improve the simulation results, measurements were made with an experimental setup.

conclusion

The present invention provides an approach to balancing seaward motion in offshore cranes. The dynamic model of the balancing actuator (hydraulically driven winch) and the load hanging on a rope are derived. Based on this model, a web follower is being developed. To compensate for the wave-induced movement of the ship / watercraft, the seaward motion is defined as a time-variant disturbance and is analyzed for decoupling conditions. With a model extension, these conditions are fulfilled and an inversion-based decoupling tax law is formulated. To stabilize the system, an observer is used to reconstruct the unknown state from a force measurement. Furthermore, the compensation can be calculated by forecasting the Seegangsbewegung be improved. A forecasting method is proposed in which no ship / watercraft models or characteristics are required. The simulation and measurement results validate the Seegangsfolgeverfahren.

Claims (15)

1. A crane control with active heave compensation for a crane arranged on a floating body, which includes a hoisting gear for lifting a load hanging on a rope, comprising
   a measuring device, which determines a current heave movement from sensor data,
   a prediction device, which determines the prevailing modes of the heave movement from the data of the measuring device, and creates a heave model using the prevailing modes determined and predicts a future vertical movement of the load suspension point with reference to the current heave movement determined and the model of the heave movement, and
   a path control of the load, which by actuating the hoisting gear of the crane on the basis of the predicted movement of the load suspension point at least partly compensates the movement of the load caused by the sea waves.
2. The crane control according to claim 1, wherein the model of the heave movement used in the prediction device is independent of the properties, and in particular of the dynamics of the floating body.
3. The crane control according to claim 1 or 2, wherein the prediction device determines the prevailing modes of the heave movement r via a frequency analysis from the data of the measuring device.
4. The crane control according to one of the preceding claims, wherein the prediction device continuously parametrizes the model with reference to the data of the measuring device, in particular via an observer, wherein in particular amplitude and phase of the modes are parametrized.
5. The crane control according to one of the preceding claims, wherein the model is updated in the case of a change in the prevailing modes of the sea waves.
6. The crane control according to any of the preceding claims, wherein the path control includes a pilot control which is stabilized on the basis of sensor data.
7. The crane control according to any of the preceding claims, wherein the path control is based on a model of crane, rope and load, which considers a change in the rope length due to an elongation of the rope.
8. The crane control according to any of the preceding claims, wherein the path control is based on a model of crane, rope and load, which considers the dynamics of the hoisting gear and/or of the rope and in particular is based on a physical model of the dynamics of the system of hoisting gear, rope and/or load.
9. The crane control according to claim 7 or 8, wherein a force sensor is provided for measuring the force acting in the rope and/or on the hoisting gear, whose measurement data are included in the path control and by means of which in particular the rope length is determined.
10. The crane control according to any of the preceding claims, wherein the measuring device comprises gyroscopes, acceleration sensors and/or GPS elements, from whose measurement data the current movement of the load suspension point is determined.
11. The crane control according to any of the preceding claims, wherein the sensors of the measuring device are arranged on the crane, in particular on the crane foundation, and wherein the measuring device advantageously determines the movement of the load suspension point with reference to a model of the crane and the relative movement of load suspension point and measurement point.
12. The crane control according to any of the preceding claims, wherein the measuring device only determines the movement of the load suspension point in the vertical.
13. A crane with a crane control according to any of the preceding claims.
14. A method for controlling a crane arranged on a floating body, which includes a hoisting gear for lifting a load hanging on a rope, with the following steps:

    determining the current heave movement from sensor data,

    determining the prevailing modes of the heave movement from the data of the measuring device and determining a model based upon the determined, prevailing modes,

    predicting a future vertical movement of the load suspension point with reference to the current heave movement determined and a model of the heave movement, and

    at least partly compensating the movement of the load caused by the sea waves by actuating the hoisting gear of the crane on the basis of the predicted movement of the load suspension point.
15. The method according to claim 14 by means of a crane control according to any of claims 1 to 12.

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