Improvement of control precision for ship movement using a multidimensional controller

ABSTRACT This paper presents the influence of a first stage inertial element on the operation of a state space controller for a third dimension object. First chapter presents a brief introduction to linear matrix inequalities as one of the methods of controller analysis and synthesis, with examples, and helps to places the below paper in this wide field of science. Chapter two presents the controlled object, which is a model of a very large crude carrier (VLCC) ship, called Blue Lady, along with its mathematical state space model. Chapter three shows state space controller design using numerical methods of linear equalities optimization, where the given values are u longitudinal, v lateral and r rotational velocities. Presented there is also the implementation method of an inertial element to a multidimensional object for space state controller synthesis. Chapter four shows computer simulation results and focuses on showing influence of the proposed design on control error of the multidimensional closed loop system. Finally, chapter five concludes and comments simulation results and shows possible direction of future studies.


Introduction
Within the last several years many new tools have been created for controller analysis and synthesis that allow solving complex control problems. Among them are systems with analytical uncertainty estimation and some classes of non-linear systems which utilize optimization algorithms [1][2][3][4][5]. A possible solution to the problem can be synthesizing a control system that assures minimum or maximum of a target function with its restrictions or confirming that no viable solution exists. Also the problem of multidimensional control, which means controlling more than two output values, has to be considered [6][7][8]. As the above methods grew more popular so did numerical methods to solve them, such as linear matrix inequalities (LMI), a method used in analysis and synthesis of control systems. These are described in detail in [9,10]. In [11], the authors show LMI use in multidimensional systems with a Fuzzy Logic Controller. While in [12] the authors go back to formulating different LMI problems using s-functions. While the authors of [13,14] claim that it is always possible to find a solution, with an acceptable precision, using high performance LMI algorithms. Controller synthesis using LMI leads to effective numerical solutions, backed up by promising computer simulation results as shown in [13] where the chapter about applications of a closed loop system shows results of computer simulations, where the space state controller for a missile, controls its flight path. It is estimated that several dozen new publications about the use of LMI are written each year, for example, in 2017 a total of 41 were registered in IEEE database. However, majority of them describe the use of LMI in land-based applications and only seldom in marine applications. Examples of the very few marine applications include papers [15][16][17] where the authors compare an LMI controller designed using H ∞ with a robust controller for a multidimensional ship model. Also paper [18] shows a stabilized platform with two degrees of freedom (2DOF) used for stabilizing equipment like satellite antennas or cameras on board ships. First controller synthesis using H 2 and H ∞ standards and pole placement restrictions for output interference compensation is presented. Later experimental results are presented of when the platform with a satellite antenna was installed on board a ship and tested by tracking satellite signal at different sea conditions. The author being closely related to marine automation and marine industry decided to focus her work on this area of LMI use. This paper presents a controller for precise ship movement control, among others at low speeds, similar to dynamic positioning systems. Inertial element proposed in this paper is used to reduce control error to the multidimensional control system.

Training ship
State space controller synthesis using LMI was done for a multidimensional model which was a ship model of a very large crude carrier (VLCC) ship called Blue Lady as shown in Figure 1. The model is isomorphous to a real VLCC ship, built with retention of geometric, kinematic and dynamic similarity laws. The Blue Lady ship model is built in 1:24 scale to a real VLCC ship, used for transportation of crude oil. Basic dimensions of the model are length 13.75 m, breadth 2.38 m, maximum speed 1.59 m/s to 3.1 kn.
The simulation model is based on papers of W. Gierusz as shown in [19,20]. It belong to 3DOF class, and takes into account movement on a single plane without including list, trim and draught during Silm Lake trials as described in [21,22].

Mathematical state space model
The controlled object is a multidimensional, non-linear real life object. When modelling ships dynamics for the use of controller synthesis, model linearization around the working point was used [23]. Object identification process included thruster operation, ship's hull construction, set of stationary Kalman filters (used for recreation of u, v and r velocities), power distribution system used for calculating three components of u velocity vector: The controlled object takes into account three input signals: τ x , τ y , τ r and three output signals:û,v,r, where τ x -required force (thrust) on the ships longitudinal axis, τ y -required force (thrust) on the ships lateral axis, τ r -required rotational force. Given signals are considered as components of input signal vectors to blocks in which forces and moments are calculated. Command signals are ng c -main propeller rotation speed, σ crudder blade angle, sstd c -relative thrust of the forward tunnel thruster, sstr c -relative thrust of the aft tunnel thruster, ssod c -relative thrust of the forward rotational tunnel thruster, α dc -rotation angle of the forward  rotational tunnel thruster, ssor c -relative thrust of the aft rotational tunnel thruster, α rc -rotation angle of the aft rotational tunnel thruster. The above signals are connected to a block that is responsible for modelling dynamics of all ship's propulsion and control elements. These are the main propeller, rudder blade, forward and aft tunnel thrusters and forward and aft rotational tunnel thrusters. Figure 2 shows a block diagram of the controlled object with all signals of the Blue Lady ship model. As a result of the identification process numerical values were calculated for the Blue Lady ship model coefficients that were later used in Equations (2) . The presented state space model is a nominal (average) model and its state space equations and output equations are as follows: ⎡ Matrix A, B and C of the controlled object, which is the Blue Lady training ship model, are as shown below:

State space control
During controller synthesis for the controlled object defined with matrices a state space controller K was received, in which output signal z for specific given values u, v, r had a control error e stat . Figure 3(a) shows a simplified block diagram of the control system with an integrating element. Achieving a static accuracy where control error reaches zero is possible by using an ideal integrating element with a transfer function of: where k i is the gain of the integrating element (equal = 1). While examining a multidimensional controlled object it was observed that adding an ideal integrating element does not provide expected results. One of the disadvantages of a control system is the fact that gain can change in a closed loop system when controlled object parameters change. To avoid adjusting gain every time a given value parameter changes, which is common in this system, an additional closed loop with an inertial element was added. This allows to a steady closed loop system static accuracy despite given value parameter changes. However, implementing and integrating element worsens stability of the whole control system. As a result of an LMI controller synthesis for a given value of longitudinal velocity u = 0.1 m/s where velocities equal to v = 0 m/s and rotational r = 0 rad/s , which in Figure 4 is the dashed line. Controlled velocity is marked in Figure 4 with the solid line. Figure 4 shows clearly that using an integrating element significantly worsens the stability of a multidimensional control system. It influences system dynamics and does not fulfil control criteria. A solution which eliminates the above relation but still keeps control error at zero (s = 0) is replacing the integrating element with a first stage inertial element with a very high gain, its transfer function as shown below: where k in is gain and T in is the time constant. Similar assumptions have been made by authors of [24,25].
Step response of a first stage inertial element is shown  Step response of a first stage inertial element.
in Figure 5, the first part of the graph is highlighted to point out how similar it is to an ideal integrating element. This method allows bringing control error down to zero with a first stage inertial element when correct values of k in and T in are selected, not equal zero. Controlled object parameters change depending on many factors like cargo weight or ship's speed. As shown in [26] to avoid adjusting static gain every time controlled object parameters change a different control system structure was used, as shown in Figure 3(b), where state space equations are as follows: State space equations of the first stage inertial element from Figure 3(a) block diagram are as shown below: Values of k in and T in were selected by experimenting and for these values the inertial element matrices have the form of: Block diagram from Figure 6 shows connections between the controlled object, Blue Lady ship model, first stage inertial element and state space controller, described as LMI gain controller. The method described in this paper required to incorporate an inertial element matrix into a multidimensional controlled object matrix. State space equations of the closed loop control system with a multidimensional controlled object with an inertial element are shown below:

Linear matrix inequalities
LMI are a tool for convex optimization used among others for robust controller synthesis [13]. According to Laypunov the necessary and sufficient condition for a linear system to be asymptotically stable is finding a symmetrical and positively defined matrix P = P T , P 0 1 (P is the unknown variable), which after taking into consideration Schur's stability condition has the from of: where matrix A is the controlled object and matrix P is the unknown. Figure 6. Block diagram of a closed loop, multidimensional control system with a state space controller, inertial element and a multidimensional controlled object with special signals described in Figure 2.
Due to the complexity of LMI as a whole, this paper will present only the stages necessary for adding a first stage inertial element to state space controller synthesis for improving system stability. Following controller synthesis was adopted for the multidimensional object of Blue Lady ship model.
The controller synthesis method for a state space regulator for a above ship model is shown in [27]. Based on simulation results controller structure based on a state space controller was chosen, where Specific stages of controller synthesis with a first stage inertial element will be described briefly below based on state space Equation (15). The first stage is to define the pole placement region located in the left half-plane of complex variable plane s in order to specify dynamic parameters of the designed closed loop system. First LMI condition: is fulfilled if and only if there exists a symmetrical positively defined matrix P reg where: • A -however in this specific case instead of A matrix we have to use the form = (A + B u K), • P reg -the unknown, symmetrical positively defined Laypunov matrix P 0, P = P T , • L, M -specific, user defined matrices, where pole placement area is selected based on closed loop system dynamics.
The second stage is H ∞ minimization related with an estimation of scalar value γ ∞ which is the upper limit of the H ∞ standard. When determining the smallest value of γ ∞ for the given pole placement region, of the closed loop system, we assume that the obtained scalar value γ ∞ is constant. Scalar value γ ∞ can be calculated based on H ∞ minimization or it can be assumed to be a constant limit. Second LMI condition: The above equation is generic and in this specific case, taking into account Equation (15), it has the below form: The third stage is H 2 standard minimization. One of the ways of obtaining that is looking for γ 2 for the value obtained in the second stage above: (20) The third LMI condition: This means that LMI conditions formulated for H ∞ by Equation (19), for H 2 by Equation (21) and for chosen pole placement ( Equation (18)) are fulfilled if there exists a symmetrical positively determined Laypunov matrix. The final form of state space controller gain matrix K, that stabilizes the whole control system and minimizes H ∞ and H 2 standards. Because matrix P is positively determined there exists its converse matrix Y and so space state controller matrix K can be described as Multitasking of LMI used for space state controller synthesis, for a multidimensional object, means that several design criteria are met at the same time, as explained in [9,13]. In this paper, the closed loop system state space controller was determined by three main stages as detailed in [27]. Based on the first stage inertial element implementation values of state space matrix K are
Two among them, Yalmip and SeDuMi, are especially useful in calculations using LMIs. Both of them are based on a "public domain" license and can be freely downloaded from:  For the above case study, computer simulations were performed using Matlab software with YALMIP and SeDuMi. Table 1 shows a comparison of mean values for deviations from given values. In order to determine the quality of control the following formula was used: where n is the number of measurements; x is the deviation value, in percentage, of measured parameter x ∈ u, v, r; x given is the reference signal value (given value) and x control is the output signal value received from the system. Computer simulations were made for the below three types of ship movement: • Manoeuver 1 Ahead movement with a given longitudinal velocity u = 0.1 m/s, with manoeuver duration of 3000 s. Given values of lateral and rotational velocities were set to v = 0 m/s, r = 0 deq/s. This type of ship maoneuver, at low velocities, is very common when navigating narrow passages like channels, rivers or entering harbours. Simulation results with the use of an integrating element are shown in Figure 7(a) while simulation results with the first stage inertial element are presented in Figure 7(b).  • Manoeuver 2 Ahead and sideways movement with given longitudinal u = 0.1 m/s and lateral v = 0.05 m/s velocities and rotational velocity equal to r = 0 deq/s velocities with manoeuver duration of 3000 s. This type of ship manoeuver, at low velocities, is a typical approach to berth which is located parallel to ships course. Simulation results with the use of an integrating element are shown in Figure 8(a) while simulation results with the first stage inertial element are presented in Figure 8(b).
• Manoeuver 3 Ahead and sideways movement with rotation, with given longitudinal u = 0.1 m/s, lateral v = 0.05 m/s and rotational r = 0.03 deq/s velocities with manoeuver duration of 1200 s. This type of ship manoeuver, at low velocities, is a typical approach to berth which is located perpendicular to ships course. Simulation results without the use of an inertial element are shown in Figure 9(a) while simulation results with the first stage inertial element are presented in Figure 9(b).
The proposed method proves to be a good tool for lowering control error down to zero in state space controller synthesis for a multidimensional controlled object. Computer simulations show that a first stage inertial element gives the benefits of an ideal integrating element without its negative influence on system stability.

Conclusions
This paper presented the benefits of implementing a first stage inertial element to a closed loop control system with a state space controller using LMI. Inertial element implementation steps were shown, followed by computer simulation results for precision ship movement control of Blue Lady VLCC ship model. The proposed method proves to be a good tool for lowering control error down to zero in state space controller synthesis for a multidimensional controlled object. Conducted computer simulations prove the validity of using a first stage inertial element. As an example deviations of controlled values from given values for longitudinal velocity u = 0.1 m/s and lateral velocity v = 0.1 m/s, while using a first stage inertial element, are several orders of magnitude lower than while using an ideal integrating element. Additionally using the inertial element doesnot have the negative impact on system stability as the integrating element does. Control error in computer simulations without the inertial element was out of range. While after including the inertial element it was within the range of 0.0001% to 0.0015%. Reducing control error had a direct impact on ship movement control precision. Thanks to the possibility of cooperation with the Foundation for Safety of Navigation and Environment Protection future will studies should include testing the validity of using a first stage inertial element in real life Silm Lake trials in Ilawa [32]. First stage of real life testing will be done on open water due to the safety of the Blue Lady ship model and its operators. If results of the first stage will prove to be satisfactory, the testing will be done while actually navigating narrow passages and berthing both parallel and perpendicular to ships course. The Silm Lake training centre has the infrastructure to train various types of ship manoeuvers including narrow passages and various harbour and berth configurations.

Note
1. The problem of finding a symmetrical, positively defined matrix P is often called the feasibility problem.