Control method based on state feedback for dynamically positioning a ship, and configured system

Abstract

The invention relates to a method in which a Q-filter LQ control law employing linear-quadratic (abbreviated LQ) control and a Q-filter is defined, the LQ control having been synthesized on the basis of a synthesis model of the ship parallel to the ship augmented by three integrators, the parameters of the synthesis model having been determined by parametric identification, the LQ control implementing a state representation with a state-feedback calculation of an augmented state vector X aug and a state-feedback gain K c , the coefficients of the LQ control having been determined by implementing an LQA or LQB adjustment methodology, and in which the position of the ship is controlled depending on the Q-filter LQ control law, the Q-filter possessing gains defined so as to create a notch at a defined frequency ƒ o in the curves of the singular values of the sensitivity transfer function complementary to the Q-filter LQ control law.

Description

[0001]DESCRIPTION Title: State Feedback Control Method for Dynamic Ship Positioning and Adapted System Field of Invention The present invention relates to the field of ship navigation control methods, particularly for dynamic ship positioning systems. More specifically, the invention relates to a state feedback control method for a dynamic ship positioning system. A system adapted to implement the method completes the invention. Prior Art A dynamic positioning system allows a ship to follow or maintain a course and position within a local geographic reference frame when the ship is moving at near-zero speed, and thus to counteract some of the disturbances acting on the ship, in particular counteracting currents.The dynamic positioning system uses information received from a navigation center, in particular a vector ^ with ^^ = [^ ^ ^] , ^, ^ being the coordinates of the ship in a local geographical frame, ^ the heading, and a vector ^ with ^^ = [^ ^ ^] , ^, ^ being the linear speeds of the ship in its own frame along the axes ^, ^ and ^ the rotation speed of the ship around its z-axis, i.e. yaw rate, according to the scheme in Figure 1. From the measurements corresponding to the vectors ^ and ^ and a setpoint vector ^. ∗The dynamic positioning system calculates, according to a control law, a vector of generalized forces (forces and torques) based on the ship's various degrees of freedom. This vector is then fed into a control system that generates commands sent to each of the ship's actuators, as shown in Figure 2. Dynamic ship positioning systems have existed since the 1960s and are of the closed-loop type. The control algorithm used in practice is still very often a multivariable and non-linear PID (Proportional, Integral, Derivative) algorithm, sometimes with the addition of a forecasting term to compensate for wind forces. The equation for this type of PID can be found in the book "Marine Craft Hydrodynamics and Control" by Thor I. Fossen (Wiley 2011), p. 394. The role of dynamic positioning systems is to compensate for disturbances acting on the ship at very low frequencies, i.e., currents.However, the disturbances induced by swell (in the context of this invention, the term "swell" also encompasses the term "wave"), which act in intermediate frequency ranges, must not be compensated and therefore must not enter the loop. This usually leads to the introduction of filters into the control law based, as explained in the same work (p. 395), on a swell observer. This observer can, in particular, be a passive nonlinear observer. The nonlinear multivariable PID has twenty-seven low-level parameters, and the passive nonlinear observer also contains a significant number.This multitude of low-level parameters presents a significant challenge for the engineer tasked with commissioning a dynamic positioning system based on this technology. This forces them to perform numerous time-consuming trial-and-error tests on-site to find a compromise between performance and the robustness of the control law, without any guarantee of achieving optimal performance. Furthermore, it is necessary to synthesize the control law from a model of the ship at sea, i.e., in its environment—also known as a synthetic model. However, a difficulty in synthesizing the control law of a dynamic positioning system lies in the inherent non-linearity of the ship model at sea, which is linked, in particular, to the change of reference frame between the ship's frame of reference and the local geographic frame of reference. In the academic literature, numerous types of control laws have been proposed to address this problem, including, among others...autres :-LQ (“Linear Quadratic”) linearized around an operating point, for example in “LQ optimal robust multivariable PID control for dynamic positioning of ships with norm bounded parametric uncertainties” by F. Gopmanda, A. Gohs, A. Kuman in Ocean Engineering, vol. 226(4), 2022 -LQG with a model linearized around an operating point, - Linearizing control (“non-linear feedback”), - Control by “backstepping”, etc… In order to circumvent this problem related to the non-linearity of the ship model in its environment, an advantageous approach is to synthesize the control law in a frame fixed with respect to the ship's axes and which uses a coordinate system parallel to the ship (vessel parallel frame coordinate in the book entitled “Marine Craft Hydrodynamics and Control” by Thor I. Fossen (2011)).This parallel model used for synthesis then exhibits only minor nonlinearities (at low speeds), thus allowing the full power and maturity of linear control to be used to establish the control law. This successful approach was notably employed in the article "Robust Dynamic Positioning of offshore vessels using -mu synthesis modeling, design and practice" by V. Hassani, AJ Sorensen, AM Pascoal, and M. Athans in Ocean Engineering, vol. 129, 2017, where an H∞-type control law with µ-synthesis is used. Unfortunately, the main drawback of the developments cited in this article remains the applicability and implementation of the control law on-site, given the difficulty of tuning H∞-type control laws. The following document is also known: an article by SOMAN ROHAN S ETAL: "MIMO PID compensation for dynamic positioning of a ship", INDIAN CONTROL CONFERENCE (ICC).IEEE, January 4, 2018 (2018-01-04), pages 137-142, XP033328006, DOI: 10.1109 / INDIANCC.2018.8307967 (Chapters II, III; Figures 1, 4, 5) and a work by Ph. De Larminat, "Applied Automation, Second Edition" (Hermès 2009). In the article by SOMAN ROHAN S ET AL, a structure is implemented in which a "feedforward gain H" is introduced into a "MIMO PID control" and which is "designed using LMI-based H^ method" and to which a set of filters must be associated. This filter set includes a wave filter to make the controller insensitive to wave disturbances. This wave filter is a notch filter (see Chapter III-C and Figure 4 of the article) and must be placed at the output of the sensors (see Figure 4 of D1). This filter set must also include additional filters: a low-pass filter ^e and a high-pass filter ^y (see Figure 6 of the article).This article therefore teaches how to implement a wave filter, which is a band-stopper, and which is placed at the output of the sensors. Summary of the invention: In the invention described in this document, a control law is proposed that reduces to the strict minimum the number of adjustment parameters of the control law of the dynamic positioner, once the parameters of the synthesis model are available. This synthesis model corresponds to a linear part derived from the model of the ship at sea, fixed with respect to the ship's axes and using a coordinate system parallel to the ship, said synthesis model being augmented by the addition of three integrators, i.e., the addition of three additional state variables to the synthesis model. On this synthesis model, a linear-quadratic (LQ) control is established, comprising twenty-seven low-level parameters determined from a methodology known as LQA or LQB, as described in the book "Applied Automation" by Ph.de Larminat (Hermès 2009), making available to the engineer in charge of synthesizing the linear quadratic (LQ) control only one or at most two high-level parameters for tuning, which greatly facilitates the implementation of the control law on site. Advantageously, in order to complement this linear quadratic (LQ) control, a device designed to make it immune to wave disturbances is added to the linear quadratic (LQ) control: This is a filter, called a Q-filter, resulting from a Youla-Kucera parameterization, which allows modification of the complementary sensitivity transfer functions of the closed loop of the linear quadratic (LQ) control by introducing a notch without changing the location of the poles of said closed loop induced by the linear quadratic (LQ) control, thus preserving its robustness.The use of the Youla-Kucera parameterization as such is known to obtain the rejection of narrowband disturbances in control laws, of variable but known frequencies: one can for example consult in the very different field of acoustic active control, the patent EP2436003 (B1) whose inventor is Mr. B. Vau (application filed in 2009), or in the field of submarine control the article "Adaptive digital disturbance rejection controller design for underwater thermal vehicles" whose authors are G. Wang, Y. Yang, S. Wang in Journal of marine science and engineering, MDPI, vol.9(4), 2021.The Youla-Kucera parameterization has also been extensively used for rejecting narrow-band disturbances of unknown frequency in the context of adaptive control (see, for example, "Adaptive rejection of narrow-band disturbances in the presence of plant uncertainties - Adaptive Youla-Kucera approach," Automatica, 2021, by B. Vau and ID Landau). However, in the typical use described in these three documents, the role of the Q-filter is to ensure the rejection of narrow-band disturbances, i.e., to create a notch in the gain curve of the direct sensitivity transfer function of the closed loop. Here, within the dynamic positioning system, the use of the Q-filter is dual, as it aims to make the linear quadratic (LQ) control insensitive to narrow-band disturbances of a specific frequency, namely wave disturbances in this case.The role of the Q-filter is therefore to create a notch in the complementary sensitivity transfer function at a selectable / variable frequency. Such use of a Q-filter in this dual framework, i.e. notch in the complementary sensitivity transfer function, appears original and brings additional advantages to the implementation of the proposed linear quadratic LQ control which in itself reduces to the bare minimum the number of adjustment parameters of the dynamic positioner, once the parameters of the synthesis model are available.Preferably, in a preliminary synthesis phase of the control law comprising the linear quadratic (LQ) control and the Q-filter (also referred to as the "LQ-Q-filter control law" in this application), the Q-filter gains are stored and tabulated for a range of frequencies and possibly for several notch widths and / or depths to allow their subsequent direct use based on the required rejection frequency. The synthesis / determination of these Q-filter gains, besides the notch frequency, depends only on two high-level parameters, determining the notch width and depth. Depending on the dominant wave frequency at a given time and, possibly, the variability of this frequency and / or the wave amplitude, the Q-filter gains appropriate to the current wave frequency are selected and included in the LQ-Q-filter control law used in real time.For example, a user of the dynamic positioning system, such as the ship's pilot, can specify a sea state, and a device or process converts this information into a dominant frequency, allowing the selection of the corresponding / appropriate Q-filter gains for the system. In any case, once the parameters of the synthesis model have been determined by parametric identification, the synthesis of the LQ to Q-filter control law of the dynamic positioning system relies at most on only four high-level parameters: - one or two parameters for the linear quadratic LQ control, - two parameters to adjust / select the width and depth of the notch in the curves of the singular values ​​of the complementary sensitivity transfer function for the Q-filter, which greatly facilitates the on-site implementation of said LQ to Q-filter control law.This synthesis of the LQ control law with a Q-filter consists of determining the coefficients of the linear quadratic LQ control, on the one hand, and determining the gains of the Q-filter, on the other. These coefficients of the linear quadratic LQ control and these gains of the Q-filter can advantageously be stored for subsequent use in the real-time LQ-Q-filter-based controller of the dynamic positioning control calculation device implementing the method of the invention. In an advanced mode, the dynamic positioning system or method further includes a device or means for estimating the dominant frequency of disturbances (e.g., waves), allowing for automatic selection of the Q-filter gains. This device or these means may consist of a recursive, online algorithm for identifying the coefficients of a generalized AR (AutoRegressive) type filter.In one embodiment, a specific measuring device that automatically measures the sea state in real time provides useful information, in particular the dominant wave frequency, e.g., via a heave sensor, and possibly a standard deviation or equivalent of the dominant frequency measurement. Advantageously, the dynamic positioning calculation device includes means for selecting, from a previously stored / memorized table of Q-filter gains, the Q-filter gains adapted to the current sea state, the table storing / memorizing a set of Q-filter gains for a set of possible dominant wave frequencies. Generally, the invention relates to a state-feedback control method for a ship's propulsion means for its dynamic positioning as described.More specifically, the invention relates to a state feedback control method for the propulsion means of a ship for its dynamic positioning, in which: - a Q-filter LQ control law is defined comprising a linear quadratic LQ control and a Q-filter, the Q-filter being derived from a Youla-Kucera parameterization, the linear quadratic LQ control having been synthesized on the basis of a synthesis model of the ship in a frame fixed with respect to the axes of the ship and using a coordinate system parallel to the ship, said synthesis model comprising parameters and being augmented with the addition of three integrators in its parameters, the parameters of the synthesis model having been determined by parametric identification, the linear quadratic LQ control implementing a state representation with a state feedback calculation of an augmented state vector ^. ^^^ and a gain ^ ^state feedback, the coefficients of the linear quadratic control LQ having been determined by implementing an LQA or LQB type tuning methodology, the LQA type tuning methodology having only two tuning parameters (^^ , ^^) and the LQB type tuning methodology having only one tuning parameter (^ ^ ), the LQ control law with a Q-filter having determinable singular value curves of its complementary sensitivity transfer function, - the ship's positioning is controlled according to said LQ control law with a Q-filter, in which the Q-filter has determined gains to create a notch at a determined frequency ^ ^in the curves of the singular values ​​of the complementary sensitivity transfer function of said LQ control law with Q-filter. Other advantageous features of the method according to the invention, taken individually or in all technically possible combinations, are as follows: - the vessel is a surface vessel, - the parallel synthesis model of the vessel is augmented by the addition of three integrators and constitutes the augmented parallel synthesis model of the vessel, - the identification of the parameters is carried out with an experimental database concerning the vessel in its maritime environment, - the state feedback control method of the propulsion means of a vessel for its dynamic positioning implements a programmable data processing device comprising a memory, in particular for data storage, - the programmable data processing device is a programmable computer,- The programmable data processing device is a computer or microcomputer, - The method implements a memory, - The memory is configured to store at least the Q-filter gains, - The memory is configured to store at least the coefficients of the linear quadratic control LQ and the Q-filter gains, - The Q-filter gains are determined by adjusting the notch frequency using a tuning parameter that is a notch frequency, - The determined frequency ^^ is the dominant frequency of a swell to which the ship is subjected, - The Q-filter gains are determined by further adjusting the depth and width of the notch, - The Q-filter gains are determined by adjusting the depth and width of the notch using two tuning parameters ^, ^ and ^ ^Additional features: - A set of Q-filter gains for a set of notch frequencies is stored in memory, and during positioning commands, the stored Q-filter gains corresponding to a notch frequency substantially equal to the dominant wave frequency to which the ship is subjected are used. - In the case where none of the notch frequencies in the set of stored Q-filter gains is substantially equal to the dominant wave frequency to which the ship is subjected, then the Q-filter gains are calculated by interpolation to the dominant wave frequency from the stored Q-filter gains. - A notch frequency is considered substantially equal to the dominant wave frequency when the dominant wave frequency is within a range of values ​​of + / - 5% of the stored notch frequency.- A dominant wave frequency estimator is implemented, estimating in real time the dominant wave frequency to which the ship is subjected by a recursive identification calculation of the coefficients of an ARMA-type filter on signals resulting from an estimation by a state observer of additive disturbances, or on a heaving signal from a heaving detector, also implemented; - The coefficients of the linear quadratic LQ control, having been determined, are stored in memory, and the stored coefficients of the linear quadratic LQ control are used during positioning control; - The ship's positioning is further controlled as a function of the forces and torques of a wind acting on the ship; - The ship's positioning is controlled by a command U produced by the implementation of the LQ control law with Q-filter and as a function of setpoints.We calculate the command U with: ^ ^ ∗ ^^ is a setpoint vector, I is an identity matrix, Kc is a state feedback gain, and the transfer matrices M, N, and Q result from the Youla-Kucera parameterization—in the case where the ship's positioning is further controlled as a function of wind forces and torques acting on the ship, and the command U is calculated ^ ^^^^is derived from wind speed measurements by at least one sensor on the ship, - the command U produced by the implementation of the LQ control law with Q-filter acts on effectors of the ship, - the parametric identification to obtain the parameters of the synthesis model parallel to the augmented ship is carried out by the programmable data processing device which implements the state feedback control method of the ship's propulsion means for its dynamic positioning, - the synthesis of the linear quadratic LQ control (i.e. determination of the coefficients of the linear quadratic LQ control) and the determination of the Q-filter gains are carried out by the programmable data processing device which implements the state feedback control method of the ship's propulsion means for its dynamic positioning,- The programmable data processing unit that implements the state-feedback control method for the ship's propulsion systems for its dynamic positioning is further configured to perform the synthesis of the linear quadratic (LQ) control and determine the Q-filter gains; - the parametric identification to obtain the parameters of the synthesis model parallel to the augmented ship is performed by equipment separate from the programmable data processing unit that implements the state-feedback control method for the ship's propulsion systems for its dynamic positioning; - the synthesis of the linear quadratic (LQ) control and the determination of the Q-filter gains are performed by equipment separate from the programmable data processing unit that implements the state-feedback control method for the ship's propulsion systems for its dynamic positioning. The invention also relates, in a general way,a dynamic positioning system for a ship by state feedback control of the ship's propulsion means, as described and / or comprising means for executing the described method. More specifically, the invention relates to a dynamic positioning system for a ship by state feedback control of the ship's propulsion means, the system being configured to control the ship's positioning by a Q-filter linear-quadratic (LQ) control law comprising a linear-quadratic (LQ) control and a Q-filter, the linear-quadratic (LQ) control having been synthesized on the basis of a synthesis model of the ship in a frame fixed with respect to the ship's axes and using a coordinate system parallel to the ship, said synthesis model comprising parameters and being augmented by the addition of three integrators to its parameters, the parameters of the synthesis model having been determined by parametric identification.the linearquadratic control LQ implementing a state representation with a state feedback calculation of an augmented state vector ^, ^^^ and a gain ^ ^ state feedback, the coefficients of the linear quadratic control LQ having been determined by implementing an LQA or LQB type tuning methodology, the LQA type tuning methodology having only two tuning parameters (^^ , ^^) and the LQB type tuning methodology having only one tuning parameter (^ ^ ), the LQ control law with a Q-filter having determinable singular value curves of its complementary sensitivity transfer function, and the Q-filter being derived from a Youla-Kucera parameterization and having determined gains to create a notch at a determined frequency ^ ^in the curves of the singular values ​​of the transfer function of the complementary sensitivity of said LQ control law with Q-filter. Other advantageous features of the system according to the invention, taken individually or in all technically possible combinations, are as follows: - the determined frequency ^^ is the dominant frequency of a swell to which the ship is subjected, - the synthesis model is a synthesis model parallel to the ship, which is further augmented by adding three integrators to form an augmented synthesis model parallel to the ship, - the parameters of the augmented synthesis model parallel to the ship are obtained by parametric identification. The method of the invention can be implemented by a computer comprising a computer program. The computer program then comprises program code which, when said program code is executed in the computer, enables the execution of the method.The computer program is advantageously in the form of a computer program product comprising a machine-readable medium containing said computer program. Brief description of the figures In the attached figures: [Fig. 1] represents a diagram of the ship in its environment, [Fig. 2] represents a general diagram of a dynamic positioning means for a ship comprising a dynamic positioning system and a dispatching system for generating commands for the ship's actuators, [Fig. 3] represents an implementation of linear quadratic control (LQ) in the case where an anticipatory action of the wind forces and torques acting on the ship is also implemented, [Fig. 4] represents a closed loop with the signals for expressing the input sensitivity functions, [Fig.[5] represents an example of singular value curves of the direct input sensitivity transfer function, in the figures "singular values" meaning "singular values" and "frequency" meaning "frequency", [Fig. 6] represents an example of singular value curves of the complementary input sensitivity transfer function, [Fig. 7] represents the overall structure of the LQ control law with Q-filter with the Q-filter nested within the linear quadratic LQ control and forming a global controller in the case where an anticipatory action of the wind forces and torques acting on the ship is also implemented, [Fig. 8] represents an overall structure of the discretized LQ control law with Q-filter gains tabulation in the case where an anticipatory action of the wind forces and torques acting on the ship is also implemented, [Fig.[Fig. 9] represents singular value curves of the direct sensitivity transfer function without Q-filter (solid line) and with Q-filter (dashed line). [Fig. 10] represents singular value curves of the complementary sensitivity transfer function without Q-filter (solid line) and with Q-filter (dashed line). [Fig. 11] represents an overall diagram of an example of a wave dominant frequency estimation device, and [Fig. 12] represents the overall structure of the control system of the invention corresponding to the concatenation of Figures 3 and 8. Detailed Description The following description, with reference to the accompanying drawings, given by way of non-limiting examples, will clearly explain what the invention consists of and how it can be implemented. The basic model of a ship at sea used to synthesize the ship's dynamic positioning system is presented in the book entitled "Marine Craft Hydrodynamics and Control" by Thor I.Fossen (Wiley 2011) is as follows: Où : ^ is the vector of current forces^ is the vector of forces / torques of the actuators^ ^^^^ : the vector of torques / perturbation forces due to wind ^ ^^^^ : the vector of disturbance forces / torques due to swell. In what follows, for the synthesis of the linear quadratic control LQ, i.e., determination of the coefficients of the linear quadratic control LQ, we consider that ^^^^^ = 0, ^^^^^ = 0, ^ = 0 such that this model becomes We observe that in (1) the matrix ^ introduces an intrinsic non-linearity to the change of reference frame between that of the ship and the local geographic reference frame. This non-linearity is a major source of difficulty for the synthesis of the control law. In what follows, we prefer to express the model of the ship at sea using a state vector ^, a matrix ^, and a vector field ^(^). We recall that in a state-space representation (in automatic control, a state-space representation allows modeling a dynamic system using state variables), a state vector contains the minimal set of state variables necessary and sufficient to determine the future evolution of a system, given the equations describing the system's operation and its inputs. The state variables are the quantities that constitute the state of the system. By setting where: N and E and the local geographic reference point, ^ the cape and U1, U2, U3 are the commands. On a donc With ^ ^^ ^ ^ ^^ 0 0 ^ ^ + ^ ^ ^^ ^ ^ ^ æ ^ ^ 0 ^ ^ ^^ ^^^ ^ ^ + ^ ^ ö æ ö ^ (^) = it ^ ^ ^^ ^ cos(^ ) ^ − sin (^÷ ^ = it 0 ^^^ ^^^ ÷ (4) ç ^ ^ ^)^^ ÷ ç 0 0 0 ÷ sin(^^) ^^ + cos (^^)^^ 0 0 0è ^ ^ ø è 0 0 0 øMany dynamic positioner control laws are established directly on this model, whether for non-linear control laws (linearization by feedback, for example), or for linear control laws established around an operating point and whose gains are interpolated as this operating point varies. Within the framework of the present invention, in order to circumvent the difficulty related to the non-linearity mentioned above, it is proposed to use a model of the ship at sea, called a parallel model, which uses a vessel parallel coordinate system as defined in the book entitled "Marine Craft Hydrodynamics and Control" by Thor I. Fossen (pp. 173, 174), by introducing the vector ^ ^ such as This leads to the introduction of a vector ^^ with And In We have ^ ^ ^ ̇ ^ ^ ^ ̇ ^ ^ = ^ ^ ^ 0 1 ^ ^^ ^^ ^ + ^ −1 0^ ^ ^^^ ^ ^^ (7) we thus obtain a “quasi-linear” model, whose non-linearities only affect ^^̇^ and ^^̇^ and only occur if the yaw rate, i.e. “yaw-rate”, ^^^ is not zero. Furthermore, in order to also ensure the rejection of static disturbances, the parallel model above is augmented with the addition of three integrators, which leads to the introduction of 3 additional state variables ^^^, ^^^, ^^^ into the model, such as The technical effect of adding three integrators, and therefore three additional state variables ^^^, ^^^, ^^^, in the model, is to ensure the rejection of static disturbances. Finally, the proposed control law is based on linear-quadratic (LQ) control, which is synthesized based on the augmented parallel model. suivant : ö ÷ ÷ ÷ ÷ ÷ ø(9) Linear quadratic control (LQ) is a state-feedback control law, that is, it can be written as ^ = −^^ ^^^^ (10) With Kc the state-feedback gain and ^^ = ^^ ^^ ^ ^^^ ^ ^ ^ (11) and where ^ ^ is the Riccati matrix solution This law minimizes an infinite horizon ^ ^ and ^ ^ being weighting matrices. In the present problem, given the size of the state vector ^ ^^^and the input vector ^, since the matrix ^^ is of size (9*9) and ^^ of size (3*3), the linear quadratic control LQ has 90 tuning parameters, which makes it very difficult to arbitrate the performance / robustness trade-off inherent in any control law, and is a source of difficulty for the implementation of this control on site. Therefore, within the scope of the invention, in order to determine the coefficients of the linear quadratic control LQ, it is proposed to use the LQA or LQB tuning methodologies proposed in the book "Applied Automation" by Philippe de Larminat, which allow the coefficients of the matrix ^^ to be determined by means of one (or at most two) high-level parameters, which are then a small number of tuning parameters. In the case of the LQA methodology, there are two tuning parameters: - The tuning parameter ^^ which controls the dynamics of the linear quadratic control LQ (the order of magnitude of ^ ^(being that of a frequency) - The tuning parameter ^^, which may allow for strengthening the stability margins. In the case of the LQB methodology, there is only one tuning parameter: - the tuning parameter ^^ which controls the dynamics of the linear quadratic LQ control. These two tuning methodologies, known as LQA and LQB type, are presented in the book by Ph. De Larminat "Applied Automation, Second Edition" (Hermès 2009) in which the author proposed tuning methods for determining the tuning parameters of a linear quadratic LQ control. These methodologies have the advantage of simplifying the determination of the coefficients of the matrix ^ ^ by means of one (or at most two) high-level parameters. Thus, on page 236 of Philippe de Larminat's book "Applied Automation, second edition," the LQA method described is such that the state feedback gain results from the minimization of a criterion. ^ ^^^^ ^^^^^^ + ^^ ^ ^^^ which corresponds to equation (13) above, where ^ ^ is a weighting or penalizing matrix of the state vector whose coefficients are determined by a high-level tuning parameter ^ ^ such as where ^ = 1 / ^^ and where ^(^) is to be ordered. A refined form uses a form factor ^ ^ where ^ ^ and ^ ^ are redefined, such that ^^ ≔ ^^ / ^^ and ^^ ≔ ^^^^ (ibid., p. 241). Furthermore, in the LQA method, ^^ is chosen arbitrarily (often diagonally). The LQB approach by the same author (ibid., p. 243) consists of determining the coefficients of ^^ that minimize a criterion ^ = ∫^^ ^ ^ ^^^ ^^^^ + ^ ^ ^ ^ ^^^ where (specific to the LQB method) there is only one high-level ^ ^ such as With ^^ = 1 / ^^. We can therefore see that LQB can be considered the dual of the LQA method, the common point of these two methods being the use of exponential grammians to determine the respective ^* weighting matrices of the LQ criterion. The instructions on ^ and ^ denoted r and ^* which come from a trajectory generator can be grouped in a vector ^ ∗ such that ^ ∗^ = [ ∗^ ^ ∗^] We can define a vector of ^ ^ ∗ such and components, in this case: and we then define a vector Thus, the quadratic LQ control with setpoints taken into account will be and its implementation is represented in Figure 3 (in this figure 3 the narrow vertical rectangle towards which the arrows of the elements represented on the left of said rectangle converge and from which ^^^^ − ^^ ∗ ^^, symbolizes the operations, additions and subtractions, performed on the elements represented on the left). In a particular embodiment, the control can be equipped with a feedforward action of the wind forces and torques acting on the ship; this action is denoted ^^^^^ and acts additively on the control, the latter then being such that (14 bis) and ^ ^^^^is calculated from wind speed measurements in different directions obtained by sensors located on the ship. During the synthesis of the linear quadratic control (LQ), starting from the augmented parallel model whose parameters were obtained by parametric identification, it is possible to determine the closed-loop sensitivity transfer functions. Recall that parametric identification is a procedure consisting of obtaining experimental data on the inputs and outputs of a system when it is subjected to excitations, and then using this data to obtain parameter estimates for the model through an optimization algorithm. In particular, it is possible to explicitly define the direct input sensitivity transfer function ^^(^), which corresponds to the transfer matrix between an additive perturbation r on the control and the signal ^ ^We can also explicitly define the complementary sensitivity transfer function ^ ^ (^), which is the transfer matrix between the additive disturbance r and the signal ^^ as shown in Figure 4 (note that Figure 4 is a generic figure intended to present the signals considered in the description of the sensitivity transfer matrices or functions). It is then possible to plot the curves of the singular values ​​of the transfer matrices corresponding to ^^, and ^^, as a function of frequency, examples of which are given in Figures 5 and 6, with the frequency in rad / s and the singular values ​​in dB. We note that the singular values ​​of ^ ^The curves tend towards 0 as the frequency approaches 0, which is the effect of the integrators introduced in the augmented model. These curves therefore show that static and very low-frequency disturbances, such as ocean currents, will be rejected by the linear quadratic LQ control. On the other hand, the complementary sensitivity transfer function has a low-pass character. It shows that pseudo-periodic disturbances due to swell, particularly swell that is generally between 0.5 rad / s and 2 rad / s, stress the actuators in frequency bands that are not far from the cutoff frequency of the complementary sensitivity transfer function. However, it turns out that these actuators, on a standard ship, do not have the necessary amplitude to effectively counteract this type of disturbance. For this reason, it is not advisable to attempt to reject this type of disturbance with the dynamic positioning system.Therefore, in the design of the control law for the dynamic positioning system, it is proposed to make it insensitive to these disturbances in a medium frequency range, for example between 0.5 rad / s and 2 rad / s. To achieve this, it is ensured that the control law proposed within the framework of the invention is capable of creating a notch around the dominant wave frequency in the curves of the singular values ​​of the complementary sensitivity transfer function. To this end, an algorithm implementing a Q-filter is added to the basic algorithm of the dynamic positioning device, which implements the linear quadratic control (LQ) described above.The resulting system, whose control law is schematically represented in figures 7 and 8, preserves the essential performance and robustness properties of the linear quadratic LQ control as such, while creating notches in the curves of the singular values ​​of the complementary sensitivity transfer function, notches whose frequency ^. ^can be adjusted / adapted to the sea state, in this case to the dominant wave frequency. Figure 12 represents the overall structure of the control system of the invention and corresponds to the concatenation of Figures 3 and 8. In another embodiment, the overall structure of the control system can also correspond to the concatenation of Figures 3 and 7. This operation constitutes a specific Youla-Kucera parameterization which is now described. The augmented system, denoted ^^^^, of the LQ control law with Q-filter, whose state matrices ^^^^, ^^^^ are defined above and ^^^^ = ^^, admits, according to the book "Optimal Control" by BDO Anderson and JB Moore, Dover 2007, a right-handed form (i.e., can be rewritten): where ^(^) and ^(^) are transfer matrices of respective sizes (9*9) and (3*9), and have the expression (cf. the previous reference p.253): It should be noted that the and are not linear quadratic LQ with its corrector LQ denoted ^^^^(^), from the start ^^^^ − ^^* ^ ^ and output ^ also admits a right-handed form: with The Q-filter is integrated into the LQ quadratic controller structure according to the diagram in Figure 7. An embodiment allowing the determination of the Q-filter gains for a given disturbance frequency is given below by way of illustration. The global LQ controller with the Q-filter, denoted ^^^^_^^^^^, represented in Figure 7, has the following form on the right: In this case, the command :^ = ^^^^^^^^^ (^)^^^^^ − ^ ∗^^^ ^ (19 bis) and if there is an anticipatory action for compensating for the effects of wind (i.e., "feedforward"), this command is written: (19 ter)and the structure of the Q-filter can be put in the form ^(^) = ^^^^^^ − ^^ + ^^^^^^^^ ^^^ + ^^ (20) We can take as an example (the Q-filter as any linear system possessing an infinite number of The term ^ ^^ is calculated by placing this placement is starting from ^ ^ ^^ ^ ^ ^ of a second-order cell ^(^) = ^ ^ ^ ^ ^^ ^ ^ ^^ ^ ^^^ ^ ^ ^ ^ =, with ^^ = 2^^^. ^ ^^ ^ ^ ^ ^^ ^ ^^^ ^ ^ ^ ^^ ^ ^ Furthermore, ^^ , ^^ are damping factors that act as parameters for adjusting the notch depth at the frequency ^ ^ in the curves of singular values ​​of the complementary sensitivity transfer function. The first tuning parameter ^ ^allows you to adjust the depth of said notch, and the second adjustment parameter ^ ^ This allows you to adjust the width of this notch. We can therefore see that the coefficients ^^, ^^, ^^, ^^ depend only on the adjustment parameters ^^, ^^, ^^. The calculation of ^ ^^ This is done by placing poles, these poles being the roots of the polynomial: ^^ + ^^^ + ^^ with a multiplicity equal to 3. We then determine ^^^. According to the expression of ^ ^^^^^^^^ ( ^ ) The notch in the curves of the singular values ​​of the complementary sensitivity transfer function will be realized if for a root ^^ of the polynomial: which implies Or even The resolution of this in imaginary terms. Furthermore, we want the Q-filter not to interfere at frequency ^ = 0 with the LQ corrector, ^^^^_^^^^^, which allows us to determine the coefficients of the matrix^ ^(Equation 20) by solving ^(0) = 0 (25) Solving these various equations allows us to find the gains of the Q-filter, which is a 6th-order dynamic system with 3 inputs and 9 outputs. Once these Q-filter gains are determined for the frequency ^^, the Q-filter can be discretized at the sampling frequency chosen for the dynamic positioning system, as can the transfer matrices ^(^) and ^(^), so that they can be implemented on the embedded global controller. In another embodiment, we can start from a right-hand form equivalent to formula (15) but expressed in terms of discrete-time transfer matrices, which allows us to directly find an expression for the Q-filter in discrete time. Furthermore, for a quasi-continuous set of frequencies ^ ^The operation can be repeated to determine the corresponding Q-filter gains for each frequency in the set. This set of Q-filter gains can be stored in memory as tables, possibly with an interpolation function for cases where the dominant wave frequency at a given instant does not correspond to one of the frequencies in the tables. The Q-filter gains are then called as a function of the dominant wave frequency at a given instant, as shown in the diagram in Figure 8, where μ(μ), μ(μ), μ(μ) are the transfer functions resulting from the discretization of μ(μ), μ(μ), and μ(μ). Figures 9 and 10 illustrate, for purely illustrative purposes, the effect of adding the Q-filter on the direct and complementary sensitivity transfer functions in the context of creating a notch at a frequency μ. ^determined (frequency of 0.63 rad / s for Figures 9 and 10). We present below, by way of illustration, an embodiment allowing the determination (or selection, or even calculation by interpolation, in the case where previously calculated gains are stored / memorized in a memory of the device) of the Q-filter gains by determining the dominant frequency of the wave disturbance. In this more advanced embodiment related to Figure 11, we therefore also propose an online / real-time estimation device for the dominant frequency of the disturbances affecting the system, and more specifically the additive disturbances on the control. Initially, it is necessary to estimate these additive perturbations, which is done using a state observer. To do this, consider the augmented parallel model according to equation (9). A reduced model whose outputs are ^^^, ^^^, ^^^ (that is, ^, ^, ^) can be extracted from it. Let ^^ the state vector of this model with (26) with An observer (block "Disturbance Observer" Figure 11) for evaluating the additive disturbances ^^, ^^, ^^ on respectively ^^, ^^, ^^ is presented in state form, the symbol (^∙) corresponding to estimated variables.^^^̇ = (^^ − ^^^^)^^^ + ^^^ + ^^^^ (27) Avec : The vector ^^ of size (3,1), which is the observer gain, is calculated, for example, by pole placement. It is therefore the dominant frequency of the estimated perturbation signals ^^̂, ^^̂, ^^̂, possibly filtered by high-pass and low-pass filters, that is estimated.5 In what follows, we describe the algorithm allowing this estimation of said dominant frequency (block "Estim. freq. Dominante" Figure 11) of a signal ^ (see below), which is therefore, optionally, ^^̂ ^^ ^^̂ ^^ ^^̂. This estimation is done by recursive identification of the coefficients of a generalized ARMA ("Autoregressive Moving-Average Model") type filter. We recall that an ARMA filter ^ has a transfer function In this example of applicationhoule, we take a deliberately simplistic model where the numerator is of order 1.15 and consists only of a differentiator, and where the denominator is of order 2, that is to say The differentiator requires that the static gain be zero, which is the case for the wave spectrum.20 The corresponding generalized ARMA filter will be of the type where the ^ ^^ ^ ( ^), ^^(^^^), … ^ an orthogonal function (GOBF). This type of parameterization was used in the context of recursive identification in the article "Closed-loop output erroridentification algorithms with predictors based on generalized orthonormal transfer functions: Convergence conditions and bias distribution" (Automatica 2021) by Bernard Vau and Henri Bourlès. To perform this online estimation of the dominant wave frequency, we consider the disturbances affecting the ship to be stochastic processes. In what follows, we define ^ as being indifferently one of the preceding variables, and we model such 35 Here ^(^) is an input component of the system to be controlled, where {^(^)} is a sequence of white noise and where ^^ and ^^ are the first and second functions of 40 where −1 < ^^ < 1 is the Laguerre pole. Given that, in general, we are faced with an estimation problem in an oversampling situation with respect to the dominant wave frequency, we will take a Laguerre pole relatively close to 1. For example, for a sampling frequency of 10 Hz, we will take ^ ^ around 0.95. In order to ensure the differentiating character of the generalized ARMA filter (having a zero at +1), it is shown that ^^ ^ ^ ^^^ ^ ^^ ^^ ^ ^ and ^^ are fixed ^^ = − ( ^^^ ^ ^) ^ ^^^ ^ and ^ ^ = ( ^^^ ^ ) ^ ^^^ ^We define a regressor ^^(^) = [{−^ ^^ ^^^(^ )^(^ + 1)} {−^^(^ )^(^ + 1)}] and a parameter vector ^^ = [^^ ^^]. The output of the ARMA model is where ^(^) is the posterior error, starting from the error of à priori ^^(^ + and the relationship between the error of at and at is and where ^(^) =^ ^(^) is an observation vector resulting from the filtering. Online estimation of the vector ^ is done, for example, using the following classical estimation algorithm, although other estimation algorithms are also possible: In one particular implementation, we can take ^ = ^ a positive-definite constant matrix, and we obtain the recursive gradient descent algorithm. In another implementation, the matrix F can be updated as follows: ^^^(^ + 1) = ^^^^^(^) + ^^^(^)^^(^) (36) where 0 < ^^ ≤ 1 and 0 ≤ ^^ < 2 are forgetting factors. It can be shown that by setting the dominant frequency ^ ^^(the index i indicating that it is from one of the estimated disturbance signals ^^̂, ^^̂, ^^̂,) of the generalized ARMA model is 1 ^ ^ 1 ^ ^ ^^ ≈ 2^^ ^ ^2 + ≈ ^2 ^1 + ^ ^ ^^ ^ 2^^ ^ 1 + ^^ (38) Once the dominant frequency ^ ^^ determined for each estimated disturbance^^̂, ^^̂, ^^̂, the "Calculate global dominant frequency" block in Figure 11 allows the global dominant frequency to be calculated ^ ^which is used for determining or selecting the Q-filter gains. In a particular embodiment, the ^ of equation (30), instead of corresponding to ^^̂^^ ^^̂ ^^ ^^̂ resulting from the implementation of the state observer, can correspond to a signal from a ship's navigation system, in particular the heaving signal, heaving referring to the vertical displacement of the ship. The dominant frequency of this heaving signal can also be obtained by recursively identifying the coefficients of an ARMA-type filter seen above. Once the dominant frequency ^^ is obtained, which corresponds to the ^^ of the notch of the filter to be used, the Q-filter gains can be determined or, more advantageously, the Q-filter gains corresponding to the frequency ^^ can be selected from a set of Q-filter gains previously determined for a set of possible and stored wave frequencies.An interpolation calculation is possible in the case where ^^ does not correspond to one of the frequencies of the set. For the implementation of the positioning control calculation device using the LQ control law with Q-filter: - in a first phase, a parametric identification of the system parameters is carried out from experimental data of the ship in its maritime environment, this data being in particular ^ ^ ^ and U1, U2, U3 previously mentioned in the description and the system being a parallel synthesis model of the ship, said parallel synthesis model of the ship being augmented by three integrators and forming an augmented parallel synthesis model of the ship, - in a second phase, the linear quadratic control LQ is synthesized to determine its coefficients using the augmented parallel synthesis model of the ship whose parameters were determined by the parametric identification of the first phase. Then,Still in this second phase, the gains of the Q-filter attached to the linear quadratic control LQ are determined to obtain a notch in the curves of the singular values ​​of the complementary sensitivity transfer function at a determined frequency ^^ corresponding to the dominant frequency ^^ of the swell or, preferably, for a set or range of frequencies that can correspond to possible dominant swell frequencies. The gains of the Q-filter and the determined coefficients of the linear quadratic control LQ, which will be necessary for the calculations of the positioning control by the device of the invention, are stored / memorized in a memory of the device of the invention. In a third phase, the stored gains of the Q-filter and the determined coefficients of the linear quadratic control LQ are implemented to constitute the global controller based on the LQ control law with Q-filter in a computer onboard the ship. In a fourth phase,The positioning command is calculated in real time / online using the global controller (i.e., the LQ control law with Q-filter) of the onboard computer, based on the setpoint and for the dominant wave frequency ^^current. This dominant wave frequency ^^current is obtained in real time or periodically through an estimation of additive disturbances by a state observer or by obtaining a heaving signal, and then, for both possibilities, by recursively identifying the coefficients of an ARMA-type filter. In this process, phase 4 is therefore performed in real time on an onboard computer that is part of the device of the invention, typically a programmable computer, for example, a microcomputer or server. Phases 1 to 3, which are performed prior to the real-time phase, are also executed in a computer, generally once everyThe computer may be the on-board computer used subsequently in phase 4 or another computer within equipment separate from the device of the invention. However, some or all of these phases 1 to 3 may be repeated at significant time intervals, for example in the event of changes concerning the ship (e.g. its structure), its operational conditions, or the sea conditions that may be encountered, in order to update the positioning control calculation device.

Claims

Claim 1. A state-feedback control method for the propulsion means of a ship for its dynamic positioning, wherein: - a Q-filter linear-quadratic (LQ) control law is defined, comprising a linear-quadratic (LQ) control and a Q-filter, the Q-filter being derived from a Youla-Kucera parameterization; the linear-quadratic (LQ) control having been synthesized on the basis of a synthesis model of the ship in a frame fixed with respect to the ship's axes and using a coordinate system parallel to the ship; said synthesis model comprising parameters and being augmented by the addition of three integrators to its parameters in order to further ensure the rejection of static disturbances; the parameters of the synthesis model having been determined by parametric identification; the linear-quadratic (LQ) control implementing a state representation with a state feedback calculation of an augmented state vector (X). ^^^ ) and a gain (K ^) of state feedback, the coefficients of the linear quadratic control LQ having been determined by implementing an LQA or LQB type tuning methodology, the LQA type tuning methodology having only two tuning parameters ω^, k^ allowing with ω^ to tune the dynamics of the linear quadratic control LQ and possibly allowing with k ^ to strengthen stability margins, and the LQB-type tuning methodology, comprising only one tuning parameter ω^, allows for adjusting the dynamics of the linear quadratic LQ control. The LQ control law with a Q-filter has determinable singular value curves for its complementary sensitivity transfer function. The ship's positioning is controlled according to said LQ control law with a Q-filter, in which the Q-filter has determined gains to create a notch at a determined frequency f. ^in the curves of the singular values ​​of the complementary sensitivity transfer function of said LQ control law to Q-filter.

2. Method according to claim 1, wherein said determined frequency ^ ^is the dominant frequency of a swell to which the ship is subjected.

3. A method according to any one of claims 1 and 2, wherein the Q-filter gains are determined with further adjustment of the notch depth and width.

4. A method according to any one of claims 2 to 3, wherein a set of Q-filter gains for a set of notch frequencies is stored in memory, and the stored Q-filter gains corresponding to a notch frequency substantially equal to the dominant frequency of the swell to which the ship is subjected are used for positioning control. 5.A method according to claim 4, wherein, if none of the notch frequencies of the stored Q-filter gain set is substantially equal to the dominant wave frequency to which the ship is subjected, then the Q-filter gains are calculated by interpolation to the dominant wave frequency from the stored Q-filter gains. A method according to any one of claims 1 to 5, wherein a dominant wave frequency estimator is implemented that estimates in real time the dominant wave frequency to which the ship is subjected by a recursive identification calculation of the coefficients of an ARMA-type filter, either on signals from an estimation by a state observer of additive disturbances, or on a heaving signal from a heaving detector.

7. A method according to any one of claims 1 to 6, wherein the coefficients of the linear quadratic LQ control, having been determined, are stored in memory, and the stored coefficients of the linear quadratic LQ control are used for positioning control in the linear quadratic LQ control.

8. A method according to any one of claims 1 to 7, wherein the positioning of the vessel is further controlled as a function of wind forces and torques acting on the vessel.

9. A method according to any one of claims 1 to 8, wherein the positioning of the vessel is controlled by a command U produced by implementing the LQ control law with Q-filter and as a function of setpoints, and the command U is calculated ^ ^ ∗ ^^ is a vector of instructions, I is an identity matrix, K cis a state feedback gain, and the transfer matrices M, N, and Q resulting from the Youla-Kucera parameterization, and in the case where the ship's positioning is further controlled as a function of wind forces and torques acting on the ship, and the command U is calculated ^ ^^^^is derived from wind speed measurements by at least one sensor on the ship.

10. Dynamic ship positioning system by state feedback control of propulsion means of said ship, the system being configured to control the ship's positioning by a Q-filter LQ control law comprising a linear quadratic LQ control and a Q-filter, the linear quadratic LQ control having been synthesized on the basis of a synthesis model of the ship in a frame fixed with respect to the ship's axes and using a coordinate system parallel to the ship, said synthesis model comprising parameters and being augmented by the addition of three integrators in its parameters in order to further ensure the rejection of static disturbances, the parameters of the synthesis model having been determined by parametric identification, the linear quadratic LQ control implementing a state representation with a state feedback calculation of an augmented state vector (^^^^ ) and a gain (^ ^ ) of state feedback, the coefficients of the linear quadratic LQ control having been determined by implementing an LQA or LQB type tuning methodology, the LQA type tuning methodology having only two tuning parameters ω^, k^ allowing with ω^ to tune the dynamics of the linear quadratic LQ control and possibly allowing with k^ to strengthen the stability margins, and the LQB type tuning methodology having only one tuning parameter ω^ allowing to tune the dynamics of the linear quadratic LQ control, the LQ control law with Q-filter having determinable singular value curves of its complementary sensitivity transfer function, and the Q-filter being derived from a Youla-Kucera parameterization and having determined gains to create a notch at a determined frequency ^ ^in the curves of the singular values ​​of the complementary sensitivity transfer function of said LQ control law to Q-filter.

11. System according to claim 10, wherein the determined frequency ^^ is the dominant frequency of a swell to which the ship is subjected.

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