State feedback control method for dynamic ship positioning and adapted system

Abstract

State feedback control method for dynamic ship positioning and adapted system. Method in which a Q-filter LQ control law is defined comprising a linear quadratic LQ control and a Q-filter, the linear quadratic LQ control having been synthesized on the basis of a ship synthesis model parallel to the ship augmented by three integrators, 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 state feedback gain, the coefficients of the linear quadratic LQ control having been determined by implementing an LQA type or LQB type tuning methodology, and in which the ship's positioning is controlled according to said Q-filter LQ control law.the Q-filter having determined gains to create a notch at a determined frequency in the curves of singular values ​​of the complementary sensitivity transfer function of said LQ control law with Q-filter. Figure for the abbreviation: Fig. 3,

Description

Title of the invention: State feedback control method for dynamic positioning of a ship and adapted system. Field of the invention

[0001] The present invention relates to the field of ship navigation control methods, particularly for ship dynamic positioning systems. More specifically, the invention relates to a state feedback control method for a ship dynamic positioning system. A system adapted to implement the method completes the invention. State of the art

[0002] A dynamic positioning system allows a ship to follow or maintain a course and position in a local geographical reference frame, when that ship is moving at almost zero speed, and thus to counteract some of the disturbances acting on the ship, in particular to counteract currents.

[0003] The dynamic positioning system uses information received from a navigation center, in particular a vector with if - \ YY ^] , X. Y being the coordinates of the ship in a local geographical frame, V the heading, and a vector v with vT — [u P r^uv being the linear velocities of the ship in its own frame along the x' y axes and the rotational speed of the ship about its z axis, i.e. yaw rate, according to the scheme of [Fig.1].

[0004] From the measurements corresponding to the vectors and v and a setpoint vector if, the dynamic positioning system calculates according to a control law a vector of generalized forces (forces and torques) according to the various degrees of freedom of the ship, this vector is then introduced into a distribution system which generates the orders sent to each of the ship actuators as shown [Fig.2].

[0005] 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 entitled "Marine Craft Hydrodynamics and Control" by Thor I. Fossen (Wiley 2011), p. 394.

[0006] The role of dynamic positioning systems is to compensate for disturbances acting on the ship at very low frequencies, i.e., currents. On the other hand, disturbances induced by swell (in the context of this invention, the term "swell" also includes the term "wave"), which act in intermediate frequency ranges, must not be compensated, and therefore should not entering the loop. This usually leads to the introduction into the control law of filters based, as explained in the same work (p. 395), on a wave observer. This observer can, in particular, be a passive nonlinear observer.

[0007] 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 major challenge for the engineer tasked with implementing a dynamic positioning system based on this technology, forcing 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 optimality.

[0008] Furthermore, it is necessary to synthesize the control law from a model of the ship at sea, i.e., in its environment, also called a synthetic model. However, a difficulty in synthesizing the control law of a dynamic positioning system is the intrinsic non-linearity of the ship model at sea, which is linked, in particular, to the change of reference frame between the ship's reference frame and the local geographic reference frame. In the academic literature, numerous types of control laws have been proposed to address this problem, and we can mention, among others:

[0009] -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

[0010] - LQG with a linearized model around an operating point,

[0011] - Linearizing control (“non-linear feedback”),

[0012] - Control by "backstepping", etc...

[0013] 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 of reference fixed relative to the ship's axes and using 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 non-linearities (at low speeds), which therefore allows 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 H30 type control law, with / . / -synthesis, is used.Unfortunately, the main flaw in the developments mentioned in this article still lies in their applicability and... implementation of the control law on site, taking into account the difficulty of adjusting H50 type control laws. Summary of the invention

[0014] In the invention described in this document, a control law is proposed which 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.

[0015] This synthesis model corresponds to a linear part derived from the ship-at-sea model, fixed with respect to the ship's axes and using a coordinate system parallel to the ship. This synthesis model is augmented by the addition of three integrators, i.e., the addition of three additional state variables to the synthesis model. A linear quadratic (LQ) control system is established on this synthesis model, comprising twenty-seven low-level parameters determined using a methodology known as LQA or LQB, as described in the book "Applied Automation" by Ph. de Larminat (Hermès 2009). This system provides the engineer responsible for synthesizing the linear quadratic (LQ) control system with only one, or at most two, high-level parameters for tuning, which greatly facilitates the implementation of the control law on-site.

[0016] Advantageously, in order to complete this linear quadratic control LQ, a device intended to make it immune to swell disturbances is added to the linear quadratic control LQ: This is a filter, called Q-filter, resulting from a Youla-Kucera parameterization and which allows the complementary sensitivity transfer functions of the closed loop of the linear quadratic control LQ to be modified by introducing a notch without modifying the location of the poles of said closed loop induced by the linear quadratic control LQ and thus preserving its robustness.

[0017] The use of the Youla-Kucera parameterization as such is known for achieving the rejection of narrowband disturbances in control laws, with variable but known frequencies: for example, in the very different field of acoustic active control, see patent EP2436003 (Bl) 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" by G. Wang, Y. Yang, and S. Wang in the Journal of Marine Science and Engineering, MDPI, vol. 9(4), 2021. The Youla-Kucera parameterization has also been extensively used for the rejection of narrowband 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 - Adual Youla-Kucera approach”, Automatica, 2021, whose authors are B. Vau and LD. Landau).

[0018] However, in the usual application described in these three documents, the role of the Q-filter is to ensure the rejection of narrowband disturbances, i.e., to create a notch in the gain curve of the direct closed-loop sensitivity transfer function. Here, within the framework of the dynamic positioning system, the use of the Q-filter is dual, since it aims to make the linear quadratic (LQ) control insensitive to narrowband disturbances of a specific frequency, namely wave noise 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.

[0019] 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.

[0020] 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 depending 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.

[0021] For example, a user of the dynamic positioning system, in particular the ship's pilot, may specify a sea state, and a device or method converts this information into a dominant frequency, allowing the selection of the corresponding / appropriate Q-filter gains for the system.

[0022] In any event, once the parameters of the synthesis model have been determined by parametric identification, the synthesis of the LQ control law with Q-filter of the dynamic positioning system is based at most on only four high-level parameters:

[0023] -one or two parameters for linear quadratic control LQ,

[0024] - two parameters to adjust / select the width and depth of the notch in the curves of singular values ​​of the complementary sensitivity transfer function for the Q-filter,

[0025] which greatly facilitates the on-site implementation of said LQ control law with Q-filter.

[0026] 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 control law-based controller of the dynamic positioning control calculation device implementing the method of the invention.

[0027] 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.

[0028] In one embodiment, a specific measuring device which automatically measures the state of the sea in real time provides useful information, in particular the dominant wave frequency, e.g. by a heaving sensor, and possibly a standard deviation or equivalent of the dominant frequency measurement.

[0029] Advantageously, the dynamic positioning calculation device includes means for selecting, from a table of Q-filter gains previously stored / memorized, the Q-filter gains adapted to the current state of the sea, the table storing / memorizing a set of Q-filter gains for a set of possible dominant swell frequencies.

[0030] In general, the invention relates to a state feedback control method for the propulsion means of a ship for its dynamic positioning as described.

[0031] More specifically, the invention relates to a state feedback control method for the propulsion means of a ship for its dynamic positioning, in which:

[0032] - a Q-filter LQ control law is defined, comprising a linear control quadratic LQ 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 synthetic 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 synthetic model comprising parameters and being augmented with the addition of three integrators to its parameters, the parameters of the synthesis model having been determined by parametric identification,

[0033] The linear quadratic control LQ implementing a state representation with a state feedback calculation of an augmented state vector Xaug and a state feedback gain Kc, 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 (ka) and the LQB type tuning methodology having only two parameters (ka) and that a single adjustment parameter, the LQ control law with a Q-filter having curves of singular values ​​of its complementary sensitivity transfer function, can be determined

[0034] - the positioning of the ship is controlled according to said control law LQ with Q-filter, wherein the Q-filter has determined gains to create a notch at a determined frequency fo in the singular value curves of the complementary sensitivity transfer function of said LQ with Q-filter control law.

[0035] Other advantageous features of the method according to the invention, taken individually or in all technically possible combinations, are as follows:

[0036] - the vessel is a surface vessel,

[0037] - the parallel synthesis model of the ship is augmented by adding three integrators and constitutes the parallel synthesis model to the augmented ship,

[0038] - parameter identification is performed using a database experimental studies concerning the ship in its marine environment

[0039] - the state feedback control method for the propulsion means of a ship for its dynamic positioning, it uses a programmable data processing device comprising a memory, in particular for data storage,

[0040] - the programmable data processing device is a programmable computer,

[0041] - the programmable data processing device is a computer or micro computer,

[0042] - the method implements a memory,

[0043] - the memory is configured to store at least the gains of the Q-filter,

[0044] - the memory is configured to store at least the coefficients of the command linear quadratic LQ and Q-filter gains,

[0045] - the Q-filter gains are determined by adjusting the notch frequency by a setting parameter which is a notch frequency,

[0046] - the determined frequency fo is the dominant frequency of a swell to which the ship is subject,

[0047] - the Q-filter gains are determined with further adjustment of the depth and the notch width,

[0048] - the Q-filter gains are determined by adjusting the depth and width of the notch by two additional adjustment parameters Ç and Cd,

[0049] - a set of Q-filter gains is stored in memory for a set of notch frequencies and, during positioning control, the Q-filter gains stored corresponding to a notch frequency substantially equal to the dominant wave frequency to which the ship is subjected are used for the Q-filter.

[0050] - in the case where none of the notch frequencies of the Q-filter gain set stored is not substantially equal to the dominant wave frequency to which the ship is subjected, so the Q-filter gains are calculated by interpolation to the dominant wave frequency from the stored Q-filter gains.

[0051] - a notch frequency is considered substantially equal to the frequency dominant wave frequency when the dominant wave frequency is within a range of + / - 5% of the stored notch frequency,

[0052] - a dominant wave frequency estimator is implemented estimating in time the actual dominant wave frequency to which the ship is subjected by a recursive identification calculation of the coefficients of an ARMA-type filter on signals from an estimation by a state observer of additive disturbances further implemented or on a pounding signal from a pounding detector further implemented,

[0053] - the coefficients of the quadratic linear control are stored in memory LQ having been determined and the coefficients of the linear quadratic LQ control being used during positioning control, for the linear quadratic LQ control,

[0054] - the ship's positioning is further controlled according to forces and pairs of wind acting on the ship,

[0055] - 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, and we calculate the control U with: u = ( - (J+MQ) (I-KcNQylKc) (Xaug-X^g)

[0056] where Xaug is an augmented vector of setpoints, 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,

[0057] - in the case where the ship's positioning is further controlled according to forces and torques of a wind acting on the ship, and we calculate the control U ^■U = ( -(I + MQ)(I-KCNQ)-,KC) +

[0058] where is derived from wind speed measurements by at least one sensor on the ship,

[0059] - the control U produced by the implementation of the LQ control law with Q-filter acts on effectors of the ship,

[0060] - parametric identification to obtain the parameters of the synthesis model Parallel to the augmented vessel, the process is carried out by the programmable data processing device which implements the state-feedback control method for the vessel's propulsion systems for its dynamic positioning.

[0061] - the synthesis of the linear quadratic control LQ (i.e., determination of the The coefficients of the linear quadratic control (LQ) and the determination of the Q-filter gains are performed by the programmable data processing device which implements the state feedback control method of the ship's propulsion means for its dynamic positioning,

[0062] - the programmable data processing device that implements the method The state-feedback control system for the ship's propulsion systems for its dynamic positioning is further configured to perform linear quadratic (LQ) control synthesis and determine the Q-filter gains.

[0063] - parametric identification to obtain the parameters of the synthesis model parallel to the augmented vessel is carried out by a separate apparatus from the programmable data processing device which implements the state-feedback control method of the vessel's propulsion means for its dynamic positioning,

[0064] - the synthesis of the linear quadratic control LQ and the determination of the gains Q-filter operations are performed by equipment separate from the programmable data processing device which implements the state feedback control method of the ship's propulsion means for its dynamic positioning.

[0065] The invention also relates, in a general way, to a dynamic positioning system of a ship by state feedback control of propulsion means of said ship, as described and / or comprising means for executing the method described.

[0066] 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 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 model of synthesis 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 Xaug and a state feedback gain Kc, 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 comprising only two tuning parameters (ka) and the LQB type tuning methodology comprising only one tuning parameter (w), 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 gains determined to create a notch at a determined frequency fo in the curves of singular values ​​of the complementary sensitivity transfer function of said LQ control law with Q-filter.

[0067] Other advantageous features of the system according to the invention, taken individually or in all technically possible combinations, are as follows:

[0068] - the determined frequency f0 is the dominant frequency of a swell to which the ship is subject,

[0069] - the synthesis model is a synthesis model parallel to the ship, which is in further enhanced by the addition of three integrators to form a synthesis model parallel to the augmented ship,

[0070] - the parameters of the parallel synthesis model for the augmented ship are obtained by parametric identification.

[0071] 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 computer-readable medium containing said computer program. Brief description of the figures

[0072] In the attached figures:

[0073] [Fig-1] represents a diagram of the ship in its environment,

[0074] [Fig.2] represents a general diagram of a means for the dynamic positioning of a vessel comprising a dynamic positioning system and a dispatching system for generating commands to actuators on the vessel,

[0075] [Fig.3] represents an implementation of the linear quadratic control LQ in the case where an anticipatory action of the wind forces and torques acting on the ship is further implemented,

[0076] [Fig.4] represents a closed loop with the signals allowing the expression of the input sensitivity functions,

[0077] [Fig.5] represents an example of singular value curves of the function of direct input sensitivity transfer, on the figures "singular values" meaning "singular values" and "frequency" meaning "frequency",

[0078] [Fig.6] represents an example of singular value curves of the function of complementary sensitivity transfer at the input,

[0079] [Fig.7] represents the overall structure of the LQ control law with Q-filter with the Q-filter nested within the linear quadratic control LQ and forming a global corrector in the case where an anticipatory action of the wind forces and torques acting on the ship is also implemented,

[0080] [Fig.8] represents an overall structure of the LQ control law with a Q-filter discretized with tabulation of the Q-filter gains in the case where an anticipatory action of the wind forces and torques acting on the ship is also implemented,

[0081] [Fig.9] represents singular value curves of the transfer function of Direct sensitivity without Q-filter (solid line) and with Q-filter (dashed line)

[0082] [Fig. 10] represents singular value curves of the complementary sensitivity transfer function without a Q-filter (solid line) and with a Q-filter (dashed line), and

[0083] [Fig. 11]: represents an overall diagram of an example of a device for estimating the dominant wave frequency. Detailed description

[0084] The following description with regard to the attached drawings, given by way of non-limiting examples, will make it clear what the invention consists of and how it can be carried out.

[0085] 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) and is as follows:

[0086] M v + Dv = RT(w)b + T + Twind + rwme

[0087] =

[0088] Where:

[0089] b is the vector of current forces

[0090] r is the vector of forces / torques of the actuators

[0091] twM; the vector of torques / perturbation forces due to wind

[0092]

[0093] rwave; the vector of forces / torques of perturbation due to swell In what follows, for the synthesis of the linear quadratic control LQ, i.e., the determination of the coefficients of the linear quadratic control LQ, we consider

[0094]

[0095]

[0096]

[0097] that wimi 0, M v + Dv = t T wave “ 0, £ = 0 such that this model becomes 7 = Xip)f (D Or M- 0 myz is the mass / inertia matrix of the ship and myz

[0098] the ship's viscous friction matrix and

[0099] Uy < dzy cosip sintp - simple cosip 01 a rotation matrix.

[0100]

[0101]

[0102] It can be seen that in equations (1) the presence of the matrix R(V?) introduces an intrinsic non-linearity in the change of reference frame between that of the ship and the local geographical 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 from a state vector x, a matrix B, and a vector field ffy.

[0103]

[0104]

[0105]

[0106] It is recalled that in a state representation (in automatic control, a state representation allows us to model 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 by knowing the equations describing the operation of the system and the inputs of this system, the state variables being the quantities which constitute the state of the system. By asking xt= (uvr NE ip) = (*i x2 x3 x4 x5 x6) U2 t / 3] (2) where: N and E give the relative position of the ship along the north and east axis of the local geographical reference, V the heading and Ul, U2, U3 are the commands. So we have

[0108] U^ (3) x = f(x) + B U2

[0109]

[0110] With "11*1 «92*2 "I" ^23^3 "32*2 + a33X3" [YES]

[0112]

[0113]

[0114]

[0115]

[0116]

[0117]

[0118]

[0119]

[0120] f(x) = cos(x6)xj - sin(x6)x2 sin(x6) xj + cos(x6)x2 x3 pii 0 0 0 ^92 ^93 0 ^32 ^33 0 0 0 0 0 0 0 0 0 I (4) Many 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 when 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 vessel 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 that This leads to the introduction of a state vector xc with xT=(Xcl Xc2 Xc3 Xc4 Xc5 = And ^■2 *c3 Xc5 । xc61 X] x2 *3 x4cosx6 + x$inxb -x^sinx^+x5cosx6 x6 Noting that X^i / COSX6 xc5 / \ - sinx6 We have (6) sinxb \ (xA / - sinx^ 11. | + COSXb I \ X^ / \ - cosx6 cos x 6 -sinx6 x4 x5

[0121] | 0 CO Ue5 / \X2 / + \-l ()H*c5r3

[0122] We therefore obtain a "quasi-linear" model, whose non-linearities only affect *c4 and xc5 and only occur if the yaw rate, i.e. "yaw-rate", x is not zero.

[0123] / ¾1 / «11 0 0 0 0 0( iXcl\ ^11 0 0 1 0 ^2 0 «22 «23 0 0 0 ^33 0 ^2 + 0 Xc5 0 1 0 0 0 0

[0124] (8)

[0125] Furthermore, in order to also ensure the rejection of static disturbances, the above parallel model is augmented with the addition of three integrators, which leads to the introduction of 3 additional state variables x'^ into the model, such as

[0126] =

[0127]

[0128] x^x^

[0129] Finally, the proposed control law is based on a linear quadratic control LQ, which is synthesized on the basis of the following augmented parallel model:

[0130] XaUg — AaugXaug + Baugü

[0131] With

[0132] xLg=(xci xc2 xc3 xe4 Xe5 Xr6 XC1 0 0 0 0 0 0 0 b32 ^33 1 0 0 0 0 0 0 0 0 0 0 0 A«ag - 0 10 0 0 0 0 0 0 Baug - 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0! 1 0 0 0

[0134] (9)

[0135] The linear quadratic control LQ is a state feedback control law, that is to say that it is written

[0136] U= -^X^IO)

[0137] With Kc the state feedback gain and

[0138]

[0139]

[0140]

[0141]

[0142]

[0143]

[0144]

[0145]

[0146]

[0147]

[0148]

[0149]

[0150]

[0151]

[0152]

[0153]

[0154]

[0155]

[0156] and where Pc is the solution of the Riccati matrix equation Pc^g + ÂLgPc + Qc - PeBaugR^B^gPc = 0 ( 12) This law minimizes a quadratic criterion with an infinite horizon J = ^xLgQXaug + UTRcUdt(13) Qc and Rc being weighting matrices. In the present problem, given the size of the state vector Xatlg and the input vector U, since the matrix Qc is of size (9*9) and Rc 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 framework 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 Qc 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 adjustment parameters: -The adjustment parameter that controls the dynamics of the linear quadratic control LQ (the order of magnitude of which is that of a frequency) -The ka adjustment parameter, which can potentially increase stability margins In the case of the LQB methodology, there is only one adjustment parameter: -the adjustment parameter (,'b which adjusts the dynamics of the linear quadratic control LQ. The setpoints on 7 and v, denoted respectively 7* and v*, which come from a trajectory generator, can be grouped into a vector x* such that x*r = [ ffT\. We can also define a setpoint vector x^ such that jh 03 cos x 6 sin x 6 0 x* = x* 03 -sinx6 cosx6 0 0 0 1 / and additional instruction components, three components in this case: Xc7 — Xc4 — -^cô

[0157] and we then define an augmented vector of instructions

[0158] x2g=(x*cl x^ x*, x*4 x*5 x^ X^)

[0159] thus the linear quadratic control LQ with consideration of setpoints will be

[0160] U — - Kc ( Xattg - Xuug ) (14)

[0161] and its implementation is shown in Figure 3 (in this figure 3 the narrow vertical rectangle towards which the arrows of the elements shown on the left of said rectangle converge and from which Xmg-XaUg comes out, symbolizes the operations, additions and subtractions, carried out on the elements shown on the left).

[0162] In a particular embodiment, the control may be equipped with a feedforward action of the wind forces and torques acting on the ship; this action is denoted T and acts additively on the control, the latter then being such that

[0163] U= -Kc (X <mg-X^g) +Twind (14 bis)

[0164] and Twimi is calculated from measurements of wind speed in different directions obtained by sensors located on the ship.

[0165] During the synthesis of the linear quadratic control LQ, from the augmented parallel model whose parameters were obtained by parametric identification. It is recalled that parametric identification is a procedure consisting of obtaining experimental data of the inputs and outputs of a system, when it is subjected to excitations, and then exploiting these data in order to obtain, by an optimization algorithm, estimates of the model parameters, it is possible to determine the sensitivity transfer functions of the closed loop. It is in particular possible to make explicit the direct input sensitivity transfer function Sjs), which corresponds to the transfer matrix between an additive disturbance r on the control and the signal. We can also make explicit the complementary sensitivity transfer function Tu(s\ 9ui cst 'a transfer matrix between the additive disturbance r and the signal wt as represented [Fig.4] (note that [Fig.4] is a generic figure intended to present the signals considered in the description of the sensitivity transfer matrices or functions).

[0166] And it is then possible to plot the curves of the singular values ​​of the transfer matrices corresponding to Scl, and as a function of the frequency, curves of which examples are given in [Fig.5] and [Fig.6], the frequency being in rad / s and the singular values ​​in dB.

[0167] It is observed that the singular values ​​of Su tend towards 0 as the frequency tends towards 0, which is the effect of the integrators introduced in the augmented model. These curves therefore show that static and very low-frequency disturbances, for example ocean currents, will be rejected by the linear quadratic control LQ.

[0168] On the other hand, the complementary sensitivity transfer function has a low-pass character. It shows that the pseudo-periodic disturbances due to swell, in particular swell which is generally between 0.5 rad / s and 2 rad / s, stress the actuators in frequency bands which are not far from the cutoff frequency of the complementary sensitivity transfer function.

[0169] However, it turns out that the said actuators, on a standard ship, do not have the necessary amplitude to effectively counteract this type of disturbance. For this reason, it is not desirable to attempt to reject this type of disturbance by the dynamic positioning system.

[0170] It is therefore proposed in the design of the control law of the dynamic positioning system to make it insensitive to these disturbances in a medium frequency area, for example between 0.5 rad / s and 2 rad / s.

[0171] For this purpose, 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.

[0172] 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 LQ control described above. The resulting system, whose control law is schematically represented in Figures 7 and 8, preserves most of the performance and robustness properties of the linear quadratic LQ control itself, while creating notches in the curves of the singular values ​​of the complementary sensitivity transfer function. The frequency of these notches, fo, can be adjusted / adapted to the sea state, in this case to the dominant wave frequency.

[0173] This operation constitutes a specific Youla-Kucera parameterization which is now described.

[0174] The augmented system denoted Gaug of the LQ control law with Q-filter whose state matrices Aaug, Baug are defined above and CaUg = Ç, admits, according to the book "Optimal Control" by BDO Anderson and JB Moore, Dover 2007, a right-hand form (i.e. can be rewritten): [°175]

[0176] where Mis) and 2V(s) are transfer matrices of respective sizes (9*9) and (3*9), and have the expression (cf. the previous reference p.253):

[0177] Mis) ^IK^sI-^ + BaugK^Baug

[0178] N(s) = (s / ~ Aaug + BaugKc) Baug

[0179] (16)

[0180] It should be noted that the expressions for M and N are not unique. The linear quadratic control LQ with its controller LQ denoted Corr(s), with input Xaug-X^ug and output L, also admits a right-hand form: Corr(s) = (17)

[0181] with

[0182] V^S) = -KC

[0183] Lq(s)=I

[0184] (18)

[0185] The Q-filter is inserted into the structure of the linear quadratic control LQ according to the scheme of [Fig.7].

[0186] The following is given, by way of illustration, an embodiment allowing the Q-filter gains to be determined for a disturbance frequency f0.

[0187] The global corrector LQ with the Q-filter denoted Corr vouja shown [Fig.7] has the following right-hand form: 101881 = - (I + MQ) (I-KJVQY'K. (W)

[0189] In this case, the command is written:

[0190] U = C^MiX^-X^g) (19bis)

[0191] and if there is an anticipatory action for compensating for the effects of wind (i.e., "feedforward"), this command is written: I o 1921 U = ( s ) ( X^g-X'^g ) + < 19 ter)

[0193] and the structure of the Q-filter can be put in the form

[0194] Q(S) ^K^sI-Ag+KgÿCgY'K^+D, (20)

[0195] We can take as an example (the Q-filter as any linear system possessing an infinite number of realizations):

[0196] ' -a} 1 0 0 0 0 ' -«2 0 0 0 0 0 0 0 -d] -a2 1 0 0 Aq~ 0 0 0 0 0 0 0 0 0 -aA 1 . 0 0 0 0 -«2 0,

[0197] ■1 0 0 0 0 0' CQ = 0 0 1 0 0 0 .0 0 0 0 1 0.

[0198] (21)

[0199] The term Koq is calculated by pole placement. This placement is carried out from a second-order cell, with = 2jrf. Wa^+a, ' 0 J o H(S) “ ^+1 " sl+^+fl2

[0200] In addition, Çn, ^d are damping factors which play the role of notch depth adjustment parameter at frequency f0 in the curves of singular values ​​of the complementary sensitivity transfer function.

[0201] The first adjustment parameter Ç allows the depth of said notch to be adjusted, and the second adjustment parameter allows the width of this notch to be adjusted. It can therefore be seen that the coefficients f1? ^2 depend only on the adjustment parameters fo, Çn, £d.

[0202] Koq is calculated by placing poles, these poles being the roots of the polynomial:

[0203] s2 + ps + p, with a multiplicity equal to 3.

[0204] Kcq is then determined.

[0205] According to the expression of Corrymla(s), the notch in the curves of the singular values ​​of the complementary sensitivity transfer function will be realized if for a root so of the polynomial:

[0206] s2 + ai« + «2, we have

[0207] I + M (s0) Q (so) = 0(22)

[0208] which implies

[0209] 0(^) = -^^)(23)

[0210] Or even

[0211] K^sJ-Ao + K^^K^

[0212] The solution to this equation is achieved by separating the real and imaginary parts.

[0213] Furthermore, it is desired that the Q-filter does not interfere at the frequency w = 0 with the LQ compensator, Corr yoidw cc, which allows the coefficients of the matrix Dq (equation 20) to be determined by solving

[0214] 0(0)-0(25)

[0215] 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.

[0216] Once these Q-filter gains are determined for the frequency fo, the Q-filter can be discretized at the sampling frequency chosen for the dynamic positioning system, as well as the transfer matrices M(s) and N(s)> so as to be implementable on the global controller on board.

[0217] In another embodiment, we can start from a right-hand form equivalent to formula (15) but expressed from discrete-time transfer matrices, which allows us to directly find an expression for the discrete-time Q-filter.

[0218] Furthermore, for a quasi-continuous set of frequencies fo, the operation can be repeated and the corresponding Q-filter gains determined for each frequency in the set. This set of Q-filter gains can be stored in memory, in the form of tables, possibly with an interpolation function for the case where the dominant wave frequency fd 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 fd at a given instant, as shown in the diagram in Figure 8, where 'cs are functions of transfer resulting from the discretization of M(s) N(s\ and

[0219] Figures 9 and 10 show, purely for 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 determined frequency fo (frequency of 0.63 rad / s for Figures 9 and 10).

[0220] We present below, by way of illustration, an embodiment allowing to determine (or select, or even calculate by interpolation, in case previously calculated gains are stored / memorized in a memory of the device) the gains of the Q-filter thanks to the determination of the dominant frequency fd of disturbance due to the swell.

[0221] In this more advanced embodiment in relation to [Fig. 11], we therefore propose in addition an online / real-time estimation device for the dominant frequency of disturbances affecting the system, and more specifically the additive disturbances on the control.

[0222] Initially, it is necessary to estimate these additive perturbations, which is done using a state observer.

[0223] To do this, let us consider the augmented parallel model according to equation (9).

[0224] A reduced model whose outputs are x<^ xe3 (that is, u' L r) can be extract. Let Xr be the state vector of this model with

[0225] Xr = ArXr + BrV

[0226] Yr = CrXr

[0227] (26)

[0228] with

[0229] «h 0 0 ■ 0 0 'Ar = 0 d'y 2 ^2 3 Br = 0 b^2 cr=i3. 0 “32 “33. 0 b yy b^

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[0247] A state observer (block "Disturbance Observer" figure 11) to evaluate the additive disturbances P? P3 on respectively UU2, U2 is presented in the form of a state, the symbol corresponding to estimated variables. " • o With : Br [°3 G=[o3 h] The Ko vector of size (3,1) which is the observer gain, being calculated for example by pole placement. Therefore, it is the dominant frequency of the estimated disturbance signals pp^ p, possibly filtered by high-pass and low-pass filters, that is estimated. The following describes the algorithm for estimating the dominant frequency (block "Dominant Frequency Estimation" Figure 11) of a signal ■J' (see below), which is therefore either p^ or p? OR p^- This estimation is performed by recursively identifying the coefficients of a generalized ARMA (Autoregressive Moving-Average Model) filter. Recall that an ARMA filter H has a transfer function (28) In this example of application to the identification of the dominant wave frequency, we take a deliberately simplistic model where the numerator is of order 1, and consists only of a differentiator, and where the denominator is of order 2, that is to say H( z) The derivatizer requires that the model have a zero static gain, which is the case for the wave spectrum. The corresponding generalized ARMA filter will be of the type B ( Z ) - (29) where Epr1), V2(r'), --APPr1) constitute a generalized orthonormal function basis (GOBF). This type of parameterization was used in the context of recursive identification in the article "Closed-loop output error identification algorithms with predictors based on generalized orthonormal transfer

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[0264] functions: Convergence conditions and bias distribution » (Automatica 2021) whose authors are Bernard Vau and Henri Bourlès. To perform this online estimation of the dominant wave frequency, the disturbances affecting the ship are considered to be stochastic processes. In what follows, any of the preceding variables are defined interchangeably, and a simplified model is adopted such that y(t) - (30) Here y( / ) is an estimate of one of the components of the additive disturbances at the input of the system to be controlled. where {¢( / )} is a sequence of white noise cl where V, cl IL are the first and second Laguerre functions with ^-pt. r1 -P +Z-1 I “ lp^ \-p^ (31) where -1 < pa < 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 Po around 0.95. To ensure the differentiating character of the generalized ARMA filter (having a zero at +1), it is shown that L and ^2 are fixed _ and _ ~ 2 '^ +p ^ We define a regressor f(t) = [{-V^q-^yft + 1)} {-V2(q^)^ + 1)}] and a parameter vector @T — [(L] The predicted output of the ARMA model is y(t) = f (t)^(t) - +y9qV2(qv) )e(t) (32) x 1 1 Z. / where ¢( / ) is the posterior prediction error, defined from the prior prediction error: + 1) = y(t+1) (t)$(t) - (y^V^q-1) +1^2(^1) )s'(t) (33) and the relationship between the prior and posterior prediction error is 6{t+ 1) -

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[0280] (34) and where 0 is an observation vector resulting from the filtering of the regressor. Online estimation of vector 6 is performed, for example, using the following classical estimation algorithm, although other estimation algorithms are also possible: 3 (t + 1) = 3 (O + F^^s (7) (35) In a particular embodiment, we can take F(t) = F, a positive definite constant matrix, and we obtain the recursive gradient algorithm. In another embodiment, the matrix F can be updated as follows: F1 (7 + 1 ) = ^F1 (O + (36) where 0 < 1 and 0 < z2 < 2 are forgetting factors. It can be shown that by setting «1 = - 2P0 + - ^2PO\ / ^ (37) the dominant frequency f di (the subscript i indicating that it is the frequency originating from one of the estimated perturbation signals pp^ p^,) of the generalized ARMA model is (38) Once the dominant frequency fdj is determined for each estimated disturbance p P y P y 'c ^oc * Calculation of global dominant frequency » of figure 11 allows to calculate the global dominant frequency which is used for the determination or selection of the gains of the filter Q. In a particular embodiment, the function of equation (30), instead of corresponding to p^otl p^ OR p, resulting from the implementation of the state observer, can correspond to a signal from the 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 as described above.

[0281] Once the dominant frequency fd is obtained, which corresponds to the f0 of the notch of the filtering to be used, we can determine the Q-filter gains or, more advantageously, select the Q-filter gains corresponding to the frequency fd within a set of Q-filter gains previously determined for a set of possible wave frequencies and stored, an interpolation calculation being possible in the case where fd does not correspond to one of the frequencies in the set.

[0282] For the implementation of the positioning control calculation device using the LQ control law with Q-filter:

[0283] - in a first phase, a parametric identification of the system parameters from experimental data of the ship in its marine environment, this data being notably wrr and Ul, 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,

[0284] - in a second phase, the linear quadratic control LQ is synthesized to determine its coefficients using the augmented parallel ship synthesis model 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 fo corresponding to the dominant frequency fd of the swell or, preferably, for a set or range of frequencies that can correspond to possible dominant swell frequencies, and 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 / remembered in a memory of the device of the invention.

[0285] - in a third phase, the Q-filter gains and coefficients are implemented determined and stored values ​​of the linear quadratic (LQ) control to constitute the global controller based on the LQ control law with a Q-filter in a computer embedded in the ship,

[0286] - in a fourth phase, calculation is carried out in real time / online, using the corrector global (i.e., the LQ control law with Q-filter) of the onboard computer, the positioning control as a function of the setpoint and for the current dominant wave frequency fd, said current dominant wave frequency fd being obtained in real time or periodically through an estimation of additive disturbances by a state observer or by obtaining a pounding signal and then, for both possibilities, by recursive identification of the coefficients of an ARMA type filter.

[0287] In this process, phase 4 is therefore in real time on an embedded computer forming part of the device of the invention, typically a programmable computer, for example of the microcomputer or server type.

[0288] Phases 1 to 3, which are performed prior to the real-time phase, are also executed in a computer, generally once each. This computer may be the onboard computer used subsequently for 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 vessel (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

Demands

1. A state feedback control method for the propulsion means of a ship for its dynamic positioning, wherein: - 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 ship's axes 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 (^aug) and a state feedback gain (Kc),The coefficients of the linear quadratic (LQ) control system having been determined by implementing an LQA or LQB type tuning methodology, the LQA type tuning methodology having only two tuning parameters (ka) and the LQB type tuning methodology having only one tuning parameter, the LQ control law with Q-filter having determinable singular value curves of its complementary sensitivity transfer function, - the positioning of the ship is controlled according to said LQ control law with Q-filter, in which the Q-filter has determined gains to create a notch at a determined frequency fo in the singular value curves of the complementary sensitivity transfer function of said LQ control law with Q-filter.

2. Method according to claim 1, wherein said determined frequency f0 is the dominant frequency of a swell to which the ship is subjected.

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

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 wave frequency to which the ship is subjected are used for positioning control.

5. Method according to claim 4, wherein, in the case where none of the notch frequencies of 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.

6. 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 on signals from an estimation by a state observer of additive disturbances further implemented or on a heaving signal from a heaving detector further implemented.

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

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

9. A method according to any one of claims 1 to 8, wherein the ship's positioning 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 with: U = ( -(I+MQ)(I-KcNQy1Kc) (Xaug-Xaug) where Xmg is an augmented vector of setpoints, 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, and in the case where the ship's positioning is also controlled based on the forces and torques of a wind acting on the ship, and the command U is calculated with: U = ( -(I + MQ')(I-KCNQ)->KC) (^-^)+ where Twind is derived from wind speed measurements by at least one sensor on the ship.

10. A dynamic ship positioning system by state feedback control of the ship's propulsion means, 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 to 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 (^aug) and a state feedback gain (Aj,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 (wa, ka) and the LQB type tuning methodology having only one tuning parameter, 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 gains determined to create a notch at a determined frequency fo in the singular value curves of the complementary sensitivity transfer function of said LQ control law with Q-filter.

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

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