A method and system for consistent tracking of a nonlinear shipboard power system

Abstract

The application provides a consistency tracking method and system of a nonlinear ship group power system, relates to the technical field of cooperative control, and performs coordinate transformation on a dynamic equation of the nonlinear ship group power system, takes a second-order strict feedback equation with time-varying input time delay obtained as a dynamic equation of a follower, determines a dynamic equation of a leader, and determines a communication relationship between the leader and the follower; a backstepping method is used to design a consistency tracking controller of the nonlinear ship group power system; and the nonlinear ship group power system is controlled based on the consistency tracking controller; the application defines an error coordinate conversion containing a compensation system based on a newly designed Lyapunov-Krasovskii function, designs an adaptive controller, processes unknown non-differentiable time-varying input delay, quantizes the controller by using a lag quantizer, optimizes the controller structure by using a command filter, and improves the stability and accuracy of the consistency tracking control.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of ship group power system cooperative control, and particularly relates to a consensus tracking method and system for a nonlinear ship group power system. BACKGROUND

[0002] The statements in this section merely provide background information related to the application and do not necessarily constitute prior art.

[0003] The consensus tracking problem originates from multi-agent systems and has been widely applied in ship group control in the past few years; on the basis of a distributed tracking protocol, the state of a follower agent is required to be synchronized with that of a leader; the leader-follower consensus tracking has advantages in terms of defining the direction of consensus control, enhancing communication, and saving energy; the consensus tracking design of a ship group power system often involves the control of large ships, high-speed ships, offshore work ships, and military vessels; due to different factors such as the size of the ship, the water environment, and the weather conditions, it is difficult to determine the modeling accuracy of the ship group power system; at the same time, the communication topology between the ships is expanded from a balanced graph to an unbalanced graph, and further to a graph containing a spanning tree, which also reduces the number of communication channels.

[0004] The above problems increase the difficulty of setting a distributed controller by considering local information, in order to simplify and solve these problems, many techniques and theories have been proposed; a neural network can be used to approximate an unknown continuous function containing neighbor agent information, and since the role of a dynamic surface is to estimate the first derivative of a virtual control, it is applied to avoid the "complexity explosion" problem; in addition, in order to reduce the computational cost and communication cost between agents, a common method is to introduce an event-triggered technology, the biggest disadvantage of which is that it is difficult to avoid potential Zeno behavior, i.e., being triggered infinitely many times in a finite time; unlike the event-triggered technology, a hysteresis quantizer can smooth the signal by eliminating potential chattering, which can reduce the communication and computational cost and ensure the performance of the system.

[0005] In the study of a ship group power system, time delay is an unavoidable problem, which can occur in state variables or input controllers, and the existence of input delay can be the source of system instability, leading to unexpected reduction in system performance; compared with a system containing state time delay, the key to designing a controller for a system containing input time delay is to compensate for the delay error in the input, in fact, global adaptive stabilization of a known or unknown input delay linear system has been achieved, but for the consensus tracking control method of a nonlinear system, although some research results have been achieved, most of the research results do not consider the problem of unknown non-differentiable time-varying input delay, which affects the stability of control and increases the tracking error. SUMMARY

[0006] To overcome the above deficiencies of the prior art, the present application provides a consistency tracking method and system for a nonlinear ship group power system, defines an error coordinate transformation containing a compensation system and a command filter based on a newly designed Lyapunov-Krasovskii function, designs an adaptive controller, processes unknown non-differentiable time-varying input delays, and uses a hysteresis quantizer to quantize the controller, uses a command filter to optimize the controller structure, and improves the stability and accuracy of the consistency tracking control.

[0007] To achieve the above object, one or more embodiments of the present application provide the following technical solutions:

[0008] The present application provides a consistency tracking method for a nonlinear ship group power system in a first aspect.

[0009] A consistency tracking method for a nonlinear ship group power system comprises the following steps:

[0010] The dynamics equation of the nonlinear ship group power system is subjected to coordinate transformation, a second-order strict feedback equation with time-varying input time delay is obtained as the dynamics equation of the follower, the dynamics equation of the leader is determined, and the communication relationship between the leader and the follower is determined.

[0011] A Lypunov-Krasovskii function is designed using the backstepping method, an error coordinate transformation containing a compensation system and a command filter is defined, a radial basis neural network and a quantization function are introduced, a virtual controller and an adaptive law of each step are designed, and finally a consistency tracking controller for the nonlinear ship group power system is obtained.

[0012] The nonlinear ship group power system is controlled based on the consistency tracking controller.

[0013] Further, the second-order strict feedback equation with time-varying input time delay is specifically as follows:

[0014]

[0015] wherein, and denote the input and output of the i-th follower, k = 1, 2, denote the state of the i-th follower, τ i (t) denotes an unknown delay at time t, and are continuous functions of unknown parameters A, B, M, K.

[0016] Further, the Lypunov-Krasovskii function is an exponential Lyapunov-Krasovskii function.

[0017] Further, the error coordinate transformation with compensation system and command filter, in particular:

[0018]

[0019] z i,2 = x i,2 - ω i + λ i

[0020]

[0021] where z i,1 , z i,2 denote tracking errors, y i , y j denote output signals of ship i and ship j, respectively, y d denotes the tracking signal output by the leader, λ i denotes the compensation signal, τ m denotes the delay, a ij denotes the weight of the amount of information sent by ship j to ship i, b i denotes the weight of the amount of information sent by the leader to ship i, k and k1 are positive design constants.

[0022] Further, the virtual controller, in particular:

[0023]

[0024] where, denotes the total weight of the amount of information received by ship i, c i,1 , p i,1 are positive design constants.

[0025] Further, the adaptive law, in particular:

[0026]

[0027]

[0028]

[0029] where γ i,1 , γ i,2 , σ i,1 , σ i,2 , a i,1 , a i,2 are positive design constants.

[0030] Further, the consensus tracking controller, in particular:

[0031]

[0032] wherein, u i represents a control signal, c i,2 , p i,2 is a positive design constant.

[0033] The second aspect of the present application provides a consensus tracking system of a nonlinear ship group power system.

[0034] A consensus tracking system of a nonlinear ship group power system comprises a first construction module, a second construction module and a system control module.

[0035] The first construction module is configured to perform coordinate transformation on a dynamic equation of the nonlinear ship group power system, take a second-order strict feedback equation with time-varying input time delay obtained as a dynamic equation of a follower, determine a dynamic equation of a leader, and determine a communication relationship between the leader and the follower.

[0036] The second construction module is configured to use a backstepping method to design a Lypunov-Krasovskii function, define an error coordinate conversion containing a compensation system and a command filter, introduce a radial basis neural network and a quantization function, design a virtual controller and an adaptive law at each step, and finally obtain a consensus tracking controller of the nonlinear ship group power system.

[0037] The system control module is configured to control the nonlinear ship group power system based on the consensus tracking controller.

[0038] The third aspect of the present application provides a computer readable storage medium having a program stored thereon, which, when executed by a processor, implements the steps in the consensus tracking method of the nonlinear ship group power system according to the first aspect of the present application.

[0039] The fourth aspect of the present application provides an electronic device comprising a memory, a processor and a program stored on the memory and executable on the processor, wherein the processor implements the steps in the consensus tracking method of the nonlinear ship group power system according to the first aspect of the present application when executing the program.

[0040] The above one or more technical solutions have the following beneficial effects:

[0041] The controller designed in the present application is continuously non-differentiable, and can ensure tracking accuracy under unknown time-varying input time delay; the communication relationship topology between the leader and the follower adopts a directed graph containing only a weighted spanning tree, thereby saving communication cost; in view of the problem that consensus control design is more difficult due to the uncertainty of delay and less local information, a neural network is applied to process unknown functions containing neighbor agent information, and a command filter is used to avoid the problem of "complexity explosion".

[0042] For unknown input delay, the application takes novel compensation measures: unlike processing known delay, the application introduces a new compensation system and processes unknown delay; a Ly punov-Krasovskii function is designed, which requires the controller to be continuous but non-differentiable, which is a non-strict condition.

[0043] For nonlinear ship group power system, the application develops a quantization consistent tracking strategy, studies using a hysteresis quantizer to quantize the controller to save communication; even if the use of quantization function increases the tracking error, in order to save communication, the hysteresis quantizer is used for the controller with unknown delay, and the control accuracy is ensured.

[0044] The advantages of the additional aspects of the application will be partially given in the following description, partially become obvious from the following description, or be known by the practice of the application. BRIEF DESCRIPTION OF DRAWINGS

[0045] The drawings constituting a part of the specification of the application are used to provide further understanding of the application, the illustrative embodiments of the application and the description thereof are used to explain the application, and do not constitute improper limitation of the application.

[0046] Figure 1 The method flowchart of the first embodiment.

[0047] Figure 2 The nonlinear ship group power system structure diagram of the first embodiment.

[0048] Figure 3 The consistent tracking effect diagram of the output signal of the first embodiment.

[0049] Figure 4 The output signal error effect diagram of the first embodiment.

[0050] Figure 5 The controller trajectory diagram of the first embodiment.

[0051] Figure 6 The adaptive law effect diagram of the first embodiment.

[0052] Figure 7 The system structure diagram of the second embodiment. DETAILED DESCRIPTION

[0053] The application will be further described below in combination with the drawings and embodiments.

[0054] The overall idea proposed by the application is:

[0055] For nonlinear systems, the Laplace transform and Pade approximation approach are usually used to estimate small unknown constant input delays, or by considering finite integral type signals, it is obtained that the signal containing time-varying time delay is bounded, or the new type of compensation system can eliminate the influence of constant input delay on the design of the controller, but how to extend these methods and results to nonlinear ship power systems with variable time and saturated input is the research focus of the present application; on the other hand, the Lypunov-Krasovskii functional design is complex and depends on the system structure, but plays an important role in dealing with input delay, and the existing literature has analyzed the stability of the system by constructing the qualitative integral and double integral type Lypunov-Krasovskii functional (where the integral function related to the controller is continuously differentiable), so the design of Lypunov-Krasovskii functional is also one of the research focuses of the present application.

[0056] The present application discusses the uniform tracking problem of nonlinear ship group power systems with weighted communication graph and unknown variable time input delay, considers the exponential Lypunov-Krasovskii type function by applying a new type of compensation system, and applies the lag quantizer to save communication cost.

[0057] Some parameters in the embodiments are explained as follows:

[0058] a ij represents the weight of the information amount sent by ship j to ship i, b i represents the weight of the information amount sent by the leader to ship i, represents the total weight of the information amount sent by other ships to ship i, k and k1 are positive design constants, c i,1 , c i,2 , p i,1 , γ i,1 , σ i,1 , γ i,2 , p i,2 , σ i,2 , a i,1 , a i,2 are all positive design constants, z i,1 , z i,2 represents the tracking error.

[0059] Embodiment one

[0060] In one or more embodiments, a uniform tracking method for a nonlinear ship group power system is disclosed, as shown in Figure 1 , comprising the following steps:

[0061] Step S1: performing coordinate transformation on the dynamic equation of the nonlinear ship group power system, using the obtained second-order strict feedback equation with time-varying input delay as the dynamic equation of the follower, determining the dynamic equation of the leader, and determining the communication relationship between the leader and the follower;

[0062] Step S2: Using the backstepping method, a Lypunov-Krasovskii function is designed to define the error coordinate transformation including the compensation system and command filter. A radial basis function neural network and quantization function are introduced to design the virtual controller and adaptive law for each step, and finally a consistent tracking controller for the nonlinear ship power system is obtained.

[0063] Step S3: Based on the consistency tracking controller, the nonlinear ship group power system is controlled.

[0064] The following describes in detail the implementation process of the consistency tracking method for the nonlinear ship group power system of this embodiment.

[0065] The nonlinear ship group dynamic system consists of a leader and N followers. The structural diagram is as follows: Figure 2 As shown in Figure 2, the dynamic equations of the nonlinear ship dynamic system are as follows:

[0066]

[0067] Where y represents the heading angle, i.e., the output signal of the nonlinear ship power system, u represents the input signal of the nonlinear ship power system, i.e., the control signal, τ is the time delay, i represents the i-th follower, represents the first-order derivative of the heading angle of the i-th follower, is the second-order derivative, M is the ship mass, K is the hydrodynamic constant, and the control input signal u of the follower is designed by inversely designing the signal y given by the leader i , so that the output of the follower is consistent with that of the leader.

[0068] Based on the above nonlinear dynamic equation of the ship group power system, by coordinate transformation x i,1 =y i , The above nonlinear dynamic equation of the ship group is transformed into a second-order strict feedback equation with time-varying input delay, that is, the dynamic equation of the follower, which is specifically:

[0069]

[0070] Among them, u i (t-τ i )∈R and y i ∈R represents the input and output signals of the i-th follower agent; is the state variable of the i-th follower agent, containing n i states, x i,k represents the kth state of the i-th follower agent; τ i is an unknown delay of the ith follower agent that varies over time, and Represents a continuous function of the unknown parameters A, B, K, and M of the i-th follower ship.

[0071] The dynamic equation of the leader is:

[0072]

[0073] y d =x d (4)

[0074] in, Assumption 2 is satisfied, y d Represents the tracking signal output by the leader, x d represents the state of the leader, f d represents a continuous function.

[0075] The following assumptions are made for the kinetic equations:

[0076] Assumption 1: The time-varying delay τ(t) has known bounds 0<τ(t)<τ m , and the derivative of τ is upper bounded to Here, β is an unknown positive constant.

[0077] Assumption 2: There exists a continuous function f(·) and a positive constant C d , so that for all t≥t0, two restrictions are satisfied, namely the inequality and the inequality |x d (t)|≤C d Both are true.

[0078] Assumption 3: Function The sign of is known, and there exists a positive constant and f , making Without loss of generality, let f i >0.

[0079] Assumption 4: Communication relationship topology between followers and leaders Contains at least one spanning tree with the leader node as its root.

[0080] Based on the dynamic equations of the follower and the leader, a consistency tracking controller is constructed. The specific steps are as follows:

[0081] I. Definition of error coordinate transformation with compensation system and command filter:

[0082]

[0083] z i,2 = x i,2 - ω i + λ i (6)

[0084] where ω i is the command filter, λ i is the compensation of the i-th follower ship, and is specifically:

[0085]

[0086]

[0087] ζ i,2 = ω i - α i,1 (9)

[0088] α i,1 (0) = ω i (0) (10)

[0089] where α i,1 represents the virtual control signal of the virtual controller, ω i is the control signal output by the filter, ζ i,2 represents the error between the control signal output by the filter and the virtual control signal, and q(·) is a quantization function; the original controller is replaced by the controller after the action of the quantization function, i.e., the original controller u(t-τ(t)) is replaced by the quantizer q(u(t-τ(t))), and is specifically:

[0090] q(u(t-τ(t))) = u(t-τ(t)) + d (11)

[0091] where q is a quantizer, u is a controller, and d is a number.

[0092] The hysteresis uniform quantizer is selected as follows:

[0093]

[0094] where u th > 0 is the switching threshold of q1(u) and q2(u), and q1(u), q2(u) are specifically:

[0095]

[0096]

[0097] where uth u i or u i = p 1-i u min , 0 < p < 1, i = 1, 2,..., u min > 0 is the dead-zone width; d = (1 - p) / (1 + p); l is the hysteresis width of q2(u), 0 < l < (1 / 2) d u th .

[0098] II. Two-step adaptive controller design procedure is realized by using backstepping method:

[0099] Step 1:

[0100] The derivative of the local tracking error (5) for the follower i is taken, which is:

[0101]

[0102] The sub-Lyapunov function is chosen as follows:

[0103]

[0104] And its first-order derivative is:

[0105]

[0106] The monotonicity and boundedness of the error are judged by the derivative, and finally it is proved that the error will become smaller and keep in a small range.

[0107] According to assumption 2, for any given e i,1 > 0, we can get:

[0108]

[0109] Therefore, combining (14) - (16), we get:

[0110]

[0111] where H i,1 (Y i,1 ) is a continuous function about Y i,1 = (x i,1 , x j,1 , x j,2 , x d ) T , which is:

[0112]

[0113] Approximating the unknown continuous function H containing neighbor follower information using radial basis function neural network (RBFNN) i,1 (Y i,1 ), the specific formula is:

[0114] H(Y)=W T S σ,a (Y)+δ(Y) (19)

[0115] Among them, W T =[W1,W2,…,W k ] represents the learned weight vector, is the Gaussian basis function, representing the k neurons of the network, and δ(Y) represents the approximation error, such that |δ(Y)|≤ε,ε>0.

[0116] Due to continuous The radial basis function neural network can be approximated by formula (19), so, combined with Young's inequality, we get:

[0117]

[0118]

[0119] Among them, p i,1 is a positive constant; at the current stage, select the virtual control input α i,1 And the adaptive law is as follows:

[0120]

[0121]

[0122] Among them, c i,1 is a positive design constant.

[0123] Substituting formulas (20)-(23) into formula (17), we obtain:

[0124]

[0125] in,

[0126] Step 2:

[0127] The local tracking error of follower i, that is, formula (5), is derived as follows:

[0128]

[0129] The selected Lyapunov function is as follows:

[0130]

[0131] where

[0132]

[0133]

[0134] V i,2 (t) is derived as

[0135]

[0136] where The adaptive controller is designed as

[0137]

[0138]

[0139]

[0140] It can be concluded that

[0141]

[0142] where the unknown constant Using the generalized Young inequality, we have

[0143]

[0144]

[0145] Applying (32) and (33), we have

[0146]

[0147] where

[0148] Combining the above two-step design process, the total Lyapunov function is defined as

[0149]

[0150] And the convergence compact set of the neural network is

[0151]

[0152] where i = 1, …, N, N is the number of followers, and C is a positive design parameter.

[0153] III. Proof that the tracking error of the system will remain within a small range under the designed controller and adaptive law:

[0154] Taking the derivative of equation (45), we get

[0155]

[0156] From equations (8)-(9), we have

[0157]

[0158] where, In the compact set Ω i is a continuous function as follows:

[0159]

[0160] Therefore, there is a constant M on Ωi, such that

[0161] Combining equations (34)-(38), we get

[0162]

[0163] It is easy to get

[0164]

[0165] Select So that

[0166]

[0167] Therefore, from the adaptive controller equations (28)-(30), (32) and (7), the tracking error and control signal u are bounded.

[0168] Theorem 1: Based on assumptions 1-4, applying control input (38) and adaptive law (33) (29) (41), the ship group power system (1)-(4) can achieve adaptive uniform tracking behavior.

[0169] To prove the effectiveness of the uniform tracking controller, the following simulation experiment is performed:

[0170] A set of ship power system models is selected, containing 1 leader and 3 followers, which are as follows:

[0171]

[0172] Where, ξ represents the heading angle and is also the output signal, u represents the input signal, A = 0.2, B = 1.85, M = 0.12 is the mass of the ship, and K = 0.28 is the fluid mechanics constant.

[0173] Let x i,1= ξ i and The dynamics equation of the follower is:

[0174]

[0175]

[0176] and the dynamics equation of the leader is defined as:

[0177]

[0178] y d = 0.5 (sint+ sin0.5t).

[0179] Assume that A, B, M, K are unknown, for analysis only.

[0180] In the simulation, the time-varying time delay τ1=0.06+0.01sin2t(s), τ2=0.07+0.01sin2t(s) and τ3=0.08+0.01sin2t(s) are selected, the maximum time delay τ m = 0.1 (s), the initial data ω i (0) = 0.1, λ i (0) = 0.1 and for all agents i = 1, 2, 3, and the design parameters are selected in Table 1.

[0181] Table 1 Design parameters

[0182]

[0183] The simulation results are as follows:

[0184] Figure 3 The performance of the outputs y i of the three ships and the output y d of the leader are shown, and the three ships can be consistent with the behavior of the leader in a short time; it can be seen from Figure 4 that the system error converges to a small bounded value in a limited time; Figures 5-6 The trajectory of the controller u i (t-τ i ) with time-varying time delay and the curve of the adaptive law are depicted, and the decay speed of the adaptive law is fast, with high stability. Although each ship has different delays, they still track the reference signal well.

[0185] Example Two

[0186] The embodiment discloses a consistency tracking control system of a nonlinear ship group power system.

[0187] As shown in the figure, a consistency tracking control system of a nonlinear ship group power system comprises a first construction module, a second construction module and a system control module: Figure 7

[0188] The first construction module is configured to perform coordinate transformation on a dynamic equation of the nonlinear ship group power system, take a second-order strict feedback equation with time-varying input time delay obtained as a dynamic equation of a follower, determine a dynamic equation of a leader, and determine a communication relationship between the leader and the follower.

[0189] The second construction module is configured to utilize a backstepping method, design a Lypunov-Krasovskii function, define an error coordinate conversion containing a compensation system and a command filter, introduce a radial basis neural network and a quantization function, design a virtual controller and an adaptive law of each step, and finally obtain a consistency tracking controller of the nonlinear ship group power system.

[0190] The system control module is configured to control the nonlinear ship group power system based on the consistency tracking controller.

[0191] Embodiment three

[0192] The purpose of the embodiment is to provide a computer-readable storage medium.

[0193] The computer-readable storage medium has a computer program stored thereon, and the program is executed by a processor to implement the steps in the consistency tracking method of the nonlinear ship group power system according to the embodiment one of the present disclosure.

[0194] Embodiment four

[0195] The purpose of the embodiment is to provide an electronic device.

[0196] The electronic device comprises a memory, a processor and a program stored in the memory and executable on the processor, and the processor executes the program to implement the steps in the consistency tracking method of the nonlinear ship group power system according to the embodiment one of the present disclosure.

[0197] The above only describes the preferred embodiments of the present disclosure and is not used to limit the present disclosure. The present disclosure can have various modifications and changes for those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present disclosure shall be included in the protection scope of the present disclosure.​

Claims

1. A consistency tracking method for a nonlinear ship dynamic system, characterized in that: include: The coordinate transformation is performed on the dynamic equations of the nonlinear ship group power system. The obtained second-order strict feedback equation with time-varying input delay is used as the dynamic equation of the follower to determine the dynamic equation of the leader and the communication relationship between the leader and followers. Using the backstepping method, a Lypunov-Krasovskii function is designed, and the error coordinate transformation including the compensation system and command filter is defined. A radial basis function neural network and quantization function are introduced, and a virtual controller and adaptive law are designed for each step. Finally, a consistent tracking controller for the nonlinear ship power system is obtained. Based on the consistency tracking controller, a nonlinear ship group power system is controlled; The error coordinate transformation including the compensation system and command filtering is specifically as follows: in, represents the tracking error, denote the output signals of ship i and ship j respectively, Represents the tracking signal output by the leader, represents the compensation signal, Indicates delay, represents the weight of the amount of information sent by ship j to ship i, represents the weight of the amount of information sent by the leader to ship i, and is a positive design constant; The virtual controller is specifically: in, represents the total weight of the information received by ship i, 、 is a positive design constant; The adaptive law is specifically: in, 、 、 、 、 、 is a positive design constant.

2. The consistency tracking method for a nonlinear ship dynamic system according to claim 1, characterized in that: The second-order strict feedback equation with time-varying input delay is specifically: in, and represents the input and output of the ith follower, , represents the state of the i-th follower, represents the unknown delay at time t, and It is a continuous function with unknown parameters A, B, M, and K.

3. The consistency tracking method for a nonlinear ship dynamic system according to claim 1, characterized in that: The Lypunov-Krasovskii function is an exponential Lyapunov-Krasovskii function.

4. The consistency tracking method for a nonlinear ship dynamic system according to claim 1, characterized in that: The consistency tracking controller is specifically: in, Indicates the control signal, 、 is a positive design constant.

5. A consistency tracking system for a nonlinear ship power system, characterized in that: The method according to any one of claims 1 to 4 comprises a first building module, a second building module and a system control module: The first building block is configured to: perform coordinate transformation on the dynamic equations of the nonlinear ship group power system, use the obtained second-order strict feedback equation with time-varying input delay as the dynamic equation of the follower, determine the dynamic equation of the leader, and determine the communication relationship between the leader and the follower; The second building block is configured to: design a Lypunov-Krasovskii function using backstepping, define an error coordinate transformation including a compensation system and command filtering, introduce a radial basis function neural network and a quantization function, design a virtual controller and adaptive law for each step, and ultimately obtain a consistent tracking controller for the nonlinear ship power system; The system control module is configured to control the nonlinear ship group power system based on the consistency tracking controller.

6. An electronic device, comprising: a memory for non-transitory storage of computer-readable instructions; as well as a processor for executing said computer-readable instructions, When the computer-readable instructions are executed by the processor, the method according to any one of claims 1 to 4 is executed.

7. A storage medium, characterized by non-transitory storage of computer-readable instructions, wherein: When the non-transitory computer-readable instructions are executed by a computer, the instructions of the method according to any one of claims 1 to 4 are executed.

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