Copper smelting process furnace temperature control method based on iterative preset performance function

Abstract

The invention discloses a copper smelting process furnace temperature control method based on an iterative preset performance function, and the method comprises the steps: defining the actual furnace temperature of a smelting furnace and a derivative thereof as state variables, so as to establish a copper smelting process smelting furnace temperature system in a state space expression form; determining the number of iterations of a preset performance function based on iteration according to the required control precision: under the condition that the number of iterations i is equal to 1, constructing the preset performance function of the first iteration according to a traditional preset performance function and a first auxiliary function; under the condition that the number i of iterations is greater than 1, constructing a preset performance function of the ith iteration according to the preset performance function of the (i-1) th iteration and a second auxiliary function; establishing a virtual controller according to a preset performance function based on iteration; and according to the virtual controller, a controller based on an iterative preset performance function is established to serve as input of the furnace temperature system of the smelting furnace in the copper smelting process, and the furnace temperature system of the smelting furnace in the copper smelting process tracks the expected furnace temperature of the smelting furnace. According to the method, the iterative improvement of the system control precision is realized, and the iterative process of precision improvement is accurately described.

Description

Technical Field

[0001] The invention relates to a furnace temperature control method for a copper smelting process based on an iterative preset performance function, and belongs to the technical field of copper smelting process control. Background Art

[0002] In recent years, non-ferrous metals, as an indispensable raw material for the modern industrial system, have been widely used in many fields such as electronics, machinery manufacturing, and aerospace. As an important basic raw material in my country, the smelting process of non-ferrous metal copper has a huge impact on the development of the national economy. At present, the methods of copper smelting in my country are mainly electrolytic and pyrometallurgical. Compared with the electrolytic method, the traditional pyrometallurgical equipment has low investment cost and high output, so it occupies a dominant position in metal smelting. However, pyrometallurgical smelting has the problems of high operating cost and serious environmental pollution, and the instability of the furnace temperature will lead to reduced product quality and production efficiency. Therefore, it has become a top priority for the non-ferrous metal industry to study the intelligent control method of furnace temperature in the copper smelting process to improve the production efficiency of pyrometallurgical smelting and reduce energy consumption.

[0003] There are various types of existing intelligent furnace temperature control methods, such as real-time control of metal smelting process based on data-driven modeling, PID furnace temperature control and adaptive furnace temperature control. However, the high process standards in the copper smelting process require accurate control of the smelting temperature to achieve the expected target when performing furnace temperature control. Among them, the superior performance of the adaptive control method in real time adjusting the control strategy according to the furnace temperature feedback information has attracted many scholars to conduct relevant research on it. For adaptive furnace temperature control, how to design adaptive laws, develop adaptive control strategies with preset furnace temperature targets, improve the accuracy and real-time performance of the control process, and ensure that the furnace temperature reaches the expected target is still a task to be solved. The control accuracy of the furnace temperature has a direct impact on the smelting output. Low control accuracy will lead to reduced output and reduced production capacity. Therefore, how to improve the control accuracy of the furnace temperature and reduce the control error has attracted widespread attention in the academic and industrial fields.

[0004] In order to improve the control accuracy of furnace temperature, reducing control errors and enhancing system transient performance are currently more commonly used methods. Some traditional adaptive furnace temperature control strategies are designed to reduce control errors, such as adaptive PID control, adaptive control based on reinforcement learning, neural network adaptive control, etc. The above control strategies reduce errors by compensating for errors caused by system disturbances or estimating unknown system dynamics. Another method is to ensure that the control error converges to the desired range and enhance the transient performance of the system by presetting the system performance. Both methods can effectively improve the control accuracy of the furnace temperature control system.

[0005] In the furnace temperature system, enhancing the transient performance of the system can achieve accurate control of the temperature. The commonly used system performance preset scheme is to use a preset performance function. The preset performance function ensures that the system control error converges to the expected range. If the control error exceeds the preset boundary, the furnace temperature control accuracy will be reduced, resulting in a decrease in smelting output. Therefore, for the furnace temperature system, it is very important to consider the performance preset during the smelting process. And whether the transient performance of the furnace temperature system can be enhanced to achieve accurate control of the smelting temperature is a key issue in improving the system control accuracy. Summary of the invention

[0006] The present invention provides a furnace temperature control method for a copper smelting process based on an iterative preset performance function. A virtual controller is established by introducing an iterative preset performance function, and then a controller based on the iterative preset performance function is constructed, so that the furnace temperature system of the copper smelting process can track the desired furnace temperature.

[0007] The technical solution of the present invention is:

[0008] According to a first aspect of the present invention, there is provided a method for controlling furnace temperature in a copper smelting process based on an iterative preset performance function, the method comprising the following steps:

[0009] The actual furnace temperature and its derivatives are defined as state variables to establish a furnace temperature system for the copper smelting process in the form of state space expression.

[0010] According to the required control accuracy, the number of iterations of the iterative preset performance function is determined: when the number of iterations i=1, the preset performance function of the first iteration is constructed according to the traditional preset performance function and the first auxiliary function; when the number of iterations i is greater than 1, the preset performance function of the i-th iteration is constructed according to the preset performance function of the i-1th iteration and the second auxiliary function;

[0011] According to the preset performance function based on iteration, a virtual controller α is established; according to the virtual controller α, a controller based on the preset performance function based on iteration is established as the input of the furnace temperature system of the copper smelting process, so as to realize the tracking of the desired furnace temperature by the furnace temperature system of the copper smelting process.

[0012] Furthermore, define the actual furnace temperature x1 and its derivative is a state variable, the furnace temperature mathematical model is expressed as the following state space expression, that is, the furnace temperature system of the copper smelting process is constructed:

[0013]

[0014] Where x = [x1, x2] T , C = [1 0], K and T1 represent the mnemonics in the mathematical model of furnace temperature; x0 is the initial state of the state space expression, x1(0), x2(0) represent the initial states of x1 and x2 respectively, and u is a controller based on an iterative preset performance function.

[0015] Furthermore, the preset performance function based on iteration is expressed as:

[0016]

[0017] Among them, ρ(t) is the traditional preset performance function, β(t) is the first auxiliary function; ρ0>ρ ∞ >0,λ>0 is an arbitrary positive design parameter, e represents an exponential constant; θ is an arbitrarily designed normal number, κ is a normal number and κ>1; is the preset performance function for the i-1th iteration; β i-1 (t) is the second auxiliary function; I represents the maximum number of iterations; t represents time.

[0018] Furthermore, the virtual controller is expressed as:

[0019]

[0020] Where k1 is an arbitrary positive design parameter, is the preset performance function for the i-1th iteration The first derivative of is the preset performance function of the i-th iteration; e1 is the controlled furnace temperature error; is the expected furnace temperature T d The first derivative of is the first-order derivative of the traditional preset performance function ρ(t).

[0021] Furthermore, the controller based on the iterative preset performance function is expressed as:

[0022]

[0023] Among them, k2 is an arbitrary positive design parameter, x1 is the actual furnace temperature; e2 is the error between the state variable x2 and the virtual controller α; K and T1 represent the mnemonics in the mathematical model of furnace temperature; τ is the constant lag time; First-order derivative of the virtual controller α.

[0024] According to a second aspect of the present invention, there is provided a terminal device, comprising a memory, a processor, and a program stored on the memory and executable by the processor, wherein when the processor executes the program, a method for controlling furnace temperature in a copper smelting process based on an iterative preset performance function as described in any one of the above is implemented.

[0025] The beneficial effects of the present invention are as follows: for the furnace temperature system, its transient performance will directly affect the accuracy of the control system; and the present invention, according to the different control accuracies, establishes an iterative preset performance function to design a controller, thereby achieving the purpose of enhancing the transient performance of the system and improving the control accuracy of the system, and realizing that the change of control accuracy is accurately characterized by a design parameter in the iterative performance function, that is, given a preset performance function, the control accuracy of the system can be iteratively enhanced by a corresponding multiple, and the control error changes from converging to the accuracy range specified by the performance function before iteration to converging to the accuracy range specified by the performance function after iteration. From the above, it can be seen that the present invention not only realizes the iterative improvement of the system control accuracy, but also accurately describes the iterative process of accuracy improvement. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 It is a flowchart of the present invention;

[0027] Figure 2 is the trajectory of β(t), 1 / β(t) about time t;

[0028] Figure 3 is the function ρ(t), Trajectory comparison;

[0029] Figure 4 It is a schematic diagram of the traditional preset performance method;

[0030] Figure 5 Schematic diagram of the furnace temperature control effect of the copper smelting process for the preset performance function of the first iteration;

[0031] Figure 6 Schematic diagram of the furnace temperature control effect of the copper smelting process for the preset performance function of the second iteration. DETAILED DESCRIPTION

[0032] In order to make the purpose, technical scheme and advantages of the embodiments of the present invention clearer, the technical scheme in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention. It should be noted that the embodiments in this application and the features in the embodiments can be combined with each other arbitrarily without conflict.

[0033] Please refer to Figure 1-Figure 6According to a first aspect of an embodiment of the present invention, a method for controlling furnace temperature of a copper smelting process based on an iterative preset performance function is provided, the method comprising the following steps: defining the actual furnace temperature of the furnace and its derivatives as state variables to establish a furnace temperature system of the copper smelting process in the form of a state space expression; determining the number of iterations of the iterative preset performance function based on the iteration according to the required control accuracy: when the number of iterations i=1 is satisfied, constructing the preset performance function of the first iteration according to the traditional preset performance function and the first auxiliary function; when the number of iterations i is satisfied greater than 1, constructing the preset performance function of the i-th iteration according to the preset performance function of the i-1th iteration and the second auxiliary function; establishing a virtual controller α based on the iterative preset performance function; establishing a controller based on the iterative preset performance function based on the virtual controller α as the input of the furnace temperature system of the copper smelting process, so as to enable the furnace temperature system of the copper smelting process to track the desired furnace temperature.

[0034] Furthermore, define the actual furnace temperature x1 and its derivative is a state variable, the furnace temperature mathematical model is expressed as the following state space expression, that is, the furnace temperature system of the copper smelting process is constructed:

[0035]

[0036] Where x = [x1, x2] T , C = [1 0], K and T1 represent the mnemonics in the mathematical model of furnace temperature; x0 is the initial state of the state space expression, x1(0), x2(0) represent the initial states of x1 and x2 respectively, and u is a controller based on an iterative preset performance function.

[0037] Furthermore, the preset performance function based on iteration is expressed as:

[0038]

[0039] Among them, ρ(t) is the traditional preset performance function, β(t) is the first auxiliary function; ρ0>ρ ∞ >0,λ>0 is an arbitrary positive design parameter, e represents an exponential constant; θ is an arbitrarily designed normal number, κ is a normal number and κ>1; is the preset performance function for the i-1th iteration; β i-1 (t) is the second auxiliary function; I represents the maximum number of iterations; t represents time.

[0040] Furthermore, the virtual controller is expressed as:

[0041]

[0042] Where k1 is an arbitrary positive design parameter, is the preset performance function for the i-1th iteration The first derivative of is the preset performance function of the i-th iteration; e1 is the controlled furnace temperature error; is the expected furnace temperature T d The first derivative of is the first-order derivative of the traditional preset performance function ρ(t).

[0043] Furthermore, the controller based on the iterative preset performance function is expressed as:

[0044]

[0045] Among them, k2 is an arbitrary positive design parameter, x1 is the actual furnace temperature; e2 is the error between the state variable x2 and the virtual controller α; K and T1 represent the mnemonics in the mathematical model of furnace temperature; τ is the constant lag time.

[0046] According to a second aspect of an embodiment of the present invention, a terminal device is provided, comprising a memory, a processor, and a program stored on the memory and executable by the processor, wherein when the processor executes the program, a furnace temperature control method for a copper smelting process based on an iterative preset performance function as described in any one of the above is implemented.

[0047] The specific process of the present invention is further described as follows:

[0048] 1. Model Building and Problem Statement

[0049] The smelting of non-ferrous metal copper is mainly achieved by controlling the temperature of the melting furnace. First, it is necessary to establish a mathematical model of the furnace temperature of the copper smelting process. The model not only needs to take into account the heating temperature of the melting furnace itself, but also needs to take into account the influence of factors such as the calorific value of the fuel, the air density, and the melting furnace material on the heating temperature. Based on this, the melting furnace used in the copper smelting process is equivalent to a dynamic thermodynamic system. According to the furnace temperature heat balance equation, the mathematical model of the furnace temperature per unit time is established as follows:

[0050]

[0051] Among them, K and T1 represent mnemonics; Q D is the calorific value of the fuel; C r ,C k ,C g ,C e ,C e ' are the average specific heat of fuel, the average specific heat of air, the average specific heat of the heated object, the average specific heat of the gas in the furnace and the specific heat of the gas outflow from the furnace; Tr ,T k ,T e ' are the average fuel temperature, the average air temperature and the furnace gas outflow temperature; G g , G e are the output of the heated object and the output of the gas in the furnace respectively; n is the air consumption coefficient, A is the gap area of ​​the furnace body; s1, s2, …s N is the thickness of the furnace material, λ1,λ2,…λ N is the thermal conductivity of different materials, N is the total number of material types; τ is the constant lag time; T(s) and B(s) are the changes in furnace temperature and fuel, respectively, and s is the Laspeyres factor.

[0052] It is known that the traditional preset performance function ρ(t) expression is:

[0053] ρ(t)=ρ0e -λt +ρ ∞ (2)

[0054] Among them, ρ0>ρ ∞ >0,λ>0 is an arbitrary positive design parameter, e represents an exponential constant; an auxiliary function β(t) is introduced, the expression is:

[0055] β(t)=κtanh(tρ(t)+θ)(3)

[0056] Among them, θ is a positive constant that can be designed arbitrarily, κ is a positive constant and κ>1; the range of ρ(t) is (ρ ∞ ,ρ0+ρ ∞ ], the range of β(t) is (-κ,κ), and the range of 1 / β(t) is are all bounded functions; in the embodiment of the present invention, κ=1000,θ=0.002,ρ0=3,λ=0.5,ρ ∞ =0.1, the range of the functions β(t) and 1 / β(t) is (-1000,1000) and Specific images such as Figure 2 shown.

[0057] In order to further improve the transient performance of the furnace temperature control system, the present invention improves on the basis of the traditional performance function ρ(t) and constructs the first iteration performance function as follows:

[0058]

[0059] Accuracy of traditional preset performance function: As time t→∞, the traditional preset performance function ρ(t) approaches ρ ∞ ,Right now It can be seen that the control accuracy of the traditional preset performance function is ε1 = ρ ∞ , the control error e1 converges to the boundary |ρ ∞ | inside, that is (-ρ ∞ <e1<ρ ∞ ).

[0060] Precision of the preset performance function of the first iteration: As time t→∞, the preset performance function of the first iteration Approaching Right now It can be seen that the control accuracy of the preset performance function of the first iteration provided by the present invention is The control error e1 converges to the boundary within, that is

[0061] The present invention innovatively designs and constructs the auxiliary function β(t) based on the traditional performance function ρ(t). On the one hand, the preset performance function of the first iteration can be Compared with the traditional performance function ρ(t), the control accuracy is improved by κ times. On the other hand, according to the preset performance function of the first iteration New auxiliary functions can be further constructed Then we can get the performance function of the second iteration with higher control accuracy: And so on, until a higher preset control accuracy is achieved. The specific expression of the performance function of the second iteration is as follows:

[0062]

[0063] Preset performance function for the second iteration After analysis, we can get the control accuracy as Compared with the traditional preset performance function ρ(t), the accuracy is improved by κ 2 times; and the performance function of the first iteration Therefore, by continuously updating the performance function and the auxiliary function, an iterative performance function with continuously improved accuracy compared to the traditional performance function can be obtained, thereby achieving iterative improvement of the system control accuracy.

[0064] The accuracy of the traditional preset performance function is determined by the parameter ρ ∞ However, in actual control problems, when the controller parameters are determined, excessively high accuracy requirements need to be adjusted by adjusting the parameter ρ ∞ To achieve this, however, ∞ If the setting is too small, the control method will fail. The iterative preset performance function proposed in the present invention can avoid this problem. It can keep the parameter ρ unchanged. ∞The required accuracy is achieved by iteratively adjusting the parameter κ.

[0065] Select parameters κ = 1000, θ = 0.002, ρ0 = 3, λ = 0.5, ρ ∞ =0.1, for the performance function of the present invention at different iteration times and the traditional preset performance function, the control accuracy of the traditional preset performance function is ε=0.1, and the control accuracy of the preset performance function of the first iteration is ε=1×10 -4 , the preset performance function control accuracy of the second iteration is ε=1×10 -7 Compared with the traditional preset performance function, the transient performance of the present invention is enhanced to 10 3 Times and 10 6 times, function ρ(t), The image is as Figure 3 shown.

[0066] Define the controlled furnace temperature error as e1 = x1-T d , x1 and its derivative are is the state variable, x1 is the actual furnace temperature, T d is the desired furnace temperature, then the furnace temperature mathematical model (1) can be expressed as the following state space expression, that is, to construct the furnace temperature system of the copper smelting process:

[0067]

[0068] Where x = [x1, x2] T , C=[1 0], x0 is the initial state of the state space expression, x1(0), x2(0) represent the initial states of x1 and x2 respectively, u is the controller, and T is the transpose.

[0069] The controller based on the above design is used as the input of the furnace temperature system of the copper smelting process to realize the actual furnace temperature x1 of the copper smelting process furnace temperature system to the expected furnace temperature T d The goal of this invention is to design a controller so that the actual furnace temperature x1 can track the desired furnace temperature T d At the same time, the closed-loop system satisfies the ultimate uniform boundedness and the convergence performance of the controlled furnace temperature error is higher than that of the traditional preset performance method, that is, e1 changes from moving inward along the boundary of the traditional preset performance function ρ(t) to moving along the iterative performance function The border moves inward represents the preset performance function of the i-th iteration.

[0070] 2. The controller design is as follows:

[0071] First, the preset performance function based on iteration is as follows:

[0072]

[0073] The controlled furnace temperature error is e1 = x1-T d , expected furnace temperature T d is a smooth and bounded temperature curve function that satisfies T d and its derivatives Continuous and bounded. Consider the following virtual controller α and controller u:

[0074]

[0075] Where k1 and k2 are arbitrary positive design parameters. The error between the virtual controller α and the state variable x2 is e2 = x2-α, is the derivative of the virtual controller α.

[0076] Select barrier function satisfy e1(0) represents the initial value of the controlled furnace temperature error. Further, the stability of equation (6) is analyzed and Theorem 1 is given as follows:

[0077] Theorem 1: Assume that there exists a given constant k0>0, for any There are |e0| <k0, represents a set of real numbers. At the same time, equation (6) is controlled by the virtual controller α and controller u of equations (8) and (9), and the closed-loop system is asymptotically stable and meets the preset system transient performance.

[0078] Consider a Lyapunov function V:

[0079]

[0080] When i=1, taking the derivative of V yields:

[0081]

[0082] When i=2,3,...I, taking the derivative of V, we get:

[0083]

[0084] Substituting virtual controller (8) and controller (9) into the equation, we get:

[0085]

[0086] Take c = min{2k1,2k2}, and we get It can be concluded that the closed-loop system is asymptotically stable, and the system control error converges to the preset iterative performance function Within the set boundaries, the control objectives of the present invention are achieved and the required controller is obtained.

[0087] In order to verify the feasibility of the method described in the present invention, the performance of the furnace temperature controller is preset using the traditional preset performance function and the iterative performance preset method proposed in the present invention. The tracking effect and transient performance curve are shown in Figure 2. Figure 4 , 5 As shown: Figure 4 The left half shows that the actual furnace temperature can track the expected furnace temperature T very well. d ; Figure 4 The right half shows that the controlled furnace temperature error e1 converges to the boundary of the traditional preset performance function ρ(t), and the preset accuracy error is ±5℃, that is, the preset accuracy is ρ ∞ =5℃. Figure 5 The left half shows that the actual furnace temperature can track the expected furnace temperature T very well. d ; Figure 5 The right half shows that the controlled furnace temperature error e1 converges to the preset performance function of the first iteration Within the boundary, the preset accuracy error is ±1°C, that is, the preset accuracy is ρ ∞ =1°C, the transient performance is enhanced by κ=5 times compared to the accuracy set by the traditional performance function.

[0088] Then, the iterative performance function proposed in the present invention is used to perform the second iteration, and the tracking effect and transient performance curve are as follows: Figure 6 As shown:

[0089] Figure 6 The left half shows that the actual furnace temperature can track the expected furnace temperature T very well. d ; Figure 6 The right half shows that the controlled furnace temperature error e1 converges to the preset performance function of the second iteration Within the boundary, the preset accuracy error is ±0.2℃, that is, the preset accuracy is ρ ∞ =0.2℃, compared with the accuracy of the traditional performance function setting, the transient performance is enhanced by κ=25 times; compared with the accuracy of the iterative preset performance function setting of the first iteration, the transient performance is enhanced by κ=5 times.

[0090] Through the above analysis, it can be seen that the actual furnace temperature of the copper smelting process will converge to the desired temperature range and the control error will not exceed the specified boundary under the constraints of the preset performance function based on iteration. At the same time, through the comparative analysis of different numbers of iterations, it can be seen that the iterative performance function designed by the present invention realizes the iterative enhancement of the system control accuracy, and the change of control accuracy can be accurately characterized by a design parameter in each iteration process.

[0091] The specific implementation modes of the present invention are described in detail above in conjunction with the accompanying drawings, but the present invention is not limited to the above implementation modes, and various changes can be made within the knowledge scope of ordinary technicians in this field without departing from the purpose of the present invention.

Claims

1. A method for controlling furnace temperature in a copper smelting process based on an iterative preset performance function, characterized in that: The method comprises the following steps: The actual furnace temperature and its derivatives are defined as state variables to establish a furnace temperature system for the copper smelting process in the form of state space expression. According to the required control accuracy, the number of iterations of the iterative preset performance function is determined: when the number of iterations i=1, the preset performance function of the first iteration is constructed according to the traditional preset performance function and the first auxiliary function; when the number of iterations i is greater than 1, the preset performance function of the i-th iteration is constructed according to the preset performance function of the i-1th iteration and the second auxiliary function; According to the preset performance function based on iteration, a virtual controller α is established; according to the virtual controller α, a controller based on the preset performance function based on iteration is established as the input of the furnace temperature system of the copper smelting process, so as to realize the tracking of the desired furnace temperature by the furnace temperature system of the copper smelting process.

2. The method for controlling furnace temperature in a copper smelting process based on an iterative preset performance function according to claim 1, characterized in that: Define the actual furnace temperature x1 and its derivative is a state variable, the furnace temperature mathematical model is expressed as the following state space expression, that is, the furnace temperature system of the copper smelting process is constructed: Where x = [x1, x2] T , C = [1 0], K and T1 represent the mnemonics in the mathematical model of furnace temperature; x0 is the initial state of the state space expression, x1(0), x2(0) represent the initial states of x1 and x2 respectively, and u is a controller based on an iterative preset performance function.

3. The method for controlling furnace temperature in a copper smelting process based on an iterative preset performance function according to claim 1, characterized in that: The preset performance function based on iteration is expressed as: Among them, ρ(t) is the traditional preset performance function, β(t) is the first auxiliary function; ρ0>ρ ∞ >0,λ>0 is an arbitrary positive design parameter, e represents an exponential constant; θ is an arbitrarily designed normal number, κ is a normal number and κ>1; is the preset performance function for the i-1th iteration; β i-1 (t) is the second auxiliary function; I represents the maximum number of iterations; t represents time.

4. The method for controlling furnace temperature in a copper smelting process based on an iterative preset performance function according to claim 1, characterized in that: The virtual controller is expressed as: Where k1 is an arbitrary positive design parameter, is the preset performance function for the i-1th iteration The first derivative of is the preset performance function of the i-th iteration; e1 is the controlled furnace temperature error; is the expected furnace temperature T d The first derivative of is the first-order derivative of the traditional preset performance function ρ(t).

5. The method for controlling furnace temperature in a copper smelting process based on an iterative preset performance function according to claim 4, characterized in that: The controller based on the iterative preset performance function is expressed as: Among them, k2 is an arbitrary positive design parameter, x1 is the actual furnace temperature; e2 is the error between the state variable x2 and the virtual controller α; K and T1 represent the mnemonics in the mathematical model of furnace temperature; τ is the constant lag time; First-order derivative of the virtual controller α.

6. A terminal device, characterized in that: It comprises a memory, a processor and a program stored in the memory and executable by the processor, wherein when the processor executes the program, the method for controlling the furnace temperature of a copper smelting process based on an iterative preset performance function as described in any one of claims 1 to 5 is implemented.

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