A method for predicting and controlling toner concentration in a copier developer bin

Abstract

The present invention proposes a method for predicting and controlling the toner concentration of a developing bin of a copier, and predicts the toner concentration at a future moment. A cost function is established to measure the prediction ability, and it is simplified in a discrete space and converted into a quadratic programming form for derivation to obtain the minimum toner replenishment amount as the optimal solution for the toner replenishment amount, and the optimal solution is taken to calculate the optimal control amount for replenishment; the optimal control amount for replenishment is converted into the duty cycle of the motor control signal. The present invention combines the toner concentration differential equation with the autoregressive model to realize the prediction of the toner concentration, and then solves the mass of toner that needs to be replenished at each moment, which can reduce the variation range of the toner concentration and avoid the problem of too dark or too light color imaging of the paper image; real-time prediction and replenishment also solves the large time lag and nonlinear problems in the toner replenishment process, improves the accuracy of toner concentration control, and reduces toner waste.

Description

Technical Field

[0001] The invention relates to the field of office automation equipment and intelligent control technology, and in particular to a method for predicting and controlling toner concentration in a developing bin of a copier. Background Art

[0002] At present, when a copier is copying or printing paper, it first needs to form an electrostatic latent image on the surface of the photosensitive drum, and then the surface of the photosensitive drum absorbs the toner particles in the developing chamber to form a toner image, and finally forms a toner image on the paper after high-voltage transfer and heating. In order to ensure the color density effect of the toner image on the paper, the copier must closely monitor the image development process, because any deviation will appear in the final result, causing the toner image on the paper to be too dark or too light. When the toner concentration is too low, the image clarity is low, the readability is poor, and it is not conducive to long-term storage; when the toner concentration is too high, the image color is darker, and the toner's own charge will also be lost as the friction time increases, causing its ability to bind with the carrier to decrease. The blank part of the paper will be contaminated by this part of toner to form a bottom ash, affecting the image quality; at the same time, it will contaminate the internal components of the copier, making it difficult to clean and causing unnecessary waste. Especially in halftone images, the developing ability is directly related to the toner concentration in the developing chamber. Toner concentration refers to the ratio of the toner mass to the carrier mass in the copier's developing chamber. During the printing process, the carrier will not be consumed and its mass remains unchanged, so the toner concentration will change with the consumption of toner in the developing chamber. In order to improve the image quality of printed products, it is necessary to replenish toner from the toner cartridge to the developing chamber in a timely manner to reduce the degree of change in toner concentration in the developing chamber.

[0003] The change of toner concentration in the copier's developing bin is a complex process, with problems such as large time lag, nonlinearity, uncertainty and noise interference. At present, the method of replenishing toner by timed quantitative method and the method of replenishing toner by proportional-integral system adopted on the market cannot overcome the disadvantages of large time lag and nonlinearity, and can no longer be well applied to the toner concentration control system of the copier's developing bin. In order to control the toner concentration in the copier's developing bin, there is a toner concentration proportional-integral controller with anti-saturation compensator and Smith predictor on the market. The disadvantage of this method is that it is sensitive to noise and requires Smith predictor, which increases the system cost.

[0004] Therefore, there is an urgent need for a method that can control the concentration of toner in the developing chamber to improve the image quality of printed products and overcome the shortcomings of large time lag, nonlinearity, uncertainty and interference in the printing process. Summary of the invention

[0005] In view of this, the present invention provides a method for predicting and controlling the toner concentration in the developing bin of a copier, which can predict and control the toner concentration in the developing bin, optimize image quality, and achieve real-time, accurate, and highly robust control effects.

[0006] In order to achieve the above-mentioned object of the invention, the technical solution of the present invention is:

[0007] A method for predicting and controlling toner concentration in a developing bin of a copier, the specific steps comprising:

[0008] Step 2.1: The toner concentration sensor detects the toner concentration value x(k) at the current moment, where k is the current moment.

[0009] Step 2.2: Based on x(k) and the toner replenishment amount Δu(k) at the current moment, calculate the toner concentration sequence X(k) consisting of the toner concentrations at p future moments starting from moment k.

[0010] Step 2.3: Based on the toner concentration sequence X(k) and the set toner concentration reference value, calculate the toner concentration prediction error sequence E(k); substitute the prediction error E(k) into the optimal solution model of the toner replenishment amount:

[0011] ΔU * (k)=-(N T ×Q×N+R) -1 ×N T ×Q×M×E(k), the optimal input sequence ΔU of toner replenishment is obtained * (k).

[0012] Among them, the matrix M=[I,A,A 2 ,…A p-1 ] T ;matrix Q is the error weight matrix of the toner concentration prediction model; R is the input weight matrix of the toner concentration prediction model; A is the state matrix of the toner concentration control system; B is the input matrix of the toner concentration control system.

[0013] Step 2.4, take ΔU * The first value in (k) is used to calculate the optimal control amount u of toner replenishment at the current moment. * (k).

[0014] Step 2.5: Set u * (k) is converted into a control signal duty cycle to control the forward rotation time of the motor to complete the replenishment of carbon powder.

[0015] Step 2.6: At the next moment, repeat steps 2.1 to 2.5, each time predicting the future p moments to achieve rolling optimization of the toner concentration control system.

[0016] Furthermore, the prediction formula of the toner concentration prediction model is:

[0017] x(k+1)=A×x(k)+B×Δu(k)

[0018] Wherein, x(k+1) is the predicted concentration of toner at the next moment.

[0019] Furthermore, the optimal control amount of toner replenishment u * The calculation formula of (k) is:

[0020] u * (k) = CΔU * (k)+u(k-1)such that lb≤u * (k)≤ub

[0021] Wherein, C is a 1×m matrix, C=[1,0,…,0]; lb is the minimum mass of carbon powder replenishment, ub is the maximum mass of carbon powder replenishment, u(k-1) is the optimal control amount at the previous moment, and the initial value of the optimal control amount is the preset value.

[0022] Furthermore, - (k) is converted into the control signal duty cycle conversion formula:

[0023] d-(k)=u-(k) / Rmax

[0024] Among them, d - (k) is the optimal solution for the duration of the motor enable signal, R max It is the maximum rate at which toner is replenished from the toner cartridge to the developer chamber.

[0025] Furthermore, based on the toner concentration sequence X(k) and the set toner concentration reference value, the toner concentration prediction error sequence E(k) is calculated by subtracting the toner concentration sequence from the toner concentration reference value.

[0026] Beneficial effects:

[0027] 1. The present invention proposes a predictive control method for the toner concentration model of the copier developing bin, which establishes a toner concentration prediction model based on the toner concentration differential equation to predict the toner concentration at the future moment; establishes a cost function to measure the predictive ability of the toner concentration prediction model; simplifies and converts the cost function into a quadratic programming form in a discrete space; derives the cost function to obtain the minimum toner replenishment amount as the optimal solution of the toner replenishment amount, and calculates the optimal replenishment control amount by taking the optimal solution; converts the optimal replenishment control amount into the duty cycle of the motor control signal, and the motor drives the toner box to rotate under the action of the control signal to put toner into the developing bin to complete the replenishment of toner. The present invention combines the toner concentration differential equation with the autoregressive model to realize the prediction of toner concentration, and then solves the mass of toner that needs to be replenished at each moment, which can reduce the range of variation of toner concentration and avoid the problem of too dark or too light color imaging of paper images; real-time predictive replenishment also solves the large time lag and nonlinear problems in the toner replenishment process, improves the accuracy of toner concentration control, and reduces toner waste.

[0028] 2. The present invention predicts the toner concentration at the next moment and its adjacent moments at each moment, and can continuously perform online rolling optimization on the toner concentration control system of the copier's developing bin, thereby effectively solving the problems of large time lag, nonlinearity and uncertainty in the toner replenishment process.

[0029] 3. When establishing the cost function, the present invention solves the cost function in a discrete space, considers the prediction error caused by noise interference and system interference on the carbon powder concentration, and eliminates irrelevant items to reduce the deviation generated in the system control process, and has strong robustness against noise, system interference and input constraints.

[0030] 4. The present invention converts the mass of carbon powder replenishment into the duty cycle of the motor control signal, which is convenient for system control. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 The figure is a flow chart of the method of the present invention.

[0032] Figure 2 It is a schematic diagram of the supplementary process of the present invention.

[0033] Figure 3 Result diagram of the simulation experiment of this embodiment. DETAILED DESCRIPTION

[0034] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0035] like Figure 2 As shown, the principle of toner replenishment is:

[0036] The amount of toner added is related to the rotation time of the motor 1. Figure 3 As shown, the motor 1 can drive the toner cartridge 5 to rotate, and the magnetic roller 4 rotates with the photosensitive drum 3. The toner cartridge 5 is filled with toner particles 7, which are sprinkled during rotation and fall into the developing chamber 2 through the toner particle 7 addition port 6 on the developing chamber 2. The developing chamber 2 contains carrier particles 8, which are mixed with the fallen toner particles 7. The mixing screw 9 in the developing chamber 2 stirs the two mixed particles, so that the toner particles 7 and the carrier particles 8 rub against each other and generate opposite charges. At this time, the smaller toner particles 7 adhere to the surface of the larger carrier particles 8 to form a mixture. The mixture adheres to the magnetic roller 4 due to the effect of magnetic force, rotates with it and contacts the photosensitive drum 3. Due to the effect of the electric field force, the toner particles 7 are adsorbed onto the electrostatic latent image of the photosensitive drum 3 with a larger potential difference. The greater the potential difference between the electrostatic latent image and the toner particles 7, the more toner particles 7 are adsorbed, and the darker the color of the image. It is known that the toner concentration is the ratio of the mass of the toner particles 7 to the mass of the carrier particles 8. When the toner concentration is higher, the amount of charge generated by friction is higher, the more toner particles 7 are adsorbed on the surface of the photosensitive drum 3, and the darker the color of the image is, and vice versa. When the toner concentration is unstable, the image is uneven in depth and the developing ability is poor. The toner concentration has a direct impact on the image quality, and in order to improve the image quality, it is very necessary to maintain the toner concentration as unchanged as possible.

[0037] Therefore, the present invention proposes Figure 1 A method for predicting and controlling the toner concentration in a developing bin of a copier is shown, which is used for Figure 2 The toner concentration control system shown in the figure. The method of the present invention is explained in detail below in combination with the principle of the present invention.

[0038] Step 1: Establish a toner concentration prediction model to calculate the optimal control amount u of toner replenishment at the current moment. - (k) by:

[0039] Step 1.1: Based on the working principle of toner replenishment, establish the differential equation for toner concentration change:

[0040] t c (k+1)=t c (k)+g×[u(k-μ)-v(k)]

[0041] Where k is the current time, k = 1, 2, ..., n, n is the total number of cycles; t c (k) is the toner concentration at the current moment, t c (k+1) is the toner concentration at the next moment; μ is the delay time for replenishing toner consumption, measured in cycles; u(k-μ) is the toner replenishment at the current moment; v(k) is the toner consumption at the current moment; g is the proportional factor, which is the inverse of the carrier mass.

[0042] Step 1.2: According to the differential equation of toner concentration change, the toner replenishment amount and toner consumption are taken as independent variables, and the toner concentration is taken as the dependent variable. The toner concentration prediction model is obtained by inputting the autoregressive system model. The formula is: A(q -1 )×x(k)=B 1 (q -1 )×u(k-μ)+B 2 (q -1 )×v(k). x(k) is the toner concentration at time k, A(q -1 ) is the denominator polynomial of the toner concentration, B 1 (q -1 ) is the molecular polynomial of the carbon powder replenishment amount, B 2 (q -1 ) is the molecular polynomial of toner consumption, q -1 is the displacement operator.

[0043] in:

[0044] A(q -1 )=1+a 1 ×q -1 +...+a n ×q -p ;

[0045] B 1 (q -1 )=b 11 +b 12 ×q -1 +...+b 1m ×q -m+1 ;

[0046] B 2 (q -1 )=b 21 +b 22 ×q -1 +...+b 2m ×q -m+1 ;

[0047] p is the prediction order, and m is the control order. 1 A(q -1 ), a n A(q -1 )’s n-order coefficient; b 11 For B 1 (q -1 ), b 12 For B 1 (q -1 )’s quadratic coefficient, b 1m For B 1 (q -1 )’s m-order coefficient; b21 For B 2 (q -1 ), b 22 For B 2 (q -1 )’s quadratic coefficient, b 2m For B 2 (q -1 ) is the m-th coefficient of .

[0048] In the embodiment of the present invention, the prediction order is set to 8 and the control order is set to 3. After system identification, the coefficients of the polynomial are obtained, and after substitution, the following is obtained:

[0049] A(q -1 )=1-1.101×q -1 +0.0949×q -2 ;

[0050] B 1 (q -1 )=-45.15×q -2 +9.136×q -3 +27.49×q -4 ;

[0051] B 2 (q -1 )=-3.609×q -1 -6.165×q -2 +2.215×q -3 .

[0052] Step 1.3: Establish the cost function of the toner concentration prediction model to measure the gap between the model's predicted value and the target value and quantify the prediction performance of the toner concentration prediction model. The formula of the cost function J is as follows:

[0053]

[0054] Wherein, r(k+i|k) represents the reference value (i.e., target value) of the toner concentration at time k+i, and in the embodiment of the present invention, it is taken as 0.95. x(k+i|k) represents the predicted value of the toner concentration at time k+i, and Δu(k+i|k) represents the toner compensation amount at time k+i. Q is the model error weight matrix, and R is the model input weight matrix, both of which are diagonal matrices. In the embodiment of the present invention:

[0055]

[0056] Step 1.4: Expand and simplify the cost function in the discretized space to ignore the terms that are irrelevant to the optimization target, i.e., the toner compensation amount Δu, specifically:

[0057] In discrete space, the carbon powder concentration x(k+1) at the next moment can be expressed as:

[0058] x(k+1)=A×x(k)+B×Δu(k)

[0059] Wherein, x(k) is the toner concentration at the current moment, Δu(k) is the toner compensation amount at the current moment; A is the state matrix of the toner concentration control system, and B is the input matrix of the toner concentration control system.

[0060] In the embodiment of the present invention,

[0061]

[0062] B=[2.647-0.3381-2.523] T .

[0063] The toner density sequence at future times is expressed as:

[0064] X(k)=[x(k|k),x(k+1|k),x(k+2|k),...,x(k+p-1|k)] T ;

[0065] Among them, x(k|k) is x(k), x(k+j|k), where j=1, 2, ..., p-1 is the predicted value of the toner concentration at the jth moment at the current moment.

[0066] The toner replenishment sequence at future times is expressed as:

[0067] ΔU(k)=[Δu(k|k),Δu(k+1|k),Δu(k+2|k),...,Δu(k+e|k)] T .

[0068] Among them, Δu(k|k) is Δu(k), Δu(k+e|k), where e=1, 2, ..., m-1 is the predicted value of the toner replenishment amount at the e-th moment at the current moment, and m is the control order.

[0069] Substituting x(k+1)=A×x(k)+B×Δu(k), the toner concentration sequence is expanded to:

[0070]

[0071] The abbreviated form is: X(k)=M×x(k)+N×ΔU(k).

[0072] Among them, M=[I,A,A 2 ,…A p-1 ] T ;

[0073] Let e(k) = x(k|k) - r(k|k) be the prediction error of toner concentration. Then the prediction error sequence is:

[0074] E(k)=[e(k),e(k+1),...,e(k+p-1)]

[0075] Wherein, e(k) is the toner concentration error at the current moment, and e(k+j|k), where j=1, 2, ..., p-1, is the predicted value of the toner concentration at the jth moment under the current moment.

[0076] The discretized form of the cost function is expressed as:

[0077] J=E T (k+1)×Q×E(k+1)+ΔU T (k)×R×ΔU(k)

[0078] Among them, x(k+1|k)=Ax(k|k)+BΔu(k|k)=Ax(k)+BΔu(k|k) in E(k+1), and the carbon powder compensation amount Δu(k) is introduced into the discretized cost function and converted into a quadratic programming form:

[0079] J=E T (k)×M T ×Q×M×E(k)+ΔU T (k)×(R+N T ×Q×N)+2E T (k)×M T ×Q×N×ΔU(k).

[0080] It can be seen that the first term E T (k)×M T ×Q×M×E(k) does not contain ΔU(k), so it can be ignored when deriving ΔU(k).

[0081] Step 1.5: Derivative the cost function in step 1.4 to make its derivative value 0. When the cost function takes the minimum value, the optimal solution ΔU of the toner replenishment amount ΔU(k) ​​at the current moment is obtained. * (k) specifically:

[0082] First, derive the cost function:

[0083]

[0084] The toner replenishment amount that minimizes the cost function at the current moment is obtained as:

[0085] ΔU * (k)=-(N T ×Q×N+R)-1 ×N T ×Q×M×E(k)

[0086] Take the first value in the optimal solution and calculate the optimal control amount u of toner replenishment at the current moment * (k).

[0087] u * (k) = CΔU * (k)+u(k-1)such that lb≤u * (k)≤ub

[0088] Wherein, C is a 1×m matrix, C=[1,0,…,0]; lb is the minimum mass of carbon powder replenishment (taken as 0 in the embodiment of the present invention), and ub is the maximum mass of carbon powder replenishment (taken as 0.042g in the embodiment of the present invention).

[0089] Step 2: Perform rolling optimization control on the current toner concentration control system, specifically:

[0090] Step 2.1. The present invention uses a toner concentration sensor to detect the toner concentration value at each moment, which is recorded as x(k), x(k+1), ..., x(k+p-1) according to the moment.

[0091] Step 2.2: Input the toner concentration value x(k) and the toner replenishment amount Δu(k) at the current moment into the prediction formula of the toner concentration prediction model: x(k+1)=A×x(k)+B×Δu(k), and obtain the toner concentration sequence X(k)=[x(k|k),x(k+1|k),x(k+2|k),...,x(k+p-1|k)] at p future moments. T .

[0092] Step 2.3, subtract the current toner concentration value from the toner concentration reference value to obtain the predicted error of the toner concentration; substitute the predicted error of the toner concentration into the optimal solution formula for the toner replenishment amount:

[0093] Δu * (k)=-(N T ×Q×N+R) -1 ×N T ×Q×M×E(k), the optimal input sequence at the current moment is obtained.

[0094] Step 2.4, take ΔU * The first value in (k) is used to calculate the optimal control amount u of toner replenishment at the current moment. * (k). When u * (k) When it exceeds the value range, the closest threshold lb or ub is taken as u at the current moment. * (k).

[0095] u * (k) = CΔU * (k)+u(k-1)such that lb≤u - (k)≤ub

[0096] Step 2.5: Set u - (k) is converted into the control signal duty cycle of motor 1 to control the forward rotation time of the motor to complete the toner replenishment work. According to the working principle of the motor, the conversion formula is:

[0097] d-(k)=u*(k) / Rmax

[0098] Among them, d * (k) is the optimal solution for the duration of the motor enable signal (in seconds), R max The maximum rate of replenishing toner from the toner cartridge 5 into the developing chamber 2 is 0.14 g / s in this embodiment.

[0099] Step 2.6, at the next moment, repeat steps 2.1 to 2.5, and predict the future p moments each time to achieve rolling optimization of the toner concentration control system. If p is 5, at the initial moment (k=0), the toner concentrations at moments 1, 2, 3, and 4 are predicted, and at the next moment (i.e., moment 1), the toner concentrations at moments 2, 3, 4, and 5 are predicted.

[0100] like Figure 3 As shown, after the conventional toner concentration control system continuously prints 140 pages, the toner concentration value has a large deviation, and the image printing effect of the second half is darker than that of the first half. The predicted value of the future system output by the method of the present invention is very close to the expected value, indicating that the color effect of the toner image imaging is very small during the entire printing process, and a constant image density is obtained, indicating that compared with the conventional toner concentration control method, the present invention can control the toner concentration to reduce fluctuations and improve image quality.

[0101] In summary, the above are only preferred embodiments of the present invention and are not intended to limit the protection scope of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for predicting and controlling the toner concentration in a copier developing bin, characterized in that: The specific steps include: Step 2.1, the toner concentration sensor detects the toner concentration value x(k) at the current moment, where k is the current moment; Step 2.2, based on x(k) and the toner replenishment amount Δu(k) at the current moment, calculate a toner concentration sequence X(k) consisting of toner concentrations at p future moments starting from moment k; Step 2.3: Based on the toner concentration sequence X(k) and the set toner concentration reference value, calculate the toner concentration prediction error sequence E(k); substitute the prediction error E(k) into the optimal solution model of the toner replenishment amount: ΔU * (k)=-(N T ×Q×N+R) -1 ×N T ×Q×M×E(k), we can get the optimal input sequence ΔU of toner replenishment. * (k); Among them, the matrix M=[I,A,A 2 ,…A p-1 ] T ;matrix Q is the error weight matrix of the toner concentration prediction model; R is the input weight matrix of the toner concentration prediction model; A is the state matrix of the toner concentration control system; B is the input matrix of the toner concentration control system; Step 2.4, take ΔU * The first value in (k) is used to calculate the optimal control amount u of toner replenishment at the current moment. * (k); Step 2.5: Set u * (k) converting into a control signal duty cycle to control the forward rotation time of the motor to complete the replenishment of carbon powder; Step 2.6: At the next moment, repeat steps 2.1 to 2.5, each time predicting the future p moments to achieve rolling optimization of the toner concentration control system.

2. The method according to claim 1, characterized in that The prediction formula of the carbon powder concentration prediction model is: x(k+1)=A×x(k)+B×Δu(k) Wherein, x(k+1) is the predicted concentration of toner at the next moment.

3. The method according to claim 1, characterized in that Toner replenishment optimal control amount * The calculation formula of (k) is: u * (k)=CΔU * (k)+u(k-1)such that lb≤u * (k)≤ub Wherein, C is a 1×m matrix, C=[1,0,…,0]; lb is the minimum mass of carbon powder replenishment, ub is the maximum mass of carbon powder replenishment, u(k-1) is the optimal control amount at the previous moment, and the initial value of the optimal control amount is the preset value.

4. The method according to claim 1 or 3, characterized in that will u * (k) is converted into the control signal duty cycle conversion formula: d*(k)=u*(k) / Rmax Among them, d * (k) is the optimal solution for the duration of the motor enable signal, R max It is the maximum rate at which toner is replenished from the toner cartridge to the developer chamber.

5. The method according to claim 1, characterized in that The toner concentration prediction error sequence E(k) is calculated based on the toner concentration sequence X(k) and the set toner concentration reference value by subtracting the toner concentration sequence from the toner concentration reference value.

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