Lithium battery parameter identification method based on innovation increment detection least square method

Abstract

The invention relates to the technical field of lithium battery parameter identification, and discloses a lithium battery parameter identification method based on an innovation increment detection least square method, and the method comprises the steps: 1, setting an active self-regression model of an identified single-input single-output system: 2, setting an error vector, and setting an evaluation function; wherein the estimated value obtained when the evaluation function is the minimum value is the least square estimated value; 3, initializing the originating regression model, initializing values of k-1 times and k-2 times from the kth time, and calculating to obtain an observation vector and an estimated vector; 4, updating the gain factor and the value of the covariance matrix of the observation matrix; 5, calculating the information and the information increment, defining a characteristic parameter for judging whether to correct the forgetting factor, and modifying the forgetting factor when the characteristic parameter is greater than zero; and step 6, calculating an estimated vector, recurring a least square estimation value calculation formula, and solving battery parameters of each step based on an innovation increment detection least square method.

Description

Technical Field

[0001] The present invention relates to the technical field of lithium battery parameter identification, and in particular to a lithium battery parameter identification method based on a least squares method for new information increment detection. Background Art

[0002] In recent years, traditional fuel vehicles that use fossil fuels as energy sources are gradually being eliminated from the market because they consume a large amount of non-renewable energy and their exhaust emissions have become the main source of urban environmental pollution. Electric vehicles, as an environmentally friendly means of transportation, have begun to gradually replace traditional fuel vehicles. The remaining battery capacity (State of charge, SOC) of electric vehicles is an important part of the development of electric vehicle technology, because it is equivalent to the fuel gauge of fuel vehicles and is an important parameter of the battery management system of electric vehicles. Its accuracy directly affects the vehicle energy management control strategy and the performance of electric vehicles, and thus affects the reliability and cost of the vehicle. A suitable battery model is the basis and key to SOC estimation. The battery model can effectively reflect the corresponding relationship between the external parameters of the battery and the internal state of the battery. By establishing an accurate battery model and adopting a precise battery SOC estimation method, the vehicle energy management control strategy and the performance of electric vehicles can be improved, which is of great significance for the simulation, design and optimization of electric vehicles. In terms of battery system modeling, the low-order RC model has too few parameters and cannot accurately describe the battery characteristics. The high-order RC model has too many parameters, the calculation is complicated and will produce overfitting. At present, the second-order Thevenin equivalent circuit model is usually used. For the parameter identification of the second-order Thevenin equivalent circuit model, the most commonly used method is the exponentially weighted least squares method (EWRLS) proposed by JD Park. However, there are several problems in its use in the parameter identification process of lithium batteries:

[0003] (1) In actual lithium battery SOC estimation, the new information is usually relatively small, which causes the forgetting factor update formula to fail and lose its correction effect on the parameters.

[0004] (2) It is difficult to obtain a suitable value for the sensitivity gain. When the sensitivity gain is too large, although it has better tracking characteristics for significant changes in the system, it will amplify the impact of noise on the system, resulting in an increase in parameter identification error. When the sensitivity gain is too small, it can obtain higher accuracy under stable continuous input, but it cannot cope with large changes in the system.

[0005] (3) The algorithm updates the forgetting factor during each recursive process, but lacks a suitable evaluation mechanism to determine whether the calculated forgetting factor plays a positive role in the convergence of the system error.

[0006] Aiming at the defects of EWRLS algorithm, the present invention starts from the incremental judgment of new information and proposes a least square method based on new information incremental detection optimization to perform parameter identification of the second-order Thevenin equivalent circuit model of the battery. Summary of the invention

[0007] The purpose of the present invention is to provide a lithium battery parameter identification method based on the least squares method of innovation increment detection. On the basis of the EWRLS algorithm, a lithium battery parameter identification method based on the least squares method of innovation threshold detection is proposed.

[0008] The purpose of the present invention can be achieved through the following technical solutions:

[0009] A lithium battery parameter identification method based on the least squares method of new information increment detection is proposed. The auto-regressive model (ARX) of the single-input single-output system to be identified is set as:

[0010] A(z -1 )y(k)=B(z -1 )u(k)+w(k) (1)

[0011] in:

[0012]

[0013] The ARX model is a time series analysis method. It is a model that estimates the output y at time k based on the k-1 input and output quantities before time k. It can also be written as follows:

[0014]

[0015] Step 2: Define the error vector e and the evaluation function J as:

[0016]

[0017] The error e is the difference between the output and the estimated value, and N is the total number of measurements. The estimated value θ obtained when the evaluation function J reaches the minimum value is the least squares estimate. Therefore, the evaluation function J is derived with respect to θ:

[0018]

[0019] when When is a regular matrix, the least squares estimate is for:

[0020]

[0021] Step 3: Initialize before starting the algorithm.

[0022] Initialize the forgetting factor λ = 1, and P(0) are assigned initial values. Generally, the initial value of the covariance matrix is ​​set to P(0) = aE, and the least squares estimate is Where E is the identity matrix and a is a large real number. Assume that the algorithm starts from the kth time and initializes the values ​​of the k-1th and k-2th times.

[0023] Step 4: Calculation and θ(k).

[0024] According to the parameters calculated in the previous step and the system described by equation (4), we have

[0025]

[0026] in, is the observation vector, θ(k) is the estimated vector, which can be expressed as:

[0027]

[0028] Among them, y(k) is the system output, u(k) is the system input, and e(k) is the error. Assuming that the measurement process is N times starting from time n, it can be rewritten as:

[0029]

[0030] Where y is the output vector, ε is the noise vector, is the observation matrix, as shown in formulas (1.8) and (1.9).

[0031]

[0032] Step 5: Update the value of the gain factor K(k) and the covariance matrix P(k) of the observation matrix.

[0033] The update formula is shown in equations (14) and (15):

[0034]

[0035] Step 6: Calculate the new information and the new information increment, and define the characteristic parameter δ(k) to determine whether to correct the forgetting factor.

[0036] The new information e(k) during the system operation is defined as the difference between the observed measurement value and the predicted measurement value, and the predicted measurement value is calculated by the current prior state estimate, which is calculated by the previous a posteriori state estimate. The definition formula is shown in (15)

[0037]

[0038] The innovation increment is defined by the following formula:

[0039] Δe(k)=e(k)-e(k-1) (16)

[0040] Δe(k) is the innovation increment, which represents the difference between the kth innovation value and the k-1th innovation value. However, it is unreasonable to judge whether the system is developing in the direction of reducing error simply by the new increment. For example, the innovation value is defined as:

[0041] e(k-1)=3,e(k)=1,e(k+1)=-1 (17)

[0042] In this example, the innovation increments are as follows:

[0043] Δe(k)=-2=Δe(k+1) (18)

[0044] It is not difficult to see that the system error decreases from the k-1th time to the kth time. From the kth time to the k+1th time, the system error first decreases to 0 and then expands to 1. However, the increment of the new information cannot reflect this change in error increase. For this reason, a characteristic parameter δ(k) is introduced, which is defined as follows:

[0045] δ(k)=e(k)*Δe(k) (19)

[0046] When the characteristic parameter δ(k)>0, we believe that the new information is increasing, whether it is positive or negative, that is, the system error is expanding, so the forgetting factor λ needs to be corrected to reduce it to improve the fast tracking of the system. On the contrary, when the characteristic parameter δ(k)<0, we believe that the new information is decreasing as the recursion proceeds, and the forgetting factor at this time is appropriate. We inherit the forgetting factor of the previous recursion without changing it.

[0047] Step 7: When the characteristic parameter δ(k) in step 6 is greater than 0, the forgetting factor is modified according to the formula.

[0048] In order to obtain a suitable forgetting factor value, the following formula is proposed to calculate it:

[0049]

[0050] Where e(k) is the new information, a0, a1, a2 are all constants, e base It is the minimum accuracy of the new information. By changing different values, it can adapt to different data and parameter identification occasions. The value of the constant in the above forgetting factor evaluation function can be changed according to different needs to improve the performance of the algorithm. The basic principle of parameter selection is analyzed as follows.

[0051] Select three different sets of parameters in Table 1, such as Figure 2 As shown, the image is drawn to analyze the influence of different values ​​of a0, a1, and a2 on the forgetting factor of the system, so that different data can be used as a reference when the system uses this method for parameter identification. It should be pointed out that e base Usually 0.001 is selected to amplify the new information to avoid it being too small and causing the algorithm to fail.

[0052] Table 1 Forgetting factor formula parameter selection values

[0053]

[0054] Figure 2 The change trend of the forgetfulness factor under three different parameters is shown, and the definition domain of the new information is limited to A relatively small forgetting factor value must be maintained to reduce the weight of historical data; Substituting the new information value into the function of different parameters to obtain different forgetting factor values: λ1=0.998, λ2=0.975, λ3=0.958; When a data point falls in the light blue area in the figure, it indicates that the AFFRLS error is small. In this case, we hope that the forgetting factor value is closer to 1. In summary, λ2 and λ3 are more in line with the requirements of the forgetting factor change.

[0055] Step 8. Calculate θ(k) and calculate R0(k), R1(k), R2(k), C1(k), C2(k) based on θ(k).

[0056] The formula for calculating the recursive least squares estimate is:

[0057]

[0058] Battery parameter identification usually uses the equivalent circuit diagram of the second-order Thevenin equivalent circuit model as shown in Figure 3:

[0059] Among them, U oc is the open circuit voltage, which represents the potential difference between the positive and negative electrodes when no current flows through the battery, that is, the equilibrium electrode potential. R0 is the ohmic internal resistance, which represents the contact resistance of the electrode material, electrolyte, and diaphragm of the power battery, and will cause a jump in the terminal voltage at the moment when the current flows. R1 and C1 are the electrochemical polarization internal resistance and electrochemical polarization capacitance respectively; R2 and C2 are the concentration difference internal resistance and concentration difference capacitance respectively; U1 is the electrochemical polarization voltage; U2 is the concentration difference polarization voltage; its mathematical model can be described as:

[0060]

[0061] U=U oc -IR0-U1-U2 (23)

[0062] Discretize it and perform Laplace transform on the above two equations to get:

[0063]

[0064] The transfer function of the system is:

[0065]

[0066] Among them, τ1=R1C1, τ2=R2C2. The transfer function is discretized using the bilinear transformation principle. The bilinear transformation expression and the discrete transfer function are:

[0067]

[0068] According to formula (24), the difference equation is calculated as:

[0069] y(k)=-α1y(k-1)-α2y(k-2)+β0x(k)+β1x(k-1)+β2x(k-2) (28)

[0070] According to formula (9) and formula (10), we have:

[0071]

[0072] The inverse bilinear transform expression is:

[0073]

[0074] The discrete transfer function is subjected to bilinear inverse transformation, and equation (25) can be transformed into:

[0075]

[0076] Comparing equation (23) and equation (25), we can obtain:

[0077]

[0078] After rearranging the above formula, we get:

[0079]

[0080] Step 9. At this point, one cycle of the algorithm ends. According to this step, the least squares method based on new information increment detection can be used to solve the battery parameters of each step.

[0081] The beneficial effects of the present invention are as follows: the charge and discharge data of the lithium iron phosphate batteries in the same group are respectively used for circuit parameter identification using the new information increment detection least squares method, the least squares method with forgetting factor (Forgetting Factor Recursive Least Square, FFRLS), and the exponentially weighted least squares method (Exponentially Weighted Recursive Least Square, EWRLS) proposed in this article, and the unstable data part with SOC lower than 20% is removed. The relationship curve between the open circuit voltage and the battery capacity is fitted to a polynomial of the SOC using a built-in function of MATLAB, and the rapidity of identification and the accuracy of the identification results are significantly improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0082] The present invention will be further described below in conjunction with the accompanying drawings.

[0083] Figure 1 It is a flow chart of a lithium battery parameter identification method based on the least squares method of new information increment detection according to the present invention;

[0084] Figure 2 It is a graph showing the change of forgetting factor with new information under different parameters in a lithium battery parameter identification method based on new information increment detection least square method of the present invention;

[0085] Figure 3 It is an equivalent circuit diagram of a second-order Thevenin model in a lithium battery parameter identification method based on the least squares method of new information increment detection of the present invention;

[0086] Figure 4 It is a graph showing the change in the value of the forgetting factor in the parameter identification process of the least squares method for detecting new information increments in a lithium battery parameter identification method based on the least squares method for detecting new information increments of the present invention;

[0087] Figure 5 It is a comparison diagram of parameter identification errors of the least squares method for detecting new information increments and the FFRLS algorithm in a lithium battery parameter identification method based on the least squares method for detecting new information increments of the present invention;

[0088] Figure 6 It is a comparison diagram of parameter identification errors of IVFFRLS algorithm and EWRLS algorithm in a lithium battery parameter identification method based on the least square method of new information increment detection of the present invention;

[0089] Figure 7 It is an open circuit voltage curve in a lithium battery parameter identification method based on the least squares method of new information increment detection of the present invention;

[0090] Figure 8It is the identification result of the ohmic internal resistance R0 in the lithium battery parameter identification method based on the least square method of new information increment detection of the present invention;

[0091] Fig. 9 It is the identification result of the polarization internal resistance R1 in the lithium battery parameter identification method based on the least square method of new information increment detection of the present invention;

[0092] Fig.10 It is the identification result of polarization internal resistance R2 in a lithium battery parameter identification method based on the least square method of new information increment detection of the present invention;

[0093] Fig.11 It is the identification result of the polarization capacitor C1 in the lithium battery parameter identification method based on the least square method of new information increment detection of the present invention;

[0094] Fig.12 It is the identification result of the polarization capacitor C2 in the lithium battery parameter identification method based on the least squares method of new information increment detection of the present invention. DETAILED DESCRIPTION

[0095] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0096] See also Figure 1-Figure 12 As shown, the present invention is a lithium battery parameter identification method based on the least squares method of new information increment detection, comprising the following steps:

[0097] Step 1: Obtain the charge and discharge data of the lithium iron phosphate battery at room temperature of 25°C, and import them into MATLAB in the order of voltage, current, time, and temperature. Take the first 80% of the data for identification. When the SOC is lower than 20%, the identification is inaccurate.

[0098] The total capacity of the battery is obtained by integrating the battery charge and discharge current with respect to time, and then the SOC at each moment is determined by the ampere-hour integration method. Finally, the relationship polynomial between the open circuit voltage and the battery capacity is obtained by function fitting.

[0099] Step 2: Establish the second-order Thevenin equivalent circuit model of the battery, such as Figure 7 As shown, where U ocis the open circuit voltage, which represents the potential difference between the positive and negative electrodes when no current flows through the battery, that is, the equilibrium electrode potential. R0 is the ohmic internal resistance, which represents the contact resistance of the electrode material, electrolyte, and diaphragm of the power battery. It will cause the terminal voltage to jump at the moment when the current flows. R1 and C1 are the electrochemical polarization internal resistance and electrochemical polarization capacitance respectively; R2 and C2 are the concentration difference internal resistance and concentration difference capacitance respectively; U1 is the electrochemical polarization voltage; U2 is the concentration difference polarization voltage;

[0100] Its mathematical modules are:

[0101]

[0102] U=U oc -IR0-U1-U2 (37)

[0103] Discretize it and perform Laplace transform on the above two equations to get:

[0104]

[0105] The transfer function of the system is:

[0106]

[0107] Among them, τ1=R1C1, τ2=R2C2, the transfer function is discretized using the bilinear transformation principle, and the bilinear transformation expression and the discrete transfer function are:

[0108]

[0109] According to formula (38), the difference equation is calculated as:

[0110] U(k)=-α1U(k-1)-α2U(k-2)+β0I(k)+β1I(k-1)+β2I(k-2) (42)

[0111] According to formula (28) and formula (29), we have:

[0112]

[0113] Step 3: Initialize before starting the algorithm.

[0114] Initialize the forgetting factor λ, assign initial values ​​to θ(k) and P(k), set the initial value of the covariance matrix P(k) = aE, where a = 10000, E is the identity matrix, and the initial value of the least squares estimate θ(k) = 0. Construct the initial data vector Fill in the initial data at time k=1;

[0115] Construct the initial data vector Fill in the initial data at time k = 2. and Connect them and expand them into a 5×15878 matrix F with zero padding, which will be used starting from time k=3. Initialize the initial values ​​of the parameters to be identified. A more accurate initial value can make the algorithm converge quickly. This time, all the parameters to be identified are set to zero.

[0116] Step 4: Start the algorithm and calculate according to formula (43) Update the gain factor K(k) and the covariance matrix P(k);

[0117] According to the voltage and current values ​​of the first two recursions, calculate and fill in the formula (43)

[0118] Update the gain factor K(k) and covariance matrix P(k) according to the following recursive formulas (44) and (45):

[0119]

[0120] Step 5: Calculate the new information and the new information increment, and define the characteristic parameter δ(k) to determine whether to correct the forgetting factor;

[0121] The new information e(k) is the difference between the observed measurement value and the predicted measurement value, and the predicted measurement value is calculated by the current prior state estimate, which is calculated by the previous a posteriori state estimate. The calculation formula is shown in (46)

[0122]

[0123] The innovation increment is defined by the following formula:

[0124] Δe(k)=e(k)-e(k-1) (47)

[0125] Δe(k) is the new information increment, which represents the difference between the kth new information value and the k-1th new information value. Calculate the characteristic parameter δ(k) according to formula (48) and determine whether to update the forgetting factor;

[0126] δ(k)=e(k)*Δe(k) (48)

[0127] Step 5: If the characteristic parameter δ(k) is less than 0, the forgetting factor of the previous time is inherited without change. On the contrary, if the characteristic parameter δ(k) is greater than 0, the forgetting factor is updated according to the following formula (49). In the formula, a0=200, a1=a2=1, e base =0.001, e base It is the minimum accuracy of the new information. By changing different values, it can adapt to different data and parameter identification occasions.

[0128]

[0129] Step 6: Calculate the value of the parameter to be identified this time according to formula (50).

[0130]

[0131] Step 7: Run the recursive algorithm in a loop to complete the battery parameter identification and output the identification results as Figure 7 Open circuit voltage curve, Figure 8 Ohmic internal resistance R0 identification result, Fig. 9 Polarization internal resistance R1 identification results, Fig.10 Polarization internal resistance R2 identification results, Fig.11 Polarized capacitor C1 identification results, Fig.12 Identification results of polarized capacitor C2.

[0132] It should be noted that: the charge and discharge data of the lithium iron phosphate battery in the same group are respectively identified by using the least square method of new information increment detection, the least square method with forgetting factor (FFRLS), and the exponentially weighted least square method (EWRLS) to remove the unstable data part with SOC lower than 20%;

[0133] like Figure 4 The figure shows the change of the forgetting factor value during the parameter identification process of the least square method for new information increment detection. During the entire parameter identification process, the forgetting factor will decide whether to update or inherit the previous forgetting factor based on the value of the characteristic parameter. And when the system error is very small, there will be no failure problem, and the forgetting factor can still be effectively updated;

[0134] like Figure 5The parameter identification error comparison diagram of the least squares method for new information increment detection and the FFRLS algorithm shown in the figure shows that the accuracy of the least squares method for new information increment detection is significantly higher than that of FFRLS for parameter identification of the same set of data. After calculation, the mean absolute error (Mean Absolute Error, MAE) of the least squares method for new information increment detection is 1.42E-4, and the MAE of the FFRLS algorithm is 1.51E-4. The root mean square error (Root Mean Square Error, RMSE) of the least squares method for new information increment detection is 8.11E-4, and the RMES of the FFRLS algorithm is 8.14E-4. Since the data involved in parameter identification is relatively stable and the unstable data part with SOC below 20% is removed, the MAE and RMES of the two algorithms are at a low level, but it can still be seen that the least squares method for new information increment detection has obvious advantages in terms of accuracy. The calculation formulas for MAE and RMES are as follows:

[0135]

[0136] like Figure 6 The parameter identification error comparison diagram of the IVFFRLS algorithm and the EWRLS algorithm shown in the figure shows that when the initial values ​​of the algorithms are set to 0, the least squares method for detecting new information increments can converge faster than the EWRLS algorithm. This illustrates the rapidity advantage of the least squares method for detecting new information increments, which is that it can track better and faster when the system changes significantly. In the second half of the algorithm, when the data changes tend to be stable, it can be seen that the least squares method for detecting new information increments and the EWRLS algorithm have similar estimation error sizes, and are even more accurate than the EWRLS algorithm.

[0137] The above is a detailed description of an embodiment of the present invention, but the content is only a preferred embodiment of the present invention and cannot be considered to limit the scope of implementation of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the scope of the patent coverage of the present invention.

Claims

1. A lithium battery parameter identification method based on the least squares method of new information increment detection, characterized in that: The following steps are involved: Step 1: Set up an organic regression model for the single-input single-output system to be identified: Among them, the model estimates the output y at time k based on the k-1 input and output quantities before time k; Step 2: Set the error vector and evaluation function; Among them, the estimated value obtained when the evaluation function is at its minimum is the least squares estimate; Step 3: Initialize the regression model, starting from the kth time, initialize the values ​​of k-1 and k-2 times, and calculate the observed vector and the estimated vector; Step 4: Update the value of the covariance matrix of the gain factor and the observation matrix; Step 5: Calculate the new information and the new information increment, define the characteristic parameter to determine whether to correct the forgetting factor, and modify the forgetting factor when the characteristic parameter is greater than zero; Step 6: Calculate the estimated vector, recursively calculate the least squares estimation formula, and solve the battery parameters for each step based on the least squares method of new information increment detection.

2. A lithium battery parameter identification method based on the least squares method of innovation increment detection according to claim 1, characterized in that: In step 1, the derived regression model is: Among them, k represents the moment.

3. A lithium battery parameter identification method based on the least squares method of new information increment detection according to claim 1, characterized in that: The evaluation function is: Among them, the error e is the difference between the output and the estimated value, N is the total number of measurements, and the θ derivative of the evaluation function is obtained for J: when When is a regular matrix, the least squares estimate is for:

4. A lithium battery parameter identification method based on the least squares method of innovation increment detection according to claim 1, characterized in that: The initialization process for the regression model is: Initialize the forgetting factor λ = 1, Assign initial values ​​to P(0), set the initial value of the covariance matrix P(0) = aE, and the least squares estimate Where E is the identity matrix and a is a large real number; According to the regression model, we get in, is the observation vector, θ(k) is the estimated vector, which can be expressed as: Among them, y(k) is the system output, u(k) is the system input, and e(k) is the error; Assume that the measurement process starts at time n and is N times in total, and rewrite it as: Where y is the output vector, ε is the noise vector, is the observation matrix, as follows:

5. The lithium battery parameter identification method based on the least squares method of innovation increment detection according to claim 1 is characterized in that: The gain factor is calculated as: The value of the covariance matrix of the observation matrix is:

6. A lithium battery parameter identification method based on the least squares method of innovation increment detection according to claim 1, characterized in that: The innovation is the difference between the observed measurement and the predicted measurement, that is: The calculation formula for the innovation increment is: Δe(k)=e(k)-e(k-1); Among them, Δe(k) is the new interest increment, which represents the difference between the kth new interest value and the k-1th new interest value.

7. A lithium battery parameter identification method based on the least squares method of new information increment detection according to claim 6, characterized in that: Set characteristic parameter δ(k); δ(k)=e(k)*Δe(k); When the characteristic parameter δ(k)>0, the new information has an increasing trend whether it is positive or negative, that is, the error of the system is constantly expanding. Therefore, it is necessary to correct the forgetting factor λ to reduce it in order to improve the rapid tracking of the system. When the characteristic parameter δ(k) is less than 0, the new information decreases as the recursion proceeds, and the forgetting factor is considered appropriate.

8. The lithium battery parameter identification method based on the least squares method of innovation increment detection according to claim 6 is characterized in that: When the characteristic parameter δ(k)>0, the forgetting factor is modified according to the formula; That is, through the formula: Where e(k) is the new information, a0, a1, a2 are all constants, e base It is the minimum accuracy of the new information. By changing different values, it can adapt to different data and parameter identification occasions.

9. The lithium battery parameter identification method based on the least squares method of innovation increment detection according to claim 1 is characterized in that: The formula for calculating the recursive least squares estimate is: The second-order Thevenin equivalent circuit model is usually used for battery parameter identification; U=U oc -IR0-U1-U2; Discretize it and perform Laplace transform on the above two equations to get: The transfer function of the system is: Among them, τ1=R1C1, τ2=R2C2, the transfer function is discretized using the bilinear transformation principle, and the bilinear transformation expression and the discrete transfer function are: The calculated difference equation is: y(k)=-α1y(k-1)-α2y(k-2)+β0x(k)+β1x(k-1)+β2x(k-2); The inverse bilinear transform expression is: The discrete transfer function can be transformed into the following bilinear inverse transformation: By comparison, we can get: After sorting:

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