A method for equivalent compensation of temperature error of MEMS accelerometer

Abstract

The present invention discloses a method for equivalent compensation of temperature error of a MEMS accelerometer, performs fractional-order modeling of temperature error on the MEMS accelerometer, establishes an equivalent compensation circuit according to the fractional-order model obtained by modeling, and uses the equivalent compensation circuit to compensate the output of the MEMS accelerometer. The present invention uses an equivalent model of the temperature error model to establish a fractional-order model of the MEMS accelerometer error, thereby avoiding a complex modeling process and having a high modeling accuracy.

Description

Technical Field

[0001] The invention relates to a method for equivalent compensation of temperature error of a MEMS accelerometer. Background Art

[0002] MEMS accelerometers are widely used in inertial navigation, military, unmanned driving, consumer electronics, robotics and other fields due to their small size, light weight and low cost. The accuracy of accelerometers directly affects the control accuracy of navigation systems, unmanned driving, robotics and other systems, and affects the performance of control systems. Therefore, reducing accelerometer errors and improving accuracy are important research topics for MEMS accelerometers.

[0003] Temperature error is an important source of accelerometer error. To solve temperature error, there are usually three methods: temperature control, device design, and algorithm compensation. Among them, algorithm compensation is widely favored due to its low cost and quick results. Algorithm compensation is to establish a temperature error model, collect the temperature of the MEMS accelerometer, and use the temperature error model for compensation. However, the causes of temperature error in MEMS accelerometers are complex, which is not only related to the material and process of the device, but also to the temperature field of the device and the law of temperature field changes. It will be extremely complicated to establish a mathematical model of temperature error based on the complex mechanism of temperature error. Even if a mathematical model is established, the model is only an approximation to a certain extent.

[0004] Based on the "intermediate process" theory, any complex process has an equivalent intermediate process. Similarly, any complex differential equation also has an equivalent equation. Based on this, there is no need to explore the complex mechanism of temperature error and establish a complex mathematical model. Instead, an equivalent model (fractional order model) of the temperature error model is established. On this basis, it is simpler and more effective to study the temperature error compensation method. Summary of the invention

[0005] Purpose of the invention: In view of the above-mentioned prior art, a method for equivalent compensation of temperature error of MEMS accelerometer is proposed, which realizes temperature error compensation by designing an equivalent circuit according to the established fractional-order model.

[0006] Technical solution: A method for equivalent compensation of temperature error of a MEMS accelerometer, comprising: performing fractional-order modeling of temperature error of the MEMS accelerometer, establishing an equivalent compensation circuit according to the fractional-order model obtained by modeling, and compensating the output of the MEMS accelerometer using the equivalent compensation circuit.

[0007] Furthermore, the fractional-order modeling of the temperature error of the MEMS accelerometer includes the following steps:

[0008] Step 1.1: Place the MEMS accelerometer on the temperature characteristic test platform, set the temperature range, and read the temperature data and the temperature error output of the MEMS accelerometer in real time;

[0009] Step 1.2: Establish the fractional-order mathematical model of the MEMS accelerometer temperature error:

[0010]

[0011] Among them, k and e0 are unknown parameters, T represents temperature, T0 represents initial condition temperature, C represents Caputo-type fractional differential, and e(T) represents temperature error. represents the β-order Caputo-type fractional differential of the temperature error e(T) with respect to temperature T, 0<β<1, is defined as:

[0012]

[0013] Among them, Γ is the gamma function, s is the integral variable;

[0014] Step 1.3: Discretize the continuous MEMS accelerometer temperature error fractional-order mathematical model:

[0015]

[0016] Among them, T l is the lth sampling of temperature, the total number of sampling is L, the sampling interval is dT, e(T l ) is T l The temperature error obtained by the acquisition;

[0017] Step 1.4: According to the discrete temperature error fractional order model, establish the temperature error iteration equation:

[0018]

[0019] in:

[0020]

[0021] In the formula, m represents the independent variable;

[0022] Step 1.5: Based on the real-time collected temperature and temperature error, the cuckoo algorithm is used to identify the unknown parameters k, e0 and differential order β.

[0023] Furthermore, an equivalent compensation circuit is established according to the fractional-order model obtained by modeling, and the output of the MEMS accelerometer is compensated by using the equivalent compensation circuit, specifically including:

[0024] Step 2.1: Perform Laplace transform on the fractional-order model after parameter identification to obtain:

[0025]

[0026] Among them, the symbol stands for Laplace operator;

[0027] Step 2.2: Use the frequency equivalence relation to convert the operator s β Rationalize and obtain the equivalent model of the fractional-order model according to the differential order after Laplace transformation in step 2.1:

[0028]

[0029] Then the equivalent model of the fractional order model is rationalized;

[0030] Step 2.3: Write the equivalent model of the rationalized fractional-order model in step 2.2 in the form of sub-network multiplication;

[0031] Step 2.4: Decompose the input link of the equivalent model according to the multiplication form of the sub-network and a number of cascaded filter units;

[0032] Step 2.5: According to the input link Design the temperature acquisition input circuit and design the corresponding circuit module according to the filter unit;

[0033] Step 2.6: Use the output of the temperature acquisition input circuit as the input of the circuit module corresponding to the first filter unit, and subtract the output of the circuit module corresponding to the last filter unit in the cascade from the output of the MEMS accelerometer to obtain the temperature compensated MEMS accelerometer output.

[0034] Furthermore, step 1.5 includes:

[0035] 1.5.1: Initialization;

[0036] 1) The parameters to be identified are p{k, e0, β}, and M bird nest locations are generated in 3D space according to the following formula:

[0037] P i,j =P j,min +r0(P j,max -P j,min ) (7)

[0038] Where i = 1, 2, ..., M, j = 1, 2, 3, P i,j represents the position of the i-th bird nest in the j-th generation, P j,min , P j,max They represent the lower and upper bounds of the j-th dimension variable of the search space, r0∈[0,1] is a random number;

[0039] 2) According to formula (4), use the parameter p corresponding to the i-th bird's nest i{k, e0, β}, get the estimated value of temperature error

[0040] 3) Calculate the objective function value at the initial position:

[0041]

[0042] Among them, e i (T l ) represents the i-th temperature error measurement value, L represents the length of the sample, Obj(P i,j ) represents the objective function corresponding to the parameters obtained by the jth update of the i-th bird's nest; the p corresponding to the minimum value of the objective function i {k, e0, β} is retained to the next generation;

[0043] 1.5.2: Global random search;

[0044] 1) Use Le′vy flight to update the nest position. In the tth iteration, in the current solution P i t Based on this, the position of the bird's nest is updated according to the following formula;

[0045]

[0046] Among them, η>0 represents the step size, which is related to the scale of p{k, e0, β}, and η is taken as 1 / 10 of the scale of the corresponding variable; Stands for vector dot product; NP i t Represents P i t A new solution of ; Le′vy(λ) represents a random walk with a step size distribution, and λ represents a distribution parameter;

[0047]

[0048] Where θ is the stability index, u represents the location parameter, 1<λ≤3;

[0049] 2) According to formula (4), use the parameter P i t and NP i t , get the corresponding estimated value of temperature error;

[0050] 3) Use formula (8) to calculate the parameter P i t and NP i t The corresponding temperature error estimate Obj(P i t ) and Obj(NP it );

[0051] 4) Optimal Update:

[0052]

[0053] 1.5.3: Local search;

[0054] 1) Use formula (12) to generate a new bird's nest location:

[0055]

[0056] Among them, JP i t Indicates P i t A new solution, They represent two random nest locations in the current population, H(·) represents the Heaviside function, r and ε are random numbers uniformly distributed in the interval [0,1];

[0057] 2) Use formula (8) to calculate the parameters P i t and JP i t The corresponding temperature error estimate Obj(P i t ) and Obj(JP i t );

[0058] 3) Optimal Update:

[0059]

[0060] P i t As the best location judge Whether the iteration stop condition is reached, if so, the global optimal solution P is obtained * , otherwise, go to the next generation and enter 1.5.2 Global Random Search.

[0061] Beneficial effects: 1. The present invention uses an equivalent model of the temperature error model to establish a MEMS accelerometer error fractional-order model, thereby avoiding a complex modeling process and achieving higher modeling accuracy.

[0062] 2. The present invention uses a fractional order model to establish a temperature error model, which has fewer modeling parameters and a better model.

[0063] 3. The temperature compensation circuit designed by the present invention using an equivalent model can compensate for temperature errors in real time, avoid complex calculations, reduce computer overhead, reduce calculation delays, and improve measurement speed. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 This is a diagram of the temperature error compensation scheme for MEMS accelerometers;

[0065] Figure 2 Flowchart for fractional-order modeling of MEMS accelerometer temperature error;

[0066] Figure 3 This is the block diagram of the equivalent circuit of the temperature error of the MEMS accelerometer;

[0067] Figure 4 It is the temperature acquisition module circuit;

[0068] Figure 5 h i (s)(i=1,2,3,4,5,6)Circuit schematic diagram. DETAILED DESCRIPTION

[0069] The present invention will be further explained below in conjunction with the accompanying drawings.

[0070] like Figure 1 As shown, a method for equivalent compensation of temperature error of a MEMS accelerometer includes: performing fractional-order modeling of temperature error of the MEMS accelerometer, establishing an equivalent compensation circuit according to the fractional-order model obtained by modeling, and compensating the output of the MEMS accelerometer by using the equivalent compensation circuit.

[0071] Among them, the fractional-order modeling of temperature error of MEMS accelerometer includes the following steps:

[0072] Step 1.1: Place the MEMS accelerometer on the temperature characteristic test platform, set the temperature range, and read the temperature data and the MEMS accelerometer temperature error output in real time.

[0073] Step 1.2: Establish the fractional-order mathematical model of the MEMS accelerometer temperature error:

[0074]

[0075] Among them, k and e0 are unknown parameters, T represents temperature, T0 represents initial condition temperature, C represents Caputo-type fractional differential, and e(T) represents temperature error. represents the β-order Caputo-type fractional differential of the temperature error e(T) with respect to temperature T, 0<β<1, is defined as:

[0076]

[0077] Where Γ is the gamma function and s is the user-defined integration variable.

[0078] Step 1.3: Since the temperature error fractional modeling is modeled using discrete sampling, the continuous fractional order model is discretized:

[0079]

[0080] Among them, T l is the lth sampling of temperature, the total number of sampling is L, the sampling interval is dT, e(T l ) is T l The temperature error obtained.

[0081] Step 1.4: According to the discrete temperature error fractional order model, establish the temperature error iteration equation:

[0082]

[0083] in:

[0084]

[0085] In the formula, m represents the independent variable. If the values ​​of parameters k and e0 can be determined, the temperature error can be solved according to step 1.4.

[0086] Step 1.5: Based on the real-time collected temperature and temperature error, the cuckoo algorithm is used to identify the unknown parameters k, e0 and differential order β.

[0087] like Figure 2 As shown, step 1.5 specifically includes:

[0088] 1.5.1: Initialization;

[0089] 1) The parameters that need to be identified are p{k, e0, β}, so M bird nest locations are generated in 3D space according to the following formula:

[0090] P i,j =P j,min +r0(P j,max -P j,min ) (5)

[0091] Where i = 1, 2, ..., M, j = 1, 2, 3, P i,j represents the position of the i-th bird nest in the j-th generation, P j,min , P j,max They represent the lower and upper bounds of the j-th dimension variable of the search space, r0∈[0,1] is a random number;

[0092] 2) According to formula (4), use the parameter p corresponding to the i-th bird's nest i {k, e0, β}, get the estimated value of temperature error

[0093] 3) Calculate the objective function value at the initial position:

[0094]

[0095] Among them, e i (T l ) represents the i-th temperature error measurement value, L represents the length of the sample, Obj(P i,j ) represents the objective function corresponding to the parameters obtained by the jth update of the i-th bird's nest; the p corresponding to the minimum value of the objective function i {k, e0, β} is retained to the next generation.

[0096] 1.5.2: Global random search;

[0097] 1) Use Le′vy flight to update the nest position. In the tth iteration, in the current solution P i t Based on this, the position of the bird's nest is updated according to the following formula;

[0098]

[0099] Among them, η>0 represents the step size, which is related to the scale of p{k, e0, β}, and η is taken as 1 / 10 of the scale of the corresponding variable; Stands for vector dot product; NP i t Represents P i t A new solution of ; Le′vy(λ) represents a random walk whose step size follows a Le′vy distribution, and λ represents the distribution parameter;

[0100]

[0101] Where θ is the stability index, u represents the location parameter, 1<λ≤3;

[0102] 2) According to formula (4), use the parameters P i t and NP i t , get the corresponding estimated value of temperature error;

[0103] 3) Use formula (6) to calculate the parameter P i t and NP i t The corresponding temperature error estimate Obj(P i t ) and Obj(NP i t );

[0104] 4) Optimal Update:

[0105]

[0106] 1.5.3: Local search;

[0107] 1) Using formula (10), generate a new bird's nest location:

[0108]

[0109] Among them, JP i t Indicates P i t A new solution, They represent two random nest locations in the current population, H(·) represents the Heaviside function, r and ε are random numbers uniformly distributed in the interval [0,1];

[0110] 2) Use formula (6) to calculate the parameter P i t and JP i t The corresponding temperature error estimate Obj(P i t ) and Obj(JP i t );

[0111] 3) Optimal Update:

[0112]

[0113] P i t As the best location judge Whether the iteration stop condition is reached, if so, the global optimal solution P is obtained * , otherwise, go to the next generation and enter 1.5.2 Global Random Search.

[0114] The equivalent compensation circuit is established based on the fractional-order model obtained by modeling, including:

[0115] Step 2.1: Fractional-order model after parameter identification Performing Laplace transform yields:

[0116]

[0117] Among them, the symbol stands for Laplace operator;

[0118] Step 2.2: Use the frequency equivalence relation to convert the operator s βRationalization, such as:

[0119]

[0120]

[0121] And according to the differential order after Laplace transformation in step 2.1, the equivalent model of the fractional order model is obtained:

[0122]

[0123] Then the equivalent model of the fractional-order model is rationalized, such as β = 0.4, and we can get:

[0124]

[0125] Step 2.3: Write the equivalent model of the rationalized fractional-order model in step 2.2 in the form of sub-network multiplication:

[0126] Step 2.4: Decompose the input link of the equivalent model according to the multiplication form of the sub-network And a number of cascaded filter units; this embodiment includes 6 filter units, and the specific expression is:

[0127]

[0128]

[0129]

[0130]

[0131]

[0132]

[0133] Step 2.5: According to the input link Design temperature acquisition input circuit, such as Figure 4 As shown: The collected temperature T and 1V voltage are output through the addition circuit as follows:

[0134] Among them, R 01 =1Ω, R 02 = kΩ, R 04 =1kΩ, k and e0 come from the model identification parameters, and substituting the above parameters into the obtained Figure 4 Output -(e0+kT).

[0135] Design the corresponding circuit module according to the filter unit, such as Figure 5 As shown, the output Voi-1 of the previous subsystem is used as the input of the current subsystem, and the output is Voi.

[0136] When the parameters meet the following conditions And take R i14 =R i24 =R i23 =R i =R o =1kΩ, the Laplace transform relationship between the current subsystem output Voi and input Voi-1 satisfies:

[0137]

[0138] Step 2.6: Use the output of the temperature acquisition input circuit as the input of the circuit module corresponding to the first filter unit, and the output of the previous filter as the input of the next filter. The two are cascaded in sequence, and finally the compensation circuit output can be obtained:

[0139] (-h1(s))(-h2(s))(-h3(s))(-h4(s))(-h5(s))(-h6(s))(-(e0+kT))

[0140] =-h1(s)h2(s)h3(s)h4(s)h5(s)h6(s)(e0+kT)

[0141] The accelerometer temperature error compensation can be achieved by adding the compensation circuit output to the MEMS accelerometer output. Figure 3 shown.

[0142] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A method for equivalent compensation of temperature error of a MEMS accelerometer, characterized in that: include: Conduct fractional-order modeling of the temperature error of the MEMS accelerometer, establish an equivalent compensation circuit based on the fractional-order model obtained by modeling, and use the equivalent compensation circuit to compensate the output of the MEMS accelerometer; The fractional-order modeling of the temperature error of a MEMS accelerometer consists of the following steps: Step 1.1: Place the MEMS accelerometer on the temperature characteristic test platform, set the temperature range, and read the temperature data and the temperature error output of the MEMS accelerometer in real time; Step 1.2: Establish the fractional-order mathematical model of the MEMS accelerometer temperature error: Among them, k and e0 are unknown parameters, T represents temperature, T0 represents initial condition temperature, C represents Caputo-type fractional differential, and e(T) represents temperature error. represents the β-order Caputo-type fractional differential of the temperature error e(T) with respect to temperature T, 0<β<1, is defined as: Among them, Γ is the gamma function, s is the integral variable; Step 1.3: Discretize the continuous MEMS accelerometer temperature error fractional-order mathematical model: Among them, T l is the lth sampling of temperature, the total number of sampling is L, the sampling interval is dT, e(T l ) is T l The temperature error obtained by the acquisition; Step 1.4: According to the discrete temperature error fractional order model, establish the temperature error iteration equation: in: In the formula, m represents the independent variable; Step 1.5: Based on the real-time collected temperature and temperature error, the cuckoo algorithm is used to identify the unknown parameters k, e0 and differential order β.

2. The method for equivalent compensation of temperature error of MEMS accelerometer according to claim 1, characterized in that: An equivalent compensation circuit is established based on the fractional-order model obtained by modeling, and the output of the MEMS accelerometer is compensated by the equivalent compensation circuit, which specifically includes: Step 2.1: Perform Laplace transform on the fractional-order model after parameter identification to obtain: Among them, the symbol stands for Laplace operator; Step 2.2: Use the frequency equivalence relation to convert the operator s β Rationalize and obtain the equivalent model of the fractional-order model according to the differential order after Laplace transformation in step 2.1: Then the equivalent model of the fractional order model is rationalized; Step 2.3: Write the equivalent model of the rationalized fractional-order model in step 2.2 in the form of sub-network multiplication; Step 2.4: Decompose the input link of the equivalent model according to the multiplication form of the sub-network and a number of cascaded filter units; Step 2.5: According to the input link Design the temperature acquisition input circuit and design the corresponding circuit module according to the filter unit; Step 2.6: Use the output of the temperature acquisition input circuit as the input of the circuit module corresponding to the first filter unit, and subtract the output of the circuit module corresponding to the last filter unit in the cascade from the output of the MEMS accelerometer to obtain the temperature compensated MEMS accelerometer output.

3. The method for equivalent compensation of temperature error of MEMS accelerometer according to claim 1, characterized in that: Step 1.5 includes: 1.5.1: Initialization; 1) The parameters to be identified are p{k, e0, β}, and M bird nest locations are generated in 3D space according to the following formula: P i,j =P j,min +r0(P j,max -P j,min ) (7) Where i = 1, 2, ..., M, j = 1, 2, 3, P i,j represents the position of the i-th bird nest in the j-th generation, P j,min , P j,max They represent the lower and upper bounds of the j-th dimension variable of the search space, r0∈[0,1] is a random number; 2) According to formula (4), use the parameter p corresponding to the i-th bird's nest i {k, e0, β}, get the estimated value of temperature error 3) Calculate the objective function value at the initial position: Among them, e i (T l ) represents the i-th temperature error measurement value, L represents the length of the sample, Obj(P i,j ) represents the objective function corresponding to the parameters obtained by the jth update of the i-th bird's nest; the p corresponding to the minimum value of the objective function i {k, e0, β} is retained to the next generation; 1.5.2: Global random search; 1) Using Le′ vy Flying updates the nest position. In the tth iteration, in the current solution P i t Based on this, the position of the bird's nest is updated according to the following formula; Among them, η>0 represents the step size, which is related to the scale of p{k, e0, β}, and η is taken as 1 / 10 of the scale of the corresponding variable; Stands for vector dot product; NP i t Represents P i t A new interpretation of Le′ vy (λ) represents a random walk with a step size distribution, and λ represents the distribution parameter; Le′ vy (λ)~u=θ -λ (10) Where θ is the stability index, u represents the location parameter, 1<λ≤3; 2) According to formula (4), use the parameters P i t and NP i t , get the corresponding estimated value of temperature error; 3) Use formula (8) to calculate the parameter P i t and NP i t The corresponding temperature error estimate Obj(P i t ) and Obj(NP i t ); 4) Optimal Update: 1.5.3: Local search; 1) Use formula (12) to generate a new bird's nest location: in, JP i t Indicates P i t A new solution, They represent two random nest locations in the current population, H(·) represents the Heaviside function, r and ε are random numbers uniformly distributed in the interval [0,1]; 2) Use formula (8) to calculate the parameters P i t and JP i t The corresponding temperature error estimate Obj(P i t ) and Obj(JP i t ); 3) Optimal Update: P i t As the best location judge Whether the iteration stop condition is reached, if so, the global optimal solution P is obtained * , otherwise, go to the next generation and enter 1.5.2 Global Random Search.

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