CN114337427A - Rotational inertia identification method of recursive least square method with forgetting factor - Google Patents
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Abstract
The invention discloses a rotational inertia identification method of a recursive least square method with forgetting factors. The method comprises the following steps: writing an asynchronous motor motion equation into a recursive least square form, and determining an output variable, a parameter to be identified and an observation matrix; defining observation quantity length, forgetting factor, initialization covariance matrix and identification parameters; calculating a gain matrix of the current moment; calculating the covariance of the current moment; updating the parameter estimation value; updating the objective function value; and comparing the calculated objective function value with a preset objective function value, continuously updating, and finally calculating to obtain the motor rotational inertia information. The invention introduces the rotational inertia identification method of the recursive least square method with forgetting factors, and improves the system control performance of the motor under the conditions of load rotational inertia change and the like.
Description
Technical Field
The invention relates to the technical field of motor rotational inertia identification, in particular to a rotational inertia identification method of a recursive least square method with forgetting factors.
Background
In a modern high-performance alternating current motor speed regulation control system, a vector control technology is widely applied to high-performance control of various alternating current motors due to the advantages of excellent performance, simple and reliable method and the like. The key point of the realization of the vector control technology is decoupling, the premise of the decoupling is accurate flux linkage estimation, and the accuracy of the flux linkage estimation greatly depends on motor parameters, so that the identification of the motor parameters plays an important basic role in the vector control technology. In addition to being limited by the accuracy of identification of motor parameters, is also affected by load characteristics. In characteristics such as torque and moment of inertia of the load, the moment of inertia has a great influence on the dynamic performance of the motor operation. In a small ac motor control system, the moment of inertia of the load is typically several times or even ten times the moment of inertia of the motor rotor, and thus variations in the moment of inertia of the load can have a significant effect on the mechanical properties of the system. For example, in a multi-axis robot widely used in the field of industrial control, the moment of inertia of a motor load may change when an object is transferred, and if the moment of inertia cannot be identified in real time, the dynamic performance of the system may be affected. Therefore, online identification of the total rotational inertia of the alternating current motor control system is an effective means for improving the performance of the control system.
The least square method is a common and most basic identification method, and the thinking of the method is to select a state variable and an observation variable of a model, calculate the sum of squares of errors between an observed value and an actual value, adjust model parameters to enable the sum of squares to be minimum, and consider the model parameters to be equal to actual system parameters at the moment. The method is widely applied, can be used for dynamic and static systems, is suitable for both off-line identification and on-line identification, and has the characteristics of unbiased, consistent, effective and the like of identification results. Based on the ordinary least square method, also the recursive least square method, the weighted least square method and the like are widely applied, wherein the recursive least square method solves the defects that the ordinary least square method is complex in calculation and occupies the memory of a processor by avoiding repeated matrix calculation and matrix inversion. In the field of motor parameter identification, a motor model is simplified and equivalent to a linear model directly related to motor parameters, and then online identification is performed by using a recursive least square method. However, the recursive least squares method, when applied to time-varying parameter identification, suffers from the problem of parameter saturation. Because the time-varying parameter identification is performed on line, new observation data is continuously obtained, and the new data and the old data have the same weight in the identification process, the result of the new data identification is affected by the old data along with the identification, thereby generating larger deviation.
Disclosure of Invention
The invention aims to provide a rotational inertia identification method of a recursive least square method with a forgetting factor, so as to improve the system control performance of a motor under the conditions of load rotational inertia change and the like.
The technical solution for realizing the purpose of the invention is as follows: a rotational inertia identification method of a recursive least square method with forgetting factors comprises the following steps:
Step 2, defining observation quantity length, forgetting factor, initializing covariance matrix P (0) and parameter theta (0) to be identified;
step 3, calculating a gain matrix K (t) at the current moment;
step 4, calculating the covariance P (t) at the current moment;
step 5, updating the parameter estimation value
Step 6, updating the objective function value Jt(θ)；
Step 7, converting the objective function value Jt(theta) and a predetermined objective function value Jset() By comparison, if Jt(θ)＞Jset() If t is t +1 and returnsAnd returning to the step 3, otherwise, outputting the rotational inertia information of the motor.
Compared with the prior art, the invention has the following remarkable advantages: (1) the forgetting factor is introduced, so that the weight occupied by old data can be reduced, when the parameter to be identified is subjected to sudden change, a new value can be identified at a higher response speed, the smaller the forgetting factor is, the smaller the weight occupied by the old data is, the higher the response speed is, and the problem of parameter saturation is effectively solved; (2) the rotational inertia identification method of the recursive least square method with forgetting factors is adopted, and the system control performance of the motor under the conditions of load rotational inertia change and the like is improved.
Drawings
Fig. 1 is a flowchart of a rotational inertia recognition method of the recursive least square method with forgetting factor in the present invention.
Fig. 2 is a system block diagram of a rotational inertia identification method of the recursive least square method with forgetting factors in the present invention.
Detailed Description
With reference to fig. 1, the rotational inertia identification method of the recursive least square method with forgetting factors of the invention comprises the following steps:
Step 2, defining observation quantity length, forgetting factor, initializing covariance matrix P (0) and parameter theta (0) to be identified;
step 3, calculating a gain matrix K (t) at the current moment;
step 4, calculating the covariance P (t) at the current moment;
step 5, updating the parameter estimation value
Step 6, updating the objective function value Jt(θ)；
Step 7, converting the objective function value Jt(theta) and a predetermined objective functionValue Jset() By comparison, if Jt(θ)＞Jset() And if the value is t +1, returning to the step 3, otherwise, outputting the motor rotational inertia information.
Further, step 1, establishing a recursive least square model of the asynchronous motor, and determining an output quantity y, a parameter theta to be identified and an observation matrix
The method comprises the following specific steps:
step 1.1, constructing a system regression equation, specifically comprising:
y(i)＝θ1u1(i)+θ2u2(i)+…+θnun(i)i＝1,2,3,…,m (32)
in the formula u1(i),u2(i),…,un(i) Is at tiThe system input observed at time, y (i) is at tiSystem output of time observation; theta ═ theta1,θ2,...,θnThe parameter matrix to be identified is a regression coefficient; n is the system order;
step 1.2, converting into a matrix form, specifically:
Y＝[y(1) y(2) … y(m)]T (33)
θ＝[θ1 θ2 … θn]T (34)
e＝[e1 e2 … em]T (36)
there is a measurement equation:
Y＝Φθ+e (37)
in the formula, Y is a system output matrix, phi is a system input matrix, theta is a parameter array to be identified, and e is a residual error;
step 1.3, establishing an error in a mode of an objective function to obtain
Where J is an objective function, also known as a cost function;
step 1.4, selecting a group of estimated values of theta
Minimizing the objective function J, so deriving J from θ, making the derivative value 0, has:
obtaining:
namely a least squares estimation formula.
Further, defining observation length, forgetting factor, initialization covariance matrix P (0), and parameter θ (0) to be identified in step 2 specifically as follows:
step 2.1, discretizing an estimation formula, and describing by using a difference equation form:
wherein y (k) is the output sample value at the k-th time of the system, u (k) is the input sample value at the k-th time of the system, and k is the k-th time; a is1...an,b0...bnIs the parameter to be identified.
Step 2.2, introducing a shift operator:
A(Z-1)y(k)＝B(Z-1)u(k) (42)
in the formula (I), the compound is shown in the specification,
are all coefficient matrices, Z-xRepresenting lag x sampling periods, u being the number of output quantities and v being the number of input quantities;
the rewrite is:
wherein e (k) is a generalized error, specifically:
e(k)＝A(Z-1)y(k)-B(Z-1)u(k) (44)
step 2.3, discrete vectors and matrixes are introduced, and the formula (12) is as follows:
Y＝[y(n+1)y(n+2)…y(n+N)]T (45)
e＝[e(n+1) e(n+2) … e(n+N)]T (46)
θ＝[a1 a2 … an b0 b1 … bn]T (47)
in the formula, Y is a system output matrix, phi is a system input matrix, theta is a parameter array to be identified, e is a residual error, N is a system order, and N is an Nth moment;
arranging into a matrix of the form Y ═ Φ θ + e to obtain:
step 2.4, changing J to eTe, a complete flat mode is prepared to obtain a difference form least square estimation formula:
wherein Y is a systemOutput matrix, phi is the system input matrix, phiTIs a transpose of the input matrix of the system,
is the identification result matrix, e is the residual error;
step 2.5, adding an observation time, wherein the observed input quantity is u (N +1), the output quantity is y (N +1), and the method comprises the following steps:
θ＝[a1 a2 … an b0 b1 … bn]T (51)
e＝[e(n+1) e(n+2) … e(n+N+1)]T (52)
increasing phi (N) by one row
Y (N) is increased by an item y (N + N +1) having:
step 2.6, obtaining a parameter estimation matrix, specifically:
step 2.7, introducing matrix inversion lemma, including:
in the formula (I), the compound is shown in the specification,
is to ask for
The obtained covariance matrix is generally initialized to be a diagonal matrix, and the value of the covariance matrix is 1e 4-1 e 10; k (N) is a correction matrix;
y (N +1) is a new output observation;
is to estimate
And (4) obtaining the output estimation value of the (N +1) th time.
The formula (26) to the formula (28) is a recursive least square method identification formula.
Step 2.8, introducing a forgetting factor, which is specifically as follows:
in the formula, lambda is a forgetting factor, lambda is more than 0 and less than or equal to 1, and usually lambda is more than or equal to 0.9 and less than or equal to 0.99; when λ is 1, the degeneration is the ordinary recursive least squares method. The initial covariance matrix is set to
The forgetting factor is set to λ 0.98.
Further, the calculating of the gain matrix k (n) at the current time is specifically as follows:
further, the calculation of the covariance p (n) at the current time in step 4 is specifically as follows:
in the formula, E is a unit diagonal matrix.
Further, updating the parameter estimation value in step 5
The method comprises the following specific steps:
further, updating the objective function value J as described in step 6t(θ), specifically as follows:
the mechanical equation of the asynchronous motor is as follows:
wherein J is the system moment of inertia, ωrIs the mechanical angular velocity, T, of the rotor of the motoreIs the motor outputting an electromagnetic torque, TLIs the load torque and B is the damping coefficient.
Neglecting the damping torque and discretizing to obtain:
in the formula, T is a sampling period and is set to 1 μ s.
Neglecting the load variation term, the equation for Y Φ θ has:
finally, the rotational inertia of the system can be identified by the output electromagnetic torque of the asynchronous motor and the mechanical angular speed of the rotor.
In conclusion, the invention adopts the identification method of the rotational inertia of the asynchronous motor based on the forgetting factor recursion least square method, and improves the system control performance of the motor under the conditions of load rotational inertia change and the like.
Claims (7)
1. The rotational inertia identification method of the recursive least square method with the forgetting factor is characterized by comprising the following steps of:
step 1, establishing a recursive least square model of an asynchronous motor, and determining an output quantity y, a parameter theta to be identified and an observation matrix
Step 2, defining observation quantity length, forgetting factor, initializing covariance matrix P (0) and parameter theta (0) to be identified;
step 3, calculating a gain matrix K (t) at the current moment;
step 4, calculating the covariance P (t) at the current moment;
step 5, updating the parameter estimation value
Step 6, updating the objective function value Jt(θ)；
Step 7, converting the objective function value Jt(theta) and a predetermined objective function value Jset() By comparison, if Jt(θ)＞Jset() And if the value is t +1, returning to the step 3, otherwise, outputting the motor rotational inertia information.
2. The method for identifying the moment of inertia by the recursive least square method with forgetting factors according to claim 1, wherein the step 1 is implemented by establishing a recursive least square model of the asynchronous motor, determining the output quantity y, the parameter theta to be identified and the observation matrix
The method comprises the following specific steps:
step 1.1, constructing a system regression equation, specifically comprising:
y(i)＝θ1u1(i)+θ2u2(i)+…+θnun(i) i＝1,2,3,…,m (1)
in the formula u1(i),u2(i),…,un(i) Is at tiThe system input observed at time, y (i) is at tiSystem output of time observation; theta ═ theta1,θ2,...,θnThe parameter matrix to be identified is a regression coefficient; n is the system order;
step 1.2, converting into a matrix form, specifically:
Y＝[y(1) y(2) … y(m)]T (2)
θ＝[θ1 θ2 … θn]T (3)
e＝[e1 e2 … em]T (5)
there is a measurement equation:
Y＝Φθ+e (6)
in the formula, Y is a system output matrix, phi is a system input matrix, theta is a parameter array to be identified, and e is a residual error;
step 1.3, establishing an error in a mode of an objective function to obtain
Where J is an objective function, also known as a cost function;
step 1.4, selecting a group of estimated values of theta
Minimizing the objective function J, so deriving J from θ, making the derivative value 0, has:
obtaining:
namely a least squares estimation formula.
3. The method for identifying rotational inertia by recursive least squares with forgetting factor according to claim 1, wherein the observation length, the forgetting factor, the initialized covariance matrix P (0), and the parameter θ (0) to be identified in step 2 are defined as follows:
step 2.1, discretizing an estimation formula, and describing by using a difference equation form:
wherein y (k) is the output sample value at the k-th time of the system, u (k) is the input sample value at the k-th time of the system, and k is the k-th time; a is1…an,b0...bnIs a parameter to be identified;
step 2.2, introducing a shift operator:
A(Z-1)y(k)＝B(Z-1)u(k) (11)
in the formula (I), the compound is shown in the specification,
are all coefficient matrices, Z-xRepresenting lag x sampling periods, u being the number of output quantities and v being the number of input quantities;
the rewrite is:
wherein e (k) is a generalized error, specifically:
e(k)＝A(Z-1)y(k)-B(Z-1)u(k) (13)
step 2.3, discrete vectors and matrixes are introduced, and the formula (12) is as follows:
Y＝[y(n+1) y(n+2) … y(n+N)]T (14)
e＝[e(n+1) e(n+2) … e(n+N)]T (15)
θ＝[a1 a2 … an b0 b1 … bn]T (16)
in the formula, Y is a system output matrix, phi is a system input matrix, theta is a parameter array to be identified, e is a residual error, N is a system order, and N is an Nth moment;
arranging into a matrix of the form Y ═ Φ θ + e to obtain:
step 2.4, changing J to eTe, a complete flat mode is prepared to obtain a difference form least square estimation formula:
where Y is the system output matrix, phi is the system input matrix, phiTIs a transpose of the input matrix of the system,
is the identification result matrix, e is the residual error;
step 2.5, adding an observation time, wherein the observed input quantity is u (N +1), the output quantity is y (N +1), and the method comprises the following steps:
θ＝[a1 a2 … an b0 b1 … bn]T (20)
e＝[e(n+1) e(n+2) … e(n+N+1)]T (21)
increasing phi (N) by one row
Y (N) is increased by an item y (N + N +1) having:
step 2.6, obtaining a parameter estimation matrix, specifically:
step 2.7, introducing matrix inversion lemma, including:
wherein P (N) ([ phi ])T(N)Φ(N)]-1Is to ask for
The obtained covariance matrix is generally initialized to be a diagonal matrix, and the value of the covariance matrix is 1e 4-1 e 10; k (N) is a correction matrix;
y (N +1) is a new output observation;
is to estimate
The (N +1) th output estimation value is obtained;
formula (26) -formula (28) is a recursive least square identification formula;
step 2.8, introducing a forgetting factor, which is specifically as follows:
in the formula, lambda is a forgetting factor, and lambda is more than or equal to 0.9 and less than or equal to 0.99.
4. The method for identifying rotational inertia by recursive least squares with forgetting factors according to claim 1, wherein the step 3 of calculating the gain matrix k (n) at the current time is as follows:
5. the method for identifying rotational inertia by recursive least squares with forgetting factors according to claim 1, wherein the covariance p (n) at the current time is calculated in step 4 as follows:
in the formula, E is a unit diagonal matrix.
6. The method for identifying rotational inertia by recursive least squares with forgetting factor as claimed in claim 1, wherein the updating of the parameter estimation value in step 5
The method comprises the following specific steps:
7. the method for identifying rotational inertia by recursive least squares with forgetting factor as claimed in claim 1, wherein the objective function value J is updated in step 6t(θ), specifically as follows:
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