Multi-axis motion system sensorless control method based on explicit model predictive control

Abstract

The invention discloses a multi-axis motion system sensorless control method based on explicit model predictive control, which comprises the following steps: firstly, in order to obtain more accurate position information, combining recursive least square identification with a strong tracking Kalman observer, so that estimation results of the recursive least square identification and the strong tracking Kalman observer are mutually transmitted; secondly, the position tracking error of each axis is brought into a state variable, and a prediction model of the motor in the multi-axis motion system is established; and then, introducing a deviation coupling control thought into a value function of explicit model prediction control, and adjusting a tracking error of each shaft motor through a penalty matrix in the value function, thereby realizing regulation and control on the tracking error and a collaborative error in an explicit model prediction control structure. And finally, carrying out linearization processing on voltage constraint and current constraint conditions of the system, as shown in Figure 2, so that the explicit model prediction controller obtains more accurate state partition and control law, and the strategy can significantly improve the tracking capability of the multi-axis motion system.

Description

Technical Field

[0001] The present invention belongs to the field of multi-motor coordinated control, and in particular relates to a position sensorless control method for a multi-axis motion system based on explicit model predictive control. Background Art

[0002] The control goal in a multi-axis motion system is mainly to achieve accurate and fast tracking of geometric paths, especially when tracking complex nonlinear trajectories, which poses a more stringent challenge to the position tracking accuracy of the system. In the prior art, if medium and high frequency reference signals are to be effectively tracked, it is usually necessary to increase the bandwidth. However, excessive bandwidth will reduce the phase margin of the system, which may cause system instability. A controller with full-pass characteristics is urgently needed to solve this problem. At the same time, the encoder speed measurement delay limits the further improvement of the multi-axis position coordination accuracy, especially when tracking high dynamic and high-frequency signals. The speed measurement delay will make the system more prone to accumulated errors when facing rapid changes, further deteriorating the stability and regulation accuracy of the system. Therefore, it is necessary to propose a control strategy with better dynamic performance to meet industrial requirements.

[0003] In view of these problems in traditional methods, the present invention proposes a position sensorless control method for a multi-axis motion system based on explicit model predictive control. Summary of the invention

[0004] In order to overcome the shortcomings of the prior art, the present invention proposes a position sensorless control method for a multi-axis motion system based on explicit model predictive control. This method calculates the control law of the system offline, and then reduces the time of online calculation through an online search method; in addition, the proposed explicit model predictive control with deviation coupling control idea realizes the regulation of tracking error and contour error in the explicit model predictive controller.

[0005] The technical solution adopted by the present invention is a position sensorless control method for a multi-axis motion system based on explicit model predictive control. The specific contents are as follows:

[0006] Step 1): In order to obtain more accurate speed and position information in a position sensorless system, we designed a strong tracking Kalman observer algorithm to estimate the speed and rotor position information. The traditional Kalman observer algorithm relies on historical data, which may affect the position estimation accuracy of the observer. To address the above problems, an attenuation factor λ is introduced based on the original Kalman observer. k , by correcting the covariance matrix of the prior estimate, the role of real-time measurement data is enhanced, thereby improving the accuracy of the estimation results.

[0007] Based on the voltage equation of the α-β axis of the permanent magnet synchronous motor and the equation of motion, the state variable x2=[iα i β i d i q ω e θ e ] T , the input variable is u2=[u α u β u d u q ] T , establish the following state space equation.

[0008]

[0009] In the formula,

[0010] u ɑ 、u β are the α and β axis voltages respectively; i α 、i β are α and β axis currents respectively; u d 、u q are d and q axis voltages respectively; i d 、i q are d-axis and q-axis currents respectively; R s is the stator resistance; L s is the stator inductance; is the pole pair number p n The square of f is the permanent magnet flux; ω e is the electrical angular velocity of the motor rotor; θ e is the rotor electrical angle; J eq is the actual moment of inertia of the motor, T s =1×10 -4 s is the control period; B is the friction coefficient; y(k) is the output variable; w(k) is the system noise; r(k) is the system measurement noise.

[0011] The Jacobian matrix F(k) of f(x2(k)) can be expressed as

[0012]

[0013] w(k) is the system noise, which indicates the influence of system parameter errors; r(k) is the system measurement noise, which indicates the measurement error during the measurement process. The variance matrix of w(k) and r(k) can be expressed as

[0014]

[0015] Where Q1(k) and R1(k) represent the covariance matrix corresponding to the system noise and the covariance matrix corresponding to the measurement noise, respectively.

[0016] The calculation steps of the Kalman observer are:

[0017]

[0018] K((k)=P-(k)C T [CP-(k)C T +R1(k)] -1

[0019]

[0020] In the formula, is the prior estimate of the Kalman observer; is the estimated value of the Kalman observer at the previous moment; is the a priori estimate of the Kalman observer; u2(k--1) is the input value of the Kalman observer at the previous moment; P-(k) is the a priori error variance matrix of the estimation result; P-(k-1) is the a priori error variance matrix of the estimation result at the previous moment; K(k) is the Kalman observer gain matrix; is the posterior error covariance matrix, and I is the identity matrix.

[0021] From the above derivation, we can see that the Kalman observer is more dependent on previous estimates, so a parameter λ is introduced k , the prior error covariance matrix is ​​adjusted to enhance the real-time estimation data, thereby enhancing the accuracy of the estimation results.

[0022] The updated error covariance matrix can be expressed as

[0023]

[0024] In the formula, λ k is the attenuation factor.

[0025] By adjusting λ in real time k , so that it satisfies the following orthogonality criteria:

[0026]

[0027] In the formula, is the prior output matrix.

[0028] If the state estimation error is much smaller than the system state x2(k), and the error term is simplified:

[0029]

[0030] Substituting K(k) into the above formula, we can get

[0031] P- (k)C T -K(k)V(k)=P - (k)C T [I-(CP-(k)C T +R1(k)) -1 ]V(k)

[0032] To make the above equation equal to 0, it must satisfy

[0033] [I-(CP-(k)C T +R1(k)) -1 ]V(k)=0

[0034] In the above formula, the matrix V(k) can be calculated by the following formula

[0035]

[0036] In the formula, ρ is a constant, 0<ρ<1.

[0037] definition:

[0038]

[0039] N(k)=V(k)-C T Q1(k)C-R1(k)

[0040] In summary, k The calculation formula is

[0041]

[0042] Where tr[N(k)] is the trace of the matrix N(k), and tr[M(k)] is the trace of the matrix M(k).

[0043] Step 2): In the process of estimating the position, there are high requirements for the inductance and moment of inertia of the motor. The accuracy of the motor parameters directly affects the estimation accuracy of the observer. Therefore, a parameter identification algorithm is used to enhance the performance of the observer.

[0044] When the identified parameters remain unchanged, the traditional recursive least squares method can track the parameters by periodically resetting the covariance matrix. When the parameters change slowly, the traditional recursive least squares method may weaken the influence of new data on parameter estimation due to the gradual decrease of the covariance matrix, which may cause the real-time parameter identification to fail. To avoid this problem, the recursive least squares method with a forgetting factor is usually used.

[0045] According to the voltage equation and motion equation of the surface-mounted permanent magnet synchronous motor in the dq coordinate system, they can be rewritten into the least squares form after discretization.

[0046]

[0047] In the formula,

[0048]

[0049] The recursive least squares iteration formula with forgetting factor is as follows

[0050]

[0051] Where K1(k) is the recursive least squares identification gain matrix, is the estimation result of recursive least squares identification, is the estimation result of the last moment of the recursive least squares identification, P1(k) is the recursive least squares identification error covariance matrix, P1(k-1) is the recursive least squares identification error covariance matrix of the last moment, P(0) and θ(0) generally need to be determined in advance, usually defined as P(0) = gI, I is the unit matrix, g is a sufficiently large positive real number, ranging from 10 4 ~10 6 Between; r For the forgetting factor.

[0052] Step 3): In order to improve the position tracking accuracy of the multi-axis motion system, the deviation coupling control concept is introduced to apply the tracking errors of other axes to this axis. According to the permanent magnet motor dq voltage equation and the motion equation, we select the state variable x1 = [i d i d ω e i d ω e i q ω e θ e θ eref err m err n ] T , where err m , err n represents the tracking error of the other two motors, θ eref is the reference electrical angle of the motor. The control variable of the system is u1=[u d u q ] T The state space equation of permanent magnet synchronous motor can be expressed as

[0053] x1(k+1)=A1x1(k)+B1u1(k)

[0054] In the formula,

[0055] Matrix O 4×5 Represents a 4×5 zero matrix; O 5×4 Represents a 5×4 zero matrix; I 4×4 Represents a 4×4 identity matrix.

[0056] In the permanent magnet synchronous motor control system, voltage constraints and current constraints need to be considered. The constraints can be expressed as

[0057]

[0058] Where U max The maximum voltage that the system can withstand is limited by the DC bus voltage. max =0.577V dc , V dc is the DC bus voltage; I max The maximum current allowed by the inverter and motor.

[0059] Since explicit model predictive control can only consider linear constraints when dealing with constraints, we need to linearize the current limit circle and the voltage limit circle. Considering the computational complexity and approximation accuracy, an inscribed octagon is used to approximate the constraint limit circle.

[0060] The value function in the control system can be expressed as

[0061]

[0062] In the formula, Q and R are the penalty matrices of the value function, n p For the prediction time domain.

[0063] In order to improve the accuracy of the system in the contour tracking process, based on the deviation coupling control idea, a value function is set to use the tracking errors of the other two motors as compensation signals to act on the motor. The Q and R matrices in the value function can be set as

[0064]

[0065] λ in matrix Q and R d , q , ω , θ , u To adjust the parameters.

[0066] The recursive least squares identification algorithm identifies the inductance and moment of inertia in real time according to the current information taken by the motor and the speed information estimated by the observer, so that it can dynamically adjust the estimation model of the strong tracking Kalman observer, improving the tracking ability and robustness of the observer; after the explicit model predictive controller calculates the optimal control law under all operating conditions offline, it uses the binary search tree algorithm to find the optimal control law applicable to the system at the current moment in real time online, outputs it and acts on the system, and realizes high-precision control of the multi-axis motion system. Compared with the existing technical solutions, the beneficial effects brought by the technical solution of the present invention are:

[0067] The present invention designs a strong tracking Kalman observer to estimate the speed and position information of the motor. At the same time, the recursive least squares identification is combined with the strong tracking Kalman observer to allow the estimation results between the two to be transmitted to each other, which can not only improve the accuracy of the observer, but also ensure the robustness of parameter estimation. In addition, the deviation coupling control idea is introduced into the value function of the explicit model predictive controller to process the tracking errors of the other two motors, thereby improving the tracking accuracy of the multi-axis motion system. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It is a diagram of a multi-axis motion mechanism of the present invention;

[0069] Figure 2 is a diagram of voltage and current constraints of the motor system of the present invention;

[0070] Figure 3 It is a block diagram of the explicit model predictive control structure of the multi-axis motion system of the present invention; DETAILED DESCRIPTION

[0071] A position sensorless control method for a multi-axis motion system based on explicit model predictive control of the present invention is described in detail below in conjunction with the accompanying drawings.

[0072] The multi-axis motion mechanism of the present invention is shown in FIG. Figure 1 As shown, the method comprises the following steps:

[0073] Step 1): In order to obtain more accurate speed and position information in a position sensorless system, we designed a strong tracking Kalman observer algorithm to estimate the speed and rotor position information. The traditional Kalman observer algorithm relies on historical data, which may affect the position estimation accuracy of the observer. To address the above problems, an attenuation factor λ is introduced based on the original Kalman observer. k , by correcting the covariance matrix of the prior estimate, the role of real-time measurement data is enhanced, thereby improving the accuracy of the estimation results.

[0074] Based on the voltage equation of the α-β axis of the permanent magnet synchronous motor and the equation of motion, the state variable x2=[iα i β i d i q ω e θ e ] T , the input variable is u2=[u α u β u d uq] T , establish the following state space equation.

[0075]

[0076] In the formula,

[0077] u α 、u β are the α and β axis voltages respectively; i α 、i β are α and β axis currents respectively; u d 、u q are d and q axis voltages respectively; i d 、i q are d-axis and q-axis currents respectively; R s is the stator resistance; L s is the stator inductance; is the pole pair number p n The square of f is the permanent magnet flux; ω e is the electrical angular velocity of the motor rotor; θ e is the rotor electrical angle; J eq is the actual moment of inertia of the motor, T s =1×10 -4 s is the control period; B is the friction coefficient; y(k) is the output variable; w(k) is the system noise; r(k) is the system measurement noise.

[0078] The Jacobian matrix F(k) of f(x2(k)) can be expressed as

[0079]

[0080] w(k) is the system noise, which indicates the influence of system parameter errors; r(k) is the system measurement noise, which indicates the measurement error during the measurement process. The variance matrix of w(k) and r(k) can be expressed as

[0081]

[0082] Where Q1(k) and R1(k) represent the covariance matrix corresponding to the system noise and the covariance matrix corresponding to the measurement noise, respectively.

[0083] The calculation steps of the Kalman observer are:

[0084]

[0085] In the formula, is the prior estimate of the Kalman observer; is the estimated value of the Kalman observer at the previous moment; is the posterior estimate of the Kalman observer; u2(k-1) is the input value of the Kalman observer at the previous moment; P - (k) is the prior error variance matrix of the estimation result; P - (k-1) is the prior error variance matrix of the estimation result at the previous moment; K(k) is the Kalman observer gain matrix; is the posterior error covariance matrix, and I is the identity matrix.

[0086] From the above derivation, we can see that the Kalman observer is more dependent on previous estimates, so a parameter λ is introduced k , the prior error covariance matrix is ​​adjusted to enhance the real-time estimation data, thereby enhancing the accuracy of the estimation results.

[0087] The updated error covariance matrix can be expressed as

[0088]

[0089] In the formula, λ k is the attenuation factor.

[0090] By adjusting λ in real time k , so that it satisfies the following orthogonality criteria:

[0091]

[0092] In the formula, is the prior output matrix.

[0093] If the state estimation error is much smaller than the system state quantity x2(k), the above formula can be rewritten as

[0094]

[0095] The error term in formula (7) can be simplified as

[0096]

[0097] Substituting the gain matrix K(kk) in equation (4) into equation (8) yields

[0098] P - (k)C T -K(k)V(k)=P- (k)C T [I-(CP - (k)C T +R1(k)) -1 ]V(k) (9)

[0099] To make the above equation equal to 0, it must satisfy

[0100] [I-(CP - (k)C T +R1(k)) -1 ]V(k)=0 (10)

[0101] Substitute equation (10) into the error covariance matrix P - (k) and simplify to get

[0102]

[0103] In formula (11), V(k) can be calculated by the following formula:

[0104]

[0105] In the formula, ρ is a constant, 0<ρ<1.

[0106] definition:

[0107]

[0108] In summary, k The calculation formula is

[0109]

[0110] Where tr[N(kk)] is the trace of the matrix N(k), and tr[M(k)] is the trace of the matrix M(k).

[0111] Step 2): In the process of estimating the position, equation (1) has high requirements on the inductance and moment of inertia of the motor. The accuracy of the motor parameters directly affects the estimation accuracy of the observer. Therefore, a parameter identification algorithm is used to enhance the performance of the observer.

[0112] The recursive least squares algorithm predicts and estimates the last estimated value with the new observed data, and then superimposes them in sequence to finally obtain the required parameter estimates. When the identified parameters remain unchanged, the traditional recursive least squares method can track the parameters by periodically resetting the covariance matrix. When the parameters change slowly, the traditional recursive least squares method may weaken the influence of new data on parameter estimation due to the gradual decrease of the covariance matrix, which may cause the real-time parameter identification to fail. To avoid this problem, the recursive least squares method with a forgetting factor is usually used.

[0113] According to the voltage equation and motion equation of the surface-mounted permanent magnet synchronous motor in the dq coordinate system, they can be rewritten into the least squares form after discretization.

[0114]

[0115] In the formula,

[0116]

[0117] The recursive least squares iteration formula with forgetting factor is as follows

[0118]

[0119] Where K1(k) is the recursive least squares identification gain matrix, is the estimation result of recursive least squares identification, is the estimation result of the last moment of the recursive least squares identification, P1(k) is the recursive least squares identification error covariance matrix, P1(k-1) is the recursive least squares identification error covariance matrix of the last moment, P(0) and θ(0) generally need to be determined in advance, usually defined as P(0) = gI, I is the unit matrix, g is a sufficiently large positive real number, ranging from 10 4 ~10 6 Between; r For the forgetting factor.

[0120] Step 3): In order to improve the position tracking accuracy of the multi-axis motion system, the deviation coupling control concept is introduced to apply the tracking errors of other axes to this axis. According to the permanent magnet motor dq voltage equation and the motion equation, we select the state variable x1 = [i d i q ω e i d ω e i q ω e θ e θ eref err m err n ] T , where err m , err n represents the tracking error of the other two motors, θ eref is the reference electrical angle of the motor. The control variable of the system is u1=[u d u q ] T The state space equation of permanent magnet synchronous motor can be expressed as

[0121] x1(k+1)=A1x1(k)+B1u1(k) (17)

[0122] In the formula,

[0123] Matrix O 4×5 Represents a 4×5 zero matrix; O 5×4 Represents a 5×4 zero matrix; I 4×4 Represents a 4×4 identity matrix.

[0124] In the permanent magnet synchronous motor control system, voltage constraints and current constraints need to be considered. The constraints can be expressed as

[0125]

[0126] Where U max The maximum voltage that the system can withstand is limited by the DC bus voltage. max =0.577V dc , V dc is the DC bus voltage; I max The maximum current allowed by the inverter and motor.

[0127] Since explicit model predictive control can only consider linear constraints when dealing with constraints, we need to linearize the current limit circle and the voltage limit circle. Considering the computational complexity and approximation accuracy, an inscribed octagon is used to approximate the constraint limit circle, such as Figure 2 After linearization, the voltage and current constraints can be expressed by the following inequalities.

[0128]

[0129] In the formula,

[0130]

[0131] The value function in the control system can be expressed as

[0132]

[0133] In the formula, Q and R are the penalty matrices of the value function, n p For the prediction time domain.

[0134] In order to improve the accuracy of the system in the contour tracking process, based on the deviation coupling control idea, a value function is set to use the tracking errors of the other two motors as compensation signals to act on the motor. The Q and R matrices in the value function can be set as

[0135]

[0136] λ in matrix Q d , q will i d 、i d Limited to the voltage and current constraints; λ ω Yes e To restrict, θ For θ e ,θ eref ,err m , err n Limit the motor position to track the reference position. u For the control quantity u d 、u q To punish.

[0137] According to the stability analysis of the closed-loop system, when all eigenvalues ​​of the coefficient matrix are within the unit circle, it can be concluded that the system is asymptotically stable. Therefore, we only need to adjust the parameters in the matrices Q and R to obtain better controller performance while ensuring the stability of the closed-loop system.

[0138] The structure diagram of the explicit model predictive control of the multi-axis motion system of the present invention is as follows: Figure 3 As shown in the figure abcx 、i abcy 、i abcz are the sampled currents ω of the three motors x, y, and z xest ,θ xest 、i dqest are the estimated speed, estimated position and estimated current θ of the strong tracking Kalman observer respectively. xref is the reference electrical angle of the x motor; θ xerr ,θ yerr ,θ zerr are the position tracking errors of the three motors x, y, and z; u dx * 、u qx * is the reference voltage output by the explicit model predictive controller; A, B, and C are the three bridge arms of the three-phase inverter respectively; MPT is an optimization toolbox for offline solving the control law of the explicit model predictive controller.

[0139] The recursive least squares identification algorithm identifies the inductance and moment of inertia in real time according to the current information taken by the motor and the speed information estimated by the observer, so that it can dynamically adjust the estimation model of the strong tracking Kalman observer and improve the tracking ability and robustness of the observer. After the explicit model predictive controller calculates the optimal control law under all operating conditions offline, it uses the binary search tree algorithm online to find the optimal control law applicable to the system at the current moment in real time, output it and act on the system, so as to achieve high-precision control of the multi-axis motion system.

[0140] From the above description, it can be seen that the present invention proposes a position sensorless control method for a multi-axis motion system based on explicit model predictive control, which is applied to the multi-axis motion system to improve the position tracking accuracy of each axis.

[0141] The above description of the functions and working processes of the present invention in combination with the accompanying drawings is only one of the preferred implementation cases, but the present invention is not limited to the above-mentioned specific functions and working processes. The above-mentioned specific implementation methods are only illustrative and not restrictive. Under the enlightenment of the present invention, ordinary technicians in this field can also make many forms without departing from the scope of protection of the purpose of the present invention and the claims, which are all within the protection scope of the present invention.

Claims

1. A position sensorless control method for a multi-axis motion system based on explicit model predictive control, characterized in that: The following steps are involved: 1) The idea of ​​deviation coupling control is introduced into the value function of the explicit model predictive controller. The collaborative error of the multi-axis motion system is adjusted through the penalty matrix in the value function. At the same time, the adjustment of the contour error and tracking error in the explicit model predictive control structure is realized; 2) A strong tracking Kalman observer is designed to estimate the speed and position information of each axis, thereby obtaining more accurate state estimation in a dynamic environment. In addition, the problem of encoder information transmission delay can also be solved; 3) Combining the recursive least squares identification algorithm with the strong tracking Kalman observer allows the estimation results between the two to be transmitted to each other, which can not only improve the estimation accuracy of the observer, but also ensure the robustness of parameter identification.

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