A predetermined performance control method for a linear motor system based on state transition

Abstract

The present invention discloses a predetermined performance control method for a linear motor system based on state transition. The method includes: establishing a predetermined performance controller for the linear motor and determining the predetermined performance boundary conditions; inputting the actual position and actual speed of the output shaft of the linear motor and the expected trajectory of the output shaft of the preset linear motor into the predetermined performance controller to obtain the control input quantity of the linear motor; obtaining the actual input of the linear motor that satisfies the input constraints under the control input quantity, and obtaining the actual position of the output shaft of the linear motor in real time, so as to satisfy the state constraints and the predetermined performance boundary conditions, thereby enabling the linear motor to achieve the desired transient and steady-state performance. The method of the present invention can realize the control of a linear motor system with state and input constraints under the conditions of parameter uncertainty and external disturbances, achieve the desired transient and steady-state performance of the system, improve the robustness of the system, and achieve the precise tracking control of the linear motor system.

Description

Technical Field

[0001] The present invention relates to a method for controlling the predetermined performance of a linear motor system, and more particularly to a method for controlling the predetermined performance of a linear motor system based on state transition. Background Art

[0002] Since linear motors eliminate mechanical transmission problems related to gears and have the potential for high-speed and high-precision motion, they have been widely used in modern mechatronic systems such as microelectronics, optical inspection systems, and advanced machine tools. However, in practical applications, linear motor systems are subject to state and input constraints, such as limited workspace, operating speed, actuator performance, etc. If the system does not comply with these constraints, problems such as reduced control accuracy and system instability may occur. In response to the state and input constraints existing in the system, various control methods have been proposed at home and abroad, including barrier Lyapunov functions, nonlinear mappings, predictive performance control, and optimal control. Among them, the barrier Lyapunov function is a widely used method for solving state constraints, but the feasibility conditions therein limit its application scope. In addition, with the improvement of industrial application requirements, the need for high-speed and high-precision motion tracking control requires the controller to have the ability to adjust the transient and steady-state of the system. However, existing control algorithms for dealing with constraints often have difficulty ensuring the transient and steady-state performance of the system. Summary of the Invention

[0003] To solve the problems existing in the background art, to solve the tracking control problem of a linear motor system with parameter uncertainty, external disturbances, state and input constraints, and at the same time ensure the transient and steady-state performance of the system, the present invention provides a method for controlling the predetermined performance of a linear motor system based on state transition. The method uses a state transition function to handle state constraints, an auxiliary boundary to handle input constraints, a time-varying barrier Lyapunov function to ensure the transient and steady-state performance of the system, and an adaptive robust control to handle parameter uncertainty and external disturbances.

[0004] The technical solution adopted by the present invention is as follows:

[0005] The predetermined performance control method of the present invention includes the following steps:

[0006] Step 1, establish a predetermined performance controller for the linear motor considering the state constraints and input constraints of the linear motor, and determine the predetermined performance boundary conditions satisfied by the position error of the output shaft of the linear motor.

[0007] Step 2, obtain the actual position and actual speed of the output shaft of the current linear motor, and input the actual position and actual speed of the output shaft of the linear motor and the preset desired trajectory of the output shaft of the linear motor into the predetermined performance controller, and the predetermined performance controller processes to obtain the control input quantity of the linear motor.

[0008] Step 3: Obtain the actual input of the linear motor that satisfies the input constraint under the control input of the linear motor through the control input of the linear motor, and input it into the linear motor to obtain the actual position of the output shaft of the linear motor in real time, that is, control the actual position and actual speed of the output shaft of the linear motor, so that the obtained actual position of the output shaft of the linear motor satisfies the state constraint and the predetermined performance boundary condition, thereby enabling the linear motor to achieve the desired transient and steady-state performance.

[0009] In the above-mentioned step 1, establish a predetermined performance controller for the linear motor considering the state constraint and input constraint of the linear motor, specifically as follows:

[0010]

[0011] where, v is the control input of the linear motor; ξ 1 is the first control variable of the predetermined performance controller, is the first derivative of the first control variable ξ 1 μ is the intermediate parameter of the first state variable ξ 1 is the first state variable ξ 1 μ; ξ 2 is the second state variable of the predetermined performance controller, is the first derivative of the second state variable ξ 2 μ is the intermediate parameter of the second state variable ξ 2 is the second state variable ξ 2 μ; τ a is the compensation term of the predetermined performance controller, τ b is the feedback term, τ c is the control term to ensure that the linear motor system has a predetermined performance boundary, τ l is the control term to compensate for the difference between the control input v of the linear motor and the actual control input; is the regressor of the actual speed of the output shaft of the linear motor; is the parameter estimation vector of the unknown parameter vector θ, and the unknown parameter vector θ is a set of parameters representing the parameter uncertainty of the linear motor, θ = [θ 1 , θ 2 , θ 3 ; T M is the load mass of the linear motor; is the first derivative of the output of the first-order filter of the predetermined performance controller; η 2 (t) is the second predetermined performance boundary function of the predetermined performance controller at time t, is the first derivative of the second predetermined performance boundary function η 2 (t) of the predetermined performance controller at time t; z 2 is the intermediate error, is an auxiliary variable used to process the input constraints of the linear motor; x 1 is the actual position of the output shaft of the linear motor, and x 2 is the actual speed of the output shaft of the linear motor, is the actual acceleration of the output shaft of the linear motor; S(x 2 ) is a smooth function with respect to the actual speed x 2 of the output shaft of the linear motor; is an auxiliary variable used to process the input constraints is the rate of change of the auxiliary variable that satisfies; g(v) is a smooth function of the linear motor under the control input v; F 12 and F 11 are the upper and lower bounds of the actual position x 1 of the output shaft of the linear motor; F 22 and F 21 are the upper and lower bounds of the actual speed of the output shaft of the linear motor respectively.

[0012] The state constraints are specifically as follows:

[0013] F 11 < x 1 < F 12

[0014] F 21 < x 2 < F 22

[0015] The input constraints are specifically as follows:

[0016]

[0017] where, u(v) is the actual input of the linear motor under the control input v; sat(v) is the saturation function of the linear motor under the control input v; κ(v) is the bounded approximation error between the smooth function g(v) and the saturation function sat(v) of the linear motor under the control input v; u max is the known maximum control input of the linear motor; exp(·) is the exponential function with the natural constant e as the base.

[0018] Approximating the saturation function sat(v) by the smooth function g(v) of the linear motor under the control input v in the input constraints can reduce the difficulty of designing the predetermined performance controller.

[0019] Before establishing the predetermined performance controller, it is necessary to first establish the state space model of the linear motor considering the state constraints and input constraints of the linear motor, specifically as follows:

[0020]

[0021] Among them, θ 1 is the viscous friction coefficient, and θ 1 x 2 is the viscous friction force received by the output shaft of the linear motor; θ 2 is the Coulomb friction coefficient, and θ 2 S(x 2 ) is the Coulomb friction force received by the output shaft of the linear motor; is the uncertainty term of the unknown disturbance of the linear motor, d is the unknown disturbance of the linear motor, and θ 3 is the nominal value of the unknown disturbance of the linear motor. The unknown disturbance includes uncertain nonlinearity and external disturbance.

[0022] The linear motor system is specifically as follows:

[0023]

[0024] Among them, θ 1 = B, θ 2 = A, θ 3 = d n .

[0025] By defining the state variable the linear motor system is converted into a state - space model.

[0026] The parameter - estimation vector needs to be calculated in advance during actual operation. When calculating the parameter - estimation vector specifically, an adaptation - rate function is first established, and the updated value of the parameter - estimation vector is obtained through the parameter - estimation vector and the adaptation - rate function. Then, the updated value of the parameter - estimation vector is directly used as the value of the parameter - estimation vector for calculation. The adaptation - rate function is specifically as follows:

[0027]

[0028] Among them, is the mapping function of the prescribed - performance controller, is the j - th term of the mapping function , j = 1, 2, 3, Φ is the adaptation function of the prescribed - performance controller; · j is the j - th term in the parameter - estimation vector input of the mapping function, · j =(ΓΦ) j ​; Parameter estimation vector The first item in Is the estimation of the viscous friction coefficient, · 1 Is the parameter estimation vector The first item in The input of the mapping function of; Parameter estimation vector The second item in Is the estimation of the Coulomb friction coefficient, · 2 Is the parameter estimation vector The second item in The input of the mapping function of; Parameter estimation vector The third item in Is the estimation of the nominal value of the unknown disturbance of the linear motor, · 3 Is the parameter estimation vector The first item in The input of the mapping function.

[0029] When the parameter estimation vector The j-th item in is equal to its minimum value θ jmin And the input of the mapping function is greater than 0, the output of the mapping function is 0; When the parameter estimation vector The j-th item in is equal to its maximum value θ jmax And the input of the mapping function is less than 0, the output of the mapping function is 0; In other cases, the output of the mapping function is equal to the input of the mapping function; Substitute the output of the mapping function into the adaptation rate function, and directly obtain the updated value of the parameter estimation vector And then directly use the updated value of the parameter estimation vector As the parameter estimation vector Value.

[0030] The predetermined performance controller contains a parameter estimation vector, the adaptation rate function is used to obtain the parameter estimation vector, and the adaptation function contains a mapping function.

[0031] The feedback term τ b Is specifically as follows:

[0032] τ b = τ b1 + τ b2

[0033]

[0034] Where, τ b1 Is the feedback term used to stabilize the nominal system, τ b2 Is the robust feedback term used to attenuate the influence of the uncertain nonlinearity and modeling error of the state space model; c 2 Is a positive constant parameter to be designed; η1 (t) is the first predetermined performance boundary function of the predetermined performance controller at time t; z 1 is the virtual error, z 1 = ξ1 1 - α 0 , α 0 is the desired trajectory of the linear motor; β is the intermediate variable of the first-order filter of the predetermined performance controller.

[0035] The desired trajectory α of the linear motor 0 , specifically as follows:

[0036]

[0037] where x 1d is the desired position of the output shaft of the linear motor.

[0038] The robust feedback term τ used to attenuate the influence of uncertain nonlinearities and modeling errors in the state-space model b2 satisfies the following conditions:

[0039] τ b2 = μ 2 τ′ b2

[0040]

[0041] where τ' b2 is the first derivative of the robust feedback term τ b2 ; is an arbitrarily small positive constant parameter to be designed for the robust feedback term τ b2 .

[0042] The first predetermined performance boundary function η 1 (t) and the second predetermined performance boundary function η 2 (t) of the predetermined performance controller at time t, specifically as follows:

[0043] η 1 (t) = η 11 exp(-r 1 t) + η 21

[0044] η 2 (t) = η 12 exp(-r 2 t) + η 22

[0045] where η 11 and η 21 are the first parameter and the second parameter of the first predetermined performance boundary function η 1 (t), η12 and η 22 are the first parameter and the second parameter of the second predetermined performance boundary function η 2 (t) respectively, η1 11 > η 21 > 0, η 12 > η 22 > 0; r 1 and r 2 are the first time constant of the first predetermined performance boundary function η 1 (t) and the second time constant of the second predetermined performance boundary function η 2 (t).

[0046] The first-order filter of the described predetermined performance controller is as follows:

[0047]

[0048] where ε is a positive constant parameter; α 1 is the virtual control quantity of the input of the first-order filter; α 1 (0) is the virtual control quantity of the input of the first-order filter at time 0, α 2 (0) is the output of the first-order filter at time 0, and β(0) is the intermediate variable of the first-order filter at time 0; α 2 is the output of the first-order filter.

[0049] Obtain its first derivative through the output of the first-order filter and input it into the predetermined performance controller.

[0050] The intermediate variable β of the first-order filter is as follows:

[0051]

[0052] where x is the actual trajectory of the output shaft of the linear motor, x = [x 1 , x 2 T ; the actual trajectory x of the output shaft of the linear motor is the state variable of the state space model; x 2d is the desired speed of the output shaft of the linear motor.

[0053] In the described step 1, the linear motor is adaptively robustly controlled by the predetermined performance controller, so that the position error φ of the output shaft of the linear motor has a predetermined performance boundary; the predetermined performance boundary conditions satisfied by the determined position error of the output shaft of the linear motor are as follows:

[0054]

[0055] |z 1 | < |η 1 ​(t)|

[0056] Among them, φ is the position error of the output shaft of the linear motor, and φ = x 1 - x 1d 。

[0057] In the said step 2, the actual position and actual speed of the output shaft of the current linear motor are obtained, specifically by directly obtaining them through the position and speed sensor preset inside the linear motor.

[0058] In the said step 3, the actual input of the linear motor satisfying the input constraint is obtained through the control input quantity of the linear motor, specifically by obtaining the actual input of the linear motor satisfying the input constraint after processing the control input quantity of the linear motor through the saturation effect of the preset saturator.

[0059] The beneficial effects of the present invention are:

[0060] 1. The present invention uses the state conversion function to process the state constraint, effectively avoiding the feasibility condition in the traditional barrier Lyapunov function and expanding the application scope of this method.

[0061] 2. The present invention uses the time-varying barrier Lyapunov function to design the prescribed performance controller, which can ensure that the system has the desired transient and steady-state performance.

[0062] 3. The method of the present invention can realize the control of the linear motor system with state and input constraints, and at the same time can improve the transient and steady-state performance of the system. In addition, the adaptive robust controller is used to effectively solve the influence of the uncertain terms in the system on the control effect, improve the robustness of the system, and achieve the precise tracking control of the linear motor system. Description of the Drawings

[0063] Figure 1 It is a schematic diagram of the state conversion function of the present invention;

[0064] Figure 2 It is a schematic diagram of the position tracking trajectory of the present invention;

[0065] Figure 3 It is a schematic diagram of the transient state of the position tracking error of the present invention;

[0066] Figure 4 It is a schematic diagram of the steady state of the position tracking error of the present invention;

[0067] Figure 5 It is a schematic diagram of the speed tracking trajectory of the present invention;

[0068] Figure 6 It is a schematic diagram of the control signal of the embodiment of the present invention;

[0069] Figure 6 (a) of this invention is the schematic diagram of the first control signal in the embodiment of the invention;

[0070] Figure 6 (b) of this invention is the schematic diagram of the second control signal in the embodiment of the invention;

[0071] Figure 6 (c) of this invention is the schematic diagram of the third control signal in the embodiment of the invention. Detailed implementation manners

[0072] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0073] The predetermined performance control method of the present invention includes the following steps:

[0074] Step 1: Establish a predetermined performance controller for the linear motor considering the state constraints and input constraints of the linear motor, and determine the predetermined performance boundary conditions satisfied by the position error of the output shaft of the linear motor.

[0075] In step 1, to establish a predetermined performance controller for the linear motor considering the state constraints and input constraints of the linear motor, the specific steps are as follows:

[0076]

[0077] Among them, v is the control input quantity of the linear motor; ξ 1 is the first control variable of the predetermined performance controller, is the first-order derivative of the first control variable ξ 1 , μ 1 is the intermediate parameter of the first state variable ξ 1 ; ξ 2 is the second state variable of the predetermined performance controller, is the first-order derivative of the second state variable ξ 2 , μ 2 is the intermediate parameter of the second state variable ξ 2 ; τ a is the compensation term of the predetermined performance controller, τ b is the feedback term, τ c is the control term to ensure that the linear motor system has a predetermined performance boundary, τ l is the control term to compensate for the difference between the control input quantity v of the linear motor and the actual control input quantity; is the regressor of the actual speed of the output shaft of the linear motor; is the parameter estimation vector of the unknown parameter vector θ, and the unknown parameter vector θ is a set of parameters representing the parameter uncertainty of the linear motor, θ = [θ 1 , θ 2 , θ 3 ​T ; M is the load mass of the linear motor; is the first derivative of the output of the first-order filter of the predetermined performance controller; η 2 (t) is the second predetermined performance boundary function of the predetermined performance controller at time t, is the second predetermined performance boundary function η of the predetermined performance controller at time t 2 (t) of the first derivative; z 2 is the intermediate error, is the auxiliary variable used to handle the input constraint of the linear motor; x 1 is the actual position of the output shaft of the linear motor, and x 2 is the actual speed of the output shaft of the linear motor, is the actual acceleration of the output shaft of the linear motor; S(x 2 ) is a smooth function of the actual speed x of the output shaft of the linear motor 2 ; is the auxiliary variable used to handle the input constraint satisfies the auxiliary variable change rate; g(v) is a smooth function of the linear motor under the control input v; F 12 and F 11 are the upper and lower boundaries of the actual position x of the output shaft of the linear motor respectively 1 ; F 22 and F 21 are the upper and lower boundaries of the actual speed of the output shaft of the linear motor respectively.

[0078] The state constraints are specifically as follows:

[0079] F 11 < x 1 < F 12

[0080] F 21 < x 2 < F 22

[0081] The input constraints are specifically as follows:

[0082]

[0083] Among them, u(v) is the actual input of the linear motor under the control input v; sat(v) is the saturation function of the linear motor under the control input v; κ(v) is the bounded approximation error between the smooth function g(v) and the saturation function sat(v) of the linear motor under the control input v; u max is the known maximum control input of the linear motor; exp(·) is the exponential function with the natural constant e as the base.

[0084] Approximating the saturation function sat(v) by the smooth function g(v) of the linear motor under the control input v in the input constraint can reduce the difficulty of designing the prescribed performance controller.

[0085] Before establishing the prescribed performance controller, it is necessary to first establish the state - space model of the linear motor considering the state constraint and input constraint of the linear motor, as follows:

[0086]

[0087] where, θ 1 is the viscous friction coefficient, and θ 1 x 2 is the viscous friction force on the output shaft of the linear motor; θ 2 is the Coulomb friction coefficient, and θ 2 S(x 2 ) is the Coulomb friction force on the output shaft of the linear motor; is the uncertain term of the unknown disturbance of the linear motor, d is the unknown disturbance of the linear motor, and θ 3 is the nominal value of the unknown disturbance of the linear motor. The unknown disturbance includes uncertain nonlinearity and external disturbance.

[0088] The linear - motor system is as follows:

[0089]

[0090] where, θ 1 = B, θ 2 = A, θ 3 = d n .

[0091] By defining the state variable the linear - motor system is converted into a state - space model.

[0092] The parameter - estimation vector needs to be calculated in advance in actual operation. When calculating the parameter - estimation vector , specifically, an adaptation - rate function is first established, and the updated value of the parameter - estimation vector is obtained through the parameter - estimation vector and the adaptation - rate function. Then, the updated value of the parameter - estimation vector is directly used as the value of the parameter - estimation vector for calculation. The adaptation - rate function is as follows:

[0093]

[0094] where, is the mapping function of the predetermined performance controller, is the mapping function for the j-th term, where j = 1, 2, 3, Φ is the adaptive function of the predetermined performance controller; · j is the parameter estimation vector for the j-th term in the input of the mapping function, · j =(ΓΦ) j ; the parameter estimation vector for the first term in is the estimation of the viscous friction coefficient, · 1 is the parameter estimation vector for the first term in the input of the mapping function; the parameter estimation vector for the second term in is the estimation of the Coulomb friction coefficient, · 2 is the parameter estimation vector for the second term in the input of the mapping function; the parameter estimation vector for the third term in is the estimation of the nominal value of the unknown disturbance of the linear motor, · 3 is the parameter estimation vector for the first term in the input of the mapping function.

[0095] When the j-th term in the parameter estimation vector is equal to its minimum value θ jmin and the input of the mapping function is greater than 0, the output of the mapping function is 0; when the j-th term in the parameter estimation vector is equal to its maximum value θ jmax and the input of the mapping function is less than 0, the output of the mapping function is 0; in other cases, the output of the mapping function is equal to the input of the mapping function; substituting the output of the mapping function into the adaptive rate function, the updated value of the parameter estimation vector is directly obtained through the adaptive rate function, and then the updated value of the parameter estimation vector is directly used as the value of the parameter estimation vector .

[0096] The predetermined performance controller contains a parameter estimation vector, the adaptive rate function is used to obtain the parameter estimation vector, and the adaptive function contains a mapping function.

[0097] The feedback term τ b is specifically as follows:

[0098] τ b =τ b1 +τb2

[0099]

[0100] Among them, τ b1 is the feedback term used to stabilize the nominal system, and τ b2 is the robust feedback term used to attenuate the influence of the uncertain nonlinearity and modeling error of the state-space model; c 2 is a positive constant parameter to be designed; η 1 (t) is the first predetermined performance boundary function of the predetermined performance controller at time t; z 1 is the virtual error, and z 1 =ξ 1 -α 0 where α 0 is the desired trajectory of the linear motor; b is the intermediate variable of the first-order filter of the predetermined performance controller.

[0101] The desired trajectory α 0 of the linear motor is as follows:

[0102]

[0103] Among them, x 1d is the desired position of the output shaft of the linear motor.

[0104] The robust feedback term τ b2 used to attenuate the influence of the uncertain nonlinearity and modeling error of the state-space model satisfies the following conditions:

[0105] τ b2 =μ 2 τ′ b2

[0106]

[0107] Among them, τ' b2 is the first derivative of the robust feedback term τ b2 ; is an arbitrarily small positive constant parameter to be designed for the robust feedback term τ b2 .

[0108] The first predetermined performance boundary function η 1 (t) and the second predetermined performance boundary function η 2 (t) of the predetermined performance controller are as follows:

[0109] η 1 (t)=η 11 exp(-r 1 t)+η 21

[0110] η 2 (t)=η 12 exp(-r 2 t)+η 22

[0111] where η 11 and η 21 are the first parameter and the second parameter of the first predefined performance boundary function η 1 (t) respectively, η 12 and η 22 are the first parameter and the second parameter of the second predefined performance boundary function η 2 (t) respectively, η 11 >η 21 >0, η 12 >η 22 >0; r 1 and r 2 are the first time constant of the first predefined performance boundary function η 1 (t) and the second time constant of the second predefined performance boundary function η 2 (t) respectively.

[0112] The first-order filter of the predefined performance controller is as follows:

[0113]

[0114] where ε is a positive constant parameter; α 1 is the virtual control quantity of the input of the first-order filter; α 1 (0) is the virtual control quantity of the input of the first-order filter at time 0, α 2 (0) is the output of the first-order filter at time 0, β(0) is the intermediate variable of the first-order filter at time 0; α 2 is the output of the first-order filter.

[0115] The first derivative is obtained from the output of the first-order filter and input into the predefined performance controller.

[0116] The intermediate variable β of the first-order filter is as follows:

[0117]

[0118] where x is the actual trajectory of the output shaft of the linear motor, x = [x 1 ,x 2 T ; the actual trajectory x of the output shaft of the linear motor is the state variable of the state space model; x 2d is the desired speed of the output shaft of the linear motor.

[0119] ​In Step 1, an adaptive robust control is performed on the linear motor by a predetermined performance controller, so that the position error φ of the output shaft of the linear motor has a predetermined performance boundary; the predetermined performance boundary conditions satisfied by the determined position error of the output shaft of the linear motor are as follows:

[0120]

[0121] |z 1 | < |η1 1 (t)|

[0122] where φ is the position error of the output shaft of the linear motor, and φ = x 1 -x 1d 。

[0123] In Step 2, the actual position and actual speed of the output shaft of the current linear motor are obtained, and the actual position and actual speed of the output shaft of the linear motor and the preset desired trajectory of the output shaft of the linear motor are input into the predetermined performance controller, and the predetermined performance controller processes to obtain the control input quantity of the linear motor.

[0124] In Step 2, the actual position and actual speed of the output shaft of the current linear motor are obtained, specifically by directly obtaining through a position and speed sensor pre - installed inside the linear motor.

[0125] In Step 3, the actual input of the linear motor satisfying the input constraint under the control input quantity is obtained through the control input quantity of the linear motor, and is input into the linear motor to obtain the actual position of the output shaft of the linear motor in real time, that is, the actual position and actual speed of the output shaft of the linear motor are controlled, so that the obtained actual position of the output shaft of the linear motor satisfies the state constraint and the predetermined performance boundary conditions, thereby enabling the linear motor to achieve the desired transient and steady - state performance.

[0126] In Step 3, the actual input of the linear motor satisfying the input constraint under the control input quantity is obtained through the control input quantity of the linear motor, specifically by obtaining the actual input of the linear motor satisfying the input constraint under the control input quantity after the saturation processing of the control input quantity of the linear motor by a preset saturator.

[0127] The specific embodiments are as follows:

[0128] To verify the effectiveness and superiority of the proposed predetermined performance control method, the following simulation comparison is carried out. The specific embodiments are as follows:

[0129] M1: Use a traditional adaptive robust controller to control the linear motor, and the designed control law is:

[0130] u r = g(v r )

[0131]

[0132] Among them, is the desired acceleration of the output shaft of the linear motor, u r is the input of the linear motor, g(v r ) is a smooth function of the linear motor at the preset control input v r . is the parameter estimation vector of the unknown parameter vector θ, obtained according to its adaptation rate , v sr2 is the robust feedback term used to attenuate the influence of the uncertain nonlinearity and modeling error of the state space model; e is the position error of the output shaft of the linear motor, is the first derivative of the position error e of the output shaft of the linear motor; c 1r and c 2r are the first control parameter and the second control parameter of the adaptive robust controller respectively, c 1r = 55, c 2r = 50.

[0133] M2: The state-transition-based controller without predetermined performance control, the designed control law is:

[0134] u p = g(v p )

[0135]

[0136] Among them, u p is the input of the linear motor, g(v p ) is a smooth function of the linear motor at the preset control input v p , μ d is the intermediate parameter of the state variable, τ b2 is the robust feedback term; c 1p and c 2p are the first control parameter and the second control parameter of the state-transition-based controller without predetermined performance control respectively, c 1p = 55, c 2p = 50.

[0137] M3: The predetermined performance control method for the linear motor system based on state transition proposed by the present invention, the first control parameter c 1 and the second control parameter c 2 of the predetermined performance controller are respectively: c 1 = 55, c 2 = 50; the parameters in the designed predetermined performance boundary are: η 11 = 5, η 21 = 0.5, r1 = 3, η 12 = 3, η 22 = 0.2, r 2 = 1。

[0138] Set the desired trajectory, system parameters, initial state, state and input constraints in the simulation experiment as follows:

[0139] Desired trajectory:

[0140] x 1d = 0.15sin[π(t + 0.85)] + 0.18

[0141] System parameters:

[0142] M = 1.03, B = 0.45, A = 0.19, d = 0.15sin(t),

[0143] Initial state:

[0144] x 1 (0) = x 2 (0) = 0

[0145] State and input constraints:

[0146] u = 10

[0147] max

[0148] -0.05 ≤ x 1 ≤ 0.4, -1.3 ≤ x 2 ≤ 1.3

[0149] Figure 2 are the schematic diagrams of the position tracking trajectories of the linear motor system under different control methods, Figure 3 and Figure 4 are respectively the transient and steady-state schematic diagrams of the position tracking error, Figure 5 is the schematic diagram of the speed tracking trajectory of the linear motor system under different control methods, Figure 6 is the schematic diagram of the control signals of different control methods. The three control methods are respectively as shown in Figure 6 (a) of Figure 6 (b) of Figure 6 (c) of

[0150] By Figure 2 and Figure 5It can be seen that the traditional adaptive robust control (M1) cannot handle state constraints; while the controller based on state transformation (M2), although it can ensure that state constraints are not violated, the transient and steady-state performance of the system cannot be guaranteed, so the convergence speed is slow and the steady-state error is large; while the predetermined performance control method for linear motor systems based on state transformation proposed in the present invention (M3) can not only ensure that the system complies with state constraints, but also has fast transient response performance and high steady-state tracking accuracy.

[0151] The above content is only the technical idea of the present invention, and the protection scope of the present invention cannot be limited thereby. Any modifications made on the basis of the technical solution according to the technical idea proposed by the present invention shall fall within the protection scope of the claims of the present invention.

Claims

1. A predetermined performance control method for a linear motor system based on state transition, characterized in that: The method includes the following steps: Step 1, establish a predetermined performance controller for the linear motor considering the state constraints and input constraints of the linear motor, and determine the predetermined performance boundary conditions satisfied by the position error of the output shaft of the linear motor; Step 2, obtain the actual position and actual speed of the output shaft of the current linear motor, and input the actual position and actual speed of the output shaft of the linear motor and the preset desired trajectory of the output shaft of the linear motor into the predetermined performance controller, and the predetermined performance controller processes to obtain the control input quantity of the linear motor; Step 3, obtain the actual input of the linear motor under the control input quantity that satisfies the input constraints through the control input quantity of the linear motor, and input it into the linear motor to obtain the actual position of the output shaft of the linear motor in real time, so that the obtained actual position of the output shaft of the linear motor satisfies the state constraints and the predetermined performance boundary conditions, thereby realizing that the linear motor achieves the desired transient and steady-state performance; In the said Step 1, to establish a predetermined performance controller for the linear motor considering the state constraints and input constraints of the linear motor, specifically as follows: where, v is the control input of the linear motor; ξ 1 is the first control variable of the prescribed performance controller, and is the first derivative of the first control variable ξ 1 ; μ 1 is the intermediate parameter of the first state variable ξ 1 ; ξ 2 is the second state variable of the prescribed performance controller, and is the first derivative of the second state variable ξ 2 ; μ 2 is the intermediate parameter of the second state variable ξ 2 ; τ a is the compensation term of the prescribed performance controller, τ b is the feedback term, τ c is the control term to ensure that the linear motor system has a prescribed performance boundary, τ l is the control term to compensate for the difference between the control input v of the linear motor and the actual control input; is the regressor of the actual speed of the output shaft of the linear motor; is the parameter estimation vector of the unknown parameter vector θ, and the unknown parameter vector θ is a set of parameters representing the parameter uncertainty of the linear motor; M is the load mass of the linear motor; is the first derivative of the output of the first-order filter of the prescribed performance controller; η 2 (t) is the second prescribed performance boundary function of the prescribed performance controller at time t, and is the first derivative of the second prescribed performance boundary function η 2 (t) of the prescribed performance controller at time t; z 2 is the intermediate error, is the auxiliary variable used to handle the input constraint of the linear motor; x 1 is the actual position of the output shaft of the linear motor, and x 2 is the actual speed of the output shaft of the linear motor, is the actual acceleration of the output shaft of the linear motor; S(x 2 ) is a smooth function of the actual speed x 2 of the output shaft of the linear motor; is the auxiliary variable used to handle the input constraint satisfied by the rate of change of the auxiliary variable; g(v) is a smooth function of the linear motor under the control input v; F 12 and F 11 are the upper and lower bounds of the actual position x 1 of the output shaft of the linear motor respectively; F 22 and F 21 They are respectively the upper and lower boundaries of the actual speed of the output shaft of the linear motor; The said state constraints are specifically as follows: F 11 <x 1 <F 12 F 21 <x 2 <F 22 The said input constraints are specifically as follows: Among them, u(v) is the actual input of the linear motor under the control input quantity v; sat(v) is the saturation function of the linear motor under the control input quantity v; κ(v) is the bounded approximation error between the smooth function g(v) and the saturation function sat(v) of the linear motor under the control input quantity v; u max is the maximum control input of the linear motor; exp(·) is the exponential function with the natural constant e as the base; The feedback item τ b is as follows: τ b = τ b1 + τ b2 where, τ b1 is a feedback term used to stabilize the nominal system, and τ b2 is a robust feedback term used to attenuate the influence of uncertain nonlinearities and modeling errors in the state-space model; c 2 is a positive constant parameter; η 1 (t) is the first predetermined performance boundary function of the predetermined performance controller at time t; z 1 is the virtual error, and z 1 = ξ 1 - α 0 where α 0 is the desired trajectory of the linear motor; β is the intermediate variable of the first-order filter of the predetermined performance controller; Desired trajectory α of the linear motor 0 , as follows: where x 1d is the desired position of the output shaft of the linear motor; The robust feedback term τ used to attenuate the influence of the uncertain nonlinearity and modeling error of the state space model b2 satisfies the following conditions: τ b2 = μ 2 τ b2 where, τ′ b2 is the first derivative of the robust feedback term τ b2 ; is the positive constant parameter of the robust feedback term τ b2 ; is the uncertainty term of the unknown disturbance of the linear motor The first predetermined performance boundary function η 1 (t) and the second predetermined performance boundary function η 2 (t) of the described predetermined performance controller at time t are specifically as follows: η 1 (t) = η 11 exp(-r 1 t) + η 21 η 2 (t) = η 12 exp(-r 2 t) + η 22 Among them, η 11 and η 21 are respectively the first parameter and the second parameter of the first predetermined performance boundary function η 1 (t), η 12 and η 22 are respectively the first parameter and the second parameter of the second predetermined performance boundary function η 2 (t), η 11 > η 21 > 0, η 12 > η 22 > 0; r 1 and r 2 are respectively the first time constant of the first predetermined performance boundary function η 1 (t) and the second time constant of the second predetermined performance boundary function η 2 (t); The first-order filter of the said predetermined performance controller is specifically as follows: where ε is a positive constant parameter; α 1 is the virtual control quantity of the input of the first-order filter; α 1 (0) is the virtual control quantity of the input of the first-order filter at time 0, α 2 (0) is the output of the first-order filter at time 0, and β(0) is the intermediate variable of the first-order filter at time 0; α 2 is the output of the first-order filter; Obtain its first derivative through the output of the first-order filter and input it into the predetermined performance controller; The intermediate variable β of the first-order filter is specifically as follows: where x is the actual trajectory of the output shaft of the linear motor, x = [x 1 , x 2 T ; x 2d is the desired speed of the output shaft of the linear motor.​ 2. A predetermined performance control method for a linear motor system based on state transition according to claim 1, characterized in that: In the said Step 1, the predetermined performance boundary conditions satisfied by the determined position error of the output shaft of the linear motor are specifically as follows: |z 1 | < | η 1 (t)| Wherein, φ is the position error of the output shaft of the linear motor, φ = x 1 -x 1d .

3. A predetermined performance control method for a linear motor system based on state transition according to claim 1, characterized in that: In the said Step 2, to obtain the actual position and actual speed of the output shaft of the current linear motor, specifically, it is directly obtained through the position and speed sensors pre-set inside the linear motor.

4. A predetermined performance control method for a linear motor system based on state transition according to claim 1, characterized in that: In the said Step 3, to obtain the actual input of the linear motor under the control input quantity that satisfies the input constraints through the control input quantity of the linear motor, specifically, it is obtained by subjecting the control input quantity of the linear motor to the saturation effect processing of a pre-set saturator to obtain the actual input of the linear motor under the control input quantity that satisfies the input constraints.

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