CN115236987A - Magnetic suspension workbench iterative learning robust control method based on error tracking - Google Patents
Magnetic suspension workbench iterative learning robust control method based on error tracking Download PDFInfo
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Abstract
A magnetic suspension workbench iterative learning robust control method based on error tracking utilizes dynamic decoupling of a magnetic suspension planar motor to convert multi-freedom control into independent single-freedom degrees, a nonlinear system model is built, an expected state track is combined, an expected error track is designed by utilizing an attenuation function, a sliding mode surface is built, an adaptive robust control item is designed to ensure stability of a magnetic suspension workbench, an uncertain item is eliminated, the adaptive robust control item and an iterative learning control item are combined through a parallel structure, a magnetic suspension workbench iterative learning robust controller based on error tracking is obtained, stability and error convergence analysis is carried out, a control algorithm is applied to an actual magnetic suspension workbench, effectiveness of the control algorithm is verified, limitation of the existing multi-freedom magnetic suspension workbench control method can be overcome, tracking accuracy is considered, and influence of initial position errors of the magnetic suspension workbench on the iterative learning control method is restrained.
Description
Technical Field
The invention relates to the field of magnetic suspension workbench control, in particular to a magnetic suspension workbench iterative learning robust control method based on error tracking.
Background
As a novel driving element, the magnetic suspension workbench is one of core components of precision machining and manufacturing equipment, and the motion performance of the magnetic suspension workbench is the key for guaranteeing the machining performance of the equipment. Compared with the traditional mechanical superposition type or air floating type driving platform, the magnetic suspension workbench can drive a single rigid body to realize multi-degree-of-freedom motion through electromagnetic force, has the advantages of no friction, vacuum operation, heat source separation and the like, and better meets the process requirements in the current precision machining and manufacturing field.
The adaptive robust control has good robustness and system parameter adaptive capacity, but the uncertain items and disturbance related to the state in the system are only inhibited through robust feedback, so that the steady-state tracking error is larger. The iterative learning control method can realize effective compensation of model uncertainty items and state follow-up interference on the premise of not depending on an accurate system model. The track tracking performance of the magnetic suspension workbench can be effectively improved by combining adaptive robust control with iterative learning control.
However, the control method based on iterative learning depends on the condition that the system has no initial position deviation, and the initial position deviation in the system is accumulated continuously along with the increase of the iteration period, so that the stability of the whole system is influenced.
Disclosure of Invention
According to the defects of the prior art, the invention aims to provide a magnetic suspension workbench iterative learning robust control method based on error tracking, which can overcome the limitations of the existing multi-degree-of-freedom magnetic suspension workbench control method, give consideration to tracking accuracy and inhibit the influence of the system initial position error on the iterative learning control method.
In order to solve the technical problems, the invention adopts the technical scheme that:
a magnetic suspension workbench iterative learning robust control method based on error tracking comprises the following steps:
step 1, converting multi-freedom control into independent single degrees of freedom for design by utilizing dynamic decoupling of a magnetic suspension planar motor, and constructing a nonlinear system model aiming at each degree of freedom;
step 2, designing an expected error track by using an attenuation function based on the nonlinear system model constructed in the step 1 and combining an expected state track, constructing a sliding mode surface based on the tracking deviation of the error and the tracking deviation derivative of the error, designing an adaptive robust control item to ensure the stability of the magnetic suspension workbench, designing an iterative learning control item to eliminate a model uncertainty item existing in the magnetic suspension workbench, and combining the adaptive robust control item and the iterative learning control item through a parallel structure to obtain an iterative learning robust controller of the magnetic suspension workbench based on error tracking;
step 3, analyzing the stability and error convergence of the magnetic suspension workbench robust iterative learning control method based on error tracking obtained in the step 2 by utilizing a Lyapunov direct method;
and 4, applying a control algorithm to an actual magnetic suspension working table, tracking the track under the condition that the magnetic suspension working table has initial position errors and interference, and verifying the effectiveness of the magnetic suspension working table iterative learning robust controller based on error tracking.
Further, in step 1, based on a dynamic decoupling strategy, each degree of freedom can be used as an independent single-input single-output system for closed-loop control, so that, under the condition of not losing generality, taking x-axis direction translation as an example (the same principle is applied to x-axis direction rotation, y-axis direction translation, y-axis direction rotation, z-axis direction translation and z-axis direction rotation), at the k-th iteration, a nonlinear system of the magnetic suspension workbench translating in the x-axis direction
Expressed as:
wherein x is k Representing the amount of displacement, u, of the table in translation along the x-axis in the k-th iteration k (t) is the control output of the kth iteration, is the theta magnetic suspension worktable parameter, and f (x) k T) is the combination of uncertainty and state-dependent disturbance associated with the state of the magnetic levitation table, delta d Representing non-repetitive external disturbances;
defining a tracking error e in a kth iteration k ，e k The calculation formula of (2) is as follows:
e k ＝x d -x k (2)
wherein x is d Representing a reference track point;
according to equation (3), equation (4) is rewritten as follows:
and similarly, constructing a nonlinear system for the rotation of the magnetic suspension workbench in the x-axis direction, a nonlinear system for the translation of the magnetic suspension workbench in the y-axis direction, a nonlinear system for the rotation of the magnetic suspension workbench in the y-axis direction, a nonlinear system for the translation of the magnetic suspension workbench in the z-axis direction, and a nonlinear system for the rotation of the magnetic suspension workbench in the z-axis direction.
Further, the tracking error e k It is required to complete the track cycle 0, T]The inner following expected error track converges, so that corresponding transition is carried out at the beginning stage to enable the actual state track x of the magnetic suspension working table k Can be in [ T ] r ，T]Follows the desired state trajectory in a time period of (T) r To correct the time, correct the time T r Satisfies the condition 0 < T r < T, correction time T r The actual requirements and the range of control input are predetermined, and the realization of good state tracking requires a correction process for initial state error and an expected error track
The design is as follows:
where t is a time and ζ (t) is a decay function.
Further, the decay function ζ (t) needs to guarantee the expected error trajectory
Continuously conductive over the time domain.
Further, the decay function ζ (t) is configured as:
further, a tracking deviation epsilon of the error is defined k (t)：
Wherein, the first and the second end of the pipe are connected with each other,
for the trajectory tracking error in the k-th iteration,
is the expected error trajectory;
the tracking offset derivative of the error
According to the formula (6) and the formula (7), the sliding mode surface s based on the error tracking deviation can be constructed k (t)：
Wherein c represents a sliding mode surface parameter, and c >0.
Further, if the control output can be such that s k (t) tends to zero, then the tracking error can also coincide with the expected error trajectory;
time derivative of sliding mode pair
Can be expressed as:
second derivative of tracking deviation of error
Substituting equation (2) into equation (9) can yield u (t) and
the relationship between them is as follows:
wherein the content of the first and second substances,
represents the model compensation at the kth iteration;
according to the formula (10), the calculation formula of the control output of the magnetic suspension workbench iterative learning robust controller based on the error tracking at the kth iteration is as follows:
wherein, the first and the second end of the pipe are connected with each other,
represents the control output quantity of the iterative learning control item,
is the output of the adaptive robust control term;
error tracking based magnet at kth iteration according to equation (10)Output of adaptive robust control item of suspension workbench iterative learning robust controller
The calculation formula of (2) is as follows:
wherein w k (t) shows an improved switching term which helps to reduce buffeting, w, of a control magnetic levitation table k The calculation formula of (t) is as follows:
w k (t)＝η k ·sgn(s k )+l s s k (t) (13)
wherein sgn (. Cndot.) is the switching function, l s For feedback of compensation coefficients, sgn(s) for simplicity k (t)) abbreviated sgn(s) k ) Magnetic levitation table estimation parameters
Generated by adaptive items in the adaptive robust control items, and parameters are estimated by adopting a parameter adaptive algorithm based on discontinuous mapping according to the boundaries of the parameters
Is derived by the following formula:
wherein gamma is a fixed parameter estimation adaptability, and the mapping correction method can ensure that
Always within the allowable range of the parameters without affecting the convergence, and furthermore, in selecting the initial value of the estimated parameters
When it is necessary to satisfy the conditions
Further, the step 3 specifically includes the following steps:
step 301, constructing energy function E of kth iteration by utilizing Lyapunov direct method k (t)：
Wherein, the first and the second end of the pipe are connected with each other,
is the learning error of iterative learning, can pass through
Calculating to obtain;
deviation of parameter estimation
Can pass through
Calculating to obtain;
step 302, deducing E between two adjacent iterations k (t) difference DeltaE k As follows:
according to V k Definition of (1), difference Δ V between two successive iterations k Expressed as: :
there is no initial tracking error bias and,
rewritable in the form of integrals
Based on the expression, substituting equation (17) with equation (12) and equation (13) may be replaced by:
in a similar manner to that described above,
the derivative in the iterative domain can be expanded into the following form:
where ξ is the learning rate;
for arbitrary constants a and
in other words, the half-saturation function
The following inequalities can be satisfied:
therefore, the integrand in equation (18) is rewritten to the following form according to the saturation function properties and the learning law of the iterative learning term:
replacing it with the corresponding integral term in equation (19) yields:
according to
Between the kth and the k-1 th iteration, the difference
Can be expressed as:
e between the k-th and k-1-th iterations according to iteration update law (11) in combination with equations (16) - (23) k The difference of (c) can be simplified as:
if a switching gain η is desired d Satisfies the condition eta d ≥Δ max Then, the inequality is ensured to be established;
according to the definition of the Lyapunov functional, E can be verified 0 (t)≥0、E k (t) ≧ 0 and E 0 (t) is a bounded value, which means when k → ∞, s k Can be asymptotically converged to zero under the definition of L2 norm because of
For a bounded value, s can be determined k (t) has continuous consistency;
based on the Barbalt theorem, the method can be obtained
Tracking error satisfaction
When T ∈ [ T ] r ，T]When the utility model is used, the water is discharged,
therefore, the magnetic suspension workbench iterative learning robust controller based on the error tracking can ensure the monotonous convergence of the magnetic suspension workbench.
Further, output of the iterative learning control item
Can effectively inhibit the uncertainty term f (x) related to the unmodeled state k T), therefore, in the iterative learning update law, a partial saturation learning strategy is introduced to ensure the boundedness of the iterative learning result, and therefore, the output of the ILC control item
Updated by the following learning rules:
wherein the content of the first and second substances,
the control output for the last iteration, ξ for the learning gain,
the specific expression of the saturation function is as follows:
wherein epsilon d Is the upper boundary value of the uncertainty item of the magnetic suspension workbench.
Further, if the effectiveness of the robust controller for the iterative learning of the magnetic suspension workbench based on the error tracking is not strong, the learning rate is adjusted or the iteration times are increased.
Compared with the prior art, the invention has the following advantages and beneficial effects:
according to the magnetic suspension workbench iterative learning robust control method based on error tracking, the expected error track is designed, the error tracking is used for replacing the traditional position tracking, the influence of the initial position error of the system on iterative learning control is eliminated, and the correction capability of the initial error and the stability of the control system are improved.
The control method can solve the tracking control problem of the multi-degree-of-freedom magnetic suspension planar motor under the condition that the system has an initial iteration position error, and has strong robustness and parameter self-adaptive capacity of the algorithm and high steady-state tracking precision.
Drawings
Fig. 1 is a schematic diagram of a multi-degree-of-freedom magnetic levitation planar motor.
FIG. 2 is a flow chart of the steps of the control method of the present invention.
Fig. 3 is a structural diagram of the magnetic levitation worktable iterative learning robust control method based on error tracking according to the present invention.
Fig. 4 is a diagram illustrating a position tracking error convergence process in a final stack of a control method according to the present invention applied to a magnetic levitation system.
Detailed Description
The technical solution in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.
An iterative learning robust control method for a magnetic suspension workbench based on error tracking, as shown in fig. 1 and fig. 2, comprises the following steps:
step 1, converting multi-freedom control into independent single degrees of freedom for design by utilizing dynamic decoupling of a magnetic suspension planar motor, and constructing a nonlinear system model aiming at each degree of freedom;
step 2, designing an expected error track by using an attenuation function based on the nonlinear system model constructed in the step 1 and combining an expected state track, constructing a sliding mode surface based on the tracking deviation of the error and the tracking deviation derivative of the error, designing an adaptive robust control item to ensure the stability of the magnetic suspension workbench, designing an iterative learning control item to eliminate a model uncertainty item existing in the magnetic suspension workbench, and combining the adaptive robust control item and the iterative learning control item through a parallel structure to obtain an iterative learning robust controller of the magnetic suspension workbench based on error tracking;
step 3, analyzing the stability and error convergence of the magnetic suspension workbench robust iterative learning control method based on error tracking obtained in the step 2 by utilizing a Lyapunov direct method;
and 4, applying the control algorithm to an actual magnetic suspension workbench, tracking the track under the condition that the magnetic suspension workbench has initial position errors and interference, and verifying the effectiveness of the control algorithm.
According to the method, the expected error track is designed, the error tracking is utilized to replace the traditional position tracking, the influence of the initial position error of the magnetic suspension workbench on iterative learning control is eliminated, and the correction capability of the initial error and the stability of a control system are improved.
The control method can solve the tracking control problem of the multi-degree-of-freedom magnetic suspension planar motor under the condition that the magnetic suspension workbench has initial iteration position errors, and has strong robustness and parameter self-adaptive capacity of the algorithm and high steady-state tracking precision.
In the invention, in step 1, based on a dynamic decoupling strategy, each degree of freedom can be used as an independent single-input single-output system for closed-loop control, so that under the condition of no loss of generality, by taking x-axis direction translation as an example (the same principle is applied to x-axis direction rotation, y-axis direction translation, y-axis direction rotation, z-axis direction translation and z-axis direction rotation), at the kth iteration, a nonlinear system of magnetic suspension workbench in x-axis direction translation is represented as
Wherein x is k Representing the x-axis direction of the table in the k-th iterationAmount of displacement to translation, u k (t) is the control output of the kth iteration, is the parameter of the theta magnetic suspension workbench, and f (x) k T) is the combination of uncertainty and state-dependent disturbance associated with the state of the magnetic levitation table, delta d Representing non-repetitive external disturbances;
defining a tracking error e in a kth iteration k ，e k The calculation formula of (2) is as follows:
e k ＝x d -x k (2)
wherein x is d Representing a reference track point;
from equation (5), equation (6) is rewritten to the following form:
and similarly, constructing a nonlinear system for the rotation of the magnetic suspension workbench in the x-axis direction, a nonlinear system for the translation of the magnetic suspension workbench in the y-axis direction, a nonlinear system for the rotation of the magnetic suspension workbench in the y-axis direction, a nonlinear system for the translation of the magnetic suspension workbench in the z-axis direction, and a nonlinear system for the rotation of the magnetic suspension workbench in the z-axis direction.
In the case of state tracking, the objective is to make the magnetic levitation table output a tracking desired state trajectory even if the tracking error converges to zero, and in the case of error tracking, the objective is to make the tracking error itself converge according to a desired error convergence trajectory, so that the tracking error converges to a desired error trajectory.
Instead of designing the controller according to the desired state trajectory, the desired error trajectory itself is independent of the desired state trajectory, and the same desired error trajectory can be applied for tasks with different desired state trajectories.
Thus, in the reference error tracking strategy, the tracking error e k It is required to complete the track cycle 0, T]The inner following expected error track converges, so that corresponding transition is carried out at the beginning stage to enable the actual state track x of the magnetic suspension working table k Can be in [ T ] r ，T]Follows the expected state trajectory over a period of time。
For non-zero initial state errors that may exist in a magnetic levitation table, achieving good state tracking requires a correction procedure for the initial state errors due to the limited amount of real control output. Correcting time T r Satisfies the condition 0 < T r < T, the size of which needs to be predetermined according to the actual demand and the range of the control input amount. The expected error track is constructed for correcting the initial error of the magnetic suspension workbench, and the expected error track
The design is as follows:
where ζ (t) is a decay function;
in practice, ζ (t) needs to guarantee the expected error trajectory
The attenuation function zeta (t) is continuously derivable in a time domain, is unique in construction mode, and only needs to meet the conditions of gradual convergence and high-order derivation.
The decay function ζ (t) is constructed as:
the present invention is only exemplified by the formula (5).
Error-defining tracking deviation epsilon k (t)：
The tracking deviation derivative of the error
According to the formula (6) and the formula (7), the sliding mode surface s based on the error tracking deviation can be constructed k (t)：
Wherein c represents a sliding mode surface parameter, and c >0.
If the control output can be such that s k (t) goes to zero, then the tracking error can also coincide with the desired error trajectory.
Sliding mode surface time derivative
Can be expressed as:
second derivative of tracking deviation of error
Substituting equation (2) into equation (9) can yield u (t) and
the relationship between them is as follows:
wherein the content of the first and second substances,
represents the model compensation at the kth iteration;
as shown in the control structure block diagram of fig. 3, according to equation (10), the calculation formula of the control output of the magnetic levitation table iterative learning robust controller based on error tracking at the kth iteration is:
wherein the content of the first and second substances,
a control output quantity representing an iterative learning control term,
is the output of the adaptive robust control term.
According to the formula (10), the output of the adaptive robust control item of the magnetic suspension workbench iterative learning robust controller based on the error tracking at the kth iteration
The calculation formula of (2) is as follows:
wherein, w k (t) shows an improved switching term which helps to reduce buffeting, w, in controlling a magnetic levitation table k The calculation formula of (t) is as follows:
ω k (t)＝η k ·sgn(s k )+l s s k (t) (13)
wherein sgn (. Cndot.) is the switching function, l s The compensation factor is fed back. For the sake of brevity, sgn(s) k (t)) abbreviated sgn(s) k ). Magnetic levitation table estimation parameters
Generated by an adaptation term in the adaptive robust control term. According to the boundary of the parameter, adopting a parameter self-adaptive algorithm based on discontinuous mapping to estimate the parameter
Is derived by the following formula:
wherein gamma is a fixed parameter estimation adaptation rate. The mapping correction method can ensure
Always within the allowable range of the parameter without affecting the convergence, and furthermore, in selecting the initial value of the estimated parameter
When it is necessary to satisfy the conditions
In step 3, the stability and the error convergence of the obtained iterative learning robust controller based on error tracking are analyzed by using a Lyapunov direct method, and the method specifically comprises the following steps:
step 301, constructing energy function E of kth iteration by utilizing Lyapunov direct method k (t)：
Wherein the content of the first and second substances,
is the learning error of iterative learning, can pass through
Calculating to obtain;
parameter estimationDeviation of (2)
Can pass through
Calculating to obtain;
step 302, deducing E between two adjacent iterations k (t) difference DeltaE k As follows:
according to V k Definition of (1), difference Δ V between two successive iterations k Expressed as: :
there is no initial tracking error bias and,
rewritable in the form of integrals
Based on the expression, substituting equation (12) and equation (13) into equation (17) for substitution results:
in a similar manner to that described above,
the derivative in the iterative domain can be expanded into the following form:
wherein xi is the learning rate;
for arbitrary constants a and
in other words, the half-saturation function
The following inequality can be satisfied:
therefore, the integrand in equation (18) is rewritten to the following form according to the saturation function properties and the learning law of the iterative learning term:
replacing it with the corresponding integral term in equation (19) yields:
according to
Between the kth and the k-1 th iteration, the difference
Can be expressed as:
e between the k-th and k-1-th iterations according to iteration update law (11) in combination with equations (16) - (23) k The difference of (a) can be simplified as:
if a switching gain η is desired d Satisfies the condition eta d ≥Δ max Then the inequality is ensured to be true.
According to the definition of the Lyapunov functional, E can be verified 0 (t)≥0、E k (t) ≧ 0 and E 0 (t) is a bounded value, which means when k → ∞, s k Can be asymptotically converged to zero under the definition of L2 norm because of
For a bounded value, s can be determined k (t) has continuous consistency.
Based on the barbat theorem, it can be obtained
Tracking error satisfaction
When T ∈ [ T ] r ，T]When the temperature of the water is higher than the set temperature,
therefore, the robust controller for the iterative learning of the magnetic suspension workbench based on the error tracking can ensure the monotonic convergence of the magnetic suspension workbench.
In the present invention, as shown in FIG. 2, the output of the control term is iteratively learned
Can effectively inhibit the uncertainty term f (x) related to the unmodeled state k T) influence. Therefore, in the iterative learning updating law, a partial saturation learning strategy is introduced to ensure the boundedness of the iterative learning result. Thus, the output of the ILC control term
Updated by the following learning rules:
wherein the content of the first and second substances,
the control output for the last iteration, ξ for the learning gain,
the specific expression of the saturation function is as follows:
wherein epsilon d Is the upper boundary value of the uncertainty item of the magnetic suspension workbench.
In one embodiment of the present invention, the process of applying the proposed control method to a magnetic levitation table is: firstly, determining structural parameters of a controller according to a magnetic suspension workbench, wherein c =10,g =60, and the parameter boundary of the magnetic suspension workbench is set to be theta min ＝2.34kg，θ max ＝4kg，ε d ＝0.158，Δ max =1.25, learning rate ξ =0.25, and correction time is set to T r ＝0.02s。
To test the tracking performance of the controller, a tracking experiment of a typical sinusoidal trajectory along the horizontal x-axis direction was developed, expecting a trajectory x d Comprises the following steps: x is a radical of a fluorine atom d And (ii) sin (π t) (mm), wherein the tracking angular velocity of the trajectory is 3.14rad/s, at a velocity of 1.57cos (π t) mm/s. x is a radical of a fluorine atom d The desired trace point. Setting the initial position of each iteration to x k (0) = (0.09 +0.02rand (k)) mm, so that at each iteration, the magnetic suspension table has an initial error of about 0.1 mm. By evaluating the error of the index root mean square e rms Maximum absolute value e of sum error max And evaluating the performance of the magnetic suspension workbench.
The actual control effect of the control method is shown in fig. 4. According to the change of the tracking error and the actual tracking track, the actual motion track of the magnetic suspension workbench is very close to the reference track through repeated iterative learning, and the tracking root mean square error is less than 1.4um.
Under the condition that the magnetic suspension workbench has an initial position error, after a limited number of iterations, the control method can effectively eliminate the influence caused by the initial position error, and can finish the correction of the initial error according to a pre-designed expected error track, thereby achieving better steady-state tracking accuracy.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.
Claims (10)
1. A magnetic suspension workbench iterative learning robust control method based on error tracking is characterized by comprising the following steps: step 1, converting multi-freedom control into independent single degrees of freedom for design by utilizing dynamic decoupling of a magnetic suspension planar motor, and constructing a nonlinear system model aiming at each degree of freedom;
step 2, designing an expected error track by using an attenuation function based on the nonlinear system model constructed in the step 1 and combining the expected state track, constructing a sliding mode surface based on the tracking deviation of the error and the tracking deviation derivative of the error, designing an adaptive robust control item to ensure the stability of the magnetic suspension workbench, designing an iterative learning control item to eliminate model uncertainty items existing in the magnetic suspension workbench, and combining the adaptive robust control item and the iterative learning control item through a parallel structure to obtain an iterative learning robust controller of the magnetic suspension workbench based on error tracking;
step 3, analyzing the stability and the convergence of errors of the magnetic suspension workbench robust iterative learning control method based on error tracking obtained in the step 2 by using a Lyapunov direct method;
and 4, applying a control algorithm to an actual magnetic suspension working table, tracking the track under the condition that the magnetic suspension working table has initial position errors and interference, and verifying the effectiveness of the magnetic suspension working table iterative learning robust controller based on error tracking.
2. The magnetic levitation workbench iterative learning robust control method based on error tracking as claimed in claim 1, characterized in that: in step 1, based on a dynamic decoupling strategy, each degree of freedom can be used as an independent single-input single-output system for closed-loop control, so that under the condition of no loss of generality, taking x-axis direction translation as an example (the same principle is applied to x-axis direction rotation, y-axis direction translation, y-axis direction rotation, z-axis direction translation and z-axis direction rotation), during the kth iteration, a nonlinear system of the magnetic suspension workbench for x-axis direction translation is adopted
Expressed as:
wherein x is k Representing the amount of displacement, u, of the table in translation along the x-axis in the k-th iteration k (t) is the control output of the kth iteration, is the theta magnetic suspension worktable parameter, and f (x) k T) is the combination of uncertainty and state-dependent disturbance associated with the state of the magnetic levitation table, delta d Representing non-repetitive external disturbances;
defining a tracking error e in a kth iteration k ，e k The calculation formula of (2) is as follows:
e k ＝x d -x k (2)
wherein x is d Representing a reference track point;
from equation (1), equation (2) is rewritten to the following form:
and similarly, constructing a nonlinear system rotating in the x-axis direction of the magnetic suspension workbench, a nonlinear system translating in the y-axis direction of the magnetic suspension workbench, a nonlinear system rotating in the y-axis direction of the magnetic suspension workbench, a nonlinear system translating in the z-axis direction of the magnetic suspension workbench and a nonlinear system rotating in the z-axis direction of the magnetic suspension workbench.
3. The magnetic levitation workbench iterative learning robust control method based on error tracking as claimed in claim 1, characterized in that: tracking error e k It is required to complete the track cycle 0, T]The inner following expected error track converges, so that corresponding transition is carried out at the beginning stage to enable the actual state track x of the magnetic suspension working table k Can be in [ T ] r ，T]Follows the desired state trajectory in a time period of (T) r To correct the time, the time T is corrected r Satisfies the condition 0 < T r < T, correction time T r The actual requirements and the range of control input are predetermined, and the realization of good state tracking requires a correction process for initial state error and an expected error track
The design is as follows:
where t is a time and ζ (t) is a decay function.
4. The magnetic levitation workbench iterative learning robust control method based on error tracking as claimed in claim 3, characterized in that: the decay function ζ (t) needs to guarantee the expected error trajectory
Continuously conductive over the time domain.
5. The magnetic levitation workbench iterative learning robust control method based on error tracking as claimed in claim 4, characterized in that: the decay function ζ (t) is constructed as:
6. the magnetic levitation workbench iterative learning robust control method based on error tracking as claimed in claim 3, characterized in that: error-defining tracking deviation epsilon k (t)：
Wherein the content of the first and second substances,
for the trajectory tracking error in the k-th iteration,
is the expected error trajectory;
the tracking deviation derivative of the error
According to the formula (6) and the formula (7), the sliding mode surface s based on the error tracking deviation can be constructed k (t)：
Wherein c represents a sliding mode surface parameter, and c is more than 0.
7. The magnetic levitation workbench iterative learning robust control method based on error tracking as claimed in claim 6, characterized in that: if the control output can be such that s k (t) tends to zero, then the tracking error can also coincide with the expected error trajectory;
time derivative of sliding mode pair
Can be expressed as:
second derivative of tracking deviation of error
And equation (2) into equation (9), u (t) and
the relationship between them is as follows:
wherein the content of the first and second substances,
representing model compensation at the kth iteration;
according to the formula (10), the calculation formula of the control output of the magnetic suspension workbench iterative learning robust controller based on the error tracking at the kth iteration is as follows:
wherein the content of the first and second substances,
represents the control output quantity of the iterative learning control item,
is the output of the adaptive robust control term;
according to the formula (10), the output of the adaptive robust control item of the magnetic suspension workbench iterative learning robust controller based on the error tracking at the kth iteration
The calculation formula of (c) is:
wherein, w k (t) shows an improved switching term which helps to reduce buffeting, w, of a control magnetic levitation table k The calculation formula of (t) is as follows:
w k (t)＝η k ·sgn(s k )+l s s k (t) (13)
wherein sgn (. Cndot.) is the switching function, l s For feedback of compensation coefficients, sgn(s) for simplicity k (t)) abbreviated sgn(s) k ) Magnetic levitation table estimation parameters
Generated by adaptive items in the adaptive robust control items, and parameters are estimated by adopting a parameter adaptive algorithm based on discontinuous mapping according to the boundaries of the parameters
Is derived by the following formula:
wherein gamma is a fixed parameter estimation adaptability, and the mapping correction method can ensure
Always within the allowable range of the parameter without affecting the convergence, and furthermore, in selecting the initial value of the estimated parameter
When it is necessary to satisfy the conditions
8. The magnetic levitation workbench iterative learning robust control method based on error tracking as claimed in claim 6, wherein said step 3 specifically comprises the following steps:
step 301, constructing an energy function Ek (t) of the kth iteration by using a Lyapunov direct method:
wherein, the first and the second end of the pipe are connected with each other,
is the learning error of iterative learning, can pass through
Calculating to obtain;
deviation of parameter estimation
Can pass through
Calculating to obtain;
step 302, deducing E between two adjacent iterations k (t) difference DeltaE k As follows:
according to V k Definition of (1), difference Δ V between two successive iterations k Expressed as:
there is no initial tracking error bias and,
rewritable as integral form
Based on the expression, substituting equation (17) with equation (12) and equation (13) may be replaced by:
in a similar manner to that described above,
the derivative in the iterative domain can be expanded into the following form:
where ξ is the learning rate;
for arbitrary constants a and
in other words, the half-saturation function
The following inequality can be satisfied:
therefore, the integrand in equation (18) is rewritten to the following form according to the saturation function properties and the learning law of the iterative learning term:
replacing it with the corresponding integral term in equation (19) yields:
according to
Between the kth and the k-1 th iteration, the difference
Can be expressed as:
according to the iteration update law (11) in combination with equations (16) - (23), the difference in Ek between the kth and the (k-1) th iterations can be simplified as:
if a switching gain η is desired d Satisfies the condition eta d ≥Δ max Then, the inequality is ensured to be established;
according to the definition of the Lyapunov functional, E can be verified 0 (t)≥0、E k (t) ≧ 0 and E 0 (t) is a bounded value, which means when k → ∞, s k Can be asymptotically converged to zero under the definition of L2 norm because of
For a bounded value, s can be determined k (t) has continuous consistency;
based on the Barbalt theorem, the method can be obtained
Tracking error satisfaction
When T ∈ [ T ] r ，T]When the temperature of the water is higher than the set temperature,
therefore, the robust controller for the iterative learning of the magnetic suspension workbench based on the error tracking can ensure the monotonic convergence of the magnetic suspension workbench.
9. The magnetic levitation workbench iterative learning robust control method based on error tracking as claimed in claim 7, characterized in that: output of iterative learning control terms
Can effectively inhibit the uncertainty term f (x) related to the unmodeled state k T), therefore, in the iterative learning update law, a partial saturation learning strategy is introduced to ensure the boundedness of the iterative learning result, and therefore, the output of the ILC control item
Updated by the following learning rules:
wherein the content of the first and second substances,
the control output for the last iteration, ξ for the learning gain,
the specific expression of the saturation function is as follows:
wherein epsilon d Is the upper boundary value of the uncertainty item of the magnetic suspension workbench.
10. The magnetic levitation workbench iterative learning robust control method based on error tracking as claimed in claim 8, characterized in that: and if the effectiveness of the robust controller for the iterative learning of the magnetic suspension workbench based on the error tracking is not strong, adjusting the learning rate or increasing the iteration times.
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