CN112558469B - Extended state observer-model prediction control method of rigid-flexible coupling motion platform - Google Patents
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Abstract
The invention discloses an extended state observer-model predictive control method of a rigid-flexible coupling motion platform, which comprises the following steps: establishing a dynamic model of the rigid-flexible coupling motion platform, and providing a prediction model suitable for the rigid-flexible coupling motion platform based on the dynamic model; designing rolling optimization suitable for the rigid-flexible coupling motion platform in the high-precision control process by utilizing the prediction model, wherein the accumulated error between the prediction output and the motion planning input is defined as an optimization objective function; and applying the extended observer to a feedback technology of the prediction model to form a closed-loop control algorithm. The invention provides an MPC control method based on ESO feedback aiming at high-precision control and combining the characteristics of a flexible hinge, and by utilizing the feedback of ESO, the model prediction can acquire the disturbance on a rigid-flexible coupling motion platform in real time, so that the optimal solution of control optimization is obtained, and the control effect is improved.
Description
Technical Field
The invention relates to the field of rigid-flexible coupling motion platform control, in particular to an extended observer-model prediction control method for a rigid-flexible coupling motion platform.
Background
Electronic manufacturing equipment plays a very important role in the electronics industry, particularly in modern manufacturing technologies. The precision motion platform is a key component of electronic manufacturing equipment, friction is a main reason for limiting precision of the precision motion platform, and the rigid-flexible coupling motion platform utilizes deformation of a flexible hinge to compensate friction, so that precision of the precision motion platform is improved.
At present, most of high-precision motion platforms adopt a classical PID control algorithm, namely a control strategy for eliminating an error is determined by the error between a control target and an actual behavior, but the control signal of the high-precision motion platform cannot be determined to be the optimal control signal based on the error control mode.
Disclosure of Invention
The invention aims to provide a model prediction control method of a rigid-flexible coupling motion platform extended observer, so that the rigid-flexible coupling motion platform can realize higher speed and higher precision at low cost.
In order to realize the task, the invention adopts the following technical scheme:
an extended state observer-model predictive control method of a rigid-flexible coupling motion platform comprises the following steps:
establishing a dynamic model of the rigid-flexible coupling motion platform, and providing a prediction model suitable for the rigid-flexible coupling motion platform based on the dynamic model;
designing rolling optimization suitable for the rigid-flexible coupling motion platform in the high-precision control process by utilizing the prediction model, wherein the accumulated error between the prediction output and the motion planning input is defined as an optimization objective function;
and applying the extended observer to a feedback technology of the prediction model to form a closed-loop control algorithm.
Furthermore, the rigid-flexible coupling motion platform is a single-degree-of-freedom linear motion platform and comprises an intermediate platform, an external frame, a flexible hinge, a mechanical guide rail, a grating ruler, a linear motor stator and a linear motor rotor, wherein the intermediate platform is connected with the external frame through the flexible hinge, and the linear motor rotor and the intermediate platform are connected and fixed at the lower part of the intermediate platform through bolts.
Further, the prediction model of the rigid-flexible coupling motion platform is represented as:
wherein, alpha is an adjustable parameter, msDenotes the mass of the intermediate platform, mbRepresenting the mass of the outer frame, y1＝ds,y2＝db,
dsIndicating the displacement of the intermediate platform, dbRepresenting the displacement of an external frame, k representing the rigidity coefficient of the flexible hinge, c representing the damping coefficient of the flexible hinge, delta t representing the sampling period, b representing the power conversion coefficient of a rigid-flexible coupling motion platform control system, u representing the signal of the control system, fbThe friction force applied to the rigid-flexible coupling motion platform is shown, and the superscript t represents the time t.
Further, the optimization objective function is represented as:
wherein, y1(i | t) predicting y at time i for current time t1，siTarget displacement, v, of the intermediate platform at time iiIs the target speed at time i;
Utthe variables that need to be solved for optimization for roll optimization are the control input signals, which are expressed as:
Ut＝{u(t|t),u(t+1|t),…,u(t+p-1|t)}
wherein u (t | t) is a control signal for predicting t moment at the current moment t, and p represents a prediction time domain of a prediction model;
the control objective is to minimize the difference between the predicted displacement and the motion planning displacement, with the motion planning displacement and velocity input as follows:
{st+1,st+2,…,st+p}
{vt+1,vt+2,…,vt+p}
and the optimization process must satisfy the control constraints of the control system:
umin≤u(t+i)≤umax,i≥0
where u (t + i) is the control signal at time t + i, uminAs a minimum value of the control signal, umaxIs the maximum value of the control signal.
Further, in the prediction model, under an unconstrained condition, an unconstrained minimum value of the target is 0, and the following equation can be obtained by solving:
substituting the formula into a prediction model to obtain a control input signal u without constraint for rigid-flexible coupling motion platform model predictionm(t)：
And further, obtaining a prediction displacement through the prediction model, substituting the prediction displacement into an optimization objective function, and solving the objective function to realize rolling optimization of model prediction.
Further, defining the rigidity and the damping of the flexible hinge in the motion process of the rigid-flexible coupling motion platform as disturbance omega (t), and simplifying the control input signal to obtain a simplified control input signal;
the control of the prediction model is performed with the latest obtained measurement values at each sampling period
And
the optimization objective function is refreshed, the optimal solution of the refreshed optimization objective function is solved, and then the optimal solution, namely u, can be obtainedm(t) acting on the system.
Further, the application of the extended observer to the feedback technique of the prediction model to form a closed-loop control algorithm includes:
actively estimating the state of the control system at the current moment by using the extended state observer and estimating disturbance; the following three-order ESO was constructed:
in the above formula, z1Is the estimation of the displacement of the system by the ESO observer; z is a radical of2Is an estimate of the system speed by the ESO observer; z is a radical of3The method is an estimation of total disturbance of the system by an ESO observer;
observer-based 20 available y1And z1The transfer function of (a) is:
the characteristic polynomial corresponding to the above formula is s3+β1s2+β2s+β3The poles of the characteristic polynomial corresponding to the observer are uniformly distributed to s-omegaoI.e. configuring the characteristic polynomial as (s + ω)o)3Then three gains β of the observer can be obtained1＝3ωo,
Wherein ω isoIs a parameter for which the bandwidth of the ESO observer is adjustable;
by means of estimation feedback of the ESO to the system, the prediction model can know the disturbance of the rigid-flexible coupling motion platform in real time, namely the z estimated by the ESO1、z2And z3Respectively replacing y in unconstrained control input signals1、y3And omega (t) to obtain an optimal control signal, so that the model prediction-ESO control algorithm realizes closed-loop control.
Further, in the actual use process, firstly, a displacement signal of the rigid-flexible coupling motion platform is acquired by using a grating ruler, the displacement signal and a control signal of the platform are substituted into an ESO observer to obtain the displacement and the speed of the middle platform and the deformation force and the damping force of the flexible hinge of the rigid-flexible coupling motion platform in the control process, then, the state parameters and the disturbance of the platform are subjected to an expression of an unconstrained control input signal to obtain an optimal control signal in the control process of the rigid-flexible coupling motion platform, then, the control signal is optimized in real time by using rolling optimization, then, the optimized control signal is input into a driver by a controller to obtain the force required by the rigid-flexible coupling motion platform in the control process, and finally, the control is realized on the control of the rigid-flexible coupling motion platform, so that the prediction control is completed.
Compared with the prior art, the invention has the following technical characteristics:
in the rigid-flexible coupling motion platform in the scheme of the invention, the middle platform is connected with the external frame by using a flexible hinge; an MPC strategy of a rigid-flexible coupling motion platform is designed; the invention provides an MPC control method based on ESO feedback aiming at high-precision control and combining the characteristics of a flexible hinge, and by utilizing the feedback of ESO, the model prediction can acquire the disturbance on a rigid-flexible coupling motion platform in real time, so that the optimal solution of control optimization is obtained, and the control effect is improved.
Drawings
FIG. 1 is a structure diagram of a rigid-flexible coupling motion platform;
FIG. 2 is an equivalent model diagram of a rigid-flexible coupling motion platform;
fig. 3 is a control block diagram of the rigid-flexible coupling motion platform.
Detailed Description
The core of the model predictive control algorithm is online constraint optimization generated by the characteristics of a predictive control method, however, online constraints aiming at problems in various research fields are difficult to satisfy. In order to solve the limitation of model prediction control on a rigid-flexible coupling motion platform, the invention introduces an Extended State Observer (ESO) of active disturbance rejection control in model prediction. The ESO is an estimation and processing of the disturbances experienced by the controlled object during the control process. Whether the controlled object is time-varying or constant, linear or nonlinear, deterministic or uncertain, after compensation based on the ESO estimation result, the system can be simplified into a basic structure of a feedback system of integral series connection, so that various complicated problems become simple and solvable.
The invention provides a model prediction control method of a rigid-flexible coupling motion platform expansion observer, which is suitable for model prediction of a rigid-flexible coupling motion platform according to a dynamic model of the rigid-flexible coupling motion platform. Therefore, the method estimates the disturbance of the deformation force and the damping force of the flexible hinge and the state of the rigid-flexible coupling motion platform by using the ESO, thereby perfecting the feedback mechanism of model prediction.
The rigid-flexible coupling motion platform structure mainly comprises a middle platform and an outer frame, wherein the middle platform is connected with the outer frame through a flexible hinge. In fig. 1 to 3:
r represents a fillet radius of the flexible hinge, w represents a width of the flexible hinge, t represents a thickness of the flexible hinge, and l represents a length of the flexible hinge; m issDenotes the mass of the intermediate platform, mbRepresenting the mass of the outer frame, fbRepresenting the friction force f experienced by the platform in rigid-flexible couplingsRepresenting the driving force applied to the rigid-flexible coupling motion platform, k representing the rigidity coefficient of the flexible hinge, c representing the damping coefficient of the flexible hinge, dsIndicating the displacement of the intermediate platform, dbRepresenting the displacement of the outer frame, b0A power conversion coefficient representing a control system; the superscript circle of the formula in this scheme represents the derivative of the parameter.
Referring to the attached drawings, the extended observer-model predictive control method of the rigid-flexible coupling motion platform comprises the following steps:
The structure of the rigid-flexible coupling motion platform is shown in figure 1 and is a single-degree-of-freedom linear motion platform, the platform comprises an external frame, an intermediate platform, a flexible hinge, a mechanical guide rail, a grating ruler, a linear motor stator and a linear motor rotor, and the linear motor rotor and the intermediate platform are connected and fixed below the intermediate platform through bolts. An equivalent model of a rigid-flexible coupling motion platform can be made according to the rigid-flexible coupling motion platform of fig. 1 as shown in fig. 2.
Based on the simplified model of the rigid-flexible coupling motion platform in fig. 2, a dynamic model of the rigid-flexible coupling motion platform is established:
order to
Equation 1 reduces to:
b is defined as the power conversion coefficient of the rigid-flexible coupling motion platform control system, u is the signal of the control system, and then:
fsbu formula 3
By substituting the above formula for formula 1, one can obtain:
order:
substituting the above formula into formula 4, and introducing an adjustable parameter α in order to adapt to uncertain factors in the experiment and improve the control effect of the experiment, and reducing the formula 4 into:
order:
then equation 6 is simplified to:
and (3) substituting the differential quotient of the state variables for the derivative to obtain an approximate difference equation under the sampling period delta t, and changing the equation 7 into the following equation:
in the formula 8, t refers to the current time, t +1 refers to the next time, and the formula 8 is simplified as follows:
Ayt+1＝(A+ΔtB)yt+Δtdtformula 9
Namely:
from this, equation 10 is a proposed prediction model for the rigid-flexible coupled motion platform MPC, and y in the prediction model is used1And y3The predicted displacement output starting at step t is noted as:
where p is the prediction time domain of the prediction model, t +1| t in parentheses represents the output at the time when t +1 is predicted at the current time t, and so on.
And 2, designing rolling optimization suitable for the rigid-flexible coupling motion platform in the high-precision control process based on the prediction model of the rigid-flexible coupling motion platform.
And recording the variable needing to be solved and optimized in the rolling optimization in the prediction model as a control input signal as Ut：
UtOr { u (t | t), u (t +1| t), …, u (t + p-1| t) } formula 12
Where u (t | t) is a control signal that predicts time t at current time t.
The control objective is to minimize the difference between the predicted displacement and the motion planning displacement, with the motion planning displacement and velocity input as follows:
and the optimization process must satisfy the control constraints of the control system:
umin≤u(t+i)≤umaxi is not less than 0 and 14
Where u (t + i) is the control signal at time t + i, uminAs a minimum value of the control signal, umaxIs the maximum value of the control signal.
In summary, that the present invention is looking at finding the best control input for rolling optimization, the more closely the predicted system output is to the motion planning input, the better, then the present invention defines the accumulated error between the predicted output and the motion planning input as the optimization objective function as follows:
in the above formula, y1(i | t) predicting y at time i for current time t1，siDisplacement of the intermediate platform at time i, viThe velocity at time i.
The optimization problem described above can then be summarized as the following optimization control problem:
problem 1:
optimizing variables: u shapet
An objective function:
constraint conditions are as follows: u. ofmin≤u(t+i|t)≤umax,i≥0,1,…,p+ 1
In the prediction model, p is taken as 1 in order to simplify calculation; thus, in the unconstrained condition, it is easy to know that the unconstrained minimum of the target is 0, and the following equation can be solved:
substituting the above formula into the third equation in the prediction model formula 10 to obtain the control input signal u without constraint for rigid-flexible coupling motion platform model predictionm(t)：
The control input signal for equation 17, including the constraint, is therefore:
therefore, in this step, the prediction displacement of the model prediction is obtained by equation 10, and substituted into equation 15, and then the rolling optimization of the model prediction is realized by solving problem 1.
And 3, applying a feedback technology of the core-extended observer of the active disturbance rejection controller and model prediction to form a closed-loop control algorithm.
From the above equation 18, it can be known that if u (t) at the time k acts on the system, the intermediate platform of the rigid-flexible coupled motion platform may receive a deformation force and a damping force, which are external forces received by the prediction model at the time k, and the resolution accuracy of the grating scale in the embodiment of the present invention is ± 0.1um of feedback accuracy, and the deformation of the flexible hinge is smaller than this value, so a more accurate measurement method is required to measure the deformation of the flexible hinge, and the rigidity and the damping of the flexible hinge are also uncertain values in the motion process of the rigid-flexible coupled motion platform. Therefore, defining the two as the disturbance ω (t), which is the amount to be compensated, equation 17 is simplified as:
in summary, the predictive model control of the present invention uses the most recently obtained measurement (feedback obtained) for each sampling period
And
) The problem 1 is refreshed, and the optimal solution of the refreshed optimization problem is solved, so that the optimal solution, i.e. u, can be obtainedm(t) acting on the system.
According to the basic principle of predictive model control, the state of the current time obtained by the state observer will be the starting point for future prediction. And because the rolling optimization decision observer of the invention needs to detect the actual output of the system at each step of control, and needs to introduce real-time tracking compensation for the interference error suffered by the controlled object. The present invention therefore uses the extended state observer to actively estimate the state of the control system at the present time and to estimate the total disturbance of the damping force and the deformation force that occurs in equation 18.
Because the rigid-flexible coupling motion platform is a second-order system, the invention constructs the following three-order ESO:
in the formula 20, z1Is the estimation of the displacement of the system by the ESO observer; z is a radical of2Is an estimate of the system speed by the ESO observer; z is a radical of3The method is an estimation of the total disturbance of the system by an ESO observer, namely the damping force and the damping variation suffered in the control process of the rigid-flexible coupling intermediate platformAnd (4) estimating the shape force.
Observer-based 20 available y1And z1The transfer function of (a) is:
the characteristic polynomial corresponding to the formula 21 is s3+β1s2+β2s+β3The poles of the characteristic polynomial corresponding to the observer are uniformly distributed to s-omegaoI.e. configuring the characteristic polynomial as (s + ω)o)3Then three gains β of the observer can be obtained1＝3ωo,
Wherein ω isoIs a tunable parameter for the bandwidth of the ESO observer.
Therefore, the estimated feedback of the ESO to the system enables the prediction model to know the disturbance of the rigid-flexible coupling motion platform in real time, namely, the z estimated by the ESO1、z2And z3By replacing y in formula 17, respectively1、y3And omega (t) to obtain an optimal control signal, so that the model prediction-ESO control algorithm realizes closed-loop control.
In the actual use process, firstly, a displacement signal of the rigid-flexible coupling motion platform is acquired by using the grating ruler, the displacement signal and the control signal of the platform are substituted into an ESO observer to obtain state parameters (displacement and speed of the platform) of the rigid-flexible coupling motion platform in the control process and deformation force and damping force of the flexible hinge, then the state parameters and disturbance of the platform are subjected to formula 17 to obtain an optimal control signal in the control process of the rigid-flexible coupling motion platform, then the control signal is optimized in real time by using rolling optimization, then the optimized control signal is input into a driver by a controller to obtain the force required by the rigid-flexible coupling motion platform in the control process, and finally the control is realized on the control of the rigid-flexible coupling motion platform, so that the prediction control is completed.
The above embodiments are only used to illustrate the technical solutions of the present application, and not to limit the same; although the present application has been described in detail with reference to the foregoing embodiments, it should be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; such modifications and substitutions do not substantially depart from the spirit and scope of the embodiments of the present application and are intended to be included within the scope of the present application.
Claims (6)
1. An extended state observer-model predictive control method of a rigid-flexible coupling motion platform is characterized by comprising the following steps:
establishing a dynamic model of the rigid-flexible coupling motion platform, and providing a prediction model suitable for the rigid-flexible coupling motion platform based on the dynamic model;
designing rolling optimization suitable for the rigid-flexible coupling motion platform in the high-precision control process by utilizing the prediction model, wherein the accumulated error between the prediction output and the motion planning input is defined as an optimization objective function;
applying the extended observer to a feedback technology of a prediction model to form a closed-loop control algorithm;
the prediction model of the rigid-flexible coupling motion platform is represented as:
wherein, alpha is an adjustable parameter, msDenotes the mass of the intermediate platform, mbRepresenting the mass of the outer frame, y1＝ds,y2＝db,
dsIndicating the displacement of the intermediate platform, dbRepresenting the displacement of the outer frame, k representing the rigidity coefficient of the flexible hinge, c representing the damping coefficient of the flexible hinge, delta t representing the sampling period, and b being of the rigid-flexible coupling motion platform control systemA power conversion coefficient, u is a signal of a control system, fbThe friction force borne by the rigid-flexible coupling motion platform is shown, and the superscript t shows the time t;
the optimization objective function is represented as:
wherein, y1(i | t) predicting y at time i for current time t1，siDisplacement of the intermediate platform at time i, viThe speed at the moment i is represented by p, and the prediction time domain of the prediction model is represented by p;
Utthe variables that need to be solved for optimization for roll optimization are the control input signals, which are expressed as:
Ut＝{u(t|t),u(t+1|t),…,u(t+p-1|t)}
wherein u (t | t) is a control signal for predicting t moment at the current moment t;
the control objective is to minimize the difference between the predicted displacement and the motion planning displacement, with the motion planning displacement and velocity input as follows:
{st+1,st+2,…,st+p}
{vt+1,vt+2,…,vt+p}
and the optimization process must satisfy the control constraints of the control system:
umin≤u(t+i)≤umax,i≥0
where u (t + i) is the control signal at time t + i, uminAs a minimum value of the control signal, umaxIs the maximum value of the control signal;
the application of the extended observer to the feedback technology of the prediction model to form a closed-loop control algorithm comprises the following steps:
actively estimating the state of the control system at the current moment by using the extended state observer and estimating disturbance;
the following three-order ESO was constructed:
in the above formula, z1Is the estimation of the displacement of the system by the ESO observer; z is a radical of2Is an estimate of the system speed by the ESO observer; z is a radical of3The method is an estimation of total disturbance of the system by an ESO observer;
observer-based 20 available y1And z1The transfer function of (a) is:
the characteristic polynomial corresponding to the above formula is s3+β1s2+β2s+β3The poles of the characteristic polynomial corresponding to the observer are uniformly distributed to s-omegaoI.e. configuring the characteristic polynomial as (s + ω)o)3Then three gains β of the observer can be obtained1＝3ωo,
Wherein ω isoIs a parameter for which the bandwidth of the ESO observer is adjustable;
by means of estimation feedback of the ESO to the system, the prediction model can know the disturbance of the rigid-flexible coupling motion platform in real time, namely the z estimated by the ESO1、z2And z3Respectively replacing y in unconstrained control input signals1、y3And ω (t) to obtain an optimal control signal, so that the model prediction-ESO control algorithm realizes closed-loop control.
2. The extended state observer-model predictive control method for a rigid-flexible coupled motion platform according to claim 1, wherein the rigid-flexible coupled motion platform is a single-degree-of-freedom linear motion platform, and comprises an intermediate platform, an external frame, a flexible hinge, a mechanical guide rail, a grating ruler, a linear motor stator and a linear motor rotor, the intermediate platform is connected with the external frame by using the flexible hinge, and the linear motor rotor and the intermediate platform are fixed at the lower part of the intermediate platform by adopting a bolt connection.
3. The extended state observer-model predictive control method for a rigid-flexible coupled motion platform according to claim 1, wherein in the predictive model, under an unconstrained condition, an unconstrained minimum value of a target is 0, and the following equation can be obtained by solving:
substituting the formula into a prediction model to obtain a control input signal u without constraint for rigid-flexible coupling motion platform model predictionm(t)：
4. The extended state observer-model predictive control method of a rigid-flexible coupled motion platform of claim 1, wherein a predictive displacement is obtained through a predictive model, the predictive displacement is substituted into an optimization objective function, and then rolling optimization of model prediction is achieved by solving the objective function.
5. The extended state observer-model predictive control method of the rigid-flexible coupled motion platform according to claim 1, characterized in that the stiffness and damping of the flexible hinge during the motion of the rigid-flexible coupled motion platform are defined as disturbance ω (t), and the control input signal is simplified to obtain a simplified control input signal;
the control of the prediction model is performed with the latest obtained measurement values at each sampling period
And
the optimization objective function is refreshed, the optimal solution of the refreshed optimization objective function is solved, and then the optimal solution, namely u, can be obtainedm(t) acting on the system.
6. The extended state observer-model predictive control method of a rigid-flexible coupled motion platform according to claim 1, wherein in the actual use process, a grating ruler is used to collect displacement signals of the rigid-flexible coupled motion platform, the displacement signals and control signals of the platform are substituted into an ESO observer to obtain the displacement and speed of the intermediate platform of the rigid-flexible coupled motion platform in the control process, the deformation force and damping force of a flexible hinge, the state parameters and disturbance of the platform are then passed through an expression of an unconstrained control input signal to obtain an optimal control signal in the rigid-flexible coupled motion platform control process, then a rolling optimization real-time optimization control signal is used, the optimized control signal is then input into a driver by a controller to obtain the force required by the rigid-flexible coupled motion platform in the control process, and finally the control is realized on the rigid-flexible coupled motion platform, thereby completing the predictive control.
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