Learning control system and method for nano-precision motion stage

Abstract

A learning control system for a nano-precision motion stage comprises a closed-loop feedback section including a motion trajectory generator, a feedback controller, a motion stage, and a first Fourier transformer; and a feedforward section including a second Fourier transformer, a learning controller, an iteration backward shift operator, and a Fourier inverse transformer. An iteration experiment count j is initialized as j=1, and a j-th frequency domain feedforward signal is initialized to 0; the system is run to collect a frequency domain error signal and a frequency domain position measurement signal; a (j+1)-th frequency domain feedforward signal is updated; and an iteration experiment count j is incremented by 1. The present disclosure can effectively suppress the influence of external noise and disturbances, and improve convergence performance. Moreover, the present disclosure requires less computation, achieves simple determination of learning gains and strong robustness, and is convenient for engineering applications.

Claims

1. A learning control system for a nano-precision motion stage, comprising a closed-loop feedback section S.sub.fb and a feedforward section S.sub.ff; wherein the closed-loop feedback section S.sub.fb comprises a motion trajectory generator C.sub.r, a feedback controller C.sub.fb a motion stage P, and a first Fourier transformer C.sub.fft1; the motion trajectory generator C.sub.r generates a desired motion trajectory r(t); the desired motion trajectory r(t) minus a position measurement signal y.sub.j(t) results in a motion error signal e.sub.j(t); the motion error signal e.sub.j(t) added to a feedforward signal u.sub.ff,j(t) results in a feedback input signal e.sub.fb,j(t); the feedback input signal e.sub.fb,j(t) is input into the feedback controller C.sub.fb to generate a feedback control signal u.sub.fb,j(t); the feedback control signal u.sub.fb,j(t) added to a disturbance signal d.sub.j(t) results in a total control signal u.sub.all,j(t); the total control signal u.sub.all,j(t) is transmitted to the motion stage P to generate an actual position signal y.sub.p,j(t); the actual position signal y.sub.p,j(t) added to a measurement noise signal v.sub.j(t) results in a position measurement signal y.sub.j(t); and the position measurement signal y.sub.j(t) is transformed into a frequency domain position measurement signal .sub.j(w) by the first Fourier transformer C.sub.fft1; the feedforward section S.sub.ff comprises a second Fourier transformer C.sub.fft2, a learning controller C.sub.ILC, an iteration backward shift operator C.sub.z, and a Fourier inverse transformer C.sub.ifft; the second Fourier transformer C.sub.fft2 is configured to transform the motion error signal e.sub.j(t) to a frequency domain error signal .sub.j(w); the frequency domain error signal .sub.j(w) and a j-th frequency domain feedforward signal .sub.ff,j(w) are jointly input to the learning controller C.sub.ILC to obtain a (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w); the (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w) is input into the iteration backward shift operator C.sub.z to generate the j-th frequency domain feedforward signal .sub.ff,j(w); and the j-th frequency domain feedforward signal .sub.ff,j(w) is transformed into the feedforward signal u.sub.ff,j(t) by the Fourier inverse transformer C.sub.fft1; and j represents an iteration experiment count, j1, t represents time, and w represents frequency.

2. A learning control method for a nano-precision motion stage, wherein in the system according to claim 1, a learning controller C.sub.ILC is designed using a frequency domain iterative learning method, to determine an iteration relation of the (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w) and the j-th frequency domain feedforward signal .sub.ff,j(w) with respect to the frequency domain error signal .sub.j(w), and the learning control method comprising the following steps: step 1: initializing an iteration experiment count j as j=1, and initializing the j-th frequency domain feedforward signal .sub.ff,j(w) to 0; step 2: transforming the j-th frequency domain feedforward signal .sub.ff,j(w) into the feedforward signal u.sub.ff,j(t) by the Fourier inverse transformer C.sub.ifft; running the closed-loop feedback section S.sub.fb to collect the motion error signal e.sub.j(t) and the frequency domain position measurement signal .sub.j(w); transforming the motion error signal e.sub.j(t) into the frequency domain error signal .sub.j(w) by the second Fourier transformer C.sub.fft2; and when the motion error signal e.sub.j(t) meets a control error requirement or the iteration experiment count j reaches a set maximum value, stopping iterations; otherwise, continuing with the following steps; step 3: updating the (j+1)-th frequency domain feedforward signal .sub.ff,j+1 (w) using a model-based approach:
.sub.ff,j+1(w)=.sub.ff,j(w)+.sub.jL(w).sub.j(w) wherein L(w) represents a learning gain, ${}_j=\frac{1}{{}_j}$ represents a learning coefficient, and .sub.j is determined as follows: ${}_j=\{\begin{array}{cc}{}_1& j=1\\ {}_{j-1}+& j2\end{array}$ wherein .sub.11, represents a conditional function; =1 when (E.sub.j).sup.TE.sub.j1<0; otherwise, =0, E.sub.j=[e.sub.j(0)e.sub.j(1)e.sub.j(2) . . . e.sub.j(N1)].sup.T, and N represents a number of sampling points; and step 4: incrementing the iteration experiment count j by 1 and returning to step 2.

3. The learning control method according to claim 2, wherein the learning gain L(w) in step 3 is determined as follows: when frequency domain models of the motion stage P and the feedback controller C.sub.fb are known, $L\left(w\right)={\left(\frac{P\left(w\right)C_{fb}\left(w\right)}{1+P\left(w\right)C_{fb}\left(w\right)}\right)}^1,$ wherein P(w) is the frequency domain model of the motion stage P and C.sub.fb (w) is the frequency domain model of the feedback controller C.sub.fb; and when the frequency domain models of the motion stage P and the feedback controller C.sub.fb are unknown, L(w) is determined through frequency sweep before step 1: setting the feedforward signal u.sub.ff,j(t) to zero, generating a white noise signal r.sub.n(t) by using the motion trajectory generator C.sub.r, with a signal length of the white noise signal r.sub.n(t) being identical to a length of the desired motion trajectory r(t), and running the closed-loop feedback section S.sub.fb to collect the frequency domain position measurement signal .sub.n(w) under the white noise signal, wherein $L\left(w\right)=\frac{r_n\left(w\right)}{{}_n\left(w\right)},$ and r.sub.n(w) is a frequency domain signal obtained after Fourier transformation of the white noise signal r.sub.n(t); and after L(w) is obtained, generating the desired motion trajectory r(t) using the motion trajectory generator C.sub.r, and then performing step 1.

4. The learning control method according to claim 2, wherein when frequency domain models of the motion stage P and/or the feedback controller C.sub.fb are unknown and frequency sweep experiments are not feasible in step 3, the (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w) is updated using a data-based approach: ${}_{\mathrm{ff},j+1}\left(w\right)=\{\begin{array}{cc}\frac{{}_j\left(w\right)}{{}_j\left(w\right)}r\left(w\right)& j=1\\ {}_{\mathrm{ff},j}\left(w\right)+{}_j\frac{{}_{\mathrm{ff},j}\left(w\right)-{}_{\mathrm{ff},j-1}\left(w\right)}{{}_j\left(w\right)-{}_{j-1}\left(w\right)}{}_j\left(w\right)& j2\end{array}$ wherein 0<.sub.j1, and r(w) is a frequency domain signal obtained after Fourier transformation of the desired motion trajectory r(t).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a topology diagram of a learning control system according to the present disclosure;

(2) FIG. 2 illustrates a desired motion trajectory of a motion stage in a simulation according to an embodiment;

(3) FIG. 3 shows simulation results of a model-based learning control method according to an embodiment;

(4) FIG. 4 shows simulation results of a data-based learning control method according to an embodiment;

(5) FIG. 5 shows a comparison between simulation results of a model-based learning control method and an existing method according to an embodiment; and

(6) FIG. 6 shows a comparison between simulation results of a data-based learning control method and an existing method according to an embodiment.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(7) The technical solutions in the embodiments of the present disclosure are clearly and completely described below with reference to the accompanying drawings in the embodiments of the present disclosure. Apparently, the described embodiments are merely a part rather than all of the embodiments of the present disclosure. All other embodiments derived from the embodiments in the present disclosure by a person of ordinary skill in the art without creative efforts should fall within the protection scope of the present disclosure.

(8) As shown in FIG. 1, a learning control system for a nano-precision motion stage is provided, including a closed-loop feedback section S.sub.fb and a feedforward section S.sub.ff.

(9) The closed-loop feedback section S.sub.fb includes a motion trajectory generator C.sub.r, a feedback controller C.sub.fb, a motion stage P, and a first Fourier transformer C.sub.fft1. The motion trajectory generator C.sub.r generates a desired motion trajectory r(t) The desired motion trajectory r(t) minus a position measurement signal y.sub.j(t) results in a motion error signal e.sub.j(t). The motion error signal e.sub.j(t) added to a feedforward signal u.sub.ff,j(t) results in a feedback input signal e.sub.fb,j(t). The feedback input signal e.sub.fb,j(t) is input into the feedback controller C.sub.fb to generate a feedback control signal u.sub.fb,j(t). The feedback control signal u.sub.fb,j(t) added to a disturbance signal d.sub.j(t) results in a total control signal u.sub.all,j(t). The total control signal u.sub.all,j(t) is input into the motion stage P to generate an actual position signal y.sub.p,j(t). The actual position signal y.sub.p,j(t) added to a measurement noise signal v.sub.j(t) results in a position measurement signal y.sub.j(t). The position measurement signal y.sub.j(t) is transformed into a frequency domain position measurement signal .sub.j(w) by the first Fourier transformer C.sub.fft1.

(10) The feedforward section S.sub.ff includes a second Fourier transformer C.sub.fft2, a learning controller C.sub.ILC, an iteration backward shift operator C.sub.z, and a Fourier inverse transformer C.sub.ifft. The second Fourier transformer C.sub.fft2 is configured to transform the motion error signal e.sub.j(t) into a frequency domain error signal .sub.j(w). The frequency domain error signal .sub.j(w) and a j-th frequency domain feedforward signal .sub.ff,j(w) are jointly input to the learning controller C.sub.ILC to obtain a (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w). The (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w) is input into the iteration backward shift operator C.sub.z to generate the j-th frequency domain feedforward signal .sub.ff,j(w). The j-th frequency domain feedforward signal .sub.ff,j (w) is transformed into the feedforward signal u.sub.ff,j(t) by the Fourier inverse transformer C.sub.ifft.

(11) The subscript j represents an iteration experiment count, j1, t represents time, and w represents frequency.

(12) As shown in FIG. 1, a learning control method for a nano-precision motion stage is provided. According to this method, a learning controller C.sub.ILC is designed using a frequency domain iterative learning method, to determine an iteration relation of a (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w) and a j-th frequency domain feedforward signal .sub.ff,j (w) with respect to a frequency domain error signal .sub.j(w). The learning control method includes the following steps:

(13) In step 1, an iteration experiment count j is initialized as j=1, and the j-th frequency domain feedforward signal .sub.ff,j (w) is initialized to 0.

(14) In step 2, the j-th frequency domain feedforward signal .sub.ff,j(w) is transformed into a feedforward signal u.sub.ff,j(t) by a Fourier inverse transformer C.sub.ifft; a closed-loop feedback section S.sub.fb is run to collect a motion error signal e.sub.j(t) and a frequency domain position measurement signal .sub.j(w); the motion error signal e.sub.j(t) is transformed into the frequency domain error signal .sub.j(w) by a second Fourier transformer C.sub.fft2; and when the motion error signal e.sub.j(t) meets a control error requirement or the iteration experiment count j reaches a set maximum value, iterations are stopped; otherwise, the following steps are performed.

(15) In step 3, the (j+1)-th frequency domain feedforward signal .sub.ff,j+1 (w) is updated using a model-based approach:
.sub.ff,j+1(w)=.sub.ff,j(w)+.sub.jL(w).sub.j(w)
where L(w) represents a learning gain,

(16) ${}_j=\frac{1}{{}_j}$
represents a learning coefficient, and .sub.j is determined as follows:

(17) ${}_j=\{\begin{array}{cc}{}_1& j=1\\ {}_{j1}+& j2\end{array}$
where .sub.11, represents a conditional function; =1 when (E.sub.j).sup.TE.sub.j1<0; otherwise, =0, E.sub.j=[e.sub.j(0)e.sub.j(1)e.sub.j(2) . . . e.sub.j(N1)].sup.T, and N represents the number of sampling points.

(18) The learning gain L(w) is determined in the following manner:

(19) When frequency domain models of the motion stage P and the feedback controller

(20) $C_{\mathrm{fb}}\mathrm{are}\mathrm{known}L\left(w\right)={\left(\frac{P\left(w\right)C_{fb}\left(w\right)}{1+P\left(w\right)C_{fb}\left(w\right)}\right)}^{-1},$
where P(w) is the frequency domain model of the motion stage P and C.sub.fb(w) is the frequency domain model of the feedback controller C.sub.fb.

(21) When the frequency domain models of the motion stage P and the feedback controller C.sub.fb are unknown, L(w) is determined through frequency sweep before step 1: setting the feedforward signal u.sub.ff,j(t) to zero, generating a white noise signal r.sub.n(t) by using the motion trajectory generator C.sub.r, with a signal length of the white noise signal r.sub.n(t) being the same as the length of the desired motion trajectory r(t), and running the closed-loop feedback section S.sub.fb to collect a frequency domain position measurement signal .sub.n(w) under the white noise signal, where

(22) $L\left(w\right)=\frac{r_n\left(w\right)}{{}_n\left(w\right)},$
and r.sub.n(w) is a frequency domain signal obtained after Fourier transformation of the white noise signal r.sub.n(t); and after L(w) is obtained, generating the desired motion trajectory r(t) using the motion trajectory generator C.sub.r, and then performing step 1.

(23) In step 4, the iteration experiment count j is incremented by 1 and step 2 is then performed.

(24) Additionally, when frequency domain models of the motion stage P and/or the feedback controller C.sub.fb are unknown and frequency sweep experiments are not feasible, the (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w) can be updated using a data-based approach in step 3:

(25) ${}_{\mathrm{ff},j+1}\left(w\right)=\{\begin{array}{cc}\frac{{}_j\left(w\right)}{{}_j\left(w\right)}r\left(w\right)& j=1\\ {}_{\mathrm{ff},j}\left(w\right)+{}_j\frac{{}_{\mathrm{ff},j}\left(w\right)-{}_{\mathrm{ff},j-1}\left(w\right)}{{}_j\left(w\right)-{}_{j-1}\left(w\right)}{}_j\left(w\right)& j2\end{array}$
where 0<.sub.j1, and r(w) is a frequency domain signal obtained after Fourier transformation of the desired motion trajectory r(t).

Embodiment

(26) In this embodiment, the motion trajectory generator C.sub.r is a 5.sup.th-order S-shaped motion trajectory generator. The desired motion trajectory r(t) generated by the motion stage P is as shown in FIG. 2. Mathematical models C.sub.fb(s) and P(s) of the feedback controller C.sub.fb and the motion stage P are as follows:

(27) $C_b\left(s\right)=\frac{1.50810^5{s}^2+2.06910^{7}s+3.94310^8}{0.001159s^2+s}$$P\left(s\right)=\frac{0.0729s^6+38.58s^5+3.8210^6{s}^4+9.9510^8{s}^3+4.1210^{13}{s}^2+3.6010^{15}s+4.410^{19}}{24s^8+1.3910^4{s}^7+1.410^9{s}^6+3.9210^{11}{s}^5+1.5510^{16}{s}^4+1.7310^{18}{s}^3+2.110^{22}{s}^2}$
where s represents a Laplace operator, and by replacing s with iw (i is an imaginary unit), frequency domain models C.sub.fb(w) and P(w) of the feedback controller C.sub.fb and the motion stage P are obtained.

(28) The measurement noise signal v.sub.j(t) is a white noise signal with a variance of 0.110.sup.9 and a mean of 0. The disturbance signal d.sub.j(t) is a signal that is periodic with respect to an actual position signal y.sub.p,j(t), used to simulate disturbances such as cable force disturbances, actuator torque fluctuations, and other disturbing forces present in real scenarios.

(29) A learning controller C.sub.ILC is designed using a frequency domain iterative learning method to correct the feedforward signal u.sub.ff,j(t) through iteration experiments, in order to gradually reduce the motion error signal e.sub.j(t), as detailed below:

(30) In step 1, an iteration experiment count j is initialized as j=1, and the j-th frequency domain feedforward signal .sub.ff,j (w) is initialized to 0. In this case, the feedforward signal u.sub.ff,j(t) is zero, the feedforward section S.sub.ff is inactive, and the learning control system consists of the closed-loop feedback section S.sub.fb only.

(31) In step 2, the j-th frequency domain feedforward signal .sub.ff,j(w) is transformed into a feedforward signal u.sub.ff,j(t) by a Fourier inverse transformer C.sub.ifft; a closed-loop feedback section S.sub.fb is run to collect a motion error signal e.sub.j(t) and a frequency domain position measurement signal y.sub.j(w); the motion error signal e.sub.j(t) is transformed into the frequency domain error signal .sub.j(w) by a second Fourier transformer C.sub.fft2; and when the motion error signal e.sub.j(t) meets a control error requirement or the iteration experiment count j reaches a set maximum value, iterations are stopped; otherwise, the following steps are performed. In this embodiment, the maximum iteration count is set to 50.

(32) In step 3, the (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w) is updated using a model-based approach:
.sub.ff,j+1(w)=.sub.ff,j(w)+.sub.jL(w).sub.j(w)
where L(w) represents a learning gain,

(33) ${}_j=\frac{1}{{}_j}$
represents a learning coefficient, and .sub.j is determined as follows:

(34) 0${}_j=\{\begin{array}{cc}{}_1& j=1\\ {}_{j-1}+& j2\end{array}$
where .sub.11, represents a conditional function; =1 when (E.sub.j).sup.T E.sub.j1<0; otherwise, =0, E.sub.j=[e.sub.j(0)e.sub.j(1)e.sub.j(2) . . . e.sub.j(N1)].sup.T, and N represents the number of sampling points; in this embodiment, .sub.1=1.5.

(35) Simulations are conducted using a model-based approach and a data-based approach, which are specifically as follows:

(36) 1. In the model-based approach, when the frequency domain models of the motion stage P and the feedback controller C.sub.fb are known, the learning gain is as follows:

(37) $L\left(w\right)={\left(\frac{P\left(w\right)C_{fb}\left(w\right)}{1+P\left(w\right)C_{fb}\left(w\right)}\right)}^{-1},$
where P(w) is the frequency domain model of the motion stage P and C.sub.fb(w) is the frequency domain model of the feedback controller C.sub.fb.

(38) Since the exact model P(w) is usually unknown in real scenarios, an approximate model of P(w) is used in this embodiment as follows:

(39) $P\left(w\right)=\frac{1.312s^2+95.5s+1.42510^6}{480s^4+4.58410^4{s}^3+6.84110^8{s}^2}{|}_{s=iw}$

(40) 2. In the data-based approach, when the frequency domain models of the motion stage P and/or the feedback controller C.sub.fb are unknown and frequency sweep experiments are not feasible, the (j+1)-th frequency domain feedforward signal .sub.ff,j+1(w) is updated as follows:

(41) ${}_{\mathrm{ff},j+1}\left(w\right)=\{\begin{array}{cc}\frac{{}_j\left(w\right)}{{}_j\left(w\right)}r\left(w\right)& j=1\\ {}_{\mathrm{ff},j}\left(w\right)+{}_j\frac{{}_{\mathrm{ff},j}\left(w\right)-{}_{\mathrm{ff},j-1}\left(w\right)}{{}_j\left(w\right)-{}_{j-1}\left(w\right)}{}_j\left(w\right)& j2\end{array}$
where 0<.sub.j1, and r(w) is a frequency domain signal obtained after Fourier transformation of the desired motion trajectory r(t); in this embodiment, .sub.j=0.9.

(42) In step 4, the iteration experiment count j is incremented by 1 and step 2 is then performed.

(43) Ultimately, simulation results of the model-based approach are shown in FIG. 3, while simulation results of the data-based approach are shown in FIG. 4. It can be observed that through multiple iterations, both the model-based learning control method and the data-based learning control method in the present disclosure significantly reduce the motion error, from the micron level to the nanometer level.

(44) Comparing the model-based learning control method of the present disclosure with the existing model-based frequency domain learning control method, as shown in FIG. 5, it is evident that beyond the 10.sup.th iteration experiment, the present disclosure can further reduce the motion error compared to the existing method. This is primarily due to the adaptive learning gain designed in the present disclosure, which can further mitigate the impact of external random noise on the learning control effect.

(45) Comparing the data-based learning control method of the present disclosure with the existing data-based frequency domain learning control method, as shown in FIG. 6, it is evident that even without using model information, the present disclosure can still significantly reduce motion errors. Moreover, under the influence of external periodic disturbances, the present disclosure can achieve smaller motion errors compared to the existing method. This is mainly because the present disclosure employs a differential approach, which can reduce the impact of periodic disturbances on the accuracy of learning gain estimation.

(46) With reference to the results shown in FIG. 3 to FIG. 6 it can be seen that under the influence of random noise and external disturbances, the present disclosure can achieve higher control precision compared to the existing methods.

(47) It will be apparent to those skilled in the art that the present disclosure is not limited to the details of the exemplary embodiments described above, but that the present disclosure can be embodied in other specific forms without departing from the spirit or essential characteristics of the present disclosure. Accordingly, the embodiments should be regarded in all points of view as exemplary and not restrictive, the scope of the present disclosure being defined by the appended claims rather than the foregoing description, and it is therefore intended that all changes falling within the meaning and scope of equivalent elements of the claims should be included in the present disclosure. Any reference numerals in the claims should not be considered as limiting the involved claims.

(48) In addition, it should be understood that although this specification is described in accordance with the implementations, not each implementation only contains an independent technical solution, and this description in the specification is only for clarity. Those skilled in the art should take the specification as a whole. The technical solutions in the embodiments can also be properly combined to form other implementations that can be understood by those skilled in the art.