A closed-loop identification method for continuous-time models of ultra-precision motion systems

Abstract

A closed-loop identification method for a continuous-time model of an ultra-precision motion system relates to the technical field of ultra-precision motion identification. A trajectory generator is used to generate the expected motion trajectory of the ultra-precision motion system at time k. The expected motion trajectory is subtracted from the actual motion trajectory output by the closed-loop displacement to obtain the position servo error. The position servo error is input into a feedback controller. The output of the feedback controller is used as the input of the ultra-precision motion system. The closed-loop identifier includes a zero-order holder and a parameter estimator. The input is the output of the feedback controller and the discrete-time signal of the actual motion trajectory at time k. The output is an identification model of the continuous-time model of the ultra-precision motion system. The parameter estimation algorithm built into the parameter estimator is used to perform parameter estimation. It is more suitable for the identification of ultra-precision motion systems and provides a safer closed-loop identification experimental environment.

Description

Technical Field

[0001] The present invention relates to the technical field of ultra-precision motion identification, and in particular to a continuous-time model closed-loop identification method for an ultra-precision motion system. Background Art

[0002] Mathematical models of dynamic systems are necessary in most areas of scientific research, and accurate identification of parametric system models is key to the application of model-based control methods. Dynamic systems in the physical world are mainly described by continuous-time differential equations. Although dynamic systems are usually described in the continuous time domain, most common and well-known system identification schemes are developed based on discrete-time models, without noticing the advantages of continuous-time model descriptions and related identification methods, which are overshadowed by the more widespread discrete-time identification methods.

[0003] Taking ultra-precision motion systems such as complex lithography precision motion platforms as an example, when continuous-time model identification methods and discrete-time model identification methods are applied to the same data set at the same time, the continuous-time model identification method performs better in both simulation and actual examples, revealing some advantages of the continuous-time model identification method. For example, the continuous-time model can provide a good understanding of the physical properties of the system, and the continuous-time model identification can immediately provide better physical insights into the nature of the dynamic system; for high-performance control, the use of high sampling rates is essential. When the sampling frequency is too high relative to the dominant frequency of the system under study, the discrete-time model identification model quality is poor, while the direct continuous-time model identification algorithm can handle the fast sampling data of the system well; the direct continuous-time model method is more convenient to combine prior knowledge and identify a more concise model. These advantages are more beneficial to the identification of lithography electromechanical positioning systems with high-performance control. For many industrial production processes, safety and production restrictions are often a strong reason for not allowing identification experiments in open loops. In this case, experimental data can only be obtained under closed-loop conditions.

[0004] In summary, a continuous-time model closed-loop identification method is proposed for ultra-precision motion systems, which is of great significance in the field of ultra-precision motion identification. Summary of the invention

[0005] In order to solve the shortcomings of the background technology, the present invention provides a continuous-time model closed-loop identification method for ultra-precision motion systems, which is more suitable for the identification of ultra-precision motion systems and provides a safer closed-loop identification experimental environment.

[0006] To achieve the above object, the present invention adopts the following technical solution: a continuous-time model closed-loop identification method for ultra-precision motion systems, comprising the following steps:

[0007] Step 1: Use the trajectory generator to generate the expected motion trajectory r(k) of the ultra-precision motion system at time k;

[0008] Step 2: The desired motion trajectory r(k) minus the actual motion trajectory y output by the closed-loop displacement obtains the position servo error e, and the position servo error e is used as the input of the feedback controller C to obtain the output u of the feedback controller C. The ultra-precision motion system uses the output u of the feedback controller C as input and outputs an ideal motion trajectory x, and the ideal motion trajectory x is added to the noise v of the closed-loop displacement output to obtain the actual motion trajectory y, and the discrete sampling data u(k) and y(k) of the output u of the feedback controller C and the actual motion trajectory y at time k are obtained respectively, and y(k)=x(k)+v(k), wherein x(k) and v(k) are the discrete sampling data of the ideal motion trajectory x and the noise v of the closed-loop displacement output at time k respectively;

[0009] Step 3: The closed-loop identifier includes a zero-order holder and a parameter estimator. The closed-loop identifier takes u(k) and y(k) as inputs. The continuous-time model to be identified of the ultra-precision motion system is: Where B(p) = b m p m +b m-1 p m-1 +…+b 1 p+b 0 and A(p)=a n p n +a n-1 p n-1 +…+a 1 p+1 are the continuous-time models to be identified The numerator and denominator polynomials, m and n are the polynomial orders, and n ≥ m, p is the Heaviside operator, and the function of the zero-order holder is that when encountering a mixed representation of a continuous-time model represented by the Heaviside operator and a discrete-time signal, the discrete-time signal includes u(k), y(k) and r(k), and the mixed representation includes:

[0010] ① The case of continuous-time model represented by Heaviside operator and mixed representation of y(k)

[0011]

[0012] The discrete-time signal y(k) is converted into a continuous-time signal through a zero-order holder, and the continuous-time signal acts on n continuous-time models respectively.

[0013] ② The case of continuous-time model represented by Heaviside operator and mixed representation of u(k)

[0014]

[0015] The discrete-time signal u(k) is converted into a continuous-time signal through a zero-order holder, and the continuous-time signal acts on m continuous-time models respectively.

[0016] ③ The case of continuous-time model represented by Heaviside operator and mixed representation of r(k)

[0017]

[0018] The discrete-time signal r(k) is converted into a continuous-time signal through a zero-order holder, and the continuous-time signal acts on m+n continuous-time models respectively. is the control system sensitivity function;

[0019] Finally, at time k, the continuous-time signal output by the continuous-time model is and Sampling is performed to obtain a discrete-time signal;

[0020] Step 4: Use the built-in parameter estimation algorithm in the parameter estimator to perform parameter estimation.

[0021] Furthermore, the parameter estimation algorithm built into the parameter estimator in step 4 includes the following steps:

[0022] S4.1, start the parameter estimator;

[0023] S4.2, given the initial value i=1 of the iteration variable i, given the data length K of u(k) and y(k), given the parameter estimation accuracy ε;

[0024] S4.3. Construction and The vector is as follows:

[0025]

[0026] Construct the filter variable y f (k,β i )as follows:

[0027]

[0028] In the formula, and are the numerator and denominator polynomials of the continuous-time model to be identified at the i-th iteration, is the estimated value of the control system sensitivity function calculated for the i-th iteration;

[0029] S4.4. Calculate the parameter estimation result of the parameter estimator after the i-th iteration The calculation is as follows:

[0030]

[0031] Expands to:

[0032]

[0033] S4.5. Determine whether If so, get the parameter estimation results Get the identification model End identification, if not, increase the iteration variable i by 1 and return to S4.3.

[0034] Compared with the prior art, the beneficial effects of the present invention are: the method of the present invention can directly obtain the continuous-time model of the ultra-precision motion system, which is more suitable for the identification of ultra-precision motion systems with high sampling frequencies, and at the same time provides a safer closed-loop identification experimental environment, which is suitable for practical engineering applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 is a system topology diagram of the present invention;

[0036] Figure 2 is a system topology diagram of the zero-order holder and the continuous-time model of the present invention;

[0037] Figure 3 It is a flow chart of the parameter estimation algorithm built into the parameter estimator of the present invention. DETAILED DESCRIPTION

[0038] The technical solution of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0039] like Figure 1 to Figure 3 As shown in the figure, a closed-loop identification method of continuous-time model for ultra-precision motion system is combined with Figure 1 As shown, the following steps are included:

[0040] Step 1: Use the trajectory generator to generate the expected motion trajectory r(k) of the ultra-precision motion system at time k;

[0041] Step 2: The desired motion trajectory r(k) minus the actual motion trajectory y output by the closed-loop displacement obtains the position servo error e, and the position servo error e is used as the input of the feedback controller C to obtain the output u of the feedback controller C. The ultra-precision motion system uses the output u of the feedback controller C as input and outputs an ideal motion trajectory x, and the ideal motion trajectory x is added to the noise v of the closed-loop displacement output to obtain the actual motion trajectory y, and the discrete sampling data u(k) and y(k) of the output u of the feedback controller C and the actual motion trajectory y at time k are obtained respectively, and y(k)=x(k)+v(k), wherein x(k) and v(k) are the discrete sampling data of the ideal motion trajectory x and the noise v of the closed-loop displacement output at time k respectively;

[0042] Step 3: The closed-loop identifier includes a zero-order holder and a parameter estimator. The closed-loop identifier takes u(k) and y(k) as inputs. The continuous-time model to be identified of the ultra-precision motion system is: Where B(p) = b m p m +b m-1 p m -1 +…+b 1 p+b 0 and A(p)=a n p n +a n-1 p n-1 +…+a 1 p+1 are the continuous-time models to be identified The numerator and denominator of are polynomials, m and n are the polynomial orders, and n ≥ m, and p is the Heaviside operator (also known as the differential operator). Figure 2 As shown, the role of the zero-order holder is that when encountering a mixed representation of a continuous-time model represented by a Heaviside operator and a discrete-time signal, the discrete-time signal includes u(k), y(k) and r(k), and the mixed representation includes:

[0043] ① The case of continuous-time model represented by Heaviside operator and mixed representation of y(k)

[0044]

[0045] The discrete-time signal y(k) is converted into a continuous-time signal through a zero-order holder, and the continuous-time signal acts on n continuous-time models respectively.

[0046] ② The case of continuous-time model represented by Heaviside operator and mixed representation of u(k)

[0047]

[0048] The discrete-time signal u(k) is converted into a continuous-time signal through a zero-order holder, and the continuous-time signal acts on m continuous-time models respectively.

[0049] ③ The case of continuous-time model represented by Heaviside operator and mixed representation of r(k)

[0050]

[0051] The discrete-time signal r(k) is converted into a continuous-time signal through a zero-order holder, and the continuous-time signal acts on m+n continuous-time models respectively. is the control system sensitivity function;

[0052] Finally, at time k, the continuous-time signal output by the continuous-time model is and Sampling is performed to obtain a discrete-time signal;

[0053] Step 4: Parameter estimation is performed using the parameter estimation algorithm built into the parameter estimator. The parameter estimation algorithm is based on the closed-loop fine instrumental variable method and combined with Figure 3 As shown, specifically including:

[0054] S4.1, start the parameter estimator;

[0055] S4.2, given the initial value of the iteration variable i = 1, given the data length K of u(k) and y(k), which determines S4.4 The computational dimension of , given the parameter estimation accuracy ε;

[0056] S4.3. Construction and The vector is as follows:

[0057]

[0058] Construct the filter variable y f (k,β i )as follows:

[0059]

[0060] In the formula, and are the numerator and denominator polynomials of the continuous-time model to be identified at the i-th iteration, is the estimated value of the control system sensitivity function calculated for the i-th iteration;

[0061] S4.4. Calculate the parameter estimation result of the parameter estimator after the i-th iteration The calculation is as follows:

[0062]

[0063] Expands to:

[0064]

[0065] S4.5. Determine whether If so, get the parameter estimation results Get the identification model End identification, if not, increase the iteration variable i by 1 and return to S4.3.

[0066] Example

[0067] In this embodiment, the feedback controller C is set as:

[0068]

[0069] The expected motion trajectory r(k) is selected as a zero-mean Gaussian white noise signal with unit variance, and the discrete sampling data v(k) of the closed-loop displacement output noise v at time k is a zero-mean Gaussian white noise with a variance of 0.01. The continuous-time model to be identified is as follows:

[0070]

[0071] The system is simulated for 4 seconds in a closed-loop setting with a sampling frequency of 5000 Hz, and the discrete sampling data u(k) and y(k) of the output u of the feedback controller C and the actual motion trajectory y at time k are collected.

[0072] Select the parameter estimates under 4 different iteration numbers, combined with the following table 1:

[0073] Table 1 Comparison of parameter estimates and true values ​​under 4 different iteration numbers

[0074]

[0075]

[0076] It can be seen from the results shown in Table 1 that the parameter estimation results of the method of the present invention converge to the true model parameters and the parameter estimation is effective.

[0077] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other forms of assembly without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered in all respects as exemplary and non-restrictive, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations within the meaning and range of equivalents of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

[0078] In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims

1. A continuous-time model closed-loop identification method for ultra-precision motion systems, characterized by: The following steps are involved: Step 1: Use the trajectory generator to generate the expected motion trajectory r(k) of the ultra-precision motion system at time k; Step 2: The desired motion trajectory r(k) minus the actual motion trajectory y output by the closed-loop displacement obtains the position servo error e, and the position servo error e is used as the input of the feedback controller C to obtain the output u of the feedback controller C. The ultra-precision motion system uses the output u of the feedback controller C as input and outputs an ideal motion trajectory x, and the ideal motion trajectory x is added to the noise v of the closed-loop displacement output to obtain the actual motion trajectory y, and the discrete sampling data u(k) and y(k) of the output u of the feedback controller C and the actual motion trajectory y at time k are obtained respectively, and y(k)=x(k)+v(k), wherein x(k) and v(k) are the discrete sampling data of the ideal motion trajectory x and the noise v of the closed-loop displacement output at time k respectively; Step 3: The closed-loop identifier includes a zero-order holder and a parameter estimator. The closed-loop identifier takes u(k) and y(k) as inputs. The continuous-time model to be identified of the ultra-precision motion system is: Where B(p) = b m p m +b m-1 p m-1 +…+b1p+b0 and A(p)=a n p n +a n-1 p n-1 +…+a1p+1 are the continuous-time models to be identified The numerator and denominator polynomials, m and n are the polynomial orders, and n ≥ m, p is the Heaviside operator, and the function of the zero-order holder is that when encountering a mixed representation of a continuous-time model represented by the Heaviside operator and a discrete-time signal, the discrete-time signal includes u(k), y(k) and r(k), and the mixed representation includes: ① The case of continuous-time model represented by Heaviside operator and mixed representation of y(k) The discrete-time signal y(k) is converted into a continuous-time signal through a zero-order holder, and the continuous-time signal acts on n continuous-time models respectively. ② The case of continuous-time model represented by Heaviside operator and mixed representation of u(k) The discrete-time signal u(k) is converted into a continuous-time signal through a zero-order holder, and the continuous-time signal acts on m continuous-time models respectively. ③ The case of continuous-time model represented by Heaviside operator and mixed representation of r(k) The discrete-time signal r(k) is converted into a continuous-time signal through a zero-order holder, and the continuous-time signal acts on m+n continuous-time models respectively. is the control system sensitivity function; Finally, at time k, the continuous-time signal output by the continuous-time model is and Sampling is performed to obtain a discrete-time signal; Step 4: Use the built-in parameter estimation algorithm in the parameter estimator to perform parameter estimation.

2. The continuous-time model closed-loop identification method for ultra-precision motion systems according to claim 1, characterized in that: The parameter estimation algorithm built into the parameter estimator in step 4 includes the following steps: S4.1, start the parameter estimator; S4.2, given the initial value i=1 of the iteration variable i, given the data length K of u(k) and y(k), given the parameter estimation accuracy ε; S4.

3. Construction and φ(k,β i ) vector is as follows: Construct the filter variable y f (k,β i )as follows: In the formula, and are the numerator and denominator polynomials of the continuous-time model to be identified at the i-th iteration, is the estimated value of the control system sensitivity function calculated for the i-th iteration; S4.

4. Calculate the parameter estimation result of the parameter estimator after the i-th iteration The calculation is as follows: Expands to: S4.

5. Determine whether If so, get the parameter estimation results Get the identification model End identification, if not, increase the iteration variable i by 1 and return to S4.3.

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