阅读说明：本技术 二阶欠阻尼系统的串联校正方法 (Series correction method of second-order under-damped system ) 是由 唐静 于 2020-12-03 设计创作，主要内容包括：一种二阶欠阻尼系统串联校正方法,具体步骤：一：测试系统阶跃响应,得超调量δ-3％,调节时间t-(s3)；二：根据步骤一得超调量δ-3％和调节时间t-(s3),建立实际开环传递函数；三：根据步骤一得超调量δ-3％和调节时间t-(s3)与步骤二推算的k-v,判断实际连续二阶欠阻尼系统静态速度误差、超调量和调节时间是否满足性能要求；如满足要求则被控对象不需要校正；如不满足要求则需要校正,进入步骤四；四：连续欠阻尼二阶系统输出波形任一性能不满足要求,则校正,在步骤四中讲述具体校正方法；五：校正后模型性能指标验算；六：实物搭建校正环节,按步骤一和步骤二测得静态速度误差、超调量和调节时间,与提出指标比较,看是否校正成功,如不成功,进一步调节参数。(A second-order under-damped system series correction method specifically comprises the following steps: firstly, the method comprises the following steps: testing the step response of the system to obtain the overshoot delta 3 % regulating time t s3 (ii) a II, secondly: obtaining the overshoot delta according to the step one 3 % and adjustment time t s3 Establishing an actual open-loop transfer function; thirdly, the method comprises the following steps: obtaining the overshoot delta according to the step one 3 % and adjustment time t s3 K derived from step two v Judging whether the static speed error, the overshoot and the adjusting time of the actual continuous second-order underdamping system meet the performance requirements or not; if the requirement is met, the controlled object does not need to be corrected; if the requirement is not met, correction is needed, and the step four is entered; fourthly, the method comprises the following steps: if any performance of the output waveform of the continuous underdamped second-order system does not meet the requirement, correcting, and explaining a specific correction method in the fourth step; fifthly: checking the performance index of the corrected model; sixthly, the method comprises the following steps: and (3) building a correction link for the object, measuring static speed error, overshoot and adjusting time according to the first step and the second step, comparing with the provided indexes, and judging whether the correction is successful or not, if not, further adjusting parameters.)

1. A series correction method of a second-order underdamped system is characterized by comprising the following specific steps:

the method comprises the following steps: testing the step response waveform of the system, reading the output response waveform to obtain the overshoot delta₃% of the total weight ofThe saving time is marked as t_s3；

Step two: overshoot delta measured according to step one₃% and adjustment time t_s3The concrete modeling process for establishing the actual open-loop transfer function model is as follows:

2.1, using overshoot formula

Calculating damping ratio Zeta₃；

The specific derivation calculation process is as follows:

ζ₃ ²(π²+(ln(δ₃％))²)＝(ln(δ₃％))²

to obtain

Using formula of adjustment time

Obtaining the undamped natural oscillation frequency of the system

Calculated damping ratio ζ₃Is provided withThe bits after decimal point are more, and the bits after decimal point with required precision are not so many, so that the zeta memory obtained according to the requirement can be obtained by omitting the bits after decimal point₄At this time

Wherein c is a constant, if the error band delta is selected to be +/-0.05, c is 3.5 or c is 3; if the error band Δ ± 0.02 is selected, c is 4.4, where ζ₄By the formula:

zeta is obtained₄(ii) a Wherein t is_s3Measuring an actual waveform in the first step;

2.2 according to the damping ratio ζ of the system₄And undamped natural oscillation frequency w_n4Establishing an open-loop transfer function of a practical continuous underdamped second-order system

Wherein due to the system damping ratio ζ₄And undamped natural oscillation frequency w_n4It is known that k can be obtained from the following formula_o1And T_o1；

By the formulaGet ζ 4, from the formulaTo obtain w_n4；

2.3 obtaining the static speed error coefficient according to the open loop transfer function of the continuous underdamping second-order system

Step three: overshoot δ according to step one test₃% and adjustment time t_s3And k obtained by modeling in step two_vJudging whether the static speed error, overshoot and adjusting time of the actual continuous underdamping second-order system meet the proposed performance indexes; the performance indexes provided by the system need to meet the following requirements: static velocity error k_v≥k_v0(s^-1) Overshoot delta₃％≤δ₀% regulating time t_s3≤t_s0If the performance index requirement is met, the controlled object does not need to be corrected and can be directly used; if the requirements are not met, correction is needed, and the step four is entered;

step four: and (3) correcting any performance index of the time domain response output waveform of the continuous underdamped second-order system if the performance index does not meet the requirement, wherein the correction steps are as follows:

4.1, if the static speed error coefficient of the actual connecting line underdamping second-order system meets the requirement, adding a proportional correction coefficient k_c＝1；

If the static speed error coefficient of the actual connecting line underdamped second-order system does not meet the requirement, adding a proportional correction coefficient k_c；

The corrected static velocity error coefficient k is required_o2The following conditions are satisfied:

wherein k is_cIs the scaling factor, k, of the correction circuit_o2Is the static velocity error of the corrected system, k_v0Is the minimum value of the static speed error coefficient and the system damping ratio zeta which are required to be achieved by the system₄And undamped natural oscillation frequency w_n4Calculating a value in the second step;

in thatWithin the range of k_cTo make it satisfy

Obtaining a new transfer function of the static speed error coefficient meeting the requirement after the proportion correction

Wherein the formula in step oneTo obtain k_o1From the formulaTo obtain T_o1Transfer function G for correcting static errors_c1(s)＝k_cAccording to the static speed error of the actual model and the performance index k provided by the system_v0By comparison, from formula k_c1 orTo obtain k_cCorrected static velocity error coefficient k_o2The following were used:

4.2 open-loop model of system after correcting static speed error coefficient

According to the open-loop model after the static speed error correction of the actual continuous underdamping second-order system is established

Wherein k is_o1、T_o1And k_cThe value is taken by the formula in the step four 4.1;

4.3, if the overshoot or the adjusting time does not meet the requirement, adding a correction link;

according to the overshootCalculating the corrected damping ratio delta₀% wherein the overshoot index value to be reached by the proposed system is a known quantity;

specific zeta₅The derivation is as follows:

ζ₅ ²(π²+(ln(δ₀％))²)≥(ln(δ₀％))²

to obtain

Wherein ζ₅Is positive and is a continuous underdamped second order system 0 < zeta₅＜1，

In thatArbitrarily take one value to obtain corrected damping ratio Zeta₅So as to overshoot

According to the zero-pole cancellation rule, make the correcting device G_c2Transfer function of(s)

4.4, obtaining the open-loop transfer function after the series correction through analyzing the static speed error, the overshoot and the adjusting time

Wherein G is_c(s) represents the total correction transfer function G_c(s)＝G_c1(s)·G_c2(s); it consists of a transfer function G for correcting static errors_c1(s) and transfer function G for correcting overshoot and adjusting time_c2(s) composition;

open-loop model with static velocity error correction

The whole correction idea is to use G_c1(s)＝k_cCorrecting static speed errors byCorrecting overshoot and settling time, both correction sequences not being mandatoryThe elements may be reversed;

the corrected open loop transfer function is:

from the corrected open loop transfer function equation:

simultaneous approximation of w by the above formula_n5Simplifying to obtain:

the left and right evolution of the above formula

Above formula T_o2Simultaneously reducing and finishing to obtain a formula

Thereon is provided withThe derivation can also be obtained using the following closed loop transfer function

By the formulaGet ζ by taking one value₅；

By the formulaTo obtain k_o2Wherein k is_cObtained by step four 4.1, k_o1Is inherent to the controlled object and is obtained by the step 2.2;

from the above has been obtained k_o2And damping ratio ζ₅Two values, using the following formula to obtain T_o2Value of

From the formula of step two 2.2To obtain T_o1From step four 4.4 equationTo obtain T_o2Correction device G_c2The parameters of the transfer function of(s) have been found;

according to the static speed error of the actual model and the performance index k provided by the system_v0By comparison, from formula k_c1 orTo obtain k_cThe parameters of the total correction model are already calculated,

step five: checking the performance index of the corrected model;

step six: building a correction link for a real object, measuring static speed error, overshoot and adjusting time according to the first step and the second step, comparing with a proposed index, and judging whether the correction is successful or not, if not, further adjusting parameters; the key of the correction implementation is that the designed correction object is necessarily adjustable in correction parameters.

2. The series correction method of the second-order under-damped system according to claim 1, characterized in that:

the first step is specifically as follows:

1.1, inputting step waveform to the controlled object, testing output response waveform, reading the output response waveform to obtain overshoot recorded as delta₃% of the total amount of the solution was recorded as t_s3；

1.2、δ₃% of:

where C (tp) is the maximum deviation of the output step response and C (∞) is the steady state quantity of the output step response;

1.3, adjusting time t_s3The calculation of (1):

firstly, determining whether the required error band is delta +/-5% or delta +/-2%;

find d₁＝(1+|Δ|)×C(∞)，d₂＝(1-|Δ|)×C(∞)

The read response arrives and remains at d in the output response waveform₁And d₂The shortest time required within the value is recorded as the adjustment time t_s3。

3. The series correction method of the second-order under-damped system according to claim 1, characterized in that: the second step is provided with:

2.4.1, calculating overshoot delta of the model before correction according to the controlled object model measured in the step two 2.1₄％，

Overshoot using formula

Calculating the adjusting time t of the model before correction according to the controlled object model actually measured in the step two 2.1_s4Adjustment time before correction:

by said formulaZeta is obtained₄From said formulaTo obtain w_n4；

2.4.2, comparing the overshoot delta calculated by the model obtained in the second step 2.1₄% and adjustment time t_s4And step one, actually testing to obtain the overshoot delta₃% and adjustment time t_s3A value of (d); since the decimal point omission possibility exists in the calculation, certain errors are allowed; if the actual measurement controlled object model obtained according to the step two 2.1Firstly, debugging a corrected object link if the corrected performance index does not meet the requirement; if the actual object is still not realized, reestablishing a correction function in the step five 5.2, and correcting the actual object; if not, consider if the modeling failed, such as considering ζ₄And w_n4Whether it is related to ζ₃And w_n3If the value difference is too large, the value is taken again to be closer to zeta according to the step two 2.1₃、w_n3ζ of₄And w_n4Re-establishing a more accurate controlled object model for correction;

2.4.3, after the verification and comparison according to the second step, the model establishment of the controlled object finishes and judges whether the static speed error coefficient of the model meets the requirement or not;

if it isThe actual second-order system meets the requirement and corrects the proportionality coefficient k_c1 is ═ 1; otherwise, the actual second-order system does not meet the requirement, the static speed error coefficient needs to be changed, and the correction scale coefficient k is increased_cCorrection of the proportionality coefficient k_cThe concrete values are shown in step four, wherein k_v0Is a known value of the performance index requirement.

4. The series correction method of the second-order under-damped system according to claim 1, characterized in that: the fifth step is specifically;

5.1、k_o2according to said formulaSelection of, delta₅% by formulaIs selected, wherein k_v0And delta₀% is the performance index value provided by the system, so the performance index requirement is met;

5.2, according to the corrected open loop transfer function:

calculating the adjustment time t of the corrected model_s5Specifically, the following is calculated:

from step four 4.4 of the formulaBecome into

Substituting into the formula of the adjustment time to obtain the adjustment time after correction

Judging t_s5And t_s0Value, if t_s5≤t_s0If the theoretical correction is finished, entering a correction circuit implementation stage; if t is_s5＞t_s0According to said formulaReselecting zeta₅From the formulaCan obtain t_s5And ζ₅In inverse proportion, the following stepsWithin range of taking ζ₅But acquire ζ more than before₅Slightly larger;

if t is_s5＞t_s0It is also possible to reselect k_cT is derived from_s5And k_cIn inverse proportion, k is taken again in the following times_cAlso, k is obtained more than before_cSlightly larger;

by the formulaAnd k_o2＝k_c×k_o1Substituting into said formula

Wherein c is a constant, if the error band delta is selected to be +/-0.05, c is 3.5 or c is 3; if the error band delta is selected to be +/-0.02, c is 4.4; k is a radical of_o1Is the value of the controlled object, is obtained by establishing a model in the second step, and can be regarded as k_o1Is a fixed value; by the formulaKnowing t_s5And k is_cAnd ζ₅Inversely proportional to the performance index t of the proposed system settling time_s0Comparing, if t of the corrected model_s5＞t_s0Then get k again_cOr (and) ζ₅Value, k_cAnd ζ₅When it is adjusted to be larger, t is_s5And (5) building a correction model again until the overshoot, the adjusting time and the static speed error meet the requirements.

5. The series correction method of the second-order under-damped system according to claim 1, characterized in that: the sixth step is specifically;

step six: according to the correction function established in the fourth step and the fifth step, the correction function is realized by a correction object; the real object correcting link is connected in series into the controlled object, then the overshoot and the adjusting time of the controlled object are tested according to the step one, the static speed error of the real object building correcting link is calculated according to the step two, whether the performance index requirement is met is judged, if the performance index requirement is not met, the parameter is adjusted in the real object correcting until the correcting is successful; the key for realizing correction is that the designed correction object is necessarily adjustable in correction parameter; if the correction function is still not available, reestablishing a correction function in the step five 5.2, then carrying out a step six real object construction correction link, measuring static speed error, overshoot and adjusting time according to the step one and the step two, comparing with the proposed performance index, and judging whether the correction is successful or not, if not, further adjusting parameters; if not, consider that the modeling isNo failure, e.g. consider ζ₄And w_n4Whether it is related to ζ₃And w_n3If the value difference is too large, the value is taken again to be closer to zeta according to the step two 2.1₃、w_n3ζ of₄And w_n4And (4) reestablishing a more accurate controlled object model for correction.

Technical Field

The invention relates to the technical field of automatic control, in particular to a series correction method of a second-order under-damping system or an approximate second-order under-damping system. When one or more poles in the transfer function denominator are far from other main poles, the far poles can be ignored, and this situation may cause that the second-order underdamping system is not analyzed by the second-order underdamping method, but the system is considered to be an approximate second-order underdamping system.

Background

A method for correcting a linear system in series, which is described in "study on a technique for correcting a linear system in series" J ", electronic technology 2016,000(006) 30-31:

and by adopting a zero pole elimination method, the derivation of the tau 1 and the tau 2 is established on the transfer function of the controlled object. (s.tau.)₁+1) is the same as the denominator of the established controlled object transfer function; (s.tau.)₂+1) is determined by the controlled object transfer function, the proposed overshoot and the adjustment time index.

The series correction method proposed in the literature is based on a controlled object transfer function obtained by theoretical estimation. The actual transfer function of the controlled object deviates from the transfer function obtained by theoretical calculation. Sometimes very different. The performance index of the controlled object after the series correction of the transfer function obtained based on theoretical calculation may not meet the requirement.

Disclosure of Invention

Aiming at the defects of the prior art, the invention provides a series correction method of a second-order under-damping system, which is different from the previous correction method, adds a correction method with a static speed error, and provides a method for modeling and establishing a correction function again when the overshoot, the adjustment time and the static speed error after correction do not meet the requirements. The specific technical scheme is as follows:

a series correction method of a second-order underdamped system comprises the following specific steps:

the method comprises the following steps: testing the step response waveform of the system, reading the output response waveform to obtain the overshoot delta₃% adjustment time t_s3；

Step two: overshoot delta measured according to step one₃% and adjustment time t_s3The concrete modeling process for establishing the actual open-loop transfer function model is as follows:

2.1, using overshoot formula

Determining dampingRatio ζ₃；

The specific derivation calculation process is as follows:

ζ₃ ²(π²+(ln(δ₃％))²)＝(ln(δ₃％))²

to obtain

Using formula of adjustment time

Obtaining the undamped natural oscillation frequency of the system

Calculated damping ratio ζ₃There is a possibility that the number of bits after the decimal point is large and the number of bits after the decimal point required for precision is not so large, so that the zeta-memory obtained as required is obtained by omitting the bits after the decimal point₄At this time

Wherein c is a constant, if an error is selectedBand Δ ± 0.05, then c ═ 3.5 or c ═ 3; if the error band Δ ± 0.02 is selected, c is 4.4, where ζ₄By the formula:

zeta is obtained₄(ii) a Wherein t is_s3Measuring an actual waveform in the first step;

2.2 according to the damping ratio ζ of the system₄And undamped natural oscillation frequency w_n4Establishing an open-loop transfer function of a practical continuous underdamped second-order system

Wherein due to the system damping ratio ζ₄And undamped natural oscillation frequency w_n4It is known that k can be obtained from the following formula_o1And T_o1；

By the formulaGet ζ 4, from the formulaTo obtain w_n4；

2.3 obtaining the static speed error coefficient according to the open loop transfer function of the continuous underdamping second-order system

Step three: overshoot δ according to step one test₃% and adjustment time t_s3And k obtained by modeling in step two_vJudging whether the static speed error, overshoot and adjusting time of the actual continuous underdamping second-order system meet the proposed performance indexes; the performance indexes provided by the system need to meet the following requirements: static velocity error k_v≥k_v0(s^-1) Overshoot delta₃％≤δ₀% regulating time t_s3≤t_s0If the performance index requirement is met, the controlled object does not need to be corrected and can be directly used; if the requirements are not met, correction is needed, and the step four is entered;

step four: and (3) correcting any performance index of the time domain response output waveform of the continuous underdamped second-order system if the performance index does not meet the requirement, wherein the correction steps are as follows:

4.1, if the static speed error coefficient of the actual connecting line underdamping second-order system meets the requirement, adding a proportional correction coefficient k_c＝1；

If the static speed error coefficient of the actual connecting line underdamped second-order system does not meet the requirement, adding a proportional correction coefficient k_c；

The corrected static velocity error coefficient k is required_o2The following conditions are satisfied:

wherein k is_cIs the scaling factor, k, of the correction circuit_o2Is the static velocity error of the corrected system, k_v0Is the minimum value of the static speed error coefficient and the system damping ratio zeta which are required to be achieved by the system₄And undamped natural oscillation frequency w_n4Calculating a value in the second step;

in thatWithin the range of k_cTo make it satisfy

Obtaining a new transfer function of the static speed error coefficient meeting the requirement after the proportion correction

Wherein the formula in step oneTo obtain k_o1From the formulaTo obtain T_o1Transfer function G for correcting static errors_c1(s)＝k_cAccording to the static speed error of the actual model and the performance index k provided by the system_v0By comparison, from formula k_c1 orTo obtain k_cCorrected static velocity error coefficient k_o2The following were used:

4.2 open-loop model of system after correcting static speed error coefficient

According to the open-loop model after the static speed error correction of the actual continuous underdamping second-order system is established

Wherein k is_o1、T_o1And k_cThe value is taken by the formula in the step four 4.1;

4.3, if the overshoot or the adjusting time does not meet the requirement, adding a correction link;

according to the overshootCalculating the corrected damping ratio delta₀% wherein the overshoot index value to be reached by the proposed system is a known quantity;

specific zeta₅The derivation is as follows:

ζ₅ ²(π²+(ln(δ₀％))²)≥(ln(δ₀％))²

to obtain

Wherein ζ₅Is positive and is a continuous underdamped second order system 0 < zeta₅＜1，

In thatArbitrarily take one value to obtain corrected damping ratio Zeta₅So as to overshoot

According to the zero-pole cancellation rule, make the correcting device G_c2Transfer function of(s)

4.4, obtaining the open-loop transfer function after the series correction through analyzing the static speed error, the overshoot and the adjusting time

Wherein G is_c(s) represents the total correction transfer function G_c(s)＝G_c1(s)·G_c2(s); it consists of a transfer function G for correcting static errors_c1(s) and transfer function G for correcting overshoot and adjusting time_c2(s) composition;

open-loop model with static velocity error correction

The whole correction idea is to use G_c1(s)＝k_cCorrecting static speed errors byThe overshoot and the adjustment time are corrected, and the two correction sequences are not mandatory and can be reversed.

The corrected open loop transfer function is:

from the corrected open loop transfer function equation:

simultaneous approximation of w by the above formula_n5Simplifying to obtain:

the left and right evolution of the above formula

Above formula T_o2Simultaneously reducing and finishing to obtain a formula

Thereon is provided withThe derivation can also be obtained using the following closed loop transfer function

By the formulaGet ζ by taking one value₅；

By the formulaTo obtain k_o2Wherein k is_cObtained in step 4.1, k_o1Is inherent to the controlled object and is obtained by the step 2.2;

from the above has been obtained k_o2And damping ratio ζ₅Two values, using the following formula to obtain T_o2Value of

From the formula of step two 2.2To obtain T_o1From step four 4.4 equationTo obtain T_o2Correction device G_c2The parameters of the transfer function of(s) have been found;

according to the static speed error of the actual model and the performance index k provided by the system_v0By comparison, from formula k_c1 orTo obtain k_cThe parameters of the total correction model are already calculated,

step five: checking the performance index of the corrected model;

step six: according to the correction function established in the fourth step and the fifth step, the correction function is realized by a correction object; the real object correcting link is connected in series into the controlled object, then the overshoot and the adjusting time of the controlled object are tested according to the step one, the static speed error of the real object building correcting link is calculated according to the step two, whether the performance index requirement is met is judged, if the performance index requirement is not met, the parameter is adjusted in the real object correcting until the correcting is successful; the key for realizing correction is that the designed correction object is necessarily adjustable in correction parameter; if the correction function is still not available, reestablishing a correction function in the step five 5.2, then carrying out a step six real object construction correction link, measuring static speed error, overshoot and adjusting time according to the step one and the step two, comparing with the proposed performance index, and judging whether the correction is successful or not, if not, further adjusting parameters; if not, consider if the modeling failed, such as considering ζ₄And w_n4Whether it is related to ζ₃And w_n3If the value difference is too large, the value is taken again to be closer to zeta according to the step two 2.1₃、w_n3ζ of₄And w_n4Value, reAnd establishing a more accurate controlled object model for correction.

Preferably, the method comprises the following steps:

the first step is specifically as follows:

1.1, inputting step waveform to the controlled object, testing output response waveform, reading the output response waveform to obtain overshoot recorded as delta₃% of the total amount of the solution was recorded as t_s3；

1.2、δ₃% of:

where C (tp) is the maximum deviation of the output step response and C (∞) is the steady state quantity of the output step response;

1.3, adjusting time t_s3The calculation of (1):

firstly, determining whether the required error band is delta +/-5% or delta +/-2%;

find d₁＝(1+|Δ|)×C(∞)，d₂＝(1-|Δ|)×C(∞)

The read response arrives and remains at d in the output response waveform₁And d₂The shortest time required within the value is recorded as the adjustment time t_s3；

Preferably, the method comprises the following steps: the second step is provided with:

2.4.1, calculating overshoot delta of the model before correction according to the controlled object model measured in the step two 2.1₄％，

Overshoot using formula

Calculating the adjusting time t of the model before correction according to the controlled object model actually measured in the step two 2.1_s4Adjustment time before correction:

by said formulaZeta is obtained₄From said formulaTo obtain w_n4；

2.4.2, comparing the overshoot delta calculated by the model obtained in the second step 2.1₄% and adjustment time t_s4And step one, actually testing to obtain the overshoot delta₃% and adjustment time t_s3A value of (d); since the decimal point omission possibility exists in the calculation, certain errors are allowed; if the actual measurement controlled object model obtained according to the step two 2.1Firstly, debugging a corrected object link if the corrected performance index does not meet the requirement; if the actual object is still not realized, reestablishing a correction function in the step five 5.2, and correcting the actual object; if not, consider if the modeling failed, such as considering ζ₄And w_n4Whether it is related to ζ₃And w_n3If the value difference is too large, the value is taken again to be closer to zeta according to the step two 2.1₃、w_n3ζ of₄And w_n4Re-establishing a more accurate controlled object model for correction;

2.4.3, after the verification and comparison according to the step two, the model of the controlled object is established and judged whether the static speed error coefficient of the model meets the requirement or not

If it isThe actual second-order system meets the requirement and corrects the proportionality coefficient k_c1 is ═ 1; otherwise, the actual second-order system does not meet the requirement, the static speed error coefficient needs to be changed, and the correction scale coefficient k is increased_cCorrection of the proportionality coefficient k_cThe concrete values are shown in step four, wherein k_v0Is a known requirement of a performance indexA value;

preferably, the method comprises the following steps: the fifth step is specifically;

5.1、k_o2according to said formulaSelection of, delta₅% by formulaIs selected, wherein k_v0And delta₀% is the performance index value provided by the system, so the performance index requirement is met;

5.2, according to the corrected open loop transfer function:

calculating the adjustment time t of the corrected model_s5Specifically, the following is calculated:

from step four 4.4 of the formulaBecome into

Substituting into the formula of the adjustment time to obtain the adjustment time after correction

Judging t_s5And t_s0Value, if t_s5≤t_s0If the theoretical correction is finished, entering a correction circuit implementation stage; if t is_s5＞t_s0According to said formulaReselecting zeta₅From the formulaCan obtain t_s5And ζ₅In inverse proportion, the following stepsWithin range of taking ζ₅But acquire ζ more than before₅Slightly larger;

if t is_s5＞t_s0It is also possible to reselect k_cT is derived from_s5And k_cIn inverse proportion, k is taken again in the following times_cAlso, k is obtained more than before_cSlightly larger;

by the formulaAnd k_o2＝k_c×k_o1Substituting into said formula

Wherein c is a constant, if the error band delta is selected to be +/-0.05, c is 3.5 or c is 3; if the error band delta is selected to be +/-0.02, c is 4.4; k is a radical of_o1Is the value of the controlled object, is obtained by establishing a model in the second step, and can be regarded as k_o1Is a fixed value; by the formulaKnowing t_s5And k is_cAnd ζ₅Inversely proportional to the performance index t of the proposed system settling time_s0Comparing, if t of the corrected model_s5＞t_s0Then get k again_cOr (and) ζ₅Value, k_cAnd ζ₅When it is adjusted to be larger, t is_s5The error rate is reduced, and a correction model is established again until the overshoot, the adjustment time and the static speed error meet the requirements;

the invention has the beneficial effects that: the invention adds a method for establishing a model according to an experimental test result, so that the serial correction can be calculated by a correct method. The model with large difference from the actual model obtained by the analytic method can also meet the requirement of control indexes through series correction. The correction effectiveness and universality are improved.

The invention adds kc on the series correction model and adds the calculation process of kc in the series correction calculation, so that the series correction method of the invention is more effective. Methods of how to modify the correction function based on overshoot, settling time, and static velocity error are also presented.

Drawings

FIG. 1 is a block diagram of a transfer function before correction in the present invention;

FIG. 2 is a block diagram of a transfer function after correction in the present invention.

Detailed Description

The following detailed description of the preferred embodiments of the present invention, taken in conjunction with the accompanying drawings, will make the advantages and features of the invention easier to understand by those skilled in the art, and thus will clearly and clearly define the scope of the invention.

As shown in fig. 1 and 2: a series correction method of a second-order underdamped system comprises the following specific steps:

the method comprises the following steps: testing the step response waveform of the system, reading the output response waveform to obtain the overshoot delta₃% adjustment time t_s3；

1.1, inputting step waveform to the controlled object, testing output response waveform, reading the output response waveform to obtain overshoot recorded as delta₃% of the total amount of the solution was recorded as t_s3；

1.2、δ₃% of:

where C (tp) is the maximum deviation of the output step response and C (∞) is the steady state quantity of the output step response;

1.3, adjusting time t_s3The calculation of (1):

firstly, determining whether the required error band is delta +/-5% or delta +/-2%;

find d₁＝(1+|Δ|)×C(∞)，d₂＝(1-|Δ|)×C(∞)

The read response arrives and remains at d in the output response waveform₁And d₂The shortest time required within the value is recorded as the adjustment time t_s3；

Step two: overshoot delta measured according to step one₃% and adjustment time t_s3The concrete modeling process for establishing the actual open-loop transfer function model is as follows:

2.1, using overshoot formula

Calculating damping ratio Zeta₃；

The specific derivation calculation process is as follows:

ζ₃ ²(π²+(ln(δ₃％))²)＝(ln(δ₃％))²

to obtain

Using formula of adjustment time

Obtaining the undamped natural oscillation frequency of the system

Calculated damping ratio ζ₃There is a possibility that the number of bits after the decimal point is large and the number of bits after the decimal point required for precision is not so large, so that the zeta-memory obtained as required is obtained by omitting the bits after the decimal point₄At this time

Wherein c is a constant, if the error band delta is selected to be +/-0.05, c is 3.5 or c is 3; if the error band Δ ± 0.02 is selected, c is 4.4, where ζ₄By the formula:

zeta is obtained₄(ii) a Wherein t is_s3Measuring an actual waveform in the first step;

2.2 according to the damping ratio ζ of the system₄And undamped natural oscillation frequency w_n4Establishing an open-loop transfer function of a practical continuous underdamped second-order system

Wherein due to the system damping ratio ζ₄And undamped natural oscillation frequency w_n4It is known that k can be obtained from the following formula_o1And T_o1；

By the formulaGet ζ 4, from the formulaTo obtain w_n4；

2.3 obtaining the static speed error coefficient according to the open loop transfer function of the continuous underdamping second-order system

The second step is also provided with:

2.4.1, calculating overshoot delta of the model before correction according to the controlled object model measured in the step two 2.1₄％，

Overshoot using formula

Calculating the adjusting time t of the model before correction according to the controlled object model actually measured in the step two 2.1_s4Adjustment time before correction:

by said formulaZeta is obtained₄From said formulaTo obtain w_n4；

2.4.2, comparing the overshoot delta calculated by the model obtained in the second step 2.1₄% and adjustment time t_s4And step one, actually testing to obtain the overshoot delta₃% and adjustment time t_s3A value of (d); since the decimal point omission possibility exists in the calculation, certain errors are allowed; if the actual measurement controlled object model obtained according to the step two 2.1Firstly, debugging a corrected object link if the corrected performance index does not meet the requirement; if the actual object is still not realized, reestablishing a correction function in the step five 5.2, and correcting the actual object; if not, consider if the modeling failed, such as considering ζ₄And w_n4Whether it is related to ζ₃And w_n3If the value difference is too large, the value is taken again to be closer to zeta according to the step two 2.1₃、w_n3ζ of₄And w_n4Re-establishing a more accurate controlled object model for correction;

2.4.3, after the verification and comparison according to the step two, the model of the controlled object is established and judged whether the static speed error coefficient of the model meets the requirement or not

If it isThe actual second-order system meets the requirement and corrects the proportionality coefficient k_c1 is ═ 1; otherwise, the actual second-order system does not meet the requirement, the static speed error coefficient needs to be changed, and the correction scale coefficient k is increased_cCorrection of the proportionality coefficient k_cThe concrete values are shown in step four, wherein k_v0Is a known value of the performance index requirement;

step three: overshoot δ according to step one test₃% and adjustment time t_s3And k obtained by modeling in step two_vJudging whether the static speed error, overshoot and regulation time of the actual continuous underdamping second-order system meet the requirementsPerformance index; the performance indexes provided by the system need to meet the following requirements: static velocity error k_v≥k_v0(s^-1) Overshoot delta₃％≤δ₀% regulating time t_s3≤t_s0If the performance index requirement is met, the controlled object does not need to be corrected and can be directly used; if the requirements are not met, correction is needed, and the step four is entered;

step four: and (3) correcting any performance index of the time domain response output waveform of the continuous underdamped second-order system if the performance index does not meet the requirement, wherein the correction steps are as follows:

4.1, if the static speed error coefficient of the actual connecting line underdamping second-order system meets the requirement, adding a proportional correction coefficient k_c＝1；

If the static speed error coefficient of the actual connecting line underdamped second-order system does not meet the requirement, adding a proportional correction coefficient k_c；

The corrected static velocity error coefficient k is required_o2The following conditions are satisfied:

wherein k is_cIs the scaling factor, k, of the correction circuit_o2Is the static velocity error of the corrected system, k_v0Is the minimum value of the static speed error coefficient and the system damping ratio zeta which are required to be achieved by the system₄And undamped natural oscillation frequency w_n4Calculating a value in the second step;

in thatWithin the range of k_cTo make it satisfy

Obtaining a new transfer function of the static speed error coefficient meeting the requirement after the proportion correction

Wherein the formula in step oneTo obtain k_o1From the formulaTo obtain T_o1Transfer function G for correcting static errors_c1(s)＝k_cAccording to the static speed error of the actual model and the performance index k provided by the system_v0By comparison, from formula k_c1 orTo obtain k_cCorrected static velocity error coefficient k_o2The following were used:

4.2 open-loop model of system after correcting static speed error coefficient

According to the open-loop model after the static speed error correction of the actual continuous underdamping second-order system is established

Wherein k is_o1、T_o1And k_cThe value is taken by the formula in the step four 4.1;

4.3, if the overshoot or the adjusting time does not meet the requirement, adding a correction link;

according to the overshootCalculating the corrected damping ratio delta₀% wherein the overshoot index value to be reached by the proposed system is a known quantity;

specific zeta₅The derivation is as follows:

ζ₅ ²(π²+(ln(δ₀％))²)≥(ln(δ₀％))²

to obtain

Wherein ζ₅Is positive and is a continuous underdamped second order system 0 < zeta₅＜1，

In thatArbitrarily take one value to obtain corrected damping ratio Zeta₅So as to overshoot

According to the zero-pole cancellation rule, make the correcting device G_c2Transfer function of(s)

4.4, obtaining the open-loop transfer function after the series correction through analyzing the static speed error, the overshoot and the adjusting time

Wherein G is_c(s) represents the total correction transfer function G_c(s)＝G_c1(s)·G_c2(s); it consists of a transfer function G for correcting static errors_c1(s) and transfer function G for correcting overshoot and adjusting time_c2(s) composition;

open-loop model with static velocity error correction

The whole correction idea is to use G_c1(s)＝k_cCorrecting static speed errors byCorrecting overshoot and adjusting time, wherein the two correction orders have no mandatory requirement and can be reversed;

the corrected open loop transfer function is:

from the corrected open loop transfer function equation:

simultaneous approximation of w by the above formula_n5Simplifying to obtain:

the left and right evolution of the above formula

Above formula T_o2Simultaneously reducing and finishing to obtain a formula

Thereon is provided withThe derivation can also be obtained using the following closed loop transfer function

By the formulaGet ζ by taking one value₅；

By the formulaTo obtain k_o2Wherein k is_cObtained by step four 4.1, k_o1Is inherent to the controlled object and is obtained by the step 2.2;

from the above has been obtained k_o2And damping ratio ζ₅Two values, using the following formula to obtain T_o2Value of

From the formula of step two 2.2To obtain T_o1From step four 4.4 equationTo obtain T_o2Correction device G_c2The parameters of the transfer function of(s) have been found;

according to the static speed error of the actual model and the system proposed performanceEnergy index k_v0By comparison, from formula k_c1 orTo obtain k_cThe parameters of the total correction model are already calculated,

step five: checking the performance index of the corrected model;

5.1、k_o2according to said formulaSelection of, delta₅% by formulaIs selected, wherein k_v0And delta₀% is the performance index value provided by the system, so the performance index requirement is met;

5.2, according to the corrected open loop transfer function:

calculating the adjustment time t of the corrected model_s5Specifically, the following is calculated:

from step four 4.4 of the formulaBecome into

Substituting into the formula of the adjustment time to obtain the adjustment time after correction

Judging t_s5And t_s0Value, if t_s5≤t_s0If the theoretical correction is finished, entering a correction circuit implementation stage; if t is_s5＞t_s0According to said formulaReselecting zeta₅From the formulaCan obtain t_s5And ζ₅In inverse proportion, the following stepsWithin range of taking ζ₅But acquire ζ more than before₅Slightly larger;

if t is_s5＞t_s0It is also possible to reselect k_cT is derived from_s5And k_cIn inverse proportion, k is taken again in the following times_cAlso, k is obtained more than before_cSlightly larger;

by the formulaAnd k_o2＝k_c×k_o1Substituting into said formula

Wherein c is a constant, if the error band delta is selected to be +/-0.05, c is 3.5 or c is 3; if the error band delta is selected to be +/-0.02, c is 4.4; k is a radical of_o1Is the value of the controlled object, is obtained by establishing a model in the second step, and can be regarded as k_o1Is a fixed value; by the formulaKnowing t_s5And k is_cAnd ζ₅Inversely proportional to the performance index t of the proposed system settling time_s0Comparing, if t of the corrected model_s5＞t_s0Then get k again_cOr (and) ζ₅Value, k_cAnd ζ₅When it is adjusted to be larger, t is_s5The error rate is reduced, and a correction model is established again until the overshoot, the adjustment time and the static speed error meet the requirements;

step six: according to the correction function established in the fourth step and the fifth step, the correction function is realized by a correction object; the real object correcting link is connected in series into the controlled object, then the overshoot and the adjusting time of the controlled object are tested according to the step one, the static speed error of the real object building correcting link is calculated according to the step two, whether the performance index requirement is met is judged, if the performance index requirement is not met, the parameter is adjusted in the real object correcting until the correcting is successful; the key for realizing correction is that the designed correction object is necessarily adjustable in correction parameter; if the correction function is still not available, reestablishing a correction function in the step five 5.2, then carrying out a step six real object construction correction link, measuring static speed error, overshoot and adjusting time according to the step one and the step two, comparing with the proposed performance index, and judging whether the correction is successful or not, if not, further adjusting parameters; if not, consider if the modeling failed, such as considering ζ₄And w_n4Whether it is related to ζ₃And w_n3If the value difference is too large, the value is taken again to be closer to zeta according to the step two 2.1₃、w_n3ζ of₄And w_n4And (4) reestablishing a more accurate controlled object model for correction.