CN106527120A - Stability margin configuration method for specified-accuracy PID control system - Google Patents

Abstract

The present invention discloses a stability margin configuration method for a specified-accuracy PID control system. The method comprises the steps of firstly, establishing a frequency characteristics model of a controlled object; secondly, according to a specified open-loop crossover frequency and a desired amplitude margin, calculating a proportional gain required by a PID controller; thirdly, according to the definition formula of a phase angle margin, searching an open-loop cut-off frequency that meets the desired amplitude margin through the optimizing control algorithm; finally, according to a given formula, calculating the integral gain and the differential gain of the PID controller. Based on above three parameter values of the PID, the specified stability margin of the system is realized. According to the technical scheme of the invention, the difficulty of solving non-linear equations is avoided through the analytic method. Therefore, the PID parameter setting is realized at the specified stability margin.

Description

PID control system stability margin collocation method with designated precision

Technical field

The invention belongs to control method technical field, more particularly to a kind of PID control system with designated precision is stably Nargin collocation method.

Background technology

As control algolithm is simple, adjustable parameter is few and can meet the performance requirement of most of control systems, so far The control loop for still having 95% adopts PID controller.Due to PID controller parameter configuration it is whether suitable, it will usually have influence on The performance of control system, therefore, the parameter tuning problem of PID controller is always the focus of academia and industrial quarters concern. In numerous setting methods of PID controller, it is broadly divided into that time domain is adjusted and frequency domain is adjusted two classes.Wherein, system stablizes abundant Degree (including magnitude margin and Phase margin) is an important indicator must being fulfilled in frequency domain setting method.However, due to nothing Method solves trigonometric function equation, thus system design index is difficult to done with high accuracy.

The content of the invention

In order to meet the design exact requirements of stability margin, the present invention provides one kind and solves cutoff frequency by optimization method The method for designing of rate：By the solving precision for arranging cut-off frequency, the cutoff frequency of designated precision is obtained using linear search algorithm Rate, so as to finally calculate three parameters of PID controller.

The technical scheme that the present invention takes is as follows：

A kind of collocation method of the PID control system stability margin with designated precision, comprises the following steps：

S1, sets up the frequency characteristic model of the frequency characteristic model and PID controller of controlled device；

S2, according to specified cross-over frequency ω_xWith magnitude margin A_mRequire, calculate the proportional gain of PID controller；

S3, based on optimizing algorithm, search meets specifies Phase margin Φ_mCut-off frequency ω_c；

S4, based on the cut-off frequency ω that the optimizing of step S3 is obtained_c, calculate the storage gain K of PID_iWith differential gain K_d。

Step S1, frequency characteristic G of controlled device_p(j ω) is：

G_p(j ω)=u (ω)+jv (ω) (1)

In formula (1), j is imaginary symbols, and ω is frequency, real parts of the u (ω) for the frequency characteristic of controlled device, and v (ω) is The imaginary part of the frequency characteristic of controlled device；

The frequency characteristic model of PID controller is：

G_c(j ω)=K_p+X(ω)j (2)

Wherein,For the imaginary part of the frequency characteristic of PID controller, K_p、K_i、K_dIt is expressed as PID controls The proportional gain of device processed, storage gain and the differential gain.

Step S2, it is intended that cross-over frequency be ω_x, then meet and specify magnitude margin A_mProportional gain by formula (3) count Calculate：

Wherein u (ω_x) for ω=ω_xWhen controlled device frequency characteristic real part, v (ω_x) for ω=ω_xWhen controlled device The imaginary part of frequency characteristic.

In step S3, it is intended that Phase margin be Φ_mMeet nonlinear equation (4)：

u(ω_c) for ω=ω_cWhen controlled device frequency characteristic real part, v (ω_c) for ω=ω_cWhen controlled device frequency The imaginary part of characteristic；Wherein X (ω_c) computing formula be formula (5)：

Solving precision ε is set, and by optimizing algorithm, search meets the cut-off frequency ω of solving precision ε_c.More preferably, pass through Two way classification is searching for ω_c。

More preferably, the cut-off frequency ω of solving precision ε is met using dichotomizing search_c, comprise the following steps：

S001:The region of search [ω is set_L ω_R] (ω during initialization_L=ω_x, ω_R=0), interval midpoint is taken for ω_c= 0.5(ω_L+ω_R)；

S002:By ω_cNonlinear equation (4) is updated to, the value on the right of nonlinear equation (4) is calculated；

S003:Judge the absolute value of difference of the value on nonlinear equation (4) left side and the value on the right whether less than default precision ε；If less than default precision ε, based on gained ω_c, turn to step S4；S004 is performed otherwise；

S004:If value is worth less than the left side on the right of nonlinear equation (4), ω is taken_L=ω_c；Conversely, ω_R=ω_c；Return S001。

The storage gain K of PID controller described in step S4_iWith differential gain K_dCalculated based on formula (6)：

Wherein,Compared with prior art, the present invention's has Beneficial effect includes：The solution of nonlinear equation is searched for by optimized algorithm, it is to avoid the difficulty of direct solution nonlinear equation, can Obtain the balance between solving precision and solving complexity, meet the needs of engineering design, can under the conditions of designated precision, PID control system stability margin configuration is carried out, the configuration required precision of PID control system stability margin is met.

Specific embodiment

PID control system stability margin collocation method with designated precision, takes following steps to calculate pid parameter, can To obtain desired stability margin (magnitude margin Am and Phase margin Φ_m)：

A kind of collocation method of the PID control system stability margin with designated precision, comprises the following steps：

S1, sets up the frequency characteristic model of the frequency characteristic model and PID controller of controlled device；

S2, according to specified cross-over frequency ω_xWith magnitude margin A_mRequire, calculate the proportional gain of PID controller；

S3, based on optimizing algorithm, search meets specifies Phase margin Φ_mCut-off frequency ω_c；

S4, based on the cut-off frequency ω that the optimizing of step S3 is obtained_c, calculate the storage gain K of PID_iWith differential gain K_d。

Step S1, frequency characteristic G of controlled device_p(j ω) is：

G_p(j ω)=u (ω)+jv (ω) (1)

In formula (1), j is imaginary symbols, and ω is frequency, real parts of the u (ω) for the frequency characteristic of controlled device, and v (ω) is The imaginary part of the frequency characteristic of controlled device；

The frequency characteristic model of PID controller is：

G_c(j ω)=K_p+X(ω)j (2)

Wherein,For the imaginary part of the frequency characteristic of PID controller, K_p、K_i、K_dIt is expressed as PID controls The proportional gain of device processed, storage gain and the differential gain.

Step S2, it is intended that cross-over frequency be ω_x, then meet and specify magnitude margin A_mProportional gain by formula (3) count Calculate：

u(ω_x) for ω=ω_xWhen controlled device frequency characteristic real part, v (ω_x) for ω=ω_xWhen controlled device frequency The imaginary part of characteristic.

In step S3, it is intended that Phase margin be Φ_mMeet nonlinear equation (4)：

u(ω_c) for ω=ω_cWhen controlled device frequency characteristic real part, v (ω_c) for ω=ω_cWhen controlled device frequency The imaginary part of characteristic；Wherein X (ω_c) computing formula be formula (5)：

Solving precision ε is set, and by optimizing algorithm, search meets the cut-off frequency ω of solving precision ε_c。

The present embodiment, searches for ω using two way classification_c：

S001:The region of search [ω is set_L ω_R] (ω during initialization_L=ω_x, ω_R=0), interval midpoint is taken for ω_c= 0.5(ω_L+ω_R)；

S002:By ω_cIt is updated to nonlinear equation (4), the value on the right of accounting equation (4)；

S003:Whether the absolute value of the difference of decisive equation (4) left and right value is less than default precision ε；Turn if condition is met To S005；S004 is performed otherwise；

S004:If value takes ω less than the left side on the right of equation (4)_L=ω_c；Conversely, ω_R=ω_c；Return S001；

S005:Using gained ω_c, based on equation below, calculate the storage gain K of the PID controller_iAnd the differential gain K_d：

In formula,

So, when three parameters of PID controller are arranged according to above-mentioned requirements, the stability margin required by design objective Required precision can just be met.

Those skilled in the art the present invention can be modified or modification design but without departing from the present invention think of Think and scope.Therefore, if these modifications of the present invention and modification belong to the technical scope of the claims in the present invention and its equivalent Within, then the present invention is also intended to comprising these changes and modification.

Claims (6)

1. a kind of collocation method of the PID control system stability margin with designated precision, its feature are comprised the following steps：

S1, sets up the frequency characteristic model of the frequency characteristic model and PID controller of controlled device；

S2, according to specified cross-over frequency ω_xWith magnitude margin A_mRequire, calculate the proportional gain of PID controller；

S3, based on optimizing algorithm, search meets specifies Phase margin Φ_mCut-off frequency ω_c；

S4, calculates the storage gain K of PID_iWith differential gain K_d。

2. the PID control system stability margin degree collocation method with designated precision according to claim 1, its feature exist In：Frequency characteristic G of controlled device_p(j ω) is：

G_p(j ω)=u (ω)+jv (ω) (1)

In formula (1), j is imaginary symbols, and ω is frequency, real parts of the u (ω) for the frequency characteristic of controlled device, and v (ω) is controlled The imaginary part of the frequency characteristic of object；

The frequency characteristic model of PID controller is：

G_c(j ω)=K_p+X(ω)j (2)

Wherein,For the imaginary part of the frequency characteristic of PID controller, K_p、K_iAnd K_dIt is expressed as PID controller Proportional gain, storage gain and the differential gain.

3. the PID control system stability margin collocation method with designated precision according to claim 1, its feature exist In：

The cross-over frequency specified is ω_x, then meet and specify magnitude margin A_mProportional gain by formula (3) calculate：

$K_p=\frac{-u\left({\mathrm{\ω}}_x\right)}{A_m\[u^2\left({\mathrm{\ω}}_x\right)+v^2\left({\mathrm{\ω}}_x\right)\]}---\left(3\right)$

u(ω_x) for ω=ω_xWhen controlled device frequency characteristic real part, v (ω_x) for ω=ω_xWhen controlled device frequency characteristic Imaginary part.

4. the PID control system stability margin collocation method with designated precision according to claim 1, its feature exist In：

In step S3, it is intended that Phase margin be Φ_mMeet nonlinear equation (4)：

Wherein X (ω_c) computing formula be formula (5)：

$X\left({\mathrm{\ω}}_c\right)=-\sqrt{\frac{1}{u^2\left({\mathrm{\ω}}_c\right)+v^2\left({\mathrm{\ω}}_c\right)}-K_p^2}---\left(5\right)$

Solving precision ε is set, and by optimizing algorithm, search meets the cut-off frequency ω of solving precision ε_c。

5. the PID control system stability margin collocation method with designated precision according to claim 4, its feature exist In：

The cut-off frequency ω of solving precision ε is met using dichotomizing search_c, comprise the following steps：

S001:The region of search [ω is set_L ω_R], interval midpoint is taken for ω_c=0.5 (ω_L+ω_R)；

S002:By ω_cNonlinear equation (4) is updated to, the value on the right of nonlinear equation (4) is calculated；

S003:Judge the absolute value of difference of the value on nonlinear equation (4) left side and the value on the right whether less than default precision ε；Such as Fruit is less than default precision ε, based on gained ω_c, then turn to step S4；S004 is performed otherwise；

S004:If value is worth less than the left side on the right of nonlinear equation (4), ω is taken_L=ω_c；Conversely, ω_R=ω_c；Return S001.

6. the PID control system stability margin collocation method with designated precision according to claim 1, its feature exist In：The storage gain K of PID controller described in step S4_iWith differential gain K_dCalculated based on formula (6)：

$\left\{\begin{array}{c}{K}_i=\mathrm{\λ}\[X\left({\mathrm{\ω}}_x\right){\mathrm{\ω}}_c-X\left({\mathrm{\ω}}_c\right){\mathrm{\ω}}_x\]\\ K_d=\mathrm{\λ}\[\frac{X\left({\mathrm{\ω}}_x\right)}{{\mathrm{\ω}}_c}-\frac{X\left({\mathrm{\ω}}_c\right)}{{\mathrm{\ω}}_x}\]\end{array}\right.---\left(6\right)$

Wherein,