CN106681150A - Fractional-order PID controller parameter optimizing and setting method based on closed-loop reference model - Google Patents

Abstract

The invention discloses a fractional-order PID controller parameter optimizing and setting method based on a closed-loop reference model. The method comprises the steps that S1, an ideal closed-loop reference model is selected, and the cut-off frequency omega c and the order alpha of the ideal closed-loop reference model are selected according to the control performance requirements of a system; S2, according to a transfer function expression of the closed-loop system and by combining with the ideal closed-loop system model H(s) and a fractional order PID controller model Gc(s), an ideal control object model (shown in the description) is derived; S3, the frequency domain response characteristics of an unknown controlled object model Gp(s) are obtained, so that the ideal control object model (shown in the description) and Gp(s) are identical in response when omega = 0 and omega = x, and a function relation of kp, kd and mu when omega = omega x is calculated; S4, by optimizing and identifying parameters in the ideal form of an unknown object (shown in the description), so that the unknown object (shown in the description) is close to the actual object Gp(s) in frequency domain response index in a cut-off frequency range, frequency domain response error indexes are established, the error indexes are optimized (shown in the description) when 0 < mu < 2, and finally parameters of a fractional-order controller are obtained. The fractional-order PID controller parameter optimizing and setting method utilizes an identification method of the system to rapidly obtain controller parameters and meanwhile can also ensure best approximation of the ideal reference model.

Description

Fractional order PID controller parameter optimization setting method based on closed-loop reference model

Technical Field

The invention relates to the field of fractional order PID control, in particular to a method for optimizing and setting parameters of a fractional order PID controller based on a closed-loop reference model.

Background

A fractional order PID controller is proposed by professor i.podlubny, whose general format is abbreviated as PI^λD^μThe method has a concept and an analysis method similar to integral differential integration, and has universality and applicability more than those of the traditional integral PID. Because the fractional order of differentiation and integration is introduced, two adjustable parameters are added to the controller, the performance adjusting range of the system is enlarged, and a better control effect can be expected. But at the same time, the difficulty of controller design and parameter setting is also increased.

In recent years, fractional order PID controllers are gradually concerned by scholars and engineering fields, and most of research is focused on direct parameter setting and performance index optimization design at present. In parameter setting, mainly aiming at a certain type of given object, deducing an analytical formula of a fractional order PID controller based on phase margin, cut-off frequency and gain robustness conditions, and determining controller parameters; in the aspect of performance index optimization design, an intelligent algorithm is mainly adopted to perform global optimization on the fractional order PID controller, such as a particle swarm algorithm, an evolutionary algorithm and the like. Although the fractional order PID controller setting method has a certain research, a rapid and effective design method is still lacked, and the controller parameters can be directly obtained according to a reference model of the closed loop expected characteristics.

Therefore, the inventor deeply researches a fractional order PID control technology and provides a method for optimizing and setting parameters of a fractional order PID controller based on a closed-loop reference model.

Disclosure of Invention

The invention aims to provide a fractional order PID controller parameter optimization setting method based on a closed-loop reference model.

In order to solve the technical problems, the technical scheme of the invention is as follows:

a fractional order PID controller parameter optimization setting method based on a closed loop reference model is applied to a closed loop feedback control structure of fractional order PID control, and the structure comprises a controlled object model G_p(s) and fractional order PID controller model G_c(s) whereink_p,k_i,k_dλ and μ are undetermined parameters of the fractional order PID controller; the method comprises the following steps:

s1: selecting ideal closed-loop reference modelSelecting a cut-off frequency omega of an ideal closed-loop reference model according to the control performance requirement of the system_cAnd order α;

s2: according to the transfer function expression of the closed-loop system, combining an ideal closed-loop system model H(s) and a fractional order PID controller model G_c(s) deriving ideal control object models

S3: obtaining unknown controlled object model G_p(s) frequency domain response characteristics ofAnd G_p(s) in ω -0 and ω - ω_xThe response is the same, i.e. the Nyquist curves of both are 0 and ω_xCrossing to obtain k_iAnd find k_p、k_dAt ω ═ ω_xIs a function of μ;

s4: identification of ideal form of unknown object by optimizationOf (2) isMaximum approach to the real object G in the cut-off frequency range_p(s) a frequency domain response indicator; establishing a frequency domain response error indicatorAnd optimizing error indexes when mu is more than 0 and less than 2And finally obtaining the parameters of the fractional order controller.

In step S1, the control performance requirement of the system is a time domain index, and the time domain index may be overshoot, adjustment time, or peak time.

The step S2 is specifically:

based on the transfer function expression of the closed-loop system, the ideal closed-loop system model H(s) and the fractional order PID controller modelDeriving ideal control object modelsThe derivation is as follows:

let λ be α, then

The lambda, mu can be a decimal number or an integer.

The step S3 is specifically:

obtaining unknown controlled object G_pFrequency domain response data of(s), let us assumeAnd G_p(s) in ω -0 and ω - ω_xHas the same frequency response, ω_xG can be selected as original system_p(s) crossover frequency of phase margin | G_p(jω_x)|＝1：

Firstly, selecting the lambda as α,meaningful at ω ═ 0, areThen according toIs provided withAccording tok_p、k_dAt ω ═ ω_xThe functional relationship between position and μ is:

wherein,

after the scheme is adopted, the invention has the beneficial effects that: the method initially determines omega according to the time domain response index of the system by selecting a reference closed-loop model_cα and lambda, optimizing to obtain the differential order of the fraction order PID by approximating the frequency response characteristic curve of the actual object model and the ideal object model, and calculating to obtain k_d，k_i，k_pThe value of (3) can obtain a fractional order PID controller approaching the ideal reference model, and the controller has robustness to the change of the gain.

The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and the detailed description.

Drawings

FIG. 1 is a block diagram of a closed loop control system comprising a fractional order PID controller and a controlled object according to the invention;

FIG. 2 is a flow chart of a closed-loop reference model based fractional order PID controller parameter optimization tuning method of the present invention;

FIG. 3 is a schematic diagram of the step response of the closed loop system controlled by the fractional order PID of the present invention;

fig. 4 is a graph of the step response for different gain conditions.

Detailed Description

The invention discloses a closed-loop reference model-based fractional order PID controller parameter optimization setting method, which is applied to a closed-loop feedback control structure of fractional order PID control, such as a closed-loop feedback control structure of fractional order PID control shown in figure 1, and comprises a stable controlled object model G_p(s) and fractional order PID controller model G_c(s)：

Wherein k is_p,k_i,k_dλ and μ are parameters to be determined by the controller, and the corresponding closed-loop transfer function is:

as shown in fig. 2, the method of the present invention specifically includes the following steps:

first, an ideal closed-loop reference model is selectedSelecting a cut-off frequency omega of an ideal closed-loop reference model according to the control performance requirement of the system_cAnd order α, the control performance requirement of the system being a time domain indicator, which may be overshoot, settling time or peak time, the ideal closed-loop reference model H(s) giving the system the desired characteristics of insensitivity to gain variations, which only cause the cut-off frequency omega when the gain varies_cThe system has strong robustness to gain change, and the overshoot of the system is only related to α and is not related to gain;

secondly, according to the transfer function expression of the closed-loop system, the ideal closed-loop system model H(s) and the fractional order PID controller modelDeriving ideal control object modelsThe derivation is as follows:

let λ be α, then

Said λ, μmay be a decimal or integer;

thirdly, acquiring unknown controlled object G_pFrequency domain response data of(s), let us assumeAnd G_p(s) in ω -0 and ω - ω_xHas the same frequency response, ω_xG can be selected as original system_p(s) crossover frequency of phase margin | G_p(jω_x)|＝1：

Firstly, selecting the lambda as α,meaningful at ω 0 (where the subject can maintain a good steady-state response, consistent with the case of a typical real-world system), there areThen according toIs provided withAccording tok_p、k_dAt ω ═ ω_xThe functional relationship between position and μ is:

wherein,

fourthly, identifying the ideal form of the unknown object by optimizingOf (2) isMaximum approach to the real object G in the cut-off frequency range_p(s) a frequency domain response indicator; establishing a frequency domain response error indicatorAnd optimizing error indexes when mu is more than 0 and less than 2And finally obtaining the parameters of the fractional order controller.

The following is an application example of the invention, and the specific steps are as follows:

identifying the motor model by using the frequency response characteristic of the object to obtain a controlled object function as shown in the following formula:

the reference model of the closed-loop transfer function is selected asWherein ω is_cWhen the response speed of the closed-loop system is 5 and the overshoot performance index is 1.1, the performance indexes such as rapidity and overshoot of the closed-loop system can be determined by H(s), an equivalent object model is obtained under the closed-loop control of fractional PID,

the PID controller is designed so that the frequency response characteristic of the motor control system is as close as possible to H(s), i.e.

The parameters of the fractional order PID controller which can be obtained according to the fractional order PID parameter setting step are as follows:

k_d＝0.000024,k_i＝0.0877,k_p＝0.0036,λ＝1.1,μ＝1.007；

the response curve of the step response of the object and the closed-loop reference model is shown in fig. 3, and it can be seen that the fractional order PID controller obtained by the method can well realize the response tracking of the closed-loop reference model, and as long as the proper closed-loop reference model parameters are selected, the method can quickly obtain the fractional order PID controller parameters and meet the specified performance requirements.

Further, the robust immunity performance is verified: keeping the parameters of the fractional order controller unchanged, and changing the objects to be: 80, 55, 40. Referring to fig. 4, under the condition that the parameters of the fractional order PID are not changed, the gain of the controlled object is changed, only the response speed is affected, and the overshoot is not affected, thereby verifying the robustness of the fractional order control system designed by the method to the gain change.

This example shows that: the invention provides a fractional order PID controller parameter optimization setting method based on a closed-loop reference model, which comprises the steps of firstly selecting a reference closed-loop model, and preliminarily determining omega according to a time domain response index of a system_cα, lambda, obtaining the differential order of the fraction order PID by approximating the frequency response characteristic curve of the actual object model and the ideal object model, and calculating to obtain k_d，k_i，k_pCan obtain a fractional order PID controller which approaches to an ideal reference model, and has robustness to the variation of gain, so the invention has the advantages ofAn effective fractional order PID controller parameter tuning method.

The above description is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily made by those skilled in the art within the technical scope of the present invention will be covered by the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (4)

1. A method for optimizing and setting parameters of a fractional order PID controller based on a closed loop reference model is applied to a closed loop feedback control structure of fractional order PID control, and the structure comprises a controlled object model G_p(s) and fractional order PID controller model G_c(s) whereink_p,k_i,k_dλ and μ are undetermined parameters of the fractional order PID controller; the method comprises the following steps:

S1：selecting ideal closed-loop reference modelSelecting a cut-off frequency omega of an ideal closed-loop reference model according to the control performance requirement of the system_cAnd order α;

s2: according to the transfer function expression of the closed-loop system, combining an ideal closed-loop system model H(s) and a fractional order PID controller model G_c(s) deriving ideal control object models

S3: obtaining unknown controlled object model G_p(s) frequency domain response characteristics ofAnd G_p(s) in ω -0 and ω - ω_xThe response is the same, i.e. the Nyquist curves of both are 0 and ω_xCrossing to obtain k_iAnd find k_p、k_dAt ω ═ ω_xIs a function of μ;

s4: identification of ideal form of unknown object by optimizationOf (2) isMaximum approach to the real object G in the cut-off frequency range_p(s) a frequency domain response indicator; establishing a frequency domain response error indicatorAnd optimizing error indexes when mu is more than 0 and less than 2And finally obtaining the parameters of the fractional order controller.

2. The closed-loop reference model-based fractional order PID controller parameter optimization tuning method of claim 1, characterized in that: in step S1, the control performance requirement of the system is a time domain index, and the time domain index may be overshoot, adjustment time, or peak time.

3. The closed-loop reference model-based fractional order PID controller parameter optimization tuning method of claim 1, wherein the step S2 specifically comprises:

based on the transfer function expression of the closed-loop system, the ideal closed-loop system model H(s) and the fractional order PID controller modelDeriving ideal control object modelsThe derivation is as follows:

${\tilde{G}}_p\left(s\right)=\frac{H}{G_c-{\mathrm{HG}}_c}=\frac{{\mathrm{\&omega;}}_c^{\mathrm{\&alpha;}}{s}^{\mathrm{\&lambda;}}}{(k_p{s}^{\mathrm{\&lambda;}}+k_i+k_d{s}^{\mathrm{\&mu;}+\mathrm{\&lambda;}})s^{\mathrm{\&alpha;}}}$

let λ be α, then

The lambda, mu can be a decimal number or an integer.

4. The closed-loop reference model-based fractional order PID controller parameter optimization tuning method of claim 1, wherein the step S3 specifically comprises:

obtaining unknown controlled object G_pFrequency domain response data of(s), let us assumeAnd G_p(s) in ω -0 and ω - ω_xHas the same frequency response, ω_xG can be selected as original system_p(s) crossover frequency of phase margin | G_p(jω_x)|＝1：

Firstly, selecting the lambda as α,meaningful at ω ═ 0, areThen according toIs provided withAccording tok_p、k_dAt ω ═ ω_xThe functional relationship between position and μ is:

$\left\{\begin{array}{c}{k}_d\left(\mathrm{\&mu;}\right)=-\frac{(dp+cq){\mathrm{\&omega;}}_x^{\mathrm{\&lambda;}}{k}_p+k_{i}q}{(pb+qa){\mathrm{\&omega;}}_x^t}\\ k_p\left(\mathrm{\&mu;}\right)=\frac{{\mathrm{bk}}_i(p^2+q^2)-(pb+qa){\mathrm{\&omega;}}_c^{\mathrm{\&lambda;}}}{(da-cb)(p^2+q^2){\mathrm{\&omega;}}_x^{\mathrm{\&lambda;}}}\end{array}\right.$

wherein,