Method For Designing PID Controller (as amended)

Abstract

Disclosed is a method for designing a PID controller, having a control model

[00001]$C\left(s\right)=K_P\left(1+\frac{K_I}{S^{\lambda }}+K_D{S}^u\right),$

wherein K.sub.D=aK.sub.I, and u=bλ, a and b are proportional coefficients, the control model is reset as

[00002]$C\left(s\right)=K_P\left(1+\frac{K_I}{S^{\lambda }}+{\mathrm{aK}}_I{S}^{b\lambda }\right),$

a transfer function of a controlled object in a control system is set as

[00003]$G\left(s\right)=\frac{K}{s^3+{\tau }_1{s}^2+{\tau }_{2}s}.$

The method comprises selecting a cut-off frequency ω.sub.c and a phase margin φ.sub.m of the control system; obtaining values of the proportional coefficients a and b, according to an optimal proportion model of control model parameters of the fractional order PID controller, and according to the cut-off frequency ω.sub.c and the phase margin φ.sub.m; calculating amplitude information and phase information of the transfer function at the cut-off frequency ω.sub.c; obtaining two equations related to an integral gain K.sub.I and a fractional order λ; solving the integral gain K.sub.I and the fractional order λ; solving a differential gain K.sub.D and a fractional order u; and calculating a proportional gain K.sub.P. According to the invention, by establishing a proportional relationship between the integral gain K.sub.I and the differential gain K.sub.D of the fractional order PID controller as well as a proportional relationship between the integral order λ and the differential order u, the freedom degree of parameters of the fractional order PID controller and consequently the difficulty in parameter setting are reduced.

Claims

1. A method for designing a PID controller, comprising: setting a control model of the PID controller, as equation 2: $\begin{array}{cc}C\left(s\right)=K_p\left(1+\frac{K_I}{s^{\lambda }}+K_D{s}^u\right),& \mathrm{equation}2\end{array}$ wherein, K.sub.P is a proportional gain, K.sub.I is an integral gain, K.sub.D is a differential gain, λ is a fractional-order, u is a fractional-order, and s is a Laplace operator; resetting the control model of the PID controller by setting K.sub.D=aK.sub.I, and u=bλ in equation 2, wherein a and b are proportional coefficients, as equation 3: $\begin{array}{cc}C\left(s\right)=K_p\left(1+\frac{K_I}{s^{\lambda }}+{\mathrm{aK}}_I{s}^{b\lambda }\right),& \mathrm{equation}3\end{array}$ setting a transfer function of a controlled object in a control system, as equation 4: $\begin{array}{cc}G\left(s\right)=\frac{K}{s^3+{\tau }_1{s}^2+{\tau }_{2}s},& \mathrm{equation}4\end{array}$ wherein τ.sub.1, τ.sub.2, and K are model parameters of the object; and the method further comprising the following steps of: step 1: selecting a cut-off frequency ω.sub.c and a phase margin φ.sub.m of the control system; step 2: obtaining values of the proportional coefficients a and b, according to an optimal proportion model of control model parameters establishing the fractional order PID controller, and according to the cut-off frequency ω.sub.c and the phase margin φ.sub.m of the control system; step 3: calculating amplitude information and phase information of the transfer function at the cut-off frequency ω.sub.c, respectively, according to equation 5 and equation 6: $\begin{array}{cc}.Math.G\left(j\omega \right).Math.=\frac{K}{\sqrt{{A\left({\omega }_c\right)}^2+{B\left({\omega }_c\right)}^2}},\mathrm{and}& \mathrm{equation}5\\ \mathrm{Arg}.Math.G\left(j{\omega }_c\right).Math.=-\arctan \left[\frac{B\left({\omega }_c\right)}{A\left({\omega }_c\right)}\right],& \mathrm{equation}6\end{array}$ wherein, A(ω)=−τ.sub.1ω.sup.2 and B(ω)=τ.sub.2ω−ω.sup.3; step 4: obtaining two equations related to the integral gain K.sub.I and the fractional-order λ according to the proportional coefficients a and b obtained in the step 2: $\begin{array}{cc}{K}_I=\hspace{1em}\hspace{1em}\frac{-M}{M{\omega }_c^{-\lambda }\cos \left(\frac{\mathrm{\lambda \pi }}{2}\right)+\mathrm{aM}{\omega }_c^{b\lambda }\cos \left(\frac{b\mathrm{\lambda \pi }}{2}\right)+\mathrm{aN}{\omega }_c^{b\lambda }\sin \left(\frac{b\mathrm{\lambda \pi }}{2}\right)-N{\omega }_c^{-\lambda }\sin \left(\frac{\mathrm{\lambda \pi }}{2}\right)},& \mathrm{equation}7\\ \mathrm{and}& \\ Q_2{K}_I^2+Q_1{K}_I+Z=0& \mathrm{equation}8\end{array}$ wherein, M=A(ω.sub.c)tan(−π+φ.sub.m)+B(ω.sub.c) and N=B(ω.sub.c)tan(−π+φ.sub.m)−A(ω.sub.c), in equation 7, and $Q_2=\frac{a\left(1+b\right)\lambda }{{\omega }_c^{1+\left(1-b\right)\lambda }}\sin \left(\frac{\left(b+1\right)\mathrm{\lambda \pi }}{2}\right)+2\mathrm{aZ}{\omega }_c^{\left(b-1\right)\mathrm{\lambda \pi }}\cos \left(\frac{\left(b+1\right)\mathrm{\lambda \pi }}{2}\right)+a^{2}Z{\omega }_c^{2b\lambda }+Z{\omega }_c^{-2\lambda },Q_1=\mathrm{ab}{\mathrm{\lambda \omega }}_c^{b\lambda -1}\sin \left(\frac{b\mathrm{\lambda \pi }}{2}\right)+\lambda {\omega }_c^{-\lambda -1}\sin \left(\frac{\mathrm{\lambda \pi }}{2}\right)+2\mathrm{aZ}{\omega }_c^{b\lambda }\cos \left(\frac{b\mathrm{\lambda \pi }}{2}\right)+2Z{\omega }_c^{-\lambda }\cos \left(\frac{\mathrm{\lambda \pi }}{2}\right)\mathrm{and},Z={.Math.\frac{d\left[\mathrm{Arg}\left[G\left(j\omega \right)\right]\right]}{d\omega }.Math.}_{\omega ={\omega }_c}\mathrm{in}\mathrm{equation}8;$ step 5: solving the integral gain K.sub.I and the fractional-order λ according to equation 7 and equation 8; step 6: solving the differential gain K.sub.D and the fractional-order u according to relationships K.sub.D=aK.sub.I and u=bλ; and step 7: calculating the proportional gain K.sub.P according to equation 9 as follows: $\begin{array}{cc}{K}_P=\frac{\sqrt{{A\left({\omega }_c\right)}^2+{B\left({\omega }_c\right)}^2}}{K\sqrt{{P\left({\omega }_c\right)}^2+{Q\left({\omega }_c\right)}^2}},& \mathrm{equation}9\\ \mathrm{wherein},P\left(\omega \right)=1+K_1{\omega }^{-\lambda }\cos \left(\frac{\mathrm{\lambda \pi }}{2}\right)+K_D{\omega }^u\cos \left(\frac{u\pi }{2}\right)\mathrm{and}Q\left(\omega \right)=K_D{\omega }^u\cos \left(\frac{u\pi }{2}\right)-K_I{\omega }^{-\lambda }\sin \left(\frac{\mathrm{\lambda \pi }}{2}\right).& \end{array}$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0027] In order to illustrate the embodiments of the invention more clearly, the drawings to be used in the description of the embodiments will be briefly described below. Obviously, the described drawings merely show some of the embodiments of the invention, rather than all the embodiments. Those skilled in the art can envisage other embodiments and drawings based on these drawings without going through any creative effort.

[0028] FIG. 1 is a flow chart of a method according to the present disclosure.

DETAILED DESCRIPTION

[0029] The concepts, the specific structures and the technical effects produced by the invention will be described in detail in conjunction with the embodiments and the accompanying drawings, for a reader to sufficiently understand the objects, features and effects of the invention. Obviously, the described embodiments are merely some of the embodiments of the invention. Other embodiments that may be envisaged by those skilled in the art without going through any creative effort shall all fall within the protection scope of the invention.

[0030] Referring to FIG. 1, in the present disclosure a method for designing a PID controller is disclosed. The PID controller has a control model which is set as equation 2:

[00013]$\begin{array}{cc}C\left(s\right)=K_P\left(1+\frac{K_I}{s^{\lambda }}+K_D{s}^u\right),& \mathrm{equation}2\end{array}$

[0031] wherein, K.sub.P is a proportional gain, K.sub.I is an integral gain, K.sub.D is a differential gain, λ is a fractional-order, u is a fractional-order, and s is a Laplace operator;

[0032] making K.sub.D=aK.sub.I, and u=bλ in equation 2, wherein a and b are proportional coefficients, and the control model of the PID controller can be modified as equation 3:

[00014]$\begin{array}{cc}C\left(s\right)=K_p\left(1+\frac{K_I}{s^{\lambda }}+{\mathrm{aK}}_I{s}^{b\lambda }\right),& \mathrm{equation}3\end{array}$

[0033] a transfer function of a controlled object in a control system is set as equation 4:

[00015]$\begin{array}{cc}G\left(s\right)=\frac{K}{s^3+{\tau }_1{s}^2+{\tau }_{2}s},& \mathrm{equation}4\end{array}$

[0034] wherein τ.sub.1, τ.sub.2, and K are model parameters of the object; and

[0035] the method comprises the following steps of:

[0036] step 1: selecting a cut-off frequency ω.sub.c and a phase margin φ.sub.m of the control system;

[0037] step 2: obtaining the values of the proportional coefficients a and b, according to an optimal proportion model of the control model parameters of the fractional order PID controller, and according to the cut-off frequency ω.sub.c and the phase margin φ.sub.m;

[0038] step 3: calculating amplitude information and phase information of the transfer function at the cut-off frequency co, respectively according to equation 5 and equation 6, wherein equation and equation 6 are as follows:

[00016]$\begin{array}{cc}.Math.G\left(j{\omega }_c\right).Math.=\frac{K}{\sqrt{{A\left({\omega }_c\right)}^2÷{B\left({\omega }_c\right)}^2}}& \mathrm{equation}5\\ \mathrm{Arg}.Math.G\left(j{\omega }_c\right).Math.=-\arctan \left[\frac{B\left({\omega }_c\right)}{A\left({\omega }_c\right)}\right]& \mathrm{equation}6\end{array}$

[0039] wherein, A(ω)=−τ.sub.1ω.sup.2 and B(ω)=τ.sub.2ω−ω.sup.3;

[0040] step 4: obtaining two equations related to the integral gain K.sub.I and the fractional order λ according to the proportional coefficients a and b obtained in step 2, which are respectively shown as equation 7 and equation 8:

[00017]$\begin{array}{cc}{K}_I=\hspace{1em}\hspace{1em}\frac{-M}{\begin{array}{c}M{\omega }_c^{-\lambda }\cos \left(\frac{\mathrm{\lambda \pi }}{2}\right)+\mathrm{aM}{\omega }_c^{b\lambda }\cos \left(\frac{b\mathrm{\lambda \pi }}{2}\right)+\\ \mathrm{aN}{\omega }_c^{b\lambda }\sin \left(\frac{b\mathrm{\lambda \pi }}{2}\right)-N{\omega }_c^{-\lambda }\sin \left(\frac{\mathrm{\lambda \pi }}{2}\right)\end{array}},\mathrm{and}& \mathrm{equation}7\\ Q_2{K}_I^2+Q_1{K}_I+Z=0& \mathrm{equation}8\\ & \end{array}$

[0041] wherein, M=A(ω.sub.c)tan(−π+φ.sub.m)+B(ω.sub.c) and N=B(ω.sub.c)tan(−π+φ.sub.m)−A(ω.sub.c) in equation 7, and

[00018]${Q_2=\frac{a\left(1+b\right)\lambda }{{\omega }_c^{1+\left(1-b\right)\lambda }}\sin \left(\frac{\left(+1\right)\mathrm{\lambda \pi }}{2}\right)+2\mathrm{aZ}{\omega }_c^{\left(b-1\right)\lambda }\cos \left(\frac{\left(b+1\right)\mathrm{\lambda \pi }}{2}\right)+a^{2}Z{\omega }_c^{2b\lambda }+Z{\omega }_c^{-2\lambda },Q_1=\mathrm{ab}{\mathrm{\lambda \omega }}_c^{b\lambda -1}\sin \left(\frac{\mathrm{b\lambda \pi }}{2}\right)+{\mathrm{\lambda \omega }}_c^{-\lambda -1}\sin \left(\frac{\mathrm{\lambda \pi }}{2}\right)+2\mathrm{aZ}{\omega }_c^{b\lambda }\cos \left(\frac{b\mathrm{\lambda \pi }}{2}\right)+2Z{\omega }_c^{-\lambda }\cos \left(\frac{\mathrm{\lambda \pi }}{2}\right)\mathrm{and}Z=\frac{d\left[\mathrm{Arg}\left[G\left(j\omega \right)\right]\right]}{d\omega }.Math.}_{\omega ={\omega }_c}\mathrm{in}\mathrm{equation}8;$

[0042] step 5: solving the integral gain K.sub.I and the fractional order λ according to equation 7 and equation 8;

[0043] step 6: solving the differential gain K.sub.D and the fractional order u according to the relationships K.sub.D=aK.sub.I and u=bλ; and

[0044] step 7: calculating the proportional gain K.sub.P according to equation 9 as follows:

[00019]$\begin{array}{cc}{K}_P=\frac{\sqrt{{A\left({\omega }_c\right)}^2+{B\left({\omega }_c\right)}^2}}{K\sqrt{{P\left({\omega }_c\right)}^2+{Q\left({\omega }_c\right)}^2}},& \mathrm{equation}9\\ \mathrm{wherein},P\left(\omega \right)=1+K_1{\omega }^{-\lambda }\cos \left(\frac{\mathrm{\lambda \pi }}{2}\right)+K_D{\omega }^u\cos \left(\frac{u\pi }{2}\right)\mathrm{and}Q\left(\omega \right)=K_D{\omega }^u\cos \left(\frac{u\pi }{2}\right)-K_I{\omega }^{-\lambda }\sin \left(\frac{\mathrm{\lambda \pi }}{2}\right).& \end{array}$

[0045] Specifically, according to the present disclosure, by establishing a proportional relationship between the integral gain K.sub.I and the differential gain K.sub.D of the fractional-order PID controller, as well as a proportional relationship between the integral order λ and the differential order u, the freedom degree of parameters of the fractional-order PID controller is reduced, and consequently the difficulty in parameter setting is reduced.

[0046] In order to more sufficiently explain the specific process of the method for establishing the optimal proportion model of parameters of the PID controller according to the present disclosure, hereafter a parameter setting process of the fractional-order PID controller applied to a permanent magnet synchronous motor servo system is described.

[0047] A transfer function of a speed loop control object of the servo system is set as

[00020]$G\left(s\right)=\frac{47979.2573}{s^3+127.38s^2+995.678s},$

a cut-off frequency is set as ω.sub.c=60 rad/s, a phase margin is set as φ.sub.m=60 deg, a proportional coefficient is set as a=7.553×10.sup.−4, and a proportional coefficient is set as b=1.253, according to the actual application conditions. Going through each of the steps of the method described above, a control model of a PID controller of the servo system obtained by calculations is shown as follows:

[00021]$C\left(s\right)=11.4944\left(1+\frac{13.0838}{s^{0.8489}}+0.0099s^{1.0637}\right)$

[0048] The foregoing describes the preferred embodiments of the invention in detail, but the invention is not limited to the embodiments, those skilled in the art can also make various equal modifications or replacements without departing from the spirit of the invention, and these equal modifications or replacements shall all fall within the scope limited by the claims of the invention.