CN105045093B - Stable Fractional Order PID parameter optimization method based on peak response index - Google Patents
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Abstract
The invention discloses a kind of stable Fractional Order PID parameter optimization method based on peak response index, including step：S1, the parametric stability region for obtaining Fractional Order PID Controller and the closed-loop system of controlled device composition, the parameter of Fractional Order PID Controller include：kp、ki、kd, λ and μ；S2, the parameter disaggregation for obtaining the Fractional Order PID Controller for meeting a default peak response index；S3, the parameter stability solution for meeting peak response is obtained by the common factor of parametric stability region and parameter disaggregation；S4, weight coefficient size is determined according to optimization aim；S5, one group of optimal solution is selected in parameter stability solution；S6：Fractional Order PID Controller is steadily controlled predetermined variable under optimal solution；Wherein, the function of optimization aim is：f(kp,ki,kd)=ξ1σ+ξ2ts, optimal solution includes：Make the functional value of optimization aim minimum, and optimal solution belongs to parameter disaggregation, σ is overshoot, tsFor regulating time, ξ1,ξ2For weight coefficient, and meet ξ1+ξ2=1.
Description
It is August in 2014 12 this application claims the applying date, is Application No. 201410395817.3, entitled based on maximum
The priority of the stable Fractional Order PID parameter optimization method of sensitivity index.
Technical field
The present invention relates to calculating, skill of deduction and calculation field, more particularly to a kind of stable fraction based on peak response index
Rank pid parameter Optimal tunning method.
Background technology
Fractional calculus has compared many advantages as a kind of tool of mathematical analysis with integer rank calculus.Fractional order
Calculus refers to differential, and the order of integration is no longer limited to integer, can be arbitrary either fraction.Due to differential, integration
The expansion of order scale, the descriptive power of fractional calculus are bigger than integer rank calculus.When description object is fraction in itself
When level is united, fractional calculus generally has more universal meaning than integer rank calculus.
Fractional order control theory is formed with the combination of fractional calculus and control theory.Subsequent fractional order control reason
Adjusted by applied to PID controller parameter, this is filled with new vitality to PID control.Fractional Order PID Controller is a kind of new
Controller, except kp,ki,kdThree parameters, it is more two adjustable parameters λ and μ so that controller design and parameter tuning
Difficulty increase, but simultaneously but also make controller parameter regulation it is more flexible, control effect is more preferable.
But most of research is concentrated on to Fractional Order PID Controller parameter tuning, for parameter optimization and and its again
Want, especially when being required for controller tracking performance, optimization is just even more important.
The content of the invention
The present invention is directed to deficiencies of the prior art, there is provided a kind of stabilization based on peak response index point
Number rank pid parameter optimization method, the present invention are achieved through the following technical solutions：
A kind of stable Fractional Order PID parameter optimization method based on peak response index, including step：
S11, the parametric stability region for obtaining Fractional Order PID Controller and the closed-loop system of controlled device composition, Fractional Order PID
The parameter of controller includes：kp、ki、kd, λ and μ；
S12, the parameter disaggregation for obtaining the Fractional Order PID Controller for meeting a default peak response index；
S13, obtained by the common factor of the parametric stability region and the parameter disaggregation and meet that the parameter of peak response is steady
Fixed solution；
S14, weight coefficient size is determined according to optimization aim；
S15, one group of optimal pid parameter solution is selected in the parameter stability solution；
S16：The Fractional Order PID Controller is put down under conditions of the optimal pid parameter solution to predetermined variable
Steady control；
Wherein, the function of optimization aim is：f(kp,ki,kd)=ξ1σ+ξ2ts, wherein, kpRepresent the Fractional Order PID control
The proportionality coefficient of device processed, kiRepresent the integral coefficient of the Fractional Order PID Controller, kdRepresent the Fractional Order PID Controller
Differential coefficient；
Optimal pid parameter solution includes：Make the functional value of optimization aim minimum, and optimal pid parameter solution belongs to parameter
Disaggregation, wherein, σ is overshoot, tsFor regulating time, ξ1,ξ2For weight coefficient, and meet ξ1+ξ2=1.
It is preferred that obtaining the stable region of the parameter of Fractional Order PID Controller using D segmentation principles, stable region is by real root
Border, without the region that roots of poverty border and compound radical border are border.
It is preferred that obtaining the parametric stability region of Fractional Order PID Controller includes：
The transmission function of Fractional Order PID Controller is：Wherein, 0 ＜ λ, μ ＜ 2；
The transmission function of controlled device is：
βn＞ Λ ＞ β1＞ β0>=0, αn＞ Λ ＞ α1＞ α0>=0, L are the pure delay time of time lag system；
The transmission function of the Fractional Order PID Controller and the closed-loop system of controlled device composition：
The transmission function of the transmission function of Fractional Order PID Controller and controlled device is substituted into the transmission letter of closed-loop system
Number, obtains proper polynomial：Ψ (s)=sλD(s)+(kpsλ+ki+kdsμ+λ)N(s)e-Ls, for one group of parameter：kp, ki, kd, λ,
μ, when the value of proper polynomial can be made to be 0, the input-output stability of Fractional Order PID Controller.
It is preferred that the parameter stability solution for meeting peak response is obtained, including：
By s=j ω,And ejx=cosx+jsinx substitutes into proper polynomial,Obtain one group of KP,KiSide on ω
Journey, in [ωmin,ωmax] interior to preset the value that step-length takes ω, obtain KP,KiParameter stability solution disaggregation.
It is preferred that peak response index is：MT≤γT, γT>=1, wherein, γTFor default peak response index.
The present invention also provides another stable Fractional Order PID parameter optimization method based on peak response index,
Fractional Order PID Controller is steadily controlled the speed of direct current generator, is comprised the following steps：
S21, the transmission function according to the Fractional Order PID ControllerWith the direct current
The transmission function of the speed of machineEstablish the closed-loop system of a direct current generatorAnd obtain the parametric stability region of the Fractional Order PID Controller；
S22, the parameter disaggregation for obtaining the Fractional Order PID Controller for meeting a default peak response index；
S23, obtained by the common factor of the parametric stability region and the parameter disaggregation and meet that the parameter of peak response is steady
Fixed solution；
S24, weight coefficient size is determined according to optimization aim；
S25, one group of optimal pid parameter solution is selected in the parameter stability solution；
S26：The Fractional Order PID Controller is under conditions of the optimal pid parameter solution to the direct current generator
Speed is steadily controlled；
Wherein, kpRepresent the proportionality coefficient of the Fractional Order PID Controller, kiRepresent the Fractional Order PID controller
Integral coefficient, kdRepresent the differential coefficient of the Fractional Order PID Controller, 0 ＜ λ,μ ＜ 2.
The function of the optimization aim is：f(kp,ki,kd)=ξ1σ+ξ2ts；
The optimal pid parameter solution includes：Make the functional value of the optimization aim minimum, and the optimal PID ginsengs
Number solutions belong to the parameter disaggregation, wherein, σ is overshoot, tsFor regulating time, ξ1,ξ2For weight coefficient, and meet ξ1+ξ2=1.
It is preferred that obtaining the stable region of the parameter of Fractional Order PID Controller using D segmentation principles, stable region is by real root
Border, without the region that roots of poverty border and compound radical border are border.
It is preferred that
Obtaining the parametric stability region of the Fractional Order PID Controller includes：
By described in the substitution of the transmission function of the transmission function of the Fractional Order PID Controller and the speed of the direct current generator
The transmission function of closed-loop system, obtains proper polynomial：Ψ (s)=sλD(s)+(kpsλ+ki+kdsμ+λ)N(s)e-Ls, for one group
Parameter：kp, ki, kd, λ, μ, when the value of proper polynomial can be made to be 0, the input-output stability of the Fractional Order PID Controller, institute
The input for stating Fractional Order PID Controller is the pre-set velocity value of the direct current generator, and the output of the Fractional Order PID Controller is
The pre-set current value of the direct current generator.
It is preferred that the parameter stability solution for meeting peak response is obtained, including：
By s=j ω,And ejx=cosx+jsinx substitutes into proper polynomial,Obtain one group of KP,KiSide on ω
Journey, in [ωmin,ωmax] interior to preset the value that step-length takes ω, obtain KP,KiParameter stability solution disaggregation.
It is preferred that peak response index is：MT≤γT, γT>=1, wherein, γTFor default peak response index.
The present invention can be met the Fractional Order PID Controller of sensitivity constraint, and controller can take into account it is multiple defeated
Enter tracking performance, this method is a kind of effective Fractional Order PID Controller parameter tuning method.
Brief description of the drawings
Shown in Fig. 1 is the structure chart of Fractional Order PID Controller and the closed-loop system of controlled device composition；
Shown in Fig. 2 is peak response index geometric representation schematic diagram；
Shown in Fig. 3 is the stable region schematic diagram of Fractional Order PID Controller；
Shown in Fig. 4 is the unit-step response schematic diagram that three parameters are randomly derived in stable region；
Shown in Fig. 5 is the Fractional Order PID Controller parameter disaggregation schematic diagram for meeting peak response index；
That shown in Fig. 6 is ξ1=0.4, ξ2Step response schematic diagram when=0.6；
That shown in Fig. 7 is ξ1=1, ξ2Step response schematic diagram when=0；
That shown in Fig. 8 is ξ1=0, ξ2Step response schematic diagram when=1.
Embodiment
Below with reference to the accompanying drawing of the present invention, clear, complete description is carried out to the technical scheme in the embodiment of the present invention
And discussion, it is clear that as described herein is only a part of example of the present invention, is not whole examples, based on the present invention
In embodiment, the every other implementation that those of ordinary skill in the art are obtained on the premise of creative work is not made
Example, belongs to protection scope of the present invention.
For the ease of the understanding to the embodiment of the present invention, make further by taking specific embodiment as an example below in conjunction with accompanying drawing
Illustrate, and each embodiment does not form the restriction to the embodiment of the present invention.
The structure of Fractional Order PID Controller that the present invention is directed to and the closed-loop system of controlled device composition as shown in figure 1,
Wherein：The input of r (t) representative fraction rank PID controllers, the output of u (t) representative fraction rank PID controllers, C (s) representative fractions
The transmission function of rank PID controller, v (t) represent controlled device, and G (s) represents the transmission function of controlled device, and y (t) is represented and closed
The output of loop system；
0 ＜ λ in formula,μ ＜ 2, kpRepresent the proportionality coefficient of the Fractional Order PID Controller, kiRepresent the Fractional Order PID
The integral coefficient of controller, kdRepresent the differential coefficient of the Fractional Order PID Controller；
Wherein, βn＞ L ＞ β1＞ β0>=0, αn＞ L ＞ α1＞ α0>=0, L are the pure delay time of time lag system；
The transmission function for the closed-loop system that Fractional Order PID Controller forms with controlled device is
Formula (1) and formula (2) are substituted into (3), the proper polynomial writing of closed-loop system：
Ψ (s)=sλD(s)+(kpsλ+ki+kdsμ+λ)N(s)e-Ls (4)
For one group of parameter (kp,ki,kd, λ, μ), should if it makes the root of characteristic equation Ψ (s)=0 have multiple real part
Subsystem is input-output stability.
By D segmentation principles (source：Serdar E H.An Algorithm for Stabilization of
Fractional-Order Time Delay Systems Using Fractional Order PID Controllers
[J]. IEEE Trans on Automatic Control,2007,52(4):1964-1969.), can be by (kp,ki,kd,λ,μ)
The parameter space Φ formed is divided into RRB, and CRB, IRB are the region D on border, and region D makes what subsystem was stablized comprising all
Point, the boundary definition of D segmentations are as follows：
Wherein,Real root border is represented,No roots of poverty border is indicated,Represent compound radical border.
S=j ω are substituted into proper polynomial (4), it is as follows to can obtain characteristic equation：
Ψ (j ω)=(j ω)λD(jω)+(kd(jω)λ+μ+kp(jω)λ+ki)N(jω)e-Ljω (7)
Principle is split according to D, to giving controlled system, fractional order PIλDμParameter space (the k of controllerp,ki,kd,λ,μ)
By possible plane (a), possible plane (b) and curved surface (c) are divided into some regions.
(a) real root border
According to first formula of formula (6), can obtain real root border is：ki=0.
(b) without roots of poverty border
(c) compound radical borderAccording toWith Euler's formula ejx=cosx+jsinx, generation
Entering characteristic equation (7) can obtain
Wherein：
By above formula (9) and (10), for given parameter kd, λ, μ, can be in (k when ω changes from 0 to ∞p,
ki) plane obtains compound radical border (CRB).
Peak response index such as Fig. 2 is represented：
Lemma 1 sets F (s)=NF(s)/DF(s) it is stable real (or multiple) rational of canonical, and DF(s) exponent number is α times
Power coefficient.Definition：
Then for a given γ ＞ 0, and if only if when meeting following two condition simultaneously, | | F (s) | |∞＜ γ：
1)|cα| ＜ γ | dα|；2)Hurwize is stable,
Wherein：cαAnd dαRespectively NFAnd DFS α power coefficients.
By sensitivity constraint index and lemma 1, corresponding N in lemma 1 can be obtainedFAnd DFRespectively：
NF(s)=N (s) e-Ls (19)
DF(s)=sλD(s)+(kpsλ+ki+kdsλ+μ)N(s)e-Ls (20)
By condition 2) formula can be obtained
Ask for the parameter disaggregation for meeting peak response index：
Wherein：f1(ω),f2(ω),f3(ω),f4(ω),f5(ω),f6(ω), h1(ω),h2(ω) is the same
By above formula (24), for given parameter kd, λ, μ, when ω changes from 0 to ∞, can obtain maximum sensitive
The Fractional Order PID Controller parameter disaggregation Q spent under indexθ。
Defining majorized function is：f(kp,ki,kd)=ξ1σ+ξ2ts(27)
Parametric optimization problem is：
Wherein：σ is overshoot, tsFor regulating time, ξ1,ξ2For weight coefficient, and meet ξ1+ξ2=1.By changing two power
The proportion of coefficient, different optimization purposes can be reached.
Parameter optimisation step：
(1) Fractional Order PID Controller parametric stability region is asked for；
(2) the parameter disaggregation Φ met under sensitivity constraint index is tried to achieve in stable region；
(3) required to determine weights ξ according to actual optimization1And ξ2Size；
(4) one group of optimal solution is obtained by solving Parametric optimization problem (28) in parameter disaggregation Φ.
DC generator speed is steadily controlled below with the present invention, using the speed of direct current generator as controlled device,
The transmission function of the relation of direct current generator electric current and speed can be expressed as：
Fractional Order PID Controller is designed, and meets sensitivity index：MT≤γT,γT=1.6.Now, for the institute of figure one
The control system loop shown, r (t) they are DC generator speed setting value, and u (t) is Fractional Order PID Controller output valve, that is,
Direct current generator given value of current value, y (t) are direct current generator values for actual speed.
Can be λ=1.2 in the hope of Fractional Order PID Controller parameter according to Fractional Order PID parameter tuning step, μ=0.62,
kdMaximum stable domain is obtained when=1.As shown in Figure 3：
Take three points, point A at random in stable region:kp=9.324, ki=2.425, point B:kp=12.532, ki=
5.562 point C:kp=18.674, ki=8.453, respectively obtain when DC generator speed is set as 160r/min with time T
Service chart, as shown in figure 4, it can be seen that control system is all stable.
Controller parameter disaggregation under being constrained further according to sensitivity index asks for strategy, obtains λ=1.5, μ=0.7, kd=
Meet that controller parameter disaggregation is as shown in Figure 5 during sensitivity constraint when 1.
Parameter optimization：
(1) optimization method according to this paper is concentrated to ginseng in the Fractional Order PID Controller parametric solution for meeting sensitivity index
Number optimizes, and chooses ξ1=0.4, ξ2=0.6.Direct current generator Fractional Order PID Controller proportionality coefficient is obtained, integral coefficient is most
Excellent parametric solution is kp=4.874, ki=0.851.Concentrated again in the parametric solution for meeting sensitivity constraint and take a point k at randomp=
12.532,ki=5.562.DC generator speed V is set as that system is with time T operation conditions such as Fig. 6 institutes during 160r/min
Show.It can be seen that before the closed-loop system step response tracking performance that optimized parameter is taken off is substantially better than optimization.
(2) ξ is worked as1=1, ξ2When=0, direct current generator Fractional Order PID Controller proportionality coefficient after now being optimized, integration
Coefficient is kp=4.325, ki=0.438.Step response is as shown in Figure 7.
(3) ξ is worked as1=0, ξ2Direct current generator Fractional Order PID Controller proportionality coefficient, integral coefficient are after being optimized when=1
kp=12.662, ki=0.454, now DC generator speed be set as system during 160r/min with time T operation conditions as schemed
Shown in 8.
The present invention is studied the parameter tuning of Fractional Order PID Controller, is given one kind and is based on peak response
The Fractional Order PID Controller parameter Optimal tunning method of index.This method first asks for Fractional Order PID Controller according to D split plot designs
Parametric stability region, then the parameter disaggregation being met under sensitivity constraint index, finally by overshoot and regulating time as excellent
Change target function parameter is optimized to obtain optimized parameter solution.End product shows that this method can be met sensitivity about
The Fractional Order PID Controller of beam, and controller can take into account multiple input tracking performances, and this method is a kind of effective fractional order
PID controller parameter setting method.
The foregoing is only a preferred embodiment of the present invention, but protection scope of the present invention be not limited thereto,
Any one skilled in the art the invention discloses technical scope in, the change or replacement that can readily occur in,
It should all be included within the scope of the present invention.Therefore, protection scope of the present invention should be with scope of the claims
It is defined.
Claims (10)
1. a kind of stable Fractional Order PID parameter optimization method based on peak response index, it is characterised in that including following step
Suddenly：
S11, the parametric stability region for obtaining Fractional Order PID Controller and the closed-loop system of controlled device composition, the Fractional Order PID
The parameter of controller includes：kp、ki、kd, λ and μ；
S12, the parameter disaggregation for obtaining the Fractional Order PID Controller for meeting a default peak response index；
S13, the parameter stability solution for meeting peak response is obtained by the common factor of the parametric stability region and the parameter disaggregation；
S14, weight coefficient size is determined according to optimization aim；
S15, one group of optimal pid parameter solution is selected in the parameter stability solution；
S16：The Fractional Order PID Controller is steadily controlled under conditions of the optimal pid parameter solution to predetermined variable
System；
Wherein, the function of the optimization aim is：f(kp,ki,kd)=ξ1σ+ξ2ts, wherein, kpRepresent the Fractional Order PID control
The proportionality coefficient of device, kiRepresent the integral coefficient of the Fractional Order PID Controller, kdRepresent the micro- of the Fractional Order PID Controller
Divide coefficient；
The optimal pid parameter solution includes：Make the functional value of the optimization aim minimum, and the optimal pid parameter solution
Belong to the parameter disaggregation, wherein, σ is overshoot, tsFor regulating time, ξ1,ξ2For weight coefficient, and meet ξ1+ξ2=1.
2. the stable Fractional Order PID parameter optimization method according to claim 1 based on peak response index, its feature
It is, the stable region of the parameter of Fractional Order PID Controller is obtained using D segmentation principles, and the stable region is by real root border, nothing
Roots of poverty border and the region that compound radical border is border.
3. the stable Fractional Order PID parameter optimization method according to claim 1 based on peak response index, its feature
It is, obtaining the parametric stability region of Fractional Order PID Controller includes：
The transmission function of the Fractional Order PID Controller is：Wherein, 0 ＜ λ, μ ＜ 2；
The transmission function of controlled device is：
βn＞ Λ ＞ β1＞ β0>=0, αn＞ Λ ＞ α1＞ α0>=0, L are the pure delay time of time lag system；
The transmission function for the closed-loop system that the Fractional Order PID Controller forms with the controlled device is：
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The transmission function of the transmission function of the Fractional Order PID Controller and the controlled device is substituted into the closed-loop system
Transmission function, obtain proper polynomial：Ψ (s)=sλD(s)+(kpsλ+ki+kdsμ+λ)N(s)e-Ls, for one group of parameter：kp,
ki, kd, λ, μ, when the value of proper polynomial can be made to be 0, the input-output stability of the Fractional Order PID Controller.
4. the stable Fractional Order PID parameter optimization method according to claim 3 based on peak response index, its feature
It is, the acquisition meets the parameter stability solution of peak response, including：
By s=j ω,And ejx=cosx+jsinx substitutes into the proper polynomial,Obtain one group of KP,KiSide on ω
Journey, in [ωmin,ωmax] interior to preset the value that step-length takes ω, obtain KP,KiParameter stability solution disaggregation.
5. the stable Fractional Order PID parameter optimization method according to claim 1 based on peak response index, its feature
It is, the peak response index is：MT≤γT, γT>=1, wherein, γTFor the default peak response index.
A kind of 6. stable Fractional Order PID parameter optimization method based on peak response index, it is characterised in that Fractional Order PID
Controller is steadily controlled the speed of direct current generator, is comprised the following steps：
S21, the transmission function according to the Fractional Order PID ControllerWith the direct current generator
Speed transmission functionEstablish the closed-loop system of a direct current generatorAnd obtain the parametric stability region of the Fractional Order PID Controller；
S22, the parameter disaggregation for obtaining the Fractional Order PID Controller for meeting a default peak response index；
S23, the parameter stability solution for meeting peak response is obtained by the common factor of the parametric stability region and the parameter disaggregation；
S24, weight coefficient size is determined according to optimization aim；
S25, one group of optimal pid parameter solution is selected in the parameter stability solution；
S26：The Fractional Order PID Controller is under conditions of the optimal pid parameter solution to the speed of the direct current generator
Steadily controlled；
Wherein, kpRepresent the proportionality coefficient of the Fractional Order PID Controller, kiRepresent the integration system of the Fractional Order PID Controller
Number, kdRepresent the differential coefficient of the Fractional Order PID Controller, 0 ＜ λ,μ ＜ 2
The function of the optimization aim is：f(kp,ki,kd)=ξ1σ+ξ2ts；
The optimal pid parameter solution includes：Make the functional value of the optimization aim minimum, and the optimal pid parameter solution
Belong to the parameter disaggregation, wherein, σ is overshoot, tsFor regulating time, ξ1,ξ2For weight coefficient, and meet ξ1+ξ2=1.
7. the stable Fractional Order PID parameter optimization method according to claim 6 based on peak response index, its feature
It is, the stable region of the parameter of Fractional Order PID Controller is obtained using D segmentation principles, and the stable region is by real root border, nothing
Roots of poverty border and the region that compound radical border is border.
8. the stable Fractional Order PID parameter optimization method according to claim 6 based on peak response index, its feature
It is, obtaining the parametric stability region of the Fractional Order PID Controller includes：
The transmission function of the transmission function of the Fractional Order PID Controller and the speed of the direct current generator is substituted into the closed loop
System transter, obtain proper polynomial：Ψ (s)=sλD(s)+(kpsλ+ki+kdsμ+λ)N(s)e-Ls, L is time lag system
Pure delay time, for one group of parameter：kp, ki, kd, λ, μ, when the value of proper polynomial can be made to be 0, the Fractional Order PID control
The input-output stability of device processed, the pre-set velocity value inputted as the direct current generator of the Fractional Order PID Controller, described point
The output of number rank PID controller is the pre-set current value of the direct current generator.
9. the stable Fractional Order PID parameter optimization method according to claim 8 based on peak response index, its feature
It is, the acquisition meets the parameter stability solution of peak response, including：
By s=j ω,And ejx=cosx+jsinx substitutes into the proper polynomial,Obtain one group of KP,KiSide on ω
Journey, in [ωmin,ωmax] interior to preset the value that step-length takes ω, obtain KP,KiParameter stability solution disaggregation.
10. the stable Fractional Order PID parameter optimization method according to claim 6 based on peak response index, it is special
Sign is that the peak response index is：MT≤γT, γT>=1, wherein, γTFor the default peak response index.
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| CN106338913A (en) * | 2016-11-04 | 2017-01-18 | 河北省科学院应用数学研究所 | Fractional-order PID control design method based on phase margin and cutoff frequency |
| CN106681150A (en) * | 2017-01-19 | 2017-05-17 | 华侨大学 | Fractional-order PID controller parameter optimizing and setting method based on closed-loop reference model |
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| CN110090728B (en) * | 2019-01-24 | 2020-11-06 | 济南大学 | Method, device and equipment for controlling feeding amount in cement raw material vertical mill |
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