CN105974798B - The multi-model fractional order weight estimation function control method of electric furnace - Google Patents
The multi-model fractional order weight estimation function control method of electric furnace Download PDFInfo
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Abstract
The invention discloses a kind of multi-model fractional order weight estimation function control methods of electric furnace.The operating temperature interval division of electric furnace is first several subintervals by the present invention, then its fractional model is established on its corresponding subinterval, recycle Oustaloup approximation method line that fractional model is converted to the integer model of high-order, the controller in each section is designed using PREDICTIVE CONTROL functional based method, the proportionality coefficient of each model is finally established according to the error between model and practical object, to obtain the controller input quantity of Multi-model MPCA.The present invention improves the control performance of system, while promoting utilization of the model predictive control method in new fractional-order system by nonlinear model conversion before for linear partial model by establishing the local state spatial model of controlled device.
Description
Technical field
The invention belongs to fields of automation technology, are related to a kind of multi-model fractional order weight estimation function control of electric furnace
Method processed.
Background technique
Temperature control system is a kind of important process procedure in Industry Control, since temperature controlled precision produces production
The quality and quality of product have important influence, and the precision for improving control is a target that we increasingly pursue.Electric furnace
It is most common device in industrial temperature control, however electric furnace this object shows strong nonlinearity, big condition range, greatly
Characteristics such as delay, and conventional control method is both for linear system, thus for this nonlinear system of electric furnace by
Concern is arrived.Although in recent years, to it is nonlinear research have been achieved for many achievements, nonlinear system modeling
Precision is but difficult to hinder the development of controller to reach.Electric furnace is to generate heat by current flowing resistance silk to control
Temperature in furnace processed.Since the variation of temperature shows non-linear, but its subrange can be approximated to be linear characteristic, therefore can
With by the division to temperature range, so that original non-linear behavior approximate transform is linear characteristic.Approximate linear special
It generallys use integer model in property to control electric furnace, since the integer model of approximate local characteristics cannot be more
Add the characteristic for accurately embodying its part, therefore we can adjust the accuracy of its part with fractional model.
Summary of the invention
The present invention is difficult to set up effective model primarily directed to Wide Range range in Industry Control, proposes one kind
Multi-fractional order model-weight forecast Control Algorithm, design apply to the temperature control of electric furnace.The advantages of this control, is
More industrial objects are utilized and establish model in different interval ranges, original single model is carried out according to different operating conditions
It divides, has obtained the fractional model under different operating conditions, Oustaloup approximation method line is recycled to convert fractional model
For the integer model of high-order, controller finally is designed using the method for Predictive function control.This method is in actual application
Compared with traditional single model, reduce the difficulty for establishing nonlinear model, the essence of model is improved using fractional model
Exactness.
The operating temperature interval division of electric furnace is first several subintervals by the present invention, then in its corresponding sub-district
Between on establish its fractional model, recycle Oustaloup approximation method line that fractional model is converted to the integer mould of high-order
Type designs the controller in each section using PREDICTIVE CONTROL functional based method, finally according to the error between model and practical object
The proportionality coefficient of each model is established, to obtain the controller input quantity of Multi-model MPCA.
The step of the method for the present invention:
1 establishes the multi-model of controlled device.
1.1 carry out i equal part working region according to the temperature region range of work, and i is the number of equal part to be carried out.
1.2 acquire the real-time step response data of real process object in the section of each equal part, are built using the data
The fractional order transfer function model M of vertical controlled devicej, form is as follows:
Wherein, MjFor j-th of submodel, α1, jFor the differential order of j-th of system, T1,j,T2, jFor corresponding coefficient, S is
Laplace transform operator, Km,jFor model scale gain coefficient, τm,jFor the lag time constant of model.
1.3 differential operator SαIt can be expressed as follows by Oustaloup approximation method:
Wherein, α is fractional order differential order, and 0 < α < 1, N are selected apparent order, Kα=Wh α, Wn'=WbWu (2n -1-α)/N, Wn=WbWu (2n-1+α)/N,WhAnd WbRespectively it is fitted the upper limit value and lower limit value of frequency.
1.4 according to Oustaloup approximation method in step 1.3, is approximately integer height by the fractional model in step 1.2
Rank model, by sampling time TSDiscretization obtains following form down plus after zero-order holder:
Y (k)=- a1y(k-1)-a2y(k-2)-…-aly(k-l)+b1u(k-1-d)+
b2u(k-2-d)+…+blu(k-l-d)
Wherein, aj, bj(j=1,2 ..., l) is the coefficient obtained after discrete approximation, the time lag d=of real process
τ/TS, l is the length of discrete model, and y (k) is the model output of the real process object at k moment, and u (k-d-1) is real process
Input value of the object at the k-d-1 moment；
In order to reduce error by obtaining following form to model progress single order backward difference:
△ y (k)=- a1△y(k-1)-a2△y(k-2)-…-al△y(k-l)+b1△u(k-1-d)+
△b2u(k-2-d)+…+△blu(k-l-d)
The state variable of 1.5 selecting systems is as follows:
△Xm(k)=[△ y (k), △ y (k-1) ..., △ y (k-l), △ u (k-1) ..., △ u (k-l+1-d)]T
In conjunction with step 1.4, the state-space model of controlled device is obtained, form is as follows:
△Xm(k+1)=Am△Xm(k)+Bm△u(k)
△ym(k+1)=Cm△Xm(k+1)
Wherein, T is the transposition symbol of matrix, △ Xm(k) dimension is (2l+d-1) × 1；
Bm=[0 ... 010 ... 0]T
Cm=[1 0 ... 00 ... 0]
The design of 2 prediction function controllers
2.1 seek the margin of error at current time:
E (k)=y (k)-r (k)
E (k) is the error at current time, and y (k) is the measured value of current time object, and r (k) is estimating for current time
Value.
By the error at current time, the error of estimating system model and practical object after P step
△ e (k+1)=△ ym(k+1)-△r(k+1)
=CmAm△Xm(k)+CmBm△um(k)-△r(k+1)
△ e (k+2)=△ ym(k+2)-△r(k+2)
=Am 2△Xm(k)+AmBm△u(k)+Bm△um(k+1)-△r(k+2)
△ e (k+P)=△ y (k+p)-△ r (k+p)
=Am p△Xm(k)+Am p-1Bm△u(k)+…+Bm△um(k+p)-△r(k+p)
Wherein △ e (k+p) is the prediction of the error after k+p step, and △ r (k+p) indicates the reference locus of k+p step adjacent moment
Difference.
2.2 choose the reference locus r (k+i) and objective function J of Predictive function controlpfc
Jfpc=min [r (k+P)-y (k+P)]2=min [e (k+P)]2
R (k+i)=βiyp(k)+(1-βi)c(k)
Wherein c (k) is setting value, and y (k+P) is to estimate to system model output at the k+P moment, and β is the soft of reference locus
Change coefficient, the reference locus that r (k+i) exports system.
2.3 Predictive function controls be with control input structure it is related, choose basic function be jump function can obtain:
U (k+i)=u (k), (i=1,2 ..., P)
Minimum value by seeking objective function can obtain:
U (k)=- M-1[y(k)-r(k)+N△x(k)+Mu(k-1)-△r]
Wherein:
M=CmAm P-1Bm+CmAm P-2Bm+…+CmBm
N=CmAm P+CmAm P-1+…+CmAm
3. the weighting coefficient of multi-model
3.1 calculate current time submodel MjModel export yj(t), and according to the mould for the submodel at this time being calculated
The deviation of the reality output of type output and current time system.
ej(t)=| yout(t)-yj(t) |, j=1,2 ... i.
Wherein yout(t) reality output for being system output channel j, ej(t) j-th of submodel of current time and reality are represented
The deviation of border output.
3.2 calculate each submodel weight coefficient.
Wherein wj(t) weighting coefficient of j-th of submodel of current time, e are indicatedi(t-k) error in history is indicated.
Therefore the input quantity of current controller can indicate are as follows:
U (t)=w1(t)u1(t)+w2(t)u2(t)+…+wi(t)ui(t)
3.3 solve the multi-model predictive function control device of fractional order in subsequent time according to the method in step 2.1 to 3.2
Control amount, then acted on controlled device, circuit sequentially operation and go down.
The invention proposes a kind of furnace temp controlling parties of multiple model predictive control based on state space fractional order
Method, this method expand to the Multi model Predictive Controllers of integer rank in the Multi model Predictive Controllers of fractional order.Pass through
The local state spatial model for establishing controlled device is improved by nonlinear model conversion before for linear partial model
The control performance of system, while promoting utilization of the model predictive control method in new fractional-order system.
Specific embodiment
Central scope of the invention: the region entirely to work is divided into several sub-regions according to some way, every
Sub-regions establish its corresponding fractional model and are combining Predictive function control.To original nonlinear model conversion
For linear fractional rank model, the precision of model avoids nonlinear complexity, and effective solution is in Industrial Engineering
Model accuracy is low and influences product quality problem.It comprises the concrete steps that:
1 establishes the Multi-model MPCA of electric furnace.
1.1 carry out i equal part its working region according to the operating temperature regional scope of electric furnace, and i is equal part to be carried out
Number.
1.2 acquire the real-time step response data of electric furnace in the section of each equal part, establish electricity using the data
The fractional order transfer function model M of heating furnacej, form is as follows:
Wherein, MjFor j-th of submodel, α1, jFor the differential order of j-th of submodel, T1,j,T2, jFor corresponding coefficient, S
For Laplace transform operator, Km,jFor model scale gain coefficient, τm,jFor the lag time constant of model.
1.3 differential operator SαIt can be expressed as follows by Oustaloup approximation method:
Wherein, α is fractional order differential order, and 0 < α < 1, N are selected apparent order, Kα=Wh α, Wn'=WbWu (2n -1-α)/N, Wn=WbWu (2n-1+α)/N,WhAnd WbRespectively it is fitted the upper limit value and lower limit value of frequency.
1.4 according to Oustaloup approximation method in step 1.3, is approximately integer height by the fractional model in step 1.2
Rank model, by sampling time TSDiscretization obtains following form down plus after zero-order holder:
Y (k)=- a1y(k-1)-a2y(k-2)-…-aly(k-l)+b1u(k-1-d)+
b2u(k-2-d)+…+blu(k-l-d)
Wherein, aj, bj(j=1,2 ..., Ls) it is the coefficient obtained after discrete approximation, the time lag d=of electric furnace
τ/TS, l is the length of discrete model, and y (k) is the output of the electric furnace model at k moment, and u (k-d-1) is electric furnace in k-
The input value at d-1 moment；
By obtaining following form to model progress single order backward difference:
△ y (k)=- a1△y(k-1)-a2△y(k-2)-…-al△y(k-l)+b1△u(k-1-d)+
△b2u(k-2-d)+…+△blu(k-l-d)
The state variable of 1.5 selecting systems is as follows:
△Xm(k)=[△ y (k), △ y (k-1) ..., △ y (k-l), △ u (k-1) ..., △ u (k-l+1-d)]T
In conjunction with step 1.4, the state-space model of controlled device is obtained, form is as follows:
△Xm(k+1)=Am△Xm(k)+Bm△u(k)
△ym(k+1)=Cm△Xm(k+1)
Wherein, T is the transposition symbol of matrix, △ Xm(k) dimension is (2l+d-1) × 1；
Bm=[0 ... 010 ... 0]T
Cm=[1 0 ... 00 ... 0]
The design of the prediction function controller of 2 electric furnaces
2.1 seek the current time margin of error of electric furnace:
E (k)=y (k)-r (k)
Wherein e (k) is the error at current time, and y (k) is the measured value of current time object, and r (k) is current time
Discreet value.
By the error at current time, the error of model and practical object after estimating P step
△ e (k+1)=△ ym(k+1)-△r(k+1)
=CmAm△Xm(k)+CmBm△um(k)-△r(k+1)
△ e (k+2)=△ ym(k+2)-△r(k+2)
=Am 2△Xm(k)+AmBm△u(k)+Bm△um(k+1)-△r(k+2)
△ e (k+P)=△ y (k+p)-△ r (k+p)
=Am p△Xm(k)+Am p-1Bm△u(k)+…+Bm△um(k+p)-△r(k+p)
Wherein △ e (k+p) is the prediction of the error after k+p step, and △ r (k+p) indicates the reference locus of k+p step adjacent moment
Difference.
2.2 choose the reference locus r (k+i) and objective function J of Predictive function controlpfc
Jfpc=min [r (k+P)-y (k+P)]2=min [e (k+P)]2
R (k+i)=βiyp(k)+(1-βi)c(k)
Wherein c (k) is setting value, and y (k+P) is to estimate to system model output at the k+P moment, and β is the soft of reference locus
Change coefficient, the reference locus that r (k+i) exports system.
2.3 Predictive function controls be with control input structure it is related, choose basic function be jump function can obtain:
U (k+i)=u (k), (i=1,2 ..., P)
Minimum value by seeking objective function can obtain:
U (k)=- M-1[y(k)-r(k)+N△x(k)+Mu(k-1)-△r]
Wherein:
M=CmAm P-1Bm+CmAm P-2Bm+…+CmBm
N=CmAm P+CmAm P-1+…+CmAm
3. the weighting coefficient of multi-model
3.1 calculate current time submodel MjModel export yj(t), and according to the mould for the submodel at this time being calculated
The deviation of the reality output of type output and current time electric furnace.
ej(t)=| yout(t)-yj(t) |, j=1,2 ... i.
Wherein yj(t) reality output for being system output channel j, ej(t) the inclined of j-th of submodel and reality output is represented
Difference.
3.2 calculate each submodel weight coefficient.
Wherein wj(t) weighting coefficient of j-th of submodel of current time, e are indicatedi(t-k) error in history is indicated.
Therefore the control amount at current time can indicate are as follows:
U (t)=w1(t)u1(t)+w2(t)u2(t)+…+wi(t)ui(t)
3.3 solve the multi-model predictive function control device of fractional order in subsequent time according to the method in step 2.1 to 3.2
Control amount, then acted on electric furnace object, circuit sequentially operation and go down.
The invention proposes a kind of furnace temp controlling parties of multiple model predictive control based on state space fractional order
Method, this method expand to the Multi model Predictive Controllers of integer rank in the Multi model Predictive Controllers of fractional order.Pass through
The local state spatial model for establishing controlled device is improved by nonlinear model conversion before for linear partial model
The control performance of system, while promoting utilization of the model predictive control method in new fractional-order system.
Claims (1)
1. the multi-model fractional order weight estimation function control method of electric furnace, it is characterised in that this method includes following step
It is rapid:
Step 1 establishes the multi-model of controlled device；
Working region is carried out i equal part according to the temperature region range of work by step 1.1, and i is the number of equal part to be carried out；
Step 1.2 acquires the real-time step response data of real process object in the section of each equal part, is built using the data
The fractional order transfer function model M of vertical controlled devicej, form is as follows:
Wherein, MjFor j-th of submodel, α1, jFor the differential order of j-th of system, T1,j,T2, jFor corresponding coefficient, s is that drawing is general
Lars transformation operator, km,jFor model scale gain coefficient, τm,jFor the lag time constant of model；
Step 1.3 differential operator SαIt is expressed as follows by Oustaloup approximation method:
Wherein, α is fractional order differential order, and 0 < α < 1, N are selected apparent order, Kα=Wh α, Wn'=WbWu (2n-1-α)/N, Wn=
WbWu (2n-1+α)/N,WhAnd WbRespectively it is fitted the upper limit value and lower limit value of frequency；
Fractional model in step 1.2 is approximately integer height according to Oustaloup approximation method in step 1.3 by step 1.4
Rank model, by sampling time TSDiscretization obtains following form down plus after zero-order holder:
Y (k)=- a1y(k-1)-a2y(k-2)-…-aly(k-l)+b1u(k-1-d)+b2u(k-2-d)+…+blu(k-l-d)
Wherein, aj, bjIt is the coefficient obtained after discrete approximation, j=1,2 ..., l, time lag d=τ/T of real processS, l
For the length of discrete model, y (k) is the model output of the real process object at k moment, and u (k-1-d) is that real process object exists
The input value at k-1-d moment；
In order to reduce error by obtaining following form to model progress single order backward difference:
Δ y (k)=- a1Δy(k-1)-a2Δy(k-2)-…-alΔy(k-l)+b1Δu(k-1-d)+b2Δu(k-2-d)+…+
blΔu(k-l-d)
The state variable of step 1.5 selecting system is as follows:
ΔXm(k)=[Δ y (k), Δ y (k-1) ..., Δ y (k-l), Δ u (k-1) ..., Δ u (k-l+1-d)]T
In conjunction with step 1.4, the state-space model of controlled device is obtained, form is as follows:
ΔXm(k+1)=AmΔXm(k)+BmΔu(k)
Δym(k+1)=CmΔXm(k+1)
Wherein, Δ Xm(k) dimension is (2l+d-1) × 1；
Bm=[0 ... 010 ... 0]T, wherein T is the transposition symbol of matrix
Cm=[1 0 ... 00 ... 0]
The design of step 2 prediction function controller
Step 2.1 seeks the margin of error at current time:
E (k)=y (k)-r (k)
E (k) is the error at current time, and y (k) is the measured value of current time object, and r (k) is the discreet value at current time；
By the error at current time, the error of estimating system model and practical object after P step
Δ e (k+1)=Δ ym(k+1)-Δr(k+1)
=CmAmΔXm(k)+CmBmΔum(k)-Δr(k+1)
Δ e (k+2)=Δ ym(k+2)-Δr(k+2)
=Am 2ΔXm(k)+AmBmΔu(k)+BmΔum(k+1)-Δr(k+2)
Δ e (k+P)=Δ y (k+P)-Δ r (k+P)
=Am pΔXm(k)+Am p-1BmΔu(k)+…+BmΔum(k+P)-Δr(k+P)
Wherein Δ e (k+P) is the prediction of the error after k+P step, and Δ r (k+P) indicates the reference locus of k+P step adjacent moment
Difference；
The reference locus r (k+i) and objective function J of step 2.2 selection Predictive function controlpfc
Jpfc=min [r (k+P)-y (k+P)]2=min [e (k+P)]2
R (k+i)=βiyp(k)+(1-βi)c(k)
Wherein c (k) is setting value, and y (k+P) is to estimate to system model output at the k+P moment, and β is the softening system of reference locus
Number, the reference locus that r (k+i) exports system；
Step 2.3 Predictive function control be with control input structure it is related, choose basic function be jump function can obtain:
U (k+i)=u (k), i=1,2 ..., P
Minimum value by seeking objective function can obtain:
U (k)=- M-1[y(k)-r(k)+NΔx(k)+Mu(k-1)-Δr]
Wherein:
M=CmAm P-1Bm+CmAm P-2Bm+…+CmBm
N=CmAm P+CmAm P-1+…+CmAm
The weighting coefficient of step 3. multi-model
Step 3.1 calculates current time submodel MjModel export yj(t), and according to the mould for the submodel at this time being calculated
The deviation of the reality output of type output and current time system；
ej(t)=| yout(t)-yj(t) |, j=1,2 ... i；
Wherein yout(t) reality output for being system output channel j, ej(t) j-th of submodel of current time and reality output are represented
Deviation；
Step 3.2 calculates each submodel weight coefficient；
Wherein wj(t) weighting coefficient of j-th of submodel of current time, e are indicatedi(t-k) error in history is indicated；
Therefore the input quantity of current controller indicates are as follows:
U (t)=w1(t)u1(t)+w2(t)u2(t)+…+wi(t)ui(t)
Step 3.3 solves the multi-model predictive function control device of fractional order in subsequent time according to the method in step 2.1 to 3.2
Control amount, then acted on controlled device, circuit sequentially operation and go down.
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| CN108897223B (en) * | 2018-08-02 | 2021-03-23 | 杭州电子科技大学 | Fractional order prediction control method for industrial heating furnace |
| CN113359467B (en) * | 2021-06-30 | 2023-05-02 | 杭州电子科技大学 | Improved internal model control method based on fractional order model in industrial process |
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