CN105334736B - A kind of temperature control method for heating furnace of fractional model PREDICTIVE CONTROL - Google Patents
A kind of temperature control method for heating furnace of fractional model PREDICTIVE CONTROL Download PDFInfo
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Abstract
The invention discloses a kind of temperature control method for heating furnace of extended mode Space Fractional Model Predictive Control, to maintain the stability of new fractional-order system and ensure good control performance.Fractional model is approximately integer rank high-order model first using Oustaloup approximation methods by the present invention, Extended state space model is established based on approximate high-order model, then fractional calculus operator is introduced into object function, and then fractional order prediction function controller is devised based on Extended state space model and the object function chosen.The present invention can apply to the real process object of fractional model description well, improve the weak point of integer rank MPC methods control new fractional-order system, the free degree of adjustment control device parameter is added simultaneously, good control performance is obtained, and the needs of actual industrial process can be met well.
Description
Technical Field
The invention belongs to the technical field of automation, and relates to a heating furnace temperature control method of fractional order model predictive control (FMPC).
Background
In the actual industrial control process, as the requirements on the control accuracy and safe operation of products are higher and higher, but many complex objects cannot be accurately described by integer order differential equations, and object characteristics and product performance can be more accurately described by fractional order differential equations. Although the application of PID control in the field of industrial process control is relatively wide, the control effect of the traditional PID control method and Model Predictive Control (MPC) method on a fractional order system cannot meet the requirement of higher and higher control precision, which requires the research of a controller with good control performance to control an actual controlled object described by a fractional order model. If we expand the state space model of the controlled object and expand the integer order model predictive control method into the fractional order model predictive control method, it will be able to effectively make up the deficiency of the integer order model predictive control method in controlling the fractional order system, and obtain better control effect, and at the same time, it will also promote the application of MPC in the fractional order system.
Disclosure of Invention
The invention aims to provide a heating furnace temperature control method for the predictive control of an extended state space fractional order model aiming at the temperature process of a heating furnace described by the fractional order model so as to maintain the stability of the fractional order system and ensure good control performance. According to the method, firstly, an Oustaloup approximation method is adopted to approximate a fractional order model to an integer order high order model, an extended state space model is established based on the approximate high order model, then a fractional order calculus operator is introduced into a target function, and a fractional order prediction function controller is designed based on the extended state space model and the selected target function.
The method can be well applied to actual process objects described by the fractional order model, overcomes the defects of controlling the fractional order system by the integer order MPC method, increases the freedom degree of adjusting the parameters of the controller, obtains good control performance and can well meet the requirements of actual industrial processes.
The technical scheme of the invention is that a heating furnace temperature control method for extended state space fractional order model predictive control is established by means of data acquisition, model establishment, prediction mechanism, optimization and the like, and the method can effectively improve the control performance of the system.
The method comprises the following steps:
step 1, establishing an extended state space model of a controlled object in an actual process, wherein the specific method comprises the following steps:
1.1, collecting real-time step response data of an actual process object, and establishing a fractional order transfer function model of a controlled object by using the data, wherein the form is as follows:
wherein alpha is 1 Is a differential order, c 0 ,c 1 For the corresponding coefficients, s is the laplace transform operator, K is the model gain, and τ is the lag time of the model.
1.2 obtaining differential operator s by Oustaloup approximation method α The approximate expression of (a) is as follows:
wherein α is a fractional order of differentiation, 0<α&1, N is a selected approximate order, w b and w h Respectively, a lower limit and an upper limit for the selected fitting frequency.
1.3 according to the method in step 1.2, the fractional order transfer function model in step 1.1 is approximated to an integer order higher order model, which is then sampled at a time T s The down plus zero order keeper discretization yields a discrete model of the form:
wherein, F j ,H j (j＝1,2,…,L S ) All the coefficients are obtained after discrete approximation, and the time lag d = tau/T of the actual process s ，L S For the length of the discrete model, y (k) is the model output for the actual process object at time k, and u (k-d-1) is the input value for the actual process object at time k-d-1.
Further taking the first order backward difference of the above model, the following form is obtained:
where Δ is the difference operator.
1.4 the following state variables were chosen:
Δx m (k)＝[Δy(k),Δy(k-1),…,Δy(k-L S +1),Δu(k-1),…,Δu(k-L S +1-d)] T
and (4) combining the step 1.3 to obtain a state space model of the controlled object, wherein the form is as follows:
Δx m (k+1)＝A m Δx m (k)+B m Δu(k)
Δy(k+1)＝C m Δx m (k+1)
where T is the transposed symbol of the matrix, Δ x m (k) Has a dimension of (2L) S +d-1)×1。
B m ＝[0 … 0 1 0 … 0] T
C m ＝[1 0 0 … 0 0 0 0]
1.5 converting the state space model obtained in step 1.4 into an extended state space model containing state variables and output tracking errors, in the following form:
z(k+1)＝Az(k)+BΔu(k)+CΔr(k+1)
wherein, the first and the second end of the pipe are connected with each other,
e(k)＝y(k)-r(k)
e(k+1)＝e(k)+C m A m Δx m (k)+C m B m Δu(k)-Δr(k+1)
r (k) is a tracking set value at k time, e (k) is an output error at k time, and 0 is (2L) S A zero matrix of + d-1) x 1 dimension, A being (2L) S +d)×(2L S + d) dimensional matrix, B, C are both (2L) S + d) x 1-dimensional matrix.
Step 2, designing a fractional order model predictive controller of the controlled object based on the extended state space model, wherein the specific method comprises the following steps:
2.1 predicting the vector form of the model output at the future k + i moment,
Z＝Gz(k)+SΔU+ΨΔR
wherein the content of the first and second substances,
ΔU＝[Δu(k) Δu(k+1) … Δu(k+M-1)] T
ΔR＝[Δr(k+1) Δr(k+2) … Δr(k+P)] T
r(k+i)＝λ i y(k)+(1-λ i )c(k)
c (k) is a set value of k time, lambda is a softening factor, P is a prediction time domain, M is a control time domain, y (k + i) is a prediction model output of the k + i time process, and i =1,2, \8230andP.
2.2, selecting an objective function J of the controlled object, wherein the form of the objective function J is as follows:
wherein, γ 1 ,γ 2 Is any real number, and is a real number,representing the function f (t) at [ t ] 1 ,t 2 ]D is the sign of the differential.
The above objective function is defined by a fractional calculus of Gr ü nwald-Letnikov at a sampling time T S Discretizing to obtain:
J＝Z T Λ(γ 1 ,T s )Z+ΔU T Λ(γ 2 ,T s )ΔU
wherein the content of the first and second substances,
when the temperature of the water is higher than the set temperature,to pair
2.3 solving according to the objective function in step 2.2The control quantity is obtained in the form:
ΔU＝-(S T Λ(γ 1 ,T s )S+Λ(γ 2 ,T s )) -1 SΛ(γ 1 ,T s )(Gz(k)+ΨΔR)
Δu(k)＝[1,0,…,0]ΔU
u(k)＝u(k-1)+Δu(k)
2.4 at the time of k + l, l =1,2,3, \ 8230, the control quantity u (k + l) of the fractional order model predictive controller is solved circularly according to the steps from 2.1 to 2.3 in turn and then acts on the controlled object.
The invention provides a heating furnace temperature control method for extended state space fractional order model predictive control, which extends an integer order model predictive control method to a fractional order model predictive control method.
Detailed Description
Taking the temperature process control of the heating furnace in the actual process as an example:
a fractional order model is obtained from real-time temperature data of the heating furnace, and heating time in a control period is adjusted by controlling a duty ratio, so that temperature control of the heating furnace is realized.
Step 1, establishing an extended state space model of a heating furnace temperature object in an actual process, wherein the specific method comprises the following steps:
1.1, acquiring real-time step response data of an actual heating furnace temperature object, and establishing a fractional order transfer function model of the temperature object by using the data, wherein the form is as follows:
wherein alpha is 1 Is a differential order, c 0 ,c 1 For the corresponding coefficients, s is the laplace transform operator, K is the model gain for the temperature object, and τ is the lag time for the temperature object model.
1.2 obtaining differential operator s by Oustaloup approximation method α The approximate expression of (a) is as follows:
wherein α is a fractional order of differentiation, 0<α&1, N is a selected approximate order, w b and w h Respectively, a lower limit and an upper limit for the selected fitting frequency.
1.3 according to the method in step 1.2, the fractional order transfer function model in step 1.1 is approximated to an integer order higher order model, which is then approximated at a sampling time T s The down plus zero order keeper discretization yields a model of the form:
wherein, F j ,H j (j＝1,2,…,L S ) All the coefficients are obtained after discrete approximation, and the time lag d = tau/T of the actual temperature control process s ，L S For the length of the discrete model, y (k) is the model output of the actual process object at time k, and u (k-d-1) is the heating time duty cycle of the actual process object at time k-d-1.
Further taking the first order backward difference of the above model, the following form is obtained:
where Δ is the difference operator.
1.4 the following state variables were chosen:
Δx m (k)＝[Δy(k),Δy(k-1),…,Δy(k-L S +1),Δu(k-1),…,Δu(k-L S +1-d)] T
and (4) combining the step 1.3 to obtain a state space model of the temperature object, wherein the form is as follows:
Δx m (k+1)＝A m Δx m (k)+B m Δu(k)
Δy(k+1)＝C m Δx m (k+1)
where T is the transposed symbol of the matrix, Δ x m (k) Has a dimension of (2L) S +d-1)×1。
B m ＝[0 … 0 1 0 … 0] T
C m ＝[1 0 0 … 0 0 0 0]
1.5 converting the state space model obtained in step 1.4 into an extended state space model containing state variables and output tracking errors, in the following form:
z(k+1)＝Az(k)+BΔu(k)+CΔr(k+1)
wherein the content of the first and second substances,
e(k)＝y(k)-r(k)
e(k+1)＝e(k)+C m A m Δx m (k)+C m B m Δu(k)-Δr(k+1)
r (k) is a tracking set value at time k, e (k) is an output error at time k, and 0 is (2L) S A zero matrix of + d-1) x 1 dimension, A being (2L) S +d)×(2L S + d) dimensional matrix, B, C are both (2L) S + d) x 1-dimensional matrix.
Step 2, designing a fractional order model predictive controller of the heating furnace temperature control process based on the extended state space model, wherein the specific method comprises the following steps:
2.1 predicting the vector form of the model output at the future k + i moment,
Z＝Gz(k)+SΔU+ΨΔR
wherein the content of the first and second substances,
ΔU＝[Δu(k) Δu(k+1) … Δu(k+M-1)] T
ΔR＝[Δr(k+1) Δr(k+2) … Δr(k+P)] T
r(k+i)＝λ i y(k)+(1-λ i )c(k)
c (k) is the set temperature at the moment k, lambda is the softening factor, P is the prediction time domain, M is the control time domain, y (k + i) is the prediction model output of the heating furnace at the moment k + i, i =1,2, \ 8230, and P.
2.2 an objective function J of the heating furnace temperature object is selected, and the form of the objective function J is as follows:
wherein, γ 1 ,γ 2 Is any real number, and is a real number,representing the function f (t) at [ t ] 1 ,t 2 ]D is the sign of the differential.
According to the fractional calculus definition of Grunwald-Letnikov, the above-mentioned objective function is subjected to sampling time T S Discretizing to obtain:
J＝Z T Λ(γ 1 ,T s )Z+ΔU T Λ(γ 2 ,T s )ΔU
wherein, the first and the second end of the pipe are connected with each other,
Λ(γ ε ,T S )＝T S γε diag(w P-1 ,w P-2 ,…,w 1 ,w 0 )
when the temperature of the water is higher than the set temperature,for is to
2.3 solving according to the objective function in step 2.2The control quantity u (k), i.e. the heating time duty cycle, is obtained in the following form:
ΔU＝-(S T Λ(γ 1 ,T s )S+Λ(γ 2 ,T s )) -1 SΛ(γ 1 ,T s )(Gz(k)+ΨΔR)
Δu(k)＝[1,0,…,0]ΔU
u(k)＝u(k-1)+Δu(k)
2.4 at the time of k + l, l =1,2,3, \8230, and the control quantity u (k + l) of the fractional order model predictive controller is solved in turn and circularly according to the steps from 2.1 to 2.3 and then is acted on the heating furnace.
Claims (1)
1. A heating furnace temperature control method of fractional order model predictive control is characterized by comprising the following specific steps:
step 1, establishing an extended state space model of a controlled object in an actual process, specifically:
1.1, collecting real-time step response data of an actual process object, and establishing a fractional order transfer function model of a controlled object by using the data, wherein the form is as follows:
wherein alpha is 1 Is a differential order, c 0 ,c 1 Is a corresponding coefficient, s is a Laplace transform operator, K is a model gain, and tau is a lag time of the model;
1.2 obtaining differential operator s by the Oustaloup approximation method α The approximate expression of (a) is as follows:
wherein α is a fractional order of differentiation, 0<α&1, N is a selected approximate order,w b and w h Lower and upper limits, respectively, for the selected fitting frequency;
1.3 according to the method in step 1.2, the fractional order transfer function model in step 1.1 is approximated to an integer order higher order model, which is then approximated at a sampling time T s The down plus zero order keeper discretization yields a discrete model of the form:
wherein, F j ,H j (j＝1,2,…,L S ) All the coefficients are obtained after discrete approximation, and the time lag d = tau/T of the actual process s ，L S Taking the length of the discrete model, y (k) is the model output of the actual process object at the moment k, and u (k-d-1) is the input value of the actual process object at the moment k-d-1;
further taking the first order backward difference of the model to obtain the following form:
where Δ is the difference operator;
1.4 the following state variables were chosen:
Δx m (k)＝[Δy(k),Δy(k-1),…,Δy(k-L S +1),Δu(k-1),…,Δu(k-L S +1-d)] T
and (4) combining the step 1.3 to obtain a state space model of the controlled object, wherein the form is as follows:
Δx m (k+1)＝A m Δx m (k)+B m Δu(k)
Δy(k+1)＝C m Δx m (k+1)
where T is the transposed symbol of the matrix, Δ x m (k) Has a dimension of (2L) S +d-1)×1；
B m ＝[0 … 0 1 0 … 0] T
C m ＝[1 0 0 … 0 0 0 0]
1.5 converting the state space model obtained in step 1.4 into an extended state space model containing state variables and output tracking errors, in the form:
z(k+1)＝Az(k)+BΔu(k)+CΔr(k+1)
wherein the content of the first and second substances,
e(k)＝y(k)-r(k)
e(k+1)＝e(k)+C m A m Δx m (k)+C m B m Δu(k)-Δr(k+1)
r (k) is a tracking set value at time k, e (k) is an output error at time k, and 0 is (2L) S A zero matrix of + d-1) x 1 dimension, A being (2L) S +d)×(2L S + d) dimensional matrix, B, C are both (2L) S + d) x 1-dimensional matrix;
step 2, designing a fractional order model predictive controller of the controlled object based on the extended state space model, which comprises the following steps:
2.1 predicting the vector form of the model output at the future k + i moment,
Z＝Gz(k)+SΔU+ΨΔR
wherein the content of the first and second substances,
ΔU＝[Δu(k) Δu(k+1) … Δu(k+M-1)] T
ΔR＝[Δr(k+1) Δr(k+2) … Δr(k+P)] T
r(k+i)＝λ i y(k)+(1-λ i )c(k)
c (k) is a set value at the moment k, lambda is a softening factor, P is a prediction time domain, M is a control time domain, r (k + i) is a tracking set value of the process at the moment k + i, i =1,2, \8230;
2.2, selecting an objective function J of the controlled object, wherein the form of the objective function J is as follows:
wherein, gamma is 1 ,γ 2 Is an arbitrary real number, and is,indicating that the function f (t) is in [ t ] 1 ,t 2 ]The above gamma integration, D is the differential sign;
the above objective function is defined by a fractional calculus of Gr ü nwald-Letnikov at a sampling time T S Discretizing to obtain:
J＝Z T Λ(γ 1 ,T s )Z+ΔU T Λ(γ 2 ,T s )ΔU
wherein, the first and the second end of the pipe are connected with each other,
when the utility model is used, the water is discharged,to q is<0，ε＝1,2；
2.3 solving according to the objective function in step 2.2To obtainThe control quantity is in the form of:
ΔU＝-(S T Λ(γ 1 ,T s )S+Λ(γ 2 ,T s )) -1 SΛ(γ 1 ,T s )(Gz(k)+ΨΔR)
Δu(k)＝[1,0,…,0]ΔU
u(k)＝u(k-1)+Δu(k)
2.4 at the time of k + l, l =1,2,3, \ 8230, the control quantity u (k + l) of the fractional order model predictive controller is solved circularly according to the steps from 2.1 to 2.3 in turn and then acts on the controlled object.
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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 |
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