CN105573355A - Liquid storage tank liquid level control method based on fractional order state spatial prediction function control - Google Patents

Abstract

The invention discloses a liquid storage tank liquid level control method based on fractional order state spatial prediction function control. The method comprises utilizing a Grunwald-Letnikov fractional calculus definition to convert a fractional order state space model into a discrete form; based on the fractional order state space model, obtaining a prediction output model, and introducing the fractional calculus in a target function; and finally designing a fractional order state spatial prediction function controller based on the fractional order state space model and the selected target function. The liquid storage tank liquid level control method can be preferably applied to a practical process object described by a fractional order model, can modify the deficiencies of a PFC method control fractional order system based on an integer order state space model, can increase the degree of freedom for adjusting the parameters of the controller, can acquire good control performance, and can preferably satisfy the requirement during a practical production process for a distillation column.

Description

The liquid tank level control method of fractional order state space Predictive function control

Technical field

The invention belongs to technical field of automation, relate to a kind of based on liquid tank level control method in the still-process of fractional order state space Predictive function control.

Background technology

Still-process is the important process process of a lot of Chemical Manufacture, and require to there is diversity and complicacy due to energy resource consumption with to production, distillation column process model building and operation optimization and control seem of crucial importance.And along with Product Precision and safe operation etc. require day by day improve, in distillation column process, the modeling process of controlled device is day by day complicated, the production run integer model of this complexity of distillation column cannot accurately describe, can the properties of product of description object characteristic sum assessment more accurately with fractional model.

In actual production process, it is widely used Industrial Process Control Methods that PID controls, but traditional PID control method and the control effects of integer model PREDICTIVE CONTROL (MPC) method to new fractional-order system model are not fine, can not meet control accuracy more and more higher in distillation column actual production process and product demand, this controls the actual controlled device described with fractional model with regard to the controller needing research and possess good control performance.Traditional state-space model PREDICTIVE CONTROL is all based on integer model, and for fractional order state-space model, if integer scalariform state space model predictive control method is expanded in fractional order state-space model forecast Control Algorithm, that effectively can make up integer model forecast Control Algorithm and control the deficiency in new fractional-order system, and can better control effects be obtained, also can promote the utilization of MPC in new fractional-order system simultaneously.Predictive function control (PFC) is a kind of control method comparatively easy in model predictive control method, there is the advantages such as calculated amount is few, control effects is good, if pfc controller can be designed based on more accurate fractional model, the performance of control system obviously can be improved.

Summary of the invention

The object of the invention is liquid tank level object in the still-process described for fractional order state-space model, there is provided a kind of based on liquid tank level control method in the still-process of fractional order state space Predictive function control, to maintain the balance of the liquid tank level that fractional order state-space model describes, ensure good control performance.First the method adopts Gr ü nwald-Letnikov fractional calculus to define and fractional order state-space model is converted into discrete form, then obtain predicting output model based on fractional order state-space model, and fractional order integration is introduced objective function, finally devise fractional order state space prediction function controller based on fractional order state-space model and the objective function chosen.

The method can apply to the real process object that fractional model describes well, improve the weak point of the PFC method control new fractional-order system based on integer rank state-space model, add the degree of freedom of adjustment control parameter simultaneously, obtain good control performance, and the needs of distillation column actual production process can be met well.

Technical scheme of the present invention is set up by data acquisition, model, predicted the means such as mechanism, optimization, establish a kind of based on liquid tank level control method in the still-process of fractional order state space Predictive function control, the method effectively can improve the control performance of system.

The step of the inventive method comprises:

Step 1, set up the fractional order state-space model of controlled device in real process, specifically:

The real-time inputoutput data of 1.1 collection real process objects, set up the fractional order state-space model of this controlled device, form is as follows:

$D_t^{\mathrm{\α}}{}_{0}x\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$

y(t)＝Cx(t)

Wherein, x, y, u are respectively the state vector of controlled device, output and input, and α is fractional order order vector, α=[α ₁, α ₂..., α _n] ^t, A, B, C are respectively system matrix, for order α _lfractional order differential symbol.

1.2 for function f (t), has been defined by Gr ü nwald-Letnikov fractional calculus,

$D_t^{{\mathrm{\α}}_l}{}_{0}f\left(t\right)\≈\frac{1}{h^{{\mathrm{\α}}_l}}\underset{j=0}{\overset{\[t/h\]}{\Σ}}{\mathrm{\ω}}_j^{\left({\mathrm{\α}}_l\right)}f(t-jh)$

${\mathrm{\ω}}_0^{\left({\mathrm{\α}}_l\right)}=1,{\mathrm{\ω}}_j^{\left({\mathrm{\α}}_l\right)}=(1-\frac{1+{\mathrm{\α}}_l}{j}){\mathrm{\ω}}_{j-1}^{\left({\mathrm{\α}}_l\right)},j=1,2,...$

Wherein, h is sampling step length, the integral part that [t/h] is t/h.

1.3 utilize the definition in step 1.2 can be the fractional order state-space model of following discrete form by the model conversion in step 1.1:

$x(k+1)=H\left(Ax\right(k)+Bu(k\left)\right)-\underset{j=1}{\overset{k+1}{\Σ}}{W}_{j}x(k+1-j)$

y(k+1)＝Cx(k+1)

Wherein, $H=diag(h^{{\mathrm{\α}}_1},h^{{\mathrm{\α}}_2},\mathrm{...}{h}^{{\mathrm{\α}}_n}),W_j=diag({\mathrm{\ω}}_j^{\left({\mathrm{\α}}_1\right)},{\mathrm{\ω}}_j^{\left({\mathrm{\α}}_2\right)},\mathrm{...}{\mathrm{\ω}}_j^{\left({\mathrm{\α}}_n\right)}).$

Step 2, based on fractional order state-space model design controlled device fractional order prediction function controller, specific as follows:

2.1 according to the state-space model in step 1.3, and obtain the model prediction output valve in following k+i moment, form is as follows:

$y(k+1)=Cx(k+1)=CHAx\left(k\right)+CHBu\left(k\right)-C\underset{j=1}{\overset{k+1}{\Σ}}{W}_{j}x(k+1-j)$

$\begin{array}{c}y(k+2)=Cx(k+2)=C{\left(HA\right)}^{2}x\left(k\right)+CHAHBu\left(k\right)+CHBu(k+1)\\ -CHA\underset{j=1}{\overset{k+1}{\Σ}}{W}_{j}x(k+1-j)-C\underset{j=1}{\overset{k+2}{\Σ}}{W}_{j}x(k+2-j)\end{array}$

.

.

.

$\begin{array}{c}y(k+P)=Cx(k+P)=C{\left(HA\right)}^{P}x\left(k\right)+C{\left(HA\right)}^{P-1}HBu\left(k\right)+\mathrm{...}+CHBu(k+P-1)\\ -C{\left(HA\right)}^{P-1}\underset{j=1}{\overset{k+1}{\Σ}}{W}_{j}x(k+1-j)-\mathrm{...}-C\underset{j=1}{\overset{k+P}{\Σ}}{W}_{j}x(k+P-j)\end{array}$

Wherein, P is prediction time domain, and y (k+i) is the model prediction output valve of k+i moment controlled device, i=1,2 ..., P.

2.2 in algorithm of predictive functional control, selects a basis function and step function, and the model prediction in step 2.1 is exported the prediction output model being converted to matrix form, form is as follows:

Y＝Gx(k)+Su(k)-Ψ

Wherein,

$G=\left[\begin{array}{c}CHA\\ C{\left(HA\right)}^2\\ .\\ .\\ .\\ C{\left(HA\right)}^P\end{array}\right],S=\left[\begin{array}{c}CHB\\ CHB+CHAHB\\ .\\ .\\ .\\ CHB+CHAHB+...+C{\left(HA\right)}^{P-1}HB\end{array}\right]$

$\mathrm{\Ψ}=\left[\begin{array}{c}C\underset{j=1}{\overset{k+1}{\Σ}}{W}_{j}x(k+1-j)\\ CHA\underset{j=1}{\overset{k+1}{\Σ}}{W}_{j}x(k+1-j)+C\underset{j=1}{\overset{k+2}{\Σ}}{W}_{j}x(k+2-j)\\ .\\ .\\ .\\ C\underset{i=1}{\overset{P}{\Σ}}{\left(HA\right)}^{P-i}\underset{j=1}{\overset{k+i}{\Σ}}{W}_{j}x(k+i-j)\end{array}\right]$

The prediction output model of 2.3 correction current time controlled devices, obtain the forecast model after correcting, form is as follows:

$\begin{array}{c}\hat{Y}=Y+E\\ =Gx\left(k\right)+Su\left(k\right)-\mathrm{\Ψ}+E\end{array}$

E＝[e(k+1),e(k+2),…,e(k+P)] ^T

e(k+i)＝y _p(k)-y(k)

Wherein, y _pk () is the real output value of k moment controlled device, y (k) is the model prediction output valve in k moment, the difference that real output value and model prediction that e (k+i) is k+i moment controlled device export.

The 2.4 reference locus y choosing predictive functional control algorithm _rand objective function J (k+i) _f, its form is as follows:

y _r(k+i)＝λ ⁱy _p(k)+(1-λ ⁱ)c(k)

$\begin{array}{c}{J}_F=I_1^P{}_{\mathrm{\γ}}{\[y_r\left(t\right)-y\left(t\right)-e\left(t\right)\]}^2\\ ={\∫}_1^P{D}^{1-\mathrm{\γ}}{\[y_r\left(t\right)-y\left(t\right)-e\left(t\right)\]}^{2}dt\end{array}$

Wherein, y _r(k+i) be the reference locus in k+i moment, λ is the softening coefficient of reference locus, and c (k) is the setting value in k moment, representative function f (t) is at [ht ₁, ht ₂] on γ integration.

According to the definition of Gr ü nwald-Letnikov fractional calculus, at sampling time h, discretize is carried out to above-mentioned objective function, and to the error amount weighting that the reference locus value after discretize exports with prediction, obtain the objective function after being weighted error term, form is as follows:

$J_F\≈{(Y_r-\hat{Y})}^{T}Q(Y_r-\hat{Y})$

Wherein,

Yr＝[y _r(k+1),y _r(k+2),…,y _r(k+P)] ^T

Q＝h ^γdiag(q ₁m _P-1,q ₂m _P-2,…,q _P-1m ₁,q _Pm ₀)

$m_q={\mathrm{\ω}}_q^{(-\mathrm{\γ})}-{\mathrm{\ω}}_{q-(P-1)}^{(-\mathrm{\γ})}$

${\mathrm{\&omega;}}_0^{(-\mathrm{\&gamma;})}=1,\&ForAll;q>0$ Time, ${\mathrm{\&omega;}}_q^{(-\mathrm{\&gamma;})}=(1-\frac{1-\mathrm{\&gamma;}}{q}){\mathrm{\&omega;}}_{q-1}^{(-\mathrm{\&gamma;})},$ To q<0, ${\mathrm{\&omega;}}_q^{(-\mathrm{\&gamma;})}=0.$ Q _ifor the error term weighting coefficient that reference locus exports with prediction.

2.5 solve controlled quentity controlled variable according to the objective function in step 2.4, and form is as follows:

u(k)＝(S ^TQS) ^-1S ^TQ(Y _r-Gx(k)+Ψ-E)

2.6 in the k+ η moment, according to the step in 2.1 to 2.5 circulate successively solve fractional order prediction function controller controlled quentity controlled variable u (k+ η) (η=1,2,3 ...), and acted on controlled device.

The present invention proposes a kind of based on liquid tank level control method in the still-process of fractional order state space Predictive function control, the method obtains predicting output model based on fractional order state-space model, and fractional order integration is introduced objective function, improve the deficiency of the PFC method control new fractional-order system based on integer rank state-space model, add the degree of freedom of adjustment control parameter, obtain good control performance, and the needs of actual production process can be met well, facilitate the utilization of predictive functional control algorithm in new fractional-order system.

Embodiment

Control for liquid tank level in distillation column actual production process:

Obtain fractional model by the real-time level data of fluid reservoir, the regulating measure of liquid tank level control system is the valve opening of the cooling water flow controlling still-process.

Step 1, set up the fractional order state-space model of liquid tank level in distillation column actual production process, specifically:

The real-time inputoutput data of 1.1 collection still-process liquid tank levels, set up the fractional order state-space model of liquid tank level:

$D_t^{\mathrm{\&alpha;}}{}_{0}x\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$

y(t)＝Cx(t)

Wherein, x, y, u are respectively the valve opening of the state vector of liquid tank level object, liquid level and controlled cooling model discharge, and α is fractional order order vector, α=[α ₁, α ₂..., α _n] ^t, A, B, C are respectively system matrix, for order α _lfractional order differential symbol.

1.2 for function f (t), has been defined by Gr ü nwald-Letnikov fractional calculus,

$D_t^{{\mathrm{\&alpha;}}_l}{}_{0}f\left(t\right)\&ap;\frac{1}{h^{{\mathrm{\&alpha;}}_l}}\underset{j=0}{\overset{\&lsqb;t/h\&rsqb;}{\&Sigma;}}{\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_l\right)}f(t-jh)$

${\mathrm{\&omega;}}_0^{\left({\mathrm{\&alpha;}}_l\right)}=1,{\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_l\right)}=(1-\frac{1+{\mathrm{\&alpha;}}_l}{j}){\mathrm{\&omega;}}_{j-1}^{\left({\mathrm{\&alpha;}}_l\right)},j=1,2,...$

Wherein, h is sampling step length, the integral part that [t/h] is t/h.

1.3 utilize the definition in step 1.2 can be the fractional order state-space model of following discrete form by the model conversion in step 1.1:

$x(k+1)=H\left(Ax\right(k)+Bu(k\left)\right)-\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)$

y(k+1)＝Cx(k+1)

Wherein, $H=diag(h^{{\mathrm{\&alpha;}}_1},h^{{\mathrm{\&alpha;}}_2},\mathrm{...}{h}^{{\mathrm{\&alpha;}}_n}),W_j=diag({\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_1\right)},{\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_2\right)},\mathrm{...}{\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_n\right)}).$

Step 2, based on the fractional order prediction function controller of liquid tank level in fractional order state-space model design still-process, specific as follows:

2.1 according to the state-space model in step 1.3, and the model prediction obtaining the following k+i moment exports, and form is as follows:

$y(k+1)=Cx(k+1)=CHAx\left(k\right)+CHBu\left(k\right)-C\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)$

$\begin{array}{c}y(k+2)=Cx(k+2)=C{\left(HA\right)}^{2}x\left(k\right)+CHAHBu\left(k\right)+CHBu(k+1)\\ -CHA\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)-C\underset{j=1}{\overset{k+2}{\&Sigma;}}{W}_{j}x(k+2-j)\end{array}$

.

.

.

$\begin{array}{c}y(k+P)=Cx(k+P)=C{\left(HA\right)}^{P}x\left(k\right)+C{\left(HA\right)}^{P-1}HBu\left(k\right)+\mathrm{...}+CHBu(k+P-1)\\ -C{\left(HA\right)}^{P-1}\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)-\mathrm{...}-C\underset{j=1}{\overset{k+P}{\&Sigma;}}{W}_{j}x(k+P-j)\end{array}$

Wherein, P is prediction time domain, and y (k+i) is the model prediction output valve of k+i moment controlled device, i=1,2 ..., P.

2.2 in algorithm of predictive functional control, selects a basis function and step function, and the model prediction in step 2.1 is exported the prediction output model being converted to matrix form, form is as follows:

Y＝Gx(k)+Su(k)-Ψ

Wherein,

$G=\left[\begin{array}{c}CHA\\ C{\left(HA\right)}^2\\ .\\ .\\ .\\ C{\left(HA\right)}^P\end{array}\right],S=\left[\begin{array}{c}CHB\\ CHB+CHAHB\\ .\\ .\\ .\\ CHB+CHAHB+...+C{\left(HA\right)}^{P-1}HB\end{array}\right]$

$\mathrm{\&Psi;}=\left[\begin{array}{c}C\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)\\ CHA\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)+C\underset{j=1}{\overset{k+2}{\&Sigma;}}{W}_{j}x(k+2-j)\\ .\\ .\\ .\\ C\underset{i=1}{\overset{P}{\&Sigma;}}{\left(HA\right)}^{P-i}\underset{j=1}{\overset{k+i}{\&Sigma;}}{W}_{j}x(k+i-j)\end{array}\right]$

The prediction output model of liquid tank level in 2.3 correction current time still-process, obtain the forecast model after correcting, form is as follows:

$\begin{array}{c}\hat{Y}=Y+E\\ =Gx\left(k\right)+Su\left(k\right)-\mathrm{\&Psi;}+E\end{array}$

E＝[e(k+1),e(k+2),…,e(k+P)] ^T

e(k+i)＝y _p(k)-y(k)

Wherein, y _pk () is the liquid level of the k moment distilling fluid reservoir in production run, y (k) is the model prediction output valve in k moment, and e (k+i) distills the difference that the liquid level of fluid reservoir in production run and model prediction export for the k+i moment.

The 2.4 reference locus y choosing predictive functional control algorithm _rand objective function J (k+i) _f, its form is as follows:

y _r(k+i)＝λ ⁱy _p(k)+(1-λ ⁱ)c(k)

$\begin{array}{c}{J}_F=I_1^P{}_{\mathrm{\&gamma;}}{\&lsqb;y_r\left(t\right)-y\left(t\right)-e\left(t\right)\&rsqb;}^2\\ ={\&Integral;}_1^P{D}^{1-\mathrm{\&gamma;}}{\&lsqb;y_r\left(t\right)-y\left(t\right)-e\left(t\right)\&rsqb;}^{2}dt\end{array}$

Wherein, y _r(k+i) be the reference locus in k+i moment, λ is the softening coefficient of reference locus, and c (k) is the setting value in k moment, representative function f (t) is at [ht ₁, ht ₂] on γ integration.

According to the definition of Gr ü nwald-Letnikov fractional calculus, at sampling time h, discretize is carried out to above-mentioned objective function, and to the error amount weighting that the reference locus value after discretize exports with prediction, obtain the objective function after being weighted error term, form is as follows:

$J_F\&ap;{(Y_r-\hat{Y})}^{T}Q(Y_r-\hat{Y})$

Wherein,

Yr＝[y _r(k+1),y _r(k+2),…,y _r(k+P)] ^T

Q＝h ^γdiag(q ₁m _P-1,q ₂m _P-2,…,q _P-1m ₁,q _Pm ₀)

$m_q={\mathrm{\&omega;}}_q^{(-\mathrm{\&gamma;})}-{\mathrm{\&omega;}}_{q-(P-1)}^{(-\mathrm{\&gamma;})}$

${\mathrm{\&omega;}}_0^{(-\mathrm{\&gamma;})}=1,\&ForAll;q>0$ Time, ${\mathrm{\&omega;}}_q^{(-\mathrm{\&gamma;})}=(1-\frac{1-\mathrm{\&gamma;}}{q}){\mathrm{\&omega;}}_{q-1}^{(-\mathrm{\&gamma;})},$ To q<0, ${\mathrm{\&omega;}}_q^{(-\mathrm{\&gamma;})}=0.$ Q _ifor the error term weighting coefficient that reference locus exports with prediction.

2.5 solve controlled quentity controlled variable according to the objective function in step 2.4, and form is as follows:

u(k)＝(S ^TQS) ^-1S ^TQ(Y _r-Gx(k)+Ψ-E)

2.6 in the k+ η moment, according to the step in 2.1 to 2.5 circulate successively solve fractional order prediction function controller controlled quentity controlled variable u (k+ η) (η=1,2,3 ...), and acted on the valve controlling fluid reservoir cooling water flow.

Claims (1)

1. the liquid tank level control method of fractional order state space Predictive function control, is characterized in that the concrete steps of the method are:

Step 1, set up the fractional order state-space model of controlled device in real process, specifically:

The real-time inputoutput data of 1.1 collection real process objects, set up the fractional order state-space model of this controlled device, form is as follows:

$D_t^{\mathrm{\&alpha;}}{}_{0}x\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$

y(t)＝Cx(t)

Wherein, x, y, u are respectively the state vector of controlled device, output and input, and α is fractional order order vector, α=[α ₁, α ₂..., α _n] ^t, A, B, C are respectively system matrix, for order α _lfractional order differential symbol.

1.2 for function f (t), has been defined by Gr ü nwald-Letnikov fractional calculus,

$D_t^{{\mathrm{\&alpha;}}_l}{}_{0}f\left(t\right)\&ap;\frac{1}{h^{{\mathrm{\&alpha;}}_l}}\underset{j=0}{\overset{\&lsqb;t/h\&rsqb;}{\&Sigma;}}{\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_l\right)}f(t-jh)$

${\mathrm{\&omega;}}_0^{\left({\mathrm{\&alpha;}}_l\right)}=1,{\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_l\right)}=(1-\frac{1+{\mathrm{\&alpha;}}_l}{j}){\mathrm{\&omega;}}_{j-1}^{\left({\mathrm{\&alpha;}}_l\right)},j=1,2,...$

Wherein, h is sampling step length, the integral part that [t/h] is t/h.

1.3 utilize the definition in step 1.2 can be the fractional order state-space model of following discrete form by the model conversion in step 1.1:

$x(k+1)=H\left(Ax\right(k)+Bu(k\left)\right)-\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)$

y(k+1)＝Cx(k+1)

Wherein, $H=diag(h^{{\mathrm{\&alpha;}}_1},h^{{\mathrm{\&alpha;}}_2},...,h^{{\mathrm{\&alpha;}}_n}),W_j=diag({\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_1\right)},{\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_2\right)},...,{\mathrm{\&omega;}}_j^{\left({\mathrm{\&alpha;}}_n\right)}).$

Step 2, based on fractional order state-space model design controlled device fractional order prediction function controller, specific as follows:

2.1 according to the state-space model in step 1.3, and obtain the model prediction output valve in following k+i moment, form is as follows:

$y(k+1)=Cx(k+1)=CHAx\left(k\right)+CHBu\left(k\right)-C\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)$

$\begin{array}{c}y(k+2)=Cx(k+2)=C{\left(HA\right)}^{2}x\left(k\right)+CHAHBu\left(k\right)+CHBu(k+1)\\ -CHA\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x\left(k+1-j\right)-C\underset{j=1}{\overset{k+2}{\&Sigma;}}{W}_{j}x\left(k+2-j\right)\end{array}$

.

.

.

$\begin{array}{c}y(k+P)=Cx(k+P)=C{\left(HA\right)}^{P}x\left(k\right)+C{\left(HA\right)}^{P-1}HBu\left(k\right)+...+CHBu(k+P-1)\\ -C{\left(HA\right)}^{P-1}\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x\left(k+1-j\right)-...-C\underset{j=1}{\overset{k+P}{\&Sigma;}}{W}_{j}x\left(k+P-j\right)\end{array}$

Wherein, P is prediction time domain, and y (k+i) is the model prediction output valve of k+i moment controlled device, i=1,2 ..., P.

2.2 in algorithm of predictive functional control, selects a basis function and step function, and the model prediction in step 2.1 is exported the prediction output model being converted to matrix form, form is as follows:

Y＝Gx(k)+Su(k)-Ψ

Wherein,

$G=\left[\begin{array}{c}CHA\\ C{\left(HA\right)}^2\\ \begin{array}{c}.\\ .\\ .\end{array}\\ C{\left(HA\right)}^P\end{array}\right],S=\left[\begin{array}{c}CHB\\ CHB+CHAHB\\ \begin{array}{c}.\\ .\\ .\end{array}\\ CHB+CHAHB+...+C{\left(HA\right)}^{P-1}HB\end{array}\right]$

$\mathrm{\&Psi;}=\left[\begin{array}{c}C\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)\\ CHA\underset{j=1}{\overset{k+1}{\&Sigma;}}{W}_{j}x(k+1-j)+C\underset{j=1}{\overset{k+2}{\&Sigma;}}{W}_{j}x(k+2-j)\\ \begin{array}{c}.\\ .\\ .\end{array}\\ C\underset{i=1}{\overset{P}{\&Sigma;}}{\left(HA\right)}^{P-i}\underset{j=1}{\overset{k+i}{\&Sigma;}}{W}_{j}x(k+i-j)\end{array}\right]$

The prediction output model of 2.3 correction current time controlled devices, obtain the forecast model after correcting, form is as follows:

$\begin{array}{c}\hat{Y}=Y+E\\ =Gx\left(k\right)+Su\left(k\right)-\mathrm{\&Psi;}+E\end{array}$

E＝[e(k+1),e(k+2),…,e(k+P)] ^T

e(k+i)＝y _p(k)-y(k)

Wherein, y _pk () is the real output value of k moment controlled device, y (k) is the model prediction output valve in k moment, the difference that real output value and model prediction that e (k+i) is k+i moment controlled device export.

The 2.4 reference locus y choosing predictive functional control algorithm _rand objective function J (k+i) _f, its form is as follows:

y _r(k+i)＝λ ⁱy _p(k)+(1-λ ⁱ)c(k)

$\begin{array}{c}{J}_F=I_1^P{}_{\mathrm{\&gamma;}}{\&lsqb;y_r\left(t\right)-y\left(t\right)-e\left(t\right)\&rsqb;}^2\\ ={\&Integral;}_1^P{D}^{1-\mathrm{\&gamma;}}{\&lsqb;y_r\left(t\right)-y\left(t\right)-e\left(t\right)\&rsqb;}^{2}dt\end{array}$

Wherein, y _r(k+i) be the reference locus in k+i moment, λ is the softening coefficient of reference locus, and c (k) is the setting value in k moment, representative function f (t) is at [ht ₁, ht ₂] on γ integration.

According to the definition of Gr ü nwald-Letnikov fractional calculus, at sampling time h, discretize is carried out to above-mentioned objective function, and to the error amount weighting that the reference locus value after discretize exports with prediction, obtain the objective function after being weighted error term, form is as follows:

$J_F\&ap;{(Y_r-\hat{Y})}^{T}Q(Y_r-\hat{Y})$

Wherein,

Yr＝[y _r(k+1),y _r(k+2),…,y _r(k+P)] ^T

Q＝h ^γdiag(q ₁m _P-1,q ₂m _P-2,…,q _P-1m ₁,q _Pm ₀)

$m_q={\mathrm{\&omega;}}_q^{(-\mathrm{\&gamma;})}-{\mathrm{\&omega;}}_{q-\left(P-1\right)}^{\left(-\mathrm{\&gamma;}\right)}$

${\mathrm{\&omega;}}_0^{(-\mathrm{\&gamma;})}=1,\&ForAll;q>0$ Time, ${\mathrm{\&omega;}}_q^{(-\mathrm{\&gamma;})}=(1-\frac{1-\mathrm{\&gamma;}}{q}){\mathrm{\&omega;}}_{q-1}^{(-\mathrm{\&gamma;})},$ To q<0, ${\mathrm{\&omega;}}_q^{(-\mathrm{\&gamma;})}=0.$ Q _ifor the error term weighting coefficient that reference locus exports with prediction.

2.5 solve controlled quentity controlled variable according to the objective function in step 2.4, and form is as follows:

u(k)＝(S ^TQS) ^-1S ^TQ(Y _r-Gx(k)+Ψ-E)

2.6 in the k+ η moment, according to the step in 2.1 to 2.5 circulate successively solve fractional order prediction function controller controlled quentity controlled variable u (k+ η) (η=1,2,3 ...), and acted on controlled device.