阅读说明：本技术 多变量ARX-Laguerre模型的PID控制算法 (PID control algorithm of multivariable ARX-Laguerre model ) 是由 郭伟 邵帅 李涛 樊茜 于 2021-01-06 设计创作，主要内容包括：本发明涉及一种多变量ARX-Laguerre模型的PID控制算法,用于分馏塔控制系统中,该方法在选取被控对象后,进一步选取Laguerre函数极点；且使用被控对象实际输入与输出对多变量ARX-Laguerre模型的参数进行在线辨识；由二者结合得到预测模型,然后得到该预测模型的输出,根据被控对象的实际输出对预测模型的输出进行校正；设置设定值根据设定值得到参考轨迹,并将校正后的预测模型输出与参考轨迹作差得到误差；通过加入FOPID控制重构性能指标式,得到新的性能指标；将误差代入性能指标,并对作极小化处理,得到最优控制增量；将最优控制增量累加得到最优控制量,将最优控制量重新输入系统,重复上述操作进行滚动优化,直到预测模型近似于实际系统。(The invention relates to a multivariable ARX-Laguerre model PID control algorithm, which is used in a fractionating tower control system, and the method further selects a Laguerre function pole after selecting a controlled object; and the parameters of the multivariable ARX-Laguerre model are identified on line by using the actual input and output of the controlled object; combining the two to obtain a prediction model, then obtaining the output of the prediction model, and correcting the output of the prediction model according to the actual output of the controlled object; setting a set value to obtain a reference track according to the set value, and subtracting the corrected output of the prediction model from the reference track to obtain an error; a new performance index is obtained by adding an FOPID control reconstruction performance index formula; substituting the error into the performance index, and carrying out minimization treatment on the performance index to obtain an optimal control increment; and accumulating the optimal control increments to obtain optimal control quantity, inputting the optimal control quantity into the system again, and repeating the operation for rolling optimization until the prediction model is similar to the actual system.)

1. A multivariable ARX-Laguerre model PID control algorithm for use in a fractionation column control system, characterized by:

firstly, selecting a controlled object;

constructing a multivariate ARX-Laguerre model, and selecting a Laguerre function pole; performing online identification on parameters of the multivariable ARX-Laguerre model by using actual input and output of a controlled object to obtain a prediction model;

computing the output of a predictive modely(ii) a Output of prediction model by using actual output y of controlled objectyCarrying out correction;

setting a set value matrix r, and obtaining a reference track y according to r_rAnd through y_rObtaining quadratic form performance indexes of model prediction control, and then subtracting the corrected prediction model output from a reference track to obtain an error E;

FOPID control is introduced to reconstruct a quadratic performance index to obtain a new performance index J;

substituting the error E into the performance index J, and carrying out minimization treatment on the J to obtain an optimal control increment delta u;

accumulating the optimal control increment to obtain an optimal control quantity u, and inputting the optimal control quantity into the multivariable ARX-Laguerre model again;

and step three, repeating the step two until the prediction model is closest to the actual system of the controlled object.

2. The multivariable ARX-Laguerre model PID control algorithm of claim 1, wherein:

respectively taking the top extraction, the side extraction and the tower bottom reflux load of the controlled object as control quantities u₁,u₂,u₃(ii) a The final distillation point at the top of the tower, the final distillation point at the bottom of the tower and the reflux temperature at the bottom of the tower are respectively controlled quantities y₁,y₂,y₃The transfer function matrix forming the controlled object is as follows:

3. the multivariable ARX-Laguerre model PID control algorithm of claim 1, wherein: the multivariable ARX-Laguerre model is provided with p inputs and m outputs; the state space expression of the model is:

in the formula: x (k) [ (X)¹(k))^T (X²(k))^T … (X^m(k))^T]^TThe state vector is, in this case,

andis the truncation order;

andfor Z transformation of an nth order Laguerre function, U_t(z) and Y_r(z) are input vectors u (k) and [ u ] respectively₁(k) u₂(k) … u_p(k)]^TAnd the model output vectory(k)＝[y₁(k) y₂(k) … y_m(k)]^TZ transformation of (1).

4. The multivariable ARX-Laguerre model PID control algorithm of claim 3, wherein: the prediction output matrix of the prediction model is

Y(k+1)＝SΔY(k+1)+Fy(k)

In the formula: f ═ diag (F)_i)，F_i＝[1 … 1]。

5. The multivariable ARX-Laguerre model PID control algorithm of claim 4, wherein: the corrected model prediction output is:

6. the multivariable ARX-Laguerre model PID control algorithm of claim 5, wherein: the quadratic performance index of the model predictive control is as follows:

in the formula: y is_r(k+1)＝[y_r(k+1),y_r(k+2),...,y_r(k+P)]^TThe expected output for future P steps;

y_r(k+i)＝βⁱy(k)+(I-βⁱ)r,i＝1,2,...,P，β＝diag(β₁,β₂,...,β_m) To soften the factor matrix, r ═ r₁ r₂ … r_m]^TFor the value matrix, I is the unit diagonal matrix and y (k) is the actual output vector.

7. The multivariable ARX-Laguerre model PID control algorithm of claim 6, wherein: the new performance index

Wherein

Technical Field

The invention relates to a FOPID predictive control algorithm based on a multivariable ARX-Laguerre model, belonging to the field of nonlinear control of a fractionating tower.

Background

The fractionating tower is an important separation and purification device and is widely applied to the fields of petrochemical industry, pharmaceutical chemical industry and the like. The energy consumption is about 40% -50% of the total energy consumption, but the energy conversion rate can only be maintained at a lower level of 5% -10%, which causes great energy waste. The irreplaceability of the fractionation tower also leads to the development of more advanced control methods to improve the control effect of the fractionation tower.

Fractional order PID control is a control method expanded on the basis of traditional PID control, and the order of integration and differentiation in the algorithm structure can be any. Compared with the traditional PID method, the adjustment freedom of the fractional order PID is better, the parameter setting range is larger, and the ideal parameter can be adjusted more easily to reach the expected control target. However, today, the control theory is continuously developed, and the control effect of the system on a complex system with characteristics of nonlinearity, time-varying property, large time lag and the like is not good, so that the control performance requirement is difficult to meet.

Model Predictive Control (MPC) is a new type of computer Control algorithm developed in the end of the seventies of the last century. MPC has been widely used in various industrial fields after several decades of theoretical research and practical application. There are three typical algorithms for conventional MPC: model Algorithm Control (MAC) and Dynamic Matrix algorithm Control (DMC) based on non-parametric models, and Generalized Predictive Control (GPC) based on parametric models. However, too many parameters of the MAC and the DMC are not beneficial to calculation, and GPC is sensitive to the order and the time delay of the system and has certain influence on the control effect.

Disclosure of Invention

The invention provides a FOPID predictive control algorithm based on a multivariable ARX-Laguerre model, aiming at making up the defect that the control precision of the traditional FOPID control on a complex system is poor and solving the defect of the model predictive control algorithm based on an intelligent control theory in the control of a fractionating tower.

In order to achieve the purpose, the technical scheme provided by the invention is as follows: a multivariable ARX-Laguerre model PID control algorithm for use in a fractionation column control system, characterized by:

firstly, selecting a controlled object;

constructing a multivariate ARX-Laguerre model, and selecting a Laguerre function pole; performing online identification on parameters of the multivariable ARX-Laguerre model by using actual input and output of a controlled object to obtain a prediction model;

computing the output of a predictive modely(ii) a Output of prediction model by using actual output y of controlled objectyCarrying out correction;

setting a set value matrix r, and obtaining a reference track y according to r_rAnd through y_rObtaining quadratic form performance indexes of model prediction control, and then subtracting the corrected prediction model output from a reference track to obtain an error E;

FOPID control is introduced to reconstruct a quadratic performance index to obtain a new performance index J;

substituting the error E into the performance index J, and carrying out minimization treatment on the J to obtain an optimal control increment delta u;

accumulating the optimal control increment to obtain an optimal control quantity u, and inputting the optimal control quantity into the multivariable ARX-Laguerre model again;

and step three, repeating the step two until the prediction model is closest to the actual system of the controlled object.

The technical scheme is further designed as follows: respectively taking the top extraction, the side extraction and the tower bottom reflux load of the controlled object as control quantities u₁,u₂,u₃(ii) a The final distillation point at the top of the tower, the final distillation point at the bottom of the tower and the reflux temperature at the bottom of the tower are respectively controlled quantities y₁,y₂,y₃The transfer function matrix forming the controlled object is as follows:

the multivariable ARX-Laguerre model is provided with p inputs and m outputs; the state space expression of the model is:

in the formula: x (k) [ (X)¹(k))^T (X²(k))^T … (X^m(k))^T]^TThe state vector is, in this case,

andis the truncation order;

andfor Z transformation of an nth order Laguerre function, U_t(z) and Y_r(z) are input vectors u (k) and [ u ] respectively₁(k) u₂(k) … u_p(k)]^TAnd the model output vectory(k)＝[y₁(k) y₂(k) … y_m(k)]^TZ transformation of (1).

The prediction output matrix of the prediction model is

Y(k+1)＝SΔY(k+1)+Fy(k)

In the formula: f ═ diag (F)_i)，F_i＝[1 … 1]。

The corrected model prediction output is:

the quadratic performance index of the model predictive control is as follows:

in the formula: y is_r(k+1)＝[y_r(k+1),y_r(k+2),...,y_r(k+P)]^TThe expected output for future P steps;

y_r(k+i)＝βⁱy(k)+(I-βⁱ)r,i＝1,2,...,P，β＝diag(β₁,β₂,...,β_m) To soften the factor matrix, r ═ r₁ r₂ … r_m]^TFor the value matrix, I is the unit diagonal matrix and y (k) is the actual output vector.

The new performance index

Wherein

Compared with the prior art, the invention has the following beneficial effects:

the invention provides a FOPID predictive control method based on a multivariable ARX-Laguerre model. Compared with the traditional FOPID algorithm and a Dynamic Matrix Control (DMC) algorithm in the traditional model predictive control, the method has more excellent control performance and high system control precision.

Drawings

FIG. 1 is a flow diagram of a fractionation column system;

FIG. 2 is a diagram of a multivariate ARX-Laguerre model architecture;

FIG. 3 is a functional block diagram of a control system according to an embodiment of the present invention;

FIG. 4 is a graph comparing the response curves of the system controlled by the MALLMPCFOPID algorithm and the FOPID algorithm;

FIG. 5 is a graph comparing the response curves of the system controlled by the MALMPCFOPID algorithm and the DMC algorithm;

FIG. 6 is a comparison graph of the response before and after the system model mismatch controlled by the simulation MALMPCFOPID algorithm.

Detailed Description

The invention is described in detail below with reference to the figures and the specific embodiments.

Examples

The PID control algorithm of the multivariable ARX-Laguerre model is used for a fractionating tower control system and comprises the following steps:

step 1: selecting a controlled object, and analyzing the system characteristics of the fractionating tower and a corresponding mathematical model;

in this embodiment, a heavy oil catalytic cracking fractionating tower control system in the Shell standard control problem is selected as a controlled object, and the flow chart is shown in fig. 1. The fractionating tower has three product outlets and three side circulation loops, and the three circulation loops are used for implementing the fractionation task. The top condenser can condense the steam which rises to the top and part of the steam flows downwards, the tower kettle is provided with a reboiler, the steam flow can be adjusted to control heat transfer, and part of liquid at the bottom of the tower is vaporized and rises, so that the next heat and mass transfer cycle is completed.

The progressive stable object in the high-order object is converted into a first-order pure hysteresis object through technologies such as fitting simplification and the like, and each channel of the whole fractionating tower system process can be described by a transfer function of first-order inertia plus pure hysteresis:

where K is the gain and θ is the pure lag.

The fractionation column in the Shell Standard control problem has practically 7 controlled quantities: the top end point, the bottom end point, the top temperature, the top reflux temperature, the side draw temperature, the middle reflux temperature and the bottom reflux temperature; 3 control variables: top extraction, side extraction and tower bottom reflux load; 2 disturbance amounts: middle section reflux load, overhead reflux load. The patent simplifies the Shell model and then processes the model transfer functionIntercepting 3 input and 3 output square matrix in the matrix, namely respectively taking top extraction, side extraction and tower bottom reflux load as control quantity u₁,u₂,u₃(ii) a The final distillation point at the top of the tower, the final distillation point at the bottom of the tower and the reflux temperature at the bottom of the tower are respectively controlled quantities y₁,y₂,y₃Forming a transfer function matrix of a controlled object:

step 2: FOPID predictive control algorithm (MALVPFOPID algorithm) based on multivariate ARX-Laguerre model, and the control algorithm principle is shown in FIG. 3.

The structure of the multivariate ARX-Laguerre model with p inputs and m outputs of the embodiment is shown in FIG. 2, and the state space expression thereof is as follows:

in the formula: x (k) [ (X)¹(k))^T (X²(k))^T … (X^m(k))^T]^TIs the state vector, k is the number of steps;

andis the truncation order;andfor Z transformation of an nth order Laguerre function, U_t(z) and Y_r(z) are input vectors u (k) and [ u ] respectively₁(k) u₂(k) … u_p(k)]^TAnd the model output vectory(k)＝[y₁(k) y₂(k) … y_m(k)]^TZ-transformation of (a);

the state space matrix is

Andis the Laguerre function pole;

g is a fourier coefficient.

Matrix A, B_u、B_yDerived from Laguerre function poles ξ. The matrix C is obtained by online identification of a minimum two-pass recursive algorithm with forgetting factors, and the identification model is as follows:

where λ is the forgetting factor.

An incremental state space expression for the multi-variable ARX-Laguerre model can be deduced from equation (1):

the future P-step output increment of the model can be deduced by the formula (2):

writing equation (3) to matrix form yields:

ΔY(k+1)＝SH_lΔX(k)+SH_uΔU(k)+SH_yΔY ^*(k) (4)

in the formula: p is the predicted number of steps, M is the control number of steps, ΔY(k+1)＝[Δy(k+1)^T,Δy(k+2)^T,…,Δy(k+P)^T]^TOutput delta matrix, Δ, for model predictionY ^*(k)＝[Δy(k)^T,Δy(k+1)^T,…,Δy(k+M-1)^T]^TOutput delta matrix, Δ, for model controlU(k)＝[Δu(k)^T,Δu(k+1)^T,…,Δu(k+M-1)^T]^TInputting an incremental matrix for model control;

S＝diag(S_i)，

obtaining a model future P-step prediction output matrix according to the formula (4):

Y(k+1)＝SΔY(k+1)+Fy(k) (5)

in the formula: f ═ diag (F)_i)，F_i＝[1… 1]。

The model has errors due to external interference, the future P-step prediction output of the formula (5) needs to be corrected by using the actual output vector y of the controlled object, and the corrected model prediction output is as follows:

the quadratic performance indexes of model prediction control are as follows:

in the formula: y is_r(k+1)＝[y_r(k+1),y_r(k+2),...,y_r(k+P)]^TFor the future P steps to expect output, namely reference track, Q is an error weighting matrix, and R is a control weighting matrix;

y_r(k+i)＝βⁱy(k)+(I-βⁱ)r,i＝1,2,...,P，β＝diag(β₁,β₂,...,β_m) To soften the factor matrix, r ═ r₁ r₂ … r_m]^TFor the value matrix, I is the unit diagonal matrix and y (k) is the actual output vector.

And (4) subtracting the corrected output of the prediction model from the reference track to obtain an error E.

FOPID control is introduced to reconstruct the model predictive control quadratic performance index to obtain a new performance index:

whereinLet D (k +1) be H_lΔX(k)+SH_yΔY ^*(k)+Fy(k)-Y_r(k +1), then

E(k+1)＝D(k+1)+SH_uΔU(k) (9)

By recursion formula and introducing a backward shift operator q^-1The following can be obtained:

substituting the error E into the performance index J, and carrying out minimization treatment on the J to obtain an optimal control increment delta u;

that is, the performance index formula (8) can be substituted with the formulas (9) and (10)

Order toAnd calculating the deviation of J, i.e. commandingCan obtain the product

Taking the first group of elements of the optimal control law as optimal control increments, and accumulating the optimal control increments to obtain an optimal control quantity u, namely

In the formula: k ═ I0 … 0.

And re-inputting the optimal control quantity into the multivariable ARX-Laguerre model.

And step three, repeating the step two, and performing rolling optimization until the prediction model is closest to the actual system of the controlled object.

FIG. 4 is a comparison graph of the response curves of the system controlled by the MALMCPFOPID algorithm and the FOPID algorithm in this embodiment; the control effect of the control algorithm provided by the invention is better than that of the traditional FOPID algorithm, the response time of three outputs is greatly improved, and y is₂、y₃Is reduced to 0, y₁The overshoot of (c) is also greatly reduced.

FIG. 5 is a comparison graph of the response curves of the system controlled by the MALMPCFOPID algorithm and the DMC algorithm in this embodiment; the control effect of the control algorithm provided by the invention is better than that of the DMC algorithm in the traditional MPC, and is shown in y₁Is obviously improved in rapidity, y₁The overshoot is greatly reduced, y₂The overshoot is reduced to 0.

FIG. 6 is a graph of response comparison before and after model mismatch for a system controlled by a simulated MALLMPCFOPID algorithm; the control algorithm provided by the invention can well complete response even if the model is mismatched in the running process, and y is increased by some overshoot₂、y₃Does not suffer much from the response speed of y₁And the regulation is also carried out rapidly, and finally, the three outputs all achieve stable control effects.

The technical solutions of the present invention are not limited to the above embodiments, and all technical solutions obtained by using equivalent substitution modes fall within the scope of the present invention.