CN103425048B - A kind of multi-model generalized predictable control system and its control method based on dynamic optimization - Google Patents

Abstract

A kind of multi-model generalized predictable control system based on dynamic optimization, which includes dynamic optimization layer, MPC layers and base control layer；The dynamic optimization layer is located at upper strata, and which calculates the optimal setting of the optimal value as MPC layers of key control variable；The MPC layers are adjusted to the variable to be optimized using the prediction algorithm of rolling optimization under conditions of variable to be optimized meets model dynamic behaviour positioned at lower floor so as to track the optimal setting drawn in S1；The base control layer is located at bottom, and the final optimization pass value of the variable to be optimized is delivered to executing agency.The present invention reduces system cost consumption, raising systematic economy benefit, and can improve the regulating power of system transient modelling performance and system model parameter saltus step, while can also effectively eliminate that the interference exported by system is disturbed.

Description

Multi-model generalized predictive control system based on dynamic optimization and control method thereof

Technical Field

The invention relates to the field of control optimization, in particular to a design method of a multi-model generalized predictive control system based on dynamic optimization.

Background

The increased market competition of production globalization makes the requirements for reducing cost consumption and improving economic benefits higher and higher. Optimization of the operational performance of a process can create tremendous benefits for enterprise production, and therefore more efficient and advanced optimization and control strategies are needed for application to associated industrial equipment. The traditional process optimization technology based on the steady-state model has achieved remarkable achievement and has good optimization effect on processes with time-varying models not strong. However, in the actual industry, the phenomenon that the model becomes strong in time often occurs, so that the traditional optimization technology based on the steady-state model is difficult to meet the requirements of the modern industry. Dynamic optimization can well process industrial processes with strong time variation and complex reaction mechanisms, and a plurality of scholars combine dynamic optimization and model predictive control to form a layered predictive control structure to optimize industrial equipment, thereby obtaining good effect.

In a layered predictive control structure formed by combining traditional dynamic optimization and model predictive control, a single model predictive controller is mostly adopted in an MPC layer, but in an actual industrial process, the situation that process parameters jump along with production operation often occurs. Because the actual production process is very complicated, a compact global control model is difficult to establish, and therefore the requirement that the system is still in a good control state when the parameters are time-varying or jumping is difficult to meet by a prediction controller of a single model. The multi-model method can effectively solve the problems of multiple working points and parameter time variation in the complex industrial process, and a plurality of scholars apply multi-model predictive control to the fields of chemical engineering, pharmacy, electric power and the like and obtain good effect. But the conventional multi-model is difficult to match with the actual process characteristics due to the existence of random noise. Therefore, how to establish a controller which can not only consider economic benefits, but also ensure the transient performance and the jump-time regulation capability of the system is a problem which needs to be solved at present.

Disclosure of Invention

1. In order to solve the above problems, the present invention provides a dynamic optimization-based multi-model generalized predictive control system, which is characterized in that the system comprises a dynamic optimization layer, an MPC layer and a basic control layer;

the dynamic optimization layer is positioned at the upper layer and dynamically optimizes the economic objective function by adopting the combination of control vector parameterization and particle swarm optimization algorithm to obtain the track of the optimal value of the key variable, and the optimal value is used as the optimization set value of the MPC layer;

the MPC layer is positioned at a lower layer and adjusts the variables to be optimized by adopting a rolling optimization prediction algorithm under the condition that the variables to be optimized meet the dynamic behavior of the model so as to track the optimal set value;

the basic control layer is positioned at the bottom layer and is used for sending the final optimized value of the variable to be optimized to the execution mechanism.

Preferably, the model dynamic behavior includes model parameter changes and interference effects.

Preferably, the MPC layer employs a plurality of fixed models and adaptive models to identify the dynamics of the system in parallel.

Preferably, the basic control layer comprises a PID controller, and the PID controller is used for restraining and eliminating the influence of disturbance entering the process on the output.

The invention also provides a working method of the multi-model generalized predictive control system based on dynamic optimization, which comprises the following steps:

s1: the dynamic optimization layer adopts a method of combining control vector parameterization and particle swarm optimization algorithm aiming at the economic objective function and the time-varying constraint of the system to obtain the optimal value track of the key variable, and the track is used as the optimal setting value reference track of the lower MPC layer;

s2: the MPC layer adopts a rolling optimization prediction algorithm to adjust the variable to be optimized in the process under the condition that the variable meets the model parameter change and the dynamic behavior is influenced by interference, so that the MPC layer tracks the optimal set value track of the variable obtained in S1, and adopts a plurality of fixed models and self-adaptive models to parallelly identify the dynamic characteristics of the system;

s3: and the base layer inhibits and eliminates the influence of disturbance entering the process on the output through PID action, and sends the final optimized value of the variable to the execution structure.

Preferably, the process of acquiring the optimal value trajectory of the key variable includes the following steps:

s11: first, the time interval t₀,t_f]Dividing the data into a plurality of sections, wherein each section is similar to the optimal track by using a section constant track to obtain a plurality of control parameters to be optimized;

s12: initializing a particle swarm: setting the number of particles m, the dimension D and two learning factors c₁,c₂Upper and lower bounds of position x_max,x_minThe maximum iteration number M, the initial position, the initial speed, the global optimal solution gBest and the local optimal solution pBest are initialized;

s13: calculating the fitness of each particle, and updating a local optimal value and a global optimal value;

s14: calculating the updating speed and the updating position, and setting the updating speed and the updating position as boundary values if the positions exceed upper and lower boundaries;

s15: and c) judging whether the maximum iteration number is reached, if not, returning to the step c) to continue the calculation. And if the condition is met, outputting the current optimal value.

Preferably, in step S2, the controlled object is: a (z)^-1)y(k)=B(z^-1) u (k-1) + ξ (k)/Δ, the multiple model set is expressed as:

Δy_i(k)=φ_i(k)^Tθ₀(k)+ξ_i(k)

i=1,2…,m,m+1,m+2

wherein

φ(k)=[-Δy(k-1)…-Δy(k-n_a)Δu(k-1)+…Δu(k-n_b-1)]；

The adaptive model adopts the following recursive least square method to identify the dynamic characteristics of the system:

K(k)=P(k-1)φ(k)[φ(k)^TP(K-1)φ(k)+μ]^-1

wherein 0< mu <1 is forgetting factor, K (k) is weight factor, P (k) is positive covariance matrix;

the multi-model switching index is expressed as:

i=1,2,...,m+2

the above equation is the performance index of model i at time k, where e_i(k) For the output error of the ith model at time k, γ and η are the error weights of the current and past times, ρ is the error forgetting factor, L is the error length of the past time, and at time k, the performance index J is used_iAnd minimally switching to the corresponding model.

Preferably, the performance optimization index of the PID is

Where E {. is a mathematical expectation, w_rTo output the desired value, N_uLambda is a control weighting coefficient, and k represents time;

the expected value of the output of the object is w_r(k + j) = w (k + j) j ∈ {1,2, …, N }, where w (k + j) is a key variable set value given by upper layer dynamic optimization, and N is an optimization time domain termination time.

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

1. the system cost consumption is reduced, and the system economic benefit is improved;

2. the transient performance of the system is improved;

3. the adjustment capability of the jump of the system model parameters is improved;

4. the interference of the disturbance to the system output can be effectively eliminated.

Drawings

Fig. 1 is a schematic structural diagram of a controller according to an embodiment of the present invention;

FIG. 2 is a schematic diagram of a multi-model switching structure according to an embodiment of the present invention;

fig. 3 is a schematic diagram of a simulation result of the controller according to the embodiment of the present invention.

DETAILED DESCRIPTION OF EMBODIMENT (S) OF INVENTION

The invention provides a multi-model generalized predictive control system based on dynamic optimization, which comprises a dynamic optimization layer, an MPC layer and a basic control layer, as shown in figure 1; the dynamic optimization layer is positioned at the upper layer and used for calculating the optimal value of the key control variable as the optimal set value of the MPC layer; the MPC layer is positioned at the lower layer, and the variable to be optimized is adjusted by adopting a rolling optimization prediction algorithm under the condition that the variable to be optimized meets the dynamic behavior of the model, so that the MPC layer tracks the optimal set value obtained in S1; and the basic control layer is positioned at the bottom layer and sends the final optimized value of the variable to be optimized to the execution mechanism.

The specific control process of the invention comprises the following steps: the method comprises the following steps:

s1: the dynamic optimization layer adopts a method of combining control vector parameterization and particle swarm optimization algorithm aiming at the economic objective function and the time-varying constraint of the system to obtain the optimal value track of the key variable, and the track is used as the reference track of the lower MPC layer;

s2: the MPC layer adopts a rolling optimization prediction algorithm to adjust the variable to be optimized in the process under the condition that the variable meets the model parameter change and the dynamic behavior is influenced by interference, so that the MPC layer tracks the optimal set value track of the variable obtained in S1, and adopts a plurality of fixed models and self-adaptive models to parallelly identify the dynamic characteristics of the system;

s3: and the base layer inhibits and eliminates the influence of disturbance entering the process on the output through PID action, and sends the final optimized value of the variable to the execution structure.

In step S1, the step of obtaining the optimal value trajectory of the key variable by dynamic optimization includes:

s11 first divides the time interval t₀,t_f]And dividing the data into N sections, wherein each section is similar to the optimal track by using a piecewise constant track to obtain N control parameters to be optimized.

S12 particle swarm initialization: setting the number of particles m, the dimension D and two learning factors c₁,c₂Upper and lower bounds of position x_max,x_minAnd maximum iteration number M, initial position, initial speed, initialized global optimal solution gBest and local optimal solution pBest

And S13, calculating the fitness of each particle, and updating the local optimal value and the global optimal value.

S14 calculates an update speed and an update position. Set to a boundary value if the position exceeds the upper and lower bounds

S15, judging whether the maximum iteration number is reached, if not, returning to c) to continue calculation. If the condition is met, outputting the current optimal value

The multi-model generalized predictive controller in step S2 is designed as follows

The controlled object is represented as:

A(z^-1)y(k)=B(z^-1)u(k-1)+ξ(k)/Δ (1)

wherein

In the formula z^-1For the post-shift operator, y (k), u (k), ξ (k) are respectively the output of the system, the input and the white noise sequence with zero mean, delta =1-z^-1Is a difference operator.

The multi-model set is represented as:

Δy_i(k)=φ_i(k)^Tθ₀(k)+ξ_i(k) (2)

i=1,2…,m,m+1,m+2

wherein

φ(k)=[-Δy(k-1)…-Δy(k-n_a)Δu(k-1)+…Δu(k-n_b-1)]

θ when i =1,2, …, m_i(k) Being a constant value of the fixed model, m fixed models may improve the transient performance of the system relative to a single model.

When i = m +1, m +2, the model is one adaptive model and one assignable adaptive model. The adaptive model online real-time identification system parameter can eliminate steady-state error, guarantees the convergence of the system and the stability of the system, and the introduction of the adaptive model capable of assigning an initial value further improves the transient performance of the system and increases the rapidity of the system. The following recursive least squares algorithm can be used for identifying model parameters

K(k)=P(k-1)φ(k)[φ(k)^TP(K-1)φ(k)+μ]^-1

Wherein 0< mu <1 is a forgetting factor; k (k) is a weight factor; p (k) is positive definite covariance matrix

The multi-model switching index is expressed as:

i=1,2,...,m+2 (4)

equation (4) is the performance index of model i at time k, where e_i(k) Is the output error of the ith model at the time k, gamma and η are the error weights of the current time and the past time, rho is the error forgetting factor, L is the error length of the past time, at the time k, the performance index J is used_iAnd minimally switching to the corresponding model.

Designing a controller:

the performance optimization index at the time k is

Wherein E {. is a mathematical expectation; w is a_rOutputting the expected value; n is a radical of_uIs a control time domain; λ is the desired value of the output of the object controlling the weighting factor

w_r(k+j)=w(k+j)j∈{1,2,…,N} (6)

In the formulaAnd (4) setting a given key variable for upper layer dynamic optimization (model output setting), wherein N is the time domain termination time of optimization.

In order to calculate the output predicted value after the ith step by using the formula (1), firstly, introducing a lost-to-the-graph equation

1=E_j(z^-1)A(z^-1)Δ+z^-jF_j(z^-1) (7)

E_j(z^-1)B(z^-1)=G_j(z^-1)+z^-jH_j(z^-1) (8)

In the formula

E_j(z^-1)=e_j,0+e_j,1z^-1+…+e_j,j-1z^-(j-1)

G_j(z^-1)=g_j,0+g_j,1z^-1+…+g_j,j-1z^-(j-1)

In the formula

F(z^-1)=[F₁(z^-1),…,F_N(z^-1)]^T

H(z^-1)=[H₁(z^-1),…H_N(z^-1)]^T

From this, the actual output control amount is obtained

u(k)=u(k-1)+Δu(k|k)。

Examples

The S1 dynamic optimization layer is set to the following process mathematical model:

x(0)=[300]^T,0≤(k)≤30,k∈[0,k_f](11)

wherein f is an economic objective function, x_A,x_BAre two parameters related to the key variable (k).

The time interval is divided into 10 equal segments, then the PSO parameter set particle number is m =10, dimension D =10, learning factor c₁=2,c₂And =2, the maximum number of iterations is 500. the set value of the key variable is obtained through dynamic optimization and is used as the reference track of the multi-model generalized predictive controller.

The controlled object of the S2MPC layer is expressed as:

y(k)+a₁y(k-1)+a₂y(k-2)=b₁u(k-1)+b₂u(k-2)+ξ(k)/Δ

the control step number is taken as 300, a₁=-1.8，a₂=1.2，b₁=1，b₂=2, the system parameter jumps in 150 steps. Jump to a₁=-0.8，a₂=-1.2，b₁，b₂ξ (remain unchanged)k) Is [0.1, -0.1 ]]The uniformly distributed white noise fixed model is taken as a₁={-2,-1}，a₂={-2,-1,1,1.5,2}，b₁=1，b₂10 in total, the initial values of the adaptive model parameters are all 0.1. The corresponding multi-model switching indexes are as follows:

the specific switching execution form is shown in FIG. 2, and the model is switched to the corresponding model according to the switching index and is output as the object

The performance optimization index at the k moment is

The expected value of the object output. That is, the reference trajectory in MPC is obtained by

w_r(k+j)=w(k+j)j∈{1,2,…,N}

Where w () is the reference trajectory (model output set value) of the key variable found at S1, and N is the optimum time domain termination time.

Introduction of lost graph

1=E_j(z^-1)A(z^-1)Δ+z^-jF_j(z^-1)

E_j(z^-1)B(z^-1)=G_j(z^-1)+z^-jH_j(z^-1)

To obtain

From this, the actual output control amount is obtained

u(k)=u(k-1)+Δu(k|k)

As can be seen from the simulation effect of fig. 3, before the steady state is reached, the transient performance of the design of the present invention is better, the output quantity change is more gradual, the fluctuation with the interference is smaller, that is, the tracking performance for the reference trajectory is better, and the control performance after the jump occurs at step 150 is also obviously improved.

Compared with the prior art, the invention reduces the cost consumption of the system, improves the economic benefit of the system, can improve the transient performance of the system and the adjustment capability of the parameter jump of the system model, and can effectively eliminate the interference of disturbance on the output of the system.

The preferred embodiments of the invention disclosed above are intended to be illustrative only. The preferred embodiments are not intended to be exhaustive or to limit the invention to the precise embodiments disclosed. Obviously, many modifications and variations are possible in light of the above teaching. The embodiments were chosen and described in order to best explain the principles of the invention and the practical application, to thereby enable others skilled in the art to best utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims (1)

1. A dynamic optimization-based optimization method of a multi-model generalized predictive control system is characterized by comprising the following steps:

s1: the dynamic optimization layer adopts a method of combining control vector parameterization and particle swarm optimization algorithm aiming at the economic objective function and the time-varying constraint of the system to obtain the optimal value track of the key variable, and the track is used as the optimal setting value reference track of the lower MPC layer;

s2: the MPC layer adopts a rolling optimization prediction algorithm to adjust the variables to be optimized in the process under the condition that the variables meet the model parameter change and the dynamic behavior is influenced by interference, so that the MPC layer tracks the optimal set value reference track of the MPC layer obtained in S1, and adopts a plurality of fixed models and self-adaptive models to parallelly identify the dynamic characteristics of the system;

s3: the base layer inhibits and eliminates the influence of disturbance entering the process on output through PID action, and sends the final optimized value of the variable to an execution structure; the process for acquiring the optimal value track of the key variable comprises the following steps:

s11: first, the time interval t₀,t_f]Dividing the data into a plurality of sections, wherein each section is similar to the optimal track by using a section constant track to obtain a plurality of control parameters to be optimized;

s12: initializing a particle swarm: setting the number of particles m, the dimension D and two learning factors c₁,c₂Upper and lower bounds of position x_max,x_minThe maximum iteration number M, the initial position, the initial speed, the global optimal solution gBest and the local optimal solution pBest are initialized;

s13: calculating the fitness of each particle, and updating a local optimal value and a global optimal value;

s14: calculating the updating speed and the updating position, and setting the updating speed and the updating position as boundary values if the positions exceed upper and lower boundaries;

s15: judging whether the maximum iteration number is reached, if not, returning to S13 for continuous calculation, and if so, outputting the current optimal value; in step S2, the controlled object is: a (z)^-1)y(k)＝B(z^-1) u (k-1) + ξ (k)/Δ, the multiple model set is expressed as:

Δy_i(k)＝φ_i(k)^Tθ₀(k)+ξ_i(k)

i＝1,2…,m,m+1,m+2

whereinTo actually optimize the system parameters of the target object model,

phi (k) is a general expression of the input and output values of the actual optimization target object

φ(k)＝[-Δy(k-1)…-Δy(k-n_a)Δu(k-1)+…Δu(k-n_b-1)]Form(s) ofSet of vectors of composition phi_i(k) For the input and output values of the i-th model, according to phi (k) [ -deltay (k-1) … -deltay (k-n)_a)Δu(k-1)+…Δu(k-n_b-1)]The vector set is formed in a form, wherein delta y (k-1) is the difference value of the output value y (k-1) of the system at the k-1 moment and the output value y (k-2) of the system at the k-2 moment, and delta u (k-1) is the difference value of the input value u (k-1) of the system at the k-1 moment and the input value u (k-2) of the system at the k-2 moment;

wherein, ξ_i(k) Represents the noise of the system at time k; n is_aRepresenting a controlled object;

A(z^-1)y(k)＝B(z^-1) A (z) in u (k-1) + ξ (k)/Δ^-1) Order of (1), n_bRepresenting controlled objects

A(z^-1)y(k)＝B(z^-1) u (k-1) + ξ (k)/Δ B (z)^-1) The order of (a);represents theta at time k-1 for system dynamics identification according to the recursive least squares method₀An estimated value; when i is 1,2, …, m, theta_i(k) Being constant values of the fixed model, m fixed models may improve the transient performance of the system relative to a single model,

wherein,

$A\left(z^{-1}\right)=1+a_1{z}^{-1}+\mathrm{......}+a_{n_a}{z}^{-n_a}$

$B\left(z^{-1}\right)=b_0+b_1{z}^{-1}+\mathrm{......}+b_{n_b}{z}^{-n_b}$

in the formula z^-1For the post-shift operator, y (k), u (k), ξ (k) are respectively the output of the system, the input and the white noise sequence with zero mean value, and Delta is 1-z^-1Is a difference operator;

the adaptive model adopts the following recursive least square method to identify the dynamic characteristics of the system:

K(k)＝P(k-1)φ(k)[φ(k)^TP(k-1)φ(k)+μ]^-1

$P\left(k\right)=\frac{1}{\mathrm{\&mu;}}\&lsqb;1-K\left(k\right)\mathrm{\&phi;}{\left(k\right)}^T\&rsqb;P(k-1)$

wherein 0< mu <1 is forgetting factor, K (k) is weight factor, P (k) is positive covariance matrix;

the multi-model switching index is expressed as:

$J_i={\mathrm{\&gamma;e}}_i^2\left(k\right)+\mathrm{\&eta;}\underset{j=1}{\overset{L}{\&Sigma;}}{\mathrm{\&rho;}}^j{e}_i^2(k-j),i=1,2,\mathrm{...},m+2$

the above equation is the performance index of model i at time k, where e_i(k) For the output error of the ith model at time k, γ and η are the error weights of the current and past times, ρ is the error forgetting factor, L is the error length of the past time, and at time k, the performance index J is used_iMinimum switching to the corresponding model; the performance optimization indexes of the PID are as follows:

$\min J\left(k\right)=E\{\underset{j=1}{\overset{N}{\&Sigma;}}{\&lsqb;y(k+j|k)-w_r(k+j)\&rsqb;}^2+\underset{j=1}{\overset{N_u}{\&Sigma;}}\mathrm{\&lambda;}{\&lsqb;\mathrm{\&Delta;}u(k+j-1|k)\&rsqb;}^2\},$

where E {. is a mathematical expectation, w_rIs w_r(k) Abbreviated and output expected value, N_uIn order to control the time domain, lambda is a control weighting coefficient, k represents the time, and y (k + j/k) is the output of k + j time obtained by prediction according to the input and the output of k time; u (k + j-1/k) is input at the moment k + j-1 which is obtained by input and output prediction at the moment k;

the expected value of the output of the object is w_r(k + j) ═ w (k + j) j ∈ {1,2, …, N }, where w (k + j) is the key variable set value given by the upper layer dynamic optimization and N is the optimization time domain termination time;

the expected values of the object outputs are:

w_r(k+j)＝w(k+j)j∈{1,2,…,N}

in the formula, w (k + j) is a key variable set value given by upper layer dynamic optimization, namely a model output set value, and N is an optimization time domain termination time;

in order to calculate the output predicted value after the ith step, firstly introducing a loss map equation:

1＝E_j(z^-1)A(z^-1)Δ+z^-jF_j(z^-1)

E_j(z^-1)B(z^-1)＝G_j(z^-1)+z^-jH_j(z^-1)

in the formula:

E_j(z^-1)＝e_j,0+e_j,1z^-1+…+e_j,j-1z^-(j-1)

$F_j\left(z^{-1}\right)=f_{j,0}+f_{j,1}{z}^{-1}+\mathrm{...}+f_{j,n_a}{z}^{-n_a}$

G_j(z^-1)＝g_j,0+g_j,1z^-1+…+g_j,j-1z^-(j-1)

$H_j\left(z^{-1}\right)=h_{j,0}+h_{j,1}{z}^{-1}+\mathrm{...}+h_{j,n_b-1}{z}^{-(n_b-1)}$

$\mathrm{\&Delta;}u\left(k\right)=\frac{Q^T(w_r-F(z^{-1}\left)y\right(k)-H\left(z^{-1}\right)\mathrm{\&Delta;}u\left(k-1\right))}{Q^{T}Q+\mathrm{\&lambda;}(1+{\mathrm{\&beta;}}^2+\mathrm{...}{\mathrm{\&beta;}}^{2(N_u-1)})}$

in the formula:

$Q=\left|\begin{array}{c}{g}_{1,0}\\ g_{2,1}+{\mathrm{\&beta;g}}_{1,0}\\ .\\ .\\ .\\ g_{i,N_u-1}+{\mathrm{\&beta;g}}_{i-1,N_u-2}+\mathrm{...}{\mathrm{\&beta;}}^{N_u-1}{g}_{1,0}\\ .\\ .\\ .\\ g_{j,N-1}+{\mathrm{\&beta;g}}_{j-1,N-2}+\mathrm{...}{\mathrm{\&beta;}}^{N_u-1}{g}_{j-N_u+1,N-N_u}\end{array}\right|$

F(z^-1)＝[F₁(z^-1),…,F_N(z^-1)]^T

H(z^-1)＝[H₁(z^-1),…H_N(z^-1)]^T

wherein, g_j,iIs a polynomial G_j(z^-1) The actual output control quantity obtained by this method is:

u(k)＝u(k-1)+Δu(k|k)；

E_j(z^-1) And F_j(z^-1) Are respectively satisfying E_j(z^-1)＝e_j,0+e_j,1z^-1+…+e_j,j-1z^-(j-1)Andbased on polynomial form and under the condition of known A (z)^-1) According to 1 ═ E_j(z^-1)A(z^-1)Δ+z^-jF_j(z^-1) Solving the obtained polynomial;

wherein n is_aRepresents a controlled object A (z)^-1)y(k)＝B(z^-1) A (z) in u (k-1) + ξ (k)/Δ^-1) J is the desired optimization step size, j ∈ {1,2, …, N } and N is the optimized time domain termination time, e_j.0……e_j.j-1Represents E_j(z^-1) The coefficient value obtained in correspondence with the calculated value,is represented by F_j(z^-1) The corresponding found coefficient value;

then, E is obtained_j(z^-1) And known B (z)^-1) According to formula E_j(z^-1)B(z^-1)＝G_j(z^-1)+z^-jH_j(z^-1) Determine G_j(z^-1)＝g_j,0+g_j,1z^-1+…+g_j,j-1z^-(j-1)、Polynomial of form G_j(z^-1)、H_j(z^-1) Wherein g is_j.0……g_j.j-1、Each represents G_j(z^-1)、H_j(z^-1) The obtained corresponding coefficient；n_bRepresents a controlled object A (z)^-1)y(k)＝B(z^-1) u (k-1) + ξ (k)/Δ B (z)^-1) The order of (a);

wherein beta is an influence coefficient and beta belongs to [0,1 ];

when j takes 1,2, … and N respectively to obtain F_j(z^-1) Then, according to the formula F (z)^-1)＝[F₁(z^-1),…,F_N(z^-1)]^TCan obtain F (z)^-1) (ii) a Similarly, when j takes 1,2, …, N respectively to obtain H_j(z^-1) Then, H (z) can be obtained^-1) (ii) a After all variables are derived, the controller output increment at time k, Δ u (k | k), which represents the difference between the input value at time k and the input value at time k-1, can be derived.