CN112947323B - Chemical process distributed control method for explicit model predictive control optimization - Google Patents

Abstract

The invention discloses a chemical process distributed control method for explicit model predictive control optimization, which comprises the following steps: step 1, establishing a novel improved model by distributed model predictive control; and 2, improving the design of the distributed model predictive control controller under the model. The invention establishes a novel explicit distributed model predictive control method under an improved model by means of data acquisition, model establishment, predictive mechanism, optimization and the like and according to related ideas of distributed control, and can well process a multivariable coupling system and improve the overall control performance of the system by utilizing the method on the premise of ensuring high control precision and stability.

Description

Chemical process distributed control method for explicit model predictive control optimization

Technical Field

The invention belongs to the technical field of automation, and relates to a chemical process distributed control method for explicit model predictive control optimization.

Background

With the development of computer network technology, centralized control cannot meet all requirements of complex process industries, and distributed control structures are occupying more and more weight. For a complex and high-dimensional large-scale system, although the distributed model predictive control can already meet basic requirements, for some industrial processes with input and output constraints and high requirements on the dynamic performance and robustness of the system, relevant standards may not be met, and particularly, under the condition of introducing interference, the output cannot better track a set value, so that the improvement of the distributed model predictive control method is particularly important.

Disclosure of Invention

The invention aims to provide a novel explicit distributed model predictive control method under an improved model aiming at the defects of distributed model predictive control in processing constraint and multivariable process control, and the method is applied to a common industrial heating furnace system in a chemical process to control the system. The method simultaneously considers the influence of output error, measurement output and input on the design of the model predictive controller, model predictive control based on the novel improved model inherits the advantages of a traditional model, simultaneously considers the important control performance index of tracking error, combines an explicit control method to process the constraint of the system, makes up the defects of the traditional distributed model predictive control in a high-precision industrial process, and improves the overall control performance of the system.

The invention establishes a novel explicit distributed model predictive control method under an improved model by means of data acquisition, model establishment, predictive mechanism, optimization and the like and according to related ideas of distributed control, and can well process a multivariable coupling system and improve the overall control performance of the system by utilizing the method on the premise of ensuring high control precision and stability. The specific technical scheme is as follows:

a chemical process distributed control method for explicit model predictive control optimization comprises the following steps:

step 1, establishing a novel improved model by distributed model predictive control;

and 2, improving the design of the distributed model predictive control controller under the model.

Further, the step 1 specifically comprises the following steps:

step 1.1 establishment of distributed model predictive control model

Considering a multivariable N-in-N-out large-scale system in an industrial process as consisting of N subsystems of first-order inertia plus pure hysteresis (FOPDT) models, an N-in-N-out multivariable FOPDT system of the form:

wherein, K_ijThe steady state gain, T, of the j (1 ≦ j ≦ n) th input to the i (1 ≦ i ≦ n) th output for the multivariable process object_ijTime constant, τ, for the jth input to ith output of a multivariable process object_ijA lag time for a jth input to an ith output of the multivariable process object;

step 1.2, dispersing an N-input N-output large-scale system into N subsystems according to a distributed predictive control idea;

in the multivariate selection process, the transfer function of the input of the jth subsystem to the output of the ith subsystem is:

step 1.3 at sampling time T_sDiscretizing the first-order plus pure lag model of the ith subsystem under the conditionThe following model was obtained:

here, let

The above equation can be simplified to:

wherein, y_i(k),y_i(k-1),y_i(k-2) the output of the ith subsystem at time k-1 and time k-2, u_i(k-1) is the input of the ith subsystem at time k-1, u_j(k-1) is the input of the jth subsystem at time k-1,

the influence of the input of other subsystems on the output of the ith subsystem at the moment of k-1 is reflected;

step 1.4, obtaining the first-order difference after the discretization model orientation:

step 1.5, selecting a state variable output by the ith subsystem:

Δx_i,j(k)＝[Δy_i(k),Δy_i(k-1),Δu_i(k-1),…,Δu_i(k-d)，Δu_j(k-1),…,Δu_j(k-d)]^T

wherein d is

The model of the ith subsystem can be found as:

Δx_i,j(k+1)＝A_i,j,mΔx_i,j(k)+B_i,j,mΔu_i(k)+D_i,j,mΔu_j(k)

Δy_i(k+1)＝C_i,j,mΔx_i,j(k+1)

wherein the content of the first and second substances,

B_i,j,m＝[0 0 1 0 0 … 0]^T；

C_i,j,m＝[1 0 0 … 0 0 0 0]；

D_i，j，m＝[0 … 0 0 1 0 … 0]；

A_i,j,mis a (2d +2) × (2d +2) matrix, B_i,j,m，C_i,j,m，D_i,j,mIs a (2d +2) -dimensional vector;

step 1.6 establishment of novel improved model for distributed model predictive control

And further converting the model of the ith subsystem into a novel improved model containing output tracking errors and state variables:

z_i(k+1)＝Az_i(k)+BΔu_i(k)+DΔu_j(k)+CΔr_i(k+1)

wherein the content of the first and second substances,

e_i(k)＝y_i(k)-r_i(k)，Δe_i(k)＝Δy_i(k)-Δr_i(k)，

e_i(k+1)＝e_i(k)+Δe_i(k+1)

＝e_i(k)+Δy_i(k+1)-Δr_i(k+1)

＝e_i(k)+C_i,j,mΔx_i,j,(k)-Δr_i(k+1)

＝e_i(k)+C_i,j,mA_i,j,mΔx_i,j(k)+C_i,j,mB_i,j,mΔu_i(k)+C_i,j,mD_i,j,mΔu_j(k)-Δr_i(k+1)

r_i(k) for reference trajectory,. DELTA.r_i(k +1) is the reference track increment, e_i(k)，e_iAnd (k +1) is the error between the system output and the reference track at the moment k and k +1 respectively.

Further, step 2 is specifically as follows:

step 2.1 according to step 1.6, a vector form of model output of the ith subsystem at the future k + i moment can be obtained, and the vector form is obtained by sorting:

Z_i＝Gz_i(k)+S₁ΔU_i+S₂ΔU_j+ΨΔR_i

wherein the content of the first and second substances,

ΔU_i＝[Δu_i(k) Δu_i(k+1) … Δu_i(k+M-1)]^T

ΔU_j＝[Δu_j(k) Δu_j(k+1) … Δu_j(k+M-1)]^T

ΔR_i＝[Δr_i(k+1) Δr_i(k+2) … Δr_i(k+P)]^T

r_i(k+m)＝λ^my_i(k)+(1-λ^m)c_i(k),m＝1,2,…,P

step 2.2 based on the idea of rolling optimization, the ith subsystem objective function is taken:

J_i＝Z_i ^TQ_i,mZ_i+ΔU_i ^TR_i,mΔU_i

wherein Q is_i,m＝block diag(Q_i,1,Q_i,2,…,Q_i,P-1,Q_i,P) λ is a reference trajectory softening factor, c_i(k) Is the set value of the ith subsystem at the moment k;

step 2.3 implements the ith subsystem objective function:

the control increment and the control quantity under the influence of other subsystems at the kth moment of the ith subsystem can be obtained as follows:

ΔU_i(k)＝-(S₁ ^TQ_i,mS₁+R_i,m)^-1S₁ ^TQ_i,m(Gz_i(k)+S₂ΔU_j+ΨΔR_i)

u_i(k)＝[1 0 … 0]ΔU_i(k)+u_i(k-1)

control increment Δ U with ith subsystem_i(k) Obtaining the actual control quantity u of the ith subsystem_i(k)＝[1 0 … 0]ΔU_i(k)+u_i(k-1) operating on the ith subsystem;

then the control quantity of other subsystems can be obtained in the same way;

step 2.4 explicit distributed model predictive control controller design based on novel improved model

For this complex large system, the constraint is assumed to exist:

Δu_i,min≤ΔU_i≤Δu_i,max

u_i,min≤U_i≤u_i,max

y_i,min≤y_i(k+1)≤y_i,max

wherein, Δ u_i,minAnd Δ u_i,maxConstrained input delta values, u, of minimum and maximum, respectively_i,minAnd u_i,maxConstraint control input values, y, being minimum and maximum, respectively_i,minAnd y_i,maxMinimum and maximum constraint output values, respectively.

Here, we further convert the constraint conditions of the ith subsystem into:

W_i,1ΔU_i≤d_i,1

W_i,2ΔU_i≤d_i,2

W_i,3ΔU_i≤d_i,3

wherein the content of the first and second substances,

d_i,1＝[-Δu_i,min,…,-Δu_i,min,Δu_i,max,…,Δu_i,max]^T

d_i,2＝[-(u_i,min-u_i(k-1)),…,-(u_i,min-u_i(k-1)),

(u_i,max-u_i(k-1)),…,(u_i,max-u_i(k-1))]^T

d_i,3＝[-{φ[y_i,min-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i},…,-{φ[y_i,min-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i},

{φ[y_i,max-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i},…,{φ[y_i,max-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i}]^T

wherein phi is from Z_iExtract Δ y therefrom_i(k) A coefficient matrix of (a);

step 2.5 thus the ith subsystem solution problem of the explicit distributed model predictive control under the novel improved model is:

it is simplified and can be further rewritten as

A_i＝S₁ ^TQ_i,m[Gz_i(k)+S₂ΔU_j+ΨΔR_i]

B_i＝S₁ ^TQ_i,mS₁+R_i,m,C_iRepresents a constant

Step 2.6 when no constraint exists, the control increment of the ith subsystem at the kth moment of the distributed predictive control can be obtained as

ΔU_i(k)＝-(S₁ ^TQ_i,mS₁+R_i,m)^-1S₁ ^TQ_i,m(Gz_i(k)+S₂ΔU_j+ΨΔR_i)

Δu_i(k)＝[1 0 … 0]ΔU_i(k)

Step 2.7, when the solved control increment analytical solution does not meet the constraint condition, activating the constraint condition; firstly, converting the activated inequality constraint condition into an equality constraint condition, wherein the converted equality constraint condition is as follows:

W_i,εΔU_i＝d_i,ε

where ε represents the collective number of constraints activated at time k, W_i,ε,d_i,εRepresenting the specifically activated constraint matrix, satisfying the following conditions:

solutions that do not satisfy the constraint condition may be passed through an activated constraint matrix W_i,εThe value range and the kernel space of (A) are solved by performing a decomposition operation, and W can be first solved_i,ε ^TTo carry out

Decomposition of

Upper type double-side ride

Can further obtain

Further transformation can obtain

It can be seen that

To be activated constraint matrix W_i,εA set of bases of the null space of (a),

is W_i,εA value domain space of;

therefore, Δ U_iIs divided into a value domain and a nuclear space for solving

Wherein, Delta U_a,iTo form W_i,εOf the value space of, Δ U_b,iIs W_i,εOf the nuclear space

General formula W_i,εΔU_i＝d_i,εTwo sides of the same ride E_i ^TIs obtained by

Because of the matrix

For the lower triangular matrix, further obtain

Will be provided with

Substituting the objective function

minJ_i＝2A_i,1 ^TΔU_a,i+2A_i,2 ^TΔU_b,i+ΔU_a,i ^TB_i,11ΔU_a,i+ΔU_a,i ^TB_i,12ΔU_b,i

+ΔU_b,i ^TB_i,21ΔU_a,i+ΔU_b,i ^TB_i,22ΔU_b,i+C_i

Wherein the content of the first and second substances,

and can be derived

Therefore, the control increment after the constraint influence of the ith subsystem of the distributed model predictive control under the novel improved model is eliminated is obtained as follows:

step 2.8, the solution of the control increment at the kth time of the ith subsystem of the distributed model predictive control under the improved model is summarized as follows:

novel distributed model predictive control improved model for new round of Nash optimal solution in ith subsystem at kth moment

When the whole system meets the convergence condition, the control increment of the whole system can be further calculated as follows:

ΔU^l+1(k)＝[ΔU₁ ^l+1(k),ΔU₂ ^l+1(k),…,ΔU_N ^l+1(k)]

then, further performing rolling optimization to solve the control increment at the next moment, so as to obtain an explicit analytic solution of the control increment at each moment of the distributed model predictive control system under the whole novel improved model;

and finally, taking the obtained control increment first term as an instant control law: Δ u_i(k)＝[1 0 … 0]ΔU_i(k) And the actual control quantity u of the ith subsystem is used_i(k)＝u_i(k-1)+Δu_i(k) Acting on the respective subsystem; and repeating the steps, performing rolling optimization on the system, and solving the optimal solution at the next moment so as to complete the control optimization of the whole process.

The invention has the beneficial effects that: the invention provides a chemical process distributed control method for explicit model predictive control optimization. The method combines the tracking error to establish a novel improved model, and designs a novel distributed model predictive controller by using an explicit model predictive control method, thereby ensuring the overall performance of the system, further perfecting the processing of the controller on the system constraint, improving the overall control performance of the system and providing technical support for the actual industrial process with higher requirements.

Detailed Description

Taking the control of the hearth pressure of a common industrial heating furnace in the chemical process as an example:

the furnace pressure control system of an industrial heating furnace is a typical multivariable coupling process and is adjusted by controlling the valve opening of a flue damper.

Step 1, establishing a hearth pressure control system model of an industrial heating furnace, wherein the specific method comprises the following steps:

1.1 establishment of furnace pressure control System model of Industrial heating furnace

Considering a multivariable N-input-N-output large-scale system in a furnace pressure control system as a furnace subsystem consisting of N first-order inertia plus pure hysteresis (FOPDT) models, the N-input-N-output multivariable FOPDT system can be obtained in the following form:

wherein, K_ijThe steady-state gain T of the jth (j is more than or equal to 1 and less than or equal to n) input to the ith (i is more than or equal to 1 and less than or equal to n) output of the furnace chamber pressure control system of the heating furnace_ijTime constant, tau, of jth input to ith output of furnace pressure control system_ijThe lag time of the jth input to the ith output of the furnace hearth pressure control system of the heating furnace.

1.2, dispersing an N-input N-output heating furnace hearth pressure control system into N hearth subsystems according to a distributed predictive control idea;

then in the furnace pressure control system of the heating furnace, the transfer function of the input of the jth furnace subsystem to the output of the ith furnace subsystem is:

1.3 at sample time T_sDiscretizing the first-order plus pure hysteresis model of the ith subsystem under the condition to obtain the following model:

here, let

The above equation can be simplified to:

wherein, y_i(k),y_i(k-1),y_i(k-2) furnace pressures at time k-1 and k-2 of the ith furnace subsystem, u_i(k-1) is the valve opening of the ith furnace subsystem at the time k-1, u_j(k-1) is the valve opening of the jth furnace subsystem at the time k-1,

the influence of the opening of the valves of other furnace subsystems on the furnace pressure of the ith furnace subsystem at the moment of k-1 is reflected.

1.4 taking the first difference after the discretized model orientation to obtain:

1.5, selecting the state variable output by the ith furnace subsystem:

Δx_i,j(k)＝[Δy_i(k),Δy_i(k-1),Δu_i(k-1),…,Δu_i(k-d)，Δu_j(k-1),…,Δu_j(k-d)]^T

wherein d is

The improved model of the ith furnace subsystem can be obtained as follows:

Δx_i,j(k+1)＝A_i,j,mΔx_i,j(k)+B_i,j,mΔu_i(k)+D_i,j,mΔu_j(k)

Δy_i(k+1)＝C_i,j,mΔx_i,j(k+1)

wherein the content of the first and second substances,

B_i,j,m＝[0 0 1 0 0 … 0]^T；

C_i,j,m＝[1 0 0 … 0 0 0 0]；

D_i,j,m＝[0 … 0 0 1 0 … 0]；

A_i,j,mis a (2d +2) × (2d +2) matrix, B_i,j,m，C_i,j,m，D_i,j,mIs a (2d +2) -dimensional vector.

1.6 establishment of novel improved model of furnace pressure control system of heating furnace

And further converting the model of the ith furnace subsystem into a novel improved model containing output tracking errors and state variables:

z_i(k+1)＝Az_i(k)+BΔu_i(k)+DΔu_j(k)+CΔr_i(k+1)

wherein the content of the first and second substances,

e_i(k)＝y_i(k)-r_i(k)，Δe_i(k)＝Δy_i(k)-Δr_i(k)，

e_i(k+1)＝e_i(k)+Δe_i(k+1)

＝e_i(k)+Δy_i(k+1)-Δr_i(k+1)

＝e_i(k)+C_i,j,mΔx_i,j,(k)-Δr_i(k+1)

＝e_i(k)+C_i,j,mA_i,j,mΔx_i,j(k)+C_i,j,mB_i,j,mΔu_i(k)+C_i,j,mD_i,j,mΔu_j(k)-Δr_i(k+1)

r_i(k) to set the reference trajectory of the pressure, Δ r_i(k +1) reference trace increment for set pressure, e_i(k)，e_iAnd (k +1) is the error between the system output and the reference track at the moment k and k +1 respectively.

Step 2, designing a controller of a furnace pressure control system of a heating furnace, which comprises the following specific steps:

2.1 according to step 1.6, one can obtain:

Z_i＝Gz_i(k)+S₁ΔU_i+S₂ΔU_j+ΨΔR_i

wherein the content of the first and second substances,

ΔU_i＝[Δu_i(k) Δu_i(k+1) … Δu_i(k+M-1)]^T

ΔU_j＝[Δu_j(k) Δu_j(k+1) … Δu_j(k+M-1)]^T

ΔR_i＝[Δr_i(k+1) Δr_i(k+2) … Δr_i(k+P)]^T

r_i(k+m)＝λ^my_i(k)+(1-λ^m)c_i(k),m＝1,2,…,P

2.2 based on the idea of rolling optimization, taking an ith furnace subsystem objective function:

J_i＝Z_i ^TQ_i,mZ_i+ΔU_i ^TR_i,mΔU_i

wherein Q is_i,m＝block diag(Q_i,1,Q_i,2,…,Q_i,P-1,Q_i,P) λ is a reference trajectory softening factor, c_i(k) The furnace pressure is set for the ith furnace subsystem at time k.

2.3, the objective function of the ith furnace subsystem is implemented as follows:

the valve opening amount and the valve opening under the influence of other furnace subsystems at the kth moment of the ith furnace subsystem can be obtained as follows:

ΔU_i(k)＝-(S₁ ^TQ_i,mS₁+R_i,m)^-1S₁ ^TQ_i,m(Gz_i(k)+S₂ΔU_j+ΨΔR_i)

u_i(k)＝[1 0 … 0]ΔU_i(k)+u_i(k-1)

valve opening increment delta U by using ith furnace subsystem_i(k) Obtaining the actual valve opening u of the ith furnace subsystem_i(k)＝[1 0 … 0]ΔU_i(k)+u_i(k-1) operating on the ith furnace subsystem;

and then obtaining the valve opening of other furnace subsystems in the same way.

2.4 related processing method of controller when there is constraint in furnace pressure control system of heating furnace

For the furnace pressure control system of the heating furnace, the constraint conditions are assumed to exist:

Δu_i,min≤ΔU_i≤Δu_i,max

u_i,min≤U_i≤u_i,max

y_i,min≤y_i(k+1)≤y_i,max

wherein, Δ u_i,minAnd Δ u_i,maxRespectively minimum and maximum valve opening incremental values u_i,minAnd u_i,maxMinimum and maximum valve opening values, y, respectively_i,minAnd y_i,maxRespectively, minimum and maximum furnace pressure values.

Here, we further convert the constraint conditions of the ith furnace subsystem into:

W_i,1ΔU_i≤d_i,1

W_i,2ΔU_i≤d_i,2

W_i,3ΔU_i≤d_i,3

wherein the content of the first and second substances,

d_i,1＝[-Δu_i,min,…,-Δu_i,min,Δu_i,max,…,Δu_i,max]^T

d_i,2＝[-(u_i,min-u_i(k-1)),…,-(u_i,min-u_i(k-1)),

(u_i,max-u_i(k-1)),…,(u_i,max-u_i(k-1))]^T

d_i,3＝[-{φ[y_i,min-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i},…,-{φ[y_i,min-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i},

{φ[y_i,max-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i},…,{φ[y_i,max-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i}]^T

wherein phi is from Z_iExtract Δ y therefrom_i(k) The coefficient matrix of (2).

2.5 obtaining the solution problem of the ith furnace subsystem of the furnace pressure control system of the heating furnace as follows:

it is simplified and can be further rewritten as

A_i＝S₁ ^TQ_i,m[Gz_i(k)+S₂ΔU_j+ΨΔR_i]

B_i＝S₁ ^TQ_i,mS₁+R_i,m,C_iRepresents a constant

2.6 when no constraint exists, the valve opening increment of the ith furnace subsystem of the furnace pressure control system of the heating furnace at the kth moment can be obtained as

ΔU_i(k)＝-(S₁ ^TQ_i,mS₁+R_i,m)^-1S₁ ^TQ_i,m(Gz_i(k)+S₂ΔU_j+ΨΔR_i)

Δu_i(k)＝[1 0 … 0]ΔU_i(k)

And 2.7 when the solved valve opening increment analytical solution does not meet the constraint condition, activating the constraint condition at the moment. Here, the activated inequality constraint is first converted into an equality constraint, and the converted equality constraint is as follows:

W_i,εΔU_i＝d_i,ε

where ε represents the collective number of constraints activated at time k, W_i,ε,d_i,εRepresenting the specifically activated constraint matrix, satisfying the following conditions:

solutions that do not satisfy the constraint condition may be passed through an activated constraint matrix W_i,εThe value range and the kernel space of (A) are solved by performing a decomposition operation, and W can be first solved_i,ε ^TTo carry out

Decomposition of

Upper type double-side ride

Can further obtain

Further transformation can obtain

It can be seen that

To be activated constraint matrix W_i,εA set of bases of the null space of (a),

is W_i,εThe value range space of (a).

Therefore, Δ U_iIs divided into a value range and a kernel spaceSolving in two parts

Wherein, Delta U_a,iTo form W_i,εOf the value space of, Δ U_b,iIs W_i,εOf the nuclear space

General formula W_i,εΔU_i＝d_i,εTwo sides of the same ride E_i ^TIs obtained by

Because of the matrix

For the lower triangular matrix, further obtain

Will be provided with

Substituting the objective function

minJ_i＝2A_i,1 ^TΔU_a,i+2A_i,2 ^TΔU_b,i+ΔU_a,i ^TB_i,11ΔU_a,i+ΔU_a,i ^TB_i,12ΔU_b,i

+ΔU_b,i ^TB_i,21ΔU_a,i+ΔU_b,i ^TB_i,22ΔU_b,i+C_i

Wherein the content of the first and second substances,

and can be derived

Thus, the valve opening increment after the ith furnace subsystem of the furnace pressure control system of the heating furnace eliminates the constraint influence is obtained as follows:

2.8 the solving summary of the valve opening increment of the ith furnace subsystem of the furnace pressure control system of the heating furnace at the kth moment is as follows:

similarly, the new valve opening increment of the ith furnace subsystem of the furnace pressure control system of the heating furnace at the kth moment can be obtained

When the whole system meets the convergence condition, the valve opening increment of the whole system can be further obtained as follows:

ΔU^l+1(k)＝[ΔU₁ ^l+1(k),ΔU₂ ^l+1(k),…,ΔU_N ^l+1(k)]

and then, the solution of the valve opening increment at the next moment can be further optimized in a rolling mode, so that an explicit analytic solution of the valve opening increment at each moment of the whole heating furnace hearth pressure control system is obtained.

And finally, taking the head term of the obtained valve opening increment as a real-time control law: Δ u_i(k)＝[1 0 … 0]ΔU_i(k) And the actual valve opening u of the ith furnace subsystem of the furnace pressure control system of the heating furnace_i(k)＝u_i(k-1)+Δu_i(k) Acting on the respective furnace subsystems. And repeating the steps, performing rolling optimization on the system, and solving the optimal valve opening at the next moment so as to complete the control optimization of the whole heating furnace hearth pressure control system.

Claims (1)

1. A chemical process distributed control method for explicit model predictive control optimization comprises the following steps:

step 1, establishing a novel improved model by distributed model predictive control;

step 2, improving the design of a distributed model predictive control controller under the model;

the step 1 is specifically as follows:

step 1.1 establishment of distributed model predictive control model

Considering a multivariable N-in-N-out large-scale system in an industrial process as consisting of N subsystems of first-order inertia plus a pure hysteresis FOPDT model, an N-in-N-out multivariable FOPDT system of the following form can be obtained:

wherein, K_ijThe steady state gain, T, of the j (1 ≦ j ≦ n) th input to the i (1 ≦ i ≦ n) th output for the multivariable process object_ijTime constant, τ, for the jth input to ith output of a multivariable process object_ijA lag time for a jth input to an ith output of the multivariable process object;

step 1.2, dispersing an N-input N-output large-scale system into N subsystems according to a distributed predictive control idea;

in the multivariate selection process, the transfer function of the input of the jth subsystem to the output of the ith subsystem is:

step 1.3 at sampling time T_sDiscretizing the first-order plus pure hysteresis model of the ith subsystem under the condition to obtain the following model:

here, let

The above equation can be simplified to:

wherein, y_i(k),y_i(k-1),y_i(k-2) the output of the ith subsystem at time k-1 and time k-2, u_i(k-1) is the input of the ith subsystem at time k-1, u_j(k-1) is the input of the jth subsystem at time k-1,

the influence of the input of other subsystems on the output of the ith subsystem at the moment of k-1 is reflected;

step 1.4, obtaining the first-order difference after the discretization model orientation:

step 1.5, selecting a state variable output by the ith subsystem:

Δx_i,j(k)＝[Δy_i(k),Δy_i(k-1),Δu_i(k-1),…,Δu_i(k-d)，Δu_j(k-1),…,Δu_j(k-d)]^T

wherein d is

The model of the ith subsystem can be found as:

Δx_i,j(k+1)＝A_i,j,mΔx_i,j(k)+B_i,j,mΔu_i(k)+D_i,j,mΔu_j(k)

Δy_i(k+1)＝C_i,j,mΔx_i,j(k+1)

wherein the content of the first and second substances,

B_i,j,m＝[0 0 1 0 0 … 0]^T；

C_i,j,m＝[1 0 0 … 0 0 0 0]；

D_i,j,m＝[0 … 0 0 1 0 … 0]；

A_i,j,mis a (2d +2) × (2d +2) matrix, B_i,j,m，C_i,j,m，D_i,j,mIs a (2d +2) -dimensional vector;

step 1.6 establishment of novel improved model for distributed model predictive control

And further converting the model of the ith subsystem into a novel improved model containing output tracking errors and state variables:

z_i(k+1)＝Az_i(k)+BΔu_i(k)+DΔu_j(k)+CΔr_i(k+1)

wherein the content of the first and second substances,

e_i(k)＝y_i(k)-r_i(k)，Δe_i(k)＝Δy_i(k)-Δr_i(k)，

e_i(k+1)＝e_i(k)+Δe_i(k+1)

＝e_i(k)+Δy_i(k+1)-Δr_i(k+1)

＝e_i(k)+C_i,j,mΔx_i,j,(k)-Δr_i(k+1)

＝e_i(k)+C_i,j,mA_i,j,mΔx_i,j(k)+C_i,j,mB_i,j,mΔu_i(k)+C_i,j,mD_i,j,mΔu_j(k)-Δr_i(k+1)

r_i(k) for reference trajectory,. DELTA.r_i(k +1) is the reference track increment, e_i(k)，e_i(k +1) are errors between the system output and the reference track at the moment k and k +1 respectively;

the step 2 is as follows:

step 2.1 according to step 1.6, a vector form of model output of the ith subsystem at the future k + i moment can be obtained, and the vector form is obtained by sorting:

Z_i＝Gz_i(k)+S₁ΔU_i+S₂ΔU_j+ΨΔR_i

wherein the content of the first and second substances,

ΔU_i＝[Δu_i(k) Δu_i(k+1) … Δu_i(k+M-1)]^T

ΔU_j＝[Δu_j(k) Δu_j(k+1) … Δu_j(k+M-1)]^T

ΔR_i＝[Δr_i(k+1) Δr_i(k+2) … Δr_i(k+P)]^T

r_i(k+m)＝λ^my_i(k)+(1-λ^m)c_i(k),m＝1,2,…,P

step 2.2 based on the idea of rolling optimization, the ith subsystem objective function is taken:

J_i＝Z_i ^TQ_i,mZ_i+ΔU_i ^TR_i,mΔU_i

wherein Q is_i,m＝blockdiag(Q_i,1,Q_i,2,…,Q_i,P-1,Q_i,P) λ is a reference trajectory softening factor, c_i(k) Is the set value of the ith subsystem at the moment k;

step 2.3 implements the ith subsystem objective function:

the control increment and the control quantity under the influence of other subsystems at the kth moment of the ith subsystem can be obtained as follows:

ΔU_i(k)＝-(S₁ ^TQ_i,mS₁+R_i,m)^-1S₁ ^TQ_i,m(Gz_i(k)+S₂ΔU_j+ΨΔR_i)

u_i(k)＝[1 0 … 0]ΔU_i(k)+u_i(k-1)

control increment Δ U with ith subsystem_i(k) Obtaining the actual control quantity u of the ith subsystem_i(k)＝[1 0 … 0]ΔU_i(k)+u_i(k-1) operating on the ith subsystem;

then the control quantity of other subsystems can be obtained in the same way;

step 2.4 explicit distributed model predictive control controller design based on novel improved model

For this N-input N-output multivariable FOPDT system, assuming that constraints exist:

Δu_i,min≤ΔU_i≤Δu_i,max

u_i,min≤U_i≤u_i,max

y_i,min≤y_i(k+1)≤y_i,max

wherein, Δ u_i,minAnd Δ u_i,maxConstrained input delta values, u, of minimum and maximum, respectively_i,minAnd u_i,maxConstraint control input values, y, being minimum and maximum, respectively_i,minAnd y_i,maxMinimum and maximum constraint output values, respectively.

Here, we further convert the constraint conditions of the ith subsystem into:

W_i,1ΔU_i≤d_i,1

W_i,2ΔU_i≤d_i,2

W_i,3ΔU_i≤d_i,3

wherein the content of the first and second substances,

d_i,1＝[-Δu_i,min,…,-Δu_i,min,Δu_i,max,…,Δu_i,max]^T

d_i,2＝[-(u_i,min-u_i(k-1)),…,-(u_i,min-u_i(k-1)),(u_i,max-u_i(k-1)),…,(u_i,max-u_i(k-1))]^T

d_i,3＝[-{φ[y_i,min-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i},…,-{φ[y_i,min-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i},{φ[y_i,max-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i},…,{φ[y_i,max-y_i(k)]-Gz_i(k)-S₂ΔU_j-ΨΔR_i}]^T

wherein phi is from Z_iExtract Δ y therefrom_i(k) A coefficient matrix of (a);

step 2.5 thus the ith subsystem solution problem of the explicit distributed model predictive control under the novel improved model is:

it is simplified and can be further rewritten as

A_i＝S₁ ^TQ_i,m[Gz_i(k)+S₂ΔU_j+ΨΔR_i]

B_i＝S₁ ^TQ_i,mS₁+R_i,m,C_iRepresents a constant

Step 2.6 when no constraint exists, the control increment of the ith subsystem at the kth moment of the distributed predictive control can be obtained as

ΔU_i(k)＝-(S₁ ^TQ_i,mS₁+R_i,m)^-1S₁ ^TQ_i,m(Gz_i(k)+S₂ΔU_j+ΨΔR_i)

Δu_i(k)＝[1 0 … 0]ΔU_i(k)

Step 2.7, when the solved control increment analytical solution does not meet the constraint condition, activating the constraint condition; firstly, converting the activated inequality constraint condition into an equality constraint condition, wherein the converted equality constraint condition is as follows:

W_i,εΔU_i＝d_i,ε

where ε represents the collective number of constraints activated at time k, W_i,ε,d_i,εRepresenting the specifically activated constraint matrix, satisfying the following conditions:

solutions that do not satisfy the constraint condition may be passed through an activated constraint matrix W_i,εThe value range and the kernel space of (A) are solved by performing a decomposition operation, and W can be first solved_i,ε ^TTo carry out

Decomposition of

Upper type double-side ride

Can further obtain

Further transformation can obtain

It can be seen that

To be activated constraint matrix W_i,εA set of bases of the null space of (a),

is W_i,εA value domain space of;

therefore, Δ U_iIs divided into a value domain and a nuclear space for solving

Wherein, Delta U_a,iTo form W_i,εOf the value space of, Δ U_b,iIs W_i,εOf the nuclear space

General formula W_i,εΔU_i＝d_i,εTwo sides of the same ride E_i ^TIs obtained by

Because of the matrix

For the lower triangular matrix, further obtain

Will be provided with

Substituting the objective function

minJ_i＝2A_i,1 ^TΔU_a,i+2A_i,2 ^TΔU_b,i+ΔU_a,i ^TB_i,11ΔU_a,i+ΔU_a,i ^TB_i,12ΔU_b,i+ΔU_b,i ^TB_i,21ΔU_a,i+ΔU_b,i ^TB_i,22ΔU_b,i+C_i

Wherein the content of the first and second substances,

and can be derived

Therefore, the control increment after the constraint influence of the ith subsystem of the distributed model predictive control under the novel improved model is eliminated is obtained as follows:

step 2.8, the solution of the control increment at the kth time of the ith subsystem of the distributed model predictive control under the improved model is summarized as follows:

novel distributed model predictive control improved model for new round of Nash optimal solution in ith subsystem at kth moment

When the whole system meets the convergence condition, the control increment of the whole system can be further calculated as follows:

ΔU^l+1(k)＝[ΔU₁ ^l+1(k),ΔU₂ ^l+1(k),…,ΔU_N ^l+1(k)]

then, further performing rolling optimization to solve the control increment at the next moment, so as to obtain an explicit analytic solution of the control increment at each moment of the distributed model predictive control system under the whole novel improved model;

and finally, taking the obtained control increment first term as an instant control law: Δ u_i(k)＝[1 0 … 0]ΔU_i(k) And the actual control quantity u of the ith subsystem is used_i(k)＝u_i(k-1)+Δu_i(k) Acting on the respective subsystem; and repeating the steps, performing rolling optimization on the system, and solving the optimal solution at the next moment so as to complete the control optimization of the whole process.