CN103605284B - The cracking waste plastics stove hearth pressure control method that dynamic matrix control is optimized - Google Patents

Abstract

The present invention discloses the cracking waste plastics stove hearth pressure control method that dynamic matrix control is optimized. First the inventive method sets up the model of furnace pressure object based on the step response data of cracking waste plastics stove furnace pressure object, excavates basic plant characteristic; Then go according to the characteristic of dynamic matrix control to adjust the parameter of corresponding PI controller; Finally cracking waste plastics stove fire box temperature object is implemented PI control. The present invention proposes a kind of cracking waste plastics stove fire box temperature PI control method optimized based on dynamic matrix control, combine the good control performance of PI control and dynamic matrix control, effectively improve the deficiency of Traditional control method, also promote the development and apply of advanced control algorithm simultaneously.

Description

Dynamic matrix control optimized waste plastic cracking furnace pressure control method

Technical Field

The invention belongs to the technical field of automation, and relates to a waste plastic cracking furnace pressure Proportional Integral (PI) control method based on Dynamic Matrix Control (DMC) optimization.

Background

With the upsizing and complication of modern industrial processes, some traditional control methods are increasingly difficult to meet the actual demands of the industry. Although some advanced process control technologies can greatly improve the production efficiency in theory, the advanced process control technologies are difficult to apply due to the aspects of hardware, cost, implementation difficulty and the like, so that the PID control still occupies the mainstream at present. The current waste plastic cracking furnace hearth pressure control usually adopts Proportional Integral (PI) control. The dynamic matrix control is one of the advanced control methods, the requirement on the model is low, the calculated amount is small, and the method for processing the time delay is simple and easy to implement.

Disclosure of Invention

The invention aims to provide a waste plastic cracking furnace pressure PI control method based on dynamic matrix control optimization aiming at the application defects of the existing advanced control method so as to obtain better actual control performance. The method combines dynamic matrix control and PI control to obtain a PI control method with dynamic matrix control performance. The method not only inherits the excellent performance of dynamic matrix control, but also has simple form and can meet the requirement of actual industrial process.

Firstly, establishing a model of a furnace pressure object based on step response data of the waste plastic cracking furnace pressure object, and excavating basic object characteristics; then, setting parameters of a corresponding PI controller according to the characteristics of dynamic matrix control; and finally, PI control is carried out on the waste plastic cracking furnace temperature object.

The technical scheme of the invention is that a PI control method based on dynamic matrix control optimization is established by means of data acquisition, dynamic matrix establishment, prediction model establishment, prediction mechanism, optimization and the like, and the method can effectively improve the control precision and stability.

The method comprises the following steps:

step (1), establishing a model of a controlled object through real-time step response data of a process object, wherein the specific method comprises the following steps:

a. and (4) giving a step input signal to the controlled object, and recording a step response curve of the controlled object.

b. C, filtering the step response curve obtained in the step a, fitting the step response curve into a smooth curve, and recording step response data corresponding to each sampling moment on the smooth curve, wherein the first sampling moment is T_sThe time interval between two adjacent sampling time is T_sSampling time is in orderSequence is T_s、2T_s、3T_s… …, respectively; the step response of the controlled object will be at a certain time t_NAfter NT, it tends to be stable when a_i(i > N) and a_NWhen the error of (a) and the measurement error are of the same order of magnitude, a can be regarded as_NApproximately equal to the steady state value of the step response. Establishing a model vector a of an object:

a＝[a₁,a₂,…a_N]^Τ

and T is a transposed symbol of the matrix, and N is a modeling time domain.

Step (2), designing a PI controller of a controlled object, wherein the specific method comprises the following steps:

a. and establishing a dynamic matrix of the controlled object by using the model vector a obtained above, wherein the dynamic matrix is in the form of:

$A=\left(\begin{array}{cccc}{a}_1& 0& ...& 0\\ a_2& a_1& ...& 0\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ a_P& a_{P-1}& ...& a_{P-M+1}\end{array}\right)$

wherein A is a dynamic matrix of P × M order of the controlled object, a_iThe method is data of step response, P is an optimized time domain of a dynamic matrix control algorithm, M is a control time domain of the dynamic matrix control algorithm, and M is more than P and less than N.

b. Establishing a model prediction initial response value y of the controlled object at the current k moment_M(k)

Firstly, obtaining a model predicted value y after adding a control increment delta u (k-1) at the moment of k-1_p(k-1)：

y_P(k-1)＝y_M(k-1)+A₀Δu(k-1)

Wherein,

$y_P(k-1)=\left[\begin{array}{c}{y}_1\left(k\right|k-1)\\ y_1(k+1|k-1)\\ .\\ .\\ .\\ y_1(k+N-1|k-1)\end{array}\right],A_0=\left[\begin{array}{c}{a}_1\\ a_2\\ .\\ .\\ .\\ a_N\end{array}\right],y_M\left(k\right)=\left[\begin{array}{c}{y}_0\left(k\right|k-1)\\ y_0(k+1|k-1)\\ .\\ .\\ .\\ y_0(k+N-1|k-1)\end{array}\right]$

y₁(k|k-1),y₁(k+1|k-1),…,y₁(k + N-1| k-1) respectively represents the model predicted value of the controlled object after adding the control increment delta u (k-1) to k, k +1, …, k + N-1 at the time k-1, y₀(k|k-1),y₀(k|k-1),…y₀(k + N-1| k-1) represents the initial predicted value at time k-1 versus time k, k +1, …, k + N-1, A₀For the matrix established for the step response data, Δ u (k-1) is the input control increment at time k-1.

Then obtaining a model prediction error value e (k) of the controlled object at the moment k:

e(k)＝y(k)-y₁(k|k-1)

where y (k) represents the actual output value of the controlled object measured at time k.

Further obtaining a correction value y output by the k-time model_cor(k):

y_cor(k)＝y_M(k-1)+h*e(k)

Wherein,

$y_{\mathrm{cor}}\left(k\right)=\left[\begin{array}{c}{y}_{\mathrm{cor}}\left(k\right|k)\\ y_{\mathrm{cor}}(k+1|k)\\ .\\ .\\ .\\ y_{\mathrm{cor}}(k+N-1|k)\end{array}\right],h=\left[\begin{array}{c}1\\ \mathrm{\α}\\ .\\ .\\ .\\ \mathrm{\α}\end{array}\right]$

y_cor(k|k),y_cor(k+1|k),…y_cor(k + N-1| k) represents the correction value of the controlled object model at the time k, h is the weight matrix for error compensation, and α is the errorA difference correction factor.

Finally obtaining the initial response value y of the model prediction at the moment k_M(k)：

y_M(k)＝Sy_cor(k)

Wherein S is a state transition matrix of NxN order,

c. calculating the predicted output value y of the controlled object under M continuous control increments delta u (k), … and delta u (k + M-1)_PMThe specific method comprises the following steps:

y_PM(k)＝y_p0(k)+AΔu_M(k)

$y_{\mathrm{PM}}\left(k\right)=\left[\begin{array}{c}{y}_M(k+1|k)\\ y_M(k+2|k)\\ .\\ .\\ .\\ y_M(k+P|k)\end{array}\right],y_{P0}\left(k\right)=\left[\begin{array}{c}{y}_0(k+1|k)\\ y_0(k+2|k)\\ .\\ .\\ .\\ y_0(k+P|k)\end{array}\right],\mathrm{\Δ}{u}_M\left(k\right)=\left[\begin{array}{c}\mathrm{\Δu}\left(k\right)\\ \mathrm{\Δu}(k+1)\\ .\\ .\\ .\\ \mathrm{\Δu}(k+M-1)\end{array}\right]$

wherein, y_P0(k) Is y_M(k) The first P term, y_M(k+1|k),y_M(k+2|k),…,y_M(k + P | k) is the model predicted output value at time k versus time k +1, k +2, …, k + P.

d. Let the control time domain M =1 of the controlled object, select the objective function j (k) of the controlled object, the form is as follows:

minJ(k)＝Q(ref(k)-y_PM(k))²+rΔu²(k)=Q(ref(k)-y_P0(k)-AΔu(k))²+rΔu²(k)

ref(k)＝[ref₁(k),ref₂(k),…,ref_P(k)]^Τ

ref_i(k)＝βⁱy(k)+(1-βⁱ)c(k),Q＝diag(q₁,q₂,…,q_P)

wherein Q is an error weighting matrix, Q₁,q₂,…,q_PThe parameter values of the weighting matrix, β softening coefficients, c (k) set values, and r ═ diag (r)₁,r₂,…r_M) To control the weighting matrix, r₁,r₂,…r_MFor controlling the parameters of the weighting matrix, ref (k) is the reference trajectory of the system, ref_i(k) Is the value of the ith reference point in the reference track.

e. Converting the control quantity u (k):

u(k)＝u(k-1)+K_p(k)(e₁(k)-e₁(k-1))+K_i(k)e₁(k)

e(k)＝c(k)-y(k)

substituting u (k) into the objective function in the step d to solve the parameters in the PI controller to obtain:

u(k)＝u(k-1)+w(k)^ΤE(k)

w(k)＝[w₁(k),w₂(k)]^Τ

w₁(k)＝K_p(k)+K_i(k)，w₂(k)＝-K_p(k)

E(k)＝[e₁(k),e₁(k-1)]^Τ

wherein Kp (K), K_i(k) Proportional and differential parameters of the PI controller at the time k, e₁(k) T is the transposed symbol of the matrix for the error between the reference trajectory value and the actual output value at time k.

By combining the above formulas, the following can be obtained:

$w\left(k\right)=\frac{{(\mathrm{ref}\left(k\right)-y_{P0}\left(k\right))}^T\mathrm{QAE}}{(A^T\mathrm{QA}+r)E^{T}E}$

further, it is possible to obtain:

K_p(k)＝-w₂(k)

K_i(k)＝w₁(k)-K_P(k)

f. obtaining parameter K of PI controller_p(k)、K_i(k) The control amount u (K) is applied to the controlled object, and u (K) is equal to u (K-1) + K_p(k)(e₁(k)-e₁(k-1))+K_i(k)e₁(k)。

h. At the next moment, the solution of the new parameter k of the PI controller is continued according to the steps from b to f_P(k+1)、k_iThe values of (k +1) are cycled through in sequence.

The invention provides a waste plastic cracking furnace temperature PI control method based on dynamic matrix control optimization, which combines good control performance of PI control and dynamic matrix control, effectively improves the defects of the traditional control method, and promotes the development and application of an advanced control algorithm.

Detailed Description

Taking the control of the pressure process of the waste plastic cracking furnace hearth as an example:

the waste plastic cracking furnace hearth pressure object is a process with hysteresis, and the adjusting means adopts the opening degree of a flue damper.

Step (1), establishing a model of a controlled object through real-time step response data of a waste plastic cracking furnace hearth pressure object, wherein the specific method comprises the following steps:

a. and (3) giving a step input signal to the waste plastic cracking furnace hearth, and recording a step response curve.

b. Filtering the corresponding step response curve, fitting the corresponding step response curve into a smooth curve, and recording the step response data corresponding to each sampling time on the smooth curve, wherein the first sampling time is T_sThe time interval between two adjacent sampling time is T_sThe sampling time sequence is T_s、2T_s、3T_s… …, respectively; response value a of the shutter opening_iWill be at a certain time t_NAfter NT, it tends to be stable when a_i(i > N) and a_NWhen the error of (a) and the measurement error are of the same order of magnitude, a can be regarded as_NApproximately equal to the step response steady state value. Establishing a model vector a of an object:

a＝[a₁,a₂,…a_N]^Τ

where T is the transposed symbol of the matrix and N is the modeled time domain.

Step (2) designing a PI controller of waste plastic cracking furnace pressure, wherein the specific method comprises the following steps:

a. and establishing a dynamic matrix of the waste plastic cracking furnace pressure by using the model vector a obtained above, wherein the form of the dynamic matrix is as follows:

$A=\left(\begin{array}{cccc}{a}_1& 0& ...& 0\\ a_2& a_1& ...& 0\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ a_P& a_{P-1}& ...& a_{P-M+1}\end{array}\right)$

wherein A is a P × M-order dynamic matrix of the waste plastic cracking furnace hearth pressure, a_iThe data is the data of the opening degree of the baffle of the waste plastic cracking furnace hearth pressure, P is the optimization time domain of the dynamic matrix control algorithm, M is the control time domain of the dynamic matrix control algorithm, and M is more than P and less than N.

b. Establishing an initial predicted value y of the current k moment of the waste plastic cracking furnace hearth pressure_M(k)

Firstly, obtaining a model predicted value y after the baffle opening is increased by delta u (k-1) at the moment of k-1_p(k-1):

y_P(k-1)＝y_M(k-1)+A₀Δu(k-1)

Wherein,

$y_P(k-1)=\left[\begin{array}{c}{y}_1\left(k\right|k-1)\\ y_1(k+1|k-1)\\ .\\ .\\ .\\ y_1(k+N-1|k-1)\end{array}\right],A_0=\left[\begin{array}{c}{a}_1\\ a_2\\ .\\ .\\ .\\ a_N\end{array}\right],y_M\left(k\right)=\left[\begin{array}{c}{y}_0\left(k\right|k-1)\\ y_0(k+1|k-1)\\ .\\ .\\ .\\ y_0(k+N-1|k-1)\end{array}\right]$

y₁(k|k-1),y₁(k+1|k-1),…,y₁(k + N-1| k-1) represents the model prediction value of the waste plastic cracking furnace hearth pressure at the time of k-1 after adding delta u (k-1) to the time of k, k +1, …, k + N-1, y₀(k|k-1),y₀(k|k-1),…y₀(k + N-1. sub. k-1) represents an initial predicted value of the waste plastic cracking furnace hearth pressure at the time of k-1 versus the time of k, k +1, …, k + N-1, A₀In order to establish a matrix from the waste plastic cracking furnace pressure step response data, Δ u (k-1) is the damper opening control increment of the waste plastic cracking furnace pressure at the time of k-1.

Then obtaining a model prediction error value e (k) of the pressure of the waste plastic cracking furnace at the time k:

e(k)＝y(k)-y₁(k|k-1)

wherein y (k) represents an actual output value of the waste plastic cracking furnace pressure measured at time k.

Further obtaining a corrected value y of model output of the waste plastic cracking furnace pressure at the time k_cor(k):

y_cor(k)＝y_M(k-1)+h*e(k)

Wherein,

$y_{\mathrm{cor}}\left(k\right)=\left[\begin{array}{c}{y}_{\mathrm{cor}}\left(k\right|k)\\ y_{\mathrm{cor}}(k+1|k)\\ .\\ .\\ .\\ y_{\mathrm{cor}}(k+N-1|k)\end{array}\right],h=\left[\begin{array}{c}1\\ \mathrm{\α}\\ .\\ .\\ .\\ \mathrm{\α}\end{array}\right]$

y_cor(k|k),y_cor(k+1|k),…y_corand (k + N-1| k) respectively represents the corrected value of the model of the waste plastic cracking furnace hearth pressure at the time k, h is a weight matrix of error compensation, and α is an error correction coefficient.

Finally obtaining the initial predicted value y of the model of the waste plastic cracking furnace hearth pressure at the time k_M(k)：

y_M(k)＝Sy_cor(k)

Wherein S is a state transition matrix of NxN order,

c. calculating the predicted output value y of the waste plastic cracking furnace hearth pressure under M continuous control increments of delta u (k), …, delta u (k + M-1)_PMThe specific method comprises the following steps:

y_PM(k)＝y_P0(k)+AΔu_M(k)

wherein,

$y_{\mathrm{PM}}\left(k\right)=\left[\begin{array}{c}{y}_M(k+1|k)\\ y_M(k+2|k)\\ .\\ .\\ .\\ y_M(k+P|k)\end{array}\right],y_{P0}\left(k\right)=\left[\begin{array}{c}{y}_0(k+1|k)\\ y_0(k+2|k)\\ .\\ .\\ .\\ y_0(k+P|k)\end{array}\right],\mathrm{\Δ}{u}_M\left(k\right)=\left[\begin{array}{c}\mathrm{\Δu}\left(k\right)\\ \mathrm{\Δu}(k+1)\\ .\\ .\\ .\\ \mathrm{\Δu}(k+M-1)\end{array}\right]$

y_P0(k) is y_M(k) The first P term, y_M(k+1|k),y_M(k+2|k),…,y_M(k + P | k) is a model prediction output value of the waste plastic cracking furnace hearth pressure at the k moment to the k +1, k +2, …, k + P moment.

d. Let the control time domain M be 1, and select an objective function J (k) of the waste plastic cracking furnace hearth pressure, the form is as follows:

minJ(k)＝Q(ref(k)-y_PM(k))²+rΔu²(k)=Q(ref(k)-y_P0(k)-AΔu(k))²+rΔu²(k)

ref(k)＝[ref₁(k),ref₂(k),…,ref_P(k)]^Τ

ref_i(k)＝βⁱy(k)+(1-βⁱ)c(k),Q＝diag(q₁,q₂,…,q_P)

wherein Q is an error weighting matrix, Q₁,q₂,…,q_PWeighting moments for errorsThe parameter value of the matrix is β is softening coefficient, c (k) is the set value of waste plastic cracking furnace hearth pressure, r is diag₁,r₂,…r_M) To control the weighting matrix, r₁,r₂,…r_MFor controlling the parameters of the weighting matrix, ref (k) is a reference trajectory of the waste plastic cracking furnace hearth pressure, ref_i(k) Is the value of the ith reference point in the reference track.

e. Changing the opening control quantity u (k) of a flue damper of a waste plastic cracking furnace hearth:

u(k)＝u(k-1)+K_p(k)(e₁(k)-e₁(k-1))+K_i(k)e₁(k)

e(k)＝c(k)-y(k)

and substituting u (k) into the objective function in the step d, further solving the parameters in the PI controller of the waste plastic cracking furnace hearth pressure, and obtaining:

u(k)＝u(k-1)+w(k)^ΤE(k)

w(k)＝[w₁(k),w₂(k)]^Τ

w1(k)＝K_p(k)+K_i(k)，w₂(k)＝-K_p(k)

E(k)＝[e₁(k),e₁(k-1)]^Τ

wherein, K_p(k)、K_i(k) Proportional and differential parameters, e, of the PI controller, respectively₁(k) T is the transposed symbol of the matrix for the error between the reference trajectory value and the actual output value at time k.

By combining the above formulas, the following can be obtained:

$w\left(k\right)=\frac{{(\mathrm{ref}\left(k\right)-y_{P0}\left(k\right))}^T\mathrm{QAE}}{(A^T\mathrm{QA}+r)E^{T}E}$

further, it is possible to obtain:

K_p(k)＝-w₂(k)

K_i(k)＝w₁(k)-K_P(k)

f. obtaining parameter K of PI controller_p(k)、K_i(k) Then, the control amount u (K) of the opening degree of the flue damper is u (K-1) + K_p(k)(e₁(k)-e₁(k-1))+K_i(k)e₁(k) Acting on the waste plastic cracking furnace hearth.

g. At the next moment, the solution of the new parameter K of the PI controller is continued according to the steps from b to f_p(k+1)、K_i(k +1) and are cycled sequentially.

Claims (1)

1. The method for controlling the pressure of the waste plastic cracking furnace hearth through dynamic matrix control optimization is characterized by comprising the following specific steps:

step (1), establishing a model of a controlled object through real-time step response data of a process object, wherein the specific method comprises the following steps:

1-a, giving a step input signal to a controlled object, and recording a step response curve of the controlled object;

1-b, filtering the step response curve obtained in the step 1-a, fitting the step response curve into a smooth curve, and recording each sample on the smooth curveStep response data corresponding to the time, wherein the first sampling time is T_sThe time interval between two adjacent sampling time is T_sThe sampling time sequence is T_s、2T_s、3T_s… …, respectively; the step response of the controlled object will be at a certain time t_N＝NT_sThen, it tends to be steady when a_iI > N, and a_NWhen the error of (a) and the measurement error are of the same order of magnitude, a can be regarded as_NApproximately equal to the steady state value of the step response; establishing a model vector a of an object:

a＝[a₁,a₂,…a_N]^T

wherein T is a transposed symbol of the matrix, and N is a modeling time domain;

step (2), designing a PI controller of a controlled object, wherein the specific method comprises the following steps:

2-a, establishing a dynamic matrix of the controlled object by using the model vector a obtained above, wherein the dynamic matrix is in the form of:

$A=\left(\begin{array}{cccc}{a}_1& 0& \mathrm{...}& 0\\ a_2& a_1& \mathrm{...}& 0\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ a_P& a_{P-1}& \mathrm{...}& a_{P-M+1}\end{array}\right)$

wherein A is a dynamic matrix of P × M order of the controlled object, a_iThe method comprises the steps of obtaining data of step response, wherein P is an optimized time domain of a dynamic matrix control algorithm, M is a control time domain of the dynamic matrix control algorithm, and M is more than P and less than N;

2-b, establishing a model prediction initial response value y of the controlled object at the current k moment_M(k)

Firstly, obtaining a model predicted value y after adding a control increment delta u (k-1) at the moment of k-1_p(k-1)：

y_P(k-1)＝y_M(k-1)+A₀Δu(k-1)

Wherein,

$y_P(k-1)=\left[\begin{array}{c}{y}_1\left(k\right|k-1)\\ y_1(k+1|k-1)\\ .\\ .\\ .\\ y_1(k+N-1|k-1)\end{array}\right],A_0=\left[\begin{array}{c}{a}_1\\ a_2\\ .\\ .\\ .\\ a_N\end{array}\right],y_M\left(k\right)=\left[\begin{array}{c}{y}_0\left(k\right|k-1)\\ y_0(k+1|k-1)\\ .\\ .\\ .\\ y_0(k+N-1|k-1)\end{array}\right]$

y₁(k|k-1),y₁(k+1|k-1),…,y₁(k + N-1| k-1) respectively represents the model predicted value of the controlled object after adding the control increment delta u (k-1) to k, k +1, …, k + N-1 at the time k-1, y₀(k|k-1),y₀(k|k-1),…y₀(k + N-1| k-1) represents the initial predicted value at time k-1 versus time k, k +1, …, k + N-1, A₀A matrix is established for the step response data, and delta u (k-1) is an input control increment at the moment of k-1;

then obtaining a model prediction error value e (k) of the controlled object at the moment k:

e(k)＝y(k)-y₁(k|k-1)

wherein y (k) represents the actual output value of the controlled object measured at the time k;

further obtaining a correction value y output by the k-time model_cor(k)∶

y_cor(k)＝y_M(k-1)+h*e(k)

Wherein,

$y_{cor}\left(r\right)=\left[\begin{array}{c}{y}_{cor}\left(k\right|k)\\ y_{cor}(k+1|k)\\ .\\ .\\ .\\ y_{cor}(k+N-1|k)\end{array}\right],h=\left[\begin{array}{c}1\\ \mathrm{\α}\\ .\\ .\\ .\\ \mathrm{\α}\end{array}\right]$

y_cor(k|k),y_cor(k+1|k),…y_cor(k + N-1| k) respectively represents the corrected value of the controlled object at the moment k, h is a weight matrix of error compensation, and α is an error correction coefficient;

finally obtaining the initial response value y of the model prediction at the moment k_M(k)：

y_M(k)＝Sy_cor(k)

Wherein S is a state transition matrix of NxN order,

2-c, calculating the predicted output value y of the controlled object under M continuous control increments delta u (k), … and delta u (k + M-1)_PMThe specific method comprises the following steps:

y_PM(k)＝y_p0(k)+AΔu_M(k)

$y_{PM}\left(k\right)=\left[\begin{array}{c}{y}_M(k+1|k)\\ y_M(k+2|k)\\ .\\ .\\ .\\ y_M(k+P|k)\end{array}\right],y_{P0}\left(k\right)=\left[\begin{array}{c}{y}_0(k+1|k)\\ y_0(k+2|k)\\ .\\ .\\ .\\ y_0(k+P|k)\end{array}\right],{\mathrm{\Δu}}_M\left(k\right)=\left[\begin{array}{c}\mathrm{\Δ}u\left(k\right)\\ \mathrm{\Δ}u(k+1)\\ .\\ .\\ .\\ \mathrm{\Δ}u(k+M-1)\end{array}\right]$

wherein, y_P0(k) Is y_M(k) The first P term, y_M(k+1|k),y_M(k+2|k),…,y_M(k + P | k) is a model prediction output value of the k moment to the k +1, k +2, … and k + P moment;

and 2-d, setting the control time domain M of the controlled object to be 1, and selecting an objective function J (k) of the controlled object, wherein the form is as follows:

minJ(k)＝Q(ref(k)-y_PM(k))²+rΔu²(k)＝Q(ref(k)-y_P0(k)-AΔu(k))²+rΔu²(k)

ref(k)＝[ref₁(k),ref₂(k),…,ref_P(k)]^T

ref_i(k)＝βⁱy(k)+(1-βⁱ)c(k),Q＝diag(q₁,q₂,…,q_P)

wherein Q is an error weighting matrix, Q₁,q₂,…,q_PThe parameter values of the weighting matrix, β softening coefficients, c (k) set values, and r ═ diag (r)₁,r₂,…r_M) To control the weighting matrix, r₁,r₂,…r_MFor controlling the parameters of the weighting matrix, ref (k) is the reference trajectory of the system, ref_i(k) Is the value of the ith reference point in the reference track;

transforming the control quantity u (k):

u(k)＝u(k-1)+K_p(k)(e₁(k)-e₁(k-1))+K_i(k)e₁(k)

e(k)＝c(k)-y(k)

substituting u (k) into the objective function in step 2-d to solve the parameters in the PI controller to obtain:

u(k)＝u(k-1)+w(k)^TE(k)

w(k)＝[w₁(k),w₂(k)]^T

w₁(k)＝K_p(k)＋K_i(k)，w₂(k)＝-K_p(k)

E(k)＝[e₁(k),e₁(k-1)]^T

wherein Kp (K), K_i(k) Proportional and differential parameters of the PI controller at the time k, e₁(k) The error between the reference track value and the actual output value at the moment k is shown, and T is a transposed symbol of the matrix;

by combining the above formulas, the following can be obtained:

$w\left(k\right)=\frac{{\left(ref\right(k)-y_{PO}(k\left)\right)}^{T}QAE\left(k\right)}{(A^{T}QA+r){\left(E\right(k\left)\right)}^{T}E\left(k\right)}$

further, it is possible to obtain:

K_p(k)＝-w₂(k)

K_i(k)＝w₁(k)-K_P(k)

2-f, obtaining parameter K of PI controller_p(k)、K_i(k) The control amount u (K) is applied to the controlled object, and u (K) is u (K-1) + K_p(k)(e₁(k)-e₁(k-1))＋K_i(k)e₁(k)；

At the next moment, continuing to solve the new parameter k of the PI controller according to the steps from 2-b to 2-f_P(k＋1)、k_iThe values of (k +1) are cycled through in sequence.