CN104317321A - Coking furnace hearth pressure control method based on state-space predictive functional control optimization - Google Patents

Abstract

The invention discloses a coking furnace hearth pressure control method based on state-space predictive functional control optimization. In industrial control, traditional PID control cannot achieve a satisfactory control effect. In the invention, first, a state-space model of a coking furnace pressure object is established based on real-time input and output data of the coking furnace pressure object, and then, an extended non-minimized state-space model is established based on the state process and output error. On the basis of the model, the parameters of a PID controller are optimized by the method of predictive functional control, and PID control on a controlled object is realized. The coking furnace hearth pressure control method of the invention has both good control performance of ENMSSPFC and the simple form of PID control.

Description

The coking furnace hearth pressure control method that state space Predictive function control is optimized

Technical field

The invention belongs to technical field of automation, relating to a kind of proportion integration differentiation (PID) control method based on expanding the coking furnace furnace pressure that non-minimum state space Predictive function control (ENMSSPFC) is optimized.

Background technology

In actual industrial controls, due to the restriction of the factor such as cost and hardware, although the method for some Dynamic matrix control obtains application to a certain degree, still cannot replace traditional PID and control.It is a large dead time, nonlinear process that coking furnace furnace pressure controls, and traditional PID controls to reach satisfied control effects.Control to have better control performance than PID in coking furnace furnace pressure controls based on expansion non-minimum state space Predictive function control, if PID can be given by the control performance of ENMSSPFC control, can ensure that the form of control structure is simple, better control performance can be obtained again.

Summary of the invention

The object of the invention is the deficiency controlled for existing PID, set up by data acquisition, model, predict the means such as mechanism, optimization, there is provided a kind of PID control method of the coking furnace furnace pressure based on ENMSSPFC optimization, to obtain control performance good in real process.The method, by controlling in conjunction with ENMSSPFC and PID, obtains the PID control method that a kind of ENMSSPFC optimizes.The method had both had the good control performance of ENMSSPFC, possessed again the simple form that PID controls simultaneously.

Step of the present invention comprises:

Step 1, set up the expansion non-minimum state-space model of controlled device, concrete steps are:

1.1, by gathering the real-time inputoutput data of controlled device, utilize least square method Modling model, form is as follows:

y _L(k)＝Ψ ^Tθ,θ＝[S ₁,-L ₁,S ₂,-L ₂,...,S _n,-L _n] ^T

Ψ＝[u(k-1),y(k-1),...,u(k-n),y(k-n)] ^T

Wherein, y _lk () represents the output valve of k moment forecast model, y (k) represents the output valve of k moment real process, and u (k) represents the controlled quentity controlled variable in k moment, and n is the order of the input/output variable of corresponding real process, L ₁, L ₂..., L _n, S ₁, S ₂..., S _nfor needing the coefficient of identification, Τ is transpose of a matrix symbol.

Utilize the real process data gathered, obtain N group sample data, form is as follows:

Y＝[y(1),y(2),...,y(j),...,y(N)] ^T

$\mathrm{\Φ}={[{\mathrm{\Ψ}}_1^T,{\mathrm{\Ψ}}_2^T,...,{\mathrm{\Ψ}}_j^T,...,{\mathrm{\Ψ}}_N^T]}^T$

Wherein, Ψ _j, y (j) represents the input data of jth group and output valve that gather, N represents total sample number.

Identification result is:

$\widehat{\mathrm{\θ}}={\left({\mathrm{\Φ}}^T\mathrm{\Φ}\right)}^{-1}{\mathrm{\Φ}}^{T}Y$

1.2, be difference model form by the model conversion obtained in step 1.1:

Δy(k+1)+L ₁Δy(k)+L ₂Δy(k-1)+...+L _nΔy(k-n+1)

＝S ₁Δu(k)+S ₂Δu(k-1)+...+S _nΔu(k-n+1)

Wherein, Δ is difference operator.

1.3, non-minimum state space variable as follows is chosen:

Δx _m(k) ^T＝[Δy(k),Δy(k-1),...,Δy(k-n+1),Δu(k-1),Δu(k-2),Δu(k-n+1)]

And then be state-space model by the model conversation in step 1.2, its form is as follows:

Δx _m(k+1)＝A _mΔx _m(k)+B _mΔu(k)

Δy(k+1)＝C _mΔx _m(k+1)

Wherein

$A_m=\left[\begin{array}{ccccccccc}-L_1& -L_2& ...& -L_{n-1}& -L_n& S_2& ...& S_{n-1}& S_n\\ 1& 0& ...& 0& 0& 0& ...& 0& 0\\ 0& 1& ...& 0& 0& 0& ...& 0& 0\\ .& .& & .& .& .& & .& .\\ .& .& ...& .& .& .& ...& .& .\\ .& .& & .& .& .& & .& .\\ 0& 0& ...& 1& 0& 0& ...& 0& 0\\ 0& 0& ...& 0& 0& 0& ...& 0& 0\\ 0& 0& ...& 0& 0& 1& ...& 0& 0\\ .& .& & .& .& & .& .& .\\ .& .& ...& .& .& ...& .& .& .\\ .& .& & .& .& & .& .& .\\ 0& 0& ...& 0& 0& 0& ...& 1& 0\end{array}\right],$

$B_m={\left[\begin{array}{cccccccc}{S}_1& 0& ...& 0& 1& 0& ...& 0\end{array}\right]}^T,$

$C_m=\left[\begin{array}{cccccccc}1& 0& 0& ...& 0& 0& 0& 0\end{array}\right],$

Δ x _mthe dimension m=2n-1 of (k).

1.4, the state-space model obtained in step 1.3 is converted to the expansion non-minimum state-space model comprising state variable and output error, form is as follows:

z(k+1)＝Az(k)+BΔu(k)+CΔr(k+1)

Wherein

$\begin{array}{c}A=\left[\begin{array}{cc}{A}_m& 0\\ C_m{A}_m& 1\end{array}\right],B=\left[\begin{array}{c}{B}_m\\ C_m{B}_m\end{array}\right],C=\left[\begin{array}{c}0\\ -1\end{array}\right]\\ z\left(k\right)=\left[\begin{array}{c}{\mathrm{\Δx}}_m\left(k\right)\\ e\left(k\right)\end{array}\right]\end{array}$

e(k)＝y(k)-r(k)

R (k) is the desired output in k moment, the difference between the real output value that e (k) is the k moment and desired output, and 0 is dimension is the null matrix of m.

The PID controller of step 2, design controlled device, concrete steps are:

2.1, calculate the k moment to the prediction output valve in kth+P moment, form is as follows:

z(k+P)＝A ^Pz(k)+ψΔu(k)+θΔR

Wherein,

$\mathrm{\θ}=\left[\begin{array}{cccc}{A}^{P-1}C& A^{P-2}C& ...& C\end{array}\right],\mathrm{\ψ}=A^{P-1}B,$

$\mathrm{\ΔR}={\left[\begin{array}{cccc}\mathrm{\Δr}(k+1)& \mathrm{\Δr}(k+2)& ...& \mathrm{\Δr}(k+P)\end{array}\right]}^T,$

r(k+i)＝α ⁱy(k)+(1-α ⁱ)c(k),i＝1,2,...,P，

P is prediction time domain, A ^prepresent that P matrix A is multiplied, α is the softening factor of reference locus, and c (k) is the setting value in k moment.

2.2, choose objective function J (k) of controlled device, form is as follows:

minJ(k)＝z(k+P) ^ΤQz(k+P)

Wherein, Q is (2n-1) × (2n-1) weight matrix, and min represents and minimizes.

2.3, solve the parameter of PID controller according to the objective function in step 2.2, concrete grammar is: first converted by controlled quentity controlled variable u (k):

u(k)＝u(k-1)+K _p(k)(e ₁(k)-e ₁(k-1))+K _i(k)e ₁(k)+K _d(k)(e ₁(k)-2e ₁(k-1)+e ₁(k-2))

e ₁(k)＝c(k)-y(k)

Wherein, K _p(k), K _i(k), K _dk () is ratio, differential, the integral parameter of k moment PID controller respectively, e ₁k () is the error between k moment setting value and real output value.

And then controlled quentity controlled variable u (k) can be simplified to matrix form:

u(k)＝u(k-1)+w(k) ^ΤE(k)

w(k)＝[w ₁(k),w ₂(k),w ₃(k)] ^Τ

E(k)＝[e ₁(k),e ₁(k-1),e ₁(k-2)] ^Τ

w ₁(k)＝K _p(k)+K _i(k)+K _d(k)

w ₂(k)＝-K _p(k)-2K _d(k)

w ₃(k)＝K _d(k)

In conjunction with the objective function in the matrix form of controlled quentity controlled variable u (k) and step 2.2, can be in the hope of:

$w\left(k\right)=-\frac{{\mathrm{\ψ}}^{T}Q(A^{P}z\left(k\right)+\mathrm{\θ\ΔR})E\left(k\right)}{{\mathrm{\ψ}}^T\mathrm{Q\ψE}{\left(k\right)}^{T}E\left(k\right)}$

Can obtain further:

K _p(k)＝-w ₂(k)-2K _d(k)

K _i(k)＝w ₁(k)-K _P(k)-K _d(k)

K _d(k)＝w ₃(k)

2.4, the parameter K of PID controller is obtained _p(k), K _i(k), K _dafter (k), form controlled quentity controlled variable: u (k)=u (k-1)+K _p(k) (e ₁(k)-e ₁(k-1))+K _i(k) e ₁(k)+K _d(k) (e ₁(k)-2e ₁(k-1)+e ₁), then acted on controlled device (k-2).

2.5, in the k+l moment, the new parameter K of PID controller is solved according to the step cycle in 2.1 to 2.4 _p(k+l), K _i(k+l), K _d(k+l), l=1,2,3 ....

First the present invention sets up the state-space model of coking furnace pressure object based on the real-time inputoutput data of coking furnace pressure object, then bonding state process and output error set up the non-minimum state-space model of expansion.On the basis of this model, the method according to Predictive function control carrys out the parameter of PID controller, finally realizes PID to controlled device and controls, effectively compensate for the deficiency of traditional control method, effectively can improve the control performance of system.

Embodiment

Control for coking furnace press process below, the invention will be further described, and in coking furnace pressure control procedure, regulating measure is the aperture regulating stack damper.

The coking furnace hearth pressure control method concrete steps that state space Predictive function control is optimized are as follows:

Step 1, set up the expansion non-minimum state-space model of controlled device, concrete steps are:

1.1, the real-time inputoutput data by gathering controlled device utilizes least square method Modling model, and form is as follows:

y _L(k)＝Ψ ^Tθ,θ＝[S ₁,-L ₁,S ₂,-L ₂,...,S _n,-L _n] ^T

Ψ＝[u(k-1),y(k-1),...,u(k-n),y(k-n)] ^T

Wherein, y _lk () represents the output valve of k moment forecast model, y (k) represents the output valve of furnace pressure object in k moment press process, and u (k) represents the controlled quentity controlled variable in k moment, and n is the order of the input/output variable of corresponding pressure process, L ₁, L ₂..., L _n, S ₁, S ₂..., S _nfor needing the coefficient of identification, Τ is transpose of a matrix symbol.

Utilize the real process data gathered, obtain N group sample data, form is as follows:

Y＝[y(1),y(2),...,y(j),...,y(N)] ^T

$\mathrm{\Φ}={[{\mathrm{\Ψ}}_1^T,{\mathrm{\Ψ}}_2^T,...,{\mathrm{\Ψ}}_j^T,...,{\mathrm{\Ψ}}_N^T]}^T$

Wherein, Ψ _j, y (j) represents the input data of jth group and output valve that gather, N represents total sample number.

Identification result is:

$\widehat{\mathrm{\θ}}={\left({\mathrm{\Φ}}^T\mathrm{\Φ}\right)}^{-1}{\mathrm{\Φ}}^{T}Y$

1.2, be difference model form by the model conversion obtaining coking furnace pressure control procedure in step 1.1:

Δy(k+1)+L ₁Δy(k)+L ₂Δy(k-1)+...+L _nΔy(k-n+1)

＝S ₁Δu(k)+S ₂Δu(k-1)+...+S _nΔu(k-n+1)

Wherein, Δ is difference operator.

1.3, non-minimum state space variable as follows is chosen:

Δx _m(k) ^T＝[Δy(k),Δy(k-1),...,Δy(k-n+1),Δu(k-1),Δu(k-2),Δu(k-n+1)]，

And then be state-space model by the model conversation in step 1.2, its form is as follows:

Δx _m(k+1)＝A _mΔx _m(k)+B _mΔu(k)

Δy(k+1)＝C _mΔx _m(k+1)

Wherein,

$B_m={\left[\begin{array}{cccccccc}{S}_1& 0& ...& 0& 1& 0& ...& 0\end{array}\right]}^T$

$C_m=\left[\begin{array}{cccccccc}1& 0& 0& ...& 0& 0& 0& 0\end{array}\right]$

$A_m=\left[\begin{array}{ccccccccc}-L_1& -L_2& ...& -L_{n-1}& -L_n& S_2& ...& S_{n-1}& S_n\\ 1& 0& ...& 0& 0& 0& ...& 0& 0\\ 0& 1& ...& 0& 0& 0& ...& 0& 0\\ .& .& & .& .& .& & .& .\\ .& .& ...& .& .& .& ...& .& .\\ .& .& & .& .& .& & .& .\\ 0& 0& ...& 1& 0& 0& ...& 0& 0\\ 0& 0& ...& 0& 0& 0& ...& 0& 0\\ 0& 0& ...& 0& 0& 1& ...& 0& 0\\ .& .& & .& .& & .& .& .\\ .& .& ...& .& .& ...& .& .& .\\ .& .& & .& .& & .& .& .\\ 0& 0& ...& 0& 0& 0& ...& 1& 0\end{array}\right]$

Δ x _mthe dimension m=2n-1 of (k).

1.4, the state-space model obtained in step 1.3 is converted to the expansion non-minimum state-space model comprising state variable and output error, form is as follows:

z(k+1)＝Az(k)+BΔu(k)+CΔr(k+1)

Wherein,

$\begin{array}{c}A=\left[\begin{array}{cc}{A}_m& 0\\ C_m{A}_m& 1\end{array}\right],B=\left[\begin{array}{c}{B}_m\\ C_m{B}_m\end{array}\right],C=\left[\begin{array}{c}0\\ -1\end{array}\right]\\ z\left(k\right)=\left[\begin{array}{c}{\mathrm{\Δx}}_m\left(k\right)\\ e\left(k\right)\end{array}\right]\end{array}$

e(k)＝y(k)-r(k)

R (k) is the desired pressure in k moment, and e (k) is the difference between k moment burner hearth actual pressure and desired pressure, and 0 is dimension is the null matrix of m.

The PID controller of step 2, design coking furnace press process, concrete steps are:

2.1, calculate the coking furnace furnace pressure prediction output valve of k moment to the kth+P moment, form is as follows:

z(k+P)＝A ^Pz(k)+ψΔu(k)+θΔR

Wherein,

$\mathrm{\θ}=\left[\begin{array}{cccc}{A}^{P-1}C& A^{P-2}C& ...& C\end{array}\right],\mathrm{\ψ}=A^{P-1}B$

$\mathrm{\ΔR}={\left[\begin{array}{cccc}\mathrm{\Δr}(k+1)& \mathrm{\Δr}(k+2)& ...& \mathrm{\Δr}(k+P)\end{array}\right]}^T$

r(k+i)＝α ⁱy(k)+(1-α ⁱ)c(k),i＝1,2,...,P

P is prediction time domain, A ^prepresent that P matrix A is multiplied, α is the softening factor of reference locus, and c (k) is the setting value of k moment press process.

2.2, choose objective function J (k) of controlled device, form is as follows:

minJ(k)＝z(k+P) ^ΤQz(k+P)

Wherein, Q is (2n-1) × (2n-1) weight matrix, and min represents and minimizes.

2.3, solve the parameter of PID controller according to the objective function in step 2.2, concrete grammar is: first converted by stack damper aperture controlled quentity controlled variable u (k) of coking furnace press process:

u(k)＝u(k-1)+K _p(k)(e ₁(k)-e ₁(k-1))+K _i(k)e ₁(k)+K _d(k)(e ₁(k)-2e ₁(k-1)+e ₁(k-2))

e ₁(k)＝c(k)-y(k)

Wherein, K _p(k), K _i(k), K _dk () is ratio, differential, the integral parameter of k moment PID controller respectively, e ₁k () is the error between the setting value of k moment controling of the pressure of the oven process and real output value.

And then controlled quentity controlled variable u (k) can be simplified to matrix form:

u(k)＝u(k-1)+w(k) ^ΤE(k)

w(k)＝[w ₁(k),w ₂(k),w ₃(k)] ^Τ

E(k)＝[e ₁(k),e ₁(k-1),e ₁(k-2)]Τ

w ₁(k)＝K _p(k)+K _i(k)+K _d(k)

w ₂(k)＝-K _p(k)-2K _d(k)

w ₃(k)＝K _d(k)

In conjunction with the objective function in the matrix form of controlled quentity controlled variable u (k) and step 2.2, can be in the hope of:

$w\left(k\right)=-\frac{{\mathrm{\ψ}}^{T}Q(A^{P}z\left(k\right)+\mathrm{\θ\ΔR})E\left(k\right)}{{\mathrm{\ψ}}^T\mathrm{Q\ψE}{\left(k\right)}^{T}E\left(k\right)}$

Can obtain further:

K _p(k)＝-w ₂(k)-2K _d(k)

K _i(k)＝w ₁(k)-K _P(k)-K _d(k)

K _d(k)＝w ₃(k)

2.4, the parameter K of PID controller is obtained _p(k), K _i(k), K _dafter (k), form controlled quentity controlled variable u (k)=u (k-1)+K _p(k) (e ₁(k)-e ₁(k-1))+K _i(k) e ₁(k)+K _d(k) (e ₁(k)-2e ₁(k-1)+e ₁), then acted on the stack damper of coking furnace press process (k-2).

2.5, in the k+l moment, the new parameter K of PID controller is solved according to the step cycle in 2.1 to 2.4 _p(k+l), K _i(k+l), K _d(k+l), l=1,2,3 ....

Claims (1)

1. the coking furnace hearth pressure control method of state space Predictive function control optimization, is characterized in that: the concrete steps of the method comprise:

Step 1, set up the expansion non-minimum state-space model of controlled device, concrete steps are:

1.1, by gathering the real-time inputoutput data of controlled device, utilize least square method Modling model, form is as follows:

y _L(k)＝Ψ ^Tθ,θ＝[S ₁,-L ₁,S ₂,-L ₂,...,S _n,-L _n] ^T

Ψ＝[u(k-1),y(k-1),...,u(k-n),y(k-n)] ^T

Wherein, y _lk () represents the output valve of k moment forecast model, y (k) represents the output valve of k moment real process, and u (k) represents the controlled quentity controlled variable in k moment, and n is the order of the input/output variable of corresponding real process, L ₁, L ₂..., L _n, S ₁, S ₂..., S _nfor needing the coefficient of identification, Τ is transpose of a matrix symbol;

Utilize the real process data gathered, obtain N group sample data, form is as follows:

Y＝[y(1),y(2),...,y(j),...,y(N)] ^T

$\mathrm{\Φ}={[{\mathrm{\Ψ}}_1^T,{\mathrm{\Ψ}}_2^T,...,{\mathrm{\Ψ}}_j^T,...,{\mathrm{\Ψ}}_N^T]}^T$

Wherein, Ψ _j, y (j) represents the input data of jth group and output valve that gather, N represents total sample number;

Identification result is:

$\widehat{\mathrm{\θ}}={\left({\mathrm{\Φ}}^T\mathrm{\Φ}\right)}^{-1}{\mathrm{\Φ}}^{T}Y$

1.2, be difference model form by the model conversion obtained in step 1.1:

Δy(k+1)+L ₁Δy(k)+L ₂Δy(k-1)+...+L _nΔy(k-n+1)

＝S ₁Δu(k)+S ₂Δu(k-1)+...+S _nΔu(k-n+1)

Wherein, Δ is difference operator;

1.3, non-minimum state space variable as follows is chosen:

Δx _m(k) ^T＝[Δy(k),Δy(k-1),...,Δy(k-n+1),Δu(k-1),Δu(k-2),Δu(k-n+1)]

And then be state-space model by the model conversation in step 1.2, its form is as follows:

Δx _m(k+1)＝A _mΔx _m(k)+B _mΔu(k)

Δy(k+1)＝C _mΔx _m(k+1)

Wherein

$A_m=\left[\begin{array}{ccccccccc}-L_1& -L_2& ...& -L_{n-1}& -L_n& S_2& ...& S_{n-1}& S_n\\ 1& 0& ...& 0& 0& 0& ...& 0& 0\\ 0& 1& ...& 0& 0& 0& ...& 0& 0\\ .& .& & .& .& .& & .& .\\ .& .& ...& .& .& .& ...& .& .\\ .& .& & .& .& .& & .& .\\ 0& 0& ...& 1& 0& 0& ...& 0& 0\\ 0& 0& ...& 0& 0& 0& ...& 0& 0\\ 0& 0& ...& 0& 0& 1& ...& 0& 0\\ .& .& & .& .& & .& .& .\\ .& .& ...& .& .& ...& .& .& .\\ .& .& & .& .& & .& .& .\\ 0& 0& ...& 0& 0& 0& ...& 1& 0\end{array}\right],$

$B_m={\left[\begin{array}{cccccccc}{S}_1& 0& ...& 0& 1& 0& ...& 0\end{array}\right]}^T,$

$C_m=\left[\begin{array}{cccccccc}1& 0& 0& ...& 0& 0& 0& 0\end{array}\right],$

Δ x _mthe dimension m=2n-1 of (k);

1.4, the state-space model obtained in step 1.3 is converted to the expansion non-minimum state-space model comprising state variable and output error, form is as follows:

z(k+1)＝Az(k)+BΔu(k)+CΔr(k+1)

Wherein

$\begin{array}{c}A=\left[\begin{array}{cc}{A}_m& 0\\ C_m{A}_m& 1\end{array}\right],B=\left[\begin{array}{c}{B}_m\\ C_m{B}_m\end{array}\right],C=\left[\begin{array}{c}0\\ -1\end{array}\right]\\ z\left(k\right)=\left[\begin{array}{c}{\mathrm{\Δx}}_m\left(k\right)\\ e\left(k\right)\end{array}\right]\end{array}$

e(k)＝y(k)-r(k)

R (k) is the desired output in k moment, the difference between the real output value that e (k) is the k moment and desired output, and 0 is dimension is the null matrix of m;

The PID controller of step 2, design controlled device, concrete steps are:

2.1, calculate the k moment to the prediction output valve in kth+P moment, form is as follows:

z(k+P)＝A ^Pz(k)+ψΔu(k)+θΔR

Wherein,

$\mathrm{\θ}=\left[\begin{array}{cccc}{A}^{P-1}C& A^{P-2}C& ...& C\end{array}\right],\mathrm{\ψ}=A^{P-1}B,$

$\mathrm{\ΔR}={\left[\begin{array}{cccc}\mathrm{\Δr}(k+1)& \mathrm{\Δr}(k+2)& ...& \mathrm{\Δr}(k+P)\end{array}\right]}^T,$

r(k+i)＝α ⁱy(k)+(1-α ⁱ)c(k),i＝1,2,...,P，

P is prediction time domain, A ^prepresent that P matrix A is multiplied, α is the softening factor of reference locus, and c (k) is the setting value in k moment;

2.2, choose objective function J (k) of controlled device, form is as follows:

minJ(k)＝z(k+P) ^ΤQz(k+P)

Wherein, Q is (2n-1) × (2n-1) weight matrix, and min represents and minimizes;

2.3, solve the parameter of PID controller according to the objective function in step 2.2, concrete grammar is:

First controlled quentity controlled variable u (k) is converted:

u(k)＝u(k-1)+K _p(k)(e ₁(k)-e ₁(k-1))+K _i(k)e ₁(k)+K _d(k)(e ₁(k)-2e ₁(k-1)+e ₁(k-2))

e ₁(k)＝c(k)-y(k)

Wherein, K _p(k), K _i(k), K _dk () is ratio, differential, the integral parameter of k moment PID controller respectively, e ₁k () is the error between k moment setting value and real output value;

And then controlled quentity controlled variable u (k) can be simplified to matrix form:

u(k)＝u(k-1)+w(k) ^ΤE(k)

w(k)＝[w ₁(k),w ₂(k),w ₃(k)] ^Τ

E(k)＝[e ₁(k),e ₁(k-1),e ₁(k-2)] ^Τ

w ₁(k)＝K _p(k)+K _i(k)+K _d(k)

w ₂(k)＝-K _p(k)-2K _d(k)

w ₃(k)＝K _d(k)

In conjunction with the objective function in the matrix form of controlled quentity controlled variable u (k) and step 2.2, can be in the hope of:

$w\left(k\right)=-\frac{{\mathrm{\ψ}}^{T}Q(A^{P}z\left(k\right)+\mathrm{\θ\ΔR})E\left(k\right)}{{\mathrm{\ψ}}^T\mathrm{Q\ψE}{\left(k\right)}^{T}E\left(k\right)}$

Can obtain further:

K _p(k)＝-w ₂(k)-2K _d(k)

K _i(k)＝w ₁(k)-K _P(k)-K _d(k)

K _d(k)＝w ₃(k)

2.4, the parameter K of PID controller is obtained _p(k), K _i(k), K _dafter (k), form controlled quentity controlled variable: u (k)=u (k-1)+K _p(k) (e ₁(k)-e ₁(k-1))+K _i(k) e ₁(k)+K _d(k) (e ₁(k)-2e ₁(k-1)+e ₁), then acted on controlled device (k-2);

2.5, in the k+l moment, the new parameter K of PID controller is solved according to the step cycle in 2.1 to 2.4 _p(k+l), K _i(k+l), K _d(k+l), l=1,2,3 ....