CN105487515A - Integrated optimization method of continuous agitated vessel reaction process technology design and control - Google Patents

Abstract

An integrated optimization method of design and control of a continuous agitated vessel reaction process technology provided by the invention comprises the following steps: the step 1, establishing a strict mechanism model aiming at the continuous agitated vessel reaction process technology on the basis of the mass balance and energy balance principle; the step 2, obtaining a linear state space model of an object in a random frame; the step 3, calculating an optimal performance boundary of the closed-loop variance of manipulated variables (MVs) and controlled variables (CVs) and an MVs/CVs variance based on the linearity quadratic gauss (LQG) control principle, and according to the LQG coordination curve, the function relation between MVs variable variance variation and the CVs variable variance variation of the system is determined; the step 4, combining the back-off strategy and process stable state model and the operation conditions and technical indexes of the process to be process constraints for setting the constraint probability permitting violation, establishing the technology design and the control of integrated optimization, and solving and obtaining optimal technology design parameters according to the optimization algorithm.

Description

The integrated optimization method of a kind of continuous stirred tank formula course of reaction technological design and control

Technical field

The present invention relates to industrial control field, be specifically related to the integrated optimization method of a kind of continuous stirred tank formula course of reaction technological design and control.

Background technology

In chemical system, reactor is the most influential cell arrangement of whole process, usually decides the character of whole technological process, also determine the economy of production and the impact on environment.CSTR (CSTR) is widely used a kind of reactor in chemical industry, be also simultaneously typical in process industrial, there is serious nonlinear dynamic system.

Traditional chemical process technological design and process control are carried out according to sequential design and control method.That is, first Kernel-based methods steady-state model and economic optimum criterion determine the technological design that meets the demands; Design con-trol system on this basis.In fact, control performance quality is largely by flowage structure, and the technological design factors such as the design specification of equipment determined.Sequential design and control method has isolated the relevance between technological design and process control performance, can not enable the optimum that designed process keeps overall under external disturbance and uncertain factor.The integrated optimization of technological design and control considers steady-state characteristic and the dynamic perfromance of process simultaneously, can obtain and meet all designs and operate the design parameter that constraint condition is issued to economic performance optimum, greatly improve operability and the economic performance of process.Conventional technological design and the integrated optimization method of control, calculation of complex, high to calculation requirement, the time of cost is many especially, strongly limit application and the popularization of integrated optimization.

Summary of the invention

The present invention will overcome the problems referred to above that prior art exists, provide a kind of continuous stirred tank formula course of reaction technological design and control integration optimization method, effectively can solve when technological design and control integration are optimized and solve complicated problem, greatly reduce computational complexity, make the process of design have good economic performance and dynamic property simultaneously simultaneously.

An integrated optimization method for the course of reaction technological design of continuous stirred tank formula and control, comprising:

Step 1, for continuous stirred tank formula course of reaction technique, according to mass balance and principle of energy balance, set up exact mechanism model, as following differential algebraic equations (DAE) form:

$f(\frac{{\mathrm{dx}}_p\left(t\right)}{dt},x_p(t),z,u(t),y(t),t)=0---\left(1\right)$

g(x _p(t),z,u(t),y(t),t)＝0(2)

In formula, x _pt () is state variable; Z is design variable; U (t) is process input; Y (t) is the output of process;

Step 2, obtains object with the linear state space model under machine frame.

Be random signal by process disturbance and noise processed, obtain augmentation process model, as follows:

$\left[\begin{array}{c}\frac{{\mathrm{dx}}_p\left(t\right)}{dt}\\ \frac{{\mathrm{dx}}_d\left(t\right)}{dt}\end{array}\right]=\left[\begin{array}{cc}f(\frac{{\mathrm{dx}}_p\left(t\right)}{dt},x_p(t),z,u(t),y(t),t)& 0\\ 0& A_d\end{array}\right]\left[\begin{array}{c}I\\ x_d\end{array}\right]+\left[\begin{array}{c}0\\ B_d\end{array}\right]w_d---\left(3\right)$

g(x _p(t),x _d(t),z,u(t),y(t),w _d,t)＝0(4)

y _m＝Y _my+v _m(5)

In formula, x _dfor disturbance variable; A _dand B _dthe model parameter of disturbance, w _dwhite Gaussian noise stochastic variable; Y _mit is measured value matrix; y _mit is measured value; v _mit is measurement noises.

Near steady state operation point, obtain object with the general linear state-space model under machine frame.As follows:

$\left[\begin{array}{c}\mathrm{\Δ}{\stackrel{\·}{x}}_p\\ \mathrm{\Δ}{\stackrel{\·}{x}}_d\end{array}\right]=\left[\begin{array}{cc}{A}_p& 0\\ 0& A_d\end{array}\right]\left[\begin{array}{c}\mathrm{\Δ}{x}_p\\ {\mathrm{\Δx}}_d\end{array}\right]+\left[\begin{array}{c}{B}_p\\ 0\end{array}\right]\mathrm{\Δ}u+\left[\begin{array}{c}0\\ B_d\end{array}\right]w_d---\left(6\right)$

${\mathrm{\Δy}}_m=C\left[\begin{array}{c}\mathrm{\Δ}{x}_p\\ {\mathrm{\Δx}}_d\end{array}\right]+v_m---\left(7\right)$

In formula, Δ x _pfor the variable quantity of state variable; Δ x _dfor the variable quantity of disturbance variable; Δ u is the variable quantity of manipulated variable; A _p, A _d, B _p, B _dfor state-space model matrix.

Above formula can be converted into state-space expression, as follows

x _t+1＝Ax _t+Bu _t-1+Ka _t

(8)

y _t＝Cx _t+a _t

Wherein, x _tfor the state variable variable quantity after conversion; A, B and C are state-space model parameter; K is Kalman state estimator; a _tfor random noise.

Step 3, design linear quadratic gaussian (LQG) controller, according to LQG control principle, calculates manipulated variable (MVs) and the closed loop variance of controlled variable (CVs) and optimal performance circle of MVs/CVs variance.Curve is coordinated, the funtcional relationship between the change of certainty annuity MVs, CVs variable variance according to LQG.

3-a, first, tries to achieve manipulated variable (MVs) and the closed loop variance of controlled variable (CVs) and optimal performance circle of MVs/CVs variance.Calculation procedure is as follows:

Step 3-a-1: Optimal state-feedback and the control law of trying to achieve system, as follows:

$\begin{array}{c}{\tilde{x}}_{t+1}=(A-KC-BL){\tilde{x}}_t+{\mathrm{Ky}}_t\\ u_t=-L{\tilde{x}}_t\end{array}---\left(9\right)$

Wherein L is optimal control law, solves by algebraic riccati equation.

Step 3-a-2: comprehensive card Kalman Filtering and Optimal state-feedback, as follows:

$\left[\begin{array}{c}{\tilde{x}}_{t+1}\\ {\hat{x}}_{t+1}\end{array}\right]=\left[\begin{array}{cc}A& -BL\\ KC& A-KC-BL\end{array}\right]\left[\begin{array}{c}{\tilde{x}}_t\\ {\hat{x}}_t\end{array}\right]+\left[\begin{array}{c}K\\ K\end{array}\right]a_t---\left(10\right)$

Step 3-a-3: try to achieve process input, export contrast, as follows:

Order $X_t=\left[\begin{array}{c}{\tilde{x}}_t\\ {\hat{x}}_t\end{array}\right],$ Then process input, output variance are respectively:

$\mathrm{var}\left(u_t\right)=\[\begin{array}{cc}0& -L\end{array}\]\mathrm{var}\left(X_t\right)\left[\begin{array}{c}0\\ -L^T\end{array}\right]---\left(11\right)$

$\mathrm{var}\left(y_t\right)=\left[\begin{array}{cc}C& 0\end{array}\right]\mathrm{var}\left(X_t\right)\left[\begin{array}{c}{C}^T\\ 0\end{array}\right]+\mathrm{var}\left(a_t\right)---\left(12\right)$

3-b, secondly, tries to achieve the funtcional relationship between the change of MVs, CVs variable variance.Calculation procedure is as follows:

Step 3-b-1: build LQG quadratic performance index function, as follows:

J(λ)＝E[Y ^TWY]+λE[U ^TRU](13)

Wherein W, R are the weighting matrix of output variable and input variable.

Step 3-b-2: build LQG quadratic performance index function, as follows:

By selecting different λ values within the specific limits, can obtain a series of under LQG control action about input variance and export the solution of contrast, based on these data, one can be made to input variance for horizontal ordinate, output variance is the performance curve of ordinate, as shown in Figure 1, this curve is illustrate linear controller performance lower bound that may reach under input and output variance characterizes.

Step 4, builds the integrated optimization problem of technological design and control, and is solved by optimized algorithm and obtain optimum process design parameter.

According to the funtcional relationship between the closed loop variance of MVs, CVs and the change of MVs/CVs variable variance, combine " retrogressings " strategy and Steady-state process model, and the operating conditions of process and technical indicator are that process constraints arranges the probability retraining permission violation; Specific as follows:

$L_Y+r_Y\×{\mathrm{\σ}}_Y\≤\stackrel{\‾}{Y}\≤H_Y-r_Y\×{\mathrm{\σ}}_Y---\left(14\right)$

$L_U+r_U\×{\mathrm{\σ}}_U\≤\stackrel{\‾}{U}\≤H_U-r_U\×{\mathrm{\σ}}_U---\left(15\right)$

In formula, H _yand L _ybe respectively the bound of output variable; H _uand L _ube respectively the bound of input variable; σ _uand σ _ybe respectively the standard deviation of input variable and output variable; with be respectively the setting value of input variable and output variable; r _yand r _urepresent that constraint allows the probability parameter violated.

The integrated optimization problem building continuous stirred tank formula course of reaction technological design and control is as follows:

$\underset{d,\stackrel{\‾}{Y},\stackrel{\‾}{U},{\mathrm{\σ}}_Y,{\mathrm{\σ}}_U}{min}CC+OC---\left(16\right)$

s.t.

$f(d,\stackrel{\‾}{X},\stackrel{\‾}{U},\stackrel{\‾}{Y})=0---\left(17\right)$

$g(d,\stackrel{\‾}{X},\stackrel{\‾}{U},\stackrel{\‾}{Y})=0---\left(18\right)$

$L_Y+r_Y\×{\mathrm{\σ}}_Y\≤\stackrel{\‾}{Y}\≤H_Y-r_Y\×{\mathrm{\σ}}_Y---\left(19\right)$

$L_U+r_U\×{\mathrm{\σ}}_U\≤\stackrel{\‾}{U}\≤H_U-r_U\×{\mathrm{\σ}}_U---\left(20\right)$

σ _Y＝f(σ _U)(21)

In above formula, CC represents equipment cost; OC represents running cost; D is design parameter;

This optimization problem can adopt NLP optimized algorithm to solve, as SQP.

Advantage of the present invention is: effectively can solve when technological design and control integration are optimized and solve complicated problem, greatly reduce computational complexity, make the process of design have good economic performance and dynamic property simultaneously simultaneously.

Accompanying drawing explanation

Constrained input variance graph of a relation under Fig. 1 LQG control action of the present invention

Fig. 2 " retrogressing " of the present invention strategy and constraint allow the schematic diagram violating probability;

The CSTR (CSTR) that Fig. 3 the present invention relates to

Embodiment

Below in conjunction with accompanying drawing, the integrated optimization method of continuous stirred tank formula course of reaction technological design of the present invention and control is described in detail.

An integrated optimization method for the course of reaction technological design of continuous stirred tank formula and control, comprising:

Step 1, for continuous stirred tank formula course of reaction technique, according to mass balance and principle of energy balance, set up exact mechanism model, as following differential algebraic equations (DAE) form:

$f(\frac{{\mathrm{dx}}_p\left(t\right)}{dt},x_p(t),z,u(t),y(t),t)=0---\left(1\right)$

g(x _p(t),z,u(t),y(t),t)＝0(2)

In formula, x _pt () is state variable; Z is design variable; U (t) is process input; Y (t) is the output of process;

Step 2, obtains object with the linear state space model under machine frame.

Be random signal by process disturbance and noise processed, obtain augmentation process model, as follows:

$\left[\begin{array}{c}\frac{{\mathrm{dx}}_p\left(t\right)}{dt}\\ \frac{{\mathrm{dx}}_d\left(t\right)}{dt}\end{array}\right]=\left[\begin{array}{cc}f(\frac{{\mathrm{dx}}_p\left(t\right)}{dt},x_p(t),z,u(t),y(t),t)& 0\\ 0& A_d\end{array}\right]\left[\begin{array}{c}I\\ x_d\end{array}\right]+\left[\begin{array}{c}0\\ B_d\end{array}\right]w_d---\left(3\right)$

g(x _p(t),x _d(t),z,u(t),y(t),w _d,t)＝0(4)

y _m＝Y _my+v _m(5)

In formula, x _dfor disturbance variable; A _dand B _dthe model parameter of disturbance, w _dwhite Gaussian noise stochastic variable; Y _mit is measured value matrix; y _mit is measured value; v _mit is measurement noises.

Near steady state operation point, obtain object with the general linear state-space model under machine frame.As follows:

$\left[\begin{array}{c}\mathrm{\Δ}{\stackrel{\·}{x}}_p\\ \mathrm{\Δ}{\stackrel{\·}{x}}_d\end{array}\right]=\left[\begin{array}{cc}{A}_p& 0\\ 0& A_d\end{array}\right]\left[\begin{array}{c}\mathrm{\Δ}{x}_p\\ {\mathrm{\Δx}}_d\end{array}\right]+\left[\begin{array}{c}{B}_p\\ 0\end{array}\right]\mathrm{\Δ}u+\left[\begin{array}{c}0\\ B_d\end{array}\right]w_d---\left(6\right)$

${\mathrm{\Δy}}_m=C\left[\begin{array}{c}\mathrm{\Δ}{x}_p\\ {\mathrm{\Δx}}_d\end{array}\right]+v_m---\left(7\right)$

In formula, Δ x _pfor the variable quantity of state variable; Δ x _dfor the variable quantity of disturbance variable; Δ u is the variable quantity of manipulated variable; A _p, A _d, B _p, B _dfor state-space model matrix.

Above formula can be converted into state-space expression, as follows

x _t+1＝Ax _t+Bu _t-1+Ka _t

(8)

y _t＝Cx _t+a _t

Wherein, x _tfor the state variable variable quantity after conversion; A, B and C are state-space model parameter; K is Kalman state estimator; a _tfor random noise.

Step 3, design linear quadratic gaussian (LQG) controller, according to LQG control principle, calculates manipulated variable (MVs) and the closed loop variance of controlled variable (CVs) and optimal performance circle of MVs/CVs variance.Curve is coordinated, the funtcional relationship between the change of certainty annuity MVs, CVs variable variance according to LQG.

3-a, first, tries to achieve manipulated variable (MVs) and the closed loop variance of controlled variable (CVs) and optimal performance circle of MVs/CVs variance.Calculation procedure is as follows:

Step 3-a-1: Optimal state-feedback and the control law of trying to achieve system, as follows:

$\begin{array}{c}{\tilde{x}}_{t+1}=(A-KC-BL){\tilde{x}}_t+{\mathrm{Ky}}_t\\ u_t=-L{\tilde{x}}_t\end{array}---\left(9\right)$

Wherein L is optimal control law, solves by algebraic riccati equation.

Step 3-a-2: comprehensive card Kalman Filtering and Optimal state-feedback, as follows:

$\left[\begin{array}{c}{\tilde{x}}_{t+1}\\ {\hat{x}}_{t+1}\end{array}\right]=\left[\begin{array}{cc}A& -BL\\ KC& A-KC-BL\end{array}\right]\left[\begin{array}{c}{\tilde{x}}_t\\ {\hat{x}}_t\end{array}\right]+\left[\begin{array}{c}K\\ K\end{array}\right]a_t---\left(10\right)$

Step 3-a-3: try to achieve process input, export contrast, as follows:

Order $X_t=\left[\begin{array}{c}{\tilde{x}}_t\\ {\hat{x}}_t\end{array}\right],$ Then process input, output variance are respectively:

$\mathrm{var}\left(u_t\right)=\[\begin{array}{cc}0& -L\end{array}\]\mathrm{var}\left(X_t\right)\left[\begin{array}{c}0\\ -L^T\end{array}\right]---\left(11\right)$

$\mathrm{var}\left(y_t\right)=\left[\begin{array}{cc}C& 0\end{array}\right]\mathrm{var}\left(X_t\right)\left[\begin{array}{c}{C}^T\\ 0\end{array}\right]+\mathrm{var}\left(a_t\right)---\left(12\right)$

3-b, secondly, tries to achieve the funtcional relationship between the change of MVs, CVs variable variance.Calculation procedure is as follows:

Step 3-b-1: build LQG quadratic performance index function, as follows:

J(λ)＝E[Y ^TWY]+λE[U ^TRU](13)

Wherein W, R are the weighting matrix of output variable and input variable.

Step 3-b-2: build LQG quadratic performance index function, as follows:

By selecting different λ values within the specific limits, can obtain a series of under LQG control action about input variance and export the solution of contrast, based on these data, one can be made to input variance for horizontal ordinate, output variance is the performance curve of ordinate, as shown in Figure 1, this curve is illustrate linear controller performance lower bound that may reach under input and output variance characterizes.

Step 4, builds the integrated optimization problem of technological design and control, and is solved by optimized algorithm and obtain optimum process design parameter.

Funtcional relationship between the closed loop variance of foundation MVs, CVs and MVs/CVs variable variance change, in conjunction with " retrogressing " strategy and Steady-state process model, and the operating conditions of process and technical indicator are the probability that process constraints arranges constraint and allows to violate, as shown in Figure 2.Specific as follows:

$L_Y+r_Y\×{\mathrm{\σ}}_Y\≤\stackrel{\‾}{Y}\≤H_Y-r_Y\×{\mathrm{\σ}}_Y---\left(14\right)$

$L_U+r_U\×{\mathrm{\σ}}_U\≤\stackrel{\‾}{U}\≤H_U-r_U\×{\mathrm{\σ}}_U---\left(15\right)$

In formula, H _yand L _ybe respectively the bound of output variable; H _uand L _ube respectively the bound of input variable; σ _uand σ _ybe respectively the standard deviation of input variable and output variable; with be respectively the setting value of input variable and output variable; r _yand r _urepresent that constraint allows the probability parameter violated.

The integrated optimization problem building continuous stirred tank formula course of reaction technological design and control is as follows:

$\underset{d,\stackrel{\‾}{Y},\stackrel{\‾}{U},{\mathrm{\σ}}_Y,{\mathrm{\σ}}_U}{min}CC+OC---\left(16\right)$

s.t.

$f(d,\stackrel{\‾}{X},\stackrel{\‾}{U},\stackrel{\‾}{Y})=0---\left(17\right)$

$g(d,\stackrel{\‾}{X},\stackrel{\‾}{U},\stackrel{\‾}{Y})=0---\left(18\right)$

$L_Y+r_Y\×{\mathrm{\σ}}_Y\≤\stackrel{\‾}{Y}\≤H_Y-r_Y\×{\mathrm{\σ}}_Y---\left(19\right)$

$L_U+r_U\×{\mathrm{\σ}}_U\≤\stackrel{\‾}{U}\≤H_U-r_U\×{\mathrm{\σ}}_U---\left(20\right)$

σ _Y＝f(σ _U)(21)

In above formula, CC represents equipment cost; OC represents running cost; D is design parameter;

This optimization problem can adopt NLP optimized algorithm to solve, as SQP.

Emulation case study on implementation

Be illustrated in figure 3 a CSTR process.Assuming that there is the irreversible themopositive reaction A → B of one-level in CSTR, the exit concentration of temperature of reaction and reactant is regulated by chuck cooling water flow.Following strong nonlinearity equations of state is had according to material balance and energy equilibrium:

$\frac{{\mathrm{dC}}_A}{dt}=\frac{F}{V}(C_{A,0}-C_A)-C_A{K}_0{e}^{-\frac{E}{RT}}---\left(22\right)$

$\frac{dT}{dt}=\left(\frac{F}{V}\right)(T_0-T)-\frac{\mathrm{\Δ}H\×C_A{K}_0}{{\mathrm{\ρC}}_p}{e}^{-\frac{E}{RT}}-\frac{{\mathrm{UA}}_H}{{\mathrm{\ρVC}}_P}(T-T_J)---\left(23\right)$

$\frac{{\mathrm{dT}}_J}{dt}=\left(\frac{F_J}{V_J}\right)(T_{J,0}-T_J)+\frac{{\mathrm{UA}}_H}{{\mathrm{\ρ}}_J{V}_J{C}_J}(T-T_J)---\left(24\right)$

A _H＝πD _Rh(25)

$V=\frac{\mathrm{\π}}{4}{D}_R^{2}h---\left(26\right)$

Wherein, C _{a, 0}for the entrance concentration of reactant, C _afor the exit concentration of reactant, F is feed rate, K ₀pre-exponential factor, E is reaction activity, and R is ideal gas constant, and T is the temperature of reaction mixture in reactor, T _jfor the temperature of chuck chilled water, T ₀for feeding temperature, T _{j, 0}for cooling water inlet temperature, Δ H is reaction heat, and U is heat transfer coefficient, A _hfor heat transfer area, C _p, C _jbe respectively the specific heat capacity of reaction mixture and the specific heat capacity of chilled water, ρ, ρ _jbe respectively the density of reaction mixture and chuck chilled water, F _jfor the inlet flow rate of chilled water, V _jfor jacket volume, h is the height of reactor, D _rfor the diameter of reactor.The model parameter of CSTR is in table 1.

Process and reactor size are subject to following constraint:

300≤T≤334(27)

0≤C _A≤801

$\frac{h}{D_R}=2$

The model parameter of table 1 CSTR

Suppose that process is subject to the disturbance of chuck import cooling water temperature.The goal in research of present case is design CSTR process, and make designed CSTR process when being subject to external disturbance, system can meet operation requirements and product quality requirement, and equipment investment cost is minimum, and control performance is good.Design variable mainly comprises: reactor diameter, highly, operation operating mode (temperature of reaction, cooling water flow etc.).

This CSTR process is designed respectively with illustrated method and dynamic optimization method.Result is as shown in table 2.

Table 2CSTR Process Design result

Can see that illustrated method and dynamic optimization method obtain substantially identical CSTR process thus, but illustrated method only adopts steady state optimization method, instead of the dynamic optimization method of complexity.

Claims (1)

1. an integrated optimization method for the technological design of continuous stirred tank formula course of reaction and control, comprising:

Step 1, for continuous stirred tank formula course of reaction technique, according to mass balance and principle of energy balance, set up exact mechanism model, as following differential algebraic equations (DAE) form:

$f(\frac{{\mathrm{dx}}_p\left(t\right)}{dt},x_p(t),z,u(t),y(t),t)=0---\left(1\right)$

g(x _p(t),z,u(t),y(t),t)＝0(2)

In formula, x _pt () is state variable; Z is design variable; U (t) is process input; Y (t) is the output of process;

Step 2, obtains object with the linear state space model under machine frame;

Be random signal by process disturbance and noise processed, obtain augmentation process model, as follows:

$\left[\begin{array}{c}\frac{{\mathrm{dx}}_p\left(t\right)}{dt}\\ \frac{{\mathrm{dx}}_d\left(t\right)}{dt}\end{array}\right]=\left[\begin{array}{cc}f\left(\frac{{\mathrm{dx}}_p\left(t\right)}{dt},x_p\left(t\right),z,u\left(t\right),y\left(t\right),t\right)& 0\\ 0& A_d\end{array}\right]\left[\begin{array}{c}I\\ x_d\end{array}\right]+\left[\begin{array}{c}0\\ B_d\end{array}\right]w_d---\left(3\right)$

g(x _p(t),x _d(t),z,u(t),y(t),w _d,t)＝0(4)

y _m＝Y _my+v _m(5)

In formula, x _dt () is disturbance variable; A _dand B _dthe model parameter of disturbance, w _dwhite Gaussian noise stochastic variable; Y _mit is measured value matrix; y _mit is measured value; v _mit is measurement noises.

Near steady state operation point, obtain object with the general linear state-space model under machine frame; As follows:

$\left[\begin{array}{c}\mathrm{\Δ}{\stackrel{\·}{x}}_p\\ \mathrm{\Δ}{\stackrel{\·}{x}}_d\end{array}\right]=\left[\begin{array}{cc}{A}_p& 0\\ 0& A_d\end{array}\right]\left[\begin{array}{c}{\mathrm{\Δx}}_p\\ {\mathrm{\Δx}}_d\end{array}\right]+\left[\begin{array}{c}{B}_p\\ 0\end{array}\right]\mathrm{\Δ}u+\left[\begin{array}{c}0\\ B_d\end{array}\right]w_d---\left(6\right)$

${\mathrm{\Δy}}_m=C\left[\begin{array}{c}\mathrm{\Δ}{x}_p\\ {\mathrm{\Δx}}_d\end{array}\right]+v_m---\left(7\right)$

In formula, Δ x _pfor the variable quantity of state variable; Δ x _dfor the variable quantity of disturbance variable; Δ u is the variable quantity of manipulated variable; A _p, A _d, B _p, B _dfor state-space model matrix.

Above formula can be converted into state-space expression, as follows

x _t+1＝Ax _t+Bu _t-1+Ka _t

(8)

y _t＝Cx _t+a _t

Wherein, x _tfor the state variable variable quantity after conversion; A, B and C are state-space model parameter; K is Kalman state estimator; a _tfor random noise.

Step 3, design linear quadratic gaussian (LQG) controller, according to LQG control principle, calculates manipulated variable (MVs) and the closed loop variance of controlled variable (CVs) and optimal performance circle of MVs/CVs variance; Curve is coordinated, the funtcional relationship between the change of certainty annuity MVs, CVs variable variance according to LQG;

3-a, first, tries to achieve manipulated variable (MVs) and the closed loop variance of controlled variable (CVs) and optimal performance circle of MVs/CVs variance; Calculation procedure is as follows:

Step 3-a-1: Optimal state-feedback and the control law of trying to achieve system, as follows:

${\tilde{x}}_{t+1}=(A-KC-BL){\tilde{x}}_t+{\mathrm{Ky}}_t$

(9)

$u_t=-L{\tilde{x}}_t$

Wherein L is optimal control law, solves by algebraic riccati equation;

Step 3-a-2: comprehensive card Kalman Filtering and Optimal state-feedback, as follows:

$\left[\begin{array}{c}{\tilde{x}}_{t+1}\\ {\hat{x}}_{t+1}\end{array}\right]=\left[\begin{array}{cc}A& -BL\\ KC& A-KC-BL\end{array}\right]\left[\begin{array}{c}{\tilde{x}}_t\\ {\hat{x}}_t\end{array}\right]+\left[\begin{array}{c}K\\ K\end{array}\right]a_t---\left(10\right)$

Step 3-a-3: try to achieve process input, export contrast, as follows:

Order $X_t=\left[\begin{array}{c}{\tilde{x}}_t\\ {\hat{x}}_t\end{array}\right],$ Then process input, output variance are respectively:

$\mathrm{var}\left(u_t\right)=\left[\begin{array}{cc}0& -L\end{array}\right]\mathrm{var}\left(X_t\right)\left[\begin{array}{c}0\\ -L^T\end{array}\right]---\left(11\right)$

$\mathrm{var}\left(y_t\right)=\left[\begin{array}{cc}C& 0\end{array}\right]\mathrm{var}\left(X_t\right)\left[\begin{array}{c}{C}^T\\ 0\end{array}\right]+\mathrm{var}\left(a_t\right)---\left(12\right)$

3-b, secondly, tries to achieve the funtcional relationship between the change of MVs, CVs variable variance; Calculation procedure is as follows:

Step 3-b-1: build LQG quadratic performance index function, as follows:

J(λ)＝E[Y ^TWY]+λE[U ^TRU](13)

Wherein, U is input variable; Y is output variable; W, R are the weighting matrix of output variable and input variable; λ is the weight factor between input variable and output variable.

Step 3-b-2: build LQG quadratic performance index function, as follows:

By selecting different λ values within the specific limits, can obtain a series of under LQG control action about input variance and export the solution of contrast, based on these data, one can be made to input variance for horizontal ordinate, output variance is the performance curve of ordinate, as shown in Figure 1, this curve is illustrate linear controller performance lower bound that may reach under input and output variance characterizes;

Step 4, builds the integrated optimization problem of technological design and control, and is solved by optimized algorithm and obtain optimum process design parameter;

According to the funtcional relationship between the closed loop variance of MVs, CVs and the change of MVs/CVs variable variance, combine " retrogressings " strategy and Steady-state process model, and the operating conditions of process and technical indicator are that process constraints arranges the probability retraining permission violation; Specific as follows:

$L_Y+r_Y\×{\mathrm{\σ}}_Y\≤\stackrel{\‾}{Y}\≤H_Y-r_Y\×{\mathrm{\σ}}_Y---\left(14\right)$

$L_U+r_U\×{\mathrm{\σ}}_U\≤\stackrel{\‾}{U}\≤H_U-r_U\×{\mathrm{\σ}}_U---\left(15\right)$

In formula, H _yand L _ybe respectively the bound of output variable; H _uand L _ube respectively the bound of input variable; σ _uand σ _ybe respectively the standard deviation of input variable and output variable; with be respectively the setting value of input variable and output variable; r _yand r _urepresent that constraint allows the probability parameter violated.

The integrated optimization problem building continuous stirred tank formula course of reaction technological design and control is as follows:

$\underset{d,\stackrel{\‾}{Y},\stackrel{\‾}{U},{\mathrm{\σ}}_Y,{\mathrm{\σ}}_U}{min}CC+OC---\left(16\right)$

s.t.

$f(d,\stackrel{\‾}{X},\stackrel{\‾}{U},\stackrel{\‾}{Y})=0---\left(17\right)$

$g(d,\stackrel{\‾}{X},\stackrel{\‾}{U},\stackrel{\‾}{Y})=0---\left(18\right)$

$L_Y+r_Y\×{\mathrm{\σ}}_Y\≤\stackrel{\‾}{Y}\≤H_Y-r_Y\×{\mathrm{\σ}}_Y---\left(19\right)$

$L_U+r_U\×{\mathrm{\σ}}_U\≤\stackrel{\‾}{U}\≤H_U-r_U\×{\mathrm{\σ}}_U---\left(20\right)$

σ _Y＝f(σ _U)(21)

In above formula, CC represents equipment cost; OC represents running cost; D is design parameter;

This optimization problem can adopt NLP optimized algorithm to solve, as SQP.