CN105487515B - A kind of integrated optimization method for continuously stirring the technological design of autoclave course of reaction and control - Google Patents

A kind of integrated optimization method for continuously stirring the technological design of autoclave course of reaction and control Download PDF

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CN105487515B
CN105487515B CN201511009070.4A CN201511009070A CN105487515B CN 105487515 B CN105487515 B CN 105487515B CN 201511009070 A CN201511009070 A CN 201511009070A CN 105487515 B CN105487515 B CN 105487515B
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CN105487515A (en
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周猛飞
蔡亦军
潘海天
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Zhejiang University of Technology ZJUT
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    • GPHYSICS
    • G05CONTROLLING; REGULATING
    • G05BCONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00Programme-control systems
    • G05B19/02Programme-control systems electric
    • G05B19/418Total factory control, i.e. centrally controlling a plurality of machines, e.g. direct or distributed numerical control [DNC], flexible manufacturing systems [FMS], integrated manufacturing systems [IMS], computer integrated manufacturing [CIM]
    • G05B19/41885Total factory control, i.e. centrally controlling a plurality of machines, e.g. direct or distributed numerical control [DNC], flexible manufacturing systems [FMS], integrated manufacturing systems [IMS], computer integrated manufacturing [CIM] characterised by modeling, simulation of the manufacturing system

Abstract

One kind continuously stirs the technological design of autoclave course of reaction and control integrated optimization method, including:Step 1, for continuously stirring autoclave course of reaction technique, according to mass balance and principle of energy balance, exact mechanism model is established;Step 2, linear state space model of the object under with machine frame is obtained;Step 3, according to linear quadratic gaussian (LQG) control principle, the closed loop variance of manipulating variable (MVs) and controlled variable (CVs) and optimal performance circle of MVs/CVs variances is calculated.Curve is coordinated according to LQG, determines the functional relation between the change of system MVs, CVs variable variance;Step 4, with reference to " retrogressing " (Back off) strategy and Steady-state process model, and the operating condition and technical indicator of process are that process constraints set constraint to allow the probability violated, technological design and the integrated optimization problem of control are built, and solves by optimized algorithm to obtain optimum process design parameter.

Description

A kind of integrated optimization method for continuously stirring the technological design of autoclave course of reaction and control
Technical field
The present invention relates to industrial control field, and in particular to one kind continuously stirs the technological design of autoclave course of reaction and control Integrated optimization method.
Background technology
In chemical system, reactor is the most influential cell arrangement of whole process, usually decides whole technique stream The property of journey, it also determine the economy of production and the influence to environment.CSTR (CSTR) is chemical industry row Widely used a kind of reactor in industry, at the same be also in process industrial it is typical, there is serious nonlinear dynamical system.
Traditional chemical process technological design and process control is carried out according to sequential design and control method.That is, first it was based on Journey steady-state model and economic optimum criterion determine to meet desired technological design;Control system is designed on this basis.In fact, Control performance quality is largely determined by the technological design factor such as flowage structure, the design specification of equipment.It is sequential step by step to set Meter method has isolated the relevance between technological design and process control performance, it is impossible to designed process is disturbed in outside Keep overall optimal under dynamic and uncertain factor.The integrated optimization of technological design and control considers the stable state of process simultaneously Characteristic and dynamic characteristic, it can obtain and meet that all design and operation constraintss are issued to the optimal design parameter of economic performance, It is greatly enhanced the operability and economic performance of process.The integrated optimization method of conventional technological design and control, calculate multiple It is miscellaneous, require high to calculating, the time of cost is especially more, strongly limit the application and popularization of integrated optimization.
The content of the invention
The present invention will overcome above mentioned problem existing for prior art, there is provided one kind continuously stirs autoclave course of reaction technique Design and control integrated optimization method, when can effectively solve the problem that technological design with controlling integrated optimization the problem of solving complexity, greatly Reduce computational complexity greatly, while make the process of design while there is preferable economic performance and dynamic property.
A kind of integrated optimization method for continuously stirring the technological design of autoclave course of reaction and control, including:
Step 1, for continuously stirring autoclave course of reaction technique, according to mass balance and principle of energy balance, establish tight Lattice mechanism model, such as following differential algebraic equations (DAE) form:
g(xp(t), z, u (t), y (t), t)=0 (2)
In formula, xp(t) it is state variable;Z is design variable;U (t) inputs for process;Y (t) is the output of process;
Step 2, linear state space model of the object under with machine frame is obtained.
It is random signal by process disturbance and noise processed, obtains augmentation process model, it is as follows:
g(xp(t),xd(t),z,u(t),y(t),wd, t) and=0 (4)
ym=Ymy+vm (5)
In formula, xdFor disturbance variable;AdAnd BdThe model parameter of disturbance, wdWhite Gaussian noise stochastic variable;YmIt is measured value Matrix;ymIt is measured value;vmIt is measurement noise.
Near steady state operation point, general linear state-space model of the object under with machine frame is obtained.It is as follows:
In formula, Δ xpFor the variable quantity of state variable;ΔxdFor the variable quantity of disturbance variable;Δ u is the change of manipulating variable Amount;Ap、Ad、Bp、BdFor state-space model matrix.
Above formula can be converted into state-space expression, as follows
xt+1=Axt+But-1+Kat
(8)
yt=Cxt+at
Wherein, xtFor the state variable variable quantity after conversion;A, B and C is state-space model parameter;K is Kalman's shape State estimator;atFor random noise.
Step 3, linear quadratic gaussian (LQG) controller is designed, according to LQG control principles, manipulating variable is calculated (MVs) with the closed loop variance of controlled variable (CVs) and optimal performance circle of MVs/CVs variances.Curve is coordinated according to LQG, it is determined that Functional relation between the change of system MVs, CVs variable variance.
3-a, first, try to achieve manipulating variable (MVs) and controlled variable (CVs) closed loop variance and MVs/CVs variances most Excellent performance bound.Calculation procedure is as follows:
Step 3-a-1:The Optimal state-feedback and control law of system are tried to achieve, it is as follows:
Wherein L is optimal control law, can be solved by algebraic riccati equation.
Step 3-a-2:Comprehensive Kalman filtering and Optimal state-feedback, it is as follows:
Step 3-a-3:Process input, output contrast are tried to achieve, it is as follows:
OrderThen process input, output variance are respectively:
3-b, secondly, try to achieve the functional relation between the change of MVs, CVs variable variance.Calculation procedure is as follows:
Step 3-b-1:LQG quadratic performance index functions are built, it is as follows:
J (λ)=E [YTWY]+λE[UTRU] (13)
Wherein W, R are the weighting matrix of output variable and input variable.
Step 3-b-2:LQG quadratic performance index functions are built, it is as follows:
By selecting different λ values within the specific limits, can obtain it is a series of under LQG control actions on input side The solution of difference and output contrast, based on these data, can make one to input variance as abscissa, and output variance is sat to be vertical Target performance curve, it is likely to be breached as shown in figure 1, the curve illustrates linear controller under input and output variance sign Performance lower bound.
Step 4, the integrated optimization problem of technological design and control is built, and solves by optimized algorithm to obtain optimum process Design parameter.
According to the functional relation between MVs, CVs closed loop variance and MVs/CVs variables variance change, with reference to " retrogressing " Tactful and Steady-state process model, and the operating condition of process and technical indicator are that process constraints set constraint to allow the general of violation Rate;It is specific as follows:
In formula, HYAnd LYThe respectively bound of output variable;HUAnd LUThe respectively bound of input variable;σUAnd σYPoint Not Wei input variable and output variable standard deviation;WithThe respectively setting value of input variable and output variable;rYAnd rUTable Show that constraint allows the probability parameter violated.
The integrated optimization problem that structure continuously stirs the technological design of autoclave course of reaction and control is as follows:
s.t.
σY=f (σU) (21)
CC represents equipment cost in above formula;OC represents running cost;D is design parameter;
The optimization problem can use NLP optimized algorithms to solve, such as SQP.
It is an advantage of the invention that:When can effectively solve the problem that technological design with controlling integrated optimization the problem of solving complexity, greatly Reduce computational complexity greatly, while make the process of design while there is preferable economic performance and dynamic property.
Brief description of the drawings
Input and output variance graph of a relation under the LQG control actions of Fig. 1 present invention
" retrogressing " strategy of Fig. 2 present invention and constraint allow the schematic diagram for violating probability;
Fig. 3 CSTRs of the present invention (CSTR)
Embodiment
Below in conjunction with the accompanying drawings, to the integrated optimization side for continuously stirring the technological design of autoclave course of reaction and control of the present invention Method is described in detail.
A kind of integrated optimization method for continuously stirring the technological design of autoclave course of reaction and control, including:
Step 1, for continuously stirring autoclave course of reaction technique, according to mass balance and principle of energy balance, establish tight Lattice mechanism model, such as following differential algebraic equations (DAE) form:
g(xp(t), z, u (t), y (t), t)=0 (2)
In formula, xp(t) it is state variable;Z is design variable;U (t) inputs for process;Y (t) is the output of process;
Step 2, linear state space model of the object under with machine frame is obtained.
It is random signal by process disturbance and noise processed, obtains augmentation process model, it is as follows:
g(xp(t),xd(t),z,u(t),y(t),wd, t) and=0 (4)
ym=Ymy+vm (5)
In formula, xdFor disturbance variable;AdAnd BdThe model parameter of disturbance, wdWhite Gaussian noise stochastic variable;YmIt is measured value Matrix;ymIt is measured value;vmIt is measurement noise.
Near steady state operation point, general linear state-space model of the object under with machine frame is obtained.It is as follows:
In formula, Δ xpFor the variable quantity of state variable;ΔxdFor the variable quantity of disturbance variable;Δ u is the change of manipulating variable Amount;Ap、Ad、Bp、BdFor state-space model matrix.
Above formula can be converted into state-space expression, as follows
xt+1=Axt+But-1+Kat
(8)
yt=Cxt+at
Wherein, xtFor the state variable variable quantity after conversion;A, B and C is state-space model parameter;K is Kalman's shape State estimator;atFor random noise.
Step 3, linear quadratic gaussian (LQG) controller is designed, according to LQG control principles, manipulating variable is calculated (MVs) with the closed loop variance of controlled variable (CVs) and optimal performance circle of MVs/CVs variances.Curve is coordinated according to LQG, it is determined that Functional relation between the change of system MVs, CVs variable variance.
3-a, first, try to achieve manipulating variable (MVs) and controlled variable (CVs) closed loop variance and MVs/CVs variances most Excellent performance bound.Calculation procedure is as follows:
Step 3-a-1:The Optimal state-feedback and control law of system are tried to achieve, it is as follows:
Wherein L is optimal control law, can be solved by algebraic riccati equation.
Step 3-a-2:Comprehensive Kalman filtering and Optimal state-feedback, it is as follows:
Step 3-a-3:Process input, output contrast are tried to achieve, it is as follows:
OrderThen process input, output variance are respectively:
3-b, secondly, try to achieve the functional relation between the change of MVs, CVs variable variance.Calculation procedure is as follows:
Step 3-b-1:LQG quadratic performance index functions are built, it is as follows:
J (λ)=E [YTWY]+λE[UTRU] (13)
Wherein W, R are the weighting matrix of output variable and input variable.
Step 3-b-2:LQG quadratic performance index functions are built, it is as follows:
By selecting different λ values within the specific limits, can obtain it is a series of under LQG control actions on input side The solution of difference and output contrast, based on these data, can make one to input variance as abscissa, and output variance is sat to be vertical Target performance curve, it is likely to be breached as shown in figure 1, the curve illustrates linear controller under input and output variance sign Performance lower bound.
Step 4, the integrated optimization problem of technological design and control is built, and solves by optimized algorithm to obtain optimum process Design parameter.
According to the functional relation between MVs, CVs closed loop variance and MVs/CVs variables variance change, with reference to " retrogressing " Tactful and Steady-state process model, and the operating condition of process and technical indicator are that process constraints set constraint to allow the general of violation Rate, as shown in Figure 2.It is specific as follows:
In formula, HYAnd LYThe respectively bound of output variable;HUAnd LUThe respectively bound of input variable;σUAnd σYPoint Not Wei input variable and output variable standard deviation;WithThe respectively setting value of input variable and output variable;rYAnd rUTable Show that constraint allows the probability parameter violated.
The integrated optimization problem that structure continuously stirs the technological design of autoclave course of reaction and control is as follows:
s.t.
σY=f (σU) (21)
CC represents equipment cost in above formula;OC represents running cost;D is design parameter;
The optimization problem can use NLP optimized algorithms to solve, such as SQP.
Simulation implementation case
It is illustrated in figure 3 a CSTR process.It is assumed that the irreversible exothermic reaction A → B of one-level occurs in CSTR, reaction The exit concentration of temperature and reactant is adjusted by chuck cooling water flow.Had according to material balance and energy balance following strong non-thread Property equations of state:
AH=π DRh (25)
Wherein, CA,0For the entrance concentration of reactant, CAFor the exit concentration of reactant, F is feed rate, K0Refer to cause Son, E are reaction activity, and R is ideal gas constant, and T is the temperature of reactant mixture in reactor, TJFor chuck cooling water Temperature, T0For feeding temperature, TJ,0For cooling water inlet temperature, Δ H is reaction heat, and U is heat transfer coefficient, AHFor heat transfer area, CP、 CJThe respectively specific heat capacity of reactant mixture and the specific heat capacity of cooling water, ρ, ρJRespectively reactant mixture and chuck cooling water Density, FJFor the inlet flow rate of cooling water, VJFor jacket volume, h is the height of reactor, DRFor the diameter of reactor.Continuously stir The model parameter for mixing tank reactor is shown in Table 1.
Process and reactor size are by following constraint:
300≤T≤334 (27)
0≤CA≤801
The model parameter of the CSTR of table 1
Assuming that process is disturbed by chuck import cooling water temperature.The goal in research of present case is one CSTR mistake of design Journey so that for designed CSTR processes when by external disturbance, system disclosure satisfy that operation requires and product quality requires, and And equipment investment cost is minimum, control performance is good.Design variable mainly includes:Reactor diameter, height, operation operating mode (reaction Temperature, cooling water flow etc.).
Method and dynamic optimization method shown in the present invention separately design the CSTR processes.As a result it is as shown in table 2.
The CSTR Process Design results of table 2
It can thus be seen that method shown in the present invention obtains essentially identical CSTR processes, but this with dynamic optimization method Method shown in invention is only with steady state optimization method, rather than complicated dynamic optimization method.

Claims (1)

1. a kind of integrated optimization method for continuously stirring the technological design of autoclave course of reaction and control, including:
Step 1, for continuously stirring autoclave course of reaction technique, according to mass balance and principle of energy balance, strict machine is established Manage model, such as following differential algebraic equations (DAE) form:
<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>dx</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>,</mo> <msub> <mi>x</mi> <mi>p</mi> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>u</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
g(xp(t), z, u (t), y (t), t)=0 (2)
In formula, xp(t) it is state variable;Z is design variable;U (t) inputs for process;Y (t) is the output of process;
Step 2, linear state space model of the object under with machine frame is obtained;
It is random signal by process disturbance and noise processed, obtains augmentation process model, it is as follows:
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mrow> <msub> <mi>dx</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msub> <mi>dx</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>dx</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>,</mo> <msub> <mi>x</mi> <mi>p</mi> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>u</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>A</mi> <mi>d</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>I</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mi>d</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mi>d</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>w</mi> <mi>d</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
g(xp(t),xd(t),z,u(t),y(t),wd, t) and=0 (4)
ym=Ymy+vm (5)
In formula, xd(t) it is disturbance variable;AdAnd BdThe model parameter of disturbance, wdWhite Gaussian noise stochastic variable;YmIt is measured value square Battle array;ymIt is measured value;vmIt is measurement noise;
Near steady state operation point, general linear state-space model of the object under with machine frame is obtained;It is as follows:
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>&amp;Delta;</mi> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>p</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&amp;Delta;</mi> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>A</mi> <mi>p</mi> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>A</mi> <mi>d</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;x</mi> <mi>p</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;x</mi> <mi>d</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>B</mi> <mi>p</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>&amp;Delta;</mi> <mi>u</mi> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mi>d</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>w</mi> <mi>d</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>&amp;Delta;y</mi> <mi>m</mi> </msub> <mo>=</mo> <mi>C</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;x</mi> <mi>p</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;x</mi> <mi>d</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msub> <mi>v</mi> <mi>m</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
In formula, Δ xpFor the variable quantity of state variable;ΔxdFor the variable quantity of disturbance variable;Δ u is the variable quantity of manipulating variable; Ap、Ad、Bp、BdFor state-space model matrix;
Above formula can be converted into state-space expression, as follows
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>Ax</mi> <mi>t</mi> </msub> <mo>+</mo> <msub> <mi>Bu</mi> <mrow> <mi>t</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Ka</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>Cx</mi> <mi>t</mi> </msub> <mo>+</mo> <msub> <mi>a</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
Wherein, xtFor the state variable variable quantity after conversion;A, B and C is state-space model parameter;K is Kalman estimator Device;atFor random noise;
Step 3, design linear quadratic gaussian (LQG) controller, according to LQG control principles, be calculated manipulating variable (MVs) with The closed loop variance of controlled variable (CVs) and optimal performance circle of MVs/CVs variances;According to LQG coordinate curve, determine system MVs, Functional relation between the change of CVs variables variance;
3-a, first, try to achieve the closed loop variance of manipulating variable (MVs) and controlled variable (CVs) and the optimality of MVs/CVs variances Can boundary;Calculation procedure is as follows:
Step 3-a-1:The Optimal state-feedback and control law of system are tried to achieve, it is as follows:
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>-</mo> <mi>K</mi> <mi>C</mi> <mo>-</mo> <mi>B</mi> <mi>L</mi> <mo>)</mo> </mrow> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mi>t</mi> </msub> <mo>+</mo> <msub> <mi>Ky</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>L</mi> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mi>t</mi> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
Wherein L is optimal control law, can be solved by algebraic riccati equation;
Step 3-a-2:Comprehensive Kalman filtering and Optimal state-feedback, it is as follows:
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mrow> <mo>-</mo> <mi>B</mi> <mi>L</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>K</mi> <mi>C</mi> </mrow> </mtd> <mtd> <mrow> <mi>A</mi> <mo>-</mo> <mi>K</mi> <mi>C</mi> <mo>-</mo> <mi>B</mi> <mi>L</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mi>t</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>t</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>K</mi> </mtd> </mtr> <mtr> <mtd> <mi>K</mi> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>a</mi> <mi>t</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
Step 3-a-3:Process input, output contrast are tried to achieve, it is as follows:
OrderThen process input, output variance are respectively:
<mrow> <mi>var</mi> <mrow> <mo>(</mo> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mi>L</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>var</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msup> <mi>L</mi> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mi>var</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>var</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>var</mi> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
3-b, secondly, try to achieve the functional relation between the change of MVs, CVs variable variance;Calculation procedure is as follows:
Step 3-b-1:LQG quadratic performance index functions are built, it is as follows:
J (λ)=E [YTWY]+λE[UTRU] (13)
Wherein, U is input variable;Y is output variable;W, R are the weighting matrix of output variable and input variable;λ becomes for input Weight factor between amount and output variable;
Step 3-b-2:LQG quadratic performance index functions are built, it is as follows:
By selecting different λ values within the specific limits, can obtain it is a series of under LQG control actions on input variance and The solution of contrast is exported, based on these data, one can be made to input variance as abscissa, output variance is ordinate Performance curve, as shown in figure 1, the curve is to illustrate the performance that linear controller is likely to be breached under input and output variance sign Lower bound;
Step 4, the integrated optimization problem of technological design and control is built, and solves to obtain optimum process by optimized algorithm and designs Parameter;
According to the functional relation between MVs, CVs closed loop variance and MVs/CVs variables variance change, with reference to " retrogressing " strategy Constraint is set to allow the probability violated for process constraints with Steady-state process model, and the operating condition of process and technical indicator; It is specific as follows:
<mrow> <msub> <mi>L</mi> <mi>Y</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mi>Y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>&amp;sigma;</mi> <mi>Y</mi> </msub> <mo>&amp;le;</mo> <mover> <mi>Y</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;le;</mo> <msub> <mi>H</mi> <mi>Y</mi> </msub> <mo>-</mo> <msub> <mi>r</mi> <mi>Y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>&amp;sigma;</mi> <mi>Y</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>L</mi> <mi>U</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mi>U</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>&amp;sigma;</mi> <mi>U</mi> </msub> <mo>&amp;le;</mo> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;le;</mo> <msub> <mi>H</mi> <mi>U</mi> </msub> <mo>-</mo> <msub> <mi>r</mi> <mi>U</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>&amp;sigma;</mi> <mi>U</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
In formula, HYAnd LYThe respectively bound of output variable;HUAnd LUThe respectively bound of input variable;σUAnd σYRespectively The standard deviation of input variable and output variable;WithThe respectively setting value of input variable and output variable;rYAnd rURepresent about Beam allows the probability parameter violated;
The integrated optimization problem that structure continuously stirs the technological design of autoclave course of reaction and control is as follows:
<mrow> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <mi>d</mi> <mo>,</mo> <mover> <mi>Y</mi> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <msub> <mi>&amp;sigma;</mi> <mi>Y</mi> </msub> <mo>,</mo> <msub> <mi>&amp;sigma;</mi> <mi>U</mi> </msub> </mrow> </munder> <mi>C</mi> <mi>C</mi> <mo>+</mo> <mi>O</mi> <mi>C</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
s.t.
<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>d</mi> <mo>,</mo> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mover> <mi>Y</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>d</mi> <mo>,</mo> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mover> <mi>Y</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>L</mi> <mi>Y</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mi>Y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>&amp;sigma;</mi> <mi>Y</mi> </msub> <mo>&amp;le;</mo> <mover> <mi>Y</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;le;</mo> <msub> <mi>H</mi> <mi>Y</mi> </msub> <mo>-</mo> <msub> <mi>r</mi> <mi>Y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>&amp;sigma;</mi> <mi>Y</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>L</mi> <mi>U</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mi>U</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>&amp;sigma;</mi> <mi>U</mi> </msub> <mo>&amp;le;</mo> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;le;</mo> <msub> <mi>H</mi> <mi>U</mi> </msub> <mo>-</mo> <msub> <mi>r</mi> <mi>U</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>&amp;sigma;</mi> <mi>U</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
σY=f (σU) (21)
CC represents equipment cost in above formula;OC represents running cost;D is design parameter;
The optimization problem can use NLP optimized algorithms to solve, such as SQP.
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