CN109814391A - A kind of Singular optimal control simultaneous solution method based on partial movement finite element node - Google Patents
A kind of Singular optimal control simultaneous solution method based on partial movement finite element node Download PDFInfo
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Abstract
The invention discloses a kind of Singular optimal control simultaneous solution methods based on partial movement finite element node.The difficult point that Singular Optimal Control solves is to be not easy to accurately find the switching point between unusual arc and nonsingular arc, this method is discrete for nonlinear programming problem by Singular Optimal Control using orthogonal collocation on finite element, determine the position for needing mobile finite element node by the switching function based on governing equation, and using this part need mobile finite element node location as optimized variable together with control variable optimizing to find the unusual switching point between nonsingular arc with very high precision.For existing Singular Optimal Controls a large amount of in production process, the present invention can obtain high-precision optimum control curves, and calculating speed is fast, suitable for the large-scale Singular Optimal Control of solution.
Description
Technical field
The present invention relates to procedures system control fields more particularly to a kind of simultaneous based on partial movement finite element node to ask
Solution method.
Background technique
Singular Optimal Control be in the production processes such as chemical industry, pharmacy it is generally existing, such as catalyst mix splitting or integrating
Criticize the optimization etc. of metabolite in feed-batch culture.For the Singular Optimal Control that often occurs in procedures system control,
Classical processing method is regularization method, and Jacobson proposition increases penalty factor in objective function, will solve it is unusual most
Excellent control problem becomes to solve a series of nonsingular optimum control subproblems [Jacobson D, Gershwin S, Lele
M.Computation of Optimal Singular Controls[J].IEEE Transactions on Automatic
Control,1970,15(1):67-73].But this method calculate it is cumbersome, it is computationally intensive, and when penalty factor is close to 0
When, it will lead to numerical value dyscalculia.Even if solving the subproblem of regularization method using commercialized pseudo- spectrometry existing at present,
The result that cannot still get well.In addition, pseudo- spectrometry is to improve computational accuracy, a large amount of grid node can be added in solution procedure
Or increase the order of interpolation polynomial, therefore its calculation amount for solving subproblem also can it is very big [Darby CL, Hager WW,
Rao AV.An hp-Adaptive Pseudospectral Method for Solving Optimal Control
Problems[J].Optimal Control Applications and Methods,2011,32(4):476-502]。
Summary of the invention
In view of the above-mentioned deficiencies in the prior art, it is an object of the present invention to propose a kind of based on partial movement finite element node
Singular optimal control simultaneous solution method.
The purpose of the present invention is achieved through the following technical solutions: a kind of surprise based on partial movement finite element node
Different optimum control simultaneous solution method, comprising the following steps:
Step 1: being divided into M finite element for control time domain, P orthogonal configuration point used in each finite element, will be former
Discrete Singular Optimal Control is nonlinear programming problem, solves nonlinear programming problem and obtains approximate optimum control song
Line；
Step 2: detecting discretization error on the non-collocation point being configured adjacently between a little in each finite element, if being less than specified
Tolerance ε1, then step 3 is carried out；Otherwise it is inserted into finite element node on non-collocation point, it is straight to solve nonlinear programming problem again
Discretization error on to non-collocation point meets tolerance ε1Until, and the final finite element number that the step is obtained is denoted as N, has
It limits first Node distribution and is denoted as a1, a2..., aN；
Step 3: based on obtained near-optimization controlling curve, switching function in each finite element is calculated, if switching function
Value be not consistent with theoretic value, be inserted into new finite element node, and by the finite element node being newly inserted into and control variable one
It rises and carries out optimizing, optimum control curve after being improved as variable to be optimized；
Step 4: based on optimum control curve after improving, if the controlling curve slope in j-th of finite element is greater than scheduled
Threshold value then deletes the finite element, and otherwise calculating terminates；
Step 5: if there is deletion finite element in step 4, remaining all finite element nodes, re-optimization control are fixed
Koji-making line, and terminate to calculate.
Further, the step 3 is realized by following sub-step:
(3.1) if switching function is not zero and controls variable again not on coboundary and lower boundary in i-th of finite element,
In the finite element [ai-1,ai] midpoint is inserted into new finite element node bi；
(3.2) fixed original node a0,a1,…,aNIf the finite element node being newly inserted into has K (to be denoted as), by node locationAs optimized variable optimizing together with control variable, and requireMeet
The invention has the advantages that being compared with the method for traditional processing Singular Optimal Control, the present invention
The switching point between unusual arc and nonsingular arc can be found more accurately, thus obtain finding better optimum control curve,
In addition calculating speed of the invention is fast, suitable for the large-scale Singular Optimal Control of solution.
Detailed description of the invention
Fig. 1 is the Singular optimal control simultaneous solution method flow diagram based on partial movement finite element node；
Fig. 2 is to solve catalyst using the Singular optimal control simultaneous solution method based on partial movement finite element node to mix
Close the optimum control curve that case obtains；
Fig. 3 is that method solves the optimum control curve that catalyst mixed case obtains in conjunction with pseudo- spectrometry using regularization；
Fig. 4 is to solve Lee- using the Singular optimal control simultaneous solution method based on partial movement finite element node
The optimum control curve that Ramirez bioreactor case obtains；
Fig. 5 is that method solution Lee-Ramirez bioreactor case obtains most in conjunction with pseudo- spectrometry using regularization
Excellent controlling curve.
Specific embodiment
Below according to attached drawing, the present invention will be described in detail.
As shown in Figure 1, the present invention is based on the Singular optimal control simultaneous solution methods of partial movement finite element node, including
Following steps:
Step 1: being divided into M finite element for control time domain, P orthogonal configuration point used in each finite element, will be former
Discrete Singular Optimal Control is nonlinear programming problem, solves nonlinear programming problem and obtains approximate optimum control song
Line；
Step 2: detecting discretization error on the non-collocation point being configured adjacently between a little in each finite element, if being less than specified
Tolerance ε1, then step 3 is carried out；Otherwise it is inserted into finite element node on non-collocation point, it is straight to solve nonlinear programming problem again
Discretization error on to non-collocation point meets tolerance ε1Until, and the final finite element number that the step is obtained is denoted as N, has
It limits first Node distribution and is denoted as a1, a2..., aN；
Step 3: based on obtained near-optimization controlling curve, switching function in each finite element is calculated, if switching function
Value be not consistent with theoretic value, be inserted into new finite element node, and by the finite element node being newly inserted into and control variable one
It rises and carries out optimizing, optimum control curve after being improved as variable to be optimized.
The step is core of the invention, is divided into following sub-step:
1) if the absolute value of switching function is greater than tolerance ε in i-th of finite element2, and variable is controlled at a distance from coboundary
Tolerance ε is both greater than with the distance of lower boundary3, then in the finite element [ai-1,ai] midpoint is inserted into new finite element node bi；
2) fixed original node a0,a1,…,aNIf the finite element node being newly inserted into has K (to be denoted as),
By node locationAs optimized variable optimizing together with control variable, and requireMeet
Step 4: based on optimum control curve after improving, if the controlling curve slope in j-th of finite element is greater than scheduled
Threshold epsilon4, then the finite element is deleted, otherwise calculating terminates；
Step 5: if there is deletion finite element in step 4, remaining all finite element nodes, re-optimization control are fixed
Koji-making line, and terminate to calculate.
Embodiment
In following two embodiment, involved parameter setting in invention are as follows: M=9, P=5, ε1=10-5, ε2=10-6, ε3=10-3, ε4=100.
Embodiment 1
Assuming that in following reaction:
It is reversible between A and B and is catalyzed by catalyst I type, B to C is irreversible to be urged by II type catalyst
Change.How we are in need of consideration mixes two kinds of catalyst so that the yield of products C is maximum, which can be described as
Following proposition:
max1-a(tf)-b(tf)
A (0)=1, b (0)=0, u (t) ∈ [0,1]
Wherein, tf indicates that the total length of reactor, t indicate the distance to reactor inlet, and a (t) and b (t) indicate reaction
The mole fraction of object A and B,WithIndicate the change rate to t；K1, k2 and k3 indicate that 1,2 and 3 reaction rate of reaction is normal
Number；U (t) indicates ratio shared by catalyst I type.
In the embodiment, tf=4, k1=k3=1, k2=10, comprising the following steps:
Step 1: control time domain is drawn and is divided into 9 finite elements, corresponding finite element node are as follows: 0.44444,
0.88889,1.33333,1.77778,2.22222,2.66667,3.11111,3.55556,4.00, using orthogonal configuration side
Method, by former Singular Optimal Control it is discrete be nonlinear programming problem, solve nonlinear programming problem obtain it is approximate optimal
Controlling curve；
Step 2: discretization error on the non-collocation point being configured adjacently between a little in each finite element is detected, it is desirable that be less than and refer to
Fixed tolerance ε1, the finally obtained finite element number of the step is 15；
Step 3: based on obtained near-optimization controlling curve, switching function in each finite element is calculated, if switching function
Value be not consistent with theoretic value, be inserted into new finite element node, and by the finite element node being newly inserted into and control variable one
It rises and carries out optimizing as variable to be optimized, optimum control curve after being improved:
1) absolute value of switching function is greater than ε in the 3,4,5,11,12,13,14th finite element2, and control variable with it is upper
The distance of boundary and lower boundary is greater than ε3, then in finite element [0.074211,0.19121], [0.19121,0.320851],
[0.320851,0.413386], [2.2222,2.66664], [2.66664,3.111084], [3.111084,3.55552],
New finite element node is inserted at the midpoint [3.55552,3.99996]；
2) fixed original finite element node, by be newly inserted into 7 finite element nodes 0.1363,0.25603,0.3672,
2.44442,2.88886,3.3333,3.72523 as optimized variable optimizing together with control variable, and the insertion that look for novelty has
It limits in first node 7 sections listed in 1) always；
Step 4: based on optimum control curve after improving, it is found that the controlling curve slope in all finite elements is both less than pre-
Fixed threshold value 100, calculating terminates, and acquired final controlling curve is as shown in Fig. 2, in the result between unusual arc and nonsingular arc
Switching point and theoretical value almost coincide.
In order to further illustrate the advantageous effect of the invention, also uses and opened based on method of the regularization in conjunction with pseudo- spectrometry
The problem is solved, as a result as shown in figure 3, differing greatly with theoretic result.
Embodiment 2
The embodiment is the optimal control problem about Lee-Ramirez bioreactor, and target is by controlling grape
Sugar and inducer feed rate maximize the amount of product, and specific embodiment is as follows:
min-x4(tf)x1(tf)
s.t.
t3(t)=0.22x7(t)/t2(t)+x6(t)
g1(t)=x3(t)t3(t)/t1(t)；
g2(t)=0.233x3(t)[0.0005+x5(t)]/[0.022+x5(t)]/t1(t)；
g3(t)=0.09x5(t)/[0.034+x5(t)]；0≤u1(t),u2(t)≤1
Wherein, t indicates the reaction moment, and tf indicates total reaction time, x1(t) volume of reactor, x are indicated2(t) it indicates
Cell density, x3(t) nutrient density, x are indicated4(t) foreign protei concentration, x are indicated5(t) inducer concentrations, x are indicated6(t) it indicates
Inducer shock factor, x7(t) indicate that inducer recycles the factor；u1(t) glucose feed rate, u are indicated2(t) inducer is indicated
Feed rate；c1Indicate the input concentration of glucose, c2It indicates to increase yield coefficient, c3Indicate the input concentration of inducer；g1
(t) specific growth rate, g are indicated2(t) foreign protei productivity, g are indicated3(t) shock rate is indicated；t1(t), t2(t) and t3(t)
For intermediate variable.
The following steps are included:
Step 1: by control temporal partitioning at 9 finite elements, corresponding finite element node are as follows: 0,1.11111,
2.22222,3.33333,4.44444,5.55556,6.66667,7.77778,8.88889,10, using orthogonal configuration method,
By former Singular Optimal Control it is discrete be nonlinear programming problem, solve nonlinear programming problem obtain approximate optimum control
Curve；
Step 2: discretization error on the non-collocation point being configured adjacently between a little in each finite element is detected, it is desirable that be less than and refer to
Fixed tolerance ε1, the finally obtained finite element number of the step is 35；
Step 3: based on obtained near-optimization controlling curve, switching function in each finite element is calculated, if switching function
Value be not consistent with theoretic value, be inserted into new finite element node, and by the finite element node being newly inserted into and control variable one
It rises and carries out optimizing as variable to be optimized, optimum control curve after being improved:
1) absolute value of switching function is greater than ε in the 1,2,3,4,15,16,17,18,19,26,27th finite element2, and control
Variable processed is greater than ε at a distance from coboundary and lower boundary3, then in finite element [0,1.11111], [1.11111,2.22222],
[2.22222,3.33333], [3.33333,4.44444], [4.72385,5.55558], [5.55558,5.58731],
[5.58731,6.66671], [6.66671,7.77781], [7.77781,7.809533], [8.26496,8.353871],
New finite element node is inserted at the midpoint [8.35387,8.7103]；
2) fixed original finite element node, by be newly inserted into 11 finite element nodes 1.10967,1.11252,
2.22367,3.33530,4.82270,5.55662,6.66563,7.26963,7.77908,8.3526,8.6585 as optimization
Variable optimizing together with control variable, and in the finite element node for the insertion that look for novelty 11 sections listed in 1) always；
Step 4: optimum control curve after improving, the controlling curve in the 2nd, 3,7,20,24,26,36 finite element are based on
Slope is greater than scheduled threshold value 100, then deletes these finite elements；
Step 5: having deletion finite element in step 4, then fix remaining all finite element nodes 39, re-optimization
Controlling curve, and terminate to calculate, acquired final controlling curve as shown in figure 4, last resulting target value be-
6.1516968538。
In order to further illustrate the advantageous effect of the invention, also uses and opened based on method of the regularization in conjunction with pseudo- spectrometry
The problem is solved, as a result as shown in figure 5, target value is -6.1515979885.In comparison the target function value wants poor.
Claims (2)
1. a kind of Singular optimal control simultaneous solution method based on partial movement finite element node, which is characterized in that including with
Lower step:
Step 1: being divided into M finite element for control time domain, P orthogonal configuration point used in each finite element, will be former unusual
Discrete optimal control problem is nonlinear programming problem, solves nonlinear programming problem and obtains approximate optimum control curve.
Step 2: detecting discretization error on the non-collocation point being configured adjacently between a little in each finite element, if being less than specified appearance
Poor ε1, then step 3 is carried out；Otherwise it is inserted into finite element node on non-collocation point, solves nonlinear programming problem again until non-
Discretization error on collocation point meets tolerance ε1Until, and the final finite element number that the step is obtained is denoted as N, finite element
Node distribution is denoted as a1, a2..., aN。
Step 3: based on obtained near-optimization controlling curve, calculating switching function in each finite element, if the value of switching function
It is not consistent with theoretic value, is inserted into new finite element node, and the finite element node being newly inserted into and control variable one are acted as
Optimizing, optimum control curve after being improved are carried out for variable to be optimized.
Step 4: based on optimum control curve after improving, if the controlling curve slope in j-th of finite element is greater than scheduled threshold
Value, then delete the finite element, and otherwise calculating terminates.
Step 5: if there is deletion finite element in step 4, remaining all finite element nodes are fixed, re-optimization control is bent
Line, and terminate to calculate.
2. special according to claim 1 based on the Singular optimal control simultaneous solution method of partial movement finite element node
Sign is that the step 3 is realized by following sub-step:
(3.1) if switching function is not zero and controls variable again not on coboundary and lower boundary, at this in i-th of finite element
Finite element [ai-1,ai] midpoint is inserted into new finite element node bi。
(3.2) fixed original node a0,a1,…,aNIf the finite element node being newly inserted into has K (to be denoted asBy node locationAs optimized variable optimizing together with control variable, and requireMeet
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