CN102636992A - Modeling based on piecewise-linear system of hybrid system theory and control method - Google Patents

Abstract

The invention discloses a modeling based on a piecewise-linear system of a hybrid system theory and a control method, and belongs to the field of automatic control. All segmented models of the piecewise linear system can be integrated in an uniform frame so as to accurately describe the interaction of a model switch event and a continuous dynamic process, thereby being beneficial to eliminating vibration caused by the model switch; and a linetype performance index is adopted to replace traditional quadratic performance index to forecast controller design, and an OPC (organic photoconductor) client end program independently developed by Matlab is utilized to realize the integration of mixed prediction control algorithm and a DCS (data communication system) control system.

Description

A kind of modeling and control method based on the theoretical piecewise-linear system of hybrid system

Technical field

The invention belongs to automation field, be used for the modeling of piecewise-linear system and, be specifically related to a kind of modeling and control method based on the theoretical piecewise-linear system of hybrid system based on the PREDICTIVE CONTROL of model.

Background technology

In the Control Theory and Control Engineering field, the problem of people's major concern is the variable of control system how, makes total system can reach certain specific target (like the stability of system, tracking performance of system or the like).Control theory that the scholar in control field proposes and method mostly are to continuous variable dynamic system.In traditional Control System Design, people near the work equilibrium point, set up the inearized model of controlled process usually, and are the system design controller according to linear control theory.When system works during in the equilibrium point working range; The controller of design can keep the better controlling effect; But some controlled process; Because it is bigger etc. former thereby shown very strong nonlinear characteristic to be operated in bigger opereating specification, set point change, also has some systems itself to have nonlinear characteristic, such as the saturation characteristic of system, dead band characteristic or the like.For these complicated nonlinear systems,, usually be similar to actual NLS with piecewise-linear system for the simplification problem.

Multi-model PID control is the modal controller of handling piecewise-linear system.The PID controller generally is made up of ratio (P), integration (I), three parts of differential (D):

$G\left(s\right)=k_p(1+\frac{1}{T_{i}s}+T_{d}s)$

K wherein _pMake proportional gain; T _iIt is integral time; T _dIt is derivative time.

Multi-model PID controlling schemes; Its basic thought is that the whole dynamic process with NLS resolves into a series of local linear models; Based on PID controller of each partial model design, design is at last switched or method of weighting constitutes whole control structure then.Multi-model PID control system commonly used is to PID controller of each local linear modelling, according to certain switching law, in each sampling instant, has only a PID controller incision backfeed loop.Because local control all is the PID controller, control structure is identical, so multi-model PID control only need be chosen different pid parameters according to different local linear models.The parameter tuning of PID is chosen parameter k exactly _p, T _iAnd T _dValue, the control performance that system is met the expectation.

The shortcoming of this control method is that controller causes disturbance even concussion when switching probably, because at the switching instant of controller, bigger saltus step may appear in the controlled quentity controlled variable of system.In addition, the problem that controller also exists the chatter phenomenon may occur exactly, promptly unlimited at short notice switch controller.

The hybrid system theory is to analyze and design the new way of the controller of piecewise-linear system.The General Definition of hybrid system is the unified dynamic system that is influenced each other, interacts and formed by continuous variable dynamic system and discrete variable dynamic system.From the angle of hybrid system, piecewise-linear system can be regarded one type of hybrid system as and study.

Summary of the invention

The object of the present invention is to provide a kind of modeling and control method based on hybrid system; Foundation can be more than traditional continuous model the accurate hybrid system model of descriptive system dynamic perfromance comprehensively; Realization avoids the model in the piecewise-linear system control to switch the jitter problem that causes based on hybrid system modelling linear prediction control.

The objective of the invention is to reach through following measure:

(1) according to the priori of controlled system etc., the continuous dynamic part of analytic system, Discrete Dynamic part and interaction between the two.To the continuous dynamic part of system, the available traditional difference equation or the differential equation are set up its model; Need simultaneously understanding system continuously dynamically between, continuously and between the Discrete Dynamic and the various logic of Discrete Dynamic self relation.

(2) various simple and all logical relations that the combination of sentences is come descriptive system of definition.According to the transformational relation of equal value of propositional logic and MIXED INTEGER inequality, obtain one group of MIXED INTEGER linear inequality.

(3) on the model basis of the continuous dynamic part of descriptive system, introduce logical variable and auxiliary variable, and be converted into linear discrete state equation form, the MIXED INTEGER inequality constrain of the various constraints of spare system and propositional logic conversion simultaneously.Obtain the mixed logic dynamic model:

x(k+1)＝Ax(k)+B ₁u(k)+B ₂δ(k)+B ₃z(k)

y(k)＝Cx(k)+D ₁u(k)+D ₂δ(k)+D ₃z(k)

E ₂δ(k)+E ₃z(k)≤E ₁u(k)+E ₄x(k)+E ₅

(4) adopt the line style performance index to replace traditional secondary type performance index to carry out the predictive controller design, be based on the PREDICTIVE CONTROL problem of the performance index of ∞ norm form:

$\min J(u_0^{T-1},x\left(t\right))=\underset{k=0}{\overset{T-1}{\mathrm{\Σ}}}\left(\right||Q_1(u\left(t\right)-u_e){\left|\right|}_{\∞}+||Q_2(\mathrm{\δ}\left(k\right|t)-{\mathrm{\δ}}_e){\left|\right|}_{\∞}+||Q_3(z\left(k\right|t)-z_e){\left|\right|}_{\∞}$

$+\left|\right|Q_4(x\left(k\right|t)-x_e)|{|}_{\∞}+|\left|Q_5(y\left(k\right|t)-y_e)\right|{|}_{\∞})$

(5) the OPC client-side program through exploitation obtains the industry spot data in real time, simultaneously the current time optimal control value that calculates is applied to field control.

The invention has the beneficial effects as follows:

Compare with traditional multi-model PID control; Utilize hybrid system theoretical modeling and control method; Can all segmented models of system be integrated in the united frame; Accurate description model handover event and the interaction of dynamic process continuously help eliminating the vibration that causes when model switches; Based on the PREDICTIVE CONTROL of hybrid system model, can handle the control problem of the system that contains a large amount of constraint conditions, what make system reaches COMPREHENSIVE OPTIMAL with discrete control continuously.

Description of drawings

Fig. 1 is the PREDICTIVE CONTROL schematic diagram.

Fig. 2 is a water tank embodiment control system structure.

Fig. 3 is independently developed OPC client-side program thought process flow diagram.

Embodiment

To combine accompanying drawing that the present invention is done further explain below.

A kind of modeling and control method of the present invention based on hybrid system, its modeling process is following:

The general expression formula of piecewise linear model is:

$\left\{\begin{array}{c}x(k+1)=A_{i}x\left(k\right)+B_{i}u\left(k\right)+b_i\\ y\left(k\right)=C_{i}x\left(k\right)+D_{i}u\left(k\right)+d_i\end{array}\right.\mathrm{\θ}\∈{\mathrm{\Ω}}_i,i=1,...,n$

Set up the mixed logic dynamic model of hybrid system; At first to set up propositional logic, then propositional logic is converted into the integer linear inequality through certain rule dynamic law, operational constraints, operating experience and the priori etc. that exist in the controlled device.Define symbol S representes a proposition, introduce bigit variable δ ∈ 1,0}, corresponding expression proposition S's is true and false, i.e. δ=1 during S=T, δ during S=F=0.Basic transformational relation between propositional logic and the MIXED INTEGER inequality is seen table 1.

Relation of equivalence between table 1 propositional logic and the MIXED INTEGER inequality

Introduce the auxiliary logic variable, δ _i(k) ∈ 0,1}, i=1 ..., n, order

$[{\mathrm{\δ}}_i=1]\↔[\mathrm{\θ}={\left[\begin{array}{cc}x& u\end{array}\right]}^{\′}\∈{\mathrm{\Ω}}_i]$

Then can be converted to

ε+(m _i-ε)δ _i(t)≤G _ix(k)+R _iu(k)-T _i≤M _i(1-δ _i(k))

Wherein

$M_i={\max }_{{\mathrm{\Ω}}_i}(G_{i}x\left(k\right)+R_{i}u\left(k\right)-T_i)$

$m_i={\min }_{{\mathrm{\Ω}}_i}(G_{i}x\left(k\right)+R_{i}u\left(k\right)-T_i)$

The model of system can be written as

$\left\{\begin{array}{c}x(k+1)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}[A_{i}x\left(k\right)+B_i\left(k\right)+b_i]{\mathrm{\δ}}_i\left(k\right)\\ y\left(k\right)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}[C_{i}x\left(k\right)+D_{i}u\left(k\right)+d_i]{\mathrm{\δ}}_i\left(k\right)\end{array}\right.i=1,...,n$

Introduce auxiliary continuous variable

$\left\{\begin{array}{c}{Z}_{\mathrm{xi}}\left(k\right)=[A_{i}x\left(k\right)+B_i\left(k\right)+b_i]{\mathrm{\δ}}_i\left(k\right)\\ Z_{\mathrm{yi}}\left(k\right)=C_{i}x\left(k\right)+D_{i}u\left(k\right)+d_i]{\mathrm{\δ}}_i\left(k\right)\end{array}\right.$

Then the model of system can further be written as

$\left\{\begin{array}{c}x(k+1)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{Z}_{\mathrm{xi}}\left(k\right)\\ y\left(k\right)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{Z}_{\mathrm{yi}}\left(k\right)\end{array}\right.$

The auxiliary continuous variable Z of conversion _XiWith auxiliary logic variable δ _iRelation, can obtain the MIXED INTEGER inequality and be:

$\left\{\begin{array}{c}{Z}_{\mathrm{xi}}\left(k\right)\≤M_{\mathrm{xi}}{\mathrm{\δ}}_i\left(k\right)\\ Z_{\mathrm{xi}}\left(k\right)\≥m_{\mathrm{xi}}{\mathrm{\δ}}_i\left(k\right)\\ Z_{\mathrm{xi}}\left(k\right)\≤A_{i}x\left(k\right)+B_{i}u\left(k\right)+b_i-m_{\mathrm{xi}}(1-{\mathrm{\δ}}_i\left(k\right))\\ Z_{\mathrm{xi}}\left(k\right)\≥A_{i}x\left(k\right)+B_{i}u\left(k\right)+b_i-M_{\mathrm{xi}}(1-{\mathrm{\δ}}_i\left(k\right))\end{array}\right.$

Similar, Z _YiAnd δ _iRelation can be expressed as:

$\left\{\begin{array}{c}{Z}_{\mathrm{yi}}\left(k\right)\≤M_{\mathrm{yi}}{\mathrm{\δ}}_i\left(k\right)\\ Z_{\mathrm{yi}}\left(k\right)\≥m_{\mathrm{yi}}{\mathrm{\δ}}_i\left(k\right)\\ Z_{\mathrm{yi}}\left(k\right)\≤A_{i}x\left(k\right)+B_{i}u\left(k\right)+b_i-m_{\mathrm{yi}}(1-{\mathrm{\δ}}_i\left(k\right))\\ Z_{\mathrm{yi}}\left(k\right)\≥A_{i}x\left(k\right)+B_{i}u\left(k\right)+b_i-M_{\mathrm{yi}}(1-{\mathrm{\δ}}_i\left(k\right))\end{array}\right.$

Two groups of inequation groups are attached to the linear equation back as constraint condition, have so just formed the MLD model of system.

Above modeling process be under the condition of the mechanism model of known system, system is obtained the linear model set at the nonlinear equation of the equilibrium point linearized system of different subspace, and then converts the MLD model to.If the mechanism model of system is unknown, equally after system decomposition is different subspace, obtain each partial model according to the inputoutput data identification of different subspace, again all mode sets are become the MLD model.

Referring to Fig. 1, PREDICTIVE CONTROL is a kind of control algolithm that combines forecast model, rolling optimization and feedback compensation in one.If the PREDICTIVE CONTROL problem based on the performance index of ∞ norm form is:

$\min J(u_0^{T-1},x\left(t\right))=\underset{k=0}{\overset{T-1}{\mathrm{\Σ}}}\left(\right||Q_1(u\left(t\right)-u_e){\left|\right|}_{\∞}+||Q_2(\mathrm{\δ}\left(k\right|t)-{\mathrm{\δ}}_e){\left|\right|}_{\∞}+||Q_3(z\left(k\right|t)-z_e){\left|\right|}_{\∞}$

$+\left|\right|Q_4(x\left(k\right|t)-x_e)|{|}_{\∞}+|\left|Q_5(y\left(k\right|t)-y_e)\right|{|}_{\∞})$

$s.t.\left\{\begin{array}{c}x\left(0\right|t)=x\left(t\right)\\ x(k+1|t)=\mathrm{Ax}\left(k\right|t)+B_{1}u\left(k\right)+B_2\mathrm{\δ}\left(k\right|t)+B_{3}z\left(k\right|t)\\ y\left(k\right|t)=\mathrm{Cx}\left(k\right|t)+D_{1}u\left(k\right)+D_2\mathrm{\δ}\left(k\right|t)+D_{3}z\left(k\right|t)\\ E_2\mathrm{\δ}\left(k\right|t)+E_{3}z\left(k\right|t)\≤E_{1}u\left(k\right)+E_{4}x\left(k\right|t)+E_5\end{array}\right.$

If t is a current time, x (t) is the state variable value of current time, and then when t=0, x (t) is the original state of system.Make and be illustrated in following k the optimum control sequence constantly that the t time optimization obtains; X (k|t) is illustrated in t constantly; Under the condition of given system state x (t) and

, based on the forecast model of system to the t+k prediction of system state constantly; The meaning of δ (k|t), z (k|t), y (k|t) is similar.

Based on the forecast model of MLD form, t+k state expression formula constantly does

$x\left(k\right|t)=A^{k}x\left(t\right)+\underset{t=0}{\overset{k-1}{\mathrm{\Σ}}}{A}^i[B_{1}u(k-1-i|t)+B_2\mathrm{\δ}(k-1-i|t)+B_{3}z(k-1-i|t)$

Can directly obtain t+k output variable constantly does

y(k|t)＝Cx(k|t)+D ₁u(k|t)+D ₂δ(k|t)+D ₃z(k|t)

Be output as y (t) in t actual measurement constantly, corresponding prediction is output as

$\hat{t}\left(t\right)=\mathrm{Cx}\left(k\right|t-k)+D_{1}u\left(t\right)+D_2\mathrm{\δ}\left(t\right)+D_{3}z\left(t\right)$

Adopt fixedly forecast model, the method for compensation prediction deviation is feedback compensation just.According to the deviation between current time actual output y (t) and the prediction output

; Following k the predicted value constantly of system carried out feedback compensation, promptly

$y\left(k\right|t)=y\left(k\right|t)+[y\left(t\right)-\hat{y}\left(t\right)]$

The definition vector

$q=[{\mathrm{\ϵ}}_0^u,...,{\mathrm{\ϵ}}_{T-1}^u,{\mathrm{\ϵ}}_0^{\mathrm{\δ}},...,{\mathrm{\ϵ}}_{T-1}^{\mathrm{\δ}},{\mathrm{\ϵ}}_{T-1}^z,...,{\mathrm{\ϵ}}_{T-1}^z,{\mathrm{\ϵ}}_0^x,...,{\mathrm{\ϵ}}_{T-1}^x,{\mathrm{\ϵ}}_0^y,...,{\mathrm{\ϵ}}_{T-1}^y{]}^{\′}$

To k=0,1 ..., T-1, each element satisfies following relationship

$\left|\right|Q_1(u\left(k\right|t)-u_e){\left|\right|}_{\∞}\≤{\mathrm{\ϵ}}_k^u$

$\left|\right|Q_2(\mathrm{\δ}\left(k\right|t)-{\mathrm{\δ}}_e){\left|\right|}_{\∞}\≤{\mathrm{\ϵ}}_k^{\mathrm{\δ}}$

$\left|\right|Q_3(z\left(k\right|t)-z_e){\left|\right|}_{\∞}\≤{\mathrm{\ϵ}}_k^z$

$\left|\right|Q_4(x\left(k\right|t)-x_e){\left|\right|}_{\∞}\≤{\mathrm{\ϵ}}_k^x$

$\left|\right|Q_5(y\left(k\right|t)-y_e){\left|\right|}_{\∞}\≤{\mathrm{\ϵ}}_k^y$

Be equivalent to

$1_m{\mathrm{\ϵ}}_k^u\≥\±Q_1(u\left(k\right|t)-u_e)$

$1_{r_i}{\mathrm{\ϵ}}_k^{\mathrm{\δ}}\≥\±Q_2(\mathrm{\δ}\left(k\right|t)-{\mathrm{\δ}}_e)$

$1_{r_c}{\mathrm{\ϵ}}_k^z\≥\±Q_3(z\left(k\right|t)-z_e)$

$1_n{\mathrm{\ϵ}}_k^x\≥\±Q_4(x\left(k\right|t)-x_e)$

$1_p{\mathrm{\ϵ}}_k^y\≥\±Q_5(y\left(k\right|t)-y_e)$

Wherein 1 _hExpression length is the column vector of h, and all elements all is 1.Definition

$J\left(q\right)={\mathrm{\ϵ}}_0^u+\·\·\·+{\mathrm{\ϵ}}_{T-1}^u+{\mathrm{\ϵ}}_0^{\mathrm{\δ}}+\·\·\·+{\mathrm{\ϵ}}_{T-1}^{\mathrm{\δ}}$

$+{\mathrm{\ϵ}}_0^z+\·\·\·+{\mathrm{\ϵ}}_{T-1}^z+{\mathrm{\ϵ}}_0^x+\·\·\·+{\mathrm{\ϵ}}_{T-1}^x+{\mathrm{\ϵ}}_0^y+\·\·\·+{\mathrm{\ϵ}}_{T-1}^y$

$=\underset{k=0}{\overset{T-1}{\mathrm{\Σ}}}{\mathrm{\ϵ}}_k^u+{\mathrm{\ϵ}}_k^{\mathrm{\δ}}+{\mathrm{\ϵ}}_k^z+{\mathrm{\ϵ}}_k^x+{\mathrm{\ϵ}}_k^y$

Then forecasting problem can be write as MILP (Mixed Integer Linear Programming, MILP) problem

$\underset{q}{\min }J\left(q\right)=\left[\begin{array}{cccc}1& 1& \·\·\·& 1\end{array}\right]q$

Find the solution this MILP problem, just can obtain t T optimum control sequence constantly

$\left\{\right[u_t^{*}\left(k\right),{\mathrm{\δ}}_t^{*}\left(k\right),z_t^{*}\left(k\right){]}^{\′}{\}}_{k=\mathrm{0,1},...,T-1}$

According to the rolling optimization principle, only get t control input constantly, ignore other control actions

$U^{*}\left(t\right)={[u_t^{*}\left(0\right),{\mathrm{\δ}}_t^{*}\left(0\right),z_t^{*}\left(0\right)]}^{\′}$

At k=k+1 constantly, repeat above step, just can realize MLD Model Predictive Control based on the line style performance index.

Referring to Fig. 2, embodiment explains embodiment with water tank.CS4000 type process control device and JX-300X control system platform based on middle control have made up the high water tank control system.The target of control system is that the liquid level of lower header remains on expectation value.

Controlled device is made up of the organic glass water tank of two identical sizes, puts into a cylindrical objects in the upper water box.When the liquid level of upper water box during above and below cylindrical objects, the liquid level dynamic perfromance of water tank is obviously different, has the characteristic of discrete event.According to experimental data, the piecewise linear model of cistern system does

$h_1(k+1)=\left\{\begin{array}{cc}0.991h_1\left(k\right)-0.001+8.6\&times;{10}^{-5}u\left(k\right)& h_1\&GreaterEqual;0.1\\ 0.988h_1\left(k\right)-0.002+1.2\&times;{10}^{-4}u\left(k\right)& h_1<0.1\end{array}\right.$

h ₂(k+1)＝0.009h ₁(k)+0.991h ₂(k)

Introduce auxiliary logic variable δ, definition

$[\mathrm{\&delta;}=1]\&LeftRightArrow;[h_1\&GreaterEqual;0.1]$

Introduce auxiliary continuous variable z ₁, z ₂, definition

if?δ?then?z ₁＝0.991h ₁-0.001

else?z ₁＝0.988h ₁-0.002

if?δ?then?z ₂＝8.6×10 ^-6u

else?z ₂＝1.2×10 ^-4u

Then system model can be described as

h ₁(k+1)＝z ₁(k)+z ₂(k)

h ₂(k+1)＝0.009h ₁(k)+0.991h ₂(k)

Simultaneity factor need satisfy constraint 0≤h _i≤0.3,0≤u≤100

If x=is [h ₁h ₂] ', z=[z ₁z ₂The MLD model that] ', can get water tank is (omission inequality constrain):

$x(k+1)=\left[\begin{array}{cc}0& 0\\ 0.009& 0.991\end{array}\right]x\left(k\right)+\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]z\left(k\right)$

y(k)＝[0?1]x(k)

Adopt the PREDICTIVE CONTROL of line style index,, make the liquid level of lower header remain on expectation value through variable valve control.Performance index are:

$\underset{u}{\min }\underset{k=0}{\overset{10}{\mathrm{\&Sigma;}}}\left|\right|y\left(k\right|t)-r_y\left(t\right){\left|\right|}_{\&infin;}$

The OPC client-side program of Matlab exploitation, the data of opc server are obtained in its effect in real time, realize mixing predictive control algorithm, and controlling value is write corresponding item.Data messages such as on-the-spot like this liquid level, flow just can be real-time transmitted to the monitoring PC based on the communication network of JX-300X, are read by independently developed client-side program through the OPC communication again; Simultaneously, the current time optimal control value of OPC client-side program calculating is applied to field control through same paths.

Fig. 3 is the process flow diagram of OPC client-side program thought.In the program there be main function:

Da=opcda (' localhost ', ' SUPCON.AdvOPCServer.1 '); % sets up object model to opc server

Connect (da); % is connected to server

Grp=addgroup (da, ' ExOPC '); % increase group

Itm1=additem (grp, ' LI_001 '); % increases an itm1, corresponding lower header liquid level item

Itm2=additem (grp, ' LI_002 '); % increases an itm2, corresponding upper water box liquid level item

Itm3=additem (grp, ' mv_opc '); % increases an itm3, corresponding control item

Disconnect (da); % breaks off OPC and connects

Delete (da); % deletion OPC DAO

V=read (itm1, ' device '); Read the itm1 data structure

H=v.Value; % reads the value of itm1, i.e. level value

Write (itm3,100); % writes data 100 in itm3, i.e. input control value is 100

The gui program that Matlab is write is compiled as executable file (.exe), in object computer hollow Matlab software can be installed like this, only needs installing M CRInstall.exe.On target machine, just can directly move independently application program.

Claims (3)

1. modeling and control method based on the theoretical piecewise-linear system of hybrid system, it is characterized in that: described modeling and control method may further comprise the steps:

(1) according to the priori of controlled system, the continuous dynamic part of analytic system, Discrete Dynamic part and interaction between the two.To the continuous dynamic part of system, set up its model with the traditional difference equation or the differential equation; Simultaneously understanding system continuously dynamically between, continuously and between the Discrete Dynamic and the various logic of Discrete Dynamic self relation;

(2) various simple and all logical relations that the combination of sentences is come descriptive system of definition.According to the transformational relation of equal value of propositional logic and MIXED INTEGER inequality, obtain one group of MIXED INTEGER linear inequality;

(3) on the model basis of the continuous dynamic part of descriptive system, introduce logical variable and auxiliary variable, and be converted into linear discrete state equation form, the MIXED INTEGER inequality constrain of the various constraints of spare system and propositional logic conversion simultaneously.

2. a kind of modeling and control method according to claim 1 based on dark assorted systemtheoretical piecewise-linear system, it is characterized in that: described modeling and control method are further comprising the steps of:

(1) forecast model is the hybrid system model according to the mixed logic dynamic-form of the said foundation of claim 1;

(2) optimization problem adopts the linear properties index based on ∞ norm form; Obtain the optimum control sequence through finding the solution MILP (MILP) problem; According to the rolling optimization principle, only get current time control input value, the online then process that is optimized repeatedly;

(3) method of feedback compensation is the actual output of computing system and the deviation between the prediction output, and following prediction compensates to system then.

3. a kind of modeling and control method according to claim 1 based on dark assorted systemtheoretical piecewise-linear system; It is characterized in that: use Matlab to develop independently 0PC client-side program; Obtain the industry spot data of the opc server that DCS provides in real time; Realization mixes predictive control algorithm, and controlling value is write corresponding item.

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