CN102778844B - Induction heating closed loop simulation method based on finite element model and system identification - Google Patents

Abstract

The invention discloses an induction heating closed loop simulation method based on finite element model and system identification. The method adopts ANSYS software to build a billet electromagnetic induction heating finite element model, and the simulation of billet induction heating process is realized through solving the model, thus obtaining the simulation data. The system input and output time history which is obtained through the finite element simulation is used for substituting the experimental data, a least square system identification method is adopted to process the input and output time history, and thus, system response characteristic parameters are obtained; then, a proportion integration differentiation (PID) feedback controller is designed on the basis of the model; and finally, an ansys parameter design language (APDL) is adopted to introduce the PID feedback controller into the finite element model, so the simulation process of an induction heating closed loop system is realized. The method takes the advantages of the induction heating finite element model, realizes the coupling of electromagnetic induction heating and a control system, can greatly improve the simulation precision, and facilitates the advanced performance analysis of induction heating closed loop control.

Description

Induction heating closed-loop simulation method based on finite element model and System Discrimination

Technical field

The invention belongs to power electronics application, relate to a kind of closed-loop simulation method of electromagnetic induction heating process, specifically a kind of closed-loop simulation method of the induction heating process based on finite element model and System Discrimination.

Background technology

Induction heating technique is to utilize electromagnetic field to produce eddy heating for heating workpiece, because its firing rate is fast, heated object is not yielding, pollute less and the feature such as energy consumption is little, is widely used in the industrial circles such as steel billet forging, rolling, thermal treatment.

By literature search, the Zhao Min of Zhejiang Polytechnical University utilizes ANSYS finite element software at Master's thesis " FEM Simulation of induction heating before 45 steel billets forgings " (2006.4), set up the computation model of induction heating finite element analogy before forging, establish the disposal route of the problems such as analytic unit, grid division, boundary condition and material temperature dependence in analog computation process, and steel billet has been carried out to sunykatuib analysis by the induction heating processes of single-unit and two joint inductive coils.Janne Nery is at article Numerical solution of 2D and 3D induction heating problems with non-linear material properties take into account (IEEE Transactions on magnetics, 2000,36 (5): 3119 ~ 3121), calculate the temperature variations of induction heating process steel billet in general direction by Finite Element Method, can obtain the data that cannot sample in experiment.The Liu Hao of the Central China University of Science and Technology is at PhD dissertation " research and development of continuous casting and rolling electromagnetic induction heating numerical simulation technology " (2007,10,30) at the inner thermometric of strand comparatively difficulty in the situation that, utilize ANSYS software to induction heating process simulation, and research affect each factor of induction heating process.There is the multiple physical change couplings such as electromagnetic induction, eddy current heat-dissipating, heat conduction, heat radiation due to electromagnetic induction heating process, adopting aforementioned Finite Element Method to set up induction heating procedures system model is a kind of effective means, but the order of finite element model is very high, cannot be directly used in control.Zhang Lifang etc. are at the Fuzzy+PID of article induction heating temperature composite control method (New Technologies, 2004,6.38-40), for the temperature control in induction heating heat treatment process, adopt the composite control method that a kind of Fuzzy controls and PID controls, for direct control.But this method must be carried out based on electromagnetic induction heating real process, cannot consider the coupling of electromagnetic induction heating and control system, can not be used for prior emulation and control performance analysis.

Summary of the invention

The object of the invention is to overcome deficiency of the prior art, proposed a kind of induction heating closed-loop simulation method based on finite element model and System Discrimination, so that the prior performance evaluation of induction heating temperature closed-loop control.The method has been utilized finite element induction heating model and system identifying method, directly Tuning PID Controller, realize the emulation of induction heating temperature closed-loop control, reflect the coupling of electromagnetic induction heating and control system, can significantly improve simulation accuracy, overcome in existing equipment heating-up temperature real-time responsiveness poor, heating-up temperature is difficult to the problems such as control.

The present invention is achieved by the following technical solutions, adopts ANSYS software to set up steel billet electromagnetic induction heating finite element model, solves the simulation that realizes the induction heating process to steel billet, obtains the data of simulation.The system input and output time history that finite element analogy is provided replaces experimental data, and the system identifying method that adopts least square is processed this input and output time history, thereby obtain the response characteristic parameter of system, then based on this modelling PID feedback controller, finally adopt the APDL language of ANSYS that PID feedback controller is introduced to finite element model, realize the induction heating closed-loop system simulation process based on finite element analogy and the combination of PID FEEDBACK CONTROL.

The inventive method concrete steps are:

Step 1. continuous casting steel billet electromagnetic induction heating process finite element modeling, adopts business finite element software ANSYS to set up finite element induction heating process model, and concrete modeling is as follows:

Steel billet is divided into 120 sections by step 1.1., get wherein any one section 1/4th carry out modeling.

Step 1.2. utilizes the front processor of ANSYS self create or read in geometric model from other modeling software.

Step 1.3. is in electromagnetic field analysis part, and setting far-field region edge magnetic potential is zero, and steel billet center applies magnetic line of force parallel boundary condition; Driving source electric current is by being equally distributed when horizontal interface inside inductive coil, and it is as the incentive condition of electromagnetic field; Steel billet, inductor coil adopt identical SOLID117 hexahedral element with the grid cell of air.

Relative permeability, the resistivity of steel billet in 750 DEG C of-1200 DEG C of temperature ranges of step 1.4. definition, the relative permeability of inductor coil, the relative permeability of air.

Step 1.5., in temperature field analysis part, is set to dummy cell by inductor coil and air, and steel billet changes SOLID97 unit into, only calculates the thermal field in steel billet region, and steel billet ambient air initial temperature is set as constant; The billet surface contacting with air, only calculates with air-grid node and carries out radiant heat exchange.

Coefficient of heat conductivity, specific heat capacity, the density of steel billet in 750 DEG C of-1200 DEG C of temperature ranges of step 1.6. definition, the heat emissivity coefficient of billet surface, Boltzmann's constant.

Closeer the closer to inductor coil when step 1.7. divides steel billet grid, mesh-density is successively decreased to center by surface.

The coupling that step 1.8. adopts sequential coupling method to carry out between electromagnetism-Re is calculated, first according to the temperature field of starting condition, determine the physical parameter of material, solve electromagnetic problems, obtain so the hot production rate of electromagnetic field output, as the required endogenous pyrogen input of thermal field, then thermal field is calculated, simultaneously according to the now distribution of steel billet temperature field, remove to revise the physical parameter of material, remove to solve electromagnetic field, so circulation is gone down, until reach the heat time of setting again.

The emulation of step 2. electromagnetic induction heating process simulation, specifically:

Step 2.1. keeps intermediate frequency power supply output power u ₁, output frequency f, output voltage U be constant, the initial temperature u of cross-sectional boundaries mid point before heating steel billet ₂as variable input signal, simulation draws the temperature y of this point after heating steel billet ₁.

Step 2.2. keeps intermediate frequency power supply output voltage U, output frequency f, initial temperature u ₂constant, to impose on the output power u of inductive coil ₁as variable input signal, simulation draws the temperature y of cross-sectional boundaries mid point after heating steel billet ₂.

Record the system input and output time history data that above-mentioned FEM numerical simulation process obtains, replace the input of experimental data as System Discrimination algorithm.

The identification of step 3. electromagnetic induction heating procedures system, specifically:

For a dual input list output TISO system, between input u, output y and process transport function, the fundamental relation of G is:

（1）

Wherein, , u ₁for output power, u ₂for initial temperature, ε is noise signal; In System Discrimination process, TISO system decomposition becomes two independently SISO systems, corresponding two transport functions; Generally, most industrial processs adopt single order or second order to add pure lag model and describe, and adopt first-order plus time delay model representation transport function to be here:

（2）

In formula, for static system gain, for pure delay time constant, for system time constant, the complex parameter that s is transport function, i={1,2}.

Suppose the input u of process ₁, u ₂with output y ₁, y ₂original state be stable state, be respectively , and y ⁰, the first step, keeps constant, a given amplitude is h ₁step input signal , system is output as , the equation that increases progressively of whole process is:

（3）

Wherein, , .

Second step, keeps constant, a given amplitude is h ₂step input signal , system is output as , the equation that increases progressively of whole process is:

（4）

Wherein, , ;

Formula (2) is converted into discrete form, is expressed as with difference equation:

（5）

Parameter to be identified in formula is respectively, , , , for the sampling time, with for input/output signal of equal value.

By input and output substitution formula (5) of equal value, the form of being write as least square is

（6）

Wherein, for output signal, for considerable measured value, for parameter to be identified, for white noise,

（7）

Solve by least square method, write as matrix form and be:

（8）

Wherein, , , ; Get criterion function

（9）

Minimization , try to achieve parameter estimated value, will make the output output of forecast system best of model;

If 's be denoted as , be called parameter least-squares estimation value, have

（10）

Solve

（11）

Have

（12）

（13）

There is said method can pick out the static gain in transfer function model and system time constant .

Suppose in scope, will value and , value is brought following formula into:

（14）

When , solve k value, basis again , obtain retardation time .

Step 4. designs PID feedback controller, specifically:

A classical PID controller, its transport function is expressed as:

（15）

In formula, for scale-up factor, for integration time constant, for derivative time constant; Suppose under least model prerequisite, the PID controller parameter Ziegler-Nichols experimental formula of adjusting is:

（16）

The emulation of step 5. closed-loop system, specifically:

Calculate steel billet in the response after the induction heating process of a time step based on Finite Element Method, the temperature-responsive of output steel billet cross-sectional boundaries mid point is as sensor signal, and this sensor signal is defined as to APDL variable, realize PID controller with APDL language, just the PID controller of realizing according to APDL program after the response of a time step of every calculating calculates the optimum control power of next time step, input using this power ratio control as finite element model, calculate again the temperature-responsive of next time step, this process realizes with APDL loop structure, so just, FEEDBACK CONTROL is introduced to finite element model, realize the closed-loop system emulation of the electromagnetic induction heating process under finite element environment.

The beneficial effect of the inventive method: the present invention utilizes the experimentation of ripe finite element software simulation steel billet induction heating, the data acquisition obtaining is obtained to system transter with the system identifying method of least square, then utilize Ziegler-Nichols experience Tuning PID Controller, for working control.

Advantage of the present invention: the advantage of drawing induction heating finite element modeling, overcome finite element model because of the high drawback that cannot be directly used in control of order, realize the coupling of electromagnetic induction heating and control system, can significantly improve simulation accuracy, so that the prior performance evaluation of induction heating temperature closed-loop control.

Brief description of the drawings

Fig. 1 is the method flow diagram of embodiment of the present invention;

Fig. 2 is steel billet induction heating process schematic diagram of the present invention;

Fig. 3 is the cross section reduced graph of steel billet induction heating process of the present invention finite element analysis;

Fig. 4 is the Coupled Electromagnetic-Thermal calculation flow chart of steel billet induction heating FEM Simulation of the present invention.

Embodiment

The principle of the invention as shown in Figure 1, a kind of induction heating closed-loop system emulation mode based on finite element and FEEDBACK CONTROL, (1) induction heating procedures system discrimination method based on finite element model, obtains the response characteristic parameter of finite element model and system; (2) for system response characteristic, design PID feedback controller; (3) adopt the APDL language of ANSYS that PID feedback controller is introduced to finite element model, realize the induction heating closed-loop system emulation based on finite element analogy and the combination of PID FEEDBACK CONTROL.

For understanding better technical scheme of the present invention, below in conjunction with drawings and Examples, the specific embodiment of the present invention is described in further detail.Following examples are only for the present invention is described, but are not used for limiting the scope of the invention.Now a certain specification steel billet is carried out to electromagnetic induction heating, use the inventive method to realize closed-loop system emulation.

1 continuous casting steel billet electromagnetic induction heating process finite element modeling

As shown in Figure 2, be continuous casting and rolling induction heating process schematic diagram, comprise 6 meters of long steel billets and a load coil.Adopt ANSYS software to set up finite element model, step is as follows:

Steel billet is divided into 120 sections by 1.1, get wherein any one section 1/4th carry out modeling, as shown in Figure 3.

1.2 utilize the front processor of ANSYS self create or read in geometric model from other modeling software.

1.3 in electromagnetic field analysis part, and setting far-field region edge magnetic potential is zero, and steel billet center applies magnetic line of force parallel boundary condition.Driving source electric current is by being equally distributed when horizontal interface inside inductive coil, and it is as the incentive condition of electromagnetic field.Steel billet, inductor coil adopt identical SOLID117 hexahedral element with the grid cell of air.

Relative permeability, the resistivity of steel billet in 750 DEG C of-1200 DEG C of temperature ranges of 1.4 definition, the relative permeability of inductor coil, the relative permeability of air.

1.5 in temperature field analysis part, and inductor coil and air are all set to dummy cell, and steel billet changes SOLID97 unit into, only calculates the thermal field in steel billet region, and steel billet ambient air initial temperature is set as constant.The billet surface contacting with air, only calculates with air-grid node and carries out radiant heat exchange.

Coefficient of heat conductivity, specific heat capacity, the density of steel billet in 750 DEG C of-1200 DEG C of temperature ranges of 1.6 definition, the heat emissivity coefficient of billet surface, Boltzmann's constant.

Closeer the closer to inductor coil when 1.7 division steel billet grid, mesh-density is successively decreased to center by surface.

1.8 couplings that adopt sequential coupling method to carry out between electromagnetism-Re are calculated, first according to the temperature field of starting condition, determine the physical parameter of material, solve electromagnetic problems, obtain so the hot production rate of electromagnetic field output, as the required endogenous pyrogen input of thermal field, then thermal field is calculated, simultaneously according to the now distribution of steel billet temperature field, remove to revise the physical parameter of material, remove to solve electromagnetic field, so circulation is gone down, until reach the heat time of setting again.Calculation flow chart as shown in Figure 4.

2 electromagnetic induction heating process simulation emulation

2.1 to set heating power supply output powers be 300000W, and output frequency is 1000HZ, and output voltage is 1600V, and using the initial temperature of cross-sectional boundaries mid point before heating steel billet as input signal, simulation draws the temperature of this point after heating steel billet.

2.2 to set heating power supply output voltages be 1600V, and output frequency is 1000HZ, and initial temperature is 800 DEG C, and using the electric current that imposes on inductive coil as input signal, simulation draws the temperature of cross-sectional boundaries mid point after heating steel billet.

Record the system input and output time history data that above-mentioned FEM numerical simulation process obtains, intend replacing the input of experimental data as System Discrimination algorithm.

3 electromagnetic induction heating procedures system identifications

Adopt the system identifying method based on least square to process this input and output time history, regard every steel billet as 120 segments, every a bit of power sum experiencing in load coil with initial temperature as the input of system, temperature rise y is as the output of system.

Adopt first-order plus time delay model representation transport function to be :

（1）

In formula, for static system gain, for pure delay time constant, for system time constant, the complex parameter that s is transport function, i={1,2}.

Suppose process input u ₁, u ₂be stable state with exporting y original state, be respectively , with , keep constant, a given amplitude is h ₁step input signal , system is output as , the equation that increases progressively of whole process is:

（2）

Wherein, , .

Second step, keeps constant, a given amplitude is h ₂step input signal , system is output as , the equation that increases progressively of whole process is:

（3）

Wherein, , .

Have above-mentionedly original TISO system decomposition to be become to two SISO systems, with wherein with input signal relevant SISO system is example, with difference equation expression (1) is:

（4）

Parameter to be identified in formula is respectively, , , , for the sampling time, with for input/output signal of equal value.

By input and output substitution formula (4) of equal value, write as least squares formalism and be

（5）

Wherein, for output signal, for considerable measured value, for parameter to be identified, for white noise,

（6）

Solve by least square method, write as matrix form and be:

（7）

Wherein, , ,

。Then adopt MATLAB software least square method to calculate least-squares estimation value be:

（8）

Have

（9）

（10）

There is said method can pick out static gain K and the system time constant T in transfer function model.

Suppose in scope, will value and , value is brought following formula into:

（11）

When , solve k value, basis again , obtain retardation time , and the process transport function that obtains on this basis system is:

（12）

In like manner also can obtain another SISO system transter thereby, for the performance that ensures induction heater provides design considerations, and then for working control.

4 design PID feedback controllers

A classical PID controller, its transport function is expressed as:

（13）

In formula, for scale-up factor, for integration time constant, for derivative time constant.Suppose under least model prerequisite, the PID controller parameter Ziegler-Nichols experimental formula of adjusting is:

（14）

5 closed-loop system emulation

Calculate steel billet in the response after the induction heating process of a time step based on Finite Element Method, the temperature-responsive of output steel billet cross-sectional boundaries mid point is as sensor signal, and this sensor signal is defined as to APDL variable, realize PID controller with APDL language, just the PID controller of realizing according to APDL program after the response of a time step of every calculating calculates the optimum control power of next time step, input using this power ratio control as finite element model, calculate again the temperature-responsive of next time step, this process realizes with APDL loop structure, so just, FEEDBACK CONTROL is introduced to finite element model, realize the closed-loop system emulation of the electromagnetic induction heating process under finite element environment.

Claims (1)

1. the induction heating closed-loop simulation method based on finite element model and System Discrimination, is characterized in that the method comprises the following steps:

Step 1. continuous casting steel billet electromagnetic induction heating process finite element modeling, adopts business finite element software ANSYS to set up finite element induction heating process model, and concrete modeling is as follows:

Steel billet is divided into 120 sections by step 1.1., get wherein any one section 1/4th carry out modeling;

Step 1.2. utilizes the front processor of ANSYS self create or read in geometric model from other modeling software;

Step 1.3. is in electromagnetic field analysis part, and setting far-field region edge magnetic potential is zero, and steel billet center applies magnetic line of force parallel boundary condition; Driving source electric current is by being equally distributed when horizontal interface inside inductor coil, and it is as the incentive condition of electromagnetic field; Steel billet, inductor coil adopt identical SOLID117 hexahedral element with the grid cell of air;

Relative permeability, the resistivity of steel billet in 750 DEG C of-1200 DEG C of temperature ranges of step 1.4. definition, the relative permeability of inductor coil, the relative permeability of air;

Step 1.5., in temperature field analysis part, is set to dummy cell by inductor coil and air, and steel billet changes SOLID97 unit into, only calculates the thermal field in steel billet region, and steel billet ambient air initial temperature is set as constant; The billet surface contacting with air, only calculates with air-grid node and carries out radiant heat exchange;

Coefficient of heat conductivity, specific heat capacity, the density of steel billet in 750 DEG C of-1200 DEG C of temperature ranges of step 1.6. definition, the heat emissivity coefficient of billet surface, Boltzmann's constant;

Closeer the closer to inductor coil when step 1.7. divides steel billet grid, mesh-density is successively decreased to center by surface;

The coupling that step 1.8. adopts sequential coupling method to carry out between electromagnetism-Re is calculated, first according to the temperature field of starting condition, determine the physical parameter of material, solve electromagnetic problems, obtain so the hot production rate of electromagnetic field output, as the required endogenous pyrogen input of thermal field, then thermal field is calculated, simultaneously according to the now distribution of steel billet temperature field, remove to revise the physical parameter of material, remove to solve electromagnetic field, so circulation is gone down, until reach the heat time of setting again;

The emulation of step 2. electromagnetic induction heating process simulation, specifically:

Step 2.1. keeps intermediate frequency power supply output power u ₁, output frequency f, output voltage U be constant, the initial temperature u of cross-sectional boundaries mid point before heating steel billet ₂as variable input signal, simulation draws the temperature y of this point after heating steel billet ₁;

Step 2.2. keeps intermediate frequency power supply output voltage U, output frequency f, initial temperature u ₂constant, to impose on the output power u of inductor coil ₁as variable input signal, simulation draws the temperature y of cross-sectional boundaries mid point after heating steel billet ₂;

Record the system input and output time history data that above-mentioned FEM numerical simulation process obtains, replace the input of experimental data as System Discrimination algorithm;

The identification of step 3. electromagnetic induction heating procedures system, specifically:

For a dual input list output TISO system, the fundamental relation between input u, output y and process transport function G is:

y＝Gu+ε (1)

Wherein, $u=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right],$ G=[G ₁g ₂], u ₁for output power, u ₂for initial temperature, ε is noise signal; In System Discrimination process, TISO system decomposition becomes two independently SISO systems, corresponding two transport functions; Generally, most industrial processs adopt single order or second order to add pure lag model and describe, and adopt first-order plus time delay model representation transport function to be here:

$G_i=\frac{K_i}{T_{i}s+1}{e}^{-{\mathrm{\δ}}_{i}s}---\left(2\right)$

In formula, K _ifor static system gain, δ _ifor pure delay time constant, T _ifor system time constant, the complex parameter that s is transport function, i={1,2};

Suppose the input u of process ₁, u ₂with output y ₁, y ₂original state be stable state, be respectively and y ⁰, the first step, keeps u ₂ constant, a given amplitude is h ₁step input signal u ₁, system is output as y ₁, the equation that increases progressively of whole process is:

Δy ₁＝G ₁Δu ₁ (3)

Wherein, Δ y ₁=y ₁-y ⁰,

Second step, keeps u ₁ constant, a given amplitude is h ₂step input signal u ₂, system is output as y ₂, the equation that increases progressively of whole process is:

Δy ₂＝G ₂Δu ₂ (4)

Wherein, Δ y ₂=y ₂-y ⁰,

Formula (2) is converted into discrete form, is expressed as with difference equation:

Δy _i(k+1)-aΔy _i(k)＝bΔu _i(k-d) (5)

Parameter to be identified in formula is respectively, b=K _i(1-a), d=δ _i/ T _s, T _sfor the sampling time, Δ u _iand Δ y (k-d) _i(k) be input/output signal of equal value;

By input and output substitution formula (5) of equal value, the form of being write as least square is

γ _i(k+1)＝[φ _i(k)] ^Tθ _i+e _i(k) (6)

Wherein, γ _i(k+1) be output signal, φ _i(k) be considerable measured value, θ _ifor parameter to be identified, e _i(k) be white noise,

$\left\{\begin{array}{c}{\mathrm{\γ}}_i(k+1)=y_i(k+1)-y^0\\ {\mathrm{\φ}}_i\left(k\right)=[y_i\left(k\right)-y^0{h}_i-u_i^0]\\ {\mathrm{\θ}}_i=\left[\begin{array}{cc}{a}_i& b_i\end{array}\right]\end{array}\right.---\left(7\right)$

Solve by least square method, write as matrix form and be:

Γ _i＝Φ _iθ _i+Ε _i (8)

Wherein, Γ _i=[γ _i(2), γ _i(3) ..., γ _i(N+1)] ^t, Ε _i=[e _i(1), e _i(2) ... e _i(N)] ^t; Get criterion function

$J\left({\mathrm{\θ}}_i\right)=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{\left[e_i\left(k\right)\right]}^2=\underset{k=1}{\overset{L}{\mathrm{\Σ}}}{[{\mathrm{\Γ}}_i\left(k\right)-{\mathrm{\Φ}}_i\left(k\right){\mathrm{\θ}}_i]}^2={({\mathrm{\Γ}}_i-{\mathrm{\Φ}}_i{\mathrm{\θ}}_i)]}^2={({\mathrm{\Γ}}_i-{\mathrm{\Φ}}_i{\mathrm{\θ}}_i)}^T({\mathrm{\Γ}}_i-{\mathrm{\Φ}}_i{\mathrm{\θ}}_i)---\left(9\right)$

Minimization J (θ _i), try to achieve parameter θ _iestimated value, will make the output output of forecast system best of model;

If obtain J (θ _ithe θ of)=min _ibe denoted as be called parameter θ _ileast-squares estimation value, have

${\frac{\∂J\left({\mathrm{\θ}}_i\right)}{\∂{\mathrm{\θ}}_i}|}_{{\widehat{\mathrm{\θ}}}_i}=\frac{\∂}{\∂{\mathrm{\θ}}_i}{({\mathrm{\Γ}}_i-{\mathrm{\Φ}}_i{\mathrm{\θ}}_i)}^T({\mathrm{\Γ}}_i-{\mathrm{\Φ}}_i{\mathrm{\θ}}_i)=0---\left(10\right)$

Solve

${\widehat{\mathrm{\θ}}}_i={\left({{\mathrm{\Φ}}_i}^T{\mathrm{\Φ}}_i\right)}^{-1}{{\mathrm{\Φ}}_i}^T{\mathrm{\Γ}}_i---\left(11\right)$

Have

$T_i=-T_s/\ln \left({\widehat{\mathrm{\θ}}}_i\left(1\right)\right)---\left(12\right)$

$K_i={\widehat{\mathrm{\θ}}}_i\left(2\right)/(1-{\widehat{\mathrm{\θ}}}_i\left(1\right))---\left(13\right)$

There is said method can pick out the static gain K in transfer function model _iwith system time constant T _i;

Suppose at k ₁<k<k ₂in scope, by y _i(k) value and a _i, b _ivalue is brought following formula into:

$(y_i(k+1)-y^0)-a_i(y_i\left(k\right)-y^0)=b_i(u_i(k-d)-u_i^0)---\left(14\right)$

Work as u _i(k-d)=h _i, solve k value, d=k-k ₁again according to d=δ _i/ T _s, obtain δ retardation time _i;

Step 4. designs PID feedback controller, specifically:

A classical PID controller, its transport function is expressed as:

$G\left(s\right)=K_p(1+\frac{1}{T_{i}s}+T_{d}s)---\left(15\right)$

In formula, K _pfor scale-up factor, T _ifor integration time constant, T _dfor derivative time constant; Suppose under least model prerequisite, the PID controller parameter Ziegler-Nichols experimental formula of adjusting is:

$\left\{\begin{array}{c}{K}_p=\frac{1.2T}{\mathrm{K\&delta;}}\\ T_i=2\mathrm{\&delta;}\\ T_d=0.5\mathrm{\&delta;}\end{array}\right.---\left(16\right)$

The emulation of step 5. closed-loop system, specifically:

Calculate steel billet in the response after the induction heating process of a time step based on Finite Element Method, the temperature-responsive of output steel billet cross-sectional boundaries mid point is as sensor signal, and this sensor signal is defined as to APDL variable, realize PID controller with APDL language, just the PID controller of realizing according to APDL program after the response of a time step of every calculating calculates the optimum control power of next time step, input using this power ratio control as finite element model, calculate again the temperature-responsive of next time step, this process realizes with APDL loop structure, so just, FEEDBACK CONTROL is introduced to finite element model, realize the closed-loop system emulation of the electromagnetic induction heating process under finite element environment.