CN106227925B - A kind of symbolic analysis method of discontinuous mode fractional order switch converters - Google Patents

A kind of symbolic analysis method of discontinuous mode fractional order switch converters Download PDF

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CN106227925B
CN106227925B CN201610556414.1A CN201610556414A CN106227925B CN 106227925 B CN106227925 B CN 106227925B CN 201610556414 A CN201610556414 A CN 201610556414A CN 106227925 B CN106227925 B CN 106227925B
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陈艳峰
胡劼
陈曦
张波
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South China University of Technology SCUT
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Abstract

The invention discloses a kind of symbolic analysis methods of discontinuous mode fractional order switch converters, this method combines the principle of harmonic balance, by the way that differential operator will be converted to about differentiating for state variable fractional-order in converter, and all differential operators are merged into diagonal sign matrix, to convert the process for solving calculus of non-integral rank operation to the process of matrix operation and linear equation (group) solution, compared to the more existing analysis method for commonly establishing Oustaloup filter approximate model in Matlab/Simulink for fractional order switch converters, the method of the present invention is in addition to being capable of analytically analytic transformation device state variable ripple peak-to-peak value size, energy-storage travelling wave tube order changes the influence to converter working condition, the stable state of state variable can also rapidly be obtained Periodic analytical solution, and can be used for analyzing the harmonic components of state variable.

Description

A kind of symbolic analysis method of discontinuous mode fractional order switch converters
Technical field
The present invention relates to the modeling of fractional order switch converters and analysis fields, refer in particular to a kind of discontinuous current mode mould The symbolic analysis method of formula (DCM, Discontinuous-conduction Mode) fractional order switch converters.
Background technique
Past for switch converters commonly model the model having with analysis method: based on State-space Averaging Principle, from It dissipates iteration map model, the piecewise linear model based on circuit theory (KCL, KVL) and combines harmonic balance and method of perturbation Equivalent small parameter method, the analysis object of these methods are the switch converters of integer rank, i.e., capacitor, inductance in converter are all It is the element of integer rank, however " the Westerlund S.Dead Matter Has Memory of existing bibliography 1！[M] .Kalmar, Sweden:Causal Consulting, 2002,2 " Podlubny of Chap.7. " and bibliography I.Fractional Differential Equations[M].San Diego:Academic Press,1999,Chap.2.” It points out actually capacitor and inductance is fractional order in itself, this just needs to establish corresponding fractional order mould for converter Type.
" Wang Faqiang, Ma Xikui are based on Boost under the discontinuous mode of fractional calculus for existing bibliography 3 The modeling of converter and analysis [J] Chinese science: technological sciences, 2013,43 (4) .368-374 " consider inductance and capacitor simultaneously Fractional order characteristic, initially set up under discontinuous mode (DCM, Discontinuous-conduction Mode) The space State Average Model of fractional order Boost, and the non-integral order frequency domain based on Oustaloup filter approach it is micro- Integral algorithm establishes simulation model (as shown in Figure 1, 2) under Matlab/Simulink environment, converts to fractional order DC-DC Device has carried out preliminary analysis with the working characteristics that order changes.According to this thinking, the prior art (such as 4 " king of bibliography Hair is strong, and the fractional order modeling and simulation of Boost analyzes [J] Acta Physica Sinica under Ma Xikui inductor current continuous mode, 2011,60 (7) .070506-1-070506-8 " etc.) it is had studied under continuous current mode and pseudo-continuous work mode respectively Fractional order switch converters, Fig. 3 and Fig. 4 are the capacitance voltage obtained respectively by the model established in Fig. 1 and Fig. 2 and inductance electricity Waveform is flowed, existing technology is to show that fractional order switchs by way of establishing modularization model in Matlab/Simulink The working characteristics of converter, and show by way of simulation waveform the ripple of stable state downconverter state variable；This method It cannot obtain analytic solutions steady-state period of state variable, it is difficult to analytically analyze ripple peak-to-peak value size.
Summary of the invention
The purpose of the present invention is to overcome the shortcomings of the existing technology and deficiency, provides a kind of discontinuous mode The symbolic analysis method of (DCM, Discontinuous-conduction Mode) fractional order switch converters, can quickly obtain Obtain work fractional order switch converters state variable analytic solutions steady-state period under DCM state.
To achieve the above object, technical solution provided by the present invention are as follows: a kind of discontinuous mode fractional order is opened Close the symbolic analysis method of converter, comprising the following steps:
1) mathematical model of fractional order switch converters is established
The system mode of fractional order DC-DC converter under discontinuous mode DCM state that works describes are as follows:
In formula, x=[iLm vCm]TThe state variable of expression system, including the electric current i on m-th of inductanceLm, m-th of capacitor On voltage vCm, p indicate corresponding inductance L, on capacitor C-element corresponding system state variables order, A0And B0Table respectively Show the coefficient matrix not influenced by switch function, A1B1And A3B3Respectively indicate the coefficient matrix influenced by switch function；
Switch function δ(1)And δ(3)Is defined as:
Wherein, converter duty ratio D when open loop works1And D3For fixed value, meanwhile, enable non-linear partial are as follows:
f(q)=δ(q)(Aqx+Bq)
τ=ω t, wherein
Then the algebraic operation about integro-differential operator will be converted to for the calculus operation of state variable, i.e.,Since there are multiple state variables in converter, therefore the corresponding integro-differential operator of each state variable is merged For the diagonal sign matrix of differential operatorThese elements of α, β are for indicating different conditions variable in matrix κ Fractional calculus order, when L, C are integer rank, κ=- I, I are unit matrix, and +/- number therein respectively indicates pair State variable is quadratured/differential；By the algebraic operation be converted to about differential operator that will differentiate, fractional order can be opened The mathematical model for closing converter is as follows:
In formula (3), G0For all G comprising the diagonal sign matrix κ of differential operatorkiThe column matrix of composition, k ∈ EirExpression is worked as It is same after the definition of overtone order k, i, k in preceding i-th rank correction amount,From GkiForm Influence of the fractional-order to state variable analytic solutions can be embodied；
By state variable x and switch function δ(q)Expand into the form of the sum of principal part and a small amount of remainders:
Above formula is substituted into f(q)=δ(q)(Aqx+Bq), it is a small amount of to merge identical order remainder, it obtains:
Wherein:
In formula, useIt indicates the principal part of the i-th rank of state variable x correction amount, usesIndicate the state variable x The remainder of i-th rank correction amount is a small amount of；
According to principle of harmonic balance, by the state variable x and switch function δ(q)Expansion (4) in principal part and the i-th rank It is as follows that remainder does Fourier expansion in a small amount:
Wherein akiIndicate the amplitude of the k subharmonic ingredient of the i-th rank correction amount, the switch function δ(q)Expansion coefficient table Up to formula are as follows:
Wherein
According to principle of harmonic balance, coefficient expansion (8) are substituted into Fourier expansion formula (7), successively solving state variable Main oscillations component and each rank correction amount；
2) main oscillations component is sought
Firstly, the main oscillations component of solving state variable, DC quantity is contained only in usual main oscillations, therefore is set as:
x0=a00
=[I00 V00]T (9)
Work as k=0, i.e. G0=G00=A0, x0In substitution formula (6)It substitutes into (4) formula, obtains again:
G00·x0+b0(A1x0+B1)+c0(A3x0+B3)+B0=0 (10)
The main oscillations component of transducer status variable is acquired by formula (10):
3) each rank correction amount is sought
According to main oscillations component remainder R1In the harmonic components that contain, if the first-order correction form of state variable is as follows:
Wherein, a11=[I11 V11]T, c.c indicates conjugation item, rear same；By the harmonic wave in the first-order correction of state variable Known to ingredientk∈E1r, substitute into f in formula (6)1, obtain first-order correction table Up to formula:
Gk1·x1+(b0A1x1+b10B1+b10A1x0+b10B1)+(c0A3x1+c10B3+c10A3x0+c10B3)+B0=0 (13)
It can be obtained according to formula (13) about harmonic amplitude a01And ak1System of linear equations；
Parameter is substituted into the expression formula of the current order correction amount of gained, if each harmonic amplitude phase of current order correction amount Compare first-order correction less than an order of magnitude, be then not required to do the amendment of higher order, conversely, continuing according to above process continuation Seek the correction amount of higher order time；
4) main oscillations component is added with each rank correction amount, obtains and is expressed about analytic solutions steady-state period of state variable Formula.
Compared with prior art, the present invention have the following advantages that with the utility model has the advantages that
Fractional order switch converters state is asked to become using this method it can be seen from the solution formula of the mentioned method of the present invention Analytic solutions steady-state period of amount are equivalent to and convert matrix operation for the complex process for solving calculus of non-integral rank operation and ask The process of linear equation (group), as long as establishing the fractional order switch converters shape such as formula (3) matrix form according to circuit theory Then coefficient expressions are substituted into each rank correction amount formula by state equation, by simple matrix multiplication and division plus and minus calculation and disappear member just It is available about fractional order transducer status variable stable state solution's expression.Compare over pure mathematics field proposition it is all kinds of The characteristics of solution procedure of the method for solving of fractional calculus equation, the mentioned method of the present invention combines switch converters, keeps away Thoroughly discussing for fractional calculus principle of operation is opened, resulting solution has apparent physical significance, according to using this The form for inventing the steady state solution that proposed method obtains, can be clearly seen that the harmonic components that state variable is included, is conducive to To the quasi-converter expansion deeper into analysis.
Detailed description of the invention
Fig. 1 is the open loop point established in Matlab/Simulink based on Oustaloup filter method in bibliography 3 Number rank Boost simulation model.
Fig. 2 is the Oustaloup filter subsystem based on fractional order frequency domain approximation method encapsulated in Fig. 1.
Fig. 3 is the Cycle by Cycle simulation waveform of open loop fractional order Boost simulation model output capacitance voltage in Fig. 1. Wherein, abscissa is the time of emulation, and ordinate indicates capacitance voltage value.
Fig. 4 is open loop fractional order Boost simulation model inductive current Cycle by Cycle simulation waveform in Fig. 1.Wherein, horizontal Coordinate is the time of emulation, and ordinate indicates inductor current value.
Fig. 5 a is method inductance when the order of inductance and capacitor is 0.8 in presently disclosed method and bibliography 3 The simulation result comparison diagram of current waveform.
Fig. 5 b is method capacitor when the order of inductance and capacitor is 0.8 in presently disclosed method and bibliography 3 The simulation result comparison diagram of voltage alternating component waveform.
Fig. 5 c is method capacitor when the order of inductance and capacitor is 0.8 in presently disclosed method and bibliography 3 The simulation result comparison diagram of voltage waveform.
Specific embodiment
The present invention is further explained in the light of specific embodiments.
The symbolic analysis method of discontinuous mode fractional order switch converters described in the present embodiment, including it is following Step:
1) mathematical model of fractional order switch converters is established
The system mode of fractional order DC-DC converter under discontinuous mode DCM state that works describes are as follows:
In formula, x=[iLm vCm]TThe state variable of expression system, including the electric current i on m-th of inductanceLm, m-th of capacitor On voltage vCm, p indicate corresponding inductance L, on capacitor C-element corresponding system state variables order, A0And B0Table respectively Show the coefficient matrix not influenced by switch function, A1B1And A3B3Respectively indicate the coefficient matrix influenced by switch function；
Switch function δ(1)And δ(3)Is defined as:
Wherein, converter duty ratio D when open loop works1And D3For fixed value, meanwhile, enable non-linear partial are as follows:
f(q)=δ(q)(Aqx+Bq)
τ=ω t, wherein
Then the algebraic operation about integro-differential operator will be converted to for the calculus operation of state variable, i.e.,Since there are multiple state variables in converter, therefore the corresponding integro-differential operator of each state variable is merged For the diagonal sign matrix of differential operatorThese elements of α, β are for indicating different conditions variable in matrix κ Fractional calculus order, when L, C are integer rank, κ=- I, I are unit matrix, and +/- number therein respectively indicates pair State variable is quadratured/differential；By the algebraic operation be converted to about differential operator that will differentiate, fractional order can be opened The mathematical model for closing converter is as follows:
In formula (3), G0For all G comprising the diagonal sign matrix κ of differential operatorkiThe column matrix of composition, k ∈ EirExpression is worked as It is same after the definition of overtone order k, i, k in preceding i-th rank correction amount,From GkiForm Influence of the fractional-order to state variable analytic solutions can be embodied；
By state variable x and switch function δ(q)Expand into the form of the sum of principal part and a small amount of remainders:
Above formula is substituted into f(q)=δ(q)(Aqx+Bq), it is a small amount of to merge identical order remainder, it obtains:
Wherein:
In formula, useIt indicates the principal part of the i-th rank of state variable x correction amount, usesIndicate the state variable x The remainder of i-th rank correction amount is a small amount of；
According to principle of harmonic balance, by the state variable x and switch function δ(q)Expansion (4) in principal part and the i-th rank It is as follows that remainder does Fourier expansion in a small amount:
Wherein akiIndicate the amplitude of the k subharmonic ingredient of the i-th rank correction amount, the switch function δ(q)Expansion coefficient table Up to formula are as follows:
Wherein
According to principle of harmonic balance, coefficient expansion (8) are substituted into Fourier expansion formula (7), successively solving state variable Main oscillations component and each rank correction amount；
2) main oscillations component is sought
Firstly, the main oscillations component of solving state variable, DC quantity is contained only in usual main oscillations, therefore is set as:
x0=a00
=[I00 V00]T (9)
Work as k=0, i.e. G0=G00=A0, x0In substitution formula (6)It substitutes into (4) formula, obtains again:
G00·x0+b0(A1x0+B1)+c0(A3x0+B3)+B0=0 (10)
The main oscillations component of transducer status variable is acquired by formula (10):
3) each rank correction amount is sought
According to main oscillations component remainder R1In the harmonic components that contain, if the first-order correction form of state variable is as follows:
Wherein, a11=[I11 V11]T, c.c indicates conjugation item, rear same；By the harmonic wave in the first-order correction of state variable Known to ingredientk∈E1r, substitute into f in formula (6)1, obtain first-order correction table Up to formula:
Gk1·x1+(b0A1x1+b10B1+b10A1x0+b10B1)+(c0A3x1+c10B3+c10A3x0+c10B3)+B0=0 (13)
It can be obtained according to formula (13) about harmonic amplitude a01And ak1System of linear equations；
Parameter is substituted into the expression formula of the current order correction amount of gained, if each harmonic amplitude phase of current order correction amount Compare first-order correction less than an order of magnitude, be then not required to do the amendment of higher order, conversely, continuing according to above process continuation Seek the correction amount of higher order time；
4) main oscillations component is added with each rank correction amount, obtains and is expressed about analytic solutions steady-state period of state variable Formula.
Below for specific example using above-mentioned discontinuous conduct mode fractional order switch converters symbolic analysis method into Row operation, for the fractional order Boost of open loop, state variable x=[iL vC]T, it is contemplated that the score of energy-storage travelling wave tube Rank characteristic and inductor loss, state equation are as follows:
Form described in corresponding (1), it is known that B1=[0 0]T、Differential operator Matrix
When using bibliography, " Wang Faqiang, Ma Xikui are based under the discontinuous mode of fractional calculus The modeling of Boost and analysis [J] Chinese science: technological sciences, 2013,43 (4), in pp.368-374 " when parameter, There are Boost switch periods fs=25kHZ, input voltage E=24V, inductance L=4mH, R is lost in inductive resistanceL=0 Ω, Capacitor C=100 μ F loads R=50 Ω, takes inductance order α=0.8, capacitor order β=0.8.
The main oscillations component, first-order correction and second order correction amount of fractional order Boost are asked according to the step of front, At this time since the Amplitude Ration main oscillations component of each harmonic in second order correction amount is much smaller, therefore do not continue to seek higher order correction amount, Fractional order Boost is as follows by the revised analytic solutions steady-state period form of two ranks:
In formula, Re (aik) indicate to take aikReal part, Im (aik) indicate to take aikImaginary part, xdcAnd xacRespectively indicate state change The direct current component and of ac of amount.
The duty ratio of each mode are as follows:
aikExpression formula it is as follows:
Parameter substitution formula (15) and (16) can be obtained into analytic solutions steady-state period are as follows:
By Symbolic Analysis Method of the present invention and 3 method therefor of bibliography state variable waveform ratio in stable state Compared with as shown in Fig. 5 a, 5b, 5c, analogous diagram is using parameter in bibliography 3.But due to resistance be 500 Ω when stablize time mistake It is long, therefore it is taken as 50 ohm.When the order of inductance and capacitor is 0.8.Method in presently disclosed method and bibliography 3 Simulation result contrast verification figure, dotted line is the time-domain simulation results according to the established model of 3 method of bibliography in figure, and solid line is The Numerical Simulation Results of method proposed by the present invention, Fig. 5 a are inductive current waveform, and Fig. 5 b is the AC compounent wave of capacitance voltage Shape, Fig. 5 c are to be superimposed the later capacitance voltage waveform of DC component.
It can be seen that two correlation curves of inductive current and capacitance voltage are fitted very well, illustrate that the present invention is mentioned Method out is effective.Fractional order switch converters state variable is asked using this method it can be seen from analytic solutions formula Analytic solutions, were equivalent to and converted matrix operation for the complex process for solving calculus of non-integral rank operation and ask linear steady-state period Then coefficient expressions are substituted into each rank and repaired by the process of equation (group) as long as establishing the fractional order converter such as formula (3) form Positive quantity formula can be obtained by the table about fractional order transducer status variable steady state solution by simple matrix operation and the member that disappears Up to formula, the harmonic components in state variable can clearly be seen that by the expression formula, by the expression formula of harmonic amplitude coefficient, It can be seen that influence of the energy-storage travelling wave tube order to transducer status variable.
Embodiment described above is only the preferred embodiments of the invention, and but not intended to limit the scope of the present invention, therefore All shapes according to the present invention change made by principle, should all be included within the scope of protection of the present invention.

Claims (1)

1. a kind of symbolic analysis method of discontinuous mode fractional order switch converters, which is characterized in that including following Step:
1) mathematical model of fractional order switch converters is established
The system mode of fractional order DC-DC converter under discontinuous mode DCM state that works describes are as follows:
In formula, x=[iLm vCm]TThe state variable of expression system, including the electric current i on m-th of inductanceLm, on m-th of capacitor Voltage vCm, p indicate corresponding inductance L, on capacitor C-element corresponding system state variables order, A0And B0It respectively indicates not The coefficient matrix influenced by switch function, A1B1And A3B3Respectively indicate the coefficient matrix influenced by switch function；
Switch function δ(1)And δ(3)Is defined as:
Wherein, converter duty ratio D when open loop works1And D3For fixed value, meanwhile, enable non-linear partial are as follows:
f(q)=δ(q)(Aqx+Bq)
τ=ω t, wherein
Then the algebraic operation about integro-differential operator will be converted to for the calculus operation of state variable, i.e., Since there are multiple state variables in converter, therefore the corresponding integro-differential operator of each state variable is merged into differential operator pair Angle sign matrixThese elements of α, β are for indicating the micro- product of the fractional order of different conditions variable in matrix κ Sublevel, when L, C are integer rank, κ=- I, I are unit matrix, and +/- number therein respectively indicates to state variable quadrature Point/differential；It, can be by the number of fractional order switch converters by the algebraic operation be converted to about differential operator that will differentiate It is as follows to learn model:
In formula (3), G0For all G comprising the diagonal sign matrix κ of differential operatorkiThe column matrix of composition, k ∈ EirIndicate current It is same after the definition of overtone order k, i, k in i rank correction amount,From GkiForm can Embody influence of the fractional-order to state variable analytic solutions；
By state variable x and switch function δ(q)Expand into the form of the sum of principal part and a small amount of remainders:
Above formula is substituted into f(q)=δ(q)(Aqx+Bq), it is a small amount of to merge identical order remainder, it obtains:
Wherein:
In formula, useIt indicates the principal part of the i-th rank of state variable x correction amount, usesIndicate i-th rank of state variable x The remainder of correction amount is a small amount of；
According to principle of harmonic balance, by the state variable x and switch function δ(q)Expansion (4) in principal part and the i-th rank remainder It is as follows to do Fourier expansion in a small amount:
Wherein akiIndicate the amplitude of the k subharmonic ingredient of the i-th rank correction amount, the switch function δ(q)Expansion coefficient expressions Are as follows:
Wherein
According to principle of harmonic balance, coefficient expansion (8) are substituted into Fourier expansion formula (7), successively the master of solving state variable Oscillating component and each rank correction amount；
2) main oscillations component is sought
Firstly, the main oscillations component of solving state variable, DC quantity is contained only in usual main oscillations, therefore is set as:
x0=a00
=[I00 V00]T (9)
Work as k=0, i.e. G0=G00=A0, x0In substitution formula (6)It substitutes into (4) formula, obtains again:
G00·x0+b0(A1x0+B1)+c0(A3x0+B3)+B0=0 (10)
The main oscillations component of transducer status variable is acquired by formula (10):
3) each rank correction amount is sought
According to main oscillations component remainder R1In the harmonic components that contain, if the first-order correction form of state variable is as follows:
Wherein, a11=[I11 V11]T, c.c indicates conjugation item, rear same；It can by the harmonic components in the first-order correction of state variable Knowk∈E1r, substitute into f in formula (6)1, obtain first-order correction expression formula:
Gk1·x1+(b0A1x1+b10B1+b10A1x0+b10B1)+(c0A3x1+c10B3+c10A3x0+c10B3)+B0=0 (13)
It can be obtained according to formula (13) about harmonic amplitude a01And ak1System of linear equations；
Parameter is substituted into the expression formula of the current order correction amount of gained, if each harmonic amplitude of current order correction amount compares Upper first-order correction is less than an order of magnitude, then is not required to do the amendment of higher order, continues to ask more conversely, continuing according to the above process The correction amount of high order；
4) main oscillations component is added with each rank correction amount, obtains analytic solutions expression formula steady-state period about state variable.
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CN104978304A (en) * 2015-07-24 2015-10-14 华南理工大学 Symbolic analysis method and apparatus of fractional order switching converter under continuous current mode

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