CN109245532A - A kind of fractional order sliding-mode control of buck-boost converter - Google Patents
A kind of fractional order sliding-mode control of buck-boost converter Download PDFInfo
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- CN109245532A CN109245532A CN201811146111.8A CN201811146111A CN109245532A CN 109245532 A CN109245532 A CN 109245532A CN 201811146111 A CN201811146111 A CN 201811146111A CN 109245532 A CN109245532 A CN 109245532A
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
- H02M3/00—Conversion of dc power input into dc power output
- H02M3/02—Conversion of dc power input into dc power output without intermediate conversion into ac
- H02M3/04—Conversion of dc power input into dc power output without intermediate conversion into ac by static converters
- H02M3/10—Conversion of dc power input into dc power output without intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode
- H02M3/145—Conversion of dc power input into dc power output without intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal
- H02M3/155—Conversion of dc power input into dc power output without intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal using semiconductor devices only
- H02M3/156—Conversion of dc power input into dc power output without intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal using semiconductor devices only with automatic control of output voltage or current, e.g. switching regulators
- H02M3/158—Conversion of dc power input into dc power output without intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal using semiconductor devices only with automatic control of output voltage or current, e.g. switching regulators including plural semiconductor devices as final control devices for a single load
- H02M3/1582—Buck-boost converters
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Abstract
The present invention proposes a kind of fractional order sliding-mode control of buck-boost converter, comprising: the foundation of fractional order mathematical model；The design of fractional order Sliding Mode Controller；And ripple analysis is carried out based on fractional model；The foundation of fractional order mathematical model has been carried out to buck-boost converter, it is set to be more in line with actual physical system, and calculating that demonstrate integer model with emulation experiment be a kind of approximate of real system by analytic expression, fractional model can more embody with hereditary capacity the bulk properties of actual physics system because of Memorability possessed by itself；Fractional order sliding mode controller is devised, compared with integer rank sliding mode controller, robustness is enhanced, and further improves the output characteristics of system, enhances the anti-interference ability of system；The output voltage stabilization of fractional order sliding mode controller it can be seen from the simulation experiment result and hardware circuit simulation result, further illustrates the necessity and reasonability of fractional order control device.
Description
Technical field
The invention belongs to the control fields of buck-boost converter in DC converter, and in particular to a kind of buck-boost converter
Fractional order sliding-mode control.
Background technique
DC/DC converter is as communication equipment, one of indispensable part in civilian service, with the hair of science and technology
Exhibition, DC/DC converter applications are in more and more equipment and require to be increasingly stringenter.Wherein One Buck-Boost converter body is made
It can be that variable DC voltage flexibly and easily can by a kind of DC voltage conversion of fixation for common DC/DC converter
Realize the controllable adjustment of output voltage.One Buck-Boost converter body has electronic device few, and structure is simple, at low cost and energy benefit
The advantages that high with rate, becomes the capital equipment of electric energy conversion and control, is widely used in data communication, office automation is set
It is standby, robot, the fields such as military aerospace.With the rapid industrial development in our country, for the transfer efficiency of electric energy, output accuracy and
Its robustness also proposed more strict requirements.
One Buck-Boost converter body is as a kind of typical time-varying, nonlinear system, and it is nonlinear for controlling, discontinuously
, and it is more sensitive to the jump of system parameter variations and load, when load has biggish change, Buck-Boost transformation
Device has the shortcomings that dynamic response is slow, output waveform distortion.In recent years, with ANN Control, fuzzy control, structure changes
Non-Linear Control Theory has been applied to DC/ by control, the development of the Non-Linear Control Theories such as chaos controlling, more and more experts
Among DC converter, and achieve better effect.Sliding mode variable structure control as a kind of Non-Linear Control Theory,
Variation and external interference to parameter have good robustness, have been achieved for more in the control of DC/DC converter
Good effect.
Popularization of the fractional calculus as integer rank calculus is to be proposed in 1695 by Leibnitz earliest, it
Order is no longer limited to integer, greatly extends the design and application of controller.It is more and more with gradually going deep into for research
Scholar fractional calculus theory and method are applied in the every field of natural science and social science.Automatically it is controlling
In the application in field processed, Oustaloup has founded CRONE control, and this theory has been applied to many practical problems
In the middle, good effect is achieved, and demonstrates CRONE controller and has more advantage than traditional PID controller.In electric power electricity
In son, inductance and capacitor are actually fractional order rather than integer rank, therefore DC converter is actually new fractional-order system,
It is all the design and analysis for being approximately integer rank progress controller in existing control mode, resulting control effect
Fruit is not conformed to the actual conditions, and there are larger differences.Fractional calculus theory is as a kind of important mathematical tool, its note
Recall and hereditary capacity, can adequately be applied in DC/DC converter.Therefore, the design is managed using fractional calculus
By in conjunction with fractional order sliding mode control theory, being set using fractional order Reaching Law in the structure of One Buck-Boost converter body
Fractional order sliding-mode surface is counted, to the output response of system, dynamic characteristic is improved, and will by way of fractional order discretization
It is applied in actual circuit.
Summary of the invention
Against the above technical problems, the present invention proposes the fractional order sliding-mode control of kind of buck-boost converter, using point
Number rank calculus theory carries out the foundation of mathematical model, the design of controller and testing for emulation experiment to buck-boost converter
Card is buck-boost converter, i.e., One Buck-Boost converter body, which provides, more accurately controls, and further improves DC converting
The output characteristics and robustness of device output voltage, while the buffeting in Sliding mode variable structure control is weakened, improve the whole of system
Body performance.
It is as follows including content:
(1) foundation of fractional order mathematical model
Based on buck-boost converter, the i.e. circuit theory of One Buck-Boost converter body, ViFor DC input voitage, T is function
The elements such as rate switching device, generally MOSFET or IGBT, it is the time of conducting and shutdown by adjusting T, adjustable defeated
Voltage V outoSize；D is freewheeling diode, and the flow direction of electric current does not change in holding circuit, and L and C are inductance and capacitor,
It is the main energy-storage travelling wave tube in circuit.When switching element T is closed, power supply is the electricity on energy transfer to inductance, on inductance
It can energy storage increase；When switching device turns off, inductance is transferred energy on capacitor and load, and the energy storage on inductance is reduced.
(1) One Buck-Boost converter body mid-score rank mathematical model is established: where One Buck-Boost converter body mid-score rank
Device is inductance L and capacitor C, mathematical model are as follows:
Wherein, iLFor inductive current, vLFor inductive drop, iCFor capacitance current, vCFor capacitance voltage, DαFor the micro- product of fractional order
Divide operator, wherein for order α between 0~1, a is lower limit of integral, and t is upper limit of integral, and L is that the size unit of inductance inductance value is H, C
Size unit for capacitor's capacity is F.
In fractional calculus theory, there are three types of main form of Definition, is Grunwald-Letnikov (G-L) respectively
Fractional calculus, Riemann-Liouville (R-L) fractional calculus and Caputo fractional calculus.
The definition of Grunwald-Letnikov fractional calculus is from the popularization that integer rank defines, by integer rank
Order be generalized in fractional order, definition are as follows:
Wherein m is positive integer and α≤m≤α+1, a are the lower limit of integral, and Γ () is Gamma function, form of Definition are as follows:
Wherein m is constant, and Re (m) > 0.
Riemann-Liouville fractional calculus is the property that should meet from fractional calculus, right
G-L type fractional calculus is improved, definition are as follows:
Wherein, n is positive integer and α≤n≤α+1.The definition of R-L type fractional calculus, can be regarded as to function u (t)
It first carries out fractional order integration and carries out integer rank differential again, form is relatively simple for the definition of G-L type, can simplify point
The calculating process of number rank calculus, in practice using relatively broad.
(2) operation is carried out using Caputo definition, calculates fractional calculus operatorThe expression formula that Caputo is defined
Are as follows:
Wherein, a is lower limit of integral, and t is upper limit of integral, and r is fractional order order, and u (t) is function to be solved, and n is fractional order
Apparent order, be greater than the smallest positive integral of fractional order, ε is integration variable.
G-L type, which defines, first defines integer rank calculus by way of the limit, is generalized in fractional order, gives
The definition of fractional calculus theory, but its expression formula is excessively complicated, be unfavorable in practice be applied to promote；R-L type is fixed
Justice and Caputo definition are improved and have been expanded, simplified the calculating of fractional calculus based on the definition of G-L type
Process, convenient for application in practice.
For order be positive integer the case where for, G-L type definition, R-L type defines and Caputo is defined all be it is of equal value,
It can mutually convert.The case where being fractional order for order, in condition: function u (t) has m+1 rank continuous derivative, and m is extremely
N-1 is got less, then n=m-1, if function u (t) meets u at this time(k)(a)=0, k=0,1 ..., n-1, then three kinds of definition equivalences
, it can mutually convert, otherwise be non-equivalence.
The definition of Caputo type remains the property of integer rank calculus due to first carrying out integer rank differential calculation, right
It is 0 in the Fractional Derivative of constant, in the modeling process in face of many actual application problems, is widely used.
(3) adoption status space average method models two switch states in One Buck-Boost converter body, obtains
The fractional order mathematical model based on switching value of One Buck-Boost converter body are as follows:
D is switching variable, < i in formulaL>, < vo>, < Vin> is inductive current, input voltage and output voltage one
Average value in a switch periods, L are that the size unit of inductance inductance value is H, and C is that the size unit of capacitor's capacity is F, and R is electricity
Hinder the size of resistance value, unit Ω.
(2) fractional order Sliding Mode Controller designs
One Buck-Boost converter body is a kind of typical switch non-linearity system, is existed on the design of controller very big
Limitation, correlative study and stability analysis and proof especially for new fractional-order system are concentrated mainly on linear system
In the middle, less for fractional order nonlinear systematic research achievement, it is unfavorable for the further analysis of controller.
(1) it converts the fractional order mathematical model of One Buck-Boost converter body: enabling [x in formula (3)1,x2]T=[iL,vo]T, u=
D, then the fractional model of original Buck-Boost is transformed to canonical form shown in formula (4), and u represents the size of duty ratio, is
One with the function of time change, be the actual control variable of whole system:
In formula, X is state variable, X=[x1,x2]T=[iL,vo]T, y is output voltage, and f (X) and g (X) are as follows:
(2) the fractional order mathematical model for further converting transformation One Buck-Boost converter body, is convenient for fractional order sliding formwork control
The design and realization of device: anti-by fractional order on the fractional order mode standard of the One Buck-Boost converter body shown in formula (4)
The mode of linearization reconfigures the form of output function, is converted into inearized model shown in formula (7)；
Wherein v, z1And z2Expression formula are as follows:
Therefore, on can be by the system after fractional order feedback linearization, the design of control system be carried out, is acted on
On raw score rank nonlinear system, the control of whole nonlinear system is realized.
(3) the control target of Buck-Boost fractional order converter is output voltage track reference voltage vref, according to
The reference of the steady operation point of One Buck-Boost converter body, available inductive current exports iLrefWith duty ratio DrefReference
Output are as follows:
Thus, it is possible to by formula (11), to calculate the inductive current i when circuit reaches stable stateLrefSize and PWM account for
The size D of empty ratiorefSize, can reference output after fractional order feedback linearization to calculate are as follows:
(4) fractional order sliding formwork is designed using fractional calculus theory for the inearized model after feedback linearization
Controller, wherein fractional order sliding-mode surface s and control law v are as follows:
Wherein s is sliding-mode surface, k1With k2For the gain coefficient of system, λ and k are slide coefficient, guarantee that system mode can be fast
Speed reaches sliding-mode surface, and sign () is sign function, as s >=0, sign (s)=1, and as s < 0, sign (s)=- 1, e1,e2
For the first derivative of output error and output error, formula are as follows:
(3) fractional model ripple is analyzed:
For Switching Power Supply, the size of output voltage ripple is to determine the important characteristic of the quality of Switching Power Supply,
It is also a kind of direct current variator foundation that component selects in the design process.Fractional order for integer rank because its
The presence of Memorability and hereditary capacity, to the output voltage ripple of One Buck-Boost converter body, there are large effects.
To the mathematical model of One Buck-Boost converter body, i.e. formula (3), in a switch device conductive, to inductance electricity
It flows and is solved with the changing value of output voltage, available:
Wherein, Δ iLFor inductive current changing value, Δ voFor output voltage changing value, Vin, L, R, C is respectively to input electricity
Pressure, inductance, resistance, capacitor value,For initial value of the output voltage within this period, ε is integration variable, Eα() is
Mittah-Leffler function, definition are as follows:
It can be seen from formula (17) and formula (18) changing value of output voltage and inductive current with fractional order order increase
It is big and reduces, therefore can be found that all integer models in analyze in the past, only to a kind of approximation of realistic model, with
Real system deviation is larger.
Advantageous effects:
The present invention is based on fractional calculus theories, have carried out fractional order mathematical modulo to buck One Buck-Boost converter body
The foundation of type makes it be more in line with actual physical system, and is calculated by analytic expression and demonstrate integer rank mould with emulation experiment
Type is a kind of approximation of real system, and fractional model can more embody reality because of Memorability possessed by itself and hereditary capacity
The bulk properties of border physical system.
It is theoretical based on fractional calculus later, fractional order sliding mode controller is devised, with integer rank sliding mode controller phase
Than robustness is enhanced.In emulation experiment, Matlab/Simulink and ninteger fractional order work are utilized
Have case, is jumped for the starting response of One Buck-Boost converter body, output voltage and load resistance jumps three kinds of situations and carries out
Simulating, verifying.Pass through the analysis of output waveform, it can be clearly seen that fractional order sliding mode controller further improves the defeated of system
Characteristic out enhances the anti-interference ability of system.
Discretization, and the reality carried out finally have been carried out to fractional order sliding mode controller by the way of Tustin+CFE herein
Designing and producing for object circuit and writing for software program.Using TMS320F28335 as core controller, four switch topologies electricity
Road has carried out the verifying of Actual Control Effect of Strong to One Buck-Boost converter body as basic circuit.It can be seen from experimental result
The output voltage stabilization of fractional order sliding mode controller, strong antijamming capability also further illustrate necessity of fractional order control device
Property and reasonability.
Detailed description of the invention
Fig. 1 is the whole functional block diagram of the embodiment of the present invention；
Fig. 2 (a) is the One Buck-Boost converter body output voltage ripple simulation comparison figure of the embodiment of the present invention；
Fig. 2 (b) is the One Buck-Boost converter body inductive current ripple simulation comparison figure of the embodiment of the present invention；
Fig. 3 (a) is output voltage analogous diagram in the fractional order Sliding Mode Controller starting response of the embodiment of the present invention；
Fig. 3 (b) is output voltage analogous diagram in the integer rank sliding mode controller starting response of the embodiment of the present invention；
Fig. 4 (a) is inductive current analogous diagram in the fractional order Sliding Mode Controller starting response of the embodiment of the present invention；
Fig. 4 (b) is inductive current analogous diagram in the integer rank sliding mode controller starting response of the embodiment of the present invention；
Output voltage emulates when Fig. 5 (a) is the fractional order Sliding mode variable structure control output voltage jump of the embodiment of the present invention
Figure；
Output voltage analogous diagram when Fig. 5 (b) is the integer rank sliding mode controller output voltage jump of the embodiment of the present invention；
Inductive current responds when Fig. 6 (a) is the fractional order Sliding mode variable structure control output voltage jump of the embodiment of the present invention
Analogous diagram
Inductive current response emulation when Fig. 6 (b) is the integer rank sliding mode controller output voltage jump of the embodiment of the present invention
Figure；
Output voltage responds when Fig. 7 (a) is the fractional order Sliding mode variable structure control load resistance jump of the embodiment of the present invention
Analogous diagram；
Output voltage response emulation when Fig. 7 (b) is the integer rank sliding mode controller load resistance jump of the embodiment of the present invention
Figure；
Inductive current responds when Fig. 8 (a) is the fractional order Sliding mode variable structure control load resistance jump of the embodiment of the present invention
Analogous diagram；
Inductive current response emulation when Fig. 8 (b) is the integer rank sliding mode controller load resistance jump of the embodiment of the present invention
Figure；
Fig. 9 is the hardware verification One Buck-Boost converter body circuit topology figure of the embodiment of the present invention；
Figure 10 is the voltage detecting circuit figure of the embodiment of the present invention；
Figure 11 is the current detection circuit figure of the embodiment of the present invention；
Figure 12 is the PWM drive circuit figure of the embodiment of the present invention；
Figure 13 is the practical main circuit diagram of the embodiment of the present invention；
Figure 14 (a) is that the principal voltage for helping power circuit of the embodiment of the present invention is converted to boost voltage；
Figure 14 (b) is the digital power voltage for helping power circuit of the embodiment of the present invention；
Figure 14 (c) is the reference voltage conversion circuit for helping power circuit of the embodiment of the present invention；
Figure 15 is the main program flow chart of the embodiment of the present invention；
Figure 16 is the Fractional Control Algorithm flow chart of the embodiment of the present invention；
Figure 17 is the control algolithm flow chart of the embodiment of the present invention；
Figure 18 is the integrated circuit figure of the One Buck-Boost converter body of the embodiment of the present invention；
Figure 19 is the hardware verification oscillograph output voltage and inductive current waveform diagram of the embodiment of the present invention；
Figure 20 is the hardware verification output voltage waveforms and PWM waveform figure of the embodiment of the present invention；
One Buck-Boost converter body simulation waveform when Figure 21 (a) is the α=0.8 of the embodiment of the present invention；
Figure 21 (b) is α=1Buck-Boost converter simulation waveform of the embodiment of the present invention；
Figure 22 is the circuit topology figure of the One Buck-Boost converter body of the embodiment of the present invention.
Specific embodiment
Invention is described further with specific implementation example with reference to the accompanying drawing, a kind of fractional order of buck-boost converter
Sliding-mode control, as shown in Figure 1, including following content:
(1) foundation of fractional order mathematical model
The circuit diagram of One Buck-Boost converter body is as shown in figure 22, ViFor DC input voitage, T is power switch device
The elements such as part, generally MOSFET or IGBT, the time of conducting and shutdown by adjusting T, adjustable output voltage Vo
Size；D is freewheeling diode, and the flow direction of electric current does not change in holding circuit, and it is in circuit that L and C, which are inductance and capacitor,
Main energy-storage travelling wave tube.When switching element T is closed, on energy transfer to inductance, the electric energy energy storage on inductance increases power supply
Add；When switching device turns off, inductance is transferred energy on capacitor and load, and the energy storage on inductance is reduced.
(1) One Buck-Boost converter body mid-score rank mathematical model is established: where One Buck-Boost converter body mid-score rank
Device is inductance L and capacitor C, mathematical model are as follows:
Wherein, iLFor inductive current, vLFor inductive drop, iCFor capacitance current, vCFor capacitance voltage, DαFor the micro- product of fractional order
Divide operator, wherein for order α between 0~1, a is lower limit of integral, and t is upper limit of integral, and L is that the size unit of inductance inductance value is H, C
Size unit for capacitor's capacity is F.
In fractional calculus theory, there are three types of main form of Definition, is Grunwald-Letnikov (G-L) respectively
Fractional calculus, Riemann-Liouville (R-L) fractional calculus and Caputo fractional calculus.
The definition of Grunwald-Letnikov fractional calculus is from the popularization that integer rank defines, by integer rank
Order be generalized in fractional order, definition are as follows:
Wherein m is positive integer and α≤m≤α+1, a are the lower limit of integral, and Γ () is Gamma function, form of Definition are as follows:
Wherein m is constant, and Re (m) > 0.
Riemann-Liouville fractional calculus is the property that should meet from fractional calculus, right
G-L type fractional calculus is improved, definition are as follows:
Wherein, n is positive integer and α≤n≤α+1.The definition of R-L type fractional calculus, can be regarded as to function u (t)
It first carries out fractional order integration and carries out integer rank differential again, form is relatively simple for the definition of G-L type, can simplify point
The calculating process of number rank calculus, in practice using relatively broad.
(2) operation is carried out using Caputo definition, calculates fractional calculus operatorThe expression formula that Caputo is defined
Are as follows:
Wherein, a is lower limit of integral, and t is upper limit of integral, and r is fractional order order, and u (t) is function to be solved, and n is fractional order
Apparent order, be greater than the smallest positive integral of fractional order, ε is integration variable.
G-L type, which defines, first defines integer rank calculus by way of the limit, is generalized in fractional order, gives
The definition of fractional calculus theory, but its expression formula is excessively complicated, be unfavorable in practice be applied to promote；R-L type is fixed
Justice and Caputo definition are improved and have been expanded, simplified the calculating of fractional calculus based on the definition of G-L type
Process, convenient for application in practice.
For order be positive integer the case where for, G-L type definition, R-L type defines and Caputo is defined all be it is of equal value,
It can mutually convert.The case where being fractional order for order, in condition: function u (t) has m+1 rank continuous derivative, and m is extremely
N-1 is got less, then n=m-1, if function u (t) meets u at this time(k)(a)=0, k=0,1 ..., n-1, then three kinds of definition equivalences
, it can mutually convert, otherwise be non-equivalence.
The definition of Caputo type remains the property of integer rank calculus due to first carrying out integer rank differential calculation, right
It is 0 in the Fractional Derivative of constant, in the modeling process in face of many actual application problems, is widely used.
(3) adoption status space average method models two switch states in One Buck-Boost converter body, obtains
The fractional order mathematical model based on switching value of One Buck-Boost converter body are as follows:
D is switching variable, < i in formulaL>, < vo>, < Vin> is inductive current, input voltage and output voltage one
Average value in a switch periods, L are that the size unit of inductance inductance value is H, and C is that the size unit of capacitor's capacity is F, and R is electricity
Hinder the size of resistance value, unit Ω.
(2) fractional order Sliding Mode Controller designs
One Buck-Boost converter body is a kind of typical switch non-linearity system, is existed on the design of controller very big
Limitation, correlative study and stability analysis and proof especially for new fractional-order system are concentrated mainly on linear system
In the middle, less for fractional order nonlinear systematic research achievement, it is unfavorable for the further analysis of controller.
(1) it converts the fractional order mathematical model of One Buck-Boost converter body: enabling [x in formula (3)1,x2]T=[iL,vo]T, u=
D, then the fractional model of original Buck-Boost is transformed to canonical form shown in formula (4), and u represents the size of duty ratio, is
One with the function of time change, be the actual control variable of whole system:
In formula, X is state variable, X=[x1,x2]T=[iL,vo]T, i.e., inductive current and output voltage, y are feedback linearization
Output voltage after change, f (X) and g (X) are as follows:
Therefore, it on the fractional order mode standard of the One Buck-Boost converter body shown in formula (4), is fed back by fractional order
The mode of linearisation reconfigures the form of output function, is converted into inearized model shown in formula (7), is convenient for score
The design and realization of rank sliding mode controller.
Wherein v, z1And z2Expression formula are as follows:
Therefore, on can be by the system after fractional order feedback linearization, the design of control system be carried out, is acted on
On raw score rank nonlinear system, the control of whole nonlinear system is realized.
To fractional order Buck-Boost fractional model, after carrying out above-mentioned feedback linearization, formula (8) can be turned to
Shown in canonical form.The control target of Buck-Boost fractional order converter is output voltage track reference voltage vref, according to
The reference of the steady operation point of One Buck-Boost converter body, available inductive current exports iLrefWith duty ratio DrefReference
Output are as follows:
Thus, it is possible to by formula (11), to calculate the inductive current i when circuit reaches stable stateLrefSize and PWM account for
The size D of empty ratiorefSize, can reference output after fractional order feedback linearization to calculate are as follows:
(4) fractional order sliding formwork is designed using fractional calculus theory for the inearized model after feedback linearization
Controller, wherein fractional order sliding-mode surface s and control law v are as follows:
S=e2+k2D1-αe1+k1D-αe1 (14)
V=-k2D1-αe2-k1e1-λs-ksign(s) (15)
Wherein s is sliding-mode surface, k1With k2For the gain coefficient of system, λ and k are slide coefficient, guarantee that system mode can be fast
Speed reaches sliding-mode surface, and sign () is sign function, as s >=0, sign (s)=1, and as s < 0, sign (s)=- 1, e1,e2
For the first derivative of output error and output error, formula are as follows:
(3) fractional model ripple is analyzed
For Switching Power Supply, the size of output voltage ripple is to determine the important characteristic of the quality of Switching Power Supply,
It is also a kind of direct current variator foundation that component selects in the design process.Fractional order for integer rank because its
The presence of Memorability and hereditary capacity, to the output voltage ripple of One Buck-Boost converter body, there are large effects.
To the mathematical model of One Buck-Boost converter body, i.e. formula (3), in a switch device conductive, to inductance electricity
It flows and is solved with the changing value of output voltage, available:
Wherein, Δ iLFor inductive current changing value, Δ voFor output voltage changing value, Vin, L, R, C is respectively to input electricity
Pressure, inductance, resistance, capacitor value,For initial value of the output voltage within this period, ε is integration variable, Eα() is
Mittah-Leffler function, definition are as follows:
It can be seen from formula (17) and formula (18) changing value of output voltage and inductive current with fractional order order increase
It is big and reduces, therefore can be found that all integer models in analyze in the past, only to a kind of approximation of realistic model, with
Real system deviation is larger.
Emulation experiment verifies control method of the present invention:
In order to verify Buck-BoostThe correctness of converter fractional order mathematical model ripple analysis, in MatlaIt uses and divides on b
The numerical simulation that number rank tool box ninteger and Simulink carries out One Buck-Boost converter body takes defeated in simulating, verifying
Enter voltage vi=20V, reference output voltage vo=15V, inductance L=1mH, capacitor C=500μF, switching frequency f=100kHz are imitated
True effect is as shown in Figure 2:
Can intuitively it be found out by Fig. 2 (a), Fig. 2 (b), fractional order mathematical model, regardless of in output voltage voRipple also
It is in inductive current iLRipple on, all significantly greater than integer rank mathematical models.The Δ v of fractional order on the output voltageo≈
0.08V, the Δ v of integer ranko≈ 0.04V or so, the Δ i of fractional order on inductive currentL≈ 0.05A, the Δ i of integer rankL≈
0.03A；And it can also be seen that when fractional order order further reduces, such as Figure 21 (a) and Figure 21 (b), output electricity from Figure 21
The ripple of pressure and inductive current further increases, and demonstrates the correctness of analysis.
(1) in terms of the starting response of One Buck-Boost converter body, the analogous diagram of output voltage and inductive current such as Fig. 3
(a), Fig. 3 (b) and Fig. 4 (a), Fig. 4 (b) are shown, it can be seen that fractional order sliding mode controller is relative to integer rank sliding mode controller
For, it is more superior on starting performance.Fractional order sliding mode controller is in 7msLeft and right i.e. can reach scheduled output voltage and
There is no the overshoot of voltage；Integer rank sliding mode controller is 15msLeft and right can trace into reference output voltage, but there are 0.5V
The overshoot of left and right.In terms of inductive current, fractional order sliding mode controller 20msLeft and right reaches stable state, and convergence rate is very fast, but
It is the overshoot that can have 0.04A or so；And integer rank sliding mode controller is 25msLeft and right reaches with reference to inductive current, convergence rate
Relatively slow compared with for fractional order, there are the overshoot of 0.16A or so.In general, in starting performance, fractional order sliding formwork control
Heredity and Memorability of the device processed since fractional order is utilized, for integer rank in terms of convergence rate and overshoot more
It is outstanding.
(2) output voltage jump experiment, 2sWhen output reference voltage give a 5V step signal, reference output voltage
20V, the output voltage and inductive current waveform such as Fig. 5 of fractional order sliding mode controller and integer rank sliding mode controller are become from 15V
(a), Fig. 5 (b) and Fig. 6 (a), Fig. 6 (b) are shown.
As can be seen that fractional order sliding mode controller comes compared to integer rank sliding mode controller from such as Fig. 5 (a), Fig. 5 (b)
It says, convergence rate is more accelerated, and overshoot is smaller, and there are the overshoot of 2V or so for integer rank sliding mode controller.In voltage jump,
All there is a degree of decline in voltage, be the increase because of output voltage, and the duty ratio of pwm control signal increases, causes out
It closes element closing time to increase, the induction charging time is elongated, and discharge time shortens, therefore it is a degree of to cause voltage that can exist
Decline.
The situation of change of inductive current is corresponding with output voltage it can be seen from Fig. 6 (a), Fig. 6 (b), and fractional order is sliding
All there is biggish advantage in terms of convergence rate and overshoot for integer rank controller in mould controller.It is overall and
Speech, on inductive current, when output voltage, which exists, to be changed, what fractional order sliding mode controller can be faster more stable reaches new
Equilibrium state.
(3) when load resistance jumps, in 2s, load resistance gives the step signal of 5 Ω, and load resistance is become by 15 Ω
For 20 Ω, the output voltage and inductive current waveform such as Fig. 7 (a), Fig. 7 of fractional order sliding mode controller and integer rank sliding mode controller
(b), shown in Fig. 8 (a), Fig. 8 (b):
In the moment of load variation it can be seen from Fig. 7 (a), Fig. 7 (b), integer rank is defeated with fractional order sliding mode controller
Voltage can all have the upward jump of 3V or so out, but fractional order sliding mode controller will reach stable state in 4ms or so again,
Resume speed is very fast, and integer rank sliding mode controller convergence rate is slower.
It can be seen from Fig. 8 (a), Fig. 8 (b) for inductive current, become known to formula (11) in load resistance
When change, the stable state of inductive current can change.For new stable state, fractional order sliding mode controller can be arrived quickly
It reaches, and is better than integer rank sliding mode controller in terms of convergence rate and overshoot.
In conclusion for can be seen that fractional order sliding mode controller compared with integer rank sliding mode controller by emulation experiment,
The heredity and memory characteristic that fractional order has are combined, sliding formwork control is carried out on the fractional model of One Buck-Boost converter body
The design of device processed is more in line with the actual characteristic of converter, when facing external interference, has better robustness, globality
It can be more superior.
Control method of the present invention is verified in hardware circuit programming:
(4) discretization and DSP programming
In actual computer control system, sampled signal and control signal are all discrete forms, therefore how
It is the key that solution Fractional Differential Equation and fractional order control device are realized by fractional order differential operator discretization.The micro- product of fractional order
Point introducing greatly extend the freedom degree and flexibility of controller design as the extension of integer rank calculus, but because
The complexity of itself is higher compared with integer rank in the implementation complexity of actual discrete.Herein by the way of Tustin+CFE
Discretization, form are carried out to designed fractional order sliding-mode surface and control law are as follows:
In formula, order α is between 0~1, PpAnd QqRespectively item number is p, the polynomial item number of q, the bigger fitting of item number
Effect is better, but calculates more complicated.In l-G simulation test, order α=0.9 is taken, takes interception item number p=q=5, sampling time T
=0.1s, using the tool box ninteger, available s0.9Discrete form are as follows:
In actual system design, for the kernel control chip used for TMS320F28335, the processing speed having is fast, locates
Reason ability is strong, quick floating-point operation and high-precision A/D conversion etc., is suitable for quick signal processing and complex control is calculated
The realization of method, development cycle and advantage of lower cost are highly suitable as the control core and current power of DC converter
The main trend of design.
(1) hardware circuit of Buck-Boost DC converter mainly completes detection and the dsp control signal of voltage and current
The function of response, in terms of following six can be divided into: main circuit, voltage detecting circuit, current detection circuit, PWM drive circuit,
Auxiliary power circuit.Each functional module is analyzed and designed separately below.
(2) main circuit design
Main circuit part shows using four switch Buck-Boost circuit topological structures, Altium Designer emulation
It is intended to as shown in figure 9, only including a main power inductance in circuit, there are four switching devices for tool on two bridge arms, can pass through
The switching sequence for controlling four switching devices realizes three kinds of buck converter, booster converter and buck-boost converter structures, spirit
It is living convenient, convenient for the application in actual switch power supply.
Four switch Buck-Boost converter topologys are compared with traditional Buck-Boost circuit, control mode and state side
Journey is essentially identical, and maximum difference is that the output voltage polarity of traditional Buck-Boost circuit with input voltage is opposite, and
Four switch topologies change the flow direction of inductive current by increasing switch, keep output voltage polarity identical as input voltage,
It is more flexible and convenient.
Relative to schematic diagram shown in Fig. 9, practical main circuit all joined greatly in input terminal and output end as shown in figure 13
Capacitor is measured, the radio-frequency component of input voltage and output voltage can be filtered, the size of input and output voltage ripple is reduced.
Current detection circuit shown in Figure 11, as shown in Fig. 4 (a), Fig. 4 (b), concatenation after the inductor, is detected by Hall effect
The size of inductive current；Voltage detecting circuit shown in Fig. 10 is also attempted by input voltage and output voltage terminal, input defeated
The detection of voltage swing out；Its PWM output end is also as shown above coupled with four by PWM drive module shown in Figure 12
On switching tube, its switch state is controlled respectively, has reached the purpose of output voltage control.
(3) voltage detecting circuit
In actual circuit, all there is fluctuation in the value moment of input voltage and output voltage, it is therefore desirable to voltage value
Moment is detected.Voltage detecting circuit is mainly the differential amplifier circuit of core composition using TLV2374, such as Figure 10 institute
Show.TLV2374 is single supply operational amplifier, and the bandwidth with 3MHz, high conversion efficiency, operating voltage range is wide, low in energy consumption,
It is encapsulated using small-sized SOT-23, it is small in size, it is suitable in the detection circuit of voltage high frequency variation.
The TLV2374 differential amplifier circuit constituted is connected in parallel on output voltage or input voltage, it can be to voltage
Signal is detected in real time.By differential amplifier circuit, the anti-interference ability of the circuit of detection can be improved, it is effective to inhibit
The interference of common-mode signal in circuit while amplifying difference mode signal, reduces the amplitude range of detection voltage, makes voltage most
Amplitude narrows down within the voltage range of the input port A/D of DSP, convenient for the use of subsequent A/D conversion.Input voltage with
The ratio column of output voltage can be adjusted by the resistance in circuit, calculation formula are as follows:
R is taken in voltage detecting circuit12=R25=1K Ω, R13=R22=10K Ω, can be calculated by formula,
It is the 1/11 of input voltage by the output voltage after difference amplifier.The port input voltage not more than 3V of DSP, then
It can learn that the maximum amplitude of input voltage and output voltage no more than 33V, otherwise will cause the input voltage mistake of the port DSP
Height leads to the damage of device.The filter capacitor C connect in circuit in voltage output end16, using capacitor itself have logical high frequency,
The characteristic for hindering low frequency can filter the High-frequency Interference in output voltage, guarantee the accuracy of detection voltage.
(4) current detection circuit
In actual circuit, DSP does not directly input the port of detection to current signal, therefore is all by will be electric
Stream signal is converted to voltage signal, is converted later by the A/D to voltage signal, Conversion and Utilization are carried out inside processor.
The method that common current signal is converted to voltage signal has: the small resistance of series connection high-precision, parallel connection RC detection circuit and Hall pass
Sensor etc..The small volume that series resistance converts current signal, precision is higher, but be easy to ground wire generate interference and
Temperature drift is larger；RC detection circuit in parallel is identical as the mode of resistance, is easy to generate larger interference to circuit；Hall sensor uses
Electromagnetic induction phenomenon is use up row conversion to electric current by different turn ratios, is not directly affected to integrated circuit, measurement result
Precision and the linearity it is all higher.In the production of actual circuit, selection is ACS758 Hall current detection chip, work
It is as shown in figure 11 to make schematic diagram.
ACS758 is the current sensing device being made of the linear hall sensor of accurate, low offset, is had
The high frequency bandwidth of 120KHz, the response time is less than 4us, more sensitive for high frequency variable signal.Its internal included voltage
The inductive current of input can be converted to voltage output, input and output calculation formula by operational amplifier circuit are as follows:
Vo=0.5*Vref+0.04*IL (26)
Wherein reference voltage VrefIt is 3.3V for the accessory power supply output valve of circuit.The circuit it can be seen from formula (26)
The middle every variation 1A of inductive current, output voltage change 0.04V, can pass through after DSP carries out A/D conversion to input voltage
Formula calculates actual inductor current value, in the calculating of control law.At the output end of ACS758 and the end A/D of DSP
Mouthful between, access a voltage follower being made of OP07 amplifier device, as between front stage buffering be isolated, mentioning
While the output impedance of high ACS758, the fluctuation of output voltage is further suppressed, makes not generating mutual shadow between front stage
It rings, guarantees the accuracy of measurement voltage.
(5) PWM drive circuit
PWM mainly has frequency conversion and fixed two kinds of frequency to the driving of metal-oxide-semiconductor, herein in programming by the way of determining frequency,
That is the frequency of pwm signal is constant, realizes the adjusting of metal-oxide-semiconductor switch time by changing the size of duty ratio.Passing through voltage
After detection circuit and current detection circuit measure electric current and voltage in Buck-Boost circuit, DSP is carried out by the port A/D
The reading of voltage carries out the calculating and reduction of data in a program, generates one by the calculating of fractional order sliding mode controller later
A new duty ratio realizes the real-time adjusting to the switch time of metal-oxide-semiconductor.The load capacity of the pwm signal output of the port DSP
It is low, it cannot be directly used to the driving of metal-oxide-semiconductor, therefore, it is necessary to pass through dedicated metal-oxide-semiconductor driving chip, to improve output signal
Voltage and electric current.Herein using UCC27211 driving chip, to carry out the driving of switching tube, circuit diagram is as shown in figure 12.
(6) auxiliary power circuit
Principal voltage is converted to boost voltage: input voltage is converted to circuit supply voltage circuit:, will by xl7035 module
The voltage of input is converted to 12V voltage output, and maximum output current 1A is that PWM drive module and other fixed voltages turn
Mold changing block is powered, as shown in Figure 14 (a).
Digital power voltage；Operational amplifier circuit is mostly important a part in measure voltage & current, the stabilization of amplifier
Property also directly determines the accuracy of collecting sample, it is therefore desirable to stable supply voltage is provided for amplifier module.Digital power
Part is then that the 12V voltage that Fig. 1 circuit is exported is converted to stable 3.3V output voltage, is powered for amplifier module,
As shown in Figure 14 (b).
Reference voltage conversion circuit: being the reference voltage that the input voltage of 3.3V is converted to 1.65V, mentions for amplifier part
For reference voltage, can be used to provide a benchmark during current detecting, as shown in Figure 14 (c).
The voltage conversion portion of the above three parts, converts input voltage into 12V, 3.3V, 1.65V, is all using solid
Constant voltage output circuit, voltage adjustability is poor, but the high stability of voltage output, is that voltage detecting and electric current are examined
Slowdown monitoring circuit provides stable power supply environment, guarantees the accuracy of detection circuit output.
(5) software design
Specifically include that ADC sampling routine designs in whole controlling software design, PWM programming, fractional order sliding formwork control
Algorithm routine design, DMA program and interrupt routine design processed etc..
The design of main program is mainly the initialization of the configuration of realization system relevant environment, variable initializer show and interruption
Deng the flow chart of global procedures is as shown in figure 15:
Step 1: stability and continuity in order to guarantee whole software program, sampling, the fractional order sliding formwork control of voltage value
The calculating of algorithm processed and the update of PWM parameter, are all placed in Interruption 1 and carry out, and the circulation of main program is avoided to wait, waste
System time.Interruption is realized using the timer module in TMS320F28335, and timer module is one
Counter built in TMS320F28335 is counted according to the size of dominant frequency.The time that timer is arranged is 10ms, i.e. 10ms
Triggering is primary to be interrupted, and carries out single treatment to all data.
Step 2: after Interruption triggering, judging whether DMA is idle.One of TMS320F28335 is special when DMA
Communication mode, the i.e. reading of data can directly be read in another register from a register, be participated in without CPU, can
To save system time.In the design can directly the data after the conversion in A/D register, directly read in memory into
The processing of row data.
Step 3:ADC sampled-data processing and the voltage value for acquiring Figure 10, Figure 11 circuit, pass through TMS320F28335 core
Included 12 analog/digital conversion modules, are converted to digital signal, are read in memory, pass through formula (25) and formula inside piece
(26) it is calculated, obtains inductor current value in actual circuit, input and output voltage values.
Step 4: discrete sliding mode control, which calculates, mainly converts inductor current value collected, output voltage values using A/D
And input voltage value, the size of the duty ratio of PWM in switch controlled is calculated using formula (17) and (18).
Step 5: duty ratio clipping is then to determine duty cycle limit in a certain range, to prevent output voltage excessive.By formula
(22) it is known that in voltage collection circuit, the port input voltage that maximum partial pressure is 11, TMS320F28335 is up to
3V, therefore the maximum voltage exported is 33V, can calculate to obtain maximum duty ratio D by formula (11)ref=0.8, therefore calculate institute
The duty ratio obtained is no more than 0.8, if being limited in 0.8 or less more than 0.8.
The update of step 6:PWM duty ratio is then, after calculating duty ratio size, to the corresponding positions of PWM register into
Row assignment makes the output PWM duty cycle in next cycle reach the resulting expectation of calculating, to adjust to output voltage
Section.
In programming, most important content is the calculating of fractional order sliding formwork control ratio.In hardware sample circuit
On, by voltage sampling circuit and current sampling circuit, can obtain the input voltage of Buck-Boost circuit, output voltage and
Inductive current can use its calculating for carrying out control law after software filtering.Pass through in (four) part
The mode of Tustin+CFE has carried out discretization to fractional order sliding mode controller, therefore also more square to the software realization of controller
Just, Fractional Control Algorithm flow chart is as shown in figure 16.
Step 1: timer interruption waits, and identical as global procedures flow chart, a data are uniformly processed in 10ms, ensure that
The real-time and stability of control system.
The acquisition of step 2:A/D value is exactly to read in main program, the practical electricity after being calculated by formula (25) and formula (26)
Electrification flow valuve, input and output voltage values, are ready for the calculating of control law.
Step 3: the calculating of error is exactly resulting in current voltage current sampling data according to formula (8) and formula (9) calculating
Under, z1,z2Size and formula (12), formula (13) resulting z1ref,z2refSize error and the e in formula (16)1,e2, it is used for
The calculating of next step control law.
Step 4: the calculating of control law is that previous step is calculated resulting e1,e2, it substitutes into formula (14), it is sliding shown in (15)
In moding structure controller, to calculate the numerical values recited of control law and duty ratio, in the control of PWM.
Step 5: the update of duty ratio is then to calculate resulting duty ratio using previous step, is compared to calculate in PWM register
Compared with the required updated value of register.
The flow chart of fractional order sliding mode controller detailed algorithm is as shown in figure 17, according to the output voltage of acquisition, input electricity
Pressure, inductor current value calculate z1,z2Value, calculate error amount and calculate actual control law.Voltage and current in program flow diagram
Reading it is identical with above flow chart, the voltage and current value after conversion is substituted into formula (8) and formula (9), calculating fractional order
Z after linear feedback linearisation1,z2Value, substitute into (14) later, in (15), calculate sliding-mode surface S and control law V；Duty ratio
Legal range is then the calculating of the comprehensive maximum output voltage analyzed above, need to guarantee duty ratio between 0.3-0.8,
If calculating dutyfactor value not within the scope of this, needs to constrain it, guarantee the safety of integrated circuit.
Hardware simulation results:
Hereinbefore the hardware circuit of One Buck-Boost converter body and software realization are analyzed, material object can be carried out
Production and experiment.Hardware components mainly include Buck-Boost main circuit, voltage detecting circuit, current detection circuit, PWM drive
Dynamic circuit, auxiliary power circuit and filter circuit etc..The integrated circuit figure of One Buck-Boost converter body is as shown in figure 18:
In this experiment, Buck-Boost circuit works under CCM (continuous current mode) mode, therefore need to guarantee inductance electricity
Stream is greater than in value perseverance, therefore the parameter in circuit need to meet by formula (17) and formula (18), guarantees that inductive current changing value is small
In the maximum value of inductance inductive current.Therefore, in physical varification, output voltage is set as 23V, input voltage 12V is loaded
R=15 Ω, switching frequency f=100kHz, inductance and capacitor be L=1mH, C=100 μ F, with oscillograph observation output voltage with
Inductive current waveform, output voltage and control PWM wave are shaped like Figure 19, shown in Figure 20.
As seen from Figure 19, the output of output voltage and inductive current is all more steady, and the ripple of output voltage is smaller.
For inductive current after Hall sensor and voltage follower, the average voltage level of output is 1.76V, be can be calculated by formula 23
Actual current value is 2.75A, smaller with calculated value error.As seen from Figure 20, inductive current and output voltage are in PWM
When low and high level switches, i.e., there are larger fluctuation when metal-oxide-semiconductor switch state converts, main cause is presence and the electricity of dead time
The change of sense and capacitor charge and discharge state.In general, the overall performance of output voltage is preferable, produces a desired effect.
Claims (2)
1. a kind of fractional order sliding-mode control of buck-boost converter, which is characterized in that including following content:
(1) foundation of fractional order mathematical model:
(1) buck-boost converter mid-score rank mathematical model is established: where buck-boost converter mid-score rank device is inductance L
With capacitor C, mathematical model are as follows:
Wherein, iLFor inductive current, vLFor inductive drop, iCFor capacitance current, vCFor capacitance voltage,For fractional calculus
Operator, wherein for order α between 0~1, a is lower limit of integral, and t is upper limit of integral, and L is that the size unit of inductance inductance value is H, and C is
The size unit of capacitor's capacity is F；
(2) operation is carried out using Caputo definition, calculates fractional calculus operatorThe expression formula that Caputo is defined are as follows:
Wherein, a is lower limit of integral, and t is upper limit of integral, and r is fractional order order, and u (t) is function to be solved, and n is the close of fractional order
Like order, it is greater than the smallest positive integral of fractional order, ε is integration variable, and Γ () is Gamma function；
(3) adoption status space average method models two switch states in buck-boost converter, obtains buck transformation
The fractional order mathematical model based on switching value of device are as follows:
Wherein, d is switching variable, < iL>, < vo>, < Vin> is respectively inductive current, input voltage and output voltage one
Average value in a switch periods, L are that the size unit of inductance inductance value is H, and C is that the size unit of capacitor's capacity is F, and R is electricity
Hinder the size of resistance value, unit Ω；
(2) fractional order Sliding Mode Controller designs:
(1) it converts the fractional order mathematical model of buck-boost converter: enabling [x in formula (3)1,x2]T=[iL,vo]T, u=d, then original rises
The fractional model of buck converter is transformed to canonical form shown in formula (4), and u represents the size of duty ratio, be one with
The function of time change, be the actual control variable of whole system:
Wherein, X is state variable, X=[x1,x2]T=[iL,vo]T, y is output voltage, and f (X) and g (X) are as follows:
(2) the fractional order mathematical model of transformation buck-boost converter is converted, further convenient for the design of fractional order sliding mode controller
With realization: on the fractional order mode standard of the buck-boost converter shown in formula (4), passing through the side of fractional order feedback linearization
Formula reconfigures the form of output function, is converted into inearized model shown in formula (7):
Wherein, v, z1And z2Expression formula are as follows:
(3) the control target of buck-boost converter fractional order converter is output voltage track reference voltage vref, according to buck
The steady operation point of converter obtains the reference output i of inductive currentLrefWith duty ratio DrefReference output are as follows:
By formula (11), the inductive current i when circuit reaches stable state is calculatedLrefSize and PWM duty cycle size Dref's
Size, and then calculate the reference output after fractional order feedback linearization are as follows:
(4) fractional order sliding formwork control is designed using fractional calculus theory for the inearized model after feedback linearization
Device, wherein fractional order sliding-mode surface s and control law v are as follows:
Wherein, k1With k2For the gain coefficient of system, λ and k are slide coefficient, and sign () is sign function, as s >=0, sign
(s)=1, as s < 0, sign (s)=- 1, e1,e2For the first derivative of output error and output error, formula are as follows:
Wherein, Z1And Z2It is defined respectively by formula (8) and formula (9), Z1refAnd Z2refRespectively by formula (12) and formula (13)
Definition.
2. the fractional order sliding-mode control of a kind of buck-boost converter according to claim 1, which is characterized in that lifting
The mathematical model of buckling parallel operation, i.e. formula (3) carry out the analysis of fractional model ripple:
Variation to the mathematical model of buck-boost converter, in a switch device conductive, to inductive current and output voltage
Value is solved, available:
Wherein, Δ iLFor inductive current changing value, Δ voFor output voltage changing value, Vin, L, R, C is respectively input voltage, electricity
Sense, resistance, capacitor value,For initial value of the output voltage within this period, ε is integration variable, Eα() is Mittah-
Leffler function, definition are as follows:
It is obtained by formula (17) and formula (18), the changing value of output voltage and inductive current subtracts as the order of fractional order increases
It is small, therefore all integer models in previous analysis are deposited compared with real system only to a kind of approximation of realistic model
In deviation, it was demonstrated that it is accurate for creating buck-boost converter mathematical model with fractional order.
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