CN109245532B - Fractional order sliding mode control method of buck-boost converter - Google Patents

Fractional order sliding mode control method of buck-boost converter Download PDF

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CN109245532B
CN109245532B CN201811146111.8A CN201811146111A CN109245532B CN 109245532 B CN109245532 B CN 109245532B CN 201811146111 A CN201811146111 A CN 201811146111A CN 109245532 B CN109245532 B CN 109245532B
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fractional order
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boost converter
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CN109245532A (en
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李志鹏
郑艳
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Northeastern University China
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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 invention provides a fractional order sliding mode control method of a buck-boost converter, which comprises the following steps: establishing a fractional order mathematical model; designing a fractional order sliding mode variable structure controller; performing ripple analysis based on the fractional order model; the fractional order mathematical model is established for the buck-boost converter, so that the buck-boost converter is more consistent with an actual physical system, and an integral order model is verified to be an approximation of the actual system through analytical calculation and simulation experiments, and the fractional order model can better reflect the internal characteristics of the actual physical system due to the memorability and the genetic characteristics of the fractional order model; a fractional order sliding mode controller is designed, compared with an integer order sliding mode controller, the robustness of the fractional order sliding mode controller is greatly improved, the output characteristic of a system is further improved, and the anti-interference capability of the system is enhanced; as can be seen from simulation experiment results and hardware circuit simulation results, the output voltage of the fractional order sliding mode controller is stable, and the necessity and the reasonability of the fractional order controller are further explained.

Description

Fractional order sliding mode control method of buck-boost converter
Technical Field
The invention belongs to the field of control over a buck-boost converter in a direct-current converter, and particularly relates to a fractional order sliding mode control method for the buck-boost converter.
Background
The DC/DC converter is one of indispensable parts in communication equipment and civil service, and as science and technology develops, the DC/DC converter is applied to more and more equipment and has stricter requirements. The Buck-Boost converter is used as a common DC/DC converter, can convert a fixed direct current voltage into a variable direct current voltage, is flexible and convenient, and can realize controllable regulation of output voltage. The Buck-Boost converter has the advantages of few electronic devices, simple structure, low cost, high energy utilization rate and the like, becomes main equipment for electric energy conversion and control, and is widely applied to the fields of data communication, office automation equipment, robots, military and aerospace and the like. With the rapid development of the industry in China, stricter requirements are also put forward on the conversion efficiency, the output precision and the robustness of electric energy.
The Buck-Boost converter is a typical time-varying and nonlinear system, the control of the Buck-Boost converter is nonlinear and discontinuous, the Buck-Boost converter is sensitive to system parameter variation and load jump, and when the load is changed greatly, the Buck-Boost converter has the defects of slow dynamic response, output waveform distortion and the like. In recent years, with the development of nonlinear control theories such as neural network control, fuzzy control, variable structure control, chaotic control and the like, more and more experts apply the nonlinear control theory to the DC/DC converter and obtain better effect. The sliding mode variable structure control is taken as a nonlinear control theory, has good robustness to parameter change and external interference, and has a good effect in the control of a DC/DC converter.
The popularization of fractional calculus as integer calculus is firstly proposed by Lebranizer in 1695, the order of the calculus is not limited to integers any more, and the design and application of the controller are greatly expanded. With the gradual and deep research, more and more scholars apply the fractional calculus theory and method to various fields of natural science and social science. In the application of the automatic control field, Oustaloup creates CRONE control, applies the theory to a plurality of practical problems, obtains good effect, and proves that the CRONE controller has more advantages than the traditional PID controller. In power electronics, an inductor and a capacitor are actually fractional orders rather than integer orders, so that a direct current converter is actually a fractional order system, and in an existing control mode, the direct current converter is approximated to integer orders to design and analyze a controller, and the obtained control effect is not in accordance with the actual control effect and has a large difference. Fractional calculus theory is an important mathematical tool, and the memory and genetic characteristics of the fractional calculus theory can be fully applied to the DC/DC converter. Therefore, the design utilizes the fractional calculus theory, combines the fractional sliding mode variable structure control theory on the structure of the Buck-Boost converter, adopts the fractional approximation law to design the fractional sliding mode surface, improves the output response and the dynamic characteristic of the system, and applies the fractional sliding mode surface to an actual circuit in a fractional discretization mode.
Disclosure of Invention
Aiming at the technical problems, the fractional order sliding mode control method of the Buck-Boost converter is provided, a mathematical model is established on the Buck-Boost converter by adopting a fractional order calculus theory, the design of a controller and the verification of a simulation experiment provide more accurate control for the Buck-Boost converter, namely a Buck-Boost converter, the output characteristic and robustness of the output voltage of the direct current converter are further improved, meanwhile, the buffeting in the sliding mode variable structure control is weakened, and the overall performance of a system is improved.
The method comprises the following steps:
fractional order mathematical model establishment
Based on the circuit principle of Buck-Boost converters, i.e. Buck-Boost converters, ViThe output voltage V can be adjusted by adjusting the on-off time of ToD is a freewheeling diode to maintain the current flow in the circuit unchanged, L and C are inductors and capacitors, which are the main energy storage elements in the circuit, when the switching device T is closed, the power supply transfers energy to the inductor, the energy storage of the electrical energy on the inductor increases, when the switching device is turned off, the inductor transfers energy to the capacitor and the load, and the energy storage on the inductor decreases.
(1) Establishing a fractional order mathematical model in the Buck-Boost converter, wherein the fractional order devices in the Buck-Boost converter are an inductor L and a capacitor C, and the mathematical model is as follows:
Figure GDA0002438604050000021
wherein iLIs an inductive current, vLIs the inductor voltage iCIs a capacitance current, vCIs the capacitor voltage, DαThe method is a fractional calculus operator, wherein the order α is between 0 and 1, a is an integration lower limit, t is an integration upper limit, L is the magnitude of an inductance value and is represented by H, and C is the magnitude of a capacitance value and is represented by F.
In the theory of fractional calculus, there are three main definitions, which are Grunwald-L etnikov (G-L) fractional calculus, Riemann-L iouville (R-L) fractional calculus and Caputo fractional calculus.
The definition of the fractional calculus of Grunwald-L etnikov is generalized from the definition of the integer order, and the order of the integer order is generalized to the fractional order, and the definition formula is as follows:
Figure GDA0002438604050000022
where m is a positive integer and α. ltoreq. m.ltoreq. α +1, a is the lower limit of the integral and is a Gamma function defined by the form:
Figure GDA0002438604050000031
wherein m is a constant and re (m) > 0.
Riemann-L iouville fractional calculus is based on the improvement of G-L fractional calculus based on the property that fractional calculus should satisfy, which is defined as:
Figure GDA0002438604050000032
wherein n is a positive integer and n is equal to or greater than α and equal to or less than α + 1. R-L type fractional calculus definition can be regarded as that the function u (t) is firstly subjected to fractional calculus and then subjected to integer differentiation, and the form of the fractional calculus definition is simpler compared with G-L type definition, can simplify the calculation process of the fractional calculus and is widely applied in practice.
(2) Calculating fractional calculus operator by using Caputo definitionaDt αThe expression defined by Caputo is:
Figure GDA0002438604050000033
wherein a is an integration lower limit, t is an integration upper limit, r is a fractional order, u (t) is a function to be solved, and n is an approximate order of the fractional order, is a minimum integer larger than the fractional order, and is an integration variable.
The G-L type definition firstly generalizes the integral order calculus definition into the fractional order through a limit form, gives the definition of the fractional order calculus theory, but the expression is too complex to be applied to popularization in practice, and the R-L type definition and the Caputo type definition both take the G-L type definition as the basis, are improved and expanded, simplify the calculation process of the fractional order calculus and are convenient to apply in practice.
For the case of order fractional order, under the condition that the function u (t) has a continuous derivative of order m +1, and m takes at least n-1, then n-m-1, when the function u (t) satisfies u (t)(k)(a) 0, k-0, 1, …, n-1, the three definitions are equivalent and can be interchanged, otherwise they are not equivalent.
The Caputo type definition retains the property of integral order calculus because integer order differential calculation is carried out firstly, and the fractional order derivative of a constant is 0, so that the Caputo type definition is widely applied to the modeling process facing many practical application problems.
(3) Modeling two switch states in the Buck-Boost converter by adopting a state space average method to obtain a fractional order mathematical model of the Buck-Boost converter based on switching value, wherein the fractional order mathematical model comprises the following steps:
Figure GDA0002438604050000041
in which d is a switching variable, iLIs an inductive current, voTo output a voltage, VinIn order to input the voltage, the voltage is,<iL>,<vo>,<Vin>the average values of the inductive current, the output voltage and the input voltage in a switching period are respectively, and R is the resistance value of the resistor and has the unit of omega.
Design of (II) fractional order sliding mode variable structure controller
The Buck-Boost converter is a typical switching nonlinear system, and has great limitation on the design of a controller, particularly, relevant research and stability analysis and verification on a fractional order system are mainly concentrated in the linear system, and research results on the fractional order nonlinear system are less, so that further analysis of the controller is not facilitated.
(1) And transforming a fractional order mathematical model of the Buck-Boost converter: in the formula (3) [ x ]1,x2]T＝[iL,vo]TAnd if u is d, the original fractional order model of the Buck-Boost is converted into a standard form shown in an expression (4), wherein u represents the size of the duty ratio, is a function changing along with time and is an actual control variable of the whole system:
Figure GDA0002438604050000042
wherein X is a state variable, and X ═ X1,x2]T＝[iL,vo]TY is the output voltage, f (X) and g (X) are shown below:
Figure GDA0002438604050000043
Figure GDA0002438604050000044
(2) and further converting a fractional order mathematical model of the Buck-Boost converter, so that the design and realization of a fractional order sliding mode controller are facilitated: reconstructing the form of an output function on the fractional order standard mode of the Buck-Boost converter shown in the formula (4) in a fractional order feedback linearization mode, and converting the output function into a linearization model shown in the formula (7);
Figure GDA0002438604050000045
wherein v, z1And z2The expression of (a) is:
Figure GDA0002438604050000046
Figure GDA0002438604050000047
Figure GDA0002438604050000051
therefore, the control system can be designed on the system after fractional order feedback linearization, and the control system acts on the original fractional order nonlinear system to realize the control of the whole nonlinear system.
(3) The control target of the Buck-Boost fractional order converter is that an output voltage tracks a reference voltage vrefAccording to the steady-state working point of the Buck-Boost converter, the reference output i of the inductive current can be obtainedLrefAnd reference output D of duty cyclerefComprises the following steps:
Figure GDA0002438604050000052
thus, the inductor current i when the circuit reaches steady state can be calculated by equation (11)LrefReference output D of magnitude and duty ratio ofrefTo calculate the reference output after fractional order feedback linearization as:
Figure GDA0002438604050000053
Figure GDA0002438604050000054
(4) aiming at a linearized model after feedback linearization, a fractional order calculus theory is adopted to design a fractional order sliding mode controller, wherein a fractional order sliding mode surface s and a control law vkComprises the following steps:
Figure GDA0002438604050000055
Figure GDA0002438604050000056
wherein s is a slip form surface, k1And k is2And (3) for a gain coefficient of the system, wherein lambda and k are sliding coefficients to ensure that the state of the system can quickly reach a sliding mode surface, sign is a sign function, when s is more than or equal to 0, sign(s) is 1, and when s is more than or equal to 0<At 0, sign(s)＝-1，e1,e2The first derivative of the output error and the output error is expressed as:
Figure GDA0002438604050000057
(III) ripple analysis of a fractional order model:
for the switching power supply, the magnitude of the output voltage ripple is an important characteristic determining the quality of the switching power supply, and is also a basis for component selection in the design process of the dc converter. Fractional order has a larger influence on the output voltage ripple of the Buck-Boost converter than the integer order due to the memory and genetic characteristics of the fractional order.
When a switching device is turned on, a mathematical model of the Buck-Boost converter, namely formula (3), is solved for a change value of the inductive current and the output voltage, so that the following can be obtained:
Figure GDA0002438604050000061
Figure GDA0002438604050000062
wherein, Δ iLIs the value of change in the inductance current, Δ voIn order to output the voltage variation value,
Figure GDA0002438604050000063
to an initial value of the output voltage in this period, EαIs a Mittah-L effler function, which is defined by the formula:
Figure GDA0002438604050000064
as can be seen from equations (17) and (18), the variation values of the output voltage and the inductor current decrease as the order of the fractional order increases, and therefore, it can be found that all the integral order models in the previous analysis are only an approximation to the actual model, and have a large deviation from the actual system.
The beneficial technical effects are as follows:
the invention establishes a fractional order mathematical model for the Buck-Boost converter based on a fractional order calculus theory, so that the Buck-Boost converter is more consistent with an actual physical system, verifies that the integer order model is an approximation of the actual system through analytical calculation and simulation experiments, and can reflect the internal characteristics of the actual physical system due to the memory and genetic characteristics of the fractional order model.
And then, a fractional order sliding mode controller is designed based on a fractional order calculus theory, and compared with an integer order sliding mode controller, the robustness of the fractional order sliding mode controller is greatly improved. In a simulation experiment, simulation verification is carried out on three conditions of starting response, output voltage jump and load resistance jump of the Buck-Boost converter by utilizing a Matlab/Simulink and ninteger fractional order toolbox. Through the analysis of the output waveform, the fractional order sliding mode controller can be obviously seen to further improve the output characteristic of the system and enhance the anti-interference capability of the system.
And finally, discretizing the fractional order sliding mode controller by adopting a Tustin + CFE mode, and designing and manufacturing a real object circuit and writing a software program. The TMS320F28335 is used as a core controller, the four-switch topological circuit is used as a basic circuit, and the Buck-Boost converter is verified in actual control effect. The experimental result shows that the fractional order sliding mode controller has stable output voltage and strong anti-interference capability, and further explains the necessity and rationality of the fractional order controller.
Drawings
FIG. 1 is an overall functional block diagram of an embodiment of the present invention;
fig. 2(a) is a simulation comparison diagram of output voltage ripple of the Buck-Boost converter according to the embodiment of the invention;
fig. 2(b) is a simulation comparison diagram of the inductance current ripple of the Buck-Boost converter according to the embodiment of the invention;
FIG. 3(a) is a simulation diagram of an output voltage in a start response of the fractional order sliding mode variable structure controller according to the embodiment of the present invention;
FIG. 3(b) is a simulation diagram of the output voltage in the start response of the integer-order sliding mode controller according to the embodiment of the present invention;
FIG. 4(a) is a simulation diagram of the inductor current in the start response of the fractional order sliding mode variable structure controller according to the embodiment of the present invention;
FIG. 4(b) is a simulation diagram of the inductor current in the start response of the integer-order sliding mode controller according to the embodiment of the present invention;
fig. 5(a) is a simulation diagram of an output voltage when the fractional order sliding mode variable structure controls the output voltage to jump according to the embodiment of the present invention;
FIG. 5(b) is a simulation diagram of the output voltage when the output voltage of the integer-order sliding mode controller jumps according to the embodiment of the present invention;
FIG. 6(a) is a simulation diagram of the response of the inductor current when the fractional order sliding mode variable structure controls the output voltage to jump according to the embodiment of the present invention
Fig. 6(b) is a simulation diagram of the response of the inductor current when the output voltage of the integer-order sliding mode controller jumps according to the embodiment of the present invention;
fig. 7(a) is a simulation diagram of an output voltage response when a fractional order sliding mode variable structure controls a load resistor to jump according to an embodiment of the present invention;
fig. 7(b) is a simulation diagram of the output voltage response when the load resistance of the integer-order sliding mode controller jumps according to the embodiment of the present invention;
fig. 8(a) is a simulation diagram of the response of the inductor current when the jump of the load resistor is controlled by the fractional order sliding mode variable structure according to the embodiment of the present invention;
fig. 8(b) is a simulation diagram of the response of the inductor current when the load resistance of the integer-order sliding mode controller jumps according to the embodiment of the present invention;
FIG. 9 is a circuit topology diagram of a hardware verification Buck-Boost converter according to an embodiment of the invention;
FIG. 10 is a circuit diagram of a voltage detection circuit according to an embodiment of the present invention;
FIG. 11 is a circuit diagram of a current detection circuit according to an embodiment of the present invention;
FIG. 12 is a circuit diagram of a PWM driving circuit according to an embodiment of the present invention;
FIG. 13 is a diagram of a practical main circuit of an embodiment of the present invention;
FIG. 14(a) shows the main voltage of the auxiliary power circuit being converted into the auxiliary voltage according to the embodiment of the present invention;
FIG. 14(b) is a digital power supply voltage of the auxiliary power supply circuit according to the embodiment of the present invention;
FIG. 14(c) shows a reference voltage converting circuit of the auxiliary power circuit according to the embodiment of the present invention;
FIG. 15 is a flowchart of a main process of an embodiment of the present invention;
FIG. 16 is a flowchart of a fractional order control algorithm according to an embodiment of the present invention;
FIG. 17 is a flow chart of a control algorithm according to an embodiment of the present invention;
fig. 18 is an overall circuit diagram of a Buck-Boost converter according to an embodiment of the present invention;
FIG. 19 is a waveform diagram of the output voltage and the inductor current of the hardware verification oscilloscope according to the embodiment of the present invention;
FIG. 20 is a diagram of a hardware verification output voltage waveform and a PWM waveform according to an embodiment of the present invention;
fig. 21(a) is a simulation waveform of the Buck-Boost converter when α is 0.8 according to the embodiment of the present invention;
fig. 21(b) is a simulation waveform of an α 1Buck-Boost converter according to an embodiment of the present invention;
fig. 22 is a circuit topology diagram of a Buck-Boost converter according to an embodiment of the present invention.
Detailed Description
The invention will be further described with reference to the accompanying drawings and specific embodiments, and a fractional order sliding mode control method of a buck-boost converter, as shown in fig. 1, includes the following steps:
fractional order mathematical model establishment
The Buck-Boost converter has a schematic circuit diagram as shown in FIG. 22, ViThe output voltage V can be adjusted by adjusting the on-off time of ToD is a freewheeling diode to keep the current flow direction in the circuit unchanged, L and C are inductors and capacitors, which are main energy storage elements in the circuit, when the switching device T is closed, the power supply transfers energy to the inductor, the energy storage of the electric energy on the inductor is increased, and when the switching device is turned off, the electricity is storedThe inductance transfers energy to the capacitor and the load, and the stored energy on the inductance is reduced.
(1) Establishing a fractional order mathematical model in the Buck-Boost converter, wherein the fractional order devices in the Buck-Boost converter are an inductor L and a capacitor C, and the mathematical model is as follows:
Figure GDA0002438604050000081
wherein iLIs an inductive current, vLIs the inductor voltage iCIs a capacitance current, vCIs the capacitor voltage, DαThe method is a fractional calculus operator, wherein the order α is between 0 and 1, a is an integration lower limit, t is an integration upper limit, L is the magnitude of an inductance value and is represented by H, and C is the magnitude of a capacitance value and is represented by F.
In the theory of fractional calculus, there are three main definitions, which are Grunwald-L etnikov (G-L) fractional calculus, Riemann-L iouville (R-L) fractional calculus and Caputo fractional calculus.
The definition of the fractional calculus of Grunwald-L etnikov is generalized from the definition of the integer order, and the order of the integer order is generalized to the fractional order, and the definition formula is as follows:
Figure GDA0002438604050000082
where m is a positive integer and α. ltoreq. m.ltoreq. α +1, a is the lower limit of the integral and is a Gamma function defined by the form:
Figure GDA0002438604050000083
wherein m is a constant and re (m) > 0.
Riemann-L iouville fractional calculus is based on the improvement of G-L fractional calculus based on the property that fractional calculus should satisfy, which is defined as:
Figure GDA0002438604050000091
wherein n is a positive integer and n is equal to or greater than α and equal to or less than α + 1. R-L type fractional calculus definition can be regarded as that the function u (t) is firstly subjected to fractional calculus and then subjected to integer differentiation, and the form of the fractional calculus definition is simpler compared with G-L type definition, can simplify the calculation process of the fractional calculus and is widely applied in practice.
(2) Calculating fractional calculus operator by using Caputo definition
Figure GDA0002438604050000092
The expression defined by Caputo is:
Figure GDA0002438604050000093
wherein a is an integration lower limit, t is an integration upper limit, r is a fractional order, u (t) is a function to be solved, and n is an approximate order of the fractional order, is a minimum integer larger than the fractional order, and is an integration variable.
The G-L type definition firstly generalizes the integral order calculus definition into the fractional order through a limit form, gives the definition of the fractional order calculus theory, but the expression is too complex to be applied to popularization in practice, and the R-L type definition and the Caputo type definition both take the G-L type definition as the basis, are improved and expanded, simplify the calculation process of the fractional order calculus and are convenient to apply in practice.
For the case of order fractional order, under the condition that the function u (t) has a continuous derivative of order m +1, and m takes at least n-1, then n-m-1, when the function u (t) satisfies u (t)(k)(a) 0, k is 0,1, …, n-1, the three definitions are equivalent and can be interchanged, otherwise they are not equivalent.
The Caputo type definition retains the property of integral order calculus because integer order differential calculation is carried out firstly, and the fractional order derivative of a constant is 0, so that the Caputo type definition is widely applied to the modeling process facing many practical application problems.
(3) Modeling two switch states in the Buck-Boost converter by adopting a state space average method to obtain a fractional order mathematical model of the Buck-Boost converter based on switching value, wherein the fractional order mathematical model comprises the following steps:
Figure GDA0002438604050000094
in which d is a switching variable, iLIs an inductive current, voTo output a voltage, VinIn order to input the voltage, the voltage is,<iL>,<vo>,<Vin>the average values of the inductive current, the output voltage and the input voltage in a switching period are respectively, and R is the resistance value of the resistor and has the unit of omega.
Design of (II) fractional order sliding mode variable structure controller
The Buck-Boost converter is a typical switching nonlinear system, and has great limitation on the design of a controller, particularly, relevant research and stability analysis and verification on a fractional order system are mainly concentrated in the linear system, and research results on the fractional order nonlinear system are less, so that further analysis of the controller is not facilitated.
(1) And transforming a fractional order mathematical model of the Buck-Boost converter: in the formula (3) [ x ]1,x2]T＝[iL,vo]TAnd if u is d, the original fractional order model of the Buck-Boost is converted into a standard form shown in an expression (4), wherein u represents the size of the duty ratio, is a function changing along with time and is an actual control variable of the whole system:
Figure GDA0002438604050000101
wherein X is a state variable, and X ═ X1,x2]T＝[iL,vo]TI.e. the inductor current and the output voltage, y is the output voltage after feedback linearization, f (x) and g (x) are as follows:
Figure GDA0002438604050000102
Figure GDA0002438604050000103
therefore, on the fractional order standard mode of the Buck-Boost converter shown in the formula (4), the form of an output function is reconstructed in a fractional order feedback linearization mode, and the output function is converted into a linearization model shown in the formula (7), so that the design and implementation of a fractional order sliding mode controller are facilitated.
Figure GDA0002438604050000104
Wherein v, z1And z2The expression of (a) is:
Figure GDA0002438604050000105
Figure GDA0002438604050000106
Figure GDA0002438604050000107
therefore, the control system can be designed on the system after fractional order feedback linearization, and the control system acts on the original fractional order nonlinear system to realize the control of the whole nonlinear system.
After the above-described feedback linearization is performed on the fractional-order Buck-Boost fractional-order model, it can be converted into a standard form shown in equation (8). The control target of the Buck-Boost fractional order converter is that an output voltage tracks a reference voltage vrefAccording to the steady-state working point of the Buck-Boost converter, the reference output i of the inductive current can be obtainedLrefAnd reference output D of duty cyclerefComprises the following steps:
Figure GDA0002438604050000111
thus, the inductor current i when the circuit reaches steady state can be calculated by equation (11)LrefReference output D of magnitude and duty ratio ofrefTo calculate the reference output after fractional order feedback linearization as:
Figure GDA0002438604050000112
Figure GDA0002438604050000113
(4) aiming at a linearized model after feedback linearization, a fractional order calculus theory is adopted to design a fractional order sliding mode controller, wherein a fractional order sliding mode surface s and a control law vkComprises the following steps:
s＝e2+k2D1-αe1+k1D-αe1(14)
Figure GDA0002438604050000114
wherein s is a slip form surface, k1And k is2And (3) for a gain coefficient of the system, wherein lambda and k are sliding coefficients to ensure that the state of the system can quickly reach a sliding mode surface, sign is a sign function, when s is more than or equal to 0, sign(s) is 1, and when s is more than or equal to 0<0, sign(s) ═ 1, e1,e2The first derivative of the output error and the output error is expressed as:
Figure GDA0002438604050000115
(III) fractional order model ripple analysis
For the switching power supply, the magnitude of the output voltage ripple is an important characteristic determining the quality of the switching power supply, and is also a basis for component selection in the design process of the dc converter. Fractional order has a larger influence on the output voltage ripple of the Buck-Boost converter than the integer order due to the memory and genetic characteristics of the fractional order.
When a switching device is turned on, a mathematical model of the Buck-Boost converter, namely formula (3), is solved for a change value of the inductive current and the output voltage, so that the following can be obtained:
Figure GDA0002438604050000121
Figure GDA0002438604050000122
wherein, Δ iLIs the value of change in the inductance current, Δ voTo output a voltage variation value, VinL, R and C are input voltage, inductance, resistance and capacitance values respectively,
Figure GDA0002438604050000123
is the initial value of the output voltage in this period, as an integral variable, EαIs a Mittah-L effler function, which is defined by the formula:
Figure GDA0002438604050000124
as can be seen from equations (17) and (18), the variation values of the output voltage and the inductor current decrease as the order of the fractional order increases, and therefore, it can be found that all the integral order models in the previous analysis are only an approximation to the actual model, and have a large deviation from the actual system.
Simulation experiments verify that the control method of the invention comprises the following steps:
in order to verify the accuracy of ripple analysis of a fractional order mathematical model of the Buck-Boost converter, a fractional order toolbox ninteger and Simulink is adopted on Matlab to carry out numerical simulation of the Buck-Boost converter, and in simulation verification, an input voltage v is takeni20V, reference output voltage Vo15V, inductor L equals 1mH, capacitor C equals 500 μ F, switching frequency F equals 100kHz, and the simulation effect is shown in fig. 2:
as can be seen from fig. 2(a) and 2(b), the fractional order mathematical model, regardless of the output voltage voIs also in the inductor current iLIs significantly larger than the integer order mathematical model. Fractional order Δ v on the output voltageo0.08V, Δ V of integer orderoΔ i of fractional order on the inductor current, about 0.04VL0.05A, Δ i of integer orderLAbout 0.03A; as can also be seen from fig. 21, when the fractional order is further decreased, as shown in fig. 21(a) and 21(b), the ripple of the output voltage and the inductor current is further increased, and the correctness of the analysis is verified.
(1) In terms of the starting response of the Buck-Boost converter, simulation graphs of the output voltage and the inductor current are shown in fig. 3(a), fig. 3(b) and fig. 4(a) and fig. 4(b), and it can be seen that the fractional order sliding mode controller is superior to the integer order sliding mode controller in starting performance. The fractional order sliding mode controller can reach the preset output voltage in about 7ms without voltage overshoot; the integer order sliding mode controller can track to the reference output voltage around 15ms, but there is overshoot around 0.5V. In the aspect of inductive current, the fractional order sliding mode controller reaches a stable state for about 20ms, the convergence speed is high, but overshoot of about 0.04A exists; and the integer order sliding mode controller reaches the reference inductance current in about 25ms, the convergence speed is slower than that of the fractional order, and the overshoot of about 0.16A exists. In general, the fractional order sliding mode controller is more excellent in convergence speed and overshoot than the integer order in terms of start-up performance because it utilizes the inheritance and memory of the fractional order.
(2) In the output voltage jump experiment, the reference voltage is output to a 5V step signal at 2s, the reference output voltage is changed from 15V to 20V, and the output voltage and the inductor current waveforms of the fractional order sliding mode controller and the integer order sliding mode controller are shown in fig. 5(a), fig. 5(b), fig. 6(a) and fig. 6 (b).
As can be seen from fig. 5(a) and 5(b), the convergence speed of the fractional order sliding mode controller is faster and the overshoot is smaller than that of the integer order sliding mode controller, and the integer order sliding mode controller has overshoot of about 2V. When the voltage jumps, the voltage drops to some extent because the output voltage increases, the duty ratio of the PWM control signal increases, the on-time of the switching element increases, the inductor charging time increases, and the discharging time decreases, and thus the voltage drops to some extent.
As can be seen from fig. 6(a) and 6(b), the change of the inductor current corresponds to the output voltage, and the fractional order sliding mode controller has a great advantage in convergence speed and overshoot compared to the integer order controller. Generally, above the inductor current, when there is a change in the output voltage, the fractional order sliding mode controller can reach a new equilibrium state more quickly and stably.
(3) When the load resistor jumps, the load resistor gives a 5 Ω step signal at 2s, the load resistor changes from 15 Ω to 20 Ω, and the output voltage and inductor current waveforms of the fractional order sliding mode controller and the integer order sliding mode controller are as shown in fig. 7(a) and 7(b), and fig. 8(a) and 8 (b):
as can be seen from fig. 7(a) and 7(b), at the moment of load change, there is an upward jump of about 3V in the output voltages of the integer-order and fractional-order sliding-mode controllers, but the fractional-order sliding-mode controller reaches a steady state again in about 4ms, the recovery speed is high, and the convergence speed of the integer-order sliding-mode controller is slow.
As can be seen from fig. 8(a) and 8(b), the inductor current changes in its steady state as the load resistance changes, as can be seen from equation (11). For a new steady state, the fractional order sliding mode controller can reach quickly, and the convergence speed and overshoot are superior to those of an integer order sliding mode controller.
In conclusion, simulation experiments show that the fractional order sliding mode controller combines the genetic and memory characteristics of the fractional order compared with an integer order sliding mode controller, the sliding mode controller is designed on the fractional order model of the Buck-Boost converter, the actual characteristics of the converter are better met, and the fractional order sliding mode controller has better robustness and more excellent overall performance when facing external interference.
The control method of the invention comprises the following steps:
(IV) discretization and DSP program compiling
In an actual computer control system, a sampling signal and a control signal are in a discrete form, so how to discretize a fractional order differential operator is a key for solving a fractional order differential equation and realizing a fractional order controller. The introduction of fractional calculus as the expansion of integral calculus greatly expands the degree of freedom and flexibility of controller design, but the complexity of realization of actual discretization is higher than that of integral due to the complexity of the controller. The method discretizes the designed fractional order sliding mode surface and the control law in a Tustin + CFE mode, and the mode is as follows:
Figure GDA0002438604050000141
wherein the order α is between 0 and 1, PpAnd QqIn the simulation test, the order α is taken to be 0.9, the number p of the intercepted terms is taken to be q to be 5, the sampling time T is 0.1s, and s can be obtained by utilizing an ninteger tool box0.9The discrete form of (a):
Figure GDA0002438604050000142
in the actual system design, the adopted core control chip is the TMS320F28335, which has the advantages of high processing speed, strong processing capability, fast floating point operation, high-precision a/D conversion and the like, is suitable for fast signal processing and the realization of complex control algorithms, has relatively low development period and cost, is very suitable for being used as the control core of a direct current converter, and is also the main trend of the current power supply design.
(1) The hardware circuit of the Buck-Boost direct current converter mainly completes the detection of voltage and current and the function of DSP control signal response, and can be divided into the following six aspects: the circuit comprises a main circuit, a voltage detection circuit, a current detection circuit, a PWM drive circuit and an auxiliary power supply circuit. Each functional module is analyzed and designed separately.
(2) Design of main circuit
The main circuit part adopts a four-switch Buck-Boost circuit topological structure, the schematic diagram of the Altium design simulation is shown in figure 9, the circuit only comprises one main power inductor, two bridge arms are provided with four switching devices, three structures of a Buck converter, a Boost converter and a Buck-Boost converter can be realized by controlling the switching time sequences of the four switching devices, and the three structures are flexible and convenient and are convenient to apply to an actual switching power supply.
Compared with a traditional Buck-Boost circuit, the four-switch Buck-Boost converter topology has the advantages that the control mode is basically the same as the state equation, the biggest difference is that the output voltage polarity of the traditional Buck-Boost circuit is opposite to the input voltage, and the four-switch topology structure changes the flow direction of inductive current by adding switches, so that the output voltage polarity is the same as the input voltage, and the four-switch Buck-Boost converter topology is more flexible and convenient.
Compared with the schematic diagram shown in fig. 9, in the actual main circuit, as shown in fig. 13, a large number of capacitors are added to both the input end and the output end, so that high-frequency components of the input voltage and the output voltage can be filtered, and the size of input-output voltage ripples can be reduced. As shown in fig. 4(a) and 4(b), the current detection circuit shown in fig. 11 is connected in series to the inductor and detects the magnitude of the inductor current by the hall effect; the voltage detection circuit shown in fig. 10 is also connected in parallel to the input voltage and output voltage terminals, and detects the magnitude of the input and output voltages; as shown in the above figure, the PWM driving module shown in fig. 12 has its PWM output terminals connected to the four switching tubes respectively, and controls the switching states of the switching tubes respectively, so as to achieve the purpose of controlling the output voltage.
(3) Voltage detection circuit
The voltage detection circuit mainly adopts a differential amplification circuit which is formed by taking T L V2374 as a core, as shown in figure 10, the T L V2374 is a single-power operational amplifier, has the bandwidth of 3MHz, high conversion efficiency, wide working voltage range and low power consumption, adopts a small-sized SOT-23 package, has small volume, and is suitable for the detection circuit of high-frequency voltage change.
The differential amplification circuit formed by T L V2374 is connected in parallel to the output voltage or the input voltage, so that the voltage signal can be detected in real time, the anti-interference capability of the detected circuit can be improved through the differential amplification circuit, the interference of common mode signals in the circuit can be effectively inhibited, the amplitude range of the detected voltage is reduced while the differential mode signals are amplified, the maximum amplitude of the voltage is reduced to the voltage range of the input A/D port of the DSP, the subsequent use of A/D conversion is convenient, the ratio of the input voltage to the output voltage can be adjusted through the resistor in the circuit, and the calculation formula is as follows:
Figure GDA0002438604050000151
in the voltage detection circuit, R is taken12＝R25＝1KΩ，R13＝R22The output voltage after passing through the differential amplifier is 1/11 times the input voltage, which can be calculated by the formula 10K Ω. If the input voltage of the port of the DSP does not exceed 3V at most, it can be known that the maximum amplitude of the input voltage and the output voltage cannot exceed 33V, otherwise, the input voltage of the port of the DSP is too high, which causes damage to the device. Filter capacitor C connected to voltage output end in circuit16The capacitor has the characteristics of high frequency passing and low frequency blocking, so that high-frequency interference in the output voltage can be filtered, and the accuracy of the detection voltage is ensured.
(4) Current detection circuit
In an actual circuit, the DSP does not have a port for directly inputting and detecting a current signal, and therefore, the current signal is converted into a voltage signal, and then the voltage signal is converted and utilized inside the processor through a/D conversion. The common method for converting the current signal into the voltage signal includes: high-precision small resistors, a parallel RC detection circuit, a Hall sensor and the like are connected in series. The series resistor has small volume for converting the current signal and high precision, but is easy to interfere the ground wire and has larger temperature drift; the parallel RC detection circuit has the same mode with the resistor, and is easy to generate larger interference on the circuit; the Hall sensor adopts an electromagnetic induction phenomenon, current is converted to the greatest extent through different turn ratios, the whole circuit is not directly influenced, and the precision and the linearity of a measuring result are high. In the actual circuit fabrication, an ACS758 hall current detection chip is selected, and the operation schematic diagram is shown in fig. 11.
The ACS758 is a current sensing device consisting of a precise, low-offset linear hall sensor, has a high frequency bandwidth of 120KHz, has a response time of less than 4us, and is sensitive to high frequency variation signals. The internal self-contained voltage operational amplifier circuit can convert input inductive current into voltage output, and the input and output calculation formula is as follows:
Vo＝0.5×Vref+0.04×IL (26)
wherein the reference voltage VrefThe output value of the auxiliary power supply of the circuit is 3.3V. As can be seen from the formula (26), when the inductance current in the circuit changes by 1A, the output voltage changes by 0.04V, and after the DSP performs a/D conversion on the input voltage, the actual inductance current value can be calculated by the formula and used in the calculation of the control law. A voltage follower formed by an OP07 operational amplifier is connected between the output end of the ACS758 and the A/D port of the DSP and is used as buffer and isolation between the front stage and the rear stage, so that the output impedance of the ACS758 is improved, the fluctuation of the output voltage is further inhibited, the mutual influence between the front stage and the rear stage is avoided, and the accuracy of the measured voltage is ensured.
(5) PWM drive circuit
The PWM mainly has two types of frequency conversion and fixed frequency for driving the MOS tube, and a fixed frequency mode is adopted in the programming, namely the frequency of a PWM signal is unchanged, and the switching time of the MOS tube is adjusted by changing the size of a duty ratio. After the current and the voltage in the Buck-Boost circuit are measured through the voltage detection circuit and the current detection circuit, the DSP reads the voltage through the A/D port, calculates and restores data in a program, and then generates a new duty ratio through the calculation of the fractional order sliding mode controller, so that the real-time adjustment of the switching time of the MOS tube is realized. The PWM signal output from the DSP port has low load capability and cannot be directly used for driving the MOS transistor, so that a dedicated MOS transistor driving chip is required to increase the voltage and current of the output signal. The UCC27211 driver chip is used to drive the switching tubes, and the circuit diagram is shown in fig. 12.
(6) Auxiliary power supply circuit
Main voltage to auxiliary voltage: input voltage converts circuit supply voltage circuit: the input voltage is converted into 12V voltage for output through an xl7035 module, the maximum output current is 1A, and power is supplied to a PWM driving module and other fixed voltage conversion modules, as shown in fig. 14 (a).
A digital supply voltage; the operational amplifier circuit is the most important part in voltage and current detection, and the stability of the operational amplifier directly determines the accuracy of sample collection, so that a stable power supply voltage needs to be provided for the operational amplifier module. The digital power supply part converts the 12V output by the circuit of fig. 1 into a stable 3.3V output voltage, and supplies power to the operational amplifier module, as shown in fig. 14 (b).
Reference voltage conversion circuit: the input voltage of 3.3V is converted into the reference voltage of 1.65V to provide the reference voltage for the operational amplifier, which can be used to provide a reference in the current detection process, as shown in fig. 14 (c).
The voltage conversion part of the three parts converts the input voltage into 12V, 3.3V and 1.65V, and all adopt fixed voltage output circuits, the voltage adjustable power is poor, but the stability of voltage output is higher, so that a stable power supply environment is provided for the voltage detection and current detection circuit, and the accuracy of the output of the detection circuit is ensured.
(V) software design
The overall control software design mainly comprises the following steps: ADC sampling program design, PWM program design, fractional order sliding mode control algorithm program design, DMA program design, interrupt program design and the like.
The main program is mainly designed to implement configuration of system-related environment, initialization of variables, initialization of interrupts, and the like, and a flowchart of the whole program is shown in fig. 15:
step 1: in order to ensure the stability and continuity of the whole software program, the sampling of the voltage value, the calculation of the fractional order sliding mode control algorithm and the updating of the PWM parameter are all carried out in the timed interruption 1, so that the cyclic waiting of the main program is avoided, and the system time is wasted. The timer interruption is realized by using a timer module in the TMS320F28335, and the timer module is a counter built in the TMS320F28335 and counts according to the size of the main frequency. The timer is set to 10ms, that is, 10ms triggers an interrupt, and all data is processed once.
Step 2: and after the timed interrupt is triggered, judging whether the DMA is idle or not. In the TMS320F28335, a special communication mode is adopted during DMA, namely, data can be directly read from one register to another register without the participation of a CPU, and the system time can be saved. In the design, the converted data in the A/D register can be directly read into the memory for data processing.
And step 3: ADC sampling data processing and voltage values acquired by circuits in the figures 10 and 11 are converted into digital signals through a TMS320F28335 chip internal analog-to-digital conversion module with 12 bits, the digital signals are read into a memory, and calculation is carried out through the formulas (25) and (26) to obtain inductance current values in an actual circuit and input and output voltage values.
And 4, step 4: the discrete sliding mode control calculation mainly uses the inductance current value, the output voltage value and the input voltage value acquired by A/D conversion, and uses the formulas (17) and (18) to calculate the duty ratio of PWM in the switch tube control.
And 5: the duty cycle limitation is to limit the duty cycle within a certain range to prevent the output voltage from being too large. As can be seen from equation (22), in the voltage acquisition circuit, the maximum voltage division is 11, the port input voltage of TMS320F28335 is 3V at the maximum, and therefore the maximum output voltage is 33V, and the maximum duty ratio D can be calculated from equation (11)refSince the duty ratio cannot exceed 0.8, if it exceeds 0.8, it is limited to 0.8 or less.
Step 6: and updating the PWM duty ratio, namely assigning a value to a corresponding bit of the PWM register after the duty ratio is calculated, so that the output PWM duty ratio in the next period reaches the expected value obtained by calculation for regulating the output voltage.
In the process of programming, the most important content is the calculation of the fractional order sliding mode control law. On the hardware sampling circuit, the input voltage, the output voltage and the inductive current of the Buck-Boost circuit can be obtained through the voltage sampling circuit and the current sampling circuit, and after filtering through software, calculation of a control law can be carried out by using the Buck-Boost circuit. In the fourth part, the fractional order sliding mode controller is discretized by a Tustin + CFE method, so that the software implementation of the controller is also convenient, and a flow chart of a fractional order control algorithm is shown in fig. 16.
Step 1: the timer is interrupted and waits, the same as the whole program flow chart, data is processed once in a unified mode within 10ms, and real-time performance and stability of the control system are guaranteed.
Step 2: the acquisition of the A/D value is to read the actual inductance current value, the input voltage value and the output voltage value after the calculation by the formula (25) and the formula (26) in the main program, and prepare for the calculation of the control law.
And step 3: the error is calculated according to the formula (8) and the formula (9), and z is obtained under the current voltage and current sampling value1,z2The size of (3) and z obtained from the formulae (12) and (13)1ref,z2refError in the magnitude of (1), and e in the formula (16)1,e2And the calculation is used for the next control law calculation.
And 4, step 4: calculation of control law, i.e. e obtained from the previous step1,e2And (3) substituting the sliding mode variable structure controllers shown in the formulas (14) and (15) to calculate the numerical values of the control law and the duty ratio, and using the numerical values in the PWM control.
And 5: the duty ratio is updated by using the duty ratio calculated in the previous step to calculate the required update value of the comparison register in the PWM register.
FIG. 17 is a flow chart of a detailed algorithm of a fractional order sliding mode controller, wherein z is calculated according to collected output voltage, input voltage and inductance current value1,z2And calculating an error value to calculate an actual control law. The voltage and current reading in the program flow chart is the same as the above flow chart, the converted voltage and current values are substituted into the formula (8) and the formula (9), and the z after fractional order linear feedback linearization is calculated1,z2After (4), (15), the sliding mode surface S and the control law v are calculatedk(ii) a The legal range of the duty ratio is the calculation of the maximum value of the output voltage analyzed by the above, the duty ratio needs to be ensured to be between 0.3 and 0.8, and if the calculated duty ratio value is not in the range, the duty ratio value needs to be restricted, so that the safety of the whole circuit is ensured.
Hardware simulation results:
in the above, the hardware circuit and software implementation of the Buck-Boost converter are analyzed, and the manufacturing and experiment of the real object can be performed. The hardware part mainly comprises a Buck-Boost main circuit, a voltage detection circuit, a current detection circuit, a PWM (pulse-width modulation) drive circuit, an auxiliary power supply circuit, a filter circuit and the like. The overall circuit diagram of the Buck-Boost converter is shown in fig. 18:
in this experiment, the Buck-Boost circuit operates in a CCM (continuous inductor current) mode, and therefore it is necessary to ensure that the inductor current is constantly greater than the maximum value, and therefore, the parameters in the circuit need to satisfy the formula (17) and the formula (18) and ensure that the inductor current variation value is smaller than the maximum value of the inductor current, and therefore, in the physical verification, the output voltage is set to be 23V, the input voltage is 12V, the load R is 15 Ω, the switching frequency F is 100kHz, the inductance and capacitance are L mH and C is 100 μ F, the waveforms of the output voltage and the inductor current are observed by an oscilloscope, and the waveforms of the output voltage and the control PWM are as shown in fig. 19 and fig. 20.
As can be seen from fig. 19, the output voltage and the output of the inductor current are stable, and the ripple of the output voltage is small. After the inductive current passes through the Hall sensor and the voltage follower, the output average voltage value is 1.76V, the actual current value is 2.75A calculated by the formula 23, and the error between the actual current value and the theoretical calculated value is small. As can be seen from fig. 20, the inductor current and the output voltage have large fluctuation when the PWM is switched between high and low levels, i.e. when the MOS transistor switches between the on and off states, mainly due to the existence of dead time and the change of the charging and discharging states of the inductor and the capacitor. In general, the overall performance of the output voltage is better, achieving the desired effect.

Claims (2)

1. A fractional order sliding mode control method of a buck-boost converter is characterized by comprising the following steps:
establishing a fractional order mathematical model:
(1) establishing a fractional order mathematical model in the buck-boost converter, wherein the fractional order devices in the buck-boost converter are an inductor L and a capacitor C, and the mathematical model is as follows:
Figure FDA0002483545890000011
wherein iLIs an inductive current, vLIs the inductor voltage iCIs a capacitance current, vCIs a voltage of the capacitor, and is,
Figure FDA0002483545890000012
the method is a fractional calculus operator, wherein the order α is between 0 and 1, a is an integration lower limit, t is an integration upper limit, L is the magnitude of an inductance value and is represented by H, C is the magnitude of a capacitance value and is represented by F;
(2) calculating fractional calculus operator by using Caputo definition
Figure FDA0002483545890000013
The expression defined by Caputo is:
Figure FDA0002483545890000014
wherein a is an integral lower limit, t is an integral upper limit, r is a fractional order, u (t) is a function to be solved, n is an approximate order of the fractional order, is a minimum integer larger than the fractional order, is an integral variable and is a Gamma function;
(3) modeling two switch states in the buck-boost converter by adopting a state space averaging method, and obtaining a fractional order mathematical model of the buck-boost converter based on switching values, wherein the fractional order mathematical model comprises the following steps:
Figure FDA0002483545890000015
wherein d is a switching variable, iLIs an inductive current, voTo output a voltage, VinIn order to input the voltage, the voltage is,<iL>,<vo>,<Vin>the average values of the inductive current, the output voltage and the input voltage in a switching period are respectively, R is the resistance value of the resistor, and the unit is omega;
(II) designing a fractional order sliding mode variable structure controller:
(1) transforming a fractional order mathematical model of the buck-boost converter: in the formula (3) [ x ]1,x2]T＝[iL,vo]TAnd if u is d, the fractional order model of the original buck-boost converter is converted into a standard form shown in an equation (4), and u represents the size of the duty ratio, is a function changing along with time and is an actual control variable of the whole system:
Figure FDA0002483545890000021
wherein X is a state variable, X ═ X1,x2]T＝[iL,vo]TY is the output voltage, f (X) and g (X) are shown below:
Figure FDA0002483545890000022
Figure FDA0002483545890000023
(2) and further converting a fractional order mathematical model of the buck-boost converter, so that the design and realization of a fractional order sliding mode controller are facilitated: on the fractional order standard mode of the buck-boost converter shown in the formula (4), the form of an output function is reconstructed in a fractional order feedback linearization mode, and the output function is converted into a linearization model shown in the formula (7):
Figure FDA0002483545890000024
wherein, v, z1And z2The expression of (a) is:
Figure FDA0002483545890000025
Figure FDA0002483545890000026
Figure FDA0002483545890000027
(3) the control target of the fractional order converter of the buck-boost converter is that the output voltage tracks the reference voltage vrefObtaining a reference output i of the inductive current according to the steady-state working point of the buck-boost converterLrefAnd reference output D of duty cyclerefComprises the following steps:
Figure FDA0002483545890000028
by equation (11), the inductor current i is calculated when the circuit reaches steady stateLrefReference output D of magnitude and duty ratio ofrefAnd then calculating the reference output after fractional order feedback linearization as follows:
Figure FDA0002483545890000029
Figure FDA0002483545890000031
(4) aiming at a linearization model after feedback linearization, a fractional calculus theory is adopted to designA fractional order sliding mode controller, wherein a fractional order sliding mode surface s and a control law vkComprises the following steps:
Figure FDA0002483545890000032
Figure FDA0002483545890000033
wherein k is1And k is2For the gain coefficient of the system, λ and k are sliding coefficients, sign is a sign function, when s ≧ 0, sign(s) is 1, when s ≧ 0<0, sign(s) ═ 1, e1,e2The first derivative of the output error and the output error is expressed as:
Figure FDA0002483545890000034
wherein z is1And z2Is defined by equation (8) and equation (9), respectively, z1refAnd z2refAre defined by equation (12) and equation (13), respectively.
2. The fractional order sliding mode control method of the buck-boost converter according to claim 1, wherein a mathematical model of the buck-boost converter, i.e. formula (3), is subjected to fractional order model ripple analysis:
for the mathematical model of the buck-boost converter, when a switching device is switched on, the change value of the inductive current and the output voltage is solved, and the following can be obtained:
Figure FDA0002483545890000035
Figure FDA0002483545890000036
wherein, △ iL△ v as a change value of an inductive currentoIn order to output the voltage variation value,
Figure FDA0002483545890000037
to an initial value of the output voltage in this period, EαIs a Mittah-L effler function, which is defined by the formula:
Figure FDA0002483545890000038
the change values of the output voltage and the inductive current are reduced along with the increase of the orders of the fractional order according to the equations (17) and (18), so that all integral order models in the prior analysis are only approximate to the actual model, and have deviation compared with the actual system, thereby proving that the mathematical model of the buck-boost converter created by the fractional order is accurate.
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