CN113872439A - CCM Buck converter differential flatness control method based on state feedback accurate linearization - Google Patents

CCM Buck converter differential flatness control method based on state feedback accurate linearization Download PDF

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CN113872439A
CN113872439A CN202111122810.0A CN202111122810A CN113872439A CN 113872439 A CN113872439 A CN 113872439A CN 202111122810 A CN202111122810 A CN 202111122810A CN 113872439 A CN113872439 A CN 113872439A
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buck converter
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皇金锋
李慧慧
谢锋
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Shaanxi University of Technology
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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

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Abstract

The invention discloses a CCM Buck converter differential flatness control method based on state feedback accurate linearization, which comprises the following steps: modeling the feedback linearization of the CCM Buck converter; designing a CCM Buck converter controller, specifically, carrying out flatness analysis on the CCM Buck converter, and designing feedback compensation of the CCM Buck converter; sensitivity analysis is carried out, and transient performance of the CCM Buck converter system is improved. The invention effectively improves the disturbance suppression of the input voltage and the load current of the CCM Buck converter.

Description

CCM Buck converter differential flatness control method based on state feedback accurate linearization
Technical Field
The invention belongs to the technical field of converters, and particularly relates to a CCM Buck converter differential flatness control method based on state feedback accurate linearization.
Background
The Buck converter is a typical time-varying nonlinear system, the traditional linear control theory has great limitation, particularly, the proportion-integral control with a simple structure cannot enable the system to obtain satisfactory dynamic response, and the system control difficulty is great due to introduction of empirical coefficients such as proportion and integral, so that the control method of the Buck converter is always a hot point problem concerned by numerous scholars.
To improve the dynamic performance of a Buck converter operating in Continuous Conduction Mode (CCM), experts and scholars at home and abroad have already made many studies. The prior document constructs a finite time feedback controller through a recursive design algorithm, then derives a novel observer to estimate the convergence speed of the finite time of the state of the Buck converter, and analyzes the effectiveness of system stability by combining a state feedback control method and a feedback control law output from the observer. Compared with the asymptotic stability control method, the convergence speed of the system after being disturbed is improved, but the calculation method is complex, and the difficulty in obtaining accurate observer values is high. In order to retain the advantages of low switching stress, small inductor size and high power density of a multi-stage Buck converter, the existing literature balances the flying capacitor voltage under a proper voltage reference value by a decoupling optimal control method of a reverse system theory. A general full-linearization output function selection method is provided on the basis of an inverse system method, the linearization and decoupling of the system are realized, and the provided control strategy has good dynamic performance. A plurality of designed optimal controllers control the subsystem, so that the design difficulty of the controllers is reduced, but the calculation process is complex. The existing literature proposes four control strategies for CCM Buck converters: the method comprises the steps of completing development of the SA and the DA by considering a nominal system with uncertain parameters of the Buck converter, completing development of the SDOB and the DDOB by considering a nominal system with uncertain parameters, analyzing and comparing advantages and disadvantages of four proposed control strategies, and having practical guiding significance for design of a controller of the Buck converter. In the prior art, a time discrete state feedback controller is designed, and can be applied to a Buck converter under the condition of wide output load to carry out high-bandwidth voltage control, discrete time dynamics of CCM and DCM are established in the control process, and modeling, design and performance of the proposed control circuit structure are proved in simulation and experiment. The existing literature provides a composite stable discretization quasi-sliding mode control (DQSMC) method to solve the stability problem caused by the negative resistance effect of constant power. The discrete integral sliding surface is designed to obtain the fast and stable dynamic response of the direct current voltage, and a second-order sliding mode interference observer is embedded in a voltage controller to carry out interference estimation and compensation, so that the combined composite DQSMC scheme has strong robustness, inherent jitter suppression and dynamic guarantee. Meanwhile, the PI current controller remains in the internal control loop to achieve current control. The prior document proposes a sequential switch control scheme based on an auxiliary parallel inductor, which introduces a small controlled inductor connected in parallel with an output inductor, increases the inductor current change rate during sudden load change, and proposes a sequential switching strategy to control the auxiliary parallel inductor, but has certain difficulty in selection and control of the controlled inductor. The control method provided in the above document can improve the dynamic performance of the CCM Buck converter, but a plurality of empirical coefficients such as proportion, integral and differential are introduced in the design process of the control method, so that the degree of freedom of system stability parameters is increased, and the system stability parameters are difficult to learn by other researchers. Moreover, most control methods run in simulation software, and it is easy to try to obtain a plurality of groups of experience coefficients, but the reproducibility in practical application is ignored.
Disclosure of Invention
The invention aims to provide a CCM Buck converter differential flatness control method based on state feedback accurate linearization, which effectively improves disturbance suppression on input voltage and load current of the CCM Buck converter and improves transient performance of a system.
The CCM Buck converter adopts the technical scheme that the CCM Buck converter comprises a switching tube S, a diode D, a filter inductor L, an output capacitor C and a load resistor R, and the input voltage is uiOutput voltage of uoThe inductor current is iLThe CCM Buck converter feedback linearization model is established, differential flattening control is carried out on the CCM Buck converter based on the model, feedback compensation design is carried out on the CCM Buck converter according to a pole allocation method, system output can accurately follow an expected value, the change trend of system characteristic values is analyzed through a sensitivity theory when circuit parameters change, the transient performance of the CCM Buck converter system is improved, and the CCM Buck converter feedback linearization model specifically comprises the following steps:
step 1: a feedback linearization model is established and is used for carrying out the feedback linearization on the target object,
step 2: the design of the flat controller is that the flat controller,
step 2.1: carrying out flatness analysis;
step 2.2: designing feedback compensation;
and step 3: sensitivity analysis is carried out, and transient performance of the CCM Buck converter system is improved.
The step 1 specifically comprises the following steps:
establishing a system state equation according to a state space average method:
Figure BDA0003277612640000031
in the formula (1), u is the duty ratio of the main power switch tube S;
selecting a system state variable x ═ x1,x2]＝[iL,uo]Formula (1) can be rewritten as:
Figure BDA0003277612640000032
in the formula (2), y is a system output variable, UrefIs a desired value of the output voltage, and
Figure BDA0003277612640000041
Figure BDA0003277612640000042
the feedback linearization converts the nonlinearity of the Buck converter into a linear model by means of coordinate transformation, and obtains an expression of the control rate. From differential geometry theory, it can be shown that equation (2) satisfies the condition of linearization and derives the following lie derivatives:
Figure BDA0003277612640000043
according to the formula (3), the relative order r of the CCM Buck converter is 2, the system dimension is n, r reflects the entire dynamic state of the system, i.e. the system is described by n-r state variables, and there is no other part, all the state variables are observable after state feedback, and there is no problem that may cause dynamic instability in the system, so as to further define the following coordinate transformation:
Figure BDA0003277612640000044
thus, it is possible to obtain:
Figure BDA0003277612640000045
since the formula (5) is a nonsingular array, the transformed global differential homomorphism is that the Brunox standard type of the system state equation after coordinate transformation is as follows:
Figure BDA0003277612640000046
in equation (6), v is a new input variable and has:
Figure BDA0003277612640000051
let the system state feedback rate control u described by equation (7) be denoted as u1I.e. by
Figure BDA0003277612640000052
The step 2.1 specifically comprises the following steps:
in order to improve the anti-interference capability of the system, the CCM Buck converter is subjected to flat control, and the method is characterized in that all state quantities and control quantities of the system are integrally described through expected flat output quantities. The basic meanings are as follows: if a flat output y can be found in the system control process and the state variable x and the control variable u can be expressed as y and a function of finite differentiation of each order of y, the system can be controlled by differential flat, namely, for an n-order nonlinear system as shown below:
Figure BDA0003277612640000053
there is y ∈ R, such that
Figure BDA0003277612640000054
In the formula (9), i and j are integers,
Figure BDA0003277612640000055
and γ is a scalar function;
before differential flattening control, appropriate flattening output quantity, state variables and control variables need to be selected again, and the analysis shows that the CCM Buck converter meets the sufficient requirements for accurate feedback linearization, so that u can be made to be v, and the following conditions are provided:
Figure BDA0003277612640000056
when formula (10) is substituted for formula (1), it is possible to obtain:
Figure BDA0003277612640000057
according to the equation (11), the system satisfies the requirement of the differential flat theory, namely:
Figure BDA0003277612640000061
Figure BDA0003277612640000062
by substituting equation (13) for equation (7), we can find the differential flat control rate based on the accurate linearization of the state feedback:
Figure BDA0003277612640000063
the step 2.2 specifically comprises the following steps:
in order to make the output of the system accurately follow the expected value, a pole allocation method is adopted for the CCM Buck converter to carry out feedback compensation design, and for the system described by the formula (1), a feedback control rate u is obtained2：
u2＝-Kx+kip (15)
In formula (15), K ═ K1K2]For the state feedback gain matrix, kiIs the integral coefficient, p ═ jeopardy (U)ref-uo)dt；
Setting system damping ratio zeta 0.707 and regulating time tsThe natural oscillation frequency of the closed loop system is calculated as:
Figure BDA0003277612640000064
the system expected closed loop pole is determined by equation (16):
Figure BDA0003277612640000065
taking p as an additional state vector, the three-order augmentation system is formed by the controlled system described by the formula (1):
Figure BDA0003277612640000066
in the formula
Figure BDA0003277612640000067
As shown in the formula (18), the system also needs to determine an expected closed-loop pole, and the expected system pole s determined according to the dynamic index of the system1,2The dominant pole of the closed-loop system is determined, and then a non-dominant pole is determined, wherein the non-dominant pole is selected as follows:
s3＝-nζωr (19)
wherein n is a constant, and n is 5;
calculating a state feedback gain matrix K of the system in MATLAB through an acker function, wherein the state feedback gain matrix K is 4.06-167.8]，ki＝1051998.5；
By combining the equations (14) and (15), the control rate of the system of equation (1) can be obtained as follows:
Figure RE-GDA0003391469960000071
the step 3 specifically comprises the following steps:
the differential flat control rate determined by equation (20) includes both the time-varying parameter uo、iLAnd constant parameter L, C, uiAnd R, wherein the invariant parameters L and C influence the transient performance of the system, and in order to clarify the influence of the L and C changes on the transient performance of the system, the transient performance is analyzed by applying a sensitivity theory: the Jacobian matrix of the CCM Buck converter closed-loop system can be obtained according to the formula (1):
Figure BDA0003277612640000072
in the formula (21)
Figure BDA0003277612640000073
The characteristic equation of the Jacobian matrix described by equation (21) is:
det(λI-J)＝0 (22)
obtaining a system Jacobian matrix J and a characteristic value lambdaiOn the basis, relevant parameters can be optimized according to the sensitivity analysis result, so that the system reliability is improved, and the expression of the characteristic root sensitivity is as follows:
Figure BDA0003277612640000074
in the formula (23), piIs the right eigenvector of J, qiIs the left eigenvector of J, σ is the real part of the eigenvalue sensitivity, and ω is the imaginary part of the eigenvalue sensitivity. According to Floquet theory, when σ>At 0, decreasing the parameter b will result in λiThe left half part of the complex plane is moved, so that the stability of the system is improved; when sigma is<At 0, increasing the parameter b will result in λiAnd the left half part of the complex plane is moved, so that the stability of the system is improved.
The invention has the beneficial effects that:
the invention relates to a CCM Buck converter differential flat control method based on state feedback accurate linearization, which comprises the steps of firstly establishing a feedback linearization model of a Continuous Conduction Mode (CCM) Buck converter, and performing state feedback accurate linearization on the nonlinear model by using a differential flat theory; then, carrying out feedback compensation design on the system according to a pole allocation method to enable the system to output accurate follow expected values; analyzing the change trend of the system characteristic value when the circuit parameter changes through a sensitivity theory; in the whole control method design process, no experience parameters such as proportional integral and the like are introduced, so that the degree of freedom of system stability caused by parameter change is small; compared with a proportional-integral control method, the control method can better inhibit the disturbance of input voltage and load current, has stronger robustness, has reference significance to the design of a CCM Buck converter controller, and can be expanded into a high-order DC-DC converter.
Drawings
FIG. 1 is a diagram of a CCM Buck converter topology;
FIG. 2 is a block diagram of system state feedback precision linearization;
FIG. 3 is a diagram of a system control architecture;
FIG. 4 is a variation trend graph of the system characteristic value, wherein FIG. 4(a) is a variation trend graph of the system characteristic value when L is decreased from 1mH to 100 μ H, and FIG. 4(b) is a variation trend graph of the system characteristic value when C is increased from 50 μ F to 90 μ F;
FIG. 5 is a graph of a steady-state output voltage waveform of a CCM Buck converter differential flatness control method based on state feedback accurate linearization according to experimental verification;
fig. 6 is an experimental waveform diagram of an input voltage surge, in which fig. 6(a) is an experimental waveform diagram of an input voltage surge corresponding to the proportional-integral control method, and fig. 6(b) is an experimental waveform diagram of an input voltage surge corresponding to the control method of the present invention;
fig. 7 is an experimental waveform diagram of input voltage sudden decrease, in which fig. 7(a) is an experimental waveform diagram of input voltage sudden decrease corresponding to the proportional-integral control method, and fig. 7(b) is an experimental waveform diagram of input voltage sudden decrease corresponding to the control method of the present invention;
fig. 8 is an experimental waveform diagram of a load current surge, in which fig. 8(a) is an experimental waveform diagram of a load current surge corresponding to the proportional-integral control method, and fig. 8(b) is an experimental waveform diagram of a load current surge corresponding to the control method of the present invention;
fig. 9 is an experimental waveform diagram of a load current sudden decrease, wherein fig. 9(a) is an experimental waveform diagram of a load current sudden decrease corresponding to a proportional-integral control method, and fig. 9(b) is an experimental waveform diagram of a load current sudden decrease corresponding to a control method of the present invention;
fig. 10 is an experimental waveform diagram of the inductance L of 200 μ H of the CCM Buck converter differential flatness control method based on the state feedback precise linearization of the present invention, wherein fig. 10(a) keeps R of 50 Ω and UiFIG. 10(b) is a waveform diagram of an experiment when the voltage is suddenly increased from 20V to 40V, and U is maintainedi＝20V，IoExperimental waveform plot when the spike from 0.125A to 1A;
fig. 11 is an experimental waveform diagram of the inductor-capacitor C of 90 μ F of the CCM Buck converter differential flatness control method based on state feedback precise linearization of the present invention, where fig. 11(a) keeps R of 50 Ω and UiFIG. 11(b) is a waveform diagram of an experiment when the voltage is suddenly increased from 20V to 40V, and U is maintainedi＝20V，IoExperimental waveform plot with a spike of 0.125A to 1A.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
The invention discloses a CCM Buck converter differential flat control method based on state feedback accurate linearization, wherein the CCM Buck converter is shown in figure 1 and comprises a switching tube S, a diode D, a filter inductor L, an output capacitor C and a load resistor R, and the input voltage is uiOutput voltage of uoThe inductor current is iLEstablishing CCM BuThe feedback linearization model of the ck converter is used for carrying out differential flattening control on the CCM Buck converter based on the feedback linearization model, and carrying out feedback compensation design on the CCM Buck converter according to a pole allocation method, so that the system output can accurately follow an expected value, the variation trend of the system characteristic value is analyzed through a sensitivity theory when circuit parameters are changed, and the transient performance of the CCM Buck converter system is improved, and the feedback linearization model comprises the following steps:
the step 1 specifically comprises the following steps: establishing a feedback linearization model
Establishing a system state equation according to a state space average method:
Figure BDA0003277612640000101
in the formula (1), u is the duty ratio of the main power switch tube S;
selecting a system state variable x ═ x1,x2]＝[iL,uo]Formula (1) can be rewritten as:
Figure BDA0003277612640000102
in the formula (2), y is a system output variable, UrefIs a desired value of the output voltage, and
Figure BDA0003277612640000103
Figure BDA0003277612640000104
the feedback linearization converts the nonlinearity of the Buck converter into a linear model by means of coordinate transformation, and obtains an expression of the control rate. From differential geometry theory, it can be shown that equation (2) satisfies the condition of linearization and derives the following lie derivatives:
Figure BDA0003277612640000105
according to the formula (3), the relative order r of the CCM Buck converter is 2, the system dimension is n, r reflects the entire dynamic state of the system, i.e. the system is described by n-r state variables, and there is no other part, all the state variables are observable after state feedback, and there is no problem that may cause dynamic instability in the system, so as to further define the following coordinate transformation:
Figure BDA0003277612640000111
thus, it is possible to obtain:
Figure BDA0003277612640000112
since the formula (5) is a nonsingular array, the transformed global differential homomorphism is that the Brunox standard type of the system state equation after coordinate transformation is as follows:
Figure BDA0003277612640000113
in equation (6), v is a new input variable and has:
Figure BDA0003277612640000114
let the system state feedback rate control u described by equation (7) be denoted as u1I.e. by
Figure BDA0003277612640000115
Step 2: flat controller design
Step 2.1, flatness analysis
In order to improve the anti-interference capability of the system, the CCM Buck converter is subjected to flat control, and the method is characterized in that all state quantities and control quantities of the system are integrally described through expected flat output quantities. The basic meanings are as follows: if a flat output y can be found in the system control process and the state variable x and the control variable u can be expressed as y and a function of finite differentiation of each order of y, the system can be controlled by differential flat, namely, for an n-order nonlinear system as shown below:
Figure BDA0003277612640000116
there is y ∈ R, such that
Figure BDA0003277612640000121
In the formula (9), i, j are integers,
Figure BDA0003277612640000122
and γ is a scalar function;
before differential flattening control, appropriate flattening output quantity, state variables and control variables need to be selected again, and the analysis shows that the CCM Buck converter meets the sufficient requirements for accurate feedback linearization, so that u can be made to be v, and the following conditions are provided:
Figure BDA0003277612640000123
when formula (10) is substituted for formula (1), it is possible to obtain:
Figure BDA0003277612640000124
according to the equation (11), the system satisfies the requirement of the differential flat theory, namely:
Figure BDA0003277612640000125
Figure BDA0003277612640000126
by substituting equation (13) for equation (7), we can find the differential flat control rate based on the accurate linearization of the state feedback:
Figure BDA0003277612640000127
step 2.2, feedback Compensation design
In order to make the output of the system accurately follow the expected value, a pole allocation method is adopted for the CCM Buck converter to carry out feedback compensation design, and for the system described by the formula (1), a feedback control rate u is obtained2：
u2＝-Kx+kip (15)
Wherein K is [ K ]1 K2]For the state feedback gain matrix, kiIs the integral coefficient, p ═ jeopardy (U)ref-uo)dt。
Setting system damping ratio zeta 0.707 and regulating time tsThe natural oscillation frequency of the closed loop system is calculated as:
Figure BDA0003277612640000131
the system expected closed loop pole is determined by equation (16):
Figure BDA0003277612640000132
taking p as an additional state vector, the three-order augmentation system is formed by the controlled system described by the formula (1):
Figure BDA0003277612640000133
in the formula
Figure BDA0003277612640000134
From equation (18), the system also needs to determine an expected closed-loop pole, rootSystem expectation pole s determined according to system dynamic index1,2The dominant pole of the closed-loop system is determined, and then a non-dominant pole is determined, wherein the non-dominant pole is selected as follows:
s3＝-nζωr (19)
wherein n is a constant, and n is 5;
calculating a state feedback gain matrix K of the system in MATLAB through an acker function, wherein the state feedback gain matrix K is 4.06-167.8]，ki＝1051998.5；
By combining the equations (14) and (15), the control rate of the system of equation (1) can be obtained as follows:
Figure RE-GDA0003391469960000135
the step 3 specifically comprises the following steps: sensitivity analysis and improvement of transient performance of CCM Buck converter system
The differential flat control rate determined by equation (20) includes both the time-varying parameter uo、iLAnd constant parameter L, C, uiAnd R, wherein the invariant parameters L and C have great influence on the transient performance (especially stability) of the system, and in order to clarify the influence of the change of L and C on the transient performance of the system, the system is analyzed by applying a sensitivity theory: the Jacobian matrix of the CCM Buck converter closed loop system can be obtained according to the formula (1):
Figure BDA0003277612640000141
in the formula (21)
Figure BDA0003277612640000142
The characteristic equation of the Jacobian matrix described by equation (21) is:
det(λI-J)＝0 (22)
obtaining a system Jacobian matrix J and a characteristic value lambdaiOn the basis, the related parameters can be optimized according to the result of sensitivity analysis, so that the reliability of the system is improved. The expression for the characteristic root sensitivity is:
Figure BDA0003277612640000143
in the formula (23), piIs the right eigenvector of J, qiIs the left eigenvector of J, σ is the real part of the eigenvalue sensitivity, and ω is the imaginary part of the eigenvalue sensitivity. According to Floquet theory, when σ>At 0, decreasing the parameter b will result in λiThe left half part of the complex plane is moved, so that the stability of the system is improved; when sigma is<At 0, increasing the parameter b will result in λiAnd the left half part of the complex plane is moved, so that the stability of the system is improved.
For ease of analysis, a set of circuit parameters for a CCM Buck converter are now given: input voltage u i20V, 20 Ω for load R, 1mH for inductance L, and 50 μ F for capacitance C. The sensitivity calculation was performed on the converter circuit parameters (L and C) according to equation (23), and the calculation results are shown in table 1.
TABLE 1 sensitivity of characteristic values to parameters
Figure BDA0003277612640000144
Figure BDA0003277612640000151
Analysis table 1 shows that the real part of the sensitivity of the inductor L is a positive value, which indicates that as L decreases, the eigenvalue moves to the left half of the complex plane, and the system stability is enhanced; and the real part of the sensitivity of C is a negative value, which shows that the characteristic value moves to the left half part of the complex plane along with the increase of C, and the stability of the system is enhanced. The variation trend of the obtained system characteristic value is shown in fig. 4 by reducing the inductance L and increasing the value of the capacitance C.
As can be seen from the analysis of FIG. 4(a), the conjugate characteristic root λ of the system is decreased from 1mH to 100 μ H in the process of the inductance L1,2And dominant feature root λ3While moving away from the virtual axis to the left half of the complex plane, transition of the systemThe process is fast and the stability is enhanced. As can be seen from the analysis of FIG. 4(b), the conjugate characteristic root λ increases as the capacitance C increases from 50 μ F to 90 μ F1,2Moving from the left half part of the complex plane to the virtual axis direction, the change trend of the real part is slow; dominant feature root λ3And the device is far away from the virtual axis, and moves to the left half part of the complex plane quickly, so that the transition process of the system becomes fast, and the stability is enhanced.
Experimental verification is performed below.
In order to accurately analyze the effectiveness and superiority of the control method, a model machine of the CCM Buck converter is built for experiment, and circuit parameters are shown in Table 2.
TABLE 2 Buck converter circuit parameters
Figure BDA0003277612640000152
Figure BDA0003277612640000161
FIG. 5 shows the steady state output voltage waveform of the control method of the present invention, and it can be seen that when the input voltage U is appliediWhen R is 20 Ω at 20V, the output voltage u isoStabilize at the given value of 10V.
In order to illustrate the superiority of the control method of the invention, an experimental platform for proportional-integral control is also built, and the circuit parameters of the two control methods are completely the same. Keeping the load R at 50 omega to make the input voltage UiJumps between 20V and 40V resulted in the experimental waveforms shown in fig. 6 and 7.
As can be seen from fig. 6(a) and 7 (a): for a proportional-integral controlled CCM Buck converter, UiWhen the voltage is suddenly increased from 20V to 40V, the adjusting time of the system is 32ms, and the overshoot is 8.5V; u shapeiDuring sudden decrease, the adjustment time of the system is 44ms, and the overshoot is 4.0V. As can be seen from fig. 6(b) and 7 (b): for CCM Buck converter, U, controlled by the control method of the inventioniDuring sudden increase, the adjustment time of the system is 16.5ms, and the overshoot is 2.0V; u shapeiAdjustment of the system during sudden decreaseThe saving time is 24.5ms, and the overshoot is 2.0V. The control method is more beneficial to the adjustment of the transient process of the system.
When U is turnediWhen the voltage is equal to 20V, the load current I is enabledoJumping between 0.125A to 1A, the experimental waveforms shown in fig. 8 and 9 can be obtained.
As can be seen from fig. 8 and 9: when I isoWhen the voltage is suddenly increased to 1A from 0.125A, the adjusting time of the CCM Buck converter controlled by the proportional-integral control is 26.0ms, and the overshoot is 6.0V; the system output regulation time controlled by the control system is 12.5ms, and the overshoot is 2.0V. I isoDuring sudden reduction, the adjusting time of the CCM Buck converter controlled by the proportional-integral is 20.5ms, and the overshoot is 6.0V; the adjustment time controlled here was 18.5ms and the overshoot was 1.5V.
Table 3 shows a comparison of the experimental results of the two control methods under different disturbance conditions, and it can be seen that the control method herein is superior to proportional-integral control in both overshoot voltage and regulation time under the same circuit parameters.
TABLE 3 comparison of two control method experiments
Tab.3 Experimental comparison of two control strategies
Figure BDA0003277612640000171
As can be seen from the system sensitivity analysis in step 3 of the present invention, decreasing the inductance L and increasing the capacitance C are beneficial to improving the transient performance of the system, and in order to verify the correctness of the theoretical analysis, on the basis of the control method herein, the influence of the inductance L ═ 200 μ H and the capacitance C ═ 90 μ F on the transient performance of the system is selected for comparative analysis.
Fig. 10(a) shows an input voltage U held at 50 μ F or 50 ΩiExperimental waveforms spiked from 20V to 40V; FIG. 10(b) shows the result of holding C50. mu.F, UiAt 20V, make the load current IoExperimental waveforms with a spike from 0.125A to 1A. As can be seen from FIG. 10, the inductances L, U are reducediWhen the voltage is suddenly increased from 20V to 40V, the system adjusting time is 8.0ms, and the overshoot is 1.0V; i isoWhen the system suddenly increases from 0.125A to 1A, the system adjustsThe time is 10.0ms and the overshoot is 2.0V.
Fig. 11(a) shows an input voltage U held at 1mH and 50 ΩiSpike from 20V to 40V experimental waveform; FIG. 11(b) shows that L is 1mH, UiAt 20V, make the load current IoExperimental waveforms with a spike from 0.125A to 1A. As can be seen from FIG. 11, the capacitances C, U are increasediWhen the voltage is suddenly increased from 20V to 40V, the system adjusting time is 10.0ms, and the overshoot is 2.0V; i isoWhen the system is suddenly increased from 0.125A to 1A, the system adjusting time is 8.0ms, and the overshoot is 2.0V. As can be seen from fig. 10 and 11, decreasing the inductance L and increasing the capacitance C is beneficial to improving the transient performance of the system, and the analysis result of the system sensitivity analysis in step 3 of the present invention is verified.
In summary, the differential flatness control method of the CCM Buck converter based on the state feedback accurate linearization firstly establishes an affine nonlinear model of the system, applies a differential geometry theory, accurately linearizes the nonlinear system, and solves the variable structure characteristic of the system; then, a differential flatness control method is adopted to design the state feedback control rate of the system, so that the anti-interference capability of the system is improved; meanwhile, a pole allocation method is utilized to carry out feedback compensation design on the system, so that the output of the system can accurately follow a given value; the influence of the circuit parameters on the transient performance of the system is analyzed through the sensitivity of the characteristic values, and the result shows that the reduction of the inductance L and the increase of the capacitance C are beneficial to improving the transient performance of the system. The final experiment result shows that the control method provided by the invention has stronger robustness and has important reference significance for the optimization of the control method of the CCM Buck converter.

Claims (5)

1. The CCM Buck converter comprises a switching tube S, a diode D, a filter inductor L, an output capacitor C and a load resistor R, and the system input voltage is uiThe system output voltage is uoThe inductor current is iLThe method is characterized in that a feedback linearization model of the CCM Buck converter is established, differential flatness control is carried out on the CCM Buck converter based on the model, and feedback compensation design is carried out on the CCM Buck converter according to a pole allocation method, so that the system is enabled to beThe output energy accurately follows the expected value, the change trend of the system characteristic value is analyzed through the sensitivity theory when the circuit parameter changes, and the transient performance of the CCM Buck converter system is improved, and the method comprises the following steps:
step 1: a feedback linearization model is established and is used for carrying out the feedback linearization on the target object,
step 2: the design of the flat controller is that the flat controller,
step 2.1: carrying out flatness analysis;
step 2.2: designing feedback compensation;
and step 3: sensitivity analysis is carried out, and transient performance of the CCM Buck converter system is improved.
2. The differential flatness control method for the CCM Buck converter based on the state feedback precise linearization as claimed in claim 1, wherein the step 1 is specifically as follows:
establishing a system state equation according to a state space average method:
Figure FDA0003277612630000011
in the formula (1), u is the duty ratio of the main power switch tube S.
Selecting a system state variable x ═ x1,x2]＝[iL,uo]Formula (1) can be rewritten as:
Figure FDA0003277612630000012
in the formula (2), y is a system output variable, UrefIs a desired value of the output voltage, and
Figure FDA0003277612630000021
Figure FDA0003277612630000022
the feedback linearization converts the nonlinearity of the Buck converter into a linear model by means of coordinate transformation, and obtains an expression of the control rate. From differential geometry theory, it can be shown that equation (2) satisfies the condition of linearization and derives the following lie derivatives:
Figure FDA0003277612630000023
according to the formula (3), the relative order r of the CCM Buck converter is 2, the system dimension is n, r reflects the entire dynamic state of the system, that is, the system is described by n-r state variables, and there is no other part, all the state variables are observable after state feedback, and there is no problem that the dynamic state in the system may be unstable, so that the following coordinate transformation is further defined:
Figure FDA0003277612630000024
thus, it is possible to obtain:
Figure FDA0003277612630000025
since the formula (5) is a nonsingular array, the transformed global differential homomorphism is as follows, and the Brunox standard type of the system state equation after coordinate transformation is as follows:
Figure FDA0003277612630000026
in equation (6), v is a new input variable and has:
Figure FDA0003277612630000031
let the system state feedback rate control u described by equation (7) be denoted as u1I.e. by
Figure FDA0003277612630000032
3. The differential flatness control method for the CCM Buck converter based on the state feedback precise linearization as claimed in claim 2, wherein the step 2.1 is specifically as follows:
in order to improve the anti-interference capability of the system, the CCM Buck converter is subjected to flat control, and the method is characterized in that all state quantities and control quantities of the system are integrally described through expected flat output quantities. The basic meanings are as follows: if a flat output y can be found in the system control process and the state variable x and the control variable u can be expressed as y and a function of finite differentiation of each order of y, the system can be controlled by differential flat, namely, for an n-order nonlinear system as shown below:
Figure FDA0003277612630000033
there is y ∈ R, such that
Figure FDA0003277612630000034
In the formula (9), i, j are integers,
Figure FDA0003277612630000035
and γ is a scalar function;
before differential flattening control, appropriate flattening output quantity, state variables and control variables need to be selected again, and the analysis shows that the CCM Buck converter meets the sufficient requirements for accurate feedback linearization, so that u can be made to be v, and the following conditions are provided:
Figure FDA0003277612630000036
when formula (10) is substituted for formula (1), it is possible to obtain:
Figure FDA0003277612630000037
according to the equation (11), the system satisfies the requirement of the differential flat theory, namely:
Figure FDA0003277612630000041
Figure FDA0003277612630000042
by substituting equation (13) for equation (7), we can find the differential flat control rate based on the accurate linearization of the state feedback:
Figure FDA0003277612630000043
4. the differential flatness control method for the CCM Buck converter based on the state feedback precise linearization according to claim 3, characterized in that the step 2.2 is specifically as follows:
in order to make the output of the system accurately follow the expected value, a pole allocation method is adopted for the CCM Buck converter to carry out feedback compensation design, and for the system described by the formula (1), a feedback control rate u is obtained2：
u2＝-Kx+kip (15)
In formula (15), K ═ K1 K2]For the state feedback gain matrix, kiIs the integral coefficient, p ═ jeopardy (U)ref-uo)dt；
Setting system damping ratio zeta 0.707 and regulating time tsThe natural oscillation frequency of the closed loop system is calculated as:
Figure RE-FDA0003391469950000044
the system expected closed loop pole is determined by equation (16):
Figure RE-FDA0003391469950000045
taking p as an additional state vector, the three-order augmentation system is formed by the controlled system described by the formula (1):
Figure RE-FDA0003391469950000046
in the formula
Figure RE-FDA0003391469950000047
As shown in the formula (18), the system also needs to determine an expected closed-loop pole, and the system expected pole s is determined according to the dynamic indexes of the system1,2The dominant pole of the closed-loop system is determined, and then a non-dominant pole is determined, wherein the non-dominant pole is selected as follows:
s3＝-nζωr (19)
wherein n is a constant, and n is 5;
calculating a state feedback gain matrix K of the system in MATLAB through an acker function, wherein the state feedback gain matrix K is 4.06-167.8]，ki＝1051998.5；
By combining the equations (14) and (15), the control rate of the system of equation (1) can be obtained as follows:
Figure RE-FDA0003391469950000051
5. the differential flatness control method for the CCM Buck converter based on the state feedback precise linearization as claimed in claim 4, wherein the step 3 is specifically as follows:
the differential flat control rate determined by equation (20) includes both time-varyingParameter uo、iLAnd constant parameter L, C, uiAnd R, wherein the invariant parameters L and C influence the transient performance of the system, and in order to clarify the influence of the L and C changes on the transient performance of the system, the transient performance is analyzed by applying a sensitivity theory: the Jacobian matrix of the CCM Buck converter closed-loop system can be obtained according to the formula (1):
Figure FDA0003277612630000052
in the formula
Figure FDA0003277612630000053
The characteristic equation of the Jacobian matrix described by equation (21) is:
det(λI-J)＝0(22)
obtaining a system Jacobian matrix J and a characteristic value lambdaiOn the basis, relevant parameters can be optimized according to the sensitivity analysis result, so that the system reliability is improved, and the expression of the characteristic root sensitivity is as follows:
Figure FDA0003277612630000061
in the formula (23), piIs the right eigenvector of J, qiIs the left eigenvector of J, σ is the real part of the eigenvalue sensitivity, and ω is the imaginary part of the eigenvalue sensitivity. According to Floquet theory, when σ>At 0, decreasing the parameter b will result in λiThe left half part of the complex plane is moved, so that the stability of the system is improved; when sigma is<At 0, increasing the parameter b will result in λiAnd the left half part of the complex plane is moved, so that the stability of the system is improved.
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115065238A (en) * 2022-08-18 2022-09-16 南京信息工程大学 Integral sliding mode control method and system for DC buck converter with constant power load

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115065238A (en) * 2022-08-18 2022-09-16 南京信息工程大学 Integral sliding mode control method and system for DC buck converter with constant power load
CN115065238B (en) * 2022-08-18 2022-11-18 南京信息工程大学 Integral sliding mode control method and system for DC buck converter with constant power load

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