CN115173701A - Power converter self-adaptive continuous sliding mode control method based on zero-crossing detection - Google Patents
Power converter self-adaptive continuous sliding mode control method based on zero-crossing detection Download PDFInfo
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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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- 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
- H02M1/00—Details of apparatus for conversion
- H02M1/08—Circuits specially adapted for the generation of control voltages for semiconductor devices incorporated in static converters
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Abstract
A self-adaptive continuous sliding mode control method for a power converter based on zero-crossing detection belongs to the technical field of sliding mode control of the power converter. The invention aims at the problem that the traditional fixed gain control technical means cannot converge to the expected steady-state error in the conventional sliding mode control method of the power converter. The method comprises the steps of introducing an integral term of a converter system state on the basis of the traditional first-order sliding mode surface design aiming at two state variables to obtain an improved sliding mode surface s; in the first stage of the motion trail of the state variable, a control law u is designed 1 Enabling the motion trail to reach a first peak position point B within a limited time; control law u 1 Has a fixed control gain; in the second stage, a control law u is designed 2 The system is gradually converged to a zero position point from a first peak position point B; control law u 2 The control device has the characteristic of variable control gain, and the motion tracks of the two state variables are in a spiral characteristic. The invention canEnsuring that the steady state error of the system converges to a given range.
Description
Technical Field
The invention relates to a zero-crossing detection-based power converter self-adaptive continuous sliding mode control method, and belongs to the technical field of sliding mode control of power converters.
Background
The sliding mode control at present mainly has the following problems:
1. the influence of the conventional first-order sliding mode buffeting problem on the electric energy quality of the power converter is as follows:
at present, in the field of sliding mode control power converters, the traditional first-order sliding mode application is mainly used. The first-order sliding mode application is mainly realized by linear sliding mode, terminal sliding mode and nonsingular terminal sliding mode application, and sign functions sgn () exist in the control law; however, in the actual system implementation, because the switching frequency of the existing switching tube is limited, and cannot be theoretically infinite, the sign function sgn (.) can induce the problem of buffeting, which is represented by signal oscillation of limited amplitude and limited frequency of voltage and current, which seriously affects the electric energy quality of the power converter, and can cause a plurality of problems of heating, accelerated aging, serious harmonic wave and the like of the power converter, and the switching tube is the focus of attention and research in the industry at present.
2. Although the high-order sliding mode is regarded as one of the most effective methods for suppressing buffeting at present, the control gain is mostly a fixed value. As the system converges to the equilibrium point, excessive control gain may reduce the dynamic and static control performance of the system, and may also reduce the power quality of the power converter:
compared with methods for weakening buffeting such as a boundary layer method and fuzzy control, a high-order sliding mode is regarded as an effective method for essentially solving the buffeting problem. The control idea is based on the relative order concept, and the actual control quantity is continuous through integration or low-pass filtering by directly adding the switching control sgn (to) to the high-order derivative of the sliding mode variable. The current common algorithms include twist algorithm, super-twist algorithm, suboptimal algorithm, etc. However, the control gain of the conventional high-order sliding mode control method is usually set to a fixed value, and the value thereof depends on the transient performance of the initial stage or the disturbance to be overcome; however, as the system trajectory approaches the equilibrium point, the fixed control gain becomes the key to destroy the steady-state performance of the system, and the excessive control gain determined in the initial stage brings large steady-state error and response time.
3. The high-order sliding mode and the self-adaptive control are combined, and the problem of fixed gain of the traditional high-order sliding mode can be solved through variable gain control. However, the existing adaptive methods are few, and most of them only consider the stability index, and cannot converge to the expected steady-state error:
in order to overcome the problem of large steady-state error of the system caused by fixed control gain, a variable control gain sliding mode control method is developed. At present, the self-adaptive mechanism mainly has two types based on stability and switching time principle. The former is based on the premise of ensuring the stability of the system, and some specific control performance indexes such as steady-state errors are less considered; the latter is based on the switching time principle, and the control idea follows the inherent switching characteristic of sliding mode control. However, the adaptive mechanism depends on the motion trajectory of the system approaching the equilibrium point, and there is no study on directly establishing the influence relationship between gain variation and some performance indexes such as steady-state error, so that the key problems of the variable gain mechanism, the switching time, the system stability, and the like need to be studied in depth.
Disclosure of Invention
Aiming at the problem that the traditional fixed gain control technical means cannot converge to the expected steady-state error in the conventional sliding mode control method of the power converter, the invention provides a self-adaptive continuous sliding mode control method of the power converter based on zero-crossing detection.
The invention relates to a self-adaptive continuous sliding mode control method of a power converter based on zero-crossing detection, which comprises the following steps,
establishing a mathematical model of the Buck type DC-DC converter;
according to the mathematical model, introducing an integral term of a converter system state on the basis of the traditional first-order sliding mode surface design aiming at two state variables to obtain a sliding mode surface s containing the integral term; the state variable is the voltage difference x between the actual output voltage and the target output voltage of the converter 1 And rate of change x of actual output voltage 2 ；
Based on a sliding mode surface s containing an integral term and a switching time principle, dividing the motion tracks of two state variables into two stages, wherein the first stage is from an initial point A to a first peak position point B; in the second stage, the first peak position point B is changed to a zero position point;
in the first stage, a control law u is designed 1 Enabling the motion tracks of the two state variables to reach a first peak position point B within limited time; control law u 1 Has a fixed control gain;
in the second stage, the control law u is designed 2 Enabling the motion tracks of the two state variables to be in a spiral characteristic and gradually converging from a first peak position point B to a zero position point; control law u 2 Has a variable control gain; and the amplitude is adaptively adjusted along with the number of zero-crossing points of the sliding mode surface s detected in the sampling interval by the variable control gain.
According to the zero-crossing detection-based power converter adaptive continuous sliding mode control method of the invention,
the initial mathematical model of a Buck-type DC-DC converter is:
in the formula i L Is the current flowing through the filter inductor, t is the time, L is the filter inductor, u is the control law, E is the DC input voltage, v c C is the actual output voltage of the converter, C is the capacitor, and R is the load resistor;
definition V ref Is the target output voltage, the state variable x 1 ＝v c -V ref ，
Deforming the initial mathematical model to obtain a deformed mathematical model:
in the formula
Is the intermediate variable(s) of the variable,
according to the zero-crossing detection-based power converter adaptive continuous sliding mode control method of the invention,
surface of conventional first order slip form s 0 Comprises the following steps:
s 0 ＝c 1 x 1 +x 2 ，
in the formula c 1 As a first design parameter, c 1 >0；
For the conventional first-order slip form surface s 0 Introducing an integral term of a converter system state to obtain a sliding mode surface s containing the integral term:
in the formula c 2 As a second design parameter, c 2 >0。
According to the zero-crossing detection-based power converter self-adaptive continuous sliding mode control method, in the first stage, a control law u is set as a control law u 1 Control law u 1 The design process of (2) comprises:
defining vectors
Wherein T is a sampling interval;
the formula of the slip form surface s is transformed into:
wherein, the first and the second end of the pipe are connected with each other,
mu is an intermediate variable which is a function of,
design control law u 1 Comprises the following steps:
where U is the fixed control gain.
According to the power converter self-adaptive continuous sliding mode control method based on zero-crossing detection, control law u 1 The process of enabling the motion tracks of the two state variables to reach the first peak position point B in a limited time comprises the following steps:
according to the formula of the deformed sliding mode surface s, solving the second derivative of the sliding mode surface s to time:
in the formula:
y 22 (μ)＝c 1 μ-μβ 1 ，
β 1 ＝1/RC，β 2 ＝ω 0 2 ＝1/LC；
define four constants ζ 1 、ζ 2 、ζ 3 、ζ 4 And function Y 21 、Y 22 The following variable substitutions were made:
the following relation is satisfied:
will control law u 1 Substituting the formula into a second derivative of the sliding mode surface s to time to obtain:
in the formula mu min Is the minimum value of the intermediate variable mu, k is a constant greater than 0;
further, obtaining:
according to
Both sides are multiplied by s, i.e. by
If true, then for the motion trajectories of the two state variables, a finite time t is reached from the initial point A 0 The position B of the first peak is reached;
wherein t is 0 ＝t B -t A ，
Wherein t is A Is the time corresponding to the initial point A, t B The time corresponding to the first peak position point B.
According to the zero-crossing detection-based power converter self-adaptive continuous sliding mode control method, the fixed control gain U is as follows:
according to the zero-crossing detection-based power converter self-adaptive continuous sliding mode control method, in the second stage, a control law u is set as a control law u 2 Control law u 2 The design is as follows:
in the formula of U j For varying the control gain, r 4 Controlling the gain for a preset fixed value r 4 >0。
According to the zero-crossing detection-based power converter self-adaptive continuous sliding mode control method, the variable control gain self-adaptive amplitude adjustment value assignment method comprises the following steps:
formula is N j Is the jth sampling interval T j The number of zero crossing points of the inner s is N, the N is a reference value of the number of the zero crossing points, and N is more than or equal to 2; lambda 1 And Λ 2 Is two positive numbers, Λ 1 <Λ 2 ；T＝{T 1 ,T 2 ,...T i J =1,2,3, … … i; i is the total number of sampling intervals; u shape 0 Is control law u 2 Initial value of (a):
U 0 ＝[Y 21 (||w|| * ,|s(t 0 )|)+Y 22 u 1max +k]，
in the formula u 1max To control law u 1 Maximum value of (d):
according to the power converter self-adaptive continuous sliding mode control method based on zero-crossing detection, control law u 2 The convergence range of the slip form surface s is:
|s|≤[U 0 +(1+r 4 )μ max U j ]T 2 ，
in the formula of max Is the maximum value of the intermediate variable μ.
According to the zero-crossing detection-based power converter self-adaptive continuous sliding mode control method, the process of obtaining the convergence range of the sliding mode surface s comprises the following steps:
will control law u 2 Substituting the expression of (b) into a second derivative expression of the sliding mode surface s to time to obtain:
the further deformation is as follows:
suppose i 0 ∈[1,N j ]，j 0 ∈[1,N j ]And i is 0 <j 0 Wherein i 0 Is [1,N j ]One of all zero-crossing points, j 0 Is [1,N j ]The other zero-crossing point of the zero-crossing points, then:
s(t i0 )＝s(t j0 )＝0；
according to N is more than or equal to 2, a time point t exists i0j0 ：t i0 <t i0j0 <t j0 So that
(ii) present;
according to the Lagrange median theorem, at t and t i0j0 Memory existence time t i0j0 ', satisfies the following relation:
t is T j At any time during the sampling interval, and t-t i0j0 |<T j ；
Similarly, at t and t i0 There is another time t i0 ' satisfies:
according to | t-t i0 |<T, upper formulaThe deformation is as follows:
and (3) simultaneously integrating the two ends of the above formula to obtain the convergence range of the sliding mode surface s:
|s|≤[U 0 +(1+r 4 )μ max U j ]T 2 。
the invention has the beneficial effects that: the method is provided based on an online zero-crossing detection self-adaptive mechanism, can effectively inhibit the buffeting problem, and can ensure that the steady-state error of the system is converged to a given range.
Firstly, establishing a mathematical model of the converter, improving the traditional sliding mode control algorithm from two aspects of a sliding mode surface and a control law, namely purposefully introducing an integral term of a system state into the design of the sliding mode surface, dividing a convergence track into two stages, and measuring the number of zero-crossing points of the system in real time; and incorporating the expected steady-state error into the design of a sliding mode control law, introducing a low-pass filtering link, deducing a continuous control law of variable gain of the sliding mode control law in stages under the constraint of a convergence track, and providing corresponding stability analysis.
Simulation and performance comparison experiments prove that the method has remarkable superiority in buffeting suppression, response speed and control precision compared with the prior art.
Drawings
FIG. 1 is a block diagram of a sliding mode control system of a Buck type DC-DC converter; the SMC controller in the figure represents a sliding mode controller; s w Is a controllable switching tube, VD is a current-limiting diode, i C In order for the current to flow through the capacitor C,
FIG. 2 is a schematic diagram of the convergence process of adaptive continuous sliding mode control corresponding to the motion trajectories of two state variables;
FIG. 3 is a schematic diagram of the number of zero crossings of s within a single sampling interval;
FIG. 4 is a simulation diagram of the actual output voltage of the converter controlled by three methods under the rated operating condition in the specific embodiment;
FIG. 5 is a schematic diagram illustrating the simulation of the inductor current for controlling the converter in three ways under the rated operating conditions in the exemplary embodiment;
FIG. 6 is a schematic diagram of the control law for controlling the converter in three ways under the rated operating conditions in the exemplary embodiment;
FIG. 7 shows the number of different zero-crossing points N when the converter is controlled by three methods under rated conditions * A lower output voltage schematic;
FIG. 8 is a simulation diagram of the actual output voltage of the converter controlled by three methods under the disturbance condition in the embodiment;
FIG. 9 is a schematic diagram of an inductor current simulation for controlling a converter by three methods under disturbance conditions in an exemplary embodiment;
FIG. 10 is a diagram illustrating the control laws for controlling the converter in three ways under disturbance conditions according to the exemplary embodiment.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
The invention is further described with reference to the following drawings and specific examples, which are not intended to be limiting.
First embodiment, referring to fig. 1, the present invention provides a power converter adaptive continuous sliding mode control method based on zero-crossing detection, including,
establishing a mathematical model of the Buck type DC-DC converter;
according to the mathematical model, in a conventional first order slip for two state variablesIntroducing an integral term of a converter system state on the basis of the design of a die surface to obtain a sliding mode surface s containing the integral term; the state variable is the voltage difference x between the actual output voltage and the target output voltage of the converter 1 And rate of change x of actual output voltage 2 ；
Based on a sliding mode surface s containing an integral term and a switching time principle, dividing the motion tracks of two state variables into two stages, wherein the first stage is from an initial point A to a first peak position point B; in the second stage, the first peak position point B is changed to a zero position point;
in the first stage, a control law u is designed 1 Enabling the motion tracks of the two state variables to reach a first peak position point B within a limited time; control law u 1 Having a fixed control gain;
in the second stage, a control law u is designed 2 Enabling the motion tracks of the two state variables to be in a spiral characteristic and gradually converging from a first peak position point B to a zero position point; control law u 2 Has a variable control gain; and the amplitude is adaptively adjusted along with the number of zero-crossing points of the sliding mode surface s detected in the sampling interval by the variable control gain.
Establishing a power converter system model:
the method of the invention can be applied to various power converters such as DC-DC, AC-AC and the like. Fig. 1 is a system block diagram for sliding mode control of a Buck-type DC-DC converter. Controllable switch tube S w The MOSFET and the IGBT are often applied mostly, and a pulse width modulation method is often adopted. The controllable switching tube is controlled by a designed control law u.
Further, in practical system applications, the power converter is operated in a continuous current mode, i.e. the inductor current i L Not equal to 0, and then establishing an initial mathematical model of the Buck type DC-DC converter shown in the figure 1 based on the Kirchoff circuit law as follows:
in the formula i L The current flowing through the filter inductor, t is time, and L is filter currentThe inductance u is the control law, E is the DC input voltage, v c C is the actual output voltage of the converter, C is a capacitor, and R is a load resistor;
definition V ref Is the target output voltage, then the state variable x 1 ＝v c -V ref ，
Deforming the initial mathematical model to obtain a deformed mathematical model:
in the formula
Is the intermediate variable(s) of the variable,
conventional first order sliding mode control: aiming at the mathematical model of the converter in the formula (2), the design of the sliding mode controller comprises a sliding mode surface and a control law. Typically, a conventional first order slip form surface s 0 Comprises the following steps:
s 0 ＝c 1 x 1 +x 2 ， (3)
in the formula c 1 As a first design parameter, c 1 >0；
x 1 And x 2 The voltage can be directly obtained by measuring the voltage and the current by utilizing the Hall sensor, and the method is simple and easy to realize. Once the converter control system converges to the slip-form surface s 0 =0, the dynamic and static performance of the system depending on
I.e. deviation of output voltage
Asymptotically converges to zero in exponential form, and design parameter c 1 The larger, the largerThe faster the system converges.
In the aspect of control law, the design of the first-order sliding mode and the second-order sliding mode control law needs to meet the sliding mode arrival condition
To ensure system stability. The buffeting inhibition mechanism based on the relative order from the high-order sliding mode is different from the buffeting inhibition mechanism based on the relative order: the first-order sliding mode SMC directly acts a switching control term sgn (.) on a first derivative of a sliding mode variable
To ensure a first-order slip mode s 0 If =0, then from equation (3), the Buck converter output voltage deviation and its derivative x are realized 1 ＝x 2 =0, but there is a buffeting problem. For a second-order sliding mode, taking a commonly used Twisting algorithm as an example, the control law of the sliding mode is usually designed as follows:
r 1 and r 2 Are all control gains, and r 1 >r 2 >0，r 1 、r 2 Is related to the response speed of the system and the steady state error. It can be seen that the first derivative of the control law occurs due to the switching of the control term sgn (.)
In the above, the actual output u is made continuous through the integration, which is the reason why the Twisting algorithm effectively solves the buffeting problem. It should be noted, however, that the gain r is controlled 1 And r 2 The entire process of the system converging to the origin remains the same, however, the closer to the origin, the larger the control gain, the worse the steady state performance of the system.
Improved adaptive second-order SMC control:
in order to improve the traditional second-order sliding mode fixed control gain problem of the formulas (3) and (4), the method is improved from two aspects of a sliding mode surface and a control law.
Designing a sliding mode surface:
for the conventional first-order slip form surface s 0 Introducing an integral term of a converter system state to obtain a sliding mode surface s containing the integral term:
in the formula c 2 As a second design parameter, c 2 >0。
In actual use, the first design parameter c 1 And a second design parameter c 2 The adjustment is carried out according to the use condition.
In the first stage, the control law u is set as the control law u 1 Control law u 1 The design process of (2) comprises:
defining vectors
Wherein T is a sampling interval;
in conjunction with equation (2), equation (5) can be transformed into:
wherein, the first and the second end of the pipe are connected with each other,
mu is an intermediate variable which is a function of,
designing a control law:
in the design of the control law u, the method of the invention is based on the switching time principle, and according to the system convergence process, as shown in fig. 2, the motion track is divided into two stages, namely, the 1 st stage is from the initial point A to the first peak position point B, and the 2 nd stage is after the point B, the 1 st stage is divided into sampling intervals with equal intervals TDenoted as { T 1 ,T 2 ,...T i }. In particular, the desired steady-state error Δ is incorporated into equation (4) as an improvement of the conventional Twisting control law, the two-phase control law u being decomposed into u 1 And u 2 Two parts, the design process is as follows.
First phase, movement of point a to point B:
in FIG. 2, assume that the time of the initial point A is t A The corresponding position is (t) A ,s A ) The time of the first peak position point B is t B The corresponding position is (t) B ,s B ) And is provided with
Comparative formula (4), design control law u 1 Comprises the following steps:
where U is the fixed control gain.
The fixed control gain U is:
wherein k is>0 is a constant; (ii) a Mu.s max And mu min Are the maximum and minimum values of μ defined by equation (7), respectively.
Control law u 1 The process of enabling the motion tracks of the two state variables to reach the first peak position point B in a limited time comprises the following steps:
similarly to equation (6), according to the formula of the sliding mode surface s after deformation, and further calculating the second derivative of the sliding mode variable s with respect to time here, the switching control term sgn (·), appears, that is, the second derivative of the sliding mode surface s with respect to time can be derived from equation (6):
in the formula:
y 22 (μ)＝c 1 μ-μβ 1 ， (10)
β 1 ＝1/RC，β 2 ＝ω 0 2 ＝1/LC；
for convenience of the following explanation, four constants ζ are defined by the formulas (9) to (10) 1 、ζ 2 、ζ 3 、ζ 4 And function Y 21 、Y 22 The following variable substitutions were made:
the following relation is satisfied:
theorem 1: for the Buck converter in the formula (2), if the sliding mode surface is designed as the formula (5), and the control law in the first stage is designed as the formulas (14) - (15), the limited time of the system reaches the point B.
First, the existence of point B in FIG. 2 is proved, because it is the first peak position point, which satisfies
For this purpose, equation (14) is substituted into equation (8), i.e. control law u 1 When the formula is substituted into the second derivative of the sliding mode surface s to time, and the combined type (12) has
In the formula mu min Is the minimum value of the intermediate variable mu, k is a constant greater than 0;
further, obtaining:
according to
Both sides are multiplied by s, i.e. by
If true, then for the motion trajectories of the two state variables, from an arbitrary initial point A, at a finite time t 0 Internally reaching a first peak position point B;
wherein t is 0 ＝t B -t A ，
Wherein t is A Is the time corresponding to the initial point A, t B The time corresponding to the first peak position point B.
Further, in the second phase, the convergent movement after point B:
in FIG. 2, when t is>t B Thereafter, the system enters a second stage of converging motion. Comparing the control law u with the control law u in the formula (5) 2 Control law u 2 The design is as follows:
in the formula of U j For varying the control gain, r 4 Controlling the gain for a predetermined fixed value r 4 >0。
The variable control gain adaptive amplitude adjustment value assignment method comprises the following steps:
according to variable control gain U j At T j The sampling interval is adaptively changed by detecting s zero crossing points, namely:
is of the formula N j For the jth sampling interval T j The number of zero crossing points of the inner s, N is a reference value of the number of the zero crossing points, and N is more than or equal to 2; lambda 1 And Λ 2 Is two positive numbers, Λ 1 <Λ 2 ；T＝{T 1 ,T 2 ,...T i J =1,2,3, … … i; i is the total number of sampling intervals; u shape 0 Is control law u 2 Corresponding to the first stage B point control law u 1 The maximum value, represented by equation (14):
U 0 ＝[Y 21 (||w|| * ,|s(t 0 )|)+Y 22 u 1max +k]， (20)
in the formula u 1max To control law u 1 The maximum value of (a) is, | w | non-woven phosphor * At the upper limit of w, there are:
theorem 2: for the Buck converter in the formula (2), if the improved sliding mode surface is designed as the formula (5), and the variable gain control law in the second stage is designed as the formulas (18) to (19), the control law u can be ensured 2 The convergence range of the slip form surface s is:
|s|≤[U 0 +(1+r 4 )μ max U j ]T 2 ， (22)
in the formula of max Is the maximum value of the intermediate variable μ.
The process of obtaining the convergence range of the sliding-mode surface s includes:
will control law u 2 Substituting expressions (18) to (19) of (a) into a second derivative expression (8) of the sliding mode surface s with respect to time, obtains:
from fig. 2, point B is the first peak position in the first stage and is also the point where the oscillation amplitude in the second stage is maximum. Thus, combining formulae (8), (12), (13) and formulae (20), (21), then (23) can be further modified to:
by T j The sampling interval is taken as an example, and the analysis condition of the zero crossing point of the system in a single sampling interval T is given. As in FIG. 3, assume i 0 ∈[1,N j ]，j 0 ∈[1,N j ]And i is 0 <j 0 Wherein i is 0 Is [1,N j ]One of all zero-crossing points, j 0 Is [1,N j ]The other zero-crossing point is shown in fig. 3:
s(t i0 )＝s(t j0 )＝0；
because the zero crossing given value N is more than or equal to 2, it means that at least two zero crossings occur in a single sampling interval T, and a certain time T must exist according to the Rohr's theorem i0j0 ：t i0 <t i0j0 <t j0 So that
(ii) present;
according to the Lagrange median theorem, at t and t i0j0 Memory existence time t i0j0 ' by the formula (24), the following relational expression is satisfied:
wherein T is T j At any time during the sampling interval, and t-t i0j0 |<T j ；
Similarly, at t and t i0 There is another time t i0 ' satisfy:
according to | t-t i0 |<T, formula (26) is modified as:
and (3) integrating the two ends of the above formula simultaneously to obtain the convergence range of the sliding mode surface s:
|s|≤[U 0 +(1+r 4 )μ max U j ]T 2 。
it should be particularly noted that during the second phase of the motion of fig. 2, as the system approaches the equilibrium point, the amplitude of s becomes smaller and the number of zero crossings in the same sampling interval increases. From the formula (18), N j The magnitude of the relation to N will influence the next sampling interval T j+1 Control gain U of j+1 Therefore, the choice of the given value N is of great importance. In practical systems, N may be measured experimentally, and N = max {2Tf may be taken j +1}, wherein f j ＝N j and/T is the frequency of the experimentally measured s zero crossing.
The specific embodiment is as follows:
in order to verify the superiority of the continuous sliding mode control method based on the online zero-crossing detection self-adaptive mechanism provided by the method in the aspects of buffeting suppression, response speed and control precision, the performance of the method is compared with that of a first-order sliding mode method and a second-order sliding mode method represented by a traditional Twisting algorithm, and table 1 shows circuit parameters of a converter. For convenience of explanation, the first-order sliding mode method is represented by "1-SMC", the second-order sliding mode method represented by "2T-SMC" representing the traditional Twisting algorithm, and the "2AT-SMC" represent the method of the invention respectively.
TABLE 1 Circuit parameters of the converter
For the Buck converter of the formula (2), the sliding mode surfaces of 1-SMC and 2T-SMC adopt the form of the formula (3), and the design parameter c 1 The control law selected as 100,1-SMC is designed as u =0.5[ sgn(s) -1 ]]In practical systems, hysteresis modulation is often used for mitigationBuffeting problem, where the hysteresis loop width is taken to be 0.01; control gain r in formula (4) of 2T-SMC 1 Take 240,r 2 Taking 120; for the 2AT-SMC method provided by the invention, the sliding mode surface parameter c of the formula (5) 1 Still take 100,c 2 0.001 is taken, 75 is taken as the control gain U in the first stage of equation (14), and r is taken as the design parameter in the second stage of equations (18) to (19) 4 ＝0.541，Λ 1 ＝2，Λ 2 ＝4，N*＝8，T＝25μs。
In the following, the control performance of the Buck converter under the action of the three methods is compared by taking two conditions of rated working condition and input voltage disturbance as examples.
(1) Rated working condition:
the control performance pairs for the three methods at nominal operation are shown in fig. 4-7 and table 2, where fig. 4 is the output voltage v c And FIG. 5 shows the inductor current i L The simulation results show that the three methods all realize convergence control of the two methods, and output voltage v c Convergence to a given value V ref =5V, with a maximum steady state error of 13.01mV for 1-SMC, followed by 6.04mV for 2T-SMC, and the best steady state performance for 2AT-SMC, with a steady state error of only 1.03mV. Comparing equations (3) and (5), the good effect of the method of the present invention is attributed to the fact that the 2AT-SMC method introduces an integral term of the system state into the design of the sliding-mode surface. Meanwhile, the convergence speed of the system under the control of the 2AT-SMC is the fastest and is only 0.042s, and in combination with the comparison of the control law u in the figure 6, the effect obtained by the method is attributed to the variable gain control effect of the 2AT-SMC method. Further from fig. 2, equation (14) and theorem 1, it can be seen that the initial motion trajectory oscillation of the system of the present invention is maximum in the first stage, which explains the reason why the amplitude of the control law u of the 2AT-SMC is maximum in this stage; and then in the second stage, as the system tends to converge, the amplitude of the system is the minimum of the three methods, and particularly, the control law u of the 1-SMC has obvious buffeting phenomenon, and even if hysteresis modulation is adopted for relieving, the second-order SMC such as 2T-SMC and 2AT-SMC has no good buffeting suppression performance. Further, in fig. 7, 2,4,8 three different zero crossing point set values N are selected, and the corresponding steady state errors of the output voltage are 12.15mv,6.43mv and 1.03mV, respectively. According to N × = max {2Tf j +1, knowing N * The larger the zero crossing point detection frequency is, the faster the variable gain control performance of the 2AT-SMC method is, and the influence of the online zero crossing point self-adaption mechanism on the system performance is further proved.
TABLE 2 comparison of Voltage and Current Performance under nominal operating conditions
(2) Disturbance operating mode
Taking the perturbation of the input voltage E as an example, assuming that the voltage jumps from 10V to 12V at t =1s and then jumps back to 10V at t =2s, the simulation is performed for example in fig. 8 to 10 and table 3.
Comparing the rated conditions of fig. 4 to 7 and the disturbance conditions of fig. 8 to 10, three methods can be seen for the output voltage v of the Buck converter c Inductor current i L The effect of the control law u is consistent due to the superiority of the sliding mode robust control. Specifically, the voltage v is output at t =1s c The disturbance is taken as an example for analysis, it can be seen that the response speed of the 2AT-SMC of the method of the invention is fastest, the 2AT-SMC is recovered to an equilibrium state AT about 1.015s, the convergence time of the 2T-SMC and the 1-SMC is 1.038s and 1.042s respectively, and oscillation occurs AT the initial stage of the former, which is caused by selecting a fixed gain of the traditional second-order sliding mode method, the reason of the oscillation can also be illustrated from the comparison of the control law u in FIG. 10, namely, when the disturbance with T =1s occurs, the control laws u of the two second-order SMC of the 2T-SMC and the 2AT-SMC are close in size, but the control gain of the 2AT-SMC method provided by the invention can be adaptively reduced along with the convergence process, and the 2T-SMC of the traditional fixed gain always maintains a larger value, so that the output voltage v is further enabled to be c Oscillations are generated during the fast convergence process.
TABLE 3 comparison of Voltage and Current Performance under disturbance conditions
Based on the performance comparison of the Buck converter under two working conditions of rating and disturbance, the advantages of the on-line zero-crossing detection adaptive mechanism-based 2AT-SMC provided by the embodiment in the aspects of buffeting suppression, response speed and control precision are all demonstrated, and the output voltage quality of the converter is improved.
Although the invention herein has been described with reference to particular embodiments, it is to be understood that these embodiments are merely illustrative of the principles and applications of the present invention. It is therefore to be understood that numerous modifications may be made to the illustrative embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that features described in different dependent claims and herein may be combined in ways different from those described in the original claims. It is also to be understood that features described in connection with individual embodiments may be used in other described embodiments.
Claims (10)
1. A self-adaptive continuous sliding mode control method of a power converter based on zero-crossing detection is characterized by comprising the following steps of,
establishing a mathematical model of the Buck type DC-DC converter;
according to the mathematical model, introducing an integral term of a converter system state on the basis of the traditional first-order sliding mode surface design aiming at two state variables to obtain a sliding mode surface s containing the integral term; the state variable is the voltage difference x between the actual output voltage and the target output voltage of the converter 1 And rate of change x of actual output voltage 2 ；
Based on a sliding mode surface s containing an integral term and a switching time principle, dividing the motion tracks of two state variables into two stages, wherein the first stage is from an initial point A to a first peak position point B; in the second stage, the first peak position point B is changed to a zero position point;
in the first stage, a control law u is designed 1 Enabling the motion tracks of the two state variables to reach a first peak position point B within a limited time; control law u 1 Having a fixed control gain;
in the second stage, a control law u is designed 2 The motion tracks of the two state variables are in spiral characteristics,gradually converging from the first peak position point B to a zero position point; control law u 2 Has a variable control gain; and the amplitude is adaptively adjusted along with the number of zero-crossing points of the sliding mode surface s detected in the sampling interval by the variable control gain.
2. The zero-crossing detection based power converter adaptive continuous sliding mode control method according to claim 1,
the initial mathematical model of a Buck-type DC-DC converter is:
in the formula i L Is the current flowing through the filter inductor, t is the time, L is the filter inductor, u is the control law, E is the DC input voltage, v c C is the actual output voltage of the converter, C is a capacitor, and R is a load resistor;
definition V ref Is the target output voltage, the state variable x 1 ＝v c -V ref ，
Deforming the initial mathematical model to obtain a deformed mathematical model:
in the formula
Is the intermediate variable(s) of the variable,
3. the zero-crossing detection based power converter adaptive continuous sliding mode control method according to claim 2,
surface of conventional first order slip form s 0 Comprises the following steps:
s 0 ＝c 1 x 1 +x 2 ，
in the formula c 1 As a first design parameter, c 1 >0；
For the conventional first-order slip form surface s 0 Introducing an integral term of a converter system state to obtain a sliding mode surface s containing the integral term:
in the formula c 2 As a second design parameter, c 2 >0。
4. The zero-crossing detection-based power converter adaptive continuous sliding-mode control method according to claim 3, wherein in the first stage, the control law u is set to be the control law u 1 Control law u 1 The design process of (2) comprises:
defining vectors
Wherein T is a sampling interval;
the formula of the slip form surface s is modified as follows:
wherein the content of the first and second substances,
mu is an intermediate variable which is a function of,
design control law u 1 Comprises the following steps:
where U is the fixed control gain.
5. The zero-crossing detection-based power converter adaptive continuous sliding-mode control method according to claim 4, characterized in that the control law u 1 The process of enabling the motion tracks of the two state variables to reach the first peak position point B in a limited time comprises the following steps:
according to the formula of the deformed sliding mode surface s, solving the second derivative of the sliding mode surface s to time:
in the formula:
y 22 (μ)＝c 1 μ-μβ 1 ，
β 1 ＝1/RC，β 2 ＝ω 0 2 ＝1/LC；
define four constants ζ 1 、ζ 2 、ζ 3 、ζ 4 And function Y 21 、Y 22 The following variable substitutions were made:
the following relation is satisfied:
will control law u 1 Substituting the formula into a second derivative of the sliding mode surface s to time to obtain:
in the formula mu min Is the minimum value of the intermediate variable mu, k is a constant greater than 0;
further, obtaining:
according to
Both sides are multiplied by s, i.e. by
If true, then for the motion trajectories of the two state variables, a finite time t is reached from the initial point A 0 Internally reaching a first peak position point B;
wherein t is 0 ＝t B -t A ，
Wherein t is A Is the time corresponding to the initial point A, t B The time corresponding to the first peak position point B.
6. The power converter adaptive continuous sliding-mode control method based on zero-crossing detection according to claim 5, wherein the first-stage fixed control gain U is:
7. the power converter adaptive continuous sliding-mode control method based on zero-crossing detection according to claim 6, wherein in the second stage, the control law u is set to be the control law u 2 Control law u 2 The design is as follows:
in the formula of U j For varying the control gain, r 4 Controlling the gain for a preset fixed value r 4 >0。
8. The zero-crossing detection based power converter adaptive continuous sliding-mode control method according to claim 7,
the variable control gain adaptive amplitude adjustment assignment method comprises the following steps:
is of the formula N j For the jth sampling interval T j The number of zero crossing points of the inner s, N is a reference value of the number of the zero crossing points, and N is more than or equal to 2; lambda 1 And Λ 2 Is two positive numbers, Λ 1 <Λ 2 ；T＝{T 1 ,T 2 ,...T i J =1,2,3, … … i; i is the total number of sampling intervals; u shape 0 Is control law u 2 Initial value of (a):
U 0 ＝[Y 21 (||w|| * ,|s(t 0 )|)+Y 22 u 1max +k]，
in the formula u 1max To control law u 1 Maximum value of (d):
9. the zero-crossing detection-based power converter adaptive continuous sliding-mode control method according to claim 8, characterized in that the control law u 2 The convergence range of the slip form surface s is:
|s|≤[U 0 +(1+r 4 )μ max U j ]T 2 ，
in the formula of max Is the maximum value of the intermediate variable μ.
10. The power converter adaptive continuous sliding-mode control method based on zero-crossing detection according to claim 9, wherein the process of obtaining the convergence range of the sliding-mode surface s comprises:
will control law u 2 Substituting the expression of (b) into a second derivative expression of the sliding mode surface s to time to obtain:
the further modification is that:
suppose i 0 ∈[1,N j ]，j 0 ∈[1,N j ]And i is 0 <j 0 Wherein i 0 Is [1,N j ]One of all zero-crossing points, j 0 Is [1,N j ]The other zero-crossing point of the zero-crossing points, then:
s(t i0 )＝s(t j0 )＝0；
according to N ≧ 2, there is a time point t i0j0 ：t i0 <t i0j0 <t j0 So that
(ii) present;
according to the Lagrange median theorem, at t and t i0j0 Memory existence time t i0j0 ', satisfies the following relation:
t is T j At any time in the sampling interval, and | t-t i0j0 |<T j ；
Similarly, at t and t i0 There is another time t i0 ' satisfy:
according to | t-t i0 |<T, the above formula is modified as follows:
and (3) integrating the two ends of the above formula simultaneously to obtain the convergence range of the sliding mode surface s:
|s|≤[U 0 +(1+r 4 )μ max U j ]T 2 。
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