Stability analysis method for digitally controlled dc-dc converters based on filippov's method

Abstract

The application discloses a stability analysis method of a digital control DC / DC converter based on a Filippov method, and the method is suitable for analyzing low-frequency and high-frequency bifurcation behaviors of any digital control DC / DC converter by establishing a general segmented smooth dynamic system of power stage state variables and a difference equation of control stage state variables. By describing the change of perturbation in a switching period, a state transition matrix and a jump matrix of the system can be obtained analytically, and the single-value matrix used for analyzing the bifurcation behavior of the system can be obtained by connecting the two matrices, and the single-value matrix is a Jacobian matrix of a Poincare mapping. Compared with the Poincare mapping, the method of the application can express the single-value matrix analytically, and therefore is easier to calculate.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of DC / DC converter, and particularly relates to a stability analysis method of a digitally controlled DC / DC converter based on a Filippov method. BACKGROUND

[0002] The digitally controlled DC / DC converter is a typical switching nonlinear system and also a hybrid system including discrete and continuous working states. The switching action leads to rich and complex nonlinear dynamic behaviors, i.e. various bifurcation behaviors and chaos. The appearance of these nonlinear phenomena indicates that the DC / DC converter loses stability, which often leads to significant decline in system performance and conversion efficiency, and even cannot work. Therefore, the research on bifurcation phenomena and chaos is crucial in the design of the DC / DC converter.

[0003] The stability analysis of the DC / DC converter is firstly researched by the average model, which is simple and accurate at the slow time scale. However, this method cannot predict the nonlinear behavior at the fast time scale. Another most common method is the Poincare map, which aims to derive an iterative function that represents the state variable at the next sampling time according to the state variable at the current sampling time. The Jacobian matrix of the Poincare map can be used to predict the stable boundary and bifurcation type of the system. However, due to the transcendental form of the state equation involved in most DC / DC converters, the Jacobian matrix cannot be expressed analytically. Therefore, the Jacobian matrix of the Poincare map is often obtained by numerical methods, and the calculation of the Jacobian matrix is quite difficult when the analyzed DC / DC converter has multiple state variables and modes.

[0004] The stability analysis based on the Filippov method is proposed by the scientist Filippov, which is aimed at the system with piecewise smooth dynamics model. This method obtains a single-valued matrix, i.e. the Jacobian matrix of the Poincare map, by combining the state transition matrix and the derived jump matrix. Compared with the Poincare map, the Filippov method can express the monotonic matrix analytically by combining the jump matrix and the state transition matrix. This method was first used to analyze the analog-controlled BUCK converter in 2008, and has become an important stability analysis method of the DC / DC converter. However, the traditional Filippov method is aimed at the jump matrix derived from the switching manifold in the analog control, in which the effects of the sample-and-hold and the controller delay (often one-step delay) are not considered.

[0005] Therefore, the traditional Filippov method cannot derive a suitable jump matrix for the digital control DC / DC converter, so that the stability analysis cannot be carried out. In order to apply the advantage of the Filippov method to the field of digital control, it has theoretical value and practical significance to extend the method to the stability analysis of the digital control DC / DC converter. SUMMARY

[0006] The purpose of the present application is to provide a stability analysis method of a digital control DC-DC converter based on the Filippov method, in order to solve the problems raised in the background art.

[0007] To achieve the above purpose, the present application provides the following technical solutions:

[0008] The stability analysis method of the digital control DC-DC converter based on the Filippov method includes power stage state variables and control stage state variables in the digital control converter, and the method steps are as follows:

[0009] Step one: describe the power stage state variable dynamics model of the DC / DC converter by piecewise differential equations, describe the change of the control law by difference equations, and establish a general system model of the digital control DC / DC converter;

[0010] Step two: extend the state vector of the system to include the power stage state variable and the control variable;

[0011] Step three: consider the effect of the digital control zero-order holder, and the differential equation of the control variable is 0 in one switching period;

[0012] Step four: consider whether the piecewise differential equation set is a linear equation set and the time of each mode, and discuss the state transition matrix according to whether it is linear and the length of the mode time;

[0013] Step five: make a small disturbance to the original periodic trajectory at the beginning of the switching period to obtain a perturbed trajectory, and assume that there is an arbitrary switching in one switching period, and each switching corresponds to a jump matrix;

[0014] Step six: Taylor expand the perturbation at the periodic trajectory, retain the low-order infinitesimal, and derive the analytical expression of the jump matrix according to the definition of the jump matrix;

[0015] Step seven: multiply the state transition matrix and the jump matrix to obtain a single-valued matrix, analyze the eigenvalues of the single-valued matrix, and obtain the stability analysis result of the digital control DC / DC converter;

[0016] Step eight: the stability analysis method of the digital control DC / DC converter based on the Filippov method is used to analyze a classical DC / DC converter, i.e. a DAB converter;

[0017] Step nine: appropriate parameters are set, one parameter is selected as a bifurcation parameter, and a corresponding bifurcation diagram is made by a numerical method to determine a bifurcation point at which the DC / DC converter occurs bifurcation;

[0018] Step ten: the equilibrium point of the DC / DC converter is obtained by the Newton-Raphson method, the equilibrium point is substituted into a state transition matrix and a jump matrix, a single-value matrix is calculated, the stability of the DC / DC converter is determined by solving the eigenvalues of the single-value matrix, if there is an eigenvalue outside the unit circle, the system is unstable, otherwise the system is stable;

[0019] Step eleven: the equilibrium point is also substituted into a Jacobian matrix of the Poincare mapping, if the obtained Jacobian matrix is consistent with the single-value matrix, the stability analysis method of the digital control DC / DC converter based on the Filippov method is correct;

[0020] Step twelve: the trajectory of the eigenvalue with the bifurcation parameter is drawn by solving the eigenvalues of the single-value matrix, the bifurcation type of the system is observed by observing how the eigenvalue crosses the unit circle, and the stability boundary of the system parameter is determined.

[0021] As a further scheme of the application: the power stage state variable can be described by the following piecewise smooth dynamics model:

[0022]

[0023] Wherein, f i is a smooth vector field and i∈[1,…,k+1]; T is a switching period, p is an arbitrary non-negative integer, the state variable x=[x1,x2,…,x n ] T ; there are k+1 subintervals in a switching period, the switching time t i is determined by the switching manifold.

[0024] As a further scheme of the application: in the digital control converter, due to the effect of the zero-order holder, the control variable remains unchanged in a switching period and is updated at the beginning of the next switching period, therefore the dynamics model of the control variable can only be expressed by a difference equation as follows:

[0025] y (p+1) =[y 1(p+1) y 2(p+1) …y m(p+1) ] T =g(x (p) ,y(p) , d (p) )

[0026] d (p+1) = [d 1(p+1) d 2(p+1) …d j(p+1) ] T = μ(x (p) , y (p) , d (p) )

[0027] where y = [y1, y2, …, y m ] T , y1, y2, …, y m are state variables in the controller; d = [d1, d2, …, d j ] T , d1, d2, …, d j are outputs of the controller, the subscripts (p) and (p+1) represent the pth and (p+1)th switching periods respectively, g(x (p) , y (p) , d (p) ) and μ(x (p) , y (p) , d (p) ) are vector functions mapping the state variables (x (p) , y (p) , d (p) ) to y (p+1) and d (p+1) respectively.

[0028] The switching instants of the digital controlled converter are determined by the switching manifold, which is expressed as

[0029] h i (t i , d) = t i - L i (d) = 0

[0030] where L i (d): R j → R maps the output d of the controller to the switching instant t i .

[0031] As a further scheme of the present application: the differential quantity in the differential equation and the difference quantity in the difference equation are set to zero to obtain a steady state working point of the system, a state transition matrix and a jump matrix of the system are obtained by exerting a small perturbation on the steady state working point, and a single value matrix of the system is obtained by multiplying the two matrices:

[0032] M(T+t0, t0) = S k+1 × Φ k+1 (T+t0, tk )×S k ×…Φ1(t1, t0)

[0033] Among them, Φ i (t i , t i-1 ) is a smooth subinterval [t i-1 , t i ] state transition matrix; S i (i∈(,1,...k+1)) is the hopping matrix corresponding to the i-th switching moment;

[0034] The stability range of each parameter of the system is determined by analyzing the modulus of the maximum eigenvalue of the single-value matrix.

[0035] As a further solution of the present invention: the hopping matrix is:

[0036]

[0037]

[0038]

[0039]

[0040] Where I is the identity matrix with the same order as the number of state variables; 1×m is the m-th order zero vector; 0 1×n is the nth-order zero vector; 0 i×j is an i×j order zero matrix.

[0041] As a further solution of the present invention, the stability result of the analyzed digitally controlled DC / DC converter is obtained by using the trajectory of the characteristic value changing with the bifurcation parameter, including:

[0042] When all the eigenvalues ​​are within the unit circle, the analyzed digital controlled DC / DC converter is determined to be in a stable state;

[0043] When the eigenvalue crosses the unit circle along the negative real axis, the analyzed digital controlled DC / DC converter is judged to have period-doubling bifurcation and is unstable.

[0044] When the eigenvalue crosses the unit circle along the positive real axis, the digital controlled DC / DC converter under analysis is judged to have a saddle-node bifurcation and is unstable;

[0045] When the conjugate eigenvalue crosses the unit circle, the analyzed digital controlled DC / DC converter is judged to have HOPF bifurcation and is unstable;

[0046] When the eigenvalue is located on the unit circle, it is determined that the analyzed digitally controlled DC / DC converter is in a critical stable state.

[0047] Compared with the prior art, the present application has the beneficial effects that:

[0048] The stability analysis method of the digital control DC / DC converter based on the Filippov method can provide an analytical expression of a single-value matrix, specifically an analytical expression of a jump matrix; the controller delay in digital control is considered, compared with the traditional Filippov method, the method can be applied to the digital control DC / DC converter; the established digital control DC / DC model is general enough, no matter how many modes the converter has, how many state variables, and whether the state space equation is linear, a single-value matrix can be obtained through the method of the present application, and then the stability of the converter is judged. BRIEF DESCRIPTION OF DRAWINGS

[0049] Figure 1 The stability analysis method of the digital control DC-DC converter based on the Filippov method is a general periodic trajectory crossing multiple switch flow forms.

[0050] Figure 2 The stability analysis method of the digital control DC-DC converter based on the Filippov method is a schematic diagram of piecewise linearization of a nonlinear vector field.

[0051] Figure 3 The stability analysis method of the digital control DC-DC converter based on the Filippov method is a schematic diagram of the perturbed trajectory after small perturbation and the original periodic trajectory crossing multiple switch flow forms.

[0052] Figure 4 The stability analysis method of the digital control DC-DC converter based on the Filippov method is a digital control DAB converter system diagram with constant power load.

[0053] Figure 5 The stability analysis method of the digital control DC-DC converter based on the Filippov method is a bifurcation diagram taking the proportional coefficient of the PI controller as the bifurcation parameter.

[0054] Figure 6 The stability analysis method of the digital control DC-DC converter based on the Filippov method is a root locus change diagram of the DAB converter bifurcation parameter made by the stability analysis method.

[0055] Figure 7 The stability analysis method of the digital control DC-DC converter based on the Filippov method is a first waveform diagram of the primary side current and the output voltage of the digital control DAB converter with constant power load.

[0056] Figure 8 The second wave pattern diagram of the primary side current and the output voltage of the digital control DAB converter with constant power load is based on the stability analysis method of the digital control DC-DC converter of the Filippov method. DETAILED DESCRIPTION

[0057] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work are within the scope of protection of the present application.

[0058] Please refer to Figures 1-8 In the embodiments of the present application, the stability analysis method of the digital control DC-DC converter based on the Filippov method includes power stage state variables and control stage state variables in the digital control converter, and the method steps are as follows:

[0059] Step one: the dynamics model of the power stage state variable of the DC / DC converter is described by piecewise differential equations, the differential equation describes the change of the control law, and the system model of the general digital control DC / DC converter is established;

[0060] Step two: the state vector of the system is extended to include the power stage state variable and the control variable;

[0061] Step three: considering the effect of the digital control zero-order holder, the differential equation of the control variable is 0 in a switching period;

[0062] Step four: whether the piecewise differential equation set is a linear equation set and the time of each mode are considered, and the state transition matrix is discussed according to whether it is linear and the length of the mode time;

[0063] Step five: a small perturbation is made to the original periodic trajectory at the beginning of the switching period to obtain a perturbation trajectory, and it is assumed that there is an arbitrary switching in a switching period, and each switching corresponds to a jump matrix;

[0064] Step six: the perturbation is Taylor expanded at the periodic trajectory, and the low-order infinitesimal term is retained, and the analytical expression of the jump matrix is derived according to the definition of the jump matrix;

[0065] Step seven: the state transition matrix and the jump matrix are multiplied to obtain a single value matrix, the eigenvalues of the single value matrix are analyzed, and the stability analysis result of the digital control DC / DC converter is obtained;

[0066] Step eight: the stability analysis method of digital controlled DC / DC converter based on Filippov method is used to analyze a classical DC / DC converter, i.e. DAB converter;

[0067] Step nine: set appropriate parameters, select one parameter as bifurcation parameter, make corresponding bifurcation diagram through numerical method, and determine the bifurcation point of DC / DC converter when bifurcation occurs;

[0068] Step ten: the equilibrium point of DC / DC converter is obtained through Newton-Raphson method, the equilibrium point is substituted into state transition matrix and jump matrix, a single value matrix is calculated, the stability of the DC / DC converter is determined by solving the eigenvalue of the single value matrix, if there is an eigenvalue outside the unit circle, it is unstable, otherwise it is stable;

[0069] Step eleven: the equilibrium point is also substituted into the Jacobian matrix of Poincare mapping, if the obtained Jacobian matrix is consistent with the single value matrix, the stability analysis method of digital controlled DC / DC converter based on Filippov method is correct;

[0070] Step twelve: the trajectory of eigenvalue with bifurcation parameter is drawn by solving the eigenvalue of the single value matrix, the bifurcation type of the system is observed by observing how the eigenvalue crosses the unit circle, and the stability boundary of the system parameter is determined.

[0071] The power stage state variable can be described by the following piecewise smooth dynamic model:

[0072]

[0073] Wherein, f i is a smooth vector field and i∈[1,..., k+1]; T is the switching period, p is any non-negative integer, state variable x=[x1, x2,..., x n ] T ; there are k+1 subintervals in a switching period, and the switching time t i is determined by the switching manifold.

[0074] In the digital controlled converter, due to the effect of zero-order holder, the control variable remains unchanged in a switching period and is updated at the beginning of the next switching period, so the dynamic model of the control variable can only be represented by difference equation as follows:

[0075] y (p+1) =[y 1(p+1) y 2(p+1) …y m(p+1) ] T =g(x (p) , y (p) , d (p) )

[0076] d (p+1) =[d 1(p+1) d 2(p+1) …d j(p+1) ] T =μ(x (p) ,y (p) , d (p) )

[0077] Where y=[y1,y2,...,y m ] T ,y1,y2,...,y m is the state variable in the controller; d=[d1,d2,…,d j ] T , d1, d2, ..., d j is the output of the controller, the subscripts (p) and (p+1) represent the pth and p+1th switching cycles respectively, g(x (p) ,y (p) , d (p) ) and μ(x (p) ,y (p) , d (p) ) are the state variables (x (p) ,y (p) , d (p) ) is mapped to y (p+1) and d (p+1) A vector function of .

[0078] The switching timing of the digital control converter is determined by the switching manifold, which is expressed as follows

[0079] h i (t i , d) = t i -L i (d) = 0

[0080] Among them, L i (d):R j →R maps the controller output d to the switching time t i .

[0081] The system's steady-state operating point can be obtained by setting the differential component in the differential equation and the difference component in the difference equation to zero. By applying a small perturbation to the steady-state operating point, the system's state transfer matrix and jump matrix are obtained. Multiplying the two matrices together yields the system's single-valued matrix:

[0082] M(T+t0,t0)=S k+1 ×Φ k+1 (T+t0,t k )×S kΦ1(t1, t0)

[0083] wherein Φ i (t i , t i-1 ) is the state transition matrix of the smooth sub-interval [t i-1 , t i ]; S i (i∈(,1,...k+1)) is the jump matrix corresponding to the i-th switching time;

[0084] The stability range of each parameter of the system is determined by analyzing the modulus of the largest eigenvalue of the single-value matrix.

[0085] The jump matrix is:

[0086]

[0087]

[0088]

[0089]

[0090] wherein I is a unit matrix with the same order as the state variable; 0 1×m is an m-order zero vector; 0 1×n is an n-order zero vector; and 0 i×j is an i×j-order zero matrix.

[0091] The stability result of the analyzed digital control DC / DC converter is obtained through the trajectory of the eigenvalue with the bifurcation parameter, including:

[0092] When the eigenvalues are all located in the unit circle, it is determined that the analyzed digital control DC / DC converter is in a stable state;

[0093] When the eigenvalues cross the unit circle along the negative real axis, it is determined that the analyzed digital control DC / DC converter occurs period-doubling bifurcation and is unstable;

[0094] When the eigenvalues cross the unit circle along the positive real axis, it is determined that the analyzed digital control DC / DC converter occurs saddle-node bifurcation and is unstable;

[0095] When the conjugate eigenvalues cross the unit circle, it is determined that the analyzed digital control DC / DC converter occurs HOPF bifurcation and is unstable;

[0096] When the eigenvalues are located on the unit circle, it is determined that the analyzed digital control DC / DC converter is in a critical stable state.

[0097] Referring to Figure 1In one switching cycle, the periodic trajectory crosses k+1 switching manifolds. The evolution of the perturbation on the smooth trajectory can be represented by the state transition matrix, while in the neighborhood of the switching manifold, the evolution of the perturbation is represented by the jump matrix. The single-value matrix used to describe the change of the perturbation in one switching cycle is expressed as:

[0098] M(T+t0,t0)=S k+1 ×Φ k+1 (T+t0,t k )×S k ×…Φ1(t1,t0) (1)

[0099] Among them, Φ i (t i , t i-1 ) is a smooth interval [t i-1 , t i ] state transition matrix; S i (i∈(, 1, ...k)) is the jump matrix corresponding to the i-th switching moment; S k+1 This will be analyzed in detail later because the jump matrix maps the perturbation before the end of the switching cycle to the perturbation after the end of the switching cycle.

[0100] The state variables of the power stage are expressed using a piecewise smooth dynamics model as

[0101]

[0102] Among them, f i is a smooth vector field, and i∈[1,...,k+1]; T is the switching period; p is a non-negative integer; the state vector x=[x1, x2,..., x n ] T ; There are k+1 modes in a switching cycle, and the switching time t i Determined by the switch manifold.

[0103] In the digital controller, it is assumed that the control rate satisfies the following difference equation

[0104] y (p+1) =[y 1(p+1) y 2(p+1) …y m(p+1) ] T =g(x (p) ,y (p) , d (p) ) (3)

[0105] d (p+1) =[d 1(p+1) d 2(p+1) …d j(p+1) ] T =μ(x (p), y (p) , d (p) ) (4)

[0106] where y = [y1, y2,..., y m ] T , y1, y2,..., y m are state variables of the digital controller (e.g. the output of the integrator in a PI controller); d = [d1, d2,..., d j ] T , d1, d2,..., d j are the outputs of the controller (typically the duty cycle and phase shift angle in DC / DC converters); the subscripts (p) and (p+1) denote the pth and (p+1)th switching period, respectively; g(x (p) , y (p) , d (p) ) and μ(x (p) , y (p) , d (p) ) are vector functions mapping the state variables (x (p) , y (p) , d (p) ) to y (p+1) and d (p+1) , respectively.

[0107] To compute the state transition matrix, the state vector is extended to ξ, containing the state variables x, y and d

[0108] ξ = [x T y T d T ] T (5)

[0109] In digital control, the control rates y and d do not change within a switching period (to, to+T), so the dynamic model of y and d within (to, to+T) can be expressed as

[0110]

[0111] By combining (2) and (6), the piecewise differential equation of the state vector ξ can be written as

[0112]

[0113] If the vector field f i in (2) is linear and has the form

[0114] f i (x(t)) = A i x(t) + B i i∈{1, 2,..., k+1} (8)

[0115] Then the state transition matrix can be calculated by exponential matrix

[0116]

[0117] where T i = t i -t i-1 is the duration of the i-th mode.

[0118] If the vector field f i in (2) is nonlinear, and the mode time (pT+t i-1 , pT+t i ] is short enough, the state transition matrix is approximated by

[0119]

[0120] where,

[0121] If the vector field f i in (2) is nonlinear, and the mode time is long, to improve the calculation accuracy of the state transition matrix, further piecewise linearization can be performed on the interval (pT+t i-1 , pT+t i ]. Divide the interval into N equal segments, such as Figure 2 It is worth noting that the choice of "N" is a trade-off between computational burden and calculation accuracy. Generally, the larger N is, the heavier the computational burden is, and the higher the accuracy is. If the switching frequency is high, N can be a small positive integer, or (10) can be used directly (N = 1).

[0122] When a suitable "N" is selected, so that each segment of the interval is short enough, the state transition matrix of each segment is

[0123]

[0124] where,

[0125] Therefore, the state transition matrix within the interval (t i-1 , t i ) can be expressed as

[0126] Φ i (t i , t i-1 ) = Φ i，N (t i , t i -Δt i ) ×…×Φ i，1 (t i-1 +Δti , t i-1 ) (12)

[0127] Next, the state transition matrix S Figure 1 in (13) is calculated i (i∈(1,...k)) which can be derived by examining the nonlinear dynamical system near the switching boundary according to the Filippov theory. Considering the periodic orbit before and after the perturbation (including the perturbation to x, y, d) as shown in Fig. 2, the variables with superscript " " represent the variables after the perturbation; Figure 3 represents the time of the i-th switching of the orbit after the perturbation. Δξ0represents the initial small perturbation at the beginning of the switching period

[0128]

[0129] In addition, due to the characteristics of digital control, the switching manifold is generally represented as

[0130] h i (t i , d) = t i - L i (d) = 0 (14)

[0131] where L i (d): R j → R maps the controller output d to the switching time t i .

[0132] Due to the perturbation of the initial perturbation δξ0to the periodic trajectory, δt i is defined as follows

[0133]

[0134] According to the definition of the jump matrix, the i-th jump matrix is used to describe the relationship between the perturbation vector δξ i+ and δξ i- .

[0135] δξ i+ = S i · δξ i- (16)

[0136] According to Figure 3 , δξ i+ and δξ i- are defined as follows

[0137]

[0138] By applying Taylor series expansion to ξ(t) at t = t i , we get ​

[0139]

[0140]

[0141] where F i- and F i+ are the vector fields before and after the switching instant, respectively, which are expressed as

[0142]

[0143] F i+ = F i+1 (ξ(t i )) (21)

[0144] Combining (17), (18), (19) gives

[0145] δξ i+ = δξ i- + (F i- - F i+ ) · δt i (22)

[0146] According to (16), (20) and (22), the jump matrix can be written as

[0147] S i · δξ i- = δξ i- + (F i (ξ(t i )) - F i+1 (ξ(t i ))) · δt i (23)

[0148] Performing Taylor series expansion on (14) after the perturbation gives

[0149]

[0150] Therefore, it is easy to get δt i

[0151]

[0152] where E j is the j-order identity matrix, and for better readability, we define

[0153]

[0154] (26) can be rewritten as

[0155] δt i = L' i• δξ0(27)

[0156] According to the definition of state transition matrix and jump matrix, δξ i- and δξ (i-1)+ can be defined as

[0157]

[0158] In particular, we can get

[0159] δξ 1- = Φ1(t1, t0) · δξ0 (29)

[0160] By analogy, we have

[0161]

[0162] Taking Taylor expansion of (30), we have

[0163]

[0164] Substituting (31) into (30), we can get

[0165] δξ i- ≈ Φ, (t i , t i-1 ) · S i-1 ····· S1· Φ1(t1, t0) · δξ0 (32)

[0166] Combining (23), (27) and (32), the general expression of state transition matrix S i can be written as

[0167]

[0168] As can be seen from (33), in order to get S i , S1should be solved first. Combining (24), (27), (29) and letting i = 1, the expression of S1is obtained as

[0169]

[0170] By combining (33) and (34), S i can be solved analytically.

[0171] It should be noted that the method of calculating the final state transition matrix S k+1 is the same as that of calculating S i ​The method is different. Since the beginning (end) of the switching period does not depend on the control law, when the perturbation crosses the end of the switching period, the evolution of δx is an identity map, the following holds

[0172]

[0173] where, and are the perturbation vectors δx and δξ before the end of the pth switching period, respectively. is the perturbation vector δx after the end of the pth switching period.

[0174] In digital control, according to (3) and (4), the discrete control variable y and d have a step change at the beginning of the (p+1)th switching period (in the interval [((p+1)T) - , ((p+1)T) + ]. According to (3) and (4), the relationship between the perturbation variables and can be calculated from the initial disturbance δξ0, as follows

[0175]

[0176] where, and are the perturbation vectors δy and δd after the end of the pth switching period, respectively. The expression of J is as follows

[0177]

[0178] According to the previous derivation, can be expressed as

[0179]

[0180] Therefore, (37) can be rewritten as

[0181]

[0182] Combining (35) and (39), the jump matrix S k+1 is given by

[0183]

[0184] Substituting (40) into (1), the single-valued matrix can be expressed as

[0185]

[0186] Due to the jump matrix in (41), the state transition matrix and J are given by the previous analytical expressions, thus the single-valued matrix obtained by the present application has an analytical expression, which is different from the Jacobian matrix of Poincare mapping.

[0187] In the embodiment of the present application, preferably, the digital controlled DC / DC converter selected by the embodiment of the present application is a DAB converter. The stability analysis method of the digital controlled DC / DC converter based on the Filippov method provided by the present example can accurately obtain the stability analysis result of the DC / DC converter. As a specific embodiment of the embodiment of the present application, the DAB converter has four modes, and the correctness of multiple jump matrices can be verified. In addition, the load of the DAB converter is set to be a constant power load, and due to the nonlinear characteristics of the constant power load, this embodiment is more general. The schematic diagram of the digital controlled DAB converter with single-phase shift modulation is shown in Figure 4 , wherein the parameters are shown in Table 1, and the proportional parameter k p is selected in the PI controller.

[0188] Table 1

[0189]

[0190] Phase shift angle is determined by the following control law:

[0191]

[0192] , wherein k p and k i are the proportional parameter and the integral parameter of the PI controller respectively; g is the output of the integrator.

[0193] According to the system model described above, in this embodiment, the state variables of the power stage are 4, which are the input filter inductor current i1, the input filter capacitor voltage u1, the transformer primary side current i2, and the output capacitor voltage u2; the state variable of the digital controller is 1, which is the output of the integrator g; the output of the controller is 1, which is the phase shift angle According to the above, n=4, m=1, j=1, and k+1=4.

[0194] Figure 5 shows the bifurcation diagram of i2 when the bifurcation parameter k p varies from 0.4 to 0.8. The diagram shows that when k p is greater than 0.55, the system is unstable.

[0195] The stability analysis method of the present application is used to analyze the stability of this embodiment. The state vector is selected as x=[i1, u1, i2, u2] T, where g is the integrator output and the controller output phase shift angle The control vector is Using Kirchhoff's voltage and current laws, the state space equations of the four modes of the DAB converter are established, and the vector field of the extended state vector is shown in Table 2 according to (7)

[0196] Table 2

[0197]

[0198] Where the switching instants t1, t2, t3 are determined by the control law, and according to (3), (4) and (42), the control law corresponding to each switching instant can be written as

[0199]

[0200] According to (26) and (43), L' is calculated as i As follows

[0201]

[0202] The mapping function of the state variable can be written by Table 2 and (43), and the periodic trajectory of the system can be solved by using Newton-Raphson method, and the periodic trajectory of the system is calculated as follows in this embodiment taking k p = 0.57 as an example

[0203]

[0204] Due to the constant power load, the vector field of the DAB converter is nonlinear, and according to the foregoing, each mode is linearized in segments. In order to improve the calculation accuracy without too much calculation, each mode is divided into two segments, and the state transition matrix is as follows

[0205]

[0206] Substitute the periodic trajectory into Table 2, and according to (33) and (34), the jump matrix is calculated as follows

[0207] According to (37) and (42), the matrix J is calculated as follows

[0208]

[0209] According to (40), the last jump matrix S k+1 is calculated as follows

[0210] According to (1), the state transition matrix and the jump matrix are multiplied to obtain a single value matrix as follows

[0211]

[0212] The modulus of the largest eigenvalue of the single-valued matrix is 1.0033. According to the Filippov theory, the DAB converter is unstable at k p = 0.57.

[0213] Similarly, the single-valued matrix under any bifurcation parameter k p can be calculated analytically. The modulus of the largest eigenvalue of the single-valued matrix is listed in Table 3. When k p = 0.55, the eigenvalue crosses the unit circle, which is consistent with the result of the bifurcation diagram shown in Figure 5 .

[0214] Table 3

[0215] k p ]]> Maximum eigenvalue modulus of a single valued matrix Stability 0.53 0.9948 Stable 0.54 0.9968 Stable 0.55 0.9990 Stable 0.56 1.0011 Unstable 0.57 1.0033 Unstable

[0216] The stability of the embodiments can also be verified by the Poincare map, since the Jacobian matrix of the Poincare map is the single-valued matrix in the Filippov method, the correctness of the method of the present application can be verified by the Jacobian matrix of the Poincare map. The eigenvalue trajectories calculated by the two methods are shown in Figure 6 . It can be found that the two methods give very similar results on the eigenvalue trajectory, proving the correctness of the method of the present application.

[0217] Referring to Figures 7-8 , the primary current i2 and the output voltage u p of the DAB converter under different bifurcation parameters k c are studied by the above experiments. According to the waveform of the primary current i2, when k p = 0.5, Figure 7 the DAB converter works in a steady state; when k p = 0.6, Figure 8 the primary current i2 begins to oscillate, which means that the DAB converter loses stability.

[0218] Although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art can make modifications to the technical solutions recorded in the foregoing embodiments, or make equivalent replacements to part of the technical features, any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present application shall be included in the protection scope of the present application.

Claims

1. A method for stability analysis of digitally controlled DC-DC converters based on the Filippov approach, including power stage state variables and control stage state variables in the digitally controlled converter, characterized in that: The method steps are as follows: Step one: the dynamics model of the state variable of the power stage of the DC / DC converter is described by piecewise differential equations, and the differential equation describes the change of the control law, and the system model of the general digital control DC / DC converter is established; Step two: the state vector of the system is extended to include the power stage state variable and the control variable, and the power stage state variable can be described by the following piecewise smooth dynamics model: where f i is a smooth vector field and i ∈ [1, …, k + 1]; T is the switching period, p is an arbitrary non-negative integer, the state variable x = [x1, x2, …, x n ] T ; there are k + 1 sub-intervals in one switching period, the switching time t i is determined by the switching manifold; In the digital control converter, due to the effect of the zero-order holder, the control variable remains unchanged in a switching period and is updated at the beginning of the next switching period, so the dynamics model of the control variable can only be represented by the following difference equation: y (p+1) = [y 1(p+1) y 2(p+1) …y m(p+1) ] T = g(x (p) ,y (p) ,d (p) ) d (p+1) = [d (p+1) d 2(p+1) …d j(p+1) ] T = μ(x (p) , y (p) , d (p) ) Where y=[y1,y2,…,y m ] T ,y1,y2,…,y m is the state variable in the controller; d=[d1,d2,…,d j ] T ,d1,d2,…,d j is the output of the controller, the subscripts (p) and (p+1) represent the pth and p+1th switching cycles respectively, g(x (p) ,y (p) ,d (p) ) and μ(x (p) ,y (p) ,d (p) ) are the state variables (x (p) ,y (p) ,d (p) ) is mapped to y (p+1) and d (p+1) Vector function of ; Step three: considering the effect of the digital control zero-order holder, the differential equation of the control variable in a switching period is 0; Step four: considering whether the piecewise differential equation group is a linear equation group and the time of each mode, the state transition matrix is discussed according to whether it is linear and the length of the mode time can be divided into three cases; Step five: a small perturbation is made to the original periodic trajectory at the beginning of the switching period, and a perturbed trajectory is obtained, assuming that there are arbitrary switching switches in a switching period, and each switching switch corresponds to a jump matrix; Step six: Taylor expansion is performed on the perturbation at the periodic trajectory, and low-order infinitesimals are retained, and the analytical expression of the jump matrix is derived according to the definition of the jump matrix; Step seven: the state transition matrix and the jump matrix are multiplied to obtain a single-value matrix, the eigenvalues of the single-value matrix are analyzed, and the stability analysis result of the digital control DC / DC converter is obtained; Step eight: the stability analysis method of the digital control DC / DC converter based on the Filippov method is used to analyze a classical DC / DC converter, namely the DAB converter; Step nine: appropriate parameters are set, one parameter is selected as a bifurcation parameter, and a corresponding bifurcation diagram is drawn by a numerical method to determine the bifurcation point of the DC / DC converter when the bifurcation occurs; Step ten: the equilibrium point of the DC / DC converter is obtained by the Newton-Raphson method, the equilibrium point is substituted into the state transition matrix and the jump matrix, the single-value matrix is calculated, and the stability of the DC / DC converter is determined by solving the eigenvalues of the single-value matrix. If there is an eigenvalue outside the unit circle, it is unstable, otherwise it is stable; Step eleven: similarly, the equilibrium point is substituted into the Jacobian matrix of the Poincare mapping, if the obtained Jacobian matrix is consistent with the single-value matrix, the derived stability analysis method of the digital control DC / DC converter based on the Filippov method is correct; Step twelve: the eigenvalue trajectory with the bifurcation parameter is drawn by solving the eigenvalues of the single-value matrix, and the type of the bifurcation of the system is observed by observing how the eigenvalue crosses the unit circle, and the stability boundary of the system parameters is determined. The switching time of the digital control converter is determined by the switching manifold, which is represented as follows 2. The method for stability analysis of digitally controlled DC-DC converters based on Filippov's method according to claim 1, characterized in that: The differential quantity in the differential equation and the difference quantity in the difference equation are set to zero to obtain the steady-state working point of the system, a small perturbation is applied to the steady-state working point, the state transition matrix and the jump matrix of the system are obtained, and the single-value matrix of the system is obtained by multiplying the two matrices: h i (t i , d) = t i -L i (d) = 0 where L i (d): R j → R maps the output d of the controller to the switching instant t i .

3. The method for stability analysis of digitally controlled DC-DC converters based on Filippov approach according to claim 1, characterized in that: ​ M(T + t0, t0) = S k+1 x Φ k+1 (T + t0, t k ) x S k x... Φ1(t1, t0) wherein, Φ i (t i ,t i-1 ) is the state transition matrix of the smooth sub-interval [t i-1 ,t i ]; S i (i∈(,1,…k+1)) is the jump matrix corresponding to the i-th switching time. The stability range of each parameter of the system is determined by analyzing the modulus of the maximum eigenvalue of the single-value matrix.

4. The method for stability analysis of digitally controlled DC-DC converters based on Filippov's approach according to claim 3, characterized in that: The jump matrix is: where I is an identity matrix of the same order as the state variable; 0 1×m is an m-order zero vector; 0 1×n is an n-order zero vector; 0 i×j is an i x j order zero matrix.

5. The method for stability analysis of digitally controlled DC-DC converters based on Filippov's approach according to claim 3, characterized in that: The closed-loop stability result of the analyzed digital control DC / DC converter is obtained through the trajectory of the eigenvalue with the bifurcation parameter, including: When the eigenvalues are all located in the unit circle, it is determined that the analyzed digital control DC / DC converter is in a stable state; When the eigenvalues cross the unit circle along the negative real axis, it is determined that the analyzed digital control DC / DC converter occurs period-doubling bifurcation and is unstable; When the eigenvalues cross the unit circle along the positive real axis, it is determined that the analyzed digital control DC / DC converter occurs saddle-node bifurcation and is unstable; When the conjugate eigenvalues cross the unit circle, it is determined that the analyzed digital control DC / DC converter occurs HOPF bifurcation and is unstable; When the eigenvalues are located on the unit circle, it is determined that the analyzed digital control DC / DC converter is in a critical stable state.

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