Fractional order PI lambda controlled DC-DC converter control and stability analysis method

Abstract

The invention discloses a control and stability analysis method for a DC-DC converter based on fractional order PI lambda control. The control and stability analysis method comprises the following steps: firstly, establishing a system state equation for the DC-DC converter; secondly, based on small signal modeling, establishing a linear analysis model of the DC-DC converter alternating current small signal; a transfer function between the duty ratio and the converter output voltage is obtained through Laplace transformation, and a fractional order closed-loop characteristic equation of the DC-DC converter system under fractional order PI lambda control is obtained; and finally, drawing an Nyquist graph of the DC-DC converter system under different parameters, and judging the stability of the system through an Nyquist criterion. According to the method, the influence of the integral order lambda in the fractional order PI lambda controller on the system stability can be simply and conveniently analyzed, and the control parameter change range of the PI lambda controller with different integral orders under the condition that the system stability is ensured.

Description

Technical Field

[0001] The invention relates to a fractional-order PIλ control and stability analysis method, which can be used for a DC-DC converter and belongs to the field of power supply control. Background Art

[0002] DC-DC converters are widely used in industrial instrumentation, military, aerospace and other fields, especially in adjustable DC switches and DC motor drives. If the control link design of the DC-DC converter is not reasonable, it will cause the output of the converter to no longer meet the performance indicators when it is disturbed by the outside world, thus causing certain harm to the system where the converter is located or even system crash. In order to improve the dynamic and steady-state performance of the DC-DC converter output and make the system have good load capacity and anti-interference ability, it is very important to choose the appropriate control method and parameters.

[0003] In actual industrial production processes, it is often difficult to establish an accurate mathematical model due to factors such as parasitic parameters, and the application of traditional PID controllers sometimes cannot achieve the desired control effect. The emergence of fractional-order PIλDμ controllers is an extension of integer-order PID controllers. Compared with integer-order PID controllers, the adjustable parameters in fractional-order PIλDμ controllers include not only the proportional coefficient Kp, the integral coefficient Ki, and the differential coefficient Kd, but also the integral order λ and the differential order μ, where the differential order and the integral order are generally real numbers that can take any value in the interval (0,2). Since the number of adjustable parameters in the fractional-order controller increases, the controlled object can be controlled more flexibly. However, at present, there is still a lack of analysis of the impact of the introduction of integral order λ and differential order μ on the stability range of control parameters in the existing literature, and there has been no stability analysis of DC-DC converters using fractional-order controllers. Summary of the invention

[0004] The technical problem to be solved by the present invention is to provide a fractional-order PIλ controlled DC-DC converter control and stability analysis method.

[0005] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0006] 1. Write the system state equation for the DC-DC converter;

[0007] 2. Calculate the average variable based on small signal modeling, separate the disturbance and linear processing to obtain the linear analytical model of the DC-DC converter AC small signal;

[0008] 3. The transfer function from duty cycle to converter output voltage is obtained by Laplace transform, and fractional-order PI is obtained λ Fractional order closed loop characteristic equation of DC-DC converter system under control;

[0009] 4. Draw the Nyquist diagram of the DC-DC converter system under different parameters and determine the system stability using the Nyquist criterion.

[0010] The beneficial effects of the present invention are as follows:

[0011] The present invention proposes a fractional-order PI DC-DC converter λ Control and stability analysis methods provide engineers and researchers with fractional-order PI λ Design methods and references for controller parameters can easily analyze fractional-order PI λ The influence of the integral order λ in the controller on the system stability, and the different integral orders, PI under the condition of ensuring system stability λ The control parameter variation range of the controller. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] The included drawings are used to provide a further understanding of the embodiments of the present invention, which constitute a part of the specification, are used to illustrate the embodiments of the present invention, and together with the text description, explain the principles of the present invention. Obviously, the drawings in the following description are only some embodiments of the present invention, and for ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0013] Figure 1 A flow chart of a fractional-order PIλ control and stability analysis method provided by an embodiment of the present invention is shown;

[0014] Figure 2 This is the schematic diagram of the DC-DC converter system structure;

[0015] Figure 3 This is the block diagram of the DC-DC converter system;

[0016] Figure 4 Nyquist diagram of the open-loop transfer function of the system when λ changes from 0 to 2;

[0017] Figure 5 Nyquist plot of the system open-loop transfer function when Kp changes from 0.01 to 0.05;

[0018] Figure 6 Nyquist diagram of the system open-loop transfer function when Ki changes from 10 to 100;

[0019] Figure 7 Nyquist plot of the system open-loop transfer function when Kp changes from 0.01 to 0.05;

[0020] Figure 8Nyquist diagram of the system open-loop transfer function when Ki changes from 5 to 20;

[0021] Fig. 9 Nyquist plot of the system open-loop transfer function when Kp changes from 0.01 to 0.05;

[0022] Fig.10 Nyquist diagram of the system open-loop transfer function when Ki varies from 10 to 600. DETAILED DESCRIPTION

[0023] It should be noted that, in the absence of conflict, the embodiments in this application and the features in the embodiments can be combined with each other. The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. The following description of at least one exemplary embodiment is actually only illustrative and is by no means intended to limit the present invention and its application or use. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0024] This paper takes Buck type DC-DC converter as the research object. Starting from the mathematical model of DC-DC converter, it analyzes the influence of integral order λ in fractional-order PIλ controller on system stability, as well as the control parameter variation range of PIλ controller with different integral orders while ensuring system stability. It provides a design method and reference for parameters of fractional-order PIλ controller for engineers and researchers. Finally, the correctness of theoretical analysis is verified through simulation and experiments.

[0025] As a fractional-order PIλ controlled DC-DC converter control and stability analysis method provided by an embodiment of the present invention, Figure 1 As shown, the following steps are included:

[0026] Step 1. Write the converter system state equation

[0027] The DC-DC converter system structure schematic diagram is as follows Figure 2 As shown in the figure, Vin is the input voltage, L is the output filter inductor, C is the output filter capacitor, Vref is the reference voltage, and Vm is the carrier amplitude, which can usually be taken as 1V.

[0028] According to whether the current in the inductor is continuous during operation, the DC-DC converter system can be divided into the following two operating modes: continuous conduction mode (CCM) of inductor current and discontinuous conduction mode (DCM) of inductor current. Compared with operating in the CCM mode, when the DC-DC converter operates in the DCM mode, problems such as high current peaks and large switching noises in the system are more serious. Therefore, the DC-DC converter is usually designed to operate in the CCM mode. In the present invention, the DC-DC converter operating in the CCM mode is modeled, and the capacitor voltage u c and the inductor current i L are taken as state variables to write the system state equations. A stable operating DC-DC converter switches periodically between the following two operating modes: Mode 1 is the stage where the switch is on and the diode is off during 0 < t < DTs; Mode 2 is the stage where the diode is on and the switch is off during DTs < t < Ts. Among them, Ts is the switching period and D is the duty cycle.

[0029] The state equation of the DC-DC converter when operating in Mode 1 is:

[0030]

[0031] where u(t) is the output voltage.

[0032] The state equation of the DC-DC converter when operating in Mode 2 is:

[0033]

[0034] Step 2. Based on small-signal modeling, establish a linear analytical model of the AC small-signal of the DC-DC converter

[0035] From the system state equations in the two operating modes in Step 1, and according to small-signal modeling, the average variables are obtained, the perturbations are separated, and linear processing is performed to obtain the linear analytical model of the AC small-signal of the DC-DC converter as shown in Equation (3).

[0036]

[0037] Step 3. Obtain the transfer function from the duty cycle to the output voltage of the converter

[0038] Corresponding to Figure 2 the block diagram of the DC-DC converter system is as shown in Figure 3 , where G(s) is the transfer function of the controller.

[0039] The transfer function Gv(s) from the duty cycle to the output voltage of the converter is obtained by Laplace transform as shown in Equation (4).

[0040]

[0041] Among them, d(s) is the representation of duty cycle D in the transfer function; s is the differential operator.

[0042] Step 4. Obtain the fractional-order closed-loop characteristic equation of the DC-DC converter system

[0043] Step 5. Draw the Nyquist diagram of the DC-DC converter system under different parameters, and determine the system stability by the Nyquist criterion.

[0044] Taking the DC-DC converter system as an example for analysis, the main circuit parameters are shown in Table 1.

[0045] Main circuit parameters of DC-DC converter

[0046] Main circuit parameters of DC-DC converter

[0047]

[0048] With a fixed proportional coefficient Kp = 0.01 and an integral order Ki = 10, the Nyquist diagram of the open-loop transfer function of the DC-DC converter system can be drawn when λ changes from 0 to 2, as shown in Figure 3 shown.

[0049] from Figure 4 It can be seen that when λ changes from 0.8 to 1.2, the phase margin of the system also increases. Therefore, in theory, increasing the integral order λ is beneficial to the stability of the DC-DC converter.

[0050] With a fixed integral coefficient Ki = 10 and an integral order λ = 1.0, the Nyquist diagram of the open-loop transfer function of the DC-DC converter system can be drawn when Kp changes from 0.01 to 0.05, as shown in Figure 5 shown.

[0051] from Figure 5 It can be seen that when Kp changes from 0.01 to 0.05, the Nyquist diagram of the open-loop transfer function of the traditional integer-order PI control system does not surround the (-l, j0) point. Therefore, according to the Nyquist criterion, the DC-DC converter is also stable at this time. As Kp increases, the phase margin of the system decreases. Since the phase margin in the open-loop frequency domain performance index corresponds to the overshoot in the time domain performance index, the larger the phase margin, the smaller the overshoot

[12] . In other words, as Kp increases, the overshoot of the traditional integer-order PI controller system also increases. Theoretically, the range of Kp is infinite, but considering the limitation of overshoot in engineering, Kp must be within a reasonable range. Generally, the overshoot is limited to 24%. From the simulation, it can be obtained that the range of Kp is [0.01, 0.04].

[0052] With a fixed proportional coefficient Kp = 0.01 and an integral order λ = 1.0, the Nyquist diagram of the open-loop transfer function of the DC-DC converter system can be drawn when Ki changes from 10 to 100, as shown in Figure 6 shown.

[0053] from Figure 6 It can be seen that when Ki changes from 10 to 100, the intersection of the Nyquist curve of the system open-loop transfer function and the real axis gradually moves to the left, and finally crosses the (-l, j0) point when Ki = 70. Therefore, as Ki increases, the traditional integer-order PI controlled DC-DC converter gradually changes from stable to unstable. Theoretically, when λ = 1.0 and Kp = 0.01, the value range of Ki is (0, 70].

[0054] With fixed integral coefficient Ki = 10 and integral order λ = 0.8, we can draw the Nyquist diagram of the open-loop transfer function of the DC-DC converter system when Kp changes from 0.01 to 0.05, as shown in Figure 7 shown.

[0055] from Figure 7 It can be seen that when Kp changes from 0.01 to 0.05, the Nyquist diagram of the open-loop transfer function of the fractional-order PIλ control system does not surround the (-l, j0) point. Therefore, according to the Nyquist criterion, the DC-DC converter is also stable at this time. However, as Kp increases, the phase margin of the system decreases, and the overshoot of the system also increases. Generally, the overshoot is limited to 24%. From the simulation, it can be obtained that the value range of Kp is [0.01, 0.04].

[0056] With a fixed proportional coefficient Kp = 0.01 and an integral order λ = 0.8, we can draw the Nyquist diagram of the open-loop transfer function of the DC-DC converter system when Ki changes from 5 to 20, as shown in Figure 8 shown.

[0057] from Figure 8 It can be seen that when Ki changes from 5 to 20, the intersection of the Nyquist curve of the system open-loop transfer function and the real axis gradually moves to the left, and finally crosses the (-l, j0) point when Ki = 20. Therefore, as Ki increases, the fractional-order PI controlled DC-DC converter gradually changes from stable to unstable. Theoretically, when λ = 0.8 and Kp = 0.01, the value range of Ki is (0, 20].

[0058] With fixed integral coefficient Ki = 8 and integral order λ = 1.2, the Nyquist diagram of the open-loop transfer function of the DC-DC converter system can be drawn when Kp changes from 0.01 to 0.05, as shown in Fig. 9shown.

[0059] from Fig. 9 It can be seen that when Kp changes from 0.01 to 0.05, the Nyquist diagram of the open-loop transfer function of the fractional-order PIλ controller system does not surround the (-l, j0) point. Therefore, according to the Nyquist criterion, the DC-DC converter is stable at this time. As Kp increases, the phase margin of the system decreases. The overshoot of the system also increases. Generally, the overshoot is limited to 24%. From the simulation, it can be obtained that the value range of Kp is [0.01, 0.04].

[0060] With a fixed proportional coefficient Kp = 0.01 and an integral order λ = 1.2, the Nyquist diagram of the open-loop transfer function of the DC-DC converter system can be drawn when Ki changes from 10 to 600, as shown in Fig.10 shown.

[0061] from Fig.10 It can be seen that when Ki changes from 10 to 600, the intersection of the Nyquist curve of the open-loop transfer function and the real axis gradually moves to the left, and finally crosses the (-l, j0) point when Ki = 500. Therefore, according to the Nyquist criterion, as Ki increases, the DC-DC converter gradually changes from stable to unstable. Theoretically, when λ = 0.8 and Kp = 0.01, the value range of Ki is (0,500].

[0062] From the above theoretical analysis, it can be obtained that as the integral order λ increases, it has little effect on the stable range of the proportional coefficient Kp, but will gradually increase the stable range of the integral coefficient Ki.

[0063] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A fractional-order PIλ controlled DC-DC converter control and stability analysis method, characterized in that: The method comprises the following steps: Step 1: Establish a system state equation for the DC-DC converter; Step 2: Based on small signal modeling, a linear analytical model of the DC-DC converter AC small signal is established; Step 3, obtain the transfer function from duty cycle to converter output voltage by Laplace transform, and obtain the fractional-order closed-loop characteristic equation of the DC-DC converter system under fractional-order PIλ control; Step 4: Draw the Nyquist diagram of the DC-DC converter system under different parameters, and determine the system stability by the Nyquist criterion.

2. The fractional-order PIλ controlled DC-DC converter control and stability analysis method according to claim 1, characterized in that: Model the DC-DC converter operating in the CCM mode, and take the capacitor voltage u c and the inductor current i L as state variables to establish the system state equation. The DC-DC converter operating stably switches periodically between the following two operating modes: Mode 1 is the stage where the switch is on and the diode is off when 0 < t < DTs; Mode 2 is the stage where the diode is on and the switch is off when DTs < t < Ts, where Ts is the switching period and D is the duty cycle; The state equation of the DC-DC converter when working in mode 1 is: Where, u(t) is the output voltage; The state equation of the DC-DC converter when working in mode 2 is:

3. The fractional-order PIλ controlled DC-DC converter control and stability analysis method according to claim 2, characterized in that: Based on the system state equations under the two working modes, the linear analytical model of the DC-DC converter AC small signal is obtained by modeling the small signal, calculating the average variable, separating the disturbance and linear processing.

4. The fractional-order PIλ controlled DC-DC converter control and stability analysis method according to claim 3, characterized in that: The transfer function Gv(s) from duty cycle to converter output voltage obtained by Laplace transform is: Among them, d(s) is the representation of duty cycle D in the transfer function; s is the differential operator, Vin is the input voltage, L is the output filter inductor, and C is the output filter capacitor.

Images