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H-bridge inverters with RLC loads have complex dynamic behavior such as bifurcation and chaos. This nonlinear behavior significantly increases the harmonic content of the output current and reduces system stability and reliability. This article expands the scope of stable system operation by adding a PI controller. Stroboscopic mapping theory is used to simulate the system, the nonlinear dynamic behavior of the inverter is studied by bifurcation diagram, folding diagram and phase orbit diagram are used for comparison and verification, and the TDFC method is introduced to suppress the chaotic behavior of the inverter, further improving the stable operating range of the system. The stability of the system was theoretically analyzed using the fast stability theorem and the correctness of the numerical simulation was verified. Therefore, the conclusion of the study provides a sound theoretical basis for inverter system design and has important theoretical and practical implications.
As a core component of power electronic systems, inverters are widely used in power electronic circuits, especially with the development of new energy sources, high voltage DC transmission and smart grids. However, the inverter is a type of highly nonlinear system with very complex dynamic characteristics. In actual system operation, complex phenomena such as electromagnetic noise, device vibration and system failure often occur, resulting in serious consequences. affects the operation of the device’s inverter. Affects system stability1,2,3,4. Therefore, it is necessary to deeply study this abnormal behavior of inverters to provide effective guidance for system design and operation.
In 2002, Reference 5 conducted the first study of chaos and bifurcation in the current control mode of an H-bridge converter with a DC reference current. In 2009, Wang Xuemei et al. studied the nonlinear dynamic behavior of an H-bridge inverter applied to a sinusoidal signal, and also studied the phenomena of bifurcation and chaos in the inverter6. In addition, the literature 7 studies single-phase SPWM inverters, and introduces and establishes two scales: fast variables and slow variables, to study fast variables and slow variables. Slow variable scales were introduced and discrete models were constructed for a deeper study of the chaotic behavior of sine wave inverters. Literature 8 takes a current-controlled photovoltaic inverter under PI control as a research object, uses a gate mapping method to obtain a discrete system model, and obtains a numerical simulation diagram of a single-phase SPWM inverter circuit under varying input voltage. . The above research results mainly take first-order inverters as the research object and conduct an in-depth study on the nonlinear behavior that occurs in typical first-order inverters. The structure of the basic circuit of a first-order system is simple, and the discrete model is easy to establish. Existing stability modeling and analysis methods require low computational effort and high speed and are suitable for nonlinear dynamic analysis of simple inverter systems. . However, real systems are usually high-order systems, and their underlying circuits are more complex than first-order systems. Therefore, when discrete iterative models are applied to high-order systems analysis, the computational effort, numerical simulation speed, and complexity will increase. This reduces the usefulness of traditional analytical methods9. The literature 10 analyzes the nonlinear dynamic behavior of an H-bridge inverter under proportional control and proposes a discrete model based on coefficient linearization that simplifies the number of algorithms involved in the simulation.
To further improve the nonlinear dynamic behavior of the system, effectively control it to suppress its chaotic behavior and improve the stability of the system, this paper uses an H-bridge inverter with an RLC load as the research object, and also adds a PI controller. , and establishes a discrete mathematical model of the system in this control mode. Observe and compare the nonlinear dynamic behavior of the system using bifurcation diagrams, folding diagrams, phase orbit diagrams, etc., study the effect of voltage parameters on the nonlinear dynamic behavior of the system, and introduce TDFC to effectively suppress the chaotic behavior of the system. system. And use the fast-varying stability theorem to carry out in-depth theoretical analysis to expand the stable operating range of H-bridge inverter with broadband RLC load and provide a reliable theoretical basis for inverter design. Theoretical foundations of inverter design.
To study the nonlinear dynamic behavior of H-bridge inverter with RLC load under PI control, discrete iterative models of H-bridge inverter with RLC load, PI control system and chaos control system were established, respectively.
The circuit diagram of the RLC load system H-bridge inverter is shown in Figure 1, which consists of voltage source E, switches D1~D4, LC filter and load resistor R. The inverter is a high-performance inverter whose output current is compared with the reference current and transferred to a proportional controller. In the operation control part, the inverter output current \(i_{L}\) is compared with the reference current \(i_{ref}\) to obtain an error signal, which is fed to the proportional controller and then sent to the PWM. Control Circuit Triangular wave modulation is used to generate a control signal that can be used to control the switching tubes on and off to monitor the operating status of each switching tube.
In the switching cycle, the system has two operating modes: D1 and D4 are on, D2 and D3 are off, which corresponds to state 1, D1 and D4 are off, and D2 and D3 are on, which corresponds to state 2; (1) and (2) are the equations corresponding to the two states of the working expression.
Among them, \(U_{C}\) and \(i_{L}\) are the state variables of the system. There are two operating states of the system, let \(\varvec{X}=\begin{bmatrix} i_L&U_ c. \end {bmatrix }^T,\) then the expression of the system state equation will be (3).
其中， \(\begin{aligned} \mathbf {A_1}=\mathbf {A_2}=\textbf{A}= \begin{bmatrix} 0& -1/_{L}\\ 1/_{C}& – 1/_{(RC)} \end{bmatrix} , \mathbf {B_1}= \begin{bmatrix} 1/_{L}\\ 0\end{bmatrix} ,\mathbf {B_2}= \begin{ bmatrix } -1/_{L}\\ 0 \end{bmatrix} \end{对齐}\)
According to the strobe mapping theory11,12, the discrete iterative model of the inverter main circuit is derived from Equation (11). (3) as shown in formula (2). (4):
In equation (4), \({p_{1}=e^{\varvec{A}T};} {~p_{2}=}{e}^{\varvec{A}(1-{d} _{ n}){T}}({e}^{\varvec{A}{d}_{n}{T}}-\varvec{{I}})\varvec{{A}^{-1 }} \varvec{{B}_{1}}+[{e}^{\varvec{A}(1-{d}_{{n}}){T}}-\varvec{{I}} ]\varvec{{A}^{-1}{B}_{2}}\), I is the third-order identity matrix, \(d_{n}\) is the duty cycle of the inverter of the nth switching cycle, This is expressed How:
In equation (5), D is a constant, k is the proportional control parameter, \(\sigma =\begin{bmatrix}1&0\end{bmatrix}\) and \(i_{refn}\) is the value of the reference current at n- th switch. During the period, the discrete model of the system consists of equation (1). (4) and (5).
Due to the complexity of the inverter system, a discrete model based on coefficient linearization is created to convert the above matrix exponential operation into a linear operation, which simplifies system modeling and other operations11, then the following arises:
Substituting equation (6) Substituting equation (7) Then the linearized discrete iterative model of the coefficients of the main circuit can be represented as:
在上式中，\({p}_{1}^{‘}=\varvec{I}+ \varvec{A}{T}, {p}_{2}^{‘}=\varvec{{ B}_{1}}\textrm{d}_{\textrm{n}}\textrm{T}+ \varvec{{A}_{1}}\varvec{{B}_{1}}\textrm {d}_{\textrm{n}}(1-\textrm{d}_{\textrm{n}})\textrm{T}^{2}+ (1-\textrm{d}_{\textrm {n}}){T}\varvec{{B}_{2}}\)
The discrete model of an H-bridge inverter with an RLC load system consists of the equations: Equations (5) and (7) are linearized using coefficients. This method greatly simplifies the computational effort of system simulation and improves numerical simulation and computational speed. analysis of system stability.
System with PI control. The circuit diagram of the H-bridge RLC load system inverter with PI control is shown in Figure 2, which consists of voltage source E, switches D1, D4, LC filter, load resistor R and PI control. Part. In the operation control part, the inverter error signal is input to the PI controller and sent to the PWM control circuit through triangular wave modulation. The control signal controls the operating state of each switch tube.
\(d_{n}\) represents the duty cycle of the nth switching cycle. The duty cycle of an H-bridge inverter with an RLC load under PI control is expressed as (8).
From Eq. According to equations (5) and (8), the discrete model of H-bridge inverter with RLC load system under PI control can be expressed as (9), (10) and (11):
In the above formula \(a_1=\left( k_\textrm{i}\frac{L}{R}-k_\textrm{p}\right) \left( \textrm{e}^{-\frac{ R }{L}T}-1\right)\), \(a_2=\left( k_\textrm{i}\frac{L}{R}-k_\textrm{p}\right)\left( \ click Crack{2}{R}\textrm{e}^{-\frac{(1-d_{n-1})RT}{L}}-\frac{1}{R}-\frac{ 1} { R}\textrm{e}^{-\frac{R}{L}T}\right) +\)\(\frac{k_\text {i}T}{R}\left( 1- 2d_{ n -1}\right)\).
The basic idea of time feedback control (TDFC) method is to differentiate a state variable with its own delay time covariate and apply the difference to a chaotic system to improve the control effect and expand the operating stability region of the system. system. . Time feedback control method, its control block diagram is shown in Figure 3. The specific implementation method is that the delay control signal \(u[i(t)-i(t-\tau )]\ ) is added to the original control signal \(k(i_{ref}-i_{n })\), where u serves as a tuning parameter and \(\tau\) is an integer multiple of the period of the target unstable periodic orbit. At this moment \(i(t)=i(t-\tau )\) the delay control signal disappears and the system enters the target cyclic orbit. In an H-bridge inverter, it represents the period of change of the control signal. Selecting τ=T_{s}\) as the delay time of the delay feedback control signal can make the sampling period of the delay controller consistent with the switching period of the system. The duty cycle of the TDFC based regulated H-bridge inverter is:
In Equation (12), Among them, D is a constant, k is the proportional control parameter, and \(i_{ref}\) is the reference current.
The circuit diagram of the system is shown in Figure 4. The process of implementing time feedback chaos control in an inverter based on PI control: first compare the load current with the reference current, compare the load current with the reference current, compare the load current with the reference current, compare the load current with the reference current. Compare, compare the load current with the reference current, multiply the load current delay time parameter and the adjustment parameter \(\begin{array}{c} \eta \\ \end{array}\) by this difference, the delay control signal \(\eta (i_{n+1}-i_{n}),\) is obtained and entered into the PI controller. Secondly, the modulation signal is obtained through a PI controller and compared with a triangle wave. Finally, a PWM control signal is obtained to control the switch tube on-off.
After introducing the TDFC method, it can be seen in Figure 4 that the expressions of the discrete chaos control model of the TDFC inverter under PI control are (13), (14) and (15):
According to the created discrete mathematical model, we will accept \(k_{\textrm{i}}=180\) and set the circuit parameters as follows: \(E=350V, R=10\Omega, L=8mH, f_{s }=5 kHz ,i_{ref}=5\sin (100\pi\)t). For different scale parameters \(k_p.\), use bifurcation diagrams, folding diagrams, and phase trajectory diagrams to analyze the nonlinear dynamic behavior generated by the system.
The bifurcation diagram method can be used to analyze the dynamic stability of a system and observe the impact of parameter changes on system performance14. Thanks to the discrete iteration model, after the iteration becomes stable, sampling is performed at several fixed switching times, the sampling points are stored, and the sampling values are plotted as different values of the bifurcation parameters. When the number of points is equal to the bifurcation period, the system is in a periodic state; when the points are chaotic, the system is in a chaotic state. Taking the proportional control parameter \(k_{p}\) as the bifurcation parameter, the bifurcation diagram of the inverter under \(k_{p}\) PI control and the output current of the inverter are drawn respectively as shown in the figure. 5. The bifurcation diagram of the inverter is shown in Figure 5a. When the system is running stably, the stable region of the proportional control parameter \(k_{p}\) is [0.2, 0.455) when it reaches the bifurcation point value. 0.45, the system is in the period 2 state, and as k_{p}\) increases, the system goes into a chaotic state.
The bifurcation diagram of the inverter system with the added PI controller during stable operation is shown in Figure 5b. The stability region of the proportional control parameter \(k_{p}\) is equal to [0.2, 0.9], and the stability limit. point \(k_{p}\)=0.9. From the bifurcation diagram it is clear that the system, having survived the process of entering a multi-period bifurcation state, continues to move from the stable state \(k_{p}\). increases and finally presents an unstable chaotic state.
The bifurcation diagram of adding time feedback PI control method to the inverter is shown in Figure 5c. When the system is running stably, the stable area of the proportional control parameter
Load current branching with proportional control parameter \(k_{p}\): (a) Inverter; (b) Inverter containing a PI controller; (c) Inverter with added TDFC integrated into PI controller converter.
k_{p}\) is [0.2,0.99], and the stability cutoff point\(k_{p}\)=0.99. From the bifurcation diagram it is clear that as \(k_{p}\) increases, the system undergoes a process of transition from a stable state to an unstable chaotic state.
Comparison of the three bifurcation diagrams shows that the H-bridge inverter with RLC load system and the addition of PI controller has a wider range of operating stability compared to the inverter. After applying the delay feedback control method, the chaotic behavior generated by the system is suppressed, the stability region of the system is further expanded, and the stability of the system is improved.
Bifurcation of load current and voltage E: (a) Inverter; (b) Inverter including PI controller; (c) Inverter with added TDFC integrated into PI controller.
Taking the voltage E as the bifurcation parameter, draw the bifurcation diagram of the output current of the inverter and the PI-controlled inverter as a function of the circuit parameter E, as shown in Figure 6. The inverter is shown in Figure 6a. When the voltage E>405V, the system is no longer stable, the sampling current begins to bifurcate, and as the voltage E continues to increase, the system enters a chaotic state. The voltage bifurcation diagram of an inverter with an added PI controller is shown in Figure 6b, in which the system ceases to be stable at E > 630 V and the system quickly becomes chaotic. The bifurcation voltage diagram of the system using the TDFC method is shown in Figure 6c. At voltage E>705V, the system begins to enter a chaotic state.
Compared with the inverter without adding PI controller, when the system operating parameter is voltage E, the stable area of the inverter system under PI control is wider. After the introduction of TDFC, the stable area of the system became even larger. expanded. method. It can be seen that adding chaos control can improve the stability of the inverter system.
The region of stability parameters of the bifurcation diagram of the specified system was analyzed and verified using the folding diagram method and the phase trajectory diagram method.
Folding graph: Replace the initial iteration value with the discrete iteration of the system, select n cycles after stable operation, align and fold according to the sampling time. In a stacked diagram, if the sample points of n sine curves completely overlap to form a sine curve, the system is stable; if the sample points are completely consistent but form multiple sine curves, the system structure is unstable but not chaotic; In densely populated areas the system is chaotic15.
The phase hodograph diagram reflects the projection of the system solution curve onto the phase space. If the phase trajectory diagram represents one closed curve, this means that the system is single-period and stable; if there are n closed curves, this means that the system is in the nth period state; the system is in a chaotic state9.
The remaining parameters remain unchanged. When the integral correction factor \(k_{i}\) is equal to 180 and the proportional correction factor \(k_{p}\) is equal to 0.46, 0.8, 0.92 and 1.2 respectively, the folded diagram and diagram are drawn in the time domain of the system. , for example, as shown in the figure. 7 and 8.
As shown in the picture. As shown in Figures 7a and 8a, when \(k_{p}\)=0.46, the folding diagram (left) of the inverter system has two sine curves, the phase path diagram has two closed curves, and the system exhibits period doubling. . The frequency division indicates that the inverter is in cycle 2 state at this time. The stacked diagram (middle) of the PI control inverter is a smooth sine wave, and the phase path diagram is also a single curve. At this time, the system operates in a stable state. Under PI control, the stacked diagram (right) of the inverter using the TDFC method is also a smooth sine curve, and the phase path diagram is also a single curve, which proves that the system is in a steady state.
As shown in the picture. As shown in Figures 7b and 8b, when k_{p}\)=0.8, the convolved diagram display area (left) of the inverter system is covered with dense sampling points, and countless chaotic curves appear on the phase trajectory. diagram And the inverter enters a chaotic state. The collapsed diagram (center) of the PI controlled inverter is still a smooth sine curve. When the inverter is in period 1 stable state, the phase path diagram is a single curve, which also indicates that the system is still stable. After applying the TDFC method, the folding diagram and phase orbit diagram (right) of the system remain unchanged and the operating state is stable.
As shown in the picture. As shown in Figures 7c and 8c, when k_{p}\)=0.92, the inverter system (left) is still in a chaotic state. The collapsed diagram (center) of the PI-controlled inverter shows two sine curves, and the phase path diagram has two closed curves. The system begins to bifurcate, indicating that the inverter enters the period 2 state at this time. Currently, after applying the TDFC method, the system folding diagram (right) is still a smooth sine curve, the phase path diagram shows a single curve , and the inverter is in a stable state.
As shown in the picture. As shown in Figures 7d and 8d, when k_{p}\)=1.2, the folding diagram and the phase path diagram (left) of the inverter remain unchanged, and the system is still in a chaotic state. The collapsed diagram (center) of the PI-controlled inverter shows two curves with dense sampling points. The phase path diagram shows a large number of chaotic curves in the system, indicating that the inverter has entered a chaotic state. The fold diagram and phase path diagram (right) of the system under TDFC control are the same as those under PI control, and the inverter is in a chaotic state.
The results show that when the parameter \(k_{i}\) is fixed and the system is varied by \(k_{p}\), the conclusions obtained from the analysis of bifurcation diagrams, folding diagrams, and phase trajectory diagrams are consistent.
Stacked diagram of various scale parameters \(k_{p}\) at \(k_{i}\)=180: inverter (left); PI-controlled inverter (center); adding a PI-controlled TDFC inverter (right);
Phase trajectories at \(k_{i}\)=180 with different scale parameters \(k_{p}\): inverter (left); PI-controlled inverter (center); adding a PI-controlled TDFC inverter converter (right);
Apply the fast stability theorem to analyze inverter stability and compare the results with bifurcation diagrams, folding diagrams, and phase orbit diagrams to check their consistency. This standard allows you to accurately determine whether a system is in stable operating condition. The basic idea of the fast-varying stability theorem method is as follows: take M switching cycles near the zero crossing point of the current drop segment, use the duty cycle of each switching cycle and the duty cycle of the next switching cycle, and then divide by two the absolute value of the difference between them, and then add the calculated number M to get P16, 17. Formula for P:
In equation (16): \(d_{n}\) is the duty cycle in the nth switching cycle; \(d_{n+1}\) — duty cycle in the n+1st switching cycle; When the system is stable, P=M, when the system is unstable, P;
Evaluation results of the rapidly changing stability theorem: (a) Inverter; (b) Inverter containing a PI controller; (c) Inverter with added TDFC integrated into PI controller.
The rapidly varying stability theorem is used to evaluate the stability of an H-bridge inverter with an RLC load system. When k_{i}\) = 180 and other parameters remain constant, the relationship between P and \(k_{p}\) is shown in Figure 9a. From the figure it is clear that at \(0.1
Post time: Oct-23-2024