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Oct 14, 2024

Research on the complex dynamical behavior of H-bridge inverter with RLC load | Scientific Reports

Scientific Reports volume 14, Article number: 23916 (2024) Cite this article Metrics details Complex dynamical behaviors such as bifurcation and chaos exist in H-bridge inverter with RLC load, and

Scientific Reports volume 14, Article number: 23916 (2024) Cite this article

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Complex dynamical behaviors such as bifurcation and chaos exist in H-bridge inverter with RLC load, and these nonlinear behaviors will greatly increase the harmonic content of the output current and reduce the stability and reliability of the system. In this paper, a PI controller is added to widen the stable operation domain of the system. The stroboscopic mapping theory is used to model the system, the nonlinear dynamic behavior of the inverter is investigated by the bifurcation diagram, the folding diagram and the phase trajectory diagram are used for comparative verification, and the TDFC method is introduced to inhibit the chaotic behavior of the inverter, which further improves the stable range of the system operation. The fast-change stability theorem is used to analyze the stability of the system theoretically and verify the correctness of the numerical simulation. Therefore, the conclusions of the study provide a reliable theoretical basis for the design of the inverter system, which has important theoretical significance and practical value.

As a core component of power electronic systems, inverters are widely used in power electronic circuits, especially with the development of new energy generation, high-voltage direct current transmission, and smart grid. However, inverters are a class of strongly nonlinear systems with very complex dynamic characteristics, and bifurcation and chaos phenomena are very common, and the system often suffers from complex phenomena such as electromagnetic noise, device oscillations, and system crashes in actual operation, which seriously affects the system stability1,2,3,4. Therefore, it is necessary to study these irregular behaviors in inverters in depth to provide effective guidance for system design and operation.

In 2002, the first study of chaos and bifurcation in a current-controlled mode H-bridge converter with reference current as dc was carried out in literature5.In 2009, Xuemei Wang et al. investigated the nonlinear dynamics behavior of H-bridge inverters with sinusoidal signals as the reference and studied the bifurcation and chaos phenomena in the inverters6. Moreover, in the literature7, a single-phase SPWM inverter was studied, and two kinds of scales, namely, fast and slow variables, were introduced and established to study the fast and slow variables. and slow-variable scales are introduced and a discrete model is established to study the chaotic behavior in sinusoidal inverters more deeply. Literature8 takes the current-controlled PV inverter under PI control as the research object, adopts the strobe mapping method to derive the discrete model of the system, and obtains the numerical simulation diagrams of the single-phase SPWM inverter circuit as the input voltage changes. In the above research results, the first-order inverter is mainly taken as the research object, and the nonlinear behaviors appearing in typical first-order inverters are studied in depth. The main circuit structure of the first-order system is simple, the discrete model is easy to establish, and the existing modeling methods and stability analysis are computationally small and fast, which are suitable for the nonlinear dynamics analysis of simple inverter systems. However, real systems generally belong to higher-order systems, whose main circuits are more complex than those of first-order systems. Therefore, when the discrete iterative model is applied to high order system analysis, the computational load, numerical simulation speed and complexity will be increased. This leads to a decline in the usefulness of traditional analytical methods9. The reference10 analyzes the nonlinear dynamic behavior of H-bridge inverter under proportional control, and proposes a discrete model based on coefficient linearization, which simplifies the amount of algorithms involved in modeling.

To make further improvements to the nonlinear dynamic behavior of the system and to impose effective control to suppress its chaotic behavior to improve the stability of the system, this paper takes the H-bridge inverter with RLC load as the research object joins the PI controller, and establishes a discrete mathematical model of the system under this control mode. We observe and compare the nonlinear dynamic behavior of the system through bifurcation diagram, folding diagram, phase trajectory diagram, etc., study the influence of voltage parameters on the nonlinear dynamic behavior of the system, introduce TDFC to effectively inhibit the chaotic behavior of the inverter, and use the fast-varying stability theorem to carry out an in-depth theoretical analysis to widen the stable working range of H-bridge inverter with RLC load, to provide reliable theoretical bases for the design of the inverters. Theoretical basis for the design of the inverter.

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, the system under PI control, and the system with the addition of chaos control are established, respectively.

system The working principle diagram of the H-bridge inverter with RLC load system, as shown in Fig. 1, consists of a voltage source E, switch D1 D4, an LC filter, and a load resistor R. The inverter is a high-performance inverter with an output current that is compared with the reference current and sent to a proportional regulator. In the operation control section, the inverter output current \(i_{L}\) is compared with the reference current \(i_{ref}\) to derive the error signal, which enters the proportional regulator and then is sent to the PWM drive circuit through triangular wave modulation to generate the drive signal that can be used for on-off control of the switch to control the operating state of each switch.

Operating schematic of H-bridge inverter with RLC load.

In a switching cycle time, the system exists in two operating modes: D1 and D4 on, D2 and D3 cutoff, this time for state 1; D1 and D4 cutoff, D2 and D3 on, this time for state 2. Eqs. (1) and (2) are the equations corresponding to the two states of the working expression.

Where \(U_{C}\) and \(i_{L}\) are the system state variables, the system exists in two operating states, let \(\varvec{X}=\begin{bmatrix} i_L&U_ c \end{bmatrix}^T,\) then the system state equation expression is Eq. (3).

Among them, \(\begin{aligned} \mathbf {A_1}=\mathbf {A_2}=\textbf{A}= \begin{bmatrix} 0& -1/_{L}\\ 1/_{C}& -1/_{(RC)} \end{bmatrix} , \mathbf {B_1}= \begin{bmatrix} 1/_{L}\\ 0\end{bmatrix} ,\mathbf {B_2}= \begin{bmatrix} -1/_{L}\\ 0 \end{bmatrix} \end{aligned}\)

According to the stroboscopic mapping theory11,12, a discrete iterative model of the main circuit of the inverter is derived from Eq. (3), as shown in Eq. (4):

In the Eq. (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 unit matrix and \(d_{n}\) is the inverter duty cycle in the nth switching cycle, which is expressed as:

In the Eq. (5), D is a constant, k is a proportional adjustment parameter, \(\sigma =\begin{bmatrix}1&0\end{bmatrix}\) and \(i_{refn}\) is the value of the reference current in the nth switching cycle, and a discrete model of the system is formed from Eqs. (4) and (5).

Due to the complexity of the inverter system, the discrete model based on linearization of coefficients is established to transform the above matrix exponential operation into linear operation, which simplifies the arithmetic process such as system modeling11, and then there are:

Substituting Eq. (6) into Eq. (7), the linearized discrete iterative model of the coefficients of the main circuit can be introduced as:

In the above equation, \({p}_{1}^{'}=\varvec{I}+ \varvec{A}{T}, {p}_{2}^{'}=\varvec{{B}_{1}}\textrm{d}_{\textrm{n}}\textrm{T}+ \varvec{{A}_{1}}\varvec{{B}_{1}}\textrm{d}_{\textrm{n}}(1-\textrm{d}_{\textrm{n}})\textrm{T}^{2}+ (1-\textrm{d}_{\textrm{n}}){T}\varvec{{B}_{2}}\)

The discrete model of the H-bridge inverter with RLC load system is composed of Eqs. (5) and (7), which are linearized by the coefficients, and this method greatly simplifies the computation of the system modeling and improves the operation speed of the numerical simulation and the system stability analysis.

system under PI control The working schematic diagram of the H-bridge inverter with RLC load system under PI control, as shown in Fig. 2, consists of voltage source E, switch D1 D4, LC filter, load resistor R, and PI control part. In the working control part, the error signal of the inverter enters into the PI controller and then is sent to the PWM driving circuit through triangular wave modulation, and the driving signal controls the working state of each switch.

\(d_{n}\) represents the duty cycle of the nth switching cycle, and the duty cycle of the H-bridge inverter with RLC load under PI control is expressed as Eq. (8).

From Eqs. (5) and (8), the discrete model of the H-bridge inverter with RLC load system under PI control can be expressed by Eqs. (9), (10) and (11):

Schematic diagram of H-bridge inverter with RLC load operation under PI control.

In the above equation, \(a_1=\left( k_\textrm{i}\frac{L}{R}-k_\textrm{p}\right) \left( \textrm{e}^{-\frac{R}{L}T}-1\right)\), \(a_2=\left( k_\textrm{i}\frac{L}{R}-k_\textrm{p}\right)\left( \frac{2}{R}\textrm{e}^{-\frac{(1-d_{n-1})RT}{L}}-\frac{1}{R}-\frac{1}{R}\textrm{e}^{-\frac{R}{L}T}\right) +\)\(\frac{k_\text {i}T}{R}\left( 1-2d_{n-1}\right)\) .

The main idea of the time-delay feedback control (TDFC) method is to differentiate the state variables from the covariates of their own delay time13 and act the difference in the chaotic system to improve the control effect and widen the operating stability domain of the system. The time-delay feedback control method, whose control block diagram is shown in Fig. 3. The specific realization method is that a time-delay control signal \(u[i(t)-i(t-\tau )]\) is attached to the original control signal \(k(i_{ref}-i_{n})\), where u is used as an adjustment parameter and \(\tau\) is an integer multiple periods of the target unstable period orbit. At that time \(i(t)=i(t-\tau )\), the time-delay control signal disappears, and the system operation enters the target cycle track. In the H-bridge inverter, represents the change period of the control signal. Choosing \(\tau =T_{s}\) as the delay time of the time-delay feedback control signal can make the sampling period of the time-delay controller consistent with the switching period of the system. The duty cycle \(d_{n}\) of the TDFC-based regulated H-bridge inverter is:

Control block diagram based on TDFC method.

In the Eq. (12) where D is a constant, k is a proportional regulation parameter and \(i_{ref}\) is the reference current.

The working schematic of the system, as shown in Fig. 4. The process of introducing time-delay feedback chaos control for the inverter based on PI regulation: first, the load current and the reference current are compared, the load current is differenced from the parameter of its own delay time, and the regulation parameter \(\begin{array}{c} \eta \\ \end{array}\) is multiplied by this difference to obtain the time-delay control signal \(\eta (i_{n+1}-i_{n}),\) input to the PI controller. Secondly, the modulating signal is obtained by the PI controller and compared with the triangle wave. Finally, a PWM drive signal is obtained for on-off control of the switch.

Schematic diagram of inverter operation with the addition of time-delay feedback chaos control.

After the introduction of the TDFC method, it can be seen from Fig. 4 that at this point the discrete model expression for the TDFC chaos control of the inverter under PI control is Eqs. (13), (14) and (15):

According to the established discrete mathematical model, \(k_{\textrm{i}}=180\) is taken and the circuit parameters are set as follows: \(E=350V,R=10\Omega ,L=8mH,f_{s}=5kHz,i_{ref}=5\sin (100\pi\)t). The nonlinear dynamic behavior of the system generated by the system is analyzed using bifurcation diagrams, folding diagrams, and phase trajectory diagrams for different scaling parameters \(k_p.\)

The bifurcation plot method can be used to analyze the dynamical stability of a system as well as to observe the effect of parameter changes on system performance14. Through a discrete iterative model, sampling is taken within a few fixed switching moments after the iteration is stabilized and the sampling points are saved to plot the sampled values into a graph for different bifurcation parameter values. When the number of points is equal to the bifurcation period, the system is in a periodic state; when the points are cluttered, the system is in a chaotic state. Taking the proportional regulation parameter \(k_{p}\) as the bifurcation parameter, the bifurcation diagrams of the inverter and the output current of the inverter under PI control with \(k_{p}\) are plotted respectively, as shown in Fig. 5. The bifurcation diagram of the inverter is shown in Fig. 5a, and the stability domain of the proportional regulation parameter \(k_{p}\) is [0.2, 0.455] during the stable operation of the system when the operation reaches the value of the bifurcation point, 0.45, the system exhibits the cycle 2 state, and the system enters into the chaotic state with the increase of \(k_{p}\).

The bifurcation diagram of the inverter with the addition of the PI controller is shown in Fig. 5b when the system is running stably, the stability domain of the proportional regulation parameter \(k_{p}\) is [0.2, 0.9], and the stability cut-off point of \(k_{p}\)=0.9, and from the bifurcation diagram, it can be seen that the system has gone through the process of entering into the multiply-periodic bifurcation state from the stable state with the increasing of \(k_{p}\) and finally presenting an unstable chaotic state.

The bifurcation diagram is shown in Fig. 5c for the time-delay feedback control method applied to the inverter with PI control added. When the system is running stably, the stability domain of the proportional regulation parameter

Bifurcation of load current with proportional regulation parameter \(k_{p}\): (a) Inverter; (b) Inverter incorporating a PI controller; (c) Inverter incorporating a PI controller after adding TDFC.

\(k_{p}\) is [0.2, 0.99], and the stability cutoff point of \(k_{p}\)=0.99. From the bifurcation diagram, it can be seen that the system undergoes the process of entering into an unstable chaotic state from a stable state with the increase of \(k_{p}\).

A comparison of the three bifurcation diagrams shows that the H-bridge inverter with RLC load system with the addition of the PI controller has a wider operating stability range compared to the inverter. After applying the time-delay feedback control method, the chaotic behavior generated by the system is suppressed, and the stability domain of the system is further extended, which improves the stability of the system.

Bifurcation of load current with voltage E: (a) Inverter; (b) Inverter incorporating a PI controller; (c) Inverter incorporating a PI controller after adding TDFC.

Taking the voltage E as the bifurcation parameter, the bifurcation diagrams of the output currents of the inverter and the inverter under PI control as a function of the circuit parameter E are plotted respectively, as shown in Fig. 6. The voltage bifurcation diagram of the inverter is shown in Fig. 6a, where the system is no longer stabilized when the voltage E> 405V, the sampling current starts to bifurcate, and the system enters into a chaotic state as the voltage E continues to increase. The voltage bifurcation diagram of the inverter with the addition of the PI controller is shown in Fig. 6b, where the system is no longer stabilized at voltage E>630V, and the system rapidly enters a chaotic state. The voltage bifurcation diagram of the system introducing the TDFC method is shown in Fig. 6c, and the system starts to enter a chaotic state when the voltage E>705V.

Compared with the inverter without adding a PI controller, when the system operating parameter is voltage E, the region when the inverter system is in the steady state under PI control is wider, and the stability domain of the system is further widened after introducing the TDFC method. It can be seen that the stability of the inverter system can be improved by adding chaos control.

The stability parameter domain of the above system bifurcation diagram is analyzed and verified by the folding diagram method and phase trajectory diagram method.

Folding diagram: replace the initial value of an iteration into the discrete iteration of the system, select the n cycles after the stable operation, and align and fold according to the sampling time. In the folding diagram, if the sampling points of n sine waves completely overlap into a sinusoidal curve, the system is stable; if the sampling points are completely consistent but form multiple sinusoidal curves, the system structure is unstable but not chaotic; if there is a dense region, the system is chaotic15.

The phase trajectory diagram reflects the projection of the solution curve of the system on the phase space. If the phase trajectory diagram is a single closed curve, it means that the system is single-period stable; if there are n closed curves, it means that the system is in the nth periodic state; numerous chaotic curves indicate that the system is in a chaotic state9.

Other parameters remain unchanged. When the integral adjustment factor \(k_{i}\) is 180 and the proportional adjustment factor \(k_{p}\) is 0.46, 0.8, 0.92 and 1.2 respectively, the folding diagram and time-domain diagram of the system are drawn, as shown in Figs. 7 and 8.

As shown in Figs. 7a and 8a, when \(k_{p}\)=0.46, the folding diagram (left) of the inverter system shows two sinusoidal curves, the phase trajectory diagram has two closed curves, and the system appears the phenomenon of period doubling and frequency division, indicating that the inverter is in the period 2 state at this time. The folding diagram (middle) of the inverter controlled by PI is a smooth sinusoidal curve, and the phase trajectory diagram also shows a single curve, at which time the system works in a stable state. Under the control of PI, the folding diagram (right) of the inverter with the TDFC method is also a smooth sine curve, and the phase trajectory diagram is also a single curve, which proves that the system is in a stable state.

As shown in Figs. 7b and 8b, when \(k_{p}\)=0.8, the display area of the folding diagram (left) of the inverter system is filled with dense sampling points, and numerous chaotic curves appear in the phase trajectory diagram, and the inverter enters a chaotic state. The folding diagram (middle) of the inverter controlled by PI is still a smooth sinusoidal curve, when the inverter is in a stable period 1 state, and the phase trajectory diagram shows a single curve, which also shows that the system is still stable. After applying the TDFC method, the folding diagram and phase track diagram (right) of the system remain unchanged, and the working state is stable.

As shown in Figs. 7c and 8c, the inverter system (left) is still in a chaotic state when \(k_{p}\)=0.92. The folding diagram (middle) of the inverter controlled by PI shows two sinusoidal curves, and there are two closed curves in the phase trajectory diagram, and the system begins to bifurcate, indicating that the inverter enters the period-2 state at this time. At this time, after applying the TDFC method, the system folding diagram (right) is still a smooth sinusoidal curve, the phase trajectory diagram shows a single curve, and the inverter is in a stable state.

As shown in Figs. 7d and 8d, the folding diagram and phase track diagram (left) of the inverter remains unchanged when \(k_{p}\)=1.2, and the system is still in a chaotic state. The folding diagram (middle) of the inverter controlled by PI shows two curves filled with dense sampling points, the phase trajectory diagram shows numerous chaotic curves, and the system appears chaotic, indicating that the inverter has entered a chaotic state. The folding diagram and phase track diagram (right) of the system under TDFC control are the same as those under PI control, and the inverter is in a chaotic state.

It is shown that the conclusions obtained from the analysis of bifurcation diagrams, folding diagrams, and phase trajectory diagrams are consistent when the parameter \(k_{i}\) is fixed and the system varies with \(k_{p}\).

Folding diagram for different scale parameters \(k_{p}\) at \(k_{i}\)=180: Inverter (left); PI-controlled inverter (center); inverter with TDFC added PI control (right).

Phase trajectories at \(k_{i}\)=180 with different scale parameters \(k_{p}\): Inverter (left); PI-controlled inverter (center); Inverter with TDFC added PI control (right).

The fast-change stability theorem is applied to analyze the stability of the inverter, and the results are compared with those of the bifurcation diagram, the folding diagram, and the phase-trajectory diagram, and their consistency is checked. This criterion can accurately determine whether the system is in a stable operation state. The core idea of the fast-variable stability theorem method is: take M switching cycles near the zero crossing point of the current descent section, respectively, use the duty cycle of each switching week and the duty cycle of the next switching cycle to make a difference and then divide by the absolute value of the difference between the two, and then add up the number of the calculated M to obtain P16,17. The formula for P is:

In the Eq. (16): \(d_{n}\) is the duty cycle in the nth switching cycle; \(d_{n+1}\) is the duty cycle in the n+1th switching cycle. When the system is stable, P=M; when the system is unstable, P<M. Taking N0=1150, M=19, and \(k_{p}\) as the regulation parameter, Eqs. (5), (9), and (11) are calculated by substituting them into Eq. (12) to obtain the judgment results based on the fast-change stability theorem, which are shown in Figs. 9a,b,c, respectively.

Judgment result of fast-change stability theorem: (a) Inverter; (b) Inverter incorporating a PI controller; (c) Inverter incorporating a PI controller after adding TDFC.

The fast-change stability theorem is utilized to evaluate the stability of the H-bridge inverter with RLC load system. When \(k_{i}\) = 180 and other parameters are constant, the relationship between P and \(k_{p}\) is shown in Fig. 9a. From the figure, when \(0.1<k_{p}<0.458,\) P=M and the system is in a stable state. When kp exceeds the range of [0.2,0.458], P<M, the system becomes unstable chaos and bifurcation state, which verifies that the stable range of kp is [0.2,0.458].

To determine the stability of the H-bridge inverter with RLC load system under PI control, when \(k_{i}\)=180, \(N_{0}\)=1150, and M=19, other parameters are kept constant and the relationship between P and \(k_{p}\) is shown in Fig. 9b. From the figure, it can be seen that when \(0.2<k_{p}<0.91,\) P=M and the system is in a stable state. When \(k_{p}\) operates outside the region [0.2,0.91], the values of P are all less than M, and the system is in an unstable state, so it can be verified that the operating domain of \(k_{p}\) stability is [0.2,0.91].

The fast-change stability theorem is applied to evaluate the stability of the H-bridge inverter with RLC load system after the introduction of TDFC chaos control. When ki=180 and other parameters are unchanged, the relationship between P and kp is shown in Fig. 9c. From the figure, it can be seen that when \(0.2<k_{p}<0.985,\) P=M and the system is in a stable state. When \(k_{p}\) exceeds [0.2,0.985], P<M and the system is in an unstable chaotic state, so the stable range of \(k_{p}\) is verified to be [0.2,0.985].

The analytical conclusions of the stable domain Figs. 7a,b,c judged based on the fast-varying stability theorem and the bifurcation diagrams shown in Figs. 5a,b,c, respectively, are in perfect agreement, which verifies the correctness of the stable range of the system.

The two-parameter stability region diagram is shown in Fig. 10. From Fig. 10a, we can see the division of the system stability region when the two control parameters – \(k_{p}\) and \(k_{i}\) have different values. Similarly, from Fig. 10b, we can also obtain the stability boundary described by two control parameters – \(k_{p}\) and E when they have different values. For example, when the coordinates (ki, kp) is (88.6, 0.5), the stability region of the inverter is on the boundary between stable and unstable. As the control parameters kp and ki increase simultaneously, the stable operating range of the inverter gradually expands. When the coordinate (E, kp) is (657, 0.5), the control parameter kp and the circuit parameter E increase at the same time, and the stability region of the inverter shrinks. Therefore, the parameter design of the inverter can be optimized through the two-parameter stability region.

Two-parameter stability region diagram: (a) kp-ki; (b) kp-E;

In this paper, the stability of the system is analyzed using bifurcation diagrams, folding diagrams and phase trajectory diagrams. The results show that the H-bridge inverter with RLC load system combined with the PI control mode can obtain a wider operating stability range and make the system more stable, and based on this, the TDFC method is added, and the results of the study show that this can further suppress the chaotic behavior of the inverter system. The findings of this paper can provide an important theoretical basis for the design and fabrication of H-bridge inverters.

All data generated or analysed during this study are included in this published article and its supplementary information files.

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This work was supported in part by the subject of educational science planning in Guangdong province under Grant 2023GXJK519, in part by the subject of Undergraduate Teaching quality Engineering Project in Guangdong province under Grant GDJX2022004, in part by innovation project of innovation and entrepreneurship fund of Wuyi University under Grant 2023111500000434, in part by science and technology plan project of education department of Jiangxi province under Grant GJJ2203702.

Caigui Zhong and Fang Yuan have contributed equally to this work.

Faculty of Electronic and Information Engineering, Wuyi University, Jiangmen, 529000, China

Mingjian Wu, Wei Jiang & Caigui Zhong

School of Mechanical and Electronic Engineering, East China University of Technology, Nanchang, 330039, China

Fang Yuan

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Correspondence to Wei Jiang.

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Wu, M., Jiang, W., Zhong, C. et al. Research on the complex dynamical behavior of H-bridge inverter with RLC load. Sci Rep 14, 23916 (2024). https://doi.org/10.1038/s41598-024-74812-8

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Received: 24 July 2024

Accepted: 30 September 2024

Published: 13 October 2024

DOI: https://doi.org/10.1038/s41598-024-74812-8

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