The proliferation of distributed energy resources has placed stringent demands on the power quality and operational stability of grid-connected interfaces. Among these, single-phase H-bridge inverters serve as a fundamental and critical power conversion stage for integrating photovoltaic arrays into the utility grid. These solar inverters are inherently nonlinear, time-varying systems due to their switching nature. When regulated by conventional linear controllers like the Proportional-Integral (PI) algorithm, these systems can exhibit complex nonlinear dynamics, including period-doubling bifurcation and chaos. Such undesirable behaviors significantly increase the harmonic distortion of the output current, elevate switching stresses, and ultimately compromise the reliability and efficiency of the entire power delivery system. Therefore, developing effective strategies to suppress chaos and expand the stable operating region in PI-controlled solar inverters is of paramount practical importance.
Existing approaches for chaos control in power electronics, such as time-delayed feedback or its variants, often face challenges including complex model dependency and the difficulty in deterministically tuning control coefficients. A promising direction is the family of delayed feedback controls. However, my research and analysis indicate that directly applying standard exponential delayed feedback control (EDFC) to a switching system like an H-bridge inverter can introduce excessive disturbance, as the feedback strength is inherently tied to the system’s state error without adequate regulation. To address this limitation, I propose an Improved Exponential Delayed Feedback Control (IEDFC) strategy. This method introduces adjustable gain parameters to manage the feedback intensity effectively, ensuring robust stabilization without causing large-signal perturbations. This paper presents a detailed first-person account of the modeling, analysis, and application of this IEDFC method for chaos suppression in single-phase H-bridge photovoltaic inverters.
System Modeling and Nonlinear Phenomena
The system under study comprises a PV array, a DC-DC boost converter with maximum power point tracking (MPPT), and a single-phase H-bridge voltage source inverter feeding an RL load. The inverter is regulated by a unipolar sinusoidal pulse-width modulation (SPWM) scheme where the duty cycle is generated by a PI controller comparing the sensed output current \(i(t)\) with a sinusoidal reference \(i_{ref}(t)\). The primary nonlinear behavior stems from this closed-loop interaction.
By applying stroboscopic mapping at the switching period \(T\), the discrete-time model of the power stage can be derived. Considering the two complementary switching states within one period, the inductor current dynamics are captured by the following iterative map:
$$
i_{n+1} = e^{-\frac{R}{L}T} i_n + \left[ \frac{2}{R}e^{-\frac{(1-d_n)RT}{L}} – \frac{1}{R}\left(1 + e^{-\frac{R}{L}T}\right) \right] E
$$
where \(i_n\) is the inductor current at the beginning of the \(n\)-th switching cycle, \(R\) and \(L\) are the load resistance and inductance, \(E\) is the DC-link voltage, and \(d_n\) is the duty cycle for that cycle (\(0 \le d_n \le 1\)).
The discrete model for the PI controller, which generates the modulation signal \(i_{con}(n)\), is given by:
$$
i_{con}(n) = p_1 i_{n-1} + i_{con}(n-1) + p_2 E + T U_{n-1}
$$
with the coefficients defined as:
$$
p_1 = \left( k_i \frac{L}{R} – k_p \right)\left( e^{-\frac{R}{L}T} – 1 \right), \quad p_2 = \left( k_i \frac{L}{R} – k_p \right)\left( \frac{2}{R}e^{-\frac{(1-d_{n-1})RT}{L}} – \frac{1}{R} – \frac{1}{R}e^{-\frac{R}{L}T} \right) + \frac{k_i T}{R}(1-2d_{n-1})
$$
$$
U_{n-1} = k_p I_m \omega \cos(\omega (n-1)T) + k_i I_m \sin(\omega (n-1)T)
$$
where \(k_p\) and \(k_i\) are the proportional and integral gains, and \(I_m\) and \(\omega\) are the amplitude and angular frequency of the reference current. The duty cycle is related to the modulation signal by \(d_n = (1 + i_{con}(n))/2\).
The complete nonlinear discrete map of the system is:
$$
\begin{cases}
i_{n+1} = e^{-\frac{R}{L}T} i_n + \left[ \frac{2}{R}e^{-\frac{(1-d_n)RT}{L}} – \frac{1}{R}\left(1 + e^{-\frac{R}{L}T}\right) \right] E, \\
i_{con}(n) = p_1 i_{n-1} + i_{con}(n-1) + p_2 E + T U_{n-1}, \\
d_n = \frac{1}{2}(1 + i_{con}(n)), \quad 0 \le d_n \le 1.
\end{cases}
$$
To analyze stability, the system’s equilibrium point \((I_Q, I_{conQ}, D_Q)\) is found, and the Jacobian matrix \(\mathbf{J}\) is evaluated at this point. The characteristic equation \(\det(\lambda \mathbf{I} – \mathbf{J}) = 0\) determines the stability: the system is stable only if all eigenvalues \(\lambda_i\) lie within the unit circle in the complex plane.
Bifurcation analysis reveals that the PI-controlled inverter loses stability via a period-doubling route. The key parameters affecting stability are the proportional gain \(k_p\) and the DC input voltage \(E\). As \(k_p\) increases beyond a critical value (approximately 1.09 for the parameters in Table 1), one eigenvalue crosses the unit circle at \(\lambda = -1\), signaling a period-doubling bifurcation. Further increase leads to a cascade of period-doublings and eventually to chaotic operation. Similarly, increasing \(E\) destabilizes the system. This chaotic regime is characterized by a distorted output current with high total harmonic distortion (THD), which is unacceptable for grid-connected solar inverters.
| Circuit Parameter | Symbol | Value |
|---|---|---|
| DC Input Voltage | \(E\) | 250 V (nominal) |
| Inductance | \(L\) | 7 mH |
| Load Resistance | \(R\) | 20 Ω |
| Reference Current Amplitude | \(I_m\) | 5 A |
| Switching Frequency | \(f_s = 1/T\) | 20 kHz |
| Fundamental Frequency | \(f\) | 50 Hz |
| Integral Gain | \(k_i\) | 180 |
Principles of the Improved Exponential Delayed Feedback Control
The proposed IEDFC strategy is designed to be a simple, additive correction to the existing PI controller, making it practical for implementation in digital signal processors commonly used in solar inverters. The core idea is to construct a stabilizing signal based on the difference between the present output and its delayed version, processed through a tunable exponential function.
The control law is formulated as follows. First, the difference between the current sample and its value one switching period earlier is computed and scaled by a gain \(k_1\):
$$
\epsilon_n = k_1 (i_n – i_{n-1}).
$$
This difference is fed into an exponential function, and its offset from unity is taken. This result is then scaled by a second gain \(k_2\) to produce the control signal \(u_c(n)\):
$$
u_c(n) = k_2 \left( e^{\epsilon_n} – 1 \right) = k_2 \left( e^{k_1 (i_n – i_{n-1})} – 1 \right).
$$
Finally, this signal is added to the original PI controller output to form the new modulation signal:
$$
i_{con}^{new}(n) = i_{con}^{PI}(n) + u_c(n).
$$
Here, \(i_{con}^{PI}(n)\) is the modulation signal generated by the standard PI controller as described in the previous section.
The discrete-time model of the system with IEDFC incorporated becomes:
$$
i_{con}(n) = p_1 i_{n-1} + i_{con}(n-1) + p_2 E + T U_{n-1} + k_2 \left( e^{k_1 (i_n – i_{n-1})} – 1 \right).
$$
The Jacobian matrix \(\mathbf{J}_{IEDFC}\) at the equilibrium point \((I_Q, I_{conQ}, D_Q)\) is derived from this augmented map. Its form is crucial for stability analysis:
$$
\mathbf{J}_{IEDFC} = \begin{bmatrix}
e^{-\frac{RT}{L}} + A k_1 k_2 & A(BC – k_1 k_2) & A\left(1 + AB – \frac{k_i T E}{R}\right) \\
1 & 0 & 0 \\
k_1 k_2 & BC – k_1 k_2 & 1 + AB – \frac{k_i T E}{R}
\end{bmatrix}
$$
where the constants \(A\), \(B\), and \(C\) are functions of circuit parameters and the equilibrium duty cycle \(D_Q\):
$$
A = \frac{ET}{L} e^{-\frac{(1-D_Q)RT}{L}}, \quad B = k_i \frac{L}{R} – k_p, \quad C = e^{-\frac{RT}{L}} – 1.
$$
Stability Analysis and Parameter Selection
The effectiveness of the IEDFC hinges on the proper selection of the feedback gains \(k_1\) and \(k_2\). Using the Jacobian \(\mathbf{J}_{IEDFC}\), the characteristic equation is obtained:
$$
\det(\lambda \mathbf{I} – \mathbf{J}_{IEDFC}) = \lambda \left[ \lambda^2 + J_1 \lambda + J_2 \right] = 0,
$$
where
$$
J_1 = \frac{k_i T E}{R} – 1 – AB – e^{-\frac{RT}{L}} – A k_1 k_2, \quad J_2 = e^{-\frac{RT}{L}}\left(1 + AB – \frac{k_i T E}{R}\right) + A(k_1 k_2 – BC).
$$
One eigenvalue is always at \(\lambda_1 = 0\). Stability is determined by the two remaining eigenvalues \(\lambda_{2,3}\) from the quadratic term. For the system to be stable, these eigenvalues must lie within the unit circle. Applying the Schur-Cohn stability criterion for a discrete-time second-order system yields explicit constraints on the combined gain term \(k_1 k_2\).
The necessary and sufficient conditions for stability are:
$$
\begin{cases}
|J_2| < 1, \\
|J_1| < 1 + J_2.
\end{cases}
$$
Solving these inequalities for \(k_1 k_2\) provides a clear, calculable range for tuning the controller. The allowable range is:
$$
\frac{ \left( \frac{k_i T E}{R} – 2 \right)\left(e^{-\frac{RT}{L}} + 1\right) + \left( AC – A e^{-\frac{RT}{L}} – A \right)B }{2A} < k_1 k_2 < \frac{1}{A} \left[ 1 + e^{-\frac{RT}{L}}\left( \frac{k_i T E}{R} – 1 \right) + \left( AC – A e^{-\frac{RT}{L}} \right)B \right].
$$
This analytical derivation is a key advantage of the IEDFC method. It removes the guesswork from parameter tuning. For a given set of circuit parameters \((R, L, E, T)\) and controller parameters \((k_p, k_i)\), one can compute the stable range for the product \(k_1 k_2\). The individual values of \(k_1\) and \(k_2\) can then be chosen for desired transient performance, often starting with \(k_1 = k_2 = \sqrt{k_1 k_2_{opt}}\). This structured approach significantly enhances the practicality of implementing chaos control in industrial solar inverters.
Simulation Results and Performance Evaluation
To validate the IEDFC method, extensive numerical simulations were performed using the discrete model and circuit-level simulations in a specialized power electronics environment. The system parameters are listed in Table 1. The performance was assessed under two major destabilizing scenarios: variation in the proportional gain \(k_p\) and variation in the DC input voltage \(E\), which is common in PV systems due to changing irradiance.
Scenario 1: High Proportional Gain (\(k_p = 1.8\)). With \(E=250\) V and \(k_p=1.8\), the uncontrolled system operates deep in the chaotic region. The output current waveform is highly erratic and its spectrum shows a very high THD of 15.54%. Applying the standard EDFC at \(t = 0.08\) s fails to stabilize the system; in fact, it increases the THD to 16.31%, demonstrating its tendency to cause excessive disturbance. In contrast, applying IEDFC with gains selected from the stability range (\(k_1=0.707, k_2=0.707\)) rapidly suppresses the chaos. The current waveform becomes a clean sinusoid, and the THD drops to an excellent 2.80%. This confirms that IEDFC successfully extends the stable range of \(k_p\).
Scenario 2: High DC Input Voltage (\(E = 500\) V). With \(k_p=1.0\) and \(E=500\) V, the uncontrolled system is again chaotic. Applying standard EDFC is ineffective. However, applying IEDFC with \(k_1=0.45\) and \(k_2=0.5\) (chosen based on the stability condition for the new \(E\)) successfully restores stable, periodic operation. This demonstrates the method’s robustness against input voltage variations, a critical feature for solar inverters connected to variable PV sources.
Comparative Analysis and Robustness. A critical test involved evaluating the dynamic response and robustness against line disturbances. For a case where \(k_p=1.4\) (a parameter where both EDFC and IEDFC can theoretically stabilize the system), IEDFC was found to have a significantly faster recovery time (~1.7 ms) compared to EDFC (~10 ms). Furthermore, when a step disturbance of \(\Delta E = +50\) V was applied to the DC link during stable operation, the system under EDFC lost stability and re-entered chaos. Remarkably, the system under IEDFC maintained stable operation, showcasing its superior disturbance rejection capability.
| Control Method | Stabilizes at \(k_p=1.8\)? | THD at \(k_p=1.8\) | Recovery Time | Robust to \(\Delta E=+50V\)? |
|---|---|---|---|---|
| No Control | No (Chaotic) | 15.54% | N/A | N/A |
| Standard EDFC | No (Worsens) | 16.31% | ~10 ms | No |
| Time-Delayed Feedback [21] | Yes (for lower \(k_p\)) | N/A | Slow | No |
| Filter-Based Control [23] | Yes (for lower \(k_p\)) | N/A | Moderate | No |
| Proposed IEDFC | Yes | 2.80% | ~1.7 ms | Yes |
The mechanism of IEDFC can be interpreted as introducing a state-dependent damping term. When the system is stable, \(i_n \approx i_{n-1}\), making \(\epsilon_n \approx 0\) and thus \(u_c(n) \approx k_2 (e^0 – 1) = 0\). The IEDFC has a negligible effect on the steady-state operation. However, when the system begins to oscillate or becomes chaotic, the difference \(|i_n – i_{n-1}|\) increases. The exponential function \(e^{k_1 (i_n – i_{n-1})}\) amplifies this difference nonlinearly. The subtraction of 1 and scaling by \(k_2\) creates a corrective signal that is always opposite in phase to the incipient instability, effectively increasing the system’s damping for large-signal disturbances. The gains \(k_1\) and \(k_2\) allow this nonlinear damping characteristic to be shaped precisely to maximize the stable region without causing overshoot or sluggish response.
Conclusion
In this work, I have addressed the critical issue of nonlinear instability in PI-controlled single-phase H-bridge photovoltaic inverters. The analysis confirmed that these solar inverters are prone to period-doubling bifurcation and chaos as key parameters like the proportional gain or DC input voltage vary, leading to degraded power quality. To mitigate this, an Improved Exponential Delayed Feedback Control strategy was proposed and thoroughly investigated.
The IEDFC method distinguishes itself by its structured design and analytical foundation. It constructs a control signal from the delayed difference of the output current, processed through an exponential function and two tunable gains. A significant contribution is the derivation of explicit, closed-form stability conditions that provide a clear range for selecting the feedback gains \(k_1\) and \(k_2\), eliminating trial-and-error tuning. The performance of the proposed controller was rigorously validated through bifurcation analysis, eigenvalue trajectory studies, and time-domain simulations.
The results conclusively demonstrate that the IEDFC method can effectively suppress chaotic oscillations and significantly extend the stable operating domain of the inverter. It outperforms the standard EDFC, especially under large parameter variations, by preventing excessive feedback disturbance. Furthermore, it exhibits faster transient recovery and superior robustness against input voltage disturbances compared to other referenced chaos control methods. By ensuring stable, low-THD operation over a wider range of conditions, this control strategy enhances the reliability and power quality of grid-connected solar inverters, contributing to the stability of modern power systems with high penetration of renewable energy.