In electromagnetic transient processes, the power rise of linear motor loads and the high C-rate discharge of lithium battery packs lead to a significant output voltage drop in the battery energy storage system (BESS), which directly affects the stable operation of electromagnetic launch systems. Additionally, the constant-gain extended state observer (ESO) deteriorates controller performance due to the peaking phenomenon caused by initial state errors. To address these issues, we propose a variable-gain active disturbance rejection control (VG-ADRC) chopping compensation strategy based on the conjugate pole method to configure nonlinear ESO (NESO) parameters. This approach reduces the peak value of total disturbance observations, improves controller compensation capability, and lowers the output voltage drop amplitude of the BESS. Frequency characteristics analysis validates that the proposed parameter configuration method reduces observer complexity and enhances disturbance observation ability. Discharge experiments on a pulsed high-power discharge platform confirm the effectiveness of the proposed control strategy and parameter configuration method.
The battery energy storage system is a core component in electromagnetic launch systems, providing instantaneous power to linear motor loads through high-rate discharge. This enables the conversion of electrical energy to mechanical energy, offering advantages such as high energy density, power density, and long cycle life. However, during high-rate discharge, the internal resistance of lithium batteries causes output voltage drops, failing to meet the voltage requirements of downstream inverters. The transient operation of linear motors further exacerbates this issue, leading to substantial voltage dips in the BESS and compromising system stability. Thus, bus voltage compensation control is essential.
Traditional methods, such as power compensation control, improve transient performance but do not analyze compensation under inverter loads. Feedforward control, combined with PID, enhances transient response but requires precise disturbance models, which are challenging to derive due to the nonlinearity of battery models and the complexity of parallel active neutral point clamped three-level inverter systems (ANPC-TLIS). Active disturbance rejection control (ADRC), as a model-free approach, compensates for disturbances through observation and rejection, with low dependency on model parameters. For instance, ADRC applied to DC/DC converters shows better output response compared to classical PI control. Adaptive linear ADRC methods for three-phase isolated AC-DC-DC power supplies improve output voltage response speed. Modified linear ADRC strategies for six-phase interleaved Buck converters enhance total disturbance observation and output voltage performance. Improved ESO designs for photovoltaic grid-connected systems address DC bus voltage fluctuations by increasing the equivalent bandwidth of disturbance observation transfer functions.
However, constant-gain ESOs suffer from peaking phenomena due to initial state errors, which can destabilize control. In the BESS, compensating peak-valued disturbances into the control loop reduces response speed and increases output voltage drops. Adaptive ESO (AESO) methods with time-varying gains mitigate peaking but increase complexity and reduce initial observation accuracy. Reducing initial observation errors is key to addressing peaking, which depends on both observer and controller gains. Thus, we propose a VG-ADRC strategy with a variable gain control law to attenuate peaking and improve controller performance, combined with a conjugate pole-based NESO parameter configuration to enhance disturbance observation capability.
The pulsed high-power system consists of a battery energy storage system, parallel ANPC-TLIS, and linear motor loads. The BESS includes two series-connected N+1 level dynamic chopping (N+1-LDC) converters, each comprising N battery groups and one battery group with a Buck converter in series. The equivalent circuit of a single N+1-LDC converter is shown below, where V_Libat is the equivalent voltage of a single battery group, N is the number of battery groups, R_r is the equivalent internal resistance, V_T and V_D are power switching devices, L_f is the filter inductance, C_bus is the bus support capacitance, R_L and R_C are parasitic resistances of the inductor and capacitor, L_X and R_X represent the linear motor load (X = A, B, C), i_L is the instantaneous inductor current, i_load is the load current, i_inv is the total current entering the inverter, R_load is the equivalent DC component, and i_tr is the equivalent triple-ripple component due to inverter modulation. The switch V_T is on during (0, D T_s) and off during (D T_s, T_s).
Assuming identical parameters for individual batteries and steady-state operation, the state-space averaging method yields the following equations for one switching cycle:
$$ L_f \frac{di_L}{dt} = D V_1 + V_2 – V_{bus} – R_\sigma i_L, $$
$$ C_{bus} \frac{du_C}{dt} \cdot R_C + u_C = V_{bus}, $$
$$ i_L – C_{bus} \frac{du_C}{dt} = \frac{V_{bus}}{R_{load}} + i_{tr}. $$
Here, u_C is the capacitor voltage, V_bus is the bus voltage, D is the steady-state duty cycle, and R_σ = D R_{r1} + R_{r2} + R_L. Introducing small-signal components, the transfer functions are derived as:
$$ G_{vi} = \frac{v_{bus}}{i_L} = \frac{R_{load} (s C_{bus} R_C + 1)}{s C_{bus} (R_C + R_{load}) + 1}, $$
$$ G_{id} = \frac{i_L}{d} = \frac{(s C_{bus} (R_C + R_{load}) + 1)(V_1 – I_L R_{r1})}{a_1 s^2 + a_2 s + a_3}, $$
where a_1 = L_f C_{bus} (R_{load} + R_C), a_2 = L_f + (D R_{r1} + R_{r2} + R_L) C_{bus} (R_{load} + R_C) + C_{bus} R_{load} R_C, a_3 = D R_{r1} + R_{r2} + R_L + R_{load}, and D = (V_{bus} + (R_{r2} + R_L) I_L – V_1) / (V_2 – I_L R_{r1}).
The VG-ADRC design focuses on the voltage outer loop, treating the plant as a first-order system after simplifying the current inner loop. For a first-order system:
$$ \dot{y} = f(y, w) + b u, $$
where y is the system output, w is the external disturbance, b is the system gain, and u is the input. Defining states x_1 = y and x_2 = f(y, w) = f_sum, and assuming bounded differentiability of f_sum, the state-space form is:
$$ \dot{x}_1 = x_2 + b_0 u, $$
$$ \dot{x}_2 = h_f, $$
where b_0 is the estimated system gain and h_f is the derivative of f_sum. The NESO observes system states using the fal(·) function:
$$ \text{fal}(e_v, \alpha, \delta) = \begin{cases}
|e_v|^\alpha \text{sign}(e_v), & |e_v| > \delta, \\
e_v / \delta^{1-\alpha}, & |e_v| \leq \delta,
\end{cases} $$
$$ e_v = U_O – z_1, $$
$$ \dot{z}_1 = z_2 + b_0 u + q_1 \text{fal}(e_v, \alpha_1, \delta_1), $$
$$ \dot{z}_2 = q_2 \text{fal}(e_v, \alpha_2, \delta_1), $$
where z_1 and z_2 are observed states, q_1 and q_2 are state feedback gains, U_O is the sampled output voltage, e_v is the observation error, α is the nonlinear factor, and δ defines the nonlinear interval. Parameters are set as q_1 = 2 ω_o and q_2 = ω_o^2, with ω_o as the observer bandwidth. Under disturbance compensation, the system becomes a integrator, and proportional control law forms a first-order inertial loop:
$$ e_r = V_{ref} – z_1, $$
$$ u_0 = k_p \text{fal}(e_r, \alpha_3, \delta_2), $$
$$ u = (u_0 – z_2) / b_0, $$
where e_r is the state feedback error, V_ref is the reference voltage, u_0 is an intermediate variable, and k_p is the controller gain set as k_p = ω_c, with ω_c as the controller bandwidth. Typically, ω_o is 4 to 10 times ω_c for stability and noise reduction.
To address peaking, a variable gain control law is designed:
$$ k_m = 2(1 + e^{-k(t – t_0)})^{-1}, $$
$$ e_r = V_{ref} – z_1, $$
$$ u_0 = k_m k_p \text{fal}(e_r, \alpha_3, \delta_2), $$
$$ u = (u_0 – z_2) / b_0, $$
where k_m is the time-varying gain, k determines convergence speed, and t_0 is set based on linear motor trajectory. This reduces initial observation errors and attenuates peaking.
Frequency analysis of the NESO total disturbance observation transfer function is conducted. The gain coefficient k_g(e) for fal(·) is:
$$ k_g(e) = \frac{\text{fal}(e, \alpha, \delta)}{e}. $$
The NESO transfer function is:
$$ \frac{z_2}{f_{sum}} = \frac{q_2 k_g2(e_v)}{s^2 + q_1 k_g1(e_v) s + q_2 k_g2(e_v)}. $$
For linear ESO (LESO), k_g1(e_v) = k_g2(e_v) = 1:
$$ \frac{z_2}{f_{sum}} = \frac{q_2}{s^2 + q_1 s + q_2}. $$
For generalized ESO (GESO):
$$ \frac{z_2}{f_{sum}} = \frac{q_3 + q_2 s}{s^3 + q_1 s^2 + q_2 s + q_3}, $$
with q_1 = 3 ω_o, q_2 = 3 ω_o^2, q_3 = ω_o^3. Bode plots show that GESO has higher equivalent bandwidth but may introduce peaking, while NESO’s performance is limited by fal(·) gain characteristics. Using conjugate pole configuration, we adjust q_1 to enhance bandwidth without increasing high-frequency gain.
Experimental validation on a pulsed high-power discharge platform uses DSP for sampling and duty cycle output, with FPGA generating interrupts and pulses. Parameters for the BESS and controllers are listed below:
| Parameter | Value |
|---|---|
| Battery 1 Voltage V1 (V) | 268 |
| Battery 2 Voltage V2 (V) | 1072 |
| Battery 1 Internal Resistance Rr1 (mΩ) | 20 |
| Battery 2 Internal Resistance Rr2 (mΩ) | 190 |
| Motor Resistance RX (mΩ) | 50 |
| Inductor Parasitic Resistance RL (mΩ) | 3.2 |
| Capacitor Parasitic Resistance RC (mΩ) | 2 |
| Motor Inductance LX (μH) | 460 |
| Filter Inductance Lf (mH) | 0.3 |
| Bus Capacitance Cbus (mF) | 44 |
| Converter Switching Frequency fs1 (kHz) | 2 |
| Inverter Switching Frequency fs2 (kHz) | 1.25 |
| Parameter | Value |
|---|---|
| Controller Bandwidth ωc (rad/s) | 250 |
| Observer Bandwidth ωo (rad/s) | 1000 |
| Gain Coefficient b0 | 45.26 |
| Voltage Proportional Coefficient kvp | 7.989 |
| Voltage Integral Coefficient kvi | 600 |
| Current Proportional Coefficient kip | 0.004 |
| Current Integral Coefficient kii | 5 |
| Nonlinear Factor α1 | 0.9 |
| Nonlinear Factor α2 | 0.7 |
| Nonlinear Factor α3 | 0.6 |
| Nonlinear Interval δ1 | 10 |
| Nonlinear Interval δ2 | 15 |
Three-phase AC current experiments show peaks of 5000 A, with current ramping up from 0.1 s to 0.3 s and frequency increasing from 0 Hz to 50 Hz. From 0.3 s to 0.6 s, current stabilizes around 5000 A at 50 Hz. Total disturbance observations under traditional LADRC exhibit peaking, while VG-ADRC attenuates this effect. Steady-state observations include fundamental, second, and third harmonic components due to load imbalance and inverter modulation. VG-ADRC shows slight phase lag but improved performance.
Full bus voltage experiments reveal that PI control results in an initial voltage drop of 88 V, while LADRC with peaking causes a 215 V drop. VG-ADRC reduces the drop and accelerates response, compensating 0.12 s earlier than gain-limited LADRC. With conjugate pole-configured NESO, the BESS output voltage transitions directly to steady state.
In conclusion, the proposed VG-ADRC strategy with conjugate pole-based NESO parameter configuration effectively attenuates total disturbance peaking, enhances controller response, and reduces voltage drops in the battery energy storage system. This ensures stable, high-quality voltage sources for downstream inverters in high-power applications. The BESS performance is significantly improved, demonstrating the viability of the approach for electromagnetic launch systems and other demanding environments.