Simultaneous improvement of control and estimation for battery management systems

1. Problem Statement

Battery Management Systems (BMS) traditionally treat control (managing charge/discharge to meet power demands) and state estimation (inferring State of Charge, SOC) as decoupled objectives.

• The Core Issue: Standard BMSs use Certainty Equivalence (CE) controllers, which assume the current state estimate is exact and optimize control inputs solely based on this estimate.
• The Blind Spot: This approach ignores the nonlinear Open-Circuit Voltage (OCV) vs. SOC relationship. In many battery chemistries, the OCV curve contains "voltage plateaus" (flat regions) where the sensitivity of voltage to SOC changes is near zero.
• Consequence: In these low-observability regions, state estimators (like Kalman Filters) degrade significantly. CE controllers may inadvertently drive the battery into these regions to satisfy tracking objectives, amplifying estimation errors precisely when accurate SOC knowledge is most critical for safety and efficiency (e.g., in grid peak shaving and valley filling).

2. Methodology

The authors propose a Dual Control approach, where the controller actively manages the system trajectory to improve future estimation quality while satisfying control objectives.

A. Stochastic Optimal Control Formulation

The problem is reformulated as a stochastic optimal control problem. The authors prove that for a broad class of cost functions (including tracking error and control effort), the expected cost can be exactly parametrized by the conditional mean ($x_{k|k}$) and covariance ($\Sigma_{k|k}$) of the SOC.

• Key Insight: The cost function includes a term dependent on the state covariance ($q^\top \Sigma_{k|k} q$). This creates a direct coupling: the control input $I_k$ influences the evolution of the covariance, which in turn affects the future cost.
• Dual Effect: By minimizing this cost, the controller naturally exhibits the "dual effect" of stochastic control: it balances caution (avoiding high-risk states) and probing (steering the system toward high-observability regions of the OCV curve to reduce uncertainty).

B. Deterministic Surrogate for Computation

Solving the full stochastic optimal control problem is computationally intractable due to the expectation over future random measurements.

• Approximation: The authors derive a deterministic surrogate cost function. They replace the random future observations with a "prediction-only" Extended Kalman Filter (eKF) recursion.
• Algorithm:
  1. Linearization: The nonlinear dynamics of the information state (mean and covariance) are linearized around the current operating point.
  2. MPC Formulation: The problem is posed as a Nonlinear Model Predictive Control (MPC) problem. To ensure computational tractability, they use a local linearization of the information state dynamics (Algorithm 1) to solve a quadratic program.
  3. Objective: The controller minimizes a cost comprising tracking error, control effort, battery uniformity, and a penalty on the state covariance (estimation uncertainty).

3. Key Contributions

1. Theoretical Reformulation: Demonstrated that battery management can be rigorously posed under the stochastic optimal control umbrella, proving that the cost is exactly parametrized by the first two moments (mean and covariance) of the SOC.
2. Dual-Control Implementation: Developed a computationally tractable MPC algorithm that explicitly couples control inputs with estimation quality, forcing the system to avoid low-observability regions.
3. Filter Independence: Showed that the improvement in estimation accuracy is not tied to a specific observer. The control strategy improves performance regardless of whether an eKF, Unscented Kalman Filter (uKF), or Moving Horizon Estimator (MHE) is used downstream.
4. Validation: Validated the approach on a multi-battery system (9 batteries) with diverse OCV-SOC characteristics.

4. Results

The proposed algorithm (Algorithm 1) was compared against a standard Certainty Equivalence Linear MPC using 100 simulation runs for a 9-battery system tracking a time-varying power demand (peak shaving/valley filling scenario).

• Control Performance:
  • Total Cost Reduction: ~17.8% reduction.
  • Tracking Cost Reduction: ~28.4% reduction.
  • The proposed method tracked the reference trajectory more accurately and maintained tighter confidence bounds ($2\sigma$) around the target.
• Estimation Performance:
  • eKF Error Reduction: ~17.8% reduction in average estimation error.
  • UKF Error Reduction: ~10.3% reduction.
  • MHE Error Reduction: ~8.5% reduction.
  • Covariance Trace: A massive 50% reduction in the trace of the covariance matrix, indicating significantly lower overall uncertainty.
• Computational Efficiency: The algorithm achieved a control frequency of 77 Hz on a standard laptop, demonstrating feasibility for real-time BMS applications despite the added complexity of covariance propagation.

5. Significance

This work addresses a critical gap in battery management by moving beyond the "separation principle" (which assumes control and estimation can be designed independently).

• Safety & Reliability: By actively steering batteries away from voltage plateaus, the system maintains high observability, reducing the risk of unsafe operations caused by SOC estimation errors.
• Grid Integration: The approach is particularly valuable for grid-connected storage (peak shaving/valley filling) where batteries must traverse wide SOC ranges, often passing through difficult-to-estimate regions.
• Generalizability: The framework is agnostic to the specific state estimator used, making it a robust upgrade for existing BMS architectures that rely on various filtering techniques.

In summary, the paper demonstrates that treating control and estimation as a coupled, dual-control problem yields simultaneous improvements in both tracking accuracy and state estimation reliability, offering a superior alternative to standard decoupled BMS designs.