KalMRACO: Unifying Kalman Filter and Model Reference Adaptive Control for Robust Control and Estimation of Uncertain Systems

A common assumption when applying the Kalman filter is a priori knowledge of the system parameters. These parameters are not necessarily known, and this may limit real-world applications of the Kalman filter. The well-established Model Reference Adaptive Controller (MRAC) utilizes a known reference model and ensures that the input-output behavior of a potentially unknown system converges to that of the reference model. We present KalMRACO, a unification of the Kalman filter and MRAC leveraging the reference model of MRAC as the Kalman filter system model, thus eliminating, to a large degree, the need for knowledge of the underlying system parameters in the application of the Kalman filter. We also introduce the concept of blending estimated states and measurements in the feedback law to handle stability issues during the initial transient. KalMRACO is validated through simulations and lab trials on an underwater vehicle. Results show superior tracking of the reference model state, observer state convergence, and noise mitigation properties.

💡 Research Summary

This paper introduces KalMRACO, a novel unified framework that integrates the Kalman filter with Model Reference Adaptive Control (MRAC) to address the combined challenges of state estimation and control in systems with uncertain parameters. The core problem stems from the Kalman filter’s fundamental requirement for an accurate system model, which is often unavailable in practice, limiting its real-world applicability. Conversely, MRAC is a robust control technique that forces an uncertain plant to behave like a predefined reference model but traditionally does not incorporate optimal state estimation.

The key innovation of KalMRACO is the direct use of the MRAC reference model as the internal system model within a conventional Kalman filter. This elegant unification means the Kalman filter performs optimal estimation based on the desired closed-loop dynamics (the reference model) rather than the unknown true plant dynamics. Consequently, the need for precise knowledge of the actual system parameters (state transition matrix, input matrix, etc.) is largely eliminated, significantly lowering the barrier to practical Kalman filter implementation.

A major technical contribution is the introduction of a “blending” control law to resolve stability issues during the initial transient period before parameter adaptation takes effect. Prior methods like MRACO required a high, constant observer gain to ensure stability initially, which amplifies measurement noise. KalMRACO’s control law blends the Kalman filter’s state estimate (ˆx) with the direct measurement (y) in a variable ratio defined by a blending function θ(e₂). This function depends on the observer error e₂ and is designed to guarantee Lyapunov stability, particularly for systems that may be unstable (a>0). This design allows the use of the time-varying, optimal Kalman gain even from low initial values, thereby maintaining the filter’s superior noise rejection properties while ensuring closed-loop stability from the start.

The paper provides a rigorous stability analysis for a scalar system, proving that with the proposed blending control law and corresponding adaptation laws for the controller parameters (ˆk, ˆl) and disturbance estimate (ˆw), the observer and tracking errors converge to zero.

Validation is conducted through comprehensive simulations and real-world lab trials using a BlueROV2 Heavy underwater vehicle. Simulation 1 demonstrates that without blending, measurement noise causes unbounded drift in the adapted parameters, whereas KalMRACO with blending maintains stable and bounded parameter estimates. Simulation 2 shows the scheme’s performance under the assumption of an unstable plant, where KalMRACO successfully tracks piecewise constant reference velocities, achieves convergence of the state estimate to the true state, and exhibits well-behaved adapted parameters. The experimental results confirm that KalMRACO delivers superior reference model tracking, observer convergence, and noise mitigation, effectively removing the high-gain requirement of previous methods and offering a robust, practical solution for control and estimation of uncertain systems.